Fractional Calder\'on problems and Poincar\'e inequalities on unbounded domains

We generalize many recent uniqueness results on the fractional Calder\'on problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calder\'on problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincar\'e inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.


Introduction
The primary objective of this article is to study the fractional Calderón problem for perturbations of the fractional Laplacian (−∆) s u := F −1 (|ξ| 2s û) where s ∈ R + \ Z.This is the nonlocal analog of the classical Calderón problem for the Schrödinger equation which appears in many imaging applications [Uhl09,Uhl14].The study of this type of fractional inverse problems was initiated by Ghosh, Salo and Uhlmann in [GSU20].In particular, we consider generalizations of the model problem where Ω e = R n \ Ω is the exterior of an open set Ω and is assumed to be nonempty.The associated Dirichlet to Neumann (DN) map to the exterior value problem (1), under certain conditions on q and Ω, is a bounded linear operator Λ q : H s (Ω e ) → (H s (Ω e )) * which acts under some stronger regularity assumptions than imposed in this article as Λ q f = (−∆) s u| Ωe (see [GSU20,Lemma A.1]).We need to impose some positivity or smallness assumptions on the perturbations q to guarantee that the exterior DN map is well-defined due to the lack of compact Sobolev embeddings on unbounded domains.The studied fractional Calderón problem can be simply formulated as follows: Given f → Λ q f , determine q.
We extend the recent uniqueness results in [CMR21,RS20a] for singular potentials and for lower order local linear perturbations [CLR20,CMRU20] to domains that are bounded in one direction.In fact, we prove a general result which applies also in the case of local higher order perturbations (i.e. the order of the perturbation is higher than the order of the fractional Laplacian) and for all domains with nonempty exterior.We also consider the setting of the fractional conductivity equation and the related Liouville transformation studied earlier in [Cov20b].The main contributions of our work are Theorem 2.8, which rather completely characterizes the uniqueness of the fractional Calderón problem for the conductivity equation in a general setting, and an extension of the usual methods on bounded domains to the case of nonlocal (or fractional) inverse problems of any order on unbounded domains (see Theorems 2.2 and 2.6).
The methods needed on unbounded domains allow us to slightly extend the uniqueness results of the fractional Calderón problem in the usual setting of bounded domains to larger classes of singular perturbations than in the scientific literature (see e.g.[CMR21,CMRU20,RS20a]).We refer to the survey [Sal17] for a more detailed but a bit outdated treatment of the topic.See also [CLL19, CLR20, Cov20a, Cov21, GLX17, GRSU20, LL19, LL22, LLR20, Li20a, Li20b, Li21, RS20b] and the references therein for other recent studies of the fractional Calderón problems.These include for example perturbations of nonlocal elliptic anisotropic operators, nonlinear perturbations and nonlocal perturbations among many other model problems as well as studies of the stability and reconstruction.Most recently, inverse problems for the fractional spectral Laplacian have been studied also on Riemannian manifolds [Fei21,FGKU21].We discuss some of these works with more details in the later sections.
The classical Calderón problem and related inverse problems have been studied on unbounded domains earlier.A historical reference is the work of Langer, which studies the inverse conductivity problem and geophysical applications for conductivities in a half-space under the assumption that the conductivity is depending only upon the depth [Lan33].The classical Calderón problem is studied in a half-space, e.g., for convex inclusions in [FI89] and in two dimensions for anisotropic conductivities in [APL05].These inverse problems are also studied in infinite slabs or similar unbounded structures in [CKS17,Ike01,KLU12,SW06] when n ≥ 3. Inverse problems for general second order elliptic equations are studied recently in [BR20].The above list is incomplete and more references can be found from the aforementioned works.In the classical Calderón problems on unbounded domains, one often demands some radiation condition for the solutions at infinity which are not present in the nonlocal formulations studied in this article.
The second goal of this article is to study Poincaré inequalities for the fractional Laplacian of any positive order s ≥ 0 in the local Bessel potential spaces.In the simplest form, we prove that for any bounded open set Ω ⊂ R n , n ≥ 1, 1 < p < ∞ and s ≥ 0 there exists a constant C(n, p, s, Ω) > 0 such that for all u ∈ C ∞ c (Ω).We obtain generalizations of this inequality to domains that are bounded in one direction.We do not impose any boundary regularity assumptions on Ω.We refer to the following surveys and expository articles on the fractional Laplacian, fractional Sobolev spaces and their applications [DNPV12,Kwa17,Kwa19,LPGea20,LSSW16].See also the books [BL76,DD12,Leo17,Ste70,Tri83,Tri95] covering nearly the whole spectra of classical methods related to the fractional order Sobolev spaces.
The Poincaré inequalities for the fractional Laplacian are important as they allow to show existence, uniqueness and boundedness theorems for weak solutions of PDEs (see e.g.[CMRU20, GSU20, ROS14a, ROS14b, RS20a]).The standard example is the fractional Laplace equation (−∆) s/2 u = F with the interior data F in some domain Ω ⊂ R n (see e.g.[RO16]).The cases when p = 2 appear naturally in the studies of nonlinear PDEs and the standard examples are different kind of p-Laplacians (see e.g.[BLP14, BS19, DNDS19, DPFBLR18, FSV15, FS08, FP14, LL14, MS21, MPSY16] and references therein).The Poincaré and other closely related inequalities for the fractional Laplacians are studied recently, for example, in [AFM20, BC18, CCRS21, DPFBLR18, HYZ12, LL14, MS21, Yer14].On recent studies and further references of the higher order fractional Laplacians, we point to [CMR21, CMRU20, ROS15].The higher order fractional Laplacian also appears in the studies of the Radon transform and imaging methods since the normal operator of the d-plane Radon transform for an integer 1 ≤ d < n can be identified with (−∆) −d/2 and the inverse operator can be defined using (−∆) d/2 (see e.g.[CMR21,Hel11,IM20]).
We state our main results on the fractional Poincaré inequalities in Section 2.1.We discuss our main results on the fractional Calderón problems in Section 2.2.We recall preliminaries in Section 3. We prove the results related to the generalized fractional Calderón problem in Section 4. We prove the results related to the fractional Poincaré inequalities in Section 5. We define some classes and decompositions of Sobolev multipliers and PDOs in Section 6, and these are used to obtain uniqueness results for concrete example models in Section 7. We study the fractional conductivity equation and the related inverse problem in Section 8. We prove some basic properties for products of functions in Bessel potential spaces in Appendix A as they are essential for our main results on the fractional conductivity equation and we could not locate all of them in the required form in the earlier literature.
Acknowledgements.The authors thank Prof. Mikko Salo for commenting an earlier version of the manuscript and suggesting changes.The authors thank Dr. Giovanni Covi for many fruitful discussions related to the topics of this article.J.R. was supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.

Main results of the article
2.1.Poincaré inequalities for the fractional Laplacians.Our first main result is the higher order fractional Poincaré inequality for domains that are bounded in one direction.We begin with the following definition.
(i) We say that an open set Ω ∞ ⊂ R n of the form Ω ∞ = R n−k × ω, where n ≥ k > 0 and ω ⊂ R k is a bounded open set, is a cylindrical domain.
(ii) We say that an open set Ω ⊂ R n is bounded in one direction if there exists a cylindrical domain Ω ∞ ⊂ R n and a rigid Euclidean motion A(x) = Lx + x 0 , where L is a linear isometry and x 0 ∈ R n , such that Ω ⊂ AΩ ∞ .
One could equivalently require that there exists a rigid Euclidean motion A and a cylindrical domain Ω ∞ ⊂ R n such that AΩ ⊂ Ω ∞ .In Section 5 we prove the following theorem.
We obtain this result and also information on the Poincaré constants using a generalization of the interpolation result of fractional Poincaré constants in [CMR21].The lower order cases 0 < t < 1 when p > 2 rely on the Poincaré inequalities for the Gagliardo seminorms (see e.g.[MS21, Theorem 1.2] and references therein).Unfortunately, we do not have a proof which would also settle the cases 0 < t < 1 and 1 < p < 2. We have stated and proved the fractional Poincaré inequality on bounded sets in Lemma 5.4 without these limitations on the integrability and Sobolev scales.See also the discussion of different methods of proof preceding the statement of Lemma 5.4.
We have the following result on the Poincaré constants.The proof is given in Section 5.
Theorem 2.3 (Interpolation of Poincaré constants).Let Ω ⊂ R n be an open set and 1 < p < ∞.Suppose that r > z, and t ≥ s ≥ r > z ≥ 0 or t ≥ r ≥ s ≥ z ≥ 0 hold.If there exists a fractional Poincaré constant C r,z > 0 with for all v ∈ H r,p (Ω), then we have for all u ∈ H t,p (Ω).
We believe that the constant in Theorem 2.3 remains optimal if C r,z is the optimal Poincaré constant.The resulting constant in (3) has the same scaling properties as the optimal constants and the constant cannot be improved by making the interpolation argument twice.Moreover, the Poincaré constants for Ht+2,p (Ω) give upper bounds for lower order Poincaré constants for all (−∆) Ht+2,p (Ω) ⊂ Ht,p (Ω), and if "(−∆) Ht+2,p (Ω) = Ht,p (Ω)" were true for all t > 0, then the optimality and the lower order cases when 1 < p < 2 would also follow by a short proof using the interpolation result.For these heuristical reasons, we have formulated the optimality of the constants as Conjecture 5.8.See also Remark 5.7 on a possibility to generalize the assumptions in Theorem 2.3.

