Mathematisch-Naturwissenschaftliche Fakultät
Lashi Bandara | Andreas Rosén
Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions
Suggested citation referring to the original publication: Communications in Partial Differential Equations 44 (2019) 12, pp. 1253–1284 DOI https://doi.org/10.1080/03605302.2019.1611847 ISSN (print) 0360-5302 ISSN (online) 1532-4133
Postprint archived at the Institutional Repository of the Potsdam University in: Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe ; 758 ISSN 1866-8372 https://nbn-resolving.org/urn:nbn:de:kobv:517-opus4-434078 DOI https://doi.org/10.25932/publishup-43407
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 2019, VOL. 44, NO. 12, 1253–1284 https://doi.org/10.1080/03605302.2019.1611847
Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions
Lashi Bandaraa and Andreas Rosen b aInstitut fur€ Mathematik, Universit€at Potsdam, Potsdam OT Golm, Germany; bMathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden
Dedicated to the memory of Alan G. R. McIntosh.
ABSTRACT ARTICLE HISTORY On a smooth complete Riemannian spin manifold with smooth com- Received 22 March 2018 pact boundary, we demonstrate that Atiyah–Singer Dirac operator Accepted 20 January 2019 D= in L2 depends Riesz continuously on L1 perturbations of local B KEYWORDS boundary conditions B: The Lipschitz bound for the map B! 1 Boundary value problems; D= 1 D= 2 2 depends on Lipschitz smoothness and ellipticity of Bð þ BÞ B Dirac operator; functional and bounds on Ricci curvature and its first derivatives as well as a calculus; real-variable lower bound on injectivity radius away from a compact neighbour- harmonic analysis; Riesz hood of the boundary. More generally, we prove perturbation esti- continuity; spectral flow mates for functional calculi of elliptic operators on manifolds with local boundary conditions. AMS SUBJECT CLASSIFICATION 58J05; 58J32; 58J37; 58J30; 42B37; 35J46; 35J56
1. Introduction The aim of this article and its companion [1] has been to prove perturbation estimates of quantities of the form
De D pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ; e2 þ 2 I þ D I D L2ðÞ!M;V L2ðÞM;V where D and De are self-adjoint elliptic first-order partial differential operators, acting on sections of a vector bundle V over a smooth manifold M: The symbol f ðfÞ¼ 1 fð1 þ f2Þ 2 is a motivating example, yielding continuity results in the Riesz sense, but our methods apply equally well to more general holomorphic symbols around R; which may be discontinuous at 1: In [1], together with Alan McIntosh, we obtained results on complete manifolds ðM; gÞ without boundary. In that case, the main example of operators D and De was the Atiyah–Singer Dirac operators on M with respect to two different metrics g and eg: The bound obtained was
CONTACT Lashi Bandara [email protected] Institut fur€ Mathematik, Universit€at Potsdam, Potsdam OT Golm D-14476, Germany. ß 2019 The Author(s). Published by Taylor & Francis Group, LLC. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way. 1254 L. BANDARA AND A. ROS EN
e pffiffiffiffiffiffiffiffiffiffiffiffiffiD ffiffiffiffiffiffiffiffiffiffiffiffiffiD Շ e ; p jjg gjj 1ðÞðÞ2;0 e2 þ 2 L T M I þ D I D L2ðÞ!M;V L2ðÞM;V where the implicit constant depends on certain geometric quantities. Note that the two Dirac operators themselves depend also on the first derivatives of the metrics. In the present paper, we consider the corresponding perturbation estimate on a mani- fold M (possibly noncompact) with smooth, compact boundary R ¼ @M: Our motivat- ing example in this case is when both D and De are the Atiyah–Singer Dirac operator, but with two different local boundary conditions, defined through two different subbun- e dles E and E of VjR: For each boundary condition we assume self-adjointness and ellip- ticity so that the domains of D and De are closed subspaces of H1ðVÞ: The bound we obtain is
e ÀÁ pffiffiffiffiffiffiffiffiffiffiffiffiffiD ffiffiffiffiffiffiffiffiffiffiffiffiffiD Շ ^ e ; ; p jjd E x Ex jjL1ðÞR (1.1) e2 þ 2 I þ D I D L2ðÞ!M;V L2ðÞM;V
^ ; e where dðEx E xÞ¼jpEðxÞ peðxÞj and pE and pe respectively are the orthogonal projec- e E E tors from VjR to E and E. Again the implicit constant in the estimate depends on a number of geometric quantities which we list completely. As described in the introduction of [1], an important application of these perturb- ation estimates is the study of spectral flow for unbounded self-adjoint operators. The study of the spectral flow was initiated by Atiyah and Singer in [2] and has important connections to particle physics. An analytic formulation of the spectral flow was given by Phillips in [3] and typically, the gap metric
i þ De i þ D i De i D L2ðÞ!M;V L2ðÞM;V is used to understand the spectral flow for unbounded operators. The Riesz topology is a preferred alternative since the spectral flow in this topology better connects to topo- logical and K-theoretic aspects of the spectral flow, which were observed in [2] for the case of bounded self-adjoint Fredholm operators. The main disadvantage is that it is typically harder to establish continuity in the Riesz topology. In particular we refer to the open problem pointed out by Lesch in the introduction of [4], namely whether a Dirac operator on a compact manifold with boundary depends Riesz continuously on pseudo-differential boundary conditions imposed on the operator. The present article answers these questions to the positive, in the special case of local boundary conditions. Self-adjoint local boundary conditions are typically physical and a very large subclass of the so-called Chiral conditions are listed in [5] by Hijazi, Montiel and Rold an as being self-adjoint boundary conditions. In particular, these exist in even dimensions or when the manifold is a space-like hypersurface in spacetime. The case of non-local boundary conditions defined by pseudo-differential projections appear to be beyond the scope of the methods used in the present paper but we anticipate they will be the object of further investigations in the future. The local nature of the boundary COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1255 conditions enter the proof in a number of instances, but the most serious occurrence concern the so-called exponential off-diagonal estimates, which relies on the domains of the operators being preserved under multiplication by smooth, bounded functions. It is important to note that the right hand sides in the perturbation estimates that we obtain, ^ e e ; ; ; namely jjg gjjL1ðT ð2 0ÞMÞ and jjdðE x ExÞjjL1ðRÞ are supremum norms, which are smaller than estimates that can be obtained from operator theoretic arguments alone. Like in [1], we use methods from operator theory and real harmonic analysis to obtain (1.1). For a self-adjoint operator, say D; the quadratic estimate ð 1 2 dt Շ 2 jjQtujj jjujj (1.2) 0 t ’ is immediate from the spectral theorem coupled with Fubini s theorem. Here Qt ¼ tDðI þ t2D2Þ 1 is a holomorphic approximation, adapted to the operator D; of the pro- jection onto frequencies in a dyadic band around 1=t: For the harmonic analyst, the estimate (1.2) yields continuity of a wavelet transform, adapted to D; and plays the same role in wavelet theory as Plancherel’s theorem does in Fourier theory. We refer to [6] by Daubechies in the case Qt is the projection onto scale t in the multiscale reso- lution. These ideas are also central in Littlewood–Paley theory. Quadratic estimates like (1.2) are a flexible tool. They can be adapted to handle non- self-adjoint operators as well as non-commuting operators. Relevant to this article is the latter extension, where we want to estimate f ðDeÞ f ðDÞ as in (1.1). By expressing these operators in terms of resolvents of De and D respectively via the Dunford functional cal- culus, such perturbation estimates can be obtained from quadratic estimates of the form ð 1 e 2 dt Շ 2: jjQtAPtujj jjujj (1.3) 0 t e e; Here Qt is like Qt above but for the operator D A typically is a bounded multi- 2 2 1 plication operator, and Pt ¼ðI þ t D Þ should be thought of as a holomorphic approximation, adapted to the operator D; to the projections onto frequencies smaller than 1=t: Just like in the non-self-adjoint case in (1.2), the estimates (1.3) are non-trivial and use the specific structure of the operators De and D: When these are differential opera- tors, allowing non-smooth coefficients, we can use methods from harmonic analysis to handle (1.3) essentially as a Carleson embedding theorem. For operators with simpler structure than our Dirac operators, it is also possible to obtain higher order perturb- ation estimates. In this case the relevant quadratic estimates look like (1.6). For our Dirac operators, (1.3) more precisely amounts to the two estimates ð 1 e 1 2 dt Շ 2 2 jjQtA1rðÞiI þ D Ptujj jjA1jj1jjujj and (1.4) 0 t ð 1 e 2 dt Շ 2 2; jjtPtdivA2Ptujj jjA2jj1jjujj (1.5) 0 t 2 1 which need to be established for u 2 L ðVÞ; where A1 and A2 are L multipliers. Through a similarity transformation of De; we can also assume that DðDeÞ¼DðDÞ: 2 2 1 e 2 e 2 1; 2 2 1 e e 2 e 2 1: Here Pt ¼ðI þ t D Þ , Pt ¼ðI þ t D Þ Qt ¼ tDðI þ t D Þ , Qt ¼ tDðI þ t D Þ 1256 L. BANDARA AND A. ROS EN
At a first glance, trying to adapt the proofs in [1] for (1.4) and (1.5) to the case of manifolds with boundary seems to be a straightforward exercise. However, closer inspection reveals an interesting dichotomy. In [1], the estimate (1.5) was standard and well known to be equivalent to a certain measure being a Carleson measure, and the 1 main new work was in establishing (1.4). Here the operator A1rðiI þ DÞ which is e ; sandwiched between Qt and Pt is not a multiplier but also incorporates a singular inte- gral operator rðiI þ DÞ 1: To estimate, a Weitzenbock-type€ inequality for D is needed. Turning to a manifold with boundary, one sees that (1.4) follows as in [1], mutatis mutandis. Instead, the presence of boundary forces (1.5) to be a non-standard estimate, since new boundary terms appear in the absence of boundary conditions for the multi- plier A2. Indeed, in order for our estimates to be useful, we need to be able to allow for ’ general A2. More precisely, by Stokes theorem ð ÀÁð ÀÁð ÀÁ e ~ e e g Pttdivu; v dl ¼ g tn u; Ptv dr u; trPtv dl: M R M
The second term on the right hand side is bounded by jjujjL2 jjvjjL2 by the ellipticity and self-adjointness of De; but clearly the first term has no such bound. This means that e in (1.5), the operators Pttdiv are not even bounded, and standard estimates break down. An important contribution of this paper lies in the new ideas needed to establish e e (1.5). Here, we observe that even though Pttdiv is unbounded, the operator PttdivA2Pt ’ as a whole is bounded by jjA2jjL1 (which is seen from Stokes theorem and the ellipti- city of D). Building on this observation, we prove (1.5)inSection 4.3 by adapting, in a non-trivial way, the standard harmonic analysis proof, usually referred to as a local T(1) argument. The inspiration for this analysis comes from [7] by Auscher, Axelsson, Hofmann and [8] by Axelsson, Keith, McIntosh. To be more precise, this allows us to reduce (1.5) for an arbitrary L2 sections instead for certain test sections which vanish near the boundary R. For this special class of test sections, we are able to adapt the boundaryless estimates and (1.5) becomes standard. The remainder of this article is organised as follows. In Section 2 we state in detail our main perturbation estimate in its general form, and show in Section 3 how it is applied to yield the motivating estimate for the Atiyah–Singer Dirac operator under perturbation of local boundary conditions. Then, Section 4 contains the proof of Theorem 2.1, as outlined above. As aforementioned, this article is a sequel to the authors’ joint paper [1] with Alan McIntosh. During our work on this project, McIntosh untimely passed away, leaving us in great sorrow. McIntosh’s great heritage to mathematics include his widely celebrated unique blend of operator theory and harmonic analysis which has lead to breakthroughs like the proof of the Calderon conjecture on the L2 boundedness of the Cauchy singular integral operator on Lipschitz curves, jointly with Coifman and Meyer in [9], and the proof of the Kato square root conjecture on the domain of the square root of elliptic second-order divergence form operators, jointly with Auscher, Hofmann, Lacey and Tchamitchian in [10]. The estimates in this article go back to the multilinear estimates pioneered by McIntosh in connection with [9]. There, expressions of the form COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1257 ð 1 2 dt jjQtA1PtA2PtA3Pt AkPtujjL2 (1.6) 0 t 2 : ; were bounded by jjujjL2 Formally, the idea is to pass a derivative from Qt through the 1 ; ; general L maps Ai, to the rightmost Pt which becomes Qt ¼ tDPt and conclude the desired estimate by (1.2). Concretely, this is achieved by harmonic analysis methods and Carleson measures. The power of this analysis is well known in real-variable har- monic analysis and, in fact, the necessary and much needed algebra of Pt and Qt opera- tors are in some circles of mathematicians referred to as McIntoshery (or in French McIntosherie). In this article, we only employ the linear case k ¼ 1 of these multilinear estimates of McIntosh, leading to first-order perturbation estimates. Even though our work is yet another successful example of McIntoshery, we have nevertheless chosen to not add his name as an author. Both authors are former students of McIntosh, and we know he had as a firm prin- ciple for omitting his name from publications unless he clearly felt that he had contributed to the novelties of the article in a substantial way. Unfortunately, he could not join us this time.
