Riesz Transform Characterizations for Multidimensional Hardy Spaces

Abstract

We study Hardy space $$H^1_L(X)$$ related to a self-adjoint operator L defined on an Euclidean subspace X of $${{\mathbb {R}}^d}$$ . We continue study from [27], where, under certain assumptions on the heat semigroup $$\exp (-tL)$$ , the atomic characterization of local type for $$H^1_L(X)$$ was proved. In this paper we provide additional assumptions that lead to another characterization of $$H^1_L(X)$$ by the Riesz transforms related to L. As an application, we prove the Riesz transform characterization of $$H^1_L(X)$$ for multidimensional Bessel and Laguerre operators, and the Dirichlet Laplacian on $${\mathbb {R}}^d_+$$ .