Harmonic analysis on quantum tori

TL;DR

This paper advances harmonic analysis on quantum tori by establishing maximal inequalities, convergence theorems, Fourier multiplier characterizations, and Littlewood-Paley theory, extending classical results to the noncommutative setting.

Contribution

It introduces noncommutative analogues of classical harmonic analysis results on quantum tori, including maximal inequalities, convergence theorems, Fourier multiplier characterizations, and Hardy space properties.

Findings

Proved maximal inequalities for summation methods on quantum tori.

Established pointwise convergence theorems for quantum harmonic analysis.

Characterized completely bounded Fourier multipliers on quantum tori.

Abstract

This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fej\'er means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded $L_p$ Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman's $\mathrm{H}_1$-BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.