STABILITY ESTIMATES FOR A MAGNETIC SCHR¨ODINGER OPERATOR WITH PARTIAL DATA

. In this paper we study local stability estimates for a magnetic Schr¨odinger operator with partial data on an open bounded set in dimension n ≥ 3. This is the corresponding stability estimates for the identiﬁability result obtained by Bukhgeim and Uhlmann [2] in the presence of a magnetic ﬁeld and when the measurements for the Dirichlet-Neumann map are taken on a neighborhood of the illuminated region of the boundary for functions supported on a neighborhood of the shadow region. We obtain log log-estimates for the magnetic ﬁelds and log log log-estimates for the electric potentials.


Introduction.
Let Ω ⊂ R n (n ≥ 3) be a simply connected open bounded set with C ∞ boundary, denoted by ∂Ω. We consider the following magnetic Schrödinger operator L A,q (x, D) := n j=1 (D j + A j (x)) 2 + q(x) = D 2 + A · D + D · A + A 2 + q, where D = −i∇, A = (A j ) n j=1 ∈ C 2 Ω; R n is a magnetic potential and q ∈ L ∞ (Ω; R) is an electric potential. Here Z 2 denotes Z · Z for operators as well as for vector-valued functions. The inverse boundary value problem, IBVP for short, under consideration in this article is to recover information (inside Ω) about the magnetic and electric potentials from measurements on subsets of the boundary. Roughly speaking, we divide the boundary ∂Ω in two open subsets, F and B. In this setting and if 0 is not an eigenvalue in L 2 (Ω) of L A,q , we define the so called Dirichlet-Neumann map (DN map) Λ B→F A,q as follows: where ν is the exterior unit normal of ∂Ω, the set H 1 2 B (∂Ω) consists of all f ∈ H 1 2 (∂Ω) such that supp f ⊂ B (we will call this condition "support constraint") and u ∈ H 1 (Ω) is the unique solution of the Dirichlet problem: Moreover, due to elliptic estimates, it follows u belongs to H 2 (Ω). Consequently, (∂ ν + iA · ν)| ∂Ω belongs to L 2 (∂Ω). This fact will be used several times later.
In the case that F or B is not equal to ∂Ω, we say that the IBVP has partial data. According to the choice of the sets F and B, we can distinguish several types of partial data results that we briefly describe.
In the case of illuminating from infinity the supporting set B could also be restricted to a neighborhood of the shadow region from infinity. In the case of A ≡ 0, stability estimates with the support constraint were derived by Caro, Dos Santos Ferreira and Ruiz [3], using Radon transform and for illumination from a point without the support constraint in [4] by using the geodesic ray transform on the sphere. In both cases, they obtained log log-estimates.
In the presence of a magnetic potential (A ≡ 0), as it was noted in [19], there exists a gauge invariance of the DN map. To be specific, if ϕ ∈ C 1 (Ω) is a real valued function with ϕ| ∂Ω = 0 then Λ A,q = Λ A+∇ϕ,q . Hence, for the identifiability problem we only expect to recover the magnetic fields, dA 1 = dA 2 , and the electric potentials, q 1 = q 2 . Here we consider the magnetical potential A as a 1-form and so dA as a 2-form, as follow: A j dx j , A = (A 1 , A 2 , . . . , A n ), For the magnetic case, we briefly mention the results concerning to full data, that is when F = B = ∂Ω. Sun [19] proved identifiability under the assumption of the smallness of the magnetic potential in a suitable space. Together with Nakamura and Uhlmann [14], they removed the smallness condition by assuming C 2 -compactly supported magnetic potential and L ∞ electric potential. For these cases, stability estimates were derived by Tzou [20]. The previous full data identifiability results were extended by Krupchyk and Uhlmann [12] for both, magnetic and electric potentials in L ∞ . The corresponding stability estimates were derived by Caro and Pohjola [5].
The aforementioned full data results were extended to the partial data case. In the case of illumination from a point, the identifiability result was obtained by Dos Santos Ferreira, Kenig, Sjöstrand and Uhlmann [8]. They considered the knowledge of the DN map on a neighborhood of the illuminated face F for functions without any support constraint on the boundary. This result was extended by Chung [6], assuming an extra condition, that is the support constraint is on a neighborhood B of the shadowed face. For both previous cases, the issue of stability still remains open. On the other hand, in the case of illumination from infinity, the identifiability result and the corresponding stability estimates were obtained by Tzou [20]. Analogously to [8], Tzou did not consider any support constraint on the boundary.
In this article we extend Tzou's result [20] in the following sense: we consider the case of illumination from infinity with an additional condition similar in spirit to [6], that is, we consider the support constraint on a neighborhood B of the shadowed face of the boundary. In this case, our main goal is to derive stability estimates when recovering dA and q.
This paper is organized as follows. In Section 2 we state the stability estimates for the magnetic and electric potentials. In Section 3 we prove the stability estimate for the magnetic potential, see Theorem 2.3. In Section 4, we prove the stability estimate for the electric potential, see Theorem 2.4. In Section 5 we deduce a new Carleman estimate with linear weight, see Lemma 5.5, and also it contains the proof of Proposition 1, which ensures the existence of special solutions for the magnetic Schrödinger equation, supported on the shadowed part of the boundary.
2. Stating our stability estimates. Before stating our results we introduce some notation following [3]. Let N be a non empty open subset of S n−1 representing the set of directions from where we are illuminating Ω, and define the sets We consider its associated operator norm defined by χΛ A,q f H −1/2 (∂Ω) .
