# Boundedness of bi-parameter Littlewood–Paley operators on product Hardy space

Research paper by **Zhengyang Li, Qingying Xue**

Indexed on: **05 May '18**Published on: **07 Mar '18**Published in: **Revista Matemática Complutense**

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#### Abstract

Let
\(n_1=n\ge 1, n_2=m\ge 1\)
and
\(\lambda _2>1\)
. For any
\(x=(x_1,x_2) \in \mathbb {R}^n\times \mathbb {R}^m\)
, let g and
\(g_{\mathbf {\lambda }}^*\)
be the bi-parameter Littlewood–Paley square functions defined by
$$\begin{aligned} g(f)(x)&= \left( \int _0^{\infty }\int _0^{\infty } \theta _{t_1,t_2} f(x_1,x_2) ^2 \frac{dt_1}{t_1} \frac{dt_2}{t_2} \right) ^{1/2}, \hbox { and} \\ g_{\mathbf {\lambda }}^*(f)(x)&= \left( \iint _{\mathbb {R}^{m+1}_{+}} \iint _{\mathbb {R}^{n+1}_{+}} \prod _{i=1}^2\Big (\frac{t_1}{t_i + x_i - y_i }\Big )^{n_i \lambda _i}\right. \\&\left. \quad \times \, \theta _{t_1,t_2} f(y_1,y_2) ^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \right) ^{1/2}, \end{aligned}$$
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
\(g_{\mathbf {\lambda }}^*\)
have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structural conditions assumed on the kernel
\(s_{t_1,t_2},\)
we show that both g and
\(g_{\mathbf {\lambda }}^*\)
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
\(g_{\mathbf {\lambda }}^*\)
will be obtained for
\(1<p<2\)
.