Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces

TL;DR

This paper establishes interior regularity results for solutions of linear elliptic integro-differential equations near the fractional Laplacian, using Nikol'skii spaces and a modified translation method.

Contribution

It introduces a novel approach combining Nikol'skii space estimates with a modified Nirenberg translation method for regularity analysis.

Findings

Proves interior $H^{2s- ext{epsilon}}$ regularity for solutions.

Develops intermediate estimates in Nikol'skii spaces.

Adapts classical translation method for non-local equations.

Abstract

We prove interior $H^{2s-\varepsilon}$ regularity for weak solutions of linear elliptic integro-differential equations close to the fractional $s$-Laplacian. The result is obtained via intermediate estimates in Nikol'skii spaces, which are in turn carried out by means of an appropriate modification of the classical translation method by Nirenberg.