Calder\'on-Zygmund operators associated to matrix-valued kernels

TL;DR

This paper investigates Calderón-Zygmund operators with noncommuting kernels, establishing weak type estimates and Lp bounds in noncommutative settings, addressing challenges in noncommutative harmonic analysis.

Contribution

It provides new weak type and Lp estimates for noncommutative Calderón-Zygmund operators and related transforms, extending the theory to noncommuting kernels.

Findings

Weak type estimates for dyadic CZOs and Haar shifts

Arbitrary CZOs satisfy H1 to L1 estimates

Row/column Lp estimates via L∞ to BMO

Abstract

Calder\'on-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$ type estimates. In conjunction with $L_\infty \to BMO$, we get certain row/column Lp estimates. Our approach also applies to noncommutative paraproducts or martingale transforms with noncommuting symbols/coefficients. Our results complement recent results of Junge, Mei, Parcet and Randrianantoanina.