The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations
type: Item
instance of: scholarly article
title: The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations
zbMATH Open document ID: 0972.46017
DOI: 10.1007/BF02511543
author: Marius Mitrea
author: Osvaldo Mendez
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-11-02T00:00:00Z
full work available at URL: https://eudml.org/doc/59657
review text: Let \(X=B_p^{s,q}(\mathbb{R}^n)\) or \(X= F_p^{s,q}(\mathbb{R}^n)\) be the Besov and Triebel-Lizorkin space, respectively. The authors deal with the case \(0<p<1\) or \(0<q<1\), when these spaces are quasi-Banach. Let \(\widehat{X}\) denote the envelope of the quasi-Banach space \(X\). The authors show that \[ \widehat{B_p^{s,q}(\mathbb{R}^n)}=\widehat{F_p^{s,q}(\mathbb{R}^n)}=B_1^{s-n\left(\frac{1}{p}-1\right), 1}(\mathbb{R}^n) \] if both \(0<p<1\) and \(0<q<1\), \[ \widehat{B_p^{s,q}(\mathbb{R}^n)}=B_1^{s-n\left(\frac{1}{p}-1\right), q}\quad \text{and} \quad \widehat{F_p^{s,q}(\mathbb{R}^n)}=B_1^{s-n\left(\frac{1}{p}-1\right), 1}(\mathbb{R}^n), \] if \(0<p<1\) and \(1\leq q < \infty\), while \[ \widehat{B_p^{s,q}(\mathbb{R}^n)}=B_p^{s, 1}\quad \text{and} \quad \widehat{F_p^{s,q}(\mathbb{R}^n)}=F_p^{s, 1}(\mathbb{R}^n), \] when \(1\leq p < \infty\) and \(0<q<1\). Some applications of these results to PDEs in non-smooth domains are also given.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 46A16
zbMATH DE Number: 1539362
zbMATH Keywords: Triebel-Lizorkin spaces
zbMATH Keywords: Banach envelope
zbMATH Keywords: quasi-Banach space
zbMATH Keywords: Lipschitz domains
zbMATH Keywords: layer potentials
zbMATH Keywords: boundary problems
reviewed by: S. G. Samko
MaRDI profile type: Publication
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OpenAlex ID: W2120678680
description: scientific article; zbMATH DE number 1539362