Symbol Calculus for Operators of Layer Potential Type on Lipschitz Surfaces with VMO Normals, and Related Pseudodifferential Operator Calculus

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1 Symbol Calculus for Operators of Layer Potential Type on Lipschitz Surfaces with VMO Normals, and Related Pseudodifferential Operator Calculus Steve Hofmann, Marius Mitrea, and Michael Taylor Steve Hofmann Department of Mathematics University of Missouri at Columbia Columbia, MO 652, USA Marius Mitrea Department of Mathematics University of Missouri at Columbia Columbia, MO 652, USA Michael Taylor Department of Mathematics University of North Carolina Chapel Hill, NC , USA Date: March, Mathematics Subject Classification. 3B0, 35J46, 35J57, 35S05, 35S5, 42B20, 42B37, 45B05, 58J05, 58J32. Key words and phrases: singular integral operator, compact operator, pseudodifferential operator, rough symbol, symbol calculus, single layer potential operator, strongly elliptic system, boundary value problem, nontangential maximal function, nontangential boundary trace, Dirichlet problem, Regularity problem, Poisson problem, Oblique derivative problem, regular elliptic boundary problem, elliptic first order system, Hodge-Dirac operator, Cauchy integral, Hardy spaces, Sobolev spaces, Besov spaces, Lipschitz domain, SKT domain. Work supported in part by grants from the US National Science Foundation, the Simons Foundation, and the University of Missouri.

2 Abstract We show that operators of layer potential type on surfaces that are locally graphs of Lipschitz functions with gradients in vmo are equal, modulo compacts, to pseudodifferential operators with rough symbols, for which a symbol calculus is available. We build further on the calculus of operators whose symbols have coefficients in L vmo, and apply these results to elliptic boundary problems on domains with such boundaries, which in turn we identify with the class of Lipschitz domains with normals in vmo. This work simultaneously extends and refines classical work of Fabes, Jodeit, and Rivière, and also work of Lewis, Salvaggi, and Sisto, in the context of C surfaces. Contents Introduction 3 2 From layer potential operators to pseudodifferential operators 6 2. General local compactness results The local compactness of the remainder A variable coefficient version of the local compactness theorem Symbol calculus 6 3. Principal symbols Transformations of operators under coordinate changes Admissible coordinate changes on a Lip vmo surface Remark on double layer potentials Cauchy integrals and their symbols Applications to elliptic boundary problems Single layers and boundary problems for elliptic systems Oblique derivative problems Regular boundary problems for first order elliptic systems Absolute and relative boundary conditions for the Hodge-Dirac operator A Auxiliary results 45 A. Spectral theory for the Dirichlet Laplacian A.2 Truncating singular integrals A.3 Background on OPL vmos 0 cl A.4 Analysis on spaces of homogeneous type A.5 On the class of Lip vmo domains

3 Introduction We produce a symbol calculus for a class of operators of layer potential type, of the form Kfx = PV kx, x yfy dσy, x Ω,.0. in the following setting. First, Ω k C R n+ R n+ \ 0,.0.2 with kx, z homogeneous of degree n in z and kx, z = kx, z. Next, Ω R n+ is a bounded Lipschitz domain, with a little extra regularity. Namely, Ω is locally the upper-graph of a function ϕ 0 : R n R satisfying ϕ 0 L R n vmor n..0.3 We say Ω is a Lip vmo domain. Since we will be dealing with a number of variants of BMO, we recall some definitions. First, where BMOR n := { f L loc Rn : f # L R n },.0.4 f # x := sup B Bx V B B fy f B dy,.0.5 with Bx := {B r x : 0 < r < }, B r x being the ball centered at x of radius r, and f B the mean value of f on B. There are variants, giving the same space. For example, one could use cubes containing x instead of balls centered at x, and one could replace f B in.0.5 by c B, chosen to minimize the integral. We set f BMO := f # L..0.6 This is not a norm, since c BMO = 0 if c is a constant; it is a seminorm. The space bmor n is defined by bmor n := { f L loc Rn : # f L R n },.0.7 where # fx := sup B B x V B B with B x := {B r x : 0 < r }. We set fy f B dy + V B x B x fy dy,.0.8 f bmo := # f L..0.9 This is a norm, and bmor n has good localization properties. Now, VMOR n is the closure in BMOR n of UCR n BMOR n, where UCR n is the space of uniformly continuous functions on R n, and vmor n is the closure in bmor n of UCR n bmor n. One can use local coordinates and partitions of unity to define bmom and vmom on a class of Riemannian manifolds M cf. [37]. See also Appendix A.3 of this paper for a discussion of BMOM and VMOM on spaces M of homogeneous type. We mention that if M is compact, BMOM coincides with bmom and VMOM coincides with vmom. With this in mind, we mention that Ω could be an open set in a compact n + -dimensional Riemannian manifold M, whose boundary, in local coordinates on M, is locally a graph as in 3

4 .0.3, and kx, x y in.0. could be the integral kernel of a pseudodifferential operator on M of order, with odd symbol. The analysis of operators of the form.0. as bounded operators on L p Ω for p,, together with nontangential maximal function estimates for Kfx = kx, x yfy dσy, x R n+ \ Ω,.0.0 Ω and nontangential convergence, was done for general Lipschitz domains in [5], carrying through the breakthrough initiated in [2], at least for k = kx y. In between [2] and [5] was another key paper, [9], which treated.0. again with k = kx y when Ω has a C boundary, and gave some applications to PDE. These applications involved looking at double layer potentials K d fx = PV νx x yex yfy dσy, x Ω,.0. Ω where νx is the unit normal to Ω, and Ez = c n z n+. Such an operator is of the form K d fx = νx Kfx, where K is as in.0., with kz = zez vector valued. In [9] it was shown that K d is compact when Ω is a bounded domain of class C. See 3.4 of this paper for a proof that K d is compact more generally when Ω is a bounded Lip vmo domain. This compactness was applied to the Dirichlet problem for the Laplace operator on bounded C domains. In fact, if K d fx = νx x yex yfy dσy, x Ω,.0.2 one has Ω K d f Ω = 2 I + K df,.0.3 so solving the Dirichlet problem u = 0 on Ω, u Ω = g, as u = K d f leads to solving 2 I + K df = g,.0.4 and the compactness of K d implies 2 I + K d is Fredholm, of index 0. For a general bounded Lipschitz domain Ω R n+, one continues to have , but K d is typically not compact. However, it was shown in [38] that 2 I + K d is still Fredholm, of index 0, using Rellich identities as a tool. This led to much work on other elliptic boundary problems, including boundary problems for the Stokes system, linear elasticity systems, and the Hodge Laplacian. In [26] a program was initiated that extended the study of.0. from k = kx y to k = kx, x y, a development that enabled the authors to work on Lipschitz domains in Riemannian manifolds. This led to a series of papers, including [27], and [22]. In these papers, variants of Rellich identities also played major roles. Meanwhile, [2] established compactness of K d in.0. when Ω R n+ is a bounded VMO domain, i.e., its boundary is locally a graph of a function ϕ 0 satisfying ϕ 0 VMOR n,.0.5 which is weaker than.0.3. This led [4] to establish compactness of a somewhat broader class of operators, not just on VMO domains, but more generally on a class of domains, introduced by 4

