  Hyt\"onen, McIntosh and Portal (J. Funct. Anal., 2008) proved two
vector-valued generalizations of the classical Carleson embedding theorem, both
of them requiring the boundedness of a new vector-valued maximal operator, and
the other one also the type p property of the underlying Banach space as an
assumption. We show that these conditions are also necessary for the respective
embedding theorems, thereby obtaining new equivalences between analytic and
geometric properties of Banach spaces.
