Matrix ApA_p weights, degenerate Sobolev spaces, and mappings of finite distortion.

We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix A_p weight. We prove that the classical Meyers-Serrin theorem, H = W, holds in this setting. As applications we prove partial regularity results for weak solutions of degenerate p-Laplacian equations, and in particular for mappings of finite distortion

The sharp weighted bound for multilinear maximal functions and Calder\'{o}n-Zygmund operators

We investigate the weighted bounds for multilinear maximal functions and Calder\'on-Zygmund operators from $L^{p_1}(w_1)\times...\times L^{p_m}(w_m)$ to $L^{p}(v_{\vec{w}})$, where $1<p_1,...,p_m<\infty$ with $1/{p_1}+...+1/{p_m}=1/p$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. We prove the sharp bound for the multilinear maximal function for all such $p_1,..., p_m$ and prove the sharp bound for $m$-linear Calder\'on-Zymund operators when $p\geq 1$