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β-dimensional sharp maximal function and applications
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β-dimensional sharp maximal function and applications
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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.
Forward citations
Cited by 2 Pith papers
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A note on new mapping properties for Wolff potential
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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A note on Sobolev inequalities in the lower limit case
New Poincaré-Sobolev inequalities are proved under Choquet δ/n-integrability of gradients w.r.t. Hausdorff content, implying a Sobolev inequality for quasicontinuous W^{1,1}_0 functions.
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