Local Regularity of very weak $s$-harmonic functions

Alessandro Carbotti, Simone Cito, Domenico Angelo La Manna, Diego Pallara

Published 2022-12-30Version 1

In this paper we prove that any very weak $s$-harmonic function $u$ in the unit ball $B$ is locally weakly $s$-harmonic. First, via localisation and bootstrap methods we prove that $u$ gains more summability. Then we prove $H^s$ Sobolev regularity using a generalisation of the Nirenberg differential quotients method to the nonlocal setting that allows to get rid of the singularity of the integral kernel. Finally, $L^2$ estimates lead us to $H^{2s}$ Sobolev regularity.