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 $\nabla u\in L_{loc}^{(n,q(n-1))}$ 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 $\nabla u\in L_{loc}^{(n,n-1)}$.

14 pages. Added in v2: the constant $\theta$ in Theorem 1.1 has been improved