Analysis on Lie Groups With Polynomial Growth

by Dungey, Nick; Ter Elst, A. F. M.; Robinson, Derek W.

• ISBN13: 9780817632250
• ISBN10: 0817632255
• Format: Hardcover
• Copyright: 2003-10-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

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