On singular integral and martingale transforms
Classical Analysis and ODEs
2008-11-05 v2 Probability
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.
Cite
@article{arxiv.math/0701516,
title = {On singular integral and martingale transforms},
author = {S. Geiss and S. Montgomery-Smith and E. Saksman},
journal= {arXiv preprint arXiv:math/0701516},
year = {2008}
}
Comments
23 pages, basically references and typos corrected in the new version