Campanato–Morrey spaces for the double phase functionals

  • Yoshihiro Mizuta
  • Eiichi Nakai
  • Takao OhnoEmail author
  • Tetsu Shimomura


We prove that the Riesz potential operator \(I_\alpha \) of order \(\alpha \) embeds from Musielak–Orlicz–Morrey spaces \(L^{\Phi ,\nu }(\mathbf{R}^N)\) of the double phase functionals \(\Phi (x,t)= t^{p} + (b(x) t)^{q}\) to Campanato–Morrey spaces, where \(1<p<q\) and \(b(\cdot )\) is non-negative, bounded and Hölder continuous of order \(\theta \in (0,1]\). We also study the continuity of Riesz potentials \(I_\alpha f\) of functions in \(L^{\Phi ,\nu }(\mathbf{R}^N)\) and show that \(I_\alpha \) embeds from \(L^{\Phi ,\nu }(\mathbf{R}^N)\) to vanishing Campanato–Morrey spaces.


Riesz potentials Morrey spaces Musielak–Orlicz–Morrey spaces Double phase functionals Campanato–Morrey spaces 

Mathematics Subject Classification

Primary 31B15 46E35 



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Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  1. 1.Higashi-HiroshimaJapan
  2. 2.Department of MathematicsIbaraki UniversityMitoJapan
  3. 3.Faculty of EducationOita UniversityDannoharu, Oita-cityJapan
  4. 4.Department of Mathematics, Graduate School of EducationHiroshima UniversityHigashi-HiroshimaJapan

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