Analysis & Applications Seminar
Singular integrals in a nondoubling setting: Riesz transforms on ax + b groups
Speaker: Alessio Martini (Politecnico di Torino)Date: Friday, 04 April 2025 - 11:30Place: Aula 520, Módulo 17, Departamento de Matemáticas, UAM
Abstract:
We prove the \(L^p\)-boundedness in the full range 1 < p < ∞ of the first-order Riesz transforms associated with the natural left-invariant Laplacian on an ax + b group with a right Haar measure. This is a neat example of singular integral operators in a nondoubling setting, as ax + b groups have exponential growth and the Riesz transforms are singular both locally and at infinity. Our result settles a question left open in previous work of Hebisch and Steger and of Gaudry and Sjögren, as we can treat the case p > 2 for the whole vector of Riesz transforms. An operator-valued Fourier multiplier theorem turns out to be key to this purpose. Our approach proves to be applicable even beyond ax + b groups, but open problems remain about endpoint results at p = ∞.
