Regularity estimates for nonlocal Schr\"odinger equations

We prove H\"older regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schr\"odinger equations subject to exterior Dirichlet conditions in an open set $\Omega\subset \mathbb{R}^N$. The class of nonlocal operators considered here are defined, via Dirichlet forms, by kernels $K(x,y)$ bounded from above and below by $|x-y|^{N+2s}$, with $s\in (0,1)$. The entries in the equations are in some Morrey spaces and the underline domain $\Omega$ satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When $K$ defines a nonlocal operator with sufficiently regular coefficients, we obtain H\"older estimates, up to the boundary of $ \Omega$, for $u$ and the ratio $u/d^s$, with $d(x)=\textrm{dist}(x,\mathbb{R}^N\setminus\Omega)$. If the kernel $K$ defines a nonlocal operator with H\"older continuous coefficients and the entries are H\"older continuous, we obtain interior $C^{2s+\beta}$ regularity estimates of the weak solutions $u$. Our argument is based on blow-up analysis and compact Sobolev embedding.


Introduction
We consider s ∈ (0, 1), N ≥ 1 and Ω an open set in R N of class C 1,γ , for some γ > 0. We are interested in interior and boundary Hölder regularity estimates for functions u solution to the equation in Ω and u = 0 in Ω c . (1.1) where Ω c := R N \ Ω and L K is a nonolocal operator defined by a symmetric kernel K ≍ |x − y| −N −2s . We refer to Section 1.1 below for more details. Our model operator is L K = (−∆) s a , the so called anisotropic fractional Laplacian, up to a sign multiple. It is defined, for all ϕ ∈ C 2 c (R N ), by |y| N +2s a (y/|y|) dy, with a : S N −1 → R satisfying a(−θ) = a(θ) and Λ ≤ a(θ) ≤ 1 Λ for all θ ∈ S N −1 , for some constant Λ > 0. Here, the entries V, f in (1.1) belongs to some Morrey spaces.
In the recent years the study of nonlocal equations have attracted a lot of interest due to their manifestations in the modeling of real-world phenomenon and their rich structures in the mathematical point of view. In this respect, regularity theory remains central questions. Interior regularity and Harncak inequality have been intensiveley investigated in last decades, see e.g. [1, 3, 4, 9-11, 16, 22, 33-37,40,50] and the references therein. On the other hand, boundary regularity and Harnack inequalities was studied in [2,5,7,13]. Results which are, in particular, most relevent to the content of this paper concern those dealing with nonlocal operator in "divergence form" with measurable coefficient, i.e. K is symmetric on R N × R N and K(x, y) ≍ |x − y| −N −2s , see [39]. In this case the de Giorgi-Nash-Moser energy methods were used to obtain interior Harnack inequality and Hölder estimates, see [16,17,38,42]. We note that the papers [38,39] deal also with more general kernels than those satisfying K(x, y) ≍ |x − y| −N −2s on R N × R N only. We also mention the work of Kuusi, Mingione and Sire in [42]

