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October 5, 2025Open Access

Calderón-Zygmund estimates for double phase problems with matrix weights

Key Points

The study proves a crucial relationship between weighted norms in double phase problems, enhancing existing theories.
It reveals that certain logarithmic properties of matrix weights are significant, enabling new results in nonuniformly elliptic contexts.
The paper also incorporates a fractional maximal-operator method, which is essential for deriving level-set estimates.
The findings extend classical Calderón-Zygmund theory to cases with matrix weights, improving understanding of elliptic equations.

Abstract

We establish an optimal Calderón-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For 11, (| F|ᵖ+a (x) | F|q) L^γ₋₎₂ \;\; (| Du|ᵖ+a (x) | Du|q) L^γ₋₎₂. Our argument combines a freezing of the logarithm of the matrix field, , with a fractional maximal-operator method governed by the Muckenhoupt-Wheeden A₏, ₒ 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.

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