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

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 loc norm of u. Finally, by applying more standard methods, such as elliptic regularity and Schauder estimates, we reach the real analyticity of u. Up to the authors’ knowledge, the difference quotient techniques are new.