Analysis on Lie Groups with Polynomial Growth

bI

Progress in Mathematics

Nick Dungey A. F. M. ter Elst Derek W. Robinson

Analysis on Lie Groups

with Polynomial Growth

Birkhauser

Progress in Mathematics Volume 214

Series Editors Hyman Bass Joseph Oesterlr Alan Weinstein

Nick Dungey A.F.M. ter Elst Derek W. Robinson

Analysis on Lie Groups with Polynomial Growth

Birkhauser Boston Basel Berlin

Nick Dungey Australian National University Centre for Mathematics and its Applications Mathematical Sciences Institute Canberra, ACT 0200 Australia

A.F.M. ter Elst Australian National University Centre for Mathematics and its Applications Mathematical Sciences Institute Canberra, ACT 0200 Australia

Derek W. Robinson Australian National University Centre for Mathematics and its Applications Mathematical Sciences Institute Canberra, ACT 0200 Australia

A.F.M. ter Elst On leave from Eindhoven University of Technology Department of Mathematics and Computer Science 5600 MB Eindhoven The Netherlands

Library of Congress Cataloging-in-Publication Data Dungey, Nick. Analysis on Lie groups with polynomial growth / Nick Dungey, A.F.M. to Elst, Derek W. Robinson.

p. cm. - (Progress in mathematics ; 214) Includes bibliographical references and index.

ISBN 0-8176-3225-5 (alk. paper) - ISBN (invalid) 3-7643-3225-5 (alk. paper) 1. Lie groups. 2. Harmonic analysis. 3. Differential equations, Partial-Asymptotic theory. 4. Homogenization (Differential equations) 1. Elst, A. F. M. ter Il. Robinson, Derek W. Ill. Title. IV. Progress in mathematics (Boston, Mass.) ; v. 214. QA387.D86 2003 512'.55-dc2l

2003049639

CIP AMS Subject Classifications: 22E30, 43A80, 35B40, 35B27, 58G1I Printed on acid-free paper

02003 Birkhauser Boston

Birkhauser

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-3225-5 ISBN 3-7643-3225-5

SPIN 10923467

Reformatted from authors' files by TTXniques. Inc., Cambridge, MA. Printed in the United States of America.

987654321 Birkhauser Boston Basel Berlin A member of BerrelsmannSpringer Science+Business Media GmbH

Contents

Preface I

vii

Introduction

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II General Formalism 11.1 Lie groups and Lie algebras .... 11.2 11.3

II.4 11.5

11.6

.. .. Subelliptic operators .. .. .. .. .. ... . .. Subelliptic kernels .. .. .. .. .. .. Growth properties .. . ... .. .. .. .. .. .. Real operators . .. .. .. .. .. Local bounds on kernels .

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11.12 Interpolation .. .. .

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Compact groups . 11.8 Transference method . . 11.9 Nilpotent groups . 11.10 De Giorgi estimates . . 11.11 Almost periodic functions .

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11.7

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III Structure Theory 111. 1 Complementary subspaces . . . . 111.2 The nilshadow; algebraic structure 111.3 Uniqueness of the nilshadow . . .

III.4 Near-nilpotent ideals

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63 64 69 78

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111.5 Stratified nilshadow 111.6 Twisted products . 111.7 The nilshadow; analytic structure Notes and Remarks . . . . . . . . . .

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105

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125 131

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141

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150 162

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IV Homogenization and Kernel Bounds IV.1 Subelliptic operators . . . . . . IV.2 Scaling . . . . IV.3 Homogenization; correctors . . . IV.4 Homogenized operators . . . IV.5 Homogenization; convergence .

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V Global Derivatives

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L2-bounds . .... ... V.1.1

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Nilpotent derivatives V.2 Gaussian bounds . . . . . . V.3 Anomalous behaviour . . . Notes and Remarks . . . . . V.1.2

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IV.6 Kernel bounds; stratified nilshadow IV.7 Kernel bounds; general case . . . Notes and Remarks . . . . . . . . .

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Appendices A.1 De Giorgi estimates . . . . . A.2 Morrey and Campanato spaces

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VI Asymptotics VI.1 Asymptotics of semigroups VI.2 Asymptotics of derivatives Notes and Remarks . . . . . .

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Proof of Theorem 1I.10.5

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A.4 Rellich lemma ...

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Notes and Remarks

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215 216

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References

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Index of Notation

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Index

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Preface

Lie groups are fascinating objects and their algebraic structure has been the subject matter of many books. The main emphasis of the present book is, however, on analytic and geometric features. There are several distinct motivations for interest

in such features: the description of symmetries in physical theories, the extension of the usual framework of Euclidean harmonic analysis, and global growth properties of groups. This text largely concerns itself with the relation between geometric properties such as global growth and asymptotic properties of the objects of harmonic analysis. The analysis is based on heat equation methods. The relation between heat kernel decay and global geometric properties has been studied previously by many authors in the context of Lie groups or arbitrary Riemannian manifolds. For example, Li and Yau derived estimates for the heat kernel on manifolds with non-negative Ricci curvature; similar estimates were obtained by Varopoulos for Lie groups and subsequently extended in the framework of Lie groups of polynomial growth by Alexopoulos. Much of the previous work focussed, however, on the Laplace-Beltrami operator in the Riemannian case or on sublaplacians in the group case. Such operators generate symmetric Markov semigroups, which have a probabilistic interpretation and for which much standard machinery is available, e.g., techniques based on the maximal principle. A novelty of this text is that for Lie groups of polynomial growth it gives a detailed description for arbitrary second-order, complex, subelliptic operators. The key to this development is the observation of Alexopoulos that (real) invariant differential operators on a Lie group of polynomial growth are essentially operators with almost periodic coefficients on a group which is close to being nilpotent.

viii

Preface

Thus, in this context it is natural to apply ideas from the homogenization theory developed for partial differential equations with (almost) periodic coefficients. The authors are particularly grateful for ideas and suggestions of Adam Sikora over an extended period of collaboration. They have also profited from past collaborations with Pascal Auscher, Charles Batty, Ola Bratteli, Palle Jorgensen and Camiel Smulders on some parts of the following material. They further acknowledge the advice of Mariano Giaquinta, Alan McIntosh and Neil Trudinger on the Morrey-De Giorgi-Campanato theory. Finally, the authors wish to thank Ann Kostant and the staff of Birkhauser for their help and cooperation in producing the final manuscript. Nick Dungey A. F. M. ter Elst Derek W. Robinson January 2003

I Introduction

Lie groups are manifolds symmetric under the group action and the symmetry places uniform constraints on the global properties of the manifold. The simplest constraint resulting from the group action is on the volume growth. There are only two possibilities. In the first case the volume of a ball grows no faster than a power of its radius. Groups with this characteristic are called Lie groups of polynomial growth. Compact Lie groups fall within this class since the volume is uniformly bounded. Nilpotent Lie groups also have polynomial growth, although this is less evident, and the rate of growth is straightforwardly determined by the nilpotent structure if the group is connected and simply connected. Moreover, all Lie groups of polynomial growth are unimodular. In the second case the volume of a ball grows exponentially with its radius. All non-unimodular Lie groups have exponential growth but non-unimodularity is not essential. For example, each non-compact semisimple Lie group is unimodular but has exponential volume growth. In this monograph we analyze the structure of connected Lie groups of polynomial growth with particular emphasis on global properties. Heat equation methods provide the main technique in this analysis. The global properties of the manifold

are intimately related to the asymptotics of the heat equation. Since the asymptotic analysis has an intrinsic interest we consider evolution problems in greater generality. In particular, we examine the asymptotics of the evolution equations associated with complex, right-invariant, second-order, subelliptic, or strongly elliptic, operators acting on the groups.

2

1. Introduction

The simplest illustration of the type of problem we analyze is the heat equation on the Euclidean group Rd. This is a parabolic partial differential equation a, t,, + Htp, = 0

(I.1)

which represents the dissipation of heat with time. The function tp, is interpreted as the distribution of heat at time t over Rd. Moreover, a, = a/at, d

H=-Eak k=1

is the usual Laplacian and the equation is considered on one of the standard func-

tion spaces over Rd. If the initial distribution co at time zero is specified the equation can be explicitly solved by Fourier analysis. One has (pt (x) = (G, * (po) (x) =

JRd

dy G1(x - y) (po(y),

where * denotes the convolution product on Rd and

G,(x) = (27r )-d

dpe-rp2eiP.x

= (47rt)-d/2e-4-'IX12,-'

fRd

is the d-dimensional Gaussian. In operator terms to, = S,tpo, where S, = e-t H is the semigroup generated by H, and G, is the semigroup kernel. The kernel G, describes the dissipation of heat in Rd and since IIGtII = (47r t)-dll the rate of dissipation is polynomial. It is proportional to V (t)-1 /2 where V (t) is the volume of the ball of radius t. This is the simplest relation between the asymptotic evolution and the global geometry; the rate of dissipation is determined by the available volume. A second simple illustration of the subsequent theory is given by the heat equation (I.1) on the compact Lie group Td, where T = (z E C : Izl = 1) is the circle group. If HT is the Laplacian given by d Dak>//)(x),

k=1

then the invariance under transwhere >[r(x) = cp(eix) and (e''' , .. , lations by Td allows one to use Fourier series to establish that

w, (z) = (GT' * (PO) (Z) = fd1'0 where the kernel GT is given implicitly by GT(eix)

= f, e-tn2ein.x nE%J

I.

Introduction

3

(2n)-d f l_ nId dx cp(e'x). Although GT cannot be explicitly and fTd dz cp(z) = evaluated in closed form, it can be related to the Gaussian G on Rd by the Jacobi identity, GT(eix) (2n)d E Gt(x +2rrn). (1.2)

=

nEZd

This relation reflects the fact that the Euclidean group Rd is the simply connected covering group of the compact, non-simply connected, group Td. Similar relations will occur in the context of general Lie groups. The asymptotic properties of the heat evolution on Td are quite different to those on Rd. The kernel GT converges to a constant as t -* oo. In fact GT(erx)

-1=

e-tn2ein.x

nEZd\10)

Therefore there is a c > 0 such that

sup IGT(z) - I I < c e'

(1.3)

ZETd

for all t > 1. Thus the evolution behaves ergodically with the heat redistributing uniformly over the compact manifold Td. In contrast to the dissipation on Rd the redistribution on Td occurs at an exponential rate. The rate of redistribution is not determined by geometric factors but by a spectral properties of the Laplacian. The Laplacian on Td has discrete spectrum (n2 : n E Zd ), the asymptotic limit of GT is the contribution of the lowest eigenstate corresponding to the zero eigenvalue, and the exponential rate of convergence to the limit is determined by the spectral gap, which is equal to one, between the two lowest eigenvalues. It is also not difficult to analyze the asymptotic behaviour of the heat equation (I.1) on a cylindrical manifold Tdi x Rd2. If the Laplacian separates as a sum of sublaplacians in the Td, and Rd2 components, then the situation is particularly simple since the solution of the heat equation factors as a product of the solutions in the two components. The heat redistributes itself exponentially fast around the cylinder and then dissipates at a polynomial rate, t-d2/2, along the cylinder. The same behaviour occurs if the Laplacian does not factor in the cylindrical directions but the analysis is somewhat more complicated as there is a coupling of the Td' and Rd2 directions. If one defines the operator H on Rd2 by

(Hcp)(x) = rte' dz (H(11 ®(p))(z,x),

(1.4)

where 1 is the constant function with value one on Td', then the asymptotic behaviour of H is determined by the projection H on Rd. Specifically, one finds that if K and k are the semigroup kernels corresponding to H and H, respectively, then there are c, k > 0 such that sup (Z.x)ETdl xRd2

I K, (z, x) - K,(x)I
1 in direct analogy with (1.3). Therefore the principal long term effect is the slow dissipation along the cylinder and the asymptotic behaviour is comparable to the evolution on the manifold Rd2. This is natural since from a global perspective the cylindrical manifold resembles Rd2. The asymptotic properties of evolution on general Lie groups of polynomial growth have many characteristics in common with the Rd and Td examples. In particular, the asymptotic behaviour gives a global description of the group in

terms of a simpler group structure. The analysis of general groups is considerably more complicated than the special examples largely because of the noncommutative group structure. There are, however, two basic elements to the analysis, local properties and structure theory.

It follows from standard structure theory that each connected, simply connected, Lie group G is the semidirect product of a semisimple Lie group M acting on a solvable Lie group Q, the radical of G. But if G is a group with polynomial growth, it follows that M is compact and Q also has polynomial growth. Then, by a lesser known result of structure theory which will be described at length in Chapter III, Q can be obtained from a nilpotent Lie group QN by modification of the group product. The group QN is called the nilshadow of Q. The groups Q and QN are identical as manifolds but differ in their group structures. Therefore, by combination of these observations, G can be identified with the `cylindrical manifold' M X QN equipped with a new product. Then, in analogy with the T" x Rd2 example, the asymptotic evolution of solutions of the heat equation is in the direction of the nilshadow QN. The coupling between the cross-sectional directions M and the directions QN is, however, more complicated in general. Nevertheless, the general behaviour is similar to that of the simple example. One can associate with each right-invariant, second-order, subelliptic operator H on G a similar operator H on QN which governs the asymptotic behaviour. The relationship between H and H is much less direct, and much more complicated, than the relationship (1.4) in the Tdi x R42 example, and involves homogenization theory. But if K and K are the semigroup kernels corresponding to H and H on G and QN, respectively, then one has an estimate sup (m,q)EMxQ

IKr(m4)-Kt(q)I

ct-'"2V(t)-1/2

(1.6)

for all t > 1 in direct analogy with (1.5). The main difference is that the estimate is polynomial and not exponential. Nevertheless it identifies the asymptotic behaviour of K since one can also establish that the kernel K on Q N has a Gaussian behaviour, e.g., one has bounds Ilk, Iloo < c v (t)-111. Thus K describes the asymptotic behaviour of K. The asymptotic heat flow is in the directions of QN and, from a global perspective, the simply connected Lie group G resembles the nilshadow QN. The asymptotic estimate (1.6) can be interpreted as the first-order of an asymptotic series for the semigroup kernel K. It can be extended and improved in several different directions which will be described in detail in Chapter VI. It is, however, essential for the asymptotic analysis to establish a priori Gaussian upper bounds

I. Introduction

5

on the kernel K, for all large t. Gaussian bounds for small t follow from the well-established local theory and for real operators large t bounds can be derived by special techniques which are inapplicable to complex operators, or systems of real operators. Gaussian upper bounds for the kernels of complex operators can, however, be derived with the help of the methods which identify the asymptotic approximant H. The construction of H and the proof of Gaussian upper bounds are described in Chapter IV.

II General Formalism

In this chapter we develop the general background information relevant to the subsequent analysis. Part of the material consists of standard results which are summarized for later reference. A second part consists of the basic definitions of subelliptic operators and the related semigroups together with the description of some preliminary results which motivate the later analysis. Thirdly, we introduce several techniques adapted to the Lie group analysis. Since most of the reference material is quite standard it is summarized in formal statements without proof. Further details and specific references to the literature are, however, given in the Notes and Remarks at the end of the chapter. In Section 11. 1 we discuss standard results of structure theory of Lie groups and

the associated Lie algebras and in Section II.11 we give some equally standard properties of almost-periodic functions over Lie groups. In Sections 11.2 and II.3 we develop the representation theory necessary to define second-order subelliptic operators, the corresponding semigroups and the semigroup kernels. The subelliptic formalism carries with it a natural geometry and this introduces a second element of structure theory, growth properties, which is discussed in Section II.4. In Sections 11.5, 11.6, 11.7 and 11.9 we recall some of the standard results established

for the subelliptic kernels. There are two types of result, local and global. The local theory is well understood and we take it as a starting point for the analysis of the global theory. The global theory is also well understood for special classes of groups such as compact groups and nilpotent groups. The global properties of these special groups also play a key role via structure theory in the later analysis of groups of polynomial growth. Finally we describe various standard techniques which are useful in the global analysis. In Section 11.8 we develop some methods of transference theory. In Section II.10 we describe a strategy for using De Giorgi

8

II. General Formalism

estimates to extend local Gaussian bounds on the subelliptic kernels to global bounds. This strategy will be adopted in Chapter IV to obtain global Gaussian bounds on the kernels corresponding to complex operators. Then in Section 11. 12 we summarize some results of interpolation theory related to Holder continuity on Lie groups.

11. 1

Lie groups and Lie algebras

Let G denote a d-dimensional Lie group with identity element e. We assume G to be connected since all analysis takes place on the connected component of the

identity. Let g = TeG, the tangent space of G at the identity, denote the Lie algebra of G. If a E TeG and a is the unique left invariant vector field on G such that ale = a, then the Lie bracket on g is defined by [a, b] = [a, b]le. A representation r of a Lie algebra g on a vector space W is a homomorphism of g into the algebra L(W) of endomorphisms of W. The adjoint representation of the algebra g on the vector space g is then defined by a i-- ada E G(g) where (ada)(b) = [a, b] for all b E g. If confusion is possible we write ado instead of ad. Then ad is a finite-dimensional representation of g since

[ada, adb] = ad([a, b]) by the Jacobi identity. The exponential map exp: g -+ G is an analytic diffeomorphism from a neighbourhood of zero in g onto a neighbourhood of the identity

in G. If a, b E g are close to zero in g then there exists a unique c E g such that

exp c = exp a exp b. Then c = a + b + 2-1 [a, b] + ... is a power series in multicommutators of a and b given by the Campbell-Baker-Hausdorff formula. We need some elements of decomposition theory of Lie algebras and Lie groups. We first consider the Lie algebras.

If 9(1) = g and g(k+t) = [g(k), 9(k)] is defined inductively, then g is called solvable if there exists a k E N such that 9(k) = (0). The radical q of g is the unique solvable ideal which contains every solvable ideal of g and g is defined to be semisimple if q = (O). Alternatively, if one sets gi = g and defines 9k+l = [9, 9k1.

(11. 1)

then the gk are ideals which form a decreasing sequence. The Lie algebra g is

called nilpotent if there exists a k E N such that 9k = (0). If g is nilpotent, then there is an ro, the rank of nilpotency, such that 9ro+l = {O} but gro 96 (0). The family (9k) is called the lower central series of g. There are four relevant subclasses of nilpotent Lie algebras with additional structure. First, a Lie algebra g is called homogeneous if there exists a family y,,, with

u > 0, of automorphisms of g such that yu = e't 1ogu for all u, where A is a diagonalizable linear transformation of g with strictly positive eigenvalues. Observe that y,,yv = y,,,, for all u, v > 0. The y,, are called dilations and we always

11.1 Lie groups and Lie algebras

9

assume that the eigenvalues of A are at least one. Secondly, the Lie algebra g is defined to be graded if it has a vector space decomposition g = ®k=t 4k such that IN, 41] c 4k+1. Then we call {4k} a grading for g. Thirdly, a graded Lie algebra is defined to be stratified if 4 t generates 9. We call NO } a stratification for g. Then each graded Lie algebra is automatically homogeneous with dilations

(y. )..o defined by y (a) = u k a for all u > 0, k E ( 1, ..., r) and a E N. Moreover, each homogeneous Lie algebra is nilpotent. Fourthly, if d', r E N, then the nilpotent Lie algebra with d' generators which Is free of step r is defined as the quotient of the free Lie algebra with d' generators by the ideal generated by the commutators of order at least r + 1. It will be denoted by 9(d', r). The algebra g(d', r) is stratified and has the following fundamental properties. II.1.1 If g is a nilpotent Lie algebra with rank at most r and el, ..., ed, E g, then there exists a unique homomorphism n: g(d', r) - 9 such that 7r(ak) = ek for all k E { 1, ... , d'}, where al, ..., ads are the generators of 9(d', r).

II.1.2 If al , ... , ad, are the generators of g(d', r) and E E G(span(a i , ... , ad')) is a bijection, then there exists a unique automorphism T: g(d', r) -* g(d', r) such that Tak = Eak for all k E { 1, ... , d'). An endomorphism L E G(9) is called nilpotent if there exists an n E N such that L" = 0. The nilradical n of a Lie algebra 9 is the maximal ideal of g such that adn is nilpotent for all n E n. Thus n is the nilpotent ideal of g which contains every nilpotent ideal of 9. Then n C q and n is also the nilradical of q.

II.13 If g is semisimple and 4 is an ideal of 9, then there exists an ideal 41 such that 9 = 4 ® 41 and [4, 41] _ (0). Moreover, both 4 and 9/11 are semisimple. This statement has a direct corollary.

II.1.4 If 9 is semisimple, then 9 = [g, 9]. A Levi subalgebra of g is a semisimple subalgebra m such that g = m ® q. A Cartan subalgebra of 9 is a nilpotent subalgebra 4 which is its own normalizer, i.e., it has the property that [a, h] c 4 with a E g implies a E 1). 11.1.5 Levi subalgebras and Cartan subalgebras always exist. In general, these subalgebras are not unique. But two distinct ones are conjugate in the following circumstances.

11.1.6 If ml and m2 are two Levi subalgebras of 9, then there exists a 4 E Gy such that 4 )(ml) = m2, where Gp is the subgroup of GL(g) generated by eadb

with bEp=[q,9]. If 41 and 42 are two Cartan subalgebras of a solvable Lie algebra g, then there exists a b E [g, 91 such that eadb4i = 42. Maximal semisimple subalgebras are Levi subalgebras by the following statement.

10

II. General Formalism

11.1.7 If m is a Levi subalgebra for g and his a semisimple subalgebra of g, then

there exists a ) E Gp such that ''(h) C m, where GP is the subgroup of GL(g) generated by eadb with b E p = [q, g]. There are several elementary algebraic results which we regularly use.

11.1.8 If a E q, then a E n if and only if ada is nilpotent.

A subset V of G(g) is called nilpotent if there exists an n E N such that

L ... L = 0 for all L1, ... , L E V. Hence adn is a nilpotent subspace of C(g), the endomorphisms of g. An endomorphism D of g is called a derivation if

D([a, b]) = [D(a), b] + [a, D(b)] for all a, b E g. In particular ada is a derivation for each a E 9. 11.1.9 Any derivation of g, or of the radical q, maps q into the nilradical it

It follows from 11.1.9 that [9, q] c n. In particular [q, q] c n and [m, q] c n for each Levi subalgebra m.

If W is a vector space and S E G(W), then the endomorphism S is called semisimple if each S-invariant subspace U, i.e., a subspace such that SU c U, has an S-invariant complementary subspace V. Thus W = U ® V and SV e V. II.1.10 Each derivation ada, with a E g, has a Jordan decomposition, ada = S(a) + K(a), in terms of a semisimple endomorphism S(a) and a nilpotent endomorphism K (a) of g which mutually commute. The S(a) and K (a) are uniquely

determined by these properties. Moreover, S(a) and K(a) are derivations and there are real polynomials s and k, without constant terms, such that S(a) = s(ada)

and

K(a) = k(ada).

Since [g, q] c n it follows that S(g)q c n and K(g)q c n. We also need some basic properties of representations.