2. Setup and statement of main theorem 2.1. Manifolds, bundles, and function spaces
Let M be a smooth manifold (possibly noncompact) with smooth boundary R ¼ @M: Throughout, we fix a smooth, Riemannian metric g on M and let r denote the associated Levi-Civita connection. We assume that g is complete, by which we mean ðM; gÞ is complete as a metric space. By M; we denote the interior Mn@M: The induced volume measure is denoted by dl on M and dr on R.Let~n be the unit outward normal vectorfield on R. The tangent, cotangent bundles are denoted by TM and T M respectively, and the ; rank (p, q)-tensor bundle by T ðp qÞM: For a smooth complex Riemannian bundle ðV; hÞ on M; let CðVÞ denote the set of ; measurable sections and Ck aðVÞ be the set of continuously k-differentiable sections with the k-th derivative being a-Holder€ continuous up to the boundary. Note that when we ; ; ; write Ck a; we do not assume Ck a with global control of the norm but rather, only Ck a ; regularity locally. We write Ck ¼ Ck 0 and C1ðVÞ ¼ \1 CkðVÞ: Moreover, define ÈÉk¼1 ; ; Ck aðÞV ¼ u 2 Ck aðÞV : spt u M compact and c no k;a k;a : : Ccc ðÞV ¼ u 2 C ðÞV spt u Mcompact Since Lipschitz maps will have special significance, we write LipðVÞ to denote sections 0;1 < : w 2 C ðVÞ with jjrwjjL1ðVÞ 1 For 1 p < 1; denote the set of p-integrable measurable sections with respect to h p : 1 C and l by L ðVÞ with norm jjnjjp The space L ðVÞ consist of n 2 ðVÞ such that > : jnj C for some C 0 almost-everywhere on M The norm jjnjj1 is then the infimum over C > 0 such that this relation holds. The spaces LpðVÞ are Banach spaces and L2ðVÞ is a Hilbert space with inner product h ; i: The latter space is what we shall be con- cerned with most in this paper and for simplicity of notation, we denote the norm : jj jj2 by jj jj The restricted bundle W¼VjR is a smooth, complex Riemannian bundle 1258 L. BANDARA AND A. ROS EN
p with metric hjR and L ðWÞ spaces are defined similarly on R with respect to the meas- ure dr: Let r be a connection on V that is compatible with h: Then, r is a closeable oper- ator in L2ðVÞ and we define the Sobolev spaces HkðVÞ as the domain of the closure of the operator ÀÁ ; 2; :::; k : 2 1 2 1 k ðÞl;0 r r r L \ C ðÞV ! L \ C l¼1T M V
2 k in L : Similarly, we obtain boundary Sobolev spaces H ðVjRÞ from rjR: By compatibility, we have that hru; vi¼hu; trrvi 2 1 ; 2 1 for u 2 L \ C ðVÞ v 2 L \ C ðT M VÞ and with either spt u M compact or spt v M compact. Thus, we obtain the divergence operator, defined as div ¼ rc as a densely-defined and closed operator with domain DðdivÞ from the oper- : 1 1 : ator rc Ccc ðVÞ ! Ccc ðT M VÞ
2.2. Main theorem
In order to phrase the main theorem as in [1], we require some assumptions on the manifold. We say that ðM; g; lÞ has exponential volume growth if there exists cE 1; j; c > 0 such that ÀÁ j cEtr 0 < lðÞBðÞx; tr ct e l BðÞx; r < 1; ðÞEloc for every t 1 and g-balls Bðx; rÞ of radius r > 0 at every x 2M: The manifold ðM; gÞ 1 satisfies a local Poincare inequality if there exists cP 1 such that for all f 2 H ðMÞ;
jjf fBjjL2ðÞB cPradðÞB jjf jjH1ðÞB ðÞPloc for all balls B in M such that the radius radðBÞ 1: We say that ðV; hÞ satisfies generalised bounded geometry,orGBG for short, if there exist q > 0 and C 1 such that, for each x 2M; there exists a continuous local trivial- : ; CN 1 ; isation wx Bðx qÞ ! pV ðBðx qÞÞ satisfying 1 1 1 ; C jwx ðÞy ujd jujhðÞy Cjwx ðÞy ujd ; ; CN 1 for all y 2 Bðx qÞ where d denotes the usual inner product in and wx ðyÞu ¼ 1 ; CN wx ðy uÞ is the pullback of the vector u 2Vy to via the local trivialisation wx at y 2 Bðx; qÞ: We call q the GBG radius. In typical application, the local trivialisations will be ; C0 1 or smooth. Letting D and De be first-order differential operators acting on a bundle V over M 1 1 and R : H ðVÞ ! H2ðVRÞ the boundary trace map, we state the following assumptions adapted to our setting from [1]:
(A1) M and V are finite dimensional, quantified by dim M < 1 and dim V < 1; ; < ; < < (A2) ðM gÞ has exponential volume growth quantified by c 1 cE 1 and j 1 in (Eloc), < ; (A3) a local Poincare inequality (Ploc) holds on M quantified by cP 1 COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1259
0;1 > < ; (A4) T M has C GBG frames j quantified by qT M 0 and CT M 1 with jr jj < CG;T M with CG;T M < 1; 0;1 > < ; < (A5) V has C GBG frames ej quantified by qV 0 and CV 1 with jrejj CG;V with CG;V < 1;
(A6) D satisfies jDejj CD;V with CD;V < 1 almost-everywhere inside each GBG frame fejg; (A7) We have gDðDÞ DðDÞ for every bounded g 2 C1ðMÞ with jjrgjj < 1; and e 1 ½D; g and ½D; g are pointwise multiplication operators on almost-every fibre Vx with a constant c > 0 such that D;De j½ D; g uxðÞj c jrgðÞx jjuxðÞj (2.1) D;De for almost-every x 2Mand the same estimate with D interchanged with De;
(A8) D and De are self-adjoint operators which are essentially self-adjoint on their restriction to ÈÉ 1 ; 1 : R ; Cc ðÞV B ¼ u 2 Cc ðÞV u 2B 1 where B¼H2ðEÞ with E VjR a smooth subbundle of VjR; and both operators have domain DðDÞ¼DðDeÞ H1ðVÞ and with C 1 the smallest constant sat- isfying 1 1 C jjujj jjujj 1 Cjjujj and C jjujj jjujj 1 Cjjujj (2.2) D H D De H De e ; for all u 2DðDÞ¼DðDÞ and where jj jjD ¼jjD jjþjj jj the operator norm, and
(A9) D satisfies the Riesz-Weitzenbock€ condition: DðD2Þ H2ðVÞ with ÀÁ 2 2 jjr ujj cW jjD ujj þ jjujj (2.3)
2 for all u 2DðD Þ with cW < 1: The implicit constants in our perturbation estimates will be allowed to depend on e CðM; V; D; DÞ¼maxfdim M; dim V; c;c ; j; c ; q ; C ; C ; ; E P T M T M G T M (2.4) ; ; ; ; ; ; < : qV CV CG;V cD CD;V C cW g 1 Our main theorem is the following. Theorem 2.1. Let M be a smooth manifold with smooth compact boundary R ¼ @M and let g be a smooth metric on M such that ðM; gÞ is complete as a metric space. Let ðV; h; rÞ be a smooth vector bundle over M with smooth metric h and connection r that are compatible. Let D; De be two first-order differential and assume the hypotheses (A1)-(A9) on M; V; D and De and that e Dw ¼ Dw þ A1rw þ div A2w þ A3w; (2.5) holds in a distributional sense for w 2DðDÞ¼DðDeÞ, where 1260 L. BANDARA AND A. ROS EN
1 A1 2 L ðÞLðÞT M V; V ; 1 A2 2 L \ LipðÞLVðÞ; T M V ; (2.6) 1 A3 2 L ðÞLVðÞ; : and let jjAjj1 ¼jjA1jj1 þjjA2jj1 þjjA3jj1 ; = ; 1 o Then, for each x 2ð0 p 2Þ and r 2ð0 1 ,wheneverf2 Hol ðSx;rÞ,wehavetheper- turbation estimate e ðÞ ðÞ 2 2 Շ 1 ; jjf D f D jjL ðÞV !L ðÞV jjf jjL ðÞSx;r jjAjj1 where the implicit constant depends on CðM; V; D; DeÞ:
o : : 2 < 2 2 2 ; 1 o Here Sx;r ¼fx þ iy y tan xx þ r g and we say that f 2 Hol ðSx;rÞ if it is o > : holomorphic on Sx;r and there exists C 0 such that jf ðfÞj C For a definition of functional calculi f ðDÞ and f ðDeÞ with symbols f bounded and holomorphic, see Section 2.3 in [1]. Remark 2.2. Self-adjointness of D and Dine Theorem 2.1 (A8) can be relaxed. Indeed, we only use self-adjointness to obtain the estimates (4.1) and (4.2). In the more general situation, that is, when the operator D or De is only similar to a self-adjoint operator 1 2 1 2 1 with similarity transform U, the constant 2 jjUjj jjU jj appears in place of 2 in (4.1) and (4.2), and also enters in CðM; V; D; DeÞ: We prove this theorem using real-variable harmonic analysis methods through the holomorphic bounded functional calculus in Section 4.
