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Commutators of singular integrals on homogeneous spaces. (English) Zbl 0913.42013

Coifman, Rochberg and Weiss showed that if $K$ is a Calderón-Zygmund operator on ${L}^{p}\left({R}^{n}\right),a\in \text{BMO}$, then the commutator $C\left[K,a\right]=aKf-K\left(af\right)$ defines a bounded map of ${L}^{P}\to {L}^{p},1. This result has been generalized to fractional integral operators, weighted estimates for real valued and vector valued operators and to higher order commutators. The authors give estimates of the Coifman-Rochberg-Weiss type for commutators of BMO functions and Calderón-Zygmund operators $K$ on homogeneous spaces.

They assume that the Calderón-Zygmund operator satisfies a pointwise Hörmander condition and that it is bounded from ${L}^{p}\to {L}^{p}$. In Section 4, they give some sufficient conditions on the kernel to guarantee this latter condition. The conditions use an extension by Christ of the David-Journé theorem to homogeneous spaces.

Applications include commutator estimates for singular integrals with mixed homogeneity, estimates for Kolmogorov type operators,

$\sum _{i,j=1}^{q}{a}_{i,j}{\partial }_{{x}_{i},{x}_{j}}+\sum _{i,j=1}^{N}{b}_{ij}{x}_{i}{\partial }_{{x}_{j}}-{\partial }_{t}$

with ${a}_{ij}$ a constant, symmetric, positive $q×q$ matrix, $q and $B=\left({b}_{ij}\right)$ is a matrix with a specified upper triangular form, ${L}^{p}$ estimates for a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, estimates for the Kohn Laplacian on the Heisenberg group, and estimates for singular kernels on Euclidean spaces with weighted measures.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 47B38 Operators on function spaces (general) 43A85 Analysis on homogeneous spaces 35K70 Ultraparabolic equations, pseudoparabolic equations, etc.