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

TL;DR

This paper proves the boundedness of the bi-parameter Littlewood-Paley $g_{oldsymbol{\lambda}}^*$-function on $L^2$ spaces using probabilistic techniques and a novel averaging identity over Whitney regions.

Contribution

It establishes the $L^2$ boundedness of the bi-parameter Littlewood-Paley $g_{oldsymbol{\lambda}}^*$-function with new methods involving probabilistic analysis and averaging over Whitney regions.

Findings

Boundedness of $g_{oldsymbol{\lambda}}^*$ on $L^2$ spaces.

Introduction of a new averaging identity over good double Whitney regions.

Application of probabilistic methods to harmonic analysis problems.

Abstract

Let $m,n\ge 1$ and $g_{\lambda_1,\lambda_2}^*$ be the bi-parameter Littlewood-Paley $g_{\lambda}^{*}$-function defined by $$ g_{\lambda_1,\lambda_2}^*(f)(x)= \bigg(\iint_{\R^{m+1}_{+}} \big(\frac{t_2}{t_2 + |x_2 - y_2|}\big)^{m \lambda_2} \iint_{\R^{n+1}_{+}} \big(\frac{t_1}{t_1 + |x_1 - y_1|}\big)^{n \lambda_1}|\theta_{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \bigg)^{1/2}, \lambda_1>1,\quad \lambda_2>1 $$ 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 $g_{\lambda_1,\lambda_2}^*$ was bounded from $L^2(\R^{n+m})$ to $L^2(\R^{n+m})$. This was done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.