Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in $L^2(\partial \Omega)$, of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper [Galkowski, M\"uller, Spence, arXiv 1608.01035].

where d is the spatial dimension. Let Ω be a bounded Lipschitz open set such that the open complement Ω + := R d \ Ω is connected (so that the scattering problem with obstacle Ω is welldefined). Recall that, for almost every x ∈ ∂Ω, there exists a unique outward-pointing unit normal vector, which we denote by n(x). For φ ∈ L 2 (∂Ω) and x ∈ ∂Ω, the single-and double-layer potential operators are defined by (recall that D ′ k is the adjoint of D k with respect to the real-valued L 2 (∂Ω) inner product; see, e.g., [7, Page 120]).
Before stating our main results, we need to make the following definitions.
where each Y ℓ is an open, relatively compact, smooth embedded manifold of dimension d − 2 in Γ, Γ lies locally on one side of Y ℓ , and Σ is closed set with d − 2 measure 0 and Σ ⊂ n l=1 Y l . We then refer to the manifold Γ as an extension of Γ.
For example, when d = 3, the interior of a 2-d polygon is a smooth hypersurface, with Y i the edges and Σ the set of corner points. Recall that the principal curvatures are the eigenvalues of the matrix of the second fundamental form in an orthonormal basis of the tangent space, and thus "curved" is equivalent to the principal curvatures being everywhere strictly positive (or everywhere strictly negative, depending on the choice of the normal).

Definition 1.3 (Piecewise smooth)
We say that a hypersurface Γ is piecewise smooth if Γ = ∪ N i=1 Γ i where Γ i are smooth hypersurfaces and Γ i ∩ Γ j = ∅.

Definition 1.4 (Piecewise curved)
We say that a piecewise smooth hypersurface Γ is piecewise curved if Γ is as in Definition 1.3 and each Γ j is curved.
The main results of this paper are contained in the following theorem. We use the notation that a b if there exists a C > 0, independent of k, such that a ≤ Cb. for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved (in the sense of Definition 1.4), then, given k 0 > 1, the following stronger estimate holds for all k ≥ k 0 S k L 2 (∂Ω)→H 1 (∂Ω) k 1/3 log k.
(c) If Ω is convex and ∂Ω is C ∞ and curved (in the sense of Definition 1.2) then, given k 0 > 0, Note that the requirement in Part (b) of Theorem 1.5 that ∂Ω is C 2,α arises since this is the regularity required of ∂Ω for D k and D ′ k to map L 2 (∂Ω) to H 1 (∂Ω); see [38,Theorem 4.2], [15,Theorem 3.6]. Remark 1.6 (Sharpness of the bounds in Theorem 1.5) In Section 3 we show that, modulo the factor log k, all of the bounds in Theorem 1.5 are sharp (i.e. the powers of k in the bounds are optimal). The sharpness (modulo the factor log k) of the L 2 (∂Ω) → L 2 (∂Ω) bounds in Theorem 2.10 was proved in [31, §A.2-A.3]. Earlier work in [6, §4] proved the sharpness of some of the L 2 (∂Ω) → L 2 (∂Ω) bounds in 2-d; we highlight that Section 3 and [31, §A.2-A.3] contain the appropriate generalisations to multidimensions of some of the arguments of [6, §4] (in particular [6,Theorems 4.2 and 4.4]).

Remark 1.7 (Comparison to previous results)
The only previously-existing bounds on the L 2 (∂Ω) → H 1 (∂Ω)-norms of S k , D k , and D ′ k are the following: when ∂Ω is Lipschitz [29, Theorem 1.6 (i)], and when ∂Ω is C 2,α [29, Theorem 1.6 (ii)]. We see that (1.7) is a factor of log k sharper than the bound (1.4) when d = 2, but otherwise all the bounds in Theorem 1.5 are sharper than (1.7) and (1.8).

