The Dirichlet Problem for L\'evy-stable operators with $L^2$-data

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of $2s$-stable processes and exterior data, inhomogeneity in weighted $L^2$-spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit $s\to 1-$ which allows us to recover the local theory.


Introduction
The study of the existence, uniqueness, and regularity of harmonic functions in domains with prescribed boundary values has a long history.In the late 1970s and 1980s significant progress was made on the Dirichlet problem with irregular boundary data.It is well known that the problem −∆u = 0 in Ω, u = g on ∂Ω, (1.1) admits a unique solution u in H 1/2 (Ω) for g in L 2 (∂Ω) and a sufficiently regular domain Ω ⊂ R d , see e.g.[JK95].One goal in this field of study is to lower the assumptions on the regularity of the boundary ∂Ω, cf.our overview in Subsection 1.3.A major difficulty in studying (1.1) in rough domains is the lack of Poisson kernel estimates near the boundary of Ω.In addition, the low regularity of the boundary datum g prohibits the use of a variational ansatz in H 1 (Ω) since the classical trace and extension theorems restrict us naturally to boundary data in H 1/2 (∂Ω).
In the last decades, nonlocal operators enjoyed notable attention.Naturally motivated by the Courrège theorem, see [Cou65], which characterizes linear operators that satisfy the positive maximum principle, these operators appear in probability theory, fluid mechanics and mathematical physics.In this work, we study the subclass of 2s-stable, nondegenerate integro-differential operators A s u(x) := p.v.
The class of 2s-stable operators includes the fractional Laplacian (−∆) s which corresponds to µ being a multiple of the surface measure on the sphere.The assumption (1.4) yields the comparability of the Fourier symbol of A s to that of the fractional Laplacian, i.e. |ξ| 2s .
These operators are translation invariant, symmetric and exhibit the scaling behavior A s [u(λ•)](x) = λ 2s A s [u](λx), x ∈ R d , λ > 0. Furthermore, elliptic second-order operators a ij ∂ ij with constant coefficients a ij appear in the limit of A s as s → 1−, cf.[FKV20].
Another motivation to consider operators of the form (1.2) arises from the field of stochastic processes.More precisely, the operators A s under the assumption (1.4) are the generators of symmetric 2s-stable nondegenerate processes, see e.g.[Sat99].
In analogy to the classical problem (1.1), we study the exterior data problems for given f : Ω → R and g : Ω c → R and a bounded C 1,α -domain Ω ⊂ R d .More precisely, we study the existence, uniqueness and Sobolev-regularity of distributional solutions to the equation (1.5).We say that a function u : R d → R is a distributional solution to (1.5) if it satisfies and u = g on Ω c , as well as some appropriate integrability properties.
Similar difficulties as in the classical setup (1.1), described above, occur.In particular, boundary estimates of the corresponding Green and Poisson kernels for the linear operators A s in general C 1,αdomains are not known.Furthermore, exterior data in some weighted L 2 -space are not sufficient to allow for a variational approach, see e.g.[GH22].
The goal of this work is twofold.Firstly, we want to provide a proof of the existence, uniqueness and Sobolev-regularity of distributional solutions to (1.5) for f, g in some weighted L 2 -spaces that is both short and easy to generalize.In particular, we will avoid any potential theory.Secondly, we want to provide a regularity theory that is continuous in the order of differentiation s, i.e. the classical result for second-order operators, as described above, is retrieved in the limit s → 1−.This entails a careful study of all estimates in terms of s.Note that the data g in the classical problem (1.1) is given on the topological boundary of Ω which is in align with the locality of the Laplacian.In turn, the nonlocal problem (1.5) requires exterior data on the complement of Ω.Thus, an appropriate choice of exterior data g is essential to achieve a 'continuous' theory.For a big subclass of the operators under consideration, like the fractional Laplacian (−∆) s , a fitting weighted L 2 -space is already introduced in the previous works [GH22] and [GK23] which deal with robust nonlocal trace and extension results.
