Abstract
We obtain a modified version of the Spanne–Peetre inequality in the context of Morrey spaces with mixed norm. The geometric structure of the mixed Morrey spaces under consideration, dictates the new definition of Morrey–Lipschitz space. The Spanne–Peetre inequality that we find ensures that if a function belongs to a suitable Morrey space with mixed norm, then the modified integral operator which characterizes the Spanne–Peetre inequality, belongs to a suitable Morrey–Lipschitz space.
Similar content being viewed by others
References
Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)
Akbulut, A., Guliyev, R.V., Celik, S., Omarova, M.N.: Fractional integral associated with Schrödinger operator on vanishing generalized Morrey spaces. J. Math. Inequal. 12, 789–805 (2018)
Barza, S., Kamińska, A., Persson, L.-E., Soria, J.: Mixed norm and multidimensional Lorentz spaces. Positivity 10(3), 539–554 (2006)
Benedek, A., Panzone, R.: The space \(L^p\), with mixed norm. Duke Math. J. 28(3), 301–324 (1961)
Gadjiev, T.S., Galandarova, S., Guliyev, V.S.: Regularity in generalized Morrey spaces of solutions to higher order nondivergence elliptic equations with VMO coefficients. Electron. J. Qual. Theory Differ. Equ. 55, 1–17 (2019)
Guliyev, V.S.: Generalized local Morrey spaces and fractional integral operators with rough kernel. J. Math. Sci. 193, 211 (2013)
Guliyev, V.S.: Local generalized Morrey spaces and singular integrals with rough kernel. Azerb. J. Math. 3(2), 79–94 (2013)
Guliyev, V.S.: Function spaces and integral operators associated with Schrdinger operators: an overview. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 40(Special Issue), 178–202 (2014)
Guliyev, V.S., Akbulut, A.: Commutator of fractional integral with Lipschitz functions associated with Schrödinger operator on local generalized Morrey spaces. Bound. Value Probl. 2018, 80 (2018)
Guliyev, V.S., Hamzayev, V.H.: Rough singular integral operators and its commutators on generalized weighted Morrey spaces. Math. Inequal. Appl. 19(3), 863–881 (2016)
Guliyev, V.S., Omarova, M.N., Ragusa, M.A., Scapellato, A.: Commutators and generalized local Morrey spaces. J. Math. Anal. Appl. 457, 1388–1402 (2018)
Guliyev, V.S., Gadjiev, T.S., Galandarova, S.: Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces. Electron. J. Qual. Theory Differ. Equ. 71, 1–17 (2017)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. I. Math. Z. 27(1), 565–606 (1928)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. II. Math. Z. 34(1), 403–439 (1932)
Jia, Y., Dong, B.-Q.: Remarks on the logarithmical regularity criterion of the supercritical surface quasi-geostrophic equation in Morrey spaces. Appl. Math. Lett. 43, 80–84 (2015)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)
Karapetyants, A., Samko, S.: Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces. Fract. Calc. Appl. Anal. 20(5), 1106–1130 (2017)
Karapetyants, A., Samko, S.: Mixed norm Bergman–Morrey-type spaces on the unit disc. Math. Notes 100, 38–48 (2016)
Karapetyants, A., Samko, S.: Mixed norm variable exponent Bergman space on the unit disc. Complex Var. Elliptic Equ. 61(8), 1090–1106 (2017)
Karapetyants, A., Samko, S.: On mixed norm Bergman–Orlicz–Morrey spaces. Georgian Math. J. 25(2), 271–282 (2018)
Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis Proceedings of a Conference held in Sendai, Japan, ICM-90 Satellite Conference Proceedings, pp. 183–189 (1991)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)
Nogayama, T.: Mixed Morrey spaces. Positivity 23, 961–1000 (2019)
Nogayama, T.: Boundedness of commutators of fractional integral operators on mixed Morrey spaces. Integral Transform. Spec. Funct. 30(10), 790–816 (2019)
Peetre, J.: On the theory of \(\cal{L}_{p, \lambda } \) spaces. J. Funct. Anal. 9, 71–87 (1969)
Ragusa, M.A., Scapellato, A.: Mixed Morrey spaces and their applications to partial differential equations. Nonlinear Anal. Theory Methods Appl. 151, 51–65 (2017)
Scapellato, A.: New perspectives in the theory of some function spaces and their applications. AIP Conf. Proc. 1978, 140002 (2018)
Scapellato, A.: Applications of singular integral operators and commutators. In: Ruzhansky, M., Dutta, H. (eds.) Advanced Topics in Mathematical Analysis, Chapter 5. CRC Press, Boca Raton (2018). ISBN:9780815350873
Sobolev, S.L.: On a theorem in functional analysis. Mat. Sb. 4, 471–497 (1938). (in Russian)
Taylor, M.E.: Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations. Commun. Partial Differ. Equ. 17, 1407–1456 (1992)
Thorin, G.O.: Convexity theorems. Matematika 1(3), 43–78 (1957)
Torchinsky, A.: Real-Variable Mathods in Harmonic Analysis. Academic Press, Cambridge (1986)
Vitanza, C.: Functions with vanishing Morrey norm and elliptic partial differential equations. In: Proceedings of the Methods of Real Analysis and Partial Differential Equations, pp. 147–150. Springer, Capri (1990)
Wheeden, R.L., Zygmund, A.: Measure and Integral: An Introduction to Real Analysis, 2nd edn. CRC Press, Boca Raton (2015)
Wu, F.: Regularity criteria for the 3D tropical climate model in Morrey–Campanato space. Electron. J. Qual. Theory Differ. Equ. 48, 1–11 (2019)
Zygmund, A.: On a theorem of Marcinkiewicz concerning interpolation of operations. J. Math. Pures Appl. 9(35), 223–248 (1956)
Acknowledgements
The author would like to express his gratitude to the anonymous referees for their useful remarks.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no conflict of interest.
Additional information
Communicated by Sorina Barza.
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
Scapellato, A. A Modified Spanne–Peetre Inequality on Mixed Morrey Spaces. Bull. Malays. Math. Sci. Soc. 43, 4197–4206 (2020). https://doi.org/10.1007/s40840-020-00914-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-00914-x