The Dirichlet problem for nonlocal elliptic operators with $C^{0,\alpha }$ exterior data

In this note we study the boundary regularity of solutions to non-local Dirichlet problems of the form Lu = 0 in Omega, u = g in R-N\Omega, in non-smooth domains Q. When g is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right-hand side for which the boundary regularity is well understood. Here, we study the case in which g is an element of C-0,C-alpha, and establish the optimal Holder regularity of u up to the boundary. Our results extend previous results of Grubb for C-infinity domains Omega.