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

Abstract

Let 0 < α < n and M α be the fractional maximal function. The nonlinear commutator of M α and a locally integrable function b is given by [b, M α ](f) = bM α (f)-M α (bf). In this paper, we mainly give some necessary and sufficient conditions for the boundedness of [b, M α ] on variable Lebesgue spaces when b belongs to Lipschitz or BMO(R n) spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(R n) spaces are obtained.

Fractional operators and their commutators on generalized Orlicz spaces

“…In this section we combine previous estimates of maximal operators to show boundedness of commutators of fractional maximal operators generalizing the ideas of Zhang, Si and Wu in [27,28].…”

“…When the operator is a maximal function instead of a Calderón-Zygmund operator, the boundedness results of commutators were studied in [3] and for fractional maximal function in [26]. These results have had their generalizations from standard L p -spaces to their nonstandard counterparts such as variable exponent Lebesgue and Orlicz spaces [13,27,28] and more refined Morrey type spaces [2,14,25].…”

Commutators of fractional maximal operator in variable Lebesgue spaces over bounded quasi‐metric measure spaces

“…The case 𝜂 ≥ 0 for both the commutators was studied in Zhang and Wu 16 for X = R n . As regards variable exponent p = p(x), the commutators M b,𝜂 and [b, M 𝜂 ] were studied in L p (•) in the Euclidean case in Zhang and Wu 18 for 𝜂 = 0 and in Zhang et al 19 for 𝜂 ≥ 0.…”

“…The case $$$ \eta \ge 0 $$$ for both the commutators was studied in Zhang and Wu 16 for $$$ X&amp;amp;amp;#x0003D;{\mathbb{R}}&amp;amp;amp;#x0005E;n $$$. As regards variable exponent $$$ p&amp;amp;amp;#x0003D;p(x) $$$, the commutators $$$ {M}_{b,\eta } $$$ and $$$ \left[b,{M}_{\eta}\right] $$$ were studied in $$$ {L}&amp;amp;amp;#x0005E;{p\left(\cdotp \right)} $$$ in the Euclidean case in Zhang and Wu 18 for $$$ \eta &amp;amp;amp;#x0003D;0 $$$ and in Zhang et al 19 for $$$ \eta \ge 0 $$$.…”

Characterization of Lipschitz Space via the Commutators of Fractional Maximal Functions on Variable Lebesgue Spaces

“…where the supremum is taken over all cubes Q ⊂ R n containing x. In 2000, Bastero, Milman and Ruiz [1] [18], Si and Wu [20] further obtained that b ≥ 0 (resp. b) belongs to Lipschitz space if and only if the commutators…”