A continuity theorem for generalized Riesz potentials
type: Item
instance of: scholarly article
title: A continuity theorem for generalized Riesz potentials
zbMATH Open document ID: 0828.46026
author: Matthias J. Stark
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-01-11T00:00:00Z
review text: Let \(0< \alpha< n\), \(1/r< 1-{\alpha \over n}\) and let \(k\in L^r (S_{n-1})\), where \(S_{n-1}\) is the unit sphere in \(\mathbb{R}^n\). The generalized Riesz kernel \(K_\alpha\) is given by \(K_\alpha (x)= k ({x\over {|x|}} )|x|^{\alpha -n}\), \(x\in \mathbb{R}^n- \{0\}\). The author proves that: (i) If \(1<p< n/\alpha\) and \(1/q= 1/p- \alpha/n\), then the map \(f\to K_\alpha *f\) is continuous from \(L^p\) into \(L^q\). (ii) The map \(f\to K_\alpha *f\) is of weak type \((1,n/ (n-\alpha))\).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 720417
zbMATH Keywords: Lebesgue spaces
zbMATH Keywords: generalized Riesz kernel
reviewed by: Bohumír Opic
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1478
OpenAlex ID: W1982052202
DOI: 10.1006/JMAA.1994.1478
description: scientific article; zbMATH DE number 720417