Optimal embeddings and compact embeddings of Bessel-potential-type spaces
type: Item
instance of: scholarly article
title: Optimal embeddings and compact embeddings of Bessel-potential-type spaces
zbMATH Open document ID: 1179.46029
author: Amiran Gogatishvili
author: Júlio S. Neves
author: Bohumír Opic
published in: Mathematische Zeitschrift
publication date: 2009-06-22T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/7768
review text: Let \(g_\sigma\), \(0 < \sigma <1\), be the Bessel kernels in \(\mathbb R^n\), \(\widehat{g_\sigma} (\xi) = c (1+ |\xi|^2)^{-\sigma/2}\). Let \(X(\mathbb R^n)\) be a rearrangement-invariant Banach function space, preferably a Lorentz-Zygmund space or, more generally, a Lorentz-Karamata space. Then the spaces \[ H^\sigma X(\mathbb R^n) = \{ u: u=f *g_\sigma, \;f\in X(\mathbb R^n) \} \] generalize the well-known classical Bessel-potential spaces \(H^\sigma_p (\mathbb R^n)\), \(1<p<\infty\), including the classical Sobolev spaces. It is the main aim of the paper to study embeddings of type \[ H^\sigma X(\mathbb R^n) \subset Y(\Omega), \] where \(Y(\Omega)\) is a suitable Banach space on the set \(\Omega\) in \(\mathbb R^n\). Here, \(Y(\Omega)\) are preferably generalized Hölder spaces. The authors provide necessary and sufficient conditions for continuous embeddings. They study their compactness.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5566982
zbMATH Keywords: Bessel-potential spaces
zbMATH Keywords: Lorentz-Zygmund spaces
zbMATH Keywords: Lorentz-Karamata spaces
zbMATH Keywords: limiting embeddings
MaRDI profile type: Publication
OpenAlex ID: W2079392085
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DOI: 10.1007/S00209-008-0395-5
description: scientific article; zbMATH DE number 5566982