For simplicity in notation, usually we ignore the subscript in the previous norm operator, that is · stands for · H 1/2 B (∂Ω)→H −1/2 (∂Ω) . It is well known that in order to obtain stability results one needs a priori bounds on the magnetic and electric potentials (conditional stability) to control oscillations. Subsequently, we introduce the class of admissible magnetic and electric potentials as follows.
Definition 2.1. Given M > 0 and γ ∈ [0, 1) , we define the class of admissible magnetic potentials A (Ω, M, γ) by Here and throughout this paper, we denote the characteristic function of Ω by χ Ω . For a function h : Ω → R (or R n ) we denote by χ Ω h its extension by zero out of Ω. According to Proposition 3.6 in [18] (see also Lemma 1.1 in [9]), we have χ Ω belongs to H σ (R n ) with σ ∈ (0, 1/2). Motivated by this fact, we also introduce the class of admissible electric potentials.

Theorem 2.4.
Let Ω ⊂ R n be a simply connected open set with smooth boundary. Consider three positive constants M , σ ∈ (0, 1/2) and γ ∈ (0, 1). Let N be a non empty open subset of S n−1 and consider F an open neighborhood of F N , where F N is defined as 3. Then there exist C > 0 (depending on n, Ω, M, σ, γ) and λ ∈ (0, 1/2) (depending on n) such that the following estimate holds true for all A 1 ∈ A (Ω, M, γ) and for all A 2 ∈ A (Ω, M, 0) satisfying A 1 = A 2 and ∂ ν A 1 = ∂ ν A 2 both on ∂Ω; and for all q 1 , q 2 ∈ Q(Ω, M, σ).
The proofs of Theorem 2.3 and Theorem 2.4, will be carried out by proving an integral identity relating the partial boundary data, i.e. the partial DN maps, with the unknown magnetic and electric potentials, by means of solutions u ∈ H 1 (Ω) of the magnetic Schrödinger equation L A,q u = 0 in Ω. In order to decode the information hidden into the integral identity, we use two classes of special solutions, the so-called complex geometric optic solutions (CGO solutions). The first class will be obtained by using a suitable Carleman estimate with a linear weight and following the arguments used in [6] in order to have the required support constraint on the boundary. In the literature, all previous results do not consider this support constraint on the boundary. We emphasize that one of the main difficulty to construct such required solutions is the derivation of a suitable Carleman estimate. The second class of solutions have already been constructed in [8] and need not have the support constraint. By plugging these solutions into the integral identity leads us to obtain two Radon transforms, one for the difference of the magnetic fields dA 1 − dA 2 and other for the difference of the electric potentials q 1 − q 2 . At this point, we apply a quantitative estimate derived in [3] to the Radon transform of dA 1 − dA 2 to end up the proof of Theorem 2.3. The quantitative estimate involves a logarithm of the difference of the DN maps. To prove Theorem 2.4, we applied the same quantitative estimate for the Radon transform now for q 1 − q 2 , the Hodge decomposition derived by Tzou [20] (here we require the connectedness hypothesis) and the gauge invariance of the DN map in order to use the already established stability estimate for the magnetic fields. This step involves two logarithms of the difference of the partial DN maps, and by the quantitative estimate for the Radon transform, an extra logarithm has to be added.
3. Stability estimate for the magnetic potential. The goal of this section is to prove Theorem 2.3. According to this, throughout this section we use the notations and hypotheses from Theorem 2.3 for the sets Ω, N and F N and the corresponding regularities for the magnetic and electric potentials.
3.1. Construction of special solutions -CGO solutions. In this section we shall establish the existence of CGO solutions u ∈ H 1 (Ω) (with and without the required boundary support constraint on B) for the magnetic Schrödinger equation L A,q u = 0. The following proposition is the analogous of Proposition 9.2 in Chung article [6]. However, we mention that there is a slight difference with respect to Chung's proof. By our method, we required a little bit more regularity than C 2 for the magnetic potential while Chung only consider C 2 . Proposition 1. Let ξ, ζ ∈ S n−1 be a pair of orthogonal vectors and let γ ∈ (0, 1). If A ∈ C 2,γ (Ω; R n ) and q ∈ L ∞ (Ω; R) then there exist three positive constants, τ 0 and C (both depending on n, Ω, A C 2,γ , q L ∞ ) and γ with 0 < γ < γ such that the equation in Ω u| ∂Ω\B = 0 has a solution u ∈ H 1 (Ω) of the form with the following properties: (i). Let A ∈ C 2,γ c (R n ) be any compactly supported extension of A in R n . If Φ := C ξ+iζ A then Φ ∈ C 3,γ (Ω) and satisfies in R n Here C ξ+iζ A denotes the Cauchy transform in R n of A, with respect to ξ and ζ.
The definition of C ξ+iζ is given in Appendix, see Definition 5.1.
(ii). The function l depends on the a priori bounds of A and q, and satisfies where the positive function k(x) dist(x, ∂Ω \ B) in G, a neighborhood of ∂Ω \ B in R n . Here l denotes the real part of l.
