Calder\'{o}n-Zygmund estimates for double phase problems with matrix weights

Abstract

We establish an optimal Calder\'{o}n-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For $1<p<q<\infty$, $a(\cdot)\in C^{0,\alpha}(\Omega)$ ($0<\alpha\le1$), and a symmetric, almost everywhere positive definite matrix weight $\M$ with $|\M(x)|\,|\M(x)^{-1}|\le\Lambda$ for some constant $\Lambda\ge 1$ and small $|\log \M|_{\mathrm{BMO}}$, we prove, for every $\gamma>1$, $$ (|\M F|^p+a(x)|\M F|^q)\in L^\gamma_{\mathrm{loc}} \;\Longrightarrow\; (|\M Du|^p+a(x)|\M Du|^q)\in L^\gamma_{\mathrm{loc}}. $$ Our argument combines a freezing of the logarithm of the matrix field, $\log \M$, with a fractional maximal-operator method governed by the Muckenhoupt-Wheeden $\mathcal{A}_{p,s}$ classes (where $1/s=1/p-\alpha/(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold $q/p\le 1+\alpha/n$. Our result recovers the identity case $\,\M\equiv {\rm I}_n\,$, i.e., the classical (unweighted) Calder\'{o}n-Zygmund theory for double-phase problems.