We establish $C^{ \sigma+\alpha}$ interior estimates for concave nonlocal fully nonlinear equations of order $\sigma\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^{ \alpha}(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(\sigma)$, then $u$ belongs to $C^{ \sigma+\alpha}(\overline{ B_{ 1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^\alpha$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.
Categories: math.AP