On singular integral and martingale transforms

Geiss, Stefan; Montgomery-Smith, Stephen; Saksman, Eero

doi:10.1090/S0002-9947-09-04953-8

On singular integral and martingale transforms
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by Stefan Geiss, Stephen Montgomery-Smith and Eero Saksman

Trans. Amer. Math. Soc. 362 (2010), 553-575

DOI: https://doi.org/10.1090/S0002-9947-09-04953-8

Published electronically: September 11, 2009

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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 $L^p_X(\mathbf {R}^2)$ 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.

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