PARTIAL DATA INVERSE PROBLEM WITH L POTENTIALS

. We construct an explicit Green’s function for the conjugated Laplacian e − ω · x/h Δ e − ω · x/h , which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schr¨odinger equation with potential q ∈ L n/ 2 . Separately, we also use this Green’s function to derive L p Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329–347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate—rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applica- tions in obtaining direct numerical reconstruction algorithms for partial data Calder´on problems which is presently only accessible with full data [Inverse Problems 27 (2011)].


Introduction
In this article we give an explicit construction of a "Dirichlet Green's function" for the conjugated Laplacian e −x·ω/h h 2 Δe x·ω/h on a bounded smooth domain Ω ⊂ R n for n ≥ 3. We apply the Green's function to solve the longstanding partial data Calderón problem with unbounded Schrödinger potential in L n/2 (Ω) for n ≥ 3.
Let Ω ⊂ R n (n ≥ 3) be a smooth domain contained in R n with outward pointing normal ν along the boundary and let ω 0 ∈ R n be a unit vector. Define Γ 0 ± := {x ∈ ∂Ω | ±ν(x) · ω 0 ≥ 0} and let F ⊂ ∂Ω be an open neighbourhood containing Γ 0 + and B ⊂ ∂Ω be an open neighbourhood containing Γ 0 − . We make the additional assumption that in the coordinate system given by (x , x n ) ∈ ω ⊥ 0 ⊕ Rω 0 , the complements of B and F are disjoint unions of an open subset of ∂Ω so that the components Γ j of the disjoint union are compactly contained in the graph x n = f j (x ) for some smooth function f j .
If zero is not an eigenvalue of the operator −Δ + q, then q ∈ L n/2 (Ω) gives rise to a well-defined Dirichlet-to-Neumann map 98 FRANCIS J. CHUNG AND LEO TZOU Theorem 1.1. Let q 1 , q 2 ∈ L n/2 (Ω) be such that Λ q 1 To date this is the only partial data Calderón problem result for unbounded potentials. The integrability assumption that q j ∈ L n/2 is optimal in the context of well-posedness theory for the Dirichlet problem for L p potentials; L n/2 is also the optimal Lebesgue space for the strong unique continuation principle to hold (see [17] for more).
Note that there are conductivities in W 2,n/2 which are not contained in the cases considered by [23]. In fact, since W 1,n ⊂ BMO but not in L ∞ this result allows one to consider partial data problems for some conductivities which are not Lipschitz. So even in the special case of the conductivity equation this gives a new result.
Traditionally the study of partial data problems are limited to bounded potentials due to their reliance on L 2 Carleman estimates on bounded domains. We circumvent this difficulty by constructing instead an explicit (conjugated) Green's function which has good L p estimates in addition to desirable boundary conditions. Let ω ∈ R n be a unit vector and let Γ ⊂ ∂Ω be an open subset which is compactly contained in {x ∈ ∂Ω | ν(x) · ω > 0}. If p = 2n n+2 < 2 < p = 2n n−2 , we have the following theorem, proved by an explicit construction via heat flow.
Furthermore, for all f ∈ L p , G Γ f ∈ H 1 (Ω) and G Γ f | Γ = 0. This Green's function possesses several new features which makes it of potential use for studying a broad range of questions. First, note that in addition to desirable asymptotic L p and L 2 estimates, this Green's function also allows us to impose the Dirichlet boundary condition on Γ. Secondly, we will see that its construction is by explicit integral kernels in contrast to the functional analysis based approach of [3,21,29]. The combination of these two features can inspire future progress in numerical algorithms for partial data reconstruction which are currently only available in the full data case [1,11]. Furthermore, this Green's function gives new Carleman estimates which may be of interest on their own (see Theorem 1.4 and the ensuing discussions).
We will provide some brief historical context for Theorems 1.1 and 1.3. The construction of the Green's function for the conjugated Laplace operator was established by Sylvester-Uhlmann [33] using Fourier multipliers with characteristic sets. They proved an L 2 estimate for their Green's function and used it to solve the Calderón problem in dimensions n ≥ 3 for bounded potentials. Chanillo in [4] showed that the Sylvester-Uhlmann Green's function also satisfies an L p → L p estimate by applying using the result of Kenig-Ruiz-Sogge [20]. This allowed Chanillo to solve the inverse Schrödinger problem with full data for small potentials in the Fefferman-Phong class (which contains L n/2 ). Related full data results were also proved by Lavine-Nachman [24] and Dos Santos Ferreira-Kenig-Salo [13]. We will follow some of the techniques developed by these authors in Section 7.1.
The drawback to the Fourier multiplier construction of the Green's function is that boundary conditions cannot be imposed. Bukhgeim-Uhlmann [3] and Kenig-Sjöstrand-Uhlmann [21] found a way to use Carleman estimates to overcome this problem and prove results for the Calderón problem with partial boundary data. Due to its versatility and robustness, this technique has since become the standard tool for solving partial data elliptic inverse problems. The review article [19] contains an excellent overview of recent work in partial data Calderón-type problems; examples for other elliptic inverse problems can be found in [31], [32], [22], [9], and [8].
This standard technique turns out to be insufficient for our purpose. The Carleman estimates in these papers are typically proved via an integration-by-parts procedure so that boundary conditions can be kept in check. The limitation of this approach is that only L 2 -type estimates can be derived; none of the available techniques adapt well to L p setting for functions with boundary conditions. Thus for q / ∈ L ∞ , there are no partial data results for the Calderón problem for Schrödinger equations-although using a different method [23] obtained a partial data result for low regularity conductivity equations.
The [3,21] approach has the additional drawback that the Green's function one "constructs" is an abstract object arising from general statements in functional analysis, like the Hahn-Banach or Riesz representation theorems. This makes partial data reconstruction procedures like the ones in [29] much more difficult to implement in a concrete setting than equivalent ones like [28] for full data.
The Green's function we construct in Theorem 1.3 has the explicit representation of the Fourier multiplier Green's function of Sylvester-Uhlmann while at the same time allowing the boundary control of the existing methods. Due to its explicit representation as a parametrix, one can easily deduce L p -type estimates as well as L 2 -type estimates. In a forthcoming article the authors intend to apply the Green's function constructed here to the problem of reconstruction. One expects that in the context of computational algorithms this Green's function would open the door to direct inversion methods for partial data Calderón problems in n ≥ 3 which is parallel to the full data case examined in [1,[10][11][12]. Theorem 1.3 also directly implies the following boundary Carleman estimates for the conjugated Laplacian. Let H 1 (Ω) denote the semiclassical Sobolev space. Define H 1 Γ (Ω) ⊂ H 1 (Ω) to be the space of functions with vanishing trace along Γ and let H −1 Γ (Ω) be its dual. Theorem 1.4. Let u ∈ C 2 (Ω) be a function which vanishes along ∂Ω and ∂ ν u | Γ c = 0. One then has the Carleman estimates The L p inequality differs from other L p Carleman estimates like the ones in Kenig-Ruiz-Sogge [20] in that it allows for u with nontrivial boundary conditions. The solution to the inverse problem does not use Theorem 1.4. We only state the theorem here because it may be of interest to those studying unique continuations in the future. To see why traditional methods do not yield the type of L p Carleman estimates we obtain with boundary terms, the reader can compare our approach to [2,20,[25][26][27].
In the remainder of the introduction we give a brief exposition of our approach to the proof of Theorem 1.3. The key observation is that there is a global ΨDO factorization of the conjugated Laplacian h 2 Δ φ := e −ω·x/h h 2 Δe ω·x/h into an elliptic operator J resembling a heat flow and a first-order operator Q which has the same characteristic set as h 2 Δ φ . One can then construct an inverse for J (and thus h 2 Δ φ ) with Dirichlet boundary conditions by solving the heat flow with zero initial condition.
This way of factoring h 2 Δ φ is in the spirit of [5]. However, in our case the factorization is global and occurs on the level of symbols so there will be error terms and they pose a challenge in the construction of the parametrix. As such this necessitates a modified factorization which differs from that of [5] (see (4.7) and the discussions which follow) to obtain the suitable estimates for the remainders of the parametrix.
This article is organized in the following way. In Section 2 we develop a ΨDO calculus which is compatible with our symbol class-proofs are given in the appendix. In Section 3 we invert a heat flow in the context of this ΨDO calculus and solve the Dirichlet problem for this heat flow. In Section 4 we restate some facts about the Sylvester-Uhlmann Green's function in the semiclassical setting and derive a factorization for the operator h 2 Δ φ involving the heat operator described in the previous section. In Section 5 we use this factorization to construct a parametrix with Dirichlet boundary conditions, and in Section 6 we turn the parametrix into a Dirichlet Green's function G Γ and prove Theorem 1.4. Section 7 is devoted to proving Theorem 1.1 using complex geometric optics solutions constructed with the help of G Γ .

