On the generalized Hardy-Rellich inequalities. Issue 2 (April 2020)

• Record Type: Journal Article
• Authors: Anoop, T.V.; Das, Ujjal; Sarkar, Abhishek
• Is Part Of: Proceedings. Volume 150:Issue 2(2020)
• Page Start: 897
• Page End: 919
• Publication Date: 2020-04
• Subjects: Generalized Hardy-Rellich inequality, -- Muckenhoupt condition, -- symmetrization, -- Lorentz spaces, -- Lorentz-Zygmund spaces, -- exterior domains; 35A23, -- 46E30, -- 46E35; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2018.128
• Languages: English
• ISSNs: 0308-2105

Abstract

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.