Calculus of Variations and Geometric Measure Theory

A. Carbotti - S. Cito - D. A. La Manna - D. Pallara

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

created by carbotti on 04 May 2024


Submitted Paper

Inserted: 4 may 2024
Last Updated: 4 may 2024

Pages: 24
Year: 2024


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.

Keywords: Sobolev regularity, Fractional Operators, $s$-harmonic functions