  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 $2\leq p<\infty$, using integration by
parts formula for radial Fourier multipliers. Moreover, we show that
$$\|R_j^*f\|_{L^p}\leq
\left({2+\frac{1}{\sqrt{2}}}\right)^{\frac{2}{p}}\|R_jf\|_{L^p},\ \ \mbox{for}\
\ p\geq2,\ \ d\geq2.$$ As by products 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:
$$\|R^t_jf\|_{L^p}\leq\|R_jf\|_{L^p}$$ and $$\|R^t_j\|_{L^p}=\|R_j\|_{L^p},$$
for all $1<p<+\infty,$ $j\in \{1,\dots,d\}$ and $t>0.$
