Boundedness of the fractional maximal operator on Orlicz-Lorentz spaces

In this paper, we prove that the fractional maximal operator M_α M α is bounded from the Orlicz-Lorentz space L,p(Rⁿ) L Ψ , p ( R n ) to the Orlicz-Lorentz space L,q(Rⁿ) L Φ , q ( R n ) , where n∈ N n ∈ N , 0≤ α <n 0 ≤ α < n , 0<p≤ q≤ ∞ 0 < p ≤ q ≤ ∞ , Φ is an Orlicz function of upper type p∈ (0,∞ ) p Φ ∈ ( 0 , ∞ ) and Ψ is an Orlicz function of lower type p∈ (1,∞ ) p Ψ ∈ ( 1 , ∞ ) . During the proof of the result, we have essentially used an inequality for the non-increasing rearrangement of the fractional maximal operator. Moreover, we also present several novel characterizations of Campanato type spaces by means of commutators of the fractional maximal operator. As an application, some conditions implying the Fefferman-Stein vector-valued inequalities for fractional maximal operator and its commutators on Orlicz-Lorentz spaces are established. The results obtained are substantial improvements and extensions of some known results.