Boundedness of Fractional Maximal Operators on Some of Homogeneous Groups

Let M α be the fractional maximal operators （0<α≤1） and （u,v） a pair of weight functions, u∈D ∞, σ=v^<sup><sup><sup><sup>-1/（p-1）∈A ∞. The boundedness of M α on some homogenous groups （G, ‖·‖, dx） and the covering Lemma of Calderon-Zygmund type are studied. Not only an adequate covering Lemma of Calderon-Zygmund type is shown, but also the boundedness of fractional maximal operators M α（0<α≤1） on some of homogeneous groups with respect to a given pair of weight functions （u,v） as above is proved. Moreover, a sufficient and necessary condition for M α∈B（u^qdx, v^<sup>pdx）, 0<α<1, 1