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E-raamat: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

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(University of Jyväskylä, Finland), (University of Cincinnati), (University of Illinois, Urbana-Champaign),
  • Formaat: PDF+DRM
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 05-Feb-2015
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316237250
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Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper GradientsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: New Mathematical Monographs
  • Ilmumisaeg: 05-Feb-2015
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316237250
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Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincaré inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincaré inequalities under GromovHausdorff convergence, and the KeithZhong self-improvement theorem for Poincaré inequalities.

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This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.
Preface xi
1 Introduction
1(6)
2 Review of basic functional analysis
7(29)
2.1 Normed and seminormed spaces
7(6)
2.2 Linear operators and dual spaces
13(3)
2.3 Convergence theorems
16(8)
2.4 Reflexive spaces
24(10)
2.5 Notes to
Chapter 2
34(2)
3 Lebesgue theory of Banach space-valued functions
36(62)
3.1 Measurability for Banach space-valued functions
36(6)
3.2 Integrable functions and the spaces Lp (X: V)
42(7)
3.3 Metric measure spaces
49(24)
3.4 Differentiation
73(17)
3.5 Maximal functions
90(6)
3.6 Notes to
Chapter 3
96(2)
4 Lipschitz functions and embeddings
98(23)
4.1 Lipschitz functions, extensions, and embeddings
98(9)
4.2 Lower semicontinuous functions
107(3)
4.3 Hausdorff measures
110(1)
4.4 Functions with bounded variation
111(8)
4.5 Notes to
Chapter 4
119(2)
5 Path integrals and modulus
121(22)
5.1 Curves in metric spaces
121(6)
5.2 Modulus of a curve family
127(7)
5.3 Estimates for modulus
134(7)
5.4 Notes to
Chapter 5
141(2)
6 Upper gradients
143(24)
6.1 Classical first-order Sobolev spaces
143(8)
6.2 Upper gradients
151(5)
6.3 Maps with p-integrable upper gradients
156(10)
6.4 Notes to
Chapter 6
166(1)
7 Sobolev spaces
167(38)
7.1 Vector-valued Sobolev functions on metric spaces
167(16)
7.2 The Sobolev p-capacity
183(7)
7.3 N1.p (X: V) is a Banach space
190(8)
7.4 The space HN1.p(X: V) and quasicontinuity
198(3)
7.5 Main equivalence classes and the MECp property
201(2)
7.6 Notes to
Chapter 7
203(2)
8 Poincare inequalities
205(40)
8.1 Poincare inequality and pointwise inequalities
205(23)
8.2 Density of Lipschitz functions
228(5)
8.3 Quasiconvexity and the Poincare inequality
233(5)
8.4 Continuous upper gradients and pointwise Lipschitz constants
238(5)
8.5 Notes to
Chapter 8
243(2)
9 Consequences of Poincare inequalities
245(40)
9.1 Sobolev--Poincare inequalities
245(16)
9.2 Lebesgue points of Sobolev functions
261(10)
9.3 Measurability of equivalence classes and MECp
271(8)
9.4 Annular quasiconvexity
279(3)
9.5 Notes to
Chapter 9
282(3)
10 Other definitions of Sobolev-type spaces
285(21)
10.1 The Cheeger--Sobolev space
285(1)
10.2 The Hajlasz--Sobolev space
286(4)
10.3 Sobolev spaces defined via Poincare inequalities
290(4)
10.4 The Korevaar--Schoen--Sobolev space
294(10)
10.5 Summary
304(1)
10.6 Notes to
Chapter 10
304(2)
11 Gromov--Hausdorff convergence and Poincare inequalities
306(31)
11.1 The Gromov--Hausdorff distance
306(6)
11.2 Gromov's compactness theorem
312(3)
11.3 Pointed Gromov--Hausdorff convergence
315(9)
11.4 Pointed measured Gromov--Hausdorff convergence
324(3)
11.5 Persistence of doubling measures under Gromov--Hausdorff convergence
327(3)
11.6 Persistence of Poincare inequalities under Gromov--Hausdorff convergence
330(5)
11.7 Notes to
Chapter 11
335(2)
12 Self-improvement of Poincare inequalities
337(27)
12.1 Geometric properties of geodesic doubling metric measure spaces
337(3)
12.2 Preliminary local arguments
340(15)
12.3 Self-improvement of the Poincare inequality
355(8)
12.4 Notes to
Chapter 12
363(1)
13 An introduction to Cheeger's differentiation theory
364(23)
13.1 Asymptotic generalized linearity
364(5)
13.2 Caccioppoli-type estimates
369(2)
13.3 Minimal weak upper gradients of distance functions are nontrivial
371(2)
13.4 The differential structure
373(4)
13.5 Comparisons between ru and Lip u, Taylor's theorem, and the reflexivity of N1.p (X)
377(8)
13.6 Notes to
Chapter 13
385(2)
14 Examples, applications, and further research directions
387(25)
14.1 Quasiconformal and quasisymmetric mappings
387(5)
14.2 Spaces supporting a Poincare inequality
392(15)
14.3 Applications and further research directions
407(5)
References 412(15)
Notation Index 427(2)
Subject Index 429
Juha Heinonen (19602007) was Professor of Mathematics at the University of Michigan. His principal areas of research interest included quasiconformal mappings, nonlinear potential theory, and analysis on metric spaces. He was the author of over 60 research articles, including several posthumously, and two textbooks. A member of the Finnish Academy of Science and Letters, Heinonen received the Excellence in Research Award from the University of Michigan in 1997 and gave an invited lecture at the International Congress of Mathematicians in Beijing in 2002. Pekka Koskela is Professor of Mathematics at the University of Jyväskylä, Finland. He works in Sobolev mappings and in the associated nonlinear analysis, and he has authored over 140 publications. He gave invited lectures at the European Congress of Mathematics in Barcelona in 2000 and at the International Congress of Mathematicians in Hyderabad in 2010. Koskela is a member of the Finnish Academy of Science and Letters. He received the Väisälä Award in 2001 and the Magnus Ehrnrooth Foundation prize in 2012. Nageswari Shanmugalingam is Professor of Mathematics at the University of Cincinnati. Her research interests include analysis in metric measure spaces, potential theory, functions of bounded variation and quasiminimal surfaces in metric setting. The foundational part of the structure of Sobolev spaces in metric setting was developed by her in her PhD thesis in 1999, and she has also contributed to the development of potential theory in metric setting. Her research contributions were recognized by the College of Arts and Sciences at the University of Cincinnati with a McMicken Dean's Award in 2008. Jeremy T. Tyson is Professor of Mathematics at the University of Illinois, Urbana-Champaign, working in analysis in metric spaces, geometric function theory and sub-Riemannian geometry. He has authored over 40 research articles and co-authored two other books. Tyson has received awards for teaching from the University of Illinois at both the departmental and college level. He is a Fellow of the American Mathematical Society.
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