Regularity for nonlocal operators with kernels of functional order

Abstract

In this thesis, we investigate the regularity results for nonlocal operators with kernels of functional order. We consider a nonlocal operator for a given jumping kernel comparable to a special function near zero. Assuming the function has a weak scaling condition one can generalize the fractional Laplace operator. Firstly, we obtain the Holder continuity of solutions to fully nonlinear nonlocal equations with respect to a certain class of linear nonlocal operators in analytic method. Secondly, we prove the Schauder estimates for nonlocal equation by introducing the generalized Holder spaces.