Local regularity of very weak -harmonic functions via fractional difference quotients
Alessandro Carbotti
Università del Salento, Lecce, ItalySimone Cito
Università del Salento, Lecce, ItalyDomenico Angelo La Manna
Università degli studi di Napoli “Federico II”, Naples, ItalyDiego Pallara
Università del Salento, and INFN, Sezione di Lecce, Lecce, Italy

Abstract
The aim of this paper is to give a new proof that any very weak -harmonic function in the unit ball is smooth. As a first step, we improve the local summability properties of . 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 norm of . Finally, by applying more standard methods, such as elliptic regularity and Schauder estimates, we reach the real analyticity of . Up to the authors’ knowledge, the difference quotient techniques are new.
Cite this article
Alessandro Carbotti, Simone Cito, Domenico Angelo La Manna, Diego Pallara, Local regularity of very weak -harmonic functions via fractional difference quotients. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (2025), published online first
DOI 10.4171/RLM/1045