Ebook: Lectures on Analysis on Metric Spaces

Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.

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Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.

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Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.
Content:
Front Matter....Pages i-x
Covering Theorems....Pages 1-9
Maximal Functions....Pages 10-13
Sobolev Spaces....Pages 14-26
Poincar? Inequality....Pages 27-33
Sobolev Spaces on Metric Spaces....Pages 34-42
Lipschitz Functions....Pages 43-48
Modulus of a Curve Family, Capacity, and Upper Gradients....Pages 49-58
Loewner Spaces....Pages 59-67
Loewner Spaces and Poincar? Inequalities....Pages 68-77
Quasisymmetric Maps: Basic Theory I....Pages 78-87
Quasisymmetric Maps: Basic Theory II....Pages 88-97
Quasisymmetric Embeddings of Metric Spaces in Euclidean Space....Pages 98-102
Existence of Doubling Measures....Pages 103-108
Doubling Measures and Quasisymmetric Maps....Pages 109-118
Conformal Gauges....Pages 119-125
Back Matter....Pages 127-141

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Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.
Content:
Front Matter....Pages i-x
Covering Theorems....Pages 1-9
Maximal Functions....Pages 10-13
Sobolev Spaces....Pages 14-26
Poincar? Inequality....Pages 27-33
Sobolev Spaces on Metric Spaces....Pages 34-42
Lipschitz Functions....Pages 43-48
Modulus of a Curve Family, Capacity, and Upper Gradients....Pages 49-58
Loewner Spaces....Pages 59-67
Loewner Spaces and Poincar? Inequalities....Pages 68-77
Quasisymmetric Maps: Basic Theory I....Pages 78-87
Quasisymmetric Maps: Basic Theory II....Pages 88-97
Quasisymmetric Embeddings of Metric Spaces in Euclidean Space....Pages 98-102
Existence of Doubling Measures....Pages 103-108
Doubling Measures and Quasisymmetric Maps....Pages 109-118
Conformal Gauges....Pages 119-125
Back Matter....Pages 127-141
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