Singular integral operators, $T1$ theorem, Littlewood-Paley theory and Hardy spaces in Dunkl Setting

TL;DR

This paper develops a comprehensive theory of singular integral operators in the Dunkl setting, establishing boundedness criteria, Littlewood-Paley theory, and Hardy spaces, with applications to Dunkl-Riesz transforms.

Contribution

It introduces a new class of singular integral operators in Dunkl analysis and proves a $T1$ theorem, along with establishing Dunkl-Hardy spaces and related boundedness results.

Findings

Boundedness of Dunkl singular integral operators on Hardy spaces

Development of Dunkl-Littlewood-Paley theory

Application to Dunkl-Riesz transforms

Abstract

The purpose of this paper is to introduce a new class of singular integral operators in the Dunkl setting involving both the Euclidean metric and the Dunkl metric. Then we provide the $T1$ theorem, the criterion for the boundedness on $L^2$ for these operators. Applying this singular integral operator theory, we establish the Littlewood-Paley theory and the Dunkl-Hardy spaces. As applications, the boundedness of singular integral operators, particularly, the Dunkl-Rieze transforms, on the Dunkl-Hardy spaces is given. The $L^2$ theory and the singular integral operator theory play crucial roles. New tools developed in this paper include the weak-type discrete Calder\'on reproducing formulae, new test functions, and distributions, the Littlewood-Paley, the wavelet-type decomposition, and molecule characterizations of the Dunkl-Hardy space, Coifman's approximation to the identity and the decomposition of the identity operator on $L^2$, Meyer's commutation Lemma, and new almost orthogonal estimates in the Dunkl setting.