Embeddings, Hardy operators and nonlinear problems
type: Item
instance of: scholarly article
title: Embeddings, Hardy operators and nonlinear problems
zbMATH Open document ID: 1208.46031
published in: Revista Matemática Complutense
publication date: 2010-10-04T00:00:00Z
review text: The author presents a survey on some recent developments involving embeddings between function spaces. He focuses first on variations of Sobolev inequalities analysing best constants and extremal functions using the technique of mass transportation which originated in the work of Brenier. The Sobolev inequality is also studied in the context of Banach function spaces covering Orlicz spaces and Lorentz spaces. Also formulations using Triebel-Lizorkin and Besov spaces and Lebesgue type with variable exponent are pointed out. A basic tool consists in reducing several embeddings to questions concerning the Hardy operator which is well understood in the setting of Banach function spaces. Also, considerations about entropy numbers and approximation numbers of compact operators are considered in the paper in terms of quantifying the compactness. These notions are used when analysing a nonlinear eigenvalue problem which leads to a series representation of compact operators acting not necessarily between Hilbert spaces but between Banach spaces with certain geometric properties. The Dirichlet problem for the \(p\)-Laplacian is also studied using some embedding maps.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 46-02
zbMATH DE Number: 5795229
zbMATH Keywords: Sobolev inequalities
zbMATH Keywords: approximation numbers
zbMATH Keywords: eigenvalues problems
zbMATH Keywords: Hardy operator
author: D. E. Edmunds
MaRDI profile type: Publication
full work available at URL: s13163-009-0016-9
OpenAlex ID: W1995947728
cites work: Regularity results for stationary electro-rheological fluids
cites work: An improved Hardy-Sobolev inequality and its application
cites work: Q4222737
cites work: Problèmes isoperimetriques et espaces de Sobolev
cites work: Weak-\(L^\infty\) and BMO
cites work: Q4039801
cites work: Approximation numbers = singular values
cites work: Basis properties of eigenfunctions of the 𝑝-Laplacian
cites work: PIECEWISE-POLYNOMIAL APPROXIMATIONS OF FUNCTIONS OF THE CLASSES $ W_{p}^{\alpha}$
cites work: Limiting embedding theorems for \(W^{s,p}\) when \(s\uparrow 1\) and applications
cites work: Q3159214
cites work: New volume ratio properties for convex symmetric bodies in \({\mathbb{R}}^ n\)
cites work: Q3834561
cites work: Polar factorization and monotone rearrangement of vector‐valued functions
cites work: Sobolev inequalities with remainder terms
cites work: Q4538008
cites work: Extremal functions for Hardy's inequality with weight
cites work: Remarks on generalized trigonometric functions
cites work: About approximation numbers in function spaces
cites work: Coverings of convex sets and volume inequalities
cites work: New covering numbers and spectral properties of operators in Banach spaces
cites work: A sharp embedding theorem for Orlicz-Sobolev spaces
cites work: Optimal Orlicz-Sobolev embeddings
cites work: Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles
cites work: The sharp Sobolev inequality in quantitative form
cites work: Q4700511
cites work: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities.
cites work: Maximal function on generalized Lebesgue spaces L^p(⋅)
cites work: On the closed solution to some nonhomogeneous eigenvalue problem with \(p\)-Laplacian
cites work: Q3764747
cites work: Q4821146
cites work: Q4347439
cites work: Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems
cites work: A spectral analysis of compact linear operators in Banach spaces
cites work: Sharp estimates of the embedding constants for Besov spaces
cites work: Sharp estimates of the embedding constants for Besov spaces \(b^s_{p,q}\), \(0<p<1\)
cites work: On a measure of non-compactness for some classical operators
cites work: Entropy numbers of limiting embeddings of logarithmic Sobolev spaces into exponential spaces
cites work: On embeddings of logarithmic Bessel potential spaces
cites work: Q4325008
cites work: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms
cites work: Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case.
cites work: Approximation numbers and Kolmogorov widths of Hardy‐type operators in a non‐homogeneous case
cites work: Bernstein widths of Hardy-type operators in a non-homogeneous case
cites work: The 𝔧‐eigenfunctions and s ‐numbers
cites work: OnLp(x)norms
cites work: Some \(s\)-numbers of an integral operator of Hardy type on \(L^{p(\cdot)}\) spaces
cites work: Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces
cites work: Sobolev embeddings with variable exponent
cites work: Q4797974
cites work: Q5690839
cites work: On the Entropy Numbers of an Operator and its Adjoint
cites work: Q3733431
cites work: Sobolev Embeddings for Generalized Ridged Domains
cites work: ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS FOR IRREGULAR DOMAINS
cites work: The approximation numbers of Hardy-type operators on trees
cites work: On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\)

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Sharp Hardy-Sobolev inequalities

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On a question of Brezis and Marcus

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Critical Hardy--Sobolev inequalities

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4440308

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Nonlinear ground state representations and sharp Hardy inequalities

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Asymptotic behaviour of the eigenvalues of the φ—laplacian

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Sobolev imbedding theorems in borderline cases

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q3778373

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4732810

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Sobolev inequalities with variable exponent attaining the values 1 and \(n\)

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Approximation numbers in some weighted function spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5494293

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On the sharpness of a limiting case of the Sobolev imbedding theorem

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Measures of non–compactness of classical embeddings of Sobolev spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: A sharp form of an embedding into exponential and double exponential spaces.

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5509729

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4120995

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Optimal Sobolev imbeddings

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On limiting embeddings of Besov spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Eigenvalue distribution of compact operators

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: COMPACT EMBEDDINGS OF BESOV SPACES IN EXPONENTIAL ORLICZ SPACES

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5290626

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5448228

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Improved estimates for the approximation numbers of Hardy-type operators.

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4821515

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4188284

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: ON A METHOD FOR ESTIMATION FROM BELOW OF DIAMETERS OF SETS IN BANACH SPACES

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: An elementary proof of sharp Sobolev embeddings

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Existence and uniqueness of monotone measure-preserving maps

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: A convexity principle for interacting gases

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: ON SHARP HIGHER ORDER SOBOLEV EMBEDDINGS

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5614914

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Orlicz spaces and modular spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Espaces d'interpolation et théorème de Soboleff

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Optimality of Function Spaces in Sobolev Embeddings

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q3873048

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4722695

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: History of Banach Spaces and Linear Operators

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Bad properties of the Bernstein numbers

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5687568

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On dilation operators in Besov spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On approximation numbers of Sobolev embeddings of weighted function spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Best constant in Sobolev inequality

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4169289

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q3336200

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4003552

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Approximation Numbers and Entropy Numbers of Embeddings of Fractional Besov-sobolev Spaces in Orlicz Spaces

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q4357054

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q2731784

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5481429

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Function spaces and wavelets on domains

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5547919

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Optimal Transport

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5544663

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: Q5520312

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: AVERAGING OF FUNCTIONALS OF THE CALCULUS OF VARIATIONS AND ELASTICITY THEORY

Embeddings, Hardy operators and nonlinear problems (cites work)
cites work: On some variational problems

Embeddings, Hardy operators and nonlinear problems (DOI)
DOI: 10.1007/S13163-009-0016-9