# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 40 (2011) pp.187-203

PDF

### 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 $\Omega \subset {\mathbb R}^n$. 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.

MSC(Primary) 46E30 42B25 variable exponent, Lebesgue space, Hardy's inequality, Sobolev embeddings