Elementary semiclassical ΨDO theory
We collect a set of facts about semiclassical pseudodifferential operators and also use this opportunity to establish some notation and conventions which we will use throughout. Proofs are contained in the appendix.

Mixed Sobolev spaces.
In this article we define the semiclassical Sobolev spaces with the norm For k ∈ N it turns out that this definition is equivalent to the one involving derivatives: (Hereafter we will drop the "scl" subscript: unless otherwise stated, all of our Sobolev spaces will be semiclassical.) Choose coordinates (x , x n ) on R n , with x ∈ R n−1 and x n ∈ R, and let (ξ , ξ n ) be the corresponding coordinates on the cotangent space. An immediate consequence of the norm equivalence stated above is that ξ is a multiplier from W 1,r (R n ) → L r (R n ). Indeed, and use these to define the mixed norm spaces W k, ,r (R n−1 , R n ). For convenience we will drop the R n−1 and R n in this notation and use the convention that the first superscript of W k, ,r denotes multiplication by hD k and the second denotes multiplication by hD .
With this definition we have that for k ≥ 0, and use the fact that hD k hD −k is a multiplier on L r by (2.2) and that hD −k hD u ∈ L r ⇐⇒ u ∈ W −k, ,r .

Tangential calculus.
We denote the Hörmander symbols by S 1 (R n ). We also consider symbols in the class S k 0 (R n ). We say that a belongs to S j (R n ) for j|β| for all multi-indices α and β. In this article we will work with product symbols of the form ba(x , ξ) then derivatives with respect to either x or ξ are a finite sum of symbols in S k 1 S j : We begin with the following Calderón-Vaillancourt-type estimate for (classical) ΨDO with symbols in S 0 1 (R n ) which can be obtained by following the argument of Theorem 9.7 in [34].
ii) Denote by k(n) to be the smallest integer for which (2.6) holds. Let a(x, ξ) be a symbol in S −k(n) 0 (R n ). Then for all 1 < r < ∞ Proof. The estimate (2.6) is Theorem 9.7 of [34]. For (2.7) we observe that if a ∈ S −k(n) 0 , then there exists a sequence {a j } ⊂ S −∞ such that p α,β (a − a j ) → 0 for |β| ≤ k(n) − 1. For u ∈ S we can conclude by using (2.6) that {a j (x, D)u} is a Cauchy sequence in L r and therefore a j (x, D)u → v in L r . On the other hand, by standard L 2 estimates a j (x, D)u → a(x, D)u in L 2 . Therefore v = a(x, D)u. Applying estimate (2.6) to a j (x, D)u and taking the limit we have (2.7).
Note that in R n there is a relation between classical and semiclassical quantization of a symbol a ∈ S ∞ given by . This identity combined with estimate (2.6) and (2.7) gives us a semiclassical version of Calderón-Vaillancourt: for all 1 < r < ∞, h > 0 sufficiently small, and a ∈ S 0 , we have the following mapping properties. In addition, we have the following compositional calculus result.
For proofs of Propositions 2.2 and 2.3, see the appendix.
Remark 2.4. We have omitted stating the mapping properties on H k δ spaces since S k 0 S 0 ⊂ S k+ 0 (R n ) and the calculus for these symbols on weighted L 2 Sobolev spaces are well documented. See for example [30,Prop 2.2] for these results and for definition of weighted semiclassical Sobolev spaces.

