Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary

conditions”

Lars Andersson

Piotr T. Chruściel

Dissertationes Mathematicae 355, 1-100 (1996)

Author address:

Department of Mathematics, Royal Institute of Technology, S 10044 Stockholm, Sweden

E-mail address: larsa@math.kth.se

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, skr. poczt. 137, PL 00–172 Warszawa, Poland and Département de Mathématiques, Université de Tours, Parc de Grandmont, F 37200 Tours, France

E-mail address: Chrusciel@Univ-Tours.fr

Contents

Chapter 1. Introduction 3

Chapter 2. The hyperboloidal initial value problem. 9 1. Conformal compactifications and Cauchy data 9 2. Some remarks on non–vacuum initial data sets 13

Chapter 3. Definitions, Preliminary Results. 17 1. Function spaces 17 2. Some embeddings 23 3. Extensions of functions defined on ∂M 25 4. Mapping properties of some integral operators 28

Chapter 4. Regularity at the boundary: the linear problem. 33 1. Tangential regularity below the threshold 33 2. Boundary regularity for a class of second order systems 41

Chapter 5. Nonlinear equations with polyhomogeneous coefficients. 55 1. Polyhomogeneity of solutions of some fully nonlinear equations 55

Chapter 6. The vector constraint equation. 61 1. Introductory remarks 61 2. (Non–weighted) Hölder spaces on the compactified manifold 62 3. Weighted Sobolev spaces 70

Chapter 7. The Lichnerowicz equation. 79 1. Introductory remarks 79 2. The linearized equation 80 3. Existence of solutions of the non–linear problem 83 4. Regularity at the boundary of the solutions 86

Appendix A. Genericity of log-terms. 93 1. The vector constraint equation 93 2. The coupled system 97

Appendix B. 101 1. “Almost Gaussian” coordinates. 101

Appendix. Bibliography 103

Index 105

Appendix. Index 107

Abstract

We prove existence of the solutions of the constraint equations satisfying “hyperboloidal boundary conditions” using the Choquet–Bruhat—Lichnerowicz—York conformal method and we analyze in detail their differentiability near the conformal boundary. We show that generic “hyperboloidal initial data” display asymptotic behaviour which is not compatible with Penrose’s hypothesis of smoothness of J^{(} . We also show that a large class of “non–generic” initial data satisfying Penrose smoothness conditions exists. The results are established by developing a theory of regularity up–to–boundary for a class of linear and semilinear elliptic systems of equations uniformly degenerating at the boundary.

1991 Mathematics Subject Classification. Primary 83C05, 35Q75, 53C21; Secondary 58G20, 35J60, 35J65.

The first author was supported in part by the Swedish Research Council under grants 4873-105 and R-RA 04873-310.

The second author was supported in part by the Alexander von Humboldt Foundation, by an Australian Research Council Fellowship, by the Polish Scientific Research Committee under grant KBN # 2 1047 9101, by the Gustafsson Foundation under grant 93:4 and by the Swedish Research Council under grant R-RA 04873-310.

CHAPTER 1 Introduction

A couple of years ago the first semi–global (i.e., global to the future of the initial data hypersurface) existence theorem in general relativity was proved by Friedrich [35], for a class of “small” initial data satisfying some asymptotic conditions (cf.

[22] for a different class of “small” data results). An important problem related to Friedrich’s theorem remained open: it was not clear “how many” initial data satisfying the required “hyperboloidal” asymptotic conditions existed. The classes of metrics which were known to satisfy the appropriate asymptotic conditions (cf. e.g.

[12] for a review of previous work on this subject), namely the boost–rotation symmetric space–times and the Robinson–Trautman space–times (cf. also [67]) were of a rather special kind, and semi–global existence results have been proved for these space–times by different methods [13, 23]. More recently, a new class of space– times satisfying the hyperboloidal asymptotic conditions has been constructed by Friedrich [34], using a clever trick of “exchanging i_{o} with i^{±}”, it is, however, still unclear how large a class of metrics can be obtained in this way, especially since the space–times constructed in [34] are analytic.

In [6] in collaboration with H. Friedrich we have constructed a large class of “hyperboloidal” initial data using the conformal Choquet–Bruhat—Lichnerowicz —York method under the assumption of smooth background fields and of pure– trace extrinsic curvature of the initial data hypersurface. In this work we generalize the results of [6] in two directions: We construct solutions of the constraint equations with an extrinsic curvature tensor which is not pure–trace. Moreover only a finite degree of differentiability of the “background fields” is assumed. We analyze exhaustively the asymptotic regularity of the fields thus constructed in the “asymptotically hyperboloidal” setting. We show that for generic “backgrounds” the Cauchy data constructed by the conformal method starting from a “conformally smoothly compact” Riemannian manifold will not possess the asymptotic regularity compatible with Penrose’s smoothness requirements for J^{(} . On the other hand, we also show that there exists a large class of non–generic “backgrounds” for which the solutions will display the required regularity.

We shall analyze the asymptotic behaviour of the solutions of the constraint equations under various differentiability conditions on the “background” metric on the conformally compactified manifold. We wish to stress that the issue of the differentiability hypotheses is not an academic one, because “physical” fields are obtained by an infinite stretching of the “unphysical” ones and thus imposing too strong differentiability conditions at the conformal boundary may result in a set–up inadequate for describing sufficiently general physical situations; some indications that it might be unrealistic to expect smoothness at the conformal boundary in generic situations may be found in [22, 21, 70, 26] (cf. e.g. [27] for a review of previous results on this subject). To obtain a real understanding of the behaviour of the gravitational field at null infinity it clearly is necessary to establish what asymptotic conditions are appropriate from a physical point of view. One might try to put forward various criteria which might be considered as physically desirable:

((i)) existence of a local in time evolution of the data, ((ii)) existence of a notion of total energy ((iii)) and finiteness thereof, ((iv)) existence of a notion of angular momentum ((v)) and finiteness thereof, ((vi)) existence of a development (^{4}M, γ) of the initial data set which admits a

J^{(} (incomplete, half–complete (i.e. complete in one time–direction only) or complete ?) of some differentiability class (C^{0}(^{4}M̄), C^{2+λ}(^{4}M̄), C^{3}(^{4}M̄), H_{6}(^{4}M̄ ∩^{3}M̄), C^{∞}(^{4}M̄) ?) ((vii)) existence of a development of the data up to i^{+}, ((viii)) one might wish to add the requirement that the function spaces con-

sidered include those data sets which arise by evolution from generic initial data which are asymptotically flat at spatial infinity, and finally

((ix)) one might ask for various mixtures of the above. C^{8}(^{3}M̄) ⊕ C^{6}(^{3}M̄) differentiability of (g, K) after compactification together with the vanishing of “shear of the conformal boundary” guarantee that most of the requirements (i)–(vii) are satisfied [5]. Recall that Friedrich’s theorem, as applied in [5], guarantees, loosely speaking, a) an incomplete J^{(} for general C^{8}(^{3}M̄)⊕C^{6}(^{3}M̄) data as above, and b) a half–complete J^{(} for those such data which are close enough to the Minkowski data in C^{8}(^{3}M̄) ⊕ C^{6}(^{3}M̄) norm. Of course, one has the known angular momentum ambiguities in the BMS group. In any case, point (viii) is not known to hold. We would like to emphasize that it is not known what degree of differentiability up to the conformal boundary is a necessary condition^{1} for any of criteria (ii)–(viii) to hold in generic situations. Let us note that given initial data (^{3}M, g, K) (satisfying the, say vacuum, constraint equations) with (g, K) locally in H_{k}(^{3}M̄) ⊕ H_{k−1}(^{3}M̄), k > 5/2, there exists a vacuum developement (^{4}M, γ) regardless of any asymptotic behaviour of the fields (this follows from the results of [48] by causality arguments as presented, e.g., in [43]). Moreover requirements of maximality and global hyperbolicity render (^{4}M, γ) unique, at least when k is large enough^{2}. It follows that perhaps the most fundamental criterion (i) above imposes no restrictions on the asymptotics at the conformal boundary. For this reason we have found it of interest to construct Cauchy data for which there exists a conformal compactification in some sense, under conditions which arise naturally from the mathematical analysis of the problem. We are planning to investigate the problem of weakest possible hypotheses on the asymptotic behaviour of (g, K) for criteria (ii)–(vi) in the future.

It should be emphasized that we construct initial data sets which admit conformal compactifications of various degree of differentiability, leaving aside the problem of compactifiability of the Cauchy developments thereof (criterion (vi) above). Under the condition that the “background fields” used to construct the Cauchy data are in, say, C^{∞}(^{3}M̄) we show that the resulting Cauchy data set is generically in C^{2+λ}(^{3}M̄), λ ∈ (0, 1), but not in C^{3}(^{3}M̄). However, we also show that there exists a “big” (infinite–dimensional) subset of non–generic smooth background fields such that the resulting Cauchy data are smoothly compactifiable. It follows from the Penrose’s vanishing Weyl curvature theorem [64] that only a small subset (still infinite dimensional) of the above smoothly compactifiable Cauchy data will give rise to a space–time with a smooth J^{(} (cf. [6] for some results concerning this problem in the pure trace extrinsic curvature case, and [5] in the general case). One can envisage the possibility that the existence of any kind of compactification of the space–time necessarily leads one to, say, C^{k}(^{3}M̄) compactifiable hyperboloidal initial data sets only with some k ≥ 3, though this seems rather unlikely to the authors.

1This should be contrasted with some of the results concerning space–times asymptotically flat at spacelike infinity, where e.g. sharp results are known for the well–posedness of the notion of energy–momentum, etc.

2cf. [17] in the smooth case, and [43] for a reasonably complete proof for k ≥ 4; for k > 5/2 this has been claimed essentially without proof in [20].

To deal with the problem of regularity at the conformal boundary we had to introduce a large number of function spaces, probably more than seems reasonable at first sight: we wish to argue that this is not the case. As usual in PDE problems the appropriate function spaces are Hölder– and Sobolev– type spaces: to capture decay of the solutions one has to consider weighted versions of those. The motivation for considering various Sobolev–type weighted spaces, as e.g. x^{α}W_{k}^{p}, is their appropriateness for studying the evolution of the initial data, a problem which we plan to consider in the future. On the other hand Hölder–type weighted spaces x^{α}C_{k+λ} are intuitively more transparent, and the results in these spaces are easier to understand for non–experts; these spaces also seem to be somewhat better suited for proving results which can be used to draw conclusions about the interesting case of a smooth–up–to–boundary background — this is especially apparent in the case of the vector constraint equation. C_{k+λ}^{α} (≠ x^{α}C_{k+λ}!) and W_{k}^{α;p} (≠ x^{α}W_{k}^{p}!) arise as spaces on which the operators we consider are isomorphisms, for appropriate ranges of the exponents α. It turns out that for some critical values μ_{±} of the exponent α the solutions of the equations we consider “pick up” log terms, even though the source term did not have any. Since in the non–linear problem we need to iterate the results of the linear theory, we are forced to consider equations where the log weighted terms appear as sources: this leads us to introduce the spaces C_{k+λ}^{α,β} and W_{k}^{α,β;p} of functions weighted by a factor x^{α}| log x|^{β}. The “conormal–type” spaces x^{α}C_{k+λ|m} appear naturally when investigating in more detail the behaviour of the solutions near the boundary. Finally the spaces A^{phg} of “polyhomogeneous” functions are the spaces to which solutions of the equations considered here belong, in the case of smooth coefficients and smooth sources.

This paper is organized as follows: in Section 1 we briefly introduce the “hyperboloidal initial data problem”; for the sake of the reader interested mainly in the C^{∞} case we state in detail our main existence and regularity theorems under the hypothesis of a smooth background; results under various different differentiability hypotheses are stated in detail and proved in chapters 6 and 7. In Section 2 we shortly mention some generalizations of our results from the vacuum case to the case of matter fields. In Chapter 3 we establish our notations, describe the various function spaces, and prove or review some function–analytic results. Lemma 3.1 and Corollary 3.2 proved in Section 3 turn out to be very useful in our applications; we believe that these results are new.

In Chapter 4 various results concerning the regularity near the boundary are proved for a class of “edge–type” (cf. [58]) linear operators. More precisely, we consider elliptic systems of equations, with the ellipticity constants degenerating in a uniform way near the boundary, cf. eqs. (1.3)–(1.4). The results of that chapter generalize similar results of Mazzeo [58] because we do not need to assume smoothness of the “background” metric; the methods seem also to be rather simpler than those of ref. [58] (note, however, that the theory of [58] applies to more general operators and addresses more issues than what we do here). The main results there are Theorems 2.6, 2.9, 2.10 and 2.11. Those theorems are stated in a somewhat abstract form, but we show that their hypotheses are satisfied in the applications we have in mind. More precisely, Theorems 2.6, 2.9 and 2.10 are used in Chapter 7 to prove (a form of) classical regularity at the boundary of solutions to a class of scalar equations, cf. Corollary 2.3. Theorem 2.11 is used in Section 3 to prove (a form of) classical regularity at the boundary for the solutions of the equation considered there, cf. Theorem 3.11.

In Chapter 1 we prove polyhomogeneity of solutions of some fully non–linear elliptic equations, under a rather heavy set of hypotheses; these hypotheses are however satisfied for our (semi–linear) problem at hand, cf. Theorem 4.1 in Chapter 7. In Chapter 6 we construct classes of solutions of the vector constraint equation under various differentiability and decay conditions. We use two different approaches to do that; these approaches give the same space of solutions of the vector constraint equation in the polyhomogeneous or smooth case. We do not know whether or not this is true in the finite differentiability case, but we suspect that this is not the case. In Section 2 the problem is formulated in terms of an equation which does not degenerate at the boundary. The main results there are Theorems 2.7, 2.8 and 2.10. In Section 3 the problem is formulated in terms of an equation uniformly degenerating at the conformal boundary, of the type considered in Chapter 4. The main results there are Theorems 3.8, 3.9, 3.10 and 3.11.

In Chapter 7 solutions of the scalar constraint equation are constructed and their asymptotic properties are established. More precisely, in Corollary 2.3 we verify that the hypotheses made in the regularity theory developed in Chapter 4 are satisfied for Laplace–type scalar equations uniformly degenerating at the boundary. In Proposition 3.1 we prove existence of solutions to a class of semilinear equations uniformly degenerating at the boundary. Existence of solutions of the Lichnerowicz equation, Theorem 3.2, is a corollary of this result. As already mentioned above, in Theorem 4.1 we apply the results of Chapter 1 to prove polyhomogeneity of solutions to a class of semi–linear equations; the Lichnerowicz equation is again a special case of the equations considered in Theorem 4.1 (cf. Corollary 4.2). The finite differentiability counterpart of Theorem 4.1 is Theorem 4.5, which is perhaps the main result of this paper. Theorem 4.7 is a specialization of Theorem 4.5 to the case of the Lichnerowicz equation. In Appendix A we prove that log terms arise generically in the solutions of the constraint equations constructed by our method. The main results there are Theorems 2.2 and 2.3. Finally, in Appendix B we show how to construct coordinate systems near the boundary ∂M of M which, for many purposes, are as useful as Gauss coordinates, and in which no differentiability of the metric is lost^{3}. This result is a rather straightforward application of the extension results proved in Section 3. Due to the large number of function spaces involved, for the convenience of the reader we have included an index (both for terminology and for notation) at the end of the paper.

3Recall that one loses two degrees of differentiability of the metric when going to Gauss coordinates.

There is a non–empty intersection between some of our results and those of refs. [42, 57, 58, 59, 60], some similar methods have been used in [42]. Some of the arguments we use are standard for this type of problems: we believe, however, that the overall approach^{4} is new. Our methods involve only elementary techniques — scaling, commutation, difference quotients, and an analysis of properties of solutions of an ODE (with parameters). A key for a relatively simple proof of boundary regularity is an extension technique for extending finitely differentiable functions on the boundary to locally smooth functions defined on the whole manifold, cf. Section 3. The method is somewhat reminiscent of that used by Hörmander [45, Vol. III, Appendix B] in a Sobolev spaces setting. It must however be admitted that the proofs of our main results are somewhat complicated: the intricacies arise mainly from the finite differentiability hypotheses, and from rather general hypotheses on α, α_{±}, μ_{±} (cf. Chapter 4 for details). The proofs can be considerably simplified if one assumes smoothness of coefficients and sources (k = ∞; cf. e.g. [6]).

In the case when the conformal background metric is smooth, a special case of our main results — the boundary regularity of solutions of the Yamabe equation — has been independently established by R. Mazzeo [59] using quite different techniques^{5}. See also [32, 62] for some related results; cf. [54] for a survey of the Yamabe problem on compact manifolds and cf. [65] and references therein for some results about the Yamabe problem with non–constant prescribed curvature.

Acknowledgements: P.T.C. is grateful to N. Trudinger and to the members of the Centre for Mathematics and its Applications of the Australian National University for their friendly hospitality during a significant part of work on this paper. Useful discussions with J. Jezierski, G. Łysik, R. Mazzeo and B. Ziemian are acknowledged. We are grateful to J. Lee for making some of his work [52] available to us prior to publication, and to Jiseong Park for useful comments about previous versions of this paper.

4Once most of this paper was written we have been informed that some of our arguments are somewhat similar to those of [53]. It should, however, be stressed that most of the work in this paper is done to handle the finite differentiability hypotheses, while in [53] smoothness of the coefficients of the equations is assumed.

5It should be pointed out, however, that the work in [59] is done using stronger a–priori assumptions, cf. [32] for a discussion.

CHAPTER 2 The hyperboloidal initial value problem.

1. Conformal compactifications and Cauchy data

In this section we will recall the basic notions of the conformal framework introduced by Penrose [64] to describe the behaviour of physical fields at null infinity.

By a spacetime, we mean a connected C^{∞} 4-dimensional manifold endowed with a Lorentzian metric. Given a “physical” space-time (M̂, γ̂) one associates to it a smooth “unphysical space-time” (M, γ) and a smooth function Ω on M, such that M̂ is a subset of M and

Ω|M̂ > 0 , γ_{μν} |M̂ = Ω^{2}γ̂_{μν} ,  (1.1)
Ω|_{∂M̂} = 0 ,  (1.2)
dΩ(p) ≠ 0 for p ∈ ∂M̂ ,  (1.3)

where ∂M̂ is the topological boundary of M̂ in M. It is common usage in general relativity to use the symbol J^{(} for ∂M̂, and we shall sometimes do so.

We will use the convention that geometric objects in the physical spacetime M̂ defined w.r.t. the metric γ̂ are decorated with a “ˆ ”, for example R̂ denotes the scalar curvature defined w.r.t. the connection ˆ∇ and the metric γ̂.

For simplicity, we will only consider the case when (M̂, γ̂) is vacuum, i.e. R̂_{μν} = 0. In general, this assumption may be replaced by fall–off conditions on the matter fields. The hypothesis of smoothness of (M, γ, Ω) and the assumption that (M̂, γ̂) is vacuum imposes severe restrictions on the geometry of (M, γ, Ω). If one defines (cf. [64])

P_{μν} = \frac{1}{2} (R_{μν} − \frac{1}{6} Rγ_{μν}) ,  (1.4)

with an analogous definition for the quantities defined w.r.t. (M̂, γ̂), one has

0 = P̂_{μν} = P_{μν} + \frac{1}{Ω}∇_{μ} ∇_{ν} Ω − \frac{1}{2Ω^{2}} ∇^{α} Ω ∇_{α} Ω γ_{μν} ,  (1.5)

where ∇_{μ} is the covariant derivative of the metric γ_{μν}. Throughout this paper the following conventions on curvature are used:

∇_{α}∇_{β}X^{γ} − ∇_{β}∇_{α}X^{γ} = R^{γ}_{μαβ}X^{μ} ,
R_{αβ} = R^{γ}_{αγβ}, R = R^{γ}_{γ} .

Note that the signs in (1.5) are opposite to those of ref. [64] because of different curvature conventions used. Equations (1.2) and (1.5) imply

∇^{α} Ω ∇_{α} Ω|_{∂M̂} = 0 ,  (1.6)
(∇_{μ} ∇_{ν} Ω − \frac{1}{4} ∇^{α} ∇_{α} Ω γ_{μν})|_{∂M̂} = 0 .  (1.7)

Let M be a spacelike hypersurface with boundary ∂M in (M, γ) and assume that M ⊂ M̂ and ∂M ⊂ ∂M̂. Then we may write M̄ = M ∪ ∂M so that M̄ is the topological closure of M in M. Let us stress that M is a spacelike hypersurface in physical space–time, and this is the only exception to our rule that physically relevant objects are denoted with a hat. Alternatively, we have M̂ = M, but we never refer to M̂, only to M.

Let g_{ij}, K_{ij}, respectively ĝ_{ij}, K̂_{ij}, be the induced metric and extrinsic curvature of M in (M, γ), respectively in (M̂, γ̂). If we denote by L^{ij} and L̂^{ij} the traceless part of K^{ij} = g^{ik}g^{jℓ}K_{kℓ}, K̂^{ij} = ĝ^{ik}ĝ^{jℓ}K̂_{kℓ},

L^{ij} = K^{ij} − \frac{1}{3} K g^{ij} , K = g^{ij} K_{ij} ,
L̂^{ij} = K̂^{ij} − \frac{1}{3} K̂ ĝ^{ij} , K̂ = ĝ^{ij} K̂_{ij} ,  (1.8)

one finds

L̂^{ij} = Ω^{3} L^{ij} , |L̂|ĝ = Ω |L|_{g} ,
K̂ = ΩK − 3 n^{α} Ω_{,α} ,  (1.9)

where n^{α} is the unit normal to M for the metric γ, and | · |_{h} denotes the tensor norm in a Riemannian metric h. Since n^{α} is timelike and, by (1.6), ∇Ω(p) is null for p ∈ ∂M we have

K̂|_{∂M} = −3n^{α} Ω_{,α}|_{∂M} ^{>}_{<} 0 ,  (1.10)

because the scalar product of two non–vanishing non–spacelike vectors cannot change sign. From (1.2) we also have

g_{ij}|_{M} = Ω^{2} ĝ_{ij} ,

and since ∇Ω is null non–vanishing at ∂M the equations (1.9)–(1.10) imply

D^{i} Ω D_{i} Ω|_{∂M} = (\frac{K̂}{3} )^{2} |_{∂M} > 0 ,  (1.11)

where D_{i} is the Riemannian connection of the metric g_{ij}. To summarize, necessary conditions for an initial data set (M, ĝ_{ij}, K̂^{ij}) to arise from an “extended initial data set (M, g_{ij}, K^{ij}) intersecting a smooth J^{(} ” are

C1. There exists a Riemannian manifold (M, g) with smooth boundary ∂M and compact closure M̄ = M ∪ ∂M and a Riemannian metric g on M̄ with g ∈ C^{k}(M̄) for some k ≥ 2 (cf. Chapter 3 for precise definitions of function spaces). Moreover there exists a function Ω ∈ C^{k}(M̄) such that

g_{ij} = Ω^{2} ĝ_{ij} ,  (1.12)
Ω|_{∂M} = 0 ,  |DΩ|_{g}|_{∂M} > 0 .  (1.13)

C2. The symmetric tensor field K̂^{ij} satisfies, for some K̂ ∈ C^{k−1}(M̄) and L^{ij} ∈ C^{k−1}(M̄),

K̂^{ij} = Ω^{3}L^{ij} + \frac{1}{3} K̂ĝ^{ij} ,  K̂ = ĝ_{ij}K̂^{ij} ,  (1.14)
K̂|_{∂M} is nowhere vanishing .  (1.15)

The above conditions are necessary but far from sufficient, cf. [5, 4] for a detailed discussion. If there existed “a lot” of space-times satisfying the Penrose conformal conditions, there should exist “a lot” of initial data satisfying C1–C2. It is therefore natural to ask the question, can one construct such data sets? This involves constructing solutions of the scalar constraint equation,

R̂ + K̂^{2} − K̂_{ij} K̂^{ij} = 0 ,  (1.16)

where R̂ denotes the Ricci scalar of the metric ĝ, and the vector constraint equation,

D̂_{i}(K̂^{ij} − K̂ ĝ^{ij}) = 0 ,  (1.17)

where D̂ is the Riemannian connection of the metric ĝ, under appropriate asymptotic conditions. No general method of producing solutions of (1.16)–(1.17) is known, unless one assumes

C3.  D̂_{i}K̂ ≡ 0  (1.18)

(cf., however, [18, 50, 28, 14] for some results). Under (1.18) the scalar and the vector constraint equations decouple, and the well known Choquet–Bruhat — Lichnerowicz — York conformal procedure [20] allows one to construct solutions of (1.16)–(1.17). An initial data set satisfying C1–C3 will be called a C^{k} hyperboloidal initial data set (smooth if k = ∞), while conditions C1– C2 will be called Penrose’s C^{k} conditions. Without loss of generality we may normalize K̂ so that

K̂ = 3 ,  (1.19)

and (1.16)–(1.17) can be rewritten as

R̂ + 6 = L̂_{ij} L̂^{ij}  (1.20)
D̂_{i} L̂^{ij} = 0 .  (1.21)

To construct solutions of (1.20)–(1.21) one can proceed as follows: fix a Riemannian manifold (M, g) with smooth boundary ∂M and compact closure, and let x be any defining function for ∂M (by definition,

x|_{∂M} = 0 , |dx|_{g}|_{∂M} > 0 ,

and x(p) = 0 ⇒ p ∈ ∂M), set

g̃_{ij} = x^{−2} g_{ij} .

Given a smooth traceless symmetric tensor field B̃^{ij} on M satisfying

D̃_{i}(B̃^{ij}) = 0 ,  (1.22)

where D̃ is the Riemannian connection of the metric g̃, it is not too difficult to check that the fields

ĝ_{ij} = ϕ^{4}g̃_{ij} ,
L̂^{ij} = ϕ^{−10}B̃^{ij}

will satisfy (1.20)–(1.21) if

8Δ̃ϕ − R̃ϕ + λϕ^{−7} − 6ϕ^{5} = 0 ,  (1.23)

where

λ ≡ |B̃|2 ; g̃ ≡ g̃_{ij} g̃_{kℓ} B̃^{ik}B̃^{jℓ} ,

and where Δ̃ = D̃^{i}D̃_{i} is the Laplacian of the metric g̃_{ij}. If ϕ and B̃ are smooth up to boundary and if moreover it holds that

ϕ|_{∂M} = 1 ,  (1.24)
λ|_{∂M} = 0 ,  (1.25)

then (M, ĝ_{ij}, K̂^{ij}) will satisfy C1–C3 (with k = ∞).

Note that (1.25) forces the coordinate components of B̃^{ik} to vanish to third order at ∂M, in coordinates regular near ∂M. It is therefore natural to introduce a new tensor field B^{ij} defined as

B^{ij} ≡ x^{−3}B̃^{ij} .  (1.26)

It turns out that the condition that ϕ and B̃ be smooth up to boundary leads to non–trivial restrictions. One of the main results of this paper is the following (cf. Corollary 4.2, Chapter 7 and Theorem 2.3 of Appendix A; cf. also [5] for some related results):

Theorem 1.1. For any smooth (M, g_{ij}, x, B^{ij}) as above there exists a unique solution of (1.23)–(1.24). Further,

((i)) For given M and x there exists an open dense set (in the C^{∞}(M̄) topology) of (g_{ij}, B^{ij})’s for which the function ϕ^{−2} can be extended to a C^{2} function from M to M̄, but not to a C^{3} function on M̄ (the third derivatives of any extension of ϕ will blow up logarithmically as one approaches ∂M); in particular for generic (in the above sense) triples (g_{ij}, B^{ij}) the initial data set (ĝ_{ij}, K̂^{ij}) will display asymptotic behaviour incompatible with Penrose’s C^{3} conditions.

((ii)) There exists a “large set” of non-generic (g_{ij}, B^{ij}) for which Ω ≡ ϕ^{−2} x satisfies Ω ∈ C^{∞}(M̄).

It should be emphasized that in Theorem 1.1 no hypotheses on the topology of M, or ∂M̂ are made, thus the resulting space-time may have a conformal boundary consisting of several connected components of varying topology (recall that e.g. some Robinson–Trautman space-times admit a smooth J^{(} the “spatial” topology of which is not a sphere [33, 24]). Let us also note that even considering only those data sets for which B = 0, or for which B^{ij} vanishes on ∂M to some desired order, point (i) above will still hold in the sense that for generic g and B’s vanishing to some prescribed order (or even e.g. identically vanishing) no C^{3} extensions of ϕ from M̂ to M will exist.

To complete the construction of initial data sets one also has to produce solutions of (1.22), the standard approach proceeds as follows: Let A^{ij} ∈ C^{∞}(M̄) be a smooth trace–free symmetric tensor field and set Ã^{ij} = x^{3}A^{ij}. Let X^{i} solve the equation

D̃_{j}(D̃^{i}X^{j} + D̃^{j}X^{i} − \frac{2}{3}D̃_{k}X^{k}g̃^{ij}) = −D̃_{j}Ã^{ij} .  (1.27)

Then the tensor field defined by

B̃^{ij} = Ã^{ij} + D̃^{i}X^{j} + D̃^{j}X^{i} − \frac{2}{3}D̃_{k}X^{k}g̃^{ij}  (1.28)

will satisfy (1.22).

In Chapter 6 the existence and regularity of solutions to the equation (1.27) is studied. In fact, in that chapter we present two different methods of constructing solutions of (1.27). In particular, the following is established:

Theorem 1.2. Let (M, g) be a smooth Riemannian manifold, with boundary ∂M and with compact closure and let x be a smooth defining function of ∂M. Then, given a trace–free symmetric two–tensor A^{ij} ∈ C^{∞}(M̄), there is a unique solution X^{i} to the equation (1.27) of the form

X^{i} = x^{2}X_{1}^{i} + x^{4} log(x) X_{2}^{i} ,
X_{1}^{i}, X_{2}^{i} ∈ C^{∞}(M̄) .

It should be pointed out that for generic A^{ij}, the source term in (1.27) will be generic and thus the corresponding solution X will have log terms, consequently B^{ij} given by (1.26) with B̃^{ij} given by (1.28) will be C^{1} but not C^{2} extendible from M to M̄. If on the other hand A^{ij} vanishes to order two or higher at the boundary, then no log terms occur in the solution of (1.27), cf. Appendix A. This condition is sufficient but not necessary for non–existence of log terms in X^{i}, cf. e.g. [5] for details.

In order to obtain initial data which can be used in Friedrich’s stability theorem [35] (cf. also [19] for a somewhat different approach) further restrictions on (ĝ_{ij}, K̂^{ij}) are needed, among others the assumption that both the tensor field

e_{αβ} ≡ ∇_{α}∇_{β}Ω − \frac{1}{4}γ^{μν}∇_{μ}∇_{ν}Ωγ_{αβ}

and the Weyl tensor C^{α}_{βγδ} vanish on ∂M. Here e_{αβ} and C^{α}_{βγδ} are evaluated formally from the Cauchy data (ĝ_{ij}, K̂^{ij}) assuming vacuum Einstein equations. The vanishing at ∂M of the tensor e_{αβ} corresponds to the condition that J^{(} is “shear free”, cf. e.g. [5]. The vanishing of C^{α}_{βγδ} on ∂M implies some further conditions on both L^{ij} and the metric g_{ij}. In the case when J^{(} is shear free and L^{ij} vanishes on ∂M, it turns out [5] that the Weyl tensor vanishes on ∂M precisely when the restriction of Ω^{−1}L^{ij} to ∂M is proportional to the induced metric on ∂M.

Point 1 of Theorem 1.1 thus shows that generic data constructed by the conformal method will not be regular enough to be used in Friedrich’s existence theorems. In fact the problem here is much more serious than just being one or two degrees of differentiability away from a threshold, because one of the fields used in Friedrich’s “conformally regular system” is d^{α}_{βγδ} ≡ x^{−1}C^{α}_{βγδ}. Whenever C^{α}_{βγδ}(p) ≢ 0 for p ∈ ∂M̂, the field d^{α}_{βγδ} blows up at ∂M̂ as 1/x, and is thus not even in L^{1}(M). It should be stressed that nevertheless point 2 of Theorem 1.1 establishes existence of a large class of non–trivial data with asymptotic behaviour compatible with the Penrose–Friedrich conditions.

2. Some remarks on non–vacuum initial data sets

In addition to the vacuum case discussed in Section 1, two matter models have been studied from the point of view of the “conformal Einstein equations”: The Einstein – Yang–Mills system has been analyzed in the conformal setting in [36].

The Einstein – scalar field model, with a massless scalar field “minimally coupled” to the metric, has been recently studied by P. Hübner [46, 47]. In those last references P. Hübner has generalized^{1} Friedrich’s results concerning existence of time–developments “with a piece of J^{(} ”, or with a semi–global J^{(} , to the scalar field case. In this section we wish to describe shortly how our results on vacuum initial data generalize to non–vacuum models.

First, it should be noted that the construction given here for the vacuum, zero cosmological constant case yields immediately initial data for Einstein equations with a non–vanishing cosmological constant with the appropriate sign. This requires only a reinterpretation of what the physical initial data are, once the solutions of the constraint equations, as described in Section 1, have been obtained. (We have, however, not analyzed in detail the question, under what conditions the initial data so obtained will satisfy all the conditions needed for the well posedness of the evolution problem in the conformal setting; cf. [37, Section 5.1] for partial results concerning this question.)

Next, let us point out that the results of Chapter 6 clearly allow for the introduction of sources in the vector constraint equation. Similarly, the existence Theorem 3.2 proved in Chapter 7 holds for rather quite a large class of couplings of the gravitational field with some matter sources (cf. e.g. [51] for a discussion of the conformal method for Einsteins equations with matter). In particular, in the Einstein – Yang–Mills case the scalar constraint equation takes the form

\frac{4(n − 1)}{n − 2}Δg̃ϕ − R̃ϕ + ∑_{i=1}^{I} ξ_{i}ϕ^{−γ_{i}} − n(n − 1)ϕ^{\frac{n+2}{n−2}} = 0 ,  (2.1)

with I = 2, γ_{1} = 3, γ_{2} = 7, ξ_{i} ≥ 0, n = 3. The method of proof of Theorem 3.2 proves existence of solutions of equation (2.1) satisfying the asymptotic condition (3.3) when the initial data for the Yang-Mills field are suitably behaved, cf. also Theorem 3.1. Similarly, the arguments of Corollary 4.2, which asserts polyhomogeneity of solutions of the Lichnerowicz equation, carry over immediately to the Einstein – Yang–Mills case, so that if the initial data for the Yang–Mills fields are smooth or polyhomogeneous on M̄, then the corresponding solution of the Lichnerowicz equation will be polyhomogeneous, cf. Theorem 4.1.

As discussed in the proof of the existence Theorem 3.2, one can always choose a conformal gauge in which R̃ = −n(n − 1) = −6. Then, as made explicit in the statement of Proposition 3.1, the existence argument applies to equations of the form

Δg̃ϕ + F (y, ϕ) = 0 ,

with a large class of functions F. In particular the assertions of Theorem 3.2 hold in the case of a scalar field ψ minimally coupled to the gravitational field, in the following sense: For this model, assuming for simplicity that the extrinsic curvature is pure trace, the Lichnerowicz equation reads

8Δg̃ϕ + (6 − \frac{κ}{2}|dψ|2 ; g̃)ϕ − (6 − \frac{κ}{2}(∇_{0}ψ)^{2})ϕ^{5} = 0 ,  (2.2)

where ∇_{0}ψ is the derivative of the scalar field ψ in the direction normal to the initial hypersurface, κ is the gravitational constant and we have assumed that R̃ = −6.

1P. Hübner [46] has also numerically analyzed the evolution problem at Scri for such a spherically symmetric model. Some related numerical results have been obtained by R. Gómez and J. Winicour [41]; cf. also M. W. Choptuik [16, 15].

Suppose that there exists c > 0 such that

6 − \frac{κ}{2}|dψ|2 ; g̃ ≥ c ,  6 − \frac{κ}{2}(∇_{0}ψ)^{2} ≥ c  (2.3)

on M̄, and suppose further that there exist constants C, α > 0 such that

|dψ|2 ; g̃ + (∇_{0}ψ)^{2} ≤ Cx^{α}

for x small enough. Then Proposition 3.1 together with the arguments of the proof of Theorem 3.2 guarantees existence of a solution ϕ of (2.2) satisfying the asymptotic boundary condition (3.3). (Here any sufficiently small constant can be used as the constant C_{−} of Proposition 3.1, and any sufficiently large constant can be used as the constant C_{+} of that Proposition.) This, together with the analysis of [63] proves existence of solutions of the scalar constraint equation with appropriate asymptotic conditions for a large class of couplings of the scalar field ψ to the gravitational field, in particular for the conformally invariant coupling. The arguments which we present in Chapter 7 can be used to infer further regularity properties of ϕ. More precisely, if the initial data for the scalar field decay at ∂M and are smooth (or polyhomogeneous) on M̄, then the corresponding solution ϕ of the Lichnerowicz equation will be polyhomogeneous. Let us also mention that (2.3), which is needed in our existence proof, is not needed for the regularity argument to go through.

CHAPTER 3 Definitions, Preliminary Results.

1. Function spaces

Let M be a smooth, paracompact, Hausdorff, n-dimensional manifold with compact smooth boundary ∂M. We will denote by M̄ = M ∪ ∂M the closure of M. Unless otherwise indicated, M̄ is not assumed to be compact. Note that with these conventions M is open.

Throughout this paper x will denote a defining function for ∂M, i.e. a smooth (up to boundary) function satisfying x|_{∂M} = 0, x ≥ 0, |dx|_{g}|_{∂M} > 0, and the implication x(p) = 0 ⇒ p ∈ ∂M holds. The symbol i_{∂M} denotes the natural embedding of ∂M in M̄, i_{∂M} : ∂M → M̄.

We can always choose a finite number of coordinate charts

ϕ_{i} : O_{i} → ℝ^{n,+} ≡ {y ∈ ℝ^{n} : y^{1} ≥ 0} , i = 1, …, I ,

covering a neighbourhood of ∂M such that ((i)) y^{1} = x. ((ii)) ϕ_{i}(O_{i}) = [0, x_{0}) × U_{i}, for some U_{i} ⊂ ℝ^{n−1}, 0 < x_{0} ≤ 1. ((iii)) The transition functions ϕ_{i}^{−1} ∘ ϕ_{j} are x-independent. ((iv)) For every p ∈ ⋃_{i=1}^{I} O_{i} such that x(p) < \frac{x_{0}}{2} there exists i(p) such that

the coordinate ball B(p, \frac{x(p)}{2}) centered at p of radius x(p)/2 is contained in O_{i(p)}.

The symbol ∂Ω will always denote the topological boundary of the set Ω: ∂Ω ≡ Ω̄ \ Ω, where Ω̄ is the closure of Ω. Ω^{∘} will sometimes be used to denote the interior of Ω. We set ˜∂Ω ≡ ∂Ω \ ∂M. We shall write Ω′⊂⊂Ω if Ω̄′ ⊂ Ω.

For 0 < σ ≤ x_{0} we set M_{σ} = {p ∈ M : 0 < x(p) < σ}, CM_{σ} = M\M̄_{σ} ≡ {p ∈ M : x(p) > σ}. We thus have ˜∂M_{σ} = {p ∈ M : x(p) = σ}, ∂M_{σ} = ˜∂M_{σ} ∪ ∂M. For 0 < σ < ρ ≤ x_{0} we also define M_{σ,ρ} ≡ {p ∈ M : σ < x(p) < ρ}. Decreasing x_{0} if necessary we may assume that for 0 < σ ≤ x_{0} the sets ˜∂M_{σ} are smooth manifolds.

When referring to coordinates on M_{x_{0}} we shall implicitly assume that points (i)–(iv) above hold; we shall use the letter v to denote the coordinates y^{2}, …, y^{n};

v^{A} = y^{A} , A = 2, …, n .

Thus

y = (x, v) .

Using the above coordinate system, we shall often identify functions on ∂M with functions on M_{x_{0}}, or on M̄_{x_{0}}, similarly for tensors, etc. If useful in the context, functions f on ∂M can also be extended to functions on M by assigning to f the function ϕ(x/x_{0})f(v), where ϕ ∈ C_{∞}(ℝ) is any function satisfying ϕ(x) ∈ [0, 1], ϕ(x) = 1 for x ∈ [0, 1/2], supp(ϕ) ⊂ [−1, 1], where here and throughout supp(f) denotes the support of f. Sometimes it is, however, useful to use better behaved extensions of f to M, cf. Lemma 3.1 and Corollary 3.2.

The standard Schwartz multi-index notation is used throughout, thus if α = (α_{1}, … α_{n}), then |α| = ∑_{i=1}^{n} α_{i} denotes the length of α and

∂^{α} = ∂_{y}^{α} = ∂_{y^{1}}^{α_{1}} ⋯ ∂_{y^{n}}^{α_{n}} = ∂_{x}^{α_{1}}∂_{y^{2}}^{α_{2}} ⋯ ∂_{y^{n}}^{α_{n}} = ∂_{x}^{α_{1}}∂_{v}^{β} ,

where β = (α_{2}, …, α_{n}). Further, we will write (x∂_{y})^{α} = x^{|α|}∂_{y}^{α}.

ℕ denotes the set of natural numbers, ℕ = {1, 2, … }, ℕ_{0} = ℕ ∪ {0}, ℕ^{∞} = ℕ ∪ {∞}, ℕ_{0}^{∞} = ℕ_{0} ∪ {∞}. An integer, without further qualification, is taken to be a number in ℕ_{0}. However, sometimes integers are defined as numbers in ℕ. For this reason when talking about an integer in ℕ_{0} we shall indicate this explicitly, whenever the distinction matters and ambiguities are likely to occur.

Let F_{1}, F_{2} be two function spaces. f will be said to belong to F_{1} + F_{2} if there exist functions f_{i} ∈ F_{i}, i = 1, 2, such that f = f_{1} + f_{2}.

1.1. (Non–weighted) Hölder spaces. Let Ω be an open subset of M. For k ∈ ℕ_{0} the spaces C_{k}(Ω), C_{k+λ}(Ω) are the spaces of k times differentiable functions on Ω; and in the C_{k+λ}(Ω) case the derivatives of order k satisfy a Hölder continuity condition with exponent λ ∈ (0, 1]. To emphasize the local character of the spaces C_{k+λ}(Ω) we shall sometimes write C_{k+λ}^{loc}(Ω) for C_{k+λ}(Ω), whenever confusion is likely to occur. Following standard notation, (cf. e.g. [1]), for k ∈ ℕ_{0} the symbol C^{k}(Ω̄) denotes the space of those functions in C_{k}(Ω) the derivatives of which up to order k can be extended by continuity to continuous functions on Ω̄; we equip C^{k}(Ω̄) with the supremum norm. The spaces C_{k}(Ω̄), C_{k+λ}(Ω̄) are the spaces of functions differentiable k-times on Ω and equipped with the supremum norm; and in the C_{k+λ}(Ω̄) case the derivatives of order k satisfy a uniform Hölder continuity condition with exponent λ ∈ (0, 1]. For λ > 0 we have C_{k+λ}(Ω̄) = C^{k+λ}(Ω̄), where C^{k+λ}(Ω̄) is defined in the usual way as in e.g. [1]; however C_{k}(Ω̄) ≠ C^{k}(Ω̄) in general, because functions in C_{k}(Ω̄) do not necessarily extend to the boundary of Ω together with all derivatives: In particular C_{0}(Ω̄) ≠ C(Ω̄), where C(Ω̄) is the usual space of continuous functions on Ω̄ (an example is given by the function sin(1/x) which is in C_{0}(M̄_{x_{0}}), while it is not in C(M̄_{x_{0}})). Our notation in which the differentiability index is put “downstairs” is mainly motivated by the fact that it is natural to put “upstairs” those indices which correspond to fall–off properties of functions near ∂M. As shown in the above discussion, this convention on the differentiability index also helps to avoid a possible confusion between the spaces C_{k}(M̄) and C^{k}(M̄). Note also that a function in C_{k}(Ω) or C_{k+λ}(Ω) is not necessarily bounded up to the boundary ∂Ω, as opposed to a function in C_{k}(Ω̄) or C_{k+λ}(Ω̄). C^{∘} _{k}(Ω), C^{∘} _{k+λ}(Ω), etc., denotes the space of C_{k}, C_{k+λ}, etc., functions on Ω which vanish in a neighbourhood of ∂Ω. When λ = 1 the space C_{k+λ} will always be denoted by “C_{k+λ}, λ = 1”, to avoid possible confusions with the space C_{k+1}. “C_{k+λ}, λ = 0” will sometimes be used to denote the space C_{k}, similarly for “C_{k+λ}^{α,β}, λ = 0” (the spaces C_{k+λ}^{α,β} are defined below), etc. We set C_{∞}(Ω̄) ≡ ∩_{k}C_{k}(Ω̄).

