Weighted norm inequalities for convolutions, differential operators, and generalized hypergeometric functions
type: Item
instance of: scholarly article
title: Weighted norm inequalities for convolutions, differential operators, and generalized hypergeometric functions
zbMATH Open document ID: 1278.26020
author: Arcadii Z. Grinshpan
published in: Integral Equations and Operator Theory
publication date: 2013-04-15T00:00:00Z
review text: A sharp weighted norm inequalities for the integral convolutions \([f*g](a,x)=\int_a^x f(t)g(x-t) dt\), \(x\in [a,b]\), and applications are discussed in this paper. The following theorem gives an inequality for the convolution on a finite interval and it can be used as a generator of various integro-differential and special function inequalities. Theorem 3. Let \(f\) and \(g\) be measurable complex-valued functions which are defined for a.e. \(x\) in \([a,b]\) and \([0,b-a]\), respectively (\(-\infty <a<b<\infty\)). Then the following inequality holds for any numbers \(\alpha, \beta, \lambda >0\), \(p>1\) (\(1/p+1/q=1\)), and \(\tau \in (0,\min(p,q))\), provided that the integrals are finite: \[ \left[\int_a^b (x-a)^{(\alpha+\beta-1)(1-\tau)} (b-x)^{\lambda-1}|[f*g](a,x)|^\tau dx \right]^{1/\tau} \] \[ \leq K (b-a)^{(\alpha+\beta+\lambda-1)(1/\tau -1)} \left[\int_a^b (x-a)^{(\alpha-1)(1-p)} (b-x)^{\beta+\lambda-1} |f(x)|^p dx \right]^{1/p} \] \[ \times \left[\int_0^{b-a} t^{(\beta-1)(1-q)} (b-a-t)^{\alpha+\lambda-1} |g(t)|^q dt \right]^{1/q}, \eqno(1) \] where \(K=K(\alpha, \beta, \lambda,p,\tau)\) is defined as \[ K=[\Gamma(\lambda)]^{1/\tau}\cdot \left[ \frac{\Gamma(\alpha)}{\Gamma(\alpha+\lambda)}\right]^{1/q} \cdot \left[ \frac{\Gamma(\beta)}{\Gamma(\beta+\lambda)}\right]^{1/p} \cdot \left[ \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+\lambda)}\right]^{1/\tau -1}. \] If \(f\) and \(g\) are not identically \(0\) and \(x^{1-\alpha}f(a+(b-a)x)\) and \(x^{1-\beta} g((b-a)x)\) are continuous functions on \([0,1]\), then equality holds if and only if \(f(x)=c_1 (x-a)^{\alpha-1} e^{i\theta x}\) \((a\leq x\leq b)\) and \(g(x)=c_2 x^{\beta-1} e^{i\theta x}\) \((0\leq x\leq b-a)\), where \(\theta\) is any real number and \(c_1,c_2\) are any non-zero complex numbers. Theorem 3 allows to obtain various solution-kernel estimates for convolution integral equations. Namely, if functions \(g\) and \(h\) are given and the right-hand side of the first kind convolution Volterr…
reviewed by: Sanja Varošanec
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 6153927
zbMATH Keywords: convolutions
zbMATH Keywords: Volterra integral equations
zbMATH Keywords: seminorms for formal power series
zbMATH Keywords: weighted inequalities for sums and integrals
zbMATH Keywords: integro-differential inequalities
zbMATH Keywords: linear differential operators
zbMATH Keywords: generalized hypergeometric series
zbMATH Keywords: confluent hypergeometric functions
MaRDI profile type: Publication
full work available at URL: s00020-012-2013-5
OpenAlex ID: W2090954874
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DOI: 10.1007/S00020-012-2013-5
description: scientific article; zbMATH DE number 6153927