Function Spaces, Entropy Numbers, Differential Operators (Cambridge Tracts in Mathematics, Series Number 120)

✍ Scribed by D. E. Edmunds, H. Triebel

• Publisher: Cambridge University Press
• Year: 1996
• Tongue: English
• Leaves: 265
• Edition: 1
• Category: Library

✦ Synopsis

The distribution of the eigenvalues of differential operators has long fascinated mathematicians. Recent advances have shed new light on classical problems in this area, and this book presents a fresh approach, largely based on the results of the authors. The emphasis here is on a topic of central importance in analysis, namely the relationship between i) function spaces on Euclidean n-space and on domains; ii) entropy numbers in quasi-Banach spaces; and iii) the distribution of the eigenvalues of degenerate elliptic (pseudo) differential operators. The treatment is largely self-contained and accessible to nonspecialists.

✦ Table of Contents

Cover
Function Spaces, Entropy Numbers, Differential Operators
Copyright
Goto 4 /FitH 555521059755
Goto 4 /FitH 555521560365
Contents
Preface
1 The Abstract Background
1.1 Introduction
1.2 Spectral theory in quasi-Banach spaces
1.3 Entropy numbers and approximation numbers
1.3.1 Definitions and elementary properties
1.3.2 Interpolation properties of entropy numbers
1.3.3 Relationships between entropy and approximation numbers
1.3.4 Connections with eigenvalues
2 Function Spaces
2.1 Introduction
2.2 The spaces B^s_{pq} and F^s_{pq} on R^n
2.2.1 Definitions
2.2.2 Concrete spaces
2.2.3 Atomic representations
2.3 Special properties
2.3.1 Dilations
2.3.2 Localisation
2.3.3 Embeddings
2.4 Holder inequalities
2.4.1 Preliminaries
2.4.2 Paramultiplication
2.4.3 The main theorem
2.4.4 Limiting cases
2.4.5 Holder inequalities for H^s_p
2.5 The spaces B^s_{pq} and F^s_{pq} on domains
2.5.1 Definitions
2.5.2 Atoms and atomic domains
2.5.3 Atomic representations
2.6 The spaces L_p(log L)_a and logarithmic Sobolev spaces
2.6.1 Definitions and preliminaries
2.6.2 Basic theorems
2.6.3 Logarithmic Sobolev spaces
2.7 Limiting embeddings
2.7.1 Extremal functions
2.7.2 Embedding constants
2.7.3 Embeddings
3 Entropy and Approximation Numbers of Embeddings
3.1 Introduction
3.2 The embedding of l^m_p in l^m_q
3.2.1 The spaces l^m_p
3.2.2 Entropy numbers
3.2.3 Approximation numbers
3.3 Embeddings between function spaces
3.3.1 Notation
3.3.2 Entropy numbers: upper estimates
3.3.3 Entropy numbers: lower estimates
3.3.4 Approximation numbers
3.3.5 Historical remarks
3.4 Limiting embeddings in spaces of Orlicz type
3.4.1 Preliminaries

3.4.3 Interior estimates
3.4.4 Duality arguments
3.5 Embeddings in non-smooth domains
4 Weighted Function Spaces and Entropy Numbers
4.1 Introduction
4.2 Weighted spaces
4.2.1 Definitions
4.2.2 Basic properties
4.2.3 Embeddings: general weights
4.2.4 Embeddings: the weights langle x
angle^a
4.2.5 Holder inequalities
4.3 Entropy numbers
4.3.1 A preparation
4.3.2 The main theorem
4.3.3 Approximation numbers
5 Elliptic Operators
5.1 Introduction
5.2 Elliptic operators in domains: non-limiting cases
5.2.1 Introduction; the Birman-Schwinger principle
5.2.2 Elliptic differential operators: mapping properties
5.2.3 Pseudodifferential operators: mapping properties
5.2.4 Elliptic operators: spectral properties
5.2.5 Elliptic operators: generalisations
5.2.6 Pseudodifferential operators: spectral properties
5.2.7 The negative spectrum
5.3 Elliptic operators in domains: limiting cases
5.3.1 Introduction
5.3.2 Orlicz spaces
5.3.3 Logarithmic Sobolev spaces
5.4 Elliptic operators in R^n
5.4.1 Introduction; the Birman-Schwinger principle revisited
5.4.2 Pseudodifferential operators: mapping properties
5.4.3 Pseudodifferential operators: spectral properties
5.4.4 Degenerate pseudodifferential operators: eigenvalue distributions
5.4.5 Degenerate pseudodifferential operators: smoothness theory
5.4.6 Degenerate pseudodifferential operators of positive order
5.4.7 The negative spectrum: basic results
5.4.8 The negative spectrum: splitting techniques
5.4.9 The negative spectrum: homogeneity arguments
References
Index of Symbols
Index