Function Spaces, Entropy Numbers, Differential Operators

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

• Publisher: Cambridge University Press
• Year: 1996
• Tongue: English
• Leaves: 265
• Series: Cambridge Tracts in Mathematics 120
• 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......Page 1
Function Spaces, Entropy Numbers, Differential Operators......Page 4
0521560365......Page 5
Contents......Page 6
Preface......Page 10
1.2 Spectral theory in quasi-Banach spaces......Page 14
1.3.1 Definitions and elementary properties......Page 20
1.3.2 Interpolation properties of entropy numbers......Page 26
1.3.3 Relationships between entropy and approximation numbers......Page 28
1.3.4 Connections with eigenvalues......Page 31
2.1 Introduction......Page 36
2.2.1 Definitions......Page 37
2.2.2 Concrete spaces......Page 38
2.2.3 Atomic representations......Page 41
2.3.1 Dilations......Page 45
2.3.2 Localisation......Page 48
2.3.3 Embeddings......Page 56
2.4.1 Preliminaries......Page 58
2.4.2 Paramultiplication......Page 60
2.4.3 The main theorem......Page 64
2.4.4 Limiting cases......Page 68
2.4.5 Holder inequalities for H^s_p......Page 69
2.5.1 Definitions......Page 70
2.5.2 Atoms and atomic domains......Page 73
2.5.3 Atomic representations......Page 76
2.6.1 Definitions and preliminaries......Page 78
2.6.2 Basic theorems......Page 82
2.6.3 Logarithmic Sobolev spaces......Page 88
2.7.1 Extremal functions......Page 94
2.7.2 Embedding constants......Page 102
2.7.3 Embeddings......Page 105
3.1 Introduction......Page 109
3.2.1 The spaces l^m_p......Page 110
3.2.2 Entropy numbers......Page 111
3.2.3 Approximation numbers......Page 114
3.3.2 Entropy numbers: upper estimates......Page 118
3.3.3 Entropy numbers: lower estimates......Page 127
3.3.4 Approximation numbers......Page 132
3.3.5 Historical remarks......Page 139
3.4.1 Preliminaries......Page 141
3.4.2 Embeddings in L_{\infty}(log L)_{-a}(Ω)......Page 142
3.4.3 Interior estimates......Page 151
3.4.4 Duality arguments......Page 163
3.5 Embeddings in non-smooth domains......Page 164
4.2.1 Definitions......Page 166
4.2.2 Basic properties......Page 169
4.2.3 Embeddings: general weights......Page 173
4.2.4 Embeddings: the weights \langle x \rangle^a......Page 175
4.2.5 Holder inequalities......Page 177
4.3.1 A preparation......Page 178
4.3.2 The main theorem......Page 181
4.3.3 Approximation numbers......Page 192
5.1 Introduction......Page 197
5.2.1 Introduction; the Birman-Schwinger principle......Page 198
5.2.2 Elliptic differential operators: mapping properties......Page 200
5.2.3 Pseudodifferential operators: mapping properties......Page 202
5.2.4 Elliptic operators: spectral properties......Page 205
5.2.5 Elliptic operators: generalisations......Page 212
5.2.6 Pseudodifferential operators: spectral properties......Page 218
5.2.7 The negative spectrum......Page 219
5.3.2 Orlicz spaces......Page 220
5.3.3 Logarithmic Sobolev spaces......Page 226
5.4.1 Introduction; the Birman-Schwinger principle revisited......Page 234
5.4.2 Pseudodifferential operators: mapping properties......Page 237
5.4.3 Pseudodifferential operators: spectral properties......Page 240
5.4.4 Degenerate pseudodifferential operators: eigenvalue distributions......Page 241
5.4.5 Degenerate pseudodifferential operators: smoothness theory......Page 245
5.4.6 Degenerate pseudodifferential operators of positive order......Page 247
5.4.7 The negative spectrum: basic results......Page 249
5.4.8 The negative spectrum: splitting techniques......Page 251
5.4.9 The negative spectrum: homogeneity arguments......Page 253
References......Page 256
Index of Symbols......Page 262
Index......Page 264