## LSU Historical Dissertations and Theses

1991

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Leonard F. Richardson

#### Abstract

This dissertation arose from efforts to prove the following conjecture, which generalizes to nilpotent Lie groups a weak form of the classical Paley-Wiener theorem for $\IR\sp{n}$: Let N be any connected, simply connected nilpotent Lie group with unitary dual N, and let $\varphi\in L\sbsp{c}{\infty}(N)$. Suppose that there exists a subset E $\subset$ N of positive Plancherel measure such that $\\varphi(\pi)$ = 0 for all $\pi\in$ E, where $\\varphi(\pi)$ is the operator-valued Fourier transform of $\varphi$. Then $\varphi$ = 0 almost everywhere on N. The writer has been able to prove a slightly weakened form of the conjecture for a large subclass of nilpotent Lie groups, and the conjecture itself for several interesting examples that lie outside this subclass. Chapter 3 contains these proofs, which make use of certain special polarizations (maximal subordinate subalgebras) of the Lie algebras there considered. Chapter 2 explains how to construct such polarizations in any nilpotent Lie algebra. Chapter 1 provides background for the work undertaken in Chapters 2 and 3.

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