Partial Fourier series on compact Lie groups

TL;DR

This paper studies the partial Fourier series on products of compact Lie groups, providing conditions for coefficients to define smooth functions or distributions, and explores applications to evolution equations on specific manifolds.

Contribution

It offers necessary and sufficient conditions for partial Fourier coefficients to correspond to smooth functions or distributions on compact Lie groups, and applies these results to evolution equations on b7b7b7.

Findings

Conditions for Fourier coefficients to define smooth functions or distributions.

Criteria for the global solvability of evolution equations on b7b7b7.

Properties of evolution equations derived from constant coefficient cases.

Abstract

In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As applications, we will study conditions for the global solvability of an evolution equation defined on $\mathbb{T}^1\times\mathbb{S}^3$ and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation.