Calderón–Zygmund operators and their commutators on generalized weighted Orlicz–Morrey spaces (original) (raw)
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Annals of the Alexandru Ioan Cuza University - Mathematics, 2014
In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,φ(w) with the weight function w(x) belonging to Muckenhoupt's class Ap. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Mp,φ 1 (w) to another Mp,φ 2 (w) for p > 1 and from M1,φ 1 (w) to the weak space W M1,φ 2 (w). In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (φ1, φ2), which do not assume any assumption on monotonicity of φ1, φ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces
Journal of Function Spaces, 2014
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.
Boundedness of the Maximal and Singular Operators on Generalized Orlicz–Morrey Spaces
Operator Theory: Advances and Applications, 2014
We consider generalized Orlicz-Morrey spaces MΦ,ϕ(R n) including their weak versions. In these generalized spaces we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. In all the cases the conditions for the boundedness are given either in terms of Zygmundtype integral inequalities on ϕ(r) without assuming any monotonicity property of ϕ(r), or in terms of supremal operators, related to ϕ(r).
Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces
Abstract and Applied Analysis, 2011
The authors study the boundedness for a large class of sublinear operatorTgenerated by Calderón-Zygmund operator on generalized Morrey spacesMp,φ. As an application of this result, the boundedness of the commutator of sublinear operatorsTaon generalized Morrey spaces is obtained. In the casea∈BMO(ℝn),1<p<∞andTais a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operatorTafrom one generalized Morrey spaceMp,φ1to anotherMp,φ2. In all cases, the conditions for the boundedness ofTaare given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity ofφ1,φ2inr. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.
Annales Academiae Scientiarum Fennicae Mathematica, 2015
We prove the boundedness of the Hardy-Littlewood maximal operator and their commutators with BMO-coefficients in vanishing generalized Orlicz-Morrey spaces V M Φ,ϕ (R n) including weak versions of these spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators involving the Young function Φ(u) and the function ϕ(x, r) defining the space. No kind of monotonicity condition on ϕ(x, r) in r is imposed.