Boundedness of Bi-parameter Littlewood-Paley operators on product Hardy space

Abstract

Let $n_1,n_2\ge 1,{\lambda }_1>1$ and ${\lambda }_2>1$. For any $x=(x_1,x_2)\in {\mathbb{R}}^n\times {\mathbb{R}}^m$, let $g$ and $g_{\vec{\lambda }}^{\ast }$ be the bi-parameter Littlewood-Paley square functions defined by \begin{align*} g(f)(x)= \Big(\int_0^{\infty}\int_0^{\infty}|θ_{t_1,t_2} f(x_1,x_2)|^2 \frac{dt_1}{t_1} \frac{dt_2}{t_2} \Big)^{1/2}, \hbox{and} \end{align*} $$g_{\vec{\lambda }}^{\ast }(f)(x)=\left({\iint }_{{\mathbb{R}}_{+}^{m+1}}{\iint }_{{\mathbb{R}}_{+}^{n+1}}\prod _{i=1}^2\right(\frac{t_1}{t_i+|x_i-y_i|}{)}^{n_i{\lambda }_i}|{\theta }_{t_1,t_2}f(y_1,y_2){|}^2\frac{d{y}_{1}d{t}_1}{t_1^{n+1}}\frac{d{y}_{2}d{t}_2}{t_2^{m+1}}{)}^{1/2},$$ \noindent where ${\theta }_{t_1,t_2}f(x_1,x_2)={\iint }_{{\mathbb{R}}^n\times {\mathbb{R}}^m}{s}_{t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2)d{y}_{1}d{y}_2$. It is known that the $L^2$ boundedness of bi-parameter $g$ and $g_{\vec{\lambda }}^{\ast }$ have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structure conditions assumed on the kernel $s_{t_1,t_2},$ we show that both $g$ and $g_{\vec{\lambda }}^{\ast }$ are bounded from product Hardy space $H^1({\mathbb{R}}^n\times {\mathbb{R}}^m)$ to $L^1({\mathbb{R}}^n\times {\mathbb{R}}^m)$. As consequences, the $L^p$ boundedness of $g$ and $g_{\vec{\lambda }}^{\ast }$ will be obtained for $1<p<2$.

26 pages