eBook ISBN:  9781470402792 
Product Code:  MEMO/145/688.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $30.00 
eBook ISBN:  9781470402792 
Product Code:  MEMO/145/688.E 
List Price:  $50.00 
MAA Member Price:  $45.00 
AMS Member Price:  $30.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 145; 2000; 101 ppMSC: Primary 46
There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot–Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms.
The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting.
We are given a metric space \(X\) equipped with a doubling measure \(\mu\). A generalization of a Sobolev function and its gradient is a pair \(u\in L^{1}_{\rm loc}(X)\), \(0\leq g\in L^{p}(X)\) such that for every ball \(B\subset X\) the Poincarétype inequality \[ ⨍_{B} uu_{B}\, d\mu \leq C r ( ⨍_{\sigma B} g^{p}\, d\mu)^{1/p}\,\] holds, where \(r\) is the radius of \(B\) and \(\sigma\geq 1\), \(C>0\) are fixed constants. Working in the above setting we show that basically all relevant results from the classical theory have their counterparts in our general setting. These include SobolevPoincaré type embeddings, RellichKondrachov compact embedding theorem, and even a version of the Sobolev embedding theorem on spheres. The second part of the paper is devoted to examples and applications in the above mentioned areas.
ReadershipGraduate students and research mathematicians interested in functional analysis.

Table of Contents

Chapters

1. Introduction

2. What are Poincaré and Sobolev inequalities?

3. Poincaré inequalities, pointwise estimates, and Sobolev classes

4. Examples and necessary conditions

5. Sobolev type inequalities by means of Riesz potentials

6. Trudinger inequality

7. A version of the Sobolev embedding theorem on spheres

8. RellichKondrachov

9. Sobolev classes in John domains

10. Poincaré inequality: examples

11. CarnotCarathéodory spaces

12. Graphs

13. Applications to P.D.E and nonlinear potential theory

14. Appendix


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot–Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms.
The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting.
We are given a metric space \(X\) equipped with a doubling measure \(\mu\). A generalization of a Sobolev function and its gradient is a pair \(u\in L^{1}_{\rm loc}(X)\), \(0\leq g\in L^{p}(X)\) such that for every ball \(B\subset X\) the Poincarétype inequality \[ ⨍_{B} uu_{B}\, d\mu \leq C r ( ⨍_{\sigma B} g^{p}\, d\mu)^{1/p}\,\] holds, where \(r\) is the radius of \(B\) and \(\sigma\geq 1\), \(C>0\) are fixed constants. Working in the above setting we show that basically all relevant results from the classical theory have their counterparts in our general setting. These include SobolevPoincaré type embeddings, RellichKondrachov compact embedding theorem, and even a version of the Sobolev embedding theorem on spheres. The second part of the paper is devoted to examples and applications in the above mentioned areas.
Graduate students and research mathematicians interested in functional analysis.

Chapters

1. Introduction

2. What are Poincaré and Sobolev inequalities?

3. Poincaré inequalities, pointwise estimates, and Sobolev classes

4. Examples and necessary conditions

5. Sobolev type inequalities by means of Riesz potentials

6. Trudinger inequality

7. A version of the Sobolev embedding theorem on spheres

8. RellichKondrachov

9. Sobolev classes in John domains

10. Poincaré inequality: examples

11. CarnotCarathéodory spaces

12. Graphs

13. Applications to P.D.E and nonlinear potential theory

14. Appendix