Heat flow
Define coordinates on R n and let R n + denote the upper half space {x n > 0}. Let F (x , ξ ) ∈ S 1 1 (R n−1 ), and define the semiclassical pseudodifferential operator on R n . It follows by considering the ξ and ξ n direction separately and applying the semiclassical Calderón-Vaillancourt theorem that j(x , hD) is a bounded operator j(x , hD) : W 1,r (R n ) → L r (R n ) for 1 < r < ∞. As we will see in the following section, one of the factors of the conjugated Laplacian has this form. In this section we will prove some basic facts about the existence and L p mapping properties of the inverse of such an operator. This extends the L 2 theory explained in [6].
To obtain an inverse, we will assume that F obeys the ellipticity condition uniformly in x for some constants c, C > 0. This ensures that the principal symbol is uniformly elliptic. We will also assume a finiteness condition on F : that there exists X > 0 such that for |x | > X , We need an extra condition to ensure that the symbol j −1 is in the suitable calculus. We assume that there exists a first-order symbol iξ n + F − (x , ξ ) with compact characteristic set, such that D x F − (x , ξ ) is supported in |x | < X , and where p(x , ξ) is a second-order polynomial in ξ with compact characteristic set and a 0 ∈ S −∞ (R n−1 ). The reason why we need this extra assumption is that (iξ n + F ) −1 is not in the class S −1 1 (R n ) (for example if F = ξ , then differentiating multiple times in ξ does not yield additonal decay in the ξ n direction). However, if χ ∈ C ∞ 0 (R n ) is identically 1 on a neighbourhood containing the characteristic sets of iξ n + F − and p, then we can derive the following expansion: .
Since χ is identically one on the characteristic set of p, it follows (1 − χ(ξ))/p(x , ξ) is a symbol in S −2 1 (R n ), and so Using the fact that p is elliptic on support of 1 − χ we can expand for all N > 0 .
where we are using S k j to represent a symbol from the class S k j (R n ). Now and the same holds for ). Finally, , so we can use (3.5) in conjunction with Proposition 2.2 to get that The operator j −1 (x , hD) also turns out to have desirable support properties. Proof. For u ∈ C ∞ c (R n ), we can write where F h is the semiclassical Fourier transform. We can write out the Fourier transform in the x n variable to get Now we can use the residue theorem to evaluate the dξ n integral explicitly. For t ≥ x n we need to take a semicircular contour in the lower half plane. Since by assumption (3.2) ReF is positive this contour yields no residue. For t ≤ x n we take a semicircular contour in the upper half plane. In this case the contour contains a pole at ξ n = iF (x , ξ ). Therefore we get For u ∈ C ∞ c (R n ), the lemma follows immediately from this representation. The lemma follows for general u ∈ L r (R n ) by (3.6) and density.
Henceforth we will refer to the support property given in Lemma 3.1 as "preserving support in the x n direction".
We can turn j −1 (x , hD) into a proper inverse. We first prove a composition-type lemma for the operator j −1 (x , hD).
Proof. The expansion (3.5) allows us to write j −1 (x , ξ) as the span of elements in We can therefore apply Proposition 2.3 to each term to obtain Using expansion (3.5) again we see that m 1 (x , ξ) is a symbol in the span of To obtain the commutator statement, we can repeat the argument for the composition j −1 (x , hD)a(x , hD ). Now we can use j −1 to build a proper inverse which preserves support in the x n direction. More generally the inversion can still be carried out even if j is perturbed by a small tangential operator hF 0 .
obeys the same finiteness condition (3.3) as F , and consider the operator For h > 0 sufficiently small there exists an inverse J −1 : L r → W 1,r of the form Furthermore, J −1 preserves support in the x n direction. The same holds for J −1 acting on H k δ spaces. Proof. We write We can apply Proposition 2.3 to the first term, using the expansion (3.5) for j −1 , and Lemma 3.2 to the second and third terms to obtain and m 2 (x , hD) : L r → L r . Using expansion (3.5) again we see that m 1 (x , ξ) is a symbol in the span of Observe that in equation (3.7) since J is a differential operator in the x n direction, it preserves support in the x n direction when acting on W 1,r . The operator j −1 (x , hD) : L r → W 1,r preserves support in x n by Lemma 3.1 and thus the left side preserves support in the x n direction. We may conclude from this that the right side preserves x n support as well and in particular hm 1 (x , hD)+h 2 m 2 (x , hD) preserves x n support. This means that inverting the right side by Neumann series preserves support in the x n direction.
One final consequence of the structure of J −1 we obtained in Proposition 3.3 is the following disjoint support property.

The analogous estimate holds as a map from weighted
. Therefore it suffices to show that where (1 + hm 1 (x , hD) + h 2 m 2 (x , hD)) −1 is given by the Neumann series Therefore, by (3.6) we can write where M : L r → W 1,r is bounded uniformly in h. Using this expression for J −1 it suffices to show that with norm bounded by O(h 2 ). We will only show (3.9) for the principal part ζ j −1 (x , hD)1 R n − and leave the lower-order term, which can be written out explicitly using (3.8), to the reader. By using (3.5) we see that the symbol j −1 belongs to We will only show (3.9) for ζ Op h (S 1 1 S −2 1 )1 R n − and the others are treated in the same way. Suppose b ∈ S 1 1 (R n−1 ) and a ∈ S −2 1 (R n ), by Proposition 2.3 we see that Since ζ is a function of x n only, it commutes with operators from S k 1 (R n−1 ), and thus estimating ζ ba( Standard disjoint support properties of ΨDO then give the desired estimates.

Green's functions on R n
The purpose of this discussion is to find a way to invert with a suitable boundary condition and good L p → L p estimates. We begin with the operator on R n given by the Fourier multiplier 1 |ξ| 2 +2iξ n −1 . We give a semiclassical formulation of an estimate established in Sylvester-Uhlmann [33].

