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 Lp(Rd) norm of the maximal truncated Riesz transform in terms of the Lp(Rd) norm of Riesz transform is dimension-free for any 2≤p<∞, using integration by parts formula for radial Fourier multipliers. Moreover, we show that (Formula presented.) As byproducts of our calculations, we infer the Lp norm contractivity of the truncated Riesz transforms Rjt in terms of Rj, and their accurate Lp norms. More precisely, we prove: (Formula presented.) and (Formula presented.) for all 1<p<+∞,j∈{1,⋯,d}, and t>0.