  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's operator. Its boundedness is obtained by a simple twist of
the proof of Carleson'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's
extension of Carleson's theorem on almost everywhere convergence of Fourier
integrals.
