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 $g_{ \vec{ \lambda}}^*$ be the bi-parameter Littlewood-Paley square functions defined by \begin{ align*} g(f)(x)= \Big(\int_0^{ \infty}\int_0^{ \infty}|\theta_{ 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*} $$ g_{ \vec{ \lambda}}^*(f)(x) = \Big(\iint_{ \mathbb{ R}^{ m+1}_{ +}} \iint_{ \mathbb{ R}^{ n+1}_{ +}} \prod_{ i=1}^2\Big(\frac{ t_1}{ t_i + |x_i - y_i|}\Big)^{ n_i \lambda_i} |\theta_{ t_1,t_2} f(y_1,y_2)|^2 \frac{ dy_1 dt_1}{ t_1^{ n+1}} \frac{ dy_2 dt_2}{ t_2^{ m+1}} \Big)^{ 1/2}, $$ \noindent where $\theta_{ t_1,t_2} f(x_1, x_2) = \iint_{ \mathbb{ R}^n\times\mathbb{ R}^m} s_{ t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2) dy_1dy_2$. It is known that the $L^2$ boundedness of bi-parameter $g$ and $g_{ \vec{ \lambda}}^*$ 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 $g_{ \vec{ \lambda}}^*$ 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 $g_{ \vec{ \lambda}}^*$ will be obtained for $1<p<2$.
Categories: math.CA
