Non-homogeneous $Tb$ Theorem for Bi-parameter $g$-Function

The main result of this paper is a bi-parameter $Tb$ theorem for Littlewood-Paley $g$-function, where $b$ is a tensor product of two pseudo-accretive functions. Instead of the doubling measure, we work with a product measure $\mu = \mu_n \times \mu_m$, where the measures $\mu_n$ and $\mu_m$ are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter $b$-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.