Harmonic analysis on 2-step stratified Lie groups without the Moore-Wolf condition

Abstract

In this thesis we investigate harmonic analysis on a particular class of sub-Riemannian manifold, namely the 2-step stratified Lie groups $\mathbb{G}$, as well as its applications in partial differential equations. This class consists a breadth of interesting geometric objects such as Heisenberg group and H-type Lie group, which can be seen as a meaningful extension of classical theories. After reviewing some main definitions and properties in Chapter 2, we start to study the most important representation of $\mathbb{G}$, the so-called Schr\"{o}dinger representation on $L^2(\mathbb{G})$, and then we prove the Stone-von Neumann theorem for the 2-step stratified Lie groups. In Chapter 3 we also study the Fourier transforms and define the $(\lambda,\nu)$-Wigner and $(\lambda,\nu)$-Weyl transform related to $\mathbb{G}$, we then show some properties of these transforms, which can help us to compute the sub-Laplacian and the $\lambda$-twisted sub-Laplacian. Moreover, in this chapter we demonstrate the beautiful interplay between the representation theory on $\mathbb{G}$ and the classical expansions in terms of Hermite functions and Lagueere functions, As applications, a global calculus of pseudo-differential operator on 2-step stratified Lie groups $\mathbb{G}$ is introduced in the fourth chapter. It relies on the explicit knowledge of the irreducible unitary representations of $\mathbb{G}$, which then allows one to reduce the analysis to study of a rescaled harmonic oscillator on unitary dual $\hat{\mathbb{G}}$. The sub-Laplacian appears as an elliptic operator in this calculus. The explicit formula for the heat kernel of the $\lambda$-twisted sub-Laplacian can be also obtained, which gives a closed formula for the heat kernel of the sub-Laplacian on $\mathbb{G}$.