Sobolev's inequality for Riesz potentials of functions in non-doubling Morrey spaces
type: Item
instance of: scholarly article
title: Sobolev's inequality for Riesz potentials of functions in non-doubling Morrey spaces
zbMATH Open document ID: 1186.31003
author: Yoshihiro Mizuta
author: Tetsu Shimomura
author: Takuya Sobukawa
published in: Osaka Journal of Mathematics
publication date: 2009-04-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1235574047
review text: Let \(G\) be an open subset of a separable metric space \(X\), and let \(\mu\) be a diffuse nonnegative Radon measure on \(X\). The Riesz potential of order \(\alpha\) for a nonnegative measurable function \(f\) on \(G\) is defined by \[ U_\alpha f(x)= \int_G \frac{|x-y|^\alpha f(y)}{\mu(B(x,4|x-y|))} d\mu(y), \] where \(|x-y|\) is the distance of \(x\) and \(y\) and \(B(x,r)\) is the open ball centered at \(x\) of radius \(r>0\). The authors prove Sobolev's inequality and Trudinger exponential integrability for Riesz potentials of functions \(f\) in non-doubling Morey spaces, i.e., \[ \sup\limits_{x\in G, 0<r<d_G} \frac{r^\nu}{\mu(B(x,kr))} \int_{G\cap B(x,r)} |f(y)|^p d\mu(y) \;< \infty, \] where \(1<p<\infty, \nu>0,k>1\), and \(d_G\) denotes the diameter of \(G\).
reviewed by: Mohamed Hmissi
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5543719
zbMATH Keywords: Riesz potential of function
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: Trudinger exponential intgrability
zbMATH Keywords: non-doubling Morey space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5543719