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

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