2.2.
Generalized fractional Calderón problems.We next describe our results for the generalized fractional Calderón problems.We first state an abstract formulation of the argument given in [GSU20] which was later modified for lower order PDO perturbations in [CLR20,CMRU20].A related question is discussed in [RS20a, Remark, p. 7] by Rüland and Salo but not studied there further.The approach outlined in [RS20a] is more related to the work accomplished in [Cov21] by Covi for quasilocal perturbations of the fractional Laplacian.We then discuss concrete model problems with our focus in the inverse problem for the fractional conductivity equation studied first in [Cov20b].
We first introduce the related definitions for the abstract formulation.
Definition 2.4 (Properties of bilinear forms).Let s ∈ R and B : Our axiomatic formulation for the uniqueness of the generalized fractional Calderón problem comes with two important improvements in comparison to the existing theorems in this area.Firstly, we replace the assumption that Ω is a bounded set by the assumption that Ω has nonempty exterior.Secondly, we allow general bilinear forms in the statement.This let us to study problems where the local perturbation q has higher order than the nonlocal bilinear form L with suitable unique continuation properties.The third benefit of this general statement is that it explicitly states what kind of unique continuation properties are really needed for the nonlocal term L. Especially, we do not have to assume that bilinear forms are symmetric.A closely related approach was taken earlier in the work [Cov21] in the case of quasilocal perturbations but these and our results are not included into each other.We also prove an exterior determination result which does not appear in the earlier literature even in the simplest case of the fractional Schrödinger equation, to the best of our knowledge.The proofs together with the necessary definitions are given in Section 4.
Theorem 2.6 (Interior determination).Let s ∈ R, and 2, be local and bounded bilinear forms.Suppose that B L,q j = L + q j are (strongly) coercive in Hs (Ω).
If the exterior data We identify functions q ∈ L 1 loc (R n ) with the bilinear form q(u, v) = ´Rn quvdx for all u, v ∈ C ∞ c (R n ) and, in particular, one of the assumptions in Corollary 2.7 is that the related bilinear form is bounded in H s (R n ).For example, q ∈ L n 2s (R n ) when 0 < s < n/2 (see Lemma A.10), or q ∈ L 2p 2−p (R n ) when s = n/2 and any 1 < p < 2 (see Proposition A.9, [Oza95, p. 261]), or q ∈ L 1 (R n ) when s > n/2.The locality of the bilinear forms of this type follows automatically.The idea behind Corollary 2.7 turns out to give strong partial data uniqueness results for the inverse fractional conductivity problem studied in details in Section 8.
Corollary 2.7 (Exterior determination).Suppose that the assumptions of Theorem 2.6 hold.If additionally In particular, we have the following full data uniqueness result: If W = Ω e and q 1 , q 2 ∈ C(R n ) (or q 1 , q 2 ∈ L 1 loc (R n ) and ∂Ω has measure zero), then q 1 = q 2 (a.e.) in R n .Theorem 2.6 can be seen as a general uniqueness lemma for nonlocal exterior value inverse problems.It might be possible to obtain more abstract versions of Theorem 2.6 in general Hilbert or Banach spaces (especially on geometric settings on manifolds) but that is out of our scope here.One can remove the assumption that B is strongly coercive by assuming that the related interior and exterior value problems have unique weak solutions for B and its adjoint B * (cf.Lemma 4.1 and Remark 4.2).See also Theorem 4.3 for a general Runge approximation result which may be of interest also in other applications.The main point of the article is to explain what Theorem 2.6 together with the Poincaré inequalities (Theorem 2.2) and some additional analysis imply for many new model problems to be discussed next.We also give minor improvements for the regularity assumptions of many existing theorems.2.2.1.Fractional conductivity equation.We highlight our result for the fractional conductivity equation which rather completely characterizes the uniqueness and nonuniqueness of the related partial data inverse problem on domains that are bounded in one direction.Theorem 2.8 would also generalize to all domains with finite fractional Poincaré constant.This problem was studied earlier in [Cov20b] by Covi for bounded domains under the assumption that conductivities are trivial in the exterior.Here we relax the assumption that the conductivities take constant values (and are thus equal) in the exterior by making suitable decay assumptions at infinity.We also define here the Liouville reduction slightly differently than in the article [Cov20b] in order to have access to exterior determination and obtain more general results.As a minor improvement, we remove the superfluous Lipschitz assumption of the domain.
The fractional gradient of order s is the bounded linear operator where C n,s > 0 is a constant.This operator appears in the studies of nonlocal diffusion (see e.g.[DGLZ12] and references therein) and is naturally associated with the L 2 Gagliardo seminorm.We denote the adjoint of ∇ s by div s .Let γ ∈ L ∞ (R n ) be a positive conductivity.We denote the conductivity matrix by Θ γ (x, y) := γ 1/2 (x)γ 1/2 (y)1 n×n for x, y ∈ R n and by m γ := γ 1/2 − 1 the background deviation.We use the notation Λ γ for the exterior DN map associated to the equation div See the beginning of Section 8 for further details of the basic definitions.We note that Lemmas 8.13 and 8.15 state some interesting special cases of Theorem 2.8 (see also Remark 8.14).Our main theorem is the following.The proof is given in the end of Section 8 and it utilizes nearly all of our other results that are proved in the earlier sections.
Theorem 2.8.Let Ω ⊂ R n be an open set which is bounded in one direction and 0 < Then the following statements hold: Remark 2.9.There were three points which seem to be overlooked in [Cov20b].However, all arguments in [Cov20b] are easily justified for C ∞ c (Ω) perturbations of the uniform background conductivity.To cover a few missing steps in the low regularity setting, we establish regularity results related to the fact that the (fractional) Liouville transformation is bijective (cf.Theorem 8.6).In the proof, we need to approximate equations with equations having regular coefficients so that the used pointwise computation with a singular integral definition of the fractional Laplacian can be applied to the coefficients coming from the conductivity equation (this particular approximation uses results in [Sil05]).The second point is that we show that the Liouville transformation can be reversed from the solutions of the associated fractional Schrödinger equation to the solutions of a conductivity equation, so that the first mentioned problem has unique solutions, by the same property of conductivity equations, and has always a well-defined exterior DN map (in particular 0 is never a Dirichlet eigenvalue for the related exterior value problem).This uses an argument showing sufficient regularity of the division by γ 1/2 (employing [AF92]).It is rather clear that this property is desired so that the result applies for all conductivities from the given class without any open questions related to the well-posedness of the reduction to the fractional Schrödinger equation.Finally, we consider multiplication results for Bessel potential spaces in Appendix A by adapting the general methods in [BM01].This is done as the reference [BM01] used in [Cov20b] establishes the multiplication results with respect to the fractional Gagliardo norms but not directly for the Bessel potential norms as actually desired in [Cov20b] and our work.Nevertheless, the main results in [Cov20b] are correct whenever 0 < s < 1 and n ≥ 2, or 0 < s < 1/2 when n = 1, as the aforesaid gaps in the proof are made rigour in our article, but the cases 1/2 ≤ s < 1 when n = 1 are actually still open in low regularity to the best of our understanding.Otherwise, the strategy of our interior determination proof follows that of [Cov20b] whereas the exterior determination requires yet another argument, given in Section 8.3.
This result is interesting for several reasons to be discussed next.Since the problem studied here is considerably more general than the one in [Cov20b], nonuniqueness may occur in some cases when W 1 ∩ W 2 = ∅.A striking difference to the earlier results for the fractional Calderón problems is that when W 1 ∩W 2 = ∅, then this actually helps to solve the inverse problem.In this case, one obtains a global uniqueness result also in Ω e without assuming that conductivities agree there in the first place.The Liouville reduction is analogous to the one for the classical Calderón problem [SU87] but we do not have to use any kind of special solutions in the end.In the proof of Theorem 2.8, we do not reduce the inverse problem only back to the Calderón problem for the fractional Schrödinger equation like in [Cov20b] but also invoke a new characterization of equal exterior data and the UCP of fractional Laplacians.This part resembles the earlier studies which avoid using the Runge approximation and rather directly rely on the UCP to show uniqueness in the fractional Calderón problem (see e.g.[GRSU20] for single measurement results).
The exterior determination result is not possible with only finitely many measurement whereas interior determination is possible with just single measurement whenever the conductivity is known in the exterior.Exterior determination can be thought as an anolog of boundary determination in the classical Calderón problem [KV84], which is an important step in the proof of uniqueness [SU87].However, we do not have a complete exterior determination result and we still need to rely to the fact that conductivities are assumed to be constant in the sets where we make our exterior measurements (this is the content of the formula (4)).A more complete answer to the exterior determination problem remains open.We remark that when the exterior conditions (fractional voltage) are set disjointly with respect to the set where the measurements (fractional current) are made, we may have the loss of uniqueness (see the article [RZ22] for constructions of counterexamples).In the classical Calderón problem there are also uniqueness results for some partial data problems which are analogous with Theorem 2.8 (i) (see [KSU07,Corollary 1.4]).
Below in Figures 2.1 and 2.2 we graphically illustrate the two possible geometric settings in Theorem 2.8 corresponding to the cases (i) and (ii), respectively.2.2.2.Fractional Schrödinger equations.Later in this article we present many applications of Theorem 2.6 in which we always fix L = (−∆) t (in the weak form) for some t ∈ R + \ Z with t ≤ s.One could also take many other bilinear forms in the place of L, for example: • One can take L to be the bilinear form related to the Bessel potential D t whenever t ∈ R \ Z [KPPV20, Theorem 1.6 and Remark 1.7].• One can take L to be the bilinear form related to the nonlocal operator L t where t ∈ (0, 1) and L is an elliptic second order operator  Remark 2.10.We note that this does not imply that the methods used in this article would solve the inverse problems studied in [Cov21] since the role of q in our work is more restricted.This only implies that the inverse problems for local perturbations of the fractional Schrödinger operators used in [Cov21] could be studied directly with Theorems 2.2 and 2.6.One would have to refine Theorem 2.6 from local perturbations into the direction of nonlocal perturbations to have access to the fractional Calderón problem for quasilocal perturbations on unbounded domains.The generalization to the case of perturbations with finite propagation should be now a rather easy problem to solve but the case of quasilocal perturbations would require one to be more careful.We do not pursue this matter here as this would put us too far away from the proof of Theorem 2.8, and our work and the work [Cov21] were put through independently around the same time.
We apply Theorems 2.2 and 2.6 to prove uniqueness results for the fractional Calderón problems on domains that are bounded in one direction.The methods needed to bypass the lack of compact Sobolev embeddings on unbounded domains turn out to be useful also in the case of bounded domains and allow us to slightly extend the recent results in [RS20a, CMR21, CMRU20] in the usual setting of bounded domains.
Rather than introducing right now all new concrete model examples where our methods apply, we only point to the following results that are stated and proved later in this article: • Remark 4.5 explains how Corollary 2.7 on exterior determination extends to the case of PDO perturbations with variable coefficients.• Proposition 4.6 gives example model classes with higher order local perturbations (i.e. the perturbations have higher order than the fractional Laplacian) for all domains that have nonempty exterior and the fractional Calderón problem is uniquely solvable.Especially, these domains are not assumed to be bounded in one direction.
Graphical illustration of the geometric setting of part (ii) of Theorem 2.8.Here the set Ω ⊂ R n is a domain which is bounded in one direction and the measurements are performed in the disjoint nonempty open subsets W 1 , W 2 ⊂ Ω e .Moreover, the supports of the background deviations m 1 , m 2 , which are represented in orange and red, respectively, do not necessarily coincide in the exterior Ω e and can even be disjoint as illustrated above.In this case uniqueness may be lost.
• Theorem 7.8 gives example models with linear local lower order perturbations on domains that are bounded in one direction.• Theorem 7.11 gives further example models on domains that are bounded in one direction in the case where the perturbations are functions rather than operators.• Theorems 7.10 and 7.13 give slight generalizations of the existing theorems on bounded domains.Needless to say, there are many other situations where the combination of existing results and the ideas presented in this article allow various generalizations for the fractional Calderón problems, the Runge approximation results and related problems.More interesting research questions, mathematically speaking, would be to understand if the single measurement, stability, quantitative Runge approximation, and nonlinear and nonlocal perturbation results [BGU21, Cov21, CGFR21, GRSU20, LL22, RS20a] have some extensions to the cases of unbounded domains or higher order perturbations, where higher order perturbations are understood in the sense that the local part has higher order than the nonlocal part of an equation.

