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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.04456 |
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Table of 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.