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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 ),aBMO, then the commutator C[K,a]=aKf-K(af) defines a bounded map of L P L p ,1<p<. 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 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,

i,j=1 q a i,j x i ,x j + i,j=1 N b ij x i x j - t

with a ij a constant, symmetric, positive q×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:
42B20Singular and oscillatory integrals, several variables
47B38Operators on function spaces (general)
43A85Analysis on homogeneous spaces
35K70Ultraparabolic equations, pseudoparabolic equations, etc.