From the Whitney extension theorem (cf. e.g. [40, Lemma 6.37]) any f ∈ C_{k+λ}(M̄), λ ∈ (0, 1] can be C_{k+λ} extended across ∂M; this is however not necessarily so if f ∈ C_{k}(M̄) (λ = 0).

1.2. Weighted Hölder spaces. Let Ω be an open subset of M. For α, β ∈ ℝ and k ∈ ℕ_{0} we denote by C_{k}^{α,β}(Ω) the space of those functions in C_{k}(Ω) for which the norm

∥f∥_{C_{k}^{α,β}(Ω)} = ∥f∥_{C_{k}(\overline{Ω\M_{x_{0}/2}})} + sup_{{0≤|γ|≤k \atop (x,v)∈Ω∩Mx_{0}}} (1 + | ln x|)^{−β} x^{−α+|γ|} |∂^{γ}f(x, v)|

is finite. For λ ∈ (0, 1] we denote by C_{k+λ}^{α,β}(Ω) the space of those functions in C_{k}^{α,β}(Ω) for which the norm

∥f∥_{C_{k+λ}^{α,β}(Ω)} = ∥f∥_{C_{k}^{α,β}(Ω)} + ∥f∥_{C_{k}+λ(\overline{Ω\M_{x_{0}/2}})}

+ sup_{{y∈Mx_{0} ∩Ω \atop |γ|=k}} sup_{y′∈B(y, \frac{x(y)}{2})∩Ω ; y′≠y} (1 + | ln x|)^{−β} x^{−α+k+λ} \frac{|∂^{γ}f(y) − ∂^{γ}f(y′)|}{|y − y′|^{λ}}

is finite. We set C_{k+λ}^{α}(Ω) = C_{k+λ}^{α,0}(Ω). Although we shall generally avoid such a notation, we shall occasionally write C_{k}^{α}, C_{k+λ}^{α}, C_{k}, C_{k+λ}, etc., for C_{k}^{α}(M), C_{k+λ}^{α}(M), C_{k}(M), C_{k+λ}(M), when confusion is unlikely to occur. Note that C_{k}^{0}(M) ≠ C_{k}(M), C_{k+λ}^{0}(M) ≠ C_{k+λ}(M), and that for σ > 0 functions in C_{k+λ}^{α,β}(M_{σ}) are, together with their derivatives up to order k, bounded near ˜∂M_{σ} (i.e. that part of the boundary of M_{σ} which does not coincide with ∂M).

Let F be a function space, and let ρ be a function. By ρF we shall denote the space of functions f such that ρ^{−1}f ∈ F. If F has a norm, we set

∥f∥_{ρF} = ∥ρ^{−1}f∥_{F} .

We have e.g. x^{α}(1 + | log x|^{2})^{β/2} C_{k+λ}^{0} = C_{k+λ}^{α,β} (this is easily seen by scaling a ball around (x, v) of radius x/2 to a ball of radius 1/2).

We shall say that f ∈ x^{∞}F if for all i ∈ ℕ, f ∈ x^{i}F.

1.3. Sobolev spaces. For p ∈ [1, ∞), L^{p}(Ω, dμ) denotes the space of μ– measurable functions defined μ-almost everywhere, the p-th power of which is integrable on Ω with the measure dμ. dμ_{g} will always denote the measure associated with a Riemannian metric g (in local coordinates, dμ_{g} = \sqrt{det g_{ij}} dy^{1} … dy^{n}). The symbol L^{p}(Ω) or L^{p} will be used to denote L^{p}(Ω, dμ) when dμ is the Lebesgue measure d^{n}y = dx d^{n−1}v in local coordinates as described at the beginning of this chapter (or a measure equivalent to it in an L^{∞} sense, e.g. dμ_{g}, where g is a metric uniformly elliptic in the above coordinates).

Let Ω be an open subset of M, let g_{ij} be a metric which is uniformly elliptic in local coordinates near ∂M, let dμ be a measure on M. For α, β ∈ ℝ, p ∈ [1, ∞),

k ∈ ℕ_{0} we define W_{k}^{p}(Ω, g, dμ) (respectively W^{∘} _{k}^{p}(Ω, g, dμ)) as the completion of C^{∘} _{∞}(M) (respectively of C^{∘} _{∞}(Ω)) in the norm

∥f∥_{W_{k}^{p}(Ω,g,dμ)}^{p} = ∑_{0≤ℓ≤k} ∫_{Ω} (|D^{ℓ}f|_{g})^{p} dμ ,

where D is the Riemannian connection of the metric g, and | |_{g} denotes the norm of a tensor with respect to the metric g. We shall sometimes write W_{k}^{p}(Ω) for W_{k}^{p}(Ω, g, dμ) when dμ is uniformly equivalent to the Lebesgue measure in local coordinates near the boundary, as described at the beginning of this chapter. We shall say that f ∈ W_{k}^{p,loc}(Ω, g, dμ) if f ∈ W_{k}^{p}(Ω′, g, dμ) for all Ω′⊂⊂Ω. Similarly we define W_{k}^{α,β;p}(Ω, g, dμ) (respectively W^{∘} _{k}^{α,β;p}(Ω, g, dμ)) as the completion of C^{∘} _{∞}(M) (respectively of C^{∘} _{∞}(Ω)) in the norm

∥f∥_{W_{k}^{α,β;p}(Ω,g,dμ)}^{p} = ∑_{0≤ℓ≤k} ∫_{Ω} (x^{−α+ℓ}(1 + | ln x|)^{−β} |D^{ℓ}f|_{g})^{p} dμ .

Note that if the metric g is in C_{k}(M̄), then in local coordinates we have

0 ≤ |γ| ≤ k ∫_{0}^{x_{0}} dx ∫ dv (x^{−α+|γ|}(1 + | ln x|)^{−β} |∂_{y}^{γ}f|)^{p} ≤ C∥f∥_{W_{k}^{α,β;p}(M_{x_{0}} ,g,dμ_{g})}^{p} ,

for some (f-independent) constant C. The space W_{k}^{α,β;p}(Ω, g, x^{−n}dμ_{g}) will some-

times be denoted by W_{k}^{α,β;p}(Ω, g), similarly W^{∘} _{k}^{α,β;p}(Ω, g) ≡ W^{∘} _{k}^{α,β;p} (Ω, g, x^{−n}dμ_{g}),

W_{k}^{α;p} ≡ W_{k}^{α,0;p} etc. We shall also write L^{α,β;p} for W_{0}^{α,β;p}.

The inclusion of the factor x^{−n} in the definition of W_{k}^{α,β;p}(M, g) has the unfortunate consequence that we have W_{0}^{α,β;p}(M, g) = x^{α+n/p}(1+| ln x|)^{β} L^{p}(M, d^{n}y) ≠ x^{α}(1+| ln x|^{β}) L^{p}(M, d^{n}y) when g is uniformly elliptic in local coordinates near the boundary. Even though the latter would seem more natural, this is however more than compensated by the simplicity of the Hölder inclusion:

W_{k}^{α,β;p}(M, g) ⊂ C_{k−n/p}^{α,β}(M) ,  0 < k − n/p ∉ ℕ  (1.1)

(this is easily seen by a scaling argument (cf. e.g. the proof of Lemma 1.1) and the standard Hölder embedding, cf. also [3]).

If g̃_{ij} = x^{−2}g_{ij}, then dμg̃ = x^{−n} dμ_{g}, so that W_{k}^{α,β;p}(M, g) = W_{k}^{α,β;p}(M, g, dμg̃). For g̃ of this form we shall also define H_{k}^{α,β}(Ω, g̃) as the completion of C^{∘} _{∞}(M) with respect to the norm

∥X∥_{H_{k}^{α,β}(Ω,g̃)} = \{ ∑_{0≤ℓ≤k} ∫_{Ω} (|D̃^{ℓ}X|g̃ x^{−α}(1 + | ln x|)^{−β})^{2} dμg̃\}^{\frac{1}{2}} ,

where D̃ is the Riemannian covariant derivative operator of the metric g̃; with

H_{k}^{α} ≡ H_{k}^{α,0}, similarly H^{∘} _{k}^{α,β}(Ω, g̃) is the completion of C^{∘} _{∞}(Ω) in this norm. It is easy to show that for functions

H_{k}^{α,β}(M, g̃) = W_{k}^{α,β;2}(M, g) ,

this will however not be the case for general tensor fields.

1.4. Nonisotropic function spaces. Let Ω be an open subset of M, let k, m ∈ ℕ_{0}, λ ∈ [0, 1]. We shall say that f ∈ C_{k+λ|m}(Ω̄), if f ∈ C_{k+λ}(Ω̄) ∩ C_{k+m+λ}^{loc}(Ω) and if for all 0 ≤ i + |γ| ≤ m we have (x∂_{x})^{i} ∂_{v}^{γ} f ∈ C_{k+λ}(\overline{Ω ∩ M_{x_{0}}}). We set

∥f∥_{C_{k+λ|m}(Ω̄)} = ∥f∥_{C_{k+λ}(Ω̄)} + ∑_{0≤i+|γ|≤m} ∥(x∂_{x})^{i} ∂_{v}^{γ} f∥_{C_{k+λ}(\overline{Ω∩M_{x_{0}}} )} .

Note that C_{k+m+λ}(Ω̄) ⊂ C_{k+λ|m}(Ω̄), C_{k+λ|0}(Ω̄) = C_{k+λ}(Ω̄).

Let Ω be an open subset of M. We define C_{k+λ,0}^{α,β}(Ω) ≡ C_{k+λ}^{α,β}(Ω), with the appropriate norm. For μ ∈ (0, 1] we shall say that f ∈ C_{k+λ,0+μ}^{α,β}(Ω) if f ∈ C_{k+λ}^{α,β}(Ω) and if moreover there exists a constant C such that for 0 ≤ |γ| ≤ k and for all (x, v), (x, v′) ∈ Ω ∩ {[0, x_{0}) × U_{i}}, |v − v′| > 0, i = 1, …, I, we have

|(x∂_{y})^{γ}(f(x, v) − f(x, v′))| ≤ Cx^{α}(1 + | ln x|)^{β} |v − v′|^{μ} .

We define

∥f∥_{C_{k+λ,0+μ}^{α,β}(Ω)} = ∥f∥_{C_{k+λ}^{α,β}(Ω)} + sup x^{−α}(1 + | ln x|)^{−β} \frac{|(x∂_{y})^{γ}(f(x, v) − f(x, v′))|}{|v − v′|^{μ}} ,

where the sup is taken over (x, v), (x, v′) ∈ Ω∩{[0, x_{0})×U_{i}}, |v−v′| > 0, i = 1, …, I, and 0 ≤ |γ| ≤ k. f will be said to belong to C_{k+λ,m+μ}^{α,β}(Ω) if f ∈ C_{k+λ,0+μ}^{α,β}(Ω) and if for 0 ≤ |γ| ≤ m we have ∂_{v}^{γ}f ∈ C_{max(0,k−|γ|+λ),0+μ}^{α,β}(Ω ∩ M_{x_{0}}). Note that if m ≤ k this is equivalent to the condition that for 0 ≤ i + |γ| ≤ m we have (x∂_{x})^{i}∂_{v}^{γ}f ∈ C_{k−|γ|−i+λ,0+μ}^{α,β}(Ω ∩ M_{x_{0}}). We set

∥f∥_{C_{k+λ,m+μ}^{α,β}(Ω)} = ∥f∥_{C_{k+λ,0+μ}^{α,β}(Ω)} + ∑_{0≤|γ|≤m} ∥∂_{v}^{γ}f∥_{C_{max(0,k−|γ|+λ),0+μ}^{α,β}(Ω∩M_{x_{0}} )} ,

and we define C_{k+λ,m+μ}^{α}(Ω) ≡ C_{k+λ,m+μ}^{α,0}(Ω), with the obvious norm.

Although they will not be used any further, we would like to point out that the following spaces of functions arise rather naturally in the context of the problems considered here: Let k, m ∈ ℕ_{0} and λ, μ ∈ (0, 1] be such that m + μ ≤ k + λ. Then f ∈ C_{k+λ}^{loc}(Ω) will be said to belong to A_{k+λ,m+μ}^{α,β}(Ω) if the norm defined by

∥f∥_{A_{k+λ,m+μ}^{α,β}(Ω)} = ∥f∥_{C_{k+λ,m}^{α,β}(Ω)} + sup \frac{∥f(x, v + z) − f(x, v)∥_{C_{k+λ−μ,m}^{α,β}(Ω∩M_{x_{0}} )}}{|z|^{μ}}

is finite. Here the supremum is taken over z ∈ ℝ^{n−1} such that

(x, v), (x, v + z) ∈ ϕ_{i}(Ω ∩ O_{i}), |z| > 0 .

[Strictly speaking, f(x, v+z)−f(x, v) is not a function on M, so that the expression ∥f(x, v + z) − f(x, v)∥_{C_{k+λ−μ,m}^{α,β}(Ω∩M_{x_{0}} )} above has to be understood in the sense of coordinate patches in the obvious way. Alternatively one could use the difference– quotients construction of the beginning of Chapter 4.] Indeed, the spaces A_{k+λ,m+μ}^{α,β} possess better interpolation properties than the spaces C_{k+λ,m+μ}^{α,β}. While the use of the latter spaces introduces some complications in some of our arguments, some other arguments get simpler. Moreover, one can prove an isomorphism theorem in the A_{k+λ,m+μ}^{α,β} spaces similar to Theorem 2.1; this is not the case in the C_{k+λ,m+μ}^{α,β} spaces. There are, however, no essential overall simplifications gained in the proof by the usage of the A_{k+λ,m+μ}^{α,β} spaces. Further, the results for C_{k+λ,m+μ}^{α,β} spaces are somewhat more general than those in the A_{k+λ,m+μ}^{α,β} spaces, the latter being proper subsets of the former. For those reasons we have decided to use the C_{k+λ,m+μ}^{α,β} spaces in our constructions.

Proposition 1.1 (Taylor formula). Let f ∈ C_{k+λ}(M̄_{x_{0}}), λ ∈ (0, 1]. There exist functions f_{i}, r_{i}, i = 0, …, k, such that for 0 ≤ ℓ ≤ k we have

f = ∑_{i=0}^{ℓ} f_{i}x^{i} + r_{ℓ} ,
f_{i} ∈ C_{k−i+λ}(∂M) ,

∀ 0 ≤ i ≤ ℓ ∂_{x}^{i}r_{ℓ} ∈ x^{ℓ−i} C_{k−ℓ+λ}(M̄_{x_{0}}) ∩ x^{ℓ−i+λ}C_{0|k−ℓ}(M̄_{x_{0}}) ,  (1.2)

Proof: We have f_{i}(v) = \frac{1}{i!} ∂_{x}^{i} f(0, v), r_{0} = f(x, v) − f(0, v), and for ℓ ≥ 1

r_{ℓ} = ∫_{0}^{x} dx_{1} ∫_{0}^{x_{1}} dx_{2} … ∫_{0}^{x_{ℓ−1}} dx_{ℓ}(∂_{x}^{ℓ}f(x_{ℓ}, v) − ∂_{x}^{ℓ}f(0, v)) .  (1.3)

For 0 ≤ i ≤ ℓ the property ∂_{x}^{i}r_{ℓ} ∈ x^{ℓ−i} C_{k−ℓ+λ} is established by straightforward estimations, ∂_{x}^{i}r_{ℓ} ∈ x^{ℓ−i+λ} C_{0|k−ℓ} follows from a change of variables x_{i} → s_{i} given by x_{i} = xs_{1} … s_{i}:

∂_{x}^{i}r_{ℓ} = x^{ℓ−i} ∫_{0}^{1} ds_{1} … ∫_{0}^{1} ds_{ℓ−i}[s_{1}^{ℓ−i−1}s_{2}^{ℓ−i−2} ⋯ s_{ℓ−i−1}(∂_{x}^{ℓ}f(s_{1} ⋯ s_{ℓ−i}x, v)−∂_{x}^{ℓ}f(0, v))] .

□

Let F be a function space over a set Ω, let B be a tensor bundle over Ω. A tensor field X = (X^{A}_{β}), where A, β are some multi-indices, |A| = r, |β| = s, will be said to belong to Γ_{F} (B) if in local coordinates as described at the beginning of this chapter the components X^{A}_{β} of X are in F. Whenever confusion is unlikely to occur we shall write X ∈ F rather than X ∈ Γ_{F} (B). Let us note that for tensor fields X ∈ C_{k...}^{α,β} or X ∈ W_{k...}^{α,β;p}(M, g), k > n/p, where g is a metric uniformly elliptic in local coordinates near ∂M, the decay index α, β always corresponds to the decay of |X|_{g} in our notation, cf. (1.1). This, in turn, corresponds to the behaviour of the components of X in the naturally preferred coordinate systems near ∂M, as discussed at the beginning of this chapter. It should be pointed out that this is not true for X ∈ H_{k}^{α}(M, g̃) unless X is a function (cf. the beginning of Section 3).

1.5. Polyhomogeneous functions. Let f_{i} be a sequence of functions, f_{i} ∈ C_{∞}(M), such that for every N ∈ ℕ and for all |α| ≤ N we have

|∂_{y}^{α}f_{i}| ≤ C_{i,N}x^{s_{i}} ,

for some sequence s_{i} →_{i→∞} ∞ and some constants C_{i,N}. We shall write

f ∼ ∑_{i=0}^{∞} f_{i}

if for every n, m ∈ ℕ there exists N ∈ ℕ and a constant C(n, m) such that for 0 < x ≤ x_{0} and for all |α| ≤ m

|∂_{y}^{α}(f − ∑_{i=0}^{N} f_{i})| ≤ C(n, m) x^{n} .

Let I ∈ ℕ_{0}^{∞}, let the sequence {(s_{i}, {N_{ij}}_{j=0}^{∞})}_{i=0}^{I}, s_{i} ∈ ℝ, N_{ij} ∈ ℕ_{0}, satisfy s_{i+1} > s_{i}. f will be said to be polyhomogeneous, f ∈ A^{{(s_{i},N_{ij})}_{i=0}^{I}}, if there exists a sequence of functions f_{ijk} ∈ C_{∞}(M̄) such that, for 0 < x ≤ min(x_{0}, \frac{1}{2}),

f ∼ ∑_{i=0}^{I} ∑_{j=0}^{∞} ∑_{k=0}^{N_{ij}} f_{ijk} x^{s_{i}+j} ln^{k} x .  (1.4)

We set A^{{s_{i}}_{i=0}^{I}} ≡ ∪_{{N_{ij}}}A^{{(s_{i},{N_{ij}})}_{i=0}^{I}}, A^{phg} ≡ ∪_{s_{i},I}A^{{s_{i}}_{i=0}^{I}}. In what follows we shall need the following Lemma, the proof of which is a straightforward generalization of the proof of Borel’s Lemma (cf. e.g. [45, Volume 1]):

Lemma 1.2 (Borel Lemma). Let I ∈ ℕ_{0}^{∞}, {(s_{i}, {N_{ij}}_{j=0}^{∞})}_{i=0}^{I}, s_{i} ∈ ℝ, N_{ij} ∈ ℕ_{0}, with s_{i+1} > s_{i}, suppose that f̃_{ijk} ∈ C_{∞}(∂M), let f_{ijk} ∈ C_{∞}(M̄) be any functions such that (f_{ijk} − f̃_{ijk})|_{O_{ijk}} = 0, for some neighbourhoods O_{ijk} ⊂ ∪_{i}O_{i} of ∂M. (Here the sets O_{i} are as defined at the beginning of this Chapter, and the f_{ijk}’s have been extended to the neighbourhood ∪_{i}O_{i} of ∂M by setting ∂f_{ijk}/∂x = 0 in some coordinate system as considered at the beginning of this Chapter.) There exists a function f ∈ C_{∞,∞}^{s_{0},N_{00}} ∩ A^{phg} such that (1.4) holds. If N_{ij} = 0, ∀i, j, then f ∈ C_{∞}(M̄).

2. Some embeddings

Proposition 2.1. Suppose that Ω = M, or Ω = M_{τ}, 0 < τ ≤ x_{0}, let f ∈ x^{α} C_{k+λ|m}(Ω̄).

((i))

0 ≤ |β_{1}| ≤ m , 0 ≤ |β_{2}| ≤ k ∂_{v}^{β_{1}+β_{2}}f ∈ x^{α} C_{k−|β_{2}|+λ|m−|β_{1}|}(Ω̄) . (2.1)

((ii)) Let g ∈ x^{α_{1}} C_{k_{1}+λ_{1}|m_{1}}(Ω̄), then

gf ∈ x^{α+α_{1}} C_{k_{2}+λ_{2}|m_{2}}(Ω̄) ,  (2.2)
where ∗_{2} = min(∗, ∗_{1}), ∗ = k, m, λ, in particular
g ∈ x^{α_{1}} C_{k+m+λ}(Ω̄) ⇒ fg ∈ x^{α+α_{1}} C_{k+λ|m}(Ω̄) .  (2.3)

Suppose that Ω = M, or Ω = M_{τ}, 0 < τ ≤ x_{0}, let f ∈ C_{k+λ,m+μ}^{α,β}(Ω).

((iii))

0 ≤ |γ| ≤ k  (x∂_{y})^{γ}f ∈ C_{k−|γ|+λ,min(m,k−|γ|)+μ}^{α,β}(Ω) .  (2.4)

((iv))

0 ≤ |γ| ≤ m  ∂_{v}^{γ}f ∈ C_{max(0,k−|γ|+λ),m−|γ|+μ}^{α,β}(Ω) .  (2.5)

((v)) Let g ∈ C_{k_{1}+λ_{1},m_{1}+μ_{1}}^{α_{1},β_{1}}(Ω), then

fg ∈ C_{k_{2}+λ_{2},m_{2}+μ_{2}}^{α+α_{1},β+β_{1}}(Ω) .  (2.6)
with ∗_{2} = min(∗, ∗_{1}), ∗ = k, m, λ, μ, in particular
g ∈ x^{α_{1}} C_{k+λ}(Ω) ⇒ fg ∈ C_{k+λ,min(k+λ,m+μ)}^{α+α_{1},β}(Ω) .  (2.7)

Proof: All the results here are elementary, let us simply point out that Eq. (2.1) and (2.5) are a consequence of

[∂_{v}^{α}, ∂_{x}^{i}] = [∂_{v}^{α}, (x∂_{x})^{i}] = 0 ,
[∂_{x}^{i}, (x∂_{x})^{j}] = ∑_{0≤ℓ≤j−1} d_{ijℓ} ∂_{x}^{i}(x∂_{x})^{ℓ} ,

with some constants d_{ijℓ}. Equation (2.3) follows from the inclusion

x^{α_{1}} C_{k+m+λ}(Ω̄) ⊂ x^{α_{1}} C_{k+λ|m}(Ω̄) .

Eq. (2.7) follows from (2.6) and the inclusion

x^{α_{1}} C_{k+λ}(Ω̄) ⊂ C_{k+λ,k+λ}^{α_{1}}(Ω) .
□

Proposition 2.2. Let Ω = M, or Ω = M_{τ}, 0 < τ ≤ x_{0}. The following inclusions are continuous

((i)) μ, λ ∈ [0, 1], α ≥ 0,

C_{0+μ,0+λ}^{α}(Ω) ↪ C_{0+σ}(Ω̄),  σ = min(α, μ, λ) .  (2.8)

((ii)) μ, λ ∈ [0, 1], α ≥ 0, k, ℓ, m ∈ ℕ_{0}, k − ℓ ≥ 0

C_{k+m+μ,k−ℓ+λ}^{ℓ+α}(M) ↪ ∩_{i=0}^{m}x^{−i}C_{ℓ+i+σ|k−ℓ}(M̄) ,  σ = min(α, μ, λ) . (2.9)

((iii)) j ∈ ℕ_{0}

x^{j} C_{k+λ|m+j}(Ω̄) ↪ C_{k+j+λ|m}(Ω̄) .  (2.10)

Proof: (2.10):

f ∈ x^{j} C_{k+λ|m+j} ⟹ 0 ≤ i + |γ| ≤ m + j ∂_{v}^{γ} x^{i}∂_{x}^{i}x^{−j}f ∈ C_{k+λ}
⟹ 0 ≤ i_{1} + |γ_{1}| ≤ m , 0 ≤ i_{2} + |γ_{2}| ≤ j ∂_{v}^{γ_{1}}x^{i_{1}}∂_{x}^{i_{1}}∂_{v}^{γ_{2}}x^{i_{2}}∂_{x}^{i_{2}}x^{−j}f ∈ C_{k+λ}
⟹ 0 ≤ i_{1} + |γ_{1}| ≤ m , 0 ≤ i_{2} + |γ_{2}| ≤ j ∂_{v}^{γ_{1}}x^{i_{1}}∂_{x}^{i_{1}}∂_{v}^{γ_{2}}∂_{x}^{i_{2}}f ∈ C_{k+λ}
⟹ f ∈ C_{k+j+λ|m}

(2.8): From the inclusion

C_{0+λ,0+μ}^{α} ⊂ C_{0+λ_{1},0+μ}^{α_{1}} , 0 ≤ λ_{1} ≤ λ , α_{1} ≤ α ,  (2.11)

it follows that without loss of generality we can suppose f ∈ C_{0+λ_{1},0+μ}^{α_{1}}, with 0 < α_{1} ≤ 1, λ_{1} = min(α_{1}, λ). Let x ≤ y, we have

|f(x, v) − f(y, w)| ≤ A + B ,
A = |f(x, v) − f(y, v)|
B = |f(y, v) − f(y, w)| .

B is estimated in a straightforward way,

B ≤ Cy^{α_{1}}|v − w|^{μ} ≤ Cx_{0}^{α_{1}}|v − w|^{μ} .

To estimate A, suppose first that x ∈ (y/2, y), which leads to

A ≤ Cx^{α_{1}−λ_{1}}|x − y|^{λ_{1}} ≤ Cx_{0}^{α_{1}−λ_{1}}|x − y|^{λ_{1}} .
Let now x ∈ [y/2^{n+1}, y/2^{n}), n ≥ 1; note that
\frac{y}{2^{n}} ≤ y(1 − \frac{1}{2^{n}} ) ≤ y − x ≤ y(1 − \frac{1}{2^{n+1}} ) .  (2.12)

We have

A = A_{1} + A_{2} ,
A_{1} = |f(x, v) − f(\frac{y}{2^{n}} , v)|
A_{2} = |f(y, v) − f(\frac{y}{2^{n}} , v)|

3. EXTENSIONS OF FUNCTIONS DEFINED ON ∂M 25

and from (2.12) it follows that

A_{1} = C(\frac{y}{2^{n}} )^{α_{1}−λ_{1}} |x − \frac{y}{2^{n}} |^{λ_{1}}

≤ C(\frac{y}{2^{n}} )^{α_{1}−λ_{1}} |\frac{y}{2^{n+1}} − \frac{y}{2^{n}} |^{λ_{1}}
= 2^{−λ_{1}}C(\frac{y}{2^{n}} )^{α_{1}} ≤ 2^{−λ_{1}}C|x − y|^{α_{1}} .

On the other hand

|f(y, v) − f(\frac{y}{2^{n}} , v)| ≤ ∑_{i=0}^{n−1} |f(\frac{y}{2^{i}} , v) − f(\frac{y}{2^{i+1}} , v)|  (2.13)

≤ C ∑_{i=0}^{n−1} (\frac{y}{2^{i}} )^{α_{1}−λ_{1}} |\frac{y}{2^{i}} − \frac{y}{2^{i+1}} |^{λ_{1}}  (2.14)

= \frac{2^{−λ_{1}}C}{1 − 2^{−α}} (1 − \frac{1}{2^{nα_{1}}} )y^{α_{1}} .  (2.15)

Recall that for α_{1} ∈ (0, 1] we have

a ≥ b ≥ 0 ⟹ a^{α_{1}} − b^{α_{1}} ≤ (a − b)^{α_{1}}  (2.16)

(a simple proof of (2.16) can be given as follows: for 0 < μ < 1 and 0 ≤ x ≤ y consider ϕ(x, y) = (y − x)^{μ} − y^{μ} + x^{μ}, we have ϕ(x, x) = 0 and for 0 < x < y, \frac{∂ϕ}{∂y} ≥ 0, thus ϕ ≥ 0). The inequality (2.16) gives

1 − \frac{1}{2^{nα_{1}}} ≤ (1 − \frac{1}{2^{n}})^{α_{1}} ,  (2.17)

which together with (2.12) and (2.15) implies

A_{2} ≤ C′(1 − \frac{1}{2^{nα_{1}}} )y^{α_{1}}
≤ C′[(1 − \frac{1}{2^{n}} )y]^{α_{1}} ≤ C′|y − x|^{α_{1}} ,

and (2.8) follows. (2.9) is a straightforward consequence of (2.8). □

3. Extensions of functions defined on ∂M

We shall use the following Lemma concerning extensions of functions defined on ∂M (cf. [45, Vol. III, Appendix B] for a similar approach in Sobolev spaces).

Lemma 3.1 (Extension Lemma:). Let k ∈ ℕ_{0}, λ ∈ [0, 1], consider ψ ∈ C^{k+λ}(∂M). For all m ∈ ℕ_{0} there exists a function ψ̃_{m} ∈ C^{k+m+λ}(M̄) ∩ C_{∞,k+λ}^{m}(M) (⟹ ψ̃_{m} ∈ C_{∞}^{loc}(M)) satisfying

0 ≤ i ≤ m ∂_{x}^{i}ψ̃_{m}|_{∂M} = \{ \begin{matrix}0, & i ≤ m, \\ ψ, & i = m. \end{matrix}

Moreover for all multi–indices α we have

(x∂_{y})^{α}ψ̃_{m} ∈ C^{k+m+λ}(M̄) ∩ C_{∞,k+λ}^{m}(M) ,  (3.1)

and it also holds that

(x∂_{y})^{α}ψ̃_{m} − ψ (x∂_{y})^{α} x^{m} ∈ C_{k,k}^{m+λ}(M) ,  (3.2)

x∂_{v}ψ̃_{m} ∈ C_{∞,k}^{m+λ}(M) .  (3.3)

Proof: A simple partition of unity argument shows that it is sufficient to establish the result for ∂M = ℝ^{n−1} and ψ — compactly supported. Let ϕ ∈ C^{∞}(ℝ^{n−1}) be any compactly supported function satisfying

∫_{ℝ^{n−1}} ϕ(v)d^{n−1}v = 1 .

Let χ ∈ C^{∞}(ℝ) be any function satisfying supp χ ⊂ (−x_{0}, x_{0}), χ|_{[−x_{0}/2,x_{0}/2]} = 1. Set

E[ψ](x, v) = x^{−(n−1)} ∫_{ℝ^{n−1}} ϕ(\frac{v − w}{x})ψ(w)d^{n−1}w,  (3.4)

ψ̃_{m}(x, v) = \frac{x^{m}}{m!} χ(x) E[ψ](x, v) .  (3.5)

From (3.4) one immediately has E[ψ] ∈ C_{∞,k+λ}^{0}(M_{x_{0}}), so that (3.5) gives ψ̃_{m} ∈ C_{∞,k+λ}^{m}(M) ⊂ C_{∞}^{loc}(M). Changing the integration variable w → z = \frac{v−w}{x} in (3.4) we have

0 ≤ i + |γ| ≤ k ∂_{x}^{i}∂_{v}^{γ}E[ψ](x, v) = ∫_{ℝ^{n−1}} ϕ(z) ∂_{x}^{i}∂_{v}^{γ}ψ(v − xz)d^{n−1}z. (3.6)

Reverting to the integration variable w in (3.6) one concludes ψ̃_{m} ∈ C_{∞,k+λ}^{m}(M), and Proposition 2.2 gives ψ̃_{m} ∈ C_{k+m}(M̄). To obtain the stronger statement ψ̃_{m} ∈ C^{k+m+λ}(M̄) some more work is needed. Assume that 0 ≤ i + |γ| ≤ m; from what has been said it follows that it is sufficient to consider the case k = 0. From (3.4)–(3.5) we obtain by direct differentiation

0 < x ≤ x_{0}/2, ∂_{x}^{i}∂_{v}^{γ} ψ̃_{m}(x, v) = ∫_{ℝ^{n−1}} x^{m−i−|γ|−(n−1)}χ_{i,γ}(\frac{v − w}{x})ψ(w)d^{n−1}w , (3.7)

for some compactly supported functions χ_{i,γ} ∈ C^{∞}(ℝ^{n−1}). Changing integration variables (3.7) becomes

∂_{x}^{i}∂_{v}^{γ} ψ̃_{m}(x, v) = x^{m−i−|γ|} ∫ χ_{i,γ}(z)ψ(v − xz)d^{n−1}z .  (3.8)

The property ∂_{x}^{i}∂_{v}^{γ} ψ̃_{m} ∈ C^{0}(M̄) follows from Lebesgue dominated convergence theorem. If λ > 0 the further property ∂_{x}^{i}∂_{v}^{γ} ψ̃_{m} ∈ C^{0+λ}(M̄) follows from (3.8) by straightforward estimations. For 0 < x ≤ x_{0}/2 we also have, for all multi–indices α,

(x∂_{y})^{α}ψ̃_{m}(x, v) = ∫_{ℝ^{n−1}} ψ(w)(x∂_{y})^{α}\{x^{m−(n−1)}ϕ(\frac{v − w}{x})\}d^{n−1}w
= ∫_{ℝ^{n−1}}\{ψ(w) − ψ(v) + ψ(v)\}(x∂_{y})^{α}\{x^{m−(n−1)}ϕ(\frac{v − w}{x})\} d^{n−1}w
= ∫_{ℝ^{n−1}}\{ψ(w) − ψ(v)\}(x∂_{y})^{α}\{x^{m−(n−1)}ϕ(\frac{v − w}{x})\} d^{n−1}w

+ψ(v) ∫_{ℝ^{n−1}}(x∂_{y})^{α}\{x^{m−(n−1)}ϕ(\frac{v − w}{x})\} d^{n−1}w

= ∫_{ℝ^{n−1}}\{ψ(w) − ψ(v)\}(x∂_{y})^{α}\{x^{m−(n−1)}ϕ(\frac{v − w}{x})\} d^{n−1}w

+ψ(v)(x∂_{y})^{α} ∫_{ℝ^{n−1}} x^{m−(n−1)}ϕ(\frac{v − w}{x}) d^{n−1}w

= ∫_{ℝ^{n−1}}\{ψ(w) − ψ(v)\}(x∂_{y})^{α}\{x^{m−(n−1)}ϕ(\frac{v − w}{x})\} d^{n−1}w

+ψ(v)(x∂_{y})^{α} x^{m} ,

and (3.2)–(3.3) easily follow. (3.1) is established by a similar simpler calculation. □

Corollary 3.2. Let k ∈ ℕ_{0}, λ ∈ [0, 1].
((i)) For 0 ≤ i ≤ k let f_{i} ∈ C^{k−i+λ}(∂M). There exists f ∈ C^{k+λ}(M̄) ∩

C_{∞}^{loc}(M) such that

∂_{x}^{i}f|_{∂M} = f_{i} .

Moreover for all multi–indices α we have

(x∂_{y})^{α}f ∈ C^{k+λ}(M̄) .  (3.9)

((ii)) Let ĝ ∈ C^{k+λ}(M̄). There exists f ∈ C^{k+λ}(M̄) ∩ C_{∞,k+λ}^{0}(M) such that

f − g ∈ C_{k+λ,0+λ}^{k}(M) ∩ C_{k+λ}^{k+λ}(M) .  (3.10)

Proof: 1. We shall proceed by induction. Suppose thus that we have proved the existence of a function f satisfying our claims for 0 ≤ i ≤ ℓ < k. To start the induction, for ℓ = −1 set f ≡ 0. For ℓ ≥ −1 define

ψ_{ℓ+1} = ∂_{x}^{ℓ+1}f|_{∂M} ∈ C^{k−ℓ−1+λ}(∂M) .

The induction step is obtained by replacing f with

f + f̃_{ℓ+1} − ψ̃_{ℓ+1} ,

where f̃_{ℓ+1}, ψ̃_{ℓ+1} are obtained from f_{ℓ+1} and of ψ_{ℓ+1} using Lemma 3.1 with m = ℓ + 1.

2. Let f be obtained from part 1 of this Corollary using the functions f_{i} = ∂_{x}^{i}g|_{∂M} ∈ C^{k−i+λ}(∂M). Now (3.10) is equivalent to

0 ≤ i + |γ| ≤ k ∂_{x}^{i}∂_{v}^{γ}(f − g) ∈ C_{0+λ,0+λ}^{k−i−|γ|}(M) ∩ C_{0+λ}^{k−i−|γ|+λ}(M) .

Choose some γ satisfying 0 ≤ |γ| ≤ k; since f − g has vanishing Taylor coefficients at ∂M the result follows for λ > 0 from eq. (1.2) with ℓ = k − |γ| of Proposition 1.1. If λ = 0 the result follows in a similar way from (1.3). □

Let us recall that Whitney’s extension Lemma is usually proved in a C^{k+λ}(M̄) context, with λ > 0. Corollary 3.2 can be used to prove the equivalent of the Whitney Lemma for functions in C^{k}(M̄) when the boundary ∂M is a sufficiently differentiable manifold (as is the case here); it also gives a rather elementary proof of that Lemma in our context.

As an application of the results here, in Appendix B we present a construction of “almost Gaussian” coordinates near ∂M. The construction leads to coordinates which for many purposes are as convenient as the “real” ones. Moreover, even^{1} in the case where finite differentiability of the metric is assumed, the construction leads to a coordinate system in which the coefficients of the metric tensor are of the same differentiability class as in the original coordinates.

4. Mapping properties of some integral operators

A significant role in our approach to boundary regularity in Section 2 is played by the mapping properties of the integral operators

I_{a}^{μ}(f)(x, v) = x^{μ} ∫_{a}^{x} s^{−1−μ}f(s, v) ds ,  (4.1)

x ∈ (0, x_{0}] , v ∈ ∂M , a ∈ [0, x_{0}] , μ ∈ ℝ ,

I^{μ}(f)(v) = x^{μ} ∫_{0}^{x_{0}} s^{−1−μ}f(s, v) ds ,

G_{a} = \frac{1}{μ_{+} − μ_{−}} (I_{a}^{μ_{+}} − I_{0}^{μ_{−}}) , μ_{−} < μ_{+} , μ_{±} ∈ ℝ ,

which arise from fundamental solutions for some ordinary differential equations, cf. the beginning of Section 2. We have the following results:

Lemma 4.1. The following maps are continuous:

((i)) α ≤ μ, λ, λ′ ∈ [0, 1]

I_{x_{0}}^{μ} : C_{0+λ,0+λ′}^{α,β}(M_{x_{o}}) → C_{0+λ,0+λ′}^{α,β′}(M_{x_{o}}) , β′ = β if α < μ;  (4.2)

β′ = β + 1 if α = μ and β ≠ −1; β′ > 0 arbitrary if α = μ and β = −1. ((ii)) α > μ , λ, λ′ ∈ [0, 1]

I_{x_{0}}^{μ} : C_{0+λ,0+λ′}^{α,β}(M_{x_{o}}) → C_{0+λ_{1},0+λ′}^{μ}(M_{x_{o}}) , λ_{1} = λ for λ ≠ α − μ , (4.3)

and if λ = α − μ then λ_{1} is arbitrary in [0, λ). ((iii)) α > μ , λ, λ′ ∈ [0, 1]

I_{0}^{μ} : C_{0+λ,0+λ′}^{α,β}(M_{x_{o}}) → C_{0+λ′′,0+λ′}^{α,β}(M_{x_{o}}) , λ′′ = min(λ, λ′) .  (4.4)

((iv)) α > μ , λ ∈ [0, 1] ,

I^{μ} : C_{0,0+λ}^{α,β}(M_{x_{o}}) → x^{μ} C_{0+λ}(∂M) .  (4.5)

((v)) μ ≥ 0 , λ ∈ [0, 1] , μ + λ > 0 ,

I_{x_{0}}^{μ} : {f ∈ C_{0+λ}(M̄_{x_{o}}), f|_{∂M} = 0} → C_{0+λ′}(M̄_{x_{o}}) ,  (4.6)

for some λ′(μ, λ) ∈ [0, λ], λ′ > 0 if λ > 0.

1Recall that one loses two degrees of differentiability of the coefficients of the metric tensor when transforming the metric to the exact Gauss coordinate system.

((vi)) μ ≤ 0 , λ ∈ [0, 1] , −μ + λ > 0 ,

I_{0}^{μ} : {f ∈ C_{0+λ}(M̄_{x_{o}}) , f|_{∂M} = 0} → C_{0+λ′}(M̄_{x_{o}}) ,  (4.7)

for some λ′(μ, λ) ∈ [0, λ], λ′ > 0 if λ > 0.

Proof: For λ = 0 the proof of (4.2)–(4.4) is a straightforward estimation; in that case continuity of I_{0}^{μ}(f) in v in (4.4) follows from Lebesgue’s dominated convergence theorem. To establish Hölder continuity when λ > 0, let us note that

I_{a}^{μ}(f)(x, v) − I_{a}^{μ}(f)(x′, v′) = A + B , a = 0, x_{0} ,  (4.8)
A = I_{a}^{μ}(f)(x, v) − I_{a}^{μ}(f)(x′, v) ,
B = I_{a}^{μ}(f)(x′, v) − I_{a}^{μ}(f)(x′, v′) .

The estimation of B is straightforward, while A can be estimated using e.g. a scaling argument. To prove (4.6), the not entirely trivial point is to estimate the term A from (4.8). We have, for 0 < x ≤ y ≤ x_{0}

I_{x_{0}}^{μ}(f)(y, v) − I_{x_{0}}^{μ}(f)(x, v) = A_{1} + A_{2}
A_{1} = −(y^{μ} − x^{μ}) ∫_{y}^{x_{0}} s^{−1−μ}f(s, v) ds ,

A_{2} = x^{μ} ∫_{x}^{y} s^{−1−μ}f(s, v) ds .  (4.9)

Since f vanishes at ∂M, we have |f(x, v)| ≤ ||f||_{C_{0+λ}(M̄_{xo})}x^{λ}. Suppose first that 0 ≤ μ ≤ 1, recall that

y^{μ} − x^{μ} ≤ (y − x)^{μ} ,  (4.10)

(cf. (2.16)). For μ = 0, A_{1} = 0 and the result follows immediately from (4.9). If 0 < μ ≤ 1, setting λ′ = min(μ, λ) we have

|A_{1}| ≤ C(x_{0}^{λ−μ} + y^{λ−μ})(y − x)^{μ}
= C(x_{0}^{λ−μ} + y^{λ−μ})(y − x)^{μ−λ′}(y − x)^{λ′}
≤ C′(y^{μ−λ′} + y^{λ−λ′})(y − x)^{λ′}
≤ C′(x_{0}^{λ−λ′} + x_{0}^{μ−λ′})(y − x)^{λ′} .

On the other hand if μ > 1 then

|A_{1}| ≤ C(x_{0}^{λ−μ} + y^{λ−μ})(y^{μ} − x^{μ})
≤ C(x_{0}^{λ−μ} + y^{λ−μ})μy^{μ−1}(y − x)
≤ μC(x_{0}^{λ−1} + y^{λ−1})(y − x)^{1−λ}(y − x)^{λ}
≤ 2μC(y − x)^{λ} .

