## Illinois Journal of Mathematics

### Bi-parameter Littlewood–Paley operators with upper doubling measures

#### Abstract

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

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

Dates
Revised: 16 August 2017
First available in Project Euclid: 3 March 2018

https://projecteuclid.org/euclid.ijm/1520046209

Digital Object Identifier
doi:10.1215/ijm/1520046209

Mathematical Reviews number (MathSciNet)
MR3770836

Zentralblatt MATH identifier
1388.42055

#### Citation

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. https://projecteuclid.org/euclid.ijm/1520046209

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