3. Application to the Atiyah–Singer Dirac operator Throughout this section, in addition to assuming that ðM; gÞ is a smooth and complete Riemannian manifold with compact boundary R ¼ @M; we assume that M is a Spin manifold. X n Xp Recall that the exterior algebra M¼ p¼0 M is a graded algebra, and it is vec- tor-space isomorphic to the Clifford algebra which we denote by DM: Fix a spin struc- n ture PSpinðMÞ and let the associated Spin bundle be denoted by D= M¼PSpin g D= R corresponding to the standard complex representation g : DRn !LðD= RnÞ: Let : CðDMÞ ! EndðD=MÞ denote Clifford multiplication on spinors. Let D= denote the Atiyah–Singer Dirac operator associated to D=M; given locally in an = k ; orthonormal frame fekg by the expression Dw ¼ e rek w where r is the Spin connec- = ; tion. Denoting feag to be an induced local orthonormal spin frameP from fekg the Spin = 2 = ; 2 1 a connection takes the local expression rea ¼ xE ea where xE ¼ 2 b < a xb eb ea is D= a the lifting of the Levi-Civita connection 2-form to M and xb is the connection 1- ; :::; : : form in E ¼ðe1 enÞ The symbol of this operator is symD= ðnÞw ¼ n w We refer the reader to Lawson and Michelsohn [11] and Ginoux [12] for a more detailed exposition on spin structures, bundles and their associated operators. To define D= as a self-adjoint elliptic operator on L2ðD=MÞ by imposing boundary con- ditions on DðD= Þ we will follow the framework developed by B€ar and Ballmann [13] and specialised to Dirac-type operators in [14]. In particular, by a local boundary condition COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1261
for D= we mean a space
1 B¼H2ðÞE with E DR= ¼ D=MjR; where E is a smooth subbundle. The operator D= with boundary condition B; denoted D= ; is the operator D= with domain B ÈÉ 2 2 DðÞD= B ¼ u 2 L ðÞD=M : D= u 2 L ðÞD=M and Ru 2B ;
where R denotes the trace map. In particular, the choice E¼0yieldD= min
and D= max ¼ D= 1 . H2D= E Two conditions we require of the local boundary condition B are as follows:
~[ (i) Self-adjointness, which by Section 3.5 in [14] occurs if and only if symD= ðn Þ maps the L2 closure of B onto its orthogonal complement. (ii) D-ellipticity,= which is defined in terms of a self-adjoint boundary operator @= = ~[ 1 ; adapted to D with principal symbol sym@=ðnÞ¼symD= ðn Þ symD= ðnÞ and for which the operator @= : 2 DR= 2 DR= pB v½Þ0;1 ðÞ L ðÞ! L ðÞ 2 is a Fredholm operator. Here, pB : L ðDR= Þ!Bis projection induced from the : DR= ; @= fibrewise orthogonal projection pE !E and v½0;1Þð Þ is the projection onto the positive spectrum of the operator @= (see Theorem 3.15 in [14]). This condition yields regularity up to the boundary, in the sense that D= u 2 k D= kþ1 D= = : Hlocð MÞ if and only if u 2 Hloc ð MÞ whenever u 2DðDBÞ For a compact ; set K M the constant CK such that 1 C jjujj= k ; jjujj k; CKjjujj= k ; K DB K H K DB K = 2 2 we call the D-ellipticity constant of order k in K. Here, jjujjT;K ¼jjvK Tujj þ 2: jjvK ujj See Section 7.3-7.4 in [13] as well as Section 3.5 in [14].
We now state our perturbation result for the Atiyah–Singer Dirac operator D= B with a local e boundary condition B: For two local boundary conditions B and B; following Section 2 in e Chapter IV in [15], we define the L1-gap between the subspaces B and B as ÀÁ ÀÁ ^ ; e ^ ; e ; d1 B B ¼jjd Ex E x jjL1ðÞR ¼ sup jpEðÞx peðÞx j x2R E e where pE and pe are the orthogonal projections from DR= to E and E respectively. We E ; e: > let jjBjjLip ¼ supx2R jrpEðxÞj and similarly for B For a set Z Mand r 0, we write : ; < ; R Zr ¼fx 2M qgðx ZÞ rg and Zr t Zr to be the double of a neighbourhood by pasting along R. Theorem 3.1. Let ðM; gÞ be a smooth, Spin manifold with smooth, compact boundary R ¼ @M that is complete as a metric space and suppose that there exists:
(i) a precompact open neighbourhood Z of R and j > 0 such that injðM n Z; gÞ > j; (ii) CR < 1 such that jRicgj CR and jrRicgj CR on MnZ, and R (iii) a smooth metric gZ on the double Z4 t Z4 obtained by pasting along and < j > ; j : CZ 1 and Z 0 with jRicgZ j CZ and injðZ2 t Z2 gZÞ Z 1262 L. BANDARA AND A. ROS EN
e Fixing CB < 1, let B and B be two local self-adjoint D= -elliptic boundary which satisfies: e iv. jjBjjLip þjjBjjLip CB, and v. D= -ellipticity constants of orders 1 and 2 in a given compact neighbourhood K of the boundary.
; = > 1 o Then, for x 2ð0 p 2Þ and r 0, whenever we have f 2 Hol ðSx;rÞ, we have the per- turbation estimate ÀÁ ÀÁ ^ e = = ~ Շ ; ; jjf ðÞ DB f DB jjL2!L2 jjf jj1d1 B B where the implicit constant depends on dim M and the constants appearing in (i)-(v). Remark 3.2. The double of a smooth manifold with boundary by pasting along that boundary is again smooth (in terms of the differentiable structure). However, the canonical reflection of the metric may fail to be smooth across the boundary. The exist-
ence of a metric gZ satisfying the assumed curvature bounds on Z2 t Z2 is always guar- anteed, but we have included this in order to quantify the dependence of the constants in the perturbation estimate. See Section 3.1 for more details.
Example 3.3 (Boundary conditions in even dimensions). For M even dimensional, the 6 Spin bundle splits D=M¼D=þM ?D= M (where D= M are the eigenspaces of u 7!~n u) and 0 D= D= ¼ ; D= þ 0 6 7 6 7 where D= 6 : D= M!D= M: Again by even dimensionality, ~n : D= R ! D= R: Let B 2 EndðD=þRÞ smooth and invertible, and define ÈÉ D= R ;~ : D=þR D= R D= R; B;x ¼ ðÞw n Bw w 2 x and B ¼tx2M B;x which is a smooth sub-bundle of DR= : The boundary condition as considered by 1 D= R : Gorokhovsky and Lesch in [16] is then given by BB ¼ H 2 ð B Þ When the boundary condition defining endomorphism B further satisfies BðxÞ ¼ ; = = 1 D= ; BðxÞ then the boundary condition BB is D-elliptic and DB on Cc ð M BBÞ is essentially self-adjoint. These facts are a consequence of Corollary 3.18 in [14], which guarantees = D= R D=?R D-ellipticity of the boundary condition BB since sym@=ðnÞ interchanges B and B for R: 0 6¼ n 2 Tx The essential self-adjointness follows from invoking Theorem 3.11 in [14], ~ 2 ? ~ ; : since symD= ðnÞ interchanges BB with its L -orthogonal complement BB ¼fðBn v vÞ 1 1 v 2 H2ðD= RÞg in H2ðDR= Þ:
Example 3.4. As noted in [5], Chiral conditions arise from an associated Chirality oper- ator G 2 C1ðLðD=MÞÞ satisfying: for all X 2 C1ðTMÞ and w; u 2 C1ðD=MÞ; 2 G ¼ I; hGu; Gwi¼hu; wi; rXðÞ¼Gw GrXw; X Gu ¼ GX u; 1 ~ : and the boundary condition is defined via the projector pGu ¼ 2 ðI n GÞ This is a self-adjoint local elliptic boundary condition which exists in any dimension (given the COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1263 map G), and has been used in the study of asymptotically flat manifolds including black holes. See Section 5.2 in [5] for more details. ^ e Proof of Theorem 3.1. Without loss of generality, we can assume that d ðB; BÞ 1=2; ^ e 1 as the estimate is trivially true from the spectral theorem for d ðB; BÞ> 1=2: Note that e 1 ~ DR= since the projectors pE and pE on to E and E respectively are orthogonal, jj2pE Ijj1 ¼ 1 and so we obtain:
1 (i) jjpE p~ jj , and E 1 2jj2pE Ijj1 ~ (ii) jjrpEjj1 þ jjrpE jj1 CB
^ e 1 We claim that there exists a U 2 LipðLðD=MÞÞ with jjU Ijj d1ðB; BÞ and e 1 2 Շ : 1 ~ jjrUjj CB such that UB¼B To see this, set U0 ¼ 2 ðI þð2pE IÞð2pE IÞÞ and it is 1 ~ : > ; R ; easy to see that pE ¼ U0 pE U0 Fix 0 such that ½0 Þ ffi N where N ¼fx 2 0 0 M : qðx; RÞ < g and note that U0 extends to a projection U ðxÞ¼U0ðx Þ for x ¼ ðt; x0Þ2½0; Þ R: Then U is given by: 8 < I x 62 N ; q ; R q ; R UðÞx ¼ : ðÞx 0 ðÞx : I U ðÞx þ I x 2 N
We verify the hypotheses (A1)–(A9) and invoke Theorem 2.1 with V¼D=M; D ¼ D= B e 1 = ~ and D ¼ U DB U to obtain the estimate ÀÁ 1 = = ~ Շ : jjf ðÞ DB f U DB U jjL2!L2 jjI Ujj1jjf jj1
The passage from this to the required estimate follows from the fact that we = 1= ~ 1 = ~ have jjI Ujj1 1 2 by noting that f ðU DB UÞ¼U f ðDB ÞU and that = ~ 1= ~ Շ : jjf ðDB Þ f ðU DB UÞjjL2!L2 jjI Ujj1jjf jj1 The first hypothesis (A1) is immediate and (A2) and (A3) are a consequence of the fact that the curvature assumptions imply that Ricg CR (c.f. Theorem 5.6.4 and 5.6.5 in [17]). The existence of GBG frames satisfying the required bounds in (A4), (A5), and (A6) follow from Proposition 3.6, which only depend on CR, j, CZ and jZ. See Section 3.1. Since we assume that B is a local boundary condition, we have that for every g 2 1 L \ LipðMÞ; the domain inclusion gDðD= BÞ DðD= BÞ holds. The commutator estimates follow from the fact that ÂÃ ½ D= ; g u ¼ dg u and U 1DU= ; g u ¼ U 1dg Uu:
This shows (A7). The hypothesis (A8) is a consequence of Proposition 3.8 and 3.9 since we assume e that B and B are D-elliptic= boundary conditions. Note that the constant arising from these propositions include the constant Cell;K in the ellipticity estimate
1 C jjujj jjujj 1 C ; jjujj ell;K D= B;K H ;K ell K D= B;K 1264 L. BANDARA AND A. ROS EN whenever u 2DðD= BÞ: The corresponding constant in the region MnK depends on the geometric bounds (i)–(iii). In addition to these constants for D= B; the corresponding esti- = ~ mate for the operator DB includes the constant CB. See Section 3.2 for details. The remaining hypothesis is the Riesz-Weitzenbock€ hypothesis (A9). This is proved similar to Proposition 3.8, using the compact set K and K1 near the boundary, along 2 with the smooth cut-off f as they appear in the proof of this proposition. The estimate 2 Շ = 2 jjr ðfuÞjj jjDBujj þ jjujj is obtained by arguing as in Proposition 3.18 in [1] via the cover provided by Lemma 3.7, and the remaining estimate jjr2ðð1 f ÞuÞjj e 2 Cell;KðjjD= ujj þ jjujjÞ is due to the boundary regularity result, Theorem 7.17 in [13]. B e Here, ellipticity constant Cell;K is the constant e 1 e C ; jjujj= k ; jjujj k; Cell;K jjujj= k ; ell K DB K H K DB K = k whenever u 2DðDBÞ for k ¼ 1, 2. The constant for the estimate in the region MnK depend on the constants in (i)-(iii). = ~ = Lastly, the decomposition of the operator DB DB ¼ A1rþdivA2 þ A3 distribution- ally proved in Proposition 3.12. See Section 3.3 for details. w Throughout the remainder of this section, we assume the hypothesis of Theorem 3.1.
