On those Weights Satisfying a Weak-Type Inequality for the Maximal Operator and Fractional Maximal Operator
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916424569585664 |
|---|---|
| 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 |