The Dirichlet problem for nonlocal operators with singular kernels: Convex and nonconvex domains

We study the interior regularity of solutions to the Dirichlet problem Lu=g in Ω, u=0 in RnØΩ, for anisotropic operators of fractional typeLu(x)=∫0+∞dρ∫Sn-1da(ω)2u(x)-u(x+ρω)-u(x-ρω)ρ1+2s. Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions).When a∈C∞(Sn-1) and g is C∞(Ω), solutions are known to be C∞ inside Ω (but not up to the boundary). However, when a is a general measure, or even when a is L∞(Sn-1), solutions are only known to be C3s inside Ω.We prove here that, for general measures a, solutions are C1+3s-ε inside Ω for all ε>0 whenever Ω is convex. When a∈L∞(Sn-1), we show that the same holds in all C1,1 domains. In particular, solutions always possess a classical first derivative.The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+ε for any ε>0 - even if g and Ω are C∞.