1.1.The main result.Throughout this work, we assume Ω ⊂ R d to be a bounded C 1,α -domain for some α ∈ (0, 1).In order to study the well-posedness of the distributional problem (1.5), it is essential that the integrals in (1.6) exist.In particular, a distributional solution u needs to be integrable against A s (η) for any η ∈ C ∞ c (Ω).A corresponding tail weight was introduced in [GH23]: For any s ∈ (0, 1) we define (1.7) This weight decays like A s (η) at infinity and is locally bounded.Thus, u ∈ L 1 (R d ; ν ⋆ s ) is a sufficient and necessary condition to study distributional solutions of (1.5).Instead of ν ⋆ s , many authors use ν ⋆ s (x) := (1 − s)/(1 + |x|) d+2s .It was proved in [GH23, Lemma A.1] that in some cases, including the fractional Laplacian, this weight is comparable to ν ⋆ s .We want to emphasize that the choice ν ⋆ s is not optimal whenever the measure µ is not supported on the whole sphere.For example the sum of Dirac measures on an orthogonal basis of R . ., d}}.Next, we introduce the weighted L 2 -spaces for the exterior data g in the problem (1.5).From the above discussion, it is clear that the weight needs to behave like ν ⋆ s at infinity.Additionally, one goal of this work is to provide a theory that is continuous in terms of s.Thereby, we need to retrieve the space L 2 (∂Ω) in the limit s → 1−.In the case of the fractional Laplacian, this was achieved in [GH22] and [GK23] where trace and extension results for a variational approach was proved.In the following definition we introduce a weight in the spirit of [GH22] that is adapted to the operators A s .
Definition 1.1.Let s ∈ (0, 1).The measure τ s on B(R d ) is defined by τ s (dx) := τ s (x)dx where ) and d x := dist(x, ∂Ω).Now, we state our main result. (1.9) This result is new, even in the case of the fractional Laplacian on C 1,α -domains.Only recently in [KR23], Theorem 1.2 was proved for (−∆) s on C 1,1 -domains but the dependency of the constant in (1.9) on s is unclear.
Remark 1.3.The first inequality in (1.9) holds true for all measurable u : Ω → R for which the right-hand side is finite, see Lemma 3.1.It is a fractional analogue of [JK95, Theorem 4.1].
Remark 1.4.The proof of Theorem 1.2 reveals that is bounded by the right-hand side of (1.9) for the solution u ∈ H s/2 (Ω).This term describes how the solution u in Ω meets the exterior data g at the boundary ∂Ω.This is a purely nonlocal phenomenon since this term vanishes in the limit s → 1−.
Remark 1.5 (Optimality).Let Ω be any bounded C 1,1 -domain, g(x) x , and u(x) := ´Ωc g(y)P s (x, y)dy, where P s is the Poisson kernel related to the fractional Laplacian.Then u solves (−∆) s u = 0 in Ω and u = g on Ω c .The exterior data g is in L 2 (Ω c ; τ s ) if and only if ε > (1 − s)/2.A short calculation, using Poisson kernel estimates, yields The right-hand side is finite if and only if ε > (1 − s)/2.
Remark 1.6.Under stronger assumptions on the exterior data g, the variational approach implies the H s (Ω)-regularity.This approach requires an appropriate function space V s (Ω|R d ), which was first introduced in [FKV15].An optimal choice of exterior data requires the study of trace and extension results, done in [DK19], [BGPR20], [GH22] and [GK23].Robust estimates like (1.9) were proved in the latter two articles.
The robustness of the constant in (1.9) in terms of s allows us to retrieve the classical H 1/2 -regularity result for the problem (1.1) in the limit s → 1−.This follows from a direct application of Theorem 1.2 and an appropriate approximation, see Theorem 5.3.
1.2.Strategy of the Proof.In the following, we sketch the proof of Theorem 1.2 in the special case of the fractional Laplacian.The first step is to prove a weighted Sobolev embedding In the second step, we introduce a particular auxiliary function φ to the right-hand side of (1.10) in place of d s x .As such it is essential to analyze the boundary behavior of this function.We choose φ to be the classical solution to This function satisfies c 2 d s x ≤ φ(x) ≤ c 3 d s x for x ∈ Ω, see [RF23, Proposition 2.6.6].Together with the properties of the carré du champ, the second term on the right-hand side of (1.10) can be estimated as follows: The first term is easily bounded from above using the Hölder inequality.A nonlocal Gauss-Green formula reveals that the second term equals Here N s is the nonlocal normal derivative, see (2.2).The first term is nonpositive and −N s φ can be estimated from above by τ s , see Lemma 2.1.To show that u is square integrable, we use a similar strategy after introducing the operator to the L 2 -norm via The main difficulty in adapting the proof for the fractional Laplacian to general 2s-stable operators is to prove the analog of (1.10), i.e. to introduce the corresponding carré du champ.This is done in Subsection 3.1.
