Illinois Journal of Mathematics

Bi-parameter Littlewood–Paley operators with upper doubling measures

Mingming Cao and Qingying Xue

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Let $\mu=\mu_{n_{1}}\times\mu_{n_{2}}$, where $\mu_{n_{1}}$ and $\mu_{n_{2}}$ are upper doubling measures on $\mathbb{R}^{n_{1}}$ and $\mathbb{R}^{n_{2}}$, respectively. Let the pseudo-accretive function $b=b_{1}\otimes b_{2}$ satisfy a bi-parameter Carleson condition. In this paper, we established the $L^{2}(\mu)$ boundedness of non-homogeneous Littlewood–Paley $g_{\lambda}^{*}$-function with non-convolution type kernels on product spaces. This was mainly done by means of dyadic analysis and non-homogenous methods. The result is new even in the setting of Lebesgue measures.

Article information

Illinois J. Math., Volume 61, Number 1-2 (2017), 53-79.

Received: 10 February 2017
Revised: 16 August 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Cao, Mingming; Xue, Qingying. Bi-parameter Littlewood–Paley operators with upper doubling measures. Illinois J. Math. 61 (2017), no. 1-2, 53--79. doi:10.1215/ijm/1520046209.

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