Maximum principle for stable operators

We prove a weak maximum principle for nonlocal symmetric stable operators including the fractional Laplacian. The main focus of this work is on minimal regularity assumptions of the functions under consideration.


Introduction
The study of maximum principles for harmonic functions can be traced back to the works [23] by Gauß and [37] by Riemann.They are a key tool in the theory of existence, particularly uniqueness, and regularity of solutions to linear second order elliptic equations.Let Ω be a sufficiently regular, bounded domain.For the Laplacian the following maximum principle is well known.If u ∈ L 1 loc (Ω) satisfies u is upper-semicontinuous in Ω and u(x) ≤ u(y) dy for all balls B(x) ⊂⊂ Ω, ( lim sup x→z u(x) ≤ 0 for all z ∈ ∂Ω, ( then u ≤ 0 in Ω.The condition (1.1) is equivalent to i.e. u is distributional subharmonic.For both statements we refer to the book [15,Chapter 27]  then u ≤ 0 a.e. in Ω.Instead of (1.5) we could assume the stronger but more accessible assumption u + ∈ L 1 (Ω, dist(x, ∂Ω) −s dx).
Analogous to the case of the Laplacian, see (1.1), (1.2) and (1.3), Silvestre proved in [46, Proposition 2.17] the following weak maximum principle.Let u ∈ L 1 (R d , (1 + |x|) −d−2s dx) be upper-semicontinuous on Ω.If u satisfies (u, (−∆) s η) L 2 (R d ) ≤ 0 for all nonnegative η ∈ C ∞ c (Ω), (1.6) u ≤ 0 in Ω c , then u ≤ 0 in Ω.We want to emphasize that the function u needs to be to upper-semicontinuous up to the boundary of Ω.In [9, Theorem 5.2] Caffarelli and Silvestre extended this result to a larger class of operators.The condition (1.5) is less restrictive than being upper-semicontinuous on Ω since upper-semicontinuous functions attain their maximum on compact sets.Cabré and Sire proved a strong maximum principle for the fractional Laplacian, using a representation as a Dirichlet-to-Neumann map, see [8,Section 4.6].In [5] Bogdan and Byczkowski used probabilistic methods to prove a strong maximum principle for supersolutions related to the Schrödinger operator (−∆) s + q.Weak and strong maximum principles for a larger class of operators of the form L + q can be found in the work [28] by Jarohs and Weth.Results for solutions to nonlinear equations and antisymmetric solutions of related linear equations can be found in [27].For the case of the fractional Laplacian we refer the reader also to the work of Chen, Li, Li [10] and Lü [34].Abatangelo proved a weak maximum principle for the fractional Laplacian in [1, Lemma 3.9].Remarkably, the only assumption on the function is u ∈ L 1 (Ω).This is a consequence of allowing for test functions η ∈ C(R d ) in (1.6) such that (−∆) s η = ψ in Ω and η = 0 on Ω c .Here ψ ∈ C ∞ c (Ω) is an arbitrary nonnegative function.Maximum principles and the failure thereof for higher order fractional Laplacians were discussed by Abatangelo, Jarohs and Saldaña in [2].
A nonlocal Green-Gauß formula motivates the following bilinear form associated to the fractional Laplacian.
In this setup the following weak maximum principle holds.Let u : Servadei and Valdinoci [45,Lemma 6].The idea is often used in the proof of maximum principles for second order elliptic operators, see Gilbarg and Trudinger [24].This approach has been applied to a larger class of nonlocal operators, including those with nonsymmetric kernels, by Felsinger, Kaßmann and Voigt in [21,Theorem 4.1].
In this article we consider generators of symmetric, stable Lévy processes.These processes play a key role in the Generalized Central Limit Theorem, e.g.see the book [43] by Samorodnitsky and Taqqu.Let s ∈ (0, 1), (X t ) a symmetric, 2s-stable process, i.e. for all t > 0 There exists a nonnegative, finite measure µ on the unit sphere S d−1 such that its generator is −A s , where with the Lévy measure given in polar coordinates via (1.8) This relation was established by Lévy in [30] and Khintchine in [29].See also Sato [44] for a proof.Additionally, we assume the nondegeneracy condition (1.9) The condition (1.9) is satisfied as soon as µ is not supported by a hyperplane.It is rooted in the work [36] of Picard as an ellipticity assumption on the The aim of this article is to find the minimal regularity of a function u : R d → R such that the following maximum principle holds.
