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

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 $2\le p<\mathrm{\infty }$, using integration by parts formula for radial Fourier multipliers. Moreover, we show that $$\mathrm{\parallel }{R}_j^{\ast }f{\mathrm{\parallel }}_{L^p}\le {\left(2+\frac{1}{\sqrt{2}}\right)}^{\frac{2}{p}}\mathrm{\parallel }{R}_{j}f{\mathrm{\parallel }}_{L^p},forp\ge 2,d\ge 2.$$ As by products of our calculations, we infer the $L^p$ norm contractivity of the truncated Riesz transforms $R_j^t$ in terms of $R_j$, and their accurate $L^p$ norms. More precisely, we prove: $$\mathrm{\parallel }{R}_j^{t}f{\mathrm{\parallel }}_{L^p}\le \mathrm{\parallel }{R}_{j}f{\mathrm{\parallel }}_{L^p}$$ and $$\mathrm{\parallel }{R}_j^t{\mathrm{\parallel }}_{L^p}=\mathrm{\parallel }{R}_j{\mathrm{\parallel }}_{L^p},$$ for all $1<p<+\mathrm{\infty },$ $j\in \{1,\dots ,d\}$ and $t>0.$