Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method
DDecompositions of high-frequency Helmholtz solutions viafunctional calculus, and application to the finite elementmethod
D. Lafontaine ∗ , E. A. Spence † , J. Wunsch ‡ February 26, 2021
Abstract
Over the last ten years, results from [51], [52], [25], and [50] decomposing high-frequencyHelmholtz solutions into “low”- and “high”-frequency components have had a large impactin the numerical analysis of the Helmholtz equation. These results have been proved for theconstant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or aninterior domain with an impedance boundary condition.Using the Helffer–Sj¨ostrand functional calculus [35], this paper proves analogous decom-positions for scattering problems fitting into the black-box scattering framework of Sj¨ostrand-Zworski [66], thus covering Helmholtz problems with variable coefficients, impenetrable obsta-cles, and penetrable obstacles all at once.In particular, these results allow us to prove new frequency-explicit convergence results for(i) the hp -finite-element method applied to the variable coefficient Helmholtz equation in theexterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the h -finite-element method applied to the Helmholtz penetrable-obstacle transmission problem. At the heart of the papers [51], [52], [25], and [50] are results that decompose solutions of thehigh-frequency Helmholtz equation, i.e., ∆ u + k u = − f (1.1)with k large, into(i) a component with H regularity, satisfying bounds with improved k -dependence comparedto those satisfied by the full Helmholtz solution, and(ii) an analytic component, satisfying bounds with the same k -dependence as those satisfied bythe full Helmholtz solution,with these components corresponding to the “high”- and “low”-frequency components of the solu-tion. In the rest of this paper, we write this decomposition as u = u H + u A .Such a decomposition was obtained for • the Helmholtz equation (1.1) posed in R d , d = 2 ,
3, with compactly-supported f , and withthe Sommerfeld radiation condition ∂u∂r ( x ) − i ku ( x ) = o (cid:18) r ( d − / (cid:19) (1.2) Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK,
[email protected] Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK,
[email protected] Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston IL 60208-2730, US, [email protected] a r X i v : . [ m a t h . A P ] F e b s r := | x | → ∞ , uniformly in (cid:98) x := x/r [51, Lemma 3.5], • the Helmholtz exterior Dirichlet problem where the obstacle has analytic boundary [52,Theorem 4.20], and • the Helmholtz interior impedance problem where the domain is either analytic ( d = 2 ,
3) [52,Theorem 4.10], [50, Theorem 4.5], or polygonal [52, Theorem 4.10], [25, Theorem 3.2],in all cases under an assumption that the solution operator grows at most polynomially in k (whichhas recently been shown to hold, for most frequencies, for a variety of scattering problems by [41]).These decompositions have had a large impact in the numerical analysis of the Helmholtzequation in that they allow one to prove convergence, explicit in the frequency k , of so-called hp -finite-element methods ( hp -FEM) applied to discretisations of the Helmholtz equation. Recall thatthe hp -FEM approximates solutions of PDEs by piecewise polynomials of degree p on a mesh withmeshwidth h and obtains convergence by both decreasing h and increasing p ; this is in contrast tothe h -FEM where p is fixed and only h varies.Indeed, these decompositions were used to prove this optimal convergence of a variety of hp methods in [51], [52], [25], [50], [73], [72], [20], [7]. These results about hp methods are particularlysignificant, since they show that, if h and p are chosen appropriately, the FEM solution is uniformlyaccurate as k → ∞ and the total number of degrees of freedom is proportional to k d ; i.e., the hp -FEM does not suffer from the so-called pollution effect (i.e. the total number of degrees of freedomneeding to be (cid:29) k d ) that plagues the h -FEM [2].These decompositions were also used to prove sharp results about the convergence of h methodswith large but fixed p [26], [19], [40]. Furthermore, analogous decompositions and analogousconvergence results were obtained for hp boundary element methods [49], [45], hp methods onHelmholtz problems with arbitrarily-small dissipation [54] and hp methods applied to formulationsof the time-harmonic Maxwell equations [53], [57]. This work has also motivated attempts toprovide simpler decompositions valid for variable-coefficient problems [14].Our recent paper [42] obtained the analogous decomposition to that in [51] for the Helmholtzproblem in R d but now for the variable-coefficient Helmholtz equation ∇ · ( A ∇ u ) + k c u = − f (1.3)with A and c ∈ C ∞ ; we discuss this result more in § The following theorem (Theorem 1.1) obtains the decomposition u = u A + u H in the frameworkof black-box scattering introduced by Sj¨ostrand–Zworski in [66]. In this framework, the operator P (cid:126) , where (cid:126) := k − is the semiclassical parameter, is the variable-coefficient Helmholtz operatoroutside B R (the ball of radius R and centre zero) for some R >
0, but is not specified inside thisball (i.e., inside the “black box”). In particular, this framework includes the Helmholtz exteriorDirichlet and transmission problems, and Theorem 1.1 is specialised to these settings in Theorems1.6 and 1.12, respectively.The following uses notation from the black-box framework, recapped in §
2. We state thetheorem and then give an informal explanation of it afterwards.
Theorem 1.1 (The decomposition in the black-box setting)
We assume that P (cid:126) is a semi-classical black-box operator on H (in the sense of Definition 2.1) and that, additionally, for some (cid:126) > , there is H ⊂ (0 , (cid:126) ] such that the following two assumptions hold.1. There is D out ⊂ D loc and M > so that for any χ ∈ C ∞ c ( R d ) equal to one near B R , thereexists C > so that if v ∈ D out is solution to ( P (cid:126) − v = χg , then (cid:107) χv (cid:107) H ≤ C (cid:126) − M − (cid:107) g (cid:107) H for all (cid:126) ∈ H. (1.4)2 . There is a function E ∈ C ( R ) , that is nowhere zero, and ρ ∈ C ∞ ( T dR (cid:93) ) equal to one near B R , so that for some α -family of black-box differentiation operators ( D ( α )) α ∈ A (in the senseof Definition 2.7 below), the following estimate holds: (cid:13)(cid:13)(cid:13) ρD ( α ) E ( P (cid:93) (cid:126) ) v (cid:13)(cid:13)(cid:13) H (cid:93) ≤ C E ( α, (cid:126) ) (cid:107) v (cid:107) H (cid:93) for all v ∈ D (cid:93), ∞ (cid:126) and (cid:126) ∈ H, (1.5) for some C E ( α, (cid:126) ) > .Then, if R < R < R (cid:93) , g ∈ H is compactly supported in B R and u ∈ D out is solution to ( P (cid:126) − u = g, (1.6) then, there exists u H ∈ D (cid:93) and u A ∈ D (cid:93), ∞ (cid:126) so that u | B R = (cid:0) u A + u H (cid:1) | B R , (1.7) where u H satisfies (cid:107) u H (cid:107) H (cid:93) + (cid:13)(cid:13) P (cid:93) (cid:126) u H (cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) g (cid:107) H for all (cid:126) ∈ H, (1.8) and, for some σ > , u A satisfies (cid:107) D ( α ) u A (cid:107) H (cid:93) (cid:46) (cid:16) max | β |≤| α | C E ( β, (cid:126) ) + (cid:126) −| α | (cid:17) σ | α | (cid:126) − M − (cid:107) g (cid:107) H for all (cid:126) ∈ H, α ∈ A , (1.9) where the omitted constants in (1.8) and (1.9) are independent of (cid:126) and α . Point 1 in Theorem 1.1 is the assumption that the solution operator is polynomially boundedin (cid:126) . In the setting of impenetrable-obstacle problems, this assumption holds with M = 0 and H = (0 , (cid:126) ] when the problem is nontrapping (see Theorem 1.5 below and the references therein).In the black-box setting, [41] proved that this assumption always holds with M > n/ H having arbitrarily-small measure.Point 2 in Theorem 1.1 is a regularity assumption that depends on the contents of the blackbox. The black-box differentiation operators, ( D ( α )) α ∈ A , are a family of operators satisfying aweak binomial-type bound and agreeing with differentiation outside the black-box (see Definition2.7 below).Regarding u H : comparing (1.4) and (1.8), we see that u H satisfies a bound with better (cid:126) -dependence than that satisfied by u ; this is the analogue of the property (i) in § u H depends on the domain of the operator ( u H ∈ D (cid:93) )but not on any other features of the black box (in particular, not on the regularity estimate (1.5)).Regarding u A : u A is in the domain of arbitrary powers of the operator ( u A ∈ D (cid:93), ∞ (cid:126) ) and sois smooth in an abstract sense. Comparing (1.4) and (1.9), we see that u A satisfies a bound withthe same (cid:126) -dependence as u , but with improved regularity. These properties are the analogue ofthe property (ii) in § u A depends on the regularity inside the black-box (from (1.5)), and, for the exterior Dirichlet problemwith analytic obstacle and coefficients analytic in a neighbourhood of the obstacle, u A is analytic. Let O − ⊂ R d , d ≥ be a bounded Lipschitzopen set such that the open complement O + := R d \ O − is connected and such that O − ⊂ B R . Let A ∈ C , ( O + , R n × n ) be such that supp( I − A ) ⊂ B R , with R > R , A is symmetric, and thereexists A min > such that (cid:0) A ( x ) ξ (cid:1) · ξ ≥ A min | ξ | for all x ∈ O + and for all ξ ∈ C d . (1.10) Let c ∈ L ∞ ( O + ) be such that supp(1 − c ) ⊂ B R , and c min ≤ c ≤ c max with c min , c max > . iven f ∈ L ( O + ) with supp f ⊂⊂ R d and k > , u ∈ H ( O + ) satisfies the exterior Dirichletproblem if c ∇ · ( A ∇ u ) + k u = − f in O + , (1.11) γu = 0 on ∂ O + , (1.12) and u satisfies the Sommerfeld radiation condition (1.2) . We highlight from Definition 1.2 that the obstacle O − is contained in B R , and the variationof the coefficients A and c is contained inside the larger ball B R .We use the standard weighted H norm, (cid:107) · (cid:107) H k ( B R ∩O + ) , defined by (cid:107) u (cid:107) H k ( B R ∩O + ) := (cid:107)∇ u (cid:107) L ( B R ∩O + ) + k (cid:107) u (cid:107) L ( B R ∩O + ) . (1.13) Definition 1.3 ( C sol ) Given f ∈ L ( O + ) supported in B R with R ≥ R , let u be the solution ofthe exterior Dirichlet problem of Definition 1.2. Given k > , let C sol = C sol ( k, A, c, R, k ) > besuch that (cid:107) u (cid:107) H k ( B R ∩O + ) ≤ C sol (cid:107) f (cid:107) L ( B R ∩O + ) for all k > . (1.14) C sol exists by standard results about uniqueness of the exterior Dirichlet problem and Fredholmtheory; see, e.g., [32, §
1] and the references therein. A key assumption in our analysis is that C sol ( k )is polynomially bounded in k in the following sense. Definition 1.4 ( C sol is polynomially bounded in k ) Given k and K ⊂ [ k , ∞ ) , C sol ( k ) ispolynomially bounded for k ∈ K if there exists C > and M > such that C sol ( k ) ≤ Ck M for all k ∈ K, (1.15) where C and M are independent of k (but depend on k and possibly also on K, A, c, d, R ). There exist C ∞ coefficients A and c such that C sol ( k j ) ≥ C exp( C k j ) for 0 < k < k < . . . with k j → ∞ as j → ∞ , see [59], but this exponential growth is the worst-possible, since C sol ( k ) ≤ c exp( c k ) for all k ≥ k by [8, Theorem 2]. We now recall results on when C sol ( k ) is polynomiallybounded in k . Theorem 1.5 (Conditions under which C sol ( k ) is polynomially bounded in k for theexterior Dirichlet problem) (i) If A and c are C ∞ and nontrapping (i.e. all the trajectories of the generalised bicharacteristicflow defined by the semiclassical principal symbol of (1.11) starting in B R leave B R after a uniformtime), then C sol ( k ) is independent of k for all sufficiently large k ; i.e., (1.15) holds for all k ≥ k with M = 0 .(ii) If A is C , and c ∈ L ∞ then, given k > and δ > there exists a set J ⊂ [ k , ∞ ) with | J | ≤ δ such that C sol ( k ) ≤ Ck d/ ε for all k ∈ [ k , ∞ ) \ J, (1.16) for any ε > , where C depends on δ, ε, d, k , and A . If A is C ,σ for some σ > then the exponentis reduced to d/ ε .References for the proof. (i) follows from either the results of [55] combined with either [69,Theorem 3]/ [70, Chapter 10, Theorem 2] or [43], or [9, Theorem 1.3 and § C sol is proportional to the length of the longest trajectory in B R ;see [28, Theorems 1 and 2, and Equation 6.32]. (ii) is proved for c = 1 in [41, Theorem 1.1 andCorollary 3.6]; the proof for more-general c follows from Lemma 2.2 below.4 .3.2 Theorem 1.1 applied to the exterior Dirichlet problemTheorem 1.6 (The main result applied to the exterior Dirichlet problem with analytic A, c , and O − ) Suppose that O − , A, c, R , and R are as in Definition 1.2. In addition, assume that O − is analytic, and that A and c are C ∞ everywhere and analytic in B R ∗ for some R < R ∗ < R .If C sol ( k ) is polynomially bounded (in the sense of Definition 1.4), then given f ∈ L ( O ) supported in B R with R ≥ R , the solution u of the exterior Dirichlet problem is such that thereexists u A , analytic in B R ∩ O + , and u H ∈ H ( B R ∩ O + ) , both with zero Dirichlet trace on ∂ O + ,such that u | B R = u A + u H . (1.17) Furthermore, there exist C , C , and C , all independent of k and α , such that (cid:107) ∂ α u H (cid:107) L ( B R ∩O + ) ≤ C k | α |− (cid:107) f (cid:107) L ( B R ∩O + ) for all k ∈ K and for all | α | ≤ , (1.18) and (cid:107) ∂ α u A (cid:107) L ( B R ∩O + ) ≤ C ( C ) | α | | α | ! k | α |− M (cid:107) f (cid:107) L ( B R ∩O + ) for all k ∈ K and for all α. (1.19) hp -FEM As discussed in § hp -FEM applied to the exterior Dirichlet problem; we now give the necessary definitions to statethis result. Recall that the FEM is based on the standard variational formulation of the exteriorDirichlet problem: given R ≥ R and F ∈ ( H ( B R ∩ O + )) ∗ ,find u ∈ H ( B R ∩ O + ) such that a ( u, v ) = F ( v ) for all v ∈ H ( B R ∩ O + ) , (1.20)where a ( u, v ) := (cid:90) B R ∩O + (cid:18) ( A ∇ u ) · ∇ v − k c uv (cid:19) − (cid:10) DtN k ( γu ) , γv (cid:11) ∂B R , (1.21)where (cid:104)· , ·(cid:105) ∂B R denotes the duality pairing on ∂B R that is linear in the first argument and antilinearin the second, and DtN k : H / ( ∂B R ) → H − / ( ∂B R ) is the Dirichlet-to-Neumann map for theequation ∆ u + k u = 0 posed in the exterior of B R with the Sommerfeld radiation condition (1.2);the definition of DtN k in terms of Hankel functions and polar coordinates (when d = 2)/sphericalpolar coordinates (when d = 3) is given in, e.g., [51, Equations 3.7 and 3.10]. We use later the factthat there exist C DtN = C DtN ( k R ) such that (cid:12)(cid:12)(cid:10) DtN k ( γu ) , γv (cid:105) ∂B R (cid:11)(cid:12)(cid:12) ≤ C DtN (cid:107) u (cid:107) H k ( B R ∩O + ) (cid:107) v (cid:107) H k ( B R ∩O + ) (1.22)for all u, v ∈ H ( B R ∩ O + ) and for all k ≥ k ; see [51, Lemma 3.3].If F ( v ) = (cid:82) B R ∩O + f v , then the solution of the variational problem (1.20) is the restriction to B R of the solution of the exterior Dirichlet problem of Definition 1.2. If F ( v ) = (cid:90) ∂B R ( ∂ n u I − DtN k γu I ) γv, (1.23)where u I is a solution of ∆ u I + k u I = 0 in B R ∩ O + , then the solution of the variational problem(1.20) is the restriction to B R ∩ O + of the sound-soft scattering problem (see, e.g, [11, Page 107]).Given a sequence, { V N } ∞ N =0 , of finite-dimensional subspaces of H ( B R ∩O + ), the finite-elementmethod for the variational problem (1.20) is the Galerkin method applied to the variational problem(1.20), i.e., find u N ∈ V N such that a ( u N , v N ) = F ( v N ) for all v N ∈ V N . (1.24) Theorem 1.7 (Quasioptimality of hp -FEM for the exterior Dirichlet problem) Let d =2 or . Suppose that O − , A, c, and R are as in Theorem 1.6. Let ( V N ) ∞ N =0 be the piecewise-polynomial approximation spaces described in [51, § § B R ∩ O + exactly), and let u N be the Galerkin solution defined by (1.24) . f C sol ( k ) is polynomially bounded (in the sense of Definition 1.4) for k ∈ K ⊂ [ k , ∞ ) then,given k > , there exist C , C > , depending on A, c, R , and d , and k , but independent of k , h ,and p , such that if hk ≤ C and p ≥ C log k, (1.25) then, for all k ∈ K , the Galerkin solution exists, is unique, and satisfies the quasi-optimal errorbound (cid:107) u − u N (cid:107) H k ( B R ∩O + ) ≤ C qo min v N ∈ V N (cid:107) u − v N (cid:107) H k ( B R ∩O + ) , (1.26) with C qo := 2 (cid:0) max { A max , c − } + C DtN (cid:1) A min . (1.27) Remark 1.8 (The significance of Theorem 1.7)
For finite-dimensional subspaces consistingof piecewise polynomials of degree p on meshes with meshwidth h , the total number of degreesof freedom ∼ ( p/h ) d . The results about Helmholtz hp -FEM in [51], [52], [25], [50], [42] showquasioptimality of the hp -FEM under the slightly weaker condition than (1.25) , namely hkp ≤ C and p ≥ C log k. These results then show that there is a choice of h and p such that the hp -FEM is quasioptimalwith the total number of degrees of freedom ∼ k d . As mentioned in § h -FEM (i.e., with p fixed) is not quasioptimal with C qo independent of k when the totalnumber of degrees of freedom ∼ k d ; this is called the pollution effect – see [2] and the referencestherein.Theorem 1.7 shows that there is a choice of h and p such that the hp -FEM applied to the exteriorDirichlet problem of Definition 1.2 (with the obstacle and coefficients analytic) is quasioptimal withthe total number of degrees of freedom ∼ ( k log k ) d .Although this result is (log k ) d away from proving that the hp -FEM applied to the exteriorDirichlet problem does not suffer from the pollution effect, it is nevertheless the first frequency-explicit result about the convergence of the hp -FEM applied the exterior Dirichlet problem (withthe obstacle and coefficients analytic). Our understanding is that the sharp result will be proved inthe forthcoming paper [5] announced in [6]. In the specific case of the plane-wave scattering problem, the recent results of [40, Theorem 9.1and Remark 9.10] allow us to bound the best approximation error on the right-hand side of (1.26)and obtain a bound on the relative error.