Preliminaries
We introduce in this subsection key definitions and basic notation.More sophisticated definitions and notation are given in later subsections when needed.

Spaces of distributions.
Throughout the article we use the convention that n ≥ 1 and Ω ⊂ R n is an open set.We denote by S (R n ) and S (R n ) Schwartz functions and tempered distributions respectively.We say that f We let D (Ω) stand for the space of distributions on Ω and by E (Ω) the space of compactly supported distributions on Ω.
We define the Fourier transform which is occasionaly also denoted by f .By duality it can be extended to the space of tempered distributions and will again be denoted by Fu = û, where u ∈ S (R n ), and we denote the inverse Fourier transform by F −1 .

and endow it with the norm
We mainly use them in Section 5 in which we recall recent results on the Poincaré inequalities on certain subspaces of W s,p (R n ) and Section 8 dealing with the fractional conductivity equation but for the sake of presentation this part of the proof is done in Appendix A.
Next we introduce the Bessel potential spaces H s,p (R n ) and various notions of local Bessel poential spaces which we introduce next.We define the Bessel potential of order s ∈ R as a Fourier multiplier We define the Bessel potential spaces for any 1 ≤ p ≤ ∞ and s ∈ R by open and closed sets respectively, 1 ≤ p ≤ ∞ and s ∈ R. We define the following local Bessel potential spaces: The spaces H s,p (Ω) and H s,p F (R n ) are equipped with the H s,p (R n ) norm.The space H s,p 0 (Ω) is equipped with the H s,p (Ω) norm.As usual we set H s := H s,2 .
Finally, we define the fractional Laplacian of order s ∈ R + as a Fourier multiplier for u ∈ S (R n ) whenever the formula is well-defined.In particular, the fractional Laplacian is a well-defined bounded mapping (−∆) s/2 :

Generalized fractional Calderón problem
In this section, we formulate and prove a uniqueness result for a generalized fractional Calderón problem.The structure of our argument follows similar ideas as introduced by Ghosh, Salo and Uhlmann [GSU20] and later modified in [CLR20,CMRU20] to obtain uniqueness results for some special cases of nonsymmetric local perturbations of the nonlocal Schrödinger equation.The axiomatic approach to be taken here has several benefits.We observe that the argument can be applied on all domains with nonempty exterior, and in general, the argument works whenever the direct problem is well-posed.One can even consider and prove uniqueness results for some higher order local perturbations of the fractional Laplacian.Either of these two examples, unbounded domains or higher order perturbations, do not appear in the earlier literature to the best of our knowledge.These general considerations in part motivate the rest of article.In later sections of this article, we will study fractional Poincaré inequalities on unbounded domains and the related decompositions of Sobolev multipliers and PDOs to be able to provide new nontrivial examples where the theory presented in this section applies.
Then the following hold: (i) For any f ∈ H s (R n ) and F ∈ ( Hs (Ω)) * there exists a unique u ∈ H s (R n ) such that u − f ∈ Hs (Ω) and B(u, φ) = F (φ) for all φ ∈ Hs (Ω).When F ≡ 0, we denote this unique solution by u f .(ii) If f 1 −f 2 ∈ Hs (Ω) and u i are solutions to the problems is a well-defined bounded linear map.(iv) The above statements hold true for the adjoint bilinear form B * and the corresponding unique solutions, when F ≡ 0, are denoted by We remark that the assumption on the coercivity of B can be replaced by the assumption that the exterior and interior value problems have unique solutions depending continuously on the data.If we similarly assume that this is true for the adjoint problem, then the presented proofs work without any changes.This applies in the whole section.
Proof.(i) Let f ∈ H s (R n ), F ∈ ( Hs (Ω)) * and set ũ := u − f .By linearity the problem is reduced to finding the unique ũ ∈ Hs (Ω) such that B(ũ, φ) = F (φ) for all φ ∈ Hs (Ω) where F := F − B(f, •).Observe that the modified functional F belongs to ( Hs (Ω)) * , since there holds for all φ ∈ Hs (Ω).The Lax-Milgram theorem implies that there exists a bounded linear operator and it is the required unique solution ũ ∈ H s (Ω).Moreover, there holds (ii) Since f 1 − f 2 ∈ Hs (Ω) there exists φ ∈ Hs (Ω) such that f 2 = f 1 + φ and the function u := u 2 − u 1 ∈ Hs (Ω) solves the homogenous problem with exterior value φ.Since w = 0 is also a solution to this problem we obtain by uniqueness of solutions that u 1 ≡ u 2 in R n .
(iii) By (i) and (ii), we have for some C > 0. This implies that for some C > 0 it holds that since the solutions are unique and u f 1 + u f 2 solves the homogeneous equation with the exterior condition Therefore we have shown that Λ B : X → X * is a bounded linear operator.
(iv) Notice that B * is also a bounded bilinear form with the same coercivity estimate.Therefore (i), (ii) and (iii) hold when B is replaced by B * .Suppose that u f solves B(u f , •) = 0 in Hs (Ω) with u f − f ∈ Hs (Ω), and be bounded bilinear forms and assume that q is local and that B L,q := L + q is (strongly) coercive in Hs (Ω).
(i) If L has the right UCP on a nonempty open set W ⊂ Ω e , then R(W We only prove here (i), since the proof of (ii) is completely analogous.
Since R(W ) ⊂ Hs (Ω) is a subspace it suffice by the Hahn-Banach theorem to show that if F ∈ ( Hs (Ω)) * vanishes on R(W ), then F is identically zero.Hence, let F ∈ ( Hs (Ω)) * and suppose that Since φ| W = 0, we have by the right UCP that φ ≡ 0 and therefore F ≡ 0.
In Figure 4.3, we give a concrete example model where Theorem 4.3 applies and the domain is not bounded in one direction.