On C 1,1 -domains, it is not necessary to introduce the auxiliary function φ to prove the bound on ´Ω d s x Γ s (u)(x)dx, because the term (−∆) s (d s x ) is bounded in that case.Note that, in contrast to the respective term with φ, the sign of (−∆) s (d s x ) in general domains is unknown and the integral cannot be dropped.For general nondegenerate stable operators A s the term A s (d s x ) is not essentially bounded even in smooth domains Ω, see [RS16b, Proposition 6.2].
In the case of the Laplacian, the solution φ 1 to −∆φ 1 = 1 in Ω and φ 1 = 0 on ∂Ω was also used in [Pet83] to prove a regularity result.
1.3.Related literature.Existence theory and Sobolev regularity for weak variational elliptic and parabolic nonlocal problems was studied in [FKV15] and a general class of integro-differential operators with possibly antisymmetric kernels were treated.The article [KR23] contains regularity results for distributional solutions to the Dirichlet problem for the fractional Laplacian on C 1,1 -domains.Exterior data and inhomogeneities in certain weighted Sobolev spaces of any order were treated.These weighted spaces were introduced in [Lot00].A careful study of these spaces shows that their result includes our result for the special case of the fractional Laplacian and C 1,1 -domains.We extend their result to C 1,α -domains with a robust estimate for s → 1−.The approach in [KR23] relies on a potential analysis of the Poisson kernel and Green function close to the boundary.Our approach allows us to consider a larger class of operators and more general domains.The article [KR23] is an extension of [CKR22], where only zero exterior data were treated, and [Gru14], where C ∞ -domains and more regular boundary data were considered.
There is also interest in nonlocal problems with boundary data being only prescribed on ∂Ω, cf.[Gru15] and references therein.
An early contribution to boundary regularity of solutions to exterior data problems, like (1.5), is [Bog97], where the author proved a boundary Harnack principle for the fractional Laplacian on Lipschitz domains.In [Sil07, Proposition 2.8, Proposition 2.9] Hölder regularity for distributional solutions to (−∆) s u = f in R d was proved.Interior Hölder regularity for weak L-harmonic functions, where L is an integrodifferential operator with a kernel comparable to the one of the fractional Laplacian, was studied in [Kas09, Theorem 1.1].This was extended to operators with kernels which have critically low singularity and do not allow for standard scaling in [KM17, Theorem 3].Optimal boundary regularity estimates for distributional solutions to (−∆) s u = f in Ω and u = 0 on Ω c were proved in [RS14, Proposition 1.1].There, Ω is a Lipschitz domain satisfying the uniform outer ball condition.Moreover, higher Hölder regularity up to the boundary of the quotient u/d s x on a C 1,1 -domain was studied, see Theorem 1.2 in [RS14].This was extended to all generators of 2s-stable processes in [RS16a, Theorem 1.2, Proposition 4.5].In [RS17, Theorem 1.2, Theorem 1.5], boundary Hölder regularity of distributional solutions in C 1,α -domains was proved.This result allows us to study the boundary behavior of our auxiliary function φ, see (2.4).The article [DK20] contains Hölder regularity estimates for weak solutions to Lu = f on R d for more general integro-differential operators that are defined via measures, which are allowed to be singular.The estimate is robust in the order of differentiability.In [DK20, Theorem 1.11], the comparability of the energy forms corresponding to singular operators like (1.2) to the energy form related to the fractional Laplacian was studied.We apply this result in Lemma 3.1.
Integro-differential operator of variable order and Hölder regularity of solutions to the respective Dirichlet problem in balls was studied in [BK05, Theorem 2.2].For Hölder regularity estimates of solutions to fully nonlinear nonlocal equations we refer to [CS09,CS11a,CS11b], and for boundary regularity to [RS16a].In [Ser15], the author proved Hölder regularity of solutions to concave nonlocal fully nonlinear elliptic equations.Therein, the operators are allowed to have non-smooth kernels and do not have to be translation invariant.In [CKW22], robust Hölder regularity estimates for minimizers of nonlocal functionals with non-standard growth were shown.Interior and boundary regularity for distributional solutions to Lu = f in C 1,α -domains and zero exterior data are proved in [DRSV22, Theorem 1.1, Theorem 1.2, Corollary 1.3], where L is a translation invariant integro-differential operator with a nonsymmetric kernel.Additionally, Theorem 1.5 contains a new integration by parts formula for these kinds of operators.In [KW22], nonsymmetric non-translation invariant nonlocal operators were treated.
