L^p bounds for a maximal dyadic sum operator

Abstract

We prove $L^p$ bounds in the range $1<p<\infty$ for a maximal dyadic sum operator on $\rn$. This maximal operator provides a discrete multidimensional model of Carleson&#39;s operator. Its boundedness is obtained by a simple twist of the proof of Carleson&#39;s theorem given by Lacey and Thiele, adapted in higher dimensions by Pramanik and Terwilleger. In dimension one, the $\lp$ boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunt&#39;s extension of Carleson&#39;s theorem on almost everywhere convergence of Fourier integrals.