(iii).] The function b belongs to C 1,γ (Ω) with supp b ⊂ G; and it depends on the a priori bounds of A and q. Moreover, we have (iv). Finally, r ∈ H 1 (Ω) satisfies r| ∂Ω\B = 0 and for all τ ≥ τ 0 the following estimates hold true For expository convenience, we leave the proof to the Appendix. We will also use the solutions constructed by Dos Santos Ferreira, Kenig, Sjöstrand and Uhlmann, see Lemma 3.4 in [8]. These solutions do not require the support constraint on B.
Proposition 2. Let ξ, ζ ∈ S n−1 be a pair of orthogonal vectors. If A ∈ C 2 (Ω; R n ) and q ∈ L ∞ (Ω; R) then there exists two positive constants, τ 0 and C (both depending on n, Ω, A C 2 , q L ∞ ) such that the equation L A,q u = 0 has a solution u ∈ H 1 (Ω) of the form u = e −τ (ξ·x−iζ·x) e Φ g + r , with the following properties: Here C ξ−iζ A denotes the Cauchy transform in R n of A, with respect to ξ and −ζ.
The definition of C ξ−iζ is given in Appendix, see Definition 5.1.

3.2.
Relating the partial DN maps with the magnetic and electric potentials. We state an integral estimate which involves a relation between the partial DN maps with the magnetic and electric potentials, see Proposition 4. From now on, for j = 1, 2, we denote the global and partial DN maps Λ Aj ,qj by Λ j and Λ Aj ,qj by Λ j , respectively. The following Carleman estimate with boundary terms was derived by Dos Santos Ferreira et al., see Proposition 2 in [8].
Remark 2. The Carleman estimate 12 is still true for all u in H 1 0 (Ω) such that L A,q u ∈ L 2 (Ω), which could be seen by a standard regularization method. The vanishing condition on the boundary of u is essential for deriving this Carleman estimate with boundary terms. In short, this estimate said us that it is possible to bound the L 2 (∂Ω +,0 (ξ))-norm by the L 2 (∂Ω −,0 (ξ))-norm plus remainder terms in L 2 (Ω)-norm. In other words, we can bound the measurements of the shadowed face of ∂Ω by measurements of the illuminated face but paying with quantities in L 2 (Ω)-norm.
The following lemma it is a well known result which relates the global DN maps with the magnetic and electric potentials, see Proposition 3.1 in [19]. Lemma 3.1. Assume that the functions u 1 , u 2 ∈ H 1 (Ω) satisfy L A1,q1 u 1 = 0 and L A2,q2 u 2 = 0. Then In particular, this identity holds true for every u 1 ∈ H 1 (Ω) satisfying both L A1,q1 u 1 = 0 and supp(u 1 )| ∂Ω ⊂ B. For technical reasons to prove Proposition 4, we need to introduce the following subsets of the boundary. Given a direction ξ ∈ S n−1 and > 0, we define the (ξ, )-illuminated face of ∂Ω as and the (ξ, )-shadowed face as Note that these sets are open neighborhoods on ∂Ω of ∂Ω −,0 (ξ) and ∂Ω +,0 (ξ), respectively; see 1 and 2. Hence, according to 3, the sets defined by are open neighborhoods on ∂Ω of F N and B N , respectively. Finally, since F is an open neighborhood of F N , it follows that the following inclusions (14) F hold true whenever > 0 is small enough. Now in order to exploit the information about A 1 − A 2 encoded into the previous integral identity, we start by obtaining an estimative of the left-hand side of 13.
Proof. We start by choosing > 0 being small enough such that 14 holds true and consider χ ∈ C ∞ (∂Ω) be a cutoff function supported in F such that it equals to 1 on F N, . We consider the decomposition We now estimate each of the right-hand side terms. By the definition of the partial DN map 4 and Cauchy-Schwarz inequality, the first term can be estimated as follows The estimation of the second term requires a more refined analysis. It can be done by using Proposition 4 as follows.
Since 15 holds for every pair of solutions u ∈ H 1 (Ω) corresponding to the magnetic Schrödinger equation L A,q u = 0, we use the CGO solutions constructed in Section 3.1. More precisely, we consider ξ ∈ N and ζ ∈ S n−1 such that ξ · ζ = 0. On the one hand, let A ext 1 be a compactly supported extension in R n of A 1 such that it still belongs to C 2,γ (R n ). By Proposition 1 there exist u 1 ∈ H 1 (Ω) satisfying and two positive constants C 1 and τ 1 such that holds true for all τ ≥ τ 1 . Moreover, we have the estimate On the other hand, we consider the following compactly supported extensions of A 2 and two positive constants C 2 and τ 2 such that the following estimate holds true for all τ ≥ τ 2 .

Remark 3. With the previous compactly supported extensions at hand, it is easy to check
We leave aside for a moment this equation. It will play a crucial role in the proof of Proposition 5.
According to Proposition 4, our next task will be to estimate the terms coming from the right-hand side of 15. For simplicity, we denote a 1 = e Φ1 , ϕ(x) = ξ · x and ψ(x) = ζ · x. Then, since l(x) = ξ · x − k(x) and taking into account 9 and 23-25, we obtain (28) We continue in this fashion to compute the Repeating similar previous arguments and using 26-27, leads us to get the following estimates for u 2 By combining the estimates 28-31 into 15, and taking into account that there exists Taking τ 0 = max(τ 1 , τ 2 , τ 3 ) and multiplying by τ −1 both sides of the previous inequality, we deduce the following estimate for all τ ≥ τ 0 which in turn can be decomposed in the following way. For simplicity, we denote ρ(x) = (ξ + iζ) · x, u r = e τ (−ρ+l) b, a 2 = e Φ2 g and recall that we have already denoted a 1 = e Φ1 . Hence the solutions u 1 and u 2 , given by 23 and 26, respectively; have the form An easy computation shows that Hence, by 13 and 34-35, we obtain (36) Now by 10 and similar analysis to that in the proof of the estimates 28-31 shows for some positive constants C 1 , C 2 and τ 4 . Thus, by combining 32, and 36-38, we get Finally, by recalling that a 1 = e Φ1 and a 2 = e Φ2 g, we obtain Next we show that it is possible to remove the exponential term e Φ1+Φ2 .