Corollary 1.9 (Bound on the relative error of the Galerkin solution)
Let the assump-tions of Theorem 1.7 hold and, furthermore, let F ( v ) be given by (1.23) with u I ( x ) = exp(i kx · a ) for some a ∈ R d with | a | = 1 (so that u is then the solution of the plane-wave scattering problem).If C sol ( k ) is polynomially bounded (in the sense of Definition 1.4) for k ∈ K ⊂ [ k , ∞ ) , then thereexists C > , independent of k , h , and p , such that if (1.25) holds, then, for all k ∈ K , (cid:107) u − u N (cid:107) H k ( B R ∩O + ) (cid:107) u (cid:107) H k ( B R ∩O + ) ≤ C qo C C C log k (cid:18) C C log k (cid:19) , (1.28) with C qo given by (1.27) ; i.e. the relative error can be made arbitrarily small by making C smaller. Let O − ⊂ R d , d ≥ be a bounded Lipschitz open set such that the open complement O + := R d \ O − isconnected and such that O − ⊂ B R . Let A = ( A − , A + ) with A ± ∈ C , ( O ± , R n × n ) be such that supp( I − A ) ⊂ B R , A is symmetric, and there exists A min > such that (1.10) holds (with O + eplaced by R d ). Let c ∈ L ∞ ( O − ) be such that c min ≤ c ≤ c max with < c min ≤ c max < ∞ . Let β > .Let ν be the unit normal vector field on ∂ O − pointing from O − into O + , and let ∂ ν,A denotethe corresponding conormal derivative defined by, e.g., [46, Lemma 4.3] (recall that this is suchthat, when v ∈ H ( O + ) , ∂ ν,A v = ν · γ ( A ∇ v ) ).Given f ∈ L ( O + ) with supp f ⊂⊂ R d and k > , u = ( u − , u + ) ∈ H ( R d ) satisfies thetransmission problem if c ∇ · ( A − ∇ u − ) + k u − = − f in O − , ∇ · ( A + ∇ u + ) + k u + = − f in O + ,γu − = γu + , ∂ ν,A − u − = β∂ ν,A + u + on ∂ O − , (1.29) and u + satisfies the Sommerfeld radiation condition (1.2) . When A − , A + , and c are constant, two of the four parameters A − , A + , c , and β are redundant.For example, by rescaling u − , u + , and f , all such transmission problems can be described by theparameters c and β (with A − = A + = 1), as in, e.g., [10], or by the parameters A − and c (with A + = β = 1); see, e.g., the discussion and examples after [56, Definition 2.3].The definition of C sol for the transmission problem is almost identical to Definition 1.3, ex-cept that the norms in (1.14) are now over B R (as opposed to B R ∩ O + ) and now C sol dependsadditionally on β Theorem 1.11 (Conditions under which C sol ( k ) is polynomially bounded in k for thetransmission problem) In each of the following conditions we assume that A , c , and O − are asin Definition 1.10.(i) If O − is smooth and strictly convex with strictly positive curvature, A = I , c is a constant ≤ , and β > , then C sol ( k ) is independent of k for all sufficiently large k ; i.e., (1.14) holds forall k ≥ k with M = 0 (ii) If O − is Lipschitz and star-shaped, A = I , and c is a constant with c ≤ β ≤ . (iii) If O − is star-shaped, β = 1 , and both A and c are monotonically non-increasing in theradial direction (in the sense of [32, Condition 2.6]).(iv) Given k > and δ > there exists a set J ⊂ [ k , ∞ ) with | J | ≤ δ such that C sol ( k ) ≤ Ck d/ ε for all k ∈ [ k , ∞ ) \ J, (1.30) for any ε > , where C depends on δ, ε, d, k , A, c , and β .References for the proof. (i) is proved in [10, Theorem 1.1] (we note that, in fact, a strongerresult with A − variable is also proved there). (ii) is proved in [56, Theorem 3.1]. (iii) is proved in[32, Theorem 2.7]. (iv) is proved for constant c and globally Lipschitz A in [41, Theorem 1.1 andCorollary 3.6]; the proof for these more-general c and A follows from Lemma 2.2 below. Suppose that
A, c, β, and O − are as in Definition 1.10 and, additionally, A , c , and O − are all C m .If C sol ( k ) is polynomially bounded (in the sense of Definition 1.4), then given f ∈ L ( R d ) supported in B R with R ≥ R , the solution u of the transmission problem is such that there exists u A = ( u + , A , u − , A ) ∈ C ∞ ( B R ∩ O + ) × C ∞ ( O − ) and u H = ( u + ,H , u − ,H ) ∈ H ( B R ∩ O + ) × H ( O − ) , satisfying (1.29), and so that u | B R = u A + u H . (1.31) Furthermore there exist C , C > , independent of k but with C = C ( p ) , such that (cid:107) ∂ α u ± ,H (cid:107) L ( B R ∩O ± ) ≤ C k | α |− (cid:107) f (cid:107) L ( B R ) for all k ∈ K and for all | α | ≤ , (1.32) and (cid:107) ∂ α u ± , A (cid:107) L ( B R ∩O ± ) ≤ C ( p ) k | α |− M (cid:107) f (cid:107) L ( B R ) for all k ∈ K and for all | α | ≤ m. (1.33)7 .4.3 Corollary about frequency-explicit convergence of the h -FEM For simplicity we consider the case where the parameter β in the transmission condition (1.29)equals one; recall from the comments below Definition 1.10 that, at least in the constant-coefficientcase, this is without loss of generality. The variational formulation of the transmission problem isthen (1.20) with B R ∩ O + replaced by B R and a ( · , · ) given by (1.21) with c understood as equalto one in B R ∩ O + Since the constant C in (1.33) depends on p , we cannot prove a result about the hp -FEM forthe transmission problem of Definition 1.10. We therefore consider the h -FEM and prove the firstsharp quasioptimality result for this problem (see Remark 1.15 below for more discussion on thenovelty of our result). Assumption 1.13 ( V N ) ∞ N =0 is a sequence of piecewise-polynomial approximation spaces on quasi-uniform meshes with mesh diameter h and polynomial degree p . Furthermore, (i) the mesh consistsof curved elements that exactly triangulate B R and O − , so that each element in the mesh is includedin either O − or B R ∩ O + , and (ii) there exists an interpolant operator I h,p such that for all ≤ j ≤ (cid:96) ≤ p , there exists C ( j, (cid:96) ) > such that (cid:12)(cid:12) v − I h,p v (cid:12)(cid:12) H j ( B R ) ≤ C ( j, (cid:96) ) h (cid:96) +1 − j (cid:16) (cid:107) v + (cid:107) H (cid:96) +1 ( B R ∩O + ) + (cid:107) v − (cid:107) H (cid:96) +1 ( O − ) (cid:17) (1.34) for all v = ( v + , v − ) ∈ H (cid:96) +1 ( B R ∩ O + ) × H (cid:96) +1 ( O − ) . Assumption 1.13 is satisfied by the hp approximation spaces described in [51, § § Theorem 1.14 (Quasioptimality of h -FEM for the transmission problem) Let d = 2 or . Suppose that β = 1 , A, c, and O − are as in Definition 1.10, and, in addition, A, c, and O − are all C p +1 with p odd. Let ( V N ) ∞ N =0 be a sequence of piecewise-polynomial approximation spacessatisfying Assumption 1.13 and let u N be the Galerkin solution defined by (1.24) .If C sol ( k ) is polynomially bounded (in the sense of Definition 1.4) for k ∈ K ⊂ [ k , ∞ ) then,given k > , there exist C > , depending on A, c, R , d , k , and p , but independent of k and h ,such that if h p k p +1+ M ≤ C (1.35) then, for all k ∈ K , the Galerkin solution exists, is unique, and satisfies the quasi-optimal errorbound (cid:107) u − u N (cid:107) H k ( B R ) ≤ C qo min v N ∈ V N (cid:107) u − v N (cid:107) H k ( B R ) , with C qo given by (1.27) . Remark 1.15 (The significance of Theorem 1.14)
The fact that “ h p k p +1 sufficiently small”is a sufficient condition for quasioptimality of the Helmholtz h -FEM in nontrapping situations(i.e. M = 0 ) was proved for a variety of Helmholtz problems for p = 1 in [47, Prop. 8.2.7], [33,Theorem 4.5], [28, Theorem 3] (building on the 1-d results of [1, Theorem 3.2], [38, Theorem 3],[37, Theorem 4.13], and [39, Theorem 3.5]) and for p > in [51, Corollary 5.6], [52, Remark 5.9],[29, Theorem 5.1], and [14, Theorem 2.15]. Numerical experiments indicate that this condition isalso necessary – see, e.g., [14, § A and c that are also allowed to be discontinuous. However, the results in [14] hold only when animpedance boundary condition is imposed on the truncation boundary (in our case ∂B R ), which isequivalent to approximating the exterior Helmholtz Dirichlet-to-Neumann map by i k . Furthermore,the proof of [14, Theorem 2.15] uses the impedance boundary condition in an essential way. Indeed,in [14, Proof of Lemma 2.13] the solution is expanded in powers of k , i.e. u = (cid:80) ∞ j =0 k j u j , and thenon ∂B R one has ∂ n u j +1 = i γu j ; this relationship between u j +1 and u j on ∂B R no longer holds if DtN k is not approximated by i k .The Helmholtz equation with an impedance boundary condition is often used as a model problemfor numerical analysis (see, e.g., the references in [27, § hat, in the limit k → ∞ with the truncation boundary fixed, the error incurred in approximating theDirichlet-to-Neumann map with i k is bounded away from zero, independently of k , even in the best-possible situation when the truncation boundary equals ∂B R for some R ; see [27, § p large), the solution of the truncated problem will not be a good approximation tothe true scattering problem when k is large. It is instructive to first recall the ideas behind the results of [51], [52], [25], and [50].
How the results of [51], [52], [25], and [50] were obtained.
The paper [51] consideredthe Helmholtz equation (1.1) posed in R d with the Sommerfeld radiation condition (1.2). Thedecomposition u = u H + u A was obtained by decomposing the data f in (1.1) into “high-” and“low-” frequency components, with u H the Helmholtz solution for the high-frequency componentof f , and u A then the Helmholtz solution for the low-frequency component of f . The frequencycut-offs were defining using the indicator function1 B λk ( ζ ) := (cid:40) | ζ | ≤ λ k, | ζ | ≥ λ k, (1.36)with λ a free parameter (see [51, Equation 3.31] and the surrounding text). In [51] the frequencycut-off (1.36) was then used with (a) the expression for u as a convolution of the fundamentalsolution and the data f , and (b) the fact that the fundamental solution is known explicitly for thePDE (1.1) to obtain the appropriate bounds on u A and u H using explicit calculation (involvingBessel and Hankel functions). The decompositions in [52], [25], [50] for the exterior Dirichletproblem and interior impedance problem were obtained using the results of [51] combined withextension operators (to go from problems with boundaries to problems on R d ).Because the proof technique in [51] does not generalise to the variable-coefficient Helmholtzequation (1.3), until the recent paper [42] there did not exist in the literature analogous decom-position results for the variable-coefficient Helmholtz equation. This was despite the increasinginterest in the numerical analysis of (1.3) see, e.g., [12], [3], [14], [30], [58], [33], [28], [40], [31]. The recent results of [42]: the decomposition for the variable-coefficient Helmholtzequation in free space.
The paper [42] obtained the analogous decomposition to that in [51] forthe Helmholtz problem in R d but now for the variable-coefficient Helmholtz equation (1.3) with A and c ∈ C ∞ . This result was obtained again using frequency cut-offs (as in [51]) but now applyingthem to the solution u as opposed to the data f . Any cut-off function that is zero for | ζ | ≥ Ck is a cutoff to a compactly supported set in phase space, and hence enjoys analytic estimates. Themain difficulty in [42], therefore, was in showing that the high-frequency component u H satisfiesa bound with one power of k improvement over the bound satisfied by u . This was achieved bychoosing the cut-off so that the (scaled) Helmholtz operator k − ∇ · A ∇ + c − is semiclassicallyelliptic on the support of the high-frequency cut-off. Then, choosing the cut-off function to besmooth (as opposed to discontinuous, as in (1.36)) allowed [42] to use basic facts about the “nice”behaviour of elliptic semiclassical pseudodifferential operators (namely, they are invertible up to asmall error) to prove the required bound on u H . The frequency decomposition achieved in Theorem 1.1.
