Vector Valued Measures of Bounded Mean Oscillation

The duality between Hl and BMO, the space of functions of bounded mean oscillation (see [JN]), was first proved by C. Fefferman (see [F], [FS]) and then other proofs of it were obtained . Using the atomic decomposition approach ([C], [L]) the author studied the problem of characterizing the dual space of Hl of vector-valued functions . In [B2] the author showed, for the case SZ = {Iz1 = 1}, that the expected duality result Hl-BMO holds in the vector valued setting if and only if X* has the Radon-Nikodym property. If we want to get a duality result valid for all Banach spaces we may consider vector valued measures (see [BT], where the vector valued L7, case is treated, for an explanation) and therefore to deal with the general case it was necessary to consider a new space of vector valued measures closely related to BMO (see [B1]) . In this paper we shall study such space in little more detail and we shall consider the H'-BMO duality for vector-valued functions in the more general setting of spaces of homogeneous type (see [CW]). Throughout the paper X will stand for a Banach space, 9 will be a space of homogeneous type (see definition in the preliminary section) and we write Lp (S2, X) for the space of measurable functions on 9 with values in X such that lif (x)jj belongs to L,(Q) . As usual C will denote a constant not necessarily the same at each occurrence .