Regularity theory for general stable operators

We establish sharp regularity estimates for solutions to Lu = f in Omega subset of R-n being the generator of any stable and symmetric Levy process. Such nonlocal operators L depend on a finite measure on Sn-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-alpha then u belong to C alpha+2s whenever alpha + 2s is not an integer. In case f is an element of L-infinity we show that the solution u is C-2s when s not equal 1/2, and C2s - is an element of for all epsilon > 0 when s =1/2. Then, we study the boundary regularity of solutions to Lu = f in Omega, u = 0 in R-n \ Omega, in C-1,C-1 domains Omega We show that solutions u satisfy u/d(s) is an element of Cs-is an element of (Omega) for all epsilon > 0, where d is the distance to partial derivative Omega. Finally, we show that our results are sharp by constructing two counterexamples. (C) 2016 Elsevier Inc. All rights reserved.