Remarks on the Rellich inequality
type: Item
instance of: scholarly article
title: Remarks on the Rellich inequality
zbMATH Open document ID: 1380.26015
author: Shuji Machihara
author: Tohru Ozawa
author: Hidemitsu Wadade
published in: Mathematische Zeitschrift
publication date: 2017-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1611.03625
review text: The aim of the paper is to study the Rellich inequality \[ \Big\|\frac{f}{|x|^2}\Big\|_{L^2(\mathbb R^n)} \leq \frac{4}{n(n-4)} \, \|\Delta f\|_{L^2(\mathbb R^n)} \] for all \(f\) from the Sobolev space \(H^2(\mathbb R^n)\), with \(n \geq 5\). To prove this inequality, the authors make use of simple equalities involving partial derivatives and then they drop some convenient terms (remainders). Such an approach also shows the nonexistence of nontrivial extremisers of the Rellich inequality in question.
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35A23
zbMATH DE Number: 6780332
zbMATH Keywords: Rellich inequality
zbMATH Keywords: Hardy inequality
zbMATH Keywords: nontrivial extremiser
zbMATH Keywords: vanishing condition of remainder
zbMATH Keywords: Schrödinger operator
reviewed by: Bohumír Opic
MaRDI profile type: Publication
OpenAlex ID: W2551896176
arXiv ID: 1611.03625
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DOI: 10.1007/S00209-016-1805-8
description: scientific article