Sobolev spaces on metric measure spaces: an approach based on upper gradients
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| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2015
|
| Series: | New mathematical monographs
27 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027757369&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Literaturverzeichnis Seite 412-426 |
| Physical Description: | xii, 434 Seiten Illustrationen |
| ISBN: | 9781107092341 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Sobolev spaces on metric measure spaces
Autor: Heinonen, Juha
Jahr: 2015
Contents
Preface page xi
1 Introduction 1
2 Review of basic functional analysis 7
2.1 Normed and seminormed spaces 7
2.2 Linear Operators and dual spaces 13
2.3 Convergence theorems 16
2.4 Reflexive spaces 24
2.5 Notes to Chapter 2 34
3 Lebesgue theory of Banach space-valued functions 36
3.1 Measurability for Banach space-valued functions 36
3.2 Integrable functions and the spaces LP(X : V) 42
3.3 Metrie measure spaces 49
3.4 Differentiation 73
3.5 Maximal functions 90
3.6 Notes to Chapter 3 96
4 Lipschitz functions and embeddings 98
4.1 Lipschitz functions, extensions, and embeddings 98
4.2 Lower semicontinuous functions 107
4.3 Hausdorff measures 110
4.4 Functions with bounded Variation 111
4.5 Notes to Chapter 4 119
5 Path integrals and modulus 121
5.1 Curves in metric spaces 121
5.2 Modulus of a curve family 127
5.3 Estimates for modulus 134
5.4 Notes to Chapter 5 141
vii
viii
Contents
6 Upper gradients 143
6.1 Classical first-order Sobolev spaces 143
6.2 Upper gradients 151
6.3 Maps with p-integrable upper gradients 156
6.4 Notes to Chapter 6 166
7 Sobolev spaces 167
7.1 Vector-valued Sobolev functions on metric spaces 167
7.2 The Sobolev p-capacity 183
7.3 N]-p(X : V) is a Banach space 190
7.4 The space HN1,p(X : V ) and quasicontinuity 198
7.5 Main equivalence classes and the MECP property 201
7.6 Notes to Chapter 7 203
8 Poincare inequalities 205
8.1 Poincare inequality and pointwise inequalities 205
8.2 Density of Lipschitz functions 228
8.3 Quasiconvexity and the Poincare inequality 233
8.4 Continuous upper gradients and pointwise
Lipschitz constants 238
8.5 Notes to Chapter 8 243
9 Consequences of Poincare inequalities 245
9.1 Sobolev-Poincare inequalities 245
9.2 Lebesgue points of Sobolev functions 261
9.3 Measurability of equivalence classes and MECP 271
9.4 Annular quasiconvexity 279
9.5 Notes to Chapter 9 282
10 Other definitions of Sobolev-type spaces 285
10.1 The Cheeger-Sobolev space 285
10.2 The Hajlasz-Sobolev space 286
10.3 Sobolev spaces defined via Poincare inequalities 290
10.4 The Korevaar-Schoen-Sobolev space 294
10.5 Summary 304
10.6 Notes to Chapter 10 304
11 Gromov-Hausdorff convergence and Poincare inequalities 306
11.1 The Gromov-Hausdorff distance 306
11.2 Gromov s compactness theorem 312
11.3 Pointed Gromov-Hausdorff convergence 315
11.4 Pointed measured Gromov-Hausdorff convergence 324
Contents ix
11.5 Persistence of doubling measures under
Gromov-Hausdorff convergence 327
1 1 .6 Persistence of Poincare inequaiities under
Gromov-Hausdorff convergence 330
11.7 Notes to Chapter 11 335
12 Self-improvement of Poincare inequaiities 337
12.1 Geometrie properties of geodesic doubling metric
measure Spaces 337
12.2 Preliminary local arguments 340
12.3 Self-improvement of the Poincare inequality 355
12.4 Notes to Chapter 12 363
13 An introduetion to Cheeger s differentiation theory 364
13.1 Asymptotic generalized linearity 364
13.2 Caccioppoli-type estimates 369
13.3 Minimal weak upper gradients of distance funetions
are nontrivial 371
13.4 The differential strueture 373
13.5 Comparisons between pu and Lip u, Taylor s theorem,
and the reflexivity of Nl p(X) 377
13.6 Notes to Chapter 13 385
14 Examples, applications, and further
research directions 387
14.1 Quasiconformal and quasisymmetric mappings 387
14.2 Spaces supporting a Poincare inequality 392
14.3 Applications and further research directions 407
References 412
Notation index 427
Subject index 429
|
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| author | Heinonen, Juha 1960-2007 Koskela, Pekka Shanmugalingam, Nageswari Tyson, Jeremy T. 1972- |
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| id | DE-604.BV042320426 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T17:07:56Z |
| institution | BVB |
| isbn | 9781107092341 |
| language | English |
| lccn | 014027794 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027757369 |
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| physical | xii, 434 Seiten Illustrationen |
| publishDate | 2015 |
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| publisher | Cambridge University Press |
| record_format | marc |
| series | New mathematical monographs |
| series2 | New mathematical monographs |
| spellingShingle | Heinonen, Juha 1960-2007 Koskela, Pekka Shanmugalingam, Nageswari Tyson, Jeremy T. 1972- Sobolev spaces on metric measure spaces an approach based on upper gradients New mathematical monographs Metric spaces Sobolev spaces Sobolev-Raum (DE-588)4055345-0 gnd Metrischer Raum (DE-588)4169745-5 gnd |
| subject_GND | (DE-588)4055345-0 (DE-588)4169745-5 |
| title | Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_auth | Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_exact_search | Sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_full | Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Late, University of Michigan; Pekka Koskela, University of Jyväskylä, Finland; Nageswari Shanmugalingam, University of Cincinnati; Jeremy T. Tyson, University of Illinois at Urbana-Champaign |
| title_fullStr | Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Late, University of Michigan; Pekka Koskela, University of Jyväskylä, Finland; Nageswari Shanmugalingam, University of Cincinnati; Jeremy T. Tyson, University of Illinois at Urbana-Champaign |
| title_full_unstemmed | Sobolev spaces on metric measure spaces an approach based on upper gradients Juha Heinonen, Late, University of Michigan; Pekka Koskela, University of Jyväskylä, Finland; Nageswari Shanmugalingam, University of Cincinnati; Jeremy T. Tyson, University of Illinois at Urbana-Champaign |
| title_short | Sobolev spaces on metric measure spaces |
| title_sort | sobolev spaces on metric measure spaces an approach based on upper gradients |
| title_sub | an approach based on upper gradients |
| topic | Metric spaces Sobolev spaces Sobolev-Raum (DE-588)4055345-0 gnd Metrischer Raum (DE-588)4169745-5 gnd |
| topic_facet | Metric spaces Sobolev spaces Sobolev-Raum Metrischer Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027757369&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035420183 |
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