If r is a representation of g on a vector space W, then a subspace U C W is called r-invariant if r(a)U e_ U for all a E g. A representation r is defined to be semisimple if each r-invariant subspace U CW has a r-invariant complementary subspace V. Thus W = V ® U for some subspace V which is invariant under r. There is one practical characterization of semisimplicity. II.1.11 A finite-dimensional representation r is semisimple if and only if the representatives r(a) are semisimple endomorphisms for every a in the radical q of g. In particular, each representation of a semisimple algebra is semisimple. Next

if g and h are Lie algebras and r: 4

G(g) is a representation of 4 in 9 by

11.1 Lie groups and Lie algebras

11

derivations the semidirect product g x Il is usually defined as the Lie algebra obtained by equipping the vector space g ® I1 with the Lie bracket

[(a, b), (a', b')]exh = ([a, a']9 + r(b)a' - r(b)a, [b, b']h)

(11.2)

where a, a' E g and b, b' E h. For our purposes it is, however, more convenient to introduce a semidirect product l x g as the vector space h ® g equipped with the Lie bracket

[(b, a), (b', a')]hxg = ([b, b']b, [a, a']0 + r(b)a' - r(b')a)

(11.3)

for a, a' E 9 and b, b' E 11. Then (0) x g is an ideal in Il a g, naturally isomorphic with g. We frequently identify a E g with (0, a) E h a g. Similarly, we frequently identify b E h with (b, 0) E ry x g. If r = 0 both definitions give the Lie algebra as a direct sum of the Lie algebras g and h. Finally we consider structure theory of Lie groups. A (connected) Lie group is called solvable, nilpotent or semisimple if its Lie algebra is solvable, nilpotent or semisimple. A Lie group is called homogeneous or stratified if it is (connected) simply connected and its Lie algebra is homogeneous or stratified. If G is homogeneous dilations, y of the Lie algebra lift to maps r,, = exp oy,, o exp-I which are group automorphisms of G, also referred to as dilations. Clearly [' r, = I',,,,

for all u, v > 0. The nilpotent Lie group on d' generators free of step r is defined to be the connected, simply connected, Lie group G(d', r) with Lie algebra g(d', r). It is a non-compact stratified Lie group since the Lie algebra g(d', r) is stratified. Homomorphisms of Lie algebras lift to homomorphisms of the corresponding Lie groups.

11.1.12 Let G' be a simply connected Lie group with Lie algebra g' and G a (connected) Lie group with Lie algebra g. If Jr: 9' -* 9 is a Lie algebra homomorphism, then there exists a unique Lie group homomorphism A: G' -+ G which lifts n, i.e., A o expG, = expG on where expG, and expG are the exponential maps g' -+ G' and 9 -+ G, respectively. Moreover, if it is surjective, then A is surjective.

In particular, 11. 1. 12 applies to the situation of 11.1.1 with G' = G(d', r) and

' = 9(d', r). 11.1.13 If g is the Lie algebra of a (connected) Lie group G and Il is a subalgebra of g, then there exists a unique connected subgroup H of G such that H has Lie algebra h.

II.1.14 If G is nilpotent, then the exponential map is surjective. If in addition, G is simply connected, then the exponential map is a diffeomorphism from g onto G. Moreover i f bi, ... , bd is a basis f o r g such that span(bk, ... , bd) is a subal-

gebra o f 9 for all k E (1, ... , d), then (ii, td) i--+ exp(t1 bi) ... exp(tdbd) is a diffeomorphism from Rd onto G.

12

II. General Formalism

11.1.15 If G is nilpotent and simply connected, then any (connected) Lie subgroup of G is closed and simply connected.

11.1.16 If G is a simply connected group and H is a (connected) normal Lie subgroup of G, then H is simply connected and closed. Moreover, the quotient group G/H is also simply connected. 11.1.17 If m is a Levi subalgebra of g and M and Q are the unique (connected) subgroups of G which have Lie algebras m and q, then the group Q is closed in G and G = M Q, where q is the radical of g. If G is simply connected, then M and Q are also simply connected, M is closed and M fl Q = {e}.

We call the groups Q and M in 11.1.17 the radical and and Levi subgroup of G. Similarly, the unique (connected) subgroup of G which has Lie algebra n is called the nilradical of G. In order to obtain results for general (connected) Lie groups from simply connected Lie groups a covering property is useful.

11.1.18 If G is a (connected) Lie group with Lie algebra g, then there exists a unique, up to an isomorphism, simply connected Lie group G with Lie algebra g 0 0 0 and a surjective homomorphism A: G -+ G. The kernel ker A is discrete. The group G is called the covering group of G. 11.1.19 A discrete normal subgroup of a Lie group G is contained in the centre

Z(G) = (g E G : gh = hg for all h E G}. Moreover, exp3(g) C Z(G), where 3(g) _ (a E g : [a, g) _ (0)} is the centre of g. 11.1.20 If G is nilpotent and simply connected, then exp3(g) = Z(G). 11.1.21 If G is nilpotent, then G contains a compact subgroup H, which is maxiasal in the sense that it contains every compact subgroup of G. Then the Lie algebra of H is contained in the centre of g, the subgroup H is a subgroup of the centre of G and the group G/H is simply connected. Moreover, if G is 0 0 the covering group of G and A: G -+ G is the natural homomorphism, then 0

1! = expG span loge, ker A.

Integration on a Lie group is possible in a basically unique manner by the next statement.

11.1.22 There exists a unique, up to a constant, non-trivial left invariant Borel measure dg on G.

We call dg the Haar measure. If G is compact, then we normalize the Haar measure such that G has total measure one. The modular function A: G -> (0, oo) is defined such that

fG

A(h)-t fdgco(g)

11.2 Subelliptic operators

13

for all h E G and tp E Cc (G), the space of all continuous complex-valued functions on G with compact support. If dg is right invariant, then A is the identity function and the group is unimodular. 11. 1.23 If (P is an automorphism of G, then w o (D E L I (G) and

=

JG

IdetnIfdgco(g)

for alltp E Ll(G), where 7r is theautomorphismofggiven byexpna = 4)(expa) for all a E 9. The modular function can be calculated as a corollary.

11.1.24 If h E G, then A(h) = detAd(h-l). II.1.25 If G is nilpotent and simply connected, then there exists a constant c > 0 such that

f dg w(g) = c J da p(exp a)

(11.4)

8

for all w E Cc (G), where da is a Lebesgue measure on 9.

11.2

Subelliptic operators

Let G be a Lie group with Haar measure dg. Most of the subsequent analysis is on function spaces over the group and is related to the representation of the group by left or right translations. The group G is usually unimodular and dg is bi-invariant. Let LP, or L p (G), denote the usual spaces L p (G ; dg), of complexvalued functions. The norm on LP is denoted by II ' Ilp. e.g., Ilwllp =

if p E [1, 00). Moreover, II

'

(fdIw8w'

l/p

)

II per denotes the norm of a bounded operator from

LP to L, e.g., IIXIIp-r = sup{IIXwIIr : IIwIIp < 1} for each bounded operator X from LP to Lr. The representation Lc of G by left translations on LP, the left regular representation, is defined by

(Lc(h)(p)(g) = w(h-1g)

for all h E G, V E LP and almost every g E G. Similarly, the right regular representation RG is defined by

(Rc(h)(p)(g) = w(gh)

14

11. General Formalism

for all h E G and almost every g E G. We make no notational distinction between the representation of LG and RG on different Lp-spaces and set L = LG and R = RG if there is no confusion about the group involved. Note that L and R are strongly continuous if p E [1, oo) and weakly' continuous if p = coo. Moreover, the representations L and R of G on LP are the dual to the corresponding

representations on Lq if p and q are dual exponents, i.e., if p-1 + q-' = 1. In addition to the LP-spaces we will also need some of the other standard function spaces such as the space of complex continuous functions C(G), the bounded continuous functions Cb(G), the infinitely-differentiable functions C' (G), the infinitely-differentiable functions with compact support C°O(G), and so on. One can use the exponential map to define representations d L and d R of g on the La-spaces by differentiation of the representations L and R of G. For example, for each a E g the left representative dL(a) is defined as the generator of the continuous one-parameter subgroup t H L(exp(-ta)). Formally, e-rdL(a)p = L(exp(-ta))tp

for all t E R. Alternatively stated, dL(a) is the derivative of t H L(exp(ta)) at t = 0 where the derivative is in the strong sense on the La-spaces with p E 11, 00) and the weak' sense on Loo. The operators are closed and densely defined in the

appropriate topologies. Moreover, each dL(a) is automatically invariant under right translations, i.e., the domain of the operator is invariant and [dL(a), R(g)] =

0 for all g E G. It follows from the definition that a H dL(a) is linear and from the multiplicative properties of the exponential map, i.e., from the BakerCampbell-Hausdorff formula, that [dL(a), dL(b)]rp = dL([a, b))cp

for all a, b E g and suitably smooth cp E LP(G). Since the operators are unbounded some care has to be taken with domains. Then, however, restricted to a

suitable domain D, it follows that a H dL(a)ID is a representation of g in D. One can define right representatives dR(a) of the Lie algebra in a similar manner but we work mainly with the right invariant left representatives.

The theory of subelliptic operators is constructed with a Lie algebraic basis al, ... , ad' of g, i.e., a finite sequence of linearly independent elements of g whose Lie algebra is equal to g. In fact the linear independence is not important and one could formulate the theory in terms of a generating basis, i.e., a finite sequence of elements of g with Lie algebra equal to g. We will, however, assume throughout that al, ... , ads is an algebraic basis. Thus there is an integer r such that al, . . . , ad', together with all commutators [ai, , [ail, ... [a;,,_, , aie }}}, with ij E ( 1, . . . , d'), where n < r, span the vector space g. The smallest integer r with this property is called the rank of the algebraic basis and in particular a vector space basis has rank one. For the sake of simplicity we adopt the notation Ai = d L(ai) for the representatives of the algebraic basis. In the sequel we need to examine multiple derivatives. Hence we immediately introduce a suitable

multi-index notation.

I1.2 Subelliptic operators

15

If n E No set n

Jn(d') = U{1, ... , d')k

U{1, ... , d'}k.

and

k=0

k=1

Further set 00

00

J(d')=U{1,...,d'}k

J+(d')=U{1,...,d'}k.

and

k=0

k=1

Then define the length of the multi-index a E { 1, ... , d'}" by lal = n and for a = in) with n > 1 set alal = [ai,, [ai2,... [ai._,, a;,,]]] and Aa = Ai, ... A;,,. We adopt the convention that Aa = /, the identity operator, if lal = 0. The Sobolev space or C"-subspace L'p;n = L'p;" (G) of the representation of G by left translations on LP is then defined as the common domain of all monomials

Aa with lal < n formed with respect to the algebraic basis a1, ... , ad'. The corresponding C"-norm is defined by Ilwllp;n = am ad,) II Aawll p

If al, ... , ad is a vector space basis, then we denote the corresponding spaces by Lp;n, II II p,n The normed spaces Lp;n are independent of the choice of the full vector space basis a 1, ... , ad, up to equivalence of norms. Finally, set L P;00 nn=1 Lp;n. It turns out that Lp;00= n,° 1 L'p;n for any algebraic basis. In the subsequent analysis it is also necessary to consider continuous representations other than the left and right regular representations. The foregoing definitions extend in a natural manner. Let (X, G, U) denote a strongly or weakly* continuous representation of G on the Banach space X by bounded operators g t-- U(g). If a E 9, then the representative dU(a) denotes the generator of

the one-parameter subgroup t ra U(exp(-ta)) of the representation. The C"subspace X;, = X,'(U) of C"-elements of the representation (X, G, U), with respect to the algebraic basis a I, ... , ad' of g, is the common domain of all monoin ), in the generators mials d U(a)a = d U (a;,) ... d U (a;,, ), where a = (i1,

dU(a1),... , dU(ad') with norm defined by IlxII', =

sup

IIdU(a)'xll

Again we delete the prime if a, , ... , ad, is a vector space basis. Set XOO = Xo.(U) = nn=1 X,,, the space of CO0-elements for U. One can associate with each such representation a dual representation. Let F denote the dual X* of X if the representation is strongly continuous and the predual X. in the case of weak* continuity. Then the dual representation (F, G, U*) is defined by setting

U*(g) =

U(g-I)*

16

II. General Formalism

where U(h)` on T denotes the adjoint of U (h) on X. If the original representation

is strongly continuous, then the dual representation is weakly' continuous and vice-versa.

The complex subelliptic operator H is now defined on the Lp-spaces in terms of the algebraic basis a 1, ... , ad, and a d' x d'-matrix C with entries ckl E C

satisfying the positive-definiteness condition 91C = 2-1(C + C') > uI with A > 0. There are two different approaches to the definition: an operator approach and a form approach. First, one can define an operator d'

Hp = - L Ckl Ak Al k.1=1

with domain D(Hp) = L'p;, on each of the Lp-spaces. The operator is densely defined and since it has a densely defined adjoint it is also closable. (These and

subsequent statements must be interpreted in the weak' sense if p = oo.) In fact with the aid of elliptic regularity arguments one can establish that H is closed on L' ;2 if p E (1, oo) and it generates a continuous semigroup S(p) on Lp (see Notes and Remarks). Moreover, the closures of H1 and Hm generate semigroups SM and S. It is a common practise, and one which we adopt, to identify the family of operators H = (Hp)pElt.ool as a single operator, the subelliptic operator corresponding to the left regular representation, the algebraic basis and the matrix of coefficients C. Since one clearly has Hpw = Hrrp for all rp E Lp;2 fl Lr;2 and p, r E [ 1, oo] this does not lead to any ambiguity. Similarly, the family S = is identified as a single semigroup, the subelliptic semigroup generated by H. If the operator and semigroup are constructed in this manner from a vector space basis a 1 , ... , ad of g, then H and S are referred to as strongly elliptic. In both cases the best possible constant µ is called the ellipticity 1

constant of H. The second way of defining the subelliptic operator is through the sesquilinear form h on L2(G) defined by setting D(h) = L2;1 and d'

h(*, (p) = L ckl(Ak*, A,(p) k.1=1

for all *. O E D(h). If h is also used to denote the associated quadratic form, h((p) = h((p, gyp), then d'

Re h(zp) > it L IIAkw11'2

(11.5)

k=1

where µ is the smallest eigenvalue of the hermitian matrix %C. It follows that the quadratic form Re h is positive and closed. In addition one has the estimates Reh((p) > A IICII-1 I Imh(w)I

(11.6)

for all (p E D(h) with IICII = sup{I k.1=1 E Cd' and I I2 < l} the usual matrix norm. Therefore h is a closed sectorial form with vertex at the origin

11.2 Subelliptic operators

17

and semiangle 9h > 9c = arctan(z IICII-') Hence there exists a closed sectorial operator H on L2 such that D(H) C L2:1 and (u', HW) = h(*, (G)

for all * E D(h) and co E D(H). Note that the adjoint form h' defined by h'((p) = h(W) satisfies the same estimates as h. The associated operator is the adjoint H' of H. It is the subelliptic operator with matrix of coefficients C' = (ckt), with ck1 = Elk, and it generates the adjoint semigroup S'. Since the form h satisfies the sectorial estimates (11.6) it follows from the gen-

eral theory of sectorial operators (see Notes and Remarks) that H generates a strongly continuous contraction semigroup S on L2 which is holomorphic in the sector A(9h) = {z E C\{0} : IargzI < 9h}. Specifically, t H S, extends to a holomorphic family z H SZ of bounded operators with the semigroup property SzjSz2 = Sz,+z2 for all Z1, Z2 E 0(9h). The subellipticity implies, however, that S is a bounded holomorphic semigroup, i.e., the norm II Sz 112-2 is uniformly bounded for all z in a non-empty subsector of the sector of holomorphy. In fact, if 46 E (-Oh, 9h), then the generator of the continuous semigroup t -+ Sfe,m is given

by HO = e''H and for 0 E (-9c, Oc) one has d'

Ree'"h(,p) =cos0Reh(cp) - Isin 0111mh((p)I > µm IIAkw112 k=1

with Ao = µ cos 0 - IICII sin 0 > 0. Therefore Hi is subelliptic for 101 < 9c 1 for all z E 0(9c). In particular S is a bounded holomorphic

and IISz112_+2

semigroup on L2. There are several useful L2-estimates for H and S which follow from general semigroup theory.

II.2.1 For each n E N there is a c, > 0, which is independent of H, such that

IIHmWII2 0 such that IIHySz112-2 0.

20

Il. General Formalism

The form approach to the definition of H immediately establishes that S is holomorphic on L2. Then it follows from complex interpolation theory that it extends from L2 fl LP to a bounded holomorphic semigroup on LP for each p E (1, oo). Thus the advantage of the form approach is that it immediately establishes the existence and holomorphy of the subelliptic semigroup on the Lu-spaces, with the exception of the end points. Nevertheless, the best estimate on the semiangle of the sector of holomorphy which can be obtained by such an abstract argument shrinks to zero as p tends to 1 or to oo.

II.3

Subelliptic kernels

Detailed analysis of the subelliptic semigroup needs more sophisticated arguments than general regularity theory or the theory of sectorial operators. It is here that the semigroup kernel plays the first important role. One can associate with each continuous, right-invariant, semigroup S on L2(G) a distribution K such that dg dh ¢(g) K,(h) r (h-I g) St(P) =

fcf

for all >G, rp E CO0(G). But the distribution character of K limits the utility of this representation in general. The particular structure of the subelliptic semigroup en-

sures, however, that the corresponding K is a bounded integrable function. The basic existence result underlying the subsequent analysis is summarized as follows.

Proposition 11.3.1 Let S be the continuous semigroup generated by the complex subelliptic operator H on the L p-spaces over the Lie group G. Then for all t > 0

there exists a function Kt E Li fl L,,, which is infinitely-often left and right differentiable, such that

(St(P)(g) = JG d h K,(h) ip(hg) for all p E [ 1, 00], tp E Lp and g E G. The K, form a semigroup with respect to the convolution product *,

Ks+t(g) = (Ks * Kt)(g) = fG&1 K(h) Kt(hg) for all s, t > 0 and g E G. The convolution semigroup K = {Kt}t>p is referred to as the kernel K of the semigroup S. It is the principal tool in analysis of the behaviour of the semigroup S. both locally and asymptotically. The adjoint semigroup S* has an analogous

kernel given by K, (g) = Kt(g-1) O(g)-1 where 0 is the modular function. In particular for unimodular groups K, (g) = Kt(g-I).

11.4 Growth properties

21

A principal element in establishing Proposition 11.3.1 is the proof that S is a continuous semigroup on the Lp-spaces and that each St with t > 0 is bounded as an operator from L 1 to Lro. The boundedness of II St II t-.oo and II St Iloo-.oo are

both necessary and sufficient to establish the existence of the kernel as a function in Lt fl Lro because IIStlll oo = IIKtlloo and IIS,Iloo-oo = IIKtlll. The boundedness and integrability of the kernel can be achieved by a variety of estimation procedures which also give estimates on growth properties. The relevant small t results will be summarized in Section 11.6. But first we note that there are some natural limitations which occur for complex operators. Although S is contractive on L2 a similar simple property cannot be expected for complex operators on the other Lp-spaces. This is a characteristic complication of the complex theory as the following result illustrates.

Proposition II.3.2 The following conditions are equivalent. The coefficients ckl of H are real.

1.

The semigroup S is positive, i.e., it naps positive functions into positive functions.

11.

III.

The semigroup S extends to a contraction semigroup on LI (G).

IV.

The semigroup S extends to a contraction semigroup on Loo(G).

V.

The kernel K, is positive for each t > 0.

Note that the proposition does not require the matrix of coefficients C = (ckt) to be symmetric. Much of the analysis of real subelliptic operators relies on the positivity and L I contractivity of the semigroup. A full understanding of the properties of complex operators requires a more detailed analysis of the kernels. The same is true for operators of higher order but that is not the subject of the current text. Example 11.3.3 If G = Rd and al, ... , ad is a Cartesian basis, then Ak = -ak = -a/axk

on Lp(Rd). Therefore H = - Ek1=t cklakal is a multiplication operator in the Fourier variables. If (o E L2 and N denotes its Fourier transform, then (IIp)(p) = w(p)ip(p) with w(p) = (p, Cp). Hence

Kr(x)= J

ddpe-(v,Cp)re-ip.x =a(4rrt)-d/2e-(x,C-tx)(ar)-t

R

where a-t = (4n)-d/2 f e-(x,C-t x)/a in particular, if the coefficients ckt are real, a = I

det9tCI-1 /2. Note that it follows from the explicit expression for the kernel that it is indeed

integrable. Moreover, it has an integrable holomorphic extension to the sector 0(0c).

II.4

Growth properties

In order to explain and discuss growth estimates on the semigroup kernel K it is necessary to introduce a metric on the group, i.e., to define a distance function. There is a canonical procedure to associate a right invariant distance with each

II. General Formalism

22

algebraic basis al, ... , ad,. First consider a full vector space basis al, ... , ad. If y: [0, 1] --> G is an absolutely continuous path from g E G to h E G, then there are tangential coordinates yk such that d

Y(t) _

Yk(t) AkIy(t) k=1

for almost every t E [0, 11. The natural definition of the distance from g to h is given in terms of these coordinates by d

I

d(g ; h)

dt

inf Y(1)=g

fo

(k=1

1/2

yk(t)2Y(0)=h

and the corresponding modulus of g E G by Ig I = d (g ; e). The distance d (g ; h) is the length of the shortest path from g to h and the modulus IgI of g is the length

of the shortest path from g to e. The distance is right invariant, d(gk ; hk) _ d(g; h), because the Ak commute with right translations. Consequently, d (g ; h) _ Igh-1 I for all g, h E G. The distance and modulus are, up to equivalence, independent of the choice of basis. 11.4.1 Let I l a and I I b denote the moduli associated with the bases al, ..., ad and b1, ... , bd, respectively. Then there is a c > 0 such that c-1181a 0 such that (11.15)

c- tIgI _< Ig1' 0 and strictly positive real numbers C, c > 0 such that

cP° < V, (P) < C

P

D

II. General Formalism

26

uniformly for all p > 1, i.e., V'(p) : p'3 for p > 1. Groups of this type are called groups of polynomial growth. The integer D is referred to as the dimension at infinity of the group. It is independent of the choice of algebraic basis and there is no direct relation between D and and the local dimension D'. The case D = 0, i.e., the case of uniformly bounded volume, corresponds to the group being compact. The second possibility is that there are A, µ > 0 and C, c > 0 such that

ceaP < V'(p) IBIgn _ (A(g))n V'(1) which grows exponentially as n - oo. This observation has an important implication.

II.4.5 Each group of polynomial growth is unimodular.

There is a characterization of polynomial growth of the group G in terms of spectral properties of the Lie algebra which is particularly useful throughout the subsequent analysis. The Lie algebra g is defined to be of type R if each operator a H ada has pure imaginary eigenvalues, i.e., there are no eigenvalues with nonzero real part. Theorem 11.4.6 The Lie group G has polynomial growth if and only if its Lie algebra 9 is of type R. Note that the modular function of the group G is given by

A(expa) = dete-ada = e - ReTr(ada) for all a in the Lie algebra g (see 11. 1.24). Thus if 9 is of type R, then Re Tr(ada) = 0 and the group is unimodular in confirmation of 11.4.5. But this calculation indicates that the converse of 11.4.5 is not valid. There are unimodular groups which have exponential growth. Example 11.4.7 Let q be the three-dimensional Lie algebra with a basis a1, a,,,, a3 such that [at , a2] = a2. [at, a3 l = -a3 and all other commutators zero. The corresponding Lie group is unimodular, because Tr(ada) = 0 for all a E q, but the algebra is not of type R, since at has eigenvalues ±1. In particular, the group must have exponential growth.

