A Hardy inequality in the half-space
type: Item
instance of: scholarly article
title: A Hardy inequality in the half-space
zbMATH Open document ID: 1077.26010
author: Jesper Tidblom
published in: Journal of Functional Analysis
publication date: 2005-06-01T00:00:00Z
review text: The author proves the following inequality of Hardy type, which is valid for \(u \in C_0^\infty(\mathbb R^n_+)\): \[ \int_{\mathbb R^n_+} |\nabla u|^2 \,dx \geq {1\over 4} \int_{\mathbb R^n_+} {|u|^2 \over x_n^2} \,dx + {1\over 8} \int_{\mathbb R^n_+} {|u|^2 \over x_n(x_{n-1}^2 + x_n^2)} \,dx. \tag{1} \] This is an improvement of a previous result by V. G. Maz'ja, who showed a similar estimate, with \({1\over 16}\) replacing \({1\over 8}\). Next, the following inequality is proved, solving in affermative a conjecture, again formulated by Maz'ja: if \(u \in C_0^\infty(\mathbb R^n_+)\), \(p \in (1, +\infty)\), \(\tau \in (0, 1]\), \[ \int_{\mathbb R^n_+} |\nabla u|^p \,dx \geq {p-1\over p} \int_{\mathbb R^n_+} {|u|^p \over x_n^p} \,dx + \alpha(p,\tau) \int_{\mathbb R^n_+} {|u|^p \over x_n^{p-\tau}(x_{n-1}^2 + x_n^2)^{\tau \over 2}} \,dx, \tag{2} \] with \(\alpha(p,\tau)\) depending only on \(p\) and \(\tau\).
reviewed by: Davide Guidetti
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26E10
zbMATH DE Number: 2172983
zbMATH Keywords: Hardy inequalities
zbMATH Keywords: halfspace
zbMATH Keywords: remainder terms
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.09.014
OpenAlex ID: W1974524244
cites work: REFINED GEOMETRIC Lp HARDY INEQUALITIES
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cites work: A geometrical version of Hardy's inequality
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cites work: A geometrical version of Hardy’s inequality for \stackrel{∘}𝑊^{1,𝑝}(Ω)
DOI: 10.1016/J.JFA.2004.09.014
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description: scientific article; zbMATH DE number 2172983