Titchmarsh theorems for Fourier transforms of H\"older-Lipschitz functions on compact homogeneous manifolds

TL;DR

This paper extends classical Fourier transform theorems to H"older-Lipschitz functions on compact homogeneous manifolds, providing new insights and tools for analysis on these spaces.

Contribution

It generalizes Titchmarsh theorems to compact homogeneous manifolds and introduces Fourier multiplier theorems for H"older-Lipschitz spaces on compact Lie groups.

Findings

Extended Titchmarsh theorems to compact homogeneous manifolds

Derived Fourier multiplier theorem for H"older-Lipschitz spaces

Characterized Dini-Lipschitz classes via Fourier coefficients

Abstract

In this paper we extend classical Titchmarsh theorems on the Fourier transform of H\"older-Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $L^2$-H\"older-Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini-Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.