who obtained local
The author's work is supported by the Alexander von Humboldt foundation. Part of this work was done while he was visiting the Goethe University in Frankfurt am Main during August-September 2017 and he thanks the Mathematics department for the kind hospitality. The author is grateful to Xavier Ros-Oton, Tobias Weth and Enrico Valdinoci for their availability and for the many useful discussions during the preparation of this work. pointwise behaviour of solutions to quasilinear nonlocal elliptic equations with measurable coefficients, provided the Wolff potential of the right hand side satisfies some qualitative properties.
In this paper, we are concerned in both interior and boundary regularity of nonlocal equations with "continuous coefficient". Let us recall that in the classical case of operators in divergence form with continuous coefficients, after scaling, the limit operator is given by the Laplace operator ∆. The meaning of "continuous coefficient" in the nonlocal framework is not immediate due to the singularity of the kernel K at the diagonal points x = y. However, under nonrestrictive continuity assumptions, detailed in Section 1.1 below, we find out that the limiting nonlocal operator is the anisotropic fractional Laplacian −(−∆) s a in many situations.
Letting d(x) := dist(x, Ω c ), the boundary regularity we are interested in here is the Hölder regularity estimates of u/d s for the nonlocal operator L K . Such regularity results for solutions to (1.1) has been studied long time ago when L K = (−∆) 1/2 . They are of interest e.g. in fracture mechnics, see [15] and the referenes therein. The general case for (−∆) s , s ∈ (0, 1), has been considered only in the recent years and it is by now merely well understood when u, V, f ∈ L ∞ (R N ) and Ω a domain of class C 1,γ , for some γ > 0. Indeed, in the case of the fractional Lapalcian (a ≡ 1) and γ = 1, the first Hölder regularity estiamte of u/d s in Ω was obtained by Ros-Oton and Serra in [49]. They sharpened and generalized this result to translation invariant operators, even to fully nonlinear equations, in their subsequent papers [45,47,48]. We refer the reader to the recent survey paper [46] for a detailed list of existing results. In the case where V , a, f and Ω are of class C ∞ , we quote the works of Grubb, [30][31][32], where it is proved that u/d s is of class C ∞ , up to the closure of Ω. Especially in [30], the enteries V, f are also allowed to belong to some L p spaces, for some large p. More precisely, when V ∈ C ∞ (Ω) and f ∈ L p (Ω), for some p > N/s, then provided Ω and a are of class C ∞ , Grubb proved in [30] that u/d s is of class C s−N/p (Ω). Here, we prove sharp Hölder regularity estimates of u/d s , for Ω an open set of class C 1,γ and V, f are in some spaces containing the Lebesgue space L p for p > N/s. Let us now recall the Morrey space which will be considered in the following of this paper. For β ∈ [0, 2s), we define the Morrey space |f (y)| dy < ∞, with M 0 := L ∞ (R N ). Such spaces introduced by Morrey in [43], are suitable for getting Hölder regularity in the study of partial differential equations. Let us now explain in an abstract form the insight in our consideration of the Morrey space. Indeed, given a function g ∈ L 1 loc (R N ), we put g r,x0 (x) := r 2s g(rx + x 0 ), for x 0 ∈ R N and r > 0. For β ≥ 0, we say that g satisfies a Rescaled Translated Coercivity Property (RTCP, for short) of order β, if there exists a constant C := C(g, N, s, β) > 0 such that for all x 0 ∈ R N and r ∈ (0, 1), we have Then what we will prove, for solutions u to (1.1) when L K is (up to a scaling) close to (−∆) s a , are the following implications: f, V satisfy a RTCP of order β ∈ [0, 2s) =⇒ u ∈ C min(1,2s−β)−ε loc (Ω), f, V satisfy a RTCP of order β ∈ [0, 2s) =⇒ u ∈ C min(s,2s−β)−ε loc (Ω), (1.4) for every ε > 0 and Ω an open set with C 1 boundary. For higher order regularity, under some regularity assumptions on K and Ω, quantified by some parameter β ′ ∈ [0, 2s), we obtain f, V satisfy a RTCP of order β ∈ (0, 2s) =⇒ u ∈ C 2s−max(β,β ′ ) loc (Ω), (Ω), (1.5) provided 2s − max(β, β ′ ) = 1. Here and in the following, it will be understood that C ν := C 1,ν−1 if ν ∈ (1, 2). It is not difficult to see that functions g satisfying a RTCP of order β belongs to M β . On the other hand the converse, which is not trivial, also holds true, and in fact, we will prove a more general inequality for the Kato class of functions which could be of independent interest, see Lemma 2.3 below.
Since our results are already new for (−∆) s a , we state first simpler versions of our main results, and postponed the generalization to L K in Section 1.1 below. To do so, we need to recall the distributional domain of the operator (−∆) s a . It is given by L 1 s , the set of functions u ∈ L 1 loc (R N ) such that u L 1 s := R N |u(x)| 1+|x| N +2s dx < ∞. Theorem 1.1. Let s ∈ (0, 1), β ∈ (0, 2s) and a satisfy (1.2). Let Ω ⊂ R N be an open set of class C 1,γ , γ > 0, in a neighborhood of 0 ∈ ∂Ω. Let u ∈ H s (B 1 ) ∩ L 1 s and f, V ∈ M β be such that (−∆) s a u + V u = f in Ω.
Provided u, V, f ∈ L ∞ (R N ), by letting β ց 0, we recover the boundary regularity in [45] for C 1,γ domains and partly the one in [47] for C 1,1 domains. We mention that in [47], a weaker ellipticity assumption (second condition in (1.2)) was considered. Obviously if f ∈ L p (R N ), with p > 1, then f ∈ M N p . For the strict inclusion of Lebesgue spaces in Morrey spaces, see e.g. [18]. An immediate consequence of Theorem 1.1 is therefore the following result. Then there exist positive constants C, ̺ > 0 such that u/d s C min(γ,s−N/p) (B̺∩Ω) ≤ C u L 2 (B1) + u L 1 s + f L p (B1) . The constants C and ̺ depend only on s, N, γ, p, Λ, Ω and V L p (B1) .
As mentioned earlier, we recall that the boundary regularity in Corollary 1.2 was known only when a ∈ C ∞ (S N −1 ) and Ω of class C ∞ , see [30]. In the classical case of the Laplace operator, the corresponding result of Corollary 1.2 is that u is of class C 1,min(γ,1−N/p) up to the boundary, see [28]. We note that interior and boundary Harnack inequalities for the operator (−∆) s + V , with V in the Kato class of potentials (larger than the Morrey space) and Ω a Lipschitz domain have been proven in [6,53]. In [19], we shall provide an explicit modulus of continuity for solutions to (1.1), when V and f belong to the Kato class of potentials.
1.1. Nonlocal operators with possibly continuous coefficients. In the following, for a function b ∈ L ∞ (S N −1 ), we define Let κ > 0 be a positive constant and λ : . We consider the class of kernels K : R N × R N → [0, +∞] satisfying the following properties: for all x = y ∈ R N , (1.6) The class of kernels satisfying (1.6) is denoted by K (λ, b, κ).
Let Ω ⊂ R N be an open set and let f, V ∈ L 1 loc (R N ). For K satisfying (1.6)(i)-(ii), we say that u ∈ H s loc (Ω) ∩ L 1 s is a (weak) solution to The class of operator L K corresponding to the kernels K satisfying (1.6)(i)-(ii) can be seen as the nonlocal version of operators in divergence form in the classical case. Here we obtain regularity estimates for K ∈ K (λ, a, κ) provided λ is small and a satisfies 1.2. We thus include, in particular, nonlocal operators with "continuous" coefficients. The meaning of continuous coefficients for nonlocal operators might be awkward, since one is dealing with kernels which are not finite at the points x = y. Using polar coordinates, we can depict an encoded limiting operator which is nothing but the anisotropic fractional Laplacian. In view of Remark 2.1 below, all results stated below remains valid if we consider kernels K : This is due to the fact that the regularity theory of the operators L K is included in those of the form L K + V , with K satisfying (2.2)(i)-(ii), for some potential V of class C ∞ .
It is worth to mention that the kernels in K (κ) appear for instance in the study of nonlocal mean curvature operator about a smooth hypersurface, see e.g. [3,20]. More generally, a typical example is when considering a C 1 -change of coordinates Φ e.g. in the kernel |x − y| −N −2s (could be defined on the product of hypersurfaces M × M). The singular part of the new kernel is then given by K Φ (x, y) = |Φ(x) − Φ(y)| −N −2s , for some local diffeomorphism Φ ∈ C 1 (B 4 ; R N ). In this case, and thus K Φ ∈ K (κ), for some κ > 0. As a consequence, λ KΦ (x, 0, θ) = |DΦ(x)θ| −N −2s , which is even in θ. We refer to Section 7.1 below in a more general setting. Thanks to theà priori estimates that we are about to state below, we shall prove in [20] optimal regularity results paralleling the regularity theory for elliptic equations in divergence form, with regular coefficients, and provide applications in the study of nonlocal geometric problems.
To obtain interior regularity in (1.5), we need to care on the kernels K for which the action of L K on affine functions can be quantified. In this respect, some regularity on K is required. More precisely, for K ∈ K (λ, a, κ) and x ∈ R N , we define where ℓ + := max(ℓ, 0), for ℓ ∈ R. Letting we see that The main regularity assumption we make on K is that j o,K is locally in M β ′ , in the sense that Here and in the following, ϕ R ∈ C ∞ c (B 2R ), with ϕ R ≡ 1 on B R . Since, β ′ depends on K, the main point will be to obtain regularity estimate by constants independent in β ′ . We observe that, for example, for a kernel K such that λ o,K (x, r, θ) ≤ r α+(2s−1)+ , for r ∈ (0, 1) and for some α > 0, then ϕ 2 j o,K ∈ L ∞ (R N ) = M 0 . Of course if s ∈ (0, 1/2] such additional regularity assumption on K is unnecessary. This is also the case if L K is a translation invariant, i.e. K(x, y) = k(x − y), for some even function k : R N → R.
Our main result for interior regularity reads as follows.
We note that the C β (R N )-norm of v in (1.11) can be replaced with v L 2 (B2) + v L 1 s , provided, we require Hölder regularity of λ K in the variable θ i.e.
We now turn to our boundary regularity estimates in (1.5). In this case, it is important to consider those kernels K for which L K d s can be quantified. Here our assumption is that L K d s is given by a function in M β ′ , for some β ′ ∈ [0, s). To be more precise, we consider all kernel K for which, there exist β ′ = β ′ (Ω, K) ∈ [0, s) and a function g Ω,K ∈ M β ′ such that in the weak sense in B r0 ∩ Ω, (1.12) where r 0 > 0, only depends on Ω, is such that ϕ 2 d s ∈ H s (B r0 ) ∩ L 1 s . We note that g Ω,K might be singular near the boundary, since we are considering only domains of class C 1,γ . In fact, see [45], for ∩ Ω, for some r 0 , only depending on Ω. This, in particular, shows that there exists a g Ω,µa ∈ M (s−γ)+ satisfying (1.12). We note that (1.12) encode both the regularity of K and of Ω.
Our next main result is the following.
and Ω an open set of class C 1,γ , γ > 0, near 0 ∈ ∂Ω. Let a satisfy (1.2), for some Λ > 0. Let K ∈ K (λ, a, κ) satisfy (1.12), with Then there exist C, ̺ > 0, only depending only on N, s, β, Λ, κ, Ω, c 0 , δ and V M β , such that if λ L ∞ (B2×B2) ≤ ε 0 , we have In the case of uniformly continuous coefficient also, we have the following boundary regularity estimates. Corollary 1.7. Let s ∈ (0, 1), β, δ ∈ (0, s) and Ω an open set of class C 1,γ , γ > 0, near 0 ∈ ∂Ω. For κ > 0, let K ∈ K (κ) satisfy: for every x 1 , x 2 ∈ B 2 , r ∈ (0, 2), θ ∈ S N −1 , Then there exist C, ̺ > 0, only depending on N, s, β, τ, κ, Ω, c 0 , δ and V M β , such that As an application of the above result together with a global diffeomorphism that locally flatten the boundary ∂Ω near 0, we get the following Theorem 1.8. Let s ∈ (0, 1), β ∈ (0, s) and Ω an open set of class C 1,1 , near 0 ∈ ∂Ω. For κ > 0, let K ∈ K (κ) satisfy: Then there exist C, ̺ > 0, only depending on N, s, β, τ, κ, Ω, c 0 , δ and V M β , such that The proof of Theorem 1.6 and Theorem 1.3 are based on some blow-up analysis argument, where normalized, rescaled and translated sequence of a solution to a PDE satisfy certain growth control and converges to a solution on a symmetric space, so that Liouville-type results allow to calssify the limiting solutions. Here, we are inpired by the work of Serra in [52], see also [45,47,48,51] for boundary regularity estimates for translation invariant nonlocal operators. Note that in the aforementioned papers, since entries and solutions are in L ∞ , the use of barriers to getà priori pointwise estimates and Arzelà-Ascoli compactness theorems were the main tools to carry out their blow-up analysis. In our situation, it is clear that there is no hope of using such tools. Our argument will be based on the estimate of the L 2 -average mean oscillation of u to getà priori pointwise estimates. Indeed, to prove (1.4), we show the growth estimates for every z ∈ B 1 ∩ ∂Ω and r ∈ (0, r 0 ), (1.16) with ψ L ∞ (B1∩∂Ω) ≤ C and the constant C does not depend on β ′ . Recall that ℓ + := max(ℓ, 0). Now using appropriate interior regularity estimates ((1.4) is enough), we translate the L 2 estimates in (1.16) to a pointwise estimate which yields the conclusion of the theorem. The proof of (1.16) uses blow-up argument that allows to estimate the growth, in r > 0, of the difference between u and its L 2 (B r (z))-projection on Rd s , the one-dimensional space generated by d s . Similarly the proof of (1.15) is achieved by estimating the growth, in r > 0, of the difference between u and its L 2 (B r (z))-projection on the finite dimensional space of affine functions t + (2s − 1) + T · (x − z) : t ∈ R and T ∈ R N . We obtain Theorem 1.5 by freezing the radial variable r at r = 0 and by using the Shauder estimates for nontranslation invariant nonlocal operators of Serra [51]. For that, we use our lower order term estimates Corollary 1.4 together with some approximation procedure and boundary regularity. Related to this work is the one of Monneau in [44] where blow-up arguments were used to estimate the modulus of mean oscillation (in L p average) for solutions to the Laplace equation with Dini-continuous right hand sides. Sharp boundary regularity in C 1,γ domains and refined Harnack inequalities in C 1 domains are useful tools to obtain sharp regularity of the free boundaries in the study of nonlocal obstacle problems, see e.g. [8]. We believe that our result and arguments might be of interest in the study of obstacle problems with non smooth obstacles and for parabolic problems.
For the organization of the paper, we put in Section 2 some notations and preliminary results related to Kato class of potentials. Section 3 is devoted to interior and boundary L 2 -growth estimates of solutions to (1.1) in C 1 domains. Statement (1.4), is proved in Section 4. Higher order boundary and interior regularity are proved in Section 5 and Section 6, respectively. The proof of the main results (in particular (1.5)) are gathered in Section 7. Finally, we prove the Liouville theorems in Appendix 8 and we put some useful technical results in Appendix 9.

