A Note on UMD Spaces and Transference in Vectorvalued Function Spaces
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\. $X\in\text{HT}$, 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 1994
 arXiv:
 arXiv:math/9406218
 Bibcode:
 1994math......6218A
 Keywords:

 Mathematics  Functional Analysis;
 43A17 42A50 60G46
 EPrint:
 Proc. Edin. Math. Soc. 39, (1996), 485490