Regularity for Fully Nonlinear Integro-differential Operators with Regularly Varying Kernels

Abstract

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre (Comm. Pure Appl. Math. 62, 597-638, 2009) are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels K-sigma,K-beta satisfying K-sigma,K-beta (y) asymptotic to 2 - sigma/|y|(n+sigma) (log 2/|y|(2))(beta(2-sigma)) with respect to sigma is an element of(0, 2) close to 2 (for a given beta is an element of R), where the regularity estimates do not blow up as the order sigma is an element of (0, 2) tends to 2.