$L^p -L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces

TL;DR

This paper investigates the boundedness of Fourier multipliers on quantum Euclidean spaces, establishing key inequalities and applications like heat semigroup estimates and Sobolev embeddings in the quantum setting.

Contribution

It introduces new $L^p -L^q$ boundedness results for Fourier multipliers on quantum Euclidean spaces and proves fundamental inequalities adapted to the quantum context.

Findings

Established $L^p -L^q$ bounds for Fourier multipliers

Proved quantum versions of classical inequalities like Paley and Hausdorff-Young-Paley

Derived applications including heat semigroup estimates and Sobolev embeddings

Abstract

In this paper, we study Fourier multipliers on quantum Euclidean spaces and obtain results on their $L^p -L^q$ boundedness. On the way to get these results, we prove Paley, Hausdorff-Young-Paley, and Hardy-Littlewood inequalities on the quantum Euclidean space. As applications, we establish the $L^p -L^q$ estimate for the heat semigroup and Sobolev embedding theorem on quantum Euclidean spaces. We also obtain quantum analogues of the logarithmic Sobolev and Nash type inequalities.