Heat kernels and theory of Hardy spaces associated to Schrödinger operators on stratified groups

Let G be a stratified group and let Δ_G be a sub-Laplacian on G. In this paper, we consider the Schrödinger operator L=Δ_G+V, where the potential V is a nonnegative polynomial. We first prove the upper bound of the higher order derivatives of the heat kernel of L. We then establish a theory of Hardy spaces H_L^p(G) associated to L for the full range p∈(0,1]. Particularly, we prove that for any p∈(0,1], the Hardy space H_L^p(G) introduced in terms of nontangential or radial maximal functions associated to the semigroup e^−tL admits a new atomic decomposition. Moreover, we provide the description of the dual spaces of these new Hardy spaces in terms of certain local Campanato spaces related to the potential V. As a byproduct, we obtain the maximal function characterizations for the local Hardy spaces associated to an arbitrary critical function.