Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind

Abstract

In the present paper, we will characterize the boundedness of the generalized fractional integral operators I ρ and the generalized fractional maximal operators M ρ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators M ρ and I ρ on generalized Orlicz-Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators M ρ and I ρ on generalized Orlicz-Morrey spaces.

Generalized fractional integral operators on Orlicz–Hardy spaces

“…Theorem 1.2 is a good contrast to the following theorem obtained in [2]. Actually, we plan to use this theorem in the proof of Theorem 1.2.…”

Generalized Fractional Integral Operators and Their Commutators with Functions in Generalized Campanato Spaces on Orlicz Spaces

“…The following theorem is an extension of the result in [20] and has been proved in [4] essentially, by using Hedberg's method in [9].…”

“…However, we need a better estimate on M ρ than I ρ to prove the boundedness of the commutator [b, I ρ ]. In this paper we give a necessary and sufficient condition of the boundedness of M ρ which sharpens the result in [4].…”

“…(iii) In [4], under the conditions that Φ, Ψ ∈ Φ Y , that ρ is increasing and that r → ρ(r)/r n is decreasing, a necessary and sufficient condition for the boundedness of M ρ has been given. In this case, "(3.7)" ⇔ r α−n/p r −n/q , r ∈ (0, ∞) ⇔ α − n/p = −n/q.…”

“…See also [21,22,23,24,26]. Recently, in [4] some necessary and sufficient conditions for the boundedness of I ρ on Orlicz spaces have been given. In this paper we consider the commutator [b, I ρ ] with a function b in generalized Campanato spaces.…”