论文标题
Gelfand三元组的Kohn-Nirenberg量子量子
Gelfand triples for the Kohn-Nirenberg quantization on homogeneous Lie groups
论文作者
论文摘要
在本文中，我们研究了对均匀谎言组的傅立叶变换和Kohn-Nirenberg量化的量子，作为某些Gelfand三元组之间的映射。为此，我们将考虑因素限制在案件中，均匀的谎言组$ g $接纳了不可约合的统一表示，它们是$ g $的中心$ z（g）$ $ g $的中心模式，而其中$ \ dim z（g）= 1 $。将Schwartz空间替换为一个子空间$ \ MATHCAL S _*（g）\ hookrightArrow \ Mathcal s（g）$，我们表征了$ \ Mathcal s _*（g）$的组傅立叶变换的范围，并在$ l^2（g，g，μ）$ l^2（g，μ）$ l^2（g）$ l^2（g，g，g^2）傅立叶变换成为Gelfand三倍异构形态。我们以大量矢量有价值的平滑函数的分布乘以乘法的结果，并使用它来建立Kohn-Nirenberg量化为我们的Gelfand Triples的同构，并为操作员的Kohn-Nirenberg符号提供明确的公式。
In this paper, we study the group Fourier transform and the Kohn-Nirenberg quantization for homogeneous Lie groups as mappings between certain Gelfand triples. For this, we restrict our considerations to the case, where the homogeneous Lie group $G$ admits irreducible unitary representations, that are square integrable modulo the center $Z(G)$ of $G$, and where $\dim Z(G)=1$. Replacing the Schwartz space by a certain subspace $\mathcal S_*(G) \hookrightarrow \mathcal S(G)$, we characterise the range of the group Fourier transform on $\mathcal S_*(G)$ and construct distributions and Gelfand triples around $L^2(G,μ)$ and its Fourier image $L^2(\hat G,\hat μ)$, such that the Fourier transform becomes a Gelfand triple isomorphism. We give results on the multiplication of distributions with a large class of vector valued smooth functions and use this to establish the Kohn-Nirenberg quantization as an isomorphism for our Gelfand triples and provide an explicit formula for the Kohn-Nirenberg symbol of an operator.