3.1. Geometric bounds in the presence of boundary
The way in which we prove Theorem 3.1 is via Theorem 2.1, which requires us to prove that under the geometric assumptions we make, the bundle D=M satisfies generalised bounded geometry and the first and second metric derivatives in each trivialisation are bounded. f We do this by considering the double of the manifold M¼MtM; which is obtained by taking two copies of M and pasting along the boundary R to obtain a manifold without boundary. Since the boundary is smooth, this manifold is again smooth (in a differential topology sense, see Theorem 9.29 in [18]). By reflection, we f obtain an extension g of the metric g to the whole of M: This metric is guaranteed ext f to be continuous everywhere and smooth on MnR; but in general, without imposing additional restrictions on the boundary, it will not be smooth. However, as we illustrate in the following lemma, we are able to construct a smooth metric sufficiently close to ; : gext that suffices to obtain the bounds we desire for ðM gÞ Lemma 3.5. e f There exists a smooth complete metric g on M with G 1 dependent on gZ and g satisfying 1 G juj~ juj Gjuj~ g gext g and for which there exists: f (i) je > 0 such that injðM; egÞ > je; e e e (ii) CR < 1 such that jRicg~j CR and jrRicg~j CR; R e f : (iii) a compact set P with P 6¼ Ø and Psuch that gext ¼ g on MnP
e; e The constants j CR and depend on the original geometric bounds j,CR, jZ,CZ. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1265
Proof. Take Z from the hypothesis of Theorem 3.1 and let P¼Z t Z: By hypothesis, since Z is precompact, we get that P is compact. As a consequence, if fxng is a Cauchy f sequence in P; then it converges to some point and if fxng is Cauchy in MnP; then f it converges to some point in MnP by the metric completeness of g: This establishes
that gext is metric complete. 1 f f f Next, let w 2 C ðMÞ be such that w ¼ 1onMnP and w ¼ 0onP3 ¼fx 2 M : 2 ; 3 : qg ðx PÞ 2g Since P is compact by construction, by the smoothness of the differen- ext f; : tiable structure of M there exists G 1 such that gext and gZ are G-close on P2 e e ; Define g ¼ wgext þð1 wÞgZ and since gext ¼ g away from P this shows that the quasi- e isometry with constant G between gext and g and also establishes (iii). Since gZ satisfies a lower bound on injectivity radius on Z2 t Z2 as well as a Ricci curva- ture bound on this set, and since g satisfies similar bounds on Z, by construction of the metric eg; we obtain (i) and (ii) with the dependency as stated in the conclusion. w Now, using this we can prove the main proposition that we require to prove the geo- metric bounds needed to prove Theorem 3.1. Proposition 3.6. > < There exist rH 0 and a constant 1 C 1 depending on j,CR, jZ ; : ; Rn and CZ such that at each x 2M wx Bðx rHÞ! corresponds to a coordinate system and inside that coordinate system with coordinate basis f@jg satisfying: 1 C juj juj Cjuj ; j@kg y j C; and j@k@lg y j C; wxdðÞy gðÞy wxdðÞy ijðÞ ijðÞ
for all y 2 Bðx; rHÞ and where d is the Euclidean metric.
Proof. Utilising the metric eg given by Lemma 3.5, we apply Theorem 1.2 in [19]to 2;a f obtain C -harmonic coordinates for the manifold ðM; egÞ with radius reH : We obtain ; e f: the same conclusions for ðM gjMÞ as it is obtained via the subspace topology on M The balls Bg and Bg~ are contained within the factor G given in the lemma, and away from the compact region P defined in the lemma, we have that Bg ¼ Bg~: So, it suffices f 2;a to set rH ¼ reH =G: On the region MnP; we have C control of the metric eg and outside of this region, by compactness, we obtain control of as many derivatives of the metric as we f like. By taking maximums of the constants appearing in the regions MnP and P; we obtain the constant C in the conclusion of this proposition. w
3.2. The domains of the operators
1 To invoke Theorem 2.1, we need to establish H regularity for the operators D= B and = ~ : DB To this end, we begin with the following covering lemma. Lemma 3.7. < > There exists CH 1,M 0 and a sequence of points xi and a smooth parti- tion of unity fgig for M that is uniformly locally finite and subordinate to fBðxi; rHÞg satisfying: P j ; :::; (i) i jr gPij CH for j ¼ 0 3, and 2: (ii) 1 M i gi
The rH > 0 here is the harmonic radius guaranteed in Proposition 3.6. 1266 L. BANDARA AND A. ROS EN
Proof. Take the double of the manifold and the smooth metric given by Lemma 3.5. f Then, by Lemma 1.1 in [19], on fixing q > 0 we find a sequence of points xi 2 M such e f that (i) fBðxi; rÞg is a uniformly locally finite cover of M for all r q and (ii) e e Bðxi; q=2Þ\Bðxj; q=2Þ¼Ø for all i 6¼ j: This relies purely on a measure counting argu- e ment since g induces a measure satisfying exponential volume growth (Eloc) by the e ; Ricci curvature lower bounds. Since gisG-close to gext the same is true for the metric g ; which is the metric guaranteed to be continuous obtained by reflection of g on M ext f f across R to the double M: Thus, a cover satisfying (i) and (ii) exists on M replacing eg e ext: balls B with gext balls B q = : Now, let rH denote the radius obtained from Proposition 3.6, and set ¼ rH 16 Let M M; R > = : M 0; 0 : fxi g Msuch that qgðxi Þ rH 16 Then fxi g MnZ where Z ¼fx 2M ; R = : R 0 qgðx Þ rH 16g Since is compact, so is Z and hence, there exists a finite number of Z0 K 0 K Z0 ; = : points fxj gj¼1 such that Z [j¼1Bðxj rH 16Þ Then, the collection of points fxig¼ M; Z0 ; = ; = fxj xk g satisfies: M¼[iBðxi rH 16Þ with fBðxi rH 16Þg uniformly locally finite. 2;a Inside each Bð xi; rH=16Þ we have C control of the metric, and therefore, the parti- tion of unity fgjg with the gradient bound in the conclusion is obtained by proceeding as in the proof of Proposition 3.2 in [19]. w With this lemma, we prove the following. 1 Proposition 3.8. The embedding DðD= BÞ ,! H ðVÞ holds along with the ellipticity estimate jjujj ’jjujj 1 for all u 2DðD= Þ: D= B H B
Proof. Let K be a compact neighbourhood of R assumed in (v) of Theorem 3.1 and let f : M!½0; 1 be smooth with f ¼ 1onMnK and f ¼ 0 on an open subset Ke K e with R K: Let u 2DðD= BÞ and we show that jjrðfuÞjj þ jjrðð1 f ÞuÞjj Շ jjD= Bujj þ jjujj: Using the cover guaranteed by Lemma 3.7, we obtain that
jjrðÞfu jj Շ jjD= BðÞfu jj þ jjfujj Շ jjD= Bujj þ jjujj; where the first inequality is from running the exact same argument as Proposition 3.6 in [1] and the second inequality is from the fact that spt rf K and hence bounded. For the remaining inequality, we note that since the boundary condition B is D-elliptic,= kþ1 D= = k D= Theorem 7.17 in [13] gives us that u 2 Hloc ð MÞ () DBu 2 Hlocð MÞ whenever u 2 DðD= Þ: Choosing k ¼ 0, and the fact that spt ð1 f Þu K; we get that B ÀÁ jjrðÞðÞ1 f u jj Cell;K jjD= BðÞðÞ1 f u jj þ jjðÞ1 f ujj Շ jjD= Bujj þ jjujj: where Cell;K < 1 is a constant that depends on K. The estimate jjujj Շ jjujj 1 for u 2DðD= Þ follows from the pointwise estimate D= B H ðVÞ B jD= uj Շ jruj (c.f. Proposition 3.6 in [1]). w Using this proposition, we prove the following.
Proposition 3.9. = = ~ The equality DðDBÞ¼DðDB UÞ holds.
