TY - JOUR
T1 - Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group
AU - Y. Liu & G. B. Tang
JO - Analysis in Theory and Applications
VL - 1
SP - 78
EP - 89
PY - 2016
DA - 2016/01
SN - 32
DO - http://doi.org/10.4208/ata.2016.v32.n1.7
UR - https://global-sci.org/intro/article_detail/ata/4656.html
KW - Heisenberg group, stratified Lie group, reverse Hölder class, Riesz transform, Schrödinger operator.
AB - <p style="text-align: justify;">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 $\Delta_{\mathbb{H}^n}$&nbsp;is
the sub-Laplacian and $V$ is the nonnegative potential belonging to the reverse Hölder
class $B_{q_1}$ 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 are
also valid on the stratified Lie group.</p>