Noncommutative Calderón-Zygmund theory, operador spaces and quantum probability
Principal investigator: Javier Parcet
Project reference: 256997
Start Date: 2010-10-01
End Date: 2015-09-30
Abstract: Von Neumann’s concept of quantization goes back to the foundations of quantum mechanics and provides a noncommutative model of integration. Over the years, von Neumann algebras have shown a profound structure and set the right framework for quantizing portions of algebra, analysis, geometry and probability.
The Fourier transform on noncommutative or even quantum groups is well-understood through both representation and Kac algebra theory. There is nevertheless no general method to face challenging problems on Fourier multipliers, Littlewood-Paley estimates or convergence of Fourier series. A fundamental part of my research is devoted to fill this gap by developing a very much expected Calderón-Zygmund theory for von Neumann algebras. The lack of natural metrics partly justifies this long standing gap in the theory.
Key new ingredients come from noncommutative Lp martingale inequalities, operator space theory and quantum/free probability. Here is short list of recent results and some potential applications related to noncommutative Calderón-Zygmund theory:
- Hörmander-Mihlin type multipliers on discrete groups
- New criteria for the Sp boundedness of Schur multipliers
- Twisted Calderón-Zygmund operators on Rn under group actions
- An extension of Calderón’s transference method for Kac algebras
- Poinwise convergence of Fourier series for matrix-valued functions
- Analysis of idempotent Fourier multipliers vs Kakeya type estimates
“This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 256997″