The critical weighted inequalities of the spherical maximal function

TL;DR

This paper investigates the weighted inequalities for the spherical maximal function, focusing on the critical case where the weight exponent equals 1 minus the dimension, a case previously not fully understood.

Contribution

The paper proves boundedness of the spherical maximal operator on weighted spaces at the critical weight exponent, filling a gap in the understanding of weighted inequalities.

Findings

Established boundedness at the critical weight exponent $oldsymbol{eta=1-d}$

Extended the understanding of weighted inequalities for the spherical maximal function

Clarified the behavior of the operator at the boundary of known weight ranges

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, $\alpha=1-d$.