Pseudo-localization of Singular Integrals and Noncommutative Calderón-zygmund Theory

After the pioneer work of Calderón and Zygmund in the 50’s, the systematic study of singular integrals has become a corner stone in harmonic analysis with deep implications in mathematical physics, partial differential equations and other mathematical disciplines. Subsequent generalizations of Calderón-Zygmund theory have essentially pursued two lines. We may either consider more general domains or ranges for the functions considered. In the first case, the Euclidean space is replaced by metric spaces equipped with a doubling or non-doubling measure of polynomial growth. In the second case, the real or complex fields are replaced by a Banach space in which martingale differences are unconditional. Historically, the study of singular integrals acting on matrix or operator valued functions has been considered part of the vector-valued theory. This is however a limited approach in the noncommutative setting and we propose to regard these functions as operators in a suitable von Neumann algebra, generalizing so the domain and not the range of classical functions. A far reaching aspect of our approach is the stability of the product fg and the absolute value |f | = √f∗f for operator-valued functions, a fundamental property not exploited in the vector theory. In this paper we follow the original Calderón-Zygmund program and present a non-commutative scalar-valued Calderón-Zygmund theory, emancipated from the vector theory.