Remark 4. Let ξ, ζ ∈ S n−1 (n ≥ 3) be unit orthogonal vectors. Then, every x ∈ R n can be written as follows it leads to parameterize R n in the following way x → (t, r, x ).
then there exist two positive constants τ 0 and C > 0 (both depending on n, Ω, M, γ) such that Here g is any smooth function depending only on x in the sense of Remark 4.
Proof. The proof is a direct consequence of Remark 3 and Lemma 5.
for any smooth function g depending only on x . In particular, this kind of smooth functions g satisfies (ξ + iζ) · ∇g = 0 in R n . Hence, by 39 and 41, we obtain Now we consider τ 0 > 0 such that for all τ ≥ τ 0 the previous inequality and e −2τ c ≤ τ −1/2 are satisfied. It is a simple matter to check that Hence, we get By a similar analysis, with (ξ + iζ) instead of (ξ − iζ), we obtain By combining the two previous inequalities, we conclude the proof.

Radon transform and its applications.
Let f be a function on R n , integrable on each hyperplane in R n . Each of these hyperplanes can be parametrized by its unit normal vector and distance to the origin: θ ∈ S n−1 and s ∈ R, respectively. Thus, each hyperplane H can be defined in the following way: In this setting, the Radon transform of f is defined by whenever the integral exists. Here θ ⊥ denotes the set of orthogonal vectors to θ. Analogously, we define the parametrized hyperplane with respect to an arbitrary y 0 ∈ R n and its corresponding Radon transform as follows respectively. It is easy to check the relation On the other hand, we define the Fourier transform with respect to the first variable of a function F : R × S n−1 → R as Thus, for α ≥ 0, we denote by H α (R × S n−1 ) the space consisting of all functions F ∈ L 2 (R × S n−1 ) such that is finite. It is well known that the Radon transform is continuous in suitable spaces and there exits a relation between its s-derivative and the partial derivatives with respect to x. More precisely, for every compactly supported function f in R n , there exists a positive constant C 0 , depending on α and n, such that and the following identity holds in the sense of distributions in C ∞ 0 (R) Here θ i denotes the i-th coordinate of θ. Both previous results can be found in [15]. The next result plays an important role in deriving stability estimates for the magnetic and electric potentials. It was proved by Caro, Dos Santos Ferreira and Ruiz, see Theorem 2.5 in [3]. Roughly speaking, it gives us a quantitative relation between a function and its Radon transform: it is enough to have knowledge of the Radon transform R R RF in a proper bounded subset of R × S n−1 to obtain information about F but we have to pay with a logarithmic term. Before stating Theorem 3.2, we introduce the set X consisting of all F ∈ L 1 (R n ) such that is finite. By means of Fubini-Tonelli theorem, a sufficient condition for a function F to be in X, is given by the following estimate where S n−1 denotes the measure of S n−1 . We also recall the distance on the unit sphere d S n−1 (x, y) = arccos( x, y ).
Theorem 3.2. Let M ≥ 1, α > 0 and β ∈ (0, 1). Given y 0 ∈ R n and θ 0 ∈ S n−1 , consider the set Γ = θ ∈ S n−1 : d S n−1 (θ 0 , θ) < arcsin β and the domain of dependence of the Radon transform given by Assume that there exist two positive constants p and λ with 1 ≤ p < ∞ and 0 < λ < p −1 , respectively; and a function F satisfying the following conditions: Then there exists a positive constant C (depending on G, M, α, β, λ), such that Remark 5. In our context, the constant β stands for the size of the set N ⊂ S n−1 and (−α, α) is the interval where we have control of the Radon transform R R RF (·, θ). Note that for fixed y 0 ∈ R n and β > 0 we can take α large enough so that Ω ⊂ G.
We take advantage of this last fact to prove Theorem 2.3.
where [ξ] ⊥ denotes the subspace of R n consisting of all the unit vectors orthogonal to ξ. Then for any g ∈ C ∞ (R) there exist two positive constants C and τ 0 (both depending on n, Ω and the a priori bounds of A j C 2 (Ω) and q j L ∞ (Ω) ) such that the following estimate holds true for all θ ∈ P and µ ∈ θ ⊥ .
Proof. We consider ξ ∈ N ⊂ S n−1 and ζ ∈ S n−1 such that ξ · ζ = 0. Since n ≥ 3, it follows that there exists an unit vector θ ∈ [ξ, ζ] ⊥ := (span {ξ, ζ}) ⊥ . Thus, every x ∈ R n can be written in the following way Hence, R n can be parametrized as Ψ : x → (t, r, s, y ). On the one hand, according to Remark 4 and Proposition 5, in terms of Ψ-coordinates, 40 holds true for every smooth function g := g • Ψ −1 depending only on s and y . For our purpose, we consider any smooth function g := g(s) depending only on s. On the other hand, since ξ, ζ and θ are unit orthogonal vectors, we have dx = dy dtdrds. Thus, for every µ ∈ [ξ, ζ] := span {ξ, ζ}, we have By combining this equality and 40, we conclude the proof.
