Capacitary Muckenhoupt Weight, BMO and BLO Spaces with Hausdorff Content, Factorization Theorems and Applications
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| Formato: | Preprint |
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2025
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| _version_ | 1866917237806333952 |
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| author | Huang, Long Zhang, Yangzhi Zhuo, Ciqiang |
| author_facet | Huang, Long Zhang, Yangzhi Zhuo, Ciqiang |
| contents | Let $δ\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ spaces for all dimension $δ\in(0,n]$, and further to comprehend the structure of these two spaces. Our main result shows that $\mathcal A_{p,δ}$ for $p\in(1,\infty)$ is equivalent to the BMO spaces, while $\mathcal A_{1,δ}$ is equivalent to the BLO spaces, and consequently yields the factorization theorems for these BMO and BLO spaces via capacitary Hardy--Littlewood maximal operators, which essentially extend main results of Coifman and Rochberg in 1980 beyond measure theory. As applications, by establishing some capacitary weighted John--Nirenberg inequalities, we obtain the equivalence between capacitary weighted BMO or BLO spaces and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ respectively. These results reveal deep connections between $\mathcal A_{p,δ}$ and BMO or BLO spaces with Hausdorff content, beyond the classical measure-theoretic settings. We develop some approaches in the proofs and using a new observation, that is, the additivity of measures and linearity of integrals are superfluous for the corresponding classical theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01161 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Capacitary Muckenhoupt Weight, BMO and BLO Spaces with Hausdorff Content, Factorization Theorems and Applications Huang, Long Zhang, Yangzhi Zhuo, Ciqiang Classical Analysis and ODEs Let $δ\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ spaces for all dimension $δ\in(0,n]$, and further to comprehend the structure of these two spaces. Our main result shows that $\mathcal A_{p,δ}$ for $p\in(1,\infty)$ is equivalent to the BMO spaces, while $\mathcal A_{1,δ}$ is equivalent to the BLO spaces, and consequently yields the factorization theorems for these BMO and BLO spaces via capacitary Hardy--Littlewood maximal operators, which essentially extend main results of Coifman and Rochberg in 1980 beyond measure theory. As applications, by establishing some capacitary weighted John--Nirenberg inequalities, we obtain the equivalence between capacitary weighted BMO or BLO spaces and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ respectively. These results reveal deep connections between $\mathcal A_{p,δ}$ and BMO or BLO spaces with Hausdorff content, beyond the classical measure-theoretic settings. We develop some approaches in the proofs and using a new observation, that is, the additivity of measures and linearity of integrals are superfluous for the corresponding classical theory. |
| title | Capacitary Muckenhoupt Weight, BMO and BLO Spaces with Hausdorff Content, Factorization Theorems and Applications |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.01161 |