$$L^{p}$$–$$L^{q}$$ 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. We will show that this delivers also some \(L^{p}\)-estimates by using some standard Lebesgue space embeddings, which come from the finite measure of the fundamental domain. Moreover, the Paley inequality allows us to prove the Hardy–Littlewood inequality. As an application we treat some Sobolev embedding results, which also indicate the sharpness of our main inequalities.