Titchmarsh theorems for H\"older-Lipschitz functions on fundamental domains of lattices in $\mathbb{R}^{d}$ with applications to boundedness of Fourier multipliers

Abstract

We extend the classical Titchmarsh theorems to the Fourier transform of two types of H\"older-Lipschitz functions - additive and multiplicative - defined on fundamental domains of lattices in $\mathbb{R}^d$. Our approach is based on generalizations of Duren's lemma, which we first illustrate in the classical Euclidean setting. As an application of the second Titchmarsh theorem, we obtain boundedness results for Fourier multipliers between H\"older-Lipschitz spaces, from which we deduce Lipschitz-Sobolev regularity for Bessel potential operators on fundamental domains of lattices in the additive case. These results provide a natural generalization of classical one-dimensional theorems on the real line and on the torus to higher dimensions.