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The Kolmogorov–Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces

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We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.

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References

  1. Aydın, I.: On variable exponent amalgam spaces. An. Ştiin ţ. Univ. “Ovidius” Constanţa Ser. Mat. 20(3), 5–20 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Aydın, I.: Weighted variable Sobolev spaces and capacity. J. Funct. Spaces Appl. (2012). https://doi.org/10.1155/2012/132690

  3. Aydın, I.: On vector-valued classical and variable exponent amalgam spaces. Commun. Fac. Sci. Univ. Ank. Series A1 66(2), 100–114 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Aydın, I., Gurkanli, A.T.: Weighted variable exponent amalgam spaces \(W\left( L^{p(x)}, L_{w}^{q}\right) \). Glas. Mat. 47(67), 165–174 (2012)

    MATH  Google Scholar 

  5. Bandaliyev, R.: Compactness criteria in weighted variable Lebesgue spaces. Miskolc Math. Notes 18(1), 95–101 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bandaliyev, R., Górka, P.: Relatively compact sets in variable-exponent Lebesgue spaces. Banach J. Math. Anal. 12(2), 331–346 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diening, L., Hästö, P.: Muckenhoupt weights in variable exponent spaces (preprint)

  8. Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics 29. American Mathematical Society, Providence (2000)

    Google Scholar 

  9. Feichtinger, H.G.: A compactness criterion for translation invariant Banach spaces of functions. Anal. Math. 8, 165–172 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fournier, J.J., Stewart, J.: Amalgams of \(L^{p}\) and \(l^{q}\). Bull. Am. Math. Soc. 13, 1–21 (1985)

    MATH  Google Scholar 

  11. Goes, S., Welland, R.: Compactness criteria for Köthe spaces. Math. Ann. 188, 251–269 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Górka, P., Macios, A.: The Riesz-Kolmogorov theorem on metric spaces. Miskolc Math. Notes 15(2), 459–465 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Górka, P., Macios, A.: Almost everything you need to know about relatively compact sets in variable Lebesgue spaces. J. Funct. Anal. 269(7), 1925–1949 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Górka, P., Rafeiro, H.: From Arzelà–Askoli to Riesz–Kolmogorov. Nonlinear Anal. 144, 23–31 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grubb, G.: Distributions and Operators. Springer, New York (2009)

    MATH  Google Scholar 

  16. Gurkanli, A.T.: The amalgam spaces \(W(L^{p(x)};L^{\left\lbrace p_{n}\right\rbrace })\) and boundedness of Hardy-Littlewood maximal operators. Current Trends in Analysis and Its Applications: Proceedings of the 9th ISAAC Congress (2013)

  17. Gurkanli, A.T., Aydın, I.: On the weighted variable exponent amalgam space \(W(L^{p(x)};L_{m}^{q})\). Acta Math. Sci. 34(4), 1098–1110 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Hanche-Olsen, H., Holden, H.: The Kolmogorov–Riesz compactness theorem. Expo. Math. 28(4), 385–394 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hanche-Olsen, H., Holden, H.: Addendum to “The Kolmogorov–Riesz compactness theorem” [Expo. Math. 28 (2010) 385–394]. Expo. Math. 34, 243–245 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hanche-Olsen, H., Holden, H., Malinnikova, E.: An improvement of the Kolmogorov–Riesz compactness theorem. Expo. Math. 37, 84–91 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext (2001)

  22. Holland, F.: Harmonic analysis on amalgams of \(L^{p}\) and \(l^{q}\). J. Lond. Math. Soc. 10(3), 295–305 (1975)

    MathSciNet  MATH  Google Scholar 

  23. Kokilashvili, V., Meskhi, A., Zaighum, M.A.: Weighted kernel operators in variable exponent amalgam spaces. J. Inequal. Appl. (2013). https://doi.org/10.1186/1029-242X-2013-173

  24. Kolmogorov, A.N.: Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel. Nachr. Ges. Wiss. Göttingen 9, 60–63 (1931)

    MATH  Google Scholar 

  25. Kováčik, O., Rákosník, J.: On spaces \( L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41(116)(4), 592–618 (1991)

    MATH  Google Scholar 

  26. Kulak, O., Gurkanli, A.T.: Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces. J. Inequal. Appl. (2014). https://doi.org/10.1186/1029-242X-2014-476

  27. Lahmi, B., Azroul, E., El Haitin, K.: Nonlinear degenerated elliptic problems with dual data and nonstandard growth. Math. Rep. 20(70)(1), 81–91 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Liu, Q.: Compact trace in weighted variable exponent Sobolev spaces \(W^{1, p(x)}\left( \varOmega ,\upsilon _{0},\upsilon _{1}\right) \). J. Math. Anal. Appl. 348(2), 760–774 (2008)

    MathSciNet  Google Scholar 

  29. Meskhi, A., Zaighum, M.A.: On the boundedness of maximal and potential operators in amalgam spaces. J. Math. Inequal. 8(1), 123–152 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  31. Pandey, S.S.: Compactness in Wiener amalgams on locally compact groups. Int. J. Math. Math. Sci. 2003(55), 3503–3517 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rafeiro, H.: Kolmogorov compactness criterion in variable exponent Lebesgue spaces. Proc. A. Razmadze Math. Inst. 150, 105–113 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Squire, M.L.T.: Amalgams of \(L^{p}\) and \(l^{q}\). Ph.D. Thesis, McMaster University (1984)

  34. Sudakov, V.N.: Criteria of compactness in function spaces. Upsekhi Math. Nauk. 12, 221–224 (1957). (in Russian)

    MathSciNet  Google Scholar 

  35. Takahashi, T.: On the compactness of the function-set by the convergence in the mean of general type. Studia Math. 5, 141–150 (1934)

    Article  MATH  Google Scholar 

  36. Weil, A.: L’integration Dans Les Groupes Topologiques et Ses Applications. Hermann et Cie, Paris (1940)

    MATH  Google Scholar 

  37. Wiener, N.: On the representation of functions by trigonometrical integrals. Math. Z. 24, 575–616 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yosida, K.: Functional Analysis. Springer, Berlin (1980)

    MATH  Google Scholar 

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The authors would like to thank the referees for the comments and suggestions that helped to improve the manuscript.

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Correspondence to Cihan Unal.

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Aydın, I., Unal, C. The Kolmogorov–Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Collect. Math. 71, 349–367 (2020). https://doi.org/10.1007/s13348-019-00262-5

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