Although our principal interest is in non-compact groups of polynomial growth

compact groups enter the analysis through structure theory (see Chapter III).

11.4 Growth properties

27

Therefore it is useful to have criteria for compactness of a Lie group and characterizations of the corresponding Lie algebras. First recall that a bilinear form

f over a Lie algebra g is defined to be symmetric if f (b, c) = f (c, b) for all b, c E g and invariant if f ((ada) (b), c) = - f (b, (ada)(c)) for all a, b, c E g.

Proposition H.4.8 Let g be a Lie algebra. 1.

There exists a compact Lie group G such that g is the Lie algebra of G if and only if there exists a positive-definite invariant symmetric bilinear form on g.

II.

If g is semisimple and G is any (connected) Lie group with Lie algebra g, then G is compact if and only if the invariant symmetric form (a, b) H -Tr((ada)(adb))

is positive-definite.

III.

If G is a semisimple Lie group with Lie algebra g, then G is compact if and only if g is of type R.

Proof For Statements I and II see Notes and Remarks. The proof of Statement III is based on the type R spectral characterization of polynomial growth, 11.4.6. If G is compact, then G has polynomial growth and g is of type R. Alternatively, if g is of type R, then the form in Statement II is positive semi-definite and hence, by semi-simplicity, it is positive-definite. Therefore G is compact by Statement 11.0 Example II.4.9 Consider the three-dimensional Lie algebra with a basis at , a2, a3 whose

non-zero commutators are [at , all = a3, [a2, a3l =a I and [a3, at I= a2. If a= a tat + a2a2 + A3a3 with a 1, 12, a3 E R, then ada =

0

-Z3

l3

0

-at

X2

at

0

-a2

This matrix is scmisimple and has cigcnvalues 0, ±i (X2 + z2 + X2)1/2. Therefore the corresponding group is compact. The group is the three-dimensional group of rotations with the aj corresponding to infinitesimal rotations.

There are a number of structural properties of groups of polynomial growth which follow straightforwardly.

II.4.10 If G has polynomial growth, then each closed subgroup has polynomial growth. Moreover if H is a closed normal subgroup of G, then G/H also has polynomial growth. More specifically one has the following useful result.

28

H. General Formalism

Proposition 11.4.11 Let M be a semisimple Lie group which has a representation by automorphisms on a Lie group H and let G denote the corresponding semidirect product. Then the following conditions are equivalent. 1.

G has polynomial growth.

II.

M is compact and H has polynomial growth.

Moreover, if these conditions are valid, then the dimensions at infinity of G and H are equal. Theorem 11.4.6 and Proposition 11.4.11 have one immediate corollary which is regularly used in the sequel.

Corollary 11.4.12 Let g be a Lie algebra with radical q and let m be a Levi subalgebra. The following conditions are equivalent. 1.

g is of type R.

II.

q and m are of type R.

All nilpotent Lie groups are groups of polynomial growth since the corresponding Lie algebras are automatically of type R. In fact the only eigenvalues of the ada are zero. This follows because nilpotency implies that for each a E g there is an n E N such that (ada)" = 0. Hence if (ada)b = xb, then (ada)"b = 7A"b = 0 and A = 0. Using 11. 1.21 we can reduce the calculation of the dimension at infinity of a general connected nilpotent Lie group to that of a simply connected group.

11.4.13 Let H be the maximal compact subgroup of the nilpotent group G, and let I I and I I 1 be moduli (associated with vector space bases) on the groups G and G/H, respectively. Let jr: G -> G/H be the natural snap. Then there is a c > 0 such that c-1ingll IgI < c(Ingll + 1) for all g E G. Therefore the dimensions at infinity of G and the simply connected group G/H are equal. The calculation of the dimension at infinity of a simply connected nilpotent Lie

group is similar to the computation of the local dimension D' of the algebraic basis al, ... , ad,. Let {gk} denote the lower central series of the corresponding nilpotent Lie algebra g. Now one has a direct sum decomposition

g = 1)1 ®... (B hro

(11.23)

of the vector space g such that gk = 1)k ® gk+1. Clearly d = Tk 1 dim hk

11.4.14 If G is simply connected and nilpotent, the dimension at infinity D is given by ro

D=

Ekdiml)k.

k=1

29

11.4 Growth properties

Therefore D > d with equality if and only if G = Rd.

The growth properties of stratified groups are particularly simple because of the dilation structure.

II.4.15 If G is a stratified group with Lie algebra g and al , ... , ad, is a basis of the generating subspace 41 C_ g, then the local dimension D' and the dimension at infinity D are equal. Moreover, I ru (g) I' = u Ig I' for all g E G and u > 0 and (II.24)

1 dg (p(ru(g)) = uD fG dg (p(g) c

The equality of the dimensions follows since the decomposition (11.21) of the Lie algebra corresponding to the basis a1, ... , ad' is identical to the decomposition (11.23) corresponding to the lower central series. The scaling property of the modulus is an easy consequence of its definition, or of the characterization (11.18),

since y a (ak) = u ak f o r all k E { 1, ... , d'}. The scaling property of the integral is a corollary of (II.4). The unique dimension D is called the homogeneous dimension of G. Example 11.4.16 The connected, simply connected, Heisenberg group 113 is topologically equivalent to R3 with the product

x Y = (xl + YI, X2 - YIx2 +

Y3)

Its Lie algebra h has a basis a I, a2, a3 with one non-zero commutator [a 1, a2) = a3. The isomorphism between R3 and H3 is given by x r. fi(x), where m(x) = exp(x1al)exp(x2a2)exp(x3a3).

The Lie algebra his nilpotent of rank 2 and hl = span(aI, a2), 112 = span(a3). The dimension at infinity D = 4. Moreover, the subspaces h,, h'2 of the algebraic basis (a,, a2) arc also given by h, = span(a1, a2) = hI. h2 = span(a3) = V2. Hcnce the corresponding local dimension D' = 4.

The Heisenberg group is a simple example of a homogeneous group. In fact it is a stratified group.

is the modulus 11.4.17 Suppose G is simply connected and nilpotent and I associated with a basis a 1, ... , ad for g passing through 41, ... , 4ro. I

1.

There exists a c > 0 such that Ila II < c I exp a I for all a E g with Ila II ?

where 11 Ed 1 ;a; II =

Ed

i

1,

l l/W' and wi = k if ai E 4k. Moreover,

there exists a c' > 0 such that I exp a 1 :5 c' Ila II for all a E g. II.

There exists a c > 0 such that II II W < c I exp 1;1 a i ... exp tdad I for all d l/ E Rd with Iexpt;iai ...expldadI > 1 , where 1 1 11

and wi = k if a;

.

= :i

1

i

i

'

4k. Moreover, there exists a c' > 0 such that Iexpl;laI...exptdadI forallt; E Rd. E

30

II. General Formalism

Proof For the proof of Statement I see Notes and Remarks. It follows from the Campbell-Baker-Hausdorff formula that there exists a c > 0 such that II log(exp a exp b) II < c p

for all p > 1 and a, b E g with Ila II < p and Ilbll < p. Then, by induction on k, it follows that for all k E (I, ... , d - 1) there exists a c > 0 such that for all for

all p > 1 and

E Rd with IJlle < p and l = ... _ k-l = 0 there exists an

r) E Rd such that d

d

i=k

i=k+I

giai),

IInIIw < cp and ni = ... = 77k = 0. Hence there exists a ci > 0 such that for all p > 1 and a E g with IlaII < p there exists a E Rd such that expa = exp ! I al ... exp l;dad and II < cl p. Note that is unique by StateII

cI Il log(expl;ial ... expldad)II for all

ment 11.1.14. Hence

E Rd with

II log(expl;isi ...expldad)II ? 1. If c > 0 is as in Statement I, then IIIIIW < cct l exp t al ... exp dad I for all l: E Rd with II log(exp I a i ... exp dad) Il ?

1.

Then the first part of Statement II follows from 11. 1. 14.

The second part of Statement II follows from the second part of Statement I and the triangle inequality. Example 11.4.18 The three-dimensional group E3 of Euclidean motions in the plane consists of a rotation and two translations. It is topologically isomorphic to T x R2, and its covering group 3 is topologically isomorphic to R3, with the product x y = (xt + yl, x2 COSY I +X3 sin yt + y2, -x2 sin yt +X3 cos yt + Y3)-

The covering map from R3 to T x R2 is given by x r. (e"1, x2, x3). The corresponding Lie algebra e3 has a basis at , a2, a3 with two non-vanishing commutators [at, a2] = a3 and [at, a3) = -a2. The at corresponds to an infinitesimal rotation and the a2, a3 to infinitesimal translations. The isomorphism between T x R2 and E3 is given by x r-. 4 (x) where 4' (x) _

exp(xtat)exp(x2a2)exp(x3a3). If a = ktat + k2a2 + k3a3 with k 1, k2. k3 E R, then

/ ada = I{\

0 0 k2

0 0

-XI

0

kI

0

)

This matrix is semisimple if and only if k2 = 0 = k3. But it has eigenvalues 0, ±i X 1. Hence 0 p is of type R and the group has polynomial growth. The dimensions at infinity of E3 and E3 are 2 and 3, respectively. The algebraic bases a 1, a2 and at , a3 both have local dimension

D'=4. 0

The covering group E3 of the Euclidean motions group is the simplest group of polynomial growth which is not nilpotent. It plays a distinguished role throughout the sequel.

11.5 Real operators

31

Example 11.4.19 Let E3 denote the three-dimensional Lie group whose Lie algebra c3 has a basis a t , a2, a3 with two non-vanishing commutators [at, azl = oat + a3 and [at, a31 =

-a2 + aa3 where a E R. Then if a = ltat + 12a2 + 13a3 with 11,12,13 E It. one has 0

ada=I 13-12a -12 - 13a

0

0

1ta

-11

1t

1to

This matrix is semisimple if and only if 12 = 0 = 13. But it has eigenvalues 0, (a ± i)1t. Hence the algebra is of type R if and only if a = 0.

11.5

Real operators

In the remainder of the chapter we describe some standard estimates on the kernels of the subelliptic semigroups. The description serves partly as preparation for the subsequent analysis of global properties of groups of polynomial growth and partly as orientation. The primary type of estimate is a comparison of the semigroup kernel with the subelliptic Gaussian function (g, t) r-+

V'(t)-1/2e-b(I811)2r-1

from G x (0, oo) into R, where b > 0. The volume factor on a group G of polynomial growth one has bounds

V'(t)-112

ensures that

dg e-b(Ixl')2r-t < c

V'(t)-1/2

fG

which are uniform for all t > 0. If the group is stratified, this can be verified by scaling since I P,, (g) I' = u lg I' by 11.4.14, and in the general case by a quadrature argument. There are two types of kernel comparison: those for special classes of operators or special types of groups, and those for general operators and groups but of a local nature.

The most widely studied subelliptic operators are those with real symmetric coefficients and in particular sublaplacians - Ek_I A. Analysis of these operators is relatively simple since the semigroup is self-adjoint and the kernel K, is positive, by Proposition 11.3.2, and integrable with total integral one. The most striking early result of the theory was the proof of global Gaussian upper and lower bounds on the kernel for all groups of polynomial growth together with Gaussian upper bounds on the subelliptic derivatives AkK,. Specifically, there exist b, c, b', c' > 0 such that

c

V'(t)-1/2e-b'(Ixl1)2r-t

< K,(g) < c

V'(t)-I/2e-b(Is11)2r-1

(11.25)

for all g E G and t > 0. Positivity of the kernel is a characteristic of real operators (see Proposition 11.3.2). Hence the Gaussian lower bounds are specific to this setting. The upper bounds on the kernel are, however, of a more general character. These are basic to many further estimates and applications.

32

II. General Formalism

Once the upper bounds are established for the kernel then a fairly straightforward argument based on ellipticity gives analogous bounds on the subelliptic derivatives of the kernel. Specifically, there is an c" > 0 such that I(AkK1)(g)I
0. They provide both local, i.e., small t, and global, i.e., large t, information. In particular they show that the subelliptic derivatives introduce an extra t-1/2singularity both for large and small t. But a similar behaviour cannot be expected for multiple derivatives. Then the local and global behaviours can differ considerably. Each subelliptic derivative contributes an additional t-I/2-singularity for small t even for complex operators. This will be discussed in detail in the next section. But the global behaviour can be quite different.

The Euclidean motion group gives a simple example for which some of the second-order subelliptic derivatives only have an asymptotic t-1/2-decrease. Example II.5.1 The left regular representation of the covering group 3 of the group E3 of Euclidean motions in the plane (see Example 11.4.18) is unitarily equivalent to the representation U of R3 on L2(R3) given by (U(x)>')(y) = *(x- y). The equivalence is given by T: L2(G) -. L2(R3) with (T>fi)(x) = >'((D(x)) and 4(x) = exp(xlal)exp(x2a2)exp(x3a3)

where al, a2, a3 is the Lie algebra basis of Example 11.4.18. Setting Xi = TAj T-1, one has

X1 =-a1

,

X2=-ty32-s133

,

X3=s1 o2-cla3,

where ct (x) = cosxl and si (x) = sinxl. Thus, if H = -Af - A2 - A3 is the Laplacian = -a - a2 - a3 is the Laplacian on R3. =

corresponding to the basis, then Therefore

THT-1

lim 11/211AIA2Sr112-»2 = lim t1/211XIX2e-1A 112-+2 1-*00

1-*00

= lim II(t-112X2'XI+X3)e-AII2..2. 1-*0 0

where

(X'2V)(x) = -(cos(x1t-112)a2(V)(x) - (sin(xit-1/2 )d3(V)(x)), (X3rp)(x) = (sin(xit-1/2 )a2w)(x) + (cos(x1t-1/2)a3(p)(x) and we have used a scaling x1 " x1 t - 1/2. A simple calculation shows that the term X3 gives the only non-zero contribution to the limit and (4e)-I/2

lim 11/211AiA2St112.2 =

100

In this example the slow decrease of the second-derivative arises because of the relation X I X2 = X2X I + X3 and the asymptotic behaviour is governed by the first-order term X3. The effect is explicit. But the non-canonical behaviour can occur in a more implicit fashion. In the next example the second-derivative X = a2 has no first-order contribution, but the subelliptic operator has first-order terms when expressed as an operator on R3. 0

Example 11.5.2 Consider the three-dimensional group E3 of Euclidean motions as in the I) previous example. Let HI = -(AI - A2)2 - A2 - A3 and set S = exp(-t III). Then lim t1/211AiS;11112 .2 = too

11. General Formalism

34

This follows by noting that the calculation of the previous example gives t

lim tt/211(Al +A2)2Sr112-2 = lira I1/211AiA2S,II2-2 =

(4e)-1/2.

r- 00

oo

But one calculates that exp(a3)at exp(-a3) = al + a2, exp(a3)a2exp(-a3) = a2 and exp(a3)a3 exP(-a3) = a3. Hence IIAISit)112-2 = II(Ai +A2)2Sr112-.2

Although these examples show that some second-order derivatives have a t

-1/2_

decrease there are others which have the canonical t-l-behaviour. One readily verifies in Example 11.5.1 that all the norms II Ai AjSt 112-.2 with i # 1 decrease like t-1 as t -+ oo. Thus the asymptotic behaviour is dependent on the direction of the derivatives. This dependence will be fully explored in Chapter V but we note in passing that the canonical t-1/2-behaviour for each derivative for large t can only occur for special groups, groups which we refer to as near-nilpotent.

Proposition II.5.3 Let G be a Lie group of polynomial growth, and H a pure second-order subelliptic operator on G and S, the associated semigroup with kernel K. Assume the coefficients ckt of H are real and symmetric. The following conditions are equivalent. I.

There exists a c > 0 such that IIAkA,Sr112-.2 0andk,1 E {1,...,d'}. II.

There exist c, b > 0 such that I (AkAI K:)(g)I
0andk,1E{1,...,d'}. III.

The group G is near-nilpotent, i.e., it is the local direct product of a compact connected Lie group Go and a connected nilpotent Lie group N, i.e., G = Go N where Go and N commute and Go fl N is discrete.

The proof of this result is not straightforward. We shall give a new and independent proof for this proposition in Theorem V.3.7 and Corollary V.3.8, which is also valid for complex operators which are not necessarily symmetric.

11.6

Local bounds on kernels

In this section we return to the discussion of complex subelliptic operators and the validity of local, i.e., small t, Gaussian bounds. Good local estimates on the semigroup kernels and their derivatives are the starting point for the detailed analysis of the operators. In the case of strongly elliptic operators these estimates can be derived by the usual methods of partial differential equations. The group G

11.6 Local bounds on kernels

35

can be locally approximated by the Euclidean group Rd and H by a second-order strongly elliptic operator with constant coefficients on Rd. Then the kernel can be calculated by parametrix methods, i.e., by perturbation around the operator with constant coefficients. An important ingredient of this method is the strong ellipticity of the approximant. If one attempts to apply a similar argument to a subelliptic operator, the approximating operator is no longer strongly elliptic and one loses control of the parametrix expansion. The difficulty arises in the subelliptic situation because the local approximation of G by Rd is inappropriate. An alternative strategy is to base the local approximation on the subelliptic geometry. This can be done as follows. Consider the direct sum decomposition (1I.21) of g determined by the algebraic basis. Then define a family of linear scaling transformations y: g -+ g such that y,, (a) = uka for all u > 0 and a E (1k. Further define a Lie bracket [ , ]u: g x

g -> g for each u > 0 by [a, b]u =

Yu(b)])

It follows that the limit [a, b]o = UI o [a, b]u

(11.27)

exists for all a, b E g and defines a Lie bracket. This process of constructing a new

Lie bracket is referred to as contraction. One can verify by a direct calculation that the Lie algebra go = (g, [ , ]o), i.e., the Lie algebra g equipped with the bracket [ , ]o, is nilpotent with rank r equal to the rank of the algebraic basis. The dilations y are automorphisms of the algebra since the contraction process automatically guarantees the property Yu([a, b]o) = [Yu(a), Yu(b)]o

for all a, b E g and u > 0. So go is homogeneous. In fact go has stronger properties of homogeneity. If go = (D,'FO f)ko) denotes the decomposition of go computed by (11. 1) and (11.23) with the new Lie bracket [ , ]o, then this coincides with the decomposition (11.21) defined by the algebraic basis. In particular h(,o) generates go. Moreover, the subspaces hk = ljk ) are eigenspaces of the dilations y. Therefore, if nk: go H go is the projection onto the subspace bro), then [nk(a), ,ri(b)] = 74+1(174(a), ni(b)]). hko

In particular, the decomposition of go into the subspaces is a grading and go is a stratified Lie algebra with dilations y. Therefore there is a unique homogeneous dimension D which coincides with the local dimension with respect to any basis of ry(o) and with the dimension at infinity of the simply connected, nilpotent Lie group Go with Lie algebra go. The Haar measure on Go has the transformation

property dyu(g) = u°dg by (11.24).

36

Il. General Formalism

The significance of this construction is that Go gives a dilation invariant local approximation of G suited to the subelliptic geometry. The construction of Go can be interpreted as a local `blowup' of G relative to the subelliptic geometry. It corresponds to analysis of the geometric microstructure enlarged to a macroscopic scale. Note that in the strongly elliptic case go is commutative and Go = Rd. In fact Go = Rd if and only if the algebraic basis is a vector space basis of the Lie algebra. Therefore the subelliptic approximation procedure is a natural extension of the classical local approximation. Example 11.6.1 The Lie algebra c3 of the three-dimensional rotation group has a basis al. a2, a3 with commutators [a I, a21 = a3, [a2 a31 =a I and [a3, al I= a2 (see Example 11.4.9). The pair [al, a2) forms an algebraic basis and the scaling corresponding to the resulting decomposition (11.21) is given by yu (al) = ua l , yu (a2) = ua2 and yu(a3) = u2 a3. Therefore [al, a2lo = a3, [a2, a310 = 0 and [a3, alto = 0. Thus go is the Heisenberg Lie algebra of Example 11.4.16.

The subelliptic semigroup kernel K of H acting on G can be estimated by a parametrix expansion around the kernel K(o) corresponding to H on Go. The argument is in three steps. First, one must establish small t Gaussian bounds on the kernel K(O). In the strongly elliptic case the approximating operator on Go = Rd has constant coefficients and the kernel can be estimated by Fourier theory. In the subelliptic case the Fourier arguments are replaced by detailed properties of the representation theory of the stratified group Go. Fourier decomposition is replaced by the decomposition of the left regular representation into irreducible components. Secondly, one uses scaling arguments to extend the local bounds to global bounds. The Haar measure on Go has the transformation property uDdg. In addition I I u (g) I' = u Ig I'. Moreover, H is second-order in the a1, ... , ad', which scale as yu(ak) = uak. Therefore a simple computation establishes that the semigroup kernel satisfies the scaling identity K(o)(g)

= u°K(0)(ru(g)) u2t

(11.28)

for all t, u > 0 and g E G. In particular, K(o)(g)

= t_D12Kio)(yr-1/2(g))

for all t > 0 and g E G. Consequently, Gaussian bounds with t = 1 translate into Gaussian bounds for all t > 0. Thirdly, one needs to adapt the parametrix method to the non-commutative Lie group setting. This is relatively straightforward since the parametrix expansion is a direct analogue of the usual expansion in 'timedependent' perturbation theory, but the argument is nevertheless technically rather complex. As a result of this line of reasoning one deduces the local Gaussian bounds in the first statement of the following proposition. Note that the proposition does not require G to have polynomial growth.

I1.6 Local bounds on kernels

37

Proposition II.6.2 Let H be a complex pure second-order subelliptic operator on a connected Lie group G with associated semigroup kernel K. The following estimates are valid. 1.

There exist b > 0, co > 0 and for all a E J (d) a c > 0 such that I (AO'Kt)(g)1

c

t-Ial/2V'(t)-!/2ewte-b(IgI')2t-i

(11.29)

for all g E G and t > 0. II.

For all p E (1, oo) and n E N one has D(H"/2) = L'p;" and there exists a c > 0 such that c-1 IIwIIP;"

11(1 + H)"/2wIIp < c Ilwllp

(11.30)

for all tp E L p;".

The kernel estimates in the proposition have the optimal t singularity as t -- 0 and the correct Gaussian distribution. But the large t behaviour is dominated by the exponential factor e" which is a simple reflection of the semigroup property. In general one cannot expect these bounds to hold with w = 0 for large t for two reasons. First, the geometric decay factor V'(t)-112 is only typical for groups of polynomial growth. The asymptotic behaviour for groups of exponential growth is not of the same simple form. Secondly, the foregoing discussion of real operators t-1/2 showed that the singularity is not characteristic of each derivative as t -+ 0o even for groups of polynomial growth. In general the decay properties of the derivatives depend on their directions. Nevertheless the w = 0 bounds do hold for some large classes of groups, as we shall see below. The second statement of the proposition follows with the help of the kernel estimates by standard arguments of singular integration theory. The operators A°`(1 + H)-I°I/2 are local analogues of Riesz transforms. The direct analogue of the Riesz transforms associated with operators on Rd would be A°H-IaI/2. The addition of the identity to the operator H effectively suppresses global effects. Note that in the case of a stratified group one can deduce from (11.30), by scaling, that II Aa H -Ial/211 p,p ca, p since the identity is scale invariant. Thus the global analogues of the Riesz transforms are bounded for stratified groups. Although the parametrix arguments used to establish Proposition 11.6.2 are very useful for deriving qualitative estimates for small t, they are not appropriate for the analysis of large t behaviour. Nor do they give particularly precise quantitative information for small t. In particular they do not give good control of the parameters entering the Gaussian bounds. For example, the dependence of these constants on the group and the operator is unclear. In Section 11.10 we will improve this aspect by the use of other techniques. In special cases one can, however, exploit the local information of Proposition 11.6.2 to obtain global information. Two important examples are for compact groups and nilpotent groups.

