A weak $L^2$ estimate for a maximal dyadic sum operator on $\mathbb{R}^n$

Abstract

Lacey and Thiele recently obtained a new proof of Carleson's theorem on almost everywhere convergence of Fourier series. This paper is a generalization of their techniques (known broadly as time-frequency analysis) to higher dimensions. In particular, a weak-type (2,2) estimate is derived for a maximal dyadic sum operator on $\mathbb R^{n}$, $n \gt 1$. As an application one obtains a new proof of Sjölin's theorem on weak $L^{2}$ estimates for the maximal conjugated Calderón-Zygmund operator on $\mathbb R^{n}$.