Hardy Spaces of Vector-Valued Functions: Duality

Abstract

We prove here that the Hardy space of $B$-valued functions ${H^1}(B)$ defined by using the conjugate function and the one defined in terms of $B$-valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on $B$. We also characterize the dual space of both spaces, the first one by using ${B^{\ast }}$-valued distributions and the second one in terms of a new space of vector-valued measures, denoted $\mathcal {B}\mathcal {M}\mathcal {O}({B^{\ast }})$, which coincides with the classical $\operatorname {BMO} ({B^{\ast }})$ of functions when ${B^{\ast }}$ has the RNP.