Norm inequalities for convolution operators
type: Item
instance of: scholarly article
title: Norm inequalities for convolution operators
zbMATH Open document ID: 1184.42005
author: N. T. Tleukhanova
author: E. D. Nursultanov
author: Sergey Tikhonov
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2009-12-22T00:00:00Z
review text: Let \(D\) and \(\Omega\) be measurable sets in \(\mathbb{R}^n\), and let \(K\) be a locally integrable function on \(D-\Omega\). There is considered the convolution operator \(A\), where \[ (Af)(y)= (K* f)(y)= \int_D K(x- y)f(x)\,dx, \] \(y\in\Omega\), both in Lebesgue spaces and Lorentz spaces. In case of Lebesgue spaces, there are extended estimations of the norm \(\| A\|_{L_p\to L_q}\), \(1< p< q<\infty\) obtained by Young and O'Neil. In particular, there are obtained estimations from below of this norm. Next, the Young-O'Neil inequalities in Lorentz spaces are extended for the limit value parameters, i.e., \(\| K* f\|_{L(p,h_1)\to L(p,h_2)}\). There are also given estimates in the case of weighted Lorentz spaces.
reviewed by: Julian Musielak
Mathematics Subject Classification ID: 42A85
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5651240
zbMATH Keywords: convolution operator
zbMATH Keywords: Lebesgue spaces
zbMATH Keywords: Lorentz spaces
MaRDI profile type: Publication
full work available at URL: j.crma.2009.10.003
OpenAlex ID: W1973876556
cites work: A note on L(\infty, q) spaces and Sobolev embeddings
cites work: Weak-\(L^\infty\) and BMO
cites work: Q4039801
cites work: On a Convolution Theorem for L(p, q) Spaces
cites work: Convolution of L(p, q) Functions
cites work: Q3310336
cites work: On L(p,q) spaces
cites work: Convolution Theorems with Weights
cites work: Convolution inequalities in Lorentz spaces
cites work: Net spaces and boundedness of integral operators
cites work: Convolution operators and L(p, q) spaces
cites work: Some remarks on convolution operators and \(L(p,q)\) spaces
DOI: 10.1016/J.CRMA.2009.10.003
description: scientific article; zbMATH DE number 5651240