Regularity for fully nonlinear integro-differential operators with kernels of variable orders

Abstract

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in Caffarelli and Silvestre (2009). Since the order of differentiability of the kernel is not characterized by a single number, we use the constant,C-phi = (integral(Rn) 1-COS y1/ vertical bar y vertical bar(n)phi (vertical bar y vertical bar) dy)(-1),instead of 2 - sigma, where phi satisfies a weak scaling condition. We obtain the uniform Harnack inequality and Holder estimates of viscosity solutions to the nonlinear integro-differential equations. (C) 2018 Elsevier Ltd. All rights reserved.,