Noncommutative Calderón-Zygmund theory, operator space geometry and quantum probability
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. A fundamental part of my research is devoted to develop 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 recent results on noncommutative martingale inequalities, operator space theory and quantum probability. This is an ambitious research project and applications include new estimates for noncommutative Riesz transforms, Fourier and Schur multipliers on arbitrary discrete groups or noncommutative ergodic theorems. Other related objectives of this project include Rubio de Francia's conjecture on the almost everywhere convergence of Fourier series for matrix valued functions or a formulation of Fefferman-Stein's maximal inequality for noncommutative martingales. Reciprocally, I will also apply new techniques from quantum probability in noncommutative Lp embedding theory and the local theory of operator spaces. I have already obtained major results in this field, which might be useful towards a noncommutative form of weighted harmonic analysis and new challenging results on quantum information theory.