Lebesgue and Sobolev spaces with variable exponents / Lars Diening ... [and others].

Format:

Book

Contributor:

Diening, Lars.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2017.

Lecture notes in mathematics ; 2017

Language:

English

Subjects (All):

Function spaces.

Physical Description:

ix, 509 pages : illustrations ; 23 cm.

Place of Publication:

Berlin ; Heidelberg ; New York : Springer, [2011]

Summary:

The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much needed accessible reference work utilizing consistent notation and terminology. Many results are provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics. Book jacket.

Contents:

1 Introduction 1

1.1 History of Variable Exponent Spaces 2

1.2 Structure of the Book 4

1.3 Summary of Central Results 7

1.4 Notation and Background 10

Part I Lebesgue Spaces

2 A Framework for Function Spaces 21

2.1 Basic Properties of Semimodular Spaces 21

2.2 Conjugate Modulars and Dual Semimodular Spaces 29

2.3 Musielak-Orlicz Spaces: Basic Properties 34

2.4 Uniform Convexity 42

2.5 Separability 48

2.6 Conjugate Φ-Functions 52

2.7 Associate Spaces and Dual Spaces 57

2.8 Embeddings and Operators 66

3 Variable Exponent Lebesgue Spaces 69

3.1 The Lebesgue Space Φ-Function 69

3.2 Basic Properties 73

3.3 Embeddings 82

3.4 Properties for Restricted Exponents 87

3.5 Limit of Exponents 92

3.6 Convolution* 94

4 The Maximal Operator 99

4.1 Logarithmic Hölder Continuity 100

4.2 Point-Wise Estimates 104

4.3 The Boundedness of the Maximal Operator 110

4.4 Weak-Type Estimates and Averaging Operators 115

4.5 Norms of Characteristic Functions 122

4.6 Mollification and Convolution 127

4.7 Necessary Conditions for Boundedness* 131

4.8 Preimage of the Maximal Operator* 138

5 The Generalized Muckenhoupt Condition* 143

5.1 Non Sufficiency of log-Hölder Continuity* 143

5.2 Class A* 149

5.3 Class A for Variable Exponent Lebesgue Spaces* 157

5.4 Class A∞* 159

5.5 A Sufficient Condition for the Boundedness of M* 168

5.6 Characterization of (Strong-)Domination* 175

5.7 The Case of Lebesgue Spaces with Variable Exponents* 180

5.8 Weighted Variable Exponent Lebesgue Spaces* 192

6 Classical Operators 199

6.1 Riesz Potentials 199

6.2 The Sharp Operator M♯f 206

6.3 Calderón-Zygmund Operators 208

7 Transfer Techniques 213

7.1 Complex Interpolation 213

7.2 Extrapolation Results 218

7.3 Local-to-Global Results 222

7.4 Ball/Cubes-to-John 237

Part II Sobolev Spaces

8 Introduction to Sobolev Spaces 247

8.1 Basic Properties 247

8.2 Poincaré Inequalities 252

8.3 Sobolev-Poincaré Inequalities and Embeddings 265

8.4 Compact Embeddings 272

8.5 Extension Operator 275

8.6 Limiting Cases of Sobolev Embeddings* 283

9 Density of Regular Functions 289

9:1 Basic Results on Density 290

9.2 Density with Continuous Exponents 293

9.3 Density with Discontinuous Exponents* 299

9.4 Density of Continuous Functions* 305

9.5 The Lipschitz Truncation Method* 310

10 Capacities 315

10.1 Sobolev Capacity 315

10.2 Relative Capacity 322

10.3 The Relationship Between the Capacities 331

10.4 Soboley Capacity and Hausdorff Measure 334

11 Fine Properties of Sobolev Functions 339

11.1 Quasicontinuity 339

11.2 Sobolev Spaces with Zero Boundary Values 345

11.3 Exceptional Sets in Variable Exponent Sobolev Spaces 350

11.4 Lebesgue Points 352

11.5 Failure of Existence of Lebesgue Points* 360

12 Other Spaces of Differentiable Functions 367

12.1 Trace Spaces 368

12.2 Homogeneous Sobolev Spaces 378

12.3 Sobolev Spaces with Negative Smoothness 383

12.4 Bessel Potential Spaces* 388

12.5 Besov and Triebel-Lizorkin Spaces* 392

Part III Applications to Partial Differential Equations

13 Dirichlet Energy Integral and Laplace Equation 401

13.1 The One Dimensional Case 402

13.2 Minimizers 412

13.3 Harmonic and Superharmonic Functions 417

13.4 Harnack's Inequality for A-harmonic Functions 421

14 PDEs and Fluid Dynamics 437

14.1 Poisson Problem 437

14.2 Stokes Problem 446

14.3 Divergence Equation and Consequences 459

14.4 Electrorheological Fluids 470.

Notes:

Includes bibliographical references (pages 483-499) and index.

ISBN:

9783642183621

364218362X

OCLC:

720430708