In this paper, we achieve thedesired decomposition into low- and high-frequency pieces in the manner best adapted to thefunctional analysis of the Helmholtz equation: by using the functional calculus for the Helmholtzoperator itself. Recall that once we realise the operator P = − c ∇ · ( A ∇ ) (1.37)with appropriate domain as a self-adjoint operator (on a space weighted by c − ), the functionalcalculus for self-adjoint operators allows us to define f ( P ) for a broad class of functions f. In9articular, given k >
0, we are interested in taking f a cutoff function of the real axis equal to 1on B (0 , µk ) for some µ >
1. Then for fixed k , (1 − f )( P ) is a high-frequency cutoff and f ( P ) alow-frequency cutoff; in the special case A = I, c = 1, these are simply Fourier multipliers of thetype used in [41].The novelty of the approach used here is to make the functional calculus approach work in themuch more general setting of semiclassical black-box scattering introduced by Sj¨ostrand-Zworski[66], which allows us to treat variable (possibly rough) media, obstacles, and partially transparentobstacles all at once. We rescale , setting (cid:126) = k − , and study operators P (cid:126) equal to a variable-coefficient Laplacian outside the “black-box” B R , and equal to − (cid:126) ∆ outside a larger ball B R .We are now interested in functions of P (cid:126) of the form ψ ( P (cid:126) ) with ψ = 1 in B (0 , µ ) and 0 in( B (0 , µ )) c . We split u = Π H u + Π L u with Π L ≡ ψ ( P (cid:126) ) , Π H u ≡ (1 − ψ )( P (cid:126) ) , and both pieces again defined by the spectral theorem. We now discuss the two pieces separately.We wish to analyze Π H u by using the semiclassical ellipticity of P (cid:126) − H were globally a pseudodifferential operator.In the broad context of the black-box theory, though, while the function ψ ( P (cid:126) ) is well-defined asan abstract operator on a Hilbert space, its structure is much less manifest than it would be forthe flat Laplacian in Euclidean space. Not much may be said in any generality about Π H on theblack-box, but this is unnecessary in any event: we use an abstract ellipticity argument basedon the Borel functional calculus, with the ellipticity in question now amounting to the boundedinvertibility of P (cid:126) − H , which just follows from the boundedness of the function( λ − − (1 − ψ ( λ )) . Since our outgoing solution u is only locally L , however, we do additionallyneed to understand the commutator of Π H with a localiser ϕ that equals one near the black-box. Fortunately, we are able to use the Helffer–Sj¨ostrand approach to the functional calculus [35]to describe this commutator explicitly. The method of [35] is a powerful tool for obtaining thestructure theorem that a decently behaved function of a self-adjoint elliptic differential operatoris, as one might hope, in fact a pseudodifferential operator [18, Chapter 8] (a result originally dueto Strichartz [68] in the setting of the homogeneous pseudodifferential calculus and Helffer–Robert[34] in the semiclassical setting used here). Additionally, Davies [16] later pointed out that infact the same method affords a novel proof of the functional calculus formulation of the spectraltheorem itself. Here, we use some refinements of Sj¨ostrand [65] to learn that away from the black-box we may in fact treat Π H as a pseudodifferential operator (see Lemma 2.6), and hence dealwith [Π H , ϕ ] as an element of the pseudodifferential calculus, solving it away by once again usingellipticity (this time in the context of pseudodifferential operators) together with our polynomialresolvent estimate.While the analysis of Π H u is insensitive to the contents of the black-box, our study of the lowfrequency piece Π L u necessarily entails “opening” the black-box and studying the local question ofelliptic or parabolic estimates within it. Intuitively the compact support in the spectral parameterof the spectral measure of P applied to Π L u should imply that strong elliptic estimates hold,but knowing uniform estimates on high derivatives is dependent on analyticity of the underlyingproblem. We therefore make an abstract regularity hypothesis (1.5) locally near the black-box,that allows us to estimate the part of Π L u spatially localised near its content. The remaining partlives in R d , thus is given, thanks to Sj¨ostrand [65] again, by a Fourier multiplier up to negligibleterms, and we can estimate it thanks to the properties of the Fourier transform as it was donein [42]. If, for instance, P is given by (1.37) exterior to a C ∞ obstacle with Dirichlet boundarycondition, we know by the functional calculus that P m Π L u is bounded for all m ∈ N . This yieldselliptic estimates which allow us to estimate all derivatives of Π L u up to the obstacle, but theresulting estimates on ∂ α Π L u grow non-optimally in α ; see Corollary 4.2 and Theorem 1.12. Suchestimates, which indeed are the only ones we have been able to obtain in the case of partiallytransparent obstacles, suffice for applications to the h -FEM but are far from optimal in dealingwith hp -FEM. In the boundary case we therefore use a stronger property of Π L u : we may run the The semiclassical parameter is usually denoted by h , but we use (cid:126) to avoid a notational clash with the meshwidthof the FEM used in Theorems 1.7 and 1.14. ackward heat equation on Π L u for as long as we like and obtain L estimates on the result. If theboundary is analytic then known heat kernel estimates [24] yield satisfactory uniform estimates on ∂ α Π L u ; see Corollary 4.1 and Theorem 1.6. § § § § hp -FEM for the exterior Dirichlet problem and the h -FEM for the transmissionproblem). § A recaps results about semiclassical pseudodifferential operators on the torus. § B and § C prove subsidiary results used to prove Lemma 2.3 and Theorem 1.1, respectively.
We now briefly recap the abstract framework of black-box scattering introduced in [66]; for moredetails, see the comprehensive presentation in [21, Chapter 4]. A brief overview of black-boxscattering with an emphasis on the counting of resonances is contained in [41, § § − (cid:126) ∆ itself as in [21, Chapter 4]. The decomposition
Let H be an Hilbert space with an orthogonal decomposition H = H R ⊕ L ( R d \ B R , ω ( x )d x ) , (BB1)where the weight-function ω : R d → R is measurable. Let B R and R d \ B R denote the corre-sponding orthogonal projections. Let P (cid:126) be a family in (cid:126) of self adjoint operators H → H withdomain
D ⊂ H independent of (cid:126) (so that, in particular, D is dense in H ). Outside the black-box H R , we assume that P (cid:126) equals Q (cid:126) defined as follows. We assume that there exists R > R and, for any multi-index | α | ≤
2, functions a (cid:126) ,α ∈ C ∞ ( R d ), uniformly bounded with respect to (cid:126) ,independent of (cid:126) for | α | = 2, and such that for some C > (cid:88) | α | =2 a (cid:126) ,α ( x ) ξ α ≥ C | ξ | for all x ∈ R d (2.1)and (cid:88) | α | =2 a (cid:126) ,α ( x ) ξ α = | ξ | for | x | ≥ R , and in such a way that the operator Q (cid:126) defined by Q (cid:126) := (cid:88) | α |≤ a (cid:126) ,α ( x )( (cid:126) D x ) α is formally self-adjoint on L ( R d , ω ( x )d x ). We require the operator P (cid:126) to be equal to Q (cid:126) outsidethe black-box H R in the sense that R d \ B R ( P (cid:126) u ) = Q (cid:126) ( u | R d \ B R ) for u ∈ D , and R d \ B R D ⊂ H ( R d \ B R ) . (BB2)We further assume that if, for some ε > v ∈ H ( R d ) and v | B R ε = 0 , then v ∈ D , (BB3)11with the restriction to B R + (cid:15) defined in terms of the projections in (BB2); see also (2.5) below)and that B R ( P (cid:126) + i) − is compact from H → H . (BB4)Under these assumptions, the semiclassical resolvent R ( z, (cid:126) ) := ( P (cid:126) − z ) − : H → D is meromorphic for Im z > H comp → D loc inthe whole complex plane when n is odd and in the logarithmic plane when n is even [21, Theorem4.4]; where H comp and D loc are defined by H comp := (cid:110) u ∈ H : R d \ B R u ∈ L ( R d \ B R ) (cid:111) , and D loc := (cid:110) u ∈ H R ⊕ L ( R d \ B R ) : if χ ∈ C ∞ c ( R d ) , χ | B R = 1then ( B R u, χ R d \ B R u ) ∈ D (cid:111) . The reference operator P (cid:93) (cid:126) Let R (cid:93) > R , and let T dR (cid:93) := R d / (2 R (cid:93) Z ) d ; we work with [ − R (cid:93) , R (cid:93) ] d as a fundamental domain forthis torus. We let H (cid:93) := H R ⊕ L ( T dR (cid:93) \ B R ) , and let B R and T dR(cid:93) \ B R denote the corresponding orthogonal projections. We define D (cid:93) := (cid:110) u ∈ H (cid:93) : if χ ∈ C ∞ ( B R (cid:93) ) , χ = 1 near B R , then ( B R u, χ T dR(cid:93) \ B R u ) ∈ D , and (1 − χ ) T dR(cid:93) \ B R u ∈ H ( T dR (cid:93) ) (cid:111) , (2.2)and, for any χ as in (2.2) and u ∈ D (cid:93) , P (cid:93) (cid:126) u := P (cid:126) ( B R u, χ T dR(cid:93) \ B R u ) + Q (cid:126) (cid:0) (1 − χ ) T dR(cid:93) \ B R u (cid:1) , (2.3)where we have identified functions supported in B (0 , R (cid:93) ) \ B (0 , R ) ⊂ T d \ B (0 , R ) with the corre-sponding functions on R d \ B (0 , R ) – see the paragraph on notation below.The idea behind these definitions is that we have glued our black box into a torus instead of R d , and then defined on the torus an operator P (cid:93) (cid:126) that can can be thought of as P (cid:126) in H R and Q (cid:126) in ( R /R (cid:93) Z ) d \ B R ; see Figure 2.1. The resolvent ( P (cid:93) (cid:126) + i ) − is compact (see [21, Lemma 4.11]), andhence the spectrum of P (cid:93) (cid:126) , denoted by Sp P (cid:93) (cid:126) , is discrete (i.e., countable and with no accumulationpoint).We assume that the eigenvalues of P (cid:93) (cid:126) satisfy the polynomial growth of eigenvalues condition N (cid:0) P (cid:93) (cid:126) , [ − C, λ ] (cid:1) = O ( (cid:126) − d (cid:93) λ d (cid:93) / ) , (BB5)for some d (cid:93) ≥ n and N ( P (cid:93) , I ) is the number of eigenvalues of P (cid:93) (cid:126) in the interval I , counted withtheir multiplicity. When d (cid:93) = d , the asymptotics (BB5) correspond to a Weyl-type upper bound,and thus (BB5) can be thought of as a weak Weyl law. We summarise with the following definition. Definition 2.1 (Semiclassical black-box operator)
We say that a family of self-adjoint oper-ators P (cid:126) on a Hilbert space H , with dense domain D , independent of (cid:126) , is a semiclassical black-boxoperator if ( P (cid:126) , H ) satisfies (BB1), (BB2), (BB3), (BB4), (BB5). Principal symbol q (cid:126) of Q (cid:126) . Let q (cid:126) ∈ S ( T dR (cid:93) ) denote the principal symbol of Q (cid:126) as an operatoracting on the torus T dR (cid:93) (see Appendix A). 12 (cid:93) ¯ h (cid:39) P ¯ h (cid:39) ? P (cid:93) ¯ h (cid:39) P ¯ h (cid:39) Q ¯ h P (cid:93) ¯ h (cid:39) P ¯ h (cid:39) − ¯ h ∆ R (cid:93) R R P ¯ h (cid:39) − ¯ h ∆ Figure 2.1: The black-box setting. The symbol (cid:39) is used to denote equality in the sense of (BB2)and (2.3).
Notation
We identify in the natural way: • the elements of { } ⊕ L ( T dR (cid:93) \ B R ) ⊂ H (cid:93) , • the elements of L ( T dR (cid:93) \ B R ), • the elements of L ( T dR (cid:93) ) essentially supported outside B R , • the elements of L ( R d ) essentially supported in [ − R (cid:93) , R (cid:93) ] d \ B R , • and the elements of { } ⊕ L ( R d \ B R ) ⊂ H whose orthogonal projection onto L ( R d \ B R )is essentially supported in [ − R (cid:93) , R (cid:93) ] d \ B R .If v ∈ H and χ ∈ C ∞ ( R d ) is equal to some constant α near B R , we define χv := ( α B R v, χ R d \ B R v ) ∈ H . (2.4)(for example, using this notation, the requirements on u in the definition of D (cid:93) are χu ∈ D and(1 − χ ) u ∈ H ( T dR (cid:93) )).If v ∈ H and R > R , we define v | B R := (cid:0) B R v, ( R d \ B R v (cid:1) | B R ) ∈ H R ⊕ L ( B R \ B R ) , (2.5)and, if v ∈ H (cid:93) , v | B R := (cid:0) B R v, ( T dR(cid:93) \ B R v (cid:1) | B R ) ∈ H R ⊕ L ( B R \ B R ) . Finally, we say that g ∈ H is compactly supported in B R if g = χ g for some χ ∈ C ∞ c ( R d ) equalto one near B R and supported in B R . The two following lemmas show that both scattering by Dirichlet obstacles with variable coefficientsand scattering by penetrable obstacles fit in the black-box framework. For other examples ofscattering problems fitting in the black-box framework, see [21, § emma 2.2 (Scattering by a Dirichlet Lipschitz obstacle fits in the black-box frame-work) Let O − , A, c, R , and R and be as in Theorem 1.6. Then the family of operators P (cid:126) v := − (cid:126) c ∇ · (cid:0) A ∇ v ) with the domain D D := (cid:110) v ∈ H ( O + ) , ∇ · (cid:0) A ∇ v (cid:1) ∈ L ( O + ) , γv = 0 (cid:111) is a semiclassical black-box operator (in the sense of Definition 2.1) with ω = c − , Q (cid:126) = − (cid:126) c ∇ · ( A ∇ ) , and H R = L (cid:0) B R ∩ O + ; c − ( x )d x (cid:1) so that H = L (cid:0) O + ; c − ( x )d x (cid:1) . Furthermore the corresponding reference operator P (cid:93) (cid:126) satisfies (BB5) with d (cid:93) = d .Proof. The non-semiclassically-scaled version of this lemma was proved for c = 1 in [41, Lemma2.1]. The proof of (BB2), (BB3), and (BB4) is essentially the same in the present semiclassically-scaled setting. The bound (BB5) follows from comparing the counting function for P (cid:93) (cid:126) to thecounting function for the problem with c = 1 by a similar argument to [41, Lemma B.2]/AppendixB, and then using the result for the problem with c = 1 proven in [41, Lemma B.1]. Lemma 2.3 (Scattering by a penetrable Lipschitz obstacle fits in the black-box frame-work)
Let O − , A, c , and R be as in Definition 1.10 Let ν be the unit normal vector field on ∂ O − pointing from O − into O + , and let ∂ ν,A the corresponding conormal derivative from either O − or O + . For D an open set, let H ( D, ∇ · ( A ∇ )) := { v : v ∈ H ( D ) , ∇ · ( A ∇ v ) ∈ L ( D ) } . Let c, α > and let H R = L (cid:0) O − , c ( x ) − α − d x (cid:1) ⊕ L (cid:0) B R \O − (cid:1) , so that H = L (cid:0) O − ; c ( x ) − α − d x (cid:1) ⊕ L (cid:0) B R \O − (cid:1) ⊕ L (cid:0) R d \ B R (cid:1) . Let, D := (cid:110) v = ( v , v , v ) where v ∈ H (cid:0) O − , ∇ · ( A − ∇ ) (cid:1) ,v ∈ H (cid:0) B R \ O − , ∇ · ( A + ∇ ) (cid:1) , v ∈ H (cid:0) R d \ B R , ∆ (cid:1) ,γv = γv and ∂ ν,A − v = α ∂ ν,A + v on ∂ O − , and γv = γv and ∂ ν v = ∂ ν v on ∂B R (cid:111) (2.6) (observe that the conditions on v and v on ∂B R in the definition of D are such that ( v , v ) ∈ H ( R d \ O − , ∇ · ( A + ∇· )) ). Then the family of operators P (cid:126) v := − (cid:126) (cid:16) c ∇ · ( A − ∇ v ) , ∇ · ( A + ∇ v ) , ∆ v (cid:17) , defined for v = ( v , v , v ) , is a semiclassical black-box operator (in the sense of Definition 2.1)on H , with Q (cid:126) = − (cid:126) ∆ , and any R > R . Furthermore, the corresponding reference operator P (cid:93) (cid:126) satisfies (BB5) with d (cid:93) = d .Proof. The non-semiclassically-scaled version of this lemma was proved for c = 1 in [41, Lemma2.3]. The proof of (BB2), (BB3), and (BB4) is essentially the same in the present semiclassically-scaled setting. The proof of the bound (BB5) is similar to the the analogous proof for c = 1 and A Lipschitz in [41, Lemma B.1]; for completeness we include the proof in § B.14 .3 A black-box functional calculus for P (cid:93) (cid:126) The operator P (cid:93) (cid:126) on the torus with domain D (cid:93) is self-adjoint with compact resolvent [21, Lemma4.11], hence we can describe the Borel functional calculus [60, Theorem VIII.6] for this opera-tor explicitly in terms of the orthonormal basis of eigenfunctions φ (cid:93)j ∈ H (cid:93) (with eigenvalues λ (cid:93)j ,appearing with multiplicity): for f a real-valued Borel function on R , f ( P (cid:93) (cid:126) ) is self-adjoint withdomain D f := (cid:26) (cid:88) a j φ (cid:93)j : (cid:88) (cid:12)(cid:12) f ( λ (cid:93)j ) a j (cid:12)(cid:12) < ∞ (cid:27) , (2.7)and if v = (cid:80) a j φ (cid:93)j ∈ D f then f ( P (cid:93) (cid:126) )( v ) := (cid:88) a j f ( λ (cid:93)j ) φ (cid:93)j . (2.8)For f a bounded Borel function, f ( P (cid:93) ) is a bounded operator, hence in this case we can dispensewith the definition of the domain and allow f to be complex-valued.Let (cid:104) ξ (cid:105) := (1 + | ξ | ) / . For m ≥
1, we then define D (cid:93),m (cid:126) as the domain of (cid:104) P (cid:93) (cid:126) (cid:105) m , equipped withthe norm (cid:107) v (cid:107) D (cid:93),m (cid:126) := (cid:107)(cid:104) P (cid:93) (cid:126) (cid:105) m v (cid:107) H (cid:93) , and D (cid:93), − m (cid:126) as its dual. In addition, we let D (cid:93), ∞ (cid:126) := ∩ m ≥ D (cid:93),m (cid:126) .The following theorem is proved in [17, Pages 23 and 24]; see also [60, Theorem VIII.5]. Theorem 2.4
The Borel functional calculus enjoys the following properties.1. f → f ( P (cid:93) (cid:126) ) is a (cid:63) -algebra homomorphism.2. for z / ∈ R , if r z ( w ) := ( w − z ) − then r z ( P (cid:93) ) = ( P (cid:93) (cid:126) − z ) − .3. If f is bounded, f ( P (cid:93) (cid:126) ) is a bounded operator for all (cid:126) , with (cid:107) f ( P (cid:93) (cid:126) ) (cid:107) L ( H (cid:93) ) ≤ sup λ ∈ R | f ( λ ) | .4. If f has disjoint support from Sp P (cid:93) (cid:126) , then f ( P (cid:93) (cid:126) ) = 0 . In describing the structure of the operators produced by the functional calculus, at least forwell-behaved functions f, it is useful to recall the Helffer–Sj¨ostrand construction of the functionalcalculus [35], [17, § f ∈ A if f ∈ C ∞ ( R ) andthere exists β < , such that, for all r > , there exists C r > | f ( r ) ( x ) | ≤ C r (cid:104) x (cid:105) β − r . Let τ ∈ C ∞ ( R ) be such that τ ( s ) = 1 for | s | ≤ τ ( s ) = 0 for | s | ≥
2. Finally, let n >
0. Wedefine an almost-analytic extension of f , denoted by (cid:101) f , by (cid:101) f ( z ) := (cid:32) n (cid:88) m =0 m ! (cid:0) ∂ m f (Re z ) (cid:1) (i Im z ) m (cid:33) τ (cid:18) Im z (cid:104) Re z (cid:105) (cid:19) (observe that (cid:101) f ( z ) = f ( z ) if z is real). For f ∈ A , we define f ( P (cid:126) ) := − π (cid:90) C ∂ (cid:101) f∂ ¯ z ( P (cid:93) (cid:126) − z ) − d x d y, (2.9)where d x d y is the Lebesgue measure on C . The integral on the right-hand side of (2.9) converges;see, e.g., [16, Lemma 1], [17, Lemma 2.2.1]. This definition can be shown to be independent of thechoices of n and τ, and to agree with the operators defined by the Borel functional calculus for f ∈ A ; see [16, Theorems 2-5], [17, Lemmas 2.2.4-2.2.7]. The construction immediately extends bycontinuity to map functions f ∈ C ( R ) to elements of L ( H (cid:93) ) , where C ( R ) = (cid:110) f ∈ C ( R ) : lim λ →±∞ f ( λ ) = 0 (cid:111) . When P is a self-adjoint elliptic semi-classical differential operator on a compact manifold,the Helffer–Sj¨ostrand construction can be used to show that f ( P ) is a pseudodifferential operator1535]. Here, in the presence of a black box, it can instead be used to show that modulo residualerrors, f ( P (cid:93) (cid:126) ) agrees with f ( Q (cid:126) ) on the region of the torus outside the black box. Furthermore,the operator wavefront-set of f ( Q (cid:126) ) can be seen to be included in the support of f ◦ q (cid:126) . We nowstate these results, obtained originally in [65].We say that E ∈ L ( H (cid:93) ) is O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) if, for any N > k >
0, there exists C N,k > (cid:107) E (cid:107) D (cid:93), − k (cid:126) →D (cid:93),k (cid:126) ≤ C N,k (cid:126) N (compare to (A.5) below). The functional calculus is pseudo-local in the following sense. Lemma 2.5 ([65, Lemma 4.1])
Suppose f ∈ C ( R ) is independent of (cid:126) , and ψ , ψ ∈ C ∞ ( T dR (cid:93) ) are constant near B R . If ψ and ψ have disjoint supports, then ψ f ( P (cid:93) (cid:126) ) ψ = O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) . Furthermore, we can show from [65, §
4] that modulo a negligible term, away from the black-box the functional calculus is given by the semiclassical pseudodifferential calculus in the followingsense. We recall in Appendix A the notion of semiclassical pseudodifferential operators on T dR (cid:93) appearing in the following Lemma. Lemma 2.6
Suppose f ∈ C ( R ) is compactly supported and independent of (cid:126) . If χ ∈ C ∞ ( T dR (cid:93) ) isequal to zero near B R , then, χf ( P (cid:93) (cid:126) ) χ = χf ( Q (cid:126) ) χ + O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) . Furthemore, f ( Q (cid:126) ) ∈ Ψ ∞ ( T dR (cid:93) ) with WF (cid:126) f ( Q (cid:126) ) ⊂ supp f ◦ q (cid:126) . Proof.