(1.10) We study the operator distributionally.Thereby, should exist for all η ∈ C ∞ c (Ω).We introduce the weighted L 1 -space L 1 (R d , ν ⋆ (x) dx) with the weight (1.12) We call it the tail space for ν and Ω.A s η is bounded in Ω and decays at infinity like (1.11) to exist, see Lemma 2.4.In the case of the fractional Laplacian this tail space coincides with the aforementioned space L 1 (R d , (1 + |x|) −d−2s ).This is proven in Lemma A.1.In our second example d j=1 (−∂ 2 j ) s the auxiliary measure ν ⋆ (x) dx only measures sets close to the coordinate axes, dependent on Ω.Here functions in L 1 (R d , ν ⋆ (x) dx) are not necessarily integrable on Ω.An example can be found in Example A.2.The weight ν ⋆ captures the behavior of ν at infinity.Foghem and Kaßmann introduced and discussed several possibilities of tail weights for a large class of Lévy measures in [22]. The satisfies (−∆) s u = 0 in B 1 (0) and u = 0 on B 1 (0) c , but it disobeys the maximum principle, see the works [26] by Hmissi, [3] by Bogdan and [18] by Dyda for a proof.Now we state the main result of this article.
Theorem 1.1.Let Ω ⊂ R d be a bounded Lipschitz domain satisfying uniform exterior ball condition, s ∈ (0, 1) and µ a nonnegative, finite measure on the unit sphere satisfying the nondegeneracy assumption (1.9),A s be as in (1.7).
Remark 1.2.Instead of (1.16), we can assume the stronger but more accessible condition This condition implies (1.16 for every δ > 0 but not for δ = 0. (iii) For the fractional Laplacian and the domain Ω = B 1 (0), Lü proved a maximum principle in [34,Theorem 6].The proof relies on the explicit representation of the Poisson kernel.[34,Theorem 6] (iv) Very recently the articles [31,32] by Li and Liu have been uploaded to arXiv.[31,Theorem 1.1] contains our Theorem 1.1 in the special case of the fractional Laplacian.They assume the same conditions on the function u.Note that the proof is rather different from ours.In contrast, our result holds for the larger class of stable, nondegenerate operators.
Let us explain the main ideas in the proof of Theorem 1.1.In an ideal situation we would use the Green function in place of η in (1.14).This is not permitted in our setup.Thereby, we approximate the Green function.We consider a sequence D ε ⊂⊂ Ω of subdomains exhausting Ω.We fix ψ ∈ C ∞ c (Ω) and solve the Dirichlet problem A s φ ε = ψ in D ε and φ ε = 0 on D c ε .The solutions φ ε approximate the Green function if we pick an approximate identity in place of ψ.Regularity results on the solution φ ε are crucial in our proof.Nonlocal operators and related Dirichlet problems are often studied in a Hilbert space setting.Following the works [21] by Felsinger, Kaßmann, Voigt and [45] by Servadei and Valdinoci, we define a bilinear form associated to A s This is motivated by a nonlocal Green-Gauß formula, see Du et al. [16] for bounded kernels, Dipierro, Ros-Oton and Vladinoci [14,Lemma 3.3] for the fractional Laplacian and Foghem and Kaßmann [22] for more general Lévy measures.
Corollary 1.5.Let Ω and µ satisfy the assumptions from Theorem 1.1 and A s be as in (1.7).
The proof of Corollary 1.5 uses u ∈ H s/2 (Ω) and fractional Hardy inequality to deduce the integrability to exist for all s ∈ (0, 1).
1.1.Outline.In Section 2 we prove basic properties of the operator A s and introduce the weak solution concept.Additionally, we state regularity results of solutions and provide technical ingredients for the proof of Theorem 1.1.Lemma 2.14 connects the Hölder regularity of solutions with the assumption (1.16).In Section 3 we prove Theorem 1.1 and Corollary 1.5.In Appendix A we compare tail spaces and in Appendix B we prove a maximum principle for the Laplacian.
Acknowledgments.Financial support by the German Research Foundation (GRK 2235 -282638148) is gratefully acknowledged.We would like to thank Moritz Kaßmann and Tobias Weth for very helpful discussions.

