Boundary regularity for general elliptic operators of order $2s$

Abstract

We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider Lévy operators that are symmetric and its Fourier symbol satisfies $\mathcal{A}(ξ)\asymp |ξ|^{2s}$ in $\mathbb{R}^d$. This was only known when the kernel of the operator (or Lévy measure) is either homogeneous or comparable to that of the fractional Laplacian, with different proofs in each case. Our new proofs extend both at the same time, and work in a very general class of domains, under a $C^1$-Dini-type condition.