Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by \(L^{p}\)-modulus of smoothness
type: Item
instance of: scholarly article
title: Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by \(L^{p}\)-modulus of smoothness
zbMATH Open document ID: 1407.26021
author: Amiran Gogatishvili
author: Júlio S. Neves
author: Bohumír Opic
published in: Journal of Functional Analysis
publication date: 2018-12-04T00:00:00Z
review text: In the last years, a lot of attention has been devoted to embeddings of Sobolev-type spaces \(W^kX(\Omega)\), where \(X(\Omega)\) is a rearrangement invariant Banach function space, \(\Omega\) is a domain in \(\mathbb{R}^n\), and \(k \in \mathbb{N}\). Methods based on rearrangement technics, which reduce Sobolev-type embeddings into rearrangement invariant Banach function spaces to one dimensional Hardy-type inequalities, have been extensively studied. Embeddings of Sobolev-type spaces \(W^kX(\Omega)\) with Hölder-type spaces as targets have been also intensively investigated. In [Ann. Mat. Pura Appl. (4) 194, No. 2, 425--450 (2015; Zbl 1318.26045)], the authors proved the reverse form of an estimate of the \(k\)-modulus of smoothness of a function \(f\) such that the norm of its distributional gradient \(| \nabla^kf|\) belongs locally to the Lorentz space \(L^{n/k,1}(\mathbb{R}^n)\), \(k \in \mathbb{N}\), \(k \leq n\). This helped to establish necessary and sufficient conditions for continuous embeddings of Sobolev type spaces, modelled upon rearrangement invariant Banach function spaces \(X(\mathbb{R}^n)\), into Hölder spaces defined by means of the \(k\)-modulus of continuity (\(L^\infty\)-modulus). A sharp estimate involving the \(L^p\)-modulus of smoothness of order \(k\), with \(p < +\infty\), was not available. The aim of this paper is to derive such a sharp estimate and use it to establish necessary and sufficient conditions for continuous embeddings of Sobolev spaces \(W^kX(\Omega)\), modelled upon rearrangement invariant Banach function spaces \(X(\Omega)\), into generalized Hölder spaces defined by means of the \(L^p\)-modulus of smoothness of order \(k\;(k \in \mathbb{N})\). The general results are illustrated with examples. In particular, a consequence is a generalization of the classical Jawerth embeddings.
reviewed by: Ioan Raşa
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 6988451
zbMATH Keywords: rearrangement-invariant Banach function spaces
zbMATH Keywords: Sobolev-type and Hölder-type spaces
zbMATH Keywords: Hardy-type operators
zbMATH Keywords: embeddings
MaRDI profile type: Publication
full work available at URL: j.jfa.2018.10.023
OpenAlex ID: W2900318900
cites work: Q3905932
cites work: Q4039801
cites work: Q4726487
cites work: Weighted $L_{Φ}$ integral inequalities for operators of Hardy type
cites work: Local growth envelopes of Besov spaces of generalized smoothness
cites work: Weak-type weights and normable Lorentz spaces
cites work: A sharp embedding theorem for Orlicz-Sobolev spaces
cites work: Optimal Orlicz-Sobolev embeddings
cites work: Sobolev inequalities in arbitrary domains
cites work: Higher-order Sobolev embeddings and isoperimetric inequalities
cites work: Q4273944
cites work: Interpolation of linear operators on Sobolev spaces
cites work: Besov Spaces on Domains in ℝ d
cites work: Weighted modular inequalities for monotone functions
cites work: Q4821146
cites work: On embeddings of logarithmic Bessel potential spaces
cites work: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms
cites work: Real Interpolation with Logarithmic Functors and Reiteration
cites work: Embeddings of Sobolev-type spaces into generalized Hölder spaces involving \(k\)-modulus of smoothness
cites work: Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces involving \(k\)-modulus of smoothness
cites work: Limiting reiteration for real interpolation with slowly varying functions
cites work: Interpolation orbits and optimal Sobolev's embeddings
cites work: Reduction theorems for operators on the cones of monotone functions
cites work: Reduction theorems for weighted integral inequalities on the cone of monotone functions
cites work: Estimates for continuity envelopes and approximation numbers of Bessel potentials
cites work: Sharp embeddings of Besov-type spaces
cites work: Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers
cites work: Q4953822
cites work: Interpolation of Quasi-Normed Spaces.
cites work: On embedding theorems.
cites work: Regular variation and differential equations
cites work: On some characterizations of Besov spaces of generalized smoothness
cites work: Lorentz–Karamata spaces, Bessel and Riesz potentials and embeddings
cites work: Reiteration theorems for real interpolation and approximation spaces
cites work: On generalized Lorentz-Zygmund spaces
cites work: Boundedness of classical operators on classical Lorentz spaces
cites work: Q4397396
cites work: Q5607484
cites work: The modular inequalities for a class of convolution operators on monotone functions
cites work: Q3336200
DOI: 10.1016/J.JFA.2018.10.023
description: scientific article; zbMATH DE number 6988451