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.