Journal of the European Mathematical Society

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Volume 20, Issue 3, 2018, pp. 529–595
DOI: 10.4171/JEMS/773

Published online: 2018-02-09

Noncommutative Riesz transforms – dimension free bounds and Fourier multipliers

Marius Junge[1], Tao Mei[2] and Javier Parcet[3]

(1) University of Illinois at Urbana-Champaign, USA
(2) Wayne State University, Detroit, USA
(3) Consejo Superior de Investigaciones Científicas, Madrid, Spain

We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry’s results in the commutative setting. Our proof is inspired by Pisier’s method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in $\mathbb R^n$ (where Meyer’s conjecture fails) or to the word length of free groups. Lust-Piquard’s work for discrete laplacians on LCA groups is also generalized in several ways. In the context of Fourier multipliers, we will prove that Hörmander–Mikhlin multipliers are Littlewood-Paley averages of our Riesz transforms. This is highly surprising in the Euclidean and (most notably) noncommutative settings. As application we provide new Sobolev/Besov type smoothness conditions. The Sobolev-type condition we give refines the classical one and yields dimension free constants. Our results hold for arbitrary unimodular groups.

Keywords: Riesz transform, group von Neumann algebra, dimension free estimates, Hörmander–Mikhlin multiplier

Junge Marius, Mei Tao, Parcet Javier: Noncommutative Riesz transforms – dimension free bounds and Fourier multipliers. J. Eur. Math. Soc. 20 (2018), 529-595. doi: 10.4171/JEMS/773