$C^{σ+α}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

Abstract

We establish $C^{σ+α}$ interior estimates for concave nonlocal fully nonlinear equations of order $σ\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^α(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(σ)$, then $u$ belongs to $C^{σ+α}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^α$ 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.