Lemma 4.2. The Fourier multiplier satisfies the estimate
In order to deal with domains with nonflat boundaries we will "flatten" boundary pieces by a coordinate change of the type γ : (y n , y ) → (x n , x ) = (y n − f (y ), y ), (4.1) where f : R n−1 → R is a smooth function which is constant outside of a compact set. Under this change of variables, the differential operator defined bỹ can be written explicitly as where K(x ) := ∇f (x ) and for convenience we will later denote 1 + hΔ x f 2 by 1 h as it is 1 in the semiclassical limit. The next proposition concerns the Green's function which is equivalent to conjugating by the operator given by pulling back by γ.
. Once this is done the corresponding statements forG φ follow via conjugation by the diffeomorphism γ whose Jacobian is identity outside of a compact set.
By Lemmas 4.2 and 4.1 the operator G φ satisfies G φ : , with χ 0 identically 1 in the ball of radius 2 and χ 1 identically 1 on the support of χ 0 . Everything commutes in the above identity since they are all constant coefficient Fourier multipliers.
Since the characteristic set of G φ is disjoint from the support of 1 − χ 0 , the The mapping properties of G c φ come from the mapping properties of G φ and the fact that χ 1 (hD) has a compactly supported symbol.
The explicit representation ofΔ φ in (4.2) shows that its characteristic set lies in the sphere |ξ | = 1, and so in particular if G c φ is multiplied by a Fourier side cutoff function supported away from that sphere, the resulting operator is well behaved. The following lemma makes this somewhat more precise.
Proof. By the construction in Proposition 4.3, vanishes in an open neighbourhood of |ξ | = 1, the symbol

Modified factorization.
To add boundary determination to the Green's function, we want to take advantage of the fact that h 2Δ φ factors into two parts, one of which is elliptic and resembles the operator described in Section 3.
Indeed, the symbol of 1 1+K 2 h 2Δ φ which appears in (4.2) factors formally as Note that the second factor here is elliptic. The problem is that the square root is not smooth at its branch cut, so this does not give a proper factorization at the operator level. The obvious thing to do is to take a smooth approximation to the square root, but for our purposes we will require something more subtle.
We take the branch of the square root that has nonnegative real part, and seek to avoid the branch cut, which happens when the argument of the square root lies on the negative real axis. From examination of the square root, we see that this occurs when K · ξ = 0 and |ξ | 2 + h Δ x f 2 ≤ |K| 2 (1 + |K| 2 ) −1 . By ensuring that ξ avoids this set, we can guarantee that the argument of the square root stays away from the branch cut.
Thus let 0 < c < c < 1 be a constant such that |K| 2 1+|K| 2 < c for all x and let ρ 0 (ξ ) be a smooth function in ξ such thatρ 0 = 1 for |ξ | 2 ≤ c and supp(ρ 0 ) B c . Introduce a second cutoffρ such that it is identically 1 on |ξ | 2 ≤ c but supp(ρ) B √ c . Observe that for h > 0 sufficiently small Since the branch cut of the square root occurs when it follows that for ξ in the support of 1 −ρ 0 and h > 0 sufficiently small, the function stays uniformly away from the branch cut of the square root. As such we may define and factor Here theã ± andã 0 are defined bỹ Observe that the support ofã 0 is compactly contained in the interior of the set whereρ = 1. We now quantize (4.5) to see that , and Q and J are the operators with symbols ξ n −ã − + hm 0 and ξ n −ã + − hm 0 , respectively. Observe that the O(h) term in the composition formula for QJ is killed by one of the O(h) terms in (4.5).
Although this decomposition still gives us an O(h) error, the symbolẽ 1 vanishes when |ξ | = 1. In particular it vanishes on the characteristic set of h 2Δ φ which is {ξ n = 0, |ξ | = 1} by (4.2). We use this observation to show that hẽ 1 (x , hD )G φ behaves one order of h better than expected.  1 (x , hD ). The operatorẼ 1Gφ is of the form Here the notation T : X → h m Y indicates that the norm of the operator T from X to Y is bounded by O(h m ).
Proof. We use the fact thatẽ 1 takes value zero on the characteristic set ofG φ . First writẽ for some compactly supported smooth function χ(ξ) which is identically 1 on the ball of radius 2. This means that can be written as the sum of a ΨDO with symbol in S −∞ (R n ) and a part containing the characteristic set (4.9) Inserting this into (4.8) would give us the lemma.
Since the characteristic set of the Fourier multiplier 1 ξ 2 n +i2ξ n +(1−|ξ | 2 ) is compactly contained in this set, let χ 2 (ξ) be a cutoff which is supported in this set and let 1 be in a neighbourhood of the characteristic set and define We now write (Op h (a − a + )(χ(hD)G φ )) as a sum of two operators The second expression is ΨDO of order −∞ since (χ(hD)(1 − χ 2 (hD))G φ ) vanishes identically near the characteristic set of G φ and is therefore a compactly supported smooth multiplier.
It remains to establish (4.9) for the part containing the characteristic set given by Since ρ 0 vanishes identically on the support of χ 2 and χ 2 is a constant coefficient Fourier multiplier, it follows from (4.10) that Note since Op h ( (1−|ξ +ξ n K| 2 ) (1+|K| 2 ) 2 ) is a differential operator, proving (4.9) amounts to proving estimates for the operators ).
Crucially, these are both bounded Fourier multipliers with compact support and therefore map L 2 → H k for all k ∈ N with norm O(1). Therefore Moving on to the L p → H k estimate we write χ(hD)G φ = χ(hD)G φ χ 100 (hD) where χ 100 (ξ) is identically 1 on the support of χ. The estimate is then a result of the L 2 estimate and the fact that χ 100 (hD) :

Parametrices on the half space
In this section we construct parametrices for h 2Δ φ on the upper half space which give a vanishing trace on the boundary. By a change of variables, we will later use these to build the Green's function of Theorem 1.3. Because the factoring in (4.7) contains a large error term A 0 at small frequencies, we will perform two separate constructions-one for the large frequency case (on supp(1 −ρ)) and one for the small frequency case (on supp(ρ)). We split the two frequency cases by using the cutoff functionρ : R n−1 → R defined above equation (4.3).

