Harmonic Analysis on Quantum Tori

Abstract

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 Fejér 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 Lp 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 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.

Function spaces on quantum tori

“…The latter is as hard as the former. Contrary to [2], the transference method plays a very limited role here. Instead, we use Fourier multipliers in a crucial way, this approach is of interest in its own right.…”

“…Instead, we use Fourier multipliers in a crucial way, this approach is of interest in its own right. We thus develop an intrinsic differential analysis on quantum tori, without frequently referring to the usual tori via transference as in [2]. This is a major advantage of the present methods over [2].…”

“…It consists in transferring problems on quantum tori to the corresponding ones in the case of operator-valued functions on the usual tori, in order to use existing results in the latter case or to adapt classical arguments. This method is efficient for several problems studied in [2]. It is still useful for some parts of the present work; for instance, Besov spaces can be investigated through the classical vector-valued Besov spaces by means of transference.…”

“…One powerful tool used in [2] is the transference method. It consists in transferring problems on quantum tori to the corresponding ones in the case of operator-valued functions on the usual tori, in order to use existing results in the latter case or to adapt classical arguments.…”

“…The aim of this note is to announce the main results of [16], which is a continuation of our previous work [2] on harmonic analysis on quantum tori. This second part is devoted to the study of Sobolev, Besov and Triebel-Lizorkin spaces.…”