A weak L2 estimate for a maximal dyadic sum operator on Rn

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 R-n, n > 1. As an application one obtains a new proof of Sjolin's theorem on weak L-2 estimates for the maximal conjugated Calderon-Zygmund operator on R-n.