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In this note, we consider the regularity of solutions of the nonlinear elliptic systems of$n$-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space$L$^{$n$,∞}. We also obtain the$a priori$global and local estimates for the$L$^{$n$,∞}-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

Let$α$_{1},$α$_{2},…,$α$_{$m$}be linear forms defined on ℂ^{$n$}and $\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$ , where$V$($α$_{$i$})={$p$∈ℂ^{$n$}$:$$α$_{$i$}($p$)=0}. The coordinate ring $O_{\mathcal{X}}$ of $\mathcal{X}$ is a holonomic$A$_{$n$}-module, where$A$_{$n$}is the$n$th Weyl algebra and since holonomic$A$_{$n$}-modules have finite length, $O_{\mathcal{X}}$ has finite length. We consider a “twisted” variant of this$A$_{$n$}-module which is also holonomic. Define M_{$α$}^{$β$}to be the free rank-1 ℂ[$x$]_{$α$}-module on the generator$α$^{$β$}(thought of as a multivalued function), where $\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ and the multi-index$β$=($β$_{1},…,$β$_{$m$})∈ℂ^{$m$}. Our main result is the computation of the number of decomposition factors of M_{$α$}^{$β$}and their description when$n$=2.