Regularity for anisotropic fully nonlinear integro-differential equations

We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality and the interior C1,γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1, \gamma }$$\end{document} regularity, extending the results of Caffarelli and Silvestre (Comm Pure Appl Math 62:597–638, 2009) to the anisotropic case.