On The Boundedness of Bi-parameter Littlewood-Paley $g_{\lambda }^{\ast }$-function

Abstract

Let $m,n\ge 1$ and $g_{{\lambda }_1,{\lambda }_2}^{\ast }$ be the bi-parameter Littlewood-Paley $g_{\lambda }^{\ast }$-function defined by $$g_{{\lambda }_1,{\lambda }_2}^{\ast }(f)(x)=\left({\iint }_{{\mathbb{R}}_{+}^{m+1}}\right(\frac{t_2}{t_2+|x_2-y_2|}{)}^{m{\lambda }_2}{\iint }_{{\mathbb{R}}_{+}^{n+1}}(\frac{t_1}{t_1+|x_1-y_1|}{)}^{n{\lambda }_1}|{\theta }_{t_1,t_2}f(y_1,y_2){|}^2\frac{d{y}_{1}d{t}_1}{t_1^{n+1}}\frac{d{y}_{2}d{t}_2}{t_2^{m+1}}{)}^{1/2},{\lambda }_1>1,{\lambda }_2>1$$ where ${\theta }_{t_1,t_2}f$ is a non-convolution kernel defined on ${\mathbb{R}}^{m+n}$. In this paper, we showed that the bi-parameter Littlewood-Paley function $g_{{\lambda }_1,{\lambda }_2}^{\ast }$ was bounded from $L^2({\mathbb{R}}^{n+m})$ to $L^2({\mathbb{R}}^{n+m})$. This was done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.

21 pages

References in corpus (1)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II