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

Abstract

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-L 2 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 L p inequalities, also unknown up to now. Our methods settle some basics for a systematic study of a noncommutative Calderón-Zygmund theory.

Weak type operator Lipschitz and commutator estimates for commuting tuples

“…Weak type inequalities for Calderón-Zygmund operators. Parcet [27] proved a non-commutative extension of Calderón-Zygmund theory. Let K be a tempered distribution on R d+1 which we refer to as the convolution kernel.…”

“…Its proof was improved/shortened very recently by Cadilhac [6]. [27]). Let K : R d+1 \{0} → C be a kernel satisfying the conditions…”

Harmonic Analysis on Quantum Tori

“…One can also include in this topic the very fresh promising direction of research on the Calderón-Zygmund singular integral operators in the noncommutative setting (cf. [31,29,18]). The concern of the present paper is directly linked to this last direction.…”

Mapping properties of operator-valued pseudo-differential operators

“…At that time, due to the fact that very little had been done about the analytic aspect, the work of Connes and his collaborators did not include L p -estimates for parametrices and error terms. Recently, inspired by the development on noncommutative harmonic analysis, a lot of progress has been made on Fourier multiplier theory and Calderón-Zygmund theory on noncommutative L p spaces, thanks the efforts of many researchers [33,40,39,7,18,30,61,63]. But so far, the mapping properties of pseudo-differential operators are rarely studied.…”