38

11.7

II. General Formalism

Compact groups

If the Lie group G is compact, one can use spectral theory to deduce global estimates on the semigroup kernel from the local estimates of Proposition 11.6.2. Compactness of G implies first that for all b > 0 the Gaussian function g i-+ V'(t)-1 /2e-b(Ixl')2t-' is bounded away from zero uniformly for t > 1. Therefore to establish good large t Gaussian bounds on K, it suffices to prove uniform bounds, II Kt Iloo < c, for all t > 1. Similarly, for the derivatives AI K,, it suffices to prove bounds II A° K, Iloo < ca t-I«1/2 for all t > 1. The crucial observation for the proof of these uniform bounds is that compactness of the group implies that the subelliptic operator H has a compact resolvent on the Lp-spaces. This is not difficult to establish (see Notes and Remarks). The spectrum of H is in fact independent of p and so one only needs to prove compactness of the resolvent on L2. But then using subellipticity this is equivalent to compactness of the set {tp E L2;1; IIw11'2;1 < k) for one k > 0. This follows, however, by a simple variation of the usual arguments on bounded subsets of Rd. Subellipticity implies that 91H is a positive self-adjoint operator on L2. Therefore the spectrum of H must lie in the right half plane. It is also evident that zero is a simple eigenvalue of H with the identity function 1 a corresponding normalized eigenfunction. Then compactness of the resolvent implies that the rest of the spectrum must be a non-zero distance oo away from the origin. Next define

P = J dg L(g) c

on each of the LP-spaces. It follow immediately that P is the projection onto the eigensubspace corresponding to the eigenvalue zero. Then P, and S, P = P, are convolution operators with kernel equal to the identity function 1. Now consider bounds on K,. One has

I K,(g) - 1(g)I 1 and g E G. In particular II K, Iloo < I + c for all t > 1. But then there

is a c' > 0 such that IK,(g) - 1(g)l < c' V(t)-1/2 e-2(Ixl')2r-' for all t > 1 and g E G. Interpolation with the bounds (11.3 1) gives

IK,(g)- 1(g)1

0 such that

H.

ct-IaIl2Vi(t)-I12e-jute-b(Is11)2t-1

I(A"Kt)(g)I
0. For each p E (1, oo) and a E J (d') there exists a c > 0 such that 11Aaw11p 0. Therefore Ho generates an exponentially decreasing semigroup and the operator (I + Ho)a Ho-a is bounded on (I - P)LP for all 6 > 0. Note that the spectral arguments give, in addition to Gaussian bounds, an asymptotic approximation for the kernel (11.32). Moreover, the derivatives AI K, decrease exponentially as t -+ oo with the rate of decrease governed by the spectral gap coo. These spectral phenomena will reoccur in the more general context of groups of polynomial growth in Chapters V and VI. One can also deduce global Gaussian bounds from local Gaussian bounds if the Lie group G is nilpotent. But this requires quite different arguments. We next digress to describe a general method that can be used in the nilpotent case and which is of wider applicability in Lie group theory.

11.8

Transference method

The idea of transference is that properties and estimates for operators associated with one representation of the Lie group G can be transferred to other representations of G. In particular estimates for convolution operators in a general Banach space representation may be inferred from estimates for the left regular representation. Transference will be applied in several different contexts in later chapters. In particular, it will be used in Chapter IV to transfer an existence property. The simplest illustration of the transference technique is given by considering a Lie subgroup H of the Lie group G. The left regular representation LG of G on Lp(G) defines a representation LH of H by restriction. This representation acts by left translations but differs from Ltt insofar it acts on Lp(G) and not on Lp(H). But one can estimate convolution operators in the representation EH by the corresponding operators in the representation Ltt. A straightforward argument with a direct integral decomposition gives Il LH(f)II Lp(G)-.Lp(G) _S II LH(f)II Lp(H)-Lp(H)

(11.33)

for all f E LI(H) and p E [1,00) where LH(f) = fitdh f(h)LH(h) and similarly LH (f) = f H dlh f (h) EH (h). One can also transfer density properties. The CO0-elements are dense in the subspace of C"-elements for the left regular representation of a Lie group but the next proposition establishes a much stronger result. Note that we do not assume that ah , ... , ad' generate the Lie algebra g. Proposition 111.8.1 Let U be a strongly continuous representation of a Lie group G with polynomial growth in a Banach space X and X"(U) the space o f Cm elements f o r U with respect to a vector space basis bt, ..., bd o f g. I f a 1 , ... , ad, E

g, with d' > 0, then XOO(U) is dense in X,,, (U) for all m E N, where X,,, (U) ( IaEJm(d') D(Aa) with norm IIxII , = max«EJm(d') IIA°xll and Ak = dU(ak). I

_

11.8 Transference method

41

Proof For all 'p E CO0(G) define the operator U('p): X -). X by U('p)x = fG dg rp(g) U(g)x. If a E g, them

U(v)dU(a)x = JG dg(dL(Ad(g)a)(p)(g) U(g)x. Hence

[dU(a), U('p)lx =

JG

dg(dL((I

-

U(g)x

= U(M(a)cp)x

(11.34)

where

(M(a)'p)(g) = (dL((! - Ad(g))a)co)(g) d

_ J(bj, (I - Ad(g))a)(Bj'p)(g)

(11.35)

j=1

with B j = d L (b j) and ( , ) an inner product on g such that bl, ... , bd is an orthonormal basis. For every multi-index a = (k1, ... , kn) E J(d) set Mace = M (ak,) ... M(ak" )cp.

By Lemma A.1.2 in Appendix A.1, applied with respect to b1 , ... , bd, there

exist a E (0, 1) and for all r E (0, 1) a function qr E C°O(B(r)) such that 0 < qr < 1 and ??r(g) = 1 for all g E B(ar). Moreover, for all a E J(d) one rE has ca = suprE(o.1)rlal IIBagrlloo < co. Set Cr = r-d fG qr. Then (cr (0, 1]} is bounded by an elementary volume estimate. Set rr = cr 1 r-d qr for all r E (0, 1]. Then (rljn : n E N) is a bounded approximation of the identity. We next prove that for all a E J(d') the set ( Marr : r E (0, 1]) is bounded in LI (G). Let a = (k1, . . , kn) E J(d') and suppose that n > 1. For all .

i E {1,...,n} and j E (1,...,d) define 'ij:G -+ R by *ij(g) = (bj,(! Ad(g))ak,). Then >yij E CO0(G) and *ij(e) = 0. Hence there exist c, c' > 0 such that I*ij(g)I c IgI and l(B,6>/iij)(g)I < c' for all g E B(1), P E Jn(d), i E (1, ... , n) and j E (1, ... , d). Then it follows from (11.35) that M(ak)ra = a=1 *ij Bjcp for all W E C°O(G). Then d

Marr =

' I jl

Bj... V/njn Bjn rr

d

E (B"'*jj,)...(B"*,,j")B"rr ji.....j"=1 f.Ni...... 6"

where the sum is over the subsets $, (31, ... , 0, E J (d) with 1011 +. - - + I Nn I +

IfI = n. Let r E (0, 1], g E G and suppose that (Marr)(g) # 0. Consider one

42

11. General Formalism

term in the sum. There are at least n - IfiI indices k such that I&l = 0. Since supp B'Or, C B(r) one has Iifkjk (g)I < c IgI < c r for such k. Then (cr)n-I0I(c')161cr I r-dcjr-I0I

I((BBiYj1j,)...(Blo"*njn)Bfrr)(g)I

C" r-d for some c" > 0, independent of r, g and )4,141, ... , Pn. So

II(BTI'G1j,)...

rrh

c"r-d

IB(r)I

is bounded, uniformly for r E (0, 1]. Hence {Mar, : r E (0, 1]} is bounded in L1(G) for all a E J(d'). In particular, the set (U(Mar1/n) : n E N) is bounded in G(X) for all a E J(d'). For all a = (k1, ... , km) E J(d') and X E Xlal(U) one has nl

U(Mar1/n)x = nl {Ak,, .... [Akm, U(r1/n)] ...]]x = 0

since the restriction of U to the space Xlal(U) is strongly continuous on Xlal(U). Hence by equicontinuity and density of Xlal(U) in X one has lim U(Mar1/n)x = 0

n-*00

(11.36)

for all x E X. Finally, let in E N, a E Jm(d') and X E X' (U). Then

AaU(r1/n)x = 1] U(Mfrl/n)Ayx

(11.37)

B.y

where the sum is over all multi-indices 0, y occurring in the Leibniz formula for the multi-derivative Aa of a product. Then

lim A0U(rl/n)x = Aax

n-*00

and the proof is complete.

O

Corollary II.8.2 Let U be a strongly continuous representation of a Lie group G with polynomial growth in a Banach space X. Let a1, ... , ad, be an algebraic basis for g. Then X00 (U) is dense in X,, (U) for all m E N.

Corollary II.8.3 Let U be a strongly continuous representation of a Lie group G with polynomial growth in a Banach space X and V the restriction of U to a Lie subgroup H. Let a 1, ... , ad, be an algebraic basis for the Lie algebra of H. Then X00 (U) is dense in X,; ,(V) for all in E N.

Since CO0(G) is dense in LP;m(G) for all m E N one deduces the following useful corollary.

11.8 Transference method

43

Corollary 11.8.4 Let H be a subgroup of a Lie group G with polynomial growth and EH the restriction of the left regular representation LG of G on L p(G), where p E [1, oo). Let al, ... , ad, be an algebraic basis for the Lie algebra of H. Then C°O(G) is dense in for all m E N. There is a second type of transference which allows one to transfer properties and estimates from one Lie group G to another related group G. This variation of the method can be used to obtain bounds on semigroup kernels from bounds on kernels on a larger group. This is of particular utility in the analysis of nilpotent groups or, more generally, groups of polynomial growth. The larger group can then be constructed to have better scaling properties. The details of this particular technique, which will be applied several times in the sequel, are as follows.

Let A: G -* G be a surjective Lie group homomorphism between two Lie groups with polynomial growth. Let a1, ... , ad" be an algebraic basis for the Lie

algebra g of G and let al,...,ad, be an algebraic basis for the Lie algebra g of G such that span(al,... , ad') = span(7ral, ... , rrad"), where n: -a g is the Lie algebra homomorphism associated to A. Note that the 7r51, irad are not necessarily linearly independent. Next for any function (p: G - C define

A'(p:G-*Cby A'V =VoA.Then _forallkE (1,...,d") letAk=dL5(ak) denote the infinitesimal generator on G. If

H=- E aktAkA1 k,l=l

is a subelliptic operator on G, then there are cki E C such that d"

d'

k,1=I

k,1=1

- E c'kl dLG(nak)dLG(7ral)

E cki Ak Al

with Ak = dLG(ak). Then d'

H=- L ck1AkAl k,1=l

is a subelliptic operator on G. For the sequel it is convenient to note that Cb(G) and Cb(G) are subspaces of LOO(G) and L.(G). If V E Cb(G), then A'V = rp o A E Cb(G) and

Ak A`rp = A`(dLG(nak)W)

for all k E (1, ... , d") and

E Cb.1(G), where C,,n(G) = (gyp E L',n(G)

A"W E Cb(G) for all a E Jn(d')} for all n E N. Consequently,

H A`rp = A`(HV)

44

ll. General Formalism

for all , E Cb,,(G). Therefore

(Al + H) A'tp = A*((Al + H)(p) and

(Al + H)-'A'(p = A'((A/ + H)-'(p) for all large k > 0. Hence, by the usual semigroup algorithms, St A*tG = A*(S, tG)

for all t > 0, where S and S are the semigroups generated by the closures of H and H. This allows one to relate the kernel K with the kernel k of the semigroup S. Let rp E Cb(G). Introduce tp by setting rp(g) = cp(g-)). Then (St A*0)(e) _ (St (A*(P) )(e) =

fdg(A*co)(g)K,(g),

because A is a homomorphism, where a is the identity element of

(St A*W)(e) _ A* (St 0))(e) = (St t)(e) =

But

dg w(g) K,(g)

since A(e) = e. Hence

JG

dgW(g) K,(g) = f dg (A*tp)(8) Kt(8) c

for all tp E Cb(G) and t > 0. Since a has polynomial growth the closed normal subgroup ker A also has polynomial growth by 11.4. 10. Note that G/ ker A is naturally isomorphic with G since A is surjective. By the Weil formula (see Notes and Remarks) there exists a normalization of the Haar measure dh on ker A such that

f d8 r(8) = G

fc dg ifb(g)

(11.38)

for all >/i E C,(G), where *5 E C,.(G) is defined by

>Gb(A8) = f

dh G(h).

(II.39)

er A

Then, by density (11. 38) and (11. 39) extend to all >' E L) (G). Moreover, 1k!'

IIL,(G) /i)b = dLG(tra)Vi'.

(II.41)

It follows by density that *b E D(dLG(na)) in the L1(G)-sense for all * E D(dLa(a)) c L1(G) and that (11.41) is valid. Therefore it follows by induction that

dh (dLa(a)°K,)(gh)

(d LG((n(a)))'Kr)(A(g)) = f

(11.42)

er A

for all a E J(d"), t > 0 and g E G. The identities (11.40) and (11.42) will be used to obtain optimal Gaussian bounds for the kernel K and its derivatives A' K. Although one cannot explicitly calculate the integral over ker A in the identities, one can estimate the integral of Gaussian

functions. Let g H Igl' denote the subelliptic modulus associated with the algebraic basis 51, ... , ad, of g and V'(t) the volume of the corresponding ball (g E G : lgl' < t ). Lemma II.8.5 For all b > 0 there are b', c', b", c" > 0 such that

c'

V'(t)-1/2e-b"(IA(x)I')2r-I
0 such that the following is valid. If a E J(d"), c, t > 0 and S > 0 and

I(dLG(a)°Kt)(8)I
0 (or t > 1), then dLG((tr(a)))aKf also satisfies Gaussian bounds for all t > 0 (or t > 1). Moreover, the additional decrease with t of dLG((tr(a)))'K, is the same as the additional decrease with t of dLa(a)aK,. 0

Example 11.8.7 Let G denote the connected, simply connected, covering group of the connected Lie group G and A: Ge -> G the natural homomorphism. Then ker A is a discrete, 0 0 central, subgroup of G. The semigroup kernels K and K corresponding to the subelliptic operator H on G and &, respectively, are related by

K,(A(K))=c

Kr(Kk)

(11.43)

kEker A

which is a discrete version of (11.40), where c is a constant depending on the Haar measures on G and 6. The Jacobi identity (1.2) is a special case of (11.43). Thus Corollary 11.8.6 establishes that Gaussian bounds transfer from the simply connected covering group to the group.

11.9

Nilpotent groups

Global Gaussian bounds can be deduced from local Gaussian bounds for the subelliptic kernels on nilpotent groups by a three step process. First one 'extends' the nilpotent group to a larger stratified group. Secondly, one uses the dilation structure on the stratified group to obtain global bounds from local. Thirdly, one transfers the bounds back to the original group by the method described in the previous section. The initial step in the argument consists of 'extending' the nilpotent group G to the stratified Lie group G = G(d', r) on d' generators which is free of step r, where r is the rank of the algebraic basis a I, ... , ad' used to define the subelliptic operator H. By II.1.1 there exists a homomorphism n: g -> g such that tr(ak) = 9 (d', r). The ak for k E ( 1 , ... , d') where a1, ... , ad, are the generators of subelliptic operator H is then lifted to the subelliptic operator d'

H=-

ckl Ak Al k.1=1

11.9 Nilpotent groups

47

with Ak = d L& (ak) acting on the spaces L ,(d). The kernel k of the correspond-

ing semigroup S is related to the kernel K of H by (11.40) where A: G - G is the group homomorphism corresponding to n by I1.1.12.

Secondly, one exploits the dilation structure on a in the manner used in the derivation of local bounds for the stratified group Go in Section II.6. The dilations

y " on g have the property yu(Ok) = u ak for all u > 0 and k E (1, ... , d'). Thus if ru denotes the corresponding dilations on G, then

(H(V)0ru' =u2H(40oru') for all u > 0 and V E CO0(G). Moreover, S,V=(S2,(poru'))oru

for all u > 0. Therefore the kernel k has the transformation property

K,(,) = u° Ku21(ru(8)), for all t, u > 0 and g E G, analogous to (11.28). In particular, if u = t-1/2, then

Kr(8) = t-D/2K1(r-11,(8))

for all t > 0 and g E G. Hence the kernel bounds for large t bounds can be deduced from the bounds for t = I. In particular, the bounds (11.29) applied to H,

K and d give e_b(jjj,)2j-1

I K,(g)i < co e' t- F)12 for all t > 0 and g E G. Finally the kernel K can be calculated from K_ by (11.40). Hence it follows from Corollary 11.8.6 that there exist b', c' > 0 such that 1 K,(g)I < c'

V(t)-1/2

e-b'(Ixl')2t-1

for all g E G and t > 0 by Lemma 11.8.5. One can estimate the derivatives A" K, in a similar manner and the first statement of the following proposition is a consequence of this line of reasoning.

Proposition 11.9.1 Let H be a complex pure second-order subelliptic operator on a connected nilpotent Lie group G with associated semigroup kernel K. The following estimates are valid. I.

There exist b > 0 and for each a E J(d') a c > 0 such that ct-1"I/2Vi(t)-1/2e-b(IgI')2r-1

(A"K,)(g)I < for all g E G and t > 0. II.

For all p E (1, oo) and a E J(d') there exists a c > 0 such that D(H1"1/2) C D(Ac) and c IIHI"I/2WIIp

for all co E D(HJ"1/2).

11. General Formalism

48

The second statement of the proposition, the boundedness of the Riesz transforms, follows from the local Riesz transforms (11.30) on stratified groups, scaling and transference. We will return to such estimates in Chapter VI. The conclusions of Proposition 11.9.1 are similar to the second and third con-

clusions of Proposition 11.7.1 for compact groups. The major difference is that the derivatives have an exponential decrease in the compact case. Similar results can be established for the direct product, or local direct product, of a compact group and a nilpotent group. More general results will, however, be derived in the analysis of groups of polynomial growth in Chapters V and VI. One can also establish an asymptotic approximation analogous to the first statement of 11.7.1 if the group G is nilpotent but not homogeneous. Then K, can be asymptotically approximated by a semigroup kernel on a related homogeneous group. The latter group is defined by a limiting process similar to (11.27) used to introduce the local approximant Go of the group G. The main difference is that the limit is a large scaling limit whose existence depends critically on the nilpotency of G. The limit does not exist for a Lie algebra which is not nilpotent.

H. 10

De Giorgi estimates

In this section we outline a method for improving the local Gaussian bounds (11.29) on the kernel K of the subelliptic semigroup S by using a family of De Giorgi estimates, i.e., energy estimates on the stationary solutions of the heat equation (1.1). This method will be applied in Chapter IV to obtain global Gaussian bounds on the semigroup kernel of a complex subelliptic operator on a group of polynomial growth. The immediate aim is to strengthen the original conclusion derived from parametrix arguments by obtaining a better control on the Gaussian parameters c, b and w. It is important for subsequent applications to establish that

these parameters can be chosen to depend on the operator H only through the ellipticity constant, the norm IICII of the matrix of coefficients and a constant occurring in the De Giorgi estimates. Moreover, the parameters depend on the group G only through a small number of essentially geometric invariants. First, we need a variation of the notation of Section 11.2. If S2 is an open subset of G, define the subelliptic Sobolev space Hz; I (S2) = {q. E L2(Q) : Akrp E L2(c2) for all k E { 1,

... , d'}},

where Akcp denotes the distributional derivative in D'(S2). This space is equipped with the norm p (ll(p ll2 n + 11V,VII2,n)1/2 where

Il0l2.0 = ( f dIp(h)121

I/2

and d'

I/2

IIV'wll2.n = (T f dh I(Akw)(h)12 k=1

n

11.10 De Giorgi estimates

49

As in the classical situation one has the following density result.

Theorem II.10.1 The space CI (Q) n H2; (0) is dense in H2: i (s2).

Proof

Since every W E L2(S2) can be extended to an element of L2(G) by setting p(g) = 0 for all g ¢ 0 the theorem follows from (11.36) and (11.37) in the proof of Proposition 11.8.1 with U the left regular representation, and a standard partition of the unity argument (see Notes and Remarks). Next, let H2:1(S2) denote the closure of CO(S2) in H2; (0). For W E L1,1 denote the average of W over S2 by (W)n. We mainly work with balls S2 = B'(g) where 1

B.(g)=(hEG:Igh0,gEGandWE f12;1(Br(g)).

(11.44)

50

I1. General Formalism

Proof We may assume that g = e. Since G is not compact one can choose an h E G such that 1h I' = 2r. Let y: [0, 1] --> G be an absolutely continuous path from e to h which satisfies the differential equation d'

Y(t)=LYk(t)Ak k=1

Y(r)

almost everywhere, with fo dt Ek=1 IYk(t)I2 < 5r2 Now suppose W E C°°(B'). Then Irp(g)I2 = IV(8) Hence IV(g)12

- W(hg)12 for all g E Br.

= fo 1 dt>Yk(t)(Akco)(Y(t)8)I2 (Y(t) 8)12 k=1

=Sr2 f dt f dg L 0

G

5r2llo'wI12,r

k=1

and the desired result follows by a density argument.

The Poincare inequality (11.44) states that the lowest non-zero eigenvalue of the sublaplacian - Ed,=1 Ak with Dirichlet boundary conditions on L2(Br) has a lower bound 5-1 r-2. This is of interest for two reasons. First the bound has the expected behaviour with r. Secondly, the constant 5-1 in the bound does not depend on the group or the algebraic basis. There is also a Poincare inequality associated with the sublaplacian with Neumann boundary conditions. But this is more delicate. In the classical setting of Rd the Dirichlet inequality is valid for quite general sets but the Neumann version requires some regularity of the boundary. This is also the situation for more general manifolds, and Lie groups in particular. We only need the following weak Lie group version of the Neumanntype Poincare inequality. Proposition 11.10.4 There exist cN > 0 and RN E (0, 11 such that IIw - (W)g,rll2,g,r 0 and RN E (0, 1) such that the Poincard inequality (11.45) is valid.

III.

There are v E (0, 1) and cDG > 0 such that H and its dual H* satisfy De Giorgi estimates of order v with De Giorgi constant cDG.

Then the semigroup kernel K associated with H satisfies the bounds IKr(g)I

0 and g E G where the values of a, b > 0 and w > 0 depend on H and G only through the parameters c, coG, cN, IICII, A > 0, D' E N and RN, V E (0, 1). The restriction to unimodular groups in the theorem is not essential but it suffices for the later applications and it somewhat simplifies the proof. This follows a standard strategy of bounding the L i -s L... crossnorm of the semigroup S. This gives a uniform bound on the kernel since II K,1100 = II St II 1X00 Then to obtain pointwise estimates one applies the same strategy to a family SP of perturbations of S as described in Section 11.5. In fact it suffices to bound II St 11200 and d SP112_+0o and one can do this by an iterative procedure. The most straight-

forward procedure is to bound successively II St 112_ P, , II Sr II,,, p2 , ... where Pt < P2 < ... with p -* coo. For operators with real coefficients this is possible, but the method breaks down if the coefficients are complex. In fact to exploit

the De Giorgi estimates it is necessary to pass through a sequence of Money and Campanato spaces. These are defined and discussed in Appendix A.2 and the proof of Theorem II. 10.5 is given in Appendix A.3. The uniform estimates of the theorem are the first important ingredient in the subsequent proof of global Gaussian bounds for the semigroup kernels associated with complex subelliptic operators on groups of polynomial growth. The second essential ingredient is the structure theory of Lie groups and Lie algebras. This is the topic of Chapter III.

