Improved Sobolev regularity for linear nonlocal equations with VMO coefficients

TL;DR

This paper demonstrates that linear nonlocal equations with irregular VMO-type coefficients exhibit higher Sobolev regularity, showing solutions belong to more regular Sobolev spaces regardless of the irregularity of coefficients.

Contribution

It establishes new regularity results for nonlocal equations with very irregular coefficients, extending previous work by removing continuity requirements and achieving greater differentiability gains.

Findings

Solutions belong to higher Sobolev spaces $W^{t,p}_{loc}$ for any $s \\leq t < \\min\{2s,1\}$

Regularity gain is independent of the amount of integrability gained

Extends previous results by allowing discontinuous coefficients and larger differentiability gains.

Abstract

This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In this paper, we follow this direction and show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions $u \in W^{s,2}$ in fact belong to $W^{t,p}_{loc}$ for any $s \leq t < \min\{2s,1\}$, where $p>2$ reflects the amount of integrability gained. In other words, our gain of differentiability does not depend on the amount of integrability we are able to gain. This extends numerous results in previous works, where either continuity of the coefficient was required or only an in general smaller gain of differentiability was proved.