Parametrix for h 2Δ
φ at large frequency. LetG φ be the Green's function from Proposition 4.3, and J + := J −1 1 R n + where J −1 is defined as in Proposition 3.3. LetΩ ⊂ R n + be a smooth bounded open subset of the upper half space (with possibly a portion of the boundary intersecting x n = 0). We show that the operator is a suitable parametrix for the operator h 2Δ φ inΩ at large frequencies. We begin by showing that P l has mapping properties like those ofG φ .
Proposition 5.1. The map P l satisfies, for δ > 0, Proof. The weighted L 2 Sobolev norms come as a direct consequence of the mapping properties ofG φ and the fact that J, J −1 has symbols in S k 0 (R n ). For the mapping property from L p (R n ) → L p (R n ), we splitG φ =G c φ +(G φ −G c φ ) following Propostion 4.3 and observe The above diagrams also show that P l v ∈ H 1 loc for all v ∈ L p by omitting the last Sobolev embedding. The trace property then comes from the definition of P l and Proposition 3.3.
In the following statement we denote 1Ω to be the indicator function ofΩ. If v ∈ L r (Ω) we use the notation 1Ωv to denote its trivial extension to a function in L r (R n ).

Proposition 5.2. LetΩ ⊂ R n + be a bounded domain with ∂Ω ∩ {x n = 0} = ∅. Denote by 1Ω the indicator function ofΩ. Then P l is a parametrix at large frequencies with vanishing trace on the boundary of the upper half space, in the sense that for all
Proof. We compute in the sense of distributions on R n + acting on C ∞ 0 (R n + ). Using (4.7) we have The first term requires some care. Applying this operator to functions v ∈ C ∞ 0 (R n ) and testing it against u ∈ C ∞ 0 (R n + ) yields . The operator Q * is a ΨDO in the ξ direction but it is only a differential operator in the ξ n direction. Therefore the support does not spread in the x n direction. The operatorρ(hD ) is an operator only in the ξ direction and therefore does not spread support in the x n direction. As such Q * (1 + K 2 )(1 −ρ) * u vanishes in an open neighbourhood containing the closure of the lower half space and therefore for all u ∈ C ∞ 0 (R n + ) and v ∈ C ∞ 0 (R n ),

FRANCIS J. CHUNG AND LEO TZOU
Therefore we may continue our computation: At this juncture we invoke the factorization (4.7) again and plug the relation as a distribution on R n + (i.e., integrating against functions in C ∞ 0 (R n + )) where

Lemma 5.3. The last term of (5.2) can be estimated by
We leave the proofs of these lemmas until the end of the section. The remainder terms in (5.2) can now be estimated using Lemmas 5.3, 5.4, and 5.5 to show that, when tested against u ∈ C ∞ 0 (Ω), 1Ωh 2Δ φ P l 1Ωv = (1 −ρ(hD ) + R l + hR l )v, where R l = 1ΩR l 1Ω and R l = 1ΩR l 1Ω have the estimates The trace property of the operator P l 1Ω on ∂Ω ∩ {x n = 0} is a result of Proposition 5.1. Note that the L 2 bounds in Proposition 5.2 are unweighted because of the conjugation with indicator functions ofΩ.

Proof of Lemma 5.3. We have
ρ]hD n J + JG φ . Some care will be needed in treating the term involving h 2 D 2 n hitting J + = J −1 1 R n + . We are only considering the expressions as maps to distributions on R n + , so for all u ∈ C ∞ 0 (R n + ) and v ∈ C ∞ 0 (R n ), hD n u, hD n J −1 1 R n Here we used the fact that J = hD n + F (x , hD ) for some F (x , ξ ) ∈ S 1 1 (R n−1 ) and the tangential operator F (x , hD ) commutes with the indicator function of the upper half space.
Combining the two expressions we obtain We decomposeG φ in (5.5) into its ΨDO part and its characteristic part as stated in Proposition 4.3. The ΨDO part of (5.5) is a bounded map from L r → L r with a gain in h obtained from the commutator. Therefore, the part containing the ΨDO belongs to the hR 0 bin.
For the part containing the characteristic set, we expand [h 2Δ whereρ 1 (ξ ) is chosen to be identically 1 in a neighbourhood compactly containing the support ofρ but supp( The only thing remaining is to treat the terms on the support ofρ 1 . We will treat the first term and the second term is dealt with in the same manner. We claim that modulo errors in the bin h 2 R 0 we can commuteρ 1 (hD ) so that it appears next toG c φ : Sinceρ 1 (ξ ) vanishes identically near |ξ | = 1, Lemma 4.4 asserts that, This proves the lemma up to verifying (5.6). It only remains to verify (5.6) by checking that all the commutator terms withρ 1 can be sorted into the h 2 R 0 bin by using Proposition 3.3, Lemma 3.2, and Proposition 2.3 in conjunction with the mapping properties ofG c φ given by Proposition 4.3. We only write out explicitly the argument for commuting with J + as it is slightly more challenging than the others. First, observe that by Proposition 3.3 where m 1 (x , hD) and m 2 (x , hD) take L r → L r and H k δ → H k δ with the inverse given by Neumann series. Therefore Standard calculus for commutators then allows us to commuteρ 1 with F 2 and j −1 (x , hD) to obtain We can commuteρ 1 with 1 R n + with no commutator sinceρ 1 is an operator in the x direction only. Commuting with J using the standard commutator calculus then gives us (5.6).
Proof of Lemma 5.4. We begin with the hẼ 1Gφ term in (5.3). By Lemma 4.5, By Proposition 4.3, the third term of (5.7) can be written as We see then that the first and second terms belong to the R 1 bin while the third and fourth terms belong to the R 1 bin.
We proceed next with the hẼ 1 J + JG φ term of (5.3): In the above calculation we commutedẼ 1 and 1 R n + sinceẼ 1 only acts in the x direction.
The first term above can be handled using (5.8)-note that there is enough regularity so that applying 1 R n + J presents no difficulty. For the first commutator term of (5.9), Lemma 3.
and therefore hJ + [J,Ẽ 1 ]G c belongs to R 1 bin.
Proof of Lemma 5.5. The terms involving h 2Ẽ 0 can be estimated directly using the estimates forG φ and P l in Propositions 4.3 and 5.1. The terms involvingÃ 0 can be estimated by observing that sinceρ(ξ ) is chosen to be identically 1 in a neighbourhood of the support ofã 0 (x , ξ ), the operator

