Hardy Operators on Variable Exponent Spaces

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.