If λ − μ > 0 the estimation |A_{2}| ≤ C(y − x)^{λ−μ} follows from (4.10); if λ − μ ≤ 0 we can without loss of generality assume λ − μ < 0. If −1 ≤ λ − μ < 0 we have, with λ′ = min(μ − λ, λ),

|A_{2}| ≤ Cx^{μ}(x^{λ−μ} − y^{λ−μ}) = Cx^{λ}y^{λ−μ}(y^{μ−λ} − x^{μ−λ})
≤ Cx^{λ}y^{λ−μ}(y − x)^{λ′}(y − x)^{μ−λ−λ′} ≤ Cx^{λ}y^{−λ′}(y − x)^{λ′}
≤ Cx_{0}^{λ−λ′}(y − x)^{λ′} .

while in the case λ − μ < −1 the estimation proceeds as follows:

|A_{2}| ≤ Cx^{λ}y^{λ−μ}(y^{μ−λ} − x^{μ−λ})
≤ C(μ − λ)x^{λ}y^{−1}(y − x)
≤ C(μ − λ)x^{λ}y^{−λ}(y − x)^{λ} ≤ C(μ − λ)(y − x)^{λ} .

(4.7) is proved by similar methods, (4.5) is straightforward. □

For the purposes of this paper only the information contained in the Lemma above and in equation (4.14) below will be needed. For completeness, and for future reference, we wish however to point out the following consequences of Lemma 4.1:

Corollary 4.2. Let i, k, m ∈ ℕ_{0}. The following maps are continuous ((i)) α ≤ μ , λ, λ′ ∈ [0, 1]

I_{x_{0}}^{μ} : C_{k+λ,m+λ′}^{α,β}(M_{x_{o}}) → C_{k+λ,m+λ′}^{α,β′}(M_{x_{o}})  (4.11)

β′ = β if α < μ; β′ = β + 1 if α = μ and β ≠ −1; β′ > 0 arbitrary if α = μ and β = −1.

((ii)) α > μ, λ, λ′ ∈ [0, 1]

I_{x_{0}}^{μ} : C_{k+λ,m+λ′}^{α,β}(M_{x_{o}}) → C_{k+λ′′,m+λ′}^{μ}(M_{x_{o}}) ,  (4.12)

with some λ′′ ∈ [0, λ], λ′′ > 0 if λ > 0. ((iii)) α > μ, λ, λ′ ∈ [0, 1], let k_{0} be any integer such that 0 ≤ k_{0} ≤ k and

k_{0} − m < α − μ,

I_{0}^{μ} : C_{k+λ,m+λ′}^{α,β}(M_{x_{o}}) → C_{k_{0}+λ_{1},m+λ′}^{α,β}(M_{x_{o}})  (4.13)

for some λ_{1} ∈ [0, min(λ, λ′)]; λ_{1} = 0 if k_{0} > m, λ_{1} > 0 if λλ′ > 0 and k_{0} ≤ m.

((iv)) α > μ, λ, λ′ ∈ [0, 1], let k_{0} be any integer such that 0 ≤ k_{0} ≤ k and k_{0} − m < α − μ,

I^{μ} : C_{k,m+λ}^{α,β}(M_{x_{o}}) → x^{μ}C_{m+λ}(∂M) ∩ x^{μ}C_{k_{0}}(∂M) .  (4.14)

((v)) α < μ, λ ∈ [0, 1] let ℓ_{0} be any integer such that μ − α > ℓ_{0} ≥ 0

I_{x_{0}}^{μ} : x^{α}C_{k+λ|m}(M̄_{x_{o}}) → x^{α}C_{min(k,ℓ_{0})+λ′|m}(M̄_{x_{o}})  (4.15)

for some λ′ ∈ [0, λ], λ′ > 0 if λ > 0. ((vi)) α > μ, λ ∈ [0, 1]

I_{0}^{μ} : x^{α}C_{k+λ|m}(M̄_{x_{o}}) → x^{α}C_{k+λ′|m}(M̄_{x_{o}}) ,  (4.16)

for some λ′ ∈ [0, λ], λ′ > 0 if λ > 0. ((vii))

I_{x_{0}}^{μ} : A^{phg} → A^{phg} .  (4.17)

((viii)) α > μ

I_{0}^{μ} : A^{phg} ∩ C_{0}^{α}(M_{x_{0}}) → A^{phg} .  (4.18)

Proof: (4.11)–(4.14) follow by straightforward induction from (4.2)–(4.5). To prove (4.15), suppose first that k ≤ ℓ_{0}, choose γ, δ satisfying 0 ≤ |γ| ≤ m, 0 ≤ |δ| ≤ k, set β = δ + γ, define f̃ ≡ ∂_{v}^{β}x^{−α}f, thus f̃ ∈ C_{k−|δ|+λ| |γ|}. Taylor expanding f̃ up to order ℓ = k − |δ| we have from Proposition 1.1

f̃(x, v) = ∑_{i=0}^{ℓ} x^{i}f̃_{i}(v) + r̃_{ℓ} , f_{i} ∈ C_{k−i−|δ|+λ}(∂M) ,

∀ 0 ≤ i ≤ ℓ ∂_{x}^{i}r_{ℓ} ∈ x^{ℓ−i}C_{0+λ} ∩ x^{ℓ−i+λ}C_{0} ,  (4.19)

so that

I_{x_{0}}^{μ−α}(f̃) = ∑_{i=0}^{ℓ} \frac{x^{i}f̃_{i}(v)}{μ − α − i} + x^{μ−α} ∑_{i=0}^{ℓ} \frac{x_{0}^{i−μ+α}f̃_{i}(v)}{i − μ + α} + I_{x_{0}}^{μ−α}(r̃_{ℓ}) .

(4.6) and (4.19) imply by induction

0 ≤ p ≤ ℓ ∂_{x}^{p}I_{x_{0}}^{μ−α}(r̃_{ℓ}) ∈ C_{0+λ′} ,

for some λ′ ∈ [0, λ], λ′ > 0 if λ > 0, which gives ∂_{x}^{p}I_{x_{0}}^{μ−α}(f̃) ∈ C_{0+λ′} for 0 ≤ p ≤ ℓ, therefore for 0 ≤ |γ| ≤ m we have ∂_{v}^{γ}I_{x_{0}}^{μ}(f) ∈ x^{α}C_{k+λ′}. From

(x∂_{x})^{j}I_{a}^{μ} = μ^{j}I_{a}^{μ} + ∑_{p+q=j−1}(x∂_{x})^{p}μ^{q}f  (4.20)

the result follows for k ≤ ℓ_{0}. The inclusion

k ≥ ℓ_{0} C_{k+λ|m}(M̄_{x_{o}}) ⊂ C_{ℓ_{0}+λ|m}(M̄_{x_{o}})

reduces the case k ≥ ℓ_{0} to the previous one. (4.16) is proved in a similar way using (4.7). (4.17) and (4.18) are elementary. □

Proposition 4.3. Let i, k ∈ ℕ_{0}, λ ∈ [0, 1]. There exists λ′ ∈ [0, λ], λ′ > 0 unless λ = 0, such that the following maps are continuous.

((i)) −k ≤ j ∈ ℤ, max(μ_{−}, μ_{−} + j) < α ≤ μ_{+},

G_{x_{0}} : C_{k+j+λ,k+λ}^{α,β}(M_{x_{o}}) → C_{k+j+λ′,k+λ}^{α,β′}(M_{x_{o}}) ,  (4.21)

β′ = β if α < μ_{+}; β′ = β + 1 if α = μ_{+} and β ≠ −1; β′ > 0 arbitrary

if α = μ_{+}, β = −1.
((ii)) α > μ_{+}, α > μ_{−} + j, j ∈ ℕ_{0},
G_{x_{0}} : C_{k+j+λ,k+λ}^{α,β}(M_{x_{o}}) → C_{k+j+λ′,k+λ}^{μ_{+}}(M_{x_{o}}) .  (4.22)
((iii)) j = 0, 1
G_{0} : C_{j+λ}^{μ_{+}+1}(M_{x_{o}}) → C_{0}^{μ_{+}+1}(M_{x_{o}}) .  (4.23)
((iv)) j = 1, 2
G_{0} : C_{k+j+λ,k+λ}^{μ_{+}+j}(M_{x_{o}}) → C_{k+j−1+λ′,k+λ}^{μ_{+}+j}(M_{x_{o}}) .  (4.24)
((v)) j ∈ ℕ_{0}, μ_{+} + j < α,
G_{0} : C_{k+j+λ,k+λ}^{α,β}(M_{x_{o}}) → C_{k+j+λ′,k+λ}^{α,β}(M_{x_{o}}) .  (4.25)
((vi)) μ_{−} < α < μ_{+}, let ℓ_{0} be any integer such that μ_{+} − α > ℓ_{0},
G_{x_{0}} : x^{α}C_{k+λ|m}(M̄_{x_{o}}) → x^{α}C_{min(k,ℓ_{0})+λ′|m}(M̄_{x_{o}}) .  (4.26)
((vii)) α > μ_{+},
G_{0} : x^{α}C_{k+λ|m}(M̄_{x_{o}}) → x^{α}C_{k+λ′|m}(M̄_{x_{o}}) .  (4.27)

32 3. DEFINITIONS, PRELIMINARY RESULTS.

((viii))

G_{a} : A^{{s_{i}}_{i=0}^{I}} → A^{phg} , a = 0, x_{0} ,  (4.28)

provided that s_{0} > μ_{−} if a = x_{0}, and s_{0} > μ_{+} if a = 0. Proof: The results follow from Corollary 4.2 and from

G_{x_{0}} = \frac{1}{μ_{+} − μ_{−}} (I_{x_{0}}^{μ_{+}} − I_{0}^{μ_{−}}) = \frac{1}{μ_{+} − μ_{−}} (I_{0}^{μ_{+}} − I_{0}^{μ_{−}} − I^{μ_{+}}) ,

x\frac{∂}{∂x} · G_{x_{0}} = \frac{1}{μ_{+} − μ_{−}} (μ_{+}I_{x_{0}}^{μ_{+}} − μ_{−}I_{x_{0}}^{μ_{−}}) .

□

CHAPTER 4 Regularity at the boundary: the linear problem.

1. Tangential regularity below the threshold

Throughout this paper the letter C denotes a constant, the value of which may change from line to line. We use the summation convention unless specified otherwise.

We shall consider systems of equations which in the local coordinates as described in Chapter 3 can be written in the form

L^{a}_{b} u^{b} = f^{a} ,  a, b = 1, …, N ,  (1.1)

Here

L^{a}_{b} = ∑_{|α|≤m} a^{a}_{bα}(y)(x∂_{y})^{α} ,  (1.2)

where we use the notation

(x∂_{y})^{α} ≡ (x∂_{x})^{α_{1}}(x∂_{y^{2}})^{α_{2}} … (x∂_{y^{n}})^{α_{n}} for α = (α_{1}, …, α_{n}) .

Dropping indices we can write

L = ∑_{|α|≤m} a_{α}(x∂_{y})^{α} ,  (1.3)

or briefly

Lu = f .

Let F be a function space. We shall say that L ∈ OP_{F}^{m} if a^{a}_{bα} ∈ F. We set

a = (a_{α}) , ∥a∥_{F} ≡ ∑_{a,b,α} ∥a^{a}_{bα}∥_{F} .

The operator L will be called elliptic on M_{x_{0}} if there exist constants c_{1}, c_{2} > 0 such that on M_{x_{0}} it holds that

c_{1}|ζ|^{mN} ≤ det ( ∑_{|α|=m} a^{a}_{bα}ζ^{α}) ≤ c_{2}|ζ|^{mN} ,  (1.4)

|ζ| = \{(ζ^{1})^{2} + … + (ζ^{n})^{2}\}^{\frac{1}{2}} .

The operator L will be called elliptic if it is elliptic on M_{x_{0}}, and if moreover it is elliptic in any standard sense in CM_{x_{0}}; no uniformity conditions on CM_{x_{0}} are assumed. In some of the results in our paper, the notion of ellipticity on M_{x_{0}} can be weakened to a suitably weighted (in the sense of (1.2)) version of the definition of Douglis–Nirenberg [30].

The main results of this chapter – Theorems 2.6, 2.9, 2.10 and 2.11 – establish some form of “boundary regularity” for solutions of (1.1), under the hypothesis of the existence of a “regularity interval” for weighted Hölder or for weighted Sobolev spaces, as defined below. Thus the results of this chapter reduce the problem of boundary behaviour to that of existence of a “regularity interval” (which may be quite difficult to prove for specific operators). In section 3 we shall prove the existence of a regularity interval for weighted Sobolev spaces for the “conformal vector Laplacian” (in fact, of a strong regularity interval), while in Chapter 7 the existence of a (strong) regularity interval for weighted Hölder spaces will be established for the Laplace operator (the existence of a regularity interval for weighted Sobolev spaces for the Laplacian follows from Corollary 3.13 of [3] and from Proposition 1.2 below).

Let F_{∗}^{α,β}(Ω) be one of the spaces C_{∗}^{α,β}, W_{∗}^{α,β;p} or H_{∗}^{α,β} introduced in Chapter 3, where ∗ stands for a regularity index, e.g. ∗ = k, or ∗ = k + λ, m + μ, etc. We shall always be interested in Ω = M or Ω = M_{x_{0}}, with x_{0} — as described at the beginning of Chapter 3; note that if Ω = M_{σ} with 0 < σ < x_{0}, one can always replace x_{0} with σ to reduce this to the case Ω = M_{x_{0}} — our results and methods are “stable” under such replacements. We shall say that an interval (α_{−}, α_{+}) ⊂ ℝ is a regularity interval for F_{∗}^{α,β}(Ω) except for β ∈ B ⊂ ℝ for an operator L of order m if the following implication holds, with F_{m}^{loc}(Ω) – space of functions which are locally in F_{m},

( \begin{matrix}Lu = f, & f ∈ F_{∗}^{α,β}(Ω) , & α ∈ (α_{−}, α_{+}), & β ∉ B, \\ u ∈ F_{0}^{α_{−}+ϵ}(Ω) ∩ F_{m}^{loc}(Ω),  ϵ > 0 \end{matrix} )

⟹

( u ∈ F_{0}^{α,β}(Ω) )  (1.5)

Usually we shall consider B = ∅, B = {−1}, or B = ℝ\{0}; in that last case we shall say that (α_{−}, α_{+}) is a regularity interval for F_{∗}^{α}.

We shall say that an interval (α_{−}, α_{+}) is a strong regularity interval for F_{∗}^{α,β}(Ω), β ∉ B, if (α_{−}, α_{+}) is a regularity interval for F_{0}^{α,β}(Ω) and if moreover there exist constants 0 < x_{1}(α, β) ≤ x_{0}, C(α, β) such that the following implication holds,

( \begin{matrix}Lu = f, & f ∈ F_{0}^{α,β}(Ω) , & α ∈ (α_{−}, α_{+}), & β ∉ B, \\ u ∈ F_{0}^{α_{−}+ϵ}(Ω) ∩ F_{m}^{loc}(Ω),  ϵ > 0 \end{matrix} )

⟹

( ∥u∥_{F_{0}^{α,β}(M_{x_{0}} )} ≤ C(∥f∥_{F_{0}^{α,β}(M_{x_{0}} )} + ∥u∥_{F_{m}(\overline{M_{x_{1},x_{0}}} )}) )  (1.6)

(recall that M_{x_{1},x_{0}} ≡ CM_{x_{1}} ∩ M_{x_{0}} ≡ {p ∈ M : x_{1} < x(p) < x_{0}}). For instance, in the weighted Hölder spaces C_{0}^{α,β}(M_{x_{0}}) case the inequality (1.6) reads

∥u∥_{C_{0}^{α,β}(M_{x_{0}} )} ≤ C_{H}(∥f∥_{C_{0}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{m}(\overline{M_{x_{1},x_{0}}} )}) .  (1.7)

In the weighted Sobolev spaces L^{α,β;p}(M_{x_{0}}, x^{−n}dμ_{g}) case the inequality (1.6) reads

∥u∥_{L^{α,β;p}(M_{x_{0}} ,x^{−n}dμ_{g})} ≤ C_{S}(∥f∥_{L^{α,β;p}(M_{x_{0}} ,x^{−n}dμ_{g})} + ∥u∥_{W_{m}^{p}(M_{x_{1},x_{0}} ,g,dμ_{g})}) .(1.8)

In what follows we shall use difference–quotient arguments, writing expressions of the type “u(x, v′) − u(x, v)”, by which we mean the following construction: we can always find a covering of ∂M by coordinate charts (U_{i}, Φ_{i}), i = 1, …, I, with U_{i} = Φ_{i}(B_{0}^{n−1}(3δ_{i})), where B_{0}^{n−1}(r) is a ball of radius r in ℝ^{n−1} centered at the origin, such that U_{i} = Φ_{i}(B_{0}^{n−1}(δ_{i})) is also a covering of ∂M, let φ_{i} be a function such that φ_{i}|_{Φ_{i}(B_{0}^{n−1}(2δ_{i}))} = 1, supp φ_{i} ⊂ U_{i}, 0 ≤ φ_{i} ≤ 1. Let Δ ∈ S(1), where S(r)

is a sphere of radius r in ℝ^{n−1}, for (x, v) ∈ M̄_{x_{0}} (v ∈ ∂M) define vector fields X_{j}, j = 1, …, I, as follows

X_{j}(x, v) = \{ \begin{matrix}φ_{j}(v)χ(x)Δ^{k}∂_{v^{k}} & v ∈ U_{j}, x(p) ≤ x_{0} , \\ 0 & otherwise, \end{matrix}

where χ ∈ C_{∞}(ℝ), χ(x) = 1 for 0 ≤ x ≤ x_{0}/2, 0 ≤ χ(x) ≤ 1, χ(0) = 0 for x ≥ 3x_{0}/4. Set

Ψ_{Δ,h,j} = exp{hX_{j}} ,

where exp{hX_{j}}, h ∈ ℝ, is the one parameter group of diffeomorphisms generated by X_{j} on M̄_{x_{0}}. Ψ_{Δ,h,j} can be extended to a smooth map from M̄ to M̄ by setting Ψ_{Δ,h,j}|_{CM_{x_{0}}} = id. Note that

0 ≤ x ≤ x_{0}/2 , v ∈ U_{j}, |h| ≤ δ_{j} :  Ψ_{Δ,h,j}(x, v) = (x, v + hΔ) . (1.9)

In all further considerations we shall always assume that the fields u^{a}, f^{a} are geometric objects, by which we mean that 1) one can define an action Ψ_{Δ,h,j}^{∗} f, Ψ_{Δ,h,j}^{∗} u, and 2) that the equation (1.1) is geometric in the sense that

Ψ_{Δ,h,j}^{∗}(Lu) = L_{Δ,h,j}Ψ_{Δ,h,j}^{∗}u = Ψ_{Δ,h,j}^{∗} f  (1.10)

with some operator L_{Δ,h,j} of the form (1.2):

L_{Δ,h,j} = ∑_{|α|≤m} a_{Δ,h,j,α}(x∂_{y})^{α} .

We shall also assume that for 0 ≤ x ≤ x_{0}/2, |h| ≤ δ_{j} and v ∈ U_{j} we have

Ψ_{Δ,h,j}^{∗} u(x, v) = u(x, v + hΔ) ,  (1.11)
a_{Δ,h,j,α}(x, v) = a_{α}(x, v + hΔ),  Ψ_{Δ,h,j}^{∗} f(x, v) = f(x, v + hΔ) , (1.12)

and that there exists a constant C such that for 0 ≤ ℓ ≤ k, |h| ≤ max_{j} δ_{j} we have

∥Ψ_{Δ,h,j}^{∗} f − f∥_{C_{k,ℓ}^{α,β}(M_{x_{0}} )} ≤ C∥f∥_{C_{k,ℓ+μ}^{α,β}(M_{x_{0}} )}h^{μ} ,  (1.13)
∥Ψ_{Δ,h,j}^{∗} f∥_{C_{k+λ,ℓ}^{α,β}(M_{x_{0}} )} ≤ C∥f∥_{C_{k+λ,ℓ}^{α,β}(M_{x_{0}} )} ,  (1.14)
∥Ψ_{Δ,h,j}^{∗} f − f∥_{W_{k,ℓ−1}^{α,β;p}(M_{x_{0}} )} ≤ C∥f∥_{W_{k,ℓ}^{α,β;p}(M_{x_{0}} )}h ,  (1.15)
∥Ψ_{Δ,h,j}^{∗} f∥_{W_{k,ℓ}^{α,β;p}(M_{x_{0}} )} ≤ C∥f∥_{W_{k,ℓ}^{α,β;p}(M_{x_{0}} )} ,  (1.16)
∥a_{Δ,h,j,α} − a_{α}∥_{C_{k,ℓ}^{α,β}(M_{x_{0}} )} ≤ C∥a∥_{C_{k,ℓ+μ}^{α,β}(M_{x_{0}} )}h^{μ} ,  (1.17)
∥a_{Δ,h,j,α}∥_{C_{k+λ,ℓ}^{α,β}(M_{x_{0}} )} ≤ C∥a∥_{C_{k+λ,ℓ}^{α,β}(M_{x_{0}} )} .  (1.18)

Note that (1.11) implies that we have

∥u∥_{C_{k+λ,ℓ+μ}^{α,β}(M_{x_{0}} )} ≤ ∥u∥_{C_{k+λ,ℓ}^{α,β}(M_{x_{0}} )} + ∑_{i=0}^{I} sup_{0<h≤δ_{i}, Δ∈S(1)} h^{−μ}∥Ψ_{Δ,h,j}^{∗} u − u∥_{C_{k,ℓ}^{α,β}(M_{x_{0}} )} . (1.19)

Similarly the well known property of difference quotients in Sobolev spaces (cf. e.g. [40, Lemma 7.24]) also gives

∥u∥_{W_{k,ℓ}^{α,β;p}(M_{x_{0}} )} ≤ ∥u∥_{W_{k,ℓ−1}^{α,β;p}(M_{x_{0}} )} + ∑_{i=0}^{I} sup_{0<h≤δ_{i}, Δ∈S(1)} h^{−1}∥Ψ_{Δ,h,j}^{∗} f − f∥_{W_{k,ℓ−1}^{α,β;p}(M_{x_{0}} )} . (1.20)

If u, f are tensor fields and L is an invariant operator, (1.10)–(1.18) are easily seen to hold (if u is e.g. a covariant tensor field, then Ψ_{Δ,h,j}^{∗} is the pull–back operation by Ψ_{Δ,h,j}; if u is e.g. a contravariant tensor field, then Ψ_{Δ,h,j}^{∗} is the push–forward operation by Ψ_{Δ,h,j}^{−1}). In all subsequent arguments involving difference quotients the reader should assume that we have this construction in mind. We shall say that L is a geometric operator, or that eq. (1.1) is a geometric equation, whenever the above “coordinate invariance” hypotheses hold.

The scaling technique, illustrated in the proof of the following Lemma, will be used throughout:

Lemma 1.1 (Scaling estimates). ((i)) Let L ∈ OP_{C_{k}^{0}(M_{x_{0}} )}^{m}, k ≥ 0, be elliptic on M_{x_{0}}, let g ∈ C_{k}^{0}(M_{x_{0}}) be a Riemannian metric on M̄. There exists a constant C such that for all u ∈ W_{0}^{α,β;p}(M_{x_{0}}, dμ_{g}) ∩ W_{m}^{p,loc}(M_{x_{0}}, g, dμ_{g}), p ∈ (1, ∞), we have

∥u∥_{W_{k+m}^{α,β;p}(M_{x_{0}/2},g,dμ_{g})} ≤ C{∥Lu∥_{W_{k}^{α,β;p}(M_{x_{0}} ,g,dμ_{g})} + ∥u∥_{W_{0}^{α,β;p}(M_{x_{0}} ,dμ_{g})}} .(1.21)

((ii)) Let L ∈ OP_{C_{k+λ}^{0}(M_{x_{0}} )}^{m}, k ≥ 0, λ ∈ (0, 1), be elliptic on M_{x_{0}}. There

exists a constant C such that for all u ∈ C_{0}^{α,β}(M_{x_{0}}) ∩ C_{m}^{loc}(M_{x_{0}}) we have

∥u∥_{C_{k+m+λ}^{α,β}(M_{x_{0}/2})} ≤ C{∥Lu∥_{C_{k+λ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{0}^{α,β}(M_{x_{0}} )}} .  (1.22)

Proof: Let L̂ be an elliptic operator of order m on B_{0}^{n}(1), the coefficients of which are k times continuously differentiable, where B_{0}^{n}(s) is a ball in ℝ^{n} of radius s centered at the origin. From [2, Theorem 10.3] and the argument of the proof of Theorem 9.11 in [40] for p ∈ (1, ∞) one has

∥û∥_{W_{k+m}^{p}(B_{0}^{n}(1/2),δ_{ij},d^{n}y)} ≤ C{∥L̂û∥_{W_{k}^{p}(B_{0}^{n}(1),δ_{ij},d^{n}y)} + ∥û∥_{L^{p}(B_{0}^{n}(1),d^{n}y)}} .(1.23)

(1.21) follows from (1.23) applied to the functions û ≡ uŷ defined by

uŷ(y) = u(\frac{x̂}{2} y + ŷ)x̂^{−α}(| ln x̂| + 1)^{−β} , y ∈ B_{0}^{n}(1) , ŷ = (x̂, v̂) ∈ M_{x_{0}/2}

by a Whitney cube (cf. e.g. [68, Chapter 6]) decomposition argument. (1.22) is obtained by applying the interior Schauder estimates [30, Theorem 1].

∥û∥_{C_{k+m+λ}(B_{0}^{n}(1/2))} ≤ C{∥L̂û∥_{C_{k+λ}(B_{0}^{n}(1))} + ∥û∥_{C_{0}(B_{0}^{n}(1))}}

to the functions uŷ. □

The following Proposition (cf. [3, Proposition 2.6]) provides a criterion for I to be a regularity interval for second order operators:

Proposition 1.2. Let k ∈ ℕ_{0}, λ ∈ (0, 1), and let x be a defining function for ∂M, with x_{0} — as described at the beginning of Chapter 3, recall that ˜∂M_{s} = {p ∈ M : x(p) = s}, suppose that B ⊂ ℝ satisfies 0 ∉ B.

((i)) Let L ∈ OP_{C_{k+λ}^{0}(M_{x_{0}} )}^{2}, suppose that for all α ∈ (α_{−}, α_{+}) and β ∉ B there exists 0 < x_{1} ≤ x_{0} (possibly depending upon α and β) such that for all ψ ∈ C_{k+2+λ}(˜∂M_{x_{1}}) L is an isomorphism between {u ∈ C_{k+2+λ}^{α,β}(M_{x_{1}}) : u|_{˜∂M_{x_{1}}} = ψ} and C_{k+λ}^{α,β}(M_{x_{1}}). Then L has (α_{−}, α_{+}) as

a regularity interval for C_{k+λ}^{α,β}(M), β ∉ B. ((ii)) Let L ∈ OP_{C_{k}^{0}(M_{x_{0}} )}^{2}, suppose that for all α ∈ (ω_{−}, ω_{+}) and β ∉ B there

exists 0 < x_{1} ≤ x_{0} (possibly depending upon α and β) such that for all ψ ∈ W_{k+2}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}), L is an isomorphism between {u : u−ψ ∈

W^{∘} _{1}^{α,β;p} ∩ W_{k+2}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g})} and W_{k}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}). Then

L has (ω_{−}, ω_{+}) as a regularity interval for W_{k}^{α,β;p}(M, g, x^{−n}dμ_{g}), β ∉ B. Moreover if the above holds with k = 0, there exists a constant C such that for u ∈ W_{0}^{ω_{−}+ϵ;p}(M_{x_{1}}, x^{−n}dμ_{g}) ∩ W_{2}^{loc}(M_{x_{1}}), ϵ > 0 for which Lu ∈ W_{0}^{α,β;p}(M_{x_{1}}, x^{−n}dμ_{g}), α ∈ (ω_{−}, ω_{+}), β ∉ B, we have

∥u∥_{W_{2}^{α,β;p}(M_{x_{1}} ,g,x^{−n}dμ_{g})} ≤ C(∥Lu∥_{W_{0}^{α,β;p}(M_{x_{1}} ,x^{−n}dμ_{g})} + ∥u∥_{W_{2}^{p}(M_{x_{1}/2,x_{1}} ,g,dμ_{g})}) (1.24)

(recall that M_{x_{1}/2,x_{1}} ≡ M_{x_{1}} \ \overline{M_{x_{1}/2}}), in particular (ω_{−}, ω_{+}) is a strong

regularity interval for W_{0}^{α,β;p}(M, g, x^{−n}dμ_{g}), β ∉ B.

Remark: Let us point out, that if the hypotheses of point (i) above hold with k = 0, then we certainly have the inequality

∥u∥_{C_{0}^{α,β}(M_{x_{0}} )} ≤ C_{H}(∥f∥_{C_{0+λ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{m+λ}(\overline{M_{x_{1},x_{0}}} )}) .  (1.25)

This does not lead to a strong regularity interval because of the Hölder continuity exponent λ in the norm of f above. It should be mentioned that at the price of some supplementary complications most of our results would go through if in the definition of a strong regularity interval the inequality (1.7) were replaced by (1.25). Since, however, (1.7) is sufficient for our purposes (cf. Chapter 7) we shall not consider that possibility.

Proof: (i): Let Lu = f, f ∈ C_{k+λ}^{α,β}(M), u = O(x^{α_{−}+ϵ}), decreasing ϵ if necessary we may assume γ = α_{−} + ϵ < α. By Lemma 1.1 u ∈ C_{k+2+λ}^{γ}(M_{x_{0}}). Set ψ = u|_{˜∂M_{x_{1}}} , thus ψ ∈ C_{k+2+λ}(˜∂M_{x_{1}}). For α ∈ (α_{−}, α_{+}), β ∉ B, L is surjective

there exists therefore χ ∈ C_{k+2+λ}^{α,β}(M_{x_{1}}), χ|_{˜∂M_{x_{1}}} = ψ, such that Lχ = f|_{M_{x_{1}}} . We have u|_{M_{x_{1}}} , χ ∈ C_{k+2+λ}^{γ}(M_{x_{1}}), γ ∈ (α_{−}, α_{+}), by injectivity of L on C_{k+2+λ}^{γ}(M_{x_{1}}) it follows that u|_{M_{x_{1}}} = χ, thus u ∈ C_{k+2+λ}^{α,β}(M_{x_{0}}).

(i): Let φ ∈ C_{∞}(M) be any function satisfying φ|_{M_{x_{1}/2}} = 1, φ|_{CM_{3x_{1}/4}} = 0, set

u_{1} = φu, u_{2} = (1 − φ)u, define Lu = f. We have u_{1} ∈ W_{0}^{ω_{−}+ϵ;p}(M_{x_{1}}, x^{−n}dμ_{g}) ∩ W_{2}^{loc}(M_{x_{1}}), u_{1} = 0 in M_{3x_{1}/4,x_{1}}, from φ|_{M_{x_{1}/2}} = 1 it follows that Lu_{1}|_{M_{x_{1}/2}} =

f|_{M_{x_{1}/2}}, and an argument as in the proof of point (i) shows that u_{1} ∈ W_{2}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}),

thus u ∈ W_{2}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) and (ω_{−}, ω_{+}) is a regularity interval. If k = 0

then L :W^{∘} _{1}^{α,β;p} ∩ W_{2}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) → W_{0}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) is a bijection,

and the open mapping theorem shows that there exists a constant C such that for

all w ∈W^{∘} _{1}^{α,β;p} ∩ W_{2}^{α,β;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) we have

∥w∥_{W_{2}^{α,β;p}(M_{x_{1}} ,g,x^{−n}dμ_{g})} ≤ C∥Lw∥_{W_{0}^{α,β;p}(M_{x_{1}} ,x^{−n}dμ_{g})} .

This inequality applied to w = u_{1} gives

∥u_{1}∥_{W_{2}^{α,β;p}(M_{x_{1}} ,g,x^{−n}dμ_{g})} ≤ C(∥f∥_{W_{0}^{α,β;p}(M_{x_{1}} ,x^{−n}dμ_{g})} + ∥u∥_{W_{1}^{p}(M_{x_{1}/2,x_{1}} ,g,dμ_{g})})

(it is easily seen that for σ > 0 we have W_{k}^{p}(M_{σ,x_{1}}, g, dμ_{g}) = W_{k}^{α,β;p}(M_{σ,x_{1}}, g, x^{−n}dμ_{g}) with equivalent norms), and (1.24) immediately follows. □

The main result of this section is the following:

Theorem 1.3 (Tangential regularity, α ∈ (α_{−}, α_{+})). ((i)) Let (α_{−}, α_{+}) be a strong regularity interval for C_{0}^{α,β}(M), β ∉ B, for a geometric elliptic operator L ∈ OP_{C_{k+λ,ℓ+μ}^{0}(M)}^{m}, λ ∈ (0, 1), μ ∈ [0, 1], 0 ≤ ℓ+μ ≤ k +λ,

let α_{−} < α < α_{+}, suppose that u ∈ C_{m}^{loc}(M) satisfies

Lu = f , f ∈ C_{k+λ,ℓ+μ}^{α,β}(M) , u = O(x^{α_{−}+ϵ}) , ϵ > 0 , β ∉ B .

Then for all λ′ ∈ (0, 1), μ′ ∈ [0, μ) if μ > 0, μ′ = 0 otherwise we have

u ∈ C_{k+m−1+λ′,ℓ+μ}^{α,β}(M_{x_{0}}) ∩ C_{k+m+λ,ℓ+μ′}^{α,β}(M_{x_{0}}) .  (1.26)

Moreover for all λ′ ∈ (0, 1), μ′ ∈ [0, μ) if μ > 0, μ′ = 0 otherwise there exists a constant C depending only upon α, β, n=dimM, N (cf. eq. (1.1)), the ellipticity constants c_{1}, c_{2} of eq. (1.4), the constant C_{H} of (1.7), the constants in (1.13)–(1.18), ∥a∥_{C_{k+λ,ℓ+μ}^{0}(M)}, λ′ and μ′ such that we have

∥u∥_{C_{k+m−1+λ′,ℓ+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ,ℓ+μ′}^{α,β} (M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ}(\overline{M_{x̃_{1},x_{0}}} )}) , (1.27)

x̃_{1} = min(x_{1}, x_{0}/2). If (α_{−}, α_{+}) is only a regularity interval for C_{0+λ}^{α,β}(M), then (1.26) holds with μ′ = μ = 0 provided that α_{−} + 1 < α < α_{+} and u = O(x^{α_{−}+1+ϵ}), ϵ > 0.

((ii)) Let (ω_{−}, ω_{+}) be a strong regularity interval for W_{0}^{α,β;p}(M, g, x^{−n}dμ_{g}), β ∉ B for a geometric elliptic operator L ∈ OP_{C_{k,ℓ}^{0}(M)}^{m}, 0 ≤ ℓ ≤ k, let

ω_{−} < α < ω_{+}, p ∈ (1, ∞), suppose that u ∈ W_{m}^{p,loc}(M, g, dμ_{g}) satisfies

Lu = f , f ∈ W_{k,ℓ}^{α,β;p}(M, g, x^{−n}dμ_{g}) , u ∈ W_{0}^{ω_{−}+ϵ;p}(M, g, x^{−n}dμ_{g}) , ϵ > 0 , β ∉ B .

Then

u ∈ W_{k+m,ℓ}^{α,β;p}(M_{x_{0}}, g, x^{−n}dμ_{g}) .  (1.28)

Moreover there exists a constant C depending only upon α, β, n=dimM, N (cf. eq. (1.1)), the ellipticity constants c_{1}, c_{2} of eq. (1.4), the constant C_{S} of (1.8), the constants in (1.13)–(1.18), ∥a∥_{C_{k,ℓ}^{0}(M)} and on the modulus of continuity of (x∂_{y})^{γ}a_{bα}^{a}(y), |γ| = k, y ∈ M_{x_{0}}, such that

∥u∥_{W_{k+m,ℓ}^{α,β;p}(M_{x_{0}} ,g,x^{−n}dμ_{g})} ≤ C(∥f∥_{W_{k,ℓ}^{α,β;p}(M_{x_{0}} ,g,x^{−n}dμ_{g})} + ∥u∥_{W_{k+m}^{p}(M_{x̃_{1},x_{0}} ,g,dμ_{g})}) , (1.29) x̃_{1} = min(x_{1}, x_{0}/2). If (ω_{−}, ω_{+}) is only a regularity interval for W_{0}^{α,β;p}(M), then (1.28) holds provided that ω_{−}+1 < α < ω_{+} and u ∈ W_{0}^{ω_{−}+1+ϵ;p}(M, g, x^{−n}dμ_{g}), ϵ > 0.

Remark: If L ∈ OP_{C_{∞}^{0}}^{m} and (α_{−}, α_{+}) is only a regularity interval, the restriction α > α_{−} + 1 can be relaxed to α > α_{−} by commuting the equation satisfied by u with a pseudodifferential operator [25], we shall however not discuss this here.

Proof: We shall prove point (i), point (ii) is proved in a similar (and simpler) way. We shall proceed by induction on ℓ. Since (α_{−}, α_{+}) is a regularity interval we have u ∈ C_{k+m+λ}^{α,β} by the scaling estimates, thus the result holds for ℓ + μ = 0. Let f ∈ C_{k+λ,ℓ_{0}+μ}^{α,β}, μ ∈ (0, 1], 0 ≤ ℓ_{0} ≤ k, or f ∈ C_{k+λ,ℓ_{0}+1+μ}^{α,β}, μ = 0, 0 ≤ ℓ_{0} + 1 ≤ k, and suppose that u ∈ C_{k+m+λ,ℓ_{0}}^{α,β}. If (α_{−}, α_{+}) is a strong regularity interval suppose moreover that we have

∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ}(\overline{M_{x_{1},x_{0}}} )}) . (1.30)

Let Y_{i}, i = 1, …, ℓ_{1}, be any smooth vector fields of the form χ(x)Y_{i}^{A}(v)\frac{∂}{∂v^{A}}, Y_{i}^{A} ∈ C_{∞}(∂M), χ ∈ C_{∞}(ℝ), χ = 1 in a neighbourhood of 0, χ = 0 outside of [0, x_{0}/2], where ℓ_{1} = ℓ_{0} + 1 when (α_{−}, α_{+}) is a regularity interval and ℓ_{1} = ℓ_{0} when (α_{−}, α_{+}) is a strong regularity interval. We have

Lũ = φ̃ , φ̃ = f̃+ ρ̃,  (1.31)
ũ = Y_{1} … Y_{ℓ_{1}}u ,
f̃ = Y_{1} … Y_{ℓ_{1}}f ∈ C_{k−ℓ_{1}+λ,0+μ}^{α,β} ,
ρ̃ = [L, Y_{1} … Y_{ℓ_{1}}]u .

If (α_{−}, α_{+}) is a regularity interval and μ = 0, by the inductive hypothesis we have ρ̃ ∈ C_{k−ℓ_{0}+λ}^{α,β} and from ũ ∈ C_{k+m−ℓ_{0}−1+λ}^{α−1,β}, α − 1 > α_{−}, it follows that ũ ∈ C_{k−ℓ_{0}−1+m+λ}^{α,β}, thus u ∈ C_{k+m+λ,ℓ_{0}+1}^{α,β} and the inductive step is completed. On the other hand if (α_{−}, α_{+}) is a strong regularity interval and μ ∈ (0, 1] then ũ ∈ C_{k−ℓ_{0}+m+λ}^{α,β}, ρ̃ ∈ C_{k−ℓ_{0}+1+λ,1}^{α,β} ⊂ C_{k−ℓ_{0}+1+λ,0+μ}^{α,β}, therefore φ̃ ∈ C_{k−ℓ_{0}+λ,0+μ}^{α,β} and for 0 < x ≤ x_{0} we have , with v′ = v + v′′, for some fixed v′′ (cf. the beginning of this chapter for the meaning of ũ(x, v) − ũ(x, v′), etc.)

L(x, v)(ũ(x, v) − ũ(x, v′)) = ψ(x, v) ,

|ψ(x, v)| ≡ |(L(x, v′) − L(x, v))ũ(x, v′) + φ̃(x, v) − φ̃(x, v′)|
≤ C(∥f∥_{C_{ℓ_{0},ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥a∥_{C_{ℓ_{0},ℓ_{0}+μ}^{0}(M_{x_{0}} )}∥u∥_{C_{m+ℓ_{0},ℓ_{0}−1+μ}^{α,β}(M_{x_{0}} )}) x^{α}(1 + | ln x|^{β})|v − v′|^{μ} ,

and similarly, for higher derivatives,

∥ψ∥_{C_{k−ℓ_{0}}^{α,β} (M_{x_{0}} )} ≤ C(∥f∥_{C_{k,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥a∥_{C_{k,ℓ_{0}+μ}^{0}(M_{x_{0}} )}∥u∥_{C_{k+m,ℓ_{0}}^{α,β} (M_{x_{0}} )}) |v − v′|^{μ} , (1.32)

∥ψ∥_{C_{k−ℓ_{0}+λ}^{α,β}(M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )} + ∥a∥_{C_{k+λ,ℓ_{0}}^{0} (M_{x_{0}} )}∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )}) (1.33)

(cf. (1.13)–(1.18)). The definition of the strong regularity interval and (1.30) imply

|ũ(x, v) − ũ(x, v′)| ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ}(\overline{M_{x_{1},x_{0}}} )}) x^{α}(1 + | ln x|)^{β} |v − v′|^{μ} . (1.34)

If ℓ_{0} < k, μ = 1, passing to the limit v → v′ one obtains from (1.34)

|∂_{v}ũ| ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ}(\overline{M_{x_{1},x_{0}}} )}) x^{α}(1 + | ln x|)^{β}

because ũ ∈ C_{k+m−ℓ_{0}}^{α,β}(M_{x_{0}}) ⊂ C_{k+m−ℓ_{0}}^{loc}(M_{x_{0}}) ⊂ C_{1}^{loc}(M_{x_{0}}), which gives ∂_{v}ũ ∈ C_{0}^{α,β}(M_{x_{0}}). The scaling estimates applied to the equation (1.31) with ℓ_{1} = ℓ_{0} + 1 give ∂_{v}ũ ∈ C_{k+m−ℓ_{0}−1+λ}^{α,β}(M_{x_{0}}), together with the inequality

∥∂_{v}ũ∥_{C_{k+m−ℓ_{0}−1+λ}^{α,β}(M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+1}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ}(\overline{M_{x_{1},x_{0}}} )}) ,

therefore u ∈ C_{k+m+λ,ℓ_{0}+1}^{α,β}(M_{x_{0}}), and it holds that

∥u∥_{C_{k+m+λ,ℓ_{0}+1}^{α,β}(M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+1}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ}(\overline{M_{x_{1},x_{0}}} )}) ,

which completes the induction step.

We have thus shown that u ∈ C_{k+m+λ,ℓ}^{α,β}(M_{x_{0}}), and it remains to show the tangential Hölder continuity of the ℓ_{0} = ℓ’th tangential derivatives of u. Set û(x, v) = ũ(x, v) − ũ(x, v + v′′), for ŷ = (x̂, v̂) ∈ M_{x_{0}/2} consider the functions

B_{0}^{n}(1/2) ∋ y ⟶ ûŷ(y) = ũ(x̂y + ŷ) x̂^{−α}(1 + | log x̂|)^{−β} ,
B_{0}^{n}(1/2) ∋ y ⟶ ψŷ(y) = ψ(x̂y + ŷ) x̂^{−α}(1 + | log x̂|)^{−β} ,

where B_{0}^{n}(r) is an open ball in ℝ^{n} of radius r centered at the origin. From (1.32)– (1.33) we have

∥ψŷ∥_{C_{k−ℓ_{0}} (\overline{B_{0}^{n}(1/2)})} ≤ C(∥f∥_{C_{k,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m,ℓ_{0}}^{α,β} (M_{x_{0}} )})|v′′|^{μ} , (1.35)

∥ψŷ∥_{C_{k−ℓ_{0}+λ}(\overline{B_{0}^{n}(1/2)}} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )} + ∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )}) . (1.36)

(1.35), (1.34) and the L^{p} interior elliptic estimates (cf. (1.23)) give

∥ûŷ∥_{W_{k−ℓ_{0}+m}^{p}(B_{0}^{n}(1/4),d^{n}y)} ≤ C(∥ψŷ∥_{W_{k−ℓ_{0}}^{p} (B_{0}^{n}(1/2),d^{n}y)} + ∥ûŷ∥_{L^{p}(B_{0}^{n}(1/2),d^{n}y)})
≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m,ℓ_{0}}^{α,β} (M_{x_{0}} )})|v′′|^{μ} ,

so that from Sobolev’s embedding one gets, for any λ′ ∈ [0, 1),

∥ûŷ∥_{C_{k−ℓ_{0}+m−1+λ′} (\overline{B_{0}^{n}(1/4)})} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m,ℓ_{0}}^{α,β} (M_{x_{0}} )})|v′′|^{μ} ⟹

∥û∥_{C_{k−ℓ_{0}+m−1+λ′}^{α,β} (M_{x_{0}/2})} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m,ℓ_{0}}^{α,β} (M_{x_{0}} )})|v′′|^{μ} ⟹

∥u∥_{C_{k+m−1+λ′,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )}+∥u∥_{C_{k+m,ℓ_{0}}^{α,β} (M_{x_{0}} )}+∥u∥_{C_{k+m−1+λ′} (\overline{M_{x_{0}/2,x_{0}}} )}) .

