Hardy spaces of vector-valued functions: duality

Blasco, Oscar

doi:10.1090/S0002-9947-1988-0951618-8

Hardy spaces of vector-valued functions: duality
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by Oscar Blasco

Trans. Amer. Math. Soc. 308 (1988), 495-507

DOI: https://doi.org/10.1090/S0002-9947-1988-0951618-8

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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.

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