On the Generalized Hardy-Rellich Inequalities

Abstract

In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: ${\int }_{\Omega }g(x)u^{2}dx\le C{\int }_{\Omega }|\Delta u{|}^{2}dx,\forall u\in {\mathcal{D}}_0^{2,2}(\Omega ),$ for some constant $C>0$, where $\Omega$ is an open set in ${\mathbb{R}}^N$ with $N\ge 1$. We find various classes of such weight functions, depending on the dimension $N$ and the geometry of $\Omega .$ Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of ${\mathcal{D}}_0^{2,2}(\Omega )$ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

24 pages

References in corpus (1)

• On the best possible remaining term in the Hardy Inequality

Cited by in corpus (1)

• Admissible function spaces for weighted Sobolev inequalities