Note on an elementary inequality and its application to the regularity of p-harmonic functions

Abstract

We study the Sobolev regularity of $p$-harmonic functions. We show that $|Du{|}^{\frac{p-2+s}{2}}Du$ belongs to the Sobolev space $W_{\mathrm{loc}}^{1,2}$, $s>-1-\frac{p-1}{n-1}$, for any $p$-harmonic function $u$. The proof is based on an elementary inequality.