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Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

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Abstract

In this paper the authors study the boundedness for a large class of sublinear operators \({T_{\alpha}, \alpha \in [0,n)}\) generated by Calderón–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces \({M_{p,\varphi}}\) . As an application of the above result, the boundeness of the commutator of sublinear operators \({T_{b,\alpha}, \alpha \in [0,n)}\) on generalized Morrey spaces is also obtained. In the case \({b \in BMO}\) and T b,α is a sublinear operator, we find the sufficient conditions on the pair \({(\varphi_1,\varphi_2)}\) which ensures the boundedness of the operators \({T_{b,\alpha}, \alpha \in [0,n)}\) from one generalized Morrey space \({M_{p,\varphi_1}}\) to another \({M_{q,\varphi_2}}\) with 1/p − 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({(\varphi_1,\varphi_2)}\) , which do not assume any assumption on monotonicity of \({\varphi_1, \, \varphi_2}\) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.

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Authors and Affiliations

  1. Department of Mathematics, Ahi Evran University, Kirsehir, Turkey

    Vagif S. Guliyev & Turhan Karaman

  2. Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, F. Agayev St. 9, Baku, AZ, 1141, Azerbaijan

    Vagif S. Guliyev, Seymur S. Aliyev & Parviz S. Shukurov

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  1. Vagif S. Guliyev
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  2. Seymur S. Aliyev
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Correspondence to Vagif S. Guliyev.

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Dedicated to 70th birthday of Prof. S. Samko

The research of V. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan project EIF-2010-1(1)-40/ 06-1. The research of V. Guliyev and T. Karaman was partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 110T695).

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Guliyev, V.S., Aliyev, S.S., Karaman, T. et al. Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces. Integr. Equ. Oper. Theory 71, 327 (2011). https://doi.org/10.1007/s00020-011-1904-1

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  • DOI: https://doi.org/10.1007/s00020-011-1904-1

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