幂零李群上的薛定谔算子的加权估计

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

中文关键词:  幂零李群  薛定谔算子  逆 H\"{o}lder 类.

英文关键词:nilpotent Lie group  Schr\"{o}dinger operators  reverse H\"{o}lder class.

中文摘要:

      在本文中考虑了幂零李群 $G$ 上非负位势 $W$ 属于逆 H\"{o}lder 类$B_{q_{_1}}$ 的薛定谔算子$-\Delta_{G} W$， 其中 $q_{_1}\geq \frac{D}{2}$ 且 $D$ 是 $G$ 在无穷远处的维数. 得到了算子 $W^\alpha(-\Delta_{G} W)^{-\beta}$ 和$W^\alpha\nabla_{G}(-\Delta_{G} W)^{-\beta}$ 的加权的 $L^p$--$L^q$ 估计.

英文摘要:

      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.