The convergence of fractional Sobolev spaces to classical Sobolev spaces was first established in [BBM01].In the article [FKV20], the nonlocal to local convergence in the sense of Mosco for quadratic forms, which appear in the variational study of nonlocal Dirichlet problems, was proved.This result was extended in [Fog20] to a larger class of integro-differential operators.In [Fog20] and [FK22], the convergence of weak solutions to nonlocal Dirichlet and Neumann problems to weak solutions of local Dirichlet and Neumann problems with nonzero boundary data was studied.The convergence of Neumann problems was extended in [GH22] to a more general class of exterior data.This was achieved by a careful study of the trace spaces related to the fractional-type Sobolev spaces which naturally appear in the setup of the weak formulation to problems like (1.5).For trace and extension results in the L p -setting for all p ≥ 1 we refer to [GK23].
Green function and Poisson kernel estimates for the fractional Laplacian on C 1,1 -domains were studied in [CS98] and [Che99].In [BS05], a Harnack inequality and on-diagonal Green function estimates for a subclass of 2s-stable Lévy processes on balls were proved.This result was extended in [BS07] to a more general class of operators.In particular, the article contains a characterization of those operators in the class of 2s-stable symmetric nonlocal operators that admit a Harnack inequality.For Dirichlet heat kernel estimates for cylindrical stable processes on C 1,1 -domains we refer to [CHZ23].
Finally, we give a short overview of results concerning the Dirichlet problem for second-order operators like (1.1).To our knowledge the first boundary regularity result with L 2 -boundary data was achieved in [Mih76] who studied the problem in C 2 -domains.This result was generalized in [CT83] to a larger class of operators.The case of C 1,α -domains and boundary data in L p , 1 < p < ∞, was treated in [Pet83].In [Lie91], the assumption on the boundary ∂Ω was reduced to C 1,Dini .In the influential articles [Ken94,JK95] the H 1/2 (Ω)-regularity result was proved in Lipschitz domains.The proof relies on fine estimates of the harmonic measure from [Dah77,Dah79] and weighted norm inequalities for the nontangential maximal function from [Dah80].
1.4.Outline.In Section 2, we introduce the notation used throughout this work as well as the auxiliary function φ.In Subsection 3.1, we prove the key estimates to relate the carré du champ, associated with the class of 2s-stable operators, with the H s/2 -seminorm.Subsection 3.2 contains a robust weighted Sobolev inequality which yields the H s/2 -regularity of distributional solutions.The proof of Theorem 1.2 is given in Section 4. As an application, we retrieve the H 1/2 -regularity of solutions to the local Dirichlet problem with boundary data in L 2 (∂Ω) by a nonlocal to local convergence of solutions in Section 5.
Acknowledgments.Financial support by the German Research Foundation (GRK 2235 -282638148) is gratefully acknowledged.We thank Moritz Kassmann for helpful discussions and Solveig Hepp for valuable comments on the manuscript.

Preliminaries
In this section, we introduce the notation we use through this work.Additionally, we define basic objects like the carré du champ and the nonlocal normal derivative.Lastly, we provide the existence of the auxiliary function φ.

Let us denote d
We use a∧b for min{a, b} as well as a∨b for max{a, b}.With a + and a − we denote the positive respectively the negative part of a.If not mentioned differently then Ω is always a bounded C 1,α -domain which means there exists a localization radius r 0 > 0 such that for any z ∈ ∂Ω there exists a rotation and translation Since ∂Ω is compact, we find a uniform constant L = L(Ω) ≥ 1 such that φ C 1,α ≤ L. In the proofs we use small c 1 , c 2 , c 3 , . . . to mark different constants and in the statements of the theorems we use the capital letter C.
