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arXiv · 2305.00703

Rearrangement inequalities of the one-dimensional maximal functions associated with general measures

Xudong Nie·Di Wu·Panwang Wang
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Abstract

We prove a rearrangement inequality for the uncentered Hardy-Littlewood maximal function $M_{\mu}$ associate to general measure $\mu$ on $\mathbb{R}$. This inequality is analogous to the Stein's result $cf^{**}(t)\leq(Mf)^{*}(t)\leq C f^{**}(t)$, where $f^*$ is the symmetric decreasing rearrangement function of $f$ and $f^{**}(t)=\int_0^tf^*(x)dx$. Moreover, we compute the best constant of $M_{\mu}$ on $L^{p,\infty}(\mathbb{R},d\mu)$.

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Xudong Nie, Di Wu, Panwang Wang. 2023-05-01. Rearrangement inequalities of the one-dimensional maximal functions associated with general measures. https://arxiv.org/abs/2305.00703

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