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

TL;DR

This paper proves optimal boundary regularity results for a broad class of nonlocal elliptic operators of order 2s, extending previous results to more general operators and domains.

Contribution

It establishes boundary regularity for general symmetric Lévy operators of order 2s, broadening the scope beyond fractional Laplacians and homogeneous kernels.

Findings

Achieved optimal $C^s$ boundary regularity for general nonlocal elliptic operators.

Extended regularity results to domains with $C^1$-Dini-type boundaries.

Provided new proofs applicable to a wider class of operators.

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 L\'evy operators that are symmetric and its Fourier symbol satisfies $\mathcal{A}(\xi)\asymp |\xi|^{2s}$ in $\mathbb{R}^d$. This was only known when the kernel of the operator (or L\'evy 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.