Characterizations of the multiple Littlewood—Paley operators on product domains

Let $m, n\ge 1$. Define the multiple Littlewood—Paley operator $\mathcal{G}_\Psi$ by $$ \mathcal{G}_\Psi(f)(x,y):=\left(\int_0^\infty\int_0^\infty|\Psi_{t,s}*f(x,y)|^2\frac{dtds}{ts}\right)^{1/2}, $$ where $\Psi(x,y)\in L^1(\mathbb{R}^m\times\mathbb{R}^n)$ and $\Psi_{t,s}(x,y)=t^{-m}s^{-n}\Psi(t^{-1}x,s^{-1}y)$. In this paper, we present several characterizations of the $L^2$-boundedness for Littlewood—Paley functions on product domains.