Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces

Abstract

When dealing with vector-valued functions, sometimes is rather difficult to give non trivial examples, meaning examples which do not come from tensoring scalar-valued functions and vectors in the Banach space, belonging to certain classes. This is the situation for vector valued BMO. One of the objectives of this paper is to look for methods to produce such examples. Our main tool will be the vector-valued extension of the following result on multipliers, proved in [MP], which says that the space of multipliers between H and BMOA can be identified with the space of Bloch functions B, i.e. (H, BMOA) = B (see Section 3 for notation), which, in particular gives that g ∗ f ∈ BMOA whenever f ∈ H and g ∈ B. Given two Banach spaces X,Y it is rather natural to define the convolution of an analytic function with values in the space of operators L(X,Y ), say F (z) = ∑∞ n=0 Tnz , and a function with values in X, say f(z) = ∑∞ n=0 xnz , as the function given by F ∗g(z) = ∑∞ n=0 Tn(xn)z . It is not difficult to see that the natural extension of the multipliers’ result to the vector valued setting does not hold for general Banach spaces. To be able to get a proof of such a result we shall be using the analogue of certain inequalities, due to Hardy and Littlewood [HL3], in the vector valued setting, namely ( ∫ 1