arXiv subjects

Arne Hendrickx

Publications and source records attributed to Arne Hendrickx.

3 recordsLinked to original sources

arXiv2025

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

We extend the classical Titchmarsh theorems to the Fourier transform of two types of Hölder-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 Hölder-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.

math.FA↗

arXiv2023

$L^{p}$-$L^{q}$ boundedness of Fourier multipliers on Fundamental domains of Lattices in $\mathbb{R}^d$

In this paper we study the $L^{p}$-$L^{q}$ boundedness of Fourier multipliers on the fundamental domain of a lattice in $\mathbb{R}^{d}$ for $1 < p,q < \infty$ under the classical Hörmander condition. First, we introduce Fourier analysis on lattices and have a look at possible generalisations. We then prove the Hausdorff-Young inequality, Paley's inequality and the Hausdorff-Young-Paley inequality in the context of lattices. This amounts to a quantitative version of the $L^{p}$-$L^{q}$ boundedness of Fourier multipliers. Moreover, the Paley inequality allows us to prove the Hardy-Littlewood inequality.

math.FA↗

arXiv2022

Global pseudo-differential operators on the Lie group $G= (-1,1)^n$

In this work we characterise the Hörmander classes $\symbClassOn{m}ρδ{\group,\textnormal{Hör}}$ on the open manifold $\group = (-1,1)^n$. We show that by endowing the open manifold $\group = (-1,1)^n$ with a group structure, the corresponding global Fourier analysis on the group allows one to define a global notion of symbol on the phase space $\group \times \R^n$. Then, the class of pseudo-differential operators associated to the global Hörmander classes $\symbClassOn{m}ρδ{\group \times \R^n}$ recovers the Hörmander classes $\symbClassOn{m}ρδ{\group,\textnormal{loc}}$ defined by local coordinate systems. The analytic and qualitative properties of the classes $\symbClassOn{m}ρδ{\group \times \R^n}$ are presented in terms of the corresponding global symbols. In particular, $L^p$-Fefferman type estimates and Calderón-Vaillancourt theorems are analysed, as well as the spectral properties of the operators.

math.AP↗