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$\beta$-dimensional sharp maximal function and applications

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arxiv 2407.04456 v3 pith:7M64FUG3 submitted 2024-07-05 math.FA

You-Wei Benson Chen , Alejandro Claros This is my paper · ORCID
classification math.FA
keywords betamathcaldimensionalinequalitymaximalsharpaligncontent
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abstract

In this paper, we study $\beta$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _\beta f(x) := \sup_{Q} \inf_{c \in \mathbb{R}} \chi_{Q}(x) \frac{1}{\ell(Q)^\beta} \int_Q |f-c| \; d \mathcal{H}^{\beta}_\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 $\mathcal{H}^{\beta}_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _\beta f$ by giving a good lambda estimate for $\beta$-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.

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Cited by 3 Pith papers

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  1. Uncentered Fractional Maximal functions and mean oscillation spaces associated with dyadic Hausdorff content

    math.FA 2025-06 conditional novelty 8.0 of 10

    Fractional maximal operators map BMO functions into Hausdorff-content BLO spaces, send BMO to VMO via uniform continuity, and preserve vanishing mean oscillation spaces adapted to dyadic Hausdorff content.

  2. The Capacitary John-Nirenberg Inequality Revisited

    math.FA 2025-01 conditional novelty 6.0 of 10

    A packing condition is shown to characterize when an outer capacity is equivalent to its induced Hausdorff content, yielding maximal estimates and John-Nirenberg inequalities for such capacities.

  3. A note on new mapping properties for Wolff potential

    math.AP 2026-07 accept novelty 5.0 of 10

    Wolff potentials map L^q into Choquet L^s spaces with respect to Hausdorff content H^{n-κq}_∞, giving corresponding integrability for p-Laplace solutions.

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