Campanato–Morrey spaces for the double phase functionals

Abstract

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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References

  1. 1.

    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(6), 1629–1663 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50(2), 201–230 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. 57, 62 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St Petersb. Math. J. 27, 347–379 (2016)

    Article  Google Scholar 

  6. 6.

    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. (7) 7(3–4), 273–279 (1987)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Colasuonno, F., Squassina, M.: Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. (4) 195(6), 1917–1959 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Colombo, M., Mingione, G.: Bounded minimizers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    De Filippis, C., Mingione, G.: On the regularity of minima of non-autonomous functionals, to appear in J. Geom. Anal

  11. 11.

    De Filippis, C., Oh, J.: Regularity for multi-phase variational problems. J. Differ. Equ. 267(3), 1631–1670 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Harjulehto, P., Hästö, P.: Boundary regularity under generalized growth conditions. Z. Anal. Anwend. 38(1), 73–96 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Harjulehto, P., Hästö, P., Karppinen, A.: Local higher integrability of the gradient of a quasiminimizer under generalized Orlicz growth conditions. Nonlinear Anal. 177, 543–552 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hästö, P.: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269(12), 4038–4048 (2015); Corrigendum to “ The maximal operator on generalized Orlicz spaces ”. J. Funct. Anal. 271(1), 240–243 (2016)

  15. 15.

    Hästö, P., Ok, J.: Calderón-Zygmund estimates in generalized Orlicz spaces. J. Differ. Equ. 267(5), 2792–2823 (2019)

    Article  Google Scholar 

  16. 16.

    Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces. Bull. Sci. Math. 137, 76–96 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Maeda, F.-Y., Mizuta, Y., Shimomura, T.: Growth properties of Musielak–Orlicz integral means for Riesz potentials. Nonlinear Anal. 112, 69–83 (2015)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s inequality inequality for double phase functionals with variable exponents. Forum Math. 31, 517–527 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent. Complex Var. Elliptic Equ. 56(7–9), 671–695 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s theorem for double phase functionals, to appear in Math. Ineq. Appl

  21. 21.

    Mizuta, Y., Ohno, T., Shimomura, T.: Herz–Morrey spaces on the unit ball with variable exponent approaching \(1\) and double phase functionals, to appear in Nagoya Math. J

  22. 22.

    Mizuta, Y., Shimomura, T.: Boundary growth of Sobolev functions for double phase functionals, to appear in Ann. Acad. Sci. Fenn. Math.

  23. 23.

    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)

    Google Scholar 

  25. 25.

    Nakai, E.: Generalized fractional integrals on Orlicz Morrey spaces. In: Kato, M., Maligranda, L. (eds.) Banach and Function Spaces, pp. 323–333. Yokohama Publishers, Yokohama (2004)

    Google Scholar 

  26. 26.

    Peetre, J.: On the theory of \(L_{p,\lambda }\) spaces. J. Funct. Anal. 4, 71–87 (1969)

    Article  Google Scholar 

  27. 27.

    Rafeiro, H., Samko, S.: BMO–VMO results for fractional integrals in variable exponent Morrey spaces. Nonlinear Anal. 184, 35–43 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Shin, P.: Calderón-Zygmund estimates for general elliptic operators with double phase, to appear in Nonlinear Anal

  29. 29.

    Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)

    MathSciNet  Google Scholar 

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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). https://doi.org/10.1007/s13163-019-00332-z

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Keywords

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

Mathematics Subject Classification

  • Primary 31B15
  • 46E35