Vector-Valued Weighted Sobolev Spaces with Variable Exponent

Year 2019, Volume: 11, 123 - 131, 30.12.2019

İsmail Aydın

Abstract

Our aim is to introduce the vector-valued weighted variable exponent Lebesgue spaces. We discuss two different type of H\"{o}lder inequalities in this spaces. We will also show that every elements of vector-valued weighted variable exponent Lebesgue spaces are locally integrable. Hence we can define vector-valued weighted variable exponent Sobolev spaces. Finally under some conditions we will investigate some basic properties of vector-valued weighted variable exponent Sobolev spaces.

Keywords

Vector-valued weighted variable Sobolev spaces, Hölder inequality, Radon-Nikodym property

References

• Amann, H., Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory, Birkh\"{a}user, Basel, 1995.
• Amann, H., {\em Operator-valued Fourier multipliers, vector-valued Besov spaces and applications}, Math. Nachr., \textbf{186}(1997), 5--56.
• Ayd\i n, I., {\em Weighted variable Sobolev spaces and capacity}, J Funct Space Appl, \textbf{2012}(2012), Article ID 132690, 17 pages, doi:10.1155/2012/132690.
• Cartan, H., Differential calculus, Hermann, Paris-France, 1971.
• Cheng, C., Xu, J., {\em Geometric properties of Banach space valued Bochner-Lebesgue spaces with variable exponent}, J Math Inequal, \textbf{7(3)}(2013), 461--475.
• Cruz-Uribe, D., Fiorenza, A., Variable Lebesgue Spaces: Foundations and Harmonic Analysis (Applied and Numerical Harmonic Analysis), Birkh\"{a}user/Springer, Heidelberg, 2013.
• Diening, L., {\em Maximal function on generalized Lebesgue spaces $L^{p(.)}$}, Math. Inequal. Appl., \textbf{7(2)}(2004), 245--253.
• Diening, L., Harjulehto, P., H\"{a}st\"{o}, P., Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011.
• Diestel, J., UHL, J.J., Vector measures, Amer Math Soc, 1977.
• Edmunds, E., Fiorenza, A., Meskhi, A., {\em On a measure of non-compactness for some classical operators}, Acta Math. Sin., \textbf{22(6)}(2006), 1847--1862.
• K\"{o}nig, H., Eigenvalue Distribution of Compact Operators, Birkh\"{a}user, Basel, 1986.
• Kov\'{a}\v{c}ik, O., R\'{a}kosnik, J., {\em On spaces $L^{p(x)}$ and $W^{k,p(x)}$}, Czech. Math. J., \textbf{41(116)}(1991), 592--618.
• Pietsch, A., Eigenvalues and S-numbers, Cambridge Univ. Press, Cambridge, 1987.
• Pr\"{u}ss, J., Evolutionary Integral Equations and Applications, Birkh\"{a}user, Basel, 1993.