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

Abstract

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 $$\|R_j^*f\|_{L^p}\leq \left({2+\frac{1}{\sqrt{2}}}\right)^{\frac{2}{p}}\|R_jf\|_{L^p},\ \ \mbox{for}\ \ p\geq2,\ \ d\geq2.$$ As by products of our calculations, we infer the $L^p$ norm contractivity of the truncated Riesz transforms $R_j^t$ in terms of $R_j$, and their accurate $L^p$ norms. More precisely, we prove: $$\Vert R_j^{t}f{\Vert }_{L^p}\le \Vert R_{j}f{\Vert }_{L^p}$$ and $$\Vert R_j^t{\Vert }_{L^p}=\Vert R_j{\Vert }_{L^p},$$ for all $1<p<+\infty ,$ $j\in \{1,\dots ,d\}$ and $t>0.$

16 pages; we add some remarks and a new proof for a limit in the Appendix, also the other referee's suggestions are incorporated