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 Hölder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{σ, β}$ satisfying $$ K_{σ,β}(y)\asymp \frac{ 2-σ}{|y|^{n+σ}}\left( \log\frac{2}{|y|^2}\right)^{β(2-σ)}\quad \mbox{near zero} $$ with respect to $σ\in(0,2)$ close to $2$ (for a given $β\in\mathbb R$), where the regularity estimates do not blow up as the order $σ\in(0,2)$ tends to $2.$

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