Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers

By Marius Junge, Tao Mei, Javier Parcet. 2014

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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 H\"ormander-Mihlin 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/405128727a6185f992b59eccb20f4fd72b5ccf5b

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In Applied Mathematics

In Mathematical Physics

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Singular integrals in quantum Euclidean spaces (2017, background): “icci curvature [3]. Noncommutative Lp methods in harmonic analysis have gained a considerable momentum in recent years. The fast development of Fourier Lpmultiplier theory on groupvon Neumann algebras[11, 32, 42, 43, 47, 48, 57] has been possible in part due to a deeper comprehension of the involved”