2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/173188Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given.23 pages, basically references and typos corrected in the new versionClassical Analysis and ODEsProbability60G46; 42B15 (Primary), 42B20; 46B09; 46B20 (Secondary)text