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In this paper, we prove that the $L^p(\mathbb{R}^d)$ norm of the maximaltruncated Riesz transform in terms of the $L^p(\mathbb{R}^d)$ norm of Riesztransform is dimension-free for any $2\leq p<\infty$, using integration byparts 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 $t>0.$

$L^p$ norm of truncated Riesz transform and an improved dimension-free $L^p$ estimate for maximal Riesz transform