Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group (Analysis in Theory and Applications, 2016)
Let $\mathbb{H}^n$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension.In this paper, we consider the Schrödinger operator $−∆_{\mathbb{H}^n} +V$, where $Δ_{\mathbb{H}^n}$ isthe sub-Laplacian and $V$ is the nonnegative potential belonging to the reverse Hölderclass $B_{q_1}$ for $q_1 ≥ Q/2$. We show that the operators $T_1 = V(−∆_{\mathbb{H}^n} +V)^{−1}$ and $T_2 =V^{1/2}(−∆_{\mathbb{H}^n} +V)^{−1/2}$ are both bounded from $H^1_L(\mathbb{H}^n)$ into $L^1(\mathbb{H}^n)$. Our results arealso valid on the stratified Lie group.
Keywords: Heisenberg group; stratified Lie group; reverse Hölder class; Riesz transform; Schrödinger operator.