Local Regularity of very weak $s$-harmonic functions via fractional difference quotients

TL;DR

This paper introduces a novel difference quotient method to prove that very weak s-harmonic functions in the unit ball are smooth and real analytic, improving local regularity and summability properties.

Contribution

The paper presents a new difference quotient technique for establishing regularity of very weak s-harmonic functions, leading to smoothness and analyticity results.

Findings

Improved local summability of very weak s-harmonic functions.

Established Sobolev regularity and local linear estimates.

Proved real analyticity of the functions.

Abstract

The aim of this paper is to give a new proof that any very weak $s$-harmonic function $u$ in the unit ball $B$ is smooth. As a first step, we improve the local summability properties of $u$. Then, we exploit a suitable version of the difference quotient method tailored to get rid of the singularity of the integral kernel and gain Sobolev regularity and local linear estimates of the $H^{s}_{\rm loc}$ norm of $u$. Finally, by applying more standard methods, such as elliptic regularity and Schauder estimates, we reach real analyticity of $u$. Up to the authors' knowledge, the difference quotient techniques are new.