Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911449827246080 |
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| author | Li, Ziwei Yang, Dachun Yuan, Wen |
| author_facet | Li, Ziwei Yang, Dachun Yuan, Wen |
| contents | Let ${\mathcal {X}}$ be a space of homogeneous type. In this article, based on the reducing operators of matrix $A_p$-weights, the authors introduce the vector-valued Hajłasz gradient sequences and establish some related matrix-weighted Poincaré-type inequalities on ${\mathcal {X}}$. As an application, the authors introduce the matrix-weighted logarithmic Besov spaces on ${\mathcal {X}}$ and establish their pointwise characterization via Hajłasz gradient sequences. The novelty of this article lies in that, by means of both the $A_p$ dimension and its properties of matrix $A_p$-weights and the wavelet reproducing formula with exponential decay of P. Auscher and T. Hytönen, all the main results get rid of the dependence on the reverse doubling conditions of both weights and ${\mathcal {X}}$ under consideration and these results are also completely new even for unweighted logarithmic Besov spaces on ${\mathcal {X}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type Li, Ziwei Yang, Dachun Yuan, Wen Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E36, Secondary 46E35, 42B35, 26D10, 26A33, 30L99 Let ${\mathcal {X}}$ be a space of homogeneous type. In this article, based on the reducing operators of matrix $A_p$-weights, the authors introduce the vector-valued Hajłasz gradient sequences and establish some related matrix-weighted Poincaré-type inequalities on ${\mathcal {X}}$. As an application, the authors introduce the matrix-weighted logarithmic Besov spaces on ${\mathcal {X}}$ and establish their pointwise characterization via Hajłasz gradient sequences. The novelty of this article lies in that, by means of both the $A_p$ dimension and its properties of matrix $A_p$-weights and the wavelet reproducing formula with exponential decay of P. Auscher and T. Hytönen, all the main results get rid of the dependence on the reverse doubling conditions of both weights and ${\mathcal {X}}$ under consideration and these results are also completely new even for unweighted logarithmic Besov spaces on ${\mathcal {X}}$. |
| title | Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 46E36, Secondary 46E35, 42B35, 26D10, 26A33, 30L99 |
| url | https://arxiv.org/abs/2602.14361 |