Constructions of representations of solvable Lie groups and non-commutative Fourier analysis

本文主要从以下几个角度研究李群上的非交换傅里叶变换。对于指数p （1<p≦2）及其共轭q，定义L^p-傅里叶变换为一个有界算子，从群上的L^p-函数空间到酉对偶上算子域的L^q空间；其范数的确定是谐波分析的主要问题之一。我们得到了由R^n的紧扩展定义的群的范数。我们也得到了一般连通幂零李群范数的估计。接下来，我们将傅里叶变换作为群的C^*-代数到酉对偶上有界算子域的C^*-代数的同态，并开始尝试分析傅里叶变换在六维可解李群上的像。

This research mainly concerns non-commutative Fourier transforms on Lie groups from the following viewpoints. For an exponent p (1＜p≦2) and its conjugate q, the L^p-Fourier transform is defined to be a bounded operator from the space of L^p-functions on the group to the L^q-space of operator fields on the unitary dual; the determination of its norm is one of the main problems of harmonic analysis. We obtained the norm for the groups defined by compact extensions of R^n. We also obtained an estimate of the norm for general connected nilpotent Lie groups. Next, we have treated the Fourier transform as a homomorphism from the C^*-algebra of the group to the C^*-algebra of bounded operator fields over the unitary dual, and began trying to analyze the image of the Fourier transform on a certain example of six dimensional solvable Lie group.