Pseudo-localization of Singular Integrals and Noncommutative Littlewood-paley Inequalities

Understood in a wide sense, square functions play a central role in classical Littlewood-Paley theory. This entails for instance dyadic type decompositions of Fourier series, Stein’s theory for symmetric diffusion semigroups or Burkholder’s martingale square function. All these topics provide a deep technique when dealing with quasi-orthogonalitymethods, sums of independent variables, Fourier multiplier estimates... The historical survey [30] is an excellent exposition. In a completely different setting, the rapid development of operator space theory and quantum probability has given rise to noncommutative analogs of several classical results in harmonic analysis. We find new results on Fourier/Schur multipliers, a settled theory of noncommutative martingale inequalities, an extension for semigroups on noncommutative Lp spaces of the Littlewood-Paley-Stein theory, a noncommutative ergodic theory and a germ for a noncommutative Calderón-Zygmund theory. We refer to [5, 7, 9, 10, 14, 20, 23] and the references therein.