Sufficient conditions for weighted inequalities of sum form
type: Item
instance of: scholarly article
title: Sufficient conditions for weighted inequalities of sum form
zbMATH Open document ID: 0587.26011
DOI: 10.1016/0022-247X(85)90263-X
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: The authors discuss inequalities of the form (for y in a class \({\mathcal D}\) of functions with domain I): (1) There is a constant K such that for all \(y\in {\mathcal D},\) \[ \int_{I}N| y^{(i)}|^ p\leq K\{\int_{I}W| y|^ p+\int_{I}P| y^{(n)}|^ p\}. \] (2) For each \(\epsilon >0\) there is a number K(\(\epsilon)\) such that for all \(y\in {\mathcal D}\), \[ \int_{I}N| y^{(i)}|^ p\leq K(\epsilon)\int_{I}W| y|^ p+\epsilon \int_{I}P| y^{(n)}|^ p. \] (3) There exist \(\xi\geq 0\) \((\xi >0\) for \(j\neq 0)\), \(\eta >0\), \(K>0\), and a set \(\Gamma\) of positive numbers such that for all \(\epsilon \in \Gamma\) and \(y\in {\mathcal D},\) \[ \int_{I}N| y^{(j)}|^ p\leq K\{\epsilon^{-\xi}\int_{I}W| y|^ p+\epsilon^{\eta}\int_{I}P| y^{(n)}|^ p\}. \] In these inequalities I is an interval of the real line, \(1\leq p<\infty,\) \(0\leq j<n,\) and N,W,P are positive measurable functions (which satisfy some additional conditions). For \(N=P=W=1,\) the \(p=\infty\) case of (3) is also considered.
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 3941833
zbMATH Keywords: weighted integral inequalities
zbMATH Keywords: sum form
author: Richard C. Brown
author: Don B. Hinton
MaRDI profile type: Publication
cites work: On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems
cites work: On integral inequalities and compact embeddings associated with ordinary differential expressions
cites work: Inequalities Associated with Regular and Singular Problems in the Calculus of Variations
cites work: ON THE INEQUALITY │f″│2≤ K│f││f(4)│
cites work: A Singular Integral Inequality on a Bounded Interval
cites work: On the minimization of singular quadratic functional
cites work: Q3715448
cites work: Q3345102
cites work: Q5672035
cites work: Norm inequalities involving derivatives
cites work: An integral inequality with an application to ordinary differential operators
cites work: Q3315815
cites work: Q5531363
cites work: Q5526484
cites work: Weighted Landau inequalities
cites work: Q5526485
cites work: Q3049377
cites work: An Extension of the Hardy-Littlewood Inequality
cites work: Ramifications of Landau's inequality
cites work: Q3913618
cites work: Weighted norm inequalities of sum form involving derivatives
cites work: Discreteness conditions for the spectrum of ordinary differential operators
cites work: Norm inequalities of product form in weighted Lp spaces
cites work: On inequalities between the powers of a linear operator
cites work: Q5634277
cites work: On inequalities for powers of linear operators and for quadratic forms
description: scientific article; zbMATH DE number 3941833