On integral inequalities of the Sobolev type
type: Item
instance of: scholarly article
title: On integral inequalities of the Sobolev type
zbMATH Open document ID: 0826.26007
DOI: 10.1007/BF01827934
author: V. Pereyra
published in: Aequationes Mathematicae
publication date: 1995-11-16T00:00:00Z
full work available at URL: https://eudml.org/doc/182477
review text: The main result states that for a rectangle \(\Omega= \prod^n_{i= 1} (a_i, b_i)\), for any functions \(f^\alpha\in C^1_0(\Omega)\), \(\alpha= 1,\dots, m\), and for any numbers \(p_\alpha\geq 1\), \(q_\alpha> 0\) with \(\sum p_\alpha/q_\alpha= 1\) the inequality \[ \int_\Omega \prod_h\alpha|f^\alpha|^{q_\alpha}\leq {M\over 2\sqrt n} \sum_\alpha \Biggl( \int_\Omega |f^\alpha|^{2(p_\alpha- 1)}\Biggr)^{1/2} \Biggl( \int_\Omega |\nabla f^\alpha|^2\Biggr)^{1/2}, \] holds with \(M= \max\{b_i- a_i: 1\leq i\leq n\}\). The elementary proof is based on the one-dimensional identity expressing the function by means of its integral of the derivative and on some analogues of the quadratic mean -- arithmetic mean -- geometric mean inequality.
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 39B72
zbMATH DE Number: 729151
zbMATH Keywords: integral inequalities
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: quadratic mean
zbMATH Keywords: arithmetic mean
zbMATH Keywords: geometric mean
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
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OpenAlex ID: W2010349620
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description: scientific article; zbMATH DE number 729151