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 $$| 1/δ(x)^α\int_Ωϕ(y) dy/|x-y|^{n-α}|_{p(\cdot)} \leqq C |ϕ|_{p(\cdot)}, \quad 0<\al<\min(1,\frac{n}{p_+})$$, where $\delta (x)=\mathrm{dist}(x,\partial \Omega )$, is equivalent to a certain property of the domain $\Om$ expressed in terms of $\al$ and $χ_\Om$.

16 pages