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 Cφ=∫Rn1−cosy1|y|nφ(|y|)dy−1instead of 2−σ, where φ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and Hölder estimates of viscosity solutions to the nonlinear integro-differential equations.