Analysis And Geometry Of Metric Measure Spaces
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| Author | : Galia Devora Dafni |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 241 |
| Release | : 2013 |
| Genre | : Mathematics |
| ISBN | : 0821894188 |
Contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation.
| Author | : Fabrice Baudoin |
| Publisher | : Springer Nature |
| Total Pages | : 312 |
| Release | : 2022-02-04 |
| Genre | : Mathematics |
| ISBN | : 3030841413 |
This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.
| Author | : Juha Heinonen |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 158 |
| Release | : 2001 |
| Genre | : Mathematics |
| ISBN | : 9780387951041 |
The purpose of this book is to communicate some of the recent advances in this field while preparing the reader for more advanced study. 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 recent and appears for the first time in book format.
| Author | : Galia Devora Dafni |
| Publisher | : |
| Total Pages | : 220 |
| Release | : 2013 |
| Genre | : Geometry, Differential |
| ISBN | : 9781470415907 |
This book contains lecture notes from most of the courses presented at the 50th anniversary edition of the Séminaire de Mathématiques Supérieures in Montréal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation. In recent decades, metric measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems, and partial differential eq.
| Author | : Juha Heinonen |
| Publisher | : Cambridge University Press |
| Total Pages | : 447 |
| Release | : 2015-02-05 |
| Genre | : Mathematics |
| ISBN | : 1316241033 |
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 Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.
| Author | : Juha Heinonen |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 149 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461301319 |
The purpose of this book is to communicate some of the recent advances in this field while preparing the reader for more advanced study. 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 recent and appears for the first time in book format.
| Author | : Xining Li |
| Publisher | : |
| Total Pages | : 138 |
| Release | : 2015 |
| Genre | : |
| ISBN | : |
In the theory of geometric analysis on metric measure spaces, two properties of a metric measure space make the theory richer. These two properties are the doubling property of the measure, and the support of a Poincare ́inequality by the metric measure space. The focus of this dissertation is to show that the doubling property of the measure and the support of a Poincare ́ inequality are preserved by two transformations of the metric measure space: sphericalization (to obtain a bounded space from an unbounded space), and flattening (to obtain an unbounded space from a bounded space). We will show that if the given metric measure space is equipped with an Ahlfors Q-regular measure, then so are the spaces obtained by the sphericalization/flattening transformations. We then show that even if the measure is not Ahlfors regular, if it is doubling, then the transformed measure is still doubling. We then show that if the given metric space satisfies an annular quaisconvexity property and the measure is doubling, and in addition if the metric measure space supports a p-Poincare ́inequality in the sense of Heinonen and Koskela's theory, then so does the transformed metric measure space (under the sphericalization/flattening procedure). Finally, we show that if we relax the annular quasiconvexity condition to an analog of the starlike condition for the metric measure space, then if the metric measure space also satisfies a p-Poincare ́inequality, the transformed space also must satisfy a q-Poincare ́inequality for some p
| Author | : Jun Kigami |
| Publisher | : Springer Nature |
| Total Pages | : 164 |
| Release | : 2020-11-16 |
| Genre | : Mathematics |
| ISBN | : 3030541541 |
The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.
| Author | : Nicola Gigli |
| Publisher | : Springer Nature |
| Total Pages | : 212 |
| Release | : 2020-02-10 |
| Genre | : Mathematics |
| ISBN | : 3030386139 |
This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces – known as RCD spaces – satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of “infinitesimally Hilbertian” metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking a comprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.
| Author | : James J Yeh |
| Publisher | : World Scientific |
| Total Pages | : 308 |
| Release | : 2019-11-18 |
| Genre | : Mathematics |
| ISBN | : 9813200421 |
Measure and metric are two fundamental concepts in measuring the size of a mathematical object. Yet there has been no systematic investigation of this relation. The book closes this gap.