Notations and Preliminary results
In this paper, the ball centered at z ∈ R N with radius r > 0 is denoted by B(z, r) and B r := B r (0). Here and in the following, we let ϕ 1 ∈ C ∞ c (B 2 ) such that ϕ 1 ≡ 1 on B 1 and 0 ≤ ϕ 1 ≤ 1 on R N . We put ϕ R (x) Recall that (see e.g. [21]), if b is even, there exists C = C(N, s, b L ∞ (S N −1 ) ), such that for all ψ ∈ C ∞ c (R N ) and for every x ∈ R N , we have where P V means that the integral is understood in the principle value sense. Throughout this paper, for the seminorm of the fractional Sobolev spaces, we adopt the notation and for the Hölder seminorm, we write for α ∈ (0, 1). Letting u ∈ L 1 loc (R N ), the mean value of u in B r (z) is denoted by u(x) dx.
2.1. The class of operators. In the following, it will be crucial to consider certain class of operators which we describe next.
2.1.1. Symmetric operators with bounded measurable coefficients. Firstly we will consider kernels K : R N × R N → (0, ∞] satisfying the following properties: Let Ω ⊂ R N be an open set and let f, V ∈ L 1 loc (R N ). For K satisfying (2.2), we say that u ∈ H s loc (Ω) ∩ L 1 s is a (weak) solution to Note, in fact, that for the first term in (2.3) to be finite, it is enough that K satisfies only the upper bound in (2.2)(ii).
[Kernels with possible compact support] In many applications, it is important to consider kernels K ′ with possible compact support. This allows to treat kernels which are only locally symmetric and locally elliptic ( (2.4) below). As a matter of fact, we note that the regularity theory of the operators L K ′ is included in those of the form L K + V , with K satisfying (2.2), for some potential V of class C ∞ . Indeed, consider a kernel K : for all x = y ∈ R N .

