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}_{\mathrm{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.