Analysis on Lie Groups With Polynomial Growth

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.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.

Preface

vii

I Introduction

1

(6)

II General Formalism

7

(56)

II.1 Lie groups and Lie algebras

8

(5)

II.2 Subelliptic operators

13

(7)

II.3 Subelliptic kernels

20

(1)

II.4 Growth properties

21

(10)

II.5 Real operators

31

(3)

II.6 Local bounds on kernels

34

(4)

II.7 Compact groups

38

(2)

II.8 Transference method

40

(6)

II.9 Nilpotent groups

46

(2)

II.10 De Giorgi estimates

48

(3)

II.11 Almost periodic functions

51

(4)

II.12 Interpolation

55

(1)

Notes and Remarks

56

(7)

III Structure Theory

63

(60)

III.1 Complementary subspaces

64

(5)

III.2 The nilshadow; algebraic structure

69

(9)

III.3 Uniqueness of the nilshadow

78

(9)

III.4 Near-nilpotent ideals

87

(3)

III.5 Stratified nilshadow

90

(4)

III.6 Twisted products

94

(11)

III.7 The nilshadow; analytic structure

105

(16)

Notes and Remarks

121

(2)

IV Homogenization and Kernel Bounds

123

IV.1 Subelliptic operators

125

(6)

IV.2 Scaling

131

(3)

IV.3 Homogenization; correctors

134

(7)

IV.4 Homogenized operators

141

(9)

IV.5 Homogenization; convergence

150

(12)

IV.6 Kernel bounds; stratified nilshadow

162

(9)

IV.7 Kernel bounds; general case

171

(4)

Notes and Remarks

175

V Global Derivatives

119

(96)

V.1 L2-bounds

180

(15)

V.1.1 Compact derivatives

184

(6)

V.1.2 Nilpotent derivatives

190

(5)

V.2 Gaussian bounds

195

(10)

V.3 Anomalous behaviour

205

(7)

Notes and Remarks

212

(3)

VI Asymptotics

215

(48)

VI.1 Asymptotics of semigroups

216

(34)

VI.2 Asymptotics of derivatives

250

(11)

Notes and Remarks

261

(2)

Appendices

263

(26)

A.1 De Giorgi estimates

263

(7)

A.2 Money and Campanato spaces

270

(3)

A.3 Proof of Theorem 11.10.5

273

(8)

A.4 Rellich lemma

281

(6)

Notes and Remarks

287

(2)

References

289

(10)

Index of Notation

299

(10)

Index

309

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Analysis on Lie Groups With Polynomial Growth

Summary

Table of Contents

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