(2.4)
We define η 1 (x) := 1 − ϕ 1 (x) and η(x, y) = η 1 (x) + η 1 (y), which satisfies Then letting u ∈ H s (B 2 ) ∩ L 1 s be a weak solution (in the sense of (2.3)) to the equation we then have that Symmetric translation invariant operators with semi-bounded measurable coefficients. The class of operators we will consider next appears as limit of rescaled operators L K , for K ∈ K (λ, b, κ) (see Section 1.1). Let (a n ) n be a sequence of functions, satisfying (1.2). Then, up to a subsequence, it converges, in the weak-star topology of L ∞ (S N −1 ), to some b ∈ L ∞ (S N −1 ). It follows that b is even on S N −1 and satisfies For such function b, we denote by L b the corresponding operator, which is given by where P V means that the integral is in the principle value sense. Here also solutions u ∈ H s loc (Ω) ∩ L 1 The following result is concerned with limiting of a sequence of operators which are close to a translation invariant operator.
Lemma 2.2. Let (a n ) n be a sequence of functions, satisfying (1.2) and converging in the weak-star for all x = y ∈ R N and for all n ∈ N.
By eveness of a n and b, Fubini's theorem and a change of variable, we can write Clearly, the function θ → ∞ 0 (ψ(x − tθ) + ψ(x + tθ) − 2ψ(x))t −1−2s dt is bounded on S N −1 and thus belongs to L 1 (S N −1 ). Therefore the sequence of functions h n (x) := (L b − L an )ψ(x) converges pointwise to zero on R N . Moreover by (2.1), we have that |h n (x)| ≤ C ψ (1 + |x|) −N −2s . Since v ∈ L 1 s , it follows from the dominated convergence theorem that | dx is bounded and converges pointwise to zero, as n → ∞. By the dominated convergence theorem, we then have that Since w n = v n − v is bounded in H s loc ∩ L 1 s and w n → 0 in L 1 s , by similar arguments as above, we get In addition, since w n → 0 in L 1 s , by (2.1), Combining the two estimates above, we conclude that Using this, (2.9) and (2.8) in (2.7), we get the conclusion in the lemma.

2.2.
Coercivity and Caccioppoli type inequality with Kato class potentials. For s > 0, we let Γ s := (−∆) −s be the Riesz potential, which satisfy (−∆) s Γ s = δ 0 in R N . Recall that for N = 2s, Γ s (z) = c N,s |z| 2s−N and for N = 2s, Γ s (z) = c N,s log(|z|), for some normalization constant c N,s . We consider the Kato class of functions given by where for N ≥ 2s, ω s (|z|) = |Γ s (z)| and if 2s > N , we set ω s ≡ 1. Here and in the following, for every V ∈ K s and r ∈ (0, 1], we define The following compactness result will be useful in the following. We also note that it holds for all s > 0, and in this case Proof. For r > 0, we consider the Bessel potential G s,r = (−∆ + r −2 ) −s/2 . See e.g. [29, Section 6.1.2], there exists a constant c = c(N, s) > 0 such that where ω s is defined in the beginning of this section.
Step 1: We assume that V ∈ L ∞ (R N ). For δ > 0, we consider the operator L : We note that the adjoint of L is given by Claim: There exists c = c(N, s) > 0 such that for every δ ∈ (0, 1/2], (2.14) By Hölder's inequality and using the fact that G s,r (x) = G s,1 (x/r), we obtain Using a change of variable and (2.13), for x ∈ R N , we get For every fixed i, we cover the annulus We then get, for every δ ∈ (0, 1/2] That is (2.14) as claimed.
Since LL * = L 2 , it then follows that Step 2: The following energy estimate is a consequence of the above coercivity result and a nonlocal Caccioppoli-type inequality proved in an appendix, Section 9.
Lemma 2.4. We consider Ω an open set with 0 ∈ ∂Ω and K satisfying (1.2). Let v ∈ H s (R N ) and V, f ∈ K s satisfy Then there exists C = C(N, s, κ) > 0 such that for every ε > 0 and every δ ∈ (0, 1], there exists C = C(ε, δ, s, N, κ) such that Proof. By Lemma 9.1 and (2.1), we get (2.17) Thanks to Lemma 2.3, (2.1) and the fact that Similarly, by Young's inequality, (2.12) and (2.1), we get Using the above estimate and (2.18) in (2.17) and using the monotonicity of η V and η f , we get the result.
We close this section with the following result. and V, f ∈ K s satisfy where C > 0 is a constant, only depending on N and s.
Proof. Testing the equation (2.19) with ψ ∈ C ∞ c (B R ∩ Ω) and using Young's inequality, we get Hence using Lemma 2.3, we conclude that which finishes the proof.

