Hardy's inequality in Orlicz-Sobolev spaces of variable exponent

Abstract

Our aim in this paper is to deal with a norm version of Hardy's inequality for Orlicz-Sobolev functions with $|\nabla u| \in L^{p(\cdot)}\log L^{p(\cdot)q(\cdot)}(\Omega)$ for an open set $\Omega \subset \R^n$. Here $p(\cdot)$ and $q(\cdot)$ are variable exponents satisfying log-H\"older and loglog-H\"older conditions, respectively. We are also concerned with the case when $p$ attains the value 1 in some parts of the domain is included in the results.