论文标题
Heisenberg Group上Schr {Ö} dinger操作员的本地分散和Strichartz估计
Local dispersive and Strichartz estimates for the Schr{ö}dinger operator on the Heisenberg group
论文作者
论文摘要
H. Bahouri，P。G{é} Rard和C.-J.证明了这一点。 XU在[9]中，Heisenberg Group $ \ Mathbb {H}^D $（涉及Sublaplacian）上的schr {Ö} dinger方程是完全非差异性进化方程的一个示例：出于这个原因，全球分散估计无法满足。本文旨在通过对Schr {Ö} dinger内核$ s_t $ on $ \ mathbb {h}^d $进行线性schr {Ö} dinger方程的$ \ mathbb {h}^d $建立本地分散估计。这些估计的清晰度通过几个示例讨论。我们的方法是基于B. Gaveau在[20]中得出的$ \ Mathbb {H}^D $上的热核的明确公式，是通过将复杂分析和傅立叶分析组合结合来实现的。作为我们结果的副产品，我们建立了本地strichartz估计，并证明了内核$ s_t $集中在$ \ mathbb {h}^d $的量化水平超平面上。
It was proved by H. Bahouri, P. G{é}rard and C.-J. Xu in [9] that the Schr{ö}dinger equation on the Heisenberg group $\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schr{ö}dinger equation, by a refined study of the Schr{ö}dinger kernel $S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d$.
