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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.

42B20Singular and oscillatory integrals, several variables
47B38Operators on function spaces (general)
43A85Analysis on homogeneous spaces
35K70Ultraparabolic equations, pseudoparabolic equations, etc.