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

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 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.

This preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in the Journal of Fourier Analysis and Applications, and is available online at https://doi.org/10.1007/s00041-022-09955-1

References in corpus (1)

• $L^p$-$L^q$ boundedness of $(k,a)$-Fourier multipliers with applications to Nonlinear equations