The dual of the Hardy space associated with the Dunkl operators

TL;DR

This paper characterizes the dual space of the Hardy space associated with Dunkl operators using weighted Carleson measures, providing an analytic proof and a Fefferman-Stein decomposition.

Contribution

It introduces a new dual space characterization for Dunkl Hardy spaces via weighted Carleson measures, avoiding atomic decomposition methods.

Findings

Dual space of Dunkl Hardy space is realized by functions linked to weighted Carleson measures.

Provides an analytic proof for the dual space characterization.

Establishes Fefferman-Stein decomposition for functions in the dual space.

Abstract

The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $R^d$ associated with a root system, that contain some non-local refection terms, and the associated Hardy space is defined by means of the Riesz transforms with respect to the Dunkl operators. The aim of the paper is to prove that its dual can be realized by a class of functions on $R^d$, denoted by $BMC_{\kappa}$ in the text, that consists of the underlying functions of a certain type of weighted Carleson measures. Our method is "purely analytic" and does not depend on the atomic decomposition. As a corollary we obtain the Fefferman-Stein decomposition of functions in $BMC_{\kappa}$.