Maximal Riesz transform in terms of Riesz transform on quantum tori and Euclidean space

TL;DR

This paper proves the boundedness of maximal Riesz transforms in quantum tori and Euclidean space, with dimension-independent constants for certain p-values, advancing harmonic analysis in noncommutative geometries.

Contribution

It establishes dimension-independent $L_p$ bounds for maximal Riesz transforms on quantum tori and Euclidean space, linking these transforms to classical Riesz transforms.

Findings

Boundedness of maximal Riesz transforms in quantum tori and Euclidean space.

Dimension-independent norm constants for $2 \,\leq\, p < \infty$.

Extension of classical harmonic analysis results to noncommutative geometries.

Abstract

For $1<p<\infty$, we establish the $L_{p}$ boundedness of the maximal Riesz transforms in terms of the Riesz transforms on quantum tori $L_{p}(\mathbb{T}^{d}_{\theta})$, and quantum Euclidean space $L_{p}(\mathbb{R}^{d}_{\theta})$. In particular, the norm constants in both cases are independent of the dimension $d$ when $2\leq p<\infty$.