For s ∈ (0, 1) we write H s (Ω) for the classical L 2 -based Sobolev-Slobodeckij space on a domain Ω.It is endowed with the norm With L p (Ω; w(x)) we denote the L p -space where integration is with respect to the weighted Lebesgue measure w(x)dx.For all α > 0 we write C α (Ω) = C ⌊α⌋,α−⌊α⌋ (Ω) for the space of all Hölder continuous function on Ω equipped with the norm u Here we write for 0 < s < 1 The set of locally Hölder continuous functions is denoted by C α loc (Ω).Furthermore, we define the carré du champ operator related to A s as The nonlocal normal derivative related to A s is defined via This nonlocal normal derivative appeared first in [DRV17].We also refer to the work [DGLZ12] who studied a similar operator in the context of peridynamics.
Lemma 2.1.Let Ω ⊂ R d be a bounded open set.There exists a constant C = C(Ω, Λ) ≥ 1 such that for any s ∈ (0, 1) In this case we calculate: (2. 3) The proof of our main result relies on a specific auxiliary function and a careful analysis thereof near the boundary of Ω.This is done in the following lemma.
Lemma 2.2.Let Ω ⊂ R d be a C 1,α -domain, α ∈ (0, 1) and s ⋆ ∈ (0, 1).For any s ∈ (s ⋆ , 1) there exists x for all x ∈ Ω. (2.5) The function φ is the expected value of the first exit time from the domain Ω of the stochastic process that is generated by A s .
Proof.The existence of a weak solution φ follows from Lax-Milgram, see e.g.[GH23,Proposition 2.11].This together with a nonlocal Gauss-Green formula and the regularity theory, see [RS17] and [RF23, Proposition 2.6.9],reveals that φ ∈ C s (R d ) ∩ C 3s loc (Ω) is a classical solution.The lower bound in (2.5) follows from the Hopf lemma [RF23, Proposition 2.6.6] and the upper bound was proved in [RF23, (2.6.3)].The dependencies of the constant C follow from carefully following the constants in the proofs.For the convenience of the reader we provide a meta analysis of the proofs of Proposition 2.6.6 and (2.6.3) from [RF23] in Remark A.1.

Functional inequalities
In this section, we provide most of the important estimates for the proof of Theorem 1.2.The necessary ingredients to lift the proof from the fractional Laplacian, as described in Subsection 1.2, to general 2s-stable operators are stated in Subsection 3.1.In particular, the necessary bound to introduce the carré du champ of A s and gain access to (1.5) is done in Lemma 3.1.The last part of this section concerns a weighted Sobolev inequality which already proves the first inequality in (1.9).Our main contribution there is the robustness of the constant.
The weighted L 2 -term on the right-hand side of (3.1) can be dropped if one proves a robust Poincaré inequality for the energy corresponding to A s .Instead, we treat this term using Proposition 3.2 to receive the L 2 (Ω c ; τ s )-norm of the exterior data, see Corollary 3.3.
Proof.We divide the proof into three steps.
Step 1: We prove that there exists a constant After an application of the triangle inequality |u 2 ) we are left with two integrals to estimate.The first one is: dx.
The second one needs a little bit more care.Let Step 2: We claim that there exists a constant c 2 = c 2 (d, s ⋆ , λ, Λ) > 0 such that for any ball B r (x 0 ) ⊂ R d we have: (3.4) By step 1, it suffices to consider the integral over (x, y) ∈ Ω × Ω such that |x − y| < d x /8.For such (x, y) and We use the Whitney covering to find: Using (3.3) for each ball B ⋆ , we estimate the right-hand side of the previous inequality by: Finally, the observation that for x ∈ B ⋆ the distance of x to the boundary ∂Ω can be estimated by d x ≥ d xB − r B ≥ r B together with the finite overlap property from (3.4) completes the proof.
Proposition 3.2.Let Ω ⊂ R d be a bounded C 1,α -domain with α ∈ (0, 1), s ⋆ ∈ (0, 1) and µ a finite measure on S d−1 that satisfies the nondegeneracy assumption (1.4).There exist two constants C = C(Ω, α, s ⋆ , λ, Λ) > 0, a = a(s ⋆ , λ, Λ) > 1 and a radius ρ = ρ(Ω, α, s ⋆ , λ, Λ) > 0 such that for any s ∈ (s ⋆ , 1) We actually do not need Ω to be bounded nor connected in this lemma.What we really need is a uniform bound on the C 1,α -norm of the maps describing the boundary of Ω locally.For the special case of cylindrical stable processes and when Ω is a ball this result is proven in [CHZ23, Lemma 2.3].Moreover, the result for the fractional Laplacian is straight forward and also true for Lipschitz domains, see e.g.[GK23].In contrast, Proposition 3.2 is not true in general Lipschitz domains and general µ even under the assumption (1.4).This is due to the fact that in a domain with a corner the support of ν s translated by x ∈ Ω may not hit the boundary close to x.An example for this phenomenon is , where e i is the i-th coordinate vector, and x = (−t, −t) for t ∈ (0, 1/2).
