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 $$1<p<2$$ .