5 [3] and [7], called chord-arc domains with vanishing constant by those authors, but called regular SKT domains in [4]. This was applied in [4] to the Dirichlet boundary problem for the Laplace operator, on regular SKT domains in Riemannian manifolds, and also to a variety of boundary problems for other second order elliptic systems. In these works on various elliptic boundary problems, both on Lipschitz domains and on regular SKT domains, each elliptic system seemed to need a separate treatment. This is in striking contrast to the now standard theory of regular elliptic boundary problems on smoothly bounded domains, for operators with smooth coefficients. Such cases yield operators of the form.0. that are pseudodifferential operators on Ω, for which a symbol calculus is effective to power the analysis. One can, for example, see the treatment of regular elliptic boundary problems in [34, Chapter 7, 2]. Our goal here is to develop a symbol calculus for operators of the form.0. in Lip vmo domains, and to apply this symbol calculus to the analysis of some elliptic boundary problems. We work in local graph coordinates, in which.0. takes the form Kfx = PV kϕx, ϕx ϕyfyσy dy, x R n,.0.6 R n where ϕx = x, ϕ 0 x, with ϕ 0 : R n R as in.0.3. In fact, we allow ϕ 0 : R n R l. The surface area element dσy = Σy dy. Our first major result is that, with K # given by K # fx = PV kϕx, Dϕxx yfyσy dy, x R n,.0.7 R n we have K K # compact on L p B,.0.8 for p,, for any ball B R n. Then, as we show, K # f = px, DΣf, with px, D OPL vmos 0 cl Rn,.0.9 a class of pseudodifferential operators studied in [36] and shown to have a viable symbol calculus. Definitions and basic results are given in Appendix A.3 of this paper. The proof of.0.8, given in 2, makes essential use of results of [2] and further material in [4]. Since.0.6 and.0.7 are given in local graph coordinates, it is important to record how operators are related when represented in two different such coordinates, and how a symbol can be associated to such an operator, independent of the coordinate representation. These matters are handled in 3. In connection with this, we mention the work [8], providing such an analysis on C manifolds. In particular,.0.8 for ϕ C plays a central role there. In [8], the function kx, z is required to be analytic in z R n+ \ {0}. The need for such analyticity arises from technical issues, which we can overcome here, thanks to the advances in [2] and [4]. One desirable effect of not requiring such analyticity is that our results readily allow for microlocalization. Though we do not pursue microlocal analysis on boundaries of Lip vmo domains here, we are pleased to advertise the potential to pursue such analysis. The structure of the rest of this paper is as follows. Section 2 is devoted to a proof of the basic result.0.8. Section 3 builds on this to produce a symbol calculus, making essential use of results on operators of the form.0.9, recalled in an appendix. Section 4 applies these results to 5

6 some boundary problems for elliptic systems on Lip vmo domains. These include the Dirichlet problem for a general class of second order, strongly elliptic systems, and a class of oblique derivative problems. We also produce a general result on regular boundary problems for first order elliptic systems, and show how this plays out for the Hodge-Dirac operator d + δ, acting on differential forms. A set of appendices deals with auxiliary results. The first gives material used in 2.. The second gives a detailed analysis of just how a principal value integral like.0. works for such domains as we consider here. The third reviews material on the class of pseudodifferential operators.0.9. The fourth reviews matters related to BMOM and VMOM when M is a space of homogeneous type. The fifth proves that a bounded domain Ω R n+ is locally the upper-graph of a function satisfying.0.3 if and only if its outward unit normal belongs to VMO Ω. 2 From layer potential operators to pseudodifferential operators The primary goal of this section is to establish the compactness of the difference between a singular integral operator K of layer potential type, as in.0. and a related operator K #, which belongs to the class of pseudodifferential operators OPL vmoscl 0, a class that is reviewed in Appendix A.3. We proceed in stages. 2. General local compactness results Below, the principal value integrals PV are understood in the sense of removing small balls centered at the singularity and passing to the limit, by letting their radii approach zero; for a more flexible view on this topic see the discussion in A.2. We begin by recalling the following local compactness result. Theorem 2.. Assume ϕ : R n R and ψ : R n R m are two locally integrable functions satisfying and set ϕ vmor n, Dψ bmor n, 2.. Γx, y := ϕx ϕy ϕxx y, x, y R n Given F : R m R smooth of a sufficiently large order M = Mm, n N, even on R m, and such that F w C + w for every w R m, 2..3 and α F L R m whenever α M, 2..4 consider the principal value integral operator ψx ψy T fx := PV x y n+ F Γx, yfy dy, R n x y x R n, 2..5 and the associated maximal operator T fx := sup x y n+ F ε>0 y R n x y >ε ψx ψy x y 6 Γx, yfy dy, x Rn. 2..6

7 Then for each p, there exists C n,p 0, such that T f L p R n C n,p α [ ] F L R m + sup + w F w w R m α M N f L ϕ BMOR n + Dψ BMOR n p R n, 2..7 for every f L p R n. Also, with B R abbreviating B0, R := {x R n : x < R}, it follows that for each R 0, and p, the operator T : L p B R L p B R is compact This result is given in [4, Theorem 4.34, p. 2725, 4.4] and [4, Theorem 4.35, p. 2726, 4.4]. As noted there, the analysis behind it is from [2]. Of course, there is a natural analogue of Theorem 2. when the function ϕ is vector-valued implied by the scalar case, by working componentwise. Here, the the goal is to prove the following version of Theorem 2.. Theorem 2.2. Suppose ϕ : R n R and ψ : R n R m are two locally integrable functions satisfying ϕ vmor n, Dψ L R n, 2..9 and let the symbol Γx, y retain the same significance as in Given an even real-valued function F C M R k for a sufficiently large M N, along with some matrix-valued function A : R n R k m, A L R n, 2..0 consider the principal value singular integral operator T A fx := PV x y n+ ψx ψy F Ax Γx, yfy dy, R n x y x R n. 2.. Then for each R 0, and p, the operator T A : L p B R L p B R is compact Once again, there is a natural analogue of Theorem 2.2 when the function ϕ is vector-valued implied by the scalar case, by working componentwise. Proof of Theorem 2.2. Fix a finite number R > Dψ L R n 2..3 and abbreviate B := {w R m : w < R }. Also, select a real-valued function χ satisfying χ C R m, χ even in R m, supp χ B, χz = whenever z Dψ L R n To proceed, let {ϑ j } j N L 2 B denote an orthonormal basis of L 2 B consisting of real-valued eigenfunctions of the Dirichlet Laplacian in B as discussed in Appendix A.. For x R n, we can write in L 2 B and a.e. z B F Axz = j N b j xϑ j z

