Regularity theory for general stable operators

We establish sharp regularity estimates for solutions to Lu=fLu=f in Ω⊂RnΩ⊂Rn, L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L   depend on a finite measure on Sn−1Sn−1, called the spectral measure.First, we study the interior regularity of solutions to Lu=fLu=f in B1B1. We prove that if f   is CαCα then u   belong to Cα+2sCα+2s whenever α+2sα+2s is not an integer. In case f∈L∞f∈L∞, we show that the solution u   is C2sC2s when s≠1/2s≠1/2, and C2s−ϵC2s−ϵ for all ϵ>0ϵ>0 when s=1/2s=1/2.Then, we study the boundary regularity of solutions to Lu=fLu=f in Ω, u=0u=0 in Rn∖ΩRn∖Ω, in C1,1C1,1 domains Ω. We show that solutions u   satisfy u/ds∈Cs−ϵ(Ω‾) for all ϵ>0ϵ>0, where d is the distance to ∂Ω.Finally, we show that our results are sharp by constructing two counterexamples.