Hardy spaces associated with Schrodinger operators on the Heisenberg group

Let $L=-{\mathrm{\Delta }}_{{\mathbb{H}}^n}+V$ be a Schr"odinger operator on the Heisenberg group ${\mathbb{H}}^n$, where ${\mathrm{\Delta }}_{{\mathbb{H}}^n}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse H"older class $B_{\frac{Q}{2}}$ and $Q$ is the homogeneous dimension of ${\mathbb{H}}^n$. The Riesz transforms associated with the Schr"odinger operator $L$ are bounded from $L^1({\mathbb{H}}^n)$ to $L^{1,\mathrm{\infty }}({\mathbb{H}}^n)$. 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 $\big {e^{-s L}:; s>0 \big}$,andgivetheatomicdecompositionof$H^1_L($\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.