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Year 2020, Volume: 8 Issue: 2, 130 - 134, 15.10.2020
https://doi.org/10.36753/mathenot.683046

Abstract

References

  • Aoyama, H., Lebesgue spaces with variable exponent on a probability space. \emph{Hiroshima Math. J.} 39 (2009), 207-216.
  • Capone, C., Formica, M. R., Giova, R., Grand Lebesgue spaces with respect to measurable functions. \emph{Nonlinear Anal.} 85 (2013), 125-131.
  • Cruz-Uribe, D. V., Fiorenza, A., Variable Lebesgue Spaces Foundations and Harmonic Analysis, Springer, New York, 2013.
  • Danelia, N., Kokilashvili, V., Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces. \emph{Georgian Math. J.} 23 (2016), no. 1, 43-53.
  • Di Fratta, G., Fiorenza, A., A direct approach to the duality of grand and small Lebesgue spaces. Nonlinear Anal. \textbf{70}(7), 2582--2592 (2009).
  • Maximal function on generalized Lebesgue spaces $L^{p(.)}$. \emph{Mathematical Inequalities and Applications} 7 (2004), 245-253.
  • Diening, L., Harjulehto, P., H\"{a}st\"{o}, P., R\r{u}\v{z}i\v{c}ka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, 2011.
  • Fan, X., Zhao, D., On the spaces $L^{p\left( x\right) }\left(\Omega \right) $ and $W^{k,p\left( x\right) }\left( \Omega \right) .$ \emph{J. Math. Anal. Appl.} 263 (2001), no. 2, 424-446.
  • Fiorenza, A., Duality and reflexivity in grand Lebesgue spaces. \emph{Collect. Math.} 51 (2000), no. 2, 131-148.
  • Fiorenza, A., Sbordone, C., Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^{1}$. \emph{Studia Math.} 127 (1998), no. 3, 223-231.
  • Fiorenza, A., Gupta, B., Jain, P., The maximal theorem in weighted grand Lebesgue spaces. \emph{Stud. Math.} 188 (2008), no. 2, 123-133.
  • Gorka, P., Ergodic theorem in variable Lebesgue spaces. \emph{Period Math. Hung.} 72 (2016), 243-247.
  • Greco, L., Iwaniec, T., Sbordone, C., Inverting the $p$-harmonic operator. \emph{Manuscripta Math.} 92 (1997), 249-258.
  • Iwaniec, T., Sbordone, C., On integrability of the Jacobien under minimal hypotheses. \emph{Arch. Rational Mechanics Anal.} 119 (1992), 129-143.
  • Kokilashvili, V., Meskhi, A., Maximal and Calderon -Zygmund operators in grand variable exponent Lebesgue spaces. \emph{Georgian Math. J.} 21 (2014), 447-461.
  • Kov\'{a}\v{c}ik, O., R\'{a}kosn\'{\i}k, J., On spaces $L^{p(x)}$ and $W^{k,p(x)}$. \emph{Czechoslovak Math. J.} 41(116) (1991), no. 4, 592-618.
  • Orlicz, W., \"{U}ber Konjugierte Exponentenfolgen. \emph{Studia Math.} 3 (1931), 200-212.
  • Rafeiro, H., Vargas, A., On the compactness in grand spaces, \emph{Georgian Math.} 22 (2015), no. 1, 141-152.

Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces

Year 2020, Volume: 8 Issue: 2, 130 - 134, 15.10.2020
https://doi.org/10.36753/mathenot.683046

Abstract

We consider several fundamental properties of grand variable exponent Lebesgue spaces. Moreover, we discuss Ergodic theorems in these spaces whenever the exponent is invariant under the transformation.

