A Note on UMD Spaces and Transference in Vector-valued Function Spaces

Details A Note on UMD Spaces and Transference in Vector-valued Function Spaces A Note on UMD Spaces and Transference in Vector-valued Function Spaces

1994-06-30

arxiv.org/abs/math/9406218v2

Proc. Edin. Math. Soc. 39, (1996), 485-490

Mathematics

Functional Analysis

Scientific paper

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, i\.e\. $X\in\text{HT}$, are ACF spaces. We then show that Bourgain's proof of $X\in\text{HT}\implies X\in\text{UMD}$ is a consequence of this result.

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