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

  • Canay AykolEmail author
  • Xayyam A. Badalov
  • Javanshir J. Hasanov
Original Paper


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.


Riesz potential Fractional maximal operator Maximal operator Local “complementary” generalized variable exponent Morrey space Hardy–Littlewood–Sobolev–Morrey type estimate BMO space 

Mathematics Subject Classification

42B20; 42B25 42B35 



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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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  • Canay Aykol
    • 1
    Email author
  • Xayyam A. Badalov
    • 2
  • Javanshir J. Hasanov
    • 3
  1. 1.Department of MathematicsAnkara UniversityTandogan, AnkaraTurkey
  2. 2.Institute of Mathematics and MechanicsBakuAzerbaijan
  3. 3.Azerbaijan State Oil and Industry UniversityBakuAzerbaijan

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