Interior and boundary growth estimates
We recall the Morrey space already introduced in the first section, for β ∈ [0, 2s), defined as Let f ∈ M β and define f r,x0 (x) = r 2s f (rx+x 0 ) for x 0 ∈ R N and r > 0. Recalling (2.11), an important property of η f we will use frequently in the following is that, for every x 0 ∈ R N and r ∈ (0, 1], we have with C a positive constant, only depending on N, s, and β. The first inequality in (3.1) can be easily checked by change of variables and using summations over annuli with small thickness. We note that (3.1) and Lemma 2.3 show that functions V, f ∈ M β satisfy RTCP of order β (see (1.3)) as mentioned in the first section.
3.1. Interior growth estimates for solutions to Schrödinger equations. The next result is merely classical but we add the proof for the sake of completeness.
with C depends only on N and α.
(ii) Suppose that 0 is a Lebesgue point of u and with C depends only on N and α.
Proof. First, to prove (i), we note that, for every ρ ≥ 1, where C is independent on m, ρ and u. Next, if m is the smallest integer for which, 2 m−1 ≤ ρ ≤ 2 m , then using (3.2) and the above estimate, we conclude that For (ii), by assumption, we have Using this and (3.3), we obtain.
where C depends only on N and α.
Let a satisfy (1.2) and K ∈ K (λ, a, κ) (satisfy (2.2)) V, f ∈ M β , we define the set of solutions to the Schrödinger equations with entries V and f by and we note that this set is nonempty thanks to Lemma 2.3 and a direct minimization argument. In fact this set is nonempty for all f, V ∈ K s for the same reason. We consider the class of (normalized) potentials Having these notations in mind, we now state the following result.
) and Λ, κ > 0. Then there exists ε 0 > 0 and C > 0 such that for every λ : Proof. The proof of (3.5) will be divided into two steps. Due to the presence of the potential V , the set S K,V,f might not be invariant when adding constants to its elements. As a way out to this difficulty, we prove first a uniform estimate of the form |u Once we get this, we complete the proof of (3.5) in the second step.
Step 1: We claim that for every ̺ ∈ (0, 1/2), there exist C > 0 and a small number ε 0 > 0 such that for every λ : Assume that (3.6) does not hold, then there exists ̺ ∈ (0, 1/2) such that for every n ∈ N, we can find λ n : R 2N → [0, κ −1 ] satisfying λ n L ∞ (B2×B2) < 1 n , a n satisfying (1.2), K an ∈ K (λ n , a n , κ), V n ∈ V β , f n ∈ M β , u n ∈ S Ka n ,Vn,fn , with u n L 2 (R N ) + f n M β ≤ 1 and r n > 0, such that We consider the (well defined, because u n L 2 (R N ) ≤ 1) nonincreasing function Θ n : where we used the monotonicity of Θ n for the last inequality, while the first inequality comes from the definition of Θ n . In particular, thanks to (3.9), Θ n (r n ) (3.10) We now define the blow-up sequence of functions for every n ≥ 2. (3.11) In view of (3.8), we have that where we have used the monotonicity of Θ n for the last inequality. Consequently, for every R ≥ 1 and n ≥ 2. (3.12) We define n K an (r n x + x n , r n y + x n ).
(3.13) Clearly K n satisfies (2.2). Therefore applying Lemma 2.4 and using (3.12), for every 1 < M < 1 2rn , we get By (3.1), we have that η V n (1) + η f n (1) ≤ Cr 2s−β n (recalling that Θ n (r n ) −1 ≤ 1). Hence, there exists a constant C(M ) independent on n ≥ 2 such that Therefore provided ε is small and n is large enough, we deduce that w n is bounded in H s loc (R N ). Hence by Sobolev embedding, up to a subsequence, w n converges strongly, in L 2 loc (R N ), to some w ∈ H s loc (R N ). In addition by (3.12), we have that v n → v in L 1 s . Moreover, by (3.11) and (3.12), we deduce that . By Lemma 2.5, (3.14) and (3.12), we get Next, we observe that K n ∈ K (λ n , a n , κ), with λ n (x, y) = λ n (r n x + x n , r n y + x n ) (see (2.2)). On the other hand In view of this and (3.16), by Lemma 2.2, as n → ∞, we have that where b is the weak-star limit of a n , which satisfy (2.5). We then conclude that L b w = 0 in R N . Now Lemma 8.3 implies that w is an affine function. This is clearly in contradiction with (3.15) since ̺ > 0.
Thanks to the choice of α < 2s, by (3.20), we get v n L 1 s ≤ C. (3.22) Using the same notations as in Step 1 for V n , f n and K n ∈ K (λ n , a n , κ), we see that where A n (r n ) := 1 |Br n | Br n (xn) u n (x) dx. Note that from Step 1 and Hölder's inequality, for every ̺ ∈ (0, (2s − β − α)/2), we can find a constant C > 0 such that for every n ≥ 2, We then define As above, we observe that η V n (1) ≤ Cr 2s−β By Lemma 9.2, for 1 < M < 1 2rn , we have Consequently, provided n is large enough and ε small, we obtain This with (3.20) imply that v n is bounded in H s loc (R N ) and, up to a subsequence, converges strongly, Since v n satisfy (3.24), by Lemma 2.5, (3.22) and (3.25), we have Letting n → ∞ in the above inequality and using Lemma 2.2, we find that L b v = 0 in R N , with b the limit of a n in the weak-star topology of L ∞ (S N −1 ). Moreover, from (3.20), we get BR |v(x)| 2 dx ≤ CR N +2α .
By Lemma 8.3, v is a constant function (because α < 1), which leads to a contradiction after passing to the limit in (3.19).

