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 − t L 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.