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Potential Analysis

, Volume 45, Issue 1, pp 55–64 | Cite as

Certain Multi(sub)linear Square Functions

  • Loukas Grafakos
  • Sha He
  • Qingying XueEmail author
Article
  • 100 Downloads

Abstract

Let \(d\ge 1, \ell \in \mathbb {Z}^{d}\), \(m\in \mathbb {Z}^{+}\) and θ i , i=1,…, m are fixed, distinct and nonzero real numbers. We show that the m-(sub)linear version below of the Ratnakumar and Shrivastava (Proc. Amer. Math. Soc. 140(12), 4285–4293 2012) Littlewood-Paley square function
$$ T(f_{1},{\dots } , f_{m})(x)=\left (\sum \limits _{\ell \in \mathbb {Z}^{d}}|{\int }_{\mathbb {R}^{d}}f_{1}(x-\theta _{1} y){\cdots } f_{m}(x-\theta _{m} y)e^{2\pi i \ell \cdot y}K (y)dy|^{2}\right )^{1/2} $$
is bounded from \(L^{p_{1}}(\mathbb {R}^{d}) \times \cdots \times L^{p_{m}}(\mathbb {R}^{d}) \) to \(L^{p}(\mathbb {R}^{d}) \) when 2 ≤ p i < satisfy 1/p=1/p 1+⋯+1/p m and 1 ≤ p < . Our proof is based on a modification of an inequality of Guliyev and Nazirova (Int. Eq. Oper. Theory 60(4), 485–497 2008) concerning multilinear convolutions.

Keywords

Multilinear operators Littlewood-Paley square functions orthogonality 

Mathematics Subject Classfication (2010)

Primary 42B20 Secondary 42B25. 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.School of Mathematical SciencesBeijing Normal University Laboratory of Mathematics and Complex Systems Ministry of EducationBeijingPeople’s Republic of China

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