Proof.Since Ω is a bounded C 1,α -domain, we find a localization radius r 0 > 0 such that for any z ∈ ∂Ω there exists a rotation and translation T z : R d → R d with T z (z) = 0 and a C 1,α -continuous map We set .
Let z x ∈ ∂Ω be a minimizer of x to the boundary ∂Ω.Since ∂Ω is continuously differentiable, the vector (z x − x)/ |z x − x| is the outer normal vector in z x , which we call for short n x .Without loss of generality we assume that T zx = Id and x = (0, −d x ).From now on we simply write φ = φ zx .This is due to A s being translation invariant and the ellipticity assumption (1.4) is invariant under rotations.Since z x = 0 minimizes the distance of x = (0, −d x ) to the graph of φ, the gradient of φ is zero in z x , i.e. ∇φ(0) = 0. Otherwise we easily find y ∈ ∂Ω which is closer to x than z x by a Taylor expansion of φ.
A simple calculation yields (3.5) We will show that the term (II) is smaller than one half of the left-hand side of (3.5) such that we can absorb the term.To achieve this, we need to find an upper bound of |n x • θ| for θ = (θ ′ , θ d ) ∈ S d−1 and any r ∈ ((a − 1)d x , ad x ) such that x + rθ and x − rθ are in Ω.We fix x + rθ with this property.Note that x + rθ ∈ B r0 (z x ) since ρ ≤ r 0 /(2(a + 1)).We distinguish two cases.
Case 2: If rθ d < 0, then we use the property x − rθ ∈ Ω and apply case 1 to x + r(−θ).By symmetry, this yields the same bound (3.6). x Here ∂ Ω is the surface ∂Ω reflected with respect to x.The grey shaded area visualizes the integration domain of the term (II).The angle β is close to π/2 by the C 1,α -boundary and the choice of a.
Finally, we insert this into the term (II) to find: x .
This puts us into the position to absorb the term (II) to the left-hand side of (3.5): Recall that Γ s is the carré du champ for the operator A s , see (2.1).
Proof.The proof follows from Lemma 3.1 with β = s, Proposition 3.2 and triangle inequality adding ±u(x + rθ) in the term u L 2 (Ω;d −s x ) with a calculation similar to the proof of Lemma 2.1.
Lemma 3.4.Let s ⋆ ∈ (0, 1) and Ω ⊂ R d be a bounded Lipschitz domain.There exists a constant C = C(d, Ω, s ⋆ ) ≥ 1 such that for any s ∈ (s ⋆ , 1) for any u : Ω → R such that the right-hand side is finite.
Proof.We divide the proof into three steps.First we prove a localized version of the statement and then pick a Whitney decomposition to prove the statement on Ω.
Step 1: We show that there exists a constant c 1 = c 1 (d, s ⋆ ) ≥ 1 such that for any ball B r with radius r > 0 and any function v ∈ H s (R d ) we find Without loss of generality we assume the ball B r to be centered at the origin by translation invariance.We prove the statement for r = 1 and then conclude by scaling.Let E : H s (B 1 ) → H s (R d ) be a Sobolev extension operator with the operator norm being bounded by Since the seminorms in the previous inequality are invariant under additive constants, we can apply it to w Now, scaling this inequality to balls with radius r yields the claim.
Step 2: It is proven in [Dyd06] that there exists a constant See also Remark A.2 for the s-dependence of the constant c 5 .