To get control of the k − ℓ_{0} + m’th derivatives of ũ some more work is required. By interpolation (cf. e.g. [44, Theorem A.5]) from (1.35)–(1.36) for any σ ∈ (0, 1) one obtains

∥ψŷ∥_{C_{k−ℓ_{0}+σλ}(\overline{B_{0}^{n}(1/2)})} ≤ C∥ψŷ∥_{C_{k−ℓ_{0}} (\overline{B_{0}^{n}(1/2)})}^{σ}∥ψŷ∥_{C_{k−ℓ_{0}+λ}(\overline{B_{0}^{n}(1/2)})}^{1−σ}

≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )})|v′′|^{(1−σ)μ} ,

and elliptic interior Hölder estimates give

∥ûŷ∥_{C_{k−ℓ_{0}+m+σλ}(\overline{B_{0}^{n}(1/4)})} ≤ C(∥ψŷ∥_{C_{k−ℓ_{0}+σλ}(\overline{B_{0}^{n}(1/2)})} + ∥ûŷ∥_{C_{0}(\overline{B_{0}^{n}(1/2)})})
≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )})|v′′|^{(1−σ)μ} ,

so that ũ ∈ C_{k+m−ℓ_{0}+σλ,(1−σ)μ}^{α,β}, and for all |γ| ≤ k + m − ℓ_{0} we have

|(x∂_{y})^{γ}(ũ(x, v)−ũ(x, v′))| ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )}+∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )}) x^{α}(1+| ln x|)^{β} |v−v′|^{(1−σ)μ} .

Since u ∈ C_{k+m+λ,ℓ_{0}}^{α,β} it follows that u ∈ C_{k+m+λ,ℓ_{0}+μ′}^{α,β}, with

∥u∥_{C_{k+m+σλ,ℓ_{0}+(1−σ)μ}^{α,β}(M_{x_{0}} )} ≤ C(∥f∥_{C_{k+λ,ℓ_{0}+μ}^{α,β}(M_{x_{0}} )} + ∥u∥_{C_{k+m+λ,ℓ_{0}}^{α,β} (M_{x_{0}} )}) ,(1.37)

and (1.27) follows by (1.30). □

Note that the argument of the last part of the proof of Theorem 1.3 proves the following Lemma, which shall be needed later on:

Lemma 1.4. Let L ∈ OP_{C_{k+λ,ℓ+μ}^{0}(M_{x_{0}} )}^{m} be a geometric elliptic operator, λ ∈

(0, 1), μ ∈ (0, 1], 0 ≤ ℓ + μ ≤ k + λ, suppose that u ∈ C_{m}^{loc}(M_{x_{0}}) ∩ C_{0,ℓ+μ}^{α,β}(M_{x_{0}}) satisfies

Lu ∈ C_{k+λ,ℓ+μ}^{α,β}(M_{x_{0}}) .

Then for all λ′ ∈ (0, 1), μ′ ∈ [0, μ) we have

u ∈ C_{k+m−1+λ′,ℓ+μ}^{α,β}(M_{x_{0}/2}) ∩ C_{k+m+λ,ℓ+μ′}^{α,β}(M_{x_{0}/2}) .

Moreover for all λ′ ∈ (0, 1), μ′ ∈ [0, μ) there exists a constant C depending only upon n=dimM, N (cf. eq. (1.1)), the ellipticity constants c_{1}, c_{2} of eq. (1.4), the constant C_{H} of (1.7), the constants in (1.13)–(1.18), ∥a∥_{C_{k+λ,ℓ+μ}^{0}(M_{x_{0}} )}, λ′ and μ′ such that we have

∥u∥_{C_{k+m−1+λ′,ℓ+μ}^{α,β}(M_{x_{0}/2})}+∥u∥_{C_{k+m+λ,ℓ+μ′}^{α,β} (M_{x_{0}/2})} ≤ C(∥Lu∥_{C_{k+λ,ℓ+μ}^{α,β}(M_{x_{0}} )}+∥u∥_{C_{0,ℓ+μ}^{α,β}(M_{x_{0}} )}) .

2. Boundary regularity for a class of second order systems

Theorem 1.3 gives some more information about the solutions than Lemma 1.1, however only for exponents α ∈ (α_{−}, α_{+}). To be able to cover α’s in the interval (α_{−}, ∞), or to obtain more regularity, some other techniques are required, and we shall from now on restrict our consideration to second order systems^{1}, for which L can be written in the form

L = L_{ab} + L̃ ,  (2.1)
L_{ab} = (x^{2}∂_{x}^{2} + ax∂_{x} + b) ⊗ id_{ℝ^{N}} ,  (2.2)
L̃ = x^{2}a^{xA} \frac{∂}{∂x} \frac{∂}{∂v^{A}} + x^{2}a^{AB} \frac{∂}{∂v^{A}} \frac{∂}{∂v^{B}} + xa^{A} \frac{∂}{∂v^{A}}
+x^{3}ψ \frac{∂^{2}}{∂x^{2}} + x^{2}φ\frac{∂}{∂x} + xχ ,  with xψ, xφ and xχ − o(1) , (2.3)

1It should, however, be noted that several of the arguments given here would apply to higher order systems as well.

where id_{ℝ^{N}} is the N × N identity matrix, and a and b are real constants satisfying

\frac{(1 − a)^{2}}{4} − b > 0 .  (2.4)

Moreover a^{xA}, a^{AB}, a^{A}, xψ, xφ and xχ are N × N matrices the entries of which are bounded functions of y. Note that one can redefine L by multiplying it by an appropriate (non–degenerate) matrix from the left (this might perhaps necessitate a decrease of x_{0}) to obtain ψ ≡ 0 in M_{x_{0}}. We shall often write L_{ab} in the form L_{ab} = x^{2}∂_{x}^{2} + ax∂_{x} + b, hoping that no confusion will occur. As will be seen in the following chapters, (2.1)–(2.3) are sufficient for the applications we have in mind — the constraint equations of general relativity. All subsequent results are based on an analysis of the ODE^{2}

L_{ab}u = f ,

solutions of which are given by

u = Ax^{μ_{−}} + Bx^{μ_{+}} + G_{x_{0}}(f) ,  (2.5)
G_{x_{0}}(f)(x) = \frac{1}{μ_{+} − μ_{−}} \{ − x^{μ_{+}} ∫_{x}^{x_{0}} s^{−1−μ_{+}}f(s) ds − x^{μ_{−}} ∫_{0}^{x} s^{−1−μ_{−}}f(s) ds\} , (2.6)

assuming that the integrals in (2.6) converge. Here the indicial roots μ_{±} are given by

μ_{±} = \frac{1 − a}{2} ± \sqrt{\frac{(1 − a)^{2}}{4} − b} ,  (2.7)

and we will assume throughout that (2.4) holds. If u = o(x^{μ_{−}}), f = O(x^{α}), α > μ_{−}, then A = 0. For α > μ_{+} one can replace G_{x_{0}}(f) in (2.5) by

G_{0}(f) = \frac{1}{μ_{+} − μ_{−}} \{x^{μ_{+}} ∫_{0}^{x} s^{−1−μ_{+}}f(s) ds − x^{μ_{−}} ∫_{0}^{x} s^{−1−μ_{−}}f(s) ds \} , (2.8)

and it will be convenient to do so.

Let us discuss shortly the relationship between μ_{±} and a weighted Hölder space regularity interval (α_{−}, α_{+}). When ψ = 0, φ = φ(x), χ = χ(x), by considering v-independent sources and solutions it follows immediately that (α_{−}, α_{+}) ⊂ (−∞, μ_{+}), and that μ_{−} ∉ (α_{−}, α_{+}). For the standard Laplacian on hyperbolic space (which is an operator in the class considered here) it is known that (α_{−}, α_{+}) ⊄ (−∞, μ_{−}) (in fact (α_{−}, α_{+}) = (μ_{−}, μ_{+}), cf. Section 7), and therefore we shall only consider the case

μ_{−} ≤ α_{−} < α_{+} ≤ μ_{+} .

It will be seen that for the equations considered in Section 7 we have α_{±} = μ_{±}, and we expect this to always hold for operators of the form (2.1)–(2.4). In fact Theorem 2.1 in the next Section shows that one may (in some sense, made precise there) assume that α_{+} = μ_{+} for sufficiently smooth sources.

Let us emphasize that the reason for considering here the possibility that α_{−} ≠ μ_{−}, or α_{+} ≠ μ_{+}, or both, is not because we expect this to be a real feature of some operators. Rather, this is motivated by the fact that for some equations it might be possible to prove the existence of a (perhaps strong) regularity interval (α_{−}, α_{+}) which is a proper subset of (μ_{−}, μ_{+}), while the known proofs do not allow one to obtain the full expected range (μ_{−}, μ_{+}) (this is actually what happens in our analysis of the vector constraint equation in weighted Sobolev spaces, Section 3). We shall nevertheless show that the existence of any strong regularity interval, or of a regularity interval of length larger than one, is sufficient to conclude some (perhaps not optimal) classical regularity of solutions at the boundary.

2In the general case the corresponding ODE is usually analyzed by Mellin transform techniques (cf. e.g. [58]); for the special class of equations considered in this paper we have found it simpler and more elementary to use the fundamental solution representation (2.5). We are grateful to G. Łysik and B. Ziemian for pointing out the Mellin transform approach.

2.1. “Tangential regularity” above the threshold. For α‘s above the threshold α_{+} we have the following equivalent of Theorem 1.3:

Theorem 2.1 (Tangential regularity, α ≥ α_{+}). Let (α_{−}, α_{+}) be a strong regularity interval for C_{0+λ}^{α,β}(M), β ∉ B, for a geometric elliptic operator L ∈ OP_{C_{k+λ,ℓ+μ}^{0}(M)}^{2}, λ ∈ (0, 1), μ ∈ (0, 1], 1 ≤ ℓ + μ ≤ k + λ. Suppose that L is of the

form (2.1)–(2.4) with ψ, ϕ, χ ∈ C_{k−1+λ,ℓ−1+μ}^{0}(M). Suppose that

μ_{−} ≤ α_{−} < α_{+} ≤ μ_{+} , α_{+} ≤ α .

Let

Lu = f , f ∈ C_{k+λ,ℓ+μ}^{α,β}(M) , u = O(x^{α_{−}+ϵ}) , ϵ > 0 , β ∉ B .

If ℓ ≥ ℓ_{0}, where ℓ_{0} ∈ ℕ is the smallest integer such that α_{+} + ℓ_{0} > min(α, μ_{+}), then there exist σ ∈ (0, 1) and δ > 0 such that for μ′ ∈ [0, μ) and ϵ > 0 we have

((i)) α < μ_{+}

u ∈ C_{k+2+λ,ℓ−ℓ_{0}+μ′}^{α,β}(M_{x_{0}}) ∩ C_{k+2+λ,ℓ−ℓ_{0}+1+σ}^{α_{+}+ℓ_{0}−1+δ}(M_{x_{0}}) .  (2.9)
((ii)) α = μ_{+} , β ≠ −1
u ∈ C_{k+2+λ,ℓ−ℓ_{0}+μ′}^{μ_{+},β+1}(M_{x_{0}}) ∩ C_{k+2+λ,ℓ−ℓ_{0}+1+σ}^{α_{+}+ℓ_{0}−1+δ}(M_{x_{0}}) .  (2.10)
((iii)) α > μ_{+}
u ∈ C_{k+2+λ,ℓ−ℓ_{0}+μ′}^{μ_{+}}(M_{x_{0}}) ∩ C_{k+2+λ,ℓ−ℓ_{0}+1+σ}^{α_{+}+ℓ_{0}−1+δ}(M_{x_{0}}) .  (2.11)

((iv)) If α > μ_{+} and ℓ ≥ ℓ_{1}, where ℓ_{1} ∈ ℕ is the smallest integer such that

α_{+} + ℓ_{1} > α, then
u = o(x^{μ_{+}}) ⟹ u ∈ C_{k+2+λ,ℓ−ℓ_{1}+μ′}^{α,β}(M_{x_{0}}) ∩ C_{k+2+λ,ℓ−ℓ_{1}+1+σ}^{α_{+}+ℓ_{1}−1+δ}(M_{x_{0}}) . (2.12)

The constants σ and δ depend only upon α_{+}, μ_{+}, α, μ, λ, ℓ and k. Moreover if F denotes any of the spaces appearing in eqs. (2.9)–(2.12), then there exists a (u and f independent) constant C such that

∥u∥_{F} ≤ C (∥f∥_{C_{k+λ,ℓ+μ}^{α,β}(M)} + ∥u∥_{C_{0}^{α_{−}+ϵ}(M)}) .  (2.13)

Remark: If (α_{−}, α_{+}) is a regularity interval for C_{0+λ}^{α,β}(M) only, then the following modifications are needed above: One has to make the supplementary assumptions that α_{−} + 1 < α_{+} and that u = O(x^{α_{−}+1+ϵ}) for some ϵ > 0; on the other hand one can also allow μ = 0. Then eqs. (2.9)–(2.12) hold with μ′ = 0 and without the claim about the appropriate decay of the transverse derivatives of order ℓ − ℓ_{∗} + 1, ℓ_{∗} = ℓ_{0} or ℓ_{∗} = ℓ_{1}.

Proof: In what follows we shall assume that μ = 0, μ′ = 0, etc., if (α_{−}, α_{+}) is not a strong regularity interval. From (2.1) it follows that u satisfies the equation

L_{ab}u = f − L̃u ,  (2.14)

and by Theorem 1.3 for any ϵ > 0 and μ′ ∈ [0, μ) if μ > 0, μ′ = 0 otherwise we have u ∈ C_{k+2+λ,ℓ+μ′}^{α_{+}−ϵ}, thus L̃u ∈ C_{k−1+λ,ℓ−1+μ′}^{1+α_{+}−ϵ} ∩ C_{k+λ,ℓ+μ′}^{α_{+}−ϵ} (cf. (2.4)–(2.6)). Solving the ODE (2.14) and choosing ϵ appropriately we obtain

u = ψx^{μ_{+}} + G_{x_{0}}(f − L̃u) = O(x^{σ}(1 + | ln x|)^{ρ}) ,  (2.15)

for some function ψ(v), with

α_{+} ≤ α < 1 + α_{+} , α < μ_{+} ⟹ σ = α, ρ = β ,
α_{+} ≤ α < 1 + α_{+} , α = μ_{+}, β ≠ −1 ⟹ σ = μ_{+}, ρ = β + 1 ,
α_{+} ≤ α < 1 + α_{+} , α > μ_{+} ⟹ σ = μ_{+}, ρ = 0 ,
1 + α_{+} ≤ α ⟹ σ = min(μ_{+}, 1 + α_{+} − ϵ), ρ = 0 .

The scaling estimates, Lemma 1.1, imply u ∈ C_{k+2+λ}^{σ,ρ}. Setting x = x_{0}/2 in (2.15) from L̃u ∈ C_{k+λ,ℓ+μ′}^{α_{+}−ϵ} we get G_{x_{0}}(f − L̃u)|_{x=x_{0}/2} ∈ C_{ℓ+μ′}(∂M) (cf. (4.14)), which together with u(x_{0}/2, ·) ∈ C_{k+2+λ}(∂M) shows that ψ ∈ C_{ℓ+μ′}(∂M). This in turn shows that we can v-differentiate (2.15): for 0 ≤ i ≤ ℓ − 1 let X_{i} = X_{i}^{A}(v) \frac{∂}{∂v^{A}} be vector fields with X_{i}^{A} ∈ C_{∞}(∂M). (2.15) gives

0 ≤ i ≤ ℓ − 1 X_{1} ⋯ X_{i}u = X_{1} ⋯ X_{i} ψ x^{μ_{+}} + G_{x_{0}}(X_{1} ⋯ X_{i}(f − L̃u))
= O(x^{σ}(1 + | ln x|)^{ρ}) ,

(2.16)

(X_{1} ⋯ X_{i}u)(x, v) − (X_{1} ⋯ X_{i}u)(x, v′) = O(x^{σ}(1 + | ln x|)^{ρ}|v − v′|^{μ′}) ,(2.17)

this last inequality being uniform in |v − v′|. The equations

LX_{1} ⋯ X_{i}u = X_{1} ⋯ X_{i}f + [L_{,}X_{1} ⋯ X_{i}] u  (2.18)

interior elliptic estimates and a scaling argument yield X_{1} ⋯ X_{i}u ∈ C_{k+2−i+λ}^{σ,ρ}, i ≤ ℓ − 1 and thus u ∈ C_{k+2+λ,ℓ−1}^{σ,ρ}. If (α_{−}, α_{+}) is a strong regularity interval, then Lemma 1.4 and eq. (2.17) show that u ∈ C_{k+2+λ,ℓ−1+μ′}^{σ,ρ}, for any μ′ ∈ [0, μ).

One can now repeat the whole argument ℓ_{0} − 1 times, where ℓ_{0} is the smallest integer such that α_{+} + ℓ_{0} > min(α, μ_{+}) (note that at each iteration one loses one degree of tangential differentiability of u) to obtain u ∈ C_{k+λ,ℓ−ℓ_{0}+μ′}^{σ̃,ρ}, with σ̃ = min(α, μ_{+}), provided ℓ ≥ ℓ_{0}. Finally if α > μ_{+} and u = o(x^{μ_{+}}) then eq. (2.15) holds with G_{x_{0}} replaced by G_{0} and with ψ = 0. Repeating the argument ℓ_{1} − ℓ_{0} times more, where ℓ_{1} is the smallest integer such that α_{+} + ℓ_{1} > α, one obtains u ∈ C_{k+2+λ,ℓ−ℓ_{1}+μ′}^{α,β}.

To obtain the claimed decay of the ℓ − i_{0} + 1 transverse derivatives in the case of a strong regularity interval, where i_{0} = ℓ_{0} or i_{0} = ℓ_{1}, note that at the last step of the iteration above we shall have, for any ϵ > 0 and μ′ ∈ (0, μ),

u ∈ C_{k+2+λ,ℓ−i_{0}+μ′}^{min(α,μ_{+}),β+1}(M_{x_{0}}) ∩ C_{k+2+λ,ℓ−i_{0}+1+μ′}^{α_{+}+i_{0}−1−ϵ}(M_{x_{0}}) .  (2.19)

We have the following:

Lemma 2.2. Under the hypotheses of Theorem 2.1, suppose that

u ∈ C_{k+2+λ,ℓ+μ′}^{σ_{1},ρ}(M_{x_{0}}) ∩ C_{k+2+λ,ℓ−1+μ′}^{σ_{0},ρ}(M_{x_{0}}) , σ_{1} < σ_{0} , 0 < μ′ < μ , with

0 < x ≤ x_{0} x^{α}(1 + | ln x|)^{β} ≤ Cx^{σ_{0}}(1 + | ln x|)^{ρ} .

Then for all 0 < θ < 1, 0 ≤ t < ℓ + μ′ − θ we have

u ∈ C_{k+2+λ,t}^{θσ_{0}+(1−θ)σ_{1},ρ}(M_{x_{0}}) .

Proof: Set u_{x}(v) = u(x, v), and for 0 ≤ t_{0} < t_{1} define t = θt_{0} + (1 − θ)t_{1}. We have the interpolation inequality (cf. e.g. [44] [Appendix A] or [69, p. 236])

∥u_{x}∥_{C_{t}(∂M)} ≤ C∥u_{x}∥_{C_{t_{0}} (∂M)}^{θ}∥u_{x}∥_{C_{t_{1}} (∂M)}^{1−θ} .

Setting t_{0} = ℓ − 1 + μ′, t_{1} = ℓ + μ′ we obtain

u ∈ C_{0,ℓ+μ′−θ}^{θσ_{0}+(1−θ)σ_{1},ρ}(M_{x_{0}}) ,

and the result follows from Lemma 1.4. □

Returning to the proof of Theorem 2.1, let θ be any number smaller than μ′ and let 2σ = μ′ − θ > 0. Eq. (2.19) and Lemma 2.2 give

u ∈ C_{k+2+λ,ℓ−i_{0}+1+σ}^{α_{+}+i_{0}−1+δ}(M_{x_{0}}) ,

with

δ = θ(α − i_{0} + 1 − α_{+} + μ′) − (1 − θ)ϵ .

As ϵ can be chosen arbitrarily small we have δ > 0, and the result follows. □

In Proposition 1.2 we have proved that if L is an isomorphism in weighted Hölder spaces “close to the boundary ∂M” for an interval of weights (α_{−}, α_{+}), then (α_{−}, α_{+}) is a regularity interval for weighted Hölder spaces. The following result shows, that an isomorphism property “close to the boundary ∂M” in weighted Sobolev spaces for an interval (ω_{−}, ω_{+}) implies that ( max(ω_{−}, μ_{−})+(n−

1)/p, min(ω_{+} + (n − 1)/p, μ_{+})) is a regularity interval for weighted Hölder spaces with tangential regularity:

Theorem 2.3. Let p ∈ (1, ∞), and let L ∈ OP_{C_{k+λ,ℓ+μ}^{0}(M)}^{2} be a geometric

elliptic operator, (n − 1)/p < ℓ ≤ k, λ ∈ (0, 1), μ ∈ [0, 1]. Suppose that (2.1)–(2.4) hold with ψ, ϕ, χ ∈ C_{k−1+λ,ℓ−1+μ}^{0}(M). Assume that for all ω ∈ (ω_{−}, ω_{+}) there exists 0 < x_{1} ≤ x_{0} (possibly depending upon ω) such that

L : W^{∘} _{1}^{ω;p}∩W _{2}^{ω;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) → W _{0}^{ω;p}(M_{x_{1}}, x^{−n}dμ_{g})

is an isomorphism, with μ_{−} ≤ ω_{−} < min (ω_{+}, μ_{+} − \frac{n−1}{p}).

((i)) Let u ∈ W_{2}^{p,loc}(M) satisfy
Lu = f ,  f ∈ C_{k+λ,ℓ+μ}^{α,β}(M) ,  (2.20)

with α ∈ (α_{−}, α_{+}), where

(α_{−}, α_{+}) = (ω_{−} + \frac{n − 1}{p} , min (μ_{+}, ω_{+} + \frac{n − 1}{p})) ,  (2.21)

suppose that either

u = O(x^{α_{−}+ϵ}) , ϵ > 0 ,  (2.22)

or

u ∈ W_{0}^{ω;p}(M_{x_{1}}, x^{−n}dμ_{g}) , ω > ω_{−} .  (2.23)

Then for all μ′ ∈ [0, μ) if μ > 0, μ′ = 0 otherwise we have

u ∈ C_{k+2+λ,ℓ_{1}+μ′}^{α,β}(M_{x_{1}}) ,  (2.24)

where ℓ_{1} ∈ ℕ_{0} is the largest integer such that ℓ_{1} < ℓ − (n − 1)/p. Moreover there exists a (u and f independent) constant C such that

∥u∥_{C_{k+2+λ,ℓ_{1}+μ}^{α,β}(M_{x_{1}} )} ≤ C(∥f∥_{C_{k+λ,ℓ+μ}^{α,β}(M)} + ∥u∥_{C_{0}(\overline{CM_{x_{1}/2})}}) .  (2.25)

((ii)) For any u_{0} ∈ C_{k+λ}(∂M) there exists a solution of (2.20) satisfying (2.24) such that

u(x_{1}, v) = u_{0}(v) .  (2.26)

u is unique in the class of solutions satisfying either (2.22) or (2.23).

Remarks:

((i)) The results proved below are unsatisfactory in two respects: 1) it would be more natural to assume μ_{−} − (n − 1)/p ≤ ω_{−} rather then μ_{−} ≤ ω_{−};

2) u is “less tangentially regular” then f. ((ii)) Let us note that under the hypotheses of Theorem 2.3 we must have

μ_{+}−\frac{n−1}{p} ∉ (ω_{−}, ω_{+}): suppose to the contrary that L :W^{∘} _{1}^{ω;p}∩W_{2}^{ω;p}(M_{x_{1}}, g, dμ_{g}) →

W_{0}^{ω;p}(M_{x_{1}}, dμ_{g}) is an isomorphism for ω ∈ (ω_{−}, ω_{+}) and that μ_{+} − \frac{n−1}{p} ∈ (ω_{−}, ω_{+}). Set ρ = L(φx^{μ_{+}}), where φ(x) is a C_{∞}(ℝ) function satisfying φ = 1 for 0 ≤ x ≤ x_{1}/2, φ = 0 for x ≥ 3x_{1}/4. If L is of the

form (2.1)–(2.4), then for any 0 ≤ ϵ < 1 we have ρ ∈ W_{k,ℓ}^{μ_{+}− \frac{n−1}{p} +ϵ;p}, and if μ_{+} − \frac{n−1}{p} ∈ (ω_{−}, ω_{+}) we can choose ϵ so that ω_{ϵ} = μ_{+} − \frac{n−1}{p} + ϵ ∈

(ω_{−}, ω_{+}), therefore there exists u_{ϵ} ∈W^{∘} _{1}^{ω;p} ∩ W_{2}^{ω_{ϵ};p} such that Lu_{ϵ} =

ρ|_{M_{x_{1}}} . Now for any ϵ > 0, φx^{μ_{+}} ∉ W_{2}^{μ_{+}− \frac{n−1}{p} +ϵ;p}, therefore u_{ϵ} ≠ φx^{μ_{+}}.

From φx^{μ_{+}} ∈W^{∘} _{1}^{μ_{+}− \frac{n−1}{p} −δ;p} ∩ W_{2}^{μ_{+}− \frac{n−1}{p} −δ;p} for all δ > 0 it follows that

0 ≠ u_{ϵ} − φx^{μ_{+}} ∈ W^{∘} _{1}^{μ_{+}− \frac{n−1}{p} −δ;p} ∩ W_{2}^{μ_{+}− \frac{n−1}{p} −δ;p}, but L(u_{ϵ} − φx^{μ_{+}}) = 0,

which contradicts injectivity of L on W^{∘} _{1}^{μ_{+}− \frac{n−1}{p} −δ;p} ∩ W_{2}^{μ_{+}− \frac{n−1}{p} −δ;p}, when δ is chosen small enough so that μ_{+} − \frac{n−1}{p} − δ ∈ (ω_{−}, ω_{+}). Similarly one can show that if e.g. L ∈ OP_{C_{k+λ}(M̄)}^{2} with k large enough, then μ_{−} − \frac{n−1}{p} cannot be larger than ω_{−}.

Proof: (i): It follows from Lemma 1.1 that for ω ∈ (ω_{−}, ω_{+}) the map L :

W^{∘} _{1}^{ω;p}∩W _{k+2}^{ω;p} → W_{k}^{ω;p} is an isomorphism, and Proposition 1.2 shows that (ω_{−}, ω_{+}) is a strong regularity interval. Let φ ∈ C_{∞}(M) be any function satisfying φ|_{M_{x_{1}/2}} =

1, φ|_{CM_{x_{1}}} = 0, replacing u by φu if necessary we may without loss of generality assume u(x_{1}, v) = 0. It is convenient to structure the proof in several steps:

Step 1: Uniqueness of solutions in weighted Hölder spaces: Let

Lu = 0 , u ∈ C_{2}^{α_{1},β_{1}}(M_{x_{1}}) , α_{1} > ω_{−} + (n − 1)/p , u(x_{1}, v) = 0 .

For all ω < α_{1} − (n − 1)/p we have u ∈W^{∘} _{1}^{ω;p} ∩ W_{2}^{ω;p}, and if ω_{−} < α_{1} − (n − 1)/p

we can choose ω so that ω ∈ (ω_{−}, ω_{+}) — u = 0 follows from injectivity of L as an

operator from W^{∘} _{1}^{ω;p} ∩ W_{2}^{ω;p} to W_{0}^{ω;p}.

Step 2: Existence of a solution in a weighted Sobolev space with some decay exponent: since f ∈ C_{k+λ}^{α,β}, for all ω < α − (n − 1)/p we have f ∈ W_{k}^{ω;p}, thus if ω_{−} + (n − 1)/p < α we can choose ω ∈ (ω_{−}, ω_{+}). Since L as an operator from

W^{∘} _{1}^{ω;p} ∩ W_{k+2}^{ω;p} to W_{k}^{ω;p} is surjective, there exists u ∈W^{∘} _{1}^{ω;p} ∩ W_{k+2}^{ω;p} which solves (2.20).

Step 3: Let u ∈W^{∘} _{1}^{ω;p} ∩ W_{2}^{p,loc}, ω_{−} < ω < ω_{+}, satisfy (2.20). By the scaling estimates of Lemma 1.1 we have u ∈ W_{k+2}^{ω;p}, and by point (ii) of Theorem 1.3 u is in

W_{k+2,ℓ}^{ω;p}. By Step 2 for any ω_{−} < ω′ < α − (n − 1)/p there exists u′ ∈W^{∘} _{1}^{ω′;p} ∩ W_{k+2}^{ω′;p}

which satisfies (2.20), uniqueness of solutions in W^{∘} _{1}^{ω′;p} ∩ W_{k+2}^{ω′;p} implies u′ = u, and therefore for any δ > 0 we can choose ω such that we have α − (n − 1)/p − δ < ω, and ω ≠ α. Similarly we may assume that the exponent ω in (2.23) satisfies ω > α − (n − 1)/p − δ, ω ≠ α. For y = (x, v) ∈ M_{x_{1}/2} consider the function

ũ(ỹ) = x^{−ω} u(y + \frac{x}{2} ỹ) , ỹ ∈ B_{0}^{n}(1) ,

where B_{0}^{n}(σ) is an open ball in ℝ^{n} of radius σ centred at the origin. From the interior C_{k+λ} and L^{p} elliptic estimates combined with Sobolev’s embedding it is easily shown that there exists a constant C such that

∥ũ∥_{C_{k+2+λ}(\overline{B_{0}^{n}(1/2)})} ≤ C\{∥L̃u∥_{C_{k+λ}(\overline{B_{0}^{n}(1)})} + ∥ũ∥_{L^{p}(B_{0}^{n}(1),dμ̃)}\} ,  (2.27)

L̃u(ỹ) = x^{−ω}Lu(y + \frac{x}{2} ỹ) , dμ̃(ỹ) = \sqrt{det g_{ij}(y + \frac{x}{2} ỹ)} d^{n}ỹ .

From the straightforward inequality

∥ũ∥_{L^{p}(B_{0}^{n}(1),dμ̃)} ≤ C∥u∥_{W_{0}^{ω;p}(M_{x_{1}} ,x^{−n}dμ_{g})}

for some constant C, combined with (2.27), it follows that u ∈ C_{k+2+λ}^{ω}. Step 4: Let us show by induction that:

(i) u ∈ C_{k+2+λ,ℓ}^{ω}(M_{x_{1}}).

(ii) Let ℓ_{∗} be an integer in [1, ℓ]. If u ∈ C_{k+2+λ}^{∗}(M_{x_{1}}) and if ∂_{v}^{γ}u = O(“x^{∗}”), 0 ≤ |γ| ≤ ℓ_{∗} (cf. Step 6), then u ∈ C_{k+2+λ,ℓ_{∗}}^{∗}(M_{x_{1}}). Here ∗ is either equal to some number ω̂ (and then “x^{∗}”= xω̂, or ∗ = α, β (and then “x^{∗}”= x^{α}(1 + | ln x|)^{β}). We also assume that x^{∗} ≥ x^{α}(1 + | ln x|)^{β}.

Indeed, suppose that u ∈ C_{k+2+λ,ℓ_{0}}^{∗}. In case (i) this holds for ∗ = ω and ℓ_{0} = 0 by Step 3; in case (ii) this holds by the inductive assumption. Let X_{1}, …, X_{i} be any smooth vector fields of the form X^{A}(v)\frac{∂}{∂v^{A}}, X^{A}(v) ∈ C_{∞}(∂M). Let u_{i} = X_{1} ⋯ X_{i}u, f_{i} = X_{1} ⋯ X_{i}f. In case (i) for ℓ_{0} + 1 ≤ ℓ and in case (ii) for ℓ_{0} + 1 ≤ ℓ_{∗} we have

Lu_{ℓ_{0}+1} = f_{ℓ_{0}+1} + [L_{,}X_{1} ⋯ X_{ℓ_{0}+1}]u ∈ C_{k−ℓ_{0}−1+λ}^{α,β} + C_{k+1−ℓ_{0}+λ}^{∗} .

Note that in case (i) we have u ∈ W_{k+2,ℓ}^{ω;p} (cf. Step 3), so that for ℓ_{0} +1 ≤ ℓ we have

u_{ℓ_{0}+1} ∈ W_{k−ℓ_{0}−1}^{ω;p}; setting ũ_{ℓ_{0}+1}(ỹ) ≡“x^{−∗}” u_{ℓ_{0}+1}(y + \frac{x}{2} ỹ), ỹ ∈ B_{0}^{n}(1) we obtain

∥ũ_{ℓ_{0}+1}∥_{L^{p}(B_{0}^{n}(1),dμ̃)} ≤ C∥u_{ℓ_{0}+1}∥_{W_{0}^{ω;p}(M_{x_{1}} ,x^{−n}dμ_{g})} .

Applying now the estimate (2.27) to the function ũ_{ℓ_{0}+1} ends the induction.

Step 5: Suppose that u = O(x^{α_{−}+ϵ}). By the scaling estimates of Lemma 1.1 we have u ∈ C_{k+2+λ}^{α_{−}+ϵ}, by existence of a solution as considered in step 4 and by uniqueness of solutions in weighted Hölder spaces, step 1, it follows that decreasing ϵ if necessary we can assume ω = α_{−} + ϵ > ω_{−} ≥ μ_{−}, ω ≠ α.

Step 6: The functions u_{i} defined in step 4 satisfy the ODE’s

L_{ab} u_{i} = f_{i} − X_{1} ⋯ X_{i} L̃u ,

and since u_{i} = O(x^{ω}), ω > μ_{−}, we have

u_{i} = ψ_{i} x^{μ_{+}} + G(f_{i}) − G(X_{1} ⋯ X_{i} L̃u) ,  (2.28)

G ≡ G_{x_{1}}, for some functions ψ_{i}(v). Since u ∈ C_{k+2+λ,ℓ}^{ω}(M_{x_{1}}) ⊂ C_{ℓ}^{loc}(M_{x_{1}}), X_{1} ⋯ X_{i} L̃u ∈ C_{ℓ−i}^{loc}(M_{x_{1}}) (cf. (2.4)–(2.6)), G(X_{1} ⋯ X_{i} L̃u) ∈ C_{ℓ−i}^{loc}(M_{x_{1}}), (cf. (4.2)–(4.4)), G(f_{i}) ∈ C_{ℓ−i}^{loc}(M_{x_{1}}), setting x = x_{1}/2 in (2.28) we obtain ψ_{i} ∈ C_{ℓ−i}(∂M), in particular ψ_{i} is bounded. We thus have

ψ_{i} x^{μ_{+}} = O(x^{μ_{+}}) ,
G(f_{i}) = O(x^{α}(1 + | ln x|)^{β}) ,
0 ≤ i ≤ ℓ − 1  G(X_{1} ⋯ X_{i}L̃u) = O(x^{ω+1}) ,

which gives for 0 ≤ i ≤ ℓ − 1

a)  ω + 1 > α ⟹ u_{i} = O(x^{α}(1 + | ln x|)^{β})  (2.29)
b)  ω + 1 < α ⟹ u_{i} = O(x^{ω+1})  (2.30)

From (2.27) we obtain u ∈ C_{k+2+λ}^{∗}, where ∗ = ω + 1 or ∗ = α, β.

Step 7 (and final): We can go through step 4 again to conclude either u ∈ C_{k+2+λ,ℓ−1}^{α,β} in case a), or u ∈ C_{k+2+λ,ℓ−1}^{ω+1} in case b). After going through steps 6 and 4 at most k_{1} times, where k_{1} is the smallest integer larger than \frac{n−1}{p}, one obtains u ∈ C_{k+2+λ,ℓ−k_{1}}^{α,β}; if μ > 0 the above considerations applied to difference quotients together with Lemma 1.4 give u ∈ C_{k+2+λ,ℓ−k_{1}+μ′}^{α,β} for all μ′ ∈ [0, μ). The inequality (2.25) follows by noting that every step above “comes equipped” with an appropriate inequality, the details are left to the reader.

(ii): Existence of some solution follows from Step 2 above in a standard way; uniqueness follows from Steps 1 and 5; regularity follows from point (i) of the Theorem. □

Combining the results of Theorem 2.3 with the arguments of Theorem 2.1 we obtain:

Theorem 2.4. Let p ∈ (1, ∞), and let ℓ̂ be the smallest integer larger than (n − 1)/p. Let L ∈ OP_{C_{k+λ,ℓ+ℓ̂+μ}^{0}(M)}^{2} be a geometric elliptic operator and suppose

that ℓ̂+ ℓ ≤ k, λ ∈ (0, 1), μ ∈ (0, 1], ℓ ≥ 1. Suppose moreover that (2.1)–(2.4) hold with ψ, ϕ, χ ∈ C_{k−1+λ,ℓ+ℓ̂−1+μ}^{0}(M), and that for all ω ∈ (ω_{−}, ω_{+}) there exists 0 < x_{1} ≤ x_{0} (possibly depending upon ω) such that

L : W^{∘} _{1}^{ω;p}∩W _{2}^{ω;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) → W _{0}^{ω;p}(M_{x_{1}}, x^{−n}dμ_{g})

is an isomorphism, with μ_{−} ≤ ω_{−} < min (ω_{+}, μ_{+} − \frac{n−1}{p}). Let u ∈ W_{2}^{p,loc}(M) satisfy

Lu = f ,  f ∈ C_{k+λ,ℓ̂+ℓ+μ}^{α,β}(M) ,  (2.31)

with α ≥ α_{+}, where

(α_{−}, α_{+}) = (ω_{−} + \frac{n − 1}{p} , min (μ_{+}, ω_{+} + \frac{n − 1}{p})) .  (2.32)

Suppose finally that either

u = O(x^{α_{−}+ϵ}) , ϵ > 0 ,  (2.33)

or

u ∈ W_{0}^{ω;p}(M_{x_{1}}, x^{−n}dμ_{g}) , ω > ω_{−} .  (2.34)

If ℓ ≥ ℓ_{0}, where ℓ_{0} is the smallest integer such that α_{+} + ℓ_{0} > min(α, μ_{+}), then there exist σ ∈ (0, 1) and δ > 0 such that for μ′ ∈ [0, μ) eqs. (2.9)–(2.12) hold. The constants σ and δ depend only upon ω_{+}, μ_{+}, α, μ, λ, ℓ, (n − 1)/p and k. Moreover the a–priori estimate (2.13) holds with a (u and f independent) constant C provided that the norm ∥f∥_{C_{k+λ,ℓ+μ}^{α,β}(M)} there is replaced by ∥f∥_{C_{k+λ,ℓ+ℓ̂+μ}^{α,β}(M)}.

2.2. Classical regularity at the boundary. In this section we shall establish boundary regularity of solutions of the problem

Lu = f,  f ∈ x^{α}C^{k+λ}(M̄) .  (2.35)

The following Lemma reduces this problem to that of regularity of solutions of

Lu = f,  f ∈ C_{k+λ,0+λ}^{k+α,β}(M) ∩ C_{k+λ}^{k+α+λ,β}(M),  β = 0 or β = 1 .

Lemma 2.5. Let k ∈ ℕ_{0}, λ ∈ [0, 1], and let an operator L ∈ OP_{C^{k+λ}(M̄)}^{2} satisfy (2.1)–(2.4). Suppose that

f ∈ x^{α}C^{k+λ}(M̄), α > μ_{−} .

((i)) Suppose moreover that α + k < μ_{+} or that μ_{+} − α ∉ ℕ_{0}. There exists

a function û ∈ C_{∞}^{α}(M) ∩_{i∈ℕ_{0}} x^{α−i}C^{k+i+λ}(M̄) such that
Lû − f ∈ C_{k+λ,0+λ}^{k+α}(M) ∩ C_{k+λ}^{k+α+λ}(M) .  (2.36)

((ii)) Suppose alternatively that α + k > μ_{+} and μ_{+} − α ∈ ℕ_{0}, set k_{1} = μ_{+} − α. There exist functions û ∈ C_{∞}^{α}(M) ∩_{i∈ℕ_{0}} x^{α−i}C^{k+i+λ}(M̄) and

û_{log} ∈ C_{∞}^{μ_{+}}(M) ∩_{i∈ℕ_{0}} x^{μ_{+}−i}C^{k−k_{1}+i+λ}(M̄) such that
L(û + û_{log} log x) − f ∈ C_{k+λ,0+λ}^{k+α,1}(M) ∩ C_{k+λ}^{k+α+λ,1}(M) .  (2.37)

Proof: We shall prove point (i) assuming λ > 0; point (i) with λ = 0 and point (ii) are proved by a similar argument. Suppose that for some ℓ satisfying −1 ≤ ℓ ≤ k − 1 we have found a function u_{ℓ} ∈ C_{∞}^{α} such that x^{−α}(Lu_{ℓ} − f) is in C_{k+λ}(M̄) and has vanishing Taylor expansion at ∂M up to order ℓ (cf. Proposition 1.1); the result is true for ℓ = −1 if one sets u_{−1} ≡ 0. Let x^{−α}w_{ℓ} be obtained from Lemma 3.1 with m = ℓ + 1 as an extension of a function ψ_{ℓ} ∈ C_{k−ℓ−1+λ}(∂M) to be specified below. By eqs. (3.1)–(3.2) and (2.1)–(2.3) we have

Lw_{ℓ} − L_{ab}w_{ℓ} ∈ C_{0}^{α+ℓ+1+λ}(M)

(note that the second term in eq. (3.2) drops out when at least one of the derivatives there is a v–derivative). By hypothesis it holds that μ_{+} − ℓ − 1 ≠ α and a straightforward calculation (cf. e.g. the proof of Theorem 2.8) shows that we can choose ψ_{ℓ} so that the function x^{−α}Lu_{ℓ+1} ≡ x^{−α}L(u_{ℓ} + w_{ℓ}) is in C_{k+λ}(M̄) and vanishes to order ℓ + 1 at the boundary. The remaining claims follow by properties of the extension operator E of Lemma 3.1 (cf. eq. (3.4)), using arguments similar to those of the proof of Corollary 3.2. □

We are ready now to prove the following:

Theorem 2.6 (Classical boundary regularity, α ∈ (α_{−}, α_{+})). Let (α_{−}, α_{+}) be a strong regularity interval for C_{0+λ}^{α}(M) for a geometric elliptic operator L ∈ OP_{C_{k+λ}(M̄)}^{2}, λ ∈ (0, 1), suppose that L is of the form (2.1)–(2.4), and that

Lu = f ∈ x^{α}C_{k+λ}(M̄),  α ∈ (α_{−}, α_{+}) ,  u = O(x^{α_{−}+ϵ}) , ϵ > 0 .

Let ℓ_{0} ∈ ℕ_{0} be the largest integer such that α + ℓ_{0} < α_{+}. Then there exists σ > 0 such that

α + k < α_{+} ⟹ u ∈ ∩_{i=0}^{2}x^{α−i}C_{k+i+σ}(M̄_{x_{0}}) ,  (2.38)
k > ℓ_{0} ⟹ u ∈ ∩_{i=0}^{2}x^{α−i}C_{ℓ_{0}+i+σ|k−ℓ_{0}}(M̄_{x_{0}}) .  (2.39)

The constant σ depends only upon α_{+}, μ_{+}, α, λ and k. Moreover when F denotes any of the spaces appearing in eqs. (2.38)–(2.39), then the a–priori estimate (2.13) holds with a (u and f independent) constant C provided that the norm ∥f∥_{C_{k+λ,ℓ+μ}^{α,β}(M)} there is replaced by ∥f∥_{x^{α}C_{k+λ}(M̄)}.