Motivation for proving Theorem 1.5
Our motivation for proving Theorem 1.5 has four parts.
1. The integral operators S k , D k , and D ′ k appear in the standard second-kind BIE formulations of the exterior Dirichlet problem for the Helmholtz equation.
2. The standard analysis of the Galerkin method applied to these second-kind BIEs is based on the fact that, when ∂Ω is C 1 , the operators S k , D k , and D ′ k are all compact, and thus A ′ k,η and A k,η are compact perturbations of 1 2 I. 3. To perform a k-explicit analysis of the Galerkin method applied to A ′ k,η or A k,η via these compact-perturbation arguments, we need to have k-explicit information about the smoothing properties of S k , D k , and D ′ k . 4. When ∂Ω is C 2,α , the operators S k , D k , and D ′ k all map L 2 (∂Ω) to H 1 (∂Ω), and k-explicit bounds on these norms therefore give the required k-explicit smoothing information.
1.3 Discussion of the results of Theorem 1.5 in the context of using semiclassical analysis in the numerical analysis of the Helmholtz equation.
In the last 10 years, there has been growing interest in using results about the k-explicit analysis of the Helmholtz equation from semiclassical analysis to design and analyse numerical methods for the Helmholtz equation 1 . The activity has occurred in, broadly speaking, four different directions: 1. The use of the results of Melrose and Taylor [41] -on the rigorous k → ∞ asymptotics of the solution of the Helmholtz equation in the exterior of a smooth convex obstacle with strictly positive curvature -to design and analyse k-dependent approximation spaces for integral-equation formulations [17], [28], [2], [21], [20], [19].
2. The use of the results of Melrose and Taylor [41], along with the work of Ikawa [37] on scattering from several convex obstacles, to analyse algorithms for multiple scattering problems [22], [1].

Outline of the paper
In §2 we prove Theorem 1.5 (the L 2 (∂Ω) → H 1 (∂Ω) bounds) and Corollary 1.9, and in §3 we show that the bounds in Theorem 1.5 are sharp in their k-dependence.
2 Proof of Theorem 1.5 and Corollary 1.9 In this section we prove Theorem 1.5 and Corollary 1.9. The vast majority of the work will be in proving Parts (a) and (b) of Theorem 1.5, with Part (c) of Theorem 1.5 following from the results in [24,Chapter 4], and Corollary 1.9 following from the results of [29].
The outline of this section is as follows: In §2.1 we discuss some preliminaries from the theory of semiclassical pseudodifferential operators, with our default references the texts [57] and [18]. In §2.2 we recap facts about function spaces on piecewise smooth hypersurfaces. In §2.3 we recap restriction bounds on quasimodes -these results are central to our proof of Theorem 1.5. In §2.4 we prove of Parts (a) and (b) of Theorem 1.5, in §2.5 we prove Part (c) of Theorem 1.5 §2.5, and in §2.6 we prove Corollary 1.9. We drop the notation in this section and state every bound with a constant C (independent of k); we do this because later in the proof it will be useful to be able to indicate whether or not the constant in our estimates depends on the order s of the Sobolev space, or on a particular hypersurface Γ (we do this via the subscript s and Γ -see, e.g., (2.20) below).

Symbols and quantization
Following [57, §3.3], for k > 0 and u ∈ S(R d ), we define the semiclassical Fourier transform F k (u) by x j ξ j . We recall the inversion formula We use the standard notation that D := −i∂, so that F k (k −1 D j u)(ξ) = ξ j F k (u)(ξ). We let ξ := (1 + |ξ| 2 ) 1/2 and, following [18, §E.1.2], we say that a(x, ξ; From here on, we follow the usual convention of suppressing the dependence of a(x, ξ; k) on k, writing instead a(x, ξ) (see, e.g., [57, Remark on Page 72]), and also writing S m (R 2d ) instead of For an element a ∈ S m , we define its quantization to be the operator for u ∈ S(R d ). These operators can be defined by duality on u ∈ S ′ (R d ). We say that an operator if it is smoothing (i.e. its Schwartz kernel K is smooth) and each seminorm of . Note that, by introducing an operator R = O Ψ −∞ (k −∞ ) as an error, we can make the operator a(x, k −1 D) properly supported (i.e. so that for any K ⋐ R d , the kernel K of a(x, k −1 D) + R has the property that both π −1 R (K) ∩ supp K and π −1 L (K) ∩ supp K are compact where π R , π L : R d × R d → R d are projection onto the right and left factors respectively). Now, we say that A(k) is a pseudodifferential operator of order m and write A(k) ∈ Ψ m (R d ) if A(k) is properly supported and for some a ∈ S m (R 2d ),

Action on semiclassical Sobolev spaces
We define the Semiclassical Sobolev spaces H s Note that for s an integer, this norm is equivalent to The definition of the semiclassical Sobolev spaces on a smooth compact manifold of dimension [39,Page 98]). Because solutions of the Helmholtz equation (−k −2 ∆ − 1)u = 0 oscillate at frequency k, scaling derivatives by k −1 makes the k-dependence of these norms uniform in the number of derivatives.
With these definitions in hand, we have the following lemma on boundedness of pseudodifferential operators.

