Weak Type Estimates for Square Functions of Dunkl Heat Flows

TL;DR

This paper proves the weak (1,1) boundedness of the Littlewood--Paley--Stein square function for Dunkl heat flows, extending previous dimension-free $L^p$ boundedness results using kernel estimates and Calderón--Zygmund techniques.

Contribution

It establishes the weak (1,1) boundedness of the Dunkl heat flow square function, complementing prior $L^p$ boundedness results and advancing harmonic analysis in Dunkl settings.

Findings

Proves weak (1,1) boundedness of Dunkl heat square function.

Uses kernel estimates and Calderón--Zygmund decomposition techniques.

Extends previous $L^p$ boundedness results to weak (1,1) case.

Abstract

The weak $(1,1)$ boundedness of the Littlewood--Paley--Stein square function for the Dunkl heat flow is proved via estimates on the Dunkl heat kernel of integral type and the Caldr\'{o}n--Zygmund decomposition, which is the continuity of the recently work [arXiv:2003.11843] where the dimension-free $L^p$ boundedness of the same square function is studied.