Calderón-Zygmund operators associated to matrix-valued kernels

Details Calderón-Zygmund operators associated to matrix-valued kernels Calderón-Zygmund operators associated to matrix-valued kernels

2012-01-20

arxiv.org/abs/1201.4351v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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