Parametrix for h 2Δ
φ at small frequency. Here we want to look for a parametrix for h 2Δ ϕ at low frequencies. We begin by defining p(x , ξ) to be the symbol of h 2Δ ϕ : . Thanks to the fact thatρ is chosen to be disjoint from the characteristic set of p(x , ξ) we may define The following proposition says that P s inverts h 2Δ φ at small frequencies, up to an O(h) error. Proposition 5.6. P s is a bounded operator P s : L r → W 2,r for all r ∈ (1, ∞). Moreover, for all r ∈ (1, ∞). h 2Δ φ P s =ρ + hR s for some R s : L r → L r bounded uniformly in h.
Proof. We want to use the symbol calculus developed in Section 2. However, we have the complication that 1/p(x , ξ) is not a proper symbol, because of the zeros of p(x , ξ). Therefore it is not immediately evident thatρ/p(x , ξ) lies in the symbol class S −∞ S −2 1 , as we would want. We can remedy this by writing where χ 100 (ξ) ∈ S −∞ (R n ) is a smooth cutoff function supported only for |ξ| < 100, and identically one in the ball |ξ| ≤ 50. Now note that by (4.3), p(x , ξ) is properly elliptic on the support ofρ(ξ ), and therefore χ 100 (ξ)ρ(ξ )/p(x , ξ) ∈ S −∞ (R n ). Moreover, since the characteristic set of p(x , ξ) lies well inside the set where χ 100 ≡ 1, we have that (1 − χ 100 (ξ))/p(x , ξ) ∈ S −2 1 (R n ). Therefore P s can be understood as the sum of two operators, one of which is in the symbol class S −∞ (R n ) and the other of which is in the symbol class S −∞ S −2 1 . Then Proposition 2.2 asserts that P s : L r → W 2,r is a bounded operator and Proposition 2.3 asserts that It turns out that our small frequency parametrix preserves support in the x n direction.

Proposition 5.7.
Suppose v ∈ L r (R n ), with 1 < r < ∞, and supp(v) is contained in the closure of R n + . Then both supp(P s v) and supp(R s v) are contained inR n + , where R s is the operator from Proposition 5.6. In particular, We split the integral on the right into x and x n variables and get

Consider the inner integral
For fixed ξ and x , we can write the Fourier transform of v in the ξ n variable explicitly to get Suppose now that supp(v) is contained in R n + so that the integral over s in (5.11) is only taken over s ≥ δ > 0. We want to show that (5.11) vanishes when x n ≤ 0. This is done by showing that the inner dξ n integral of (5.11) vanishes if x n < 0 and s > 0. We do this by using residue calculus.
To evaluate the dξ n integral of (5.11) when x n < 0 and s > 0 we should take a contour on the lower half plane. The integral vanishes if we can verify that the zeros of p(x , ξ) as a polynomial in ξ n for values of ξ on supp(ρ(ξ )) {|ξ | < 1} all belong to the upper half plane.
Factoring p(x , ξ) as a quadratic function in ξ n , we have and the square root is defined by choosing angles between (−π, π]. With this choice we see that a + has a positive imaginary part when h > 0 is sufficiently small. We will now argue that the same holds for a − (x , ξ ) for ξ on the support ofρ. Note that K(x ) is compactly supported so this clearly holds for h > 0 small and x outside the support of K. DefineD : supp(K(x )) × supp(ρ(ξ )) → C by and one easily sees that D(x , ξ ) ∈ R − if and only if LetN to be a small neighbourhood containing N . On the connected set supp(ρ)\N , D(x , ξ ) is a continuous function if h > 0 is small enough. If the imaginary part of a − vanishes on supp(ρ) \N , then by an appropriate choice of ξ n ∈ R the factor (ξ n − a − ) in (5.12) can be made to vanish. But p(x , ξ) is elliptic on support of ρ(ξ ) by (4.3) so the imaginary part of a − cannot vanish on the support ofρ(ξ ). On the other hand, by choosingN small enough we will have that the imaginary part of a − takes on positive value somewhere on supp(ρ) \N . By connectedness the imaginary part of a − must be positive everywhere on supp(ρ) \N .
Meanwhile onN the function D(x , ξ ) takes on value sufficiently close to R − . Therefore by our chosen branch of the square root, D(x , ξ has small real part. So a − (x , ξ ) has positive imaginary part on here as well.
We are now able to conclude, at least in the case Now from Proposition 5.6 we have and it follows from the trace theorem that for any fixed x n ≤ 0, If v ∈ L r (R n ) is supported in the closure of R n + , we can approximate it with C ∞ 0 (R n ) functions supported in R n + . The trace property (5.14) then allows us to conclude Op h ρ p v(x , x n ) = 0 for x n ≤ 0. This shows that P s has the desired support property. The support property for R s then follows from writing h 2Δ φ P s −ρ(hD ) = hR s and noting that every operator on the left hand side of this equation has the desired support property. 6. Dirichlet Green's function and Carleman estimates 6.1. Green's function for single graph domains. By combining Sections 5.1 and 5.2 we see that 1Ω(P s + P l )1Ω is a parametrix for the operator h 2Δ φ in the domainΩ. As one expects, this parametrix can be modified into a Green's function.
In this section we consider domains with a component of the boundary which coincides with the graph of a function. In particular, let Ω be a bounded domain in R n , and suppose f ∈ C ∞ 0 (R n−1 ) such that Ω lies in the set {x n > f(x )} with a portion of the boundary Γ ⊂ ∂Ω lying on the graph {x n = f (x )}. Denote by γ the change of variable (x , x n ) → (x , x n − f (x )). SetΩ andΓ to be the image of Ω and Γ under this change of variables.