11. 11

Almost periodic functions

In the subsequent study of subelliptic operators, in Section IV.3, we require some elementary properties of almost periodic functions. The theory of almost periodic

52

II. General Formalism

functions originates with functions over Rd but with the appropriate choice of definitions can be extended to functions over a general group G, even without a topology on G. The principal elements of the theory related to the subsequent analysis concern finite-dimensional representations of G, but we first recall some basic representation independent properties. Further details can be found in the references cited in the Notes and Remarks. The space A(Rd) fl C(Rd) of continuous almost periodic functions over Rd is defined to be the closure with respect to the supremum norm of the vector space T(Rd) of complex-valued trigonometric polynomials, i.e., functions of the form Ecaeil.x

fA(x) = aEA

where A is a finite subset of Rd and ca E C. This definition is clearly not appropriate for extension to more general spaces such as Lie groups but there are several other equivalent definitions that are suited to this purpose. Let G be a general group.

II.11.1 Let >!': G -+ C be a bounded function. The following are equivalent. 1.

The closure of the orbit {L(g)* : g E G) under left translations is compact with respect to the uniform topology.

II.

III.

The closure of the orbit {R(g)>f : g E G} under right translations is compact with respect to the uniform topology. The closure of the orbit {L(g) R(h)>f : g, h E G} is compact with respect to the uniform topology.

Therefore, one can define a bounded function over G to be almost periodic if it satisfies these equivalent conditions. We denote by A(G) the space of all almost periodic functions. It is evident that the almost periodic functions A(G) over G form a uniformly closed subspace of LA(G) which contains the constant functions and is closed under left translations, right translations and complex conjugation. It is also clear that if G = Rd, then each trigonometric function satisfies the equivalent conditions of Statement II.11.1. In fact, if G = Rd, the previous definition of continuous almost periodic functions coincides with the general one. The definition of almost periodicity can be clarified by noting that it is equivalent to the statement that a bounded function >G E A(G) if and only if for each

e > 0 there exists a finite set (gl, ... , g,) c G such that sup

inf

gEG iEII....,nl

II L(g)'G - L(gi)*II00 < E.

Thusforeachg E G thereisag' E e, i.e., the function is close to being periodic with period g-Ig'. It follows easily that A(G) is closed under multiplication, i.e., if co, ' E A(G), then (P * E A(G). Therefore A(G) is a C'-subalgebra of LA(G) with identity. The important feature of A(G) for the sequel is the existence of a unique in-

variant mean M.

I1.11 Almost periodic functions

53

11.11.2 There exists a unique linear functional M over A(G) with the following three properties. 1.

II.

III.

M is positive, i.e., if >G > 0, then M(VI) > 0. M is normalized, i.e., M(1) = 1 where 1 is the unit function. M is both left and right invariant, i.e.,

M(L(g)'G) = M(Vi) = M(R(g)'/') for all tli E A(G) and g E G. Moreover, M satisfies the following properties. IV.

M commutes with complex conjugation, i.e., M(T) = M(*) for all i/r E A(G).

V.

M is faithful, i.e., if >;(i > 0 and M(t/i) = 0, then >G = 0.

VI.

For all i/i E A(G) and E > 0 there exist n E N and gl, ... , g E G such that

MW - I E'G(g gk h)

< E

k=l

for all g, h E G. The uniqueness of M immediately implies that it has strong invariance properties.

Corollary II.11.3 If 4> is an automorphism of G, then M('p o (D) = M(ep) for all cp E A(G).

Proof The linear functional cp H M(cp o (P) from A(G) into C is positive, normalized and both left and right invariant. For example, the left invariance follows since L(g)(cp o (D) = (L((D (g))cp) o (D. Hence by the uniqueness of the mean one has M(cp o 4>) = M(cp) for alI rp E A(G).

It is noteworthy that the existence, uniqueness and invariance of the mean M does not require any topological restrictions on the group G or continuity assumptions on the functions A(G). Nevertheless under stronger hypotheses one can obtain more detailed representations of the mean as an average over the group. Example 11.11.4 Let G be a Lie group of polynomial growth. Then

M(,P) =,lioc V(r)-' f dg '(g) B,

for all i' E A(G) fl C(G) where Br is the ball of radius r, measured with respect to an arbitrary modulus, centred at the identity e and V (r) = I Br I is the volume of the ball. More generally. if G is a locally compact group with left Haar measure and t21 , 02.... are measurable subsets of G such that 0 < IS2nI < oo for all n E N and limn.. IS2nI-I I(gi2n)\f2nI = 0 for all g E G, then

- I f dg 5k(g) M(V,) = n l i m 00 Ii2n I nn

for all l' E A(G)nC(G).

54

II. General Formalism Example 11.11.5 Let G be a stratified Lie group with Haar measure dg and family of dilations ru, with u > 0. Then

f dg (p (g)

dg co(g) ('lr o I'u)(g) = M(+G) ulim00fG G for all >G E A(G) fl C(G) and V ELI (G). It suffices, by a straightforward density argument, to establish the identification for V = XB,h and >' E A(G), where XB,h is the characteristic function of the ball Brh with centre h and radius r, and the modulus is with respect to a vector space basis for the first layer of the Lie algebra of G. For all u > 0 one has IBrI-1 f

W ('G o ru) =

* = IBurl-1 f

u-DIBrI-1 f

ru(B,h) Let e > 0. There exist n E N and k1, ... , k E G such that inf IIR(k)Vr - R(ki)''IIoo O such that

I M(P) -

IBurI-1

f

<e

'Bur Bur

f o r all u > N and i E ( 1 , ... , n). Let u > N. Then there exists an i E { 1, ... , n) such that IIR(I'u(h))V, - R(ki)Jrlloo < e. Therefore I M(+G) -

IBurI-1 f

Bur

R(ru(h)),P

<s+

l R(ki)1k - R(ru(h))*) Bur \

IBurI-1 f

< 2e

from which the limit follows.

A finite-dimensional representation M of the group G is defined to be bounded if mki E Loo(G) for all k,1 E {1, ... , m}, where (mkl(g)) is the matrix of M with

respect to some fixed basis, almost periodic if the mkt E A(G) and unitary if each matrix (mkl(g)) is unitary. Two finite-dimensional representations M and N are defined to be (unitarily) equivalent if they have the same dimension and there is a (unitary) non-singular matrix W such that M(g) = WN(g)W-I for all g E G. These concepts are closely related. 11.11.6 Let M be afinite-dimensional representation of the group G. The following conditions are equivalent. I.

M is bounded.

II.

M is almost periodic.

III.

M is equivalent to a unitary representation.

It follows from this observation that one can define the mean value of the matrix elements of each finite-dimensional unitary representation, or of products of such matrix elements. These means have many useful properties. Finally, the mean ergodic theorem gives a different perspective on the invariant mean.

111.11.7 If M is a finite-dimensional unitary representation of G in a Hilbert space l and P(O) is the orthogonal projection onto the M-invariant subspace {x E R : M(g)x = x for all g E G}, then M(t/i) = (x, P(0)y) for all x, y E where >G(g) = (x, M(g)y).

11.12 Interpolation

11. 12

55

Interpolation

Interpolation theory is a standard method of analysis for functions and operators over the Euclidean space Rd. Many of the standard results extend to continuous representations (X, G, U) of Lie groups (see Notes and Remarks). The C"subspaces X"(U), or the comparable subelliptic spaces X, (U), act as a scale of interpolation spaces. One can then use the corresponding intermediate spaces to analyze the Lipschitz spaces of the representation or of the subelliptic semigroups. First, let Yo and Yj be two normed spaces with Yi continuously embedded in Yo. Define the interpolation space of the K-method of Peetre by (Yo, YI)y,p;K = {y E Yo :

00

f

dtt-'It-'Ky(t)IP < oo},

0

where Ky(t) = inf{Ily - yi Ilyo + t Ilyi lly, : yi E Yi } for all y E Yo. The norm on the space (Yo, YI)y,p;K is defined by 00

IlYll(Yo.YI),..o;K = IlYllyo+(

Jo

\UP

dtt-1It-yKy(t)IPI

.

/

The main properties of the interpolation spaces are summarized in the next statement. II.12.1 Let Yo and Yi be normed spaces with Yi continuously embedded in Yo. 1.

If y E (0, 1) and p E [ 1, oo], then Yi c (Yo, Yl )y, p; K c Yo and the embeddings are continuous.

II.

IfyE(0,1)and 1 1 (see also [EIR2], Theorem 3.2). Statement II follows from (11) in [EIR2].

III Structure Theory

Structure theory is an essential ingredient in the analysis of subelliptic semigroup kernels on groups of polynomial growth. In Chapter II we summarized most of the relevant standard results but it is also necessary to establish a number of other results adapted to the analysis. There are several aspects of this adaptation which will be described in detail

in this chapter. First, the group can be replaced by its covering group because estimates on the covering group can be transferred to the original group by the methods of Section 11.8. Since the covering group is simply connected this is an immediate simplification. Secondly, the standard structure theory establishes that each simply connected Lie group G has a decomposition as a semidirect product of a Levi subgroup M acting on the radical Q. Both these component groups are simply connected by I1.1.17. Moreover, if G has polynomial growth, M must be compact and Q must have polynomial growth by Proposition II.4.11. Thirdly, in this chapter we shall show that each solvable Lie group Q can be obtained from a nilpotent Lie group QN, the nilshadow of Q, by modification of the group product. The two groups Q and QN are identical as manifolds but differ in their algebraic properties. By combination of these observations one may conclude that the original simply connected group G with polynomial growth can be obtained from the direct product GN = M X QN of the compact Levi subgroup and the nilshadow of the radical by equipping GN with a new product. Each subelliptic operator on G with constant coefficients can then be viewed as a subelliptic operator on GN with variable coefficients, albeit coefficients of a particularly simple character. This viewpoint is the key to the subsequent analysis of the subelliptic kernels. Finally, although the nilshadow is nilpotent, it is not necessarily homogeneous, or

64

III. Structure Theory

stratified. Nevertheless, G can be realized as a quotient of a larger group a whose radical Q has a stratified _nilshadow QN. Therefore it is possible to analyze the subelliptic operators on GN = M X QN and transfer estimates to GN by another application of the methods of Section 11.8. In this chapter we describe in detail the construction of the nilshadow and the corresponding group product. In addition we construct the larger group with the stratified nilshadow. The theory separates naturally into two parts, the Lie algebraic constructions and the Lie group structures. Although we are eventually interested in groups of polynomial growth, much of the analysis is not restricted to this setting.

111. 1

Complementary subspaces

Let g be a Lie algebra with radical q and nilradical n. Our aim is to describe how q can be equipped with a new Lie product which agrees with the original product on the nilradical n but is also nilpotent on q. The Lie algebra with the new product is called the nilshadow. Its definition depends on the properties of the complement

uofninq. Let m be a Levi subalgebra of g. Since the adjoint representation of the Levi subalgebra m in the vector space q leaves the nilradical n invariant, there is a complementary subspace u of n in q which is invariant under this action, i.e., [m, o] C u. This is a consequence of the semisimplicity of m (see Il. 1. 11). But [m, o] C n and since n fl u = {0} it follows that [m, uJ = {0}. Our initial aim is to establish that n has a complementary subspace u within q which commutes with the Levi subalgebra m and has the key property that S(u)u = (0) with S the semisimple component in the Jordan decomposition.

Proposition III.1.1 There exists a vector subspace u of g satisfying 1.

[u, m] _ {0},

II.

q=un.

III.

S(u)u = (0).

Moreover, if u is a vector subspace of q satisfying Property III and K is the nilpotent component in the Jordan decomposition, then IV.

[S(u), S(u)] _ [S(u), K(u)] = (0),

V.

The maps a r- S(a) and a ra K(a) are linearfrom u into L(g).

Proof The proof of the first statement follows using the theory of Cartan subalgebras. Define a subalgebra qo of q by

qo=(aEq: [a,m]=(0)}

IlI.I Complementary subspaces

65

and let 11 be a Cartan subalgebra of qo. If a E , then S(a) leaves l invariant since S(a) is a polynomial in ada and h is a subalgebra of g. Hence S(a)sh is semisimple. Moreover, h is nilpotent, by definition. Therefore in the Jordan decomposition

adba = adaIh = S(a)Ib + K(a)Ib of a E h one must have S(a) I h = 0 and adha = K (a) I h, by uniqueness of the decomposition. But this means S(a)b = 0 for all a, b E 4. Next we argue that

q=h+n

(111. 1)

where the sum is a vector space sum which is not necessarily a direct sum. Once this is established the first statement of Proposition 111. 1. 1 follows by choosing a subspace v c_ ll such that q = n ® n. Then [u, m] = {0} because u c_ qo and

S(a)b=0for all a,bEnbecause ttC . The proof of (11I.1) is based on the following standard result.

Lemma 111.1.2 Let n: g -+ g' be a surjective homomorphism of Lie algebras. If

h is a Cartan subalgebra of g, then h' = tr(fl) is a Cartan subalgebra of g' _ n(o). Proof See Notes and Remarks. It remains to justify (111. 1).

First, it follows from the discussion preceding the proposition that there is a subspace u such that q = u ® n and [u, m] = (0). Since u c qo, one immediately has q = qo + n. Therefore, if n': q H q/n is the projection onto the quotient, then 7r'(qo) = q/n. It follows from Lemma 111. 1.2 that ir'(1l) is a Cartan subalgebra of q/n. But q/n is abelian, since [q, q] C n, and hence ir'(I)) = q/n. This establishes (Ill. 1) and concludes the proof of the first statement of Proposition Ill. 1.1. It remains to prove the second statement of the proposition. The proof depends on three relatively elementary algebraic properties of a derivation D of a general Lie algebra 9. Lemma 111.13 Let D be a derivation of the Lie algebra g. 1.

II. III.

If a E g, then [D, ada] = adDa. If Da = O for some a E 9, then [ D, S(a) ] = 0 = [ D, K (a) ]. If D is nilpotent and V is a nilpotent subset of r(9) such that [ D, V ] c V, then D + V is nilpotent.

Proof Statement I follows from the identity [D, ada]b = D[a, b] - [a, Db] _ [Da, b] which is a consequence of the derivation property. If Da = 0, then one has [D, ada] = 0 and Statement II follows immediately since S(a) = s(ada) and K(a) = k(ada) where s and k are polynomials with no constant terms. Thirdly, if n, m E N are such that Dm = 0 and V" = (0), then (D + V ),n = {0}. Indeed, (D + V)m" is a set of linear maps on g and each one is a sum of products of

III. Structure Theory

66

the form L I ... L,,,,, where Li = D or Li E V for all i E { 1, ... , inn). Using [D, V] C V this product is contained in Ek_"o DkV' where 1 is the number of indices i for which Li E V. But V' = (0) unless 1 < n - 1. Then, however, there must be an index i for which L;+t ... Li+m = D' = 0, since there are ! + 1 < n intervals containing elements of V and n(m - 1) + (n - 1) < mn. The implication III=IV of Proposition 111. 1. 1 is an immediate consequence of the second statement of Lemma 111. 1.3. Finally we prove the implication III=V of the proposition.

First, if n is the nilradical of q, then K(a)n c_ n for all a E q. It follows that [K(a), adn] c_ adK(a)n c_ adn by Statement I of Lemma 111.1.3. Therefore K(a) + adn is nilpotent by Statement III of the same lemma.

Secondly, fix a, b E v. Define V as the subspace spanned by K(b) and n E N), where b" = (ada)"b E n. Then V is nilpotent by the previ-

{adb,,

:

ous observation. Thirdly,

[K(a), K(b)] = [S(a) + K(a), S(b) + K(b)) = [ada, adb] = adbi E V

since [S(a), S(b)] = 0 and [S(a), K(b)] = 0 = [S(b), K(a)]. Next S(a) is a derivation and S(a)a = 0 = S(a)b. Therefore S(a)b,, = 0 and [S(a), adb,,] _ adS(a)b,, = 0. Hence

[K(a), adb,,] = [ada, adb,,] = ad((ada)b,,) = adb,,+i E V.

Consequently, [K(a), V] c V. Then K(a) + V is nilpotent by Statement III of Lemma 111. 1.3. In particular K(a) + K(b) is nilpotent. Fourthly, S(a) and S(b) are commuting semisimple operators. Therefore S(a)+ S(b) is a semisimple operator. Finally,

ad(a + b) = ada + adb = (S(a) + S(b)) + (K(a) + K(b)) and S(a) + S(b) commutes with K(a) + K(b). Hence, by the uniqueness of the Jordan decomposition, S(a + b) = S(a) + S(b) and K(a + b) = K(a) + K(b). Since this is valid for all a, b E u, the maps a -+ S(a) and a H K(a) are linear on v. This completes the proof of Proposition 1I1.1.1.

The preceding arguments relate the commutativity and linearity properties of the semisimple operators S(v) with the relation S(u)v = (0). There are three corollaries of the arguments which are of subsequent utility.

Corollary 111.1.4 If a E g and b E n, then K(a) + adb is nilpotent. Moreover the following are equivalent. 1.

S(a)b belongs to the centre of g.

II.

S(a) and adb commute.

Ill.1 Complementary subspaces

67

S(a) = S(a + b) and K (a + b) = K (a) + adb.

III.

Proof The nilpotency of K (a)+adb follows from Statement III of Lemma 111. 1.3,

with V = adn and D = K(a), and the equivalence I..II is a consequence of Statement I of the lemma. But III=>II since the semisimple and nilpotent compo-

nents in the Jordan decomposition commute. Hence 0 = [S(a + b), K(a + b)] _ [S(a), K(a) + adb] = [S(a), adb]. Finally

S(a) + (K(a) + adb) = ada + adb = S(a + b) + K(a + b). But K(a) + adb is nilpotent. Therefore II=:>III by the uniqueness of the Jordan decomposition.

Corollary 111.1.5 If g is solvable, then for each a E g one may choose v such that g = v ® n, S(u)v = {0} and S(a)n = (0).

Proof

Define a subalgebra of g by ga = (b E g

:

S(a)b = 0). Choosing

a Cartan subalgebra h of ga, we have as in the proof of Proposition 111.1.1 that

S(a)b = 0 for all a, b E lj. Since S(a) is semisimple and S(a)g g n, there is a subspace u such that g = u ® n and S(a)u = {0}. Thus u c ga. Now a repetition of the reasoning used to prove the first statement of Proposition 111. 1. 1 shows that

g = I) + n. Therefore we can choose n c l with g = v ® n to reach the desired conclusion.

Corollary 111.1.6 Let v be a subspace of q satisfying Properties I-III of Proposition III.1.1. If b C a are subspaces of q which are both invariant under adm and S(v), then there exists a vector subspace c of a such that a = b ® c and c is invariant under adm and S(u).

Proof Define p: m®q -> C(a) by p(m+a) = admla+S(a0)Ia form E m and a E q where an denotes the component of a in v. Using the inclusion [q, q] c n it follows that p is a representation of m ® q in a. Moreover, the radical of m ® q equals q. If a E q, then p(a) = S(an)Ia is semisimple. So the representation p is semisimple and the existence of the complementary subspace c follows from 11.1.11.

The possibility of choosing the complementary subspace v of n in q with the property S(v)v = (0) will be of fundamental importance in the sequel. It should be emphasized that this property is a type of nilpotency condition. Example 111.1.7 If v is a subspace of a solvable Lie algebra q, then S(u)u = 101 if and only if the Lie subalgebra ro of q generated by u is nilpotent. To see this, first let adro denote the adjoint representation of m. But adroa = ada l ro = S(a) I ro + K (a)15. Moreover, S(a)lm and K(a)Iro are semisimple and nilpotent derivations, respectively, which mutually commute. So

S(a)Iro + K(a)hro is the Jordan decomposition of adroa. If ro is nilpotent, then adroa is nilpotent for all a E sand S(a)Im = 0. Hence S(ro)m = (0). In particular, S(u)n = (0).

111. Structure Theory

68

Conversely, if S(u)n = (0) and v E u, then S(v)m = (0) because n generates to and S(v) is a derivation. Hence adro v = K(v)Iro is nilpotent. Note that (to, ro] C [q, qJ C n so that adro[ro, rol is a nilpotent subset of C(ro). Since (ro, rol is an ideal of to, one has [adrov, adro[ro, roll c adro[ro, to)

for each v E n. Then, by the third statement of Lemma 111.1.3. adrov + adro [ro, roJ is a nilpotent subset of L (w). But to = n + [tu, ro] because u generates to. Therefore adro to is a nilpotent set, and to is nilpotent.

The foregoing structural results were independent of any assumption of polynomial growth of the Lie group G with Lie algebra g, and there is no direct link between solvability of the algebra and the growth condition. The next example gives three classes of Lie algebra which are solvable but not of type R. (i) The two-dimensional Lie group of affine transformations of the real line, the (ax + b)-group, has a Lie algebra q with a basis al, a2 satisfying [al, a2] = aa2. Example 111.1.8

The algebra is solvable. The nilradical is the one-dimensional span of a2 and the complementary subspace can be chosen as the span of al . The algebra is not of type R if Cr 96 0, since adaI has eigenvalues 0 and a. (ii) The three-dimensional Lie algebra e3 of Example 11.4.19 is solvable and has a twodimensional abelian nilradical n = span(a2, a3). One may choose u as the one-dimensional span of al. The algebra is not of type R if a # 0 (see Example 11.4.19 ). (iii) The four-dimensional Lie algebra f4a with basis al, a2, a3, a4 and non-vanishing products

[a2 a4l = aa4

[al, a4] = a3

[al. a3] = -a4

,

[a2. a31 = aa3

has an abelian nilradical, n = span{a3, a4) if Or # 0 and n = span(a2, a3. a4) if a = 0. The complementary subspace u can be chosen as span(al, a2) if Or 96 0 and span(al) if or = 0. The algebra is not of type R if a 96 0, since ada2 has eigenvalues 0 and a.

The surprising observation is that these three classes of Lie algebra essentially characterize the solvable algebras which are not of type R. Example 111.1.9 Let g be a solvable Lie algebra. The following conditions are equivalent. 1.

g is not of type R.

11.

There is a surjective Lie algebra homomorphism of g onto one of the algebras aZ ,

t3, [y with a 0 0. (See Notes and Remarks.)

There is an analogous characterization of the solvable type R algebras which are not nilpotent. But then there is only one algebraic obstruction. Example 111.1.10 Let g be a solvable Lie algebra of type R. The following conditions are equivalent. 1.

11.

g is not nilpotent.

There is a surjcctive Lie algebra homomorphism of g onto e3 (= e3), the Lie algebra of the group of Euclidean motions in the plane.

(See Notes and Remarks.)