It is easy to see that for any θ ∈ P ⊂ S n−1 , the vectors of the form µ jk := θ j e k − θ k e j (j, k = 1, 2, . . . , n) belong to θ ⊥ . Here (e j ) n j=1 denotes the canonical basis of R n and θ j the j-th component of θ. For simplicity in computations, we denote A = χ Ω (A 1 − A 2 ) and A j = A · e j . Hence, by 43, the following equality holds true for every compactly supported smooth function h in R. This equality and applying Proposition 6 with µ := µ j,k and g = ∂ s h, give us and then where L ∞ (P; H −3 (R)) denotes the functions on P which are vector valued in H −3 (R), whose norm is given by On the other hand, by Remark 3, A ∈ C 2 c (R n ). Hence, applying 42 with α = 1 and where we have used the following identity in the notation of vector valued spaces H n+1 2 (R × S n−1 ) = L 2 (S n−1 ; H n+1 2 (R)). Now a complex interpolation, see Theorem 3 in [1], between L p0 (P; H −3 (R)) (with p 0 > 0 large enough) and L 2 (P; H n+1 2 (R)), give us , for any p 1 < (n + 7)/3 and s 1 < 1 2 (n + 1)/(n + 7). Notice that in [1] the spaces L p1 (P; H s1 (R)) are denoted as L p1 (H s1 (R)) with the measure space P ⊂ S n−1 omited in the notation. Once again, for simplicity we denote F j,k := ∂ xj A k − ∂ x k A j with j, k = 1, 2, . . . , n. By Remark 3, F j,k ∈ C 1 c (R n ). Our next objective is to apply Theorem 3.2 in order to extract information about F j,k by means of its Radon transform estimate 47. To do that, F j,k has to verify three conditions. Condition (a). It is clear that χ E F j,k ∈ L 1 ∩ L ∞ (R n ). Thus, from 44 and the estimate it follows that χ E F j,k ∈ X ∩ L ∞ (R n ) and then the condition (a) is verified. Note that to verify this condition we are only using the compactness of the support of F j,k . In this way, we are able to consider E with α > 0 large enough. This property is closely related with 47 because we have the control of the Radon transform of F j,k in the whole real line in the s-variable.
Condition (b). One sees immediately that for every fixed θ 0 ∈ P, by translation there exists y 0 ∈ supp F j,k such that the hyperplane H y0 (0, θ 0 ) stands on one side of supp F j,k .
Condition (c). It is well known that a function Y has (λ, 2, 2)-Besov regularity if and only if Y ∈ H λ (R n ) with 0 < λ < 1. According to Proposition 3.6 in [18] (see also Lemma 1.1 in [9]), we have χ E belongs to H λ (R n ) with λ ∈ (0, 1/2). Hence χ E F j,k ∈ H λ (R n ) and then it has (λ, 2, 2)-Besov regularity. So the condition (c) is satisfied with p = 2 and λ ∈ (0, 1/2). As we have discussed already and recalling Remark 5, although β ∈ (0, 1), which is related closely with the size of the set N , could be very small; we can take α > 0 large enough such that supp F i,j ⊂ G, where G is defined by 45. Thus, Theorem 3.2 with Γ = P, ensures that there exists C > 0 such that and denote by |P| the Lebesgue measure of P. Then 47, Fubini-Tonelli theorem and Hölder's inequality, give us We conclude the proof by taking logarithm to both sides of this inequality and taking into account 48.
4. Stability estimate for the electric potential. The goal of this section is to prove Theorem 2.4, we use its notations and hypotheses for the sets Ω, N and F N and the corresponding regularities for the magnetic and electric potentials. The idea will be to combine the gauge invariance of the DN map, a Hodge type decomposition and the stability result already proved in Theorem 2.3. Although the following result has not been explicitly declared by Tzou [20], it can be easily deduced by combining his Lemma 6.2 (applied to A 1 − A 2 ) with the discussion just after of its proof. See also estimate 23 in [20].
