On singular integral and martingale transforms

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-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 $L^p_X(\mathbf{R}^2)$ with $p\in (1,\infty).$ Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals $p^*-1$ with $p^*:= \max \{p, (p/(p-1))\}$, where the novelty is the lower bound.