By [65, Lemma 4.2 and the subsequent paragraph], χf ( P (cid:93) (cid:126) ) χ = χf ( Q (cid:126) ) χ + O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) . The results of Helffer-Robert [34] (see the account in [62]) imply that f ( Q (cid:126) ) is a pseudodifferentialoperator on T dR (cid:93) . It remains to show that WF (cid:126) f ( Q (cid:126) ) ⊂ supp f ◦ q (cid:126) . To do so, let K (cid:15) be definedfor (cid:15) > K (cid:15) := (cid:110) z ∈ T ∗ T dR (cid:93) : dist( q (cid:126) ( z ) , supp f ) ≤ (cid:15) (cid:111) . We show that WF (cid:126) f ( Q (cid:126) ) ⊂ K (cid:15) for any (cid:15) >
0, from which the result follows. To do so, let b ∈ C ∞ ( T ∗ T dR (cid:93) ) be such that b = 1 on ( K (cid:15) ) c , and supp b ⊂ ( K (cid:15)/ ) c , and let B := Op T dR(cid:93) (cid:126) ( b ). Itsuffices to show that Bf ( Q (cid:126) ) = O ( (cid:126) ∞ ) Ψ −∞ . Indeed, if this is the case, then WF (cid:126) Bf ( Q (cid:126) ) = ∅ by(A.10). Then, by (A.11) and (A.12), WF (cid:126) f ( Q (cid:126) ) ⊂ WF (cid:126) ( I − B ) f ( Q (cid:126) ) ⊂ WF (cid:126) ( I − B ) ⊂ K (cid:15) .It therefore remains to prove that Bf ( Q (cid:126) ) = O ( (cid:126) ∞ ) Ψ −∞ . Let g ∈ C ∞ ( R ) be such that g = 0 on supp f and g = 1 on q (cid:126) (( K (cid:15)/ ) c ); such a g exists since, by the definition of K (cid:15)/ ,dist(supp f, q (cid:126) (( K (cid:15)/ ) c )) ≥ (cid:15)/ >
0. This definition then implies that g ◦ q (cid:126) = 1 on ( K (cid:15)/ ) c . ByPart 1 of Theorem 2.4, g ( Q (cid:126) ) f ( Q (cid:126) ) = ( gf )( Q (cid:126) ) = 0 . (2.10)By [62] again, g ( Q (cid:126) ) is a pseudodifferential operator with principal symbol g ◦ q (cid:126) , which is oneon ( K (cid:15)/ ) c and hence on WF (cid:126) B . Therefore, by the microlocal elliptic parametrix, Theorem A.2,there exists a pseudodifferential operator S such that B = Sg ( Q (cid:126) ) + O ( (cid:126) ∞ ) Ψ −∞ . Using this and(2.10), we obtain that Bf ( Q (cid:126) ) = Sg ( Q (cid:126) ) f ( Q (cid:126) ) + O ( (cid:126) ∞ ) Ψ −∞ = O ( (cid:126) ∞ ) Ψ −∞ , and the proof is complete.Finally, we define a family of black-box differentiation operators a family of operators satisfyinga weak binomial-type bound and agreeing with differentiation outside the black-box.16 efinition 2.7 (Black-box differentiation operator) ( D ( α )) α ∈ A is a family of black-box dif-ferentiation operators if A is a family of n –multi-indices, and the following two properties.1. For any ψ ∈ C ∞ ( T dR (cid:93) ) constant near B R , if ψ ∈ C ∞ ( T dR (cid:93) ) is so that ˜ ψ = 1 near supp ψ , wehave for all u ∈ D (cid:93), ∞ (cid:126) , (cid:107) D ( α ) ψu (cid:107) H (cid:93) ≤ (cid:88) | β | = | α | β = β + β sup | ∂ β ψ | (cid:13)(cid:13) (cid:101) ψD ( β ) u (cid:13)(cid:13) H (cid:93) .
2. For any α and any v ∈ L ( T dR (cid:93) \ B R ) , D ( α ) v = ∂ α v. The decomposition (1.7) is defined in § § Let ϕ ∈ C ∞ c ( R d ) be equal to to one in B R and supported in B R (cid:93) . For v ∈ H , we define M ϕ v := ϕv, where the multiplication is in the sense of (2.4). Now, let u ∈ D out be solution to( P (cid:126) − u = g, and let w := M ϕ u. We view w as an element of H (cid:93) and work in the torus T dR (cid:93) .We now define our frequency cut-offs. By (2.1), there exists µ > c ell > q (cid:126) ( x, ξ ) ≥ µ implies that (cid:104) ξ (cid:105) − ( q (cid:126) ( x, ξ ) − ≥ c ell > . (3.1)Now, let ψ ∈ C ∞ c ( R ) be so that ψ = (cid:40) B (0 , , B (0 , c . (3.2)We now fix ≤ µ (cid:48) ≤ µ , and define ψ µ := ψ (cid:18) · µ (cid:19) , ψ µ (cid:48) := ψ (cid:18) · µ (cid:48) (cid:19) . (3.3)Observe that ψ µ = 1 on supp ψ µ (cid:48) (3.4)(from µ (cid:48) ≤ µ ), and 1 / ∈ supp(1 − ψ µ (cid:48) ) (3.5)(from ≤ µ (cid:48) ).We define, by the Borel functional calculus for P (cid:93) (cid:126) (Theorem 2.4), in L ( H (cid:93) )Π L := ψ µ ( P (cid:93) (cid:126) ) , (3.6)and additionally Π H := (1 − ψ µ )( P (cid:93) (cid:126) ) = Id − Π L , Π (cid:48) H := (1 − ψ µ (cid:48) )( P (cid:93) (cid:126) ) . (3.7)17sing (3.4), since the Borel functional calculus is an algebra homomorphism (Part 1 of Theorem2.4), we have Π (cid:48) H Π H = Π H . (3.8)By Part 3 of Theorem 2.4, these operators are bounded on H (cid:93) , with (cid:107) Π L (cid:107) L ( H (cid:93) ) , (cid:107) Π H (cid:107) L ( H (cid:93) ) , (cid:107) Π (cid:48) H (cid:107) L ( H (cid:93) ) ≤ , (3.9)and they commute with P (cid:93) (cid:126) by Part 1 of Theorem 2.4.Observe that, since u ∈ D loc , by the definition of D (cid:93) (2.2), (BB2), and the fact that ϕ iscompactly supported, w ∈ D (cid:93) . On the other hand, in view of (2.7), as Π L w projects non-triviallyonly on a finite number of eigenspaces of P (cid:93) (cid:126) (by the definition of ψ µ (3.3), (2.8), and the fact thatSp P (cid:93) (cid:126) is discrete), Π L w ∈ D (cid:93), ∞ (cid:126) . Therefore Π H w = w − Π L w is in D (cid:93) as well. We now define u H := Π H w ∈ D (cid:93) , u L := Π L w ∈ D (cid:93), ∞ (cid:126) . We write u L in the form u L = u A + u (cid:15) , (3.10)where u A ∈ D (cid:93), ∞ (cid:126) satisfies (1.9), u H and u (cid:15) satisfy (cid:107) u H (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) P (cid:93) (cid:126) u H (cid:13)(cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) g (cid:107) H , (3.11)and (cid:107) u (cid:15) (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) P (cid:93) (cid:126) u (cid:15) (cid:13)(cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) g (cid:107) H , (3.12)and we define u H := u H + u (cid:15) so that the decomposition (1.7), (1.8) and (1.9) hold.We prove the estimate (3.11) for u H in § u A satisfying(1.9) and u (cid:15) satsifying (3.12) in § (cid:126) ∈ H . u H (high-frequency component) We proceed in three steps: we first use the abstract information we have about P (cid:93) to boundΠ H w by (cid:107) g (cid:107) H modulo a commutator term living away from the black box B R . We then use theresults of [65] Lemma 2.5, 2.6 to show that this commutator is given, up to negligible terms, bythe semiclassical pseudodifferential calculus on the torus T dR (cid:93) . Finally, we work in the torus anduse a semiclassical parametrix construction to estimate this commutator, seen as a semiclassicalpseudodifferential operator on T dR (cid:93) . Step 1: An abstract estimate in H (cid:93) Observe that, as Π H commutes with P (cid:93) (cid:126) ( P (cid:93) (cid:126) − H w ) = Π H ( P (cid:93) (cid:126) − w )= Π H ( P (cid:126) − w ) = Π H ϕg + Π H [ P (cid:126) , M ϕ ] u = Π H ϕg + Π H [ P (cid:93) (cid:126) , M ϕ ] u, (3.13)where we replaced P (cid:93) (cid:126) by P (cid:126) and conversely using (BB2) and (2.3). For λ ∈ R , we let g ( λ ) := ( λ − − (1 − ψ µ (cid:48) )( λ ) , where g is in C ( R ) by (3.5). Using (3.8), the fact that the Borel calculus in an algebra homomor-phism (Part 1 of Theorem 2.4), and finally (3.13), we getΠ H w = Π (cid:48) H Π H w = (cid:0) (1 − ψ µ (cid:48) )( · − − ( · − (cid:1) ( P (cid:93) (cid:126) )Π H w = g ( P (cid:93) (cid:126) )( P (cid:93) (cid:126) − H w = g ( P (cid:93) (cid:126) ) (cid:0) Π H ϕg + Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:1) . (3.14)18ince g is in C ( R ) , g ( P (cid:93) (cid:126) ) is uniformly bounded in (cid:126) ∈ H in H (cid:93) → H (cid:93) by Part 3 of Theorem 2.4.Thus, we obtain from (3.14) (cid:107) Π H w (cid:107) H (cid:93) (cid:46) (cid:107) Π H ϕg (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:13)(cid:13)(cid:13) H (cid:93) . Writing P (cid:93) (cid:126) Π H w = Π H w + ( P (cid:93) (cid:126) − H w and using (3.13) again, we obtain that (cid:107) Π H w (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) P (cid:93) (cid:126) Π H w (cid:13)(cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) Π H ϕg (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:13)(cid:13)(cid:13) H (cid:93) . Hence, by (3.9) (cid:107) Π H w (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) P (cid:93) (cid:126) Π H w (cid:13)(cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) ϕg (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:13)(cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) g (cid:107) H + (cid:13)(cid:13)(cid:13) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:13)(cid:13)(cid:13) H (cid:93) . (3.15) Step 2: Π H [ P (cid:93) (cid:126) , M ϕ ] as a semiclassical pseudodifferential operator on T dR (cid:93) It therefore remains to deal with the commutator term Π H [ P (cid:93) (cid:126) , M ϕ ] u . To do so, as [ P (cid:93) (cid:126) , M ϕ ] livesaway from H R , we consider the high-frequency cut-off in terms of the semiclassical pseudodiffer-ential calculus thanks to Lemma 2.6.Observe that, as ϕ is compactly supported in B R (cid:93) and equal to one near B R , [ P (cid:93) (cid:126) , M ϕ ] decom-poses in H (cid:93) as [ P (cid:93) (cid:126) , M ϕ ] = (0 , [ Q (cid:126) , ϕ ]) = (0 , φ [ Q (cid:126) , ϕ ] φ ) = (0 , [ Q (cid:126) , ϕ ] φ ) (3.16)where φ ∈ C ∞ c ( R d ) is supported in B R (cid:93) , equal to zero near B R , and so that φ = 1 near supp ∇ ϕ. (3.17)Hence, by the pseudo-locality of the functional calculus given by Lemma 2.5, taking χ ∈ C ∞ c ( R d )supported in B R (cid:93) , equal to zero near B R , and equal to one near supp φ we getΠ H [ P (cid:93) (cid:126) , M ϕ ] = χ Π H χφ [ P (cid:93) (cid:126) , M ϕ ] φ + O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) = χ Π H χ [ P (cid:93) (cid:126) , M ϕ ] φ + O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) , (3.18)where we used the last equality in (3.16) to obtain the second line. Observe that, by Lemma 2.6used for f ( P (cid:93) (cid:126) ) = ψ µ ( P (cid:93) (cid:126) ) = Π L , there is Π Ψ L ∈ Ψ ∞ ( T dR (cid:93) ) so that χ Π L χ = χ Π Ψ L χ + O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) , WF (cid:126) Π Ψ L ⊂ supp ψ µ ◦ q (cid:126) , and hence, taking Π Ψ H := I − Π Ψ L ∈ Ψ (cid:126) ( T dR (cid:93) ), χ Π H χ = χ Π Ψ H χ + O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) , WF (cid:126) Π Ψ H ⊂ supp(1 − ψ µ ) ◦ q (cid:126) , (3.19)in other words, modulo negligible terms, χ Π H χ is a high-frequency cut-off defined from the semi-classical pseudodifferential calculus. We here emphasise the fact that, since χ is supported in B R (cid:93) and vanishes near B R , Π Ψ H can be seen both as an element of L ( H (cid:93) ) and of Ψ (cid:126) ( T dR (cid:93) ). By (3.18)and (3.19), for any N and any k , (cid:107) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:107) H (cid:93) ≤ (cid:107) Π Ψ H [ P (cid:93) (cid:126) , M ϕ ] φu (cid:107) H (cid:93) + C N,k (cid:126) N (cid:107) [ P (cid:93) (cid:126) , M ϕ ] φu (cid:107) D (cid:93), − k (cid:126) + C (cid:48) N (cid:126) N (cid:107) ˜ φu (cid:107) H (cid:93) , with ˜ φ compactly supported in B R (cid:93) \ B R and equal to one on supp φ . Taking k = 1, then N = M and using the resolvent estimate (1.4) we get (cid:107) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:107) H (cid:93) ≤ (cid:107) Π Ψ H [ P (cid:93) (cid:126) , M ϕ ] φu (cid:107) H (cid:93) + C (cid:48)(cid:48) N (cid:126) N (cid:107) ˜ φu (cid:107) H (cid:93) = (cid:107) Π Ψ H [ P (cid:93) (cid:126) , M ϕ ] φu (cid:107) H (cid:93) + C (cid:48)(cid:48) N (cid:126) N (cid:107) ˜ φu (cid:107) H (cid:46) (cid:107) Π Ψ H [ P (cid:93) (cid:126) , M ϕ ] φu (cid:107) H (cid:93) + (cid:107) g (cid:107) H . (3.20)Finally, observe that (cid:107) Π Ψ H [ P (cid:93) (cid:126) , M ϕ ] φu (cid:107) H (cid:93) = (cid:107) Π Ψ H [ Q (cid:126) − , ϕ ] φu (cid:107) L ( T dR(cid:93) ) , hence by (3.20), (cid:107) Π H [ P (cid:93) (cid:126) , M ϕ ] u (cid:107) H (cid:93) (cid:46) (cid:107) Π Ψ H [ Q (cid:126) − , ϕ ] φu (cid:107) L ( T dR(cid:93) ) + (cid:107) g (cid:107) H . (3.21)19 tep 3: A semi-classical elliptic estimate in T dR (cid:93) We are thus reduced to estimate Π Ψ H [ Q (cid:126) − , ϕ ] φu in L ( T dR (cid:93) ). To do so, we use the semiclassicalparametrix construction given by Theorem A.2. Lemma 3.1
The operator Q (cid:126) − is semiclassicaly elliptic on the semiclassical wavefront-set of (cid:126) − Π Ψ H [ Q (cid:126) − , ϕ ] .Proof. By (A.12), (A.14), (3.19) and the support properties of ψ µ given by (3.2), (3.3),WF (cid:126) ( (cid:126) − Π Ψ H [ Q (cid:126) − , ϕ ]) ⊂ WF (cid:126) Π Ψ H ⊂ supp(1 − ψ µ ) ◦ q (cid:126) ⊂ { q (cid:126) ≥ µ } . But, on { q (cid:126) ≥ µ } , by definition of µ (3.1), (cid:104) ξ (cid:105) − ( q (cid:126) ( x, ξ ) − ≥ c ell > , and the proof is complete.Since (cid:126) − Π Ψ H [ Q (cid:126) − , ϕ ] ∈ Ψ (cid:126) ( T dR (cid:93) ) by Theorem A.1, we can therefore apply the ellipticparametrix construction given by Theorem A.2 with A = (cid:126) − Π Ψ H [ Q (cid:126) − , ϕ ], B = Q (cid:126) −
1, and m = 1, k = 2. Hence, there exists S ∈ Ψ − (cid:126) ( T dR (cid:93) ) and R = O ( (cid:126) ∞ ) Ψ −∞ withWF (cid:126) S ⊂ WF (cid:126) (cid:0) (cid:126) − Π Ψ H [ Q (cid:126) − , ϕ ] (cid:1) , (3.22)and so that Π Ψ H [ Q (cid:126) − , ϕ ] = (cid:126) S ( Q (cid:126) −
1) + R. Applying both sides of this identity to φu , we getΠ Ψ H [ Q (cid:126) − , ϕ ] φu = (cid:126) S ( Q (cid:126) − φu + Rφu = (cid:126) Sφ ( Q (cid:126) − u + (cid:126) S [ Q (cid:126) − , φ ] u + Rφu. (3.23)We now show that the commutator term arising in the right-hand side of this identity is negligible;this follows from the following lemma.