B ((−∆)
. This means heuristically speaking that all functions in H s (R n ) are almost s-harmonic up to an error depending on δ, > 0. This observation is analogous to the celebrated result on bounded domains [DSV17] where one may take δ = 0, but this possible choice of δ is also true on domains that are bounded in one direction.We do not perform a quantitative analysis of this phenomenon in this article.
Lemma 4.4 (Alessandrini identity).Let s ∈ R, and Ω ⊂ R n be an open set such that Ω e = ∅.
2, be bounded bilinear forms that are (strongly) coercive in Hs (Ω).Then Notice that the solutions u f and u * g exist by Lemma 4.1.We can directly calculate using (iv) of Lemma 4.1 that We can now prove the uniqueness theorem for the generalized fractional Calderón problem (Theorem 2.6).
Proof of Theorem 2.6.Using the Alessandrini identity (Lemma 4.4) we have . By the Runge approximation property (Lemma 4.3), we have that there exists sequences ) and v, v * ∈ Hs (Ω).Since q 1 , q 2 are bounded bilinear forms and local, we find that where the first identity follows from the boundedness of q 1 − q 2 , the second identity follows from the support conditions and the locality of q 1 − q 2 , and the last identity follows from the assumption that the exterior DN maps are equal.This implies that q 1 = q 2 in Hs (Ω) × Hs (Ω).
Let us now take f, g ∈ C ∞ c (W ).We have by the Alessandrini identity (Lemma 4.4) and the assumption that Λ L,q The rest is a simple calculation and an approximation argument.Let us write that B := q 1 − q 2 and calculate where we used that u f − f, u * g − g ∈ Hs (Ω) so that B(u f − f, u * g − g) vanishes by interior determination, and B(u f − f, g) and B(f, u * g − g) vanish by the locality of B. This shows that q 1 (f, g) = q 2 (f, g) for all f, g ∈ C ∞ c (W ).Approximation gives that q 1 = q 2 in Hs (W ) × Hs (W ).This completess the proof of the first part of the statement.
We next prove the conclusions about the full data case.We fix now that W = Ω e .Suppose first that q 1 , q 2 ∈ L 1 loc (R n ).Now we have that ´Rn q 1 φdx = ´Rn q 2 φdx for all φ ∈ C ∞ c (Ω ∪ Ω e ) by interior and exterior determination.This implies that q 1 = q 2 a.e. in Ω ∪ Ω e .If ∂Ω has measure zero, then q 1 = q 2 a.e. in R n .On the other hand, if q 1 , q 2 ∈ C(R n ), then we find first that q 1 = q 2 in Ω ∪ Ω e , but this must then also hold in R n = Ω ∪ Ω e by the continuity without assuming anything about the regularity or measure of ∂Ω.
Remark 4.5.Corollary 2.7 also applies in the case of local linear perturbations studied in Section 7. One could obtain similar exterior and full data results as stated in the case of zeroth order perturbations in Corollary 2.7.The argument first uses this general result for bilinear forms, and then similar test functions as used in the proof of Theorem 7.8 (but now for the exterior conditions).Finally, the full data case can be concluded either using continuity or basic measure theory.This would not involve any new ideas beyond these and the details are thus omitted.
We give the following simple examples as corollaries right now.Both results given in the examples are new but we do not aim to write the most general examples into this direction here.We also remark that the generalized Runge approximation result in Lemma 4.3 may find applications in other problems on unbounded domains.We study more sophisticated examples in Section 7. (i) (Domains without Poincaré inequalities) For (−∆) s + q in Ω where s ∈ R + \ Z and the potential q is uniformly positive and bounded, i.e. q ∈ L ∞ (R n ) is such that q ≥ c > 0 a.e. in Ω for some c(q) > 0. (ii) (Higher order perturbations) For (−∆) t + (−∆) s/2 (γ(−∆) s/2 ) + q in Ω where t ∈ R + \ Z, s ∈ 2Z and t < s, and γ, q are uniformly positive and bounded in Ω.
Remark 4.7.In the domains, with finite fractional Poincaré constants, one could formulate results for more general local higher order perturbations.If additionally the domain is bounded, then the spectral theory would allow to consider a large class of local higher order perturbations as long as a weaker coercivity estimate can be established.We note that the problem must then be set in the space whose Sobolev scale depends on the local perturbation rather than the order of the fractional Laplacian.The key observation is that still the fractional Laplacian permits the uniqueness results for the related inverse problems whenever the forward problems are well-posed.
Proof.(i) The associated bilinear forms are L(u, v) = (−∆) s/2 u, (−∆) s/2 v and q(u, v) = qu, v .It follows that the related bilinear form L + q is equivalent with H s (R n ) norm in Hs (Ω).It follows from [CMR21, Theorem 1.2] and [GSU20, Theorem 1.2] that L has the symmetric UCP on any open set, and by the definition q is local.Hence, Theorem 2.6 applies.
(ii) We have now that L = (−∆) t/2 u, (−∆) t/2 v has again the symmetric UCP on any open set for the functions in The bilinear form related to (−∆) s/2 (γ(−∆) s/2 ) + q is given by It is clear that this bilinear form is also bounded in H s (R n ) × H s (R n ).The coercivity estimate is also easy to argue since Q is (strongly) coercive in Hs (Ω) and L(v, v) ≥ 0 for any v ∈ H s (R n ).Since s/2 ∈ Z, we have that the bilinear form Q is local, and L has the symmetric UCP on any open set as earlier.Again, the rest follows from Theorem 2.6.

