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 \cite{CS}. Since the order of differentiability of the kernel is not characterized by a single number, we use the constant \begin{align*} C_\varphi = \left( \int_{\mathbb{R}^n} \frac{1-\cos y_1}{\vert y \vert^n \varphi (\vert y \vert)} \, dy \right)^{-1} \end{align*} instead of $2-\sigma$, where $\varphi$ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and H\"older estimates of viscosity solutions to the nonlinear integro-differential equations.