Regularity theory for general stable operators: Parabolic equations

We establish sharp interior and boundary regularity estimates for solutions to partial derivative(t)u - Lu = f(t,x) in I x ohm, with I C R and ohm C R-n. The operators L we consider are infinitesimal generators of stable Levy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that u is C2s+alpha in x and C1+alpha/2s in t, whenever f is C-alpha in x and C(alpha/2s)in t. In the case f is an element of L-infinity ,we prove that u is C2s-epsilon in x and C1-epsilon in t, for any epsilon > 0. On the other hand, we study the boundary regularity of solutions in C-1,C-1 domains. We prove that for solutions u to the Dirichlet problem the quotient u/d(s) is H (o) over bar lder continuous in space and time up to the boundary partial derivative ohm, where d is the distance to partial derivative ohm. This is new even when L is the fractional Laplacian. (C) 2017 Elsevier Inc. All rights reserved.