On some differential properties of functions in generalized grand Sobolev-Morrey spaces

Abstract

In this paper we introduce a generalized grand Sobolev-Morrey spaces. Some differential and differential-difference properties of functions from this spaces are proved by means of the integral representation.

Keywords

• Generalized grand Sobolev-Morrey spaces
• integral representation
• flexible-horn condition
• Hölder condition

References

1. Besov, O. V., Ilyin, V. P., Nikolskii, S. M., Integralnye Predstavleniya Funktsi i Teoremy Vlozheniya (Russian) (Integral Representations of Functions and Embedding Theorems), Fizmatlit ”Nauka”, Moscow, 1996.
2. Fiorenza, A., Formica, M., Gogatishvili, A., On grand and small Lebesgue equations and applications, Differ. Equ. Appl., 10(1) (2018), 21-46. dx.doi.org/10.7153/dea-2018-10-03
3. He, S., Tao, Sh., Boundedness of some operators on grand generalized Morrey spaces over nonhomogeneous spaces, AIMS Mathematics, 7(1) (2022), 1000-1014. doi:10.39341 math.2022060
4. Iwaniec, T., Sbordone, C., On the integrability of the Jacobian under minimal hypoteses, Arch. Rational Mech. Anal., 119(2) (1992), 129-143. https://doi.org/10.1007/BF00375119
5. Kokilashvili, V. M., Meskhi, A., Rafeiro, H., Riesz type potential operators in generalized grand Morrey spaces, Georgian Math. J., 20(1) (2013), 43-64. https://doi.org/10.1515/gmj-2013-0009
6. Liu, Y., Yuan, W., Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces, Czechoslovak Math. J., 67(3) (2017), 715-732. DOI:10.21136/CMJ.2017.0081-16
7. Meskhi, A., Maximal functions potentials and singular integrals in grand Morrey spaces, Comp. Var. Ellip. Equations, 56(10-11) (2011), 1003-1019. https://doi.org/10.1080/17476933.2010.534793
8. Mizuta, Y., Ohno, T., Trudingers exponential integrability for Riesz potentials of function in generalized grand Morrey spaces, J. Math. Anal. Appl., 420(1) (2014). https://doi.org/10.1016/j.jmaa.2014.05.082

9. Najafov, A. M., Alekberli, S. T., On properties of functions from grand Sobolev-Morrey spaces, J. Baku Engineering Univ., 2(1) (2018), 27-36.
10. Najafov, A. M., Rustamova, N. R., Some differential properties of anisotropic grand Sobolev Morrey spaces, Trans. A. Razmadze Math. Inst., 172(1) (2018), 82-89. https://doi.org/10.1016/j.trmi.2017.10.001
11. Najafov, A. M., Gasimova, A. M., On embedding theorems in grand grand Nikolski-Morrey spaces, Eur. J. Pure Appl. Math., 12(4) (2019), 1602-1611. https://doi.org/10.29020/nybg.ejpam.v12i4.3567
12. Najafov, A. M., On some properties differential properties of small small Sobolev-Morrey spaces, Eurasian Math. J., 12(1) (2021), 57-67. https://doi.org/10.32523/2077-9879-2021-12-1-57-67
13. Rafeiro, H., A note on boundedness of operators in grand grand Morrey spaces, Operator Theory: Advances and Applications, 229 (2013) 349–356. DOI:10.1007/978-3-0348-0516-2 19
14. Umarkhadzhiev, S., The boundedness of the Riesz potential operator from generalized grand Lebesgue spaces to generalized grand Morrey spaces, Operator Theory, Operator Algebras and Applications, 363-373, Oper. Theory Adv. Appl., 242, Birkh¨auser, Springer, Basel, 2014. DOI:10.1007/978-3-0348-0816-3 22