Calderón-Zygmund estimates for double phase problems with matrix weights
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912860453470208 |
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| author | Byun, Sun-Sig Cho, Yumi Ryu, Seungjin |
| author_facet | Byun, Sun-Sig Cho, Yumi Ryu, Seungjin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20856 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Calderón-Zygmund estimates for double phase problems with matrix weights Byun, Sun-Sig Cho, Yumi Ryu, Seungjin Analysis of PDEs 35B65, 35J70, 35J75 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. |
| title | Calderón-Zygmund estimates for double phase problems with matrix weights |
| topic | Analysis of PDEs 35B65, 35J70, 35J75 |
| url | https://arxiv.org/abs/2505.20856 |