Sum-intersection property of Sobolev spaces
type: Item
instance of: scholarly article
title: Sum-intersection property of Sobolev spaces
zbMATH Open document ID: 1416.46038
DOI: 10.1007/978-3-319-89800-1_8
author: Petru Mironescu
publication date: 2019-01-24T00:00:00Z
review text: Let $1\le p \le \infty$. If $s \ge 0$ is an integer, then $W^s_p = W^s_p (\mathbb{R}^n)$ are the classical Sobolev spaces. If $s>0$ is not an integer, then $W^s_p = W^s_p (\mathbb{R}^n) = B^s_{p,p} (\mathbb{R}^n)$ are the special Besov or Slobodeckij spaces. A triple $(W^{s_1}_{p_1}, W^s_p, W^{s_2}_{p_2})$ is called admissible if \[ s= \theta s_1 + (1-\theta) s_2, \qquad \frac{1}{p} = \frac{\theta}{p_1} + \frac{1-\theta}{p_2}, \] for some $0 \le \theta \le 1$. It is the main aim of the paper to discuss for which admissible triples any $f\in W^s_p$ can be decomposed as \[ f= f_1 + f_2, \qquad f_j \in W^{s_j}_{p_j} \cap W^s_p, \quad j=1,2. \] This is mainly the case (regular triples, Theorem 1). Exceptional limiting cases, called irregular triples, are discussed in detail (Theorem 2) where the nowadays fashionable (though existing for some 40 years) spaces $F^s_{\infty,q}$ occur (with $bmo =F^0_{\infty,2}$). For the entire collection see [Zbl 1403.46004].
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 7006525
zbMATH Keywords: Sobolev spaces
MaRDI profile type: Publication
full work available at URL: 978-3-319-89800-1 8
OpenAlex ID: W2770443392
description: scientific article; zbMATH DE number 7006525