Positivity 4: 339–356, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Remarks on Vector-valued BMOA and
Universidad De Valencia, Departamento Analisis Matematico, 46100 Burjassot (Valencia), Spain
(Received 24 November 1998; accepted 25 August 1999)
Abstract. In this paper we consider the vector-valued interpretation of the space BMOA deﬁned in
terms of Carleson measures and analyze the relationship with the one deﬁned in terms of oscillation.
We study the space of multipliers between H
and BMOA in the vector-valued setting. This leads us
to the consideration of some geometric properties depending upon the validity of certain inequalities
due to Littlewood and Paley on the g-function for vector-valued functions.
Mathematics Subject Classiﬁcation: 46B20, 46E40
Key words: vector valued Hardy spaces, B.M.O and Bloch functions, vector valued multipliers, type,
In [6, 7] the author considered the vector-valued situation of the result by M.
Mateljevich and M. Pavlovic  which establishes that the space of multipiers
and BMOA can be identiﬁed with the space of Bloch functions, i.e.
,BMOA) = Bloch. For such a purpose it was introduced the notion of pairs
(X, Y ) having the (H
,BMO)-property for those where the space of multipliers
(X), BMOA(Y )), with its natural deﬁnition (see Section 3), coincides with
Bloch(L(X, Y)) .
It was observed there that the validity of (H
(X), BMOA(Y )) = Bloch(L(X,
Y))depends on the fact that X and Y satisfy the vector-valued formulation of some
inequalites due to Hardy and Littlewood (see [20, 21, 22]) in the scalar-valued case.
In this paper we consider the vector-valued interpretation of the space BMOA
deﬁned in terms of Carleson measures (see Deﬁnition 1.2 below) instead of the one
considered in  and analyze the relationship with the previous one, studying the
result on vector-valued multipliers for this formulation of BMOA.
This leads us to the consideration of some other geometric properties coming
from other inequalities due to Littlewood and Paley on the g-function which have
been already considered in  and more recently in [11, 12, 35].
The author has been partially supported by the Spanish DGICYT, Proyecto PB95-0261.