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

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