Titchmarsh theorems for Fourier transforms of H\xf6lder\u2013Lipschitz functions on compact homogeneous manifolds

Julio Delgado,Michael Ruzhansky,Radouan Daher

doi:10.1007/s00605-018-1253-0

Abstract

In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder–Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for L^2-Hölder–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.

Highlights

The studies of the convergence and of the rate of decay of Fourier coefficients are among the most classical problems in Fourier analysis
Starting from the Riemann– Lebesgue theorem relating the integrability of a function on the torus T1 and the convergence of its Fourier coefficients, through the Hausdorff–Young inequality relating the integrability of a function and of its Fourier transform
One can relate the smoothness of a function on the torus and the rate of decay of its Fourier coefficients: if f ∈ Ck(T1) for some k ≥ 1, one has f ( j ) = o(| j |−k), that is, | j|k| f ( j)| → 0 as j → ∞

Summary

Introduction

The studies of the convergence and of the rate of decay of Fourier coefficients are among the most classical problems in Fourier analysis. This is a natural family of spaces on G0, extending the 2-norm appearing in the Plancherel identity in (1.9), satisfying the Hausdorff–Young inequalities on G/K and many other functional analytic properties These spaces were analysed in [20, Section 2] in the setting of compact homogeneous manifolds, extending the corresponding definition on the compact Lie groups introduced earlier in [30, Section 10.3.3], in which case we would have G0 = G and kξ = dξ : see Definition 2.2 for this case. As an application of the obtained characterisations, we will show a Fourier multiplier theorem for Hölder/Lipschitz spaces on compact Lie groups. Our analysis in this paper is based on a logarithmic extension of the Duren lemma allowing inclusion of the log-terms in the characterisations

Preliminaries

Titchmarsh theorems for Fourier transforms of Lipschitz functions

Dini–Lipschitz functions

Findings

Background