1 D= ; Proof. On fixing u 2 Cc ð MÞ we compute at a point x 2Mwith a frame satisfying : rei ejðxÞ¼0 COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1267 = i a = a i = a b = ; DUðÞu ¼ e rei ðÞu Uea ¼ ðÞeiu e U ea þ u eiUa eb from which it follows directly that jD= ðUuÞj2 jUj2jruj2 þjrUj2juj2: Now, for u 2 e = ~ ; 1 D= ; DðDB Þ we have from Theorem 3.10 in [14] that there is a sequence un 2 Cc ð M BÞ ; = ~ : 0 1 D= ; = such that un ! u in the graph norm of DB Moreover, Uun 2 Cc ð M BÞ DðDBÞ : and by Proposition 3.8, jjrðun uÞjj ! 0 Hence, combining this with our pointwise estimate and integrating, we obtain that = Շ jjDUðÞun Uum jj jjUjj1jjrðÞun um jj þ jjrUjj1jjun umjj ! 0 as m; n !1: By the closedness of D= B; we have that Uu 2DðD= BÞ: The reverse containment is obtained similarly. w
3.3. Decomposition of the difference of operators
A crucial assumption in Theorem 2.1 is to be able to write the difference of our opera- = 1= ~ tors DB and U DB Uas 1 = = ~ DB U DB U ¼ A1rþdivA2 þ A3 : with jjAijj1 controlled by jjU Ijj1 Our computations here are similar to those in Section 3 of [1], with the key observa- tion being that the last term in Lemma 3.10 cannot be used as A3, since it would yield Շ Շ : only a bound jjA3jj1 1 and not jjA3jj1 jjU Ijj1 Instead, we proceed via an appli- cation of the product rule for derivatives as in Lemma 3.11. Throughout this subsection, unless otherwise stated, we fix an open set X Mand let feig and fe=ag be orthonormal frames for TM and D=M respectively inside X. Lemma 3.10. For u 2 C1ðD=MÞ we have the following pointwise equality almost-every- where inside X: ÀÁ = 1 = X a b 1 j = D U DU u ¼ Xru þ Z u þ u rej Ua U e eb with X : CðT M D=MÞ ! CðD=MÞ and ZX : CðDX= Þ!CðDX= Þ with almost-everywhere pointwise estimates Շ X Շ ; jXj jjI Ujj1 and jZ j jjI Ujj1 where the implicit constants depends on the constants in Theorem 3.1. Proof. A direction calculation yields that ÀÁ = a j = a b j = a b j = : DUðÞu ¼rej u e Uea þ u rej Ua e eb þ u Uae rej eb
= 2 = ; 1 Since the term rej eb ¼ xEðejÞ eb multiplying this expression by U on the left, and then subtracting it from the expression for D= u; we obtain that ÀÁÀÁ D= U 1DU= u ¼r ua ej e= U 1ej Ue= ej a a j 2 1 j 2 a b 1 j = : þ e xEðÞej U e xEðÞej U u þ u rej Ua U e eb 1268 L. BANDARA AND A. ROS EN
To obtain a bound on the first expression to the right of this, we note that j 1 j j 1 j e e=a U e Ue=a ¼ e ðÞI U e=a þ ðÞI U e Ue=a; and we can write a j = a j 2 = ; rej u e ðÞI U ea ¼ X1ru u e ðÞI U xEðÞej ea a k = a k = : a k = where X1ðwke eaÞ¼wke ðI UÞea Now, similarly, writing X2ðwke eaÞ¼ a 1 j = ; wkðI U Þe Uea we obtain that aðÞ 1 j = aðÞ 1 j 2 = : rej u I U e Uea ¼ X2ru u I U e U xEðÞej ea Letting X ¼ X þ X ; we obtain that ÀÁ1 2 a j = 1 j = rej u e ea U e Uea ¼ Xru j 2 ðÞ 1 j 2 : e ðÞI U xEðÞej u I U e U xEðÞej u Now, note that j 2 1 j 2 j 2 ðÞ 1 j 2 ; e xEðÞej U e xEðÞej U ¼ e xEðÞej ðÞI U þ I U e xEðÞej U and on setting X j 2 ðÞ 1 j 2 Z ¼ e xEðÞej ðÞI U þ I U e xEðÞej U j 2 ðÞ 1 j 2 ; e ðÞI U xEðÞej I U e U xEðÞej we obtain the conclusion. w This lemma illustrates that the main term to analyse is the last term a b 1 j = : u ðrej UaÞU e eb This is the content of the following lemma. Lemma 3.11. For u 2 C1ðD=MÞ, we have the following decomposition pointwise almost- everywhere inside X: a b 1 j = X X X : u rej Ua U e eb ¼ L ru þ div M u þ N u
The coefficients satisfy the estimates X X X Շ X Շ ; jjL jj1 þjjM jj1 þjjN jj1 jjI Ujj1 and jjrM jj1 1 where the implicit constants depend on the constants listed in Theorem 3.1.
b b b; a b 1 j = Proof. First note that on letting ea ¼ da Ua we have u ðrej UaÞU e eb ¼ a b 1 j = : X : C DX= C X DX= X u ðrej eaÞU e eb Let M ð Þ! ðT Þ written inside as X a h k 1 k = M w ¼ u Ma;ke U e eh with the coefficients to be determined later. Note that: ÀÁ ÀÁ X h a j k 1 k r M u ¼ M ; re u e e U e e=h a k j ÀÁ a h j k 1 k = a h j k 1 k = : þ u rej Ma;k e e U e eh þ u Ma;ke rej e U e eh COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1269
On taking the trace, and rearranging the equation, ÀÁ a h 1 j = X u rej Ma;j U e eh ¼ trgr M u ÀÁ ÀÁ a h 1 j = a h j k 1 k = : rej u Ma;jU e eh u Ma;ktr e rej e U e eh
h h; a b 1 j = : So set Ma;k ¼ a which gives us an expression for u ðrej aÞU e eb It remains to show that the remaining terms in this expression can be decomposed to X X j 1 j h L ru þ Nu: Let L ðe e= Þ¼U e M ; e= ; then we have that a a j h a h j k 1 k = X 1 j X 2 : trg u rej Ma;k e e U e eh ¼ L ru U e M xEðÞej u
Absorbing the error term in this computation along with the remaining term from the former expression, we can set ÀÁ X a h j k 1 k = 1 j X 2 : N u ¼ u atr e rej e U e eh U e M xEðÞej u The estimates in the conclusion for LX; MX; NX and rMX follows from the defini- tions of these maps. w Using these two lemmata, arguing in a similar way to Proposition 3.16 in [1], we obtain the following decomposition globally on M: Proposition 3.12. WeÀÁ have that: 1 = = ~ DB U DB U u ¼ A1ru þ divA2u þ A3u distributionally for all u 2DðD= BÞ where the coefficients Ai satisfy: 1 A1 2 L ðÞLðÞT M D=M ; 1 A2 2 L \ LipðÞLðÞD=M; T M D=M ; 1 A3 2 L ðÞLðÞD=M ; Շ with jjA1jj1 þjjA2jj1 þjjA3jj1 jjI Ujj1. The implicit constants depend on the con- stants listed in Theorem 3.1.
Proof. Following the proof of Proposition 3.16 in [1], it suffices to show that there exists a cover fBjg of balls with a fixed radius r > 0 with orthonormal frames fej;lg inside Bj, and a Lipschitz partition of unity fgjg subordinate to fBjg satisfying: jrej;lj C1 and ; jrgjj C2 where C1 and C2 are finite constants independent of j and l. The covering with the gradient bound on the partition of unity is given in Lemma 3.7 and the uni- form control of jrei;kj C1 is a consequence of the fact that each Bj corresponds to a ; ball in which we have C2 a uniform control of the metric. Then, as in Proposition 3.16 in [1], using Lemma 3.10 and LemmaX 3.11, we set Bj A1u ¼ Xu þ L gju X j Bj A2u ¼ M gju j X X ÀÁ Bj Bj : A3u ¼ ðÞN þ Z gju tr rgj u j j It is readily verified that this yields the desired decomposition. w 1270 L. BANDARA AND A. ROS EN
4. Operator theory and harmonic analysis Throughout this section, we assume the hypothesis of Theorem 2.1. Moreover, we assume that the reader is familiar with the holomorphic functional calculus via the Riesz-Dunford integral and how to estimate functional calculus of non-smooth opera- tors with harmonic analysis. A brief description of this framework is included in Section 2.1 in [1], but [20] is a more detailed reference. For t > 0, define the operators 1 e 1 Rt ¼ ; Rt ¼ ; I þ itD I þ itDe 1 1 ¼ ; e ¼ ; Pt 2 2 Pt 2 I þ t D I þ t2De ; e e e : Qt ¼ tDPt and Qt ¼ tDPt Due to self-adjointness, we have the bounds ð ð 1 1 e 2 dt 1 2 2 dt 1 2; jjQtujj jjujj and jjQtujj jjujj (4.1) 0 t 2 0 t 2 and ; e ; ; e ; ; e 1 : sup jjRtjj sup jjRtjj sup jjPtjj sup jjPtjj sup jjQtjj sup jjQtjj (4.2) t t t t t t 2 Each of these operators are also self-adjoint. We note the identities
e e e ; Rt ¼ Pt iQt and Rt ¼ Pt iQt (4.3) as well as
e e e e e e e e : Rt Rt ¼ Rt½ itðÞD D Rt and Qt Qt ¼ Pt½ tðÞD D Pt Qt½ tðÞD D Qt (4.4)
Using the hypothesis that D De ¼ A rþdiv A þ A ; ÀÁ 1 2 3 e jj Qt Qt f jj e e e jjPtðÞtA1r Ptf jj þ jjPtðÞt div A2 Ptf jj þ jjPtðÞtA3 Ptf jj (4.5) e e e : þjjQtðÞtA1r Qtf jj þ jjQtðÞt div A2 Qtf jj þ jjQtðÞtA3 Qtf jj
4.1. Reduction to quadratic estimates
The goal of this subsection is to prove the following reduction of the main estimate in Theorem 2.1 to the two quadratic estimates appearing the hypothesis of the following proposition. It is these two quadratic estimates that allow us to access real-variable har- monic analysis methods. The proofs of these estimates are given in Sections 4.2 and 4.3 respectively. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1271
Proposition 4.1. Suppose that ð 1 dt jjQe A rðÞiI þ D 1P ujj2 C jjAjj2 jjujj2and t 1 t t 1 1 0 ð 1 e 2 dt 2 2 jjtPt div A2Ptujj C2jjAjj1jjujj 0 t 2 ; = ; 1 o for all u 2 L ðVÞ. Then, for x 2ð0 p 2Þ and r 2ð0 1Þ, whenever f 2 Hol ðSx;rÞ,we obtain that ðÞe Շ jjf D f ðÞD jj jjf jj1jjAjj1 e where the implicit constant depends on C1; C2 and CðM; V; D; DÞ: First, we show that f ðDÞf ðDeÞ can be reduced to a quadratic estimate involving e : the difference of Qt and Qt This is done via (4.5) and we estimate each of these terms using Proposition 4.5 and Proposition 4.7 in [1]. Unlike in the situation of [1] where the boundary was empty, we use the following trace lemma to control the 1 1 estimate on the boundary. In what is to follow, R : H ðVÞ ! H2ðWÞ is the boundary trace map. Proposition 4.2. e e ; e e ; ; Let Ut be one of Rt Pt or Qt and Ut be one of Rt Pt Qt. Then, e Շ : sup jjtUt div A2Utjj jjA2jj1 t > 0
