On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916424569585664 |
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| author | Sweeting, Brandon |
| author_facet | Sweeting, Brandon |
| contents | In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood maximal operator and the Hilbert transform for weights in the class $A_p$, while also deriving necessary conditions to characterize the weights for which these estimates hold. In this paper, we establish the sufficiency of these conditions for the maximal operator when $p > 1$ and present corresponding results for the fractional maximal operators. This completes the characterization and resolves the open problem posed by Muckenhoupt and Wheeden for $p > 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04031 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator Sweeting, Brandon Classical Analysis and ODEs In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood maximal operator and the Hilbert transform for weights in the class $A_p$, while also deriving necessary conditions to characterize the weights for which these estimates hold. In this paper, we establish the sufficiency of these conditions for the maximal operator when $p > 1$ and present corresponding results for the fractional maximal operators. This completes the characterization and resolves the open problem posed by Muckenhoupt and Wheeden for $p > 1$. |
| title | On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2410.04031 |