Ellipticity
For A ∈ Ψ m (R d ), we say that (x, ξ) ∈ R 2d is in the elliptic set of A, denoted ell(A), if there exists U a neighborhood of (x, ξ) such that for some δ > 0, We then have the following lemma then the same conclusions hold with R i ∈ Ψ m2−m1 (R d ).

Pseudodifferential operators on manifolds
Since we only use the notion of a pseudodifferential operator on a manifold in passing (in Lemma 2.15 and §2.5 below), we simply note that it is possible to define pseudodifferential operators on manifolds (see, e.g., [57,Chapter 14]). The analogues of Lemmas 2.1, 2.2, and 2.3 all hold in this setting. Moreover, the principal symbol map can still be defined although its definition is somewhat more involved.

Function spaces on piecewise smooth hypersurfaces
We now define the spaces H s (Γ) andḢ s (Γ) (with the notation for these spaces taken from [36, §B.2]).

Definition 2.4 (Extendable Sobolev space H s (Γ) on a smooth hypersurface)
Let Γ be a smooth hypersurface of R d (in the sense of Definition 1.1) and let Γ be an extension of Γ. Given We make two remarks: 1. The definition of the norm H s (Γ) depends on Γ, χ, and the choice of charts (U j , ψ j ) and partition of unity (χ j ). One can however prove that two different choices of charts (U j , ψ j ) and partition of unity (χ j ) lead to equivalent norms H s (Γ). In what follows, (U j , ψ j , χ j ) will be traces on Γ of charts and partition of unity on R d .
2. This definition is the same as, e.g., the definition of H s (Γ) for Γ ⊂ R d any non-empty open set in [39,Page 77]. However, we use the specific notation H s (Γ) for the following two reasons: (i) parallelism with the space H s (∂Ω) in Definition 2.6 below, and (ii) the fact that, without using the overline, H s (·) would be defined differently depending on whether the · is a smooth hypersurface or the boundary of a Lipschitz domain. Since Γ has C 0 boundary, one can show [10, Theorem 3.3, Lemma 3.15] that the dual of H s (Γ) is given byḢ −s (Γ) with the dual pairing inherited from that of H s comp ( Γ) and H −s comp ( Γ). For piecewise smooth ∂Ω, it is useful to consider the following "piecewise-H s " spaces.
We similarly define the norms H s Then, and the proof is complete.

7)
and where ∂ ν is a choice of normal derivative to Γ.   .7) is valid when Γ is only C 1,1 , and the k 1/6 -bound is valid when Γ is C 2,1 and curved. Therefore, with some extra work it should be possible to prove that the bounds on S k in Theorem 1.5 hold with the assumption "piecewise smooth" replaced by "piecewise C 1,1 " and "piecewise C 2,1 and curved" respectively. On the other hand, the bound (2.8) is not known in the literature for lower regularity Γ.