From 23 and 26, we have where l, b and Φ j , r j (j = 1, 2) satisfy the equations, regularities and estimates discussed in Section 3.2. With these solutions at hand, we use the gauge invariance and 13 with A j and U j instead of A j and u j , respectively; to get From this identity, we will now try to isolate q 1 − q 2 and hence, we have to deal with terms of the form ( A 1 − A 2 ) · (DU 1 U 2 + U 1 DU 2 ) and ( A 2 1 − A 2 2 )U 1 U 2 . Both terms have the common factor A 1 − A 2 = A 1 − A 2 − ∇ω, which can be managed by using Lemma 4.1 and an elementary interpolation as follows where n < p < z and t ∈ (0, 1) such that 1/p = t/2 + (1 − t)/z. Thus, by Theorem 2.3 and 49, we get By an analysis similar to that in Section 3.2, we deduce that From identity 51, we get (54) and 52-54, give us The next step will be to remove the exponential term e Φ1+Φ2+iω . To do this, we start with the following identity (56) By Lemma 4.1 and Morrey's inequality we have ω ∈ C 2,1− n p (Ω) and ω| ∂Ω = 0. Combining Remark 3 and Lemma 5.2, we obtain in R n Taking into account the following well know inequality for complex numbers e a − e b ≤ |a − b| e max{ a, b} , a, b ∈ C, and Hölder's inequality, we deduce that Hence, 52 and 55-56, imply that for all τ ≥ τ 0 . Choosing τ as follows which is true whenever

Proof of Theorem 2.4. Analysis similar to that in the proof of Theorem 2.3
shows that estimate 57 can be written in term of the Radon transform as for all θ ∈ P and g ∈ C ∞ c (R n ), where P is defined by 46. Hence, we have Applying 42 with α = 0 and f = χ Ω (q 1 − q 2 ), we get We can now proceed analogously to the proof of the magnetic case. Consider p 0 > 0 large enough and a complex interpolation between L p0 (P; H −2 (R)) and L 2 (P; H n−1 2 (R)), give us , for any p 2 < (n + 3)/2 and s 2 < tλ 3 (n − 1)/(n + 3). For simplicity, we now denote q = χ Ω (q 1 − q 2 ). Our next objective is to apply Theorem 3.2 in order to extract information about q by means of its Radon transform estimate 58. The conditions (a) and (b) are verified similar to the magnetic case. The condition (c) is also satisfied since by hypothesis we have q ∈ H σ (R n ). So q has (σ, 2, 2)-Besov regularity. Hence, we can take α > 0 large enough such that supp q ⊂ G, where G is defined by 45. Thus, Theorem 3.2 with Γ = P, ensures that there exists C 6 > 0 such that Then 58, Fubini-Tonelli theorem and Hölder's inequality, give us We conclude the proof by taking logarithms in both sides of the above inequality and taking into account the estimate 59.

5.
Appendix. This section is mainly focused on proving Proposition 1. For more details, see also Appendix in [16].
5.1. The Cauchy transform. We first discuss the existence of solutions Φ in R n of the following equation: where ξ, ζ ∈ S n−1 are orthogonal vectors and W is a prescribed function.
Definition 5.1. Let ξ, ζ ∈ S n−1 be orthogonal vectors. The Cauchy transform in R n of W , with respect to ξ and ζ, is defined by whenever the integral exists.
This transform allows us to obtain a solution for the equation 60. More precisely, we have the following result. See Lemma 2.1 in [19] and also Lemma 4.6 in [17].
The following is an already well known result in the case g ≡ 1, see Proposition 3.3 in [12] and also Lemma 2.6 in [20].
Moreover, the following identity: holds true for all smooth function g depending only on x in the sense of Remark 4.
Proof. Without loss of generality we can assume that ξ = e 1 and ζ = e 2 . Let g be a smooth function depending only on x . Now Lemma 5.2 ensures that Φ := C ξ+iζ ((ξ + iζ) · W ) satisfies 62 and so we have Green's theorem give us By combining this identity and 63, the lemma follows.
To construct solutions for the magnetic Schrödinger operator with the support constraint on the shadowed face of the boundary, we need a more refined estimate than 61. The proof of the following result can be found in Proposition B.3.1 in [16].

5.2.
Proof of Proposition 1. We only give the main ideas to prove Proposition 1 in order to convince the reader that it is essentially similar to Proposition 9.2 in [6] with a linear limiting Carleman weight ϕ(x) = ξ · x with ξ ∈ S n−1 , instead of the logarithmic weight ϕ(x) = log |x − x 0 |. To construct solutions with the desired support constraint on the shadowed face of the boundary, Chung derived a novel Carleman estimate with logarithmic weight in an extended domain of Ω, see Theorem 1.4 in [6]; which for us is the heart of his article. Actually, he dedicated several sections to prove this estimate, employing original, elegant and sophisticated arguments which can be easily adapted to our case. The key point in his paper is to prove that one can pass from a Carleman estimate L 2 → H 1 to a Carleman estimate H −1 → L 2 with some condition on the support of the functions adapted by duality to the case with the support constrains. Fortunately there is a result in Chung's works, not explicitly stated, that avoid a repetition of his argument and computations for our linear limiting Carleman weight. We state this result in Lemma 5.9. This lemma could be interesting by itself for future applications. After deriving such estimate, a standard Hahn-Banach argument completes our proof. For a complete version of the proof of Proposition 1, we refer the reader to Appendix C in [16].
We mention that the limiting Carleman weights defined in any open subset of the R n are completely characterized. There are infinitely many of them, which can be classified in 6 categories as it was shown in [7]. Aside from this, for a large parameter τ > 0 and a limiting Carleman weight ϕ in Ω, we set L A,q,ϕ = τ −2 e τ ϕ L A,q e −τ ϕ .
For a bounded open subset V ⊂ R n , we denote by H 1 scl (V ) the H 1 -Sobolev space with semiclassical parameter τ −1 and its dual space by H −1 scl (V ). Their norms are respectively defined by Here ·, · V denotes the distribution duality in V . From here we divide the proof into two steps. For expository convenience, in the first step, we only state our Carleman estimate with linear weight. As a consequence, we prove the existence of CGO solutions with the desired support constraint on the boundary. In the second step, we formulate the hidden implicit result in [6] and how to use it in order to prove the Carleman estimate aforementioned in the first step.
