Transference of certain maximal Hilbert transforms on the torus

Dashan Fan, Huoxiong Wu and Fayou Zhao

Vol. 284 (2016), No. 2, 343–364

DOI: 10.2140/pjm.2016.284.343

Abstract

Using transference techniques, we show that $L^p\left({ℝ}^n\right)$ estimates for many operators may be transferred to the $L^p\left({\mathbb{T}}^n\right)$ estimates on the $n$-torus ${\mathbb{T}}^n$ via measure-preserving actions of ${ℝ}^n$. These operators include the maximal bilinear Hilbert transform, the oscillation, and the variation and short variation operators of the Hilbert transform on the torus $\mathbb{T}$. As an extension, we study the (maximal) bilinear Riesz transforms on the $n$-torus ${\mathbb{T}}^n$.

Keywords

bilinear Hilbert transform, maximal operator, Riesz transform, oscillation, variation, torus, transference

Mathematical Subject Classification 2010

Primary: 42B20, 42B25

Secondary: 42A50, 40A30

Milestones

Received: 25 January 2015

Accepted: 28 January 2016

Published: 30 August 2016