H\"ormander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications

TL;DR

This paper extends classical harmonic analysis inequalities and theorems to quantum tori, establishing new embedding, interpolation, and decay results for function spaces and differential equations in this noncommutative setting.

Contribution

It introduces H"ormander type Fourier multiplier theorems and Nikolskii inequalities on quantum tori, along with classical inequalities and embedding results in this noncommutative framework.

Findings

Established Fourier multiplier theorems on quantum tori

Proved embedding theorems between various function spaces on quantum tori

Derived decay estimates for heat equation solutions in this setting

Abstract

In this paper, we study H\"ormander type Fourier multiplier theorem and the Nikolskii inequality on quantum tori. On the way to obtain these results, we also prove some classical inequalities such as Paley type, Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities on quantum tori. As applications we establish embedding theorems between Sobolev, Besov spaces as well as embeddings between Besov and Wiener and Beurling spaces on quantum tori. We also analyse $\beta$-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an applications of the analysis, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.