Riesz transform characterization of Hardy spaces associated with Dunkl--Schr\"odinger operators

Abstract

Let $L=-\Delta_k+V$ be a Schr\"odinger operator associated with the Dunkl Laplacian and a nonnegative potential. We characterize the Hardy space $H_L^1$, defined by the maximal operator of the semigroup generated by $-L$, in terms of the Riesz transforms associated with $L$. Under a local integral condition on the potential and a decay condition on the semigroup, formulated relative to an admissible covering by cubes, we prove that \[ \|f\|_{H_L^1} \asymp \|f\|_{L^1(dw)} +\sum_{j=1}^N \|\widetilde{R}_j f\|_{L^1(dw)}. \] The proof combines an atomic characterization of $H_L^1$ with local Hardy space theory and weighted integral estimates for Dunkl derivatives of the heat kernels. The result applies to arbitrary root systems and nonnegative multiplicity functions.