论文标题
对恩格尔和卡坦群的量化
Quantizations on the Engel and the Cartan groups
论文作者
论文摘要
这项工作旨在在两个分级的三步式谎言组的具体环境中开发全球量化；即恩格尔集团和卡坦集团。我们对上述群体的结构和表示及其相应的代数提供了初步分析。此外，得出了两个设置中差异运算符的显式公式，构成了两种情况下$ψ^{m} _ {ρ，δ} $类符号类的构造的必要先决条件。在恩格尔组的情况下，科恩·尼伦贝格量化与恩格尔组的表示之间的关系使我们能够以欧几里得空间中符号的量化来表达运营商。我们说明了光谱乘数定理的$ l^p-l^q $有界性，两组中具有特定的运算符示例，这些示例会产生适当的sobolev型不平等。作为上述内容的进一步应用，我们在两种设置中这些特定操作员的特定示例的热内核的$ l^p-l^q $规范会产生一些后果。
This work aims to develop a global quantization in the concrete settings of two graded nilpotent Lie groups of 3-step; namely of the Engel group and the Cartan group. We provide a preliminary analysis on the structure and the representations of the aforementioned groups, and their corresponding Lie algebras. In addition, the explicit formulas for the difference operators in the two settings are derived, constituting the necessary prerequisites for the constructions of the $Ψ^{m}_{ρ,δ}$ classes of symbols in both cases. In the case of the Engel group, the relation between the Kohn-Nirenberg quantization and the representations of the Engel group enables us to express operators in this setting in terms of quantization of symbols in the Euclidean space. We illustrate the $L^p-L^q$ boundedness for spectral multipliers theorems with particular examples of operators in both groups, that yield appropriate Sobolev-type inequalities. As a further application of the above, we get some consequences for the $L^p-L^q$ norm for the heat kernel of those particular examples of operators in both settings.
