In this paper, we prove that the \(L^p(\mathbb {R}^d)\) norm of the maximal truncated Riesz transform in terms of the \(L^p(\mathbb {R}^d)\) norm of Riesz transform is dimension-free for any \(2\le p<\infty \), using integration by parts formula for radial Fourier multipliers. Moreover, we show that  

　　$$\begin{aligned} \Vert R_j^*f\Vert _{L^p}\le \left( {2+\frac{1}{\sqrt{2}}}\right) ^{\frac{2}{p}}\Vert R_jf\Vert _{L^p},\ \ \text{ for }\ \ p\ge 2,\ \ d\ge 2. \end{aligned}$$  

　　As byproducts of our calculations, we infer the \(L^p\) norm contractivity of the truncated Riesz transforms \(R^t_j\) in terms of \(R_j\), and their accurate \(L^p\) norms. More precisely, we prove:  

　　$$\begin{aligned} \Vert R^t_jf\Vert _{L^p}\le \Vert R_jf\Vert _{L^p} \end{aligned}$$  

　　and  

　　$$\begin{aligned} \Vert R^t_j\Vert _{L^p}=\Vert R_j\Vert _{L^p}, \end{aligned}$$  

　　for all \(1<p<+\infty ,\) \(j\in \{1,\dots ,d\}\), and \(t>0.\)  

　　Publication:  

　　Mathematische Annalen (03 October 2023)  

　　http://dx.doi.org/10.1007/s00208-023-02736-1  

　　Author:  

　　Jinsong Liu  

　　HLM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China  

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China  

　　Email: liujsong@math.ac.cn  

　　Petar Melentijevi？  

　　Matemati？ki fakultet, University of Belgrade, Beograd, Serbia  

　　Jian-Feng Zhu  

　　School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, China