Runlian Xia

University of Glasgow


Cotlar identities for groups acting on tree-like structures


Analysis and Geometry Seminar


9th May 2024, 3:30 pm – 4:30 pm
Fry Building, 2.04


The Hilbert transform H is a basic example of Fourier multipliers. Its behaviour on Fourier series is the following:
n∈ℤ aneinx → ∑n∈ℤ m(n)aneinx,
with m(n) = −i sgn(n). Riesz proved that H is a bounded operator on Lp(T) for all 1 < p < ∞. We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative Lp spaces. The pioneering work in this direction is due to Mei and Ricard who proved Lp-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on tree-like structures. This class of groups includes amalgamated free products, HNN extensions, left orderable groups and many others.
Joint work with Adrián González and Javier Parcet.






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