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.
Similar content being viewed by others
References
Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)
Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(6), 1629–1663 (2004)
Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50(2), 201–230 (2012)
Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. 57, 62 (2018)
Baroni, P., Colombo, M., Mingione, G.: Non-autonomous functionals, borderline cases and related function classes. St Petersb. Math. J. 27, 347–379 (2016)
Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. (7) 7(3–4), 273–279 (1987)
Colasuonno, F., Squassina, M.: Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. (4) 195(6), 1917–1959 (2016)
Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)
Colombo, M., Mingione, G.: Bounded minimizers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)
De Filippis, C., Mingione, G.: On the regularity of minima of non-autonomous functionals, to appear in J. Geom. Anal
De Filippis, C., Oh, J.: Regularity for multi-phase variational problems. J. Differ. Equ. 267(3), 1631–1670 (2019)
Harjulehto, P., Hästö, P.: Boundary regularity under generalized growth conditions. Z. Anal. Anwend. 38(1), 73–96 (2019)
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)
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)
Hästö, P., Ok, J.: Calderón-Zygmund estimates in generalized Orlicz spaces. J. Differ. Equ. 267(5), 2792–2823 (2019)
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)
Maeda, F.-Y., Mizuta, Y., Shimomura, T.: Growth properties of Musielak–Orlicz integral means for Riesz potentials. Nonlinear Anal. 112, 69–83 (2015)
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)
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)
Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s theorem for double phase functionals, to appear in Math. Ineq. Appl
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
Mizuta, Y., Shimomura, T.: Boundary growth of Sobolev functions for double phase functionals, to appear in Ann. Acad. Sci. Fenn. Math.
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)
Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
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)
Peetre, J.: On the theory of \(L_{p,\lambda }\) spaces. J. Funct. Anal. 4, 71–87 (1969)
Rafeiro, H., Samko, S.: BMO–VMO results for fractional integrals in variable exponent Morrey spaces. Nonlinear Anal. 184, 35–43 (2019)
Shin, P.: Calderón-Zygmund estimates for general elliptic operators with double phase, to appear in Nonlinear Anal
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-019-00332-z
Keywords
- Riesz potentials
- Morrey spaces
- Musielak–Orlicz–Morrey spaces
- Double phase functionals
- Campanato–Morrey spaces