8 where, for each j N, we have set b j x := F Axzϑ j z dz, B x R n To estimate the b j s, fix j N, x R n, and observe that for each N N we may write λ N j bj x = F Axz B N ϑ j z dz = z N[ F Axz ] ϑ j z dz B { C N A 2N L R n sup w R A L R n α =2N α F w } ϑ j L B C A,F,R,Nj /2+2/n, 2..7 by A..9. In light of A..8 this ultimately shows that for each N N there exists a constant C N 0, such that b j L R n C N j N, j N Moving on, we note that combining 2..5 with its version written for z in place of z, and keeping in mind that F is even, yields F Axz = j N b j x ϑ j z 2..9 where, for each j N, we have set In particular, for each j N, ϑ j z := ϑ jz + ϑ j z, z B ϑ j C loc B is even, vanishes on B, and satisfies ϑ j = λ j ϑj in B Multiplying both sides of 2..9 with the cut-off function χ from 2..4 then finally yields χzf Axz = j N b j xf j z, x R n, z R m, where, for each j N, we have set F j z := χz ϑ j z, z R m, naturally viewed as zero outside B. Hence, for each j N, F j C R m is an even function supported in B,

9 and A.. implies that for every multi-index α N m 0 there exists a constant C m,α 0, such that α F j L R m C m,α j /2+2/n Since z = we deduce from that ψx ψy x y = z Dψ L R m = χz =, T A fx = j N b j xt j fx, where for each j N we have set ψx ψy T j fx := PV x y n+ F j Γx, yfy dy, R n x y x R n At this stage, Theorem 2. applies to each operator T j. In concert, estimates 2..7 and yield a polynomial bound in j N on the operator norms of T j on L p R n. Then, in the context of the expansion 2..27, the rapid decrease 2..8 implies the desired compactness on L p B R for T A, for each R 0, and p,. It is possible to prove Theorem 2.2 using the Fourier transform in place of spectral methods, based on Dirichlet eigenfunction decompositions. We shall do so below and, in the process, derive further information about the family of truncated operators indexed by ε > 0 T A,ε fx := {y R n : x y >ε} x y n+ F where x R n, including the pointwise a.e. integral operator. Ax ψx ψy x y Γx, yfy dy, existence of the associated principal value singular Theorem 2.3. For each ε > 0 let T A,ε be as in 2..29, where Γx, y is defined as in 2..2 for a function ϕ : R n R satisfying ϕ BMOR n, A L R n is a k m matrix-valued function, ψ : R n R m is Lipschitz, and F C M R k is even. Then, if M = Mm, n N is large enough, there is a positive M 0 < such that for < p <, sup T A,ε f L p R n ε>0 sup ε>0 TA,ε f L p R n C 0 + ψ L R n M0 ϕ BMOR n f L p R n, where the constant C 0 depends on A, p, n, m, k, and F C M B0, A R with Moreover, ϕ VMOR n = R := 2 ψ lim T A,εfx exists for a.e. x R n, f L p R n ε 0 + In fact, a more general result of this nature holds. Specifically, if B : R n R m is a bi-lipschitz function and if for each ε > 0 we set T A,B,ε fx := x y n+ ψx ψy F Ax Γx, yfy dy, x y {y R n : Bx By >ε} 9

10 where x R n, then ϕ VMOR n = lim T A,B,εfx exists for a.e. x R n, f L p R n ε 0 + We shall prove estimate by reducing it to the scalar valued case k = m =, with A, which is Theorem.0 in [2]. Note that given 2..30, for ϕ vmor n one then gets local compactness as in the statement of Theorem 2.2; cf of the associated principal value operator, by the usual methods. Proof of Theorem 2.3. For z R m, set F x z := F Ax z. Note that since A L, we have that F x C M, with sup j F x L B controlled uniformly in x, for every ball B R m j M Moreover, as before, we may suppose that F x is supported in the ball B0, R R m, for every x R n, where R is as in For notational convenience, we normalize F so that We may write sup j F L B0, A R = j M F x z = c Fx ξ cosz ξ dξ, R m where F x is the Fourier transform of F x, and we observe that by standard estimates for the Fourier transform, and our normalization of F from 2..37, ess sup x R n F x ξ C R m + ξ M Let η C 0 2, 2 be an even function, with η on [, ], and for ξ Rm, t R, set t E ξ t := cost η ξ R Observe that for z B0, R R m, we may replace cosz ξ by E ξ z ξ in In concert with and 2..38, this permits us to write T A,ε fx = x y n+ ψx ψy F Ax Γx, yfy dy {y R n : x y >ε} x y { } = c Fx ξ x y n+ ψx ψy E ξ ξ Γx, yfy dy dξ R m {y R n : x y >ε} x y = c + ξ M N Fx ξt ξ,ε fx dξ, 2..4 R m where T ξ,ε fx := {y R n : x y >ε} x y n+ Ẽ ξ ξ 0 ψx ψy Γx, yfy dy x y

11 and, with N a large number to be chosen later, Ẽ ξ t := + ξ N M E ξ t, t R In turn, from 2..4 and we deduce that sup TA,ε f L ε>0 p R n CRm + ξ N R m We now set sup ε>0 Tξ,ε f L dξ, p R n N := M 2, and note that this choice ensures that for all non-negative integers j, d j Ẽξ t dt C j + ξ 2 2 C j R 2 + t / + ξ R + t 2, where the constant C j may depend on j, but is independent of ξ. By [2, Theorem.0, p. 470], applied to the scalar-valued Lipschitz function ξ ψ, we then have that for some M <, sup T ξ,ε fx L p ε>0 x R n CR2 + ξ R M ϕ BMOR n f L p R n Plugging the latter estimate into 2..44, and finally choosing M := M + m + 3, we obtain thanks to Finally, there remains to consider the issue of the existence of the limits in and We treat in detail the former, since the argument for the latter is similar, granted our results in A.2. To justify 2..32, make the standing assumption that ϕ VMOR n, and recall from 2..4, that T A,ε fx = c + ξ 2 Fx ξt ξ,ε fx dξ, R m where T ξ,ε fx is as in To proceed, observe that for each f L p R n there holds + ξ 2 Fx ξt ξ,ε fx L ξ R m, for a.e. fixed x R n sup ε>0 To see that this is the case, use Minkowski s inequality along with and to estimate { + ξ 2 Fx ξt ξ,ε fx } /p p dξ dx R n R m sup ε>0 sup + ξ 2 Fx ξt ξ,ε fx L dξ 2..5 R m p ε>0 x R n [ + ξ 2 esssup x R n F x ξ ] sup Tξ,ε fx L dξ R m p ε>0 x R n CR m+2 ϕ BMOR n f L p R n + ξ 2 M + ξ R M dξ < +, R m

