Finite Riesz products and Ornstein non-inequalities on quantum tori

TL;DR

This paper constructs non-commutative analogues of Riesz products on quantum tori and proves a non-commutative Ornstein non-inequality, extending classical harmonic analysis results to quantum settings.

Contribution

It introduces a new construction of finite Riesz products on quantum tori and establishes a non-commutative Ornstein non-inequality, bridging classical and quantum harmonic analysis.

Findings

Constructed finite Riesz products on quantum tori.

Proved a non-commutative Ornstein non-inequality.

Extended classical harmonic analysis results to non-commutative geometry.

Abstract

We demonstrate a construction of products on the quantum torus $\mathbb{T}_\theta^2$ that generalises the usual construction of finite Riesz products on the commutative torus $\mathbb{T}^2$. We explain why the former constitutes a natural analogue of the latter in the non-commutative setting and, based on this construction, as well as on previous results by K. Kazaniecki and the second author, we prove a non-commutative version of an Ornstein non-inequality.