Riesz transform characterizations for multidimensional Hardy spaces

We study Hardy space $H^1_L(X)$ related to a self-adjoint operator $L$ defined on Euclidean domain $X \subseteq \mathbb{R}^d$. Under certain assumptions on the heat semigroup $\exp(-tL)$ we prove characterization of $H^1_L(X)$ by the Riesz transforms related to $L$. As an application, we prove the Riesz transform characterization for multidimensional Bessel and Laguerre operators.