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

TL;DR

This paper investigates the boundedness of Fourier multipliers between different L^p spaces on fundamental domains of lattices in R^d, extending classical inequalities to this setting.

Contribution

It introduces Fourier analysis on lattices and establishes key inequalities, providing a quantitative framework for Fourier multiplier boundedness on lattice domains.

Findings

Proves Hausdorff-Young inequality for lattice domains

Establishes Paley's inequality in the lattice context

Derives Hardy-Littlewood inequality from Paley's inequality

Abstract

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 H\"ormander 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.