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 (Comm. Pure Appl. Math. 62, 597–638, 2009) 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 Kσ,β satisfying Kσ,β(y)≍2−σ|y|n+σlog2|y|2β(2−σ)near zero\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{\sigma,\beta}(y)\asymp\frac{ 2-\sigma}{|y|^{n+\sigma}}\left( \log\frac{2}{|y|^{2}}\right)^{\beta(2-\sigma)} \text{near zero} $$\end{document} with respect to σ ∈ (0, 2) close to 2 (for a given β∈ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta \in \mathbb R$\end{document}), where the regularity estimates do not blow up as the order σ ∈ (0, 2) tends to 2.