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

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 $2\leq p<\infty$.