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Singular integrals and commutators on homogeneous groups. (English) Zbl 1026.43007
The authors using the basic idea from the paper of F. Soria and G. Weiss [Indiana Univ. Math. J. 43, 187-204 (1994; Zbl 0803.42004)], prove several general theorems for the boundedness of sublinear operators and commutators generated by linear operators and $$\text{BMO}({\mathbf G})$$ functions on the weighted Lebesgue space on the homogeneous group $${\mathbf G}$$. The conditions of these theorems are so general that many important operators in analysis satisfy these conditions.
Some of these theorems are best possible even for $$n$$-dimensional Euclidean spaces. They also establish some new applications of these theorems on weighted inequalities for singular integrals (including rough singular integral operators, oscillatory integrals, parabolic singular integrals) and their commutators and the maximal operators associated with them. These methods are more direct than previously used methods.

##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory
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