Uniform growth estimates at the boundary for solutions to Schrödinger equations.
Let Ω be an open subset of R N such that ∂Ω ∩ B 2 is a C 1 hypersurface. We will assume that 0 ∈ ∂Ω and that ∂Ω separates B 2 into two domains. As before, for K ∈ K (λ, a, κ), V ∈ V β and f ∈ M β , we consider the (nonempty) set of solutions: We have the following result. Proposition 3.3. Let s ∈ (0, 1), β ∈ [0, 2s), α ∈ (0, min(s, 2s − β)) and Λ, κ > 0. Then there exist ε 1 > 0 small and C > 0 such that for every λ : Proof. As in the proof of Proposition 3.2, if (3.27) does not hold, then we can a find sequence of real numbers r n → 0, sequence of points z n ∈ B 1 ∩ ∂Ω, sequences of functions λ n satisfying λ n L ∞ (B2×B2) < 1 n , a n satisfying (1.2), K an ∈ K (λ n , a n , κ), V n ∈ V β , f n ∈ M β u n ∈ S Ka n ,Vn,fn;Ω , with u n L 2 (R N ) + f n M β ≤ 1, such that for every r > 0 and n ≥ 2 (3.28) and Θ n (r n ) ≥ n/2 for all integer n ≥ 2. We define v n (x) = Θ n (r n ) −1/2 r −α n u n (r n x + z n ), so that We also let f n (x) := Θ n (r n ) −1/2 r 2s n f n (r n x + z n ) and V n (x) := r 2s n V n (r n x + z n ). By (3.1), we get with a constant C = C(N, s, β, α). It is clear that where K n (x, y) = r N +2s n K an (r n x + z n , r n y + z n ).
Next, by the monotonicity of Θ n and (3.28), we get Hence, for every n ≥ 2, v n 2 L 2 (BR) ≤ R N +2α for every R ≥ 1. (3.32) This with Hölder's inequality imply that BR |v n (x)|dx ≤ CR N +α for every R ≥ 1. . We then deduce that v n is bounded in H s loc (R N ). Hence by Sobolev embedding, (3.33) and since α < 2s, we may assume that the sequence v n converges strongly, in Next, we note that 1 Since Ω is of class C 1 , provided n is large enough, we have that ψ ∈ C ∞ c (Ω n ). Therefore by Lemma 2.5, (3.34), (3.32) and (3.30), we obtain with C(M ) a constant not depending on n ≥ n 0 and large. Denoting by b the weak-star limit of a n , then by Lemma 2.2, we get Furthermore by (3.32), v 2 L 2 (BR) ≤ CR N +2α , for every R ≥ 1. Since α < s, it follows from Lemma 8.3 that v = 0, which is impossible by (3.35).

Interior and boundary Hölder regularity estimates
4.1. Interior Hölder regularity. We have the following regularity estimates.
As a consequence of Corollary 4.1, we obtain regularity estimates for nonlocal operators with "uniformly continuous" coefficient. For K ∈ K (κ), we define the functions If there is no ambiguity, we will simply write λ e and λ o in the place of λ e,K and λ o,K , respectively.

4.2.
Hölder regularity estimates up to the boundary. Hölder regularity up to the boundary for the linear second order partial differential equations with coefficients in Morrey space was obtained in [14]. Coupling the interior regularity in Corollary 4.1 and the uniform L 2 growth estimates up to the boundary given by Proposition 3.3 together with some scaling arguments, we get the following result.
We consider the cut-off of the distance function d denoted by d s 2 := ϕ 2 d s . For u ∈ L 2 (R N ), z ∈ ∂Ω∩B 1 and r > 0, we let P r,z (u) be the L 2 loc (B r (z))-projection of u on d s 2 = Rd s 2 , the one-dimensional space spanned by d s 2 . Therefore For z ∈ B 1 ∩ ∂Ω and r > 0, we define Before going on, we explain the arguments in the next two main results of this section. Observe that by Hölder's inequality, for every r ∈ (0, r 0 ] and z ∈ B 1 ∩ ∂Ω, L 2 (Br (z)) u L 2 (Br (z)) = d s −1 L 2 (Br (z)) u L 2 (Br(z)) . Hence by Proposition 3.3 and (5.1), for every δ 0 ∈ (0, min(s, 2s − β)), there exist constants C, ε 0 > 0 such that for every λ ∈ L ∞ (B 2 ×B 2 ) satisfying λ L ∞ (B2×B2) < ε 0 , for every f ∈ M β and u ∈ S K,0,f ;Ω (recall the notation (3.26)), r ∈ (0, r 0 ] and z ∈ B 1 ∩ ∂Ω, we have Our objective is to get u/d s ∈ C s−β (B r0 ∩ Ω), whenever β ∈ (0, s) and Ω regular enough. This requires, at least, we already know that |u| ≤ d s , or equivalently |Q u,z (r)| ≤ C. For this purpose, we will use a bootstrap argument in two steps to obtain (5.5) with δ 0 = s, as long as β < s and under more regularity assumption on K and ∂Ω. This will be the content of the next two results.
In order to get the sharp boundary regularity, it will be crucial to quantify the action of the operators L K on d s for K ∈ K (λ, a, κ). To this end, we first note that by Lemma 9.3, up to decreasing r 0 if necessary, we may assume that d s 2 ∈ H s (B 2r0 ) ∩ L 1 s . Next, we introduce K (λ, a, κ, Ω), the class of kernels K ∈ K (λ, a, κ) such that: there exist β ′ = β ′ (Ω, K) ∈ [0, s) and a function g Ω,K ∈ M β ′ such that We note that the class of kernels K (λ, a, κ, Ω) is not empty. This is the case for K = µ a , with a satisfying (1.2), see Section 7 below. We have the following result.
We further define K n (x, y) := r N +2s n K an (r n x + z n , r n y + z n ) for every x, y ∈ R N , f n (x) := r 2s n f n (r n x + z n ) and g n (x) := r 2s n g Ω,Kn (r n x + z n ). (5.13) Since u n ∈ S Ka n ,0,f ;Ω , by (5.6), it is plain that (5.14) Claim: Let us put α ′ = α n − ̺ > 0, and we note that 0 < s − ̺ < α ′ < 2s − β, for every n ≥ 2. By a change of variable, we have ≤ 2Θ n (r n ) −1 r −N −2α ′ n u − P rnR,zn (u n ) 2 L 2 (Br n R (zn)) + 2Θ n (r n ) −1 r −N −2α ′ n P rnR,zn (u n ) − P rn,zn (u n ) 2 L 2 (Br n R (zn)) . Hence by (5.11) and the monotonicity of Θ n , we get v n 2 L 2 (BR) ≤ 2R 2α ′ + 2Θ n (r n ) −1 r −N −2α ′ n P rnR,zn (u n ) − P rn,zn (u n ) 2 L 2 (Br n R (zn)) . (5.16) Now by (5.1), for all r ∈ (0, r 0 ] and z ∈ B 1 ∩ ∂Ω, we have Now using the monotonicity of Θ n , we then deduce that there exists a constant C > 0 such that for every n ≥ 2, r ∈ (0, r 0 ] and z ∈ B 1 ∩ ∂Ω, Hence, for m ≥ 0, with 2 m ≤ r0 r , using (5.2) and the monotonicity of Θ n , we get As a consequence, we find that Now using this in (5.16), we then get, with C is a positive constant depending neither on n nor on m. We then conclude that v n 2 L 2 (BR) ≤ CR N +2α ′ for every R ≥ 1, with Rr n ≤ r 0 . (5.17) We now consider the case R ≥ 1 and Rr n ≥ r 0 . Using the fact that Θ n (r n ) −1 ≤ 1 and α ′ = α n − ̺ > 0 together with (5.3) and (5.2), we obtain v n with C > 0 independent on n. From now on, we let n large, so that B r0/(2rn) ⊂ B 2r0/rn (−z n ). Since v n satisfy (5.14), then letting v n,M = ϕ M v n , we can apply Lemma 9.2, for 1 < M < r0 2rn , to get where F n L ∞ (R N ) ≤ C 0 v n L 1 s . Using (5.15) and Hölder's inequality, we get F n L ∞ (R N ) ≤ C. It then follows that η Fv n (1) ≤ C for every n ≥ 2.
Since Θ n (r n ) −1/2 → 0 as n → ∞, by (5.15), we then deduce that v n is bounded in H s loc (R N ). Hence by Sobolev embedding, v n → v in L 2 loc (R N ), for some v ∈ H s loc (R N ). In addition, by (5.18) and since α n = 2s − max(β, β ′ n ) < 2s, we deduce that v n → v in L 1 s . We also have that 1 Ωn∩B 1/(2rn ) → 1 H in L 1 loc as n → ∞, where H is a half-space, with 0 ∈ ∂H. Moreover, passing to the limit in (5.12), we get since Ω is of class C 1 , for n large enough, we obtain ψ ∈ C ∞ c (Ω n ). Since v n satisfy (5.14), then by Lemma 2.5, (5.21) and (5.19), we obtain Thanks to Lemma 2.2, letting n → ∞, we thus get Here b denotes the weak limit of a n . Letting α := lim n→∞ α n ∈ [0, 2s), by (5.15), we have that v 2 L 2 (BR) ≤ CR N +2α for every R ≥ 1. It follows from Lemma 8.3 that v does not change sign on R N , which is in contradiction with (5.22). The next, result finalizes the two-step bootstrap argument mentioned earlier.
Combining Lemma 5.2 and the interior estimates in Theorem 4.2, we get the following result.