Remarks: 1. The above results are sharp when α_{+} = μ_{+}, except perhaps for the value of the modulus of Hölder continuity which we leave unspecified.

2. For α_{+} < μ_{+} define ℓ̂_{0} ∈ ℕ_{0} to be the largest integer such that α + ℓ̂_{0} < μ_{+}. The reader will notice that for α ≤ α_{+} the arguments of the proof of Theorem 2.9 can be used to show that the right–hand–side of the implication (2.38) holds for all k ≤ ℓ̂_{0}; similarly for k > ℓ̂_{0} the right–hand–side of the implication (2.39) will hold with ℓ_{0} replaced by ℓ̂_{0}. This is again sharp except for the value of the modulus of Hölder continuity of u.

3. Under the conditions of (2.39) in the special case μ_{+} −α ∈ ℕ_{0} one can obtain some more information about u. This, however, requires different techniques and will be considered separately in Theorem 2.10 below.

Proof: Suppose first that α + k < α_{+}, let û be given by Lemma 2.5, by point (i) of that Lemma and by interpolation (cf. the arguments of proof of Lemma 2.2) there exists 0 < 2σ < λ such that

L(u − û) ∈ C_{k+2σ,0+2σ}^{α+k+2σ}(M) .

By Theorem 1.3 point (i) we have u−û ∈ C_{k+2+2σ,0+σ}^{α+k+2σ}(M) ⊂ ∩_{i=0}^{2}x^{α−i}C_{k+i+σ}(M̄_{x_{0}}) (this last inclusion following from Proposition 2.2), and (2.38) follows. Equation (2.39) follows from the inclusions C_{k+2σ,0+2σ}^{α+k+2σ}(M) ⊂ C_{k+2σ,k−ℓ_{0}+2σ}^{α+ℓ_{0}+2σ}(M) and C_{k+2+2σ,k−ℓ_{0}+σ}^{α+ℓ_{0}+2σ}(M) ⊂ ∩_{i=0}^{2}x^{α−i}C_{ℓ_{0}+i+σ|k−ℓ_{0}}(M̄_{x_{0}}) by the same argument. □

To obtain classical regularity at the boundary for α’s beyond the threshold α_{+} some more work is required. The next Lemma will allow us to reduce the problem to an analysis of those solutions, the decay rate of which is faster than the one corresponding to the larger critical exponent. The proof proceeds along the lines of proof of Lemma 2.5.

Lemma 2.7. Under the hypotheses of Lemma 2.5 let ψ ∈ C^{k+λ}(∂M). There exists ψ̂ ∈ ∩_{i∈ℕ_{0}}x^{μ_{+}−i} C^{k+i+λ}(M̄) such that

x^{−μ_{+}} ψ̂|_{∂M} = ψ ,
Lψ̂ ∈ C_{k+λ,0+λ}^{k+μ_{+}}(M) ∩ C_{k+λ}^{k+μ_{+}+λ}(M) .

Arguing as in the proof of Theorem 2.1 one obtains the following:

Lemma 2.8. Let L ∈ OP_{C_{k+λ,k+λ}^{0}(M)}^{2}, λ ∈ (0, 1), be a geometric elliptic operator

with non–empty strong regularity interval (α_{−}, α_{+}) for C_{0+λ}^{α}(M). Suppose that L is of the form (2.1)–(2.4) with ψ, ϕ, χ ∈ C_{k−1+λ,k−1+λ}^{0}(M). Let m, ℓ ∈ ℕ_{0} and assume that β ≥ μ_{−} and m ≤ ℓ ≤ k. If β+m ≥ μ_{+} assume further that u = o(x^{μ_{+}}). If

Lu = f , f ∈ C_{k+λ,0+λ}^{β+m+λ}(M) , u ∈ C_{k+2+λ,ℓ+λ}^{β+λ}(M) ,

then there exists σ > 0 such that

u ∈ C_{k+2+σ,ℓ−m+σ}^{β+m+σ}(M_{x_{0}}) .

We are ready now to prove classical regularity at the boundary above the α_{+} treshhold in the presence of a regularity interval, or of a strong regularity interval. To avoid a somewhat tedious and not very enlightning discussion of various cases we shall only give the proofs when α_{+} = μ_{+}. It should be clear from the proofs below how to generalize the argument to the case α_{+} < μ_{+}, cf. also Remark 1 below.

Theorem 2.9 (Classical boundary regularity, α > α_{+}). Let (α_{−}, α_{+}) be a strong regularity interval for C_{0+λ}^{α}(M) for a geometric elliptic operator L ∈ OP_{C_{k+λ}(M̄)}^{2}, λ ∈ (0, 1), k ≥ 1, suppose that L is of the form (2.1)–(2.4), and that

Lu = f ∈ x^{α}C_{k+λ}(M̄),  α > α_{+} = μ_{+} ,  u = O(x^{α_{−}+ϵ}) , ϵ > 0 .

Then there exists σ > 0 such that

u ∈ ∩_{i=0}^{2}x^{μ_{+}−i}C_{k+i+σ}(M̄_{x_{0}}) + ∩_{i∈ℕ_{0}}x^{α−i}C_{k+i+λ}(M̄_{x_{0}}) .  (2.40)

In particular, if we denote by ℓ_{0} ∈ ℕ_{0} the largest integer such that μ_{+} + ℓ_{0} < α, then we have

((i)) If μ_{+} + k < α or if α − μ_{+} ∈ ℕ (or both)

u ∈ ∩_{i=0}^{2}x^{μ_{+}−i}C_{k+i+σ}(M̄_{x_{0}}) ,  (2.41)
((ii)) k > ℓ_{0}
u ∈ ∩_{i=0}^{2}x^{μ_{+}−i}C_{ℓ_{0}+i+σ|k−ℓ_{0}}(M̄_{x_{0}}) .  (2.42)

If moreover u = o(x^{μ_{+}}) then it also holds that

u ∈ ∩_{i=−k}^{2}x^{μ_{+}−i}C_{k+i+σ}(M̄_{x_{0}}) + ∩_{i∈ℕ_{0}}x^{α−i}C_{k+i+λ}(M̄_{x_{0}}) ,  (2.43)

and for k ≥ ℓ̃, where ℓ̃ is the smallest integer such that ℓ̃≥ α − μ_{+}, we also have

u ∈ ∩_{i=0}^{ℓ̃+2} x^{α−i}C_{k−ℓ̃+i+σ}(M̄_{x_{0}}) .  (2.44)

The constant σ depends only upon α_{+}, μ_{+}, α, λ and k. Moreover when F denotes any of the spaces appearing in eqs. (2.41)–(2.42) or (2.44), then the a–priori estimate (2.13) holds with a (u and f independent) constant C provided that the norm ∥f∥_{C_{k+λ,ℓ+μ}^{α,β}(M)} there is replaced by ∥f∥_{x^{α}C_{k+λ}(M̄)}. Similarly if F_{1}, F_{2} denote any of the summand spaces in eqs. (2.40) and (2.43), we have the a priori estimate

∥u∥_{F_{1}⊕F_{2}} ≤ C (∥f∥_{C_{x^{α}C_{k+λ}(M̄)}} + ∥u∥_{C_{0}^{α_{−}+ϵ}(M)}) .  (2.45)

Remarks: 1. If α_{+} < μ_{+} one can still obtain some classical regularity of the solutions, the results being somewhat worse than in the case α_{+} = μ_{+}. More precisely, we have the following: If α_{+} < α < μ_{+} and if k ≥ ℓ_{1}, where ℓ_{1} is the smallest integer larger than α − α_{+}, then the implications (2.38)–(2.39) hold when k is replaced by k − ℓ_{1} at both sides of those implications. Similarly if α > μ_{+} and k ≥ ℓ_{2} + 1, where ℓ_{2} is the smallest integer larger than or equal to μ_{+} − α_{+}, then eqs. (2.40)–(2.44) hold with, however, k replaced by k − ℓ_{2}; this substitution should be done in both members of the implication (2.41). For the validity of (2.42), respectively of (2.44), the condition k > ℓ_{0}, respectively k ≥ ℓ̃, should of course be replaced by k > ℓ_{0} + ℓ_{2}, respectively k ≥ ℓ̃+ ℓ_{2}. In all cases discussed here one has the corresponding a–priori estimates.

2. If (α_{−}, α_{+}) is only a regularity interval for C_{0+λ}^{α}(M) and if α_{+} = μ_{+}, then eqs. (2.40)–(2.44) hold if we moreover assume α_{−} + 1 < α_{+}, u = o(x^{α_{−}+1+ϵ}) for some ϵ > 0 with, however, σ = 0 and k replaced by k − 1; in (2.42) the condition k > ℓ_{0} should of course be replaced by k > ℓ_{0} + 1; similarly for (2.44). If α_{+} < μ_{+} then the results discussed in Remark 1 will hold with σ = 0 and with a value of k further decreased by 1 as compared to the value there.

Proof: In what follows the symbol σ denotes a real number in the interval (0, λ) the value of which may vary from one expression to the next. Let û be given by Lemma 2.5, by point (i) of that Lemma and by interpolation (cf. the arguments of proof of Lemma 2.2) there exists 0 < σ < λ such that

f̃ ≡ L(u − û) ∈ C_{k+σ,0+σ}^{α+k+σ}(M) ⊂ C_{k+σ,k+σ}^{μ_{+}+σ}(M) .

By Theorem 1.3 point (i) for all ϵ > 0 we have u − û ∈ C_{k+2+σ,k+σ}^{μ_{+}−ϵ}(M_{x_{0}}) so that L̃(u − û) ∈ C_{k−1+σ,k−1+σ}^{μ_{+}+1−ϵ}(M_{x_{0}}) ∩ C_{k+σ,k+σ}^{μ_{+}−ϵ}(M_{x_{0}}); interpolation gives L̃(u − û) ∈ C_{k+σ,k+σ}^{μ_{+}+σ}(M_{x_{0}}). We have

u − û = ψx^{μ_{+}} + G_{0}[f̃] − G_{0}[L̃(u − û)]  (2.46)

with some function ψ(v), and it follows from this equation at x = x_{0}/2 and from (4.14) that ψ ∈ C_{k+σ}(∂M). Let ψ̂ be given by Lemma 2.7, set ũ ≡ u − û − ψ̂, we then have

Lũ ∈ C_{k+σ,0+σ}^{μ_{+}+k+σ}(M_{x_{0}}),  ũ = o(x^{μ_{+}}) .

so that we can write

ũ = G_{0}[Lũ] − G_{0}[L̃ũ] .  (2.47)

It follows that ũ ∈ C_{0,k+σ}^{μ_{+}+σ}(M_{x_{0}}) and Lemma 1.4 gives ũ ∈ C_{k+2+σ,k+σ}^{μ_{+}+σ}(M_{x_{0}}). Lemma 2.8 with β = μ_{+}, λ there replaced by σ, and ℓ = m = k yields

u − û − ψ̂ ∈ C_{k+2+σ,0+σ}^{μ_{+}+k+σ}(M_{x_{0}}) .

Eqs. (2.40)–(2.43) follow now from Proposition 2.2. □

Similar arguments give the following, no details will be given (note that below we are not assuming that α_{+} = μ_{+}):

Theorem 2.10 (Classical boundary regularity, μ_{+} − α ∈ ℕ_{0}). Let (α_{−}, α_{+}) be a strong regularity interval for C_{0+λ}^{α}(M) for a geometric elliptic operator L ∈ OP_{C_{k+λ}(M̄)}^{2}, λ ∈ (0, 1), suppose that L is of the form (2.1)–(2.4) and that

α_{+} ≤ μ_{+} ,  α_{−} < α ≤ μ_{+} ,  μ_{+} − α ∈ ℕ_{0} ,
Lu = f ∈ x^{α}C_{k+λ}(M̄), u = O(x^{α_{−}+ϵ}) , ϵ > 0 .

For α ≤ α_{+} set ℓ = 0, otherwise let ℓ be the smallest integer such that α_{+}+ℓ > α. If α+k ≥ μ_{+}+ℓ+1, then there exists σ > 0 and u_{log} ∈ ∩_{i=0}^{μ_{+}−α+2}x^{μ_{+}−i} C_{k−(μ_{+}−α)+i+λ}(M̄) such that

u − u_{log} log x ∈ ∩_{i=0}^{2}x^{α−i}C_{k−ℓ+i+σ}(M̄_{x_{0}}) .  (2.48)

If u_{log}|_{∂M} = 0, then u_{log} ≡ 0. The constant σ depends only upon α_{+}, μ_{+}, α, λ and k. Moreover when F_{1}, F_{2} denote any of the summand spaces in eq. (2.48), then the a–priori estimate (2.45) holds with a (u and f independent) constant C.

Remark: If (α_{−}, α_{+}) is only a regularity interval for C_{0+λ}^{α}(M) and if we moreover assume that α_{−} + 1 < α_{+} and u = O(x^{α_{−}+1+ϵ}) for some ϵ > 0, then eq. (2.48) holds with σ = 0 and with k replaced by k − 1.

Theorems 2.6, 2.9 and 2.10 were proved assuming the existence of a (strong) regularity interval in weighted Hölder spaces. In weighted Sobolev spaces we have the following corresponding result:

Theorem 2.11. Let p ∈ (1, ∞), and let ℓ̂ be the smallest integer larger than or equal to (n − 1)/p (recall that n = dim M). Let L ∈ OP_{C_{k+ℓ̂+λ}(M̄)}^{2}, λ ∈ (0, 1) be a geometric elliptic operator of the form (2.1)–(2.4). Suppose that for all ω ∈ (ω_{−}, ω_{+}) there exists 0 < x_{1} ≤ x_{0} (possibly depending upon ω) such that

L : W^{∘} _{1}^{ω;p}∩W _{2}^{ω;p}(M_{x_{1}}, g, x^{−n}dμ_{g}) → W _{0}^{ω;p}(M_{x_{1}}, x^{−n}dμ_{g})

is an isomorphism, with μ_{−} ≤ ω_{−} < min (ω_{+}, μ_{+} − \frac{n−1}{p}). Define

(α_{−}, α_{+}) = (ω_{−} + \frac{n − 1}{p} , min (μ_{+}, ω_{+} + \frac{n − 1}{p})) .  (2.49)

Let u ∈ W_{2}^{p,loc}(M) satisfy

Lu = f ,  f ∈ x^{α}C_{k+ℓ̂+λ}(M̄) ,  (2.50)

with α > α_{−}. Suppose finally that either

u = O(x^{α_{−}+ϵ}) , ϵ > 0 ,  (2.51)

or

u ∈ W_{0}^{ω;p}(M_{x_{1}}, x^{−n}dμ_{g}) , ω > ω_{−} .  (2.52)

Then:

((i)) Consider α ∈ (α_{−}, α_{+}), let ℓ_{0} ∈ ℕ_{0} be the largest integer such that

α + ℓ_{0} < μ_{+}. There exists σ > 0 such that
α + k < μ_{+} ⟹ u ∈ ∩_{i=0}^{2+ℓ̂}x^{α−i}C_{k+i+σ}(M̄_{x_{0}}) ,  (2.53)
k > ℓ_{0} ⟹ u ∈ ∩_{i=0}^{2+ℓ̂}x^{α−i}C_{ℓ_{0}+i+σ|k−ℓ_{0}}(M̄_{x_{0}}) .  (2.54)

((ii)) If α_{+} < μ_{+} consider α_{+} ≤ α < μ_{+}. Define ℓ_{1} to be the smallest integer larger than α − α_{+}. If k − ℓ_{1} ≥ 1, then the implications (2.53)–(2.53) hold with k − ℓ_{1} substituted for k at both sides of those implications.

((iii)) If μ_{+} − α ∈ ℕ_{0}, then the conclusions of Theorem 2.10 hold.

((iv)) Let ℓ_{2} ∈ ℕ_{0} be the smallest number larger than or equal to μ_{+} − α_{+}. If α > μ_{+} and if in Theorem 2.9 we replace the conditions on k by conditions on k − ℓ_{2} (in particular the condition k − ℓ_{2} ≥ 1 must hold), then the conclusions of Theorem 2.9 will hold with k replaced by k −ℓ_{2}.

Moreover the a–priori estimates of those theorems hold modified in the obvious way.

Proof: When (n−1)/p is not an integer the result follows from Theorems 2.3, 2.4 and Lemma 2.8 using the arguments given in the proofs of Theorems 2.6 and 2.9. Here Lemma 2.8 is used with ℓ = k or ℓ = k − ℓ_{1}, and with k there replaced by k + ℓ̂. Note, however, that for (n − 1)/p ∈ ℕ a verbatim application of Theorems 2.3 and 2.4 would lead to a result worse by one as far as differentiability of u is concerned. This can be improved by noting that, in the notation^{3} of Theorem 2.3 and under the hypotheses there, for α < α_{+} the inclusion (2.24) in Theorem 2.3 can be replaced by

∀ ϵ > 0  u − û ∈ C_{k+2+λ,k−ℓ̂+λ}^{α−ϵ}(M_{x_{1}}) .  (2.55)

In eq. (2.55) the constant ℓ̂ is as defined in the statement of Theorem 2.11. This is sufficient for our purposes and leads to the result here with ℓ̂ — as described in the statement of Theorem 2.11. Similarly for (n − 1)/p ∈ ℕ the constant δ in Theorem 2.4 can be chosen so that α_{+} + ℓ_{0} − 1 + δ is arbitrarily close to α. The result follows then by the same arguments as indicated above. □

3It should be stressed that k in eq. (2.55) does not coincide with k used in the statement of Theorem 2.11.

CHAPTER 5 Nonlinear equations with polyhomogeneous coefficients.

1. Polyhomogeneity of solutions of some fully nonlinear equations

In this chapter we shall show, under appropriate hypotheses, that solutions of a class of (uniformly degenerating) fully nonlinear second order systems of equations are polyhomogeneous near ∂M. Roughly speaking, to prove polyhomogeneity of solutions we shall need

((i)) existence of a polyhomogeneous “approximate solution” ϕ_{0}, ((ii)) existence of a “regularity interval” for the equation linearized at ϕ_{0}, ((iii)) boundedness of the solution in some weighted Hölder class.

More precisely, consider a system of equations of the form

F^{A}[y, ϕ^{B}, x∂_{i}ϕ^{B}, x^{2}∂_{i}∂_{j}ϕ^{B}] = 0,  A = 1, …, N,  (1.1)

or, shortly,

F [ϕ] = 0.

We shall assume that both the fields ϕ and the equations (1.1) are geometric in the sense of Chapter 4, we leave to the reader the easy task of formalizing this notion along the lines of Chapter 4. We shall assume that we have an approximate solution ϕ_{0} of equation (1.1) satisfying

ϕ_{0} ∈ A_{phg} ∩ C_{0}(M̄) ,  (1.2)
F [ϕ_{0}] ∈ A_{phg} ∩ C_{0}^{α_{0}}(M),  α_{0} > 0 .  (1.3)

The functions F^{A}(y, z^{B}, p_{i}^{B}, q_{ij}^{B}) will be assumed to be continuous with respect to all the variables, smooth with respect to the variables z, p, q, and polyhomogeneous w.r.t. x in the following sense: for all multiindices ν_{1}, ν_{2}, ν_{3} we have

[∂_{z}^{ν_{1}}∂_{p}^{ν_{2}}∂_{q}^{ν_{3}}F]|_{ϕ=ϕ_{0}} ∈ A_{phg} ∩ C_{0}(M̄) ,  (1.4)

and for all i and for all multiindices μ, ν_{1}, ν_{2}, ν_{3} it holds that

(x∂_{x})^{i}∂_{v}^{μ}∂_{z}^{ν_{1}}∂_{p}^{ν_{2}}∂_{q}^{ν_{3}}F ∈ C^{0}(M̄ × ℝ^{N(1+n+n^{2})}).  (1.5)

Remark: It should be clear from the proof below that it is sufficient to assume (1.5) in a neighbourhood of the graph of ϕ_{0} over M̄_{x_{0}} defined as

{(y, ϕ_{0}, x∂ϕ_{0}, x^{2}∂_{i}∂_{j}ϕ_{0}), y ∈ M̄_{x_{0}}} ⊂ M̄_{x_{0}} × ℝ^{N(1+n+n^{2})} .

Let F′[χ] be the linearization of (1.1) at ϕ = χ:

F′[χ]ψ = \frac{∂F^{A}}{∂z^{B}} |_{ϕ=χ}ψ^{B} + x\frac{∂F^{A}}{∂p_{i}^{A}} |_{ϕ=χ} \frac{∂ψ^{B}}{∂y^{i}} + x^{2} \frac{∂F^{A}}{∂q_{ij}^{B}} |_{ϕ=χ} \frac{∂^{2}ψ^{B}}{∂y^{i}∂y^{j}} . (1.6)

Our hypotheses above are motivated by the harmonic map equation (under appropriate conditions; cf. [55, 56, 31] for some related results), and by the equation for hypersurfaces with prescribed mean curvature (cf. [49, 7]). Indeed, for metrics on the source space of the form g̃_{ij} = x^{−2}g_{ij} as considered in Chapter 3 or in Chapter 7 the harmonic map equation will be of the form considered here. Similarly, in a space–time with a smooth or polyhomogeneous Scri, spacelike hypersurfaces intersecting Scri transversally will satisfy an equation of the form (1.1). Our theorem below together with the remarks following it reduce the question of boundary regularity of solutions to those equations to an a–priori estimate in C_{1}^{α}(M_{x_{0}}) spaces, together with the proof of existence of the strong regularity interval. Note that for the prescribed mean curvature equation the existence of a strong regularity interval for appropriate approximate solutions required in Theorem 1.1 below follows from the results in Section 2.

We have the following:

Theorem 1.1. Suppose that (1.2)–(1.5) hold, assume that F′[ϕ_{0}] is a geometric elliptic operator of the form (2.1)–(2.4) considered in Section 2 and that F′[ϕ_{0}] has a strong regularity interval (α_{−}, α_{+}) with

α_{+} > 0,  α_{0} > α_{−} .  (1.7)

Let ϕ ∈ C_{∞}^{loc}(M) be a solution of (1.1) satisfying

ϕ − ϕ_{0} ∈ C_{∞}^{α}(M_{x_{0}}),  α > max(0, α_{−}) .  (1.8)

Then

ϕ ∈ A_{phg} ∩ C^{0}(M̄_{x_{0}}) .

Remarks:

((i)) In the case when eq. (1.1) is quasi–linear, i.e., linear in second derivatives, then eq. (1.8) can be further weakened to ϕ − ϕ_{0} ∈ C_{1}^{α}(M_{x_{0}}), α > max(0, α_{−}). Finally, if eq. (1.1) is linear both in second and first derivatives of ϕ, then eq. (1.8) can be replaced by ϕ − ϕ_{0} ∈ C_{0}^{α}(M_{x_{0}}), α > max(0, α_{−}); cf. Proposition 1.4 below.

((ii)) If (α_{−}, α_{+}) is a weak regularity interval, the result remains true if we add the conditions α_{+} − α_{−} > 1, α_{0} > α_{−} + 1, α > max(0, α_{−} + 1), cf. the Remark following Lemma 1.3 below.

Proof: We shall prove that there exists a sequence s_{i} →_{i→∞} ∞ and a sequence of functions ϕ_{i} ∈ A_{phg} ∩ C_{0}(M̄) such that

ϕ_{i} − ϕ ∈ C_{∞,∞}^{s_{i}}(M) + A_{phg} ∩ C_{0}^{α_{0}}(M)  (1.9)

from which the result follows. We shall proceed recursively; suppose thus that ϕ_{i} has already been defined for some i ≥ 0: Then ϕ satisfies the equation

L_{i}(ϕ − ϕ_{i}) ≡ F′[ϕ_{i}](ϕ − ϕ_{i}) = G_{i}(ϕ − ϕ_{i}) + F (ϕ_{i}),  (1.10)

where

G_{i}(ψ) := F [ϕ_{i} + ψ] − F′[ϕ_{i}]ψ − F [ϕ_{i}]  (1.11)

We shall need the following two Lemmata:

Lemma 1.2. Let ϕ_{i} − ϕ_{0} ∈ A_{phg} ∩ C_{0}^{α_{0}}(M). For k ∈ ℕ_{0}^{∞}, λ ∈ [0, 1], δ > 0 we have:

1. POLYHOMOGENEITY OF SOLUTIONS OF SOME FULLY NONLINEAR EQUATIONS 57

((i)) k ∈ ℕ_{0}^{∞}, λ ∈ [0, 1], ψ ∈ C_{∞,k+λ}^{δ}(M_{x_{0}}),

G_{i}(ψ) ∈ C_{∞,k+λ}^{2δ}(M_{x_{0}}) .

((ii)) ψ ∈ A_{phg} ∩ C_{0}^{δ}(M_{x_{0}}),

F (ϕ_{0} + ψ) ∈ A_{phg} ∩ C_{0}^{min(α_{0},δ)}(M_{x_{0}})

((iii)) ψ ∈ A_{phg} ∩ C_{0}^{δ}(M_{x_{0}}), χ ∈ C_{∞,k+λ}^{γ}(M_{x_{0}}),

(F′[ϕ_{0} + ψ] − F′[ϕ_{0}])χ ∈ C_{∞,k+λ}^{γ+min(α_{0},δ)}(M_{x_{0}}) .

Proof: Point (ii) follows from (1.4)–(1.5) using the Taylor expansion. Points (i) and (iii) follow by a straightforward analysis of the remainder term (1.3) in the Taylor expansion. □

We can find s_{0} > max(0, α_{−}) satisfying s_{0} ≤ min(α, α_{0}) and s_{0} < α_{+}. Decreasing α_{0} if necessary, we may assume that s_{0} = α_{0}. We have

Lemma 1.3. ϕ − ϕ_{0} ∈ C_{∞,∞}^{s_{0}}(M_{x_{0}}).

Proof: Let u = ϕ − ϕ_{0} and consider

G[u] = F [ϕ_{0} + u] − F′[ϕ_{0}]u − F [ϕ_{0}].  (1.12)

We will argue by induction. Fix an irrational θ ∈ (0, 1) so that 2s_{0} − θ ≥ s_{0}. It follows from our assumptions that u ∈ C_{∞,0}^{s_{0}} which is the starting point of the induction.

Now assume that u ∈ C_{∞,kθ}^{s_{0}} for some k ∈ ℕ. By point (i) of Lemma 1.2, G[u] ∈ C_{∞,kθ}^{2s_{0}}. For any f ∈ C_{∞,kθ}^{2s_{0}} and any multiindex γ we have

(x∂_{y})^{γ}f ∈ C_{∞,kθ}^{2s_{0}} ∩ C_{∞,kθ+1}^{2s_{0}−1}.

Applying interpolation as in the proof of Lemma 2.2 gives with the above choice of θ,

(x∂_{y})^{γ}f ∈ C_{0,(k+1)θ}^{s_{0}}

but γ was arbitrary so we get f ∈ C_{∞,(k+1)θ}^{s_{0}}. Applying this result to G[u] and noting that F [ϕ_{0}] ∈ C_{∞,∞}^{s_{0}} and F [ϕ_{0} + u] = 0 by assumption, we find

F′[ϕ_{0}]u ∈ C_{∞,(k+1)θ}^{s_{0}}.

By assumption (α_{−}, α_{+}) is a strong regularity interval for F′[ϕ_{0}] and s_{0} ∈ (α_{−}, α_{+}). We can now apply Theorem 1.3 to conclude that u ∈ C_{∞,(k+1)θ}^{s_{0}}. It follows by induction that u ∈ C_{∞,∞}^{s_{0}}. □

Remark: In the case when (α_{−}, α_{+}) is only a weak regularity interval, the above argument is valid only under restrictions on s_{0}. Instead the appropriate version of Lemma 1.3 is proved under the conditions outlined under point (ii) of the remark following Theorem 1.1 by considering the identity

F′[ϕ_{0}]\frac{∂ϕ}{∂v^{A}} = \underbrace{[F′[ϕ_{0}] − F′[ϕ]]\frac{∂ϕ}{∂v^{A}}} + \underbrace{\frac{∂F}{∂v^{A}} [ϕ_{0}] − \frac{∂F}{∂v^{A}} [ϕ]} −\frac{∂F}{∂v^{A}} [ϕ_{0}] .

By eq. (1.3) and by point (iii) of Lemma 1.2 each group at the right hand side of this equation is in C_{∞}^{s_{0}}. From ϕ ∈ C_{∞}^{s_{0}} we have ∂ϕ/∂v^{A} ∈ C_{∞}^{s_{0}−1}, and since (α_{−}, α_{+}) is a regularity interval and both s_{0} and s_{0} − 1 are in (α_{−}, α_{+}) we find ∂ϕ/∂v^{A} ∈ C_{∞}^{s_{0}}, hence ϕ ∈ C_{∞,1}^{s_{0}}. The result follows by an inductive repetition of this argument; cf. the proof of Theorem 4.4 in Chapter 7 below for a more detailed exposition in a similar context. □

Returning to the proof of Theorem 1.1, Lemma 1.3 shows that (1.9) is satisfied with i = 0. Consider again equation (1.10). By Lemma 1.2, point (i), we have G_{i}(ϕ−ϕ_{i}) ∈ C_{∞,∞}^{2s_{i}}(M). Now point (ii) of Lemma 1.2 gives F [ϕ_{i}] ∈ A_{phg} ∩C_{0}^{α_{0}}(M), thus

L_{i}(ϕ − ϕ_{i}) ∈ C_{∞,∞}^{2s_{i}} + A_{phg} ∩ C_{0}^{α_{0}}(M).

This and point (iii) of Lemma 1.2 imply

L_{0}(ϕ − ϕ_{i}) = (L_{0} − L_{i})(ϕ − ϕ_{i}) + L_{i}(ϕ − ϕ_{i}) ∈ C_{∞,∞}^{s_{i}+α_{0}} + C_{∞,∞}^{2s_{i}} + A_{phg} ∩ C_{0}^{α_{0}}(M) .

A straightforward extension of Lemma 2.5 shows^{1} that we can find ϕ̂ ∈ A_{phg} ∩ C_{0}^{α_{0}}(M) such that

L_{0}(ϕ − ϕ_{i} − ϕ̂) ∈ C_{∞,∞}^{s_{i}+α_{0}} + C_{∞,∞}^{2s_{i}} .  (1.13)

Similarly, for s_{i} + α_{0} > μ_{+} it follows from a straightforward extension^{1} of Lemma 2.7 that ϕ̂ can moreover be chosen so that

ϕ − ϕ_{i} − ϕ̂ = o(x^{μ_{+}}) .

Here μ_{+} is the larger indicial exponent of L_{0}, as defined at the beginning of Chapter 1. Lemma 2.8 gives

ϕ − ϕ_{i} ∈ A_{phg} ∩ C_{0}^{α}(M) + C_{∞,∞}^{s_{i}+δ}, δ = min(s_{i}, α_{0}) ≥ s_{0} > 0 .

ϕ_{i+1} is now defined as ϕ_{i} plus the sum of those terms in ϕ−ϕ_{i} which are polyhomogeneous and which decay slower than s_{i+1} = s_{i} + δ. This completes the induction step, and the result follows. □

Now we specialize to consider the cases when F is quasi– or semi–linear. For such equations we shall show that the hypothesis ϕ − ϕ_{0} ∈ C_{∞}^{α}(M_{x_{0}}) can be weakened.

Definition 1.1. • We will say that the system F is quasi–linear if F is of the form

F [ϕ] = x^{2}a^{ij}∂_{i}∂_{j}ϕ + c  (1.14)

where a^{ij} ≡ a^{ij}[y, ϕ, x∂_{k}ϕ] and similarly for c.

• We will say that the system F is semi–linear if F is of the form

F [ϕ] = x^{2}a^{ij}[y, ϕ]∂_{i}∂_{j}ϕ + xb^{i}[y, ϕ]∂_{i}ϕ + c[y, ϕ]  (1.15)

Proposition 1.4. Assume that F is of the form (1.14) with F satisfying (1.4)– 1.5). Further let ϕ_{0} be an approximate solution to F [ϕ] = 0 satisfying (1.2)–(1.3). Assume that F′[ϕ_{0}] is an elliptic operator on M in the sense of Section 1, that

F [ϕ] = 0 ,

and that

ϕ − ϕ_{0} ∈ C_{1}^{α}(M_{x_{0}}) ∩ C_{2}^{loc}(M),  α > 0 .  (1.16)

1The result needed here can actually be proved in a rather simpler way as compared to the proof of Lemma 2.7: Indeed, all the functions appearing in the polyhomogeneous expansions are smooth in the v variable, so that in the proof one does not need to use the extension operators of Lemma 3.1.

Then there exists x_{1} > 0 such that ϕ − ϕ_{0} ∈ C_{∞}^{s_{0}}(M_{x_{1}}), where s_{0} = min(α_{0}, α). If moreover F is of the form (1.15), then (1.16) can be replaced by ϕ − ϕ_{0} ∈ C_{0}^{α}(M_{x_{0}}) ∩ C_{2}^{loc}(M), α > 0.

Proof: Let L_{ϕ} be the operator defined by

L_{ϕ}ψ = x^{2}a^{ij}(y, ϕ, x∂_{k}ϕ)∂_{i}∂_{j}ψ + xb^{i}(y, ϕ, x∂_{k}ϕ)∂_{i}ψ

and let

u = ϕ − ϕ_{0} .

From the regularity assumptions on F, a straightforward argument using Taylors theorem shows the existence of some x_{1} > 0 so that L_{ϕ} ∈ OP_{C_{0}^{0}(M_{x_{1}} )}^{2} is elliptic in the sense of Chapter 1 and

L_{ϕ}u ∈ C_{0}^{s_{0}}(M_{x_{1}}) .

Now an application of interior estimates and scaling gives u ∈ C_{1+λ}^{s_{0}}(M_{x_{1}/2}) for λ ∈ (0, 1). From the above we have that u ∈ C_{k+λ}^{s_{0}}(M_{x_{1}/2}) with k = 1 and, since by assumption u ∈ C_{2}^{loc}(M), it follows that u ∈ C_{k+λ}^{s_{0}}(M_{x_{1}}) with k = 1.

We wish to show by induction that u ∈ C_{k+λ}^{s_{0}}(M_{x_{1}}) for all k. We have shown this to be true for k = 1. Assuming that u ∈ C_{k+λ}^{s_{0}}(M_{x_{1}}) it follows from F [ϕ] = 0 and (1.14), using Taylors theorem, that

F′[ϕ_{0}]u ∈ C_{k−1+λ}^{s_{0}}(M_{x_{1}}).

An application of the scaling estimate Lemma 1.1 gives u ∈ C_{k+1+λ}^{s_{0}}(M_{x_{1}}). This achieves the induction step and proves the Proposition. □

CHAPTER 6 The vector constraint equation.

1. Introductory remarks

Let (M̄, g) be a compact Riemannian manifold with boundary ∂M, let x be any defining function for ∂M, for x ∈ M ≡ M̄ \ ∂M let

g̃_{ij} = x^{−2} g_{ij} .  (1.1)

As discussed in Chapter 2, to construct solutions of the vector constraint equation one makes use of the conformal invariance of the system

D_{i} P^{ij} = 0 , D_{k} (g_{ij} P^{ij}) = 0 .

There are at least two ways to proceed:

Method 1. Let A^{ij} be a symmetric, traceless tensor field (g_{ij} A^{ij} = 0), suppose that β ∈ ℝ, let the vector field X^{i} satisfy

D_{i} (D^{i}X^{j} + D^{j}X^{i} − \frac{2}{n} (D_{k}X^{k}) g^{ij}) = −D_{i} (x^{β} A^{ij}) ,  (1.2)

where D is the Riemannian connection of g_{ij}. Then the tensor field

B^{ij} ≡ D^{i}X^{j} + D^{j}X^{i} − \frac{2}{n} (D_{k}X^{k}) g^{ij} + x^{β} A^{ij}

is symmetric, traceless and transverse for the metric g_{ij} (D_{j}B^{ij} = 0). Moreover the tensor field B̃^{ij} ≡ x^{−(n+2)}B^{ij} is transverse for the metric g̃_{ij} (D̃_{j}B̃^{ij} = 0).

Method 2. Let Ã^{ij} be a symmetric, traceless tensor field (g̃_{ij} Ã^{ij} = 0), suppose that β ∈ ℝ, let the vector field X̃^{i} satisfy

D̃_{i} (D̃^{i}X̃^{j} + D̃^{j}X̃^{i} − \frac{2}{n} (D̃_{k}X̃^{k}) g̃^{ij}) = −D̃_{i} (x^{β} Ã^{ij})  (1.3)

where D̃ is the Riemannian connection of g̃_{ij}. Then the tensor field

B̃^{ij} = D̃^{i}X̃^{j} + D̃^{j}X̃^{i} − \frac{2}{n} (D̃_{k}X̃^{k}) g^{ij} + x^{β} Ã^{ij}

is symmetric, traceless and transverse for the metric g̃_{ij} (D̃_{j}B̃^{ij} = 0). Moreover the tensor field B^{ij} ≡ x^{(n+2)}B̃^{ij} is transverse for the metric g_{ij} (D_{j}B^{ij} = 0).

Because of the singular character (recall that x = 0 on ∂M), of the relationship (1.1) between g_{ij} and g̃_{ij}, the methods turn out to be quite different. Since the operator on the right-hand-side of (1.2) is a regular elliptic operator on M (i.e., elliptic up–to–boundary in the standard sense), the first method turns out to be more convenient for proving existence of solutions of the vector constraint which are smoothly extendable across ∂M; this is most easily achieved using (standard) non–weighted Hölder spaces on the compactified manifold, as shown in Section 2.

On the other hand the evolution problem is most directly formulated in terms of Sobolev-type spaces, in the non-compactified picture, it is therefore natural to use the second method to do that: this analysis is carried out in Section 3.

2. (Non–weighted) Hölder spaces on the compactified manifold

In this section we shall prove existence of solutions of the system of equation

Δ_{L,g} X = Y ,  (2.1)
(Δ_{L,g} X)^{i} ≡ D_{j} L^{ij}(X) ,  (2.2)
L^{ij}(X) ≡ \frac{1}{2}(D^{i}X^{j} + D^{j}X^{i}) − \frac{1}{n} D_{k}X^{k} g^{ij} ,  (2.3)

under various hypotheses on the source term Y, where g_{ij} is a Riemannian metric on a compact manifold M̄ with boundary ∂M; by this we mean that (M̄\∂M, g|_{M̄\∂M}) is a Riemannian manifold and in local coordinates {y^{i}} near ∂M as described at the beginnig of Chapter 3 the matrix g_{ij} ≡ g (\frac{∂}{∂y^{i}} , \frac{∂}{∂y^{j}} ) is strictly positive definite up to ∂M, thus:

∀ X^{i} ∈ ℝ^{n} C^{−1} ∑_{i}(X^{i})^{2} ≤ g_{ij}X^{i}X^{j} ≤ C ∑_{i}(X^{i})^{2}  (2.4)

for some constant C. D_{i} always denotes the Riemannian connection of g_{ij}.

Definition 2.1. The metric g_{ij} will be said to be of class M_{k+λ}^{α}, α ≥ 0, if (2.4) holds and if for 0 ≤ |γ| ≤ k we have ∂_{y}^{γ}g_{ij} ∈ C_{0+λ}^{min(0,α−|γ|)}(M) (thus g_{ij} ∈ C_{k+λ}^{loc}(M), and in local coordinates near ∂M (cf. the beginning of Chapter 3) we have

|g_{ij}| + |g^{ij}| + (1 + x^{α−1})^{−1} |\frac{∂}{∂y^{k}} g_{ij}| + … + (1 + x^{α−k})^{−1} |\frac{∂}{∂y^{i_{1}}} … \frac{∂}{∂y^{i_{k}}} g_{ij}| ≤ C , (2.5)

for some constant C, together with an appropriate weighted Hölder condition if λ > 0).

Note that a metric g ∈ C_{k+λ}(M̄) is in M_{k+λ}^{α} for all α > 0.

Proposition 2.2. Let g be a metric of M_{k+λ}^{α} class on M, 2 < dim M, λ ∈ [0, 1],

α > 0, k ≥ 3, let X ∈ Cα̃ ; 1(M) ∩ W_{3}^{p;loc}(M, g, dμ_{g}), α̃ > 0, p ∈ [1, ∞), satisfy

L(X) = \frac{1}{2}(D^{i}X^{j} + D^{j}X^{i}) − \frac{1}{n} (D_{k}X^{k}) g^{ij} = 0 .  (2.6)

Then

X = 0 .

To prove Proposition 2.2 we shall need the following Lemmata:

Lemma 2.3. Let g be of class M_{k}^{α}, k ≥ 3, then for x ≤ x_{0} in local coordinates we have

|Γ_{jk}^{i}|_{δ} + x|R|_{g} + x^{2}|DR|_{g} ≤ C(1 + x^{α−1}) ,

for some constant C, where R = (R^{i}_{jkℓ}) is the Riemann tensor, DR = (D_{k}R^{i}_{jmn}), | |_{h} stands for the Riemannian norm of a tensor with respect to the metric h.

Proof: Simple computation. □ The result that follows is well known (we use the convention D_{k}D_{j}X^{i}−D_{j}D_{k}X^{i} =

R^{i}_{ℓkj}X^{ℓ}):

Lemma 2.4. Let δX ≡ D_{i}X^{i}, X_{i} ≡ g_{ij}X^{j}. Under the hypotheses of Proposition 2.2 we have X ∈ C_{3}^{loc}(M), and the following hold (n = dim M > 2):

D_{m}D_{k}D_{i}X_{j} = D_{m}R^{ℓ}_{kij}X_{ℓ} + R^{ℓ}_{kij}D_{m}X_{ℓ}

+\frac{1}{n} \{D_{m}D_{k}(δX)g_{ij} + D_{m}D_{i}(δX)g_{jk} − D_{m}D_{j}(δX)g_{ik}\} , (2.7)

D_{i}D_{k}δX = \frac{n}{n − 2}\{ − 2R_{ℓ(i}D_{k)}X^{ℓ} + \frac{R}{n(n − 1)} δXg_{ik}
+X^{ℓ}D_{ℓ} (\frac{1}{2(n − 1)} Rg_{ik} − R_{ik}) \} ,  (2.8)

where R_{ij} = R^{k}_{ikj} is the Ricci tensor of g_{ij}, R = g^{ij}R_{ij} the Ricci scalar.

Proof: It is an easy exercise in distributional differentiation to show from (2.6) and X ∈ W_{3}^{p,loc}(M, g, dμ_{g}) that the equations (2.7)–(2.8) hold in a distributional sense, which together with X ∈ C_{1}^{loc}(M) implies that X ∈ C_{3}^{loc}(M), and shows that (2.7)–(2.8) hold pointwise. □

Lemma 2.5. Under the hypotheses of Proposition 2.2 for any σ > 0 there exists a constant C(σ) such that

|X|_{g} + x|DX|_{g} + x^{2}|DDX|_{g} ≤ Cx^{σ} ,  (2.9)

where DX = (D_{i}X_{j}), DDX = (D_{i}D_{j}X_{k}).

