Smoothness of Densities on Compact Lie Groups

TL;DR

This paper establishes precise conditions under which probability densities on compact Lie groups are smooth or square integrable, enhancing understanding of their regularity properties.

Contribution

It provides the first complete characterization of when densities on compact Lie groups are smooth or square integrable, filling a gap in harmonic analysis on these groups.

Findings

Necessary and sufficient conditions for smoothness.

Criteria for square integrability of densities.

Advances in harmonic analysis on Lie groups.

Abstract

We give necessary and sufficient conditions for both square integrability and smoothness for densities of a probability measure on a compact connected Lie group.