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References

  • Aoyama, H., Lebesgue spaces with variable exponent on a probability space. \emph{Hiroshima Math. J.} 39 (2009), 207-216.
  • Capone, C., Formica, M. R., Giova, R., Grand Lebesgue spaces with respect to measurable functions. \emph{Nonlinear Anal.} 85 (2013), 125-131.
  • Cruz-Uribe, D. V., Fiorenza, A., Variable Lebesgue Spaces Foundations and Harmonic Analysis, Springer, New York, 2013.
  • Danelia, N., Kokilashvili, V., Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces. \emph{Georgian Math. J.} 23 (2016), no. 1, 43-53.
  • Di Fratta, G., Fiorenza, A., A direct approach to the duality of grand and small Lebesgue spaces. Nonlinear Anal. \textbf{70}(7), 2582--2592 (2009).
  • Maximal function on generalized Lebesgue spaces $L^{p(.)}$. \emph{Mathematical Inequalities and Applications} 7 (2004), 245-253.
  • Diening, L., Harjulehto, P., H\"{a}st\"{o}, P., R\r{u}\v{z}i\v{c}ka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, 2011.
  • Fan, X., Zhao, D., On the spaces $L^{p\left( x\right) }\left(\Omega \right) $ and $W^{k,p\left( x\right) }\left( \Omega \right) .$ \emph{J. Math. Anal. Appl.} 263 (2001), no. 2, 424-446.
  • Fiorenza, A., Duality and reflexivity in grand Lebesgue spaces. \emph{Collect. Math.} 51 (2000), no. 2, 131-148.
  • Fiorenza, A., Sbordone, C., Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^{1}$. \emph{Studia Math.} 127 (1998), no. 3, 223-231.
  • Fiorenza, A., Gupta, B., Jain, P., The maximal theorem in weighted grand Lebesgue spaces. \emph{Stud. Math.} 188 (2008), no. 2, 123-133.
  • Gorka, P., Ergodic theorem in variable Lebesgue spaces. \emph{Period Math. Hung.} 72 (2016), 243-247.
  • Greco, L., Iwaniec, T., Sbordone, C., Inverting the $p$-harmonic operator. \emph{Manuscripta Math.} 92 (1997), 249-258.
  • Iwaniec, T., Sbordone, C., On integrability of the Jacobien under minimal hypotheses. \emph{Arch. Rational Mechanics Anal.} 119 (1992), 129-143.
  • Kokilashvili, V., Meskhi, A., Maximal and Calderon -Zygmund operators in grand variable exponent Lebesgue spaces. \emph{Georgian Math. J.} 21 (2014), 447-461.
  • Kov\'{a}\v{c}ik, O., R\'{a}kosn\'{\i}k, J., On spaces $L^{p(x)}$ and $W^{k,p(x)}$. \emph{Czechoslovak Math. J.} 41(116) (1991), no. 4, 592-618.
  • Orlicz, W., \"{U}ber Konjugierte Exponentenfolgen. \emph{Studia Math.} 3 (1931), 200-212.
  • Rafeiro, H., Vargas, A., On the compactness in grand spaces, \emph{Georgian Math.} 22 (2015), no. 1, 141-152.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Cihan Unal 0000-0002-7242-393X

Publication Date October 15, 2020
Submission Date January 31, 2020
Acceptance Date July 5, 2020
Published in Issue Year 2020 Volume: 8 Issue: 2

Cite

APA Unal, C. (2020). Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces. Mathematical Sciences and Applications E-Notes, 8(2), 130-134. https://doi.org/10.36753/mathenot.683046
AMA Unal C. Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces. Math. Sci. Appl. E-Notes. October 2020;8(2):130-134. doi:10.36753/mathenot.683046
Chicago Unal, Cihan. “Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces”. Mathematical Sciences and Applications E-Notes 8, no. 2 (October 2020): 130-34. https://doi.org/10.36753/mathenot.683046.
EndNote Unal C (October 1, 2020) Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces. Mathematical Sciences and Applications E-Notes 8 2 130–134.
IEEE C. Unal, “Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces”, Math. Sci. Appl. E-Notes, vol. 8, no. 2, pp. 130–134, 2020, doi: 10.36753/mathenot.683046.
ISNAD Unal, Cihan. “Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces”. Mathematical Sciences and Applications E-Notes 8/2 (October 2020), 130-134. https://doi.org/10.36753/mathenot.683046.
JAMA Unal C. Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces. Math. Sci. Appl. E-Notes. 2020;8:130–134.
MLA Unal, Cihan. “Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, 2020, pp. 130-4, doi:10.36753/mathenot.683046.
Vancouver Unal C. Ergodic Theorem in Grand Variable Exponent Lebesgue Spaces. Math. Sci. Appl. E-Notes. 2020;8(2):130-4.

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