Preliminaries
We set a ∧ b := min{a, b}, a ∨ b := max{a, b} and a + := max{a, 0} for a, b ∈ R. We denote the set of Hölder continuous functions on Ω ⊂ R d by C α (Ω) = C ⌊α⌋,α−⌊α⌋ (Ω) for all α > 0. In case that Ω is bounded, we equip the space C α (Ω) with the usual norm with the Hölder semi norm for 0 < s < 1 We say that a domain Ω ⊂ R d satisfies uniform exterior ball condition if there exists a radius ρ > 0 such that for every x ∈ ∂Ω there exists a ball B ⊂ Ω c of radius ρ such that B ∩ Ω = {x}.
The following lemma ensures the existence of a sequence of C ∞ subdomains exhausting a bounded Lipschitz domain satisfying uniform exterior ball condition.This type of domain exhaustion is classical.We are particularly interested in a uniform bound of the exterior ball radius.The result is taken from the work by Mitrea, see [35,Lemma 6.4].Lemma 2.1.[35,Lemma 6.4] Let Ω ⊂ R d be a bounded Lipschitz domain satisfying uniform exterior ball condition.There exists a sequence of ii) D ε satisfies uniform exterior ball condition with a radius independent of ε, (iii) there exists a constant λ > 1 such that Proof.The existence of the sequence {D ε } ε>0 satisfying the properties (i) and (ii) follow from [35,Lemma 6.4].Property (iii) can be ensured by choosing the constants in the construction of D ε in the proof of [35,Lemma 6.4] accordingly.
The symmetry of the Lévy measure ν allows us to rewrite A s to a double difference.

Proposition 2.2. Fix an open set
The term A s u(x) exists and Proof.Without loss of generality. 2 s + α < 2 and 2 s + α < 1 for s < 1  2 .Take x ∈ Ω and any ball B δ (x) ⊂⊂ Ω.We define for κ > 0 This term exists by the assumption and use the transformation r → −r in its second occurrence. (2.4) For s ≥ 1 2 , θ ∈ S d−1 and |r| < δ the fundamental theorem of calculus yields Lastly, for all s ∈ (0, 1), θ ∈ S d−1 and |r| ≥ δ Additionally, we know using transformation theorem and Fubini's theorem that u Therefore, the right-hand side of (2.3) is finite for a.e.x ∈ Ω.Since u is continuous on R d , this property holds for every x ∈ Ω.By the dominated convergence theorem with the previous bounds, the limit κ → 0 in (2.3) exists.Lastly, we show that A s u is continuous in Ω.We fix x ∈ Ω and write A s u(x) as The dominated convergence theorem, the estimates (2.4), (2.5) and (2.6) and the continuity of u yield the continuity of A s u at x.

Definition 2.6 (Weak solution).
Let Ω ⊂ R d be a bounded domain and ψ ∈ L ∞ (Ω).We say that φ is a weak solution to the problem There is a rich theory on existence and uniqueness of weak solutions.We refer the reader to Bogdan et al. [7], Felsinger, Kaßmann and Voigt [21], Grzywny, Kaßmann and Leżaj [25] and Rutkowski [42].In the following proposition we use the existence theorem from [42] to deduce the existence of solutions in the sense of Definition 2.6.
Hölder regularity up to the boundary of solutions to Dirichlet problems was proven by Ros-Oton and Serra in [39].The following proposition is taken from their work.For the fractional Laplacian see also Ros-Oton and Serra [38].
Additional regularity inside of the domain was proven in the same work [39,Theorem 1.1].We reduce their result to fit our purposes.Theorem 2.10.[39,Theorem 1.1] Let s ∈ (0, 1) and ψ ∈ C α (B 1 ) for some α > 0 and φ be any bounded weak solution to whenever 2s + α is not an integer.
Now we combine the regularity results of Ros-Oton and Serra for solutions to (2.9) with Proposition 2.8 and the weak existence from Proposition 2.7 to prove the following existence and uniqueness of classical solutions.
Proof.The second claim follows from (1.15) and supp(η For the first property we fix x ∈ Ω ε .Now we use Fubini's theorem, the radiality of η and the transformation theorem to conclude The second equality is true by (2.7) and dominated convergence.The last inequality is true because The next lemma show the following.If u is a subsolution, then u + = max{u, 0} is again a subsolution.Silvestre proved this result for the fractional Laplacian in [46,Lemma 2.18].We follow the technique in [33, Lemma 2] by Li, Wu and Xu as it generalizes easier to our class of operators.(1.14) and (1.15), then u + = max{u, 0} satisfies (1.14) and u + = 0 on Ω c .Proof.The proof of the second claim is immediate.We divide the proof into two steps.

