The Weighted Estimates of the Schr\"{o}dinger Operators on the Nilpotent Lie Group

Abstract:

      In this paper we consider the Schr\"{o}dinger operator $-\Delta_{G} W$ on the nilpotent Lie group $G$ where the nonnegative potential $W$ belongs to the reverse H\"{o}lder class $B_{q_{_1}}$ for some $q_{_1}\geq \frac{D}{2}$ and $D$ is the dimension at infinity of $G$. The weighted $L^p-L^q$ estimates for the operators $W^\alpha(-\Delta_{G} W)^{-\beta}$ and $W^\alpha\nabla_{G}(-\Delta_{G} W)^{-\beta}$ are obtained.