The Characterizations of the Hardy Space Associated with Schrödinger Operators via Square Functions on Heisenberg Groups

Abstract

In this paper we assume that $L = −∆_{\mathbb{H}^n}+ V$ is a Schrödinger operator on the Heisenberg group $\mathbb{H}^n,$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_{Q/2}.$ We introduce the Littlewood-Paley $\mathfrak{g}$-functions, the Lusin area functions and the $\mathfrak{g}^∗_λ$-functions generated by the heat semigroup $\{e^{−tL}\}_{t>0}$ and the Poisson semigroup $\{e^{−t\sqrt{L}}\}_{t>0},$ respectively. By means of the reproducing formulas and the regularity properties of semigroups, we establish several square function characterizations of the Hardy space $H_L^1(\mathbb{H}^n)$ associated with $L.$