On bilinear Littlewood-Paley square functions

Abstract

In this paper, we study the bilinear Littlewood-Paley square function introduced by M. Lacey. We give an easy proof of its boundedness from $L^p(\mathbb {R}^d) \times L^q(\mathbb {R}^d)$ into $L^r(\mathbb {R}^d),~d\geq 1,$ for all possible values of exponents $p,q,r,$ i.e. for $2\leq p,q\leq \infty ,~1\leq r\leq \infty$ satisfying $\frac {1}{p}+\frac {1}{q}= \frac {1}{r}$. We also prove analogous results for bilinear square functions on the torus group $\mathbb {T}^d.$