# Hardy Operators On Variable Exponent Spaces

Jan Lang, D. Edmunds
Published 2011 · Mathematics

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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