Pseudo-localization of singular integrals and noncommutative Calderón–Zygmund theory

The weak type (1,1) boundedness of singular integrals acting on matrix-valued functions has remained open since the 1980s, mainly because the methods provided by the vector-valued theory are not strong enough. In fact, we can also consider the action of generalized Calderón–Zygmund operators on functions taking values in any other von Neumann algebra. Our main tools for its solution are two. First, the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases—L2 sense—outside of the support of the function on which it acts. This gives rise to a pseudo-localization principle which is of independent interest, even in the classical theory. Second, we construct a noncommutative form of Calderón–Zygmund decomposition by means of the recent theory of noncommutative martingales. This is a corner stone in the theory. As application, we obtain the sharp asymptotic behavior of the constants for the strong Lp inequalities, also unknown up to now. Our methods settle some basics for a systematic study of a noncommutative Calderón–Zygmund theory.