Hardy spaces associated with Schrodinger operators on the Heisenberg group

Abstract

Let $L=-{\Delta }_{{\mathbb{H}}^n}+V$ be a Schrödinger operator on the Heisenberg group ${\mathbb{H}}^n$, where ${\Delta }_{{\mathbb{H}}^n}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse Hölder class $B_{\frac{Q}{2}}$ and $Q$ is the homogeneous dimension of ${\mathbb{H}}^n$. The Riesz transforms associated with the Schrödinger operator $L$ are bounded from $L^1({\mathbb{H}}^n)$ to $L^{1,\infty }({\mathbb{H}}^n)$. The $L^1$ integrability of the Riesz transforms associated with $L$ characterizes a certain Hardy type space denoted by $H_L^1({\mathbb{H}}^n)$ which is larger than the usual Hardy space $H^1({\mathbb{H}}^n)$. We define $H_L^1({\mathbb{H}}^n)$ in terms of the maximal function with respect to the semigroup $\{e^{-sL}:s>0\}$, and give the atomic decomposition of $H_L^1({\mathbb{H}}^n)$. As an application of the atomic decomposition theorem, we prove that $H_L^1({\mathbb{H}}^n)$ can be characterized by the Riesz transforms associated with $L$. All results hold for stratified groups as well.

42 pages