First step. With the above notation and definition at hand, we have: Theorem 5.5. Let A ∈ C 2 (Ω; R n ) and q ∈ L ∞ (Ω; R). Let ξ ∈ S n−1 and set ϕ(x) = ξ · x. Suppose that Ω is a smooth domain with Ω ⊂ Ω such that ∂Ω ∩ ∂Ω = E, where E is a compact subset of ∂Ω −,0 (ξ) on ∂Ω. Then there exist two positive constants C and τ 0 (depending on n, Ω and priori bounds on A and q) such that the following estimate By a standard Hahn-Banach argument (see for instance Proposition 9.1 in [6]), it follows immediately the next existence result.
Proposition 8. Assume the hypotheses of Theorem 5.5. Then for every v ∈ L 2 (Ω), there exists u ∈ H 1 (Ω) satisfying Moreover, there exist two positive constants C and τ 0 such that . This result will be used to ensure the existence of a remainder term r, which it is part of our solution. More precisely, our next task is to provide the necessary conditions for the functions a, r, l and b such that u of the form Applying this formula twice, first with ρ(x) = τ (ξ · x + iζ · x) and v = a + r and later with ρ(x) = τ l(x) and v = b, we deduce that u defined by 64 satisfies L A,q u = 0 in Ω whenever the following identity holds true: This naturally suggests the following construction.
Equation for a. In order to reduce the right-hand side of the above expression, we start by despising the first term of τ −1 order. We consider a satisfying in Ω (66) (ξ + iζ) · ∇a + i(ξ + iζ) · A a = 0.
We try solutions of the form a = e Φ , so the function Φ must satisfy This can be solved by means of the Cauchy transform as follows. Let A be a compactly supported extension in R n of A such that belongs to C 2,γ c (R n ) and satisfying Then, applying Proposition 7 with m = 2 and F = −i(ξ + iζ) · A, it follows that Φ := C ξ+iζ (−i(ξ + iζ) · A) belongs to ∈ C 3,γ(R n ) with 0 < γ < γ, satisfy in R n (ξ + iζ) · ∇Φ + i(ξ + iζ) · A = 0, and the following estimate .
In particular, the restriction Φ| Ω ∈ C 3,γ (Ω), it still denotes by Φ, is a solution of the equation 67. Hence, we deduce that a = e Φ also belongs to C 3,γ (Ω) and satisfies 66. The estimates 6-7 follow by combining 68-69 and Proposition 7. Note that a is independent of τ . The next paragraph is intended to motivate the equations that have to satisfy both l and b.
By 65 and once we had proved the existence of a, we deduce that R := e iτ ζ·x r has to satisfy (70) This equation for R can be solved by using Proposition 8 with E = ∂Ω \ B. In particular, R| ∂Ω\B = 0 and so r| ∂Ω\B = 0 (these facts will be verified later on).
One way to achieve this condition is considering the functions l and b with the boundary conditions: l(x)| ∂Ω\B = (ξ · x + iζ · x)| ∂Ω\B and b| ∂Ω\B = a| ∂Ω\B . Moreover, in order to have a decay of R in τ , we have to ensure the decay in τ of the righthand side of 70. It can be done by assuming that Dl·Dl and 2Dl·Db+(2Dl·A+ D 2 l)b are small enough in a suitable sense.
Equation for l. We claim that for every p ∈ N there exist a function l satisfying Indeed, first note that the function (ξ + iζ) · x satisfies the first two conditions of 71. The reason to consider the third boundary condition is to avoid the repetition of this solution. We start by picking coordinates (t, s) in a neighborhood of ∂Ω \ B, where t is the coordinate over ∂Ω \ B and s is perpendicular to ∂Ω \ B and then stands for dist(x, ∂Ω \ B). Formally, we consider l as follows where the smooth functions a j can be determined by imposing From the boundary conditions, it follows immediately that By a recursive relation, we can determine a j for j ≥ 2. In (t, s)-coordinates, the gradient of l has the form From this recursive formula, we have for m ≥ 1 Thus, to determine a m+1 we need to verify that a 1 = 0 on ∂Ω \ B. On the one hand, ν(x) · ξ > 0 on ∂Ω \ B, since ∂Ω \ B ⊂ ∂Ω −,0 (ξ). On the other hand, ∂Ω \ B is a compact subset of the boundary, thus taking into account 72, we deduce that |a 1 | > 0 > 0. Hence, we can divide by a 1 the last above recursive identity to know a m for all m ∈ N. From here, it is immediate to see that the p-truncation of the serie corresponding to l give us a solution of 71. Indeed, consider the p-truncation of l defined by a j (t)s j .

Equation for b.
Once proved the existence of the function l, we consider b being a solution of the equation We try a solution of the form where the functions b j will be determined later on. By boundary condition, we immediately deduce that b 0 (t) = a| ∂Ω\B . It remains to determine b 1 and b 2 . At this point, there is a slight difference with respect to the construction of l. This is because the magnetic potential A has only integer derivative until the second order, so its Taylor series is not well-defined. For this reason, we consider its residual approximation until the second order in the (t, s)-coordinates as follows A(t, s) = (A 0 (t) + A 1 (t)s + R A (t, s), A n 0 (t) + A n 1 (t)s + R n A (t, s)) , where A j and R A are vector-valued functions in R n−1 , A n j and R n A are real-valued functions with j = 0, 1. Since A ∈ C 2,γ (Ω), in particular belongs to C 2 (Ω), we deduce that . Moreover, by residual approximation, we have (77) (R A (t, s); R n A (t, s)) ≤ Cs 2 , where the constant C > 0 only depends on Ω and A C 2,γ (Ω) . Now, for a fixed p ∈ N, we consider l = l p defined by 74. Thus, by an easy computation, we obtain where we have used 77 to get the term O(s 2 ). Here, the function d 0 and d 1 are defined by According to 78, the function b defined by 76 satisfies 75 whenever d 0 = d 1 = 0. In this way, since |a 1 | > 0 > 0 on ∂Ω \ Ω, we can divide by a 1 both sides of 79 to obtain b 1 ∈ C 2,γ . Once known b 1 , we divide by a 1 now both sides of 80 to obtain b 2 ∈ C 1,γ (Ω).

