On weighted estimates for a class of integral operators
type: Item
instance of: scholarly article
title: On weighted estimates for a class of integral operators
zbMATH Open document ID: 0815.47064
DOI: 10.1007/BF00975180
published in: Siberian Mathematical Journal
publication date: 1994-11-08T00:00:00Z
review text: Let \(\varphi(t)\), \(t\in (0,1)\), be a measurable function satisfying the following conditions: a) \(\varphi(t) \geq 0\) and \(\varphi(t)\) is nonincreasing; b) \(\varphi (t_ 1+ t_ 2)\leq D(\varphi(t_ 1)+ \varphi(t_ 2))\), \(0<t_ 1,t_ 2< 1\), where \(D\) is independent of \(t_ 1\), \(t_ 2\). For the integral operator \[ (T_ \varphi f) (x)= \int_ 0^ x \varphi(t/x) f(t) dt \] weighted norm inequalities of the form \[ \| u T_ \varphi f \|_{L^ q (0,\infty)} \leq c\| vf \|_{L^ p (0,\infty)} \] are studied in the case \(0<q<p <\infty\), \(p>1\). Moreover, necessary and sufficient conditions for the operator \(T_ \varphi\) to be compact are given in the case \(1<p, q<\infty\).
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 652187
zbMATH Keywords: compactness
zbMATH Keywords: weighted norm inequalities
author: V. D. Stepanov
reviewed by: Boris Rubin
MaRDI profile type: Publication
cites work: Q3997067
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cites work: Weighted Inequalities for Riemann-Liouville Fractional Integrals of Order One and Greater
cites work: Weighted Hardy and Opial-type inequalities
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description: scientific article; zbMATH DE number 652187