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In this work we extend the theory of the classical Hardy space $H^1$ to therational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on aEuclidean 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 anappropriate uniform $L^1$ condition. We prove that the boundary values of suchharmonic functions, which constitute the real Hardy space $H^1$, can becharacterized in several different ways, namely by means of atoms, Riesztransforms, maximal functions or Littlewood-Paley square functions.

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