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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 $L^p(R^n), a \in \text{BMO}$, then the commutator $C[K,a] = aKf - K(af)$ defines a bounded map of $L^P \rightarrow L^p, 1<p< \infty$. 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 $L^p \rightarrow L^p$. 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 $q\times q$ matrix, $q<N$ and $B =(b_{ij})$ 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.