Regularity theory for general stable operators

Abstract

Abstract. We establish sharp regularity estimates for solutions to Lu = f in Ω ⊂ R n , being L the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on S n−1 , called the spectral measure.First, we study the interior regularity of solutions to Lu = f in B 1 . We prove that if f is C α then u belong to C α+2s whenever α + 2s is not an integer. In case f ∈ L ∞ , we show that the solution u is C 2s when s = 1/2, and C 2s−ǫ for all ǫ > 0 when s = 1/2.Then, we study the boundary regularity of solutions to Lu = f in Ω, u = 0 in R n \ Ω, in C 1,1 domains Ω. We show that solutions u satisfy u/d s ∈ C s−ǫ (Ω) for all ǫ > 0, where d is the distance to ∂Ω.Finally, we show that our results are sharp by constructing two counterexamples.

Regularity of spectral fractional Dirichlet and Neumann problems

“…e.g. , and their references; in the case where K is outside 0, this defines an operator of the type P mentioned above.) More generally, K can be subject to estimates comparing with .…”

“…For the restricted Dirichlet fractional Laplacian, detailed regularity properties of solutions of in Hölder spaces and Sobolev spaces have just recently been shown, in Ros‐Oton and Serra –, Grubb , .…”

Schauder Estimates for Equations Associated with Lévy Generators

“…Since then, several extensions and refinements of these estimates have been obtained. Ros-Oton & Serra [24] studied the interior Hölder regularity of solutions to equations Af = g associated with symmetric α-stable operators and established under a mild degeneracy condition on the spectral measure estimates of the form f C α+κ (B(0,1)) ≤ c f C κ (R d ) + g C κ (B(0,2)) for κ ≥ 0 such that α + κ is not an integer. In the recent paper [14], global Schauder estimates…”

Liouville Theorems for a General Class of Nonlocal Operators

“…In [13], the authors consider anisotropic operators where the function a ∈ L ∞ (S N −1 ) above is replaced by an even nonnegative measure on S N −1 . In this case, it is in general not possible to define the corresponding operator on the space L 1 s (R N ) in distributional sense, and instead [13, Theorem 2.1] relies on the stronger a priori assumption u L ∞ (B R (0)) ≤ CR β for R ≥ 1 with some constants β < 2s and C > 0.…”

“…In this case, it is in general not possible to define the corresponding operator on the space L 1 s (R N ) in distributional sense, and instead [13, Theorem 2.1] relies on the stronger a priori assumption u L ∞ (B R (0)) ≤ CR β for R ≥ 1 with some constants β < 2s and C > 0. The argument in [13] relies on this pointwise growth restriction and does not apply to functions in L 1 s (R N ). Our proof of Theorem 1.4 relies on the well known fact that the real part of the corresponding symbol η is homogeneous of degree 2s, and for ξ ∈ S N −1 it is given up to a constant by (1.15), see Section 3 below.…”