Hokkaido Mathematical Journal

MIZUTA Yoshihiro, NAKAI Eiichi, OHNO Takao, SHIMOMURA Tetsu,

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

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

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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
MSC(Secondary)42B25
Uncontrolled Keywordsvariable exponent, Lebesgue space, Hardy's inequality, Sobolev embeddings