On the generalized Hardy-Rellich inequalities

Abstract

AbstractIn this paper, we look for the weight functions (sayg) that admit the following generalized Hardy-Rellich type inequality:$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$for some constantC> 0, whereΩis an open set in ℝNwithN⩾ 1. We find various classes of such weight functions, depending on the dimensionNand the geometry of Ω. 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${\cal D}_0^{2,2} $into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.