12 thanks to With 2..5 in hand, the claim in readily follows. Next, granted 2..48, we claim that for each fixed function f L p R n the following holds: for each fixed ξ R m, the limit lim ε 0 + T ξ,εfx exists for a.e. x R n Given that we have already established 2..30, this may be justified along the lines of the proof of Theorem 5., pp in [2], based on Proposition A.3 and keeping in mind that VMO functions may be approximated in the BMO norm by continuous functions with compact support which, in turn, are uniformly approximable by functions in C0. In concert with the uniform integrability property 2..50, the existence of the limit in makes it possible to use Lebesgue s Dominated Convergence Theorem in order to write that, for a.e. x R n, lim T A,εfx = c lim + ξ 2 Fx ξt ξ,ε fx dξ ε 0 + ε 0 + R m = c + ξ 2 Fx ξ lim T ξ,εfx dξ R m ε 0 + This proves the claim in and finishes the proof of the theorem. 2.2 The local compactness of the remainder Let ϕ : R n R n+l be a Lipschitz map, of graph type, i.e., assume that for some Note that this implies Let Then ϕx = x, ϕ 0 x, x R n, 2.2. ϕ 0 : R n R l Lipschitz ϕx ϕy x y, x, y R n k : R n+l \ {0} R be a smooth function, positive homogeneous of degree n, and satisfying k w = kw for all w R n+l \ {0}. Kfx := PV k ϕx ϕy fy dy R n = PV x y n k R n x y ϕx ϕy fy dy, x R n, defines a bounded operator on L p R n, for each p,. We aim to establish a finer structure when ϕ C R n or, more generally, when the Jacobian Dϕ of ϕ satisfies Dϕ L R n vmor n Namely, we set R := K K 0,

13 with K 0 fx := PV k Dϕxx y fy dy, x R n R n Note that implies Dϕxz z for all z R n. We have ϕ C R n = K 0 OPC 0 S 0 cl, Dϕ L R n vmor n = K 0 OPL vmos 0 cl The latter class is studied in [36, Chapter, ] and, for reader s convenience, useful background material on this topic is presented in A.3. See Theorem 2.6 for a derivation of the second part of 2.2.9, in a more general setting. As for the remainder R in we have Rfx = PV rx, yfy dy, x R n, R n where with rx, y := k ϕx ϕy k Dϕxx y = The following is our first major result. r τ x, y := k ϕx ϕy + τγx, y Γx, y, Γx, y := ϕx ϕy Dϕxx y. 0 r τ x, y dτ, Theorem 2.4. Let ϕ be as in , suppose k is as in 2.2.4, and define R as in 2.2.7, where K, K 0 are as in and 2.2.8, respectively. Finally, assume that holds. Then for each ball B R n and p,, the operator R : L p B L p B is compact In the case when ϕ C R n and Dϕ has a modulus of continuity satisfying a Dini condition, the compactness result is straightforward. See [36, Chapter 3, 4]. Proof of Theorem 2.4. Note that R = 0 R τ dτ, interpreted as a Bochner integral, with R τ fx := PV r τ x, yfy dy, R n x R n, and the integral kernel r τ x, y as in Given this, and bearing in mind that the collection of compact operators on L p B is a closed linear subspace of L L p B, L p B, it suffices to show that each operator R τ has the compactness property With this goal in mind, for each τ [0, ] observe that the operator R τ has the form R τ fx = PV x y n+ F R n Dϕxx y + τγx, y x y Γx, yfy dy,

14 with Γx, y as in and F := k. Note that the argument of F in is Dϕxx y + τγx, y = x y, Dϕ 0 xx y + τγ 0 x, y, with ϕ 0 as in and Γ 0 x, y as in 2.2.2, but with ϕ replaced by ϕ 0. In particular, there exists a constant C, such that Dϕxx y + τγx, y x y C, for all x, y R n and all τ [0, ]. As such, we can alter the function F w at will off the set {w R n+l : w C}, and arrange that F C 0 R n+l, while keeping F even. Moving on, observe that another way of looking at the argument of F in is to write with and Dϕxx y + τγx, y = τ ϕx ϕy + τdϕxx y = [ τϕx + τdϕxx ] [ τϕy + τdϕxy ] = A τ x ψx ψy, A τ x := τi τdϕx, ϕx ψx :=, ψ : R n R 2n+l x The bottom line is that for each τ [0, ] we have R τ fx = PV x y n+ ψx ψy F A τ x Γx, yfy dy, R n x y x R n, where A τ, ψ are as in and we can assume F is even and satisfies Granted this, Theorem 2.2 applies and yields that each R τ has the compactness property A variable coefficient version of the local compactness theorem Here the goal is to work out a variable coefficient version of Theorem 2.4, by treating the following class of operators. Let k C R n+l R n+l \ 0. Suppose kw, z is odd in z and homogeneous of degree n in z. In addition, assume bounds D α wd β z kw, z C αβ z n β 2.3. We take ϕ : R n R n+l as in , 2.2.6, and consider Kfx := PV k ϕx, ϕx ϕy fy dy, x R n R n 4

15 To analyze this type of singular integral operator with variable coefficient kernel, it is convenient to expand kw, z = a j wω n,j z, j where, starting with orthonormal, real-valued, spherical harmonics Ω j on S n, we have set z Ω n,j z := Ω j z n, z R n \ {0}, z and where the coefficient functions a j are given by a j w := S n kw, zω j z dz We can arrange that all the functions Ω n,j z in are odd. There is a polynomial bound in j on the C m norm of Ω S n,j n, for each m N, and the coefficients a j are rapidly decreasing in C m norm, for each m N. We have K = K j, j where, for each j, K j fx := a j ϕx PV Ω n,j ϕx ϕy fy dy, x R n R n The series converges rapidly in L p -operator norm, for each p,. Let us compare K with K #, defined as K # fx := PV k ϕx, Dϕxx y fy dy, x R n R n This time yields K # = j K # j, with K # j given by K # j fx := a jϕx PV R n Ω n,j Dϕxx y fy dy, x R n We claim that the series is rapidly convergent in L p -operator norm for each p,. Indeed, Theorem 2.4 directly implies that, for each j, K j K # j is compact on L p B, 2.3. for each ball B R n, and each p,. The operator norm convergence of and then yield the following variable coefficient counterpart to Theorem 2.4. Theorem 2.5. Given K as in and K # as in 2.3.8, K K # is compact on L p B

