Integro-Differential equations with nonlinear directional dependence Moritz KassmannMarcus RangRussell Schwab 35B6535J6035R0960J75Integro-Differential EquationsJump ProcessesLevy ProcessesNonlocal Elliptic EquationsRegularity Theory We prove H\"older regularity results for a class of nonlinear elliptic integro-differential operators with integration kernels whose ellipticity bounds are strongly directionally dependent. These results extend those in [Luis \'Angel Caffarelli and Luis Silvestre, \textit{Regularity theory for fully nonlinear integro-differential equations}, Comm. Pure Appl. Math. \textbf{62} (2009), no. 5, 597--638], and are also uniform as the order of operators approaches $2$. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5394 10.1512/iumj.2014.63.5394 en Indiana Univ. Math. J. 63 (2014) 1467 - 1498 state-of-the-art mathematics http://iumj.org/access/