Classical Analysis and Nilpotent Lie Groups

TL;DR

This paper explores the extension of classical Fourier analysis to nilpotent Lie groups and related Riemannian manifolds, highlighting the similarities with traditional Fourier theories and discussing some infinite-dimensional analogs.

Contribution

It provides a detailed description of how Fourier analysis concepts extend to nilpotent Lie groups and certain Riemannian manifolds, connecting classical analysis with geometric group theory.

Findings

Fourier analysis extends to nilpotent Lie groups with similar principles as classical Fourier theory.

The paper discusses analogs in infinite-dimensional settings.

Connections between Fourier analysis, group theory, and geometry are elucidated.

Abstract

Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I'll describe how this goes for nilpotent Lie groups and for a class of Riemannian manifolds closely related to a nilpotent Lie group structure. There are also some infinite dimensional analogs but I won't go into that here. The analytic ideas are not so different from those of the classical Fourier transform and Fourier inversion theories in one real variable.