16 Moving on, we propose to further analyze and show that again, see the discussion in A.3 for relevant definitions K # OPL vmos 0 cl To this end, it is convenient to write kw, Az = j b j w, AΩ n,j z, for A : R n R n+l of the form with b j w, A := I A =, A 0 S n kw, AzΩ j z dσz Again, we can arrange that only odd functions Ω n,j arise in As A 0 runs over a compact subset of LR n, R l, the space of linear transformations from R n to R l. we have uniform rapid decay of b j w, A and each of its derivatives. We have the following conclusion. Theorem 2.6. The operator K # defined by satisfies K # fx = b j ϕx, Dϕx PV Ω n,j x yfy dy, j R n x R n, hence with Consequently, and follows. K # fx = px, Dfx, x R n, px, ξ := b j ϕx, Dϕx Ωn,j ξ. j p L vmoscl Symbol calculus Our goals here are to associate symbols to the operators studied in Section 2 and to examine how these operators behave under coordinate changes. 3. Principal symbols Let Ω R n+ be a bounded Lip vmo domain, so Ω is locally a graph of the form , 2.2.6, with l =. Let Ω denote the subset of Ω of the form ϕx such that x is an L p -Lebesgue point of Dϕ with p > n so in particular ϕ is differentiable at x. Then we set T ϕx Ω := { Dϕxv : v R n}, whenever ϕx Ω

17 In this fashion, we can talk about the tangent bundle and cotangent bundle over Ω, in the latter case, the fiber T ϕx Ω being the dual space to 3... T Ω, T Ω, 3..2 Let kw, z be smooth on R n+ R n+ \ 0, odd in z, and homogeneous of degree n in z. Consider Kfx := PV kx, x yfy dσy, K : L p Ω L p Ω, p, Ω In a local coordinate system described above, Kfx = PV k ϕx, ϕx ϕy fyσy dy, 3..4 O with O R n and dσy = Σy dy. Note that Σ L vmo. As we have seen in 2.3, K = px, D mod compact, 3..5 with px, ξ L vmoscl 0, odd and homogeneous of degree 0 in ξ. We want to associate to K a principal symbol σ K, defined on T Ω. We propose σ K ϕx, ξ := p x, Dϕx ξ, 3..6 for x O, ϕx Ω, with p as in If Ω is smooth, this coincides with the classical transformation formula for the symbol of a pseudodifferential operator. Now K = K # mod compact, with K # given by 2.3.8, with a factor of Σy thrown in. This factor can be changed to Σx, mod compact, so we can take px, Dfx = PV kϕx, Dϕxx yσxfy dy The standard formula connecting a pseudodifferential operator and its symbol yields px, ζ = k ϕx, Dϕxz e iz ζ Σx dz, 3..8 R n so compare p x, Dϕx ξ = k ϕx, Dϕxz e idϕxz ξ Σx dz R n = k ϕx, z 0 e iz0 ξ dz 0, 3..9 T ϕx Ω since the area element of Ω at w Ω coincides with that of T w Ω. Hence σ K w, ξ = kw, z 0 e iz0 ξ dz 0, w Ω T w Ω 7

18 This last formula is independent of the choice of local coordinates on Ω. In case Ω is smooth, 3..0 is the standard formula. We note that T w Ω inherits an inner product, hence a volume form, as a linear subspace of R n+, and dz 0 = Σx dz, when w = ϕx. Suppose K is an l l system of singular integral operators. We say K is elliptic on Ω if there exists a constant C > 0 such that σ K w, ξv C v, v C l, for σ-a.e. w Ω. 3.. In such a case, by 3..6, the operator px, D OPL vmos 0 cl associated to K in a local graph coordinate system, is elliptic, i.e., its symbol px, ξ satisfies the analogue of 3... We can hence prove the following. Theorem 3.. Let Ω R n+ be a bounded Lip vmo domain. If K is an l l elliptic system of singular integral operators of the form 3..3 and satisfies the ellipticity condition 3.., then K : L p Ω L p Ω is Fredholm, p, Moreover, the index of K in 3..2 is independent of p,, and the following regularity result holds: if < p < q < and f L p Ω, Kf L q Ω = f L q Ω Proof. Let {O j } j be an open cover of Ω on which we have graph coordinates. We also identify each O j with an open subset of R n. Let {ψ j } j be a Lipschitz partition of unity on Ω subordinate to this cover. Let ϕ j LipO j have compact support and satisfy ϕ j on a neighborhood of supp ψ j. Then K = KM ψj = M ϕj KM ψj, mod compacts, 3..4 j j where, generally speaking, M ψ f := ψf. Now we have cf M ϕj KM ψj = M ϕj p j x, DM ψj, mod compacts, 3..5 with p j x, D OPL vmos 0 cl, elliptic. We have a parametrix e jx, D OPL vmos 0 cl, satisfying M ϕi e i x, DM ψi M ϕj KM ψj = M ψi ψ j, mod compacts Set E := i M ϕi e i x, DM ψi Then EK = i,j M ϕi e i x, DM ψi M ϕj KM ψj, mod compacts = i,j M ψi ψ j, mod compacts = I, mod compacts Similarly, E is a right Fredholm inverse of K, and we have

19 Going further, for each p, let ι p K denote the index of K on L p Ω. Then, if < p < q < and N p denotes the null space of K on L p Ω, N p that of K on L p Ω, we have N q N p, N p N q, hence ι p K ι q K The same type of argument applies to E, yielding ι p E ι q E, hence as wanted. Note that, together with 3..9, this actually forces ι p K = ι q K, N q = N p and N p = N q Finally, for 3..3, if f L p Ω and Kf = g L q Ω, then g annihilates N p. Since N q = N p, g annihilates N q, so g = K f for some f L q Ω. Given p < q, we have f f N p. Hence f f N q, and thus f L q Ω, as asserted in Transformations of operators under coordinate changes Let ϕ : R n R n be a bi-lipschitz map, so there exist a, b 0, such that In addition, we assume Given we set Let us also set a x y ϕx ϕy b x y, x, y R n Dϕ vmor n k C R n \ 0, homogeneous of degree n, k z = kz, Kfx := PV kx yfy dy, R n x R n K ϕ fx := PV k ϕx ϕy fy dy, x R n R n As in the past, we let M χ denote the operator of pointwise multiplication by χ. Definition 3.. Say that ϕ TR n provided that hold and, in addition, whenever holds, then the singular integral operator K ϕ associated with ϕ as in may be decomposed K ϕ fx = PV k Dϕxx y fy dy + R ϕ fx, x R n, R n for a remainder with the property that for each cut-off function χ C0 Rn one has M χ R ϕ M χ : L p R n L p R n compact, p, By Theorem 2.6, the principal value integral on the right-hand side of defines an operator K ϕ OPL vmos 0 cl, which is bounded on L p R n for each p,. The following is a variant of Theorem 2.4, proven by the same sort of arguments. 9

