Riesz potential and its commutators on Orlicz spaces

Abstract

In the present paper, we shall give necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential operator on Orlicz spaces. Cianchi (J. Lond. Math. Soc. 60(1):247-286, 2011) found necessary and sufficient conditions on general Young functions Φ and Ψ ensuring that this operator is of weak or strong type from into . Our characterizations for the boundedness of the above-mentioned operator are different from the ones in (Cianchi in J. Lond. Math. Soc. 60(1):247-286, 2011). As an application of these results, we consider the boundedness of the commutators of Riesz potential operator on Orlicz spaces when b belongs to the BMO and Lipschitz spaces, respectively.

Boundedness of the Riesz potential in central Morrey–Orlicz spaces

“…If λ = μ = 0, then inequalities ( 15) and ( 16) are the same. Moreover, condition (15) in this case is a sufficient condition for boundedness of I α from Orlicz space L (R n ) to weak Orlicz space W L (R n ), and if additionally * ∈ 2 then I α is bounded from Orlicz space L (R n ) to Orlicz space L (R n ) (proof we can find, for example, in [20,Theorem 3.3]).…”

“…Another sufficient conditions for boundedness of the Riesz operator I α (and even for a generalized fractional operator I ρ ) were given in 2001 by Nakai [32,33]. Then in 2017, Guliyev-Deringoz-Hasanov in [20,Theorem 3.3], gave more readable necessary and sufficient conditions for the boundedness of…”

“…Then we define the Morrey-Orlicz spaces M ,λ (R n ) and central Morrey-Orlicz spaces M ,λ (0). In the 2000s, several authors (for example, F. Deringoz, V. S. Guliyev, J. J. Hasanov, T. Mizuhara, E. Nakai, S. Samko, Y. Sawano, H. Tanaka and others) defined Orlicz versions of the Morrey space, i.e., Morrey-Orlicz spaces, and investigated the boundedness for the Hardy-Littlewood maximal operator and other operators on them (see, for example, [11,19,20,34,39] and the references therein). The Orlicz version of central Morrey spaces, i.e., central Morrey-Orlicz spaces were defined in papers by the second and third authors.…”

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

“…Chanillo [2] considerd the commutator [b, I α ]f = bI α f − I α (bf ), with b ∈ BMO and proved that [b, I α ] has the same boundedness as I α . The result was also extended to Orlicz spaces by Fu, Yang and Yuan [6] and Guliyev, Deringoz and Hasanov [8].…”

A characterization for fractional integral and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

“…Also, in [14] the authors extended the Adams type boundedness of Riesz potential and its commutators to the generalized Orlicz-Morrey spaces on the n-dimensional Euclidean space R n . Moreover, the authors find criteria for the boundedness of Riesz potential and its commutators on Orlicz spaces on the n-dimensional Euclidean space R n in [20]. The purpose of this paper is to extend these results to the spaces of homogeneous type.…”