Inequalities for Differential Forms (eBook)

Ravi P. Agarwal, Shusen Ding, Craig Nolder (Autoren)

eBook Download: PDF

2009 | 2010
XVI, 388 Seiten
Springer New York (Verlag)
978-0-387-68417-8 (ISBN)

Infos

This monograph is the first one to systematically present a series of local and global estimates and inequalities for differential forms, in particular the ones that satisfy the A-harmonic equations. The presentation focuses on the Hardy-Littlewood, Poincare, Cacciooli, imbedded and reverse Holder inequalities.  Integral estimates for operators, such as homotopy operator, the Laplace-Beltrami operator, and the gradient operator are discussed next.  Additionally, some related topics such as BMO inequalities, Lipschitz classes, Orlicz spaces and inequalities in Carnot groups are discussed in the concluding chapter. An abundance of bibliographical references and historical material supplement the text throughout.

This rigorous presentation requires a familiarity with topics such as differential forms, topology and Sobolev space theory. It will serve as an invaluable reference for researchers, instructors and graduate students in analysis and partial differential equations and could be used as additional material for specific courses in these fields.

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Differential forms satisfying the A-harmonic equations have found wide applications in fields such as general relativity, theory of elasticity, quasiconformal analysis, differential geometry, and nonlinear differential equations in domains on manifolds.This monograph is the first one to systematically present a series of local and global estimates and inequalities for such differential forms in particular. It concentrates on the Hardy-Littlewood, Poincare, Cacciooli, imbedded and reverse Holder inequalities. Integral estimates for operators, such as homotopy operator, the Laplace-Beltrami operator, and the gradient operator are also presented. Additionally, some related topics such as BMO inequalities, Lipschitz classes, Orlicz spaces and inequalities in Carnot groups are discussed in the concluding chapter. An abundance of bibliographical references and historical material supplement the text throughout.This book will serve as an invaluable reference for researchers, instructors and graduate students in analysis and partial differential equations and could be used as additional material for specific courses in these fields.

Inhalt

Preface 6
Contents 9
Hardy–Littlewood inequalities 15
1.1 Differential forms 15
1.2 A-harmonic equations 22
1.3 p-Harmonic equations 28
1.4 Some weight classes 35
1.5 Inequalities in John domains 43
1.6 Inequalities in averaging domains 51
1.7 Two-weight cases 57
1.8 The best integrable condition 59
1.9 Inequalities with Orlicz norms 61
Norm comparison theorems 71
2.1 Introduction 71
2.2 The local unweighted estimates 72
2.3 The local weighted estimates 75
2.4 The global estimates 81
2.5 Applications 86
Poincare-type inequalities 88
3.1 Introduction 88
3.2 Inequalities for differential forms 88
3.3 Inequalities for Green’s operator 99
3.4 Inequalities with Orlicz norms 105
3.5 Two-weight inequalities 113
3.6 Inequalities for Jacobians 120
3.7 Inequalities for the projection operator 124
3.8 Other Poincare-type inequalities 129
Caccioppoli inequalities 131
4.1 Preliminary results 131
4.2 Local and global weighted cases 132
4.3 Local and global two-weight cases 139
4.4 Inequalities with Orlicz norms 145
4.5 Inequalities with the codifferential operator 152
Imbedding theorems 156
5.1 Introduction 156
5.2 Quasiconformal mappings 156
5.3 Solutions to the nonhomogeneous equation 157
5.4 Imbedding inequalities for operators 158
5.5 Other weighted cases 165
5.6 Compositions of operators 176
5.7 Two-weight cases 183
Reverse Hölder inequalities 197
6.1 Preliminaries 197
6.2 The first weighted case 199
6.3 The second weighted case 211
6.4 The third weighted case 219
6.5 Two-weight inequalities 224
6.6 Inequalities with Orlicz norms 229
Inequalities for operators 234
7.1 Introduction 234
7.2 Some basic estimates 235
7.3 Compositions of operators 245
7.4 Poincare-type inequalities for operators 265
7.5 The homotopy operator 290
7.6 Homotopy and projection operators 297
7.7 Compositions of three operators 313
7.8 The maximal operators 322
7.9 Singular integrals 327
Estimates for Jacobians 331
8.1 Introduction 331
8.2 Global integrability 332
Lipschitz and BMO norms 346
9.1 Introduction 346
9.2 BMO spaces and Lipschitz classes 347
9.3 Global integrability 351
9.4 Lipschitz and BMO norms 353
References 375
Index 390

Produkt-Details

• Erscheint lt. Verlag: 19.9.2009
• Zusatzinfo: XVI, 387 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Schlagworte: a-harmonic equations • Differential Equations • Differential Geometry • General relativity • manifold • partial differential equation • Partial differential equations • quasiconformal analysis • Sobolev Space • Theory of elasticity
• ISBN-10: 0-387-68417-4 / 0387684174
• ISBN-13: 978-0-387-68417-8 / 9780387684178