Lemma 3.2 WF (cid:126) S ∩ WF (cid:126) [ Q (cid:126) − , φ ] = ∅ . Proof.
By (3.22), we get using (A.12) and (A.14)WF (cid:126) S ⊂ WF (cid:126) [ Q (cid:126) − , ϕ ] ⊂ supp ∇ ϕ × R d On the other hand, by (A.12) and (A.14) againWF (cid:126) [ Q (cid:126) − , φ ] ⊂ supp ∇ φ × R d , but by (3.17), supp ∇ ϕ and supp ∇ φ are disjoint, hence the result follows.We therefore obtain using (A.13) S [ Q (cid:126) − , φ ] = O ( (cid:126) ∞ ) Ψ −∞ . (3.24)Therefore, by (3.23), (3.24) and the definition of O ( (cid:126) ∞ ) Ψ −∞ (A.5), for any N , there exists C N , C (cid:48) N > (cid:107) Π Ψ H [ Q (cid:126) − , ϕ ] φu (cid:107) L ( T dR(cid:93) ) ≤ (cid:126) (cid:107) Sφ ( Q (cid:126) − u (cid:107) L ( T dR(cid:93) ) + C N (cid:126) N (cid:107) ˜ φu (cid:107) L ( T dR(cid:93) ) + C (cid:48) N (cid:126) N (cid:107) φu (cid:107) L ( T dR(cid:93) ) = (cid:126) (cid:107) Sφ ( Q (cid:126) − u (cid:107) L ( T dR(cid:93) ) + C N (cid:126) N (cid:107) ˜ φu (cid:107) H + C (cid:48) N (cid:126) N (cid:107) φu (cid:107) H , where ˜ φ is compactly supported in B R (cid:93) \ B R and equal to one on supp φ . Taking N := M + 1 andusing the resolvent estimate (1.4) we get from the above (cid:107) Π Ψ H [ Q (cid:126) − , ϕ ] φu (cid:107) L ( T dR(cid:93) ) (cid:46) (cid:126) (cid:107) Sφ ( Q (cid:126) − u (cid:107) L ( T dR(cid:93) ) + (cid:126) (cid:107) g (cid:107) H (cid:126) (cid:107) φ ( Q (cid:126) − u (cid:107) L ( T dR(cid:93) ) + (cid:126) (cid:107) g (cid:107) H , (3.25)where we used in the second line the fact that S ∈ Ψ − ( T dR (cid:93) ) ⊂ Ψ ( T dR (cid:93) ) together with TheoremA.1, (iii). But now, since φ is equal to zero near B R and supported in B R (cid:93) , (cid:107) φ ( Q (cid:126) − u (cid:107) L ( T dR(cid:93) ) = (cid:107) φ ( P (cid:126) − u (cid:107) H = (cid:107) φg (cid:107) H ≤ (cid:107) g (cid:107) H . Thus, we obtain from (3.25) (cid:107) Π Ψ H [ Q (cid:126) − , ϕ ] φu (cid:107) L ( T dR(cid:93) ) (cid:46) (cid:126) (cid:107) g (cid:107) H . Combining this last estimate with (3.15) and (3.21) we conclude that (cid:107) Π H w (cid:107) H (cid:93) + (cid:13)(cid:13)(cid:13) P (cid:93) (cid:126) Π H w (cid:13)(cid:13)(cid:13) H (cid:93) (cid:46) (cid:107) g (cid:107) H , hence (3.11) holds. u L , and proof the bounds (1.9) and (3.12)(low frequency component) We decompose u L in three parts: a part u R A living near the black-box H R , that we estimatethanks to (1.5), one u ∞A living away from it, for which Π L is given, modulo negligible terms thatwe gather in a an error term u (cid:15) , by a pseudodifferential operator; and we define u A as u R A + u ∞A .Let ρ , ρ , ρ ∈ C ∞ ( T dR (cid:93) ) be equal to one near B R and so thatsupp ρ j ⊂ (cid:8) ρ j − = 1 (cid:9) , j = 0 , , , with ρ − := ρ . In addition, let ρ ∈ C ∞ ( T dR (cid:93) ) be equal to one near supp ρ , in such a way thatsupp(1 − ρ ) ⊂ (cid:8) − ρ = 1 (cid:9) . Furthermore, we construct ρ in such a way that | ∂ β ρ | ≤ A | β | , (3.26)for some A >
1. By the pseudo-locality of the functional calculus given by Lemma 2.5, we haveΠ L ρ = ρ Π L ρ + E , Π L (1 − ρ ) = (1 − ρ )Π L (1 − ρ ) + E , where E , E = O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) . Observe further that (1 − ρ )Π L (1 − ρ ) = (1 − ρ )Π L (1 − ρ )(1 − ρ ), hence, by Lemma 2.6,(1 − ρ )Π L (1 − ρ ) = (1 − ρ )Π Ψ L (1 − ρ )(1 − ρ ) + E , where E = O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) , Π Ψ L ∈ Ψ ∞ ( T dR (cid:93) ) andWF (cid:126) Π Ψ L ⊂ supp ψ µ ◦ q h . (3.27)We set u R A := ρ Π L ρ w, u ∞ L := (1 − ρ )Π Ψ L (1 − ρ )(1 − ρ ) w, u (cid:15), := ( E + E + E ) w. (3.28)We further split u ∞ L as u ∞ L := u ∞A + u (cid:15), , (3.29)and we decompose u L = Π L w asΠ L w = u A + u (cid:15) , u A := u R A + u ∞A , u (cid:15) := u (cid:15), + u (cid:15), , (3.30)so that (3.10) holds, and we show (1.9) and (3.12).21 he localised term u R A . We first estimate u R A . Since ρ = 1 near supp ρ , by Part 1 ofDefinition 2.7, and using (3.26) (cid:107) D ( α ) u R A (cid:107) H (cid:93) = (cid:107) D ( α ) ρ Π L ρ w (cid:107) H (cid:93) ≤ (cid:88) | β | = | α | β = β + β sup | ∂ β ρ |(cid:107) ρ D ( β )Π L ρ w (cid:107) H (cid:93) ≤ A | α | (cid:88) | β | = | α | β = β + β (cid:107) ρ D ( β )Π L ρ w (cid:107) H (cid:93) ≤ A | α | max | β |≤| α | (cid:107) ρ D ( β )Π L ρ w (cid:107) H (cid:93) (cid:16) (cid:88) | β | = | α | β = β + β (cid:17) . By the binomial theorem (cid:80) | β | = | α | β = β + β n | α | , so that (cid:107) D ( α ) u R A (cid:107) H (cid:93) ≤ A | α | n | α | max | β |≤| α | (cid:107) ρ D ( β )Π L ρ w (cid:107) H (cid:93) . (3.31)Now, observe that, since E is nowhere zero, the function E ψ µ is well-defined and in C ( R ). Thedefinition of Π L (3.6) and Part 1 of Theorem 2.4 imply that D ( β )Π L = D ( β ) ψ µ ( P (cid:93) (cid:126) ) = D ( β ) E ( P (cid:93) (cid:126) )( 1 E ψ µ )( P (cid:93) (cid:126) ) . (3.32)Using (3.32), the low-energy estimate (1.5), Part 3 of Theorem 2.4, and finally the resolventestimate (1.4), we therefore get, using in the first line the fact that ρ = 1 on supp ρ : (cid:107) ρ D ( β )Π L ρ w (cid:107) H (cid:93) = (cid:107) ρ ρD ( β )Π L ρ w (cid:107) H (cid:93) (cid:46) (cid:107) ρD ( β )Π L ρ w (cid:107) H (cid:93) ≤ C E ( β, (cid:126) ) (cid:107) ( 1 E ψ µ )( P (cid:93) (cid:126) ) ρ w (cid:107) H (cid:93) ≤ C E ( β, (cid:126) ) sup λ ∈ R (cid:12)(cid:12) E ( λ ) ψ µ ( λ ) (cid:12)(cid:12) (cid:107) w (cid:107) H (cid:93) = C E ( β, (cid:126) ) sup λ ∈ R (cid:12)(cid:12) E ( λ ) ψ µ ( λ ) (cid:12)(cid:12) (cid:107) w (cid:107) H (cid:46) C E ( β, (cid:126) ) sup λ ∈ R (cid:12)(cid:12) E ( λ ) ψ µ ( λ ) (cid:12)(cid:12) (cid:126) − M − (cid:107) g (cid:107) H , and combining the above with (3.31) we conclude (cid:107) D ( α ) u R A (cid:107) H (cid:93) (cid:46) (cid:16) max | β |≤| α | C E ( β, (cid:126) ) (cid:17) A | α | n | α | (cid:126) − M − (cid:107) g (cid:107) H . (3.33) The term away from the black-box u ∞ L . We now split u ∞ L as u ∞A + u (cid:15), and estimate bothterms. To do so, we work on the torus T dR (cid:93) and use semi-classical pseudodifferential operatorsthere, as recapped in Appendix A. By (2.1), as ψ µ is compactly supported, there exists λ > ψ µ ◦ q (cid:126) ⊂ T dR (cid:93) × B (0 , λ . (3.34)Now, let ϕ ∈ C ∞ c be compactly supported in B (0 , λ ) and equal to one on B (0 , λ ). By (3.34)and (3.27) together with (A.14), WF (cid:126) (cid:0) − Op T dR(cid:93) (cid:126) ( ϕ ( | ξ | )) (cid:1) ∩ WF (cid:126) (cid:0) (1 − ρ )Π Ψ L (cid:1) = ∅ , therefore, by(A.13) (1 − ρ )Π Ψ L = Op T dR(cid:93) (cid:126) ( ϕ ( | ξ | ))(1 − ρ )Π Ψ L + E , where E = O ( (cid:126) ∞ ) Ψ −∞ . We now take ρ ∈ C ∞ ( T dR (cid:93) ) equal to one near B R and so that supp(1 − ρ ) ⊂ { − ρ = 1 } , in such a way that 1 − ρ = (1 − ρ )(1 − ρ ), and we set u ∞A := (1 − ρ ) Op T dR(cid:93) (cid:126) ( ϕ ( | ξ | ))(1 − ρ )Π Ψ L (1 − ρ )(1 − ρ ) w, u (cid:15), := (1 − ρ ) E (1 − ρ ) w, so that (3.29) holds. Furthermore, we construct ρ in such a way that, for some B > | ∂ β (1 − ρ ) | ≤ B | β | .
22e first deal with u ∞A . Observe that, by the above (in the same way as we did it for (3.31)) (cid:107) ∂ α u ∞A (cid:107) L ( T dR(cid:93) ) ≤ B | α | n | α | max | β |≤| α | (cid:13)(cid:13)(cid:13) ∂ β (cid:16) Op T dR(cid:93) (cid:126) ( ϕ ( | ξ | ))(1 − ρ )Π Ψ L (1 − ρ )(1 − ρ ) w (cid:17)(cid:13)(cid:13)(cid:13) L ( T dR(cid:93) ) . (3.35)Now, by Lemma A.3, Op T dR(cid:93) (cid:126) ( ϕ ( | ξ | )) is given as a Fourier multiplierOp T dR(cid:93) (cid:126) ( ϕ ( | ξ | )) = ϕ ( − (cid:126) ∆) . (3.36)Let v ∈ L ( T dR (cid:93) ) be arbitrary. Denoting ˆ v ( j ) the Fourier coefficients of v , we have, by definition ofa Fourier multiplier on the torus (A.16) ϕ ( − (cid:126) ∆) v = (cid:88) j ∈ Z d ˆ v ( j ) ϕ ( (cid:126) | j | π /R (cid:93) ) e j , where the normalised eigenvalues e j are defined by (A.1). Hence, for any multi-indice β∂ β ϕ ( − (cid:126) ∆) v = (cid:88) j ∈ Z d ˆ v ( j ) ϕ ( (cid:126) | j | π /R (cid:93) ) (cid:18) i πjR (cid:93) (cid:19) β e j = (cid:88) j ∈ Z d , | j |≤ λR(cid:93) (cid:126) π ˆ v ( j ) ϕ ( (cid:126) | j | π /R (cid:93) ) (cid:18) i πjR (cid:93) (cid:19) β e j . Therefore (cid:107) ∂ β ϕ ( − (cid:126) ∆) v (cid:107) L ( T dR(cid:93) ) = (cid:88) j ∈ Z d , | j |≤ λR(cid:93) (cid:126) π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ v ( j ) ϕ ( (cid:126) | j | π /R (cid:93) ) (cid:18) i πjR (cid:93) (cid:19) β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ | β | (cid:126) − | β | (cid:88) j ∈ Z d | ˆ v ( j ) | = λ | β | (cid:126) − | β | (cid:107) v (cid:107) L ( T dR (cid:93) ) . (3.37)We now use (3.37) with v := (1 − ρ )Π Ψ L (1 − ρ )(1 − ρ ) w, and combine the resulting estimate with (3.35) and (3.36); this yields (cid:107) ∂ α u ∞A (cid:107) L ( T dR(cid:93) ) (cid:46) B | α | n | α | λ | α | (cid:126) −| α | (cid:107) (1 − ρ )Π Ψ L (1 − ρ )(1 − ρ ) w (cid:107) L ( T dR(cid:93) ) . Observe that (1 − ρ )Π Ψ L (1 − ρ ) ∈ Ψ ∞ (cid:126) , and hence is in particular bounded L → L uniformly in (cid:126) , thus we obtain (cid:107) ∂ α u ∞A (cid:107) L ( T dR(cid:93) ) (cid:46) B | α | n | α | λ | α | (cid:126) −| α | (cid:107) (1 − ρ ) w (cid:107) L ( T dR(cid:93) ) = B | α | n | α | λ | α | (cid:126) −| α | (cid:107) (1 − ρ ) w (cid:107) H , where we used in the second step the fact that (1 − ρ ) ϕ is supported away from B R , and in B R (cid:93) .From the fact that (1 − ρ ) is zero near B R , Part 2 of Definition 2.7, and the resolvent estimate(1.4), we conclude that (cid:107) D ( α ) u ∞A (cid:107) H (cid:93) = (cid:107) ∂ α u ∞A (cid:107) L ( T dR(cid:93) ) (cid:46) B | α | n | α | λ | α | (cid:126) −| α | (cid:126) − M − (cid:107) g (cid:107) H . (3.38)We now deal with u (cid:15), . Since (1 − ρ ) is zero near B R , using E = O ( (cid:126) ∞ ) Ψ −∞ , we have (cid:107) u (cid:15), (cid:107) H (cid:93) + (cid:107) P (cid:93) (cid:126) u (cid:15), (cid:107) H (cid:93) = (cid:107) u (cid:15), (cid:107) L ( T dR(cid:93) ) + (cid:107) Q (cid:126) u (cid:15), (cid:107) L ( T dR(cid:93) ) (cid:46) M (cid:126) M +1 (cid:107) (1 − ρ ) w (cid:107) L ( T dR(cid:93) ) = (cid:126) M +1 (cid:107) (1 − ρ ) w (cid:107) H (cid:93) . The resolvent estimate (1.4) then implies that (cid:107) u (cid:15), (cid:107) H (cid:93) + (cid:107) P (cid:93) (cid:126) u (cid:15), (cid:107) H (cid:93) (cid:46) (cid:107) g (cid:107) H . (3.39)23 onclusion. Since E + E + E is O ( (cid:126) ∞ ) D (cid:93), −∞ (cid:126) →D (cid:93), ∞ (cid:126) , so is P (cid:93) (cid:126) ( E + E + E ), and we thereforehave, using the resolvent estimate (1.4) (cid:107) u (cid:15), (cid:107) H (cid:93) + (cid:107) P (cid:93) (cid:126) u (cid:15), (cid:107) H (cid:93) (cid:46) M (cid:126) M +1 (cid:107) w (cid:107) H (cid:93) (cid:46) (cid:107) g (cid:107) H . Combining the above with (3.39) and (3.30), we get the estimate (3.12) on u (cid:15) . Finally, combining(3.30) with (3.33) and (3.38), we get the estimate (1.9) on u A with σ := max( A, n ) +max( B, n, λ ) ,which ends the proof of Theorem 1.1. Theorems 1.6 and 1.12 are proved using the following two corollaries of Theorem 1.1. In the firstcorollary (Corollary 4.1), the low-energy estimate (1.5) comes from a heat-flow estimate, and inthe second (Corollary 4.2) from an elliptic-regularity estimate.
