Abstract:
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.
