Nakhlé Asmar, Brian Kelly and Stephen Montgomery-Smith, A Note on UMD Spaces and Transference in Vector-valued Function Spaces. Proc. Edin. Math. Soc. 39, (1996), 485-490. We introduce the notion of an \(\text{ACF}\) space, that is, a space for which a generalized version of M. Riesz's theorem for conjugate functions with values in the Banach space is bounded. We use transference to prove that spaces for which the Hilbert transform is bounded, i.e. \(X\in \text{HT}\), are \(\text{ACF}\) spaces. We then show that Bourgain's proof of \(X \in \text{HT} \Rightarrow X \in \text{UMD}\) is a consequence of this result. (tex, dvi, ps, pdf.)


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