Lectures on Analysis on Metric Spaces

1. Front Matter

   Pages i-x

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2. Covering Theorems

   • Juha Heinonen

   Pages 1-9

3. Maximal Functions

   • Juha Heinonen

   Pages 10-13

4. Sobolev Spaces

   • Juha Heinonen

   Pages 14-26

5. Poincaré Inequality

   • Juha Heinonen

   Pages 27-33

6. Sobolev Spaces on Metric Spaces

   • Juha Heinonen

   Pages 34-42

7. Lipschitz Functions

   • Juha Heinonen

   Pages 43-48

8. Modulus of a Curve Family, Capacity, and Upper Gradients

   • Juha Heinonen

   Pages 49-58

9. Loewner Spaces

   • Juha Heinonen

   Pages 59-67

10. Loewner Spaces and Poincaré Inequalities

    • Juha Heinonen

    Pages 68-77

11. Quasisymmetric Maps: Basic Theory I

    • Juha Heinonen

    Pages 78-87

12. Quasisymmetric Maps: Basic Theory II

    • Juha Heinonen

    Pages 88-97

13. Quasisymmetric Embeddings of Metric Spaces in Euclidean Space

    • Juha Heinonen

    Pages 98-102

14. Existence of Doubling Measures

    • Juha Heinonen

    Pages 103-108

15. Doubling Measures and Quasisymmetric Maps

    • Juha Heinonen

    Pages 109-118

16. Conformal Gauges

    • Juha Heinonen

    Pages 119-125

17. 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.

• Area
• Finite
• Sobolev space
• behavior
• calculus
• development
• eXist
• equality
• form
• inequality
• mapping
• maximum
• quasiconformal mapping
• theorem
• types

• Mathematics Department East Hall, University of Michigan, Ann Arbor, USA

  Juha Heinonen

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