We establish an optimal Calderón-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For \(1<p<q<\infty \) , \(a(\cdot )\in C^{0,\alpha }(\Omega )\) ( \(0<\alpha \le 1\) ), and a symmetric, almost everywhere positive definite matrix weight \(\mathbb {M}\) with \(|\mathbb {M}(x)|\,|\mathbb {M}(x)^{-1}|\le \Lambda \) for some constant \(\Lambda \ge 1\) and small \(|\log \mathbb {M}|_{\textrm{BMO}}\) , we prove, for every \(\gamma >1\) , \( (|\mathbb {M}F|^p+a(x)|\mathbb {M}F|^q)\in L^\gamma _{\textrm{loc}} \;\Longrightarrow \; (|\mathbb {M}Du|^p+a(x)|\mathbb {M}Du|^q)\in L^\gamma _{\textrm{loc}}. \) Our argument combines a freezing of the logarithm of the matrix field, \(\log \mathbb {M}\) , 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 \(\,\mathbb {M}\equiv \textrm{I}_n\,\) , i.e., the classical (unweighted) Calderón-Zygmund theory for double-phase problems.