Interpolation of fractional Poincaré constants on unbounded domains
The purpose of this section is to extend the fractional Poincaré inequality to open sets which are bounded in one direction as introduced in Definition 2.1.The main result of this section is the proof of Theorem 2.2.The Poincaré inequalities on these sets allow us to come up with concrete nontrivial example models that satisfy the assumptions of Theorem 2.6.
We first recall a recent result on the fractional Poincaré inequalities when 1 ≤ p < ∞ and 0 < s < 1 in [CCRS21,MS21].They study the Poincaré constant via the variational problem , where in the notation used in this article.The fractional Poincaré inequalities with respect to the Gagliardo seminorms are not equivalent with the Poincaré inequalities for the fractional Laplacian when p = 2.However, there exist the following embeddings when 0 < s < 1 (see e.g.[Tri83,p. 47 Proposition 2 (iii)/(9), pp.51-52, p. 242 Theorem 2 and Remark 3, and Section 5.2.5] and [BL76, Theorems 6.2.4,6.3.2 and 6.4.4]where one identifies W s,p (R n ) = B s pp (R n ) as a Besov space and with some constants C > 0 independent of the choice of a function from the left hand side spaces.
The last two embeddings are the same as continuity of the associated fractional Laplacian operators, in H s,p (R n ) the operator does not depend on a particular choice of p and always coincides with the Fourier multiplier definition of (−∆) s/2 .
We remark that it is well-known that in open bounded sets one has finite Poincaré constants for the Gagliardo seminorms.See also [AFM20, Theorem 1.1] where the result of Theorem 5.1 was obtained when p = 2 by considering asymptotics of the fractional Poincaré constants for domains that are bounded in one direction.The fractional Hardy inequalities on (possibly unbounded) convex sets [BC18, Theorem 1.2] also imply that P n,s,p (Ω) > 0 whenever Ω is bounded in one direction.See also the related work [Yer14] proving that P n,s,2 (Ω) > 0.
Theorem 2.3 is a generalization of a recent interpolation result for the fractional Poincaré constants of the fractional Laplacians in [CMR21, Theorem 3.17].Their result relates the classical Poincaré constant for the gradient in L 2 to the Poincaré constants for pairs different order fractional Laplacians.In comparison to the result in [CMR21], we observe here that a similar result holds also on unbounded domains, without reducing the problem to the classical Poincaré constants, and for any 1 < p < ∞.The proof of Theorem 2.3 is given in the end of this section.
We can now prove Theorem 2.2 using Theorems 5.1 and 2.3.The case when s ≥ 1 uses the boundedness of Riesz transformation.We do not know a proof which would settle the cases 1 < p < 2 and 0 < s < 1.We also give one alternative proof when p = 2 as its structure may be relevant for the future attempts to prove the missing special cases.
We first recall a simple lemma due to the absence of a reference.
Lemma 5.2.Let A : R n → R n be a rigid Euclidean motion A(x) = Lx + x 0 where L is a linear isometry and Proof.By a simple change of variables it is easy to see that (6) implies On the other hand, the formula (6) follows by a direct calculation using the symbols and a change of variables.In fact, and Therefore, without loss of generality, we will often work directly with the cylindrical domains in the proofs.
Proof of Theorem 2.2 when p ≥ 2. By Remark 5.3 it suffices to prove the assertion for cylindrical domains Ω = R n−k × ω, where ω ⊂ R k is an open bounded set and n ≥ k > 0.Moreover, by Theorem 5.1 and the fact that P k,r,p (ω) > 0 we know that the assumptions of Theorem 2.3 hold for the parameters z = 0 and 0 < r < 1.This follows rather directly from the embeddings (5) between the (in)homogenenous Besov and Triebel spaces when p ≥ 2. We are free to choose any t ≥ s ≥ r to find that where C r,0 := CP k,r,p (ω) −1/p > 0, where C > 0 is the operator norm of the embedding Ḣr,p (R n ) → Ẇ r,p (R n ).Since r > 0 can be taken as small as possible, this proves Theorem 2.2 for any t, s > 0 with t ≥ s.
On the other hand, when s = 0, we have that r > s = z and we are free to choose any t ≥ r in Theorem 2.3.This proves Theorem 2.2 in the case t > 0 and s = 0.
Proof of Theorem 2.2 when 1 < p < 2, s ≥ 1.Notice that the boundedness of the Riesz transform gives ∇u . By the classical Poincaré inequality, we know that there exists some C > 0 such that Now the proof continues as the earlier proof of Theorem 2.2.
Next we recall and prove the fractional Poincaré inequality on bounded sets.We make a few remarks about it now.One could replace the argument using the Hardy-Littlewood-Sobolev inequality by the embedding theorem of homogeneous Triebel-Lizorkin spaces [Han95, Theorem A].One could replace the interpolation argument by an iteration with integer orders and monotonicity arguments, using certain special inequalities for the critical cases (tp = n) [Oza95, p. 261] and Hölder estimates for homogeneous convolution kernels in the supcritical cases (tp > n) [H 03, Theorem 4.5.10](see [CMR21, Theorem 3.14] for this proof in the simpler special case p = 2).One can also prove the fractional Poincaré inequality on bounded sets using the compact Sobolev embeddings (see e.g.[ARS21, Lemma 3.3] and the related discussion for the outline) but we are not sure if this directly works for all possible parameters (i.e. in the critical and supcritical cases) without interpolation or iteration.One can also obtain some stronger statements, at least in many special cases, using the maximum principle (see [ROS14b,Proposition 1.4] and [ROS14a, Claim 2.8]).
Proof.Let us first assume that 0 < s < n and 1 < p < n/s.It follows from the Hardy-Littlewood-Sobolev inequality and Hölder's inequality that for any u ∈ C ∞ c (Ω) we have for some C (p, q, Ω), C(n, s, p) > 0 where q = np n−sp .This implies that for any 1 < p < ∞, there exists some > 0 such that for all s ∈ (0, ).It follows now from the interpolation of fractional Poincaré constants (Theorem 2.3) that there exists C(n, s, p, Ω) > 0 such that holds for any s ≥ 0 and u ∈ Hs (Ω).This shows the fractional Poincaré inequality on bounded sets.
The rest follows from Theorem 2.3, or alternatively from the monotonicity of Bessel potential spaces and boundedness of the fractional Laplacian.
We can now give an alternative proof of Theorem 2.2 when p = 2.We find this to be a quite natural approach for the problem but there is an issue which we do not know how to overcome when p = 2.
Alternative proof of Theorem 2.2 when p = 2. Suppose for simplicity that Ω = R n−1 × (a, b) for some a < b and let us denote points x in Ω by x = (x , x n ).By Fubini's theorem and the fractional Poincaré inequality for bounded domains we deduce that there exists C(s, p, a, b) > 0 such that for any s ≥ 0 and u ∈ C ∞ c (Ω).We have that ˆRn−1 We need an estimate for some C(n, s, p) > 0. This holds with C = 1 when p = 2, since the related multiplier A slight modification of this argument also applies for cylindrical domains Ω = R n−k × ω, which in turn gives the desired result for open sets which are bounded in one direction.In fact, the above proof establishes an estimate for the optimal Poincaré constants, which reads C n (s, Ω) ≤ C k (s, ω).(We know actually that equality holds because of Theorem 5.1 but this gives an alternative proof for the upper bound when p = 2.) Remark 5.5.The multiplier (7) is also homogeneous of degree zero but fails to be smooth when restricted to S n−1 , which is the main obstruction to generalize this natural argument to cover also the cases p = 2.
We now move on towards proving Theorem 2.3 but first we recall an interpolation result for the fractional Laplacians in [BL76].The result is stated also when p = 2 in [BCD11, Proposition 1.32] but we apply the following more general version in the proof of Theorem 2.3.Lemma 5.6 ([BL76, Exercise 6.8.31]).Let n ≥ 1, 1 < p 0 , p 1 < ∞, s 0 , s 1 ∈ R and 0 ≤ θ ≤ 1.If s θ = (1 − θ)s 0 + θs 1 and 1 Sketch of the proof.As this formula is stated without a proof in [BL76], we give a brief explanation here.See also [AF92, Lemma 2.1] for this statement and a brief outline for an alternative proof.
By the complex interpolation method in homogeneous Triebel-Lizorkin spaces Ḟ s p,q we know [ Ḣs 0 ,p 0 We remark that Lemma 5.6 in particular holds for all u ∈ H s Proof of Theorem 2.3.We set p = p 1 = p 2 in Lemma 5.6 so that the other condition is automatically satisfied.We assume in the proof that u ∈ C ∞ c (Ω).The result follows from this by approximation.
The first case: t ≥ s ≥ r > z ≥ 0. The case s = t is trivial and so we can assume t > s.First of all recall that u Ḣs,p (R n ) = (−∆) s/2 u L p (R n ) for all u ∈ Ḣs,p (R n ), s ≥ 0.
(i) If we define s 0 = z, s 1 = t, θ = r−z t−z ∈ (0, 1), then By Lemma 5.6 and (2) we obtain (ii) Now, let s 0 = r, s 1 = t, θ = s−r t−r ∈ [0, 1) and therefore As before we get the estimate and therefore we have shown (3) for u ∈ C ∞ c (Ω).It follows from approximation that the same estimate holds in H t,p (R n ) by the continuity of the fractional Laplacian.
The second case: t ≥ r ≥ s ≥ z ≥ 0 and r > z.Again without loss of generality we can assume t > s.We can additionally assume that r > s since when r = s, we would have that t > r = s > z and this case was already proved in the first part of the proof.This observation is important for the following proof to be valid.
(i) We apply the interpolation result to s 0 = s, s 1 = t, θ = r−s t−s ∈ (0, 1) and By Lemma 5.6 we obtain (ii) This time we take s 0 = z, s 1 = r, θ = s−z r−z ∈ [0, 1) and therefore Hence, we obtain and we can conclude for u ∈ C ∞ c (Ω).The general case again follows by approximation.Remark 5.7.Theorem 2.3 and Lemma 5.6 have their natural counterparts for the Gagliardo seminorms when 0 Sections 2.4.7 and 5.2.5]).The proof of Theorem 2.3 works just as well in this case.In particular, this implies that it is enough to establish the Gagliardo seminorm Poincaré inequality only for all t ∈ (0, ) with some > 0 as this already implies the Gagliardo seminorm inequalities for the rest of the cases ≤ t < 1.The argument works on arbitrary subsets of W t,p (R n ) having some a priori given inequality with respect to the Gagliardo seminorms as the proof only applies generic properties of all functions in the homogeneous spaces, the monotonicity of the inhomogeneous spaces and the boundedness of the seminorms when t ≥ 0. The same applies for the spaces H t,p (R n ).Only for the sake of simplicity, we use the spaces Ht,p (Ω) in the statement of Theorem 2.3.Also the Poincaré-Sobolev type estimates in the mixed integrability cases (p 1 , p 2 ) might be of interest in other applications but not for us.These are quite possible modifications of the argument as the interpolation result is actually stronger than what we use in the proofs (since we set p = p 1 = p 2 ).
Theorem 2.3 makes us to suggest the following conjecture on the equivalence of the fractional Poincaré constants.
Conjecture 5.8 (Equivalence of the optimal fractional Poincaré constants).Let Ω ⊂ R n be an open (bounded) domain and 1 < p < ∞.If C r,z is the optimal fractional Poincaré constant for r > z ≥ 0, then C t,s = C t−s r−z r,z is the optimal Poincaré constant for t > s ≥ 0.