Proposition 6.1. There exists a Green's function G Γ which satisfies the relation
with R obeying the estimates The Green's function G Γ satisfies the estimates Proof. Change coordinates (x , x n ) → (x , x n − f (x )) so thatΓ ⊂ {x n = 0} and letΔ φ be the pulled-back conjugated Laplacian described in (4.2). All equalities below are in the sense of distributions inΩ. By Propositions 5.2 and 5.6, for any v ∈ L p (Ω), Let S : L r → L r denote the inverse of (1 + hR l + hR s ) by Neumann series. Then inΩ we have h 2Δ φ 1Ω(P s + P l )1ΩS = I + R l S with R l S : L 2 → h L 2 while R l S : L p → h 0 L 2 . Therefore, for all v ∈ L p (Ω) the Neumann series is well-defined and the series converge in L 2 (Ω). Then we have the operator Direct computation verifies that this is a Green's function in the original coordinates.
For verifying the estimates of G Γ v and its trace along Γ it is more convenient to work with the operator 1Ω(P s + P l )1ΩS(1 + R l S) −1 and deduce the analogous properties for G Γ . We first check that 1Ω(P s + P l )1ΩS(1 + R l S) −1 v ∈ H 1 (Ω) for all v ∈ L p and that the trace vanishes onΓ ⊂ {x n = 0}.
By Proposition 5.1 the operator P l maps L p into H 1 loc has vanishing trace on {x n = 0}. By Proposition 5.6 P s w is an element of W 2,p (R n ) → H 1 (R n ) which vanishes in {x n ≤ 0} if w ∈ L p (R n ) is supported only on the closure of R n + . Therefore we conclude that 1Ω(P s + P l )1ΩS(1 + R l S) −1 v ∈ H 1 (Ω) has trace zero onΓ for all v ∈ L p (Ω) and thus G Γ has vanishing trace on Γ.
To verify the mapping properties of 1Ω(P s + P l )1ΩS(1 + R l S) −1 write Since S : L r → L r , inserting an L 2 (Ω) function would yield, by Propositions 5.1 and 5.6, an H 1 function with a loss of h −1 in the first term and no loss in the second. For mappings from L p we only need to concern ourselves with the first term since the Neumann sum maps L p → L 2 with no loss in h and we can refer to the L 2 estimate for 1Ω(P s + P l )1ΩS.
To analyze the mapping properties of the first term of (6.1) observe that due to Propositions 5.1 and 5.6, This finishes the proof of Theorem 1.3 in the case when Γ lies in a single graph. In the next section we move on to the general case where Γ is a disjoint union of graphs. Let χ ∈ C ∞ 0 (R n ) be supported inside Ω Γ with χ = 1 near Γ. We can arrange that supp(χ) ∩ ∂Ω ⊂ Γ . We can also arrange for the derivatives of χ to have the following support property: In this setting choose an open subset O ⊂ Ω ∩ {(x , x n ) | x n > f(x )} which contains Γ as a part of its boundary and whose closure contains the support of χ1 Ω . Set G Γ to be the Green's function constructed in Proposition 6.1 for the domain O with vanishing trace on Γ. We may then define Note that G Γ is not defined on the portion of Ω that lies below the graph of f , but this point is rendered moot when we multiply by χ. Observe that by Proposition 6.1 one has the trace identity

Lemma 6.2. One has the estimates
With this lemma we are in a position to construct a general Green's function for the h 2 Δ φ on a general domain Ω. Let ω ∈ R n be a unit vector and let Γ ⊂ ∂Ω be compactly contained in {x ∈ ∂Ω | ω · ν(x) > 0}. Without loss of generality we may assume as before that ω = (0 , 1). Assume in addition that Γ as a union of its connected components Γ j each of which lies in the graph of x n = f j (x ) for some smooth compactly supported function f j . For each Γ j construct χ j and Π Γ j as earlier. One then, by (6.4), has that Note that as before we can invert by Neumann series since L p gets mapped by R to L 2 with no loss and the Neumann series converge in L 2 . Theorem 1.3 is now complete by the estimates of (6.3), Lemma 4.1, and Lemma 4.2. All that remains is to give a proof of Lemma 6.2.
Proof of Lemma 6.2. By Proposition 6.1, G Γ is by construction a right inverse for h 2 Δ φ in Ω, and χ1 Ω is supported only on Ω, so χh 2 Δ φ 1 Ω G Γ v(x) = χv(x) as distributions on Ω. Meanwhile G φ is an honest right inverse for h 2 Δ φ on R n , so h 2 Δ φ 1 Ω G φ = I as distributions on Ω. Therefore as distributions on Ω, To analyze this term we will change coordinates by (x , x n ) → (x , x n − f (x )) and mark the pushed-forward domains, functions, and operators with a tilde. Then by the push-forward expression for the operator G Γ stated in Proposition 6.1, the right side of (6.5) is Computing the commutator [h 2Δ φ ,χ] explicitly in conjunction with the operator estimates in Propositions 5.6 and 5.1 we have that (6.6) . Returning to (6.6), we see that E has the desired estimates, so it remains only to analyze the first term of (6.6) 1Ω. Since we are only doing the computation as distributions onΩ, the first-order differential operator [h 2Δ φ ,χ] commutes with the indicator function 1Ω, and we have φ ,χ] maps H 1 to L 2 with the gain of h, so the term involving P s has the desired behaviour. Therefore the only term of difficulty is (6.7) By (6.2) the term 1Ω[h 2Δ φ ,χ] is a first-order differential operator whose coefficients are supported in {x n ≥ > 0}. This allows us to apply Lemma 3.4 to obtain the estimate Inserting this estimate back into (6.7) and splittingG φ by using Proposition 4.3 we see that . Therefore we see that every term in (6.6) has the desired form. Proof of Theorem 1.4. Let u ∈ C 2 (Ω) be a function which vanishes along ∂Ω and ∂ ν u | Γ c = 0, and let v ∈ C ∞ 0 (Ω). Integrating by parts, we have (6.8) v Ω with the boundary terms vanishing because of the boundary conditions on u and the boundary behaviour of G Γ v. Equation (6.8) implies that Applying the boundedness results for G Γ and taking the supremum over v ∈ C ∞ 0 (Ω) completes the proof.

