Von Neumann's concept of quantization goes back to the foundations of quantum mechanicsand provides a noncommutative model of integration. Over the years, von Neumann algebrashave 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 avery much expected Calderón-Zygmund theory for von Neumann algebras. The lack of naturalmetrics partly justifies this long standing gap in the theory. Key new ingredients come fromrecent results on noncommutative martingale inequalities, operator space theory and quantumprobability. This is an ambitious research project and applications include new estimates fornoncommutative Riesz transforms, Fourier and Schur multipliers on arbitrary discrete groupsor noncommutative ergodic theorems. Other related objectives of this project include Rubiode Francia's conjecture on the almost everywhere convergence of Fourier series for matrixvalued functions or a formulation of Fefferman-Stein's maximal inequality for noncommutativemartingales. Reciprocally, I will also apply new techniques from quantum probability innoncommutative Lp embedding theory and the local theory of operator spaces. I have alreadyobtained major results in this field, which might be useful towards a noncommutative form ofweighted harmonic analysis and new challenging results on quantum information theory.