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

TL;DR

The article extends dimension-free $L_p$ bounds for maximal Riesz transforms to noncommutative spaces by establishing, for $1<p<\infty$, bounds of the form $\|\sup_{\varepsilon>0}R_j^{\varepsilon}x\|_{L_p} \le C_{d,p}\,\|R_j x\|_{L_p}$ on both quantum tori $L_p(\mathbb{T}^d_{\theta})$ and quantum Euclidean space $L_p(\mathbb{R}^d_{\theta})$. The approach relies on transference principles: on quantum tori, a semi-commutative transference to $L_p(L_{\infty}(\mathbb{T}^d)\overline{\otimes}\mathcal{M})$ and a factorization $R_j^{t}=A^{t}(R_j)$ link the maximal operator to the Riesz transform; on quantum Euclidean space, a noncommutative Calderón transference combined with amenability of $\mathbb{R}^d$ enables a parallel reduction. The main results yield dimension-free constants for $2\le p<\infty$, namely $C_{d,p} \le (96+2\sqrt{2})^{2/p}$, providing a robust noncommutative analogue of the classical, dimension-free estimates. These findings bridge classical maximal-CZ theory with noncommutative harmonic analysis on deformed spaces and highlight the power of transference techniques in quantum settings.

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 $L_{p}(\mathbb{T}^{d}_θ)$, and quantum Euclidean space $L_{p}(\mathbb{R}^{d}_θ)$. In particular, the norm constants in both cases are independent of the dimension $d$ when $2\leq p<\infty$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Proposition 3.1