Corollary 4.1
Let P (cid:126) be a semiclassical black-box operator on H satisfying the polynomial resol-vent estimate (1.4) in H ⊂ (0 , (cid:126) ] . Assume further that (i) P (cid:93) (cid:126) ≥ a ( (cid:126) ) > for some a ( (cid:126) ) > ,and (ii) for some α -family of black-box differentiation operators ( D ( α )) α ∈ A (Definition 2.7), thereexists ρ ∈ C ∞ ( T dR (cid:93) ) equal to one near B R such that the following localised heat-flow estimateholds, (cid:13)(cid:13)(cid:13) ρD ( α )e − t (cid:126) − P (cid:126) (cid:13)(cid:13)(cid:13) H (cid:93) →H (cid:93) ≤ C ( α ) t −| α | / for all α ∈ A , < t < , (cid:126) ∈ H. (4.1) Then, if R < R < R (cid:93) , g ∈ H is compactly supported in B R , and u ( (cid:126) ) ∈ D out satisfies (1.6),there exists u A ∈ D (cid:93), ∞ (cid:126) , u H ∈ D (cid:93) such that u can be written as (1.7), u H satisfies (1.8), and u A satisfies, for some σ > , (cid:107) D ( α ) u A (cid:107) H (cid:93) (cid:46) (cid:16) max | β |≤| α | C ( β ) + 1 (cid:17) σ | α | (cid:126) −| α |− M − (cid:107) g (cid:107) H for all α ∈ A , h ∈ H, (4.2) where the omitted constant is independent of (cid:126) and α .Proof. Since P (cid:93) (cid:126) ≥ a ( (cid:126) ) >
0, Sp P (cid:93) (cid:126) ⊂ [ a ( (cid:126) ) , ∞ ). Therefore, by Parts 4 and 3 of Theorem 2.4, e − P (cid:126) = E ( P (cid:126) ), where E ( λ ) := e −| λ | . Such an E is in C ( R ), never vanishes, and satisfies (1.5) with C E ( α, (cid:126) ) := C ( α ) (cid:126) −| α | by (4.1) with t := (cid:126) . The result therefore follows from Theorem 1.1. Corollary 4.2
Let P (cid:126) be a semiclassical black-box operator on H satisfying the polynomial resol-vent estimate (1.4) in H ⊂ (0 , (cid:126) ] . Assume further that, for some α -family of black-box differentia-tion operators ( D ( α )) α ∈ A (in the sense of Definition 2.7), there exists L ( α ) > and ρ ∈ C ∞ ( T dR (cid:93) ) equal to one near B R so that the following elliptic-regularity estimate holds, (cid:107) ρD ( α ) w (cid:107) H (cid:93) ≤ L ( α ) (cid:88) (cid:96) =0 C (cid:96) ( α, (cid:126) ) (cid:13)(cid:13) ( P (cid:93) (cid:126) ) (cid:96) w (cid:13)(cid:13) H (cid:93) for all α ∈ A , w ∈ D (cid:93), ∞ (cid:126) , (cid:126) ∈ H, (4.3) for some C (cid:96) ( α, (cid:126) ) > , (cid:96) = 0 , . . . , L ( α ) .Then, if R < R < R (cid:93) , g ∈ H is compactly supported in B R and u ( (cid:126) ) ∈ D out satisfies (1.6),there exists u A ∈ D (cid:93), ∞ (cid:126) , u H ∈ D (cid:93) such that u can be written as (1.7), u H satisfies (1.8), and u A satisfies, for some σ > , (cid:107) D ( α ) u A (cid:107) H (cid:93) (cid:46) (cid:16) max | β |≤| α | max (cid:96) =0 ,...,L ( α ) C (cid:96) ( β, (cid:126) )+ (cid:126) −| α | (cid:17) σ | α | (cid:126) − M − (cid:107) g (cid:107) H for all α ∈ A , (cid:126) ∈ H, (4.4) where the omitted constant is independent of (cid:126) and α . roof. Let E ( λ ) := (cid:104) λ (cid:105) − L ( α ) and C E ( α, (cid:126) ) := max (cid:96) =0 ,...,L ( α ) C (cid:96) ( α, (cid:126) ). We now need to show thatthe bound (4.3) implies that the bound (1.5) holds with these choices of E and C E . Given v ∈ D (cid:93), ∞ (cid:126) ,let w := (cid:104) P (cid:93) (cid:126) (cid:105) − L ( α ) v ∈ D (cid:93), ∞ (cid:126) , the bound (4.3) implies that (cid:13)(cid:13)(cid:13) ρD ( α ) (cid:104) P (cid:93) (cid:126) (cid:105) − L ( α ) v (cid:13)(cid:13)(cid:13) H (cid:93) ≤ L ( α ) (cid:88) (cid:96) =0 C (cid:96) ( α, (cid:126) ) (cid:13)(cid:13) ( P (cid:93) (cid:126) ) (cid:96) (cid:104) P (cid:93) (cid:126) (cid:105) − L ( α ) v (cid:13)(cid:13) H (cid:93) for all α ∈ A , (cid:126) ∈ H. (4.5)Since (cid:104) λ (cid:105) − L ( α ) λ (cid:96) ≤
1, by Part 3 of Theorem 2.4, the term in brackets on the right-hand side of(4.5) is bounded by C E ( α, (cid:126) ) (cid:107) v (cid:107) H (cid:93) , and then (1.5) follows. The result (4.4) then follows from thebound (1.9) in Theorem 1.1. The plan is to apply Corollary 4.1. We let (cid:126) := k − , g := (cid:126) f , and define H and P (cid:126) as in Lemma2.2. By Lemma 2.2, P (cid:126) is a semiclassical black-box operator on H .As in Corollary 4.1, we choose ρ to be equal to one near B R , and further assume that ρ issupported in B ( R + R ∗ ) / (i.e., in a region where A and c are known to be analytic). Since O − isanalytic, and A and c are analytic on a domain including the support of ρ , the heat-flow estimate(4.1) is satisfied with D ( α ) := ∂ α and C ( α ) = ν − −| α | / | α | ! , for some ν >
0, and, without loss of generality, <
1, by [24, 23] (note that the heat-flow given bythe functional calculus, appearing in (4.1), is indeed the solution of the heat equation; see, e.g.,[60, Theorem VIII.7]).We can therefore apply Corollary 4.1 with an arbitrary R (cid:93) > R , and we obtain u H , u A ∈ L ( T dR (cid:93) \O ) satisfying (1.7), (1.8) and (4.2). Observe that u H and u A satisfy ther Dirichlet bound-ary condition (1.12) since they are in D (cid:93) . For J := { k ∈ [ k , ∞ ) | k − / ∈ H } , the low-frequencyestimate (4.2) gives directly the bound (1.19). Having proved (1.19), Lemma C.1 shows that u A is analytic.The bound (1.8) implies that (cid:107) u H (cid:107) L ( T dR(cid:93) \O ) + k − (cid:107) ∆ u H (cid:107) L ( T dR(cid:93) \O ) (cid:46) k − (cid:107) f (cid:107) L ( B R ∩O + ) , and then Green’s identity (see, e.g., [46, Lemma 4.3]) implies that (cid:107) u H (cid:107) L ( T dR(cid:93) \O ) + k − (cid:107)∇ u H (cid:107) L ( T dR(cid:93) \O ) + k − (cid:107) ∆ u H (cid:107) L ( T dR(cid:93) \O ) (cid:46) k − (cid:107) f (cid:107) L ( B R ∩O + ) ; (4.6)i.e., (1.18) holds for | α | = 0 and 1. Finally, to obtain (1.18) for | α | = 2, we combine (4.6)with the H regularity result of, e.g., [46, Part (i) of Theorem 4.18, pages 137-138], applied withΩ = B R ∩ O + and Ω = B ( R + R (cid:93) ) / ∩ O + . The plan is to apply Corollary 4.2 with ρ = 1. We let (cid:126) := k − , g := (cid:126) f , and define H and P (cid:126) asin Lemma 2.2. By Lemma 2.2, P (cid:126) is a semiclassical black-box operator on H .We now need to check that the regularity estimate (4.3) is satisfied with (i) D ( α ) :=( ∂ α | O − , ∂ α | O + ), (ii) A consisting of multi-indices α such that | α | is even and | α | ≤ m , (iii) L ( α ) = | α | / , and (iv) an appropriate choice of C (cid:96) ( α, (cid:126) ). Let O − (cid:98) Ω (cid:98) Ω (cid:98) Ω ⊂ T dR (cid:93) and χ ∈ C ∞ c ( R d ) to be equal to one on Ω and supported in Ω . For v satisfying (1.29), we apply theregularity result of [15, Theorem 5.2.1, Part (i)] to χv with k = 2, L + = ∇· ( A ∇· ), L − = c ∇· ( A ∇ ),and g = h = h I = g I = . We obtain, where the omitted constant depends on Ω , A , and c , (cid:107) v (cid:107) H ( O − ) ⊕ H (Ω ∩O + ) (cid:46) (cid:107)∇ · ( A ∇ v ) (cid:107) L ( O − ) ⊕ L (Ω ∩O + ) + (cid:107) v (cid:107) H ( O − ) ⊕ H (Ω ∩O + ) ≤ (cid:107)∇ · ( A ∇ v ) (cid:107) L ( O − ) ⊕ L ( T dR(cid:93) ∩O + ) + (cid:107) v (cid:107) H ( O − ) ⊕ H ( T dR(cid:93) ∩O + ) , (cid:107)∇ · ( A ∇ v ) (cid:107) L ( O − ) ⊕ L ( T dR(cid:93) ∩O + ) + (cid:107) v (cid:107) L ( O − ) ⊕ L ( T dR(cid:93) ∩O + ) , using Green’s identity. Using this last inequality with [15, Theorem 5.2.1, Part (i)] with k =4 , , . . . m , we obtain that, for any v satisfying (1.29), (cid:107) v (cid:107) H | α | ( O − ) ⊕ H | α | (Ω ∩O + ) ≤ | α | / (cid:88) (cid:96) =0 (cid:101) C (cid:96) ( α ) (cid:13)(cid:13) ( ∇ · ( A ∇ )) (cid:96) v (cid:13)(cid:13) L ( O − ) ⊕ L ( T dR(cid:93) ∩O + ) (4.7)where (cid:101) C (cid:96) ( α ) also depends on Ω , A , and c . Since the torus is compact (and is thus covered by a fi-nite number of Ω s), (4.7) holds with the left-hand side replaced by (cid:107) v (cid:107) H | α | ( O − ) ⊕ H | α | ( O + ∩ T dR(cid:93) ) .Therefore, the regularity estimate (4.3) holds with L = | α | / D ( α ) := ( ∂ α | O − , ∂ α | O + ), and C (cid:96) ( α, (cid:126) ) = (cid:126) − (cid:96) (cid:101) C (cid:96) ( α ). Therefore, there exist C = C ( m ) > | α | ≤ m ,max | β |≤| α | max (cid:96) =0 ,..., | α | / C (cid:96) ( β, (cid:126) ) + (cid:126) −| α | ≤ C ( m ) (cid:126) −| α | . We can therefore apply Corollary 4.2. We obtain u H , u A satisfying (1.7), (1.8), and (4.4).Observe that u H and u A satisfy the transmission conditions (1.29) since they are in D (cid:93) . If J := { k ∈ [ k , ∞ ) | k − / ∈ H } , then the low-frequency estimate (4.2) gives (1.33) for all α ∈ A ,i.e., for all α with | α | even and ≤ m . By interpolation, the bound (1.33) then holds for all α with | α | ≤ m . Finally, (1.32) follows from the high frequency estimate (1.8), together with Green’sidentity and (4.7) applied with (cid:96) = 1 (similar to the end of the proof of Theorem 1.6). The two ingredients for the proof of Theorems 1.7 and 1.14 are • Lemma 5.4, which is the standard duality argument giving a condition for quasi-optimalityto hold in terms of how well the solution of the adjoint problem is approximated by thefinite-element space (measured by the quantity η ( V N ) defined by (5.4)), and • Lemma 5.5 that bounds η ( V N ) using the decomposition from Theorems 1.6 and 1.12.Regarding Lemma 5.4: this argument came out of ideas introduced in [64], was then formalised in[63], and has been used extensively in the analysis of the Helmholtz FEM; see, e.g., [1, 38, 47, 63,51, 52, 73, 71, 19, 13, 44, 14, 29, 33, 28, 42].Before stating Lemma 5.4 we need to introduce some notation. Let C cont = C cont ( A, c − , R, k )be the continuity constant of the sesquilinear form a ( · , · ) (defined in (1.21)) in the norm (cid:107) · (cid:107) H k ( B R ∩O + ) ; i.e. a ( u, v ) ≤ C cont (cid:107) u (cid:107) H k ( B R ∩O + ) (cid:107) v (cid:107) H k ( B R ∩O + ) for all u, v ∈ H ( B R ∩ O + ) . By the Cauchy-Schwarz inequality and (1.22), C cont ≤ max { A max , c − } + C DtN . (5.1)The following definitions are stated for the sesquilinear form of the Dirichlet problem (1.21). Forthe sesquilinear form of the transmission problem with the transmission parameter β = 1, one onlyneeds to replace B R ∩ O + by B R and define c to be equal to one in B R ∩ O + . Definition 5.1 (The adjoint sesquilinear form a ∗ ( · , · ) ) The adjoint sesquilinear form, a ∗ ( u, v ) , to the sesquilinear form a ( · , · ) defined in (1.21) is given by a ∗ ( u, v ) := a ( v, u ) = (cid:90) B R ∩O + (cid:18) ( A ∇ u ) · ∇ v − k c uv (cid:19) − (cid:10) γu, T R ( γv ) (cid:11) ∂B R . efinition 5.2 (Adjoint solution operator S ∗ ) Given f ∈ L ( B R ) , let S ∗ f be defined as thesolution of the variational problemfind S ∗ f ∈ H ( B R ∩ O + ) such that a ∗ ( S ∗ f, v ) = (cid:90) B R ∩O + f v for all v ∈ H ( B R ∩ O + ) . (5.2)Green’s second identity applied to solutions of the Helmholtz equation satisfying the Sommer-feld radiation condition (1.2) implies that (cid:10) DtN k ψ, φ (cid:11) ∂B R = (cid:10) DtN k φ, ψ (cid:11) ∂B R (see, e.g., [67, Lemma6.13]); thus a ( v, u ) = a ( u, v ) and so the definition (5.2) implies that a ( S ∗ f , v ) = ( f , v ) L ( B R ) for all v ∈ H ( B R ∩ O + ) . (5.3) Definition 5.3 ( η ( V N ) ) Given a sequence ( V N ) ∞ N =0 of finite-dimensional subspaces of let η ( V N ) := sup (cid:54) = f ∈ L ( B R ∩O + ) min v N ∈ V N (cid:107) S ∗ f − v N (cid:107) H k ( B R ∩O + ) (cid:13)(cid:13) f (cid:13)(cid:13) L ( B R ∩O + ) . (5.4) Lemma 5.4 (Conditions for quasi-optimality) If k η ( V N ) ≤ C cont (cid:115) A min (cid:0) n max + A min (cid:1) , then the Galerkin equations (1.24) have a unique solution which satisfies (cid:107) u − u h (cid:107) H k ( B R ∩O + ) ≤ C cont A min (cid:18) min v N ∈ V N (cid:107) u − v N (cid:107) H k ( B R ∩O + ) (cid:19) . References for the proof.
