Campanato–Morrey spaces for the double phase functionals


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.

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Correspondence to Takao Ohno.

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Mizuta, Y., Nakai, E., Ohno, T. et al. Campanato–Morrey spaces for the double phase functionals. Rev Mat Complut 33, 817–834 (2020).

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

Mathematics Subject Classification

  • Primary 31B15
  • 46E35