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On singular integral and martingale transforms

S. Geiss·S. Montgomery-Smith·E. Saksman
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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(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.

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S. Geiss, S. Montgomery-Smith, E. Saksman. 2008-11-05. On singular integral and martingale transforms. https://arxiv.org/abs/math/0701516

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