Calderón-Zygmund estimates for double phase problems with matrix weights
Sun-Sig Byun, Yumi Cho, Seungjin Ryu
TL;DR
The paper advances Calderón-Zygmund theory for double-phase, nonuniformly elliptic problems with matrix weights. By freezing the logarithm of the matrix field $\mathbb{M}$ and employing a fractional Muckenhoupt-Wheeden framework, it constructs scale-invariant comparison and level-set estimates that avoid Lavrentiev gaps at the sharp threshold $q/p\le 1+\alpha/n$. The main result shows that for any $\gamma>1$, $H(x,\mathbb{M}F)\in L^\gamma_{\text{loc}}(\Omega)$ implies $H(x,\mathbb{M}Du)\in L^\gamma_{\text{loc}}(\Omega)$ with explicit dependence on $\Lambda$ and $|\log\mathbb{M}|_{\mathrm{BMO}}$, generalizing the identity-weight case and bridging single-phase and anisotropic matrix-weighted models. The work introduces a novel combination of $\mathcal{A}_{p,s}$-type fractional control and $\log\mathrm{BMO}$ perturbation, enabling a robust, scale-invariant, and quantitatively sharp regularity theory for a broad class of anisotropic double-phase problems. This framework broadens applications to anisotropic materials and variable-coefficient elasticity while aligning with classical results as $\mathbb{M}\to I_n$.
Abstract
We establish an optimal Calderón-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For $1<p<q<\infty$, $a(\cdot)\in C^{0,α}(Ω)$ ($0<α\le1$), and a symmetric, almost everywhere positive definite matrix weight $\M$ with $|\M(x)|\,|\M(x)^{-1}|\leΛ$ for some constant $Λ\ge 1$ and small $|\log \M|_{\mathrm{BMO}}$, we prove, for every $γ>1$, $$ (|\M F|^p+a(x)|\M F|^q)\in L^γ_{\mathrm{loc}} \;\Longrightarrow\; (|\M Du|^p+a(x)|\M Du|^q)\in L^γ_{\mathrm{loc}}. $$ 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 $1/s=1/p-α/(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold $q/p\le 1+α/n$. Our result recovers the identity case $\,\M\equiv {\rm I}_n\,$, i.e., the classical (unweighted) Calderón-Zygmund theory for double-phase problems.