Sharp constants in the Hardy-Rellich inequalities
type: Item
instance of: scholarly article
title: Sharp constants in the Hardy-Rellich inequalities
zbMATH Open document ID: 0981.26016
published in: Journal of Functional Analysis
publication date: 2002-02-25T00:00:00Z
review text: Optimal constants are found in the inequalities \[ \int_{\mathbb R^d}|x|^{-2l}|u(x)|^2dx\leq C_l\int_{\mathbb R^d}|\xi|^{2l}|\hat u(\xi)|^2d\xi \] with \(l<d/2\) and \[ \int_{\mathbb R^d}|x|^{-2l}\Bigl|u(x)-\sum_{|\alpha|\leq [l-d/2]}(\alpha!)^{-1}(\partial^\alpha u)(0)x^\alpha\Bigr|^2dx \leq C_l\int_{\mathbb R^d}|\xi|^{2l}|\hat u(\xi)|^2d\xi \] with \(l>d/2\), \(l-d/2\notin\mathbb Z\), to hold for all \(u\) such that \(\hat u\in C^\infty_0(\mathbb R^d\setminus\{0\})\). The result is then extended to larger classes of functions and the case \(l-d/2\in\mathbb Z\) is discussed.
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1374984
zbMATH Keywords: optimal constant
zbMATH Keywords: Hardy inequality
zbMATH Keywords: Rellich inequality
zbMATH Keywords: Fourier transform
author: Dmitri Yafaev
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
full work available at URL: jfan.1999.3462
OpenAlex ID: W1981151781
cites work: Nonoscillation theory of elliptic equations of order \(2n\)
cites work: Q5820721
cites work: The spectrum of singular boundary problems
cites work: Q4011162
cites work: Explicit constants for Rellich inequalities in \(L_p(\Omega)\)
cites work: ON THE NEGATIVE SPECTRUM OF AN ELLIPTIC OPERATOR
cites work: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
cites work: Q5571936
cites work: A remark on the Hardy inequalities
cites work: Q5640160
cites work: Q4241850
DOI: 10.1006/JFAN.1999.3462
description: scientific article; zbMATH DE number 1374984