Proof: Equations (2.7)–(2.8) and X ∈ Cα̃ ; 1 show that DDDX ∈ C_{0}^{α̃−3}, which easily (e.g.,by scaled interpolation) implies X ∈ Cα̃ ; 3. Let

I = {σ ∈ ℝ : ∃ C such that (2.9) holds}

since X ∈ Cα̃ ; 3, α̃ > 0, it follows that (−∞, α̃] ⊂ I, thus I ≠ ∅. Let

σ̃ = lim sup I ,

suppose that σ̃ ≠ ∞, set σ_{1} = σ̃ − \frac{1}{2} min(α, 1); by definition of σ̃ we have σ_{1} ∈ I, thus

|X|_{g} + x|DX|_{g} + x^{2}|DDX|_{g} ≤ Cx^{σ_{1}} .  (2.10)

Decreasing σ_{1} slightly if necessary we may assume −σ_{1} ∉ ℕ. Equations (2.7)–(2.8) imply an equation of the form

∂_{x} DDX = “Γ DDX + RDX + DR X”
= O(x^{σ_{1}+min(1,α)−3}) .  (2.11)

Suppose first that σ_{1} + min(1, α) < 1; from

f(x_{1}, v) = f(x_{2}, v) − ∫_{x_{1}}^{x_{2}} \frac{∂f}{∂x} (s, v) ds  (2.12)

one has, setting f = DDX, x_{1} = x, x_{2} = x_{0},

DDX = O(x^{σ_{1}+min(1,α)−2}) ,  (2.13)

similarly from

∂_{x}DX = “ΓDX + DDX”  (2.14)

and from (2.12)–(2.13) it follows

DX = O(x^{σ_{1}+min(1,α)−1}) .  (2.15)

Passing to the limit x_{1} → 0 in the equation

X(x, v) = X(x_{1}, v) + ∫_{x_{1}}^{x} “(ΓX + DX)”(s, v) ds  (2.16)

(the integral at the right-hand-side of (2.16) converges because σ_{1} + min(1, α) > σ̃ > α̃ > 0), making use of X(0, v) = 0, one obtains

X = O(x^{σ_{1}+min(1,α)}) ,  (2.17)

which together with (2.13) and (2.15) contradicts the definition of σ̃, thus σ_{1} + min(1, α) > 1; suppose that σ_{1} + min(1, α) < 2. (2.13) still holds so that (2.12) with f = DX shows (cf. (2.14)) that the limit

A_{ij}(v) = lim_{x→0} D_{i}X_{j}(x, v)

exists, and we have

D_{i}X_{j} − A_{ij} = O(x^{σ_{1}+min(1,α)−1}) ,  (2.18)

while (2.16) gives

X_{j} = O(x) .  (2.19)

(2.13), (2.18) and (2.19) yield

∂X = O(1) ,
∂∂X = O(x^{ϵ−1}), ϵ = min(0, σ_{1} − 1) + min(1, α) > 0 .  (2.20)

Let v_{0} ∈ ∂M; (2.12) with f = g_{ij} together with g ∈ M_{3}^{α} implies that f_{ij} ≡ lim_{x→0} g_{ij}(x, v_{0}) exists. Passing to the limit x → 0 in (2.6) one has

A_{ij} + A_{ji} = \frac{2}{n} f^{ℓm} A_{ℓm} f_{ij} ,  (2.21)

where f^{ℓm} ≡ (f_{ij})^{−1} (cf. (2.4)). The interpolation inequality (cf. e.g. [44, Appendix A]:

∥f∥_{C_{1}(∂M)} ≤ C∥f∥_{C_{2}(∂M)}^{1/2}∥f∥_{C_{0}(∂M)}^{1/2}  (2.22)

applied to f(v) = X_{i}(x, v) together with (2.19) and (2.20) gives

|\frac{∂X_{i}}{∂v} | (v, x) ≤ Cx^{ϵ/2} ,  (2.23)

thus A_{Ai} = 0. Let e_{(i)} = e_{(i)}^{j} ∂_{j} be an orthonormal basis at (0, v_{0}) for f_{ij}, with e_{(A)} = e_{(A)}^{B} ∂_{v^{B}}, e_{(1)}⊥∂M, set A_{(i)(j)} = A_{ij} e^{i}_{(i)} e^{j}_{(j)}; we have, for A = 2, …, n,

e_{(A)}^{1} = 0 ⟹ A_{(A)(j)} ≡ e_{(A)}^{k} e_{(j)}^{ℓ} A_{kℓ} = 0 ⟹ f^{ij} A_{ij} = A_{(1)(1)} .

(2.21) rewritten in the orthonormal frame e_{(i)},

A_{(i)(j)} + A_{(j)(i)} = \frac{2}{n} A_{(1)(1)} δ_{(i)(j)} ,

with (i)(j) = (1)(1), (1)(A) gives

A_{ij} = 0 ,

thus (2.15) and (2.17) hold, contradicting again the definition of σ̃, thus σ_{1} + min(1, α) > 2. Going through the whole argument once again one similarly shows that the limit

A_{ijk} = lim_{x→0} D_{i}D_{j}X_{k}(x, v_{0})

exists, and A_{Ajk} = 0. Algebraic manipulations with the equations

A_{kij} + A_{kji} − \frac{2}{n} f^{ℓm} A_{ℓmk} f_{ij} = lim_{x→0} D_{k} (D_{i}X_{j} + D_{j}X_{i} − \frac{2}{n} g^{kℓ} D_{k}X_{ℓ} g_{ij}) = 0

lead to

A_{ijk} = 0 .

This implies that (2.13), (2.15) and (2.17) hold again, contradicting the definition of σ̃, thus σ̃ = ∞, which had to be established. □

Proof of Proposition 2.2: Let

f(x) = (|X|_{g}^{2} + x^{2}|DX|_{g}^{2} + x^{4}|DDX|_{g}^{2}) .

We have

\frac{∂f}{∂x} = \frac{2}{x}⟨X, xD_{x}X⟩ + \frac{2}{x} x^{2}|DX|_{g}^{2} + \frac{2}{x}⟨xDX, x^{2}D_{x}DX⟩ + \frac{4}{x} x^{4}|DDX|_{g}^{2} + F ,

where F is given by

F = 2⟨x^{2}DDX, x^{2}D_{x}DDX⟩ .

Thus, using 2ab ≤ a^{2} + b^{2},

\frac{∂f}{∂x} ≤ \frac{4 + |\frac{∂}{∂x} |_{g}}{x} f + F .

From Lemmata 2.3 and 2.4 it follows

|F | ≤ Cx^{2}|DDX|(x^{min(1,α)−1}|X| + x^{min(1,α)−1}x|DX|)
≤ \frac{Cx^{min(1,α)}}{x} (x^{2}|DDX| |X| + x^{2}|DDX| x|DX|)
≤ \frac{Cx^{min(1,α)}}{x} f .

Therefore, for 0 ≤ x ≤ x_{0}, we have

\frac{∂f}{∂x} ≤ \frac{4 + |\frac{∂}{∂x} |_{g} + C}{x} f ≤ \frac{σf}{x} ,

for some constant σ, so that for 0 < x_{1} ≤ x ≤ x_{0} one has

ln [\frac{f(x)x_{1}^{σ}}{f(x_{1})x^{σ}} ] ≤ 0 ,

which implies

f(x) ≤ \frac{f(x_{1})}{x_{1}^{σ}} x^{σ} .

Passing to the limit x_{1} → 0 one obtains from Lemma 2.5

f(x) ≡ 0 ,

thus X vanishes in a neighbourhood of ∂M, which by well known results on conformal Killing vector fields (cf. Lemma 2.4) implies X ≡ 0. □

Proposition 2.6. Let g be a C_{ℓ+λ}(M̄) ∩ M_{3}^{α} metric, ℓ ≥ 2, α > 0.

((i)) For p ∈ [2, ∞) if k = 0 and p ∈ (1, ∞) if 1 ≤ k ≤ ℓ − 2 there exists a

constant C such that for all X ∈ W^{∘} _{1}^{p}∩W _{k+2}^{p}(M, g, dμ_{g}) we have
∥X∥_{W_{k+2}^{p}(M,g,dμ_{g})} ≤ C∥Δ_{L,g} X∥_{W_{k}^{p}(M,g,dμ_{g})} .  (2.24)

((ii)) For 0 ≤ k ≤ ℓ − 2, λ ∈ (0, 1), there exists a constant C such that for all X ∈ C_{k+2+λ}(M̄) satisfying X(p) = 0 for p ∈ ∂M,

∥X∥_{C_{k+2+λ}(M̄)} ≤ C∥Δ_{L,g} X∥_{C_{k+λ}(M̄)} .  (2.25)

Proof: To prove (2.24), recall that since Δ_{L,g} is elliptic we have the estimate [2, Theorem 10.5]

∥X∥_{W_{k+2}^{p}} ≤ C (∥Δ_{L,g} X∥_{W_{k}^{p}} + ∥X∥_{L^{1}}) .  (2.26)

Suppose that (2.24) does not hold; thus for i ∈ ℕ there exists X_{i} such that ∥X_{i}∥_{L^{1}} = 1 and

∥Δ_{L,g} X_{i}∥_{W_{k}^{p}} ≤ \frac{1}{i} ∥X∥_{W_{k+2}^{p}} .

(2.26) implies

∥X_{i}∥_{W_{k+2}^{p}} ≤ \frac{C}{i} ∥X_{i}∥_{W_{k+2}^{p}} + C ⟹ ∥X_{i}∥_{W_{k+2}^{p}} ≤ 2C for i ≥ 2C ,

therefore

∥Δ_{L,g} X_{i}∥_{W_{k+2}^{p}} ≤ \frac{2C}{i} .

By the Rellich-Kondrakov theorem a subsequence, still denoted by X_{i}, can be chosen converging strongly in W_{1}^{2} to X_{∞} ∈ W_{1}^{2}, and we have

∀ Y ∈W^{∘} _{1}^{2}(M, g, dμ_{g}) ∫_{M}(D^{i}X_{∞}^{j} + D^{j}X_{∞}^{i} − \frac{2}{n}D_{k}X_{∞}^{k}g^{ij})D_{j}Y_{i} dμ_{g} = 0 . (2.27)

Elliptic regularity (cf. e.g. [39]) and g ∈ C_{2}(M̄) give X_{∞} ∈ C_{1+α̃}(M̄) for some α̃ ∈ (0, 1), while g ∈ M_{3}^{α} implies that X_{∞} ∈ W_{3}^{p,loc}(M, g, dμ_{g}) for all p ∈ (1, ∞). Setting Y = X_{∞} in (2.27) it follows that X_{∞} is a conformal Killing vector. Proposition 2.2 implies X_{∞} = 0, which contradicts ∥X_{∞}∥_{L^{1}} = 1, and proves (2.24).

(2.25) is proved in a similar (and simpler) way, using

∥X∥_{C_{k+2+λ}(M̄)} ≤ C (∥Δ_{L,g} X∥_{C_{k+λ}(M̄)} + ∥X∥_{L^{1}(M,dμ_{g})})

(cf. [2, Theorem 9.2]) and the Arzela-Arcoli theorem. □

Theorem 2.7. Let g be a C_{ℓ+λ}(M̄) ∩ M_{3}^{α} metric on a compact manifold M̄ (with boundary), ℓ ≥ 2, λ ∈ [0, 1], α > 0.

((i)) Suppose that λ ∈ (0, 1). For 0 ≤ k ≤ ℓ − 2 the map

Δ_{L,g} : {X ∈ C_{k+2+λ}(M̄) : X(p) = 0 for p ∈ ∂M} → C_{k+λ}(M̄) (2.28)

is an isomorphism. ((ii)) For p ∈ [2, ∞), k = 0, or p ∈ (1, ∞), 1 ≤ k ≤ ℓ − 2 the map

Δ_{L,g} : W^{∘} _{1}^{p}∩W _{k+2}^{p}(M, g, dμ_{g}) → W_{k}^{p}(M, g, dμ_{g})  (2.29)

is an isomorphism.

Remark: If ℓ ≥ 3 then g ∈ C_{ℓ}(M̄) implies g ∈ M_{3}^{α} for all α > 0, cf. Definition 2.1.

Proof: (2.24) with p = 2, k = 0, shows that the problem

Δ_{L,g} X = Y ,  Y ∈ L^{2}, X ∈W^{∘} _{1}^{2}  (2.30)

satisfies the coerciveness condition, existence of a weak solution follows from the Lax–Milgram theorem [40, Theorem 5.8], regularity follows from e.g. [39], uniqueness of solutions follows from (2.24). □

Theorem 2.8. Let M̄ be a compact manifold (with boundary), suppose that g is a metric on M ≡ M̄ \ ∂M̄ which can be C_{∞} extended across ∂M, let x be any smooth defining function for ∂M. Consider the equation

Δ_{L,g} X = x^{α} Y ,  Y ∈ C_{∞}(M̄) .  (2.31)

((i)) For α ∈ ℕ_{0} and for any X_{0} ∈ C_{∞}(M̄) there exists a solution of (2.31) of the form

X = X_{0} + X_{1},  X_{1} ∈ C_{∞}(M̄) , X_{1} = O(x) .  (2.32)

X is uniquely determined by Y and by ∂M ∋ p → X_{0}(p), in the class of W_{1}^{2}(M, g, dμ_{g}) solutions of (2.31).

((ii)) For α ∉ ℤ there exist X_{α+2}, X_{0} ∈ C_{∞}(M̄) such that the vector field X given by

X = x^{α+2} X_{α+2} + X_{0}  (2.33)

is a solution of (2.31). X is determined by Y, in the class of solutions having the described properties, up to the addition of a C_{∞}(M̄) solution of the homogeneous equation (cf. point 1).

((iii)) For −α ∈ ℕ there exist X_{α+2}, X_{0}, X_{log} ∈ C_{∞}(M̄) such that the vector field X given by

X = x^{α+2} X_{α+2} + X_{0} + log x X_{log}  (2.34)

is a solution of (2.31). If α = −1 then there exists X_{log,1} ∈ C_{∞}(M̄) such that X_{log} = x X_{log,1}. X is determined by Y, in the class of solutions having the described properties, up to the addition of a C_{∞}(M̄) solution of the homogeneous equation (cf. point 1).

Remarks:

((i)) In (2.33) one can replace X_{α+2} by X_{α+2} +X̂_{α+2}, with X̂_{α+2} ∈ C_{∞}^{∞}(M) without changing the form of (2.33), thus X_{α+2} is not unique. A similar remark applies to (2.34).

((ii)) For α + 2 > 1/2, X in (2.33) is unique in the class of solutions which are in C_{2}^{1/2+ϵ}, ϵ > 0, as follows from Proposition 2.2.

Proof: Point 1 folows immediately from Theorem 2.7. To prove the remaining claims, let

X̃_{α+2} = Z_{α+2} + V_{α+2} , Z_{α+2} ⊥ Dx , V_{α+2} ∼ Dx .

A straightforward calculation gives

2Δ_{L,g} (x^{α+2}X̃_{α+2}) = (α + 1)(α + 2) x^{α} |Dx|_{g}^{2} \{Z_{α+2} + \frac{2(n − 1)}{n} V_{α+2}\} + W_{α+2} , (2.35)

W_{α+2} = x^{α+1} W′_{α+2} ,

with W′_{α+2} ∈ C_{∞}(M̄) if Z_{α+2}, V_{α+2} ∈ C_{∞}(M̄). For α ≠ −2, −1 we can choose Z_{α+2} and V_{α+2} ∈ C_{∞}(M̄) so that

(α + 1)(α + 2) lim_{x→0} \{(Z_{α+2} + \frac{2(n − 1)}{n} V_{α+2}) |Dx|_{g}^{2}\} = lim_{x→0} Y , (2.36)

and if we set

Y_{1} = \frac{1}{x} \{Y − (α + 1)(α + 2) |Dx|_{g}^{2} (Z_{α+2} + \frac{2(n − 1)}{n} V_{α+2})\} − W′_{α+2} ∈ C_{∞}(M̄)
X̂_{1} = X − x^{α+2}X̃_{α+2} ,

then X will be a solution of (2.31) if X̂_{1} satisfies

2Δ_{L,g} X̂_{1} = x^{α+1} Y_{1} .  (2.37)

Suppose first that −α ∉ ℕ_{0}; continuing as above we can find a sequence {X̃_{α+2+i}}_{i=0}^{∞} of vector fields X̃_{α+2+i} ∈ C_{∞}(M̄) such that

2Δ_{L,g} (∑_{i=0}^{N} x^{α+2+i} X̃_{α+2+i}) = x^{α+N} Y_{N} ,  (2.38)

for some Y_{N} ∈ C_{∞}(M̄). By Borel’s Lemma 1.2 there exists a smooth vector field X_{α+2} such that

X_{α+2} ∼ ∑_{i=0}^{∞} x^{i} X̃_{α+2+i} ,

and by (2.38) we have

2Δ_{L,g} (x^{α+2}X_{α+2}) − x^{α} Y ∈ C_{∞}^{∞}(M) ⊂ C_{∞}(M̄) .

By Theorem 2.7 there exists a vector field X_{1} ∈ C_{∞}(M̄), X_{1}(p) = 0 for p ∈ ∂M, such that

2Δ_{L,g} X_{1} = x^{α} Y − Δ_{L,g} (x^{α+2} X_{α+2}) .

Setting X = x^{α+2} X_{α+2} + X_{1} one obtains the required solution.

For −α ∈ ℕ after a finite number of steps in the procedure leading from (2.31) to (2.37) one obtains, for some Y_{−2}, X̃_{i} ∈ C_{∞}(M̄),

2Δ_{L,g} (X − ∑_{i=α+2}^{−1} x^{i} X̃_{i}) = x^{−2} Y_{−2} ,  (2.39)

which can be solved to leading order by setting

X̃_{log} = (Z_{log} + V_{log}) Z_{log}, V_{log} ∈ C_{∞}(M̄) , Z_{log} ⊥ Dx , V_{log} ∼ Dx ,

lim_{x→0} \{(Z_{log} + \frac{2(n − 1)}{n} V_{log}) |Dx|_{g}^{2}\} = − lim_{x→0} Y_{−2} ,

and we obtain

2Δ_{L,g} (X − ∑_{i=α+2}^{−1} x^{i} X̃_{i} − log x X̃_{log}) = x^{−1} Y_{−1} + log x Y_{log}

with some Y_{−1}, Y_{log} ∈ C_{∞}(M̄). For i ∈ ℕ consider

X̃_{log,i} = (Z_{log,i} + V_{log,i}) , Z_{log,i}, V_{log,i} ∈ C_{∞}(M̄) , Z_{log,i} ⊥ Dx , V_{log,i} ∼ Dx .

We have

2Δ_{L,g} (X̃_{log,i}x^{i} log x) = x^{i−2}\{i(i−1) log x+2i−1\} |Dx|_{g}^{2} \{Z_{log,i} + \frac{2(n − 1)}{n} V_{log,i}\}

+x^{i−1}(W_{log,i} log x + W_{i}) ,

with W_{i}, W_{log,i} ∈ C_{∞}(M̄), and proceeding as before one shows that there exists X_{log} ∈ C_{∞}(M̄), with X_{log} ∼ ∑_{i=0}^{∞} x^{i}X̃_{log,i} such that

Δ_{L,g} (X_{log} log x) − x^{−1} Y_{−1} − log x Y_{log} ∈ C_{∞}(M̄)

(note that if α = −1 then X_{log} = xX_{log,1}, for some X_{log,1} ∈ C_{∞}(M̄)), and the result follows by point 1. Uniqueness, up to a smooth solution of the homogeneous equation, follows by construction. □

Theorem 2.8 shows that a natural space in which solutions of equation (2.31) “live” is the space of polyhomogeneous vector fields. This leads us to introduce the following:

Definition 2.9. A metric g on M will be said polyhomogeneous, and we shall write g ∈ M^{phg}, if g is strictly positive definite up to ∂M, in the sense of (2.4), and if in local coordinates near ∂M as described in Chapter 3 the components g_{ij} of the metric tensor are polyhomogeneous functions. We thus have (cf. Chapter 3)

g_{ij}(x, v) ∼ ∑_{m=0}^{I} ∑_{n=0}^{∞} ∑_{k=0}^{N_{mn}} g_{ijmnk}(v) x^{s_{m}+n} ln^{k} x ,

for some functions g_{ijmnk}(v) ∈ C_{∞}(∂M), and some sequence {(s_{i}, {N_{ij}}_{j=0}^{∞})}_{i=0}^{I}, with s_{i} ∈ ℝ, N_{ij} ∈ ℕ_{0}, I ∈ ℕ_{0}^{∞}, s_{i} > s_{j} for i > j.

It should be remarked that (2.4) implies that s_{0} = 0, ∑_{ij} (g_{ij000}(v))^{2} > 0, and that N_{00} = 0. A metric g ∈ C_{∞}(M̄) is necessarily polyhomogeneous, and we also have M^{phg} ⊂ M_{∞}^{α} ≡ ∩_{k}M_{k}^{α} (cf. Definition 2.1), for some α > 0 (we have e.g. α = s_{1} if s_{1} < 1 and N_{10} = 0, etc.).

The proof of the following result is a straightforward repetition of the proof of Theorem 2.8, the details are left to the reader:

Theorem 2.10. Let g be a polyhomogeneous metric on a compact manifold M̄ with boundary, which can be C_{2} extended across ∂M, consider the equation

Δ_{L,g} X = Y ,  Y ∈ A^{phg} .  (2.40)

((i)) For any Y = o(x^{−2}) and X_{0} ∈ C_{∞}(M̄) there exists a solution of (2.40) of the form

X = X_{0} + X_{∗},  X_{∗} ∈ A^{phg} , X_{∗}(p) = 0 for p ∈ ∂M .  (2.41)

X is uniquely determined by Y and by ∂M ∋ p → X_{0}(p), in the class of polyhomogeneous solutions of (2.40).

((ii)) There exists a solution X ∈ A^{phg} of (2.40). Any two polyhomogeneous solutions of (2.40) differ by a solution of the homogeneous equation, as described in point (i) above.

3. Weighted Sobolev spaces

In this section we shall establish existence of solutions of the equation

Δ_{L̃,g̃} X = Y ,  (3.1)

with g̃_{ij} = x^{−2} g_{ij}, g_{ij} — Riemannian metric on a compact manifold M with boundary ∂M, x — a defining function for ∂M, under various conditions on Y. Let us set

L̃(X)^{ij} = \frac{1}{2}(D̃^{i}X^{j} + D̃^{j}X^{i}) − \frac{1}{n} D̃_{k}X^{k} g̃^{ij} ,  (3.2)

where D̃_{i} is the Riemannian connection of the metric g̃_{ij}, let L̃^{∗} be the formal L^{2} adjoint of L̃, we have

Δ_{L̃,g̃} X^{j} = D̃_{i} L̃(X)^{ij} = −L̃^{∗} L̃(X)^{j} .  (3.3)

The operator Δ_{L̃,g̃} is of the form considered in Chapter 4, in fact we have

Δ_{L̃,g̃} X^{j} = x^{n+2} D_{i}(x^{−n} L^{ij}(X))
= x^{2} Δ_{L,g} X^{j} − nx D_{i}x L^{ij}(X) ,

with Δ_{L,g}, L^{ij} — defined in (2.1)–(2.3). In local coordinates as described at the beginning of Chapter 3 this can be written as

Δ_{L̃,g̃} X = |Dx|_{g}^{2} [(x^{2} ∂_{x}^{2} − nx∂_{x}) ϕ + L̃ϕ] ,  (3.4)

where ϕ^{1} = (1 − \frac{1}{n}) X^{1}, ϕ^{A} = X^{A}/2, with L̃ of the form (2.3), and the theory of Chapter 4 will apply if we can show the existence of some regularity interval, either in weighted Hölder or in weighted Sobolev spaces. Let us note that the indicial roots for Δ_{L̃,g̃} are

(μ_{−}, μ_{+}) = (0, n + 1) .  (3.5)

In this section we shall make appeal to some results of [3], let us shortly discuss the notational correspondence between [3] and this paper. We have g̃ here ↔ g in [3], x here ↔ ρ in [3], g here ↔ h in [3], thus g̃ = x^{−2}g here while g = ρ^{−2}h in [3]. Let X be an s-covariant r-contravariant tensor field, we have

X ∈ C_{k+λ}^{α} here ↔ X ∈ C_{−α+s−r}^{k,λ} in [3]
X ∈ H_{k}^{α}(M, g̃) here ↔ X ∈ H_{−α}^{2,k}(M, g) in [3]
↔ X ∈ W_{k}^{α−s+r;2}(M, g) as defined in Chapter 3 .

We will also use spaces

H_{k}^{α,β}(M, g̃) ↔ W_{k}^{α−s+r,β;2}(M, g) as defined in Chapter 3.  (3.6)

Let us start with a McKean–type inequality (cf. [61]):

Proposition 3.1. Let g be a M_{1}^{σ}, σ > 0, metric on M and suppose that f is a C_{1} tensor field with suppf ⊂ M_{ϵ}, with some ϵ > 0. For 1 ≤ p and α > −(n − 1)/p there exists ϵ_{0} > 0 such that for 0 < ϵ ≤ ϵ_{0} we have

\{[\frac{n − 1}{p}] + α − o(1)\} ∥ |Dx|_{g} |f|g̃∥_{x^{α}L^{p}(M_{ϵ},dμg̃)} ≤ ∥ |D̃f|g̃∥_{x^{α}L^{p}(M_{ϵ},dμg̃)} ,(3.7)

with o(1) → 0 as ϵ → 0.

Proof: Suppose first that f is a function. The identity

0 = ∫_{M_{ϵ}} D̃_{i} (|f|^{p}x^{−pα−1}|Dx|_{g}^{p−2}D̃^{i}x) dμg̃

written out in detail gives, after some rearrangements,

(\frac{n − 1}{p} + α − o(1)) ∫_{M_{ϵ}} |Dx|_{g}^{p}|f|^{p}x^{−pα}dμg̃

≤ ∫_{M_{ϵ}} |Dx|_{g}^{p−1}|D̃f|g̃|f|^{p−1}x^{−pα}dμg̃ .(3.8)

From Hölder’s inequality we obtain

∫_{M_{ϵ}} |Dx|_{g}^{p−1}|D̃f|g̃|f|^{p−1}x^{−pα}dμg̃

≤ (∫_{M_{ϵ}} |D̃f|p ; g̃x^{−pα}dμg̃)^{1/p} (∫_{M_{ϵ}} |Dx|_{g}^{p}|f|^{p}x^{−pα}dμg̃)^{(p−1)/p}

which together with (3.8) gives (3.7).

If f is a tensor, the result follows by applying (3.7) to the function ϕ(x) (δ + |f|2 ; g̃)^{\frac{1}{2}} and passing to the limit δ → 0, where δ is a positive constant and ϕ is any function satisfying ϕ ∈ C^{∘} _{∞}(M_{ϵ}), ϕ|_{suppf} = 1. □

We now improve this for p = 2 and for r–forms (we shall need the result for vectors, which are of course naturally identified with 1–forms). The following computational Lemma is useful:

Lemma 3.2. Let g be a M_{2+λ}^{σ} metric on M with 0 < σ ≤ 1 and let H̃_{log(x)} denote the Hessian of log(x) with respect to the metric g̃, where x is the defining function of ∂M as above and let s ∈ ℝ. Then the eigenvalues of H̃_{log(x)} satisfy

λ_{i}(H̃_{log(x^{s})}) ≥ −s|Dx|_{g}^{2} + O(x^{σ}) .

Moreover, the Laplacian Δg̃x^{s}, with Δg̃ = D̃_{i}D̃^{i}, satisfies

Δg̃x^{s} = −x^{s} [s(n − 1 − s)|Dx|_{g}^{2} + O(x^{σ})] .

Further, if we let d˜∗ denote the exterior co–derivative w.r.t. g̃, then

d˜∗(D̃x/|D̃x|g̃) = −D̃_{i}(D̃^{i}x/|D̃x|g̃) = (n − 1)|Dx|_{g} + O(x^{σ}) .

In regular coordinates near ∂M the Riemann tensor R̃_{ijkl} of g̃ satisfies

R̃_{ijkl} = x^{−4}[|Dx|_{g}^{2}(g_{ij}g_{kl} − g_{il}g_{jk}) + O(x^{σ})] ,

and the Ricci tensor R̃_{ij} of g̃ satisfies

R̃_{ij} = x^{−2}[−(n − 1)|Dx|_{g}^{2}g_{ij} + O(x^{σ})] .

The following result was originally proved for manifolds close to hyperbolic space by Donnelly and Xavier [29]. The following argument, due to Lee [52], gives the result in the general conformally compact case, cf. also [3] for a proof based on the argument of [29]. We give the proof here for completeness; by definition the norm || · ||_{H_{k}(M_{ϵ},g̃)} here is the norm in H_{k}^{0,0}(M_{ϵ}, g̃), as defined in Section 1.3, page

Lemma 3.3. Let ξ be a C^{∞} r–form with supp|ξ|g̃ ⊂ M_{ϵ}, let g be a M_{1}^{σ} ∩ C_{2}^{loc} metric on M, σ > 0. Then, for r ≤ (n − 1)/2 or r ≥ (n + 1)/2,

[\frac{(n − 1 − 2r)^{2}}{4} + o(1)] ∥|Dx|_{g}ξ∥_{H_{0}(M_{ϵ},g̃)}^{2} ≤ ∥dξ∥_{H_{0}(M_{ϵ},g̃)}^{2} + 2∥d˜∗ξ∥_{H_{0}(M_{ϵ},g̃)}^{2}(3.9)

Proof: To avoid a proliferation of tilde’s, we momentarily suspend the convention that geometric quantities refering to the metric g̃ are decorated with tildes. Thus, until specified otherwise all geometric quantities refer to the g̃ metric. Let {e_{j}} denote any local orthonormal frame for T M and {e^{j}} the corresponding dual coframe. Let D denote the covariant derivative of the metric acting on tensors, with D_{j} ≡ D_{e_{j}}. Recall the identities

dξ = ∑ e^{j} ∧ D_{j}ξ ,
d^{∗}ξ = ∑ −e^{j} ∨ D_{j}ξ ,
d(e^{−u}ξ) = e^{−u}(dξ − du ∧ ξ) ,
d^{∗}(e^{u}ξ) = e^{u}(d^{∗}ξ − du ∨ ξ) ,
d(du ∨ ξ) = −du ∨ dξ + H_{u}ξ + D_{Du}ξ ,
|du|^{2}|ξ|^{2} − (Δ_{H}u)|ξ|^{2} = −e^{−u}Δ_{H}(e^{u})|ξ|^{2} .

Here ∨ is the contraction operator^{1}, H_{u} = Ddu denotes the Hessian of u and H_{u}ξ denotes the induced operator on Λ^{r},

H_{u}ξ = D_{i}du ∧ (e^{i} ∨ ξ) .

In the calculation below, Δ_{H} = dd^{∗} + d^{∗}d.

∫_{M} |e^{u}d(e^{−u}ξ)|^{2} + |e^{−u}d^{∗}(e^{u}ξ)|^{2}
= ∫_{M} |dξ − du ∧ ξ|^{2} + |d^{∗}ξ − du ∨ ξ|^{2}
= ∫_{M} |dξ|^{2} − 2⟨dξ, du ∧ ξ⟩ + |du ∨ ξ|^{2} + |d^{∗}ξ|^{2} − 2⟨d^{∗}ξ, du ∨ ξ⟩ + |du ∧ ξ|^{2}
= ∫_{M}⟨ξ, Δ_{H}ξ⟩ − 2⟨du ∨ dξ, ξ⟩ + ⟨du ∨ (du ∧ ξ), ξ⟩ − 2⟨ξ, d(du ∨ ξ)⟩ + ⟨du ∧ (du ∨ ξ), ξ⟩
= ∫_{M}⟨ξ, Δ_{H}ξ⟩ + |du|^{2}|ξ|^{2} − 2⟨ξ, H_{u}ξ⟩ − 2⟨ξ, D_{gradu}ξ⟩
= ∫_{M}⟨ξ, Δ_{H}ξ⟩ + |du|^{2}|ξ|^{2} − 2⟨ξ, H_{u}ξ⟩ − (Δ_{H}u)|ξ|^{2}
= ∫_{M}⟨ξ, Δ_{H}ξ⟩ − e^{−u}Δ_{H}(e^{u})|ξ|^{2} − 2⟨ξ, H_{u}ξ⟩

We can state this in the form

∫_{M}⟨ξ, Δ_{H}ξ⟩ = ∫_{M} ϕ^{−1}Δ_{H}ϕ|ξ|^{2} + 2⟨ξ, H_{log ϕ}ξ⟩ + |ϕd(ϕ^{−1}ξ)|^{2} + |ϕ^{−1}d^{∗}(ϕξ)|^{2}
≥ ∫_{M} ϕ^{−1}Δ_{H}ϕ|ξ|^{2} + 2⟨ξ, H_{log ϕ}ξ⟩ .

1If α is a q–form and β a p + q–form, in local coordinates we have (α ∨ β)_{i_{1}...i_{p}} = α^{j_{1}...j_{q}} β_{j_{1}...j_{q}i_{1}...i_{p}}, where the indices on α_{j_{1}...j_{q}} have been raised with the appropriate metric.

Let us now return to our convention, that quantities referring to the g̃ metric are decorated with tildes. Choosing ϕ = x^{s} with s ≠ 0, n − 1, a straightforward computation gives (recall that Δ_{H} = −Δg̃ when acting on functions)

ϕ^{−1}Δg̃ϕ = −s(n − 1 − s)|Dx|_{g}^{2} + o(1) ,
H̃_{log ϕ} ≡ D̃_{i}D̃_{j} log ϕ = sx^{−2}|Dx|_{g}^{2}(n_{i}n_{j} − g_{ij} + o(1)) ,

with n_{i} = D_{i}x/|Dx|_{g}. Note that if A ∈ End(T M) is symmetric with eigenvalues bounded from below by λ then ⟨Aξ, ξ⟩ ≥ λr|ξ|^{2} when A is considered in End(Λ^{r}). This gives the inequality

[s(n − 1 − s − 2r) + o(1)] ∫_{M} |ξ|2 ; g̃|Dx|_{g}^{2}dμg̃ ≤ ∫_{M}(|dξ|2 ; g̃ + |d˜∗ξ|2 ; g̃)dμg̃ . (3.10)

Here s is a free parameter and we maximize the left hand side of (3.10) w.r.t. s. This gives s = (n − 1 − 2r)/2 which inserted into (3.10) gives the result. □

Remark: Note that in the case r = (n ± 1)/2 the above result does not give any information better that λ_{r} ≥ 0 and this is in fact sharp, as can be shown for the case of constant sectional curvature.

Proposition 3.4. Let ξ be a C^{∞} r–form with supp|ξ|g̃ ⊂ M_{ϵ}, let g be a M_{2}^{σ} metric on M with σ > 0. Then

[\frac{(n − 1)^{2}}{4} + r′ − o(1)] ∥ |Dx|_{g} ξ∥_{H_{0}(M_{ϵ},g̃)}^{2} ≤ ∥ D̃ξ∥_{H_{0}(M_{ϵ},g̃)}^{2} .  (3.11)

where r′ = r if r < n/2 and r′ = n − r if r > n/2.

Proof: This is immediate from the Weitzenbock formula (cf., e.g., [38]),

< (dd˜∗ + d˜∗d)ξ, ξ >g̃= |D̃ξ|2 ; g̃+ < R̃_{r}ξ, ξ >g̃ + pure divergence, (3.12)

and the asymptotic behaviour of the Riemann tensor, cf. Lemma 3.2. cf. [11] or [38] for the precise form of R̃_{r}. □

Proposition 3.5. Let g ∈ M_{2}^{σ}, σ > 0, and suppose that

|α| < \frac{n + 1}{2}\sqrt{\frac{n}{2(n − 1)}} .

For every β ∈ ℝ there exists ϵ > 0 and a constant C such that for every X ∈ C^{∘} _{2}(M_{ϵ}) we have

∥ |X|g̃∥_{x^{α}(1+| ln x|)^{β} L^{2}(M_{ϵ},dμg̃)} ≤ C∥ |Δ_{L̃,g̃} X|g̃∥_{x^{α}(1+| ln x|)^{β} L^{2}(M_{ϵ},dμg̃)} . (3.13)

Remark: We believe that the above bound on α is not sharp, and we expect the inequality to hold for |α| < (n + 1)/2.

To prove Proposition 3.5 we need still another Lemma:

Lemma 3.6. Let ϕ ∈ C_{2}(M), X ∈ C^{∘} _{2}(M_{x_{0}}), set Y = ϕX. Under the hypotheses of Proposition 3.5 we have

∗ ≡ − ∫_{M_{x_{0}}} Y_{i} Δ_{L̃,g̃} Y^{i} dμg̃ ≤ \frac{(n − 1)}{n} ∫_{M_{x_{0}}} |D̃ ϕ|2 ; g̃ |X|2 ; g̃ dμg̃
− ∫_{M_{x_{0}}} ϕ^{2} X_{i} Δ_{L̃,g̃} X^{i} dμg̃ .  (3.14)

Proof: From the Ricci identity we have

2Δ_{L̃,g̃} Y^{i} = Δg̃ Y^{i} + R̃^{i}_{j} Y^{j} + \frac{n − 2}{n} D̃^{i} δ̃Y ,  (3.15)

where Δg̃ = D̃_{i} D̃^{i} is the standard Laplacian, R̃_{ij} is the Ricci tensor of g̃_{ij}, and δ̃Y = D̃_{i} Y^{i}. This implies

∗ = \frac{1}{2}(A + B + C) ,  (3.16)
A = ∫ |D̃ Y |2 ; g̃ ,  (3.17)
B = − ∫ R̃_{ij} Y^{i} Y^{j} ,  (3.18)
C = \frac{n − 2}{n} ∫ (D̃_{i} Y^{i})^{2} .  (3.19)

Setting Y = ϕ X the identity

0 = ∫ D̃^{i} (ϕ^{2} X^{j} D̃_{i} X_{j})

written out in detail and inserted in A gives

A = ∫ |D̃ ϕ|2 ; g̃ |X|2 ; g̃ − ∫ ϕ^{2} X_{j} Δg̃ X^{j} .  (3.20)

Similarly the identity

0 = ∫ D̃_{i} (ϕ^{2} X^{i} D̃_{k} X^{k})

yields

C = \frac{n − 2}{n} ∫ [(X^{i} D̃_{i} ϕ)^{2} − ϕ^{2} X^{i} D̃_{i} D̃_{k} X^{k}] .  (3.21)

Substituting (3.20)–(3.21) in (3.16) one gets, using (3.15),

∗ = ∫ \{\frac{1}{2}|D̃ ϕ|2 ; g̃|X|2 ; g̃ + \frac{n − 2}{2n} (X^{i} D̃_{i} ϕ)^{2} − ϕ^{2} X_{i} Δ_{L,g̃} X^{i}\} ,

which implies (3.14). □ Proof of Proposition 3.5. Consider again the equation (3.16). We have

−R̃_{ij} Y^{i} Y^{j} = (n − 1) |Dx|_{g}^{2} (1 + o(1)) ϕ^{2} |X|2 ; g̃ ,

so that using neglecting C in (3.16) and using (3.11) to estimate A one has

[\frac{(n + 1)^{2}}{8} − o(1)] ∫ |Dx|_{g}^{2} ϕ^{2} |X|2 ; g̃ dμg̃ ≤ ∗ ,

which together with (3.14) yields

∫ \{(\frac{(n + 1)^{2}}{8} − o(1)) ϕ^{2} |Dx|_{g}^{2} − \frac{(n − 1)}{n}|D̃ ϕ|2 ; g̃\} |X|2 ; g̃ ≤ − ∫ ϕ^{2} X_{i} Δ_{L̃,g̃} X^{i} .

Let ϵ < 1, set ϕ = x^{−α}(1 + | ln x|)^{−β}, for α ≠ 0 we have

|D̃ ϕ|g̃ = |α| x^{−α}(1 + | ln x|)^{−β}) |Dx|_{g} (1 + O (\frac{1}{| ln ϵ|})) ,

while if α = 0

|D̃ ϕ|g̃ = β | ln x|^{β−1} |Dx|_{g} ,

so that in either case

|D̃ ϕ|g̃ = ϕ |Dx|_{g}(|α| + o(1)) ,

thus

\{\frac{(n + 1)^{2}}{8} − \frac{(n − 1)}{n} α^{2} − o(1)\} ∥ϕ|Dx|_{g} |X|g̃∥_{L^{2}}^{2} ≤ − ∫ ϕ^{2} X_{i} Δ_{L,g̃} X^{i} dμg̃
≤ ∥ϕ|X|g̃∥_{L^{2}} ∥ϕ|Δ_{L,g̃} X|g̃∥_{L^{2}}

which implies (3.13). □

Proposition 3.5 shows that Proposition 2.6 of [3] applies, and since Δ_{L̃,g̃} is formally self-adjoint, Corollary 2.7 of [3] shows:

Proposition 3.7. Let M be a manifold with boundary with M̄ — compact. Suppose that g ∈ M_{k+2+λ}^{σ}, and k ≥ 0. For

|α| < \frac{n + 1}{2}\sqrt{\frac{n}{2(n − 1)}} , β ∈ ℝ ,

the operator

Δ_{L̃,g̃} : H_{k+2}^{α,β}(M, g̃) → H_{k}^{α,β}(M, g̃)

is Fredholm.

Remark: The spaces H_{k}^{α,β}(M, g̃) are defined in (3.6).

Theorem 3.8. Let M be a manifold with boundary with M̄ — compact. Let g ∈ M_{k+2+λ}^{σ}, k ≥ 1, λ ∈ [0, 1), σ > 0, k + λ > 1 (cf. (2.5)) be a Riemannian metric on M, let g̃ = x^{−2} g where x is a defining function for ∂M. For

|α| < \frac{n + 1}{2}\sqrt{\frac{n}{2(n − 1)}} ,  β ∈ ℝ ,  0 ≤ ℓ ≤ k ,

the maps

Δ_{L̃,g̃} : H_{ℓ+2}^{α,β}(M, g̃) → H_{ℓ}^{α,β}(M, g̃) ,  (3.22)
Δ_{L̃,g̃} : H^{∘} _{1}^{α,β} ∩ H_{ℓ+2}^{α,β}(M_{x_{0}}, g̃) → H_{ℓ}^{α,β}(M_{x_{0}}, g̃) ,

are isomorphisms.

Remarks:

((i)) As noted at the beginning of this section, H_{m}^{α,β}(M, g̃) = W_{m}^{α+1,β;2}(M, g)

for vector fields. ((ii)) In the physically interesting case n = 3 we get α ∈ (−\sqrt{3}, \sqrt{3}). In

local coordinates near ∂M this corresponds, roughly speaking, to vector fields the components of which behave as x^{β}, β ∈ (2 − \sqrt{3}, 2 + \sqrt{3}).

Proof: By Proposition 3.7 Δ_{L̃,g̃} is Fredholm, and formal self-adjointness of Δ_{L̃,g̃} together with duality arguments imply (cf. e.g. [10, 3]) that it is sufficient to show that the kernel of Δ_{L̃,g̃} on H_{2}^{α,β}(M, g̃) is zero for α’s in the desired interval. Let thus

Δ_{L̃,g̃} X = 0 , X ∈ H_{2}^{α,β}(M, g̃) .

The proof of Proposition 3.10 of [3] shows that we have X ∈ H_{2}^{α′,β}(M, g̃) for any α′ ∈ (−ω_{+}, ω_{+}) and β in ℝ, where ω_{+} = \frac{n+1}{2} \sqrt{\frac{n}{2(n−1)}}. In particular we have

X ∈ H_{2}^{0,0}(M, g̃), the standard H_{2} Sobolev space on M equipped with the metric g̃. For all Y ∈ H_{2}^{0,0}(M, g̃) which are compactly supported we have the integration by parts identity

∫_{M} Y_{i} Δ_{L̃,g̃} Y^{i} = ∫_{M} |L̃(Y)|2 ; g̃ dμg̃ ,  (3.23)

where L̃ has been defined in (3.2), and a standard density argument shows that (3.23) holds for all (not necessarily compactly supported) Y ∈ H_{2}^{0,0}(M, g̃). It follows that

0 = ∫_{M} X_{i} Δ_{L̃,g̃} X^{i} = ∫_{M} |L̃(X)|2 ; g̃ dμg̃ ,  (3.24)

and we conclude that

D̃_{i} X_{j} + D̃_{j} X_{i} − \frac{2}{n} D̃_{k} X^{k} g̃_{ij} = 0 .  (3.25)

Elliptic regularity implies X ∈ H_{k+2}^{α,β}(M, g̃), and from Proposition 3.10 of [3] it follows that X ∈ C_{0+σ}^{1+α′}, for any

|α′| ≤ \frac{n + 1}{2}\sqrt{\frac{n}{2(n − 1)}} ,  (3.26)

and for some σ > 0. If λ > 0 the scaling estimates (1.22) imply X ∈ C_{k+min(λ,σ)}^{1+α′},

while if λ = 0 then by hypothesis k > 1 and X ∈ C_{k−1+σ}^{1+α′}, thus in either case X ∈ C_{1}^{1+α′}. Conformal invariance of (3.25) implies that (2.6) holds as well, and the vanishing of X follows by Proposition 2.2. □

The two Theorems that follow can be proved by arguments similar to those used in the proof of Theorem 2.8, using Theorem 3.8 (cf. also [6] or Section 2.2 here), the details will be omitted. Let us start with an equivalent of Theorem 2.8:

Theorem 3.9. Let M̄ be a compact manifold (with boundary), suppose that g is a metric on M ≡ M̄ \ ∂M̄ which can be C_{∞} extended across ∂M, let x be any smooth defining function for ∂M. Consider the equation

Δ_{L̃,g̃} X = x^{α} Y ,  Y ∈ C_{∞}(M̄) .  (3.27)

((i)) For any α and Y there exists a solution of (3.27) of the form

X = x^{α}X_{α} + log xX_{0,1} + X_{0} + x^{n+1} log^{2} xX_{n+1,2} ,
X_{α}, X_{0,1}, X_{0}, X_{n+1,2} ∈ C_{∞}(M̄) .

