**S. Geiss, Stephen Montgomery-Smith and E. Saksman, On singular integral and martingale transforms.**
*Transactions of the American Math Society, ***362**, (2010), 553-575.
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 \(L^p_X(\mathbb R^2)\) with \(p\in (1,\infty).\) 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.
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