Nonlinear Stein theorem for differential forms

TL;DR

This paper extends the classical Stein theorem to nonlinear differential forms, proving continuity of the form's derivative under certain integrability and regularity conditions, generalizing previous scalar and system results.

Contribution

It introduces a nonlinear Stein theorem for differential forms, broadening the scope of regularity results from scalar functions to differential forms with nonlinear operators.

Findings

Proves continuity of $du$ under specified conditions.

Generalizes Stein's scalar result to differential forms.

Discusses H"older, BMO, and VMO regularity for $p \\geq 2$.

Abstract

We prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ differential $k$-form with $\delta \left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k \leq 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 H\"{o}lder, BMO and VMO regularity estimates for such systems when $p \geq 2.$