Variable Muckenhoupt $A_\infty$ Weights
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914562562850816 |
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| author | Yang, Dachun Yuan, Wen Zeng, Zongze |
| author_facet | Yang, Dachun Yuan, Wen Zeng, Zongze |
| contents | In this article, with introducing concepts of variable scalar $\mathcal{A}_{p(\cdot),\infty}$ weights and variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we seek a comprehensive theory of $A_\infty$ weights within the framework of variable exponent spaces. We first show that a weight belongs to $\mathcal{A}_{p(\cdot),\infty}$ if and only if its $p(\cdot)$-th power is an $A_\infty$ weight. Using this, we characterize the $\mathcal{A}_{p(\cdot),\infty}$ condition by the minimal operator. Then we establish the reverse Hölder's inequality for $\mathcal{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces with explicit constants and, combining this with the previously established relationship between $\mathcal{A}_{p(\cdot),\infty}$ weights and $A_\infty$ weights, we prove that, for any weight $w$, the reverse Hölder's inequality holds in variable Lebesgue spaces if and only if $w$ is an $\mathcal{A}_{p(\cdot),\infty}$ weight. For the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we first show the existence of the reducing operators for matrix $\mathscr{A}_{p(\cdot),\infty}$ weights and then, combining the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights with the scalar $\mathcal{A}_{p(\cdot),\infty}$ weights, we establish the reverse Hölder's inequality for $\mathscr{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces. Finally, for further applications to variable matrix-weighted function spaces, we introduce the upper and the lower dimensions for $\mathscr{A}_{p(\cdot),\infty}$ weights and use these concepts to establish the sharp estimate involving reducing operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variable Muckenhoupt $A_\infty$ Weights Yang, Dachun Yuan, Wen Zeng, Zongze Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E30, Secondary 47A56, 15A15, 46E40, 42B99 In this article, with introducing concepts of variable scalar $\mathcal{A}_{p(\cdot),\infty}$ weights and variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we seek a comprehensive theory of $A_\infty$ weights within the framework of variable exponent spaces. We first show that a weight belongs to $\mathcal{A}_{p(\cdot),\infty}$ if and only if its $p(\cdot)$-th power is an $A_\infty$ weight. Using this, we characterize the $\mathcal{A}_{p(\cdot),\infty}$ condition by the minimal operator. Then we establish the reverse Hölder's inequality for $\mathcal{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces with explicit constants and, combining this with the previously established relationship between $\mathcal{A}_{p(\cdot),\infty}$ weights and $A_\infty$ weights, we prove that, for any weight $w$, the reverse Hölder's inequality holds in variable Lebesgue spaces if and only if $w$ is an $\mathcal{A}_{p(\cdot),\infty}$ weight. For the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we first show the existence of the reducing operators for matrix $\mathscr{A}_{p(\cdot),\infty}$ weights and then, combining the matrix $\mathscr{A}_{p(\cdot),\infty}$ weights with the scalar $\mathcal{A}_{p(\cdot),\infty}$ weights, we establish the reverse Hölder's inequality for $\mathscr{A}_{p(\cdot),\infty}$ weights in variable Lebesgue spaces. Finally, for further applications to variable matrix-weighted function spaces, we introduce the upper and the lower dimensions for $\mathscr{A}_{p(\cdot),\infty}$ weights and use these concepts to establish the sharp estimate involving reducing operators. |
| title | Variable Muckenhoupt $A_\infty$ Weights |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E30, Secondary 47A56, 15A15, 46E40, 42B99 |
| url | https://arxiv.org/abs/2605.12941 |