Higher order interior regularity
For K a kernel satisfying (2.2), we define the functions We suppose in the following in this section that, for 2s > 1, the function x → P V R N yJ o,K (x; y) dy belongs to L 1 loc (B 2 ; R N ). We then consider the map j o,K : B 2 → R N defined as where ℓ + := max(ℓ, 0) for all ℓ ∈ R. We note that if u ∈ C 2s+ε (Ω) ∩ L 1 s , for some ε > 0 and an open set Ω, then for every ψ ∈ C ∞ c (Ω), we have Moreover for every x ∈ Ω, we have We consider the family of affine functions q t,T (x) = t + (2s − 1) + T · x t ∈ R and T ∈ R N .
Hence, since 2s − β n ≥ 1 if 2s > 1, for every integer m ≥ 1, we get with C > 0 a constant independent on m and on n ≥ 2, since 2s − β n − 1 ≥ min(2s − 1 − β, d) > 0. Similarly, we also have that |t(2 m r) − t(r)| ≤ CΘ n (r)(2 m r) 2s−βn . Now for R ≥ 1, letting m be the smallest integer such that 2 m−1 ≤ R ≤ 2 m , we then get By the monotonicity of Θ n , we get the claim. It follows from (6.10) that w n L 1 s ≤ C for every n ≥ 2. (6.11) We define K n (x, y) := r N +2s n K an (r n x + z n , r n y + z n ), and we note that J o,Kn (x; y) = r N +2s n J o,Ka n (r n x + z n ; r n y).
We put P n (x) := P rn,zn (u n )(r n x + z n ) and let ψ ∈ C ∞ c (R N ). We use (6.3), to get Therefore writing P rn,zn (u n )(x) = t n + (2s − 1) + T n · (x − z n ), we see that P V R N (P n (x) − P n (x + y))J o,Kn (x; y) dy = (2s − 1) + (r n T n ) · r 2s n j o,Ka n (r n x + z n ) .
We then conclude that L Kn w n = r −(2s−βn) n Θ n (r n ) −1 f n + (2s − 1) + h n in B 1/2rn , (6.12) where, noting that ϕ 2 ≡ 1 on B 2 and recalling (6.1), f n (x) := r 2s n f n (r n x + z n ) and h n (x) = (r n T n ) · r 2s n j o,Ka n (r n x + z n ) ϕ 2 (r n x + z n ). Since u n L ∞ (R N ) ≤ 1, then |T n | ≤ r −1 n . Therefore, since by assumption, Next, we note that K n ∈ K ( λ n , a n , κ), with λ n (x, y) = λ n (r n x + z n , r n y + z n ). By assumption, λ n L ∞ (B 1/(2rn ) ×B 1/(2rn ) ) ≤ 1 n . Now by Corollary 4.1, (6.11) and (6.13), we deduce that w n is bounded in C δ loc (R N ), for some δ > 0. In addition thanks to (6.10), up to a subsequence, it converges in L 1 s ∩ C δ/2 loc (R N ) to some w ∈ C δ loc (R N ) ∩ L 1 s . Moreover, by (6.8) and (6.9), we deduce that w L ∞ (B1) ≥ 1 16 (6.14) and B1 w(x)p(x) dx = 0 for every p ∈ H 0 . (6.15) We apply Lemma 2.4 (after a cut-off argument as in the proof of Proposition 3.2), use (6.10) and (6.13) to get where we used the fact that Θ n (r n ) −1 ≤ 1, for every n ≥ 2. Therefore, provided ε is small enough, we find that w n is bounded in H s loc (R N ) and thus w ∈ H s loc (R N ). Now by Lemma 2.5 and (6.13), we have Letting n → ∞ in the above inequality and using Lemma 2.2, we find that L b w = 0 in R N , with b the limit of a n in the weak-star topology of L ∞ (S N −1 ). By (6.10) and Lemma 8.3, w ∈ H 0 , which contradicts (6.15) and (6.14).