Decompositions of Sobolev multipliers and PDOs
In this section, we define classes of local perturbations to the fractional Schrödinger equations using the Poincaré inequalities.These classes of perturbations will naturally satisfy the assumptions of Theorem 2.6 in different settings.
6.1.Basic notation and definitions.We define the spaces of Sobolev multipliers between Bessel potential spaces H s in this section.For more details, we point to the article [CMRU20, Section 2].For r, t ∈ R, we define the space of multipliers M (H r → H t ) between pairs of Bessel potential spaces according to [MS09,Ch. 3].Eventually we will study a generalization of the problem (1) where the potential q is replaced by a partial differential operator P (x, D) having its coefficients in the spaces of Sobolev multipliers. If We can analogously define the unique adjoint multiplication map m * f : It is easy to verify that the adjoint of m f is m * f .We will just write f u for both m f (u) and m * f (u).
6.2.Decompositions of Sobolev multipliers.Now we introduce some subspaces of Sobolev multipliers M (H r → H t ) to be used later in the context of fractional Calderón problems.We define for any open set Ω ⊂ R n which is bounded in one direction and s > 0 the quantity , where C(s, Ω) is the optimal Poincaré constant on Ω (see Theorem 2.2).We do not write the dependence of δ on s > 0 explicitly, but it must be noted.We have if we choose t = 1 C(s,Ω)+1 ∈ (0, 1).This calculation was presented here to motivate the definition of suitable classes of perturbations for the fractional Schrödinger equation such that the fractional Laplacian dominates the perturbations in the estimates, especially, from below.This allows to show the existence and uniqueness of weak solutions for the related perturbed fractional Schrödinger equations even when one lacks of compact Sobolev embeddings.
where ω k is the volume of the unit ball in R k .This gives a quantitative lower bound δ(Ω) 2 ≥ 2 −s (1 + c(s, Ω)) −1 > 0. This estimate could be used to define more concrete subclasses of Sobolev multipliers and PDOs within the classes which we let to depend directly on the Poincaré constants.
Definition 6.2 (Classes of Sobolev multipliers).Let C > 0, s > 0 and κ ≥ 0. We define the space of small multipliers as and the spaces of nonnegative multipliers as Remark 6.3.One can generalize the example models in Proposition 4.6 to the settings q ∈ M ≥κ (H s → H −s ) or γ ∈ M ≥κ (L 2 → L 2 ) for any κ > 0. Definition 6.4 (Multipliers close to M 0 ).Given a domain Ω ⊂ R n with a finite Poincaré constant of order s and integrability scale p = 2.We define the δ-neighbourhood of M 0 as follows It follows from the definitions that In particular, if there were a decomposition "M = M 0 + M ≥0 + M δ(Ω) " for a domain Ω, then one would not have any restrictions in the class of multipliers on bounded sets in the theorems of Section 7.This is true if one can show that every Sobolev multiplier in M (H s → H −s ) can be approximated by C ∞ c (R n ) after changing the multiplier a little bit and possibly removing an arbitrary nonnegative part.See the following remarks for further discussion.
Remark 6.5.The characterization of M 0 (H r → H t ) is an open problem and the authors are not even aware if M 0 (H r → H t ) M (H r → H t ) holds (see also [RS20a, Remark 2.5] and the related results in the special cases t = −r.).Remark 6.6.Let s > 0. The following are equivalent: (i) M (H s → H −s ) = M Ω (H s → H −s ) for all balls, i.e.Ω = B r (0), r > 0; (ii) M (H s → H −s ) = M 0 (H s → H −s ).This follows since the Poincaré constant on bounded domains grows like d s when d = 2r → ∞, and thus δ(B r (0)) → 0 as r → ∞.
If for some R ≥ 1 there exists . This follows as in the best possible case 1 > δ(Ω) ≈ 1 when C(s, Ω) ≈ 0. Remark 6.7.In mathematical terms, to have unique weak solutions (modulo the discrete spectrum) for the fractional Schrödinger equation with a potential q ∈ M (H s → H −s ) studied in Lemma 7.9 of Section 7.2, it would be sufficient to show that for any a ∈ M (H s → H −s ) there exists a ∈ M 0 (H s → H −s ) such that a − a s,−s < δ(Ω), which is certainly a weaker condition than M (H s → H −s ) = M 0 (H s → H −s ).In the works [CMR21,RS20a], one restricted to take the potentials from M 0 (H s → H −s ) rather than from the larger class M Ω (H s → H −s ).The benefit of this is of course that the class of perturbations in [CMR21,RS20a] do not depend on the domain Ω. 6.3.PDOs with Sobolev multiplier coefficients.Let m ∈ N 0 .We denote PDOs of order m as where the coefficients are chosen appropriately.In order to prove existence and uniqueness of weak solutions on domains that are bounded in one direction, we make a suitable smallness assumption on P due to the lack of compact Sobolev embeddings on unbounded domains Ω = R m × ω.
More precisely, we first calculate that Using this calculation as our guide, we know how to choose the coefficients a α .This motivates the next definition, which gives the smallness condition that guarantees the existence of unique weak solutions on domains that have finite Poincaré constants.
Definition 6.8 (Multiplier norms of PDOs).Let Ω ⊂ R n be open with a finite fractional Poincaré constant of any order s > 0. We define the subspace of small PDOs of order (m, s) ∈ N 0 × R + as follows where the norm of PDOs is Remark 6.9.Note that by Theorem 2.2 the set P m,s (Ω) is well-defined if s > 0 and Ω is an open set which is bounded in one direction.Moreover, we remark that in the articles [CMRU20], where the fractional Calderón problems were studied, one had to restrict to coefficients of PDOs or potentials from M 0 (H r → H t ).In our case of unbounded domains, the necessary smallness assumption avoids making this additional assumption and sufficiently small Bessel potential H s,∞ coefficients in [CMRU20, Theorem 1.2] belong the class we are using here (see the related estimate [CMRU20, eq. ( 22) in the proof of lemma 4.1]).

Fractional Calderón problems on unbounded and bounded domains
In this section, we study fractional Calderón problems on domains that are bounded in one direction and begin to employ the tools introduced in the earlier sections.We remark that the arguments in this section follow closely those of [CLR20, CMR21, CMRU20, GSU20] and the emphasis is to explain the points were the differences occur.In Section 7.1, we briefly recall the basic structure of the argument for the sake of completeness and relate this to the abstract framework of Section 4. We note that this approach simplifies and generalizes the original uniqueness proofs in [CLR20,CMRU20].
7.1.Small local linear perturbations.Consider the problem and the corresponding adjoint problem The problems ( 9) and ( 10) are associated with the bilinear forms Notice that definition of the bilinear forms do not explicitly depend on the particular choice of Ω, but their quantitative properties like coercivity and boundedness will depend on Ω.Later the domain Ω only shows up in the class of test functions Hs (Ω), the interior data F, F * ∈ ( Hs (Ω)) * , and the exterior data u − f, u * − f * ∈ Hs (Ω) of the associated exterior value problem.Next we introduce the used notion of weak solutions of (9) and (10).
We next begin the preparation to show the existence and uniqueness of weak solutions in order to finally define the exterior DN maps.Lemma 7.2 (Boundedness of the bilinear forms).Let s ∈ R + \ Z and m ∈ N be such that 2s ≥ m, and let a α ∈ M (H s−|α| → H −s ).Then B P and B * P extend as bounded bilinear forms to Proof.We will verify this only for the adjoint bilinear form, since the proof for the original problem is analogous.
where in the last step we used a r,s = a −s,−r .Now the claim follows from the density of Let Ω ⊂ R n be an open set which is bounded in one direction, assume that s ∈ R + \ Z, m ∈ N satisfy 2s > m and let P ∈ P m,s (Ω).Then for any and One has the estimate The function u is the unique u ∈ H s (R n ) satisfying Proof.We get from the definition of B P that where c = δ(Ω) − P m,s > 0. Therefore the assumptions of Lemma 4.1 are valid, and the statements follow.
We define the exterior DN maps associated to the problems (9) and (10) next.We recall that the abstract trace space is X := H s (R n )/ H s (Ω) and for simplicity we denote in the rest of the article its elements by f instead of [f ].
Definition 7.4.Let Ω ⊂ R n be an open set which is bounded in one direction, assume that s ∈ R + \ Z, m ∈ N satisfy 2s > m and let P ∈ P m,s (Ω).The exterior DN maps Λ P and Λ * P are Λ P : X → X * defined by Λ P f, g := B P (u f , g) and Λ * P : X → X * defined by Λ * P f, g := B * P (u * f , g) where u f , u * f are the unique solutions to the equations and We state in the next lemma that the exterior DN maps Λ P and Λ * P are well-defined.The proof is an elementary consequence of Lemma 7.3, and the proof given in [CMRU20, Lemma 3.6] holds identically in our setting regardless of having the problem defined in unbounded domains.It also follows from Lemma 4.1.
Lemma 7.5 (Exterior DN maps).The exterior DN maps Λ P and Λ * P are well-defined, linear and continuous.One has the identity Λ P f, g = f, Λ * P g .We note that Lemma 4.4 implies the following. and where f 1 , f 2 ∈ H s (R n ), then we have the identity The Runge approximation property follows directly from Theorem 4.3 and the unique continuation property of the higher order fractional Laplacian (see [CMR21,GSU20]).
Lemma 7.7 (Runge approximation property).Let Ω ⊂ R n be an open set which is bounded in one direction, assume that s ∈ R + \ Z, m ∈ N satisfy 2s > m and let respectively.Then R and R * are dense in H s (Ω).
We are ready to prove our main result on the fractional Calderón problem.One could follow the proof given in [GSU20] using the Alessandrini identity and the Runge approximation property stated above.We however directly rely on Theorem 2.6.Theorem 7.8 (Fractional Calderón problem).Let Ω ⊂ R n be an open set which is bounded in one direction, assume that s ∈ R + \ Z, m ∈ N satisfy 2s > m and let be open sets.If the exterior DN maps for the equations Proof of Theorem 7.8.Using Theorem 2.6 we deduce (11 For α = 0 it immediately follows that a 1,α | Ω = a 2,α | Ω .We finish the proof by induction using the identity (11).Suppose that a 1,β | Ω = a 2,β | Ω for all β such that |β| < N for some This also implies that 7.2.On generalizations of the uniqueness results for the fractional Calderón problems on bounded and unbounded domains.We consider the problem (9) and the related inverse problem of Theorem 7.8 in the special case when m = 0, the PDOs P 1 , P 2 are replaced by the zeroth order potentials q 1 , q 2 ∈ M (H s → H −s ) and consider also the case where Ω ⊂ R n is a bounded domain.It was shown in [CMR21,RS20a] that the result of Theorem 7.8 holds when m = 0 and q 1 , q 2 ∈ M 0 (H s → H −s ) and s ∈ R + \ Z on all bounded domains.We extend this result on bounded sets to potentials in M Ω (H s → H −s ) + M ≥0 (H s → H −s ) as this small improvement comes only with a little additional work using the same methods that are needed for unbounded domains.In the cases of unbounded domains, we restrict to potentials in The proofs remain almost identical to the proof of Theorem 7.8.It is also possible to obtain similar generalizations on bounded sets in the case of lower order linear perturbations studied in Section 7.1.Rather than giving full details of the proofs in this section, which are very similar to the earlier proofs in [CMR21, CMRU20, RS20a] and the proof of Theorem 7.8, we only explicitly prove here the related well-posedness results and conclude the uniqueness results for the fractional Calderón problems.The proof of Theorem 7.8 with obvious changes can be used as a guide to complete the missing details.
We assume that the potential q ∈ M (H s → H −s ) is such that 0 is not the Dirichlet eigenvalue of the operator (−∆) s + q, namely: If u ∈ H s (R n ) solves ((−∆) s + q)u = 0 in Ω and u| Ωe = 0, then u = 0.
We define the associated bilinear form Then the following statements hold.(i) There exist a real number µ > 0 and a countable set Σ ⊂ (−µ, ∞) of eigenvalues with with the norm estimate where C is independent of F and f .
Proof.Using Lemma 7.2 we see that it is enough to solve the problem with zero exterior condition.By the assumption there is a decomposition q = q s + q 0 , q s ∈ M Ω (H s → H −s ), q 0 ∈ M ≥0 (H s → H −s ).
First, we may calculate q, v 2 = q s , v 2 + q 0 , v 2 ≥ q s , v 2 for any v ∈ H s (R n ).Using (8) we have q s = q s,1 + q s,2 , where q s,1 ∈ M 0 (H s → H −s ) and q s,2 ∈ M δ(Ω) (H s → H −s ).The small part has the estimate for any v ∈ H s (R n ).Next suppose that v ∈ Hs (Ω).We may now estimate as in the proof of Lemma 7.3 to get where where φ − denotes the negative part of φ.When we write q s,1 = φ + (q s,1 − φ), we have by (13) for any v ∈ Hs (Ω) that , where c = c − q s,1 − φ s,−s > 0 is independent of v.This coercivity estimate and the boundedness of the bilinear form B q implies that B q,µ defines an equivalent inner product on Hs (Ω).
The proof is now completed as in [GSU20, Lemma 2.3].The Riesz representation theorem implies that for every F ∈ ( H s (Ω)) * there is unique u = G µ F ∈ H s (Ω) such that B q (u, v) + µ u, v = F (v) for all v ∈ H s (Ω).Next note that there is a unique u ∈ Hs (Ω) such that B q (u, v) − λ u, v = F (v) for all v ∈ H s (Ω) if and only if there holds u = G µ [(µ + λ)u + F ].The map G µ : ( H s (Ω)) * → H s (Ω) induces a compact, self-adjoint and positive definite operator G µ : L 2 (Ω) → L 2 (Ω) by the compact Sobolev embedding theorem.The Fredholm alternative together with the spectral theorem for the self-adjoint, compact, positive definite operator G µ implies the first part of the claim in (i).Moreover, the estimate follows from the Riesz representation theorem and the fact that if λ / ∈ Σ, then solutions u ∈ H s (Ω) to The assertion in (iii) follows from the Fredholm alternative.
Lemma 7.9 and the same proof as given for Theorem 7.8 let us conclude the following theorem.Note that the exterior DN maps are defined according to Definition 7.4 and the other properties such as the Alessandrini identity and the Runge approximation property remains valid.See also e.g.[CMR21, GSU20, RS20a] for a simpler proof for the case of just a potential q rather than an operator P .
Theorem 7.10 (Fractional Calderón problem with M Ω + M ≥0 potentials).Let Ω ⊂ R n be a bounded open set, s ∈ R + \ Z, and q 1 , q 2 ∈ M Ω (H s → H −s ) + M ≥0 (H s → H −s ) satisfy the condition (12).Let W 1 , W 2 ⊂ Ω e be open sets.If the exterior DN maps for the equations We get similarly the following theorem on unbounded domains.The proof of well-posedness and uniqueness of solutions is similar to the proof of Lemma 7.3 and only relies on the Riesz representation theorem without the spectral theorem.Therefore, we do not have to assume additionally anything about the Dirichlet eigenvalues due to the smallness and positivity assumptions on the potentials.
Theorem 7.11 (Fractional Calderón problem with M δ(Ω) + M ≥0 potentials on unbounded domains).Let Ω ⊂ R n be an open set which is bounded in one direction, s ∈ R + \ Z, and be open sets.If the exterior DN maps for the equations (−∆) s u + q j u = 0 in Ω, j = 1, 2, satisfy We finally move on to the case of local linear perturbations on bounded domains.It has a generalization from the usual coefficients in M 0 (H r → H t ) to more general coefficients, analogously to Theorem 7.10.We will state and prove the related well-posedness result as it is less trivial than in any of our earlier cases.Then the uniqueness result for the related fractional Calderón problem can be proved following the proof of Theorem 7.8 (given originally in [CMRU20] in the context of bounded domains but for slightly less general coefficients).
In the uniqueness theorem, we assume that 0 is not a Dirichlet eigenvalue of the operator ((−∆) s + P (x, D)), namely: Then the following statements hold.
(i) There exist a real number µ > 0 and a countable set Σ ⊂ (−µ, ∞) of eigenvalues with λ 1 ≤ λ 2 ≤ . . .→ ∞ having the following property: If λ / ∈ Σ, then for any f ∈ H s (R n ) and F ∈ ( H s (Ω)) * there is a unique u ∈ H s (R n ) satisfying with the norm estimate where C is independent of F and f .(ii) The function u in (i) is also the unique u ∈ H s (R n ) satisfying Proof.The proof is a combination of the ideas presented already in the proof of Lemmas 7.3 and 7.9, and the proof given in [CMRU20, Lemma 3.4].We describe here how the problem is reduced back to the estimates used in [CMRU20, Lemma 3.4].The crucial part is to establish the coercivity estimate.
Let v ∈ Hs (Ω).We get from the definition of B P that where c = δ(Ω) − P m,s > 0. This now leads to the estimate (15) which is of the form analyzed in [CMRU20, Lemma 3.4].
The role of the additional c > 0 is the same as having a larger but still a finite Poincaré constant when compared to the situation in [CMRU20, Lemma 3.4].It follows from (15) and the more involved estimates calculated in [CMRU20, Lemma 3.4] that there exists some c 0 , µ > 0 independent of v such that From here, the proof is completed similarly as in the proof of Lemma 7.9 but this time using the Lax-Milgram theorem and the spectral theorem for the compact, positive definite operators.
Lemma 7.12 allows to set up the fractional Calderón problem and to prove the following theorem using the exactly same steps as in the proof of Theorem 7.8 (or [CMRU20, Section 3]).
Remark 7.14.We do not have uniqueness for the decomposition P j = (P j − P j,s ) + P j,s but only for the coefficients of the resulting operator.
Remark 7.15.This is a small step towards the unification of the two different approaches in [CMRU20, Theorem 1.1 and Theorem 1.4]) since when Ω is a Lipschitz domain we can now include H s,∞ (Ω) type coefficients for the small term P j,s .The issue is that it seems to be difficult (if even possible) to approximate H s,∞ (Ω) coefficients with C ∞ c (R n ) in the Sobolev multiplier norms.However, the relevant bounded Bessel potential coefficients belong to the spaces of all multipliers on Lipschitz domains as proved in [CMRU20, Lemma 4.1].See also [CMRU20, Propositions 1.2 and 1.3] for the inclusion of H s,∞ (Ω) type coefficients to the spaces M 0 when additional vanishing on the boundary is assumed.One can therefore interpret the assumptions of Theorem 7.13 as a smallness assumption on the boundary for the bounded Bessel coefficients in the case of Lipschitz domains.

