Abstract
In this final chapter we introduce the spaces \(L_{p(\cdot)}\) with variable exponent p and establish their basic properties. When I is a bounded interval (a,b) in \(\mathbb{R}\) the Hardy operator \(T_a:L_P(.)(I)\longrightarrow L_{p(.)}(I)\) given by \(T_af(x)=\int_{a}^{x}f(t)dt\) is studied: the asymptotic behaviour of its approximation, Bernstein, Gelfand and Kolmogorov numbers is determined. To conclude, a version of the \({p(\cdot)}\)-Laplacian is presented and the existence established of a countable family of eigenfunctions and eigenvalues of the corresponding Dirichlet problem.
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Keywords
- Variable Exponent
- Banach Function Space
- Hardy Operator
- Littlewood Maximal Operator
- Generalise Lebesgue Space
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© 2011 Springer-Verlag Berlin Heidelberg
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Lang, J., Edmunds, D. (2011). Hardy Operators on Variable Exponent Spaces. In: Eigenvalues, Embeddings and Generalised Trigonometric Functions. Lecture Notes in Mathematics(), vol 2016. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18429-1_9
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DOI: https://doi.org/10.1007/978-3-642-18429-1_9
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18267-9
Online ISBN: 978-3-642-18429-1
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