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Acta Mathematica Scientia

, Volume 39, Issue 1, pp 11–25 | Cite as

Limiting Weak-Type Behaviors for Certain Littlewood-Paley Functions

  • Xianming Hou (侯宪明)
  • Huoxiong Wu (吴火熊)Email author
Article

Abstract

In this paper, we establish the following limiting weak-type behaviors of Littlewood-Paley g-function g': for nonnegative function fL1(Rn), \(\mathop {\lim }\limits_{\lambda \to {0_ + }} \lambda m\left( {\left\{ {x \in {\mathbb{R}^n}:|g\varphi f\left( x \right)| > \lambda } \right\}} \right) = m\left( {\left\{ {x \in {\mathbb{R}^n}:{{\left( {\int_0^\infty {{{\left| {\varphi r\left( x \right)} \right|}^2}\frac{{dr}}{r}} } \right)}^{1/2}} > 1} \right\}} \right){\left\| f \right\|_1}\) and \(\mathop {\lim }\limits_{t \to {0_ + }} m\left( {\left\{ {x \in {\mathbb{R}^n}:|g\varphi {f_t}\left( x \right) - {{\left( {\int_0^\infty {{{\left| {\varphi r\left( x \right)} \right|}^2}\frac{{dr}}{r}} } \right)}^{1/2}}| > 1} \right\}} \right) = 0\), where ft(x) = tnf(t−1x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying Lαq -Dini condition are also given.

Key words

limiting behaviors weak-type bounds Littlewood-Paley g-functions Marcinkiewicz integrals Lαq-Dini conditions 

2010 MR Subject Classification

42B20 42B25 

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Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Xianming Hou (侯宪明)
    • 1
  • Huoxiong Wu (吴火熊)
    • 1
  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina

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