On singular integral and martingale transforms

Abstract

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 Lp X(R2) with p ∈ (1,∞). 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. © 2009 American Mathematical Society.