; 1 ; Proof. Fix u v 2 Cc ðV BÞ and note that
hðÞ div A2u; v ¼ hðÞA2u; rv þ divWuðÞ; v ; ; j i l k where Wðu vÞ¼ðA2Þiku djlv dx inside an orthonormal frame, readily checked to be a well-defined covectorfield. By Stokes’ theorem, ð ÀÁ ~ hdivA2u; vi hA2u; rvi¼ g WuðÞ; v jR; n dr: R By Cauchy-Schwartz, compactness of R and smoothness of ~n; we obtain that ð ÀÁ ; ;~ Շ R R : g WuðÞv jR n dr jjA2jj1jj ujjjj vjj R
Next, note that whenever u 2DðDÞ we have that u 2Dðdiv A2Þ and there exists a 1 ; sequence un 2 Cc ðV BÞ such that un ! u in DðDÞ by the essential self-adjointness : : : 1 of D We prove that un ! u in DðdivA2Þ To prove this, note that A2 C ðVÞ ! 0;1 C ðT M MÞ and fix a point x 2M; choose an orthonormal frame feig for V and i i 1 ; jk i k ; fdx g for T M with rei ¼rdx ¼ 0atx. For w 2 C ðVÞ A2w ¼ðA2Þi w dx ej and X X jk i k @ j i j @ i : div A2w ¼ trr ðÞA2 i w dx ej ¼ kðÞA2 ik w þ ðÞA2 ik kw Þej k k 2 Շ 2 2 2 2: ; Thus, jdiv A2wj jjrA2jj1jwj þjjA2jj1jrwj Now, writing w ¼ un um we obtain that 1272 L. BANDARA AND A. ROS EN
2 Շ 2 2 2 2: jjdiv A2ðÞun um jj jjrA2jj1jjun umjj þjjA2jj1jjrðÞun um jj ; Շ : Since un 2DðDÞ we have that jjrðun umÞjj jjDðun umÞjj þ jjun umjj Thus, we have that un ! u and divA2un ! v and since divA2 is closed as A2 is bounded, we obtain u 2DðdivA2Þ and v ¼ divA2u: ; 2 : 1 ; ; Now, let u v 2 L ðVÞ Since we assume that D is essentially self-adjoint on Cc ðV BÞ ; 1 ; e ; there exist sequences un vm 2 Cc ðV BÞ such that un ! Utu and vm ! Utv with con- vergence in DðDÞ; DðrÞ and DðdivA2Þ by what we have already established. Thus, e jhtUt div A2Utu; vij ¼ j lim ht div A2un; vmij m;n!1 jh ; r ij þ jj jj jjR jjjjR jj ;lim tA2un vm ;lim A2 1t un vm m n!1 ÀÁm n!1 e Շ lim jjA2jj jjunjj jjtDvmjj þ tjjvmjj m;n!1 1 e þjjA2jj tjjRUtujjjjRUtvjj ÀÁ1 pffiffi Շ R ; jjA2jj1 jjujj þ tjj Utujj jjvjj whereffiffi the last inequality follows from the standard boundary trace inequality. on p e e e e e tjjRUtvjj and from the uniform bounds on jjtrUtvjj Շ jjtDUtvjj þ jjtUtvjj and e tjjUtvjj: We obtain the conclusion by estimating jjRUtujj similarly. w As a consequence of this proposition and (4.5), we obtain e Շ : sup jjUt Utjj jjAjj1 t2ð 0;1 Using this, arguing exactly as in Section 4.2 in [1], we can reduce the required esti- mate in the conclusion of Proposition 4.1 to proving a quadratic estimate: ð 1 ÀÁ e 2 dt Շ 2 2 jj Qt Qt ujj jjAjj1jjujj 0 t for all u 2 L2ðVÞ: From (4.5), we obtain that ! ð 1 1 ÀÁ2 e 2 dt jj Qt Qt ujj 0 t ! ! ð 1 ð 1 1 2 1 2 e 2 dt e 2 dt jjPttA1rPtujj þ jjPttdivA2Ptujj 0 t 0 t ! ð 1 1 2 e 2 dt (4.6) þ jjPttA3Ptujj 0 t ! ! ð 1 ð 1 1 2 1 2 e 2 dt e 2 dt þ jjQttA1rQtujj þ jjQttdivA2Qtujj 0 t 0 t 1 dt 2 þjjQe tA Q ujj2 : t 3 t t COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1273
Estimating as in Proposition 4.7 in [1], we bound the first, third and sixth term by 2 2: jjAjj1jjf jj The second and forth terms are controlled by the hypothesis of Proposition 4.1. The only term that remains to be bounded is the penultimate term in this expression for which the estimate in Proposition 4.7 in [1] does not work. The way in which we estimate this term requires a slight excursion into interpolation theory. Let H1ðVÞ denote the first-order Sobolev space on V and define ÂÃ s 2 ; 1 ; H ðÞV ¼ L ðÞV H ðÞV h¼s ; ; for s 2½0 1 where ½ h represents complex interpolation. Also, let jj jj ÂÃ s 1 Hs ; s s ; s 2 ; 1 : H0ðÞV ¼ Ccc ðÞV H ðÞV ¼ H0ðÞV and H00ðÞV ¼ L ðÞV H0ðÞV h¼s In order to gain an explicit expression for the norms in these interpolation scales, we connect these spaces to domains of operators. Let rN ¼ r2 and rD ¼ r0 ; where r2 : 1 2 1 2 : 1 1 : C \ L ðVÞ ! C \ L ðT M VÞ and r0 Ccc ðVÞ ! Ccc ðT M VÞ The subscripts “ ” “ ” 1ðVÞ ¼ N and Dpffiffiffiffiffiffiffiare chosen for Neumann andpffiffiffiffiffiffi Dirichlet respectively since H D 1 D ; D D DðrNÞ¼Dð NÞ and H0 ¼ DðrDffiffiffiffiffiffiffiÞ¼Dð DÞ where N ¼rffiffiffiffiffiffiN rN and D ¼ p p rD rD: Moreover, jj jj 1 ’ jjðI þ DNÞ jjand jj jj 1 ’ jjðI þ DDÞ jj: H H0 Consequently, by Theorem 6.6.9 in [21], we have that: ÂÃÀÁ pffiffiffiffiffiffiffi s HsðÞV ¼ L2ðÞV ; H1ðÞV ¼D I þ D ; h¼s ÀÁN ÂÃ pffiffiffiffiffiffi s s ðÞ 2ðÞ; 1ðÞ D ; H00 V ¼ L V H0 V h¼s ¼D I þ D and in particular for s 2½0; 1 ; pffiffiffiffiffiffiffi s pffiffiffiffiffiffiffi s D D : jj jjHs ’jj I þ N jj and jj jjH s ’jj I þ N jj 1 , 1 0 , 0 ; Since the identity map embeds H00ðVÞ ! H ðVÞ and H00ðVÞ ! H ðVÞ we have by interpolation that pffiffiffiffiffiffi s pffiffiffiffiffiffiffi s D s , s D D I þ D ¼ H00ðÞV ! H ðÞV ¼D I þ N pffiffiffiffiffiffi for s 2ð0; 1Þ: Similarly, since DðDÞ ¼ DðjDjÞ; where jDj¼ D2 and jjðI þjDjÞujj ’ jjujj þ jjDujj; by the same Theorem 6.6.9 in [21], ÂÃÀÁ ÀÁ 2 ; s s : L ðÞV DðÞD h¼s ¼D jDj ¼D ðÞI þjDj The following key result is well known in the case of functions on the upper half space and smooth Euclidean domains by the work of Bergh and Lofstr€ om€ in [22]or Triebel in [23]. The following is a vector bundle version which, to our knowledge, does not seem to have been treated previously in the literature. Lemma 4.3. s s s < = : The equality H ðVÞ ¼ H0ðVÞ ¼ H00ðVÞ holds whenever 0 s 1 2
Proof. Now let U0 ¼MnZ; where Z is a smooth precompact open neighbourhood of R @ ; ; : Rn ; :::; ; ¼ M and ðuj wj UjÞ trivialisations wj inside charts uj Uj ! þ for j ¼ 1 M M : : so that M ¼[j¼0Uj Let fgjg be a smooth partition of unity subordinate to fUjg We > can choose gj such that jrgjj C for some C 0. 1274 L. BANDARA AND A. ROS EN
Define: M B ¼ L2ðÞV ; A ¼ L2ðÞV L2 Rn ; CN 0 0 þ M B ¼ H1ðÞV ; A ¼ H1ðÞV H1 Rn ; CN 1 1 0 þ M 0 1 ; 0 1 1 Rn ; CN : B1 ¼ H0ðÞV A1 ¼ H0ðÞV H0 þ
Now, define S : B ! A by 0 0 ; 1; :::; 1 ; Su ¼ g0 w1ðÞg1u u1 wMðÞgMu wM with j-th coordinate map extended to 0 outside of the support of gj, and note S is an 0 0: : injection. Moreover, it is also a map B1 7! A1 and B1 7! A1 Also, define R A0 ! B0 by ; ; :::; 1 ::: 1 : RuðÞ0 u1 uM ¼ u0 þ g1w1 ðÞu1 u1 þ þ gMwM ðÞuM uM 0 0: It is also easy to see that this is a map A1 7! B1 and A1 7! B1 ; 0; 0 : Now, note that RS ¼ IonLðBj BjÞ for j ¼ 0, 1 and LðB1 B1Þ That is, R is a retrac- tion and S is a coretraction associated to R. By Theorem ( ) in Section 1.2.4 of [23]we get that S is an isomorphic mapping from HsðVÞ ffi W for s 2ð0; 1Þ where W is a s s Rn ; CN M: s closed subspace of H00ðVÞ H ð þ Þ Similarly, we have that H00ðVÞ ffi W0 with s s Rn ; CN M: W0 is a closed subspace of H00ðVÞ H00ð þ Þ The subspace W is the range of SR s s Rn ; CN M restricted to H00ðVÞ H ð þ Þ and similarly W0 is the range of SR restricted to s s Rn ; CN M: H00ðVÞ H00ð þ Þ But by Theorems 11.1 and 11.2 in [22], we obtain s Rn ; CN s Rn ; CN s Rn ; CN < = ; H0ð þ Þ¼H00ð þ Þ¼H ð þ Þ for 0 s 1 2 and therefore, W0 ¼ W for < = : sðVÞ ¼ s ðVÞ < = : 0 s 1 2 This shows that H H00 pffiffiffiffiffiffiffifor 0 s 1 2pffiffiffiffiffiffi D Շ D To finish off the proof,pffiffiffiffiffiffiffi note that jjðIpþffiffiffiffiffiffi NÞujj jjðI þ DÞujj so through inter- D s Շ D s : 1 s polationp weffiffiffiffiffiffi get jjðI þ NÞ ujj jjðI þ DÞ ujj Since Ccc ðVÞ is dense in H00ðVÞ ¼ D s ; s , s : s , s DððI þ DÞ Þ we have that H00ðVÞ ! H0ðVÞ But we have H0ðVÞ ! H ðVÞ and since s s < = ; w we have already proved H ðVÞ ¼ H00ðVÞ for 0 s 1 2 we obtain the conclusion. With the aid of this lemma, we obtain the following. Proposition 4.4. The quadratic estimate ð 1 e 2 dt Շ 2 jjQtt div A2Qtf jj jjf jj 0 t holds for f 2 L2ðVÞ:
Proof. Fix u 2 L2ðVÞ and estimate e ; ; e R ; Re : hQtt div A2Qtf ui¼ hA2Qtf trQtuiþthA2 Qtf QtuiL2ðÞW It is easy to see that ; e Շ ; jhA2Qtf trQtuij jjA2jj1jjujjjjQtf jj so it remains to consider the boundary term. Note that COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1275