((ii)) If moreover −α ∉ ℕ_{0} or if X_{0,1}|_{∂M} = 0, then there exists a solution of (3.27) of the form

X = x^{α}X_{α} + X_{0} + x^{n+1} log xX_{n+1,1} ,
X_{α}, X_{0}, X_{n+1,1} ∈ C_{∞}(M̄) .

If X_{n+1,1}|_{∂M} = 0, then X_{n+1,1} ≡ 0. ((iii)) Any solution X ∈ W_{1}^{loc} of the homogeneous equation satisfying either

1) X ∈ H_{0}^{α}(M, g̃) with α > −\frac{n+1}{2} \sqrt{\frac{n}{2(n−1)}}, or 2) X = O(x^{α}) for some

α > \frac{n+1}{2}(1 − \sqrt{\frac{n}{2(n−1)}}), vanishes.

We also have the following equivalent of Theorem 2.10 (cf. Definition 2.9 for the definition of a polymogeneous metric). The reader should note that the result below holds for any polyhomogeneous metric g — this should be contrasted with Theorem 2.10 in which the metric g is assumed to be polyhomogeneous and uniformly C_{2} up to boundary.

Theorem 3.10. Let g be a polyhomogeneous metric on a compact manifold M̄ with boundary, ∂M, consider the equation

Δ_{L̃,g̃} X = Y ,  Y ∈ A^{phg} .  (3.28)

There exists a solution X ∈ A^{phg} of (3.28). Any solution of the homogeneous equation (3.28) satisfying either 1) X ∈ H_{0}^{α}(M, g̃) with α > −\frac{n+1}{2} \sqrt{\frac{n}{2(n−1)}}, or 2)

X = O(x^{α}) for some α > \frac{n+1}{2}(1 − \sqrt{\frac{n}{2(n−1)}}), vanishes.

As far as classical regularity of solutions of eq. (3.1) is concerned, we have the following Corollary of Theorems 3.8 and 2.11 (here the case dim M = 3 only will be considered, other cases can be obtained in a similar way):

Theorem 3.11. Suppose that k ≥ 2, λ ∈ (0, 1], and let g ∈ C_{k+1+λ}(M̄) be a Riemannian metric on a three dimensional manifold with boundary M, with M̄ – compact. Define

(α_{−}, α_{+}) = (2 − \sqrt{3}, 2 + \sqrt{3}) .

Consider the equation

Δ_{L̃,g̃} X = Y , Y ∈ x^{α} C_{k̂+λ}(M̄) ,
α > α_{−}, 1 ≤ k̂ ≤ k .

Suppose that either

u = O(x^{α_{−}+ϵ}) , ϵ > 0 ,

or

u ∈ W_{0}^{ω;2}(M_{x_{1}}, x^{−n}dμ_{g}) , ω > 1 + \sqrt{3} .

Then there exists σ > 0 such that the following hold:

((i)) Let α < 2 + \sqrt{3} and let ℓ_{0} ∈ ℕ_{0} be the largest number such that

α + ℓ_{0} < 4.
(a) α + k̂ < 5 ⟹ X ∈ ∩_{i=0}^{3}x^{α−i} C_{k̂−1+i+σ}(M̄).
(b) k̂ = ℓ_{0} + k_{0} + 1 ⟹ X ∈ ∩_{i=0}^{3}x^{α−i} C_{ℓ_{0}+i+σ|k_{0}}(M̄).

((ii)) For α ∈ [2+\sqrt{3}, 4) the conclusions of point (i) hold with k there replaced by k − 1.

((iii)) Let α ∈ ℕ_{0}∩[1, 4]. If k̂ ≥ 6−α, then there exists X_{log} ∈ ∩_{i=−(4−α)}^{3}x^{α−i} log x C_{k̂−1+i+λ}(M̄)

such that

X − X_{log} log x ∈ ∩_{i=0}^{3}x^{α−i}C_{k̂−1+i+σ}(M̄) .

If X_{log}|_{∂M} = 0, then X_{log} ≡ 0. ((iv)) Let α > 4, suppose that k̂ ≥ 3, and let ℓ_{0} ∈ ℕ_{0} be the largest integer

such that 4 + ℓ_{0} ≤ α.

(a) If 2+k̂ < α or if α ∈ ℕ (or both) ⟹ X ∈ ∩_{i=0}^{4}x^{4−i}C_{k̂−2+i+σ}(M̄) .

(b) k̂ > ℓ_{0} + 2 ⟹ X ∈ ∩_{i=0}^{4}x^{4−i}C_{ℓ_{0}+i+σ|k̂−2−ℓ_{0}}(M̄) .

CHAPTER 7 The Lichnerowicz equation.

1. Introductory remarks

In this chapter we shall examine the boundary behaviour of the solutions of the scalar constraint equation, under various hypotheses on the differentiability of the metric: this is obtained using the methods of Chapter 4 together with a fairly standard “bootstrap” procedure. We shall construct solutions of the Lichnerowicz– type equation,

\frac{4(n − 1)}{n − 2} Δg̃ ϕ − R̃ϕ + ζϕ^{−κ} − n(n − 1) ϕ^{\frac{n+2}{n−2}} = 0 ,  κ > 0 ,  (1.1)

with the boundary condition

ϕ ⟶_{x→0} |Dx|g^{\frac{n−2}{2}} .  (1.2)

Let us point out that under suitable conditions on the metric any solution of (1.1) which is uniformly bounded above and uniformly bounded away from zero has to satisfy (1.2), cf. Theorem 3.2. Here κ is a positive constant^{1}, ζ is a function satisfying ζ ≥ 0. Δg̃ is the Laplace operator of the metric g̃_{ij} = x^{−2} g_{ij}, Δg̃ = D̃_{i} D̃^{i}, and R̃ is the scalar curvature of g̃, n = dim M ≥ 3. If n = 3 and κ = 7 the equation (1.1) is of the form (1.23). Note also that if ζ = 0 then (1.1) is the Yamabe equation (with the scalar curvature of the metric ϕ^{4/(n−2)} g̃_{ij} equal to −n(n − 1)).

We have the equality

R̃ = −n(n − 1) |Dx|_{g}^{2} + 2(n − 1)xΔ_{g}x + x^{2} R ,  (1.3)

where Δ_{g} = D_{i}D^{i} is the Laplace operator of the metric g, and R is the Ricci scalar of g. If we assume that ζ = o(1), Δ_{g}x = o(x^{−1}) and R = o(x^{−2}) (which holds e.g. if g ∈ C_{k+2}(M̄) or if^{2} g ∈ M_{k+2+λ}^{ϵ}, with ϵ > 0, k ∈ ℕ_{0}, λ ∈ [0, 1]), then from (1.1)–(1.3) it follows that Δg̃ ϕ ⟶_{x→0} 0 if and only if (1.2) holds, except perhaps when ϕ → 0. In that last case one does not expect (M, ϕ^{4/(n−2)} g̃) to be complete; it certainly will not if ϕ ∈ C_{1}(M̄). This shows that the boundary condition (1.2) is necessary if the solution ϕ has to have some degree of regularity at ∂M (e.g., if we look for solutions ϕ ∈ C_{2}(M̄)), and if we require completeness of (M, ϕ^{4/(n−2)} g̃)) .

Note that by carrying out a conformal transformation g → ψ^{2}g, where ψ is any strictly positive function of the same differentiability as the metric and which coincides with |Dx|_{g} near ∂M, we get

|Dx|_{g} ≡ 1  (1.4)

1Most of the results of this chapter remain true when κ < −1 and ζ ≤ 0. When κ ∈ (−1, 0] some further restrictions than non–positivity of ζ are needed for the existence theorem, cf. e.g.

section 2 for a discussion of an example with κ = −1.

2Recall that the space M_{k+2+λ}^{ϵ} has been defined in Definition 2.1, page 62.

in a neighbourhood of ∂M. Assuming (1.4), the linearization of (1.1) at ϕ = 1 gives an operator which in a neighbourhood of ∂M takes the form

L = Δg̃ + ξ ,  (1.5)

If we make the same hypotheses about the metric and ζ as in the discussion following eq. (1.3), we obtain

ξ = −\frac{(n − 2)}{4(n − 1)} [R̃ + \frac{(n − 1)n(n + 2)}{(n − 2)} + κζ] ⟶_{x→0} − n .  (1.6)

We can now rewrite (1.1) in the form

Lu = F (y, u) + S ,  F (y, 0) = \frac{∂F (y, u)}{∂u}|_{u=0} = 0 ,  (1.7)

with

u ≡ ϕ − 1 ,

L as in (1.5) and, assuming (1.4),

S = \frac{(n − 2)}{4(n − 1)} (R̃ + n(n − 1) − ζ) ,  (1.8)
F (y, u) = \frac{n(n − 2)}{4} \{(1 + u)^{(n+2)/(n−2)} − 1 − \frac{n + 2}{n − 2} u\}
−\frac{(n − 2)}{4(n − 1)} ζ\{ (1 + u)^{−κ} − 1 + κu\} .  (1.9)

Let us point out that the form (1.7) of the Lichnerowicz equation is sufficiently general to cover many other matter models than the vacuum case considered above, cf. the discussion in Section 2. For this reason, and with future applications in mind, we will in the rest of this chapter discuss equations of this general form.

2. The linearized equation

Before passing to the non-linear equation (1.1) it will be necessary to give a detailed discussion of the properties of the linear operator (1.5). In order to be able to apply the results of Chapter 4 to that equation we have to prove existence of a regularity interval for L. It has been shown in [3] that (α_{−}, α_{+}) = (−(n+1)/2, (n+1)/2) is a strong regularity interval for L for H_{0}^{α}(M, g̃) ≡ W_{k}^{0,0;2}(M, g), with W_{k}^{0,0;2}(M, g) defined in Section 1.3. This and Theorem 2.3 show that (α_{−}, α_{+}) = (−1, n) is a regularity interval for C_{k+λ,ℓ}^{α}, with (n − 1)/2 < ℓ ≤ k. We shall show that a direct analysis in the framework of weighted Hölder spaces gives (α_{−}, α_{+}) = (−1, n) as a strong regularity interval for C_{k+λ}^{α}.

Let us start with the observation, that for g ∈ C_{k+2+λ}(M̄), k ∈ ℕ_{0}, λ ∈ (0, 1), we can write

L = x^{2} (Δ_{g} − \frac{(n − 2)}{x} D^{i}x ∂_{i}) + ξ
= |Dx|_{g}^{2}{ (x^{2}∂_{x}^{2} − x (n − 2) ∂_{x} − n) + L̃} .  (2.1)

It follows that after a trivial rescaling L can be written in the form (2.1)–(2.2), with L̃ of the form (2.3). The indicial roots for (2.1) are (cf. (2.7))

μ_{−} = −1 , μ_{+} = n .

To motivate the hypotheses of the theorem to follow, let us note that if g ∈ M_{k+2+λ}^{ϵ}, k ∈ ℕ_{0}, λ ∈ (0, 1) for some ϵ > 0, then we can still formally define L̃ using eq. (2.1). Doing so and assuming (1.4) we obtain

∀α ∈ ℝ  L̃x^{α} = O(x^{α+ϵ}) .  (2.2)

In the result that follows, a scalar operator is defined as an operator which maps functions on M to functions on M.

Theorem 2.1. Let L ∈ OP_{C_{k+λ}^{0}(M)}^{2}, k ∈ ℕ_{0}, λ ∈ (0, 1), be a scalar geometric

elliptic operator satisfying (2.1)–(2.2) and (2.4), with L̃ satisfying (2.2) and with indicial exponents μ_{−} < μ_{+} (cf. eq. (2.7)). Consider the equation

Lu = f .  (2.3)

Then

((i)) For every β ∈ ℝ and α ∈ (μ_{−}, μ_{+}) there exists 0 < x_{1} ≤ x_{0} such that for every f ∈ C_{k+λ}^{α,β}(M_{x_{1}}), ψ ∈ C_{k+2+λ}(˜∂M_{x_{1}}), there exists a solution u ∈ C_{k+2+λ}^{α,β}(M_{x_{1}}) of (2.3) satisfying u|_{˜∂M_{x_{1}}} = ψ. Moreover L has a

strong regularity interval (α_{−}, α_{+}) for C_{k+λ}^{α,β}(M) with α_{±} = μ_{±}, and with no restrictions on β.

((ii)) Assume further that M̄ is compact, with μ_{+} > 0 and with L – elliptic in the standard sense on CM_{x_{0}}, with coefficients in C^{k+λ}(CM_{x_{0}}) and with a_{0} ≤ 0, where a_{0} is the 0–order coefficient of L in (1.3) (this condition on a_{0} is supposed to hold throughout M, not only near ∂M). Then L : C_{k+2+λ}^{α,β}(M) → C_{k+λ}^{α,β}(M) is an isomorphism.

Remarks:

((i)) Assume that g is a M_{k+1+λ}^{ϵ} metric on a compact manifold M̄ (with boundary), k ≥ 0, λ ∈ (0, 1), ϵ > 0, and g̃_{ij} = x^{−2} g_{ij}, where x is a defining function for ∂M. Then Theorem 2.1 applies to equations of the form

Lu ≡ Δg̃u + x^{−1}ξ_{1} g̃(dx, du) + ξ_{2}u = f ,  (2.4)

provided that ξ_{1} = p + ψ_{1}, ξ_{2} = q|Dx|_{g}^{2} + ψ_{2}, with ψ_{a} ∈ C_{k+λ}^{ϵ}, a=1,2,

and with δ ≡ (\frac{n−1−p}{2})^{2} − q > 0. In that case the indicial exponents

are μ_{±} = α_{±} = \frac{n−1−p}{2} ± \sqrt{δ}. ((ii)) Graham and Lee [42, Theorem 3.10] have proved the following related

result (we use here the notation of Remark (i) above): If (M, g̃) is the hyperbolic space (M ≈ ℝ^{n}, with g̃ — the maximally symmetric metric of sectional curvature equal to −1) and ξ_{1} = ψ_{2} = 0, then for α ∈ (μ_{−}, μ_{+}) the operator L defined in (2.4) is an isomorphism between C_{k+2+λ}^{α}(M) and C_{k+λ}^{α}(M).

Proof: For σ ∈ ℝ, let L_{σ} be the operator defined by L_{σ}u = x^{σ}L(x^{−σ}u). Then L_{σ} is of the above form with the indicial roots μ̃_{±} = μ_{±} + σ (cf. eqs. (2.7)–(2.4)). Let α̃ = α + σ and f_{σ} = x^{σ}f. By choosing σ > −μ_{−} and considering the equivalent equation L_{σ}u = f_{σ} we may without loss of generality assume that α̃ > μ̃_{−} > 0 in the following. Now L̃_{σ} can be written in the form

L̃_{σ} = x^{2}a^{xx} \frac{∂^{2}}{∂x^{2}} + x^{2}a^{xA} \frac{∂}{∂x} \frac{∂}{∂v^{A}} + x^{2}a^{AB} \frac{∂}{∂v^{A}} \frac{∂}{∂v^{B}} + xa^{A} \frac{∂}{∂v^{A}} + xa^{x} \frac{∂}{∂x} + a_{0} .

Setting α = 0 in (2.2) we obtain a_{0} = O(x^{ϵ}). Note moreover that the constant b of eq. (2.2) for L_{σ} so defined must be negative as μ̃_{−} > 0. It follows that the comparison principle applies for L_{σ} in M_{x_{0}}, provided that x_{0} is chosen small enough.

Consider (2.3) with f ∈ C_{k+λ}^{α,β}(M). A straightforward calculation shows that there exist constants C, 0 < x_{1} = x_{1}(α, β, L) ≤ x_{0} such that for x ≤ x_{1} the functions

ϕ_{±} = ±C ∥f_{σ}∥_{C_{0}^{α̃,β}} xα̃ (1 + ln \frac{1}{x})^{β}

are super– and sub–solutions of (2.3):

(L_{σ}ϕ_{+} − f_{σ})|_{M_{x_{1}}} ≤ 0 , (L_{σ}ϕ_{−} − f_{σ})|_{M_{x_{1}}} ≥ 0 .

Choose a decreasing sequence {x_{k}}_{k=0}^{∞} ⊂ ℝ so that x_{k} → 0 as k → ∞. As the 0–th order coefficient of L_{σ} is negative on M_{x_{0}}, there is a unique solution u_{k} of the equation L_{σ}u_{k} = f_{σ} in CM_{x_{k}} satisfying u_{k}|_{˜∂M_{x_{k}}} = 0 and u_{k}|_{˜∂M_{x_{1}}} = x_{1}^{−σ}ψ. By a stan-

dard diagonalization procedure we can find a subsequence of {u_{k}} which converges to some u_{σ} ∈ C_{0}^{α̃,β}(M_{x_{1}}) ∩ C_{2}^{loc}(M_{x_{1}}). The scaling estimates of Lemma 1.1 near the boundary give u_{σ} ∈ C_{k+2+λ}^{α̃,β}(M_{x_{1}}). This proves existence of a solution in the appropriate space, uniqueness follows from the maximum principle. It follows that L_{σ} is an isomorphism between {u ∈ C_{k+2+λ}^{α̃,β}(M_{x_{1}}) : u|_{˜∂M_{x_{1}}} = ψ} and C_{k+λ}^{α̃,β}(M_{x_{1}}).

Consequently L is an isomorphism between {u ∈ C_{k+2+λ}^{α,β}(M_{x_{1}}) : u|_{˜∂M_{x_{1}}} = ψ} and

C_{k+λ}^{α,β}(M_{x_{1}}). Now the existence of a regularity interval with α_{±} = μ_{±} follows from point 1 of Proposition 1.2. The fact that (μ_{−}, μ_{+}) is a strong regularity interval follows by construction from the form of the barriers ϕ_{±}.

To prove point 2, note that if Lu = 0 in M and u = O(x^{μ_{−}+ϵ}) for some ϵ > 0, then it follows by the definition of a regularity interval and by point 1 of this Theorem that u = O(x^{μ_{+}−ϵ}) for any ϵ > 0. As μ_{+} > 0 by hypothesis, it follows that u goes to zero at ∂M, and then u ≡ 0 by the maximum principle. Also note that there exists a constant C such that the functions

ϕ̃_{±} = ±C min (1, Cx^{α}(1 + ln \frac{1}{x})^{β}) ∥f∥_{C_{0}^{α,β}}

are (weak) super– and sub–solutions of (2.3). The result follows by similar arguments as in the proof of point 1. □

What has been said so far can be summarized in the following:

Proposition 2.2. Let g ∈ C_{k+2+λ}(M̄) and let L be as in (2.1). Then L is a geometric elliptic operator in the sense of Chapter 4. Moreover L is of the form (2.1)–(2.4) with (μ_{−}, μ_{+}) = (−1, n), and L has a strong regularity interval (α_{−}, α_{+}) for C_{0+λ}^{α,β}, with α_{±} = μ_{±}. The same holds for g ∈ M_{k+2+λ}^{ϵ}, ϵ > 0.

Proposition 2.2 shows that we can use Theorems 1.3 and 2.1 to obtain tangential regularity of solutions u of the linearized counterpart of eq. (1.1) whenever the metric and the source f are tangentially regular. Similarly it follows that Theorems 2.6, 2.9 and 2.10 can be applied to give information about the classical regularity of the solutions to equation (2.3):

Corollary 2.3. Let M̄ be compact and assume that the hypotheses of Theorem 2.1 hold. Assume further that L ∈ OP_{C_{k+λ}(M̄)}^{2} is elliptic in a standard sense

in CM_{x_{0}}, and that L̃ is of the form (2.3). Consider the equation (2.3) with

f ∈ x^{α}C_{k+λ}(M̄),  α > μ_{−},  u = O(x^{μ_{−}+ϵ}),  ϵ > 0 .

Then there exists σ ∈ (0, 1) such that the following results hold:

((i)) Let α < μ_{+} and let ℓ_{0} ∈ ℕ_{0} be the largest integer such that α+ℓ_{0} < μ_{+}.

(a) α + k < μ_{+} ⟹ u ∈ ∩_{i=0}^{2}x^{α−i} C_{k+i+σ}(M̄).
(b) k = ℓ_{0} + k_{0} ⟹ u ∈ ∩_{i=0}^{2}x^{α−i} C_{ℓ_{0}+i+σ|k_{0}}(M̄).

((ii)) Let α ≤ μ_{+}, μ_{+} − α ∈ ℕ_{0}. If k ≥ μ_{+} − α + 1, then there exists

u_{log} ∈ ∩_{i=0}^{μ_{+}−α+2}x^{μ_{+}−i} C_{k−(μ_{+}−α)+i+λ}(M̄) such that
u − u_{log} log x ∈ ∩_{i=0}^{2}x^{α−i}C_{k+i+σ}(M̄) .
If u_{log}|_{∂M} = 0, then u_{log} ≡ 0.

((iii)) Let α > μ_{+}, suppose that k ≥ 1, and let ℓ_{0} ∈ ℕ_{0} be the largest integer such that μ_{+} + ℓ_{0} < α.

(a) If μ_{+}+k < α or if α−μ_{+} ∈ ℕ (or both) ⟹ u ∈ ∩_{i=0}^{2}x^{μ_{+}−i}C_{k+i+σ}(M̄) .

(b) k > ℓ_{0} ⟹ u ∈ ∩_{i=0}^{2}x^{μ_{+}−i}C_{ℓ_{0}+i+σ|k−ℓ_{0}}(M̄) .

(c) If u = o(x^{μ_{+}}) and if k ≥ ℓ̃, where ℓ̃ is the smallest integer such that ℓ̃≥ α − μ_{+}, then

u ∈ ∩_{i=0}^{ℓ̃+2} x^{α−i}C_{k−ℓ̃+i+σ}(M̄) .

The above results provide rather detailed information about the linear counterpart of eq. (1.1).

3. Existence of solutions of the non–linear problem

In order to prove existence of solutions to equation (1.1) we will first consider a somewhat more general class of equations.

Proposition 3.1. Consider the equation for a scalar function u

Δg̃u + X(u) + F (y, u) = 0 ,  (3.1)

where g̃ = x^{−2}g with x a defining function for ∂M, g a (locally) C_{1} metric on M, X a continuous vector field on M, F (y, u) a continuous function in both arguments. Suppose that

((i)) There exists a function u_{0} ∈ C_{0}(M̄)∩C_{2}^{loc}(M) and constants x_{0}, C_{1}, α > 0 such that

∀ 0 < x < x_{0} |Δg̃u_{0} + X(u_{0})| ≤ C_{1}x^{α} .

We shall moreover require that du_{0} ∈ C_{0}^{α−1}(M_{x_{0})}. ((ii)) There exist constants C_{±}, L satisfying C_{−} ≤ u_{0}|_{M_{x_{0}}} ≤ C_{+} such that

∀y ∈ M, F (y, C_{−}) ≥ 0, F (y, C_{+}) ≤ 0 ,

∀z_{1}, z_{2} ∈ [C_{−}, C_{+}], y ∈ M |F (y, z_{1}) − F (y, z_{2})| ≤ L|z_{1} − z_{2}| .

((iii)) There exist constants C_{2} > 0, x_{1} > 0 and β satisfying 0 < β ≤ α such that for all 0 ≤ x ≤ x_{1} we have

Δg̃x^{β} + X(x^{β}) ≤ −C_{2}x^{β} .

((iv)) There exists a constant C_{3} > 0 such that for 0 < x < x_{0} we have

u_{0} ≤ u ≤ C_{+} ⟹ F (y, u) ≤ C_{3}x^{α} ,
C_{−} ≤ u ≤ u_{0} ⟹ F (y, u) ≥ −C_{3}x^{α} .

Then there exists a solution u ∈ W_{2}^{loc;p}(M) for any 1 < p ≤ ∞ of equation (3.1) satisfying

C_{−} ≤ u ≤ C_{+},  |u − u_{0}||_{M_{x_{0}}} ≤ C_{4}x^{β}  (3.2)

for some constant C_{4}.

Remarks: 1. The solution u needs not be unique.

2. Note that point (iv) above will be satisfied if |F (y, u_{0})| ≤ C_{3}x^{α} and if F (y, u) is monotonously decreasing in u for 0 < x < x_{0} and C_{−} < u < C_{+}.

Proof: We shall show that there exist weak barriers u_{±} ∈ W_{1}^{loc;∞}(M), the result then follows by a well known method using e.g. the monotone iteration scheme (cf. e.g. [66, Theorem 2.3.1]). Let thus

u_{+} = min(u_{0} + Bx^{β}, C_{+}),
u_{−} = max(u_{0} − Bx^{β}, C_{−}),

with some constant B which shall be specified later. Let x_{+} = min(x_{0}, x_{1}). For those x ≤ x_{+} for which u_{+} = u_{0} + Bx^{β} we have

Δg̃u_{+} + X(u_{+}) + F (y, u_{+}) = Δg̃u_{0} + X(u_{0})
+B(Δg̃x^{β} + X(x^{β})) + F (y, u_{+})
≤ (C_{1} + C_{3})x^{α} − BC_{2}x^{β}

Let us set B_{+} = max (\frac{(C_{1}+C_{3})x_{+}^{α−β}}{C_{2}} , (C_{+} − minM̄ u_{0})x_{+}^{−β}); for all B ≥ B_{+} we then

have

Δg̃u_{+} + X(u_{+}) + F (y, u_{+}) ≤ 0.

Define Γ to be that connected component of the set {u_{0} + Bx^{β} = C_{+}} which intersects {0 < x < x_{+}}. By the definition of B_{+}, if B ≥ B_{+} we have Γ ⊂ {0 ≤ x ≤ x_{+}}. Increasing B if necessary, we can assume that D(u_{0} + Bx^{β}) does not vanish anywhere in {0 ≤ x ≤ x_{+}} so that Γ is a C_{2} submanifold of M. An integration by parts calculation shows that u_{+} is a weak supersolution of equation (3.1). Similarly one shows that increasing B if necessary u_{−} will be a weak subsolution of equation (3.1) and the result follows. □

Existence of bounded solutions of (1.1) (with ζ = 0) has been proved under various conditions on (M, g̃) in [8, 9] and [6]. As far as existence of solutions of (1.1) satisfying (1.2) is concerned, we have the following theorem:

Theorem 3.2. Let g ∈ M_{k+2+λ}^{ϵ} be a metric on a compact manifold M̄ with boundary, 0 ≤ ζ ∈ C_{k+λ}^{ϵ}(M), 0 < ϵ ≤ 1, λ ∈ (0, 1), k ≥ 0. There exists a solution ϕ of (1.1) such that

ϕ − |Dx|g^{\frac{n−2}{2}} ∈ C_{k+2+λ}^{ϵ}(M) .  (3.3)

ϕ is unique in the class of uniformly bounded, uniformly bounded away from zero, locally C_{2} solutions of (1.1).

Remark: For ζ = 0, it has been proved in [6, § 3] that, for g ∈ C_{∞}(M̄), the condition x^{−(n−2)/2}ϕ →_{x→∂M} ∞ implies uniqueness. (Although the uniqueness question is not discussed in [62] in the same context as in [6], it should be pointed out that the a–priori estimate which is the key for the uniqueness proof of [6] has also been obtained in [62].) It is not hard to see that this argument holds also when ζ ∈ xC_{∞}(M̄), ζ ≥ 0. For ζ = 0 and g ∈ C_{∞}(M̄) it has been announced without proof in [59] that u is unique provided that the manifold M̄ equipped with the topology induced by the metric ϕ^{4(n−2)}g is complete and is not a compact manifold with boundary.

Proof: We may assume without loss of generality that |Dx|_{g}^{2} ≡ 1 in a neighbourhood of ∂M. The argument of the proof of theorem 3.4 of [6] shows that there exists a function ϕ_{1} ∈ C_{k−2+λ}^{0}(M) satisfying 0 < C_{1} ≤ ϕ_{1} ≤ C_{2} for some constants

C_{1}, C_{2} > 0, and satisfying ϕ_{1} − 1 ∈ C_{k+2+λ}^{ϵ}(M), such that the metric ϕ1^{\frac{4}{n−2}} g̃ has

constant scalar curvature equal to −n(n − 1). Replacing g̃ by ϕ1^{\frac{4}{n−2}} g̃ we may without loss of generality assume that R̃ = R(g̃) = −n(n−1). Replacing ϕ by ϕϕ_{1}^{−1} one obtains an equation of the form (1.1) with possibly a different ζ ≥ 0. One readily checks that the hypotheses of Proposition 3.1 are satisfied with u_{0} = 1, α = β = ϵ, C_{−} = 1 and C_{+} – a sufficiently large positive constant. Further regularity than that asserted by Proposition 3.1 follows by elliptic regularity and a scaling estimate (Lemma 1.1).

To prove uniqueness, let ϕ_{2} be the solution constructed above and let ϕ_{1} be any other solution bounded from above and bounded away from zero. Conformally rescaling the metric we can choose a “conformal gauge” in which ϕ_{2} = 1. In this gauge eq. (1.1) reads

R̃ = −n(n − 1) + ζ .

Here ζ ≥ 0 is an appropriately rescaled counterpart of the function ζ of eq. (1.1). The eq. satisfied by ϕ ≡ ϕ_{1}/ϕ_{2} is eq. (1.1) with R̃ as above, so that we have

\frac{4(n − 1)}{n − 2} Δg̃ ϕ + ζ(ϕ^{−κ} − 1) + n(n − 1)(ϕ − ϕ^{\frac{n+2}{n−2}} ) = 0 .  (3.4)

Set

α ≡ inf_{M} ϕ .

Suppose first that α is attained at some point p ∈ M, in that case the maximum principle shows that α cannot be smaller than one. Suppose next that α is not attained on M, then by [42, Theorem 3.5] there exists a sequence of points p_{k} ∈ M, p_{k} → p ∈ ∂M, such that ^{3}

ϕ(p_{k}) → α , lim inf_{k→∞} Δg̃ ϕ(p_{k}) ≥ 0 .

Evaluating eq. (3.4) at p_{k} and passing to the limit k → ∞ one obtains

ζ(α^{−κ} − 1) + n(n − 1)(α − α^{\frac{n+2}{n−2}} ) ≤ 0 .

It follows again that α cannot be smaller than one, so that we must have

inf_{M} ϕ ≥ 1 .

3Here lim inf is taken in ℝ ∪ {±∞}.

One similarly shows that

sup_{M} ϕ ≤ 1 ,

so that ϕ_{1} = ϕ_{2} follows. □

4. Regularity at the boundary of the solutions

4.1. Polyhomogeneous or smooth backgrounds. The result that follows establishes polyhomogeneity for a class of semilinear equations when the metric is polyhomogeneous and when a polyhomogeneous approximate solution can be found.

Theorem 4.1. Let g be a polyhomogeneous metric on M (cf. Definition 2.9) and suppose that u ∈ C_{2}^{loc}(M) is a solution to the scalar equation

Δg̃u + xX(u) + F (x, u) = 0,  (4.1)

where X is a vector field on M satisfying X −pDx ∈ A^{phg} ∩C_{0}^{β}(M) for some β > 0 and p ∈ ℝ. Let u_{0} ∈ A^{phg} ∩ C_{0}(M̄) and assume that there exists q ∈ ℝ such that

\frac{∂F}{∂u} (x, u_{0}) − q|Dx|_{g}^{2} ∈ C_{0}^{β}(M),
q < (\frac{n−1−p}{2})^{2} , μ_{+} > 0 ,

where

μ_{±} = \frac{n − 1 − p}{2} ± \sqrt{(\frac{n − 1 − p}{2})^{2} − q} .  (4.2)

Further assume that for all i we have (∂_{u}^{i}F)(x, u_{0}) ∈ C_{0}(M̄) ∩ A^{phg}, and that for all i, j and all multiindices μ the functions (x∂x)^{i}∂_{v}^{μ}∂_{u}^{j}F are continuous in a neighbourhood of the graph of u_{0} in M̄_{x_{0}} × ℝ. If

Δg̃u_{0} + xX(u_{0}) + F (x, u_{0}) ∈ C_{0}^{α_{0}}(M) , α_{0} > max(0, μ_{−}) ,
u − u_{0} ∈ C_{0}^{α}(M) , α > max(0, μ_{−}) ,

then

u ∈ A^{phg} ∩ C_{0}(M̄_{x_{0}}) .

Remarks:

((i)) The condition on the function F will be satisfied if e.g. F is a rational function of u with coefficients which are polyhomogeneous, appropriately bounded functions of x.

((ii)) Let L and M satisfy the conditions of Corollary 2.3 and consider the equation

Lu = F (y, u) ,  (4.3)

with F as considered above. Assumer further that the coefficients a_{α} of L as defined in eq. (1.3) are bounded polyhomogeneous functions. Then we can define a metric g and a vector field X on M so that equation (4.3) takes the form (4.1), with all the hypotheses of Theorem 4.1 being satisfied.

Proof: Decreasing α if necessary without loss of generality we may assume α < μ_{+}. By a scaling argument one easily shows that u − u_{0} ∈ C_{∞}^{min(α,α_{0})}(M). By Theorem 2.1 the linearized equation has a regularity interval (α_{−}, α_{+}) with α_{±} = μ_{±} given by (4.2), and the result follows from Proposition 1.4 and Theorem 1.1. □

Corollary 4.2. Let g be a polyhomogeneous metric on a compact manifold with boundary M, let 0 ≤ ζ ∈ A^{phg} ∩ C_{0}^{ϵ}(M), ϵ > 0. There exists a solution ϕ of (1.1) such that

ϕ − |Dx|g^{\frac{n−1}{2}} ∈ A^{phg} ∩ C_{0}^{ϵ}(M) .

The solution ϕ is unique in the class of uniformly bounded, uniformly bounded away from zero, locally C_{2} solutions of (1.1).

Proof: Existence and uniqueness follows from Theorem 3.2; regularity from Theorem 4.1 – the appropriate approximate solution is ϕ_{0} = |Dx|g^{\frac{n−1}{2}} . □

When g ∈ C_{∞}(M̄), the solution ϕ of (1.1) given by Corollary 4.2 will be polyhomogeneous rather than smooth–up–to–boundary in general, cf. Appendix A and also [5, 4].

4.2. Background metrics with finite differentiability at the boundary. In the remainder of this chapter we shall prove various results concerning the regularity at the boundary of the solution of the Lichnerowicz equation, when finite differentiability only of the metric is assumed. As intermediate steps in the proof of the final results we shall need to prove various weighted regularity results. Our next result shows tangential regularity of the solutions of (1.1), under hypotheses weaker than those of Theorem 4.1. It is convenient to introduce first the following space of metrics:

Definition 4.3. We shall say that a metric g is of M_{k+λ,m+μ}^{ϵ} class, 0 ≤ ϵ ≤ 1, k ≥ max(1, m), if g ∈ M_{k+λ}^{ϵ} (cf. Definition 2.1), g_{ij} ∈ C_{k+λ,m+μ}^{0}, and if for |γ| = 1 we have x∂_{y}^{γ}g ∈ C_{k−1+λ,min(k−1,m)+μ}^{ϵ} (cf. (2.4)).

It can be noted that for g ∈ M_{k+1+λ,m+μ}^{ϵ} we have Δg̃ ∈ OP_{C_{k+λ,min(k,m)+μ}^{0}}^{2}. If

g ∈ M_{k+2+λ,m+μ}^{ϵ}, then the functions ξ of eq. (1.5) and S of eq. (1.7) satisfy

S ∈ C_{k+λ,m+μ}^{ϵ} , ξ + n|Dx|_{g}^{2} ∈ C_{k+λ,m+μ}^{ϵ} .

Theorem 4.4 (Tangential regularity). Let g ∈ M_{k+2+λ,m+μ}^{ϵ}, 0 ≤ ζ ∈ C_{k+λ,m+μ}^{ϵ}, ϵ ∈ (0, 1], λ ∈ (0, 1), μ ∈ [0, 1], 1 ≤ m ≤ k + 2. Then the solution ϕ of (1.1) given by Theorem 3.2 satisfies

ϕ − |Dx|_{g}^{(n−2)/2} ∈ C_{k+2+λ,m+μ′}^{ϵ}(M) ,

with μ′ = 0 if μ = 0, while if μ > 0 then μ′ is an arbitrary number in [0, μ).

Proof: The proof is a bootstrap-commutation argument similar to that of Theorem 1.3. For 1 ≤ i ≤ m let X_{i} = X_{i}^{A}(v) \frac{∂}{∂v^{A}}, X_{i}^{A} ∈ C_{∞}(∂M) be smooth vector fields on M_{x_{0}}. (1.7) implies that X_{i}(u) satisfies the equation

L̄X_{i}(u) ≡ [Δg̃ + ξ − \frac{∂F}{∂u} ] X_{i}(u) = ρ_{i} ,
ρ_{i} ≡ X_{i}(S) − X_{i}(ξ)u + [Δg̃, X_{i}]u + X_{i}^{A} \frac{∂F}{∂v^{A}} .

From u ∈ C_{k+2+λ}^{ϵ} it follows that

ξ − \frac{∂F}{∂u} + n|Dx|_{g}^{2} ∈ C_{k+λ}^{ϵ} , ρ_{i} ∈ C_{k−1+λ}^{ϵ} .

By Theorem 2.1 (−1, n) is a regularity interval for C_{0+λ}^{α} for L̄, and from X_{i}(u) ∈ C_{k+1+λ}^{ϵ−1} and from scaling estimates one obtains X_{i}(u) ∈ C_{k+1+λ}^{ϵ}, thus u ∈ C_{k+2+λ,1}^{ϵ}. For 0 ≤ μ ≤ 1 the inclusion C_{k+2+λ,1}^{ϵ} ⊂ C_{k+2+λ,0+μ}^{ϵ} implies u ∈ C_{k+2+λ,0+μ}^{ϵ}, which in turn yields

ξ − \frac{∂F}{∂u} + n|Dx|_{g}^{2} ∈ C_{k+λ,0+μ}^{ϵ} , ρ_{i} ∈ C_{k−1+λ,0+μ}^{ϵ} .

The claim that u ∈ C_{k+2+λ,1+μ}^{ϵ} follows by a difference quotient argument as in the proof of Theorem 1.3, because by Proposition 2.2 (−1, n) is a strong regularity interval for L̄. For m ≥ 1 the result follows by induction as in Theorem 1.3 from the equation

L̄X_{1} … X_{i}(u) = [L̄, X_{1} … X_{i−1}]X_{i}(u) + X_{1} … X_{i−1}ρ_{i} .
□

Our next result proves classical regularity at the boundary of solutions of semilinear equations under appropriate conditions – this is the main result of this chapter or, indeed, of this paper. To avoid a tedious but otherwise straightforward discussion of various possibilities we only consider α, μ_{+} ∈ ℕ, α ≤ μ_{+} and μ_{+} = α_{+} — these conditions hold in our applications to the general relativistic constraint equations. Similar results can be obtained without those restrictions using the same arguments. Let us emphasize that the following result holds for semilinear systems, and we are not assuming that (4.4) is a scalar equation.

Theorem 4.5 (Classical boundary regularity for semilinear systems). Let (α_{−}, α_{+}), α_{+} > 0, be a strong regularity interval for C_{0+λ}^{α}(M) for a geometric elliptic operator L ∈ OP_{C_{k+λ}(M̄)}^{2}, λ ∈ (0, 1). Let L be of the form (2.1)–(2.4) and assume that μ_{+} = α_{+}. Suppose that u ∈ C_{loc}^{2}(M), let S ∈ x^{α}C_{k+λ}(M̄) for some α > α_{−}, and assume further that

α, μ_{+} ∈ ℕ, k_{1} := μ_{+} − α ≥ 0 .

Let u ∈ C_{2}^{loc}(M), u = O(x^{α_{−}+ϵ}), α_{−} + ϵ ≥ 0, ϵ > 0, solve the semilinear system of equations

Lu = F (y, u) + S ,  F (y, 0) = ∂_{s}F (y, s)|_{s=0} = 0 .  (4.4)

Let N be the smallest integer such that Nα > k. Assume that F (y, s) ∈ C_{k+λ}(M̄ × ℝ) and that for all 0 ≤ i ≤ N +k +2 and for s ∈ ℝ satisfying |s| ≤ K ≡ sup_{M_{xo}} |u^{a}| there are constants C_{i} such that

||∂_{s}^{i}F (·, s)||_{C_{k+λ}(M̄)} ≤ C_{i}.  (4.5)

Then there exists σ > 0 such that ((i)) If α + k < μ_{+} then u ∈ ∩_{i=0}^{2}x^{α−i}C_{k+i+σ}(M̄_{x_{0}}).

((ii)) If α + k ≥ μ_{+} then
u ∈ ∩_{i=0}^{2}x^{α−i}C_{μ_{+}−α−1+i+σ|k−(μ_{+}−α−1)}(M̄_{x_{0}}) .

(If α = μ_{+}, the case i = 0 should be excluded in the equation above.)

Moreover for α + k ≥ μ_{+} + 1 there exist functions u_{i}, i = 1, …, N̂, such that u = ∑N̂ ; j=0 u_{j} log^{j}(x), with

u_{0} ∈ ∩_{i=0}^{2}x^{α−i}C_{k+i+σ}(M̄_{x_{0}}) ,  (4.6)

and for j ≥ 1,

u_{j} ∈ ∩_{i=0}^{k_{1}+2}x^{jμ_{+}−i}C_{k−k_{1}+i+λ}(M̄_{x_{0}}) .  (4.7)

Here N̂ is the smallest integer such that N̂ > (k + 2)/μ_{+}. If, finally, u_{1}|_{∂M} = 0, then u ∈ ∩_{i=0}^{2}x^{α−i}C_{k+i+σ}(M̄_{x_{0}}).

Remarks:

((i)) We have imposed the somewhat unnatural condition (4.5) on F because it is satisfied by our problem at hand, namely the general relativistic constraint equations. A similar but somewhat worse result as far as differentiability of the functions u_{i} is concerned can be established by the same methods assuming only that F (y, s) ∈ C_{k+λ}(M̄ × ℝ).

((ii)) For ℕ ∋ α ≥ μ_{+} and k ≥ 1, and assuming that the remaining hypotheses of Theorem 4.5 remain unchanged, we obtain u ∈ ∩_{i=0}^{2}x^{μ_{+}−i}C_{k+i+σ}(M̄_{x_{0}}). Similarly, for ℕ ∌ α ≥ μ_{+} and k ≥ 1 the regularity of the solution of the non–linear problem will be the same as the for the solution of the linear problem with the corresponding value of α, cf. Theorem 2.9 and the remarks following that theorem. Clearly similar results can be obtained under the hypotheses of Theorem 2.11.

Proof: In this proof, σ ∈ (0, 1) will be a constant which may vary from line to line. Let F_{m}(x) = ∂_{s}^{m}F (x, s)|_{s=0}. From Taylor’s Theorem we obtain

F (y, s) = ∑_{m=2}^{N} \frac{F_{m}(y)}{m!}s^{m} + s^{N+1}F_{R}(y, s) ,  (4.8)

with F_{m} ∈ C_{k+λ}(M̄) and F_{R} ∈ C_{k+λ}(M̄ × ℝ). Clearly, if v ∈ C_{k+λ,ℓ+σ}^{β}(M) for

ℓ + σ ≤ k + λ, then v^{N+1}F_{R}(x, v) ∈ C_{k+λ,ℓ+σ}^{(N+1)β}.

Let v be a function of the form given in point (ii) of the Theorem and satisfying (4.6)–(4.7) with σ = λ. Then from (4.8) we get

F (y, v) = ∑_{j=0}^{N} F̃_{j}(y) log^{j}(x) + ξ .

