Two results on the Dunkl maximal operator

TL;DR

This paper advances the understanding of the Dunkl maximal operator by refining its scalar maximal theorem with detailed constant behavior and completing the vector-valued theorem for a specific reflection group, including exponential integrability results.

Contribution

It improves the scalar maximal theorem for Dunkl operators with detailed constant analysis and completes the vector-valued theorem for the ^d reflection group, adding exponential integrability insights.

Findings

Refined the scalar maximal theorem with precise constant behavior.

Completed the vector-valued theorem for ^d reflection group.

Established exponential integrability for the case p=+ in the vector-valued setting.

Abstract

In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the case $\mathbb Z_2^d$ by establishing a result of exponential integrability corresponding to the case $p=+\infty$.