Lipschitz Functions

1. Front Matter

   Pages i-xiv

   PDF

2. Prerequisites

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 1-97

3. Basic Facts Concerning Lipschitz Functions

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 99-142

4. Relations with Other Classes of Functions

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 143-210

5. Extension Results for Lipschitz Mappings

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 211-251

6. Extension Results for Lipschitz Mappings in Geodesic Spaces

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 253-316

7. Approximations Involving Lipschitz Functions

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 317-334

8. Lipschitz Isomorphisms of Metric Spaces

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 335-363

9. Banach Spaces of Lipschitz Functions

   • Ştefan Cobzaş, Radu Miculescu, Adriana Nicolae

   Pages 365-556

10. Back Matter

    Pages 557-593

    PDF

The aim of this book is to present various facets of the theory and applications of Lipschitz functions, starting with classical and culminating with some recent results. Among the included topics we mention: characterizations of Lipschitz functions and relations with other classes of functions, extension results for Lipschitz functions and Lipschitz partitions of unity, Lipschitz free Banach spaces and their applications, compactness properties of Lipschitz operators, Bishop-Phelps type results for Lipschitz functionals, applications to best approximation in metric and in metric linear spaces, Kantorovich-Rubinstein norm and applications to duality in the optimal transport problem, Lipschitz mappings on geodesic spaces.


The prerequisites are basic results in real analysis, functional analysis, measure theory (including vector measures) and topology, which, for reader's convenience, are surveyed in the first chapter of the book.

• Banach Spaces of Lipschitz Functions
• Bishop-Phelps Property
• Compact Operators
• Composition Operators
• Extension Properties for Lipschitz Operators
• Geodesic Metric Spaces
• Holder Functions
• Lipschitz Free Banach Spaces
• Lipschitz Functions
• Lipschitz Operators
• Metric Spaces
• Nonlinear Embeddings

"This is an excellent source for all aspects of the notion of Lipschitz continuity and will undoubtedly become a standard reference.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 196 (1), 2021)

“The book is accessible to graduate students, but it also contains recent results of interest to researchers in various domains as metric geometry, mathematical analysis, and functional analysis. My opinion this book will be of interest to everyone whose domain of interest is mathematical analysis and its applications.” (Andrey Zahariev, zbMATH 1431.26002, 2020)

“This book provides a very large amount of interesting information. … it is very useful that they are brought to readers' attention.” (Mikhail Ostrovskii, Mathematical Reviews, December, 2019)

• Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

  Ştefan Cobzaş, Adriana Nicolae

• Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Brasov, Romania

  Radu Miculescu

Stefan Cobzas is Emeritus Professor at the Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania. He graduated from the same university in 1968 and obtained a Ph.D. in 1979. His scientific interests concern mainly applied functional analysis -- optimization and best approximation in Banach spaces. In the last years he worked on some problems in asymmetric functional analysis and published several papers and a book (in the series Frontiers in Mathematics, Birkhauser-Springer, 2013) on this topic.


Radu Miculescu is Professor at Transilvania University of Brasov, Romania. He graduated from Bucharest University, Romania, in 1992 and obtained his Ph.D in 1999 from the same university with a thesis concerning Lipschitz functions. In the last period his scientific interest includes Hutchinson-Barnsley fractals.


Adriana Nicolae is Associate Professor at the Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania. She graduated in 2007 from the same university and focused during her Ph.D. on various aspects in metric fixed point and best approximation theory mainly in the setting of geodesic metric spaces. In the last years she also addressed problems in areas such as geometry and analysis in metric spaces, optimization, or proof mining.

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