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Year 2020, , 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, , 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

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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