Hardy type inequality in variable Lebesgue spaces

Abstract

We prove that in variable exponent spaces $L^{p(\cdot)}(\Omega)$, where $p(\cdot)$ satisfies the log-condition and $\Omega$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \bar{\Omega}$ has the cone property, the validity of the Hardy type inequality $$| 1/\delta(x)^\alpha \int_\Omega \phi(y) dy/|x-y|^{n-\alpha}|_{p(\cdot)} \leqq C |\phi|_{p(\cdot)}, \quad 0<\al<\min(1,\frac{n}{p_+})$$, where $\delta(x)=\mathrm{dist}(x,\partial\Omega)$, is equivalent to a certain property of the domain $\Om$ expressed in terms of $\al$ and $\chi_\Om$.