@article{7cc343902e2f4ef4b31a7ce767b99007,
title = "Smooth Fourier multipliers on group von Neumann algebras",
abstract = "We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a H{\"o}rmander–Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood–Paley type inequalities in group von Neumann algebras, prove Lp estimates for noncommutative Riesz transforms and characterize L∞ → BMO boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calder{\'o}n–Zygmund theory.",
author = "Marius Junge and Tao Mei and Javier Parcet",
note = "Funding Information: Over the last years, we have discussed our results with many colleagues. We thank the interesting comments and bibliographic references from A. Carbery, G. Garrig{\'o}s, D. M{\"u}ller, N. Ozawa, J. Peterson, {\'E}. Ricard, A. Seeger, A. Thom and J. Wright. We are also indebted to the referee for his comments, which led to a significantly more transparent presentation. Junge is partially supported by the NSF DMS-0901457 and DMS-1201886, Mei by the NSF DMS-1266042 and Parcet by the ERC Grant StG-256997-CZOSQP and MTM2010-16518. Junge and Parcet are also supported in part by ICMAT Severo Ochoa Grant SEV-2011-0087. Publisher Copyright: {\textcopyright} 2014, Springer Basel.",
year = "2014",
month = dec,
day = "2",
doi = "10.1007/s00039-014-0307-2",
language = "English (US)",
volume = "24",
pages = "1913--1980",
journal = "Geometric and Functional Analysis",
issn = "1016-443X",
publisher = "Birkhauser Verlag Basel",
number = "6",
}