On the generalized Hardy-Rellich inequalities. Issue 2 (April 2020) - Anoop, T.V.
Abstract: In this paper, we look for the weight functions (say g ) 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 constant C > 0, where Ω is an open set in ℝ N with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and 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.