Proof of Parts (a) and (b) of Theorem 1.5
When proving these results, it is more convenient to work in semiclassical Sobolev spaces, i.e. to prove the bounds from L 2 (∂Ω) to H 1 k (∂Ω), where (following §2.1.2), where ∇ ∂Ω is the surface gradient on ∂Ω (defined by, e.g., [7,Equations (A.13) and (A.14)]). We therefore now restate Theorem 1.5 as Theorem 2.10 below, working in these spaces.
for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved (in the sense of Definition 1.4), then, given k 0 > 1, the following stronger estimate holds for all k ≥ k 0 Moreover, if ∂Ω is piecewise curved, then, given k 0 > 1, there exists C > 0 (independent of k) such that the following stronger estimates hold for all k ≥ k 0 (c) If Ω is convex and ∂Ω is C ∞ and curved (in the sense of Definition 1.2) then, given k 0 > 1, there exists C such that, for k ≥ k 0 , In §2.4.2 below, we give an outline of the proof of Parts (a) and (b). This outline, however, requires the definitions of S k , D k , and D ′ k in terms of the free resolvent (a.k.a. the Newtonian, or volume, potential), given in the next subsection.
2.4.1 S k , D k , and D ′ k written in terms of the free resolvent R 0 (k) We now recall the definitions of S k , D k , and D ′ k in terms of the free resolvent R 0 (k), these expressions are well-known in the theory of BIEs on Lipschitz domains [16], [39,Chapters 6 and 7]. We then specialise these to the case when ∂Ω is a piecewise smooth hypersurface (in the sense of Definition 1.3) Let R 0 (k) be the free (outgoing) resolvent at k; i.e. for ψ ∈ C ∞ comp (R d ) we have . With ∂ * n denoting the adjoint of the normal derivative trace (see, e.g., [39,Equation 6.14]), we have that the double-layer potential, D k , is defined by Recalling that the normal vector n points out of Ω and into Ω + , we have that the traces of D k from Ω ± to Γ are given by Similarly, results about the normal-derivative traces of the single-layer potential S k imply that . We now consider the case when ∂Ω is a piecewise smooth hypersurface (in the sense of Definition 1.3) and use the notation that Γ i are the compact embedded smooth manifolds of R d such that, for each i, Γ i is an open subset of Γ i . Let L i be a vector field whose restriction to Γ i is equal to ∂ νi , the unit normal to Γ i that is outward pointing with respect to ∂Ω. Let γ i : H s loc (R d ) → H s−1/2 (Γ i ) denote restriction to Γ i . We note that γ * i is the inclusion map f → f δ Γi where δ Γi is d − 1 dimensional Hausdorff measure on Γ. Finally, we let γ ± i denote restrictions from the interior and exterior respectively, where "interior" and "exterior" are defined via considering Γ i as a subset of ∂Ω. With these notations, we have that and the advantage of these last two expressions over (2.14) and (2.15) is that they involve γ i and L i instead of ∂ * n and ∂ ± n . In the rest of this section, we use the formulae (2.13), (2.16), and (2.17) as the definitions of S k , D k , and D ′ k . Note that we slightly abuse notation by omitting the sums in (2.16) and (2.17) and instead writing

Outline of the proof of Parts (a) and (b) of Theorem 2.10
The proof of Parts (a) and (b) of Theorem 2.10 will follow in two steps. In Lemma 2.11, we obtain estimates on frequencies ≤ M k and in Lemma 2.20 we complete the proof by estimating the high frequencies (≥ M k).
To estimate the low frequency components, we spectrally decompose the resolvent using the Fourier transform. We are then able to reduce the proof of the low-frequency estimates to the estimates on the restriction of eigenfunctions (or more precisely quasimodes) to ∂Ω that we recalled in §2.3. To understand this reduction, we proceed formally. From the description of S k in terms of the free resolvent, (2.13), the spectral decomposition of S k via the Fourier transform is formally where u(r) is a generalized eigenfunction of −∆ with eigenvalue r 2 , and k + i0 denotes the limit of k + iε as ε → 0 + . Observe that the integral in (2.19) is not well-defined (hence why this calculation is only formal), but (2.19) nevertheless indicates that estimating S k amounts to estimating the restriction of the generalized eigenfunction u(r) to ∂Ω. At very high frequency, we compare the operators S k , D k , and D ′ k with the corresponding operators when k = 1 (recall that the mapping properties of boundary integral operators with k = 1 have been extensively studied on rough domains; see, e.g. [42,Chapter 15], [39], [43]). By using a description of the resolvent at very high frequency as a pseudodifferential operator, we are able to see that these differences gain additional regularity and hence obtain estimates on them easily.
The new ingredients in our proof compared to [26] and [31] are that we have H s norms in Lemma 2.11 and Lemma 2.20 rather than the L 2 norms appearing in the previous work.

Proof of Parts (a) and (b) of Theorem 2.10
Low-frequency estimates. Following the outline in §2.4.2, our first task is to estimate frequencies ≤ kM . We start by proving a conditional result that assumes a certain estimate on restriction of the Fourier transform of surface measures to the sphere of radius r (Lemma 2.11). In Lemma 2.13 we then show that the hypotheses in Lemma 2.11 are a consequence of restriction estimates for quasimodes. In Lemma 2.17 we show how the low-frequency estimates on S k , D k , and D ′ k follow from Lemma 2.11.
In this section we denote the sphere of radius r by S d−1 r and we denote the surface measure on S d−1 r by dσ. We also use · to denote the non-semiclassical Fourier transform, i.e. u(ξ) is defined by the right-hand side of (2.1) with k = 1.

