The dual of the Hardy space associated with the Dunkl operators

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 $BMC_{\kappa}$.