Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations

We study a series of regularity results for solutions to a degenerate or singular fully nonlinear integro-differential equation of the form $$- \big( \sigma_{1}(|Du|) + a(x) \sigma_{2}(|Du|) \big) \mathcal{I}_{\tau}(u,x) = f(x).$$ In the degenerate case, we establish borderline regularity, provided the inverse of the degeneracy law $\sigma_{2}$ is Dini-continuous. In addition, we show Schauder-type higher regularity at local extremum points for a specific non-local degenerate equation. In the singular case, we establish H\"{o}lder continuity of the gradient for solutions to a general non-local equation. It is noteworthy that these results are new even in the case $a(x) \equiv 0$. Finally, as a byproduct of the borderline regularity analysis, we demonstrate how our methods can be applied to study of the corresponding regularity for a class of degenerate non-local normalized $p$-Laplacian equations.