2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149475We 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.Functional Analysis43A17 42A50 60G46text