On bilinear Littlewood-Paley square functions

Ratnakumar, P.; Shrivastava, Saurabh

doi:10.1090/S0002-9939-2012-11349-8

On bilinear Littlewood-Paley square functions
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by P. K. Ratnakumar and Saurabh Shrivastava PDF

Proc. Amer. Math. Soc. 140 (2012), 4285-4293 Request permission

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 $\mathbb {T}^d.$

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