$H^1$ BOUNDEDNESS FOR RIESZ TRANSFORM RELATED TO SCHRÖDINGER OPERATOR ON NILPOTENT GROUPS

Abstract

Let $\mathbb{G}$ be a nilpotent Lie groups equipped with a Hörmander system of vector fields $X = (X_1,\ldots,X_m)$ and $\Delta = \sum_{i=1}^m X_i^2$ be the sub-Laplacians associated with $X$. Let $A = -\Delta + W$ be the Schrödinger operator with the potential function $W$ belongs to the reverse Hölder class $B_q$ for some $q \ge D/2$, where $D$ denote the dimension at infinity. In this paper, we prove that the Riesz transform $\nabla A^{-1/2}$ related to Schrödinger operator $A$ is bounded from the Hardy space $H^1(\mathbb{G})$ to itself.