20 Theorem 3.2. Assume ϕ satisfies Assume also that there exists κ > 0 such that, for all τ [0, ], τ[ϕx ϕy] + τdϕxx y κ x y, x, y R n Then ϕ TR n. In fact, given a function χ C0 for all points x, y suppχ. Rn, one has provided the estimate in holds Note the similarity of and In this connection, if Σ R n+l is an n-dimensional graph over R n, as introduced in 2.2, and if it is also represented as a graph over a nearby n- dimensional linear space V, then one gets a bi-lipschitz map from R n to V R n, satisfying In such a way, one can represent Σ as a Lip vmo manifold, whose transition maps satisfy the conditions of Theorem 3.2. See the next section for more on this. We proceed to a variable coefficient version of Take k measurable on R n R n, satisfying kx, z homogeneous of degree n in z, kx, z = kx, z Assume kx, z is smooth in z 0, and that for each multiindex α there exists a finite constant C α > 0 such that α z k, z L vmo C α z n α, 3.2. where, for f L R n, Then we can write f L vmo := kx, z = j 0 { f L if f vmo, if f / vmo z k j x z n Ω j, z where {Ω j } j is an orthonormal set of spherical harmonics on S n, all odd, and for each j N we have k j L vmo C N j N, for every N N In place of , we take Kfx := PV kx, x yfy dy, x R n, R n K ϕ fx := PV k ϕx, ϕx ϕy fy dy, x R n, R n and write K ϕ fx = PV k ϕx, Dϕxx y fy dy + R ϕ fx, R n x R n Using , we can write these as rapidly convergent series, and deduce that ϕ TR n = M χ R ϕ M χ : L p R n L p R n compact, p,, whenever χ C 0 Rn. Implementing this for involves using the following result. 20

21 Lemma 3.3. The function spaces bmor n and vmor n are invariant under u u ϕ, provided ϕ : R n R n is a bi-lipschitz map. Proof. This has the same proof as Proposition A.5 cf. also [37, Proposition 3.3] and [, Theorem 2, p. 56]. As in 3.2.8, the integral on the right side of defines an operator K ϕ OPL vmos 0 cl We use these results to analyze how an operator P = px, D OPL vmos 0 cl transforms under a map ϕ TR n. In more detail, given P : L p R n L p R n, set P ϕ gx := P fϕx, f L p R n, gx = fϕx Our hypothesis 3.2. implies g L p f L p, so P ϕ : L p R n L p R n. We claim that P ϕ OPL vmoscl 0, at least modulo an operator with the compactness property Furthermore, we obtain a formula for its principal symbol. We take px, ξ to be homogeneous of degree 0 in ξ. To start, we assume px, ξ = px, ξ Now with so Note that P fx = PV kx, x yfy dy, R n x R n, kx, z = 2π n R n px, ξe iz ξ dξ, px, ξ = kx, ze iz ξ dz. R n px, ξ = j 0 ξ p j xω j, ξ with {Ω j } j as in again, all odd, and p j L vmo C N j N, N N It follows that kx, z satisfies Hence apply. Consequently, with J ϕ y := det Dϕy, P ϕ gx = P fϕx = PV k ϕx, ϕx y fy dy R n = PV k ϕx, ϕx ϕy fϕyj ϕ y dy R n = PV k ϕx, ϕx ϕy gyj ϕ y dy R n 2

22 Applying , we have P ϕ gx = PV k ϕx, Dϕxx y gyj ϕ y dy + R ϕ, R n where R ϕ has the compactness property Also, J ϕ L vmo, so we can use the commutator estimate from [6] to replace J ϕ y by J ϕ x in 3.2.3, replacing R ϕ by R 2ϕ, also satisfying Consequently, we have P ϕ gx = 2π n p ϕ x, ξe ix y ξ gy dy dξ + R 2ϕ, R n R n and 2π n p ϕ x, ξ e iz ξ dξ = J ϕ xk ϕx, Dϕxz R n Taking ξ = Dϕx ξ gives dξ = J ϕ x dξ. We have cancellation of the factors J ϕ x, hence 2π n R n p ϕ x, Dϕx ξe i ϕxz ξ dξ = k ϕx, Dϕxz Hence, with we have σx, ξ = p ϕ x, Dϕx ξ, z = Dϕxz, π n R n σx, ξe iz ξ dξ = k ϕx, z, so Comparison with yields the formula σx, ξ = k ϕx, z e iz ξ dz R n p ϕ x, Dϕx ξ = pϕx, ξ This has been derived for px, ξ satisfying We now address the general case. Theorem 3.4. Assume ϕ TR n. Given P OPL vmoscl 0, with principal symbol px, ξ, and P ϕ defined by , one can decompose P ϕ = p ϕ x, D + R ϕ, with R ϕ satisfying and p ϕ x, D OPL vmos 0 cl satisfying Proof. We have this when px, ξ satisfies It remains to treat the case px, ξ = px, ξ. For this, we can write n px, D = q j x, Ds j x, D, j= s j x, ξ = ξ j ξ, q jx, ξ = px, ξ ξ j ξ. The previous analysis holds for the factors q j x, D and s j x, D, and our conclusion follows by basic operator calculus for OPL vmos 0 cl. 22

23 3.3 Admissible coordinate changes on a Lip vmo surface Let ϕ : R n R n+l have the form ϕx = x, ϕ 0 x, with Dϕ 0 x L R n vmor n, as in 2.2. Thus ϕ maps R n onto an n-dimensional surface Σ. Let V R n+l be an n-dimensional linear space. If V is not too far from R n depending on Dϕ 0 L, then Σ is also a graph over V, and we have the coordinate change map ψ : R n V, ψx = Qϕx, 3.3. where Q : R n+l V is the orthogonal projection. Consequently, x v ψx = Q, Dψxv = Q ϕ 0 x Dϕ 0 xv Consequently, τ[ψx ψy] + τdψxx y x y = Q τ[ϕ 0 x ϕ 0 y] + τdϕ 0 xx y Recall that the condition for Theorem 3.2 to apply is that has norm κ x y, for some κ > 0, for x, y R n, τ [0, ]. We see that the norm of is Qx y γx, y, where, with Q 0 denoting the orthogonal projection of R n+l onto R n, γx, y = QI Q 0 τ[ϕ 0 x ϕ 0 y] + τdϕ 0 xx y Dϕ 0 L QI Q 0 x y Since Qx y = x y + I QQ 0 x y, we deduce that the norm of is I QQ 0 I Q 0 Q Dϕ 0 L x y Consequently, Theorem 3.2 applies as long as This in turn holds provided We have the following conclusion. I QQ 0 + I Q 0 Q Dϕ 0 L < Q Q 0 < + Dϕ 0 L Proposition 3.5. Let ψ : R n V be as constructed in Assume holds, where Q and Q 0 are the orthogonal projections of R n+l onto V and R n, respectively. Take a linear isomorphism J : V R n. Then J ψ belongs to TR n. 23