Step 3: Let W be a Whitney ball covering of Ω as in step 3 in the proof of Lemma 3.1.In particular, recall that for any ball B ∈ W we denote by r B its radius, by B ⋆ a slightly bigger ball with radius (7/4)r B and by x B its center.These slightly bigger balls satisfy a bounded overlap property with a constant N ∈ N, see (3.4).Now, we use step 2 and decompose the integral over Ω into these Whitney balls: In the beginning of step 3 in the proof of Lemma 3.1 we proved that for any B ∈ W, x ∈ B and y ∈ B dx/8 (x) both x and, more importantly, y are in the slightly bigger ball B ⋆ .We apply this and step 1 to find: By the properties of the Whitney covering, see (3.4), we find that r s B ≤ 4 s (d x ∧ d y ) s for x, y ∈ B ⋆ .Thus, summing over all balls yields Here we used the finite overlap property of the family of slightly bigger balls.

Proof of Theorem 1.2
We structure the proof as follows: First we show that solutions to (1.5) for regular f, g satisfy the bound (1.9).This will be done in the following two lemmata.Then we conclude the result by approximating rough f, g as in Theorem 1.2 by a sequence of regular functions f n , g n .
Since φ and Γ s (u) are nonnegative, the second term is nonpositive.In addition, we use a b ≤ 1/8a 2 +2 5 b 2 , for any real a, b, in the first term and −N s (φ) ≤ c 2 τ s in the last term to find Absorbing the L 2 -norm of u and using the upper distance bound on φ, we find the desired estimate: For any s ∈ (s ⋆ , 1) and any classical solution u ∈ C s (R d ) ∩ C 3s loc (Ω) to (1.5) the estimate (1.9) is satisfied.
Proof.By Lemma 4.1, u satisfies the correct L 2 -bound (4.1).Thus, it suffices to estimate the seminorm on the left-hand side of (1.9).Now let φ ∈ C s (R d )∩C 3s loc (Ω) be the classical solution to A s φ = 1 in Ω and φ = 0 on Ω c from Lemma 2.2.Further, let c 1 = c 1 (d, Ω, α, s ⋆ , λ, Λ) ≥ 1 such that c −1 1 d s x ≤ φ(x) ≤ c 1 d s x .By Lemma 3.4 and Corollary 3.3 there exists a constant c 2 = c 2 (d, Ω, α, s ⋆ , λ, Λ) ≥ 1 such that Again, after estimating d s x by φ, we use Thus, the first term on the right-hand side of (4.2) is bounded by Since u and φ are classical solution with smooth data, we can use the nonlocal Gauss-Green formula to find: The first term is nonpositive and for the second term we use Lemma 2.1 to estimate (Ω;τs) .
Proof of Theorem 1.2.Let f ∈ L 2 (Ω; d 2s x ) and g ∈ L 2 (Ω c ; τ s ).By the density of C ∞ c -functions in L 2spaces with Radon measures on R d , there exist sequences {f m } and {g m } where f m ∈ C ∞ (Ω) and Let u m ∈ C s (R d ) ∩ C 3s loc (Ω) be the classical solution to (1.5) with the data f m and g m .The existence of such u m follows as in Lemma 2.2.By Lemma 4.1 and Lemma 4.2, we find a constant . By the linearity of (1.5), the sequence {u m } is a Cauchy sequence in H s/2 (Ω) and there exists a limit ũ ∈ H s/2 (Ω).For every φ Thus, an application of Hölder's inequality yields As in Lemma 2.1, the operator A s acting on the test function φ on Ω c is bounded by a multiple of τ s , i.e.N s (φ)(x) ≤ c 2 τ s (x) for x ∈ Ω c with c 2 = c 2 (d, Ω, α, s ⋆ , λ, Λ).Since τ s is a finite measure on Ω c , we conclude lim m→∞ (g m − g, A s (φ)) L 2 (Ω c ) = 0.
Lastly, we prove the uniqueness of distributional solutions in H s/2 (Ω).Suppose there is another distributional solution v ∈ H s/2 (Ω).Then u − v is a distributional solution to the problem (1.5) with f = 0 and g = 0. Since u − v ∈ H s/2 (Ω), the fractional Hardy inequality, see [Dyd04, Theorem 1.1 (17)] or [CS03, Theorem 2.3], yields a constant c 4 such that for ε > 0  it was shown that, for g = 0, solutions in the sense of (5.2) satisfy Ω) and (5.3) is a solution in the sense of (5.2) for g = 0.