Fractional conductivity equation and the Liouville transformation
In this last section, we will study the fractional conductivity equation and the related inverse problem, which was introduced in [Cov20b], on arbitrary bounded open sets and domains bounded in one direction.It was shown in [Cov20b,Theorem 1 (Ω) and the associated DN maps restricted to open subsets W 1 , W 2 ⊂ Ω e agree, then γ 1 = γ 2 in Ω.We will not impose any regularity assumption on the boundary of Ω and weaken the assumption m 1 , m 2 ∈ H2s, n 2s (Ω) besides generalizing the setting to domains only bounded in one direction.Our proofs are based on certain continuity properties of the multiplication map (f, g) → f g in Bessel potential spaces, which will be established in Appendix A, and on our results on the generalized Calderón problem from Section 4 as well as Section 7.2.Our main result in this section is the proof of Theorem 2.8.8.1.Fractional gradient and divergence.We start by recalling the notion of fractional gradient and divergence as introduced in [Cov20b, DGLZ12].Let 0 < s < 1, then the fractional gradient of order s is the bounded linear operator ∇ s : H where and there holds (cf.[DNPV12, Propositions 3.4 and 3.6]) Here we have used that the Slobodeckij spaces W s,2 (R n ) coincide with the Bessel potential spaces H s (R n ) for 0 < s < 1, the continuity of the fractional Laplacian and we will write for simplicity . Therefore, we can define the fractional divergence of order s as the formal adjoint operator of the fractional gradient, that is, it is the map div s : One can show the following relation between these fractional operators and the fractional Laplacian: Lemma 8.1 ([Cov20b, Lemma 2.1]).Let 0 < s < 1.Then there holds in the weak sense Remark 8.2.In [CMR21, Section 6.2] it is shown that this lemma can be generalized to s ∈ (0, ∞) \ N but the definition of the higher order fractional gradient is rather technical as it involves tensor fields.We have not considered the question to what extent the theory presented in this section extends to the higher order cases.8.2.Liouville transformation, well-posedness of the fractional conductivity equation and DN map.First we introduce bilinear forms associated to the fractional conductivity equation and fractional Schrödinger equation when the potential q belongs to L n/2s (R n ).

