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On the generalized Hardy-Rellich inequalities
T.V. Anoop, U. Das,
Published in Cambridge University Press
2020
Volume: 150

Issue: 2
Pages: 897 - 919
Abstract
In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: for some constant C > 0, where Ω is an open set in ℝN 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 into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger. © Copyright Royal Society of Edinburgh 2019.