Step 1: We equip the space of test-functions C 1,∆ 0 (Ω) with the norm Step 2: We reduce the claim to g ∈ H 1/2 (∂Ω) by approximating g ∈ L 2 (∂Ω) with a sequence of elements g n ∈ H 1/2 (∂Ω).If the solutions u n satisfy the bound u n H 1/2 (Ω) ≤ c( f L 2 (Ω;d 2 x ) + g n L 2 (∂Ω) ), then the limit u satisfies the same bound with g n replaced by g. and supremum of c(s, ε ′ ) w.r.t.s ∈ (s ⋆ , 1) to get bounds independent of s.For s close to 1, the idea is to separate dt into two integrals over (0, 1) and (1, ∞) and treat them separately.We will call the integration variable in both terms t from now on.Estimating the (1, ∞) integral is easy, as it does not contain the singularity at t = 0.For t ∈ (0, 1) one can use Taylor's formula to rewrite (1 ± t) s+ε ′ .This will lead to the factor s + ε ′ − 1, which explains the choice of s > 1 − ε ′ .Moreover, one has to take care of the singularity appearing in t = 1, as the Taylor expansion yields a term (1 − t) s+ε−2 .This can be treated by additionally separating the integral into 0 < t < 1/2 and 1/2 < t < 1 to split the singularities.Finally, there will appear a (1 − s) −1 -term, which gets canceled by the (1 − s)-term from the definition of (−∆) s R .Remark A. for a Lipschitz domain Ω and any η ∈ (0, 1).We will now discuss that the constant can be chosen depending on s only through a lower bound 0 < s ⋆ < s < 1.The main ingredients are [Dyd06, Lemma 3, Proposition 5].In Lemma 3 it is proven that for any p > 0, β > 1 there exists a constant c > 0 such that for every nonnegative sequence (a n ) n one has More precisely, the author shows that for any m ≥ 1 we have Thus, for β > 1 any choice of the integer m such that m ≥ (p − 1) ln 2 ln β gives the desired estimate.In the proof of (A.3), Lemma 3 is used for β = (1 + η M ) s , where M > 1 is a number depending only on the Lipschitz constant of Ω.This will lead to the fact that the constant c depends on s −1 .In the last step of Proposition 5, the constant C depends on s −1 from the use of Lemma 3, some geometric quantities and the sum The remaining steps in the proof of Proposition 5 and (A.3) are purely geometric and therefore do not lead to a factor depending on s.

½
Br (x0) (x + tθ)µ(dθ)dtdx.(3.3)This result follows from [DK20, Theorem 1.11] together with a scaling argument.Note that (1.4) implies the nondegeneracy assumption lin supp µ = R d in [DK20, Theorem 1.11].Step 3: We localize the problem and apply step 2. Let W(Ω) = W be a Whitney covering of the open set Ω by balls, see e.g.[CW77, Theorem 3.2].We denote for B ∈ W by x B the center, by r B the radius of the ball B and write B ⋆ = 7/4 B for the ball with 7/4-times the radius of B but the same center.There exists a number N = N (d, Ω) ∈ N such that
2. Within the proof of [Dyd06, Theorem 1], more precisely equation (13), the author proves Ω Ω |u(x) − u(y)| |x − y| d+s dydx ≤ C(d, Ω, η, s) ) s in all coordinate directions and the measure ν ⋆ s (x)dx is supported only in a neighborhood of the coordinate axes depending on Ω, i.e. supp ν ⋆ s d , µ = [GH22]ximum principle for distributional solutions, see [GH23, Theorem 1.1], yields u = v.Note that this is applicable as described in [GH23, Remark 1.4] together with [RF23, Proposition 2.6.9].5.Nonlocal to LocalAs an application of the robust regularity result Theorem 1.2 and the convergence of trace spaces from[GH22], we can approximate distributional solutions of the Dirichlet problem for the Laplacian with distributional solutions to (1.5) and receive H 1/2 (Ω)-regularity for solutions to (1.1) with L 2 -boundary data.One difficulty in the setup of the local Dirichlet problem with data g ∈ L 2 (∂Ω) is the question of how to prescribe the boundary data.The obstruction is that a distributional solution does not have sufficient regularity a priori such that it allows for a description of its boundary values.There are multiple ways of solving this problem.One of them is to describe boundary values by nontangential convergence, cf.[HW68, Dah77, JK81].Another way is to include the boundary data into the equation by enlarging the test function space.