A Note on UMD Spaces and Transference in Vector-valued Function Spaces
N. Asmar, B. Kelly, Stephen J. Montgomery-Smith
Abstract
We introduce the notion of an 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, iė\. X∈HT, are ACF spaces. We then show that Bourgain's proof of X∈HT X∈UMD is a consequence of this result.
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