Nonlinear Stein theorem for differential forms

Abstract

We prove that if $u$ is an ${\mathbb{R}}^N$-valued $W_{loc}^{1,p}$ differential $k$-form with $\delta \left(a(x)|du{|}^{p-2}du\right)\in L_{loc}^{(n,1)}$ in a domain of ${\mathbb{R}}^n$ for $N\ge 1,$ $n\ge 2,$ $0\le k\le n-1,$ $1<p<\infty ,$ with uniformly positive, bounded, Dini continuous scalar function $a$, then $du$ is continuous. This generalizes the classical result by Stein in the scalar case and the work of Kuusi-Mingione for the $p$-Laplacian type systems. We also discuss Hölder, BMO and VMO regularity estimates for such systems when $p\ge 2.$

arXiv commentSome typos are fixed, presentations are changed a bit from the previous version