On a Closed Subspace of L^(p(.)(Ω))

Year 2020, Volume: 9 Issue: 2, 682 - 688, 15.06.2020

Yasin Kaya

https://doi.org/10.17798/bitlisfen.651211

Cited By: 1

Abstract

In this study, we first give a description of L^(p(.)(Ω)) spaces. These spaces are an important generalization of
classical Lebesgue spaces. We mention 
their various applications in engineering and physics fields. Thereafter,
as it is naturally,  one of the main task
in L^(p(.)(Ω)) spaces is to generalize known properties classical Lebesgue
spaces L^p(Ω))  to L^(p(.)(Ω)) spaces.  Provided that measure of the set Ω  is finite, we extend a
theorem which about a closed subspace of  space, from constant exponent
to variable exponent. Our proof method based on embedding between L^(p(.)(Ω)) - L^p(Ω)) spaces and the proof
of constant case. The essence of the method is to take advantage
of properties of Hilbert space L^2(Ω)), and also based on the use of the closed
graph theorem and finite measure of the set Ω.

Keywords

Variable Exponent, Lebesgue Space, Closed Subspace

References

• 1. Acerbi E., Mingione G. 2002. Regularity Results for Stationary Electro-rheological Fluids, Archive for Rational Mechanics and Analysis,164(3): 213-259.
• 2. Cruz-Uribe DV., Fiorenza A. 2013.Variable Lebesgue Spaces: Foundations and Harmonic Aanalysis. Springer Science & Business Media.
• 3. Lars D., Harjulehto P., Hästö P., Růžička M. 2011. Lebesgue and Sobolev Spaces with Variable Eexponents, Springer.
• 4. Růžička M. 2000. Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin.
• 5. Zhikov V.V. 1997. Meyer-Type Estimates for Solving the nonlinear Stokes System, Differential Equations 33(1): 108-115.
• 6. Amaziane B., Pankratov L., Piatnitski A. 2009. Nonlinear Flow Through Double Porosity Media in Variable Exponent Sobolev Spaces, Nonlinear Analysis: Real World Applications,10(4): 2521-30.
• 7. Cekic B., Kalinin AV., Mashiyev RA., Avci M. 2012. -Estimates of Vector Fields and Some Aapplications to Magnetostatics Problems, Journal of Mathematical Analysis and Applications, 389(2):838-51.
• 8. Blomgren P., Chan TF., Mulet P., Wong CK. 1997. Total Variation Image Restoration: Numerical methods and Extensions. InProceedings of International Conference on Image Processing (Vol. 3, pp. 384-387). IEEE.
• 9. Kováčik O., Rákosník J. 1991. On Spaces and , Czechoslovak Math. J., 41: 592-618.
• 10. Fan X., Zhao D. 2001. On the Spaces ) and , Journal of Mathematical Analysis and Applications, 263(2):424-46.
• 11. Bruckner A., Bruckner J., Thomson B. 1997. Real analysis, Prentice-Hall, N.J.
• 12. Grothendieck A. 1954. Sur Certains Sous-espaces Vectoriels de , Canadian Journal of Mathematics, 6:158-60.