Characterization of the Boundedness of a Generalized Fractional Maximal Operator in Weighted Lorentz Spaces for Certain Parameter Values

The object of this study is generalized fractional-maximal operators acting in weighted Lorentz spaces. The goal of this paper is to obtain boundedness criteria for a generalized fractional-maximal operator between weighted Lorentz spaces and to describe these criteria in terms of weighted integral inequalities. To achieve this goal, methods from the theory of function rearrangements, estimates for nonincreasing rearrangements of operators, and methods for studying weighted inequalities for Hardy-type operators and supremal operators are used. Sharp estimates are obtained for a nonincreasing rearrangement of a generalized fractional-maximal operator. An equivalence is established between the boundedness of this operator in weighted Lorentz spaces and the boundedness of the corresponding supremal operator. Based on reduction theorems, the problem is reduced to the study of weighted integral inequalities of Hardy type. Necessary and sufficient conditions for the boundedness of the generalized fractional maximal operator from Λ1(v) to Λq(w) in the case p=1,0<q<∞ and from Λp(v) to Λ1(w) in the case q=1<p<∞.