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E-raamat: Lebesgue and Sobolev Spaces with Variable Exponents

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  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2017
  • Ilmumisaeg: 29-Mar-2011
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642183638
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Lebesgue and Sobolev Spaces with Variable ExponentsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2017
  • Ilmumisaeg: 29-Mar-2011
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642183638
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This timely monograph collects all the basic properties of variable exponent Lebesgue and Sobolev spaces. It provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results have new and improved proofs.



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 also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.

Arvustused

From the reviews:

The authors provide a comprehensive survey of the state of the art concerning Lebesgue and Sobolev spaces with variable exponents. The book is also having a rich bibliography of 399 entries, a long list of symbols and an index. It will certainly become a standard reference in this field and stimulate further work in this direction. (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 165 (1), January, 2012)

The book is devoted to Lebesgue and Soboley spaces with variable exponents. The present book consists of the introduction and three parts. The majority of the results presented in the monograph were obtained by the authors and their collaborators. the books is a useful source of unified information on Lebesgue and Soboley spaces with variable exponents. (Alexei Yu. Karlovich, Zentralblatt MATH, Vol. 1222, 2011)

This book consists of three parts of different lengths and intentions, sub-divided into several chapters. There is a nice figure at the very beginning of the monograph explaining the dependencies among the chapters, together with some recommendations on which parts should be used for first reading or when teaching the subject in a graduate course. the presentation can thus also be considered as a textbook and extremely useful reference for graduate students and researchers working in related fields . (Dorothee D. Haroske, Mathematical Reviews, January, 2013)

1 Introduction
1(20)
1.1 History of Variable Exponent Spaces
2(2)
1.2 Structure of the Book
4(3)
1.3 Summary of Central Results
7(3)
1.4 Notation and Background
10(11)
Part I Lebesgue Spaces
2 A Framework for Function Spaces
21(48)
2.1 Basic Properties of Semimodular Spaces
21(8)
2.2 Conjugate Modulars and Dual Semimodular Spaces
29(5)
2.3 Musielak-Orlicz Spaces: Basic Properties
34(8)
2.4 Uniform Convexity
42(6)
2.5 Separability
48(4)
2.6 Conjugate Φ-Functions
52(5)
2.7 Associate Spaces and Dual Spaces
57(9)
2.8 Embeddings and Operators
66(3)
3 Variable Exponent Lebesgue Spaces
69(30)
3.1 The Lebesgue Space Φ-Function
69(4)
3.2 Basic Properties
73(9)
3.3 Embeddings
82(5)
3.4 Properties for Restricted Exponents
87(5)
3.5 Limit of Exponents
92(2)
3.6 Convolution
94(5)
4 The Maximal Operator
99(44)
4.1 Logarithmic Holder Continuity
100(4)
4.2 Point-Wise Estimates
104(6)
4.3 The Boundedness of the Maximal Operator
110(5)
4.4 Weak-Type Estimates and Averaging Operators
115(7)
4.5 Norms of Characteristic Functions
122(5)
4.6 Mollification and Convolution
127(4)
4.7 Necessary Conditions for Boundedness
131(7)
4.8 Preimage of the Maximal Operator
138(5)
5 The Generalized Muckenhoupt Condition
143(56)
5.1 Non Sufficiency of log-Holder Continuity
143(6)
5.2 Class A
149(8)
5.3 Class A for Variable Exponent Lebesgue Spaces
157(2)
5.4 Class A∞
159(9)
5.5 A Sufficient Condition for the Boundedness of M
168(7)
5.6 Characterization of (Strong-)Domination
175(5)
5.7 The Case of Lebesgue Spaces with Variable Exponents
180(12)
5.8 Weighted Variable Exponent Lebesgue Space
192(7)
6 Classical Operators
199(14)
6.1 Riesz Potentials
199(7)
6.2 The Sharp Operator M# ƒ
206(2)
6.3 Calderon-Zygmund Operators
208(5)
7 Transfer Techniques
213(34)
7.1 Complex Interpolation
213(5)
7.2 Extrapolation Results
218(4)
7.3 Local-to-Global Results
222(15)
7.4 Ball/Cubes-to-John
237(10)
Part II Sobolev Spaces
8 Introduction to Sobolev Spaces
247(42)
8.1 Basic Properties
247(5)
8.2 Poincare Inequalities
252(13)
8.3 Sobolev-Poincare Inequalities and Embeddings
265(7)
8.4 Compact Embeddings
272(3)
8.5 Extension Operator
275(8)
8.6 Limiting Cases of Sobolev Embeddings
283(6)
9 Density of Regular Functions
289(26)
9.1 Basic Results on Density
290(3)
9.2 Density with Continuous Exponents
293(6)
9.3 Density with Discontinuous Exponents
299(6)
9.4 Density of Continuous Functions
305(5)
9.5 The Lipschitz Truncation Method
310(5)
10 Capacities
315(24)
10.1 Sobolev Capacity
315(7)
10.2 Relative Capacity
322(9)
10.3 The Relationship Between the Capacities
331(3)
10.4 Sobolev Capacity and Hausdorff Measure
334(5)
11 Fine Properties of Sobolev Functions
339(28)
11.1 Quasicontinuity
339(6)
11.2 Sobolev Spaces with Zero Boundary Values
345(5)
11.3 Exceptional Sets in Variable Exponent Sobolev Spaces
350(2)
11.4 Lebesgue Points
352(8)
11.5 Failure of Existence of Lebesgue Points
360(7)
12 Other Spaces of Differentiable Functions
367(34)
12.1 Trace Spaces
368(10)
12.2 Homogeneous Sobolev Spaces
378(5)
12.3 Sobolev Spaces with Negative Smoothness
383(5)
12.4 Bessel Potential Spaces
388(4)
12.5 Besov and Triebel-Lizorkin Spaces
392(9)
Part III Applications to Partial Differential Equations
13 Dirichlet Energy Integral and Laplace Equation
401(36)
13.1 The One Dimensional Case
402(10)
13.2 Minimizers
412(5)
13.3 Harmonic and Superharmonic Functions
417(4)
13.4 Harnack's Inequality for A-harmonic Functions
421(16)
14 PDEs and Fluid Dynamics
437(46)
14.1 Poisson Problem
437(9)
14.2 Stokes Problem
446(13)
14.3 Divergence Equation and Consequences
459(11)
14.4 Electrorheological Fluids
470(13)
References 483(18)
List of Symbols 501(4)
Index 505
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