Remark 2.12
The estimate (2.29) is the only term where the log |k| appears, which leads to the log k factors in the bounds of Theorem 1.5 (without which these bounds would be sharp). To prove this lemma, we need to understand certain properties of the operator T r defined by with T r defined by (2.30). Before proving Lemma 2.13 we prove two lemmas (Lemma 2.14 and 2.15) collecting properties of T r . Lemma 2.14 Let T r be defined by (2.30) and χ ∈ C ∞ c (R d ). Then, Proof of Lemma 2.14. We estimate B := (χT r ) * χT r : ). This operator has kernel B(ξ, η) = Now, for η ∈ S d−1 r , and any N > 0, Thus, by Schur's inequality, B is bounded on L 2 (S d−1 r ) uniformly in r. Therefore, In the next lemma, we use r (the radius of S d−1 r ) as a semiclassical parameter, with the space H s r (Γ) defined in exactly the same way as H s k (Γ) is defined in §2.2.

Lemma 2.15
With T r be defined by (2.30), let Γ denote an extension of Γ, χ ∈ C ∞ c (R d ) and A ∈ Ψ 1 (R d ) with χ ≡ 1 in a neighborhood of Γ. Then for s ∈ R, Proof of Lemma 2.15. Since T r φ is supported on |ξ| ≤ 2r, χT r φ is compactly microlocalized in the sense that for (Note that ψ(r −1 |D|) can be defined using (2.2) since ψ(t) is constant near t = 0.) Let γ Γ denote restriction to Γ, and γ| Γ restriction to Γ. Let is a pseudodifferential operator on Γ with symbol ψ 1 (|ξ ′ | g ) and | · | g denotes the metric induced on T * Γ from R d (see Remark 2.16 below). Hence, for r > 1, Remark 2.16 (The definition of | · | g used in the proof of Lemma 2.15) We now briefly review the definition of | · | g from Riemannian geometry. Observe that the metric on R d is given by g e = d i=1 (dy i ) 2 where y i are standard coordinates on R d . To induce a metric on Γ, at a point By doing this at each point x 0 ∈ Γ, we obtain a metric on Γ, Next, choose coordinates x i on Γ and write the metric g as g( a i ∂ x i , b j ∂ x j ) = ij g ij (x)a i b j . Then, for the corresponding dual coordinates ξ on T * Γ , we have |ξ| g = ij g ij ξ i ξ j where g ij denotes the inverse matrix of g ij . Note that this definition is independent of all of the choices of coordinates.
We are now in a position to prove Lemma 2.13.
Proof of Lemma 2.13. The key observation for the proof of Lemma 2.13 is that for χ ∈ C ∞ c (R d ), with χ ≡ 1 in a neighborhood of Γ, χT r φ is a quasimode of the Laplacian with k = r in the sense of (2.6) in Theorem 2.8. To see this, observe first that −∆T r φ = r 2 T r φ by the definition (2.30). Therefore, Therefore, taking such aχ withχ ≡ 1 in a neighborhood, U of Γ shows that χT r φ is a quasimode.
To prove (2.21), we let A = I. Then, by the bounds (2.7) in Theorem 2.8 together with Lemmas 2.14 and 2.15, for s ≥ 0,

32)
and if Γ is curved then χT r φ H s (Γ) ≤ C r To prove (2.20), we take A = L. Observe that Hence, using the fact that L = ∂ ν on Γ together with the bound (2.8) in Theorem 2.8, we can estimate LT r φ.
High frequency estimates. Next, we obtain an estimate on the high frequency (≥ kM ) components of S k , D k , and D ′ k . We start by analyzing the high frequency components of the free resolvent, proving two lemmata on the structure of the free resolvent there.
Proof of Lemma 2.18.

Now, by Lemma 2.3 there exists
and A 0 has (Indeed, since we are working on R d , with no remainder.) Composing (2.44) on the right with A 0 , we have where E 1 ∈ Ψ −1 (R d ) and we have used that ϕ 1 ≡ 1 on supp ϕ 0 and hence Now, applying the same arguments, but with A n such that and assume that for some N . Then, and thus by induction, for all N ≥ 1, The proof of the statement for B 2 is identical.
Next, we prove an estimate on the difference between the resolvent at high energy and that at fixed energy.
With Lemma 2.18 and 2.19 in hand, we obtain the high-frequency estimates of the boundaryintegral operators by comparing them to those at fixed frequency. If, in addition, ∂Ω is C 2,α for some α > 0, then (2.52)

Remark 2.21
The factors of log k in the bounds of Lemma 2.20 are likely artifacts of our proof, but since they do not affect our final results, we do not attempt to remove them here. In fact, if ∂Ω is smooth (rather than piecewise smooth), then one can show that the logarithmic factors can be removed from the bounds in Lemma 2.20 using the analysis in [24,Section 4.4].

