Regularity for fully nonlinear integro-differential operators with regularly varying kernels

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre \cite{CS1} 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 H\"older estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{\sigma, \beta}$ satisfying $$K_{\sigma,\beta}(y)\asymp \frac{ 2-\sigma}{|y|^{n+\sigma}}\left( \log\frac{2}{|y|^2}\right)^{\beta(2-\sigma)}\quad \mbox{near zero}$$ with respect to $\sigma\in(0,2)$ close to $2$ (for a given $\beta\in\mathbb R$), where the regularity estimates do not blow up as the order $\sigma\in(0,2)$ tends to $2.$