Fractional maximal function and its commutators on Orlicz spaces

Abstract

In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator M α on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator M b,α and nonlinear commutator of fractional maximal operator [b, M α ] on Orlicz spaces, when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

“…More precisely, we will give some new necessary and sufficient conditions for the boundedness of [b, M α ] on variable Lebesgue spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(R n ) spaces are obtained. Moreover, our results also give affirmative answers to the questions mentioned in [16] and [29] (see Remarks 1.4 and 1.5 below, respectively). We would like to note that some of our results are new even in the case of Lebesgue spaces with constant exponents.…”

“…Similar to (1.1), we can define two different kinds of commutator of the fractional maximal function as follows. The mapping property of [b, M α ] has been extensively studied; see [1,2,8,12,13,16,23,[25][26][27][28][29][30][31], for instance. There are some applications of nonlinear commutators in analysis.…”

“…Some necessary and sufficient conditions for the boundedness of [b, M] on Lebesgue and Morrey spaces are given. Some of the results were extended to variable Lebesgue spaces in [27] and to the context of Orlicz spaces in [15,16] and [31].…”

“…Remark 1.8 The equivalence of (1), (2) and (3) was proved in [26] (for α = 0) and in [16] (for 0 < α < n). The equivalence of (1), (4) and (5) is contained in Lemma 2.1 below.…”

Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

“…We refer to [2] for a detailed investigation of the operators M b and the commutator of the maximal operator [b, M] and references therein. For the boundedness of these operators in Orlicz space L Φ (R n ) see for instance [5,7].…”