111.2 The nilshadow; algebraic structure

111.2

69

The nilshadow; algebraic structure

The decomposition of Proposition 111. 1. 1 is the key to the definition of the nilshadow qN of the solvable Lie algebra q. The nilshadow is a nilpotent Lie algebra

which is constructed from q by the introduction of a new nilpotent Lie bracket. Although the adjoint representation of g maps q into the nilradical n, the algebra q fails to be nilpotent since the adjoint representation of o on n is not nilpotent if o $ {0}. The idea behind the construction of qN is to modify the action of ado by subtraction of the semisimple component and then the modified action is nilpotent. This process can be expressed as a modification of the Lie product which renders the resulting algebra qN nilpotent. First, consider the radical q and fix a vector subspace o of q such that Properties II and III of Proposition 111. 1. 1 are satisfied. Thus q = u ® n and S(o)o = {0}.

Secondly, if a E g = m ® o ® n, let au denote the component of a in v. Then the nilshadow is the Lie algebra qN defined as the vector space q equipped with the Lie bracket [ , IN defined by [a, b]N = [a, b] - S(an)b + S(bn)a

(111.2)

for all a, b E q. One needs to verify that [ , IN is indeed a Lie bracket. The linearity and anti-symmetry properties are straightforward. The projection a H at, is linear and the map v r-a S(v) from o into L (q) is linear by Proposition 111. 1. 1. It is somewhat more complicated to verify the Jacobi identity. This can be achieved by direct calculation, but more is true. If A E R and one sets

[a, b]N.A = [a, b] - AS(an)b + AS(bn)a

(111.3)

for all a, b E q, then [ , ]N,a is a Lie bracket. The Jacobi identity is established as follows. A straightforward calculation gives

[a, [b, c]N.a]N,x = [a, [b, c]] - ,LS(an)[b, c] - A[a, S(bn)c] + A[a, S(cn)b] + A2S(an)S(bu)c - AZS(an)S(cn)b

for all a, b, c E q. Note that additional terms such as S(S(b)c) vanish since S(q)q c n and S(n) = (0). Therefore the double commutator is quadratic in k and summing over cyclic permutations of a, b and c one again obtains a quadratic expression. But the zero-order term vanishes by the Jacobi identity for the original Lie bracket [ , ], the first-order term is zero by the derivation property of S(ae), S(b0), S(ce) and the second-order term is zero because of the commutativity of S(o). Thus we have a one-parameter family of Lie brackets [ , IN.), on q and a corresponding family qNd, of Lie algebras. The value A = 0 corresponds to the original algebra and X = 1 corresponds to the nilshadow. The nilpotency of the Lie bracket (111.2) of the nilshadow is established in two

steps. First, the corresponding adjoint representation adN maps q into n since

70

III. Structure Theory

(adq)q e n and S(q)q g n. In particular, qN is solvable. Secondly, if a = v + n with v E t) and n E n, then (adNa)b = (ad(v + n))b - S(v)b

= (S(v) + K(v) + adn))b - S(v)b = (K(v) + adn))b E it for all b E n. But K(v) + adn is nilpotent by Corollary 111. 1.4. Hence adNa is nilpotent for all a E q. Therefore qN is nilpotent by 11. 1.8. Example 111.2.1 The two-dimensional Lie group of of ine transformations of the real line,

the (ax + b)-group, has a Lie algebra q with basis at, a2 satisfying [at, a21 = oat (see Example 111.1.8). If a 54 0, the algebra is solvable with nilradical the one-dimensional span of a2. One may choose o as the one-dimensional span of at . The nilshadow q.N is abelian

since adat is semisimple and ada2 is nilpotent, e.g., [at. a21N = (ada, )a2 - S(at )a2 = 0. Similarly, the nilshadows of the Lie algebras e3 and l; of Example 111.1.8 are abelian since in each case the adjoint representation of o on n is semisimple.

Although the new product [ , IN changes the algebraic structure of q, it does not affect the algebraic character of some of the key operators such as adv, S(v) and K(v) for V E o and adni with m E m if o satisfies in addition Property I of Proposition 111. 1. 1. This is used to establish the following observation. Let qN; j denotes the lower central series of qN, i.e., qN;l = q and qN; j + I [q, qN; j IN for all j E N. Let r be the rank of qN.

=

Lemma 111.2.2 If o satisfies Properties II and III of Proposition 111. 1. 1, then for all v E o the operators adv, S(v) and K(v) are derivations of qN, e.g.,

(adv)([a, bIN) = [(adv)(a), b1N + [a, (adu)(b)]N

for v E o and a, b E q. Moreover each ideal qN;j is invariant under adv, S(v) and K(v). Finally, if o also satisfies Property I of Proposition III.1.1, then for all ni E m the operator adni is a derivation of qN which leaves the ideals qN;j invariant.

Proof Consider the last statement. Property I states that (adm)(o) = {0}. Hence adm commutes with ado, S(o) and K(o) by Lemma 111. 1.3. It follows from the definition of the Lie bracket [ , IN that

(adm)([a, b1N) = [(adrn)a, b] + [a, (adm)b] - (adm)(S(an)b) + (adm)(S(bo)a) = [(adm)a, bJN + [a, (adm)b1 N

- S(bt,)(adm)a + S(an)(adm)b

- (adm)(S(an)b) + (adm)(S(bv)a) = [(adm)a, b]N + [a, (adm)b]N

111.2 The nilshadow; algebraic structure

71

for all m E m and a, b E q, where the second equality uses (adm)(q) c n and S(n) = (0), whilst the third uses the commutativity of adm and S(u). Then the invariance of the ideals qN; j under adm follows from the derivation property by induction. The proofs for the other operators follow by similar calculations which depend on Property IV of Proposition 111. 1. 1, e.g., [S(u), S(u)] = (0). Corollary 111.2.3 If o satisfies Properties II and III of Proposition 111. 1. 1, then

K(v)gN;j C qN;j+1

for all v E vand j E (1,...,r). Proof If V' E u, then (adv)v' = [u, v']N, and since K(v) is a polynomial in adv without a constant term and adv is a derivation of qN, it follows that K(v)n C_

qN;2 Clearly K(v)n = [v, n]N C qN;2. So K(v)q g qN:2. Since K(v) is a derivation of qN the corollary follows.

The commuting semisimple operators S(o) and the adm will play a significant role in what follows. It is worthwhile noting at this point, and we shall prove this in Corollary 111.2.4, that one can choose S(u)-invariant subspaces h1, ... , hr of

q such that qN;j = h j ® gN;j_1. Moreover, these subspaces can be chosen to be invariant under the adm and in addition u C h 1. Therefore g has a direct sum decomposition g = m ®h1 ® ... (D hr into subspaces invariant under the families S(u) and adm. The subspace 41 plays a special role. There is an S(o)-invariant subspace t c_ n which is complementary to u in h 1. It follows from the construction of o that S(o)o = (0) and (adm)(n) = (0) but t also separates into a subspace to which is annihilated by these operators and a complementary subspace t1. There are two possibilities t1 = (0) and t1 54 (0). This distinction is crucial in the subsequent structural analysis of Lie groups with polynomial growth. Corollary 111.2.4 If o satisfies Properties I-III of Proposition 111. 1. 1, then there e x i s t vector subspaces h 1, ... , hr, t, to, t1 of q such that the following are valid.

II.

qN;j =hj ED ...ED hrforall j E (1,...,r). h1 =to ®tand t=to ®t1.

III.

n=t®qN;2.

IV.

S(u)hj g hj and [m, l)j] C hj forall j E (1,...,r).

V.

to = (a E t : S(o)a = (0) and [m, a] = (0)).

VI.

S(O)t1 C t1 and [m, t1 ] C P1.

1.

Proof

First, note that any ideal of g is invariant under adm and S(a) for all

a E u, since S(a) is a polynomial in ada without constant term. Secondly, if b C a are subspaces of q which are both invariant under adm and

S(u), then there exists a vector subspace c of a such that a = b ® c and c is invariant under adm and S(u), by Corollary 111.1.6.

III. Structure Theory

72

For all j E (2, ... , r + 1) apply the above to the ideals qNJ+I C qN;j. Then there exists a vector subspace lIj of qNJ which is invariant under adm and the S(a), with a E v, such that qN;j = 4; ® qN; j+I I. Thirdly, for the construction of 41 one has to make one additional step. Now n and qN;2 are ideals so there is an invariant subspace t of g such that n = t (D qN;2. Then set l)I = v ® t. Define to = (a E t : S(v)a = (0) and [m, a] = (0)}. The subspaces t and to of q are both invariant under adm and S(v). Hence there is an invariant subspace tl of t O

such that t = to E D

The subspace 1II is automatically generating.

Lemma 111.2.5 Let 1l1 be any subspace of qN such that qN = ICI + qN;2 Then hl generates the Lie algebra qN. Proof Let a be the Lie subalgebra of qN generated by 1)1 . If n E N, al I ,(IN and one writes ak = bk + ck with bk E f I and ck E qN;2, then

[al, [... [an - I, an]N

.

an E

I N]N - [bl, I... [bn-I , bn]N ...]N]N E qN;n+I

Since qN;r+I = (0), it follows by backwards induction on j that qN;j c a for all j E (1, ... , r). Hence qN = qN;1 c a and l)I generates qN. 0 The definition of the product on the nilshadow (111.2) is clearly analogous to the definition of the Lie bracket (11.3) on the semidirect product. There are two distinct ways in which one can pursue this analogy. First, the space v can be viewed as a

commutative Lie algebra by equipping it with a Lie bracket which is identically zero. Then Proposition 111. 1. 1 establishes that S: v -+ £(q) is a representation of v in q by derivations. Let v x q be the semidirect product with Lie bracket (11.3)

[(vi, al), (v2,a2)]s = (0, [at,a2l+ S(vi)a2 - S(v2)al). Then note that

[(-au, a), (-b0, b)]s = (0, [a, b]N)

for all a, b E q. Thus if P denotes the projection onto the first component, one has

[a, b]N = P([(-an, a), (-bn, b)]s).

(111.4)

This relation allows one to identify the nilshadow qN of q with the nilradical of

vxq. Proposition 111.2.6 The semidirect product u x q is a solvable Lie algebra. Moreover [u x q, u x q] C (0, n). The nilradical ns of u x q is given by ns = ((-ad, a) a E q).

Proof Since S(u)n = (0) and S(u)n c_ n it follows that [v x q, v x q] c (0, n). Then the solvability of u x q is a direct consequence of the nilpotence of n. Next set

p = ((-an, a) : a E q).

111.2 The nilshadow; algebraic structure

73

Then [p, n a q]s c (0, n) c p and p is an ideal. But

[(-au, a), (0, n)]s = (0, (ada - S(ae))n) = (0, (K(an) + adan)n) for all n E n, where an is the component of a inn, and Klan) + adan is nilpotent by Corollary 111. 1.4. Therefore p c_ ns. Finally, let (w, b) E ns\p. Then, by linearity, there exists a v E o such that (v, 0) E ns\p and v A 0. But v ¢ n, so adv is not nilpotent and therefore S(v) 54 0. Moreover, S(v) is semisimple. Hence there exists a b E q, and even a b E n, such that S(v)"b 54 0 for all n E N. Then [(v, 0), (0, b)]s = (0, S(v)b) and by induction one has (ads(v, 0))"(0, b) = (0, S(v)"b) 0 0.

It follows that ads(v, 0) is not nilpotent. Thus p = ns. Proposition 111.2.6 and the relation (111.4) show that the nilshadow qN can be characterized as the nilradical ns of the semidirect product o a q. This characterization is close to the original definition of the nilshadow (see Notes and Remarks). The second way of relating the construction of the nilshadow and the definition of the semidirect product is by noting that a r-> S(an) is an abelian representation of q in itself by derivations. This follows because the maps a ra an from q into

n and v r- S(v) from o into £(q) are linear by Proposition 111.1.1. Moreover, [S(vi ), S(v2)] = 0 = S([vi, v2]) for all V1, v2 E o by Proposition Ill. 1.1 and the property S(n) = (0). Since S(q)q c n the latter condition also ensures that the representation satisfies S(S(a)b) = 0 for all a, b E q. Then qN can be viewed as a kind of semidirect product of q with itself constructed via the representation S. This is a particular example of the following general construction.

Let g be a Lie algebra and r: g -+ L (q) a representation of g in itself by derivations with the property

r(r(a)b) = 0

(111.5)

[a, b]T = [a, b] + r(a)b - r(b)a

(I11.6)

for all a, b E g. Then

defines a twisted Lie bracket [ , ]T on the vector space g. The verification of the Jacobi identity for [ , It is very similar to the verification of the identity for the Lie brackets [ , ]N a defined by (111.3). In fact there is a converse to this statement. Explicitly, the bracket [ , ]T defined by (111.6) satisfies the Jacobi identity if and only if the representation r satisfies (111.5). The converse will be deduced as a corollary of properties of the group of automorphisms of G generated by r (see Corollary 111.6.2).

If gt = (g, [ , ]T) is the Lie algebra with the new Lie bracket, then a simple calculation, using (111.5), establishes that r is a representation of gr in itself. But (111.5) is not sufficient to ensure that the r(a) are derivations of gt. One has r(a)[b, c]t = [r(a)b, c]r + [b, r(a)c]T + [r(a), r(b)]c - [r(a), r(c)]b.

(111.7)

III. Structure Theory

74

Hence if the r(a) commute, then they are derivations of Or. Again there is a converse which is not quite so evident. The operators r(a), with a E g, are derivations of 9, if and only if the r(a) are mutually commuting. This will be deduced in the subsequent discussion of the group structure in Section 111.6 (see Corollary 111.6.5). It is related to the invertibility of the transformation g r-> Or Solvability of g does not suffice to prove that Or is solvable but if (111.5) is replaced by the stronger assumption that

r(n) = {0),

(111.8)

where n is the nilradical of g, then gr = (g, [

, ]r) is solvable. This follows directly from (111.6) and (111.8) because solvability of g implies that [g, g] c n

and any derivation maps g into n (see 11. 1.9). Therefore [g, g]r c n and the solvability of Or follows from the nilpotence of n. If, in addition to (111.8), one assumes that r(a) + ro(a) is nilpotent on n for each a E g, where to denotes the adjoint representation with respect to the original Lie bracket, then one can also conclude that gr is nilpotent. This follows since [a, it], = ro(a)n + r(a)n for all

nEn. Returning to the particular case discussed earlier, let q be solvable and let n be a subspace satisfying Properties II and III of Proposition 111.1.1. If one defines

rq: q -> £(q) by rq(a) = -S(at,), with a0 the component of a in n, then rq is a representation of q in itself by derivations and rq satisfies (111.5). One has [a, b]rq = [a, b]N for all a, b E q. So q,v = qrq as Lie algebras. Moreover, Tq satisfies (111.8). and rq(a) + ro(a) = -S(au) + adan + adan = K(ae) + adan is nilpotent by Corollary Ill. 1.4. So one rediscovers that qN = qrq is nilpotent. Note that qN and rq depend on the choice of n, but this is not expressed in the notation. Example 111.2.7 If oqN: qN -. C(qN) is defined by agN(a) = S(ad) for all a E

q,

then oqN is a representation of qN in itself by derivations, by Lemma 111.2.2. Moreover, [qN. qNI c n. Therefore oqN is a representation of qN in itself by derivations, which is again abelian, and aqN (aqN (a)b) = 0 for all a, b E qN. Then q = (gN)ogN as Lie algebras.

Example 111.2.8 The three Lie algebras n2 , ej , f of Example 111.1.8 all have an abelian nilshadow (see Example 111.2.1). If one defines a derivation r(a) = -adav. then the Lie

bracket of each algebra is given by (a, b] = [an, b] + [a, bb] = -r(a)b + r(b)a. Hence [a, b]r = [a, b]+r (a)b - r (b)a = 0. This calculation also has a converse. The Lie bracket of the algebra is constructed from the trivial bracket on the abelian nilshadow with the derivation

r(a) = S(ab).

The next example shows that the construction of [ the semidirect product of two Lie algebras.

,

]r is a generalization of

Example 111.2.9 Let 91, 92 be Lie algebras and is gt -. £(82) be a representation of 91 in 02 by derivations. Let g = 91 x 92 be the direct product of 91 and 92 and define r: g G(9) by r(al , a2)(b1, b2) = (0, i(a, )b2). Then r is a representation of gin itself by derivations and r(r(a)b) = 0. Moreover, (g, [ - ]r) equals the semidirect product 91 x 92 of 91 and 92 as Lie algebras since the Lie bracket

[(al. a2), (bt , b2)]r = ([al, bt ]g,, [a2, b2]92 + f (al )b2 - i(bt )a2) coincides with the definition (11.3).

111.2 The nilshadow; algebraic structure

75

One can also obtain each rank 2 nilpotent Lie algebra, i.e., each Lie algebra with [9, [9, 9] ] = [0}, from an abelian Lie algebra by the r construction.

9

Example 111.2.10 Let g = (g, [ , J) be a rank 2 nilpotent Lie algebra, and gab the abelian Lie algebra with underlying vector space g. Define r: g -+ £(g) by r(a) = 2-1 ada for a E g. Clearly r(a) is a derivation of gab. But since [g, [g, gJ) = 10), it follows that r(a)r(b) =

0 and r(r(a)b) = 0 for all a, b E g. In particular r(a) commutes with r(b) and r is a representation of gab. Then, however, the r construction applied to gab gives

[a, blr = r(a)b - r(b)a = [a, bJ. SO g = (gab)r

The solvable Lie algebras q with abelian nilshadow have a particularly simple structure. First, if v, v' E 0, then by definition

[v, v']N = [v, v'] = K(v)v' because S(v)v' = 0. Alternatively, if n, n' E n, then [n, n'] Iv = [n,

n'] = K(n)n'

because adn = K(n). Hence commutativity of the nilshadow qN implies that v and n are both abelian subalgebras of q. Secondly, by the definition of the nilshadow

[v, n]N = [v, n] - S(v)n = K(v)n for V E n and n E it. Therefore, if qN is abelian, then [v, n] = S(v)n and the adjoint representation of n on it must be semisimple. Clearly the nilshadow associated with a solvable algebra with non-abelian nilradical is not abelian. One can also construct examples for which the nilradical is abelian but adv has a nilpotent component for some v E U. Example 111.2.11 Consider the four-dimensional solvable Lie algebra q with non-vanishing

commutators [at, a2] = a3, [at, a3] = -a2 and [a2, a31 = a4. The nilradical n, the Lie subalgebra generated by a2. a3, a4, is isomorphic to the Heisenberg algebra (see Example 11.4.16). The subspace o is the linear span of a 1 and the algebra is the semidirect product

o x n where the representation S of o is semisimple, an infinitesimal rotation of a2 and

a3. Then the nilshadow is the direct product o x n and (a,,a21N = 0 = [al, a3lN but

=a4. Example 111.2.12 Consider the five-dimensional solvable Lie algebra q with non-vanishing commutators [a 1 , a2 l = a3 + a4, [a, , a3l = -a2 + a5, [al, a41 = a5 and [a 1, a5 l = -a4-

The nilradical n is the abelian Lie subalgebra generated by a2, a3, a4. a5 and the subspace o is the span of at. The algebra is the semidirect product of n acting on n but the representation is no longer semisimple. The semisimple component of the representation of o gives infinitesimal rotations on the subspaces spanned by (a2, a3) and (a4, a5). The nilshadow qN, which is given by eliminating the semisimple part of the representation on n, then has nonzero commutators [al, a2)N = a4 and [al, a31N = a5. Thus qN is the semidirect product of o and n with the residual nilpotent representation.

III. Structure Theory

76

One can apply the foregoing construction to a general Lie algebra g. Let u be a subspace of q satisfying Properties II and III and m a Levi subalgebra of g.

Define r9: g -+ £(g) by r9(a) = -S(an). Since [m, m] c m and [g, q] c n it follows that [g, g] c m ® n. Therefore Tg is again an abelian representation of g in itself by derivations. Moreover, S(v)g c n. This follows because S(v) is a polynomial in adv without constant term. As a consequence re satisfies the condition r9(r9(a)b) = 0 for all a, b E g. Therefore (111.6) defines a Lie bracket ]iv on the vector space g by the previous reasoning by

[

[a,b]N = [a, b] - S(ati)b + S(bu)a.

We call gN = (g, [

] ,) the semidirect shadow of g. But if m, m' E m, then [m, m']N = [m, in']

because both in and in' have component zero in v. Alternatively, if q, q' E q, then

[q, q']N = [q, q'] - S(g0)q' + S(q,)q and of course the bracket coincides with the previously given definition of [ , IN on the radical q. Finally, if, in addition, v satisfies Property I of Proposition 111. 1. 1, then

[m, qIN = [m, q]

for all m E m and q E q because in has zero component in u and S(gd)m = 0. Therefore

[m + q, m' + q'I'N = [m, m'] + ([q, q'l N +

(adm')q)

for all q, q' E q and m, m' E m. Since adm is a derivation of the nilshadow qrv = (q, [ , IN) of q for all m E m, by Lemma 111.2.2, the Lie algebra gN = N) is naturally isomorphic with m x (IN, where the action of m is by the , adjoint representation. (g, [

Thus the algorithm g r+ gra gives the transformation m x q H m x U. The essential action of S(v) is on q. Therefore rg is a representation of g by derivations on q. For this reason the modification of the Lie product only affects the q component. The foregoing process can be inverted in the manner of Example 111.2.7. The latter example describes the passage from a solvable Lie algebra q to its nilshadow qN and the converse the passage from qN to q. This inversion is not affected by the semidirect product with the semisimple Lie algebra m.

Let aqN = -rg. Thus aqN (a) = S(an). Then aqN is an abelian representation of gN in itself satisfying the condition aqN (aqN (a)b) = 0 for all a, b E gN. A straightforward calculation analogous to (111.7) establishes that the aqN (a) are derivations of gN. But the aqN modification of the bracket [ , IN gives

[m+q, in'+q'] ,

= [ni+q, m'+q']'N+S(q)q' -S(q,)q = [in +q, m'+q'],

111.2 The nilshadow; algebraic structure

77

i.e., the modified bracket is the original Lie bracket. Therefore the foregoing general procedure of redefining the Lie product allows one to pass from g to gN and

back again. In particular g = m x q = (m x gN)ogN = (9N)ogN where aqr, is given by

aq, (a) = S(an),

(111.9)

with au the component of a in v. There is also a method for constructing the semidirect product g = m x q in the Levi decomposition from the direct product m x qN by modifying the Lie bracket. This will be of fundamental importance in the analysis of semigroup kernels in Chapter IV. In contrast to the foregoing construction of m x q from m x qN, the construction of g = m x q from m x qN is not invertible.

