A note on limiting Calderon-Zygmund theory for transformed $n$-Laplace systems in divergence form

Abstract

We consider rotated $n$-Laplace systems on the unit ball $B_1 \subset \mathbb{R}^n$ of the form \begin{align*} -\mathrm{div}\left( Q|\nabla u|^{n-2} \nabla u\right) = \mathrm{div}(G), \end{align*} where $u\in W^{1,n}(B_1;\mathbb{R}^N)$, $Q\in W^{1,n}(B_1;SO(N))$ and $G\in L^{\left( \frac{n}{n-1},q \right)}(B_1;\mathbb{R}^n\otimes \mathbb{R}^N)$ for some $0<q<\frac{n}{n-1}$. We prove that $\nabla u\in L^{(n,q(n-1))}_{loc}$ with estimates. As a corollary, we obtain that solutions to $\Delta_n u \in \mathcal{H}^1$, where $\mathcal{H}^1$ is the Hardy space, have a higher integrability, namely $\nabla u \in L^{(n,n-1)}_{loc}$.