Regularity of solutions for degenerate/singular fully nonlinear integro-differential equation

We study a series of regularity results for solutions of a degenerate/singular fully nonlinear integro-differential equation $$- \bigg( \sigma_{1}(|Du|) + a(x) \sigma_{2}(|Du|) \bigg) I_{\tau}(u,x) = f(x).$$ In the degenerate case, we establish borderline regularity provided the inverse of degeneracy law \( \sigma_{2}\) is Dini-continuous. In addition, we show Schauder-type higher regularity at local extrema point for a concrete non-local degenerate equation. In the singular case, we prove gradient H\"{o}lder regularity of solutions to general non-local equation. It is noteworthy that these results are new even for the case \( a(x) \equiv 0 \). Finally, as a byproduct of the borderline regularity, we show how to apply our strategies in the study of the corresponding regularity for a class of degenerate non-local normalized \( p\)-Laplacian equation.