Proof of Part (c) of Theorem 2.10
Proof of Part (c) of Theorem 2.10.
First, recall that D ′ k and S k have wavefront set relation given by the billiard ball relation (see for example [24,Chapter 4]). Let B * ∂Ω and S * ∂Ω denote respectively the unit coball and cosphere bundles in ∂Ω. That is, Denote the relation by C β ⊂ B * ∂Ω × B * ∂Ω i.e.
3 Sharpness of the bounds in Theorem 1.5 We now prove that the powers of k in the S k L 2 (∂Ω)→H 1 (∂Ω) bounds in Theorem 2.10 are optimal. The analysis in [31, §A.3] proves that the powers of k in the D k L 2 (∂Ω)→L 2 (∂Ω) bounds are optimal, but can be adapted in a similar way to below to prove the sharpness of the D k L 2 (∂Ω)→H 1 (∂Ω) bounds.
In this section we write x ∈ R d as x = (x ′ , x d ) for x ′ ∈ R d−1 , and x ′ = (x 1 , x ′′ ) (in the case d = 2, the x ′′ variable is superfluous). for some δ > 0 and is C 2 in a neighborhood thereof (i.e. ∂Ω contains a line segment), then there exists k 0 > 0 and C > 0 (independent of k), such that, for all k ≥ k 0 , S k L 2 (∂Ω)→L 2 (∂Ω) ≥ Ck −1/2 and S k L 2 (∂Ω)→H 1 (∂Ω) ≥ Ck 1/2 . Figure 1: A recap of the billiard ball map. Let q = (x, ξ) ∈ B * ∂Ω (the unit ball in the cotangent bundle of ∂Ω). The solid black arrow on the left denotes the covector ξ ∈ B * x ∂Ω, with the dashed arrow denoting the unique inward-pointing unit vector whose tangential component is ξ. The dashed arrow on the right is the continuation of the dashed arrow on the left, and the solid black arrow on the right is ξ(β(q)) ∈ B * πx(β(q)) ∂Ω. The center of the left circle is x and that of the right is π x (β(q)). If this process is repeated, then the dashed arrow on the right is reflected in the tangent plane at π x (β(q)): the standard "angle of incidence equals angle of reflection" rule. Lemma 3.1 shows that the bound (1.4), when ∂Ω is piecewise smooth, is sharp up to a factor of log k. Lemma 3.2 (General lower bound on S k L 2 (∂Ω)→H 1 (∂Ω) ) If ∂Ω is C 2 in a neighborhood of a point then there exists k 0 > 0 and C > 0 (independent of k), such that, for all k ≥ k 0 , S k L 2 (∂Ω)→L 2 (∂Ω) ≥ Ck −2/3 and S k L 2 (∂Ω)→H 1 (∂Ω) ≥ Ck 1/3 . Lemma 3.2 shows that the bound (1.5), when ∂Ω is piecewise curved, is sharp up to a factor of log k and that the bound (1.6), when ∂Ω is smooth and curved, is sharp.
Similar to the proof of Lemma 3.1, it is sufficient to prove that there exists u k ∈ L 2 (∂Ω) with supp u k ⊂ Γ, k 0 > 0, and C > 0 (independent of k), such that S k u k L 2 (Γ) ≥ Ck −1/2 u k L 2 (Γ) and ∂ x1 S k u L 2 (Γ) ≥ Ck 1/2 u L 2 (Γ) (3.10) for all k ≥ k 0 . The idea in the curved case is the same as in the flat case: choose u concentrating as close as possible to a glancing point and measure near the point given by the billiard map. More practically, this amounts to ensuring that |x − y| looks like x 1 − y 1 modulo terms that are much smaller than k −1 . The fact that Γ may be curved will force us to choose u differently and cause our estimates to be worse than in the flat case (leading to the weaker -but still sharp -lower bound).