From (4.6) with i = 0 and (4.7) with i = k_{1} we obtain ξ ∈ C_{k+λ,k+λ}^{k+α+λ}(M) and F̃_{0} ∈ x^{2α}C_{k+λ}(M̄). Similarly the choice i = 0 in (4.7) gives F̃_{j} ∈ x^{jμ_{+}}C_{k−k_{1}+λ}(M̄) for j ≥ 1. Expressions of the form F (y, v_{1} + v_{2}) − F (y, v_{1}) can be analyzed in a similar way. Finally we note that if v_{1} is of the form given in point (ii) and v_{2} ∈ C_{k+λ,ℓ+σ}^{β} with ℓ + σ ≤ k + λ, then

F (y, v_{1} + v_{2}) − F (y, v_{1}) ∈ C_{k+λ,ℓ+σ}^{α+β} ,  (4.9)

which is easily seen by the mean value Theorem.

After those preliminary remarks, let us by induction construct an approximate solution û of eq. (4.4). Set w_{−1} = 0 and let w_{0} ∈ ∩_{i∈ℕ_{0}}x^{α−i}C_{k+i+λ}(M̄) be given by Lemma 2.5 so that Lw_{0} − S ∈ C_{k+λ,0+λ}^{k+α,β_{0}} ∩ C_{k+α+λ}^{k+α+λ,β_{0}}, β_{0} = 0 or β_{0} = 1. By interpolation for β ∈ ℕ_{0} we have C_{k+λ,0+λ}^{k+α,β}(M) ∩ C_{k+λ}^{k+α+λ,β}(M) ↪ C_{k+σ,0+σ}^{k+α+σ}(M) for some σ > 0.

Suppose then that for some m ≤ N we have found functions w_{i}, i = −1, …, m, of the form

w_{i} = ∑_{j=0}^{N̂} w_{i,j} log^{j}(x) ,  (4.10)

with coefficients w_{i,j} satisfying (4.6)–(4.7). We shall moreover require that w_{i} − w_{i−1} ∈ C_{k+2+λ}^{(i+1)α}. Then a function w_{m+1} of the form (4.10) with w_{m+1} − w_{m} ∈ C_{k+2+λ}^{(m+2)α} can be constructed as a solution of the equation

L(w_{m+1} − w_{m}) − (F (y, w_{m}) − F (y, w_{m−1})) ∈ C_{k+σ,0+σ}^{k+α+σ}(M) ,

by using the argument of the proof of Lemma 2.5. This completes the induction. Note that for α + k < μ_{+} we will have w_{i,j} ≡ 0 for j > 1.

Letting û ≡ w_{N}, we have

ξ := F (y, û) + S − Lû ∈ C_{k+σ,0+σ}^{k+α+σ} .  (4.11)

Setting ũ = u − û we obtain

L(ũ) = F (y, ũ + û) − F (y, û) + ξ ,  ξ ∈ C_{k+σ,0+σ}^{k+α+σ} .  (4.12)

We have u ∈ C_{0}^{α_{−}+ϵ}, so that decreasing ϵ if necessary we obtain ũ ∈ C_{0}^{α_{−}+ϵ}. Equations (4.9) and (4.12) give Lũ ∈ C_{0}^{α}. By point (i) of Lemma 1.1 and the weighted Sobolev embedding we obtain ũ ∈ C_{1+λ}^{α_{−}+ϵ}. Iterating this argument using point (ii) of Lemma 1.1 and (4.9) gives ũ ∈ C_{k+2+λ}^{α_{−}+ϵ}. We have:

Lemma 4.6. For 0 ≤ ℓ ≤ k and γ < min(k − ℓ + α + σ, μ_{+}) we have

ũ ∈ C_{k+2+λ,ℓ+σ}^{γ} .

Proof: The argument below is rather similar to that of Lemma 1.3. Some care, however, must be taken because of the finite degree of differentiability of the functions here. Let ℓ ≥ 0, choose β so that 0 < β ≤ α_{−} + ϵ, β < μ_{+} and 2β ≤ α, and let σ ∈ (0, 1) satisfy ℓ + σ < k + λ. Let θ ∈ (0, β) and K ∈ ℕ be chosen so that Kθ = ℓ + σ. Let ϵ_{κ}, κ = 0, …, K be any increasing sequence of numbers such that ϵ_{0} = 0, and ϵ_{κ} < σ. Note that for 0 ≤ ℓ ≤ k eq. (4.11) implies that

ξ ∈ C_{k+λ,ℓ+σ}^{k−ℓ+α+σ} .

We will argue by induction. Assume that ũ ∈ C_{k+λ,κθ−ϵ_{κ}}^{β}; we have shown that this holds with κ = 0. Then (4.9) and (4.12) imply

Lũ ∈ C_{k+λ,κθ−ϵ_{κ}}^{2β} ∩ C_{k+λ,κθ+1−ϵ_{κ}}^{2β−1} .

Lemma 2.2 gives

Lũ ∈ C_{k+λ,κθ+1−τ−ϵ_{κ+1}}^{2β−(1−τ)} ,

for any τ ∈ (0, 1). Choosing τ = 1 − θ we get

Lũ ∈ C_{k+λ,(κ+1)θ−ϵ_{κ+1}}^{β} ,

and, increasing slightly ϵ_{k+1}, Theorem 1.3 gives

ũ ∈ C_{k+2+λ,(κ+1)θ−ϵ_{κ+1}}^{β} .

This completes the induction step.

We have thus shown that ũ ∈ C_{k+2+λ,ℓ+σ}^{β} with some β > 0. An iterative argument based on (4.9), (4.12) and Theorem 1.3 completes the proof. □

Returning to the proof of Theorem 4.5, suppose first that α + k < μ_{+}. In this case Lemma 4.6 with ℓ = 0 gives ũ ∈ C_{k+2+σ,0+σ}^{k+α+σ}(M_{x_{0}}), and the result follows from Proposition 2.2.

In the case α+k ≥ μ_{+} some more work is needed. If k = μ_{+} −α our conclusion follows again from Lemma 4.6 and Proposition 2.2. (Note that in this case our result is weaker than for k > μ_{+} − α, cf. point (ii).) For k > 0 set f̃ = L(ũ) and rewrite equation (4.12) in the form (2.46). Then arguing as in the proof of Theorem 2.9 we find, using Lemma 2.7, a function ψ̂ ∈ ∩_{i∈ℕ_{0}}x^{μ_{+}−i}C_{k+i+λ}(M̄) with Lψ̂ ∈ C_{k+σ,0+σ}^{k+μ_{+}+σ}(M) and ũ ≡ u − û − ψ̂ = o(x^{μ_{+}}). Now we can change w_{0} as defined above to w_{0} + ψ̂ and repeat the induction argument described previously to obtain a new approximate solution, still denoted û, of the form (4.10) and satisfying (4.11). A direct analysis of the identity (2.47) of the proof of Theorem 2.9 gives ũ ≡ u − û ∈ C_{k+2+σ,k−(μ_{+}−α)+σ}^{μ_{+}+σ}(M_{x_{0}}). We wish to show that in fact we must have ũ ∈ C_{k+2+σ,0+σ}^{k+α+σ}(M_{x_{0}}). For suppose first that k ≤ μ_{+}. In this case Lũ ∈ C_{k+σ,0+σ}^{k+α+σ}(M) by (4.9), and Lemma 2.8 with β = α, m = ℓ = k yields ũ ∈ C_{k+2+σ,0+σ}^{k+α+σ}(M_{x_{0}}). To handle the case k > μ_{+}, suppose that ũ ∈ C_{k+2+σ,ℓ+σ}^{β+σ}(M_{x_{0}}) with some β ∈ ℕ satisfying β < k + α and some ℓ ≥ m := min(k + α − β, α). Then (4.9) and Lemma 2.8 yield ũ ∈ C_{k+2+σ,ℓ−m+σ}^{β+m+σ}(M_{x_{0}}). Applying this argument repeatedly leads indeed after a finite number of steps to ũ ∈ C_{k+2+σ,0+σ}^{k+α+σ}(M_{x_{0}}). The proof of point (ii) is completed by Proposition 2.2. □

The classical regularity up–to–boundary of solutions of the Lichnerowicz equation up to the threshold n − 1 = dim M − 1 is a straightforward corollary of the above:

Theorem 4.7 (Boundary regularity for solutions of the Lichnerowicz equation). Let M be an n–dimensional manifold with boundary with M̄ – compact. Let g ∈ C_{k+2+λ}(M̄), suppose that ζ ≥ 0 and suppose moreover that ζ ∈ C_{k+2+λ}(M̄) ∩ C_{0}^{1}(M), or ζ ∈ x C_{k+1+λ}(M̄), or ζ ∈ x^{2}C_{k+λ}(M̄), λ ∈ (0, 1), k ∈ ℕ_{0}. There exists σ ∈ (0, 1) such that the solution ϕ of (1.1) given by Theorem 3.2 satisfies

k + 2 ≤ n − 1 ⟹ ϕ ∈ C_{k+2+σ}(M̄) ,
k + 2 = n + k_{1}, k_{1} ∈ ℕ_{0} ⟹ ϕ ∈ C_{n−1+σ|k_{1}+1}(M̄) .

Moreover in the case k+2 > n there exist functions ϕ_{j} ∈ ∩_{i=0}^{n}x^{jn−i}C_{k+2−n+i+λ}(M̄), j = 1, …, N, such that

ϕ − ∑_{i=1}^{N} ϕ_{i} log^{i} x ∈ C_{k+2+σ}(M̄) .

Here N is the smallest integer such that N > (k + 2)/n. If finally ϕ_{1}|_{∂M} = 0, then ϕ ∈ C_{k+2+σ}(M̄).

Remark: The results of Theorem 4.7 are optimal (except perhaps for the modulus of Hölder continuity σ which we leave unspecified), as the x^{n} log x term generically arises in the solution of (1.1), cf. Appendix A. The reader is also

92 7. THE LICHNEROWICZ EQUATION.

referred to [6, 5] for an analysis under which conditions on the geometry it holds that ϕ_{1}|_{∂M} = 0; in these references dim M = 3 is assumed.

APPENDIX A Genericity of log-terms.

In this section we shall show that for generic “background fields”, in a sense to be made precise in the statements of Proposition 1.1 and Theorems 2.2, 2.3, the Cauchy data constructed by the conformal method will “pick-up” log-terms, even though the background fields are smooth up to boundary. The results of this section are somewhat related to those of [5]. In [5] rather more geometric results can be found; on the other hand, the proofs and the discussion here are much simpler.

To avoid a tedious differentiability–chase we have stated all the results in a C_{∞} setting. However, the calculations presented here carry immediately over to the finite differentiability case. Some of the equations below can be somewhat simplified if one makes use of the “almost Gaussian coordinates” of Appendix B.

Throughout this section M is a manifold with boundary with M̄ – compact, and with dim M ≥ 3. In some results we shall specialize to the physically relevant case n = 3, but this condition is not assumed unless specified otherwise. The reader is referred to Chapter 3 for any notation which is not defined here.

Let us start with the vector constraint equation. As explained at the beginning of Chapter 6, there are at least two methods of constructing solutions to this equation. Here we shall analyze in detail the method described in section 2; assuming smoothness of the background fields the method of section 3 gives equivalent results. Note, however, that the discussion of decay rates of various fields of section 2 has been tailored to fit the construction of section 3. This explains the difference between the decay rates of section 2 and those of section 1 below.

1. The vector constraint equation

Proposition 1.1. Let g ∈ C_{∞}(M̄), denote by i_{∂M} the embedding i_{∂M} : ∂M → M̄, and by Γ_{C_{∞}(∂M)}(i_{∂M}^{∗}T^{2}M̄) the space of smooth sections of the bundle i_{∂M}^{∗}T^{2}M̄ over ∂M obtained by pull-back by i_{∂M} of the bundle T^{2}M̄ of two-contravariant tensors over M̄, suppose that α = −2 or α = −1. For A_{ij} ∈ C_{∞}(M̄) let X be any solution of the equation

D_{i} L^{ij} ≡ \frac{1}{2}D_{i} (D^{i}X^{j} + D^{i}X^{i} − \frac{2}{n} D_{k}X^{k} g^{ij} + x^{α} A^{ij}) = 0  (1.1)

given by Theorem 2.8, point 3 thus

L^{ij} = x^{α} L_{α}^{ij} + log x L_{log}^{ij} , L_{α}^{ij}, L_{log}^{ij} ∈ C_{∞}(M̄) .  (1.2)

((i)) Let α = −1. There exists a closed subspace A ⊂ Γ_{C_{∞}(∂M)}(i_{∂M}^{∗}T M̄) of finite dimension not larger than (dim ∂M +1)(dim ∂M +2)/2 = n(n+ 1)/2 such that for all A^{ij} satisfying D_{i}x(|Dx|_{g}^{2}A^{ij}−Dx_{k}Dx^{j}A^{ik})(0, v) ∉ A we have

L_{log}^{ij}(0, v) ≢ 0 .

((ii)) Let α = −2. For any A^{∘} ∈ Γ_{C_{∞}(∂M̄)}(i_{∂M}^{∗}T^{2} M) and ψ ∈ C_{∞}(∂M̄) there exists a closed subspace A_{A^{∘} ,ψ} ⊂ Γ_{C_{∞}(∂M)}(i_{∂M}^{∗}T M̄), of finite dimension not larger than (dim ∂M + 1)(dim ∂M + 2)/2, such that for A^{ij} satisfying A^{ij}(0, v) = A^{∘} ^{ij}(v), D_{i}x D_{j}x\frac{∂A^{ij}}{∂x} (0, v) = ψ and D_{i}x(|Dx|_{g}^{2} \frac{∂A^{ij}}{∂x} −

Dx_{k}Dx^{j} \frac{∂A^{ik}}{∂x} )(0, v) ∉ A_{A^{∘} ,ψ} we have

L_{log}^{ij}(0, v) ≢ 0 .

Moreover for all A_{ij} for which a solution with L_{log}^{ij}(0, v) ≢ 0 exists, there exists no solution of (1.1) satisfying

X ∈ x^{α+1} (C_{∞}(M̄) + x log x C_{∞}(M̄)) ,

such that x^{−α} L^{ij} ∈ C_{−α}(M̄).

Proof: Uniqueness up to the addition of a smooth (up to boundary) solution of the homogeneous equation follows from Theorem 2.8, thus the only thing to show is the non-vanishing of L_{log}^{ij}. Let thus

α = −1  X = X_{0} + log xX_{log} ,  (1.3)
α = −2  X = x^{−1} X_{−1} + log xX_{log} ,  (1.4)

X_{0}, X_{−1}, X_{log} ∈ C_{∞}(M̄). X solves the equation

Δ_{L,g} X^{j} ≡ \frac{1}{2}D_{i} (D^{i}X^{j} + D^{j}X^{i} − \frac{2}{n} D_{k}X^{k} g^{ij})
= x^{α−1} Z^{j} ,  (1.5)

with

Z^{j} = −x^{−α+1} D_{i}(x^{α} A^{ij}) = − (αA^{xj} + xD_{i} A^{ij}) .  (1.6)

Inserting (1.3) in (1.5) one finds, with m = |Dx|_{g}^{−1} Dx (thus m is the inward pointing unit normal to ∂M),

α = −1  Δ_{L,g} X^{j} = −\frac{|Dx|_{g}^{2}}{2x^{2}} (X_{log}^{j} + \frac{n − 2}{n} g(m, X_{log}) m^{j}) + O (x^{−1}) ,
α = −2  Δ_{L,g} X^{j} = \frac{|Dx|_{g}^{2}}{x^{3}} \{X_{−1}^{j} + \frac{n − 2}{n} g(m, X_{−1}) m^{j}\} (0, v)

−\frac{|Dx|_{g}^{2}}{2x^{2}} \{X_{log}^{j} + \frac{n − 2}{n} g(m, X_{log}) m^{j} + L_{1}(X_{−1})\} + O(x^{−1}) ,

where L_{1} is a homogeneous linear first order differential operator which has the property that L_{1}(X_{−1})(0, v) does not depend upon ∂_{x}X_{−1}(0, v). For α = −1 we thus must have (cf. the beginning of Chapter 3 for conventions on coordinate systems)

(X_{log}^{j} + \frac{n − 2}{n} g(m, X_{log}) m^{j}) (0, v) = − (|Dx|_{g}^{−2} A^{xj}) (0, v) ,

which can be algebraically solved for X_{log}^{j}(0, v) in terms of A^{xj}(0, v), and which shows that X_{log}^{j}(0, v) runs over Γ_{C_{∞}(∂M)}(i_{∂M}^{∗}T M̄) as A^{xj}(0, v) does. For α = −2 one obtains

(X_{−1}^{j} + \frac{n − 2}{n} g(m, X_{−1}) m^{j}) (0, v) = (|Dx|_{g}^{−2} A^{xj}) (0, v) ,  (1.7)
(X_{log}^{j} + \frac{n − 2}{n} g(m, X) m^{j}) (0, v) = − (|Dx|_{g}^{−2} \frac{∂A^{xj}}{∂x} ) (0, v) + F , (1.8)
F = (|Dx|_{g}^{−2} [D_{i} A^{ij} − ∂_{x} A^{xj}]) (0, v) − L_{1}(X_{−1})(0, v) .  (1.9)

From (1.7) one can algebraically determine X_{−1}^{j}(0, v) in terms of A^{xj}(0, v) and since

(D_{i} A^{ij} − \frac{∂A^{ij}}{∂x} ) (0, v) depends upon A^{ij}(0, v) and its tangential derivatives only,

one finds that F in (1.8) is uniquely determined by A^{ij}(0, v); (1.8) implies then that for any fixed A^{ij}(0, v) the vector field X_{log}^{j}(0, v) runs over Γ_{C_{∞}(∂M)}(i_{∂M}^{∗}T M̄)

as \frac{∂A^{xj}}{∂x} (0, v) does. Now a simple calculation shows that

2L_{log}^{ij} ≡ D^{i} X_{log}^{j} + D^{j} X_{log}^{i} − \frac{2}{n} D_{k} X_{log}^{k} g^{ij} ,

and to finish the proof we have to show that the vanishing of L_{log}^{ij}(0, v) implies that X_{log}^{j} is of rather special form. Let h_{ij} be the induced metric on ∂M,

h_{ij} = g_{ij} − m_{i}m_{j} ,  (1.10)

let D_{i} denote the covariant Riemannian derivative operator of the metric h_{ij}, set β_{i} = m^{j} D_{j} m_{i}. Setting

ϕ = X_{log}^{i} m_{i} ,  Y^{i} = X_{log}^{i} − ϕm^{i} ,  (1.11)

one finds

2m_{i}m_{j} L_{log}^{ij} = \frac{2}{n} [(n − 1) (m^{k}D_{k}ϕ − Y^{k}β_{k}) − D_{k}Y^{k} + ϕλ] ,  (1.12)

2h_{k}^{i}m_{j} L_{log}^{kj} = D_{m}Y^{i} + D^{i}ϕ + λ^{i}_{k} Y^{k} + ϕβ^{i} ,  (1.13)

2(h_{k}^{i}h_{ℓ}^{j}L_{log}^{kℓ} + \frac{1}{n − 1}m_{k}m_{ℓ} L_{log}^{kℓ}h^{ij}) = D^{i}Y^{j} + D^{j}Y^{i} − \frac{2}{n − 1} D_{k}Y^{k}h^{ij}

−2ϕ (λ^{ij} − \frac{1}{n − 1} λh^{ij}) ,  (1.14)

where λ_{ij} is the extrinsic curvature of ∂M (λ_{ij} = −h_{i}^{k}h_{j}^{ℓ}D_{k}m_{ℓ}), λ = h^{ij} λ_{ij}, and

D_{m}Y^{i} = h_{j}^{i} m^{k} D_{k} Y^{j} .

For dim ∂M ≥ 3 eq. (1.14) shows that the vanishing of L_{log}^{kℓ}h_{k}^{i}h_{ℓ}^{j} leads to an overdetermined system of equations for X_{log}^{i}−m_{k}X_{log}^{k}m^{i}, and a straightforward repetition of the argument which shows that the space of conformal Killing vectors cannot exceed (dim ∂M + 1)(dim ∂M + 2)/2 yields the same upper bound for the dimension of the space of Y ’s for which we have L_{log}^{ij} = 0. In the physically relevant case dim ∂M = 2, the operator

Y^{i} ⟶ D^{i}Y^{j} + D^{j}Y^{i} − \frac{2}{n − 1} D_{k}Y^{k}h^{ij}

(which maps vectors to symmetric traceless tensors) is elliptic, with (as is well known)

((i)) trivial kernel if g(∂M) > 1, where g(∂M) is the genus of ∂M; ((ii)) two dimensional kernel if ∂M ≈ S^{1} × S^{1}; ((iii)) six dimensional kernel if ∂M ≈ S^{2}.

It follows again that the condition L_{log}^{kℓ} = 0 will hold for at most a (dim ∂M + 1)(dim ∂M + 2)/2 dimensional space of Y ’s, and the result follows. □

For the proof of Theorem 2.2 we shall need the following proposition:

Proposition 1.2. Let m_{i} be the unit normal to ∂M, define T ⊂ Γ_{C_{∞}(∂M)} (i_{∂M}^{∗}T^{2}M̄) as the space of tensors A^{ij}(v) over ∂M satisfying m_{i}A^{ij} = g_{ij}A^{ij} = 0, let P be the projection on T : (P A)^{ij} = h^{i}_{k}h^{j}_{ℓ} (A^{kℓ} − h^{mn}A_{mn}h^{kℓ}/(n − 1)). (Here h is

the metric induced by g on ∂M,cf. eq. (1.10).) For any A^{∘} , Â ∈ T there exists a transverse traceless tensor field L^{ij} ∈ x^{−2}C_{∞}(M̄) + ln x C_{∞}(M̄) such that

L^{ij}(x, v) = A^{∘} (v)x^{−2} + L_{−1}^{ij}(v)x^{−1} + O(ln x) ,  (1.15)
(P L_{−1})^{ij} = Â^{ij} .  (1.16)

Remarks:

((i)) Note that for a transverse traceless tensor of the form (1.15) the con-

dition A^{∘} (v) ∈ T is necessary. ((ii)) The question of existence of transverse traceless tensors L^{ij} ∈ x^{−α} C_{∞}(M̄), α = 1, 2, such that x^{α}L^{ij}|_{∂M} ≢ 0 is considered in [5], and is shown to be related to the vanishing of the space–time Weyl tensor at ∂M.

Proof: Let A_{ij} ∈ C_{∞}(M̄), we have

D_{i}(x^{−2}A^{ij})(x, v) = −2x^{−3}A^{xj}(0, v) + x^{−2} [D_{i}A^{ij}(x, v) − 2x^{−1} (A^{xj}(x, v) − A^{xj}(0, v))] , (1.17)

where A^{xj} = D_{i}xA^{ij}. Let A^{ij}(0, v) = A^{∘} ^{ij}(v), so that A^{xj}(0, v) = 0. Taylor expanding there exist R^{j} ∈ C_{∞}(M̄), S^{j} ∈ C_{∞}(M̄) such that

A^{xj}(x, v) = ∂_{x}A^{xj}(0, v)x + \frac{1}{2!} ∂_{x}^{2}A^{xj}(0, v)x^{2} + x^{3}R^{j} ,
∂_{x}A^{xj}(x, v) = ∂_{x}A^{xj}(0, v) + ∂_{x}^{2}A^{xj}(0, v)x + x^{2}S^{j} .

Define

“ΓA” = D_{x}A^{xj} − ∂_{x}A^{xj} ;

(1.17) can be rewritten as

D_{i}(x^{−2}A^{ij})(x, v) = x^{−2} [−∂_{x}A^{xj}(0, v) + (“ΓA” + D_{A}A^{Aj}) (x, v)] + S^{j} − 2R^{j} .

If we choose A^{ij}(x, v) so that

∂_{x}A^{xj}(0, v) = (“Γ A^{∘} ” + D_{A}A^{∘} ^{Aj}) (0, v) ,
P ∂_{x}A^{ij}(0, v) = Â^{ij}(v) ,

we obtain

D_{i}(x^{−2}A^{ij}) ∈ x^{−1}C_{∞}(M̄) ,

therefore by Theorem 2.8 there exists a solution X ∈ C_{∞}(M̄) + x ln xC_{∞}(M̄) of equation (1.1). □

2. The coupled system

In the remainder of this section we shall restrict our considerations to n = dim M = 3. Before analysing the presence or absence of log-terms in the solutions of the Lichnerowicz equation, recall that we are mainly interested in conformal classes of “background fields” [(g, L)], where L is a T T -tensor, in which the pair (g, L) is identified with the pair (ψ^{2}g, ψ^{−5}L) for any positive function ψ ∈ C_{∞}(M̄): (g, L) and (ψ^{2}g, ψ^{−5}L) lead to the same solution of the constraint equations. We shall assume that L ∈ x^{−2}C_{∞}(M̄)+log xC_{∞}(M̄), as forced upon us by Proposition 1.1 in generic situations, it should be stressed that the log terms in L will not affect the conclusion of Theorem 2.2, because they do not affect the x^{3} log x terms in the expansion of solutions of the Lichnerowicz equation.

Lemma 2.1. Let x, x_{1} ∈ C_{∞}(M̄) be two defining functions for ∂M and let L^{ij} be a transverse traceless tensor for the metric g. Suppose that (g_{ij}, L^{ij}) =

(ψ^{4}g_{kj}^{1}, ψ^{−10}L_{1}^{ij}), g ∈ C_{∞}(M̄), L ∈ x^{−2}C_{∞}(M̄)+ln x C_{∞}(M̄), with some bounded

positive function ψ ∈ C_{3}(M̄), ψ uniformly bounded away from zero. Set

g̃_{ij} = x^{−2}g_{ij} , g̃_{ij}^{1} = x_{1}^{−2}g_{ij}^{1} , L̃^{ij} = x^{3}L^{ij} , L̃_{1}^{ij} = x_{1}^{3}L_{1}^{ij} .

Let ϕ, ϕ_{1} be uniformly bounded, locally C_{2} solutions of

8Δg̃ϕ − R̃ϕ + |L̃|2 ; g̃ϕ^{−7} − 6ϕ^{5} = 0

8Δg̃_{1}ϕ_{1} − R̃_{1}ϕ_{1} + |L̃_{1}|2 ; g̃_{1}ϕ_{1}^{−7} − 6ϕ_{1}^{5} = 0

ϕ has a x^{3} log x term in the asymptotic expansion at ∂M if and only if ϕ_{1} has one.

Proof: By the uniqueness part of Theorem 1.1 we have

ϕ_{1} = ψ (\frac{x}{x_{1}} )^{\frac{1}{2}} ϕ .  (2.1)

Since \frac{x}{x_{1}} is in C_{∞}(M̄), the result follows by comparing coefficients in an asymptotic expansion of both sides of (2.1). □

Theorem 2.2. Let X be the collection of pairs (g, L), where g ∈ C_{∞}(M̄) is a Riemannian metric and L^{ij} ∈ x^{−2}C_{∞}(M̄) + log x C_{∞}(M̄) is transverse traceless.

((i)) The subset X^{∘} of X consisting of pairs for which the log terms in the expansion of the solutions of the Lichnerowicz equation do not vanish is open and dense in the C_{∞}(M̄) topology on X.

((ii)) The set X̂ = X\X^{∘} is an infinite dimensional closed subspace of X.

Proof: Choose some defining function x for ∂M, by Lemma 2.1 the vanishing or not of the log terms does not depend upon this choice. Let ḡ_{ij} = |Dx|_{g}^{2}g_{ij}, in the metric ḡ_{ij} we have ḡ^{ij}x_{,i}x_{,j} = 1, set g̃_{ij} = x^{−2}ḡ_{ij}. Let ϕ be the unique solution of

8Δg̃ϕ − R̃ϕ + |L̃|2 ; g̃ϕ^{−7} − 6ϕ^{5} = 0  (2.2)

satisfying ϕ|_{∂M} = 1. By the results of Chapter 7 there exist ϕ_{1}, ϕ_{2}, ϕ_{log} ∈ C_{∞}(∂M) such that

ϕ − 1 − ϕ_{1}x − ϕ_{2}x^{2} − ϕ_{log}x^{3} log x ∈ C_{3}^{3+ϵ} , ϵ > 0 .  (2.3)

Let m̄^{i} = ḡ^{ij}x_{,j}, h̄_{ij} = ḡ_{ij} − m̄_{i}m̄_{j}, let λ̄_{ij} be the extrinsic curvature (in the metric ḡ_{ij}) of the sets x = const., λ̄ = h̄^{ij}λ̄_{ij}, define the expansion functions λ̄_{0}, etc., by

λ̄(x, v) = λ̄_{0}(v) + xλ̄_{1}(v) + x^{2}λ̄_{2}(v) + O(x^{3})  (2.4)
|Dx|_{g}^{−5} L^{ij}(x, v) = \frac{A^{∘} ^{ij}(v)}{x^{2}} + \frac{Â^{ij}(v)}{x} + O(ln x)  (2.5)
R(ḡ)(x, v) = R̄_{0}(v) + xR̄_{1}(v) + O(x^{2}) ,  (2.6)
ḡ^{xA}(x, v) = ḡ^{xA}(0, v) + x˙ḡ^{xA}(v) + O(x^{2}) ,  (2.7)

where R(ḡ) is the scalar curvature of the metric ḡ_{ij}. Inserting (2.3)–(2.7) in (2.2) one finds with the help of a simple REDUCE code

ϕ_{1} = λ̄_{0}/8 ,  (2.8)
ϕ_{2}(v) = \frac{1}{24} (ḡ_{ij}ḡ_{kℓ})(0, v)A^{∘} ^{ij}(v)A^{∘} ^{jℓ}(v) + \frac{1}{6} [λ̄_{1}(v) − R̄_{0}(v) + (D̄_{i}xD̄^{i}λ̄_{0})(0, v)]
−\frac{23}{384} λ̄_{0}^{2}(v) ,  (2.9)
ϕ_{log}(v) = −\frac{1}{16} (ḡ_{ij}ḡ_{kℓ})(0, v) [A^{∘} ^{ik}(v)Â^{jℓ}(v) + A^{∘} ^{ik}(v)A^{∘} ^{jℓ}(v)λ̄_{0}(v)] + ψ(v) , (2.10)
ψ(v) = \{\frac{1}{32} [R̄_{1} − 4λ̄_{2} + 5λ̄_{0}λ̄_{1} − λ̄_{0}R̄_{0} − Δ_{g}λ̄_{0} + ˙ ḡ^{xA}∂_{A}λ̄_{0}] − \frac{207}{4096}λ̄_{0}^{3}

+\frac{9}{256} λ̄_{0}D̄_{i}xD̄^{i}λ̄_{0} − \frac{3}{4} D̄_{i}xD̄^{i}ϕ_{2}\}(0, v) . (2.11)

It follows from (2.10)–(2.11) that the condition ϕ_{log} = 0 defines a closed subspace of the set of all (g, L), thus X^{∘} is an open subset.

Suppose that (g, L) ∈ X̂, thus ϕ_{log} = 0. By Proposition 1.2 A^{∘} ^{ij}(v) and B^{ij} = h̄_{k}^{i}h̄_{ℓ}^{j} (Â^{kℓ} −h̄^{mn}Â_{mn}h̄^{kℓ}) can assume arbitrary values satisfying A^{∘} ^{ij}m_{j} = B^{ij}m_{j} = h̄_{ij}A^{∘} ^{ij} = h̄_{ij}B^{ij} = 0, and one can thus find a sequence (g, L_{k}^{ij}) ⟶_{k→∞}(g, L^{ij})

such that ϕ_{log}(v) ≢ 0 by choosing A^{∘} _{k}^{ij} and B_{k}^{ij} suitably, and the theorem follows.□

In Theorem 2.2 it was essential that the space of T T -tensors considered included those for which (x^{2}L^{ij})|_{∂M} ≢ 0. When x^{2}L^{ij}|_{∂M} = 0 we have the following:

Theorem 2.3. Let g ∈ C_{∞}(M̄) be a Riemannian metric, and let L be a transverse traceless tensor with respect to g. Consider the sets X_{a}, a = 1, 2, 3 of pairs (g, L) such that

X_{1} :  L ∈ x^{−1}C_{∞}(M̄) + log xC_{∞}(M̄)
X_{2} :  L ∈ C_{∞}(M̄)
X_{3} :  L ≡ 0 .

((i)) The set X^{∘} _{a}, a = 1, 2, 3, of pairs (g, L) for which the solution of the Lichnerowicz equation has a x^{3} log x term is open and dense in X_{a}. ((ii)) In each case the sets X̂_{a} = X_{a}\X^{∘} _{a} are closed infinite dimensional

subsets of X_{a}.

Remark: Note that for (g, L) ∈ X̂_{a}, a = 2, 3, the solution ϕ of the Lichnerowicz equation is smooth. On the other hand for generic (g, L) ∈ X̂_{1}, ϕ is expected to be C_{4}(M̄) but not C_{5}(M̄) because of x^{5} log x terms coming from |L̃|2 ; g̃. The blow up of the fifth derivatives is in this case mild enough so that one gets ϕ ∈ H_{5}(M, g, dμ_{g}), xϕ^{−1/2} ∈ H_{6}(M, g, dμ_{g}).

Proof: In the notation of the proof of Theorem 2.2, let D̄_{i} be the Riemannian connection of the metric h̄_{ij} = ḡ_{ij} − m̄_{i}m̄_{j}. A 2+1 decomposition calculation gives (note that m̄^{i}D̄_{i}m̄^{j} = 0 because m̄_{i}dx^{i} = dx)

R̄ = 2m̄^{k}D̄_{k}λ̄ − λ̄^{ij}λ̄_{ij} − λ̄^{2} + R̄ ,  (2.12)

where R̄ is the curvature scalar of h̄_{AB}, so that if the coordinates v^{A} are chosen to satisfy ḡ^{xA} ≡ 0 one finds

R̄_{1} ≡ \frac{∂R̄}{∂x}|_{∂M} = m̄^{k}D̄_{k}R|_{∂M}
= (2m̄^{i}m̄^{j}D̄_{i}D̄_{j}λ̄ − 2λ̄^{ij}m̄^{k}D̄_{k}λ̄_{ij} − 2λ̄m̄^{k}D̄_{k}λ̄ + m̄^{k}D̄_{k}R̄) + 2λ^{ik}λ_{i}^{j}λ_{jk}|_{∂M}

and (2.10) gives

ϕ_{log}(v) = \frac{1}{32} [−2λ̄^{ij}m̄^{k}D̄_{k}λ̄_{ij} + λ̄_{0}m̄^{k}D̄_{k}λ̄] (0, v) + ψ̄(v) ,  (2.13)

Here ψ̄ depends only upon g_{ij}|_{∂M}, \frac{∂g_{ij}}{∂x} |_{∂M} and their derivatives in directions tan-

gent to ∂M. Suppose that (g, L) is such that ϕ_{log} = 0. If λ̄_{ij}|_{∂M} = 0, in an

arbitrary neighbourhood of g one can find a metric g′ such that λ̄′_{ij}|_{∂M} ≠ 0. An

appropriate small perturbation of \frac{∂^{2}g′_{ij}}{∂x^{2}} which does not change g′_{ij}|_{∂M} and \frac{∂g′_{ij}}{∂x} |_{∂M}

will then make ϕ_{log} non-zero, which shows density of X^{∘} _{a}. □

APPENDIX B

1. “Almost Gaussian” coordinates.

Consider a metric g ∈ C^{k+λ}(M̄); if k ≥ 2 Gauss coordinates near ∂M can be introduced. Namely, 1) there exists a finite cover {U_{i}}_{i=0}^{I} of ∂M and x_{0} > 0 such that ∪_{i=1}^{I}[0, x_{0}) × U_{i} is a neighbourhood of ∂M and 2) in local coordinates y = (x, v^{A}), x ∈ [0, x_{0}), v^{A} ∈ U_{i} we have

g^{ij} = g(dy^{i}, dy^{j}) ∈ C^{k−2+λ}(\overline{(0, x_{0}) × U_{i}}),  (1.1)
g^{xx} ≡ 1, g^{xA} ≡ 0.  (1.2)

The problem is, that the coordinates (x, v) above are obtained through a solution of the geodesic equation, which leads to the threshold k ≥ 2, and which for k < ∞ leads to the loss of differentiability of g in the new coordinate system, as emphasized in eq. (1.1). This loss of differentiability is quite annoying, as then the use of exact Gauss coordinates in various applications leads to stronger restrictions on the degree of differentiability of the metric. On the other hand, for many applications it is sufficient for (1.2) to hold only approximately near ∂M. The aim of this Appendix is to show that this can be achieved without losing differentiability of g^{ij}. More precisely, we have the following:

Proposition 1.1 (“Almost Gaussian Coordinates at ∂M”). Let 0 ≤ k ≤ ∞, λ ∈ [0, 1], consider a Riemannian metric g ∈ C^{k+λ}(M̄) on M̄, with ∂M — compact. There exists x_{0} > 0 and a finite cover {U_{i}}_{i=1}^{I} of ∂M together with coordinates (y^{i}) = (x, v^{A}) ∈ [0, x_{0}] × U_{i}, such that ∪_{i=1}^{I}[o, x_{0}] × U_{i} is a neighbourhood of ∂M, and such that

g(dy^{i}, dy^{j}) = g^{ij} ∈ C^{k+λ}(\overline{(0, x_{0}) × U_{i}}),  (1.3)
g(dx, dx) − 1 = o(x^{k}),  (1.4)
g(dx, dv^{A}) = o(x^{k}).  (1.5)

Remark: If λ > 0, (1.4) – (1.5) can be strengthened to g^{xx} − 1 = O(x^{k+λ}), g^{xA} = O(x^{k+λ}).

Proof: If k = ∞ we can use Gauss coordinates near ∂M, and the result follows. Suppose thus that k < ∞, let p ∈ ∂M, let O be any conditionally compact coordinate neighbourhood of ∂M in which O ∩ ∂M = {x = 0}, with g^{ij} ≡ g(dy^{i}, dy^{j}) ∈ C^{k+λ}(Ō). Passing to a subset of O if necessary without loss of generality we may assume O ≈ [0, x_{0}) × U, U ⊂ ∂M. Coordinate systems of this form will be called cylindrical. By Lemma 3.1 there exists x̄ ∈ C^{k+1+λ}(Ū) satisfying

\frac{∂x̄}{∂x}|_{x=0} = (g^{xx})^{−1/2}, x̄|_{x=0} = 0.

The implicit function theorem implies that there exists O_{1} ⊂ O such that (x̄, v) are coordinates on O_{1}, O_{1} being cylindrical in the coordinates (x̄, v). We also have ḡ^{xx}|_{x̄=0} = g(dx̄, dx̄)|_{x=0} = g^{xx}(\frac{∂x̄}{∂x})^{2}|_{x=0} = 1. Passing to the coordinates (x̄, v), dropping the “1” on O_{1} and dropping bars on x̄ and on ḡ^{xx} we thus have

g^{xx}|_{x=0} = 1.  (1.6)

Let ϕ, f^{A} ∈ C^{k+1+λ}(Ō), consider

x̄ = x + ϕ(x, v),  (1.7)
v̄^{A} = v^{A} + f^{A}(x, v),  (1.8)

hence

ḡ^{xx} = g^{xx} + 2g^{xi} \frac{∂ϕ}{∂x^{i}} + g^{ij} \frac{∂ϕ}{∂x^{i}} \frac{∂ϕ}{∂x^{j}} ,  (1.9)
ḡ^{xA} ≡ g(dx̄, dv̄^{A}) = g^{xA} + g^{xi} \frac{∂f^{A}}{∂x^{i}} + g^{Ai} \frac{∂ϕ}{∂x^{i}} + g^{ij} \frac{∂ϕ}{∂x^{i}} \frac{∂f^{A}}{∂x^{j}} .  (1.10)

Suppose that for some ℓ ≥ 0 we have

g^{xx} − 1 = o(x^{ℓ}), g^{xA} = o(x^{ℓ−1}) .  (1.11)

It follows from (1.6) that (1.11) holds with ℓ = 0. By Lemma 3.1 there exists f^{A} ∈ C^{k+1+λ}(Ō) satisfying

\frac{∂^{ℓ+1}f^{A}}{∂x^{ℓ+1}} |_{x=0} = −\frac{∂^{ℓ}g^{xA}}{∂x^{ℓ}} |_{x=0},

while all the lower order x–derivatives of f^{A} vanish at x = 0. Passing to coordinates (x, v̄) on a (possibly smaller) cylindrical neighbourhood O, and dropping bars, one finds from (1.9)–(1.10) that (1.10) still holds with, moreover

g^{xA} = o(x^{ℓ}),  (1.12)

so that for ℓ = k = 0 the proof is completed. If 0 < ℓ ≤ k by Lemma 3.1 there exists ϕ ∈ C^{k+1+λ}(Ω̄) such that

\frac{∂^{ℓ+1}ϕ}{∂x}|_{x=0} = −\frac{∂^{ℓ}g^{xA}}{∂x} |_{x=0},

with all the lower order x–derivatives of ϕ vanishing at x = 0. Passing to the coordinates (x̄, v) and dropping the bar on x̄ one finds that (1.11) holds with ℓ replaced by ℓ + 1, and the induction step is completed. □

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Index

(x∂_{y})^{α}, 18 A_{k+λ,m+μ}^{α,β}(Ω), 21 CM_{σ}, 17 C^{k}(Ω̄), 18 C^{k+λ}(Ω̄), 18 C_{∞}(Ω̄), 18 C_{k}(Ω), 18 C_{k}(Ω̄), 18 C_{k+λ,0+μ}^{α,β}(Ω), 20 C_{k+λ,m+μ}^{α}(Ω), 21

C_{k+λ,m+μ}^{α,β}(Ω), 21 C_{k+λ|m}(Ω̄), 20 C_{k+λ}(Ω), 18 C_{k+λ}(Ω̄), 18 C_{k+λ}^{α}(Ω), 19 C_{k+λ}^{α,β}(Ω), 19 C_{k+λ}^{loc}(Ω), 18 C_{k}^{α,β}(Ω), 19 D, 19 F_{1} + F_{2}, 18 F_{m}^{loc}(Ω), 34 H_{k}^{α,β}(Ω, g̃), 20 L^{p}(Ω, dμ), 19 L^{α,β;p}, 20 M^{phg}, 69 M_{σ}, 17 M_{σ,ρ}, 17 M_{k+λ,m+μ}^{ϵ}, 87 M_{k+λ}^{α}, 62 OP_{F}^{m}, 33 W_{k}^{α,β;p}(Ω, g), 20 W_{k}^{α,β;p}(Ω, g, dμ), 20 W_{k}^{α;p}, 20 W_{k}^{p,loc}(Ω, g, dμ), 19 W_{k}^{p}(Ω), 19 W_{k}^{p}(Ω, g, dμ), 19 Γ_{F} (B), 22 ℕ^{∞}, 18 ℕ_{0}, 18 ℕ_{0}^{∞}, 18 ⊂⊂, 17 Ω̄, 17 μ_{±}, 42 ∂Ω, 17 ∂^{α}, 18 ∂_{v}^{β}, 18 ∂_{y}^{α}, 18 ρF, 19 ∼, 22

Ω^{∘}, 17

supp(f), 18

D̃, 20 dμ_{g}, 19 i_{∂M}, 17 x^{∞}F, 19 x_{0}, 17 A^{phg}, 22 A^{{(s_{i},N_{ij})}_{i=0}^{I}}, 22 A^{{s_{i}}_{i=0}^{I}}, 22 C^{∘} _{k}(Ω), 18 C^{∘} _{k+λ}(Ω), 18

W^{∘} _{k}^{α,β;p}(Ω, g), 20

W^{∘} _{k}^{α,β;p}(Ω, g, dμ), 20

W^{∘} _{k}^{p}(Ω, g, dμ), 19

H^{∘} _{k}^{α,β}(Ω, g̃), 20

˜∂Ω, 17 “x^{∗}”, 47

Borel Lemma, 23

coordinate systems, 17

curvature tensors conventions, 9

defining function for ∂M, 17

difference quotients on a manifold, 34

elliptic on M_{x_{0}}, 33 elliptic operator, 33

function spaces of tensors, 22

geometric equation, 36 geometric operator, 36 indicial roots, 42

polyhomogeneous metric, 69

regularity interval, 34 Riemann tensor convention, 62 Riemannian metric on a manifold with bound-

ary, 62

scalar operator, 81 scaling estimates, 36 strong regularity interval

for weighted Sobolev spaces, 34 strong regularity interval, 34

for weighted Hölder spaces, 34

Taylor formula, 21

weighted Hölder embedding, 20