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On Sobolev norms for Lie group representations
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.jfa.2020.108882
Heiko Gimperlein , Bernhard Krötz

We define a continuous scale of Sobolev norms for a Banach representation $(\pi, E)$ of a Lie group, with regard to a single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is a Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for $D$ on the space of smooth vectors of $E$. The main tool is a novel factorization of the delta distribution on a Lie group.

中文翻译:

李群表示的Sobolev范数

我们为 Lie 群的 Banach 表示 $(\pi, E)$ 定义了 Sobolev 范数的连续尺度,关于单个微分算子 $D=d\pi(R^2+\Delta)$。这里,$\Delta$ 是泛包络代数中的拉普拉斯元素,$R>0$ 明确取决于表示的增长率。特别是,我们在 $E$ 的平滑向量空间上获得了 $D$ 的谱间隙。主要工具是李群上 delta 分布的新因式分解。
更新日期:2021-02-01
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