ON THE BOUNDEDNESS OF A GENERALIZED FRACTIONAL-MAXIMAL OPERATOR IN LORENTZ SPACES

In this paper considers a generalized fractional-maximal operator, a special case of which is the classical fractional-maximal function. Conditions for the function Φ, which defines a generalized fractional-maximal function, and for the weight functions w and v, which determine the weighted Lorentz spaces Λp(v) and Λq(w) (1 < p ≤ q < ∞) under which the generalized maximal-fractional operator is bounded from one Lorentz space Λp(v) to another Lorentz space Λq(w) are obtained. For the classical fractional maximal operator and the classical maximal Hardy-Littlewood function such results were previously known. When proving the main result, we make essential use of an estimate for a nonincreasing rearrangement of a generalized fractional-maximal operator. In addition, we introduce a supremal operator for which conditions of boundedness in weighted Lebesgue spaces are obtained. This result is also essentially used in the proof of the main theorem.