Complex geometrical optics and the inverse problem
Let Ω ⊂ R n , ω ∈ S n−1 and Γ ⊂ ∂Ω be an open subset of the boundary compactly contained in {x ∈ ∂Ω | ν(x) · ω > 0} where ν denotes the normal vector. Assume in addition that in coordinates given by (x , x n ) ∈ ω ⊥ ⊕ Rω that Γ is the disjoint union of open subsets Γ j such that Γ j is the graph of x n = f j (x ). By Theorem 1.3 there exists a Green's function G Γ for h 2 Δ φ with vanishing trace on Γ and 7.1. Semiclassical solvability. Let ω be a unit vector and let Γ ⊂ ∂Ω be as before. We have the following solvability result, resembling the one in [24] (see the explanation of this method in [13]), but with an additional term.
Proof. We try solutions of the form r = G Γ ( |q|v + L) for v ∈ L 2 with v L 2 ≤ Ch 2 . Supposing this can be accomplished, then using L L 2 ≤ Ch 2 , where for any > 0 we decompose |q| = |q| + |q| with |q| ∈ L ∞ and |q| L n ≤ . Therefore, by taking h → 0 and using that v 2 ≤ Ch 2 . For the L p norm, observe that The mapping property of G Γ from L p → h −2 L p then gives the result. We now show that we can indeed construct such a v. Inserting the ansatz into (7.1) and writing q = e iθ |q| for some θ(·) : Ω → R we see that it suffices to construct v ∈ L 2 solving the integral equation with v L 2 ≤ Ch 2 . Observe that the right side is O(h 2 ) in L 2 norm due to the fact that L L 2 ≤ Ch 2 so it suffices to show that h 2 e iθ |q|G Γ |q| : L 2 → L 2 is bounded by o(1) as h → 0 and invert by Neumann series. Indeed, writing |q| = |q| + |q| we have Each of the three pieces have the following mapping properties: Therefore we have that h 2 e iθ |q|G Γ |q| :

Ansatz for the Schrödinger equation.
We briefly summarize the ansatz construction procedure given in [21]; see also the explanation in [5]. Let φ(x) and ψ(x) be linear functions satisfying D(φ + iψ) · D(φ + iψ) = 0. If Γ ⊂ ∂Ω is an open subset of the boundary satisfying Dφ · ν(x) ≥ 0 > 0 for all x ∈Γ, we first look to construct a solution to with L L 2 ≤ Ch 2 and a h ∈ L ∞ . By the fact that ∇φ · ν(x) ≥ 0 > 0 for all x ∈ Γ, we can apply Borel's lemma to construct ∈ C ∞ such that Since we are working with linear weights we will need a slightly more general h-dependent phase function than φ + iψ. Let ξ ∈ R n be a fixed vector which is orthogonal to both Dφ and Dψ, and let ψ h (x) be a linear function defined by is a vector of length O(h). Observe that in this setting the linear function φ + i(ψ + hψ h ) still solves the eikonal equation We now construct b ∈ C ∞ (Ω) supported close to Γ such that Using the fact that D · D = d(x, Γ) ∞ and Dψ h = ξ − ω h with |ω h | ≤ Ch we see that this amounts to solving the transport equation Taking advantage of the fact that −∂ ν Re( ) | Γ = ∂ ν φ | Γ ≥ 0 > 0 we can again solve the iterative equation and use Borel's lemma to construct b ∈ C ∞ (Ω) supported in an arbitrarily small neighbourhood of Γ satisfying this approximate equation. We have therefore constructed b ∈ C ∞ solving (7.3).
Inserting the expressions for u ± gives 0 = Ω e 2iξ·x q(1 + a − h a + h + a − h + a + h + a − h r + + a + h r − + r − + r + + r + r − ), where q = q 1 − q 2 . The function q ∈ L n/2 ⊂ L 1 and a ± h L ∞ ≤ C, lim h→0 a ± h (x) = 0 ∀x ∈ Ω by (7.4). Therefore, terms lim h→0 Ω e 2iξ·x q(a − h a + h + a − h + a + h ) = 0. For the terms involving Ω e 2iξ qa ± h r ∓ , we note that for all > 0 we may split q = q + q where q ∈ L ∞ while q L n/2 ≤ . Then, using the fact that a ± h L ∞ ≤ C, where p = 2n n−2 . By the estimates on r ∓ given in Proposition 7.2 we have that lim h→0 r ∓ L 2 = 0 and r ∓ L p ≤ C. Therefore, the limit lim h→0 Ω e 2iξ·x qa ± h r ∓ ≤ C for all > 0 and therefore the limit vanishes. The terms Ω e 2iξ q(r − + r + ) can be estimated the same way. For the last term, we again decompose, for all > 0, q = q + q . The integral | Ω e 2iξ·x qr − r + | is then estimated by Ω |q r − r + | + |q r − r + | ≤ q L ∞ r + L 2 r − L 2 + q L n/2 r − L p r + L p .
The L p norms of r ± stay uniformly bounded while the L 2 norms vanish when h → 0. Therefore the limit lim h→0 Ω e 2iξ·x qr − r + ≤ C q L n/2 ≤ C for all > 0 and therefore vanishes. This means that F(q)(ξ) = 0 for all ξ which are orthogonal to ω. Note that varying ω in a small neighbourhood does not change the fact that Γ lies in the set {x ∈ ∂Ω|ω · ν(x) > 0}, and so the construction in Proposition 7.2 still applies. Then varying ω in a small neighbourhood and using the analyticity of the Fourier transform for q compactly supported we have that q = q 1 − q 2 = 0.

Appendix
Here we will provide proofs for Propositions 2.2 and 2.3 from Section 2. . The last term Op h (m) takes L r → L r by (2.8). The first term is a composition of an operator taking L r (R n−1 ) → L r (R n−1 ) (leaving the x n direction untouched) and an operator from L r (R n ) → L r (R n ) by (2.8 (R n ) and apply (2.8). To derive (8.1) simply use the standard methods as in [35]. First we have that where c(x , ξ) = 1 (2πh) n e −iy·(ξ−η) a(x , η)b(y + x , ξ )dηdy.
Standard computation then yields that where m(x , ξ) has the explicit representation dηdy.