Calderón-Zygmund estimates for double phase problems with matrix weights

Abstract

We establish an optimal Calderón-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For $1<p<q<\infty$, $a(\cdot )\in C^{0,\hat{I}\pm }(\hat{I}\text{c◯})$ ($0<\hat{I}\pm \le 1$), and a symmetric, almost everywhere positive definite matrix weight $\M$ with $|\M(x)|\,|\M(x)^{-1}|\leΛ$ for some constant $\hat{I}›\ge 1$ and small $|\log \M|_{\mathrm{BMO}}$, we prove, for every ${\hat{I}}^3>1$, $$ (|\M F|^p+a(x)|\M F|^q)\in L^γ_{\mathrm{loc}} \;\Longrightarrow\; (|\M Du|^p+a(x)|\M Du|^q)\in L^γ_{\mathrm{loc}}. $$ 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 $1/s=1/p-\hat{I}\pm /(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold $q/p\le 1+\hat{I}\pm /n$. Our result recovers the identity case $\,\M\equiv {\rm I}_n\,$, i.e., the classical (unweighted) Calderón-Zygmund theory for double-phase problems.