Identify q E q with (0, q) E g= m x q and in E m with (m, 0) E g. Hence m+q = (m, q) for all (m, q) E m®q. Let v be a subspace satisfying Properties IIII of Proposition I11.1.1. Further, let ON = m x qN be the direct product of the Lie algebra m and the nilshadow qN of q. W e denote by [ , IN the Lie bracket on qN and call (ON, [ , ]N) the shadow of the Lie algebra g. Define Cr: ON -* £(ON) by a (m, q) (m', q') = (0, (adgm + S(qu))q'), (III.10)

where qt, is again the component of q in v. By Lemma 111.2.2 the operators adgm and S(qu) are derivations of qN, for all m E m and q E q. So a(m, q) is a derivation of ON for all (m, q) E ON. Also, a is a representation since v satisfies Prop-

erties I and IV of Proposition I11.1.1. Moreover, a(a(m, q)(m', q')) = 0 for all (m, q), (m', q') E ON because [g, q] c n, which implies that a(m, q)(m', q') E {0} x n. Then

[(m, q), (m', q')],, = ([m, m'], [q, q']v + S(g0)q' - S((q')n)q + (adgin)q' - (adgn:')q) = ([m, m'], [q, q'] + (adgm)q' - (adgm')q) = [(nt, q), (in', q')]g

(III.11)

for all (in, q), (in', q') E g. So g = m x q = (m x qN)a = (ON), as Lie algebras. Again ON and a depend on the choice of m and v. Thus, a general Lie algebra is equal to the Lie algebra obtained by a twisted Lie bracket on a direct product of a semisimple Lie algebra and a nilpotent Lie algebra. In the previous construction of the solvable Lie algebra q from its nilshadow qN the twist agr,(a) = S(an) is semisimple. If the general Lie algebra is of type R, then this property is still valid.

Proposition 111.2.13 If g is type R, then a(a)I q = S(am + au)Iq for all a E g, where am and au are the components of a in m and v, respectively. In particular or (a) is semisimple for all a E g.

Proof First, let G be the simply connected Lie group with Lie algebra g and let M denote the simply connected subgroup with Lie algebra m. Since 9 is type R

III. Structure Theory

78

the group M is compact by Proposition 11.4.8 and Corollary 11.4.12. Therefore the

adjoint action m H Ad(m)a of M on g is bounded for all a E 9, where Ad is the adjoint representation of G on g. Secondly, the eigenvalues of adb are purely imaginary, for all b E g, because g is type R. But the eigenvalues of S(b) are the same as those of adb, as a consequence of the Jordan decomposition. Therefore the eigenvalues of S(b) are purely imaginary.

Thirdly, it follows from the identity erK(b)a = e-1S(b)Ad(exp(tb))a that the function t r-> e`K(b)a is bounded from R into g for all a E g and b E m. Hence

K(b)a = 0 for all a E g and b E m. Therefore K(b) = 0 and S(b) = adb for all b E m. In particular adblq = S(b)Iq for all b E m. Therefore a(a) _ (S(am) + S(an))I q for all a E 9. Fourthly, since [m, u] = {0} it follows that adam and adan commute. Since S(am) and S(an) are polynomials in adam and adan, respectively, it follows that [S(am), S(an)] = 0. Hence S(am)+S(an) is semisimple. Moreover, since K(au) is a polynomial in adan it also follows that S(am) and S(an) commute with K (ad). Hence S(am) + S(an) commutes with the nilpotent endomorphism K(a0). Finally, since K(am) = 0,

(S(am) + S(an)) + K (an) = adam + adan = ad(am + an). Hence, by the uniqueness of the Jordan decomposition, S(am) + S(a0) = S(am + an) and K(ae) = K(am + a0). Therefore using the previous identification of or one deduces that a(a)I q = (S(am) + S(au))Iq = S(am + an)Iq for all a E g. We continue the analysis of spectral properties of a in the next section.

111.3

Uniqueness of the nilshadow

The construction of the nilshadow qN of the radical q of the Lie algebra g depends on the choice of a complementary subspace n satisfying Properties II and III of Proposition 111. 1. 1. Although there is no unique choice of o, we next argue that the nilshadow is essentially unique. Different choices of t7 lead to isomorphic nilshadows. Although this result is not essential for the later discussion of asymptotic behaviour it is of structural interest. Moreover, analysis of the isomorphism problem gives additional useful spectral information. As a first step to an isomorphy result define the linear space

ro(t)) = {b E q : S(o)b = (0)). Note that if aqN is defined by (111.9) with respect to u, then

W(V) = n keragN(a). aEg

111.3 Uniqueness of the nilshadow

79

Clearly o c_ m(o) and q = to (n)+n. Now suppose w E ro (o) and write w = v+n

with v E n and n E n. Then n E ro(o) and S(v)n = 0. Hence S(v) = S(w) by Corollary 111. 1.4. This relation immediately implies the following identification.

Lemma 111.3.1 Let o I and 02 be two subspaces satisfying Properties II and III of Proposition I11.1.1. Let qN) = (q, [ , ]N)) and qN) = (q, [ , ]N)) denote the corresponding nilshadows. If ro(of) = m(02), then qN) = as Lie algebras. q(2N)

Proof Let a I , a2 be the components of a E q in the subspace o l, 02, respectively.

Then at, a2 E ro(nl) = ro(02) and one has a decomposition at = a2 + n with tl E n fl ro(ot). Hence S(al) = S(a2) by the argument preceding the lemma and the Lie brackets of the two possible nilshadows are equal, by definition.

0

Although the condition ro(of) = tv(o2) is sufficient to ensure equality of the corresponding nilshadows, it is not a necessary condition. This is most clearly seen in the case of an abelian nilshadow. Example 111.3.2 The Lie algebra e3 of the three-dimensional group E3 of Euclidean motions (see Examples 11.4.18, 11.5.1 and 11.5.2) has a basis at , a2, a3 with two non-vanishing commutators [al. a2] = a3 and [al, a3 l = -a2. The nilradical n is the abelian subalgebra generated by a2. a3. For X2, A3 E R one can choose complementary subspaces of and 02 as the one-dimensional spans of aI and at + 12a2 + X3a3, respectively. Both choices lead to an abelian nilshadow. But ro(ot) = nt and ro(ut) = 02. Therefore ro(ot) = ro(o2) if and only

if A2=0=A3.

An alternative uniqueness statement follows for the complementary subspaces which are equal up to isomorphism.

Proposition 111.3.3 Let 4) be an automorphism of the Lie algebra g. If u is a subspace satisfying Properties II and III of Proposition 111.1.1 then o' = c(o) satisfies the same properties. Moreover, if qN and qN, denote the nilshadows cor-

responding to 0 and o', respectively, then the restriction OIq E £(q) of 0 to the radical q is an isomorphism front qN onto qN,. The essential ingredient in the proof of the proposition is a general result on surjective homomorphisms. Lemma 111.3.4 Let g and 1) be Lie algebras and tr: g -+ I) a surjective Lie homontorphisnt. Then the components in the Jordan decomposition satisfy zr(S(a)b) = S(n(a))zr(b)

and

7r(K(a)b) = K(tr(a))n(b)

for alla,bE0. Proof Fix a E g. The kernel t = 7r ((0)) of n is an ideal of g. Since S(a), K(a) are polynomials in ada by (11.1.10), then S(a)t c_ t and K(a)f c_ t. Therefore there exist transformations TI (a), T2(a) E G(I)) such that

T1(a)(nb) = tr(S(a)b)

,

T2(a)(7rb) = tr(K(a)b)

80

III. Structure Theory

for all b E g, and clearly ad(,ra) = Ti (a) + T2(a). From the corresponding properties of S(a) and K(a), it is straightforward to deduce that Ti (a), T2(a) are respectively semisimple and nilpotent, and [T1 (a), T2(a)] = 0. The uniqueness of the Jordan decomposition for ad(7ra) implies that Tt (a) = S(ira) and T2(a) _ K(7ra). Now the proof of the proposition is straightforward. Proof of Proposition 111.3.3 It follows immediately from Lemma 111.3.4 that one

has 4 (S(a)b) = S((D(a))(D(b) and 1(K(a)b) = K((D(a))(b(b) for all a, b E g. Hence o' satisfies Properties II and III of Proposition Ill. 1.1. Moreover, 4'(a,) _ ((D (a)) v, and CD(bu) = ((D (b)),,. Therefore

4)([a, b1 N) _'([a, b] - S(ae)b + S(bu)a)

= [c(a), 0(b)] - S((D(an))4 (b) + S((D(bn))4(a)

_ [c(a), c(b)] - S(((D(a))d)' (b) + S(((V(b))n')(V(a) _ [(D(a), 4)(b)]N' which yields the proposition.

Proposition 111.3.3 gives a case in which different choices of the subspace n in Proposition III.1.1 imply an isomorphism of the two nilshadows. Next we argue that in any case the two nilshadows are equivalent. First we claim that ro(o) is a Cartan subalgebra of q. Obviously ro(o) is a subalgebra. Then since

tv(o)=oED (b En:S(a)b=0 forallaE0), it follows that [a, b]N = [a, b] for all a, b E ro(o). Hence m(o) is nilpotent. To prove that ro(o) is its own normalizer in q we must show that if c E q and [c, to(o)] c to(o), then c E tu(o). Since S(a) is a polynomial in ada the conditions on c imply that S(a)c E to(o) for all a E ro(u). Hence S(a)2c = 0 for all a E o. Then S(a)c = 0, by semisimplicity of S(a), and C E ro(o). Next, if m is a Levi subalgebra of g and o is a subspace of q which satisfies Properties II and III of Proposition 111. 1. 1, then set

ro(m, o) = m ® {b E q : S(o)b = {0} and [m, bJ = (0)) = m ® (b E ro(o) : [m, b] = (0)).

Note that ro(m, o) = laEg kera(a), where a is defined by (Ill. 10) with respect to the spaces m and o. Moreover, note that to(m, o) is the Lie algebra direct sum of m and the nilpotent Lie algebra {b E to(o) : [m, b] = {0}}. We have the following conjugacy results for to(n) and ro(m, u).

111.3 Uniqueness of the nilshadow

81

Theorem 111.3.5 1.

Suppose of and 02 are two subspaces of q which satisfy Properties II and III of Proposition 111. 1. 1. Then there exists a c E n such that ead`(ro(0t)) = MOD.

II.

Suppose m1, 0t and m2, 02 are two pairs which satisfy Properties I-III of Proposition 111. 1. 1. Then there exists a c E n such that eadr(ro(mI

, ot)) = 10(m2, 02)

Proof Statement I follows because any two Cartan subalgebras of the solvable Lie algebra q are conjugate to each other by an inner automorphism ead` with c E n (see 11. 1.6). Next we prove Statement II. First choose c E n satisfying Statement I. It follows

from the definition of w(oi) and the commutation property [mi, oi] = (0) that [mi, ro(oi)] c w(of) for each i E (1, 2). Therefore we may define subalgebras of g by to, = mi ® ro(of). Note that ro(of) and mi are the radical and a Levi subalgebra of to, respectively. Define m3 = eadc(ml)

and

103 = m3 ® w(02) =

eadc(rot)

Then 103 is a subalgebra of g and [m3, m(02)] c MODWe claim that r02 = to3. To prove this, let a E M3. Since g = m2 ® q we can write a = a2 + b with a2 E m2 and b E q. Because [mi, m(02)] c m(o) for each

i E (2, 3) it follows that [b, m(02)] c ro(02). Then b E w(02) since t,002) is a Cartan subalgebra of q. This proves that m3 c_ to2. Hence ro3 c_ 102. Similarly o2 c ro3 and the subalgebras are equal. Now m2 and m3 are Levi subalgebras of r02. By the Mal'cev-Harish-Chandra conjugacy theorem, Statement 11.1.6, there exists a cV E Gp such that 4)(M3) = m2 where Gp denotes the Lie subgroup of GL(m2) whose Lie algebra consists

of the transformations adm2b, with b E p = [m2, m(02)] c_ m(b2). Since p c [g, q] c n it follows that Gp is nilpotent. Therefore the exponential map for Gp is surjective by 11. and it follows that there exists a c' E p such that (D equals the a-ad`. 10(02) c 10(02). One restriction to 102 of eau`'. Since C' E 10(02) one has deduces that m(m2, 02) = m2 ® (b E m(02) : [m2, b] = (0)}

=

eadc'(m3 ® (b E ro(ot) : [m3, b]

=

eadc'eadr(m, ® (b E 10(01) : [ml, b] _ (0)) 1.

= (0}}

Because the Lie subgroup of GL(g) generated by the eaua, with a E n, is nilpotent there exists a c" E n such that eadc'eadc = eadc" and the proof is complete. 0

III. Structure Theory

82

Corollary 111.3.6 Let 01, 02 satisfy Properties II and III of Proposition 111. 1.1 and let qN) = (q, [ , ]N and qN) = (q, [ , ]N)) denote the corresponding nilshadows. Then there exists a c E n such that (D = ead`Iq E £(q) is an isomorphism from q(1) onto q(2).

Proof Choose C E n as in Statement I of Theorem 111.3.5 and define (D as above. Then tIq is an isomorphism from qN) onto q(3) by Proposition 111.3.3, where q(3)

is the nilshadow associated to 4(u). But ro((D(01)) = 4 (ro(ut)) = ro(172). So q(3)

= q(2) by Lemma 111.3. 1.

Thus up to isomorphism the nilshadow is unique. Example 1113.7 Let b1 and n2 be the two choices of complementary subspace in Example 111.3.2, i.e., the one-dimensional spans of a1 and b1 = at + 1qa2 + x3a3, respectively. If c = Z2a3 - a3a2, then [r, all = k2a2 + A3a3 and eadral = b1. Hence m = eadr gives the isomorphism between the two nilshadows. Then the equality of the nilshadows follows by noting that r E n. Therefore 4'(n) = n for all n E n, because the nilradical is abelian, [m(a), fi(b)] = 4>([a, b]) = [a, b], since [a, b] E n, and S((D(a 1))4>(b) = since m maps the component a in bt into the component a,7 in 02.

Remark 111.3.8 Let n 1, 02 satisfy Properties II and III of Proposition 111. 1. 1 and let c E n be as in Statement I of Theorem 111.3.5. So ead`,(ro(o1)) = m(02). Let

a E 02. Since ro(01) c q = n1 ® n there exist v E pl and n E n such that eadra = v + n. But v + n E eadr(ro(u2)) = ro(ut). Hence S(n1)n = S(01)(v + n) = (0) and n E ro(ut) nn. Conversely, let o satisfy Properties II and III of Proposition 111. 1. 1 and let 01 c v + (ro(u) fl n) be a subspace of q such that q = ul ® n. Then uI satisfies Properties II and III of Proposition 111. 1. 1 by Corollary 111. 1.4 and ro(o1) = ro(o).

Moreover, if c E n, then Proposition 111.3.3 establishes that 02 = e-adc(01) is a subspace of q which also satisfies Properties II and III of Proposition Ill. 1.1. Theorem 111.3.5 also gives a characterization of the semisimple part of ada.

Corollary 111.3.9 If g is solvable, then for all a E g there exist v E u and c E n such that S(a) = eadr S(v) a-adr

Proof It follows from Corollary 111. 1.5 that one can choose a subspace u' such that g = o' ® n, S(u')u' = {0} and S(n')a = {0}. Then if a = v' + n with v' E 0' and n E n, one has S(a) = S(v') by Corollary 111. 1.4.

But it follows from Remark 111.3.8 that there exist c E n and n' E {n E n S(0')n = {0} }, such that v = e-adc(v' + n') E U. Then S(v')n' = 0 and S(v' + n') = S(v') again by Corollary 111.1.4. Combining these conclusions one has

S(v' + n') = S(a). So S(v) = S(e-adr(v' + n')) = e-adc S(v' + n') eadr = e-adc S(a)

eadr.

This corollary gives a characterization of solvable Lie algebras of type R.

Lemma 111.3.10 Let q be a solvable Lie algebra. The following conditions are equivalent.

111.3 Uniqueness of the nilshadow I.

q is of type R.

II.

The eigenvalues of S(a) are purely imaginaryfor all a E q.

III.

The eigenvalues of S(v) are purely imaginaryfor all v E V.

83

Proof I..II. The Lie algebra q is of type R if and only if ada has purely imaginary eigenvalues for each a E q. But this is equivalent to the operators S(a) having purely imaginary eigenvalues, since the eigenvalues of S(a) are the same as those of ada as a consequence of the Jordan decomposition. II=>III. This is evident. III=II. If a E q, then by Corollary 111.3.9 there exist v E u and c E n such that

S(a) = ead`S(v)e-ad. Since S(v) has purely imaginary eigenvalues S(a) also has purely imaginary eigenvalues.

One can give another criterion for a general Lie algebra to be of type R using spectral analysis. The operators S(v) can be analyzed by spectral theory. In fact, the spectral analysis can be carried out in a manner compatible with the lower central series (qN;k) of qN. By Corollary 111.2.4 there are subspaces Ilt, ..., f)r

of q, all invariant under S(n) and adm, such that qN;j = hi ® ® hr for all j E (1, ... , r) and n c 41. Therefore it suffices to decompose each of the spaces Il j.

Lemma 111.3.11 If u satisfies Properties II and III of Proposition 111. 1. 1 and

is

a subspace of q satisfying S(u)ll c , then there are s, s' E No, two-dimensional subspaces Et , ... , Es and one-dimensional subspaces E'1,..., E', of Il, and for all I E (1, ... , s) there are independent P1, at E Et, a non-zero linear function A1: u -+ R and a linear function Al: u -> R, and for all 1 E (1, ... , s'} there is a linear function js : u -+ R such that the following are valid. 1.

H.

h=El

El®...®E'T,.

S(a)p, = µ,(a) pt + A,(a) at and S(a)o, = µ1(a) at - A1(a) p1 for all

a E uandl E {1,...,s}. III.

Proof

S(a)p=p.(a)pfor all aE0,IE(1,...,s')andpEE'. For a E u consider the complexification Sc(a) of the operator S(a) on

the complexification 4C of the space 1). Since the SC(a) commute they can be put in diagonal form simultaneously, with complex eigenvalues.

If b E bC is a joint eigenvector, write b = p + is with p, a E h and p 54 0. If a # 0, then p - is is also an eigenvector. Let A, .s: u --> R be the linear function

such that Sc(a)b = (µ(a) - i A(a))b for all a E D. Then ), 54 0 since a # 0. Moreover,

S(a)p =A(a) p +),(a) or

,

S(a)a = µ.(a) a - X(a) p

for all a E u. Then span(p, a) contributes a two-dimensional subspace El. The case a = 0 can be dealt with similarly to contribute a one-dimensional subspace E'. The completion of the lemma is obvious.

III. Structure Theory

84

The decomposition leads to the following characterization of Lie algebras of type R.

Proposition 111.3.12 Let g be a Lie algebra, m a Levi subalgebra of g and v a subspace satisfying Properties I-III of Proposition Ill. 1.1 The following are equivalent. 1.

g is of type R.

IT.

There exists a (real) inner product ( , ) on g such that ada and S(v) are skew-symmetric for all a E m and v E U.

Moreover, i f these conditions are valid and f 1, . hr, to. pt are subspaces of q satisfying the properties of Corollary 111.2.4, then one can choose the inner prod-

uct ( , ) on g such that the spaces m, 0, to, t1, h2.... , hr are mutually orthogonal.

Proof I=II. If g is of type R, then the eigenvalues of S(v) are purely imaginary for all v E v by Lemma 111.3.10. Let ht . . , hr, po, Ct be subspaces of q satisfying the properties of Corollary III.2.4.Then g =m® v®to ®P1 ED hr and each of these components is invariant under S(v). Therefore one can apply Lemma 111.3.11 to each component, except for the m component, and one has Al = A' = 0 for all l E ( 1 , ... , s) and k E (1, ... , s'). One can then find a basis

passing through Et ® ... ® E. ® E, ®... ® Econtaining pt, of, ...

, ps, ors,

where we use the notation of Lemma 111.3.11. Joining the bases of all components, and choosing a basis for m, one obtains a basis bi, ... , bd for g. Let (- , ) be the real inner product on g such that bi, ... , bd is an orthonormal basis. Then S(v) is skew-symmetric with respect to (- , - ) by Lemma 111.3.11 and the spaces m, 0, to, P t . 42. . hr are mutually orthogonal by construction. Let M be the (connected), simply connected, Lie group with Lie algebra m. Since m is semisimple and type R the group M is compact by Statement III of Proposition II.4.8. Now define the symmetric bilinear form ( , ) on g by

(a, b) =

Jm

dm (Ad(m)a, Ad(m)b),

where din is the normalized Haar measure on M. Then (- , ) is positive-definite, so it is an inner product on g. The spaces m, v , to, t j, 42. , hr are again mutually orthogonal by construction. Since S(v) commutes with ada for all a E m, and therefore with Ad(m) for all rn E M, the operators S(v) are skew-symmetric with respect to (- , ). Moreover, if a, b E g and c E m, then (a, b) =

dm (Ad(,n expM(tc))a, Ad(m expM(tc))b) _ (etadra etadrb)

for all t E R, by right invariance of dm. Differentiation with respect tot gives the identity 0 = ((adc)a, b) + (a, (adc)b). Hence adc is skew-symmetric.

III. First if S(v) is skew-symmetric, then S(v) has purely imaginary eigenvalues and q is of type R by Lemma 111.3.10. Secondly, the restriction of (- , - ) to

111.3 Uniqueness of the nilshadow

85

m gives a positive-definite invariant symmetric bilinear form on in. The invariance is a weakening of the skew-symmetry of the adgc with c E in. So in is isomorphic

to the Lie algebra of a compact group and in is of type R by Proposition 11.4.8. Now the implication follows from Corollary II.4.12. Example II13.13 The Lie algebra e3 of the three-dimensional group E3 of Euclidean motions, Example 111.3.2, has a basis at , a2, a3 with two non-vanishing commutators [at , a21 =

a3 and [at , a31 = -a2. The nilradical n is the abelian subalgebra generated by a2, a3. Choose the complementary subspace n as the one-dimensional span of at and the inner product (- . ) such that a 1, a2, a3 are orthonormal. Then S(a t) = ada 1 and if ax = x t at + Z2a2 + a3a3, then

(x2143 - ),3142) = -(ax, S(al)aµ) so the S(v) are skew-symmetric. Clearly o and n arc orthogonal.

Next we derive a uniqueness statement concerning the lower central series of the nilshadow of the solvable Lie algebra q. Let r be the rank of the nilshadow , ]N) and qN;j the lower central series, i.e., qN; I = q and qN;j+1 = [q, qN;j]N with qN;r+1 = {0} but qN;r # {0}. Then q = qN;1 2 n D qN;2 D 3 qN;r 3 qN;r+1 = (0). In principle the subspaces qN; j could depend on the .

(q, [

choice of subspace v. But this is not the case. Lemma 111.3.14 The subspaces qN;j are invariants of g. Specifically, if qN; j and qN';j are the lower central series of qN and qN', the nilshadows corresponding to subspace v and o' satisfying Properties II and III of Proposition 111. 1. 1, then

qN;j = qN';j for all j E N.

Proof

The statement is obvious for j = 1, so we may assume j > 2. By

Corollary 111.3.6, there exists a c E n such that eaddlq maps qN isomorphically onto qN'. Then ead`'(gN;j) = qN;j for all j E N. But it follows from the definition of [ , IN that qN; j c [q, q] c n for all j > 2. Since the Lie brackets [ , ] and [ , IN agree on n, by definition, and since c E n, one has eadr(gN;j)

= eadN"(qN;j) = qN;j.

The last identity follows because qN;j is an ideal of qN. Thus q' ;j = eadr (qN;j) _ qN;j and the qN;j are invariants of g.

Recall that a characteristic ideal of a Lie algebra is an ideal which is invariant under any automorphism of the Lie algebra. For example, q and n are characteristic ideals of 9. Obviously the qN;j are characteristic ideals of qN. Less obviously, one has the following useful result.

Corollary 111.3.15 The subspaces qN;j are characteristic ideals of g.

Proof Let' be an automorphism of g. Set o' = c(u) and define the nilshadow qN, and lower central series qN';j corresponding to v' (see Proposition 111.3.3). Then 1(gN;j) = qN';j = qN;j by Lemma 111.3.14. We next analyze the action of surjective homomorphisms on 9.

