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<∞, a(·)∈ C0,α(Ω) (0<α≤1), and a symmetric, almost everywhere positive definite matrix weight with |(x)|\,|(x)⁻¹|≤Λ for some constant Λ≥ 1 and small |log |BMO, we prove, for every $γ>1$, (| F|ᵖ+a(x)| F|q)∈ L^γloc \;\; (| Du|ᵖ+a(x)| Du|q)∈ L^γloc. Our argument combines a freezing of the logarithm of the matrix field, log, with a fractional maximal-operator method governed by the Muckenhoupt-Wheeden Ap,s classes (where $1/s=1/p-α/(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold q/p≤ 1+α/n. Our result recovers the identity case \,≡ Iₙ\,, i.e., the classical (unweighted) Calderón-Zygmund theory for double-phase problems.