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http://dx.doi.org/10.4134/JKMS.2010.47.2.425

Lp ESTIMATES FOR SCHRÖDINGER TYPE OPERATORS ON THE HEISENBERG GROUP  

Yu, Liu (Department of Mathematics and Mechanics, University of Science and Technology Beijing)
Publication Information
Journal of the Korean Mathematical Society / v.47, no.2, 2010 , pp. 425-443 More about this Journal
Abstract
We investigate the Schr$\ddot{o}$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$\ddot{o}$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 $\nabla^4_{\mathbb{H}^n}H^{-1}_2$, where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n$.
Keywords
Heisenberg group; Schr$\ddot{o}$dinger operators; reverse H$\ddot{o}$lder class;
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