Harmonic functions, conjugate harmonic functions and the Hardy space $H^1$ in the rational Dunkl setting

Abstract

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space ${\mathbb{R}}^N$. On the half-space ${\mathbb{R}}_{+}\times {\mathbb{R}}^N$, we consider systems of conjugate $({\partial }_t^2+{\Delta }_{\mathbf{x}})$-harmonic functions satisfying an appropriate uniform $L^1$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $H^1$, can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood-Paley square functions.