Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces

Abstract

The boundedness of multilinear commutators of Calderón-Zygmund operatorTb→on generalized weighted Morrey spacesMp,φ(w)with the weight functionwbelonging to Muckenhoupt's classApis studied. When1<p<∞andb→=(b1,…,bm),bi∈BMO,i=1,…,m, the sufficient conditions on the pair(φ1,φ2)which ensure the boundedness of the operatorTb→fromMp,φ1(w)toMp,φ2(w)are found. In all cases the conditions for the boundedness ofTb→are given in terms of Zygmund-type integral inequalities on(φ1,φ2), which do not assume any assumption on monotonicity ofφ1(x,r), φ2(x,r)inr.

Parameter θ -Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

“…Lemma 14. [30] Let ρ, p ∈ ½1, ∞Þ, ω ∈ A ρ p ðωÞ and η ∈ ½5ρ, ∞Þ. Then, there exist positive constants C 1 , C 2 ∈ ½1, ∞Þ such that (i) for any ball B and μ-measurable…”

“…Based on this, Nakamura and Sawano established the boundedness of singular integral operator and its commutator on weighted Morrey space (see [28]). In 2012, Guliyev [29] first introduced the generalized weighted Morrey spaces M p,ρ ðωÞ and studied the boundedness of the sublinear operators and their higher order commutators which is generated by Calderón-Zygmund operators and Riese potentials on these spaces (see also [30,31]). In 2016, Nakamura defined another definition of generalized weighted Morrey space and established the boundedness of classical operators on this space (see [32]).…”

Calderón-Zygmund operators and their commutators on generalized weighted Orlicz-Morrey spaces

“…Mizuhara [30] and Nakai [32] introduced generalized Morrey spaces M p,ϕ (R n ) (see, also [12]); Komori and Shirai [27] defined weighted Morrey spaces L p,κ (w); Guliyev [13] gave a concept of the generalized weighted Morrey spaces M p,ϕ w (R n ) which could be viewed as extension of both M p,ϕ (R n ) and L p,κ (w). In [13], the boundedness of the classical operators and their commutators in spaces M p,ϕ w was also studied, see also [14,16,17,19,22].…”