Calderón-Zygmund estimates for double phase problems with matrix weights

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