Proof of the main results
Proof of Theorem 1.1. Suppose that Ω is domain of class C 1,γ , with 0 ∈ ∂Ω. We consider Ω ′ a bounded domain of class C 1,γ which coincides with ∂Ω in a neighborhood of 0. We let r > 0 small so that the distance function d = dist(·, R N \ Ω) is of class C 1,γ in Ω ∩ B 4r and d Ω ′ We define g Ω,µa (x) = (−∆) s a (ϕ 2r d s )(x) for x ∈ Ω ∩ B r and g Ω,µa (x) = 0 for x ∈ R N \ (Ω ∩ B r ). Then, there exists a constant C = C(Ω, N, s, Λ) > 0 such that for every x ∈ R N .
Proof of Corollary 1.7. Using a scaling and a covering argument as in the proof of Theorem 4.2 and applying Theorem 1.6, we get the result.
7.1. Proof of Theorem 1.8. We start with the following result which provides a global diffeomorphism that locally flattens the boundary of ∂Ω near the origin.
Let Ω be an open set with boundary of class C k,γ , for some k ≥ 1 and γ ∈ [0, 1]. Suppose that 0 ∈ ∂Ω and that the interior unit normal of ∂Ω at 0 coincides with e N . Then there exists ρ 0 > 0 such that for every ρ ∈ (0, ρ 0 ), there exists a (global) diffeomorphism Φ ρ : R N → R N with the following properties Here B ′ ρ denotes the ball in R N −1 centered at 0, with radius ρ > 0.
Proof of Theorem 1.8 (completed). We assume that the interior unit normal of ∂Ω at 0 coincides with e N . Consider Φ ρ ∈ C 1,1 (R N ; R N ), given by Lemma 7.1. In the following, we fix ρ > 0 small, so that By Theorem 4.4, there exists C > 0, only depending on N, s, Ω, β, Λ, γ, δ and V M β , such that We then have that L K u = f − uV on Ω. Letting U (x) = u(Φ ρ (x)) and F (x) = f (Φ ρ (x)) − U (x)V (Φ ρ (x)), then by a change of variable, we have and we note that by (7.20), ϕ ρ/4 F ∈ M β . We observe that .

Appendix 1: Liouville theorems
In this section we consider H being either R N or the half-space R N + = {x ∈ R N : x N > 0}. We prove a classification result for all functions u ∈ H s loc (R N ) ∩ L 1 s satisfying L b u = 0 in H and u = 0 in H c , provided b is a weak limit of a n satisying (1.2) and u satisfying some growth conditions. We note that in the case u ∈ L ∞ loc (R N ) such classification results (for more general nonolcal operators L b ) are proved in [47]. We will need the following result for the proof of the Liouville theorems. Then there exists C = C(N, s, Λ, Ω) > 0 such that u L ∞ (B1∩Ω) ≤ C u L 1 s + u L 2 (B2) . Proof. The interior L ∞ loc (B 2 ∩ Ω) estimate follows from [17], where the authors used the De Giorgi iteration argument. We note that in [17], it is assumed that u ∈ H s (R N ) but by carefully looking at their arguments, we see that this can be weakened to u ∈ H s loc (R N ) ∩ L 1 s . For the L ∞ (B 1 ∩ Ω) estimate, the proof is precisely the same.
In the following, for b satisfying (2.5) and f ∈ H s (R N ), we put Recall the Poincaré-type inequality related to this seminorm, see [24,27], 2), we deduce that the sequence (v n,M ) n is bounded in H s (R N ). Hence, up to a subsequence, (v n,M ) n converges in L 2 (B 2 ∩Ω) and in L 1 s to some function v M . Passing to the limit in (8.2) as n → ∞ and using Lemma 2.2, we find that We then deduce that w M → 0 in L 2 (R N ) as M → ∞, by (8.1). In addition, we have We conclude that v M → u in L 1 s as M → ∞. Letting M → ∞ in (8.5), we get u L ∞ (B1∩Ω) ≤ C u L 1 s + u L 2 (B2) .
Lemma 8.3. Suppose that there exists a sequence of functions a n satisfying (1.2) and a n * for some γ < 2s and for every R ≥ 1.

(8.7)
Then u is an affine function if H = R N , while u is proportional to max(x N , 0) s if H = R N + . Proof. We put v R (z) = u(Rz), for R ≥ 1 and z ∈ R N . Since L b v R = 0 in H and v R = 0 on H c , by Lemma 8.2, v R L ∞ (B1) ≤ C v R L 1 s + v R L 2 (B2) . Scaling back, we get u L ∞ (BR) ≤ C R 2s R N |u(x)| R N +2s + |x| N +2s dx + 2 γ R γ . Now, using Hölder's inequality and (8.7), we get u L 1 (BR) ≤ CR N +γ . We then have We deduce that u L ∞ (BR) ≤ CR γ for all R ≥ 1.
It follows from the Liouville theorems in [45], that u is an affine function if H = R N , while u is proportional to max(x N , 0) s when H = R N + .
Using the above two estimates above in (9.2), we get the result.
The following result provides a localization of solutions for nonlocal equations.
Therefore letting The proof is thus finished.
We close this section with the following result.

Lemma 9.3.
Let Ω be C 1 open set with 0 ∈ ∂Ω and such that the interior normal of ∂Ω at 0 coincides with e N . Then d s ∈ H s (B r ), for some r > 0.
This implies that d s + ∈ H s loc (R N ). To conclude, we use the parameterization Φ ρ given by Lemma 7.1 and make changes of variables, to get  provided ρ is small enough. The proof is thus finished.