11) and dominated convergence yield
The application of dominated convergence is justified.By Proposition 2.2, A s u is continuous in Ω and, thus, A s u ∈ L ∞ (supp η).Therefore, the first term ´Ω A s u(x)η(x) dx exists.E As (u, η) exists by Lemma 2.5.By Lemma 2.4 and These integrals exist since η has compact support and u ∈ L 1 (R d , ν ⋆ (x)dx).By symmetry, (II) is zero and (III) is nonpositive by the definition of P .We conclude By Proposition 2.2 and dominated convergence, we conclude 14) and (1.15).By Lemma 2.12 and step 1, holds for any two real numbers a, b, we notice for any 0 < ε < ε 0 This term converges to zero for Because u ∈ L 1 (Ω) and by assumption (1.15) u + ∈ L 1 (R d ).Therefore, the right-hand side of the previous inequality converges to zero as ).Thus, we conclude by Hölder's inequality The following lemma is the key technical estimate in the proof of Theorem There exists a constant C > 0 such that Proof.Fix any 0 < ε < ε 0 and any x ∈ Ω satisfying dist(x, ∂Ω) < (1 + λ)ε.Now we estimate the convoluted distance function.
Now we apply coarea formula, see Federer [20], to the right-hand side of (2.12) with the Lipschitz continuous function dist(•, ∂D ε ), which satisfies Here H (d−1) is the (d−1)-dimensional Hausdorff measure.The Hausdorff measure of a ball B r intersecting a hyperplane scales like r d−1 .Thus, there exists a constant c > 0 such that

Proof of maximum principles
In this section we provide the proofs of Theorem 1.1 and Corollary 1.5.
By the previous estimate and Lemma 2.14, there exists a constant C > 0 such that This converges to zero as ε → 0+ by assumption (1.16).Now we finish the proof.(3.1), (3.2) and the choice of In the previous calculation we used dominated convergence and Proposition 2.2.(3.4) implies that R ε,κ converges to zero as ε → 0+ uniformly in κ.We take the limit ε → 0+ in (3.5).Thereby, An application of Theorem 1.1 finishes the proof, see Remark 1.2.

Remark 1 . 4 .
Assumptions on the regularity of the boundary of the domain Ω in Theorem 1.1 seem unnatural.They are only needed for boundary regularity of solutions to the Dirichlet problem in Proposition 2.8, see Ros-Oton and Serra [39, Proposition 4.5].

Proposition 2 . 3 .Lemma 2 . 4 .
Let Ω ⊂ R d be open and bounded, s ∈ (0, 1) and α > 0. There exists a constant C > 0 such thatA s φ L ∞ (Ω) ≤ C φ C 2s+α (Ω) (2.7) for all φ ∈ C 2s+α c (Ω) extended by zero to R d .Proof.The claim follows by using the arguments in Proposition 2.2 uniformly for any x ∈ Ω.Instead of picking a small ball B δ (x) in the beginning of the proof of Proposition 2.2, we choose a ball B ⊃⊃ Ω.Let Ω ⊂ R d be open and bounded.If
[12] measure ν.Another motivation to study operators like (1.7) is Courrége's theorem, which characterizes the operators satisfying a maximum principle, see the work[12]by Courrége.We emphasize two examples in this class of operators.If we pick µ as a uniform distribution on the sphere, then the resulting operator is the aforementioned fractional Laplacian (−∆) s .In this case µ is a sum of Dirac measures δ ei , where e i are basis vectors of R d .