Equation for
r. Finally, we prove the existence of r ∈ H 1 (Ω). We claim that w defined by belongs to L 2 (Ω). More precisely, w L 2 (Ω) = O(τ −2 ). Indeed, we divide the analysis in two cases.
Analogously to the previous case, we also have On the one hand, since k(x) s(x) = dist(x, ∂Ω \ B) and s > τ −1/2 , we have On the other hand, we have Combining these inequalities, we deduce that |w(x)| ≤ C 18 τ −2 . This proves the claim. Hence, according to Proposition 8, there exists R ∈ H 1 (Ω) satisfying L A,q,−ϕ R = w, R| ∂Ω\B = 0 and the following estimate It is a simple matter to verify that r = e −iτ ξ·x R belongs to H 1 (Ω) and satisfies 11. This completes the proof of Proposition 1. As it was already mentioned, we now turn to give the main ideas to prove Theorem 5.5.
Second step. We briefly outline the main steps followed by Chung [6] to prove a Carleman estimate with a logarithmic weight, see his Theorem 1.4.

A1.
By means of a change to spherical coordinates, the starting point is the derivation of a Carleman estimate for a particular case, when Ω lies entirely in the upper part of a region determined by the graph of a positive real-valued smooth function f defined in S n−1 and such that E is a subset of the graph of f . Moreover, f is small enough in a suitable sense, see his Proposition 3.1. A2. The smallness condition over f is removed while still maintaining that E is a subset of its graph. This is done by means of a partition of unity argument applied to a finite open cover of Ω, see his Proposition 8.1. A3. Finally, by combining the previous results with arguments of compactness and cutoff functions, the graph condition over E is removed and so the proof of Theorem 1.4 is completed, see Section 8 in [6]. Let us explain with more details the step A1. Under a standard change of coordinates, Chung first flatten out the boundary of Ω, in particular the set E, to work in R n +1 = (θ, r) ∈ R n−1 × R : r ≥ 1 . Thus, he obtained a new second order operator defined by 3.5 in [6], instead of L 0,0,ϕ, . Due to the following result obtained by Dos Santos Ferreira et al. [8], see their Equation 2.12; the estimate 3.4 is satisfied for every limiting Carleman weight, in particular it holds for our linear case.
Proposition 9. If ϕ is a limiting Carleman weight in V , an open subset of R n , then there exist two positive constant τ 0 and C such that the following estimate τ −1 √ w H 1 scl (V ) ≤ C L 0,0,ϕ, w L 2 (V ) , τ ≥ τ 0 holds true for every w ∈ C ∞ 0 (V ). Now we return to [6]. We have noted that Lemma 4.1 and Lemma 4.2 hold true whenever the conditions 3.4-3.8 are satisfied. Combining these lemmas and changing coordinates back to the original ones, Chung ends the proof of Proposition 3.1, see pages 128-130. Just before changing back coordinates, we have detected the implicit result. This is a Carleman estimate in R n +1 contained on page 130. It can be stated as follows.
Lemma 5.6. Let L τ, be a second-order semiclassical operator on R n +1 = (θ, r) ∈ R n−1 × R : r ≥ 1 of the form (82) where F and G are smooth vector fields, ∂ r denotes the partial derivative with respect to r, ∇ θ denotes the gradient operator with respect to θ, a τ, is a smooth real-valued function and L θ is a second-order differential operator of the form L θ = a 1 ∂ 2 θ1 + a 2 ∂ 2 θ2 + . . . + a n−1 ∂ 2 θn−1 + first and zero order terms.
Here θ = (θ 1 , . . . , θ n−1 ) and (a j ) n−1 j=1 are smooth real-valued functions. Let U and U 2 be two bounded open sets on R n +1 \{e n } (e n denotes the n-th canonical unit vector of R n ) with smooth boundaries such that U U 2 and ∅ = ∂U ∩ ∂U 2 ⊂ ∂R n +1 . Finally, assume that there exist positive constants C, τ 0 , δ small enough; and K ∈ R n such that: (i) The operator L τ, satisfies the following estimate (83) τ −1 √ w H 1 scl (U2) ≤ C L τ, w L 2 (U2) , τ ≥ τ 0 , for all w ∈ C ∞ 0 (U 2 ). (ii) The coefficients of the operator L τ, satisfy for all τ ≥ τ 0 . Then there exist two positive constants C 1 and τ 1 such that , τ ≥ τ 1 , holds true for all w ∈ C ∞ 0 (U ). Using this result and following Chung's ideas according to step A1, we first obtain an initial Carleman estimate for our linear case, which is analogous to Proposition 3.1 in [6].
Changing back coordinates to the original ones, the proof is complete.
Finally, we mention that similar arguments employed in steps A2 and A3 also work in our linear case. That is, we can first remove the smallness condition for the function f and later, the graph condition for the set E. This completes the proof of Theorem 5.5.