Harmonic Functions, Conjugate Harmonic Functions and the Hardy Space \(H^1\) in the Rational Dunkl Setting

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 \(H^1_{\Delta }\), can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood–Paley square functions.