  This paper studies a new maximal operator introduced by Hyt\"onen, McIntosh
and Portal in 2008 for functions taking values in a Banach space. The
L^p-boundedness of this operator depends on the range space; certain
requirements on type and cotype are present for instance. The original
Euclidean definition of the maximal function is generalized to sigma-finite
measure spaces with filtrations and the L^p-boundedness is shown not to depend
on the underlying measure space or the filtration. Martingale techniques are
applied to prove that a weak type inequality is sufficient for L^p-boundedness
and also to provide a characterization by concave functions.
