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arXiv · 2309.00068

The critical weighted inequalities of the spherical maximal function

Juyoung Lee
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Abstract

Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $|x|^{\alpha}$, it is known that the spherical maximal operator on $\mathbb{R}^d$ is bounded on $L^p(|x|^{\alpha})$ only if $1-d\leq \alpha<(d-1)(p-1)-d$ and under this condition, it is known to be bounded except $\alpha=1-d$. In this paper, we prove the case of the critical order, $\alpha=1-d$.

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Juyoung Lee. 2023-08-31. The critical weighted inequalities of the spherical maximal function. https://arxiv.org/abs/2309.00068

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