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

Research paper by Zhengyang Li, Qingying Xue

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



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 \(1<p<2\) .