R. R. Coifman, R. Rochberg and G. Weiss [Ann. Math., II. Ser. 103, 611-635 (1976; Zbl 0326.32011)] proved that the commutator operator, , where is a Calderón-Zygmund singular integral operator is bounded on some , , if and only if . Various generalizations of this type of boundedness result for commutator operators have been studied where one varies either the operator class of or the class of functions .
In this paper, the author presents two very nice theorems of this nature and then follows with generalizations of his own results. Here we describe the notation involved and summarize the first two results alone. Let be the Riesz potential of order . We define the commutator . The function space is the homogeneous Lipschitz space defined in terms of the th-difference operator , where . We say that if . The homogeneous Triebel-Lizorkin space is denoted by . The characterization of , of fundamental importance to this paper, is that for and , . In the first theorem when , the author establishes the equivalence of the three conditions (i) , (ii) is a bounded operator from to and (iii) is a bounded operator from to , if . The second theorem concerns the operator and the setting , , . In that scenario, he proves that conditions (i) , (ii) is a bounded operator from to and (iii) is a bounded operator from to , if are equivalent.