$L^p$ 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 $2\leq p<\infty$, 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 $t>0.$
Comment: 16 pages; we add some remarks and a new proof for a limit in the Appendix, also the other referee's suggestions are incorporated