Coifman, Rochberg and Weiss showed that if is a Calderón-Zygmund operator on , then the commutator defines a bounded map of . 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 on homogeneous spaces.
They assume that the Calderón-Zygmund operator satisfies a pointwise Hörmander condition and that it is bounded from . 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,
with a constant, symmetric, positive matrix, and is a matrix with a specified upper triangular form, 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.