See, e.g., [42, Lemma 6.4].
Lemma 5.5 (Bound on η ( V N ) for the exterior Dirichlet problem) Suppose that d = 2 or and A, c, and O − are as in Theorem 1.6. Let ( V N ) ∞ N =0 be the piecewise-polynomial approximationspaces described in [51, § k > , there exist C , C , σ > , depending on A, c, R , n , and k , but independent of k , h , and p , such that k η ( V N ) ≤ C hkp (cid:18) hkp (cid:19) + C k M +1 (cid:18) hkσ (cid:19) p for all k ≥ k . (5.5) Lemma 5.6 (Bound on η ( V N ) for the transmission problem) Suppose that d = 2 or , β =1 , and A, c, and O − are as in Theorem 1.14. Let ( V N ) ∞ N =0 be a sequence of piecewise-polynomialapproximation spaces satisfying Assumption 1.13.Given k > , there exist (cid:101) C , (cid:101) C , depending on A, c, R , n , k , and p , but independent of k and h , such that k η ( V N ) ≤ (cid:0) hk (cid:1)(cid:16) (cid:101) C hk + (cid:101) C k M +1 ( hk ) p (cid:17) for all k ≥ k . (5.6) Proof of Theorems 1.7/1.14 assuming Lemmas 5.5/5.6.
Theorem 1.7 follows by combiningLemmas 5.4 and 5.5 and the inequality (5.1). Theorem 1.14 follows similarly, but replacing Lemma5.5 with Lemma 5.6.
Given f ∈ L , let v = S ∗ f . By (5.3) and Theorem 1.6, v = v H + v A , where v H and v A satisfythe bounds (1.18) and (1.19) with u replaced by v .Let (cid:12)(cid:12) ∇ n u ( x ) (cid:12)(cid:12) := (cid:88) | α | = n n ! α ! (cid:12)(cid:12) ∂ α u ( x ) (cid:12)(cid:12) . (cid:88) | α | = n n ! α ! = d n , (5.7)the bound (1.19) on v A implies that (cid:107)∇ p v A (cid:107) L ( B R ∩O + ) ≤ C (cid:0) √ d C (cid:1) p p ! k p − M (cid:107) f (cid:107) L ( B R ∩O + ) for all k ∈ K and for all p ∈ Z + . (5.8)The proof of Lemma 5.5 is very similar to the proofs of [51, Theorem 5.5] and [52, Proposition5.3]. The only difference is that the results in [51], [52] are proved under the assumption that (cid:107)∇ p v A (cid:107) L ( B R ∩O + ) (cid:46) γ p max (cid:8) p, k (cid:9) p k − k M (cid:107) f (cid:107) L ( B R ∩O + ) , (5.9)for some γ > k and p , whereas we have (5.8), i.e., max (cid:8) p, k } p is replaced by p ! k p .The end result is that we have k ( kh/σ ) p on the right-hand side of (5.5), instead of the stronger k ( kh/ ( σp )) p on the right-hand sides of the corresponding bounds in [51, Theorem 5.5] and [52,Proposition 5.3] (see also [42, Equation 6.5]. We now go through the proof of [51, Theorem 5.5],outlining the necessary changes to prove (5.5).Exactly as in the proof of [51, Theorem 5.5], there exists C > w N ∈ V N (cid:107) v − w N (cid:107) H k ( B R ∩O + ) ≤ C hp (cid:18) hkp (cid:19) | v | H ( B R ∩O + ) , (5.10)for all v ∈ H ( B R ∩ O + ); recall that this result follows from the polynomial-approximation resultof [51, Theorem B.4] and the definition (1.13) of the norm (cid:107) · (cid:107) H k . Applying the bound (5.10) to v H and using (1.18) with | α | = 2, we obtainmin w N ∈ V N (cid:107) v H − w N (cid:107) H k ( B R ∩O + ) ≤ C C hp (cid:18) hkp (cid:19) (cid:107) f (cid:107) L ( B R ∩O + ) ;we then let C := C C .To prove (5.5), therefore, we only need to show thatmin w N ∈ V N (cid:107) v A − w N (cid:107) H k ( B R ∩O + ) ≤ C k M (cid:18) hkσ (cid:19) p (cid:107) f (cid:107) L ( B R ∩O + ) . The proof proceeds as in the proof of [51, Theorem 5.5], but with • [51, Equation 5.8] holding with ( k M ) on the right-hand side, • the bounds on (cid:107)∇ p (cid:101) v (cid:107) L ( (cid:101) K ) and (cid:107)∇ p (cid:98) v (cid:107) L ( (cid:101) K ) holding with max { p, k } p replaced by p ! k p , and • the last equation on [51, Page 1896] holding with k M on the right-hand side.The rest of the proof of [51, Theorem 5.5] consists of applying three lemmas: [51, Lemmas C.1,C.2, and C.3]. We now discuss the necessary modifications to each of these.The analogue of [51, Lemma C.1] holds with max { p, k } p replaced by p ! k p in both bounds inthe statement of the lemma. The proof of this modified result is very similar to the proof of [51,Lemma C.1]; indeed, both follow from the arguments in [48, Lemma 4.3.1], however the proofis easier in our case since there is no parameter ε , and so [48, Equation 4.3.38] holds with theright-hand side equal to C instead of C e γ (cid:48) δr/ε . Furthermore, in the calculations at the top of [48,Page 166] we only need to consider the second of the two terms.The result [51, Lemma C.2] is replaced by Lemma 5.7 below. Once this is established, thenatural analogue of [51, Lemma C.3] follows immediately from the reasoning in the proof of [51,Lemma C.3]. We highlight that [51, Lemma C.2] contains an arbitrary parameter R >
0; however,the choice R = 1 is made in the proof of [51, Theorem 5.5]. Therefore, for simplicity, we make thechoice R = 1 from the outset here. 28 emma 5.7 (Analogue of Lemma C.2 in [51]) Let d ∈ { , , } , and let (cid:98) K ⊂ R d be the ref-erence simplex. Let γ, (cid:101) C > be given. Then there exist constants C, σ > that depend solely on γ and (cid:101) C such that the following is true: For any function u that satisfies for some C u , h , κ ≥ theconditions (cid:107)∇ n u (cid:107) L ( (cid:98) K ) ≤ C u ( γh ) n n ! κ n , for n = 2 , , . . . , if κh ≤ (cid:101) C (5.11) and p ∈ N then inf π ∈P p (cid:107) u − π (cid:107) W , ∞ ( (cid:98) K ) ≤ CC u (cid:18) κhσ (cid:19) p +1 , (5.12) where P p denotes the space of polynomials on (cid:98) K of degree ≤ p . We make the following remarks: • Both [51, Lemma C.2] and Lemma 5.7 are applied with κ = k , but we keep the κ notationhere for consistency with [51]. • The analogues of (5.11) and (5.12) in [51, Lemma C.2] are, respectively, h + κhp ≤ (cid:101) C and (cid:18) hσ + h (cid:19) p +1 + (cid:18) κhσp (cid:19) p +1 . The fact that Lemma 5.7 has replaced hκ/p by hκ is the reason why we do not quite obtainthe optimal conditions for quasioptimality. • As h → κ fixed, the approximation error in (5.12) is O ( h p +1 ), as expected.It therefore remains to prove Lemma 5.7. Proof of Lemma 5.7.
We proceed as in the proof of [51, Lemma C.2], replacing max { n, κ } n by n ! κ n in the third and fourth displayed equations in this proof. The analogue of [51, Equation C.6]is now (cid:107)∇ n (cid:101) u (cid:107) L ∞ ( (cid:98) K ) ≤ CC u p ( γγh ) n ( n + 2)! κ n +2 , for n = 0 , , , . . . , and the analogue of [51, Equation C.7] is now µ := γγ √ d. The Taylor series of (cid:101) u then converges in the L ∞ norm in a ball of radius 1 / ( µhκ ). As in [51,Lemma C.2], we let r := diam( (cid:98) K ) and consider two cases: µκh > / (2 r ) and µκh ≥ / (2 r ).(We note that in [51, Lemma C.2] the ball has radius 1 / ( µh ) and the two cases are µh ≤ / (2 r )and µh > / (2 r ).)For the case µκh ≤ / (2 r ) we approximate (cid:101) u by its truncated Taylor series T p u . With b (cid:98) K thebarycenter of (cid:98) K , (cid:107) (cid:101) u − T p u (cid:107) L ∞ ( B r ( b (cid:99) K )) ≤ S, where S := CC u p ∞ (cid:88) n = p +1 ( n + 2)( n + 1)( µκhr ) n . (5.13)To compare with [51, Lemma C.2]: because of the presence of the max in (5.9), there are two sumsin [51, Lemma C.2], denoted by S and S . We deal with S (5.13) in a similar way to how the sum S is dealt with in [51, Lemma C.3].The series in (5.13) converges if µκhr <
1, and this is the case because we’re in the situationthat µκh ≤ / (2 r ). If 0 < a ≤ a <
1, then ∞ (cid:88) n = p +1 ( n + 2)( n + 1) a n = ∂ ∂a (cid:18) a p +3 − a (cid:19) ≤ C (cid:48) a p +1 p C (cid:48) depends on a but is independent of p . Therefore S ≤ CC u C (cid:48) p ( hκµr ) p +1 ≤ CC u C (cid:48) C (cid:48)(cid:48) ( hκµr ) p +1 , for some C (cid:48)(cid:48) >
0, independent of all parameters, where we have used the exponential decay of( hκµr ) p +1 as p → ∞ to absorb the p term.We now let σ = 1 / ( µr ) and proceed exactly as in [51, Lemma C.3] (i.e., using the Cauchyintegral formula) to obtain that (cid:107) (cid:101) u − T p u (cid:107) W , ∞ ( (cid:98) K ) ≤ CC u C (cid:48) C (cid:48)(cid:48) (cid:18) hκσ (cid:19) p +1 , which concludes the proof in the case µκh ≤ / (2 r ).For the case µκh > / (2 r ), we proceed as in [51, Lemma C.2], observing that (5.11) impliesthat 1 / ( µκh ) ≥ / ( (cid:101) Cµ ) =: 2 r . Then, exactly as in [51, Lemma C.2], (cid:101) u is analytic on (cid:98) U r := ∪ x ∈ (cid:98) K B r ( x ) ⊂ C d . The analogue of the fifth displayed equation on [51, Page 1910] is then (cid:107) (cid:101) u (cid:107) L ∞ ( U r ) ≤ CC u p ( θκh ) , for some θ >
0, independent of p, κ, and h , and where (cid:98) U r := ∪ x ∈ (cid:98) K B r ( x ). We proceed as in [51,Lemma C.2], using approximation results for analytic functions on triangles/tetrahedra from [48,Prop. 3.2.16] and [22, Theorem 1]. We use the fact that given r , µ, and b , there exists σ, C (cid:48)(cid:48)(cid:48) > r , µ, and b ) such that p e − bp ≤ C (cid:48)(cid:48)(cid:48) (cid:18) σ r µ (cid:19) p +1 for all p ≥ , and then obtain, using that 1 / (2 r µ ) < κh in this case, thatinf π ∈P p (cid:107) u − π (cid:107) W , ∞ ( (cid:98) K ) ≤ CC u C (cid:48)(cid:48)(cid:48) (cid:18) hκσ (cid:19) p +1 , which completes the proof in this case. Given f ∈ L , let v = S ∗ f . By (5.3) and Theorem 1.12, v = v H + v A , where v H and v A satisfythe bounds (1.32) and (1.33) with u replaced by v .By the definition of the H k norm (1.13) and the bound (1.34), there exists C int = C int ( (cid:96) ) > w N ∈ V N (cid:107) w − w N (cid:107) H k ( B R ) ≤ C int ( (cid:96) )(1 + kh ) h (cid:96) (cid:16) (cid:107) w + (cid:107) H (cid:96) +1 ( B R ∩O + ) + (cid:107) w − (cid:107) H (cid:96) +1 ( O − ) (cid:17) (5.14)for all w = ( w + , w − ) ∈ H (cid:96) +1 ( B R ∩ O + ) × H (cid:96) +1 ( O − ). Applying (5.14) with (cid:96) = 1 to v H and using(1.32) with | α | = 2, we obtain thatmin w N ∈ V N (cid:107) v H − w N (cid:107) H k ( B R ) ≤ C int (1)(1 + kh ) h C (cid:107) f (cid:107) L ( B R ) . (5.15)Applying (5.14) with (cid:96) = p to v A and using (1.33) with | α | = p + 1 (which is allowed since A, c ,and O − are all C p +1 ), we obtain thatmin w N ∈ V N (cid:107) v A − w N (cid:107) H k ( B R ) ≤ C int ( p )(1 + kh ) h p C ( p ) k M + p (cid:107) f (cid:107) L ( B R ) . (5.16)The bound on η ( V N ) in (5.6) then follows from combining (5.15) and (5.16), with then (cid:101) C := C int (1) C and (cid:101) C := C int ( p ) C . 30 .4 Proof of Corollary 1.9 If u is the solution of the plane-wave scattering problem, then | u | H ( B R ) ≤ C osc k (cid:107) u (cid:107) H k ( B R ) (5.17)by [40, Theorem 9.1 and Remark 9.10], where C osc depends on A , c, d, and R , but is independentof k . The combination of (5.17) and (5.10) then imply thatmin v N ∈ V N (cid:107) u − v N (cid:107) H k ( B R ) ≤ C C osc hkp (cid:18) hkp (cid:19) (cid:107) u (cid:107) H k ( B R ) . (5.18)Combining (1.26), (5.18), and (1.25), we obtain the result (1.28) with C := C C osc . A Semiclassical pseudodifferential operators on the torus
Recall that for R (cid:93) > T dR (cid:93) := R d / (2 R (cid:93) Z ) d . The purpose of this Appendix is to review the material about semiclassical pseudodifferentialoperators on T dR (cid:93) used in § Semiclassical Sobolev spaces.
We consider functions or distributions on the torus as periodicfunctions or distributions on R d . To eliminate confusion between Fourier series and integrals, for f ∈ L ( T dR (cid:93) ) we define the Fourier coefficientsˆ f ( j ) := (cid:90) T dR(cid:93) f ( x ) e j ( x ) dx, where j ∈ Z d and the integral is over the cube of side 2 R (cid:93) , and where the Fourier basis given bythe L -normalized functions e j ( x ) = (2 R (cid:93) ) − n/ exp (cid:0) i πj · x/R (cid:93) (cid:1) . (A.1)The Fourier inversion formula is then f = (cid:88) j ∈ Z d ˆ f ( j ) e j . The action of the operator ( (cid:126) D ) α on the torus is therefore( (cid:126) D ) α f = (cid:88) j ∈ Z d ( (cid:126) j ) α ˆ f ( j ) e j . We work on the spaces defined by the boundedness of these operators: H m (cid:126) ( T dR (cid:93) ) := (cid:110) u ∈ L ( T dR (cid:93) ) , (cid:104) j (cid:105) m ˆ f ( j ) ∈ (cid:96) ( Z d ) (cid:111) . and use the norm (cid:107) u (cid:107) H m (cid:126) ( T dR(cid:93) ) := (cid:88) | ˆ f ( j ) | (cid:104) (cid:126) j (cid:105) m ; (A.2)see [74, § § E.1.8]. We often abbreviate H m (cid:126) ( T dR (cid:93) ) to H m (cid:126) and L ( T dR (cid:93) ) to L in thisAppendix.Since these spaces are defined for positive integer m by boundedness of ( hD ) α with | α | = m (and can be extended to m ∈ R by interpolation and duality), they agree with localized versions of31he corresponding spaces on R d defined by semiclassical Fourier transform: we let the semiclassicalFourier transform F (cid:126) (see [74, § (cid:126) > F (cid:126) φ ( ξ ) := (cid:90) T dR(cid:93) exp (cid:0) − i x · ξ/ (cid:126) (cid:1) φ ( x ) d x, and for a function on R d , we set (cid:107) u (cid:107) H m (cid:126) ( R d ) := (2 π (cid:126) ) − d (cid:90) R d (cid:104) ξ (cid:105) m |F (cid:126) u ( ξ ) | d ξ. Note for later use that the inverse semiclassical Fourier transform has a pre-factor of (2 π (cid:126) ) − n inthis normalization. Phase space.