R ; Re Շ R Re : jthA2 Qtf QtuiL2ðÞR j jjA2jj1tjj Qtf jjL2ðÞW jj QtujjL2ðÞW pffiffi Re Շ : By the standard boundary trace inequality, we obtain that tjj QtujjL2ðWÞ jjujj ; ~ ~ To bound Qtf let N be an extension of the normal vectorfield n on a compact neighbourhood around R. Then, ð R 2 2~ tjj Qtf jjL2ðWÞ ¼ t divðjQtf j NÞdl ð M Շ ; 2 t Re gðrN~ Qtf Qtf Þdl þ tjjQtf jj M Շ ; 2: tjhrN~ Qtf Qtf ij þ tjjQtf jj On fixing 0 < s < 1=2, we note that ; Շ ; jhrN~ Qtf Qtf ij jjrN~ Qtf jjH s jjQtf jjHs (4.7) : 1 2 ; Now, note that rN~ H ðVÞ ! L ðVÞ and on defining ðrN~ uÞðvÞ¼ hu rN~ vi for v 2 1 ; : 2 1 1 Cc ðVÞ we obtain that rN~ L ðVÞ ! H0ðVÞ ¼ H ðVÞ boundedly. By interpolation, : 1 ; 2 2 ; 1 we obtain that rN~ ½H ðVÞ L ðVÞ h¼s !½L ðVÞ H ðVÞ h¼s boundedly. Note, however, that ÂÃÂÃ 1 ; 2 2 ; 1 1 s ; H ðÞV L ðÞV h¼s ¼ L ðÞV H ðÞV h¼1 s ¼ H ðÞV and that ÂÃÂÃ 2 ; 1 2 ; 1 s s s ; L ðÞV H ðÞV h¼s ¼ L ðÞV H0ðÞV h¼s ¼ H00ðÞV ¼ H0ðÞV ¼ H ðÞV where we have used that L2ðVÞ is reflexive and Corollary 4.5.2 in [22] in the first equal- ity and that s < 1=2 and Lemma 4.3 in the penultimate equality. On combining these facts, we obtain that Շ : jjrN~ Qtf jjH s jjQtf jjH1 s Moreover, since DðjDjÞ ,! H1ðVÞ and DðjDj0Þ¼L2ðVÞ ,! H0ðVÞ ¼ L2ðVÞ; we have DðjDjqÞ ,! HqðVÞ for q 2½0; 1 by interpolation and hence, ÀÁ q Շ q q ; t jjQtf jjHq jjt I þjDj Qtf jj jjwqðÞtD f jj þ jjQtf jj q 2 1: where wqðfÞ¼fjfj ð1 þ f Þ Thus, tjhr~ Q f ; Q f ij N t ÀÁt ÀÁ Շ 1 s s Շ 2 2 2; t jjQtf jjH1 s t jjQtf jjHs jjw1 sðÞtD f jj þjjwsðÞtD f jj þjjQtf jj and therefore, R Շ 2 2 2: tjj Qtf jjL2ðÞW jjw1 sðÞtD f jj þjjwsðÞtD f jj þ ðÞ1 þ t jjQtf jj Noting that ð 1 2 dt 2 jjwqðÞtD f jj Cqjjf jj 0 t for q 2½0; 1Þ completes the proof. w 1276 L. BANDARA AND A. ROS EN
Remark 4.5. The equation (4.7) demonstrates the necessity of the interpolation methods since we can only conclude the desired quadratic estimates provided a derivative of : order strictly less than 1 is applied to Qtf
4.2. Harmonic analysis I
In this subsection, on drawing from the estimates in Section 5 in [1], we demonstrate how to handle the first quadratic estimate term ð 1 e 1 2 dt Շ 2 2 jjQtA1rðÞiI þ D Ptf jj jjAjj1jjf jj 0 t appearing in the hypothesis of Proposition 4.1. In order to avoid repetition, we encour- age the reader to keep a copy of [1] handy to navigate through the remainder of this article. The following is an itemisation of the notation that we will require from Section 5 of [1]:
k : ; N ; k k; k Dyadic cubes fQa M a 2 Ik k 2 g with centres za 2 Qa where [kQa cover > ; k l k l : M almost everywhere, and when b a Qa \ Qb ¼ ØorQa Qb The cubes are j j of a fixed “length” d 2ð0; 1Þ; and a d cube contains an a0d ball and has diam- j eter at most C1d : The length of a cube Q is denoted ‘ðQÞ: The constant g > 0is an exponent that measures smallness of the volume toward the edge of a cube with constant C2 > 0: See Theorem 5.1 in [1]. J J = ; ; ; The scale is defined as tS ¼ d where C1d q 5 with q ¼ maxfqT M qVg the maximum of the GBG radii of T M and V: j j The collection of dyadic cubes Q ; Q ¼[j JQ ; and Qt for t tS: The unique ancestor Q^ 2 QJ for a dyadic cube Q, the set of GBG coordinates C; which for a cube Q 2 Qj is the GBG trivialisation pertaining to the unique GBG J ball containing the cube in Q containing Q, and dyadic GBG coordinates CJ which is the restriction of this GBG ball toÐ the cube which contains it. The cube integral Bðx^ ; qÞ Q ʯ ðx; QÞ 7!ð ÞðxÞ defined on L1 ðVÞ by Qð ð Q loc i u ðÞx ¼ u ðÞy dlðÞy eiðÞx Q Q Ð where ei is the GBG coordinates of Q, and cube average uQ ¼ 6 Qu inside the GBG coordinate ball of Q and 0 outside it.
> E : 1 1 E ForÐ t 0, the dyadic averaging operator t LlocðVÞ ! LlocðVÞ given by tðxÞ¼ ð 6 QuÞðxÞ where x ʯ Q: i CN CN; For a w ¼ w ei 2 the locally constant extension inside the GBG coordi- c i nates of Q are given by x ðxÞ¼w eiðxÞ and zero outside of this coordinate ball. Q Given a t-uniformly bounded family of operators t, define the principal part cQ : CN Q cQ Q xc : t ðxÞ ffiVx !Vx of t by by t ðxÞw ¼ð t ÞðxÞ COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1277
The following is a key lemma that is necessary in order to adapt the arguments of Section 5 of [1] to our manifold with boundary. It allows us to ensure that we can use a cut-off that restricts the estimates away from the boundary. e k Lemma 4.6. There exist constants k0; eg; C3 > 0 such that for all cubes Q 2 Q with k > k0 and Q \ R 6¼;, we have ÈÉ e ~g l x 2 Q : qðÞx; R s‘ðÞQ C3s lðÞQ :
k In particular, for every Q 2 Q with k > k0; ÈÉ e ~g l x 2 Q : qðÞx; MnðÞQ n R s‘ðÞQ C3s lðÞQ : e e The constants g and C3 depends on g; a0 and C1 from Theorem 5.1 in [1]. Proof. Let Z ¼fx 2M: qðx; RÞ eg with e < 1 chosen sufficiently small so that Z is a e smooth compact submanifold of M with smooth boundary R. Let Z be the smooth compact manifold without boundary obtained by taking two copies of Z and identifying the boundaries, and extending the metric appropriately. This metric is C0 and there exists a smooth C1 metric G-close to g for some G 1: Consequently, without loss of e generality, we assume that the metric extension is smooth. Let kR ¼ injðZÞ > 0: e By the compactness of Z; we use Theorem 1.2 in [19]toobtainCR 1 such that for each e ; ; 1 R; x 2 Z ðwx Bð2 k xÞÞ is a coordinate chart with 1 CR juj juj CRjuj ; wxdðÞy gðÞy wxdðÞy for each y 2 Bð1 kR; xÞ; and where d is the Euclidean metric in that chart. In particular, e 2 e since Z Z and the topology of Z is the subspace topology inherited from Z; we get ; 1 R; ; that this holds for balls B(x, r)inZ as well. From this, inside ðwx Bð2 k xÞÞ on letting ; L L; q ðx yÞ¼jwxðxÞ wyðyÞj and ¼ wx n n 1 2 2 CR q ðÞx; y qðÞx; y CR q ðÞx; y and CR dL dl CRdL : (4.8) k > 0 < 1 R: > ; Now, fix k0 0 such that so that C1d 10 k Then, for all k k0 whenever k Q ; ; 1 R ; Q 2 we have that Q BðxQ 2 k Þ which corresponds to a coordinate system with control on the metric and measure as we have describe before. k Fix such a cube Q 2 Q and define QR;s ¼fx 2 Q : qðx; RÞ s‘ðQÞg and note that on using (4.8), n o : ; Rn 1 k : wQðÞQR;s ER;s ¼ x 2 wQðÞQ qRn x \ wQðÞQ CRsd ; k ; k ; k Similarly, we have that wQðBðxQ C1d ÞÞ BRn ðxQ CRC1d Þ BoxRn ðxQ CRC1d Þ where x ¼ w ðx Þ and BoxRn ðx; lÞ is a Euclidean box centred at x of length l. Then, Q Q Q n 1 n k k LðÞER;s L R \ BoxRn xQ; CRC1d CRsd ÀÁ k n 1 k n n 1 nk : CRC1d CRsd ¼ CRC1 d s ; k ; 1 k Similarly, we have that wQ BxQ a0d BRn xQ CR a0d , and 1278 L. BANDARA AND A. ROS EN
n 2 lðQR;sÞ lðQR;sÞ CRLðER;sÞ n l ; k 2 1 k ðQÞ lðBðxQ a0d Þ CR LðBRn ð xQ; CR a0d ÞÞ n n 1 nk 3n n 1 n CRC1 d s CR C1 CR ¼ s; 1 k n n xnðCR a0d Þ xna0 where the first estimate follows from Theorem 5.1 (v) in [1], the second estimate from our previous calculation combined with (4.8), and where xn is the volume of the ball of unit radius in Rn: no C3nCn 1 e ; e ; R 1 ; Set g ¼ maxf1 gg and C3 ¼ max C3 x an and noting ÈÉn 0ÈÉ x 2 Q : qðÞx; MnðÞQ n R s‘ðÞQ ¼ x 2 Q : qðÞx; MnQ s‘ðÞQ [ QR;s; completes the proof. w
Proposition 4.7. The quadratic estimate ð 1 e 1 2 dt Շ 2 2 jjQtA1rðÞiI þ D Ptujj jjAjj1jjujj 0 t holds for all u 2 L2ðVÞ, with the implicit constant depending on CðM; V; D; DeÞ:
Proof. We split the estimate as follows: ð ð 1 dt 1 dt jjQe A rðÞiI þ D 1P ujj2 Շ jj Qe c E A rðÞiI þ D 1P ujj2 t 1 t t t t t 1 t t 0 0ð 1 dt þ jjc E A rðÞiI þ D 1ðÞI P ujj2 t t 1 t t ð0 1 E 1 2 dt : þ jjct tA1rðÞiI þ D ujj 0 t Now, we note that the off-diagonal decay given in Lemma 5.9 in [1] is valid for our e operator QtA1 due to the local boundary conditions encoded in assumption (A7). Thus, we can apply Propositions 5.4, Lemma 5.8 and Proposition 5.12 in [1] to estimate the terms appearing in this decomposition. We give a brief description of how this is done. The first term is estimated by using an argument similar to the proof of Proposition : 5.4 and Theorem 2.4 in [1], with W¼T M V It suffices to note that since jjujjD ’ ; jjujjH1 for u 2DðDÞ this argument can be run in verbatim. It simply remains to prove Շ 1: jjrSujj jjujjH1 for S ¼rðiI þ DÞ This argument is included in the proof of Theorem 2.4 in [1] on noting that the argument runs in verbatim due to assumption (A9). For the middle term in the estimate, we use the argument in proving Proposition 5.10 in [1]. This argument is straightforward from establishing the cancellation lemma, Lemma 5.8 in [1]. To prove this lemma, we note that for each dyadic cube Q, and for each u 2DðDÞ with spt u Q \M; we have that ð ð
1 1 Dudl Շ lðÞQ 2jjujj and rudl Շ lðÞQ 2jjujj; Q Q where the implicit constants depends on CðM; V; D; DeÞ: On coupling these estimates with Lemma 4.6, we obtain the statement of Lemma 5.8 in [1] in our present context. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1279
