Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces

Turhan Karaman,Vagif S. Guliyev,Seymur S. Aliyev

doi:10.1155/2011/356041

Turhan Karaman, Vagif S. Guliyev + Show 1 more

Open Access

https://doi.org/10.1155/2011/356041

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Abstract

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.

Highlights

For x ∈ Rn and r > 0, we denote by B x, r the open ball centered at x of radius r, and by B x, r denote its complement
We prove the boundedness of the sublinear operator T satisfies the condition 1.5 generated by Calderon-Zygmund operator from one generalized Morrey space Mp,φ1 to another Mp,φ2, 1 < p < ∞, and from M1,φ1 to the weak space WM1,φ2
In the case a ∈ BMO Rn, 1 < p < ∞ and the commutator operator Ta is a sublinear operator, we find the sufficient conditions on the pair φ1, φ2 which ensures the boundedness of the operators Ta from Mp,φ1 to Mp,φ2

Summary

Introduction

For x ∈ Rn and r > 0, we denote by B x, r the open ball centered at x of radius r, and by B x, r denote its complement. Suppose that T represents a linear or a sublinear operator, which satisfies that for any f ∈ L1 Rn with compact support and x ∈/ supp f fy. Suppose that the commutator operator Ta represents a linear or a sublinear operator, which satisfies that for any f ∈ L1 Rn with compact support and x ∈/ supp f. The condition 1.5 are satisfied by many interesting operators in harmonic analysis, such as the Calderon-Zygmund operators, Carleson’s maximal operator, Hardy-Littlewood maximal operator, C. We prove the boundedness of the sublinear operator T satisfies the condition 1.5 generated by Calderon-Zygmund operator from one generalized Morrey space Mp,φ1 to another Mp,φ2 , 1 < p < ∞, and from M1,φ1 to the weak space WM1,φ2. If A B and B A, we write A ≈ B and say that A and B are equivalent

Morrey Spaces

Generalized Morrey Spaces

The John-Nirenberg inequality implies that

Pseudodifferential Operators

Littlewood-Paley Operator

Marcinkiewicz Operator

Bochner-Riesz Operator