The set of all possible positions x and momenta (i.e. Fourier variables) ξ is denotedby T ∗ T dR (cid:93) ; this is known informally as “phase space”. Strictly, T ∗ T dR (cid:93) := T dR (cid:93) × ( R d ) ∗ , but for ourpurposes, we can consider T ∗ T dR (cid:93) as { ( x, ξ ) : x ∈ T dR (cid:93) , ξ ∈ R d } . We also use the analogous notationfor T ∗ R d where appropriate.To deal uniformly near fiber-infinity with the behavior of functions on phase space, we alsoconsider the radial compactification in the fibers of this space, T ∗ T dR (cid:93) := R d × B d . Here B d denotes the closed unit ball, considered as the closure of the image of R d under the radialcompactification map RC : ξ (cid:55)→ ξ/ (1 + (cid:104) ξ (cid:105) )(cf. [21, § E.1.3]). Near the boundary of the ball | ξ | − ◦ RC − becomes a smooth function, vanishingto first order at the boundary, with ( | ξ | − ◦ RC − , ˆ ξ ◦ RC − ) thus furnishing local coordinates onthe ball near its boundary. The boundary of the ball should be considered as a sphere at infinityconsisting of all possible directions of the momentum variable. Where appropriate (e.g., in dealingwith finite values of ξ only), we abuse notation by dropping the composition with RC from ournotation and simply identifying R d with the interior of B d . Symbols, quantisation, and semiclassical pseudodifferential operators.
A symbol on R d is a function on T ∗ R d that is also allowed to depend on (cid:126) , and thus can be considered as an (cid:126) -dependent family of functions. Such a family a = ( a (cid:126) ) < (cid:126) ≤ (cid:126) , with a (cid:126) ∈ C ∞ ( R d ), is a symbol oforder m on the R d , written as a ∈ S m ( R d ), if for any multiindices α, β | ∂ αx ∂ βξ a ( x, ξ ) | ≤ C α,β (cid:104) ξ (cid:105) m −| β | for all ( x, ξ ) ∈ T ∗ R d and for all 0 < (cid:126) ≤ (cid:126) , (A.3)where C α,β does not depend on (cid:126) ; see [74, p. 207], [21, § E.1.2].For a ∈ S m ( R d ), we define the semiclassical quantisation of a on R d , denoted by Op (cid:126) ( a ) (cid:0) Op (cid:126) ( a ) v (cid:1) ( x ) := (2 π (cid:126) ) − n (cid:90) ξ ∈ R d (cid:90) y ∈ R d exp (cid:0) i( x − y ) · ξ/ (cid:126) (cid:1) a ( x, ξ ) v ( y ) d y d ξ ; (A.4)[74, § either as an oscillatory integral in the sense of [74, § § or as an iterated integral, with the y integration performed first; see [21, Page 543]. It can be shown that for any symbol a, Op (cid:126) ( a )preserves Schwartz functions, and extends by duality to act on tempered distributions [74, § a = a ( ξ ) depends only on ξ , thenOp (cid:126) ( a ) = F − (cid:126) M a F (cid:126) , where M a denotes multiplication by a ; i.e., in this case Op (cid:126) ( a ) is simply a Fourier multiplier on R d .
32e now return to considering the torus: if a ( x, ξ ) ∈ S m ( R d ) and is periodic, and if v is adistribution on the torus, we can view v as a periodic (hence, tempered) distribution on R d , anddefine (cid:0) Op T dR(cid:93) (cid:126) ( a ) v (cid:1) = (cid:0) Op (cid:126) ( a ) v (cid:1) , since the right side is again periodic [74, § T dR (cid:93) in this appendix.If A can be written in the form above, i. e. A = Op (cid:126) ( a ) with a ∈ S m , we say that A is a semiclassical pseudodifferential operator of order m on the torus and we write A ∈ Ψ m (cid:126) ( T dR (cid:93) );furthermore that we often abbreviate Ψ m (cid:126) ( T dR (cid:93) ) to Ψ m (cid:126) in this Appendix. We use the notation a ∈ (cid:126) l S m if (cid:126) − l a ∈ S m ; similarly A ∈ (cid:126) l Ψ m (cid:126) if (cid:126) − l A ∈ Ψ m (cid:126) . Theorem A.1 (Composition and mapping properties of semiclassical pseudodifferen-tial operators [74, Theorem 8.10], [21, Proposition E.17 and Proposition E.19]) If A ∈ Ψ m (cid:126) and B ∈ Ψ m (cid:126) , then(i) AB ∈ Ψ m + m (cid:126) ,(ii) [ A, B ] ∈ (cid:126) Ψ m + m − (cid:126) ,(iii) For any s ∈ R , A is bounded uniformly in (cid:126) as an operator from H s (cid:126) to H s − m (cid:126) . Residual class.
We say that A = O ( (cid:126) ∞ ) Ψ −∞ if, for any s > N ≥
1, there exists C s,N > (cid:107) A (cid:107) H − s (cid:126) → H s (cid:126) ≤ C N,s (cid:126) N ; (A.5)i.e. A ∈ Ψ −∞ (cid:126) and furthermore all of its operator norms are bounded by any algebraic power of (cid:126) . Principal symbol σ (cid:126) . Let the quotient space S m / (cid:126) S m − be defined by identifying elements of S m that differ only by an element of (cid:126) S m − . For any m , there is a linear, surjective map σ m (cid:126) : Ψ m (cid:126) → S m / (cid:126) S m − , called the principal symbol map , such that, for a ∈ S m , σ m (cid:126) (cid:0) Op (cid:126) ( a ) (cid:1) = a mod (cid:126) S m − ; (A.6)see [74, Page 213], [21, Proposition E.14] (observe that (A.6) implies that ker( σ m (cid:126) ) = (cid:126) Ψ m − (cid:126) ).When applying the map σ m (cid:126) to elements of Ψ m (cid:126) , we denote it by σ (cid:126) (i.e. we omit the m dependence) and we use σ (cid:126) ( A ) to denote one the representatives in S m (with the results weuse then independent of the choice of representative). Two key properties of the principal symbolthat we use in § σ (cid:126) ( AB ) = σ (cid:126) ( A ) σ (cid:126) ( B ) , (A.7) σ (cid:126) ( − (cid:126) ∆) = | ξ | . (A.8) Operator wavefront set WF (cid:126) . We say that ( x , ζ ) ∈ T ∗ T dR (cid:93) is not in the semiclassical operatorwavefront set of A = Op (cid:126) ( a ) ∈ Ψ m (cid:126) , denoted by WF (cid:126) A , if there exists a neighbourhood U of ( x , ζ )such that for all multiindices α, β and all N ≥ C α,β,U,N > (cid:126) ) sothat, for all 0 < (cid:126) ≤ (cid:126) , | ∂ αx ∂ βξ a ( x, ξ ) | ≤ C α,β,U,N (cid:126) N (cid:104) ξ (cid:105) − N for all ( x, RC ( ξ )) ∈ U. (A.9)For ζ = RC ( ξ ) in the interior of B d , the factor (cid:104) ξ (cid:105) − N is moot, and the definition merely saysthat outside its semiclassical operator wavefront set an operator is the quantization of a symbolthat vanishes faster than any algebraic power of (cid:126) ; see [74, Page 194], [21, Definition E.27]. For33 ∈ ∂B d = S d − , by contrast, the definition says that the symbol decays rapidly in a conicneighborhood of the direction ζ , in addition to decaying in (cid:126) . Three properties of the semiclassical operator wavefront set that we use in § (cid:126) A = ∅ if and only if A = O ( (cid:126) ∞ ) Ψ −∞ , (A.10)(see [21, E.2.2]), WF (cid:126) ( A + B ) ⊂ WF (cid:126) A ∪ WF (cid:126) A, (A.11)(see [21, E.2.4]), WF (cid:126) ( AB ) ⊂ WF (cid:126) A ∩ WF (cid:126) B, (A.12)(see [74, § (cid:126) ( A ) ∩ WF (cid:126) ( B ) = ∅ = ⇒ AB = O ( (cid:126) ∞ ) Ψ −∞ , (A.13)(by, e.g. (A.12) together with [21, E.2.3]), andWF (cid:126) (cid:0) Op (cid:126) ( a ) (cid:1) ⊂ supp a (A.14)(since (supp a ) c ⊂ (WF (cid:126) (Op (cid:126) ( a ))) c by (A.9)). Ellipticity.
We say that B ∈ Ψ m (cid:126) is elliptic at ( x , ζ ) ∈ T ∗ T dR (cid:93) if there exists a neighborhood U of ( x , ζ ) and c >
0, independent of (cid:126) , such that (cid:104) ξ (cid:105) − m (cid:12)(cid:12) σ (cid:126) ( B )( x, ξ ) (cid:12)(cid:12) ≥ c for all ( x, RC ( ξ )) ∈ U and for all 0 < (cid:126) ≤ (cid:126) . (A.15)A key feature of elliptic operators is that they are microlocally invertible; this is reflected inthe following result. Theorem A.2 (Elliptic parametrix [21, Proposition E.32]) Let A ∈ Ψ m (cid:126) ( T dR (cid:93) ) and B ∈ Ψ k (cid:126) ( T dR (cid:93) ) be such that B is elliptic on WF (cid:126) ( A ) . Then there exist S, S (cid:48) ∈ Ψ m − k (cid:126) ( T dR (cid:93) ) such that A = BS + O ( (cid:126) ∞ ) Ψ −∞ = S (cid:48) B + O ( (cid:126) ∞ ) Ψ −∞ , with WF (cid:126) S ⊂ WF (cid:126) A, WF (cid:126) S (cid:48) ⊂ WF (cid:126) A. Functional Calculus.
For f a Borel function, the operator f ( − (cid:126) ∆) is defined on smoothfunctions on the torus (and indeed on distributions if f has polynomial growth) by the functionalcalculus for the flat Laplacian, i.e., by the Fourier multiplier f ( − (cid:126) ∆) v = (cid:88) j ∈ Z d ˆ v ( j ) f ( (cid:126) | j | π /R (cid:93) ) e j . (A.16)It is reassuring to discover that indeed it is precisely the quantization of f ( | ξ | ) . Since our quanti-zation procedure was defined in terms of Fourier transform rather than Fourier series, this is notobvious a priori.
Lemma A.3
For f ∈ S m ( R ) , f ( − (cid:126) ∆) = Op (cid:126) f ( | ξ | ) . Proof.
First note that for v ∈ C ∞ ( T dR (cid:93) ) ,v = (cid:88) ˆ v ( j ) e j = (2 R (cid:93) ) − n/ (cid:90) R d (cid:88) j ∈ Z d ˆ v ( j ) δ ( ξ − (cid:126) πj/R (cid:93) ) exp(i ξx/ (cid:126) ) d ξ We highlight that working in a compact manifold allows us to dispense with the proper support assumptionappearing in [42, §
34 (2 π (cid:126) ) d (2 R (cid:93) ) − n/ F − (cid:126) (cid:88) j ∈ Z d ˆ v ( j ) δ ( ξ − (cid:126) πj/R (cid:93) ) . (A.17)Thus, if we take the semi-classical Fourier transform of v, regarded as a periodic function, F (cid:126) v ( ξ ) = (2 π (cid:126) ) d (2 R (cid:93) ) − n/ (cid:88) j ∈ Z d ˆ v ( j ) δ ( ξ − (cid:126) πj/R (cid:93) ) . Consequently, F (cid:126) (cid:2) f ( − (cid:126) ∆) v (cid:3) ( ξ ) = (2 π (cid:126) ) d (2 R (cid:93) ) − n/ (cid:88) j ∈ Z d f ( (cid:126) π | j | /R (cid:93) )ˆ v ( j ) δ ( ξ − (cid:126) πj/R (cid:93) )= (2 π (cid:126) ) d (2 R (cid:93) ) − n/ (cid:88) j ∈ Z d f ( | ξ | )ˆ v ( j ) δ ( ξ − (cid:126) πj/R (cid:93) )= f ( | ξ | ) F (cid:126) [ v ]( ξ ) , by (A.17), from which f ( − (cid:126) ∆) v = Op (cid:126) f ( | ξ | )( v ) . B Proof of (BB5) for the transmission problem
By the min-max principle for self-adjoint operators with compact resolvent (see, e.g., [61, Page 76,Theorem 13.1]) λ n = inf X ∈ Φ n ( D ) sup u ∈ X (cid:104) P u, u (cid:105) α,c (cid:107) u + (cid:107) L ( out ) + α − (cid:107) u − /c (cid:107) L ( O − ) , (B.1)where ( λ n ) n ≥ denotes the ordered eigenvalues of P , D is the domain of P defined by (2.6),Φ n ( D ) the set of all n -dimensional subspaces of D , and (cid:104)· , ·(cid:105) α,c is the scalar product definedimplicitly by the norm in the denominator (which is the norm in Lemma 2.3).By Green’s identity and the definition of D , (cid:104) P u, u (cid:105) α,c = (cid:126) (cid:104) A + ∇ u + , ∇ u + (cid:105) L ( out ) + α − (cid:126) (cid:104) A − ∇ u − , ∇ u − (cid:105) L ( O − ) . (B.2)Furthemore, (cid:104) A + ∇ u + , ∇ u + (cid:105) L ( out ) + α − (cid:104) A − ∇ u − , ∇ u − (cid:105) L ( O − ) (cid:107) u + (cid:107) L ( out ) + α − (cid:107) u − /c (cid:107) L ( O − ) ≥ min (cid:0) ( A + ) min , α − ( A − ) min (cid:1) max (cid:0) , α − ( c min ) − (cid:1) (cid:107)∇ u (cid:107) L (cid:107) u (cid:107) L (B.3)Observe that D ⊂ (cid:8) ( u , u ) ∈ H ( T d \O ) ⊕ H ( O ) s.t. u = u on ∂ O (cid:9) = H ( T d ) . (B.4)Using (B.2), (B.3), and (B.4) in (B.1), we have λ n ≥ min (cid:0) ( A + ) min , α − ( A − ) min (cid:1) max (cid:0) , α − ( c min ) − (cid:1) (cid:32) inf X ∈ Φ n ( H ( T d )) sup u ∈ X (cid:126) (cid:107)∇ u (cid:107) L (cid:107) u (cid:107) L (cid:33) . (B.5)The result then hold using the min-max principle for the eigenvalues of the Laplacian on the torus. C Real analyticity from derivative bounds of the form (1.19)
Lemma C.1 If u ∈ C ∞ ( D ) with (cid:107) ∂ α u (cid:107) L ( D ) ≤ C u ( C ) | α | | α | ! for all α, (C.1) then u is real analytic in D . roof. It is sufficient to prove that there exists n ∈ Z + such that (cid:107) ∂ α u (cid:107) L ∞ ( D ) ≤ (cid:101) C u ( (cid:101) C ) | α | ( | α | + n )! . (C.2)Indeed, the Lagrange form of the remainder in the Taylor-series up to n − c ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) | α | = n ( x − x (cid:48) ) α α ! (cid:0) ∂ α u (cid:0) x (cid:48) + c ( x − x (cid:48) ) (cid:1)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) | α | = n n ! α ! ( n + 1) . . . ( n + n ) (cid:101) C u ( (cid:101) C ) n | x − x (cid:48) | n , = (cid:101) C u ( n + 1) . . . ( n + n ) (cid:0) d (cid:101) C | x − x (cid:48) | (cid:1) n , which → n → ∞ if | x − x (cid:48) | < ( d (cid:101) C ) − .We therefore only need to prove (C.2). Let n := (cid:100) ( d + 1) / (cid:101) . Then, by the Sobolev embeddingtheorem (see, e.g., [46, Theorem 3.26]), (C.1), and the fact that | α + β | = | α | + | β | , (cid:107) ∂ α u (cid:107) L ∞ ( D ) ≤ C (cid:88) | β |≤ n (cid:13)(cid:13) ∂ α + β u (cid:13)(cid:13) L ( D ) ≤ CC u C | α | (cid:18) (cid:88) | β |≤ n C | β | ( | α | + | β | )! (cid:19) , ≤ CC u C | α | ( | α | + n )! (cid:18) (cid:88) | β |≤ n C | β | (cid:19) , so that (C.2) holds with (cid:101) C := C and (cid:101) C u := C C u (cid:0) (cid:80) | β |≤ n C | β | (cid:1) . Acknowledgements
The authors thank Luis Escauriaza (Universidad del Pa´ıs Vasco/Euskal Herriko Unibertsitatea)for useful discussions related to the paper [24].
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