$H^p$ spaces in the Dunkl setting meet Coifman--Weiss--type atoms

Abstract

Let $Δ_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which is originally defined by a relevant square function, coincides with the space generated by $({\rm CW},p,2)$-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where \[ s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor \] and $\mathbf N$ is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the $L^2$ size condition in the definition of atoms is replaced by the $L^\infty$ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of $({\rm CW},p,2)$-atoms.