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On the generalized Hardy-Rellich inequalities

Published online by Cambridge University Press:  26 January 2019

T.V. Anoop
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai600036, India (anoop@iitm.ac.in)
Ujjal Das
Affiliation:
The Institute of Mathematical Sciences, Chennai600113, India (ujjaldas@imsc.res.in, ujjal.rupam.das@gmail.com)
Abhishek Sarkar
Affiliation:
NTIS, University of West Bohemia Technická 8, 306 14 Plzeň, Czech Republic (sarkara@ntis.zcu.cz)

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 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 ${\cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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