The last term is obtained by a straightforward application of Proposition 5.12 in [1]. w
4.3. Harmonic analysis II
In this subsection, we prove the remaining estimate ð 1 e 2 dt Շ 2 2 jjtPtdivA2Ptujj jjAjj1jjujj 0 t for all u 2 L2ðVÞ: It is in the proof of this estimate where the main novelty of the har- monic analysis in this article can be found. A key difficulty here is that the off-diagonal 2 e decay - and even L -boundedness - of tPtdivA2; which holds when M has no bound- ary, is not valid due to the fact that A2 does not preserve boundary conditions. Despite e this obstacle, on considering the operator tPtdivA2Pt instead as a whole, we are able to prove the required quadratic estimate. Our approach here is motivated by a similar argument in [7] by Auscher, Axelsson (Ros en) and Hofmann. For the remainder of this subsection, let e Ht ¼ tPtdivA2Pt H H H c ; c and let ct denote the principal part of t we recall is ct ðxÞw ¼ð tx ÞðxÞ where x is N the constant section related to w 2Vx ffi C : Lemma 4.8. H > The operators t are uniformly bounded in t 0 and have the off-diagonal decay estimate: there exists CH > 0 such that, for each M > 0, there exists a constant CD;M > 0 with qðÞE; F M qðÞE; F jjvEH ðÞvFu jj 2 CD; jjAjj exp CH jjvFujj 2 ; t L ðÞV M 1 t t L ðÞV for every Borel set E; F M; u 2 L2ðVÞ, and where hai¼maxf1; ag: H Proof. Uniform bounds for t were proved in Proposition 4.2. Building on this, we prove the off-diagonal estimates in the conclusion by reduction to corresponding such e estimates for the resolvents Rt and Rt; which are immediate by replicating the argument of Lemma 5.3 in [9] in light of (A7). Given E; F M Borel with qðE; FÞ > 0; pick g 2 C1ðMÞ such that gðxÞ¼1 when qðx; EÞ < 1=3 qðE; FÞ and gðxÞ¼0 when qðx; FÞ < 1=3 qðE; FÞ so that Շ = ; : e jjrgjj1 1 qðE FÞ It suffices to prove the required estimates for RttdivA2Rt since by replacing t by – t in the estimates below and noting Pt ¼ðRt þ R tÞ=2 and similarly Pe ¼ðRe þ Re Þ=2 yields the bound for H . Now, note that t t t t Âà e ; e jjvERttdivA2RtðÞvFu jj ¼ jjvE g RttdivA2Rt vFujj and Âà g; Re tdivA R t 2 t Âà ÀÁ e e e e e e ¼ Rt g; itD RttdivA2Rt þ Rt½ g; tdiv A2Rt RttdivA2Rt ½ g; itD Rt: Since ½g; De ; ½g; div ; ½g; D are multiplication operators whose L1 norm is bounded by
jjrgjj1 and supported on 1280 L. BANDARA AND A. ROS EN 1 1 G ¼ x 2M: qðÞx; E qðÞE; F and qðÞx; F qðÞE; F ; 3 3 e 2 2 we obtain the conclusion from off-diagonal estimates for Rt : L ðG; VÞ ! L ðE; VÞ and 2 2 e Rt : L ðF; VÞ ! L ðG; VÞ; and from uniform bounds on RttdivA2Rt from Proposition 4.2. w Next, we split the required estimate in the following way: ð ð ð 1 dt 1 dt 1 dt jjH ujj2 jjH ðÞI P ujj2 þ jjðÞH c E P ujj2 t t t t t t t t t t 0 0 ð0 ð 1 1 (4.9) E 2 dt E 2 dt þ jjct tðÞPt I ujj þ jjct tujj 0 t 0 t The first three terms to the right of this expression can be handled relatively easily as the following lemma demonstrates. Lemma 4.9. We have that: ð ð ð 1 1 1 H 2 dt H E 2 dt E 2 dt Շ 2 2: jj tðÞI Pt ujj þ jjðÞt ct t Ptujj þ jjct tðÞPt I ujj jjAjj1jjujj 0 t 0 t 0 t
Proof. For the first term, we estimate by noting that ÀÁ H H e ; tðÞ¼I Pt ttDQt ¼ tPtdivA2Qt Qt we obtain the required quadratic estimate using Proposition 4.2 to assert uniform e bounds for tPtdivA2Qt and by noting that Qt satisfies quadratic estimates (4.1). The two remaining estimates are handled via Propositions 5.4 and Proposition 5.10 in [1] with S ¼ I: The versions of these propositions in our current context can be obtained exactly the way described in the proof of Proposition 4.7. w Thus, we have left with the last term in this expression, which we reduce to a Carleson measure estimate. That is, by Carleson’s Theorem, the estimate of this term is obtained by proving that
dlðÞx dt d ðÞx; t ¼jc ðÞx j2 t t is a Carleson measure. This is obtained if we prove for each cube Q 2 Q; and for Carleson regions RQ ¼ Q ð0;‘ðQÞÞ; ðð 2 dlðÞx dt Շ 2 : jctðÞx j jjAjj1lðÞQ (4.10) RQ t
The estimate we perform here is more intricate and involved than the Carleson meas- ure estimate in Proposition 5.12 in [1], and we provide full details. First, observe the following important reduction. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1281
Lemma 4.10. Suppose that for every cube Q 2 Q with ‘ðQÞ qðQ; RÞ the Carleson esti- mate (4.10) holds. Then, (4.10) holds for every cube Q 2 Q: Qj; ; Proof. Fix Q 2 with j ¼ maxfk0 Jg (with k0 coming from Lemma 4.6), and define the two sets ÈÉ A¼ÈÉQ0 2 Q : Q0 Q and qðÞQ0; R ‘ðÞQ0 ; B¼ Q0 2 Q : Q0 Q and qðÞQ0; R <‘ðÞQ0 :
0 0 Now, consider the dyadic Whitney region W 0 ¼ Q ðd‘ðQ Þ;‘ðQÞÞ so that Q RQ ¼[WQ0 [[WQ0 : Q02A Q02B
00 0 0 00 Note that Q Q and Q 2Aimplies that Q 2A: Setting Amax to be the maximal cubes in A; we obtain that
[ WQ0 ¼[RQ0 : 0 0 Q 2A Q 2Amax On using the hypothesis, we obtain that ðð X l X 2 d dt Շ 2 0 Շ 2 jctj jjAjj1 lðÞQ jjAjj1lðÞQ 0 R 0 t 0 Q 2Amax Q Q 2Amax by the disjointedness of the cubes in Amax: H H : 1 2 ; Next, note that from the off-diagonal decay of t, we obtain that t L ðVÞ ! LlocðVÞ and reasoning as in Section 5.2 in [1], which comes from Corollary 5.3 in [8], we have that ð = 2 Շ 0 jctj dl lðÞQ Q0 and therefore, ð ð‘ dldt ðÞQ0 dt jc j2 Շ lðÞQ0 Շ lðÞQ0 : t ‘ 0 t ðÞQ0 t Q 2 k 0 k Now, fix k > j and note that d ‘ðQÞ and for every cube Q 2Bk ¼B\Q ; we 0 k have that Q fx 2 Q : qðx; RÞ ðC1 þ 1Þd g: On invoking Lemma 4.6 with s ¼ k 1 d ðC1 þ 1Þ‘ðQÞ ; we obtain that ~ ÈÉdkg 0 Շ : ; R ‘ Շ Շ ; lðÞQ l x 2 Q qðÞx s ðÞQ ~ lðÞQ lðÞQ ‘ðQÞg where the second inequality follows from dk ‘ðQÞ: Note now that if Q0 2B and Q00ˆQ0 then ‘ðQ00Þ d‘ðQ0Þ and therefore, ÀÁÀÁ 0 0 0 00 00 00 WQ0 ¼ Q d‘ðÞQ ;‘ðÞQ \ Q d‘ðÞQ ;‘ðÞQ ¼ WQ00 ¼;; and therefore X ðð X X ðð 2 dldt Շ 2 dldt Շ ; jctj jctj lðÞQ 0 W 0 t 0 W 0 t Q 2B Q k > j Q 2Bk Q which completes the proof. w We finally prove (4.10) for the remaining cubes Q bounded away from R. 1282 L. BANDARA AND A. ROS EN
Proposition 4.11. Suppose that qðQ; RÞ ‘ðQÞ. Then, the Carleson measure estimate (4.10) holds.
CN; : ; Proof. Fix w 2 let fQ M!½0 1 with spt fQ compact, and fQ ¼ 1onQ and 0 out- 1 c i side BðxQ; 2‘ðQÞÞ with jrfQj Շ ‘ðQÞ : Define wQðxÞ¼fQðxÞw ðxÞ¼fQðxÞw eiðxÞ inside ; ; ; E c: BðxQ^ qÞ the GBG trivialisation of Q. Note that, for x 2 Q and t tS twQðxÞ¼w Since the metric h is uniformly comparable to the trivial metric inside this trivialisation, and using the facts we have just mentioned, ðð ðð dldt dldt jc j2 Շ sup jc E w ðÞx j2 : t t t t Q t RQ jwjd¼1 RQ We split ðð dldt jc E w ðÞx j2 t t Q t ððRQ ðð E H 2 dldt H 2 dldt : jðÞct t t wQðÞx j þ j twQðÞx j RQ t RQ t On following the exact same argument as in Proposition 5.11 in [1], noting that this proof only requires that H satisfies the off-diagonal estimates, we obtain that ðð t E H 2 dldt Շ 2 : jðÞct t t wQðÞx j jjAjj1lðÞQ RQ t For the remaining part, let e e HtwQ ¼ tPtdivA2ðÞPt I wQ þ tPtdivA2wQ: R We first obtain the required estimate on the second term. For that, observe wQ ¼ 0 near : e e e e 1 ; and hence, A2xQ 2DðdivminÞ Using the identity tPtdivmin ¼ðQt þ itPtÞðrðiI DÞ Þ we estimate ð ‘ðÞQ e 2 dt Շ e 1 2 Շ 2 : jjtPt div A2wQjj jjðrðiI DÞ Þ A2wQjj jjAjj1lðQÞ 0 t e To estimate the remaining term, we note that tPtdivA2ðPt IÞwQ ¼ e tPt div A2QtðtDwQÞ and so by Proposition 4.2 ÀÁ e e 2 2 2 2 2 2 2 2 2 1 jjtPt div A2 Pt I wQjj Շ t jjAjj jjDwQjj Շ t jjAjj jjrwQjj Շ t jjAjj lðÞQ : 1 1 1 ‘ðÞQ 2 Therefore, ðð ð ÀÁ ‘ðÞQ e e 2 dldt e 2 dt jtPtdivA2 Pt I wQj jjtPtdivA2ðÞPt I wQjj R t 0 t Q ð ‘ðÞQ Շ 2 t Շ 2 ; jjAjj1lðÞQ 2 dt jjAjj1lðÞQ 0 ‘ðÞQ which establishes the conclusion. w COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS 1283
Proof of Theorem 2.1. On combining the estimates in Section 4.3 and Proposition 4.7, the hypothesis of Proposition 4.1 is satisfied. This proves Theorem 2.1. w
Acknowledgements The authors thank Moritz Egert (Paris 11) and Magnus Goffeng (Gothenburg University) for use- ful discussions. The authors thank the anonymous referee for a detailed examination of the paper and for useful feedback.
Funding The first author was supported by the Knut and Alice Wallenberg foundation, KAW 2013.0322 postdoctoral programme in Mathematics for researchers from outside Sweden as well as SPP2026 from the German Research Foundation (DFG). ORCID Lashi Bandara http://orcid.org/0000-0003-1160-2261 Andreas Ros en http://orcid.org/0000-0003-2393-1804
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