Agarwal / Ding / Nolder Inequalities for Differential Forms

1. Auflage 2009
ISBN: 978-0-387-68417-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 387 Seiten

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.

Autoren/Hrsg.

Agarwal, Ravi P.

Ding, Shusen

Nolder, Craig

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;Hardy–Littlewood inequalities;15
3.1;1.1 Differential forms;15
3.2;1.2 A-harmonic equations;22
3.3;1.3 p-Harmonic equations;28
3.4;1.4 Some weight classes;35
3.5;1.5 Inequalities in John domains;43
3.6;1.6 Inequalities in averaging domains;51
3.7;1.7 Two-weight cases;57
3.8;1.8 The best integrable condition;59
3.9;1.9 Inequalities with Orlicz norms;61
4;Norm comparison theorems;71
4.1;2.1 Introduction;71
4.2;2.2 The local unweighted estimates;72
4.3;2.3 The local weighted estimates;75
4.4;2.4 The global estimates;81
4.5;2.5 Applications;86
5;Poincare-type inequalities;88
5.1;3.1 Introduction;88
5.2;3.2 Inequalities for differential forms;88
5.3;3.3 Inequalities for Green’s operator;99
5.4;3.4 Inequalities with Orlicz norms;105
5.5;3.5 Two-weight inequalities;113
5.6;3.6 Inequalities for Jacobians;120
5.7;3.7 Inequalities for the projection operator;124
5.8;3.8 Other Poincare-type inequalities;129
6;Caccioppoli inequalities;131
6.1;4.1 Preliminary results;131
6.2;4.2 Local and global weighted cases;132
6.3;4.3 Local and global two-weight cases;139
6.4;4.4 Inequalities with Orlicz norms;145
6.5;4.5 Inequalities with the codifferential operator;152
7;Imbedding theorems;156
7.1;5.1 Introduction;156
7.2;5.2 Quasiconformal mappings;156
7.3;5.3 Solutions to the nonhomogeneous equation;157
7.4;5.4 Imbedding inequalities for operators;158
7.5;5.5 Other weighted cases;165
7.6;5.6 Compositions of operators;176
7.7;5.7 Two-weight cases;183
8;Reverse Hölder inequalities;197
8.1;6.1 Preliminaries;197
8.2;6.2 The first weighted case;199
8.3;6.3 The second weighted case;211
8.4;6.4 The third weighted case;219
8.5;6.5 Two-weight inequalities;224
8.6;6.6 Inequalities with Orlicz norms;229
9;Inequalities for operators;234
9.1;7.1 Introduction;234
9.2;7.2 Some basic estimates;235
9.3;7.3 Compositions of operators;245
9.4;7.4 Poincare-type inequalities for operators;265
9.5;7.5 The homotopy operator;290
9.6;7.6 Homotopy and projection operators;297
9.7;7.7 Compositions of three operators;313
9.8;7.8 The maximal operators;322
9.9;7.9 Singular integrals;327
10;Estimates for Jacobians;331
10.1;8.1 Introduction;331
10.2;8.2 Global integrability;332
11;Lipschitz and BMO norms;346
11.1;9.1 Introduction;346
11.2;9.2 BMO spaces and Lipschitz classes;347
11.3;9.3 Global integrability;351
11.4;9.4 Lipschitz and BMO norms;353
12;References;375
13;Index;390