Let n1=n≥1,n2=m≥1\documentclass[12pt]{minimal}
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\begin{document}$$n_1=n\ge 1, n_2=m\ge 1$$\end{document} and λ2>1\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _2>1$$\end{document}. For any x=(x1,x2)∈Rn×Rm\documentclass[12pt]{minimal}
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\begin{document}$$x=(x_1,x_2) \in \mathbb {R}^n\times \mathbb {R}^m$$\end{document}, let g and gλ∗\documentclass[12pt]{minimal}
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\begin{document}$$g_{\mathbf {\lambda }}^*$$\end{document} be the bi-parameter Littlewood–Paley square functions defined by g(f)(x)=∫0∞∫0∞|θt1,t2f(x1,x2)|2dt1t1dt2t21/2,andgλ∗(f)(x)=∬R+m+1∬R+n+1∏i=12(t1ti+|xi-yi|)niλi×|θt1,t2f(y1,y2)|2dy1dt1t1n+1dy2dt2t2m+11/2,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g(f)(x)&= \left( \int _0^{\infty }\int _0^{\infty }|\theta _{t_1,t_2} f(x_1,x_2)|^2 \frac{dt_1}{t_1} \frac{dt_2}{t_2} \right) ^{1/2}, \hbox { and} \\ g_{\mathbf {\lambda }}^*(f)(x)&= \left( \iint _{\mathbb {R}^{m+1}_{+}} \iint _{\mathbb {R}^{n+1}_{+}} \prod _{i=1}^2\Big (\frac{t_1}{t_i + |x_i - y_i|}\Big )^{n_i \lambda _i}\right. \\&\left. \quad \times \, |\theta _{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \right) ^{1/2}, \end{aligned}$$\end{document}where θt1,t2f(x1,x2)=∬Rn×Rmst1,t2(x1,x2,y1,y2)f(y1,y2)dy1dy2\documentclass[12pt]{minimal}
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\begin{document}$$\theta _{t_1,t_2} f(x_1, x_2) = \iint _{\mathbb {R}^n\times \mathbb {R}^m} s_{t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2) dy_1dy_2$$\end{document}. It is known that the L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document} boundedness of bi-parameter g and gλ∗\documentclass[12pt]{minimal}
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\begin{document}$$g_{\mathbf {\lambda }}^*$$\end{document} have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structural conditions assumed on the kernel st1,t2,\documentclass[12pt]{minimal}
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\begin{document}$$s_{t_1,t_2},$$\end{document} we show that both g and gλ∗\documentclass[12pt]{minimal}
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\begin{document}$$g_{\mathbf {\lambda }}^*$$\end{document} are bounded from product Hardy space H1(Rn×Rm)\documentclass[12pt]{minimal}
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\begin{document}$$H^1(\mathbb {R}^n\times \mathbb {R}^m)$$\end{document} to L1(Rn×Rm)\documentclass[12pt]{minimal}
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\begin{document}$$L^1(\mathbb {R}^n\times \mathbb {R}^m)$$\end{document}. As consequences, the Lp\documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document} boundedness of g and gλ∗\documentclass[12pt]{minimal}
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\begin{document}$$g_{\mathbf {\lambda }}^*$$\end{document} will be obtained for 1<p<2\documentclass[12pt]{minimal}
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\begin{document}$$1<p<2$$\end{document}.