24 3.4 Remark on double layer potentials Assume that a kernel E : R n+ \ {0} R be a smooth function, positive homogeneous of degree n +, and satisfying E X = EX for all X R n+ \ {0} 3.4. has been given. Also, let Ω R n+ be a bounded Lip vmo domain, and consider the singular integral operator of the type KfX := PV νx, X Y EX Y fy dσy, X Ω, Ω where ν and σ are, respectively, the outward unit normal and surface measure on Ω. To study this, focus on a local version of of the following sort. Let ϕ 0 : O R Lipschitz, with ϕ 0 vmo where O R n is open, be such that its graph is contained in Ω, and define the Lipschitz map ϕ : O R n+ by setting ϕx := x, ϕ 0 x, x O Then in these local coordinates, K takes the form K ϕ fx = PV ϕ0 x,, ϕx ϕy E ϕx ϕy fy dy O Its sharp form, obtained by replacing ϕx ϕy with Dϕxx y is then K ϕ # fx := PV ϕ0 x,, Dϕxx y E Dϕxx y fy dy = PV = 0, O O Dϕx ϕ 0 x,, x y E Dϕxx y fy dy since In n Dϕx = Dϕx ϕ 0 x ϕ 0 x, = I n n ϕ 0 x = ϕ 0 x In concert with our local compactness result, according to which K ϕ K ϕ # each p,, this ultimately gives that if Ω R n+ is a bounded Lip vmo domain and E is as in 3.4. then K from is compact on L p Ω, for each p,. is compact on L p for Of course, the above result contains as a particular case the fact which is a key result in the work of Fabes, Jodeit, and Riviére in [9] that the principal-value harmonic double layer operator νy, Y X KfX := lim ε 0 + ω n X Y n+ fy dσy, X Ω, Y Ω X Y >ε is compact on L p Ω, for each p,, if Ω R n+ is a bounded C domain. 24

25 3.5 Cauchy integrals and their symbols Given l N, we let Ml, C denote the collection of l l matrices with complex entries. Let D be a first order elliptic l l system of differential operators on R n+, Dux = j A j j u, A j Ml, C Thus σ D ζ = i j A jζ j is invertible for each nonzero ζ R n+, and D has a fundamental solution kz = 2π n+ R n+ Eζe iz ζ dζ, Eζ = σ D ζ, odd and homogeneous of degree n in z. If Ω R n+ is a bounded UR acronym for uniformly rectifiable domain, we can form Bfx = kx yfy dσy, x Ω, with nontangential limits cf Bf n.t. Ω Ω z := lim Bfz = Γ κx z x 2i σ Dνx fx + Bfx, for σ-a.e. x Ω, where Γ κ x Ω is a region of nontangential approach to x Ω cf. 4..2, and Bfx := PV kx yfy dσy, x Ω Ω One is hence motivated to consider the Cauchy integral C D fx = i kx yσ D νyfy dσy, x Ω, Ω with nontangential limits n.t. C D f x = Ω 2 fx + C Dfx, for σ-a.e. x Ω, where C D fx := ipv kx yσ D νyfy dσy, x Ω As shown in [23], a reproducing formula yields Ω P D = 2 I + C D = P 2 D = P D This is studied in [23] in the setting of UR domains and also for variable coefficient situations, which for simplicity we do not take up here in detail. The operator P D is a Calderón projector. 25

26 Here, we take Ω to be a Lip vmo domain and analyze the principal symbol of P D, as a projection-valued function on T Ω \ 0. To start, we recall from 3..0 that, for B in 3.5.5, σ B w, ξ = kz 0 e iz0 ξ dz 0, w Ω T w Ω Plugging in and using basic Fourier analysis, we obtain We then have σ CD w, ξ = σ B w, ξ = 2π PV Now σ D ξ + sνw = σd ξ + sσ D νw, so with E ξ + sνw ds i 2π PV σd σ D ξ + isνw νw ds σ D ξ + sνw σd νw = The invertibility of σ D ξ + sνw and of σ D νw imply that We have σ CD w, ξ = Mw, ξ + si, Mw, ξ = σ D νw σ D ξ Spec Mw, ξ R = i 2π PV Lemma 3.6. Assume A Ml, C and Spec A R =. Then si + Mw, ξ ds πi s A e iεs ds = { e iεa P + A if ε > 0, e iεa P A if ε < 0, where P + A is the projection of C l onto the linear span of the generalized eigenvectors of A associated to eigenvalues in SpecA with positive imaginary part, annihilating those associated to eigenvectors with negative imaginary part, and P A = I P + A. Hence 2πi PV s A ds = P + A 2I Proof. In case ε > 0, the left-hand side of is equal to lim s A ds, R 2πi D + R where D R := {s C : s < R} and D + R := D R {s C : Is > 0}. This path integral stabilizes when R > A, and the desired conclusion in this case follows from the Riesz functional calculus. The treatment of the case when ε < 0 is similar. Then follows readily from

27 We apply Lemma 3.6 to with A := Mw, ξ. Making use of the identity P + M = P M, we have the following conclusion. Proposition 3.7. The operator C D and the associated Calderón projector, derived from the Cauchy integral via , have symbols given by σ CD w, ξ = P Mw, ξ 2 I and = 2 I P Mw, ξ, σ PD w, ξ = P + Mw, ξ, respectively, with Mw, ξ as in and P + A as described in Lemma 3.6. Remark 3.. Extensions of the results in this section to variable coefficient operators acting between vector bundles and to domains on manifolds can be worked out using the formalism developed in [23], [24]. 4 Applications to elliptic boundary problems Here we apply the results of Sections 2 3 to several classes of elliptic boundary problems, including the Dirichlet problem for general strongly elliptic, second order systems and general regular boundary problems for first order elliptic systems of differential operators. 4. Single layers and boundary problems for elliptic systems Let M be a smooth, compact, n + -dimensional manifold, equipped with a Riemannian metric tensor g = g jk dx j dx k, with g jk C j,k Also, consider a Lip vmo domain Ω M cf. the discussion in the last part of A.5. Having some fixed κ 0,, for each x Ω, define the nontangential approach region with vertex at x by setting Γ κ x := { y Ω : distx, y < + κ dist y, Ω } Next, given an arbitrary u : Ω C, define its nontangential maximal function and its pointwise nontangential boundary trace at x Ω, respectively, as Nκ u x := sup { uy : y Γ κ x }, u n.t. x := lim uy, 4..3 Ω Γ κx y x whenever the limit exists. The parameter κ plays a somewhat secondary role in the proceedings, since for any κ, κ 2 0, and p 0, there exists C = Cκ, κ 2, p, with the property that C Nκ u L p Ω Nκ2 u L p Ω C Nκ u L p Ω, 4..4 for each u : Ω C. Given this, we will simplify notation and write N in place of N κ. 27

