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The aim of this paper is to give a new proof that any very weak $s$-harmonicfunction $u$ in the unit ball $B$ is smooth. As a first step, we improve thelocal summability properties of $u$. Then, we exploit a suitable version of thedifference quotient method tailored to get rid of the singularity of theintegral kernel and gain Sobolev regularity and local linear estimates of the$H^{s}_{\rm loc}$ norm of $u$. Finally, by applying more standard methods, suchas elliptic regularity and Schauder estimates, we reach real analyticity of$u$. Up to the authors' knowledge, the difference quotient techniques are new.

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