L p bounds for a maximal dyadic sum operator

Abstract

The authors prove L p bounds in the range 1<p<∞ for a maximal dyadic sum operator on R n . This maximal operator provides a discrete multidimensional model of Carleson’s operator. Its boundedness is obtained by a simple twist of the proof of Carleson’s theorem given by Lacey and Thiele [7] adapted in higher dimensions [9]. In dimension one, the L p boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunt’s extension [4] of Carleson’s theorem on almost everywhere convergence of Fourier integrals.