Mathematische Annalen

, Volume 351, Issue 1, pp 1–49

Lp estimates for non-smooth bilinear Littlewood–Paley square functions on \({\mathbb{R}}\)

Article

DOI: 10.1007/s00208-010-0588-1

Cite this article as:
Bernicot, F. Math. Ann. (2011) 351: 1. doi:10.1007/s00208-010-0588-1

Abstract

In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that
$$\left\| \left( \sum_{n\in \mathbb{Z}}\left|S_n(f,g) \right|^2 \right)^{1/2}\right\|_{L^{r}(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})}\|g\|_{L^q(\mathbb{R})}.$$

Mathematics Subject Classification (2000)

42B2042B25

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.CNRS, Université Lille 1Villeneuve d’Ascq CedexFrance