$β$-dimensional sharp maximal function and applications
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909577410248704 |
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| author | Chen, You-Wei Benson Claros, Alejandro |
| author_facet | Chen, You-Wei Benson Claros, Alejandro |
| contents | In this paper, we study $β$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _βf(x) := \sup_{Q} \inf_{c \in \mathbb{R}} χ_{Q}(x) \frac{1}{\ell(Q)^β} \int_Q |f-c| \; d \mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all cubes in $\mathbb{R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{H}^β_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _βf$ by giving a good lambda estimate for $β$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04456 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $β$-dimensional sharp maximal function and applications Chen, You-Wei Benson Claros, Alejandro Functional Analysis 42B25, 46E35, 42B35 In this paper, we study $β$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _βf(x) := \sup_{Q} \inf_{c \in \mathbb{R}} χ_{Q}(x) \frac{1}{\ell(Q)^β} \int_Q |f-c| \; d \mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all cubes in $\mathbb{R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{H}^β_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _βf$ by giving a good lambda estimate for $β$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest. |
| title | $β$-dimensional sharp maximal function and applications |
| topic | Functional Analysis 42B25, 46E35, 42B35 |
| url | https://arxiv.org/abs/2407.04456 |