Lemma 8.3 (Definition of bilinear forms and conductivity matrix). Let
) and define the conductivity matrix associated to γ by Θ γ : R 2n → R n×n , Θ γ (x, y) := γ 1/2 (x)γ 1/2 (y)1 n×n for x, y ∈ R n .Then the maps defined by are continuous bilinear forms.Moreover, we have q ∈ M 0 (H s → H −s ) with Remark 8.4.If no confusion can arise we will drop the subscript γ in the definition for the conductivity matrix Θ γ .
Proof.By Hölder's inequality, γ ∈ L ∞ (R n ) and ( 16) the map B γ is well-defined and we have ) and therefore we obtain by Hölder's inequality quv ∈ L 1 (R n ).Using the continuity of the fractional Laplacian we get the estimate This result now allows us to introduce in both cases a notion of weak solutions.
for all φ ∈ Hs (Ω).Similarly, we say that v ∈ H s (R n ) is a weak solution to the fractional Schrödinger equation for all φ ∈ Hs (Ω).Now we proof a similar result as [Cov20b, Theorem 3.1] but which also holds on possibly unbdounded domains Ω ⊂ R n .Theorem 8.6 (Liouville transformation).Let Ω ⊂ R n be an open set, 0 < s < min(1, n/2), γ ∈ L ∞ (R n ) with conductivity matrix Θ satisfies γ(x) ≥ γ 0 > 0 and define the background deviation m γ : R n → R by m γ := γ 1/2 − 1.Moreover, assume that m γ ∈ H 2s, n 2s (R n ), then the following assertions holds: if and only if v is a weak solution of the fractional Schrödinger equation weak solution of (18) if and only if u is a weak solution of (17).
Remark 8.7.If it is clear to which conductivity γ the background deviation m γ corresponds, we will denote it by m.Moreover, from now on we will always set q γ := − and refer to it simply as electric potential.As for the background deviation we will drop the subscript γ if the dependence is clear from the context.
Proof.Throughout the proof we will write for simplicity m, q instead of m γ , q γ .(i) First note that by Corollary A.7 we have Moreover, if g − h ∈ Hs (Ω) then Corollary A.8 shows that f − γ 1/2 h ∈ Hs (Ω) and hence f = γ 1/2 g ∈ H s (R n )/ Hs (Ω) is well-defined.Moreover, by the assumptions m ∈ H 2s, n 2s (R n ), γ ≥ γ 0 > 0 and the mapping properties of the fractional Laplacian we have q ∈ L n 2s (R n ).Hence, Lemma 8.3 shows that the introduced notions of weak solutions to both equations make sense.
Remark 8.9.If f ∈ L ∞ (R n ) and ρ ∈ C ∞ c (R n ), then f = f * ρ satisfies ∂ α f = f * (∂ α ρ ) for any α ∈ N n 0 and is bounded and uniformly continuous.If the first and second order differences are denoted by

Therefore we have
|h| for all α ∈ N n 0 with |α| ≤ k.This shows that f belongs to any Hölder space C t (R n ) with t ∈ (0, ∞) \ N (see e. Next we show that the fractional conductivity equation and the fractional Schrödinger equation are well-posed when the potential comes from a conductivity.Lemma 8.10 (Well-posedness and DN maps).Let Ω ⊂ R n be an open set which is bounded in one direction and 0 < s < min(1, n/2).Assume that γ ∈ L ∞ (R n ) with conductivity matrix Θ, background deviation m and electric potential q satisfies γ(x) ≥ γ 0 > 0 and m ∈ H 2s, n 2s (R n ).Then the following assertions hold: (i) For all f ∈ X := H s (R n )/ Hs (Ω) there are unique weak solutions u f , v f ∈ H s (R n ) of the fractional conductivity equation div s (Θ • ∇ s u) = 0 in Ω, u = f in Ω e and the the fractional Schrödinger equation (ii) The exterior DN maps Λ γ : X → X * , Λ q : X → X * given by Λ γ f, g := B γ (u f , g), Λ q f, g := B q (v f , g), where u f , v f ∈ H s (R n ) are the unique solutions to the conductivity equation and Schrödinger equation with exterior value f , are well-defined bounded linear maps.
Remark 8.11.Note that we are integrating over all of R n in the term involving the potential q in our definition of the bilinear form for the fractional Schrödinger equation, whereas in [Cov20b] the integral is restricted to Ω.
Proof.The bilinear form B γ is continuous by Lemma 8.3.Moreover, we have by the properties of the fractional gradient and the fractional Poincaré inequality on domains bounded in one direction (cf.Theorem 2.2) for all u ∈ Hs (Ω) and therefore B γ is (strongly) coercive.Now by Lemma 4.1 the assertions for the conductivity equation follow.
Next consider the Schrödinger equation with exterior value f ∈ X.By part (ii) of Theorem 8.6 we have g := γ −1/2 f ∈ X and we already know that there is a unique weak solution v g ∈ H s (R n ) to the conductivity equation with exterior value g.Therefore, part (i) of Theorem 8.6 demonstrates that u f := γ 1/2 v g ∈ H s (R n ) has exterior value f and solves the Schrödinger equation as desired.If u 1 , u 2 ∈ H s (R n ) are weak solutions to the Schrödinger equation with exterior values f ∈ X, then u := u 1 −u 2 ∈ Hs (Ω) solves again the Schrödinger equation but this time with exterior value zero and hence by Theorem 8.6, part (ii) the function v := γ −1/2 u ∈ Hs (Ω) is a weak solution to div s (Θ • ∇ s v) = 0 in Ω, v = 0 in Ω e .
Since solutions to the conductivity equation are unique we obtain v ≡ 0 which in turn implies u 1 = u 2 .Therefore solutions to the Schrödinger equation are unique as well.The statements for the DN map associated to the Schrödinger equation Λ q follow from Lemma 8.3 and standard arguments.
Proof.If supp(m) = R n , there is nothing to prove and hence we can assume supp(m) = R n .By assumption we have γ 1/2 h = (m + 1)h = mh + h = h for h = f, g.By Theorem 8.6 we have γ 1/2 u f = v γ 1/2 f = v f , where u f , v f ∈ H s (R n ) are the unique solutions to the homogenous fractional conductivity equation and Schrödinger equation with exterior value f ∈ H s (R n ), respectively.Hence, by Remark 8.8 and Lemma 8.10 we obtain Λ γ f, g = B γ (u f , g) = B q (γ 1/2 u f , γ 1/2 g) = B q (v f , g) = Λ q f, g .

8.3.
Uniqueness results for the inverse problem.
be a bounded bilinear form.(i) We say that B has the left UCP on an open nonempty set W ⊂ R n when the following holds: If u ∈ H s (R n ), u| W = 0 and B(u, φ) = 0 for all φ ∈ C ∞ c (W ), then u ≡ 0. (ii) We say that B has the right UCP on an open nonempty set W ⊂ R n when the following holds: If u ∈ H s (R n ), u| W = 0 and B(φ, u) = 0 for all φ ∈ C ∞ c (W ), then u ≡ 0. (iii) We say that B has the symmetric UCP on an open nonempty set W ⊂ R n when it has the left UCP and the right UCP on W . (iv) We say that B is local when the following holds: If u, v ∈ H s (R n ) and supp(u) ∩ supp(v) = ∅, then B(u, v) = 0. (v) We denote by B * (u, v) := B(v, u) the adjoint (or transpose) bilinear form of B. Remark 2.5.If B is local, then B * is local.If B has the left UCP, then B * has the right UCP and vice versa.If B is symmetric, that is B = B * , then B has the symmetric UCP if and only if it has the left or the right UCP.

Ω
Figure 2.1.Graphical illustration of the geometric setting of part (i) of Theorem 2.8.Here the set Ω ⊂ R n is a domain which is bounded in one direction and the measurements are performed in the nonempty open subsets W 1 , W 2 ⊂ Ω e that have a nonempty intersection.The support of m is represented in red.The DN map Λ γ determines m, and hence γ, uniquely.

Figure 4 . 3 .
Figure 4.3.An example illustrating Theorem 4.3.Let us denote B = B(0; ) ⊂ R n for any > 0 and n ≥ 1.For any , δ > 0 and Ω := R n \ B , we have that the restriction to R n \B of the unique solutions u f to the equation ((−∆) s +δ)u = 0 in R n \ B are dense in Hs (R n \ B ) with exterior conditions f ∈ C ∞ c (B ).This means heuristically speaking that all functions in H s (R n ) are almost s-harmonic up to an error depending on δ, > 0. This observation is analogous to the celebrated result on bounded domains[DSV17] where one may take δ = 0, but this possible choice of δ is also true on domains that are bounded in one direction.We do not perform a quantitative analysis of this phenomenon in this article.
Proposition 4.6 (Example models).Let s ≥ 0, and Ω ⊂ R n be an open set such that Ω e = ∅.The generalized fractional Calderón problem is well-defined and has the uniqueness property (in the weak sense) in the following cases: Now we only need to notice that |Lξ| = |ξ| and F(u • L)(ξ) = û(Lξ) hold.This completes the proof.Remark 5.3.Lemma 5.2 implies that Ω and AΩ have the same fractional Poincaré constants.It is clear that if Ω has a finite fractional Poincaré constant then all open subsets of Ω also have finite fractional Poincaré constants which are at most the fractional Poincaré constant of Ω.
Sections 2.4.7,5.2.3 and 5.2.5] or [BL76, Exercise 6.8.29]) and the complex interpolation functor is exact of exponent θ ∈ (0, 1) (see [BL76, Theorem 4.1.2]),since the spaces in question are Banach spaces [Tri83, Section 5.1.5].The end point cases of Lemma 5.6 hold trivially.Now the conclusion follows from the definition of exactness of exponent θ (see [BL76, p. 27 (6)]) by considering the operator T Remark 6.1.Let Ω ⊂ R n be an open set which is bounded in one direction.Then by assumption there exists a rigid Euclidean motion A such that AΩ ⊂ Ω ∞ = R n−k × ω, where ω ⊂ R k is a bounded open set and n ≥ k > 0. Using Remark 5.3 and Theorem 5.1, we can prove the following estimate (the last inequality is obtained by optimizing the constant in [CMR21, Theorem 3.7])
be a bounded bilinear form with the following properties: (i) There exists a nonempty open set W 1 ⊂ Ω e such that L has the right UCP on W 1 .
(ii) There exists a nonempty open set W 2 ⊂ Ω e such that L has the left UCP on W 2