Analysis on Lie Groups with Polynomial Growth is the first book to present a method for examining the surprising connection between invariant differential operators and almost periodic operators on a suitable nilpotent Lie group. It deals with the theory of second-order, right invariant, elliptic operators on a large class of manifolds: Lie groups with polynomial growth. In systematically developing the analytic and algebraic background on Lie groups with polynomial growth, it is possible to describe the large time behavior for the semigroup generated by a complex second-order operator with the aid of homogenization theory and to present an asymptotic expansion. Further, the text goes beyond the classical homogenization theory by converting an analytical problem into an algebraic one.
Key features:
* Completely self-contained work, including a review of well-established local theory for elliptic operators and a summary of the essential aspects of Lie group theory
* Numerous illustrative examples
* Appendices covering technical subtleties
* An exhaustive bibliography and index
This work is aimed at graduate students as well as researchers in the above areas. Prerequisites include knowledge of basic results from semigroup theory and Lie group theory.
Author(s): Nick Dungey, Derek Robinson
Publisher: Birkhauser
Year: 2003
Language: English
Pages: 322
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
I Introduction......Page 10
II General Formalism......Page 16
II.1 Lie groups and Lie algebras......Page 17
II.2 Subelliptic operators......Page 22
II.3 Subelliptic kernels......Page 29
II.4 Growth properties......Page 30
II.5 Real operators......Page 40
II.6 Local bounds on kernels......Page 43
II.7 Compact groups......Page 47
II.8 Transference method......Page 49
II.9 Nilpotent groups......Page 55
II.10 De Giorgi estimates......Page 57
II.11 Almost periodic functions......Page 60
II.12 Interpolation......Page 64
Notes and Remarks......Page 65
III Structure Theory......Page 72
III. 1 Complementary subspaces......Page 73
III.2 The nilshadow; algebraic structure......Page 78
III.3 Uniqueness of the nilshadow......Page 87
III.4 Near-nilpotent ideals......Page 96
III.5 Stratified nilshadow......Page 99
III.6 Twisted products......Page 103
III.7 The nilshadow; analytic structure......Page 114
Notes and Remarks......Page 130
IV Homogenization and Kernel Bounds......Page 132
IV.1 Subelliptic operators......Page 134
IV.2 Scaling......Page 140
IV.3 Homogenization; correctors......Page 143
IV.4 Homogenized operators......Page 150
IV.5 Homogenization; convergence......Page 159
IV.6 Kernel bounds; stratified nilshadow......Page 171
IV.7 Kernel bounds; general case......Page 180
Notes and Remarks......Page 184
V Global Derivatives......Page 188
V.1 L^2-bounds......Page 189
V.1.1 Compact derivatives......Page 193
V.1.2 Nilpotent derivatives......Page 199
V.2 Gaussian bounds......Page 204
V.3 Anomalous behaviour......Page 214
Notes and Remarks......Page 221
VI Asymptotics......Page 224
VI.1 Asymptotics of semigroups......Page 225
VI.2 Asymptotics of derivatives......Page 259
Notes and Remarks......Page 270
A.1 De Giorgi estimates......Page 272
A.2 Morrey and Campanato spaces......Page 279
A.3 Proof of Theorem 1I.10.5......Page 282
A.4 Rellich lemma......Page 290
Notes and Remarks......Page 296
References......Page 298
Index of Notation......Page 308
Index......Page 318
Back Cover......Page 322