28 Moving on, let L be a second-order, strongly elliptic, k k system of differential operators on M. Assume that, locally, Lu = i,j j A ij x j u + j B j x j u + V xu, 4..5 where Also, suppose A ij C 2, B j C, V L L : H,p M H,p M is an isomorphism, for < p < We want to solve the Dirichlet boundary problem Lu = 0 on Ω, u n.t. Ω = f Lp Ω, N u L p Ω, 4..8 via the layer potential method. To this end, let E denote the Schwartz kernel of L, so that L vx = Ex, yvy dvoly, x M, 4..9 M where dvol stands for the volume element on M. Then, with σ denoting the surface measure on Ω, define the single layer potential operator and its boundary version by Sgx := Ex, ygy dσy, x M \ Ω, Ω 4..0 We want to solve 4..8 in the form and Sg := Sg n.t. Ω on Ω. u = Sg, where g is chosen so that Sg = f. 4.. As such, if H s,p Ω, with < p < and s, denotes the L p -based scale of Sobolev spaces of fractional order s on Ω, we would like to show S : H,p Ω L p Ω is Fredholm, of index Since the adjoint of S is the single layer associated with L which continues to be a second-order, strongly elliptic, k k system of differential operators on M, this is further equivalent with q := p the Hölder conjugate exponent of p to the condition that S : L q Ω H,q Ω is Fredholm, of index Such a result was established, for q close to 2, when Ω is a Lipschitz domain, in Chapter 3 of [22]. The argument made use of a Rellich type identity. In the scalar case the result was established in the setting of regular SKT domains in [4, Section 6.4], and applied in 7. of that paper to the Dirichlet problem. In case Ω is smooth, it is standard that S OPS Ω and it is strongly elliptic, from which 4..2 and 4..3 follow. Here is what we propose. Proposition 4.. Let Ω be a Lip vmo domain and let L be a second-order, strongly elliptic, k k system of differential operators on M, as in , and satisfying Then 4..2 holds for all p, and 4..3 holds for all q,. 28

29 Proof. We start with the proof of Pick L vmo vector fields X j, j N, tangent to Ω, such that N X j x A > 0 for a.e. x Ω j= Then set T f := {X j f : j N}. We have T S : L q Ω L q Ω for all q,. Theorem 2.4 or rather its standard variable coefficient extension implies T S = k 0 x, D + R, k 0 x, D OPL vmos 0 cl, 4..5 with R compact on L q Ω. At this point we make the following Claim. We have the overdetermined ellipticity property k 0 x, ξv A 0 v, A 0 > Assuming for now this claim whose proof will be provided later, we obtain that k 0x, Dk 0 x, D OPL vmos 0 cl mod compacts 4..7 is a determined elliptic operator, so it has a parametrix Q OPL vmoscl 0 cf. A.3. Hence, This implies that Qk 0x, D T S = I + R, with R compact on L q Ω S : L q Ω H,q Ω is semi-fredholm, 4..9 namely, it has closed range and finite dimensional null-space. To complete the argument, we take a continuous family L τ of second order, strongly elliptic operators on M, τ [0, ], such that L = L and L 0 is scalar. This gives a norm continuous family S τ : L q Ω H,q Ω, all semi-fredholm We know that S 0 is Fredholm of index 0. Hence so are all the operators S τ in This gives 4..3 which, by duality, also yields Now we return to the proof of the claim made in That is, we shall establish the overdetermined ellipticity of k 0 x, D OPL vmoscl 0, arising in 4..5 which is equal modulo a compact operator to T S. To begin, we discuss the smooth case. If Ω is smooth, and L is strongly elliptic of second order with smooth coefficients, then actually S OPS Ω, and this operator is strongly elliptic. In fact, given x, ξ T Ω \ 0, and with ν Tx Ω the outward unit conormal to Ω, we have + + σ S x, ξ = C n σ E x, ξ + tν dt = C n σ L x, ξ + tν dt This is seen as in [34,.-.2 in Chapter 7, Vol. 2], where we take m = 2, x n = 0. Strong ellipticity of S then follows from 4..2, keeping in mind the strong ellipticity of L. Specifically, note that σ S x, ξ is positive homogeneous of degree in ξ and the integrals in 4..2 are 29

30 absolutely convergent since σ L x, ξ + tνx C ξ 2 + t 2. Keeping this in mind, for any section η and any 0 ξ T x Ω T x M, we may estimate σs x, ξη, η + x = C n σl x, ξ + tνx η, η dt + C η 2 ξ 2 + t 2 dt C η 2 ξ, for some C > 0. This yields the strong ellipticity of S. Next, since σ Xj S = σ Xj σ S, the ellipticity of T S is an immediate consequence of what we have just proved and To tackle the case when Ω is a Lip vmo domain, we take local graph coordinates ϕx = x, ϕ 0 x, and arrange that the vector fields {X j } j N include those associated with coordinate differentiation. The integral kernel Ex, y has the form Ex, y = E 0 x, x y + rx, y, where E 0 x, z is smooth on {z 0} and homogeneous of degree n in z note that dim Ω = n, and rx, y has lower order. See the analysis in [22]. Locally, the operator S has the form Sgx = ϕx, ϕx ϕy gyσy dy + Rgx, x R n, R n E 0 where dσy = Σy dy and R denotes the integral operator with kernel rx, y. Hence, for each j {,..., n}, j Sgx = PV j ϕx 2 E 0 ϕx, ϕx ϕy gyσy dy + Rj gx, x R n, R n where here and below R j will denote perhaps different operators that are compact on L p, for < p <. Theorem 2.4 or rather its natural variable coefficient extension from 2.3 gives j Sgx = PV j ϕx 2 E 0 ϕx, Dϕxx y gyσy dy + Rj gx, x R n, R n i.e., j Sgx = T j x, DΣgx + R j gx, x R n, where T j x, DΣgx is given by the principal value integral in We therefore have T j x, D OPL vmoscl 0, with symbol T j x, ξ = e iz ξ j ϕx 2 E 0 ϕx, Dϕxz dz R n Given that L is a k k system, T j x, ξ is a k k matrix, i.e., T j x, ξ Mk, C, for ξ 0, and a.e. x. We need to show that there exists C > 0 such that, for all ξ 0 and v C k, T j x, ξv C v, for a.e. x j 30