J. Korean Math. Soc. 2010; 47(2): 425-443
Printed March 1, 2010
https://doi.org/10.4134/JKMS.2010.47.2.425
Copyright © The Korean Mathematical Society.
Liu Yu
University of Science and Technology Beijing
We investigate the Schr\"{o}dinger type operator $H_2=(-\Delta_{\mathbb{H}^n})^2+V^2$ on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sublaplacian and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_q$ for $q\geq \frac{Q}{2}$, where $Q$ is the homogeneous dimension of $\mathbb{H}^n$. We shall establish the estimates of the fundamental solution for the operator $H_2$ and obtain the $L^p$ estimates for the operator $ \nabla_{\mathbb H^n}^4H^{-1}_2$, where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n$.
Keywords: Heisenberg group, Schr\"{o}dinger operators, reverse H\"{o}lder class
MSC numbers: 22E30, 35J10, 42B20
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