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Boundedness of the potential operators and their commutators in the local “complementary” generalized variable exponent Morrey spaces on unbounded sets

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Abstract

In this paper we prove a Sobolev–Spanne type \({\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega } (\varOmega )\rightarrow {\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{q(\cdot ),\omega } (\varOmega )\)-theorem for the potential operators \(I^{\alpha }\), where \({\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega }(\varOmega )\) is local “complementary” generalized Morrey spaces with variable exponent p(x), \(\omega (r)\) is a general function defining the Morrey-type norm and \(\varOmega \) is an open unbounded subset of \({{\mathbb {R}}^n}\). In addition, we prove the boundedness of the commutator of potential operators \([b,I^{\alpha }]\) in these spaces. In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \(\omega (x,r)\), which do not assume any assumption on monotonicity of \(\omega (x,r)\) in r.

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Acknowledgements

The authors would like to thank Prof. Dr. Vagif S. Guliyev (Dumlupinar University, Turkey and Institute of Mathematics and Mechanics of NAS of Azerbaijan, Azerbaijan) and Prof. Dr. Ayhan Serbetci (Ankara University, Turkey) for their helpful advice and comments. The research of Canay Aykol was partially supported by the Grant of Ankara University Scientific Research Project (BAP.17B0430003).

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Authors and Affiliations

  1. Department of Mathematics, Ankara University, 06100, Tandogan, Ankara, Turkey

    Canay Aykol

  2. Institute of Mathematics and Mechanics, B.Vahabzadeh Str. 9, AZ1141, Baku, Azerbaijan

    Xayyam A. Badalov

  3. Azerbaijan State Oil and Industry University, Azadlig Av. 20, AZ1601, Baku, Azerbaijan

    Javanshir J. Hasanov

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  1. Canay Aykol
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  2. Xayyam A. Badalov
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  3. Javanshir J. Hasanov
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Correspondence to Canay Aykol.

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Communicated by Sorina Barza.

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Aykol, C., Badalov, X.A. & Hasanov, J.J. Boundedness of the potential operators and their commutators in the local “complementary” generalized variable exponent Morrey spaces on unbounded sets. Ann. Funct. Anal. 11, 423–438 (2020). https://doi.org/10.1007/s43034-019-00012-5

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  • DOI: https://doi.org/10.1007/s43034-019-00012-5

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