Sharp embeddings of Besov spaces involving only logarithmic smoothness
type: Item
instance of: scholarly article
title: Sharp embeddings of Besov spaces involving only logarithmic smoothness
zbMATH Open document ID: 1161.46017
author: António M. Caetano
author: Amiran Gogatishvili
author: Bohumír Opic
published in: Journal of Approximation Theory
publication date: 2008-07-31T00:00:00Z
review text: The paper contains sharp embedding assertions for spaces \(B^{0,\beta}_{p,r}(\mathbb{R}^n)\) of Besov type into spaces \(L^{\mathrm{loc}}_{p,q;\gamma}(\mathbb{R}^n)\) of Lorentz-Zygmund type. The Besov spaces are defined by means of the modulus of continuity \(\omega_1(f,t)_p\), \[ \| f| B^{0,\beta}_{p,r}(\mathbb{R}^n)\| = \| f| L_p(\mathbb{R}^n)\| + \left(\int_0^1 (1+| \ln t| )^{\beta r } \omega_1^r(f,t)_p \frac{dt}{t}\right)^{1/r}, \] whereas the Lorentz-Zygmund spaces \(L^{\mathrm{loc}}_{p,q;\gamma}(\mathbb{R}^n)\) collect all \(f\) with finite quasi-norm \[ \left(\int_0^1 t^{q/p} (1+| \ln t| )^{\gamma q } f^\ast(t)^q \frac{dt}{t}\right)^{1/q}, \] where \(1\leq p<\infty\), \(1\leq r,q\leq \infty\), \(\beta, \gamma\in\mathbb{R}\) with \(\beta+1/r>0\), and \(f^\ast\) denotes the non-increasing rearrangement, as usual. The essential point here is that the Besov spaces have zero classical smoothness and differ from their counterparts defined by Fourier analytical methods, say. Thus many of the standard approaches to prove optimality or sharpness of embeddings do not work and the essential benefit of this paper is to close this gap and, moreover, find new phenomena in connection with growth envelopes and limiting embeddings. The main results, listed in Section~3 of this interesting paper, contain criteria for the above embedding of the form that \(B^{0,\beta}_{p,r}(\mathbb{R}^n)\) is embedded into \(L^{\mathrm{loc}}_{p,q;\gamma}(\mathbb{R}^n)\) with \(\gamma=\beta+\frac1r + \frac{1}{\max(p,q)}-\frac1q\) if, and only if, \(q\geq r\) (Theorem~3.1). Furthermore, the growth envelope of the space \(B^{0,\beta}_{p,r}(\mathbb{R}^n)\) is determined by the pair \((t^{-\frac1p} (1+| \ln t| )^{-\beta-\frac1r}, \max(p,r))\). The proofs, presented in Section~5-9, rely on an inequality by \textit{V. I. Kolyada} [Math. USSR, Sb. 64, No. 1, 1--21 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 1(5), 3--23 (1988; Zbl 0693.46030)] and some inverse of it, stated as Propositio…
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 5306482
zbMATH Keywords: Besov spaces with generalized smoothness
zbMATH Keywords: Lorentz-Zygmund spaces
zbMATH Keywords: sharp embeddings
zbMATH Keywords: growth envelopes
reviewed by: Dorothee D. Haroske
MaRDI profile type: Publication
full work available at URL: j.jat.2007.12.003
OpenAlex ID: W2041410718
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DOI: 10.1016/J.JAT.2007.12.003
description: scientific article; zbMATH DE number 5306482