III. Structure Theory

86

Lemma 111.3.16 Let g and II be Lie solvable algebras and n: g -+ h a surjective homomorphism. Denote the nilradical of f) by n(l)). Let

to = (v E u

tr(v) E n(())}

:

and let o' be a subspace complementary to to in u, that is, u = o' ® to. Then the following are valid. I.

,r(n) C n(ry).

II.

S(a)b = O for all a, b E 7r (V) and (III.12)

g=0'ED to ED n, ,r

(n(f))) = to ® n,

(III.13)

h = n(0') ® n(l)). III.

(111.14)

If ( , ]N denotes the nilpotent Lie bracket in f) defined from n(0'), then n is a homomorphism f r o m (g, [

,

]N) onto (h, [

,

]N).

Proof Since n is surjective, n(n) is a nilpotent ideal of h. Then 7r(n) g n(4) by the definition of the nilradical n(()).

Next we prove Statement II. Since S(a)b = 0 for all a, b E o' one deduces from Lemma 111.3.4 that S(a')b' = 0 for all a', b' E n(0'). The decomposition (111. 12) is immediate. Obviously ir(ro ® n) = n(tv) +n(n) C_ n(f)), which means

that to ® n C n-1(n(E))). Conversely let a E n-1(n(h)) and write a = v + n with V E 0 and n E n. Then ir(v) = 7r(a) - ir(n) E n(b), so v E tv. Therefore a E to ® n, which completes the proof of (111. 13).

Since 9 = u'®n-1(n(l )) and n is surjective one has f) = 7r(g) = tr(u')+n(h) To prove (111. 14) it remains to prove that the sum is direct. Let V' E 0' and suppose that 7r (V') E n(l ). Then

v' E v' fl it ' (n(I))) = (0), and 7r(v') = 0.

Finally we prove Statement III. Let a, b E 0 = o' ® to. Then a = a' + all, b = b' + b" with a', b' E o' and a", b" E to. Further let a"', b"' E n. Then it is sufficient to establish identities

ir([a', b']N) = [ir(a'), n(b')]N ir([a, b,,,]N)

_ [ir(a') n(b. )]N

,

n([a', b"]N) = [n(a') n(b")]N

,

etc.

for all choices of pairs of the components. We will only establish the relation ir([a', b"]N) = [ir(a'), ir(b")]N as the proofs of the remaining cases are sim-

ilar. Since a', b" E u one has [a', b"]N = [a', b"] = K(a')b". Hence it follows from Lemma 111.3.4 that ir([a', b"]N) = K(rr(a'))ir(b"). But ir(a') E n(0') and n(b") E n(ro) C n(I)). Therefore one then has K(ir(a'))ir(b") = (tr(a'), n(b")]N by definition of [

,

IN. This establishes the second identity.

III.4 Near-nilpotent ideals

87

Corollary 111.3.17 Let g and Il be solvable Lie algebras and n: g --> h be a surjective homomorphism. Then n(gN; j) = 4N; j for all j E N. Proof By Lemma 111.3.14 the subspaces ON; j and f)N; j are independent of the choice of [ , ]N. So the result follows from Statement III of Lemma 111.3.16. O

It is useful to consider homomorphisms and the connection between the twist on the two Lie algebras. Corollary 111.3.18 For i E (1, 2) let mi be a Levi subalgebra of a Lie algebra gi with radical q; and nilradical n;. Let ti be a subalgebra of q; satisfying Properties I-III of Proposition 111. 1. 1 and let a;: giN - G(giN) be the representation by derivations given by (111. 10). Next let n: 91 --> 92 be a surjective Lie algebra homomorphism such that n(m1) = m2, n(01) = 02 and n(nt) = n2. Then it is a Lie algebra homomorphism from the shadow 9IN into 92N. Moreover, nab (a)b = a2(na)nb for all a, b E 91N.

Proof It follows from Lemma 111.3.16 that the restriction nIq, is a Lie algebra homomorphism from q I N into Q2A. Moreover, 7r (a, b) = (7r (a), 7r (b)) for all

a E ml and b E qt by the identification of a with (a, 0) and b with (0, b) in the semidirect products, since tr(ml) = m2 and n(qt) = q2. So it is also a Lie algebra homomorphism from 9IN into 92N. Finally, if a, b E qt, then

(na)12 because n(o1) = 02 and ir(nt) = n2. Hence nai(a)b = a2(na)nb for all a, b E 91N. The corollary follows immediby Lemma 111.3.4, since ately.

III.4

Near-nilpotent ideals

The Lie algebra 9 has, by 11.1.5, a direct sum vector space decomposition g = m ® q in terms of a semisimple Levi subalgebra m and the radical q of g. The radical has a further decomposition q = v ® n into a vector subspace n and the nilradical n. The Levi subalgebra m and the complementary subspace 0 are far from unique but can be chosen to satisfy the properties of Proposition 111. 1. 1. In particular [v, m] = (0). The radical and nilradical are maximal in the sense that q is the unique solvable ideal which contains every solvable ideal of g and n is the unique nilpotent ideal which contains every nilpotent ideal of g. Now we define an ideal to of g to be near-nilpotent if it has the form to = to, ® ro as vector spaces where w, and ron are semisimple and nilpotent ideals of g, respectively. Then to = to, ® tv as Lie

algebras since both tv, and to are ideals and therefore [ro ron] c rv, n tv c_ in n q = (0). We shall show below, in Proposition 111.4.1, that there exists a

88

III. Structure Theory

unique maximal near-nilpotent ideal n,,, of g. This ideal, which we call the near-

nilradical, contains every near-nilpotent ideal of g. The Lie algebra g is then defined to be near-nilpotent if g = n, which is equivalent tog being the direct product of the Levi subalgebra m and the nilradical n. In particular the Lie algebra g is near-nilpotent if and only if it is equal to its shadow. If g is of type R, then m is the Lie algebra of a compact group by Proposition II.4.8 and this motivates the terminology near-nilpotent. It is not evident that ns exists but this is established by the following proposition.

Proposition III.4.1 There exists a semisimple ideal s of g which contains every semisimple ideal of g and a near-nilpotent ideal n,,.,, which contains every nearnilpotent ideal of 9. Moreover, s and n, are characteristic ideals of g, s n q = (0) and n,,, = s ® n as Lie algebras. Finally s = {m E m : [m, q] = (0))

(111.15)

for any Levi subalgebra m of g.

Proof Fix a Levi subalgebra m and define s by (111. 15). It readily follows that s is an ideal of m. But each ideal of a semisimple Lie algebra is semisimple by 11. 1.3. Hence s is semisimple. Then s is an ideal of 9, because 9 = m ® q and [s, q] = {0}. Since s c m, the intersection s n q is trivial. Next suppose that a is a semisimple ideal of 9. We argue that a e_ s. By 11. 1.7 there exists a 4 E Gp such that 0(a) c m, where Gp is the subgroup of GL(g) with b E p = [q, g]. Since a is an ideal, 0(a) = a so a e m. generated by But as a and q are ideals we deduce that

la,q]cangcmnq=(0). Therefore a c s. Thus s contains every semisimple ideal. It follows straightforwardly that s is independent of the choice of Levi subalgebra used in its definition and that s is a characteristic ideal of g. Define n,,, = s ® n. It is clear that ns is a near-nilpotent ideal and a characteristic ideal of g. Next let a be a near-nilpotent ideal of g. Then a = a, ® a, where a, and a are semisimple and nilpotent ideals of g, respectively. But a, e s because s is maximal among the semisimple ideals of g and a c n by maximality of n among the nilpotent ideals. Hence a c_ ns and n,,, is maximal among the near-nilpotent ideals.

Note that by 11.1.3 there exists an ideal sl- of m such that m = s ® sl and [s, sl] = (0). Set go = sl ® q. Then since (s, q] = (0) it follows that 9 = s ® go

and

[s, go] = {0},

i.e., the Lie algebra g is the direct product of the semisimple ideal s and the Lie algebra go. Moreover, the Lie algebra go does not have any non-trivial semisimple

III.4 Near-nilpotent ideals

89

ideals. The nilradical it of g is also the nilradical of go and in addition is the nearnilradical of go. Despite this factorization, the ideal s plays a complicating role in the subsequent analysis of subelliptic operators because the operators do not necessarily factor across s and go. There are several useful alternative characterizations of nsn. Define a: ON G(9N) by (111.10). Recall ON = m x qN with (IN the nilshadow of the radical q. Since a is a representation it follows that {a(a) : a E gN} is a Lie subalgebra of ,C(gN). But g = ON as sets and a(g) decomposes as a direct sum

-

Cr (g) = a (M) ED a (U)

(111.16)

with a(m) and a(v) semisimple and abelian ideals of a(g), respectively. This is established by the following reasoning. First, g = m ®v ® it and since a (n) = (0) one has a(g) = a(m) + a(u). But [a(m), a(u)] = (0) because v satisfies Property I of Proposition 111. 1. 1. Secondly, a(a) = S(a) for a E v. Hence or (o) is abelian, by Property IV of Proposition 111.1.1. Thirdly, a(m) is isomorphic to m/(keralm). But keraIm is an ideal of m and hence is semisimple by 11.1.3. Therefore m/(ker a I m) is semisimple, again by 1I.1.3, and or (m) is also semisimple. Then a (m) fla (o) is both semisimple and abelian. Hence a (m) nor (v) = (0). The decomposition (111. 16) immediately implies that the near-nilradica is equal to the kernel of a.

Proposition III.4.2 The following identities and inclusions are valid:

nsn = kera = (a E g : a(a)t1 = (0))

={aE9:[a,tjI

U;2)

= (a E 9 : dfEN(ada)ntI C qN:2)

c (a E g : adalq is nilpotent) C (a E g : dbEt,3fENo(ada)nb E qN;2),

where tj is a subspace as in Corollary 111.2.4. Moreover if g is of type R or g is solvable, then all the above sets are equal.

Proof Since a(n) = (0) and a(g) = a(m) ® a(o) by (111. 16) it follows that the kernel kera also decomposes

kera =((kera)flm)®((kera)flo)ED n = {m E m: [m, q] = (0)} ®{v E o: S(v) = 0} ® n

=s®n=nsn Hence nsn = kera and the first equality is valid. Fix a E g. Then

adalq = a(a)Iq + K(ati)Iq + adanlq,

(111.17)

90

III. Structure Theory

where at, and an are the components of a in n and n, respectively. The operator K(an)Iq + adanlq maps q into n and qNJ fl n into qN;j+t by the definition of [ , ]N and Corollary 111.2.3. Hence K(ad)Iq + adanlq is nilpotent. Moreover,

(ada)'b = a(a)"b mod qN;2

(111.18)

for all b E tt and n E N since a(a) leaves t, invariant by Corollary 111.2.4.

Now kera c (a E g

:

o(a)ts = {0}}. Conversely, if a(a)tl = (0), then

a(a)hl = {0} with Ill as in Corollary 111.2.4. Since or is a derivation of qN it follows from Lemma 111.2.5 that a(a)q = {0}. Therefore a(a) = 0. Hence the second and third sets in the proposition are equal.

Next if a(a)ti = (0), then [a, tt] c qN;2 by (111. 18). Conversely, if [a, tj] c qN;2 then a(a)tl qN;2 by (111.18). But a(a)ti c tj and ei fl qN;2 = {0}. Therefore a(a)tl _ {0}. Hence the third and fourth sets in the proposition are equal. But the fourth and fifth sets are equal since [a, qN;21 c qN;2.

Next, if a E kera, then it follows from (Ill. 17) that adalq = K(ae)Iq + adanlq is nilpotent. Therefore the first five sets are included in the sixth which is clearly included in the seventh.

Finally assume g is type R and a is an element of the last set. Then a(a) is semisimple by Proposition 111.2.13. Let b E ti. Then there exists an n E No such that (ada)'b E qN;2. Then either n = 0, from which it follows that b = 0, or n E N, from which it follows that a(a)"b = 0, by (111. 18). Since a(a) is semisimple one deduces that a(a)b = 0. Hence a(a)ti = (0). This completes the proof of the proposition.

Corollary 111.4.3 If El is a subspace as in Corollary 111.2.4, then g is nearnilpotent if and only if ti = {0}.

Proof If ti = (0), then a(a)ti = (0) for all a E g and g = n," by Proposition 111.4.2. Conversely, if g = nS then S(n)11 = (0) and (adm)(tl) = (0) by Proposition III.4.2. Therefore ej = (0) by definition.

The corollary establishes that the Lie algebras which do not have the simple product structure, compact times nilpotent, are characterized by the condition tt : (0). The directions f, will play a distinguished role in the asymptotic analysis of Chapters IV-VI.

111.5

Stratified nilshadow

The nilshadow of the radical of a Lie algebra is by definition nilpotent but it is not necessarily stratified or even homogeneous. Nevertheless, we next argue that one can 'extend' each Lie algebra g to a Lie algebra g such that the radical q of the extension has a nilshadow qN which is a stratified Lie algebra and there is a projection from (g, qN) onto the original pair (g, qN). The relation between b

111.5 Stratified nilshadow

91

and g is such that the transference methods of Section 11.8 are applicable. This is of great utility in the subsequent analysis of semigroup kernels. One can exploit the dilation invariance of the stratified nilshadow of the enlarged algebra to obtain estimates which can then be transferred to the original algebra. The idea behind the construction of the enlarged system is quite simple. First,

the nilshadow qN is constructed from the radical q of g by subtraction of the semisimple part of the adjoint representation on the nilradical by the procedure described in Section 111.2. Secondly, qN is embedded in a larger stratified Lie algebra qN by the method briefly described in the discussion of nilpotent groups in Section 11.9. Thirdly, q is constructed from qN by addition of a semisimple part to the adjoint representation. This third step is a reversal of the process used to pass from q to qN. Finally, if g = m x q, where m is a Levi subalgebra, then g is constructed as an analogous semidirect product m x q. Recall that a Lie algebra r is stratified if there is a vector space decomposition r = ®j=1 h j such that [4j, t)k ] C f) j+k for all j, k E (1, ... , r) and 41 generates

r. Then we call (4j) the stratification of r and the lower central series is given by rj = ®k_j N. Next let g be a Lie algebra with Levi subalgebra m, radical q and vector subspace u satisfying Properties II and III of Proposition 111. 1. 1. If

the nilshadow qN of q is stratified, we define the stratification 14j) to be com-

patible with u if n c 41 and S(u)b c t) j for all j E (1, ..., r). If, in addition, u satisfies Property I of Proposition III.1.1, then the stratification (4j) is called compatible with m if (adm)() j c 4j for all j E {1, ..., r}. So the stratification (4j) is compatible with u and m if and only if l)1, ... , br satisfy the properties of Corollary 111.2.4. Finally, we say that g has a stratified nilshadow if qN has a stratification compatible with a suitable choice of m and u.

Proposition 111.5.1 Let g be a Lie algebra with radical q and Levi subalgebra m. Further let qN denote the nilshadow of q with respect to a vector subspace u satisfying Properties I-III of Proposition III. 1.1. Let b1 be a subspace of q as in Corollary 111.2.4-

Then there exists a Lie algebra b with radical q, a Levi subalgebra m and a vector subspace 6 satisfying Properties I-III of Proposition 111.1.1, together with a surjective homomorphism n: g -+ g, such that the nilshadow qN has a stratification (6j) which is compatible with 6 and m, the restriction of n to qN is a surjective homomorphism from qN onto qN and 7r-1(q) = q. Moreover, 7r (6) = v, 7r (t) = m, n(61) = 41 and 7r-1(n) = n, where n and

n denote the nilradicals of g and g. Also, the restrictions "I61: 61 - 41 and n I m: m --> mare bijections. I n addition, i f a I , --- ad, is an algebraic basis for g, then there exist d" > d' and a 1 , ... , ad,, E p such that a I , -- ad is an algebraic basis for j, 7r(iii) = ai

for all i E (1,...,d')and7r(5;)=Oforalli E{d'+1,...,d"}. Finally, if g is of type R, then g is of type R.

Proof Let do = dim u and dj = dim 41 and fix a basis b1.... , bd, of bt such that b1, ... , bda is a basis of u and bdo+1, ... , bd, a basis of e = 41 nn. Since

92

III. Structure Theory

qN = 41 ® qN;2 it follows from Lemma 111.2.5 that b1, ..., bd, is an algebraic basis of qN. Next, let r = g(dt, r) denote the nilpotent Lie algebra free of step r generated by d1 elements b1, ... , bd, with Lie bracket [ , ]T. By the fundamental property 11. 1. 1 of r there exists a unique homomorphism n: r -+ qN such that tr(bk) = bk f o r all k E (1, ... , dl). Because bl, ... , bd, is an algebraic basis of qN this homomorphism is surjective. For all t > 0 there is a unique automorphism y, of r such that y,(bk) = tbk for all k E (1, ..., d1). Moreover, ys o y, = y for all s, t > 0 and r is stratified with the stratification {6j}, where

hj =(aEr:y,(a)=t3aforallt>0). Alternatively, j, is the span of the multicommutators of order j in b,, ... , bd,, with respect to the Lie bracket [ , ]T. In particular 6 = 61 = span(bl .... , bd, ). Set 6 = span(b1, ... , bda) and E = span(bda+1, ... , bd, ). Then

r=h®r(2)=o®ED r(2)-6 ii where n = e ®r(2) and r(2) = [r, r],. Consider the linear bijection nt: 6 -> b = u ®1, defined by nl (bk) = bk for all k E (1, ... , d1). Thus nl is obtained by restricting n. For each v E 6 define a linear transformation 1(v) E £(6) by f (v)b = it 1(S(nt (v))nl (b)) for all b E h. Then the 1(v) are mutually commuting, semisimple, linear transformations of I with 1(v)v' = 0 for all v, v' E 6. Moreover, the map v H r(v) is linear from 6 into £((j). Let V E 6. Because r is free nilpotent, the bijective operator ett(t) E £(l)) extends uniquely to an automorphism A(t) of r, for each t E R. (See 11. 1.2.) Then

t r+ A(t) is a one-parameter group of automorphisms of t. Therefore i(v) _ lim,_.o 1-1(A(t) - 1) is a derivation of r which extends 1(v). Using the derivation property and the fact that bt, ... , bd, generate r, one verifies that the ?(v) are mutually commuting, semisimple derivations of t, i(v)4j C

4j for all j E { 1, ..., r}, and the map v r-+ i(v) is linear from 6 into £(r). Moreover,

in o i(v) = S(n(v)) o in

(111.19)

for all v E 6.

Next define r: r -+ L (r) by setting r(v + n) = i(v) for v E 6 and n E n. In particular, r(n) = 0 for all n E n. It follows from the properties of the i(v) that r is a representation of r in itself by derivations. Moreover, since r(a)b E n for all a, b E r, we have c(r(a)b) = 0. Thus (111.5) is satisfied and we may define a new Lie bracket [ , ] on t by setting

[a, b] = [a, b], + r(a)b - r(b)a

93

111.5 Stratified nilshadow

for a, b E t. Let 4 denote the Lie algebra (r, [ we find [vl, v2] = [VI, v2]t

,

,

]). From the definition of [

,

]

[vt, nt] = [vt, n(]t+ i(vi)nI,

[n 1, n2] = [n 1, n2]t

(111.20)

for all v1, v2 E 6 and nI, n2 E n. It is now readily verified from these relations, together with (111.19), that n is a homomorphism from r into qN. Moreover, n maps 6 onto v and n onto n. Hence n is a homomorphism from ij onto q. Note that [a, b] E n for all a, b E 4 and that the brackets [ , ] and [ , ]t agree on ii. Therefore n is a nilpotent ideal of 4 containing [4, q]. In particular [4, 4] is nilpotent, and so 4 is solvable. Let nq be the nilradical of q. Since Yr (n4) is

a nilpotent ideal of q we have n(nq) c n, that is, nq C n-((n). Since n-I (n) = n and n is a nilpotent ideal, we conclude that nq = n. Let adg and adt denote the adjoint representations of 4 and t respectively, and let adga = K4(a) + S4(a) denote the Jordan decomposition of adga for a E 4. Then it follows from (111.20) and the property i(n)n = (0) that adgv = adty + ?(v) for each v E 6. Moreover, adty and i(v) are nilpotent and semisimple transformations, respectively. Since i(v)v = 0 it follows from Statement I of Lemma Ill. 1.3 that ?(v) and adty commute. Therefore

K4(v) = adty

and

S4(v) = i(v)

(111.21)

for all v E 6 by the uniqueness of the Jordan decomposition. In particular one has

S4(6)6 = (0). Since 4 = 6 ® n and n is the nilradical of 4, we have established that 6 satisfies Properties II and III of Proposition III.1.1 relative to 4. Hence the nilshadow 4N of 4 is well defined, with Lie bracket given by the relations (111.2) relative to 6 and 4. It follows from (111.20) and (111.21) that 4N = t as Lie algebras. Moreover, the stratification (1)k) of 4N = r is compatible with 6 since 6 C I) = ryl and S4(v) = F(v) leaves each hk invariant. Thus we have constructed a solvable Lie algebra 4 with stratified nilshadow 4N and a surjective

homomorphism n: 4 -+ q such that the restriction of r to 4N is a surjective homomorphism onto the nilshadow qN of q. If a l , . . . , ad' is an algebraic basis for g, then by surjectivity of n there exist 61, ... , ad' E g such that 7r (6i) = a; for all i E (1, ... , d'). The elements 51, ... , ad, are independent but do not always form an algebraic basis. Hence we enlarge the basis by adjoining additional elements. Let ad'+t , , ad" be a basis for kerjr, the kernel of n. W e claim that a 1 , ... , ad" is an algebraic basis f o r g. Clearly a1, ... , ad" are independent. Let b E p. Then ir(b) E g, and since ai,...,ad' is an algebraic basis for g, there are ca E R such that .

n(b) = YaEJ+(d') ca, al,,). Then rr(b) = 7r(F-aEJ+(d,) ca alai) and

b-

ca a[al E ker r = span (ad'+1 , ... , ad")aEJ+(d')

So a(, ... , ad" is an algebraic basis for g.

94

III. Structure Theory

Finally, we construct g as a semidirect product of m and 4. If g is solvable this last step is unnecessary since it suffices to set g = q. Define a representation p of m in the vector space h by setting

p(m)b = ni 1([m, nI (b)])

for each m E m and b E h. Each p(m) extends uniquely to a derivation of r, which we continue to denote by p(m). Then p(m) leaves the spaces 6j invariant. Because p(m)v = 0 for all v E 6 it follows from Lemma 111.1.3 that [p(m), Sy(v)] = 0. Then it is readily verified that p(m) is also a derivation of 4. Moreover, m ra p(m) is a representation of m in 4 by derivations, with the property that

n(p(m)a) = [m, Jr(a)]

(111.22)

for allmEmandaE4. m x q corresponding to the action p. We Let g be the semidirect product identify m and 4 with the subspaces m x {0}, (0) x 4 of g. Then the Lie bracket on g is given by

[mI +ql, m2 + q2] = [m1, m2] + [q1, q2] + P(ml )g2 - P(m2)ql for all m1, m2 E m and all qj, q2 E q. Next extend n: q -+ q to a map n: g -s g by setting n(m + q) = m + n(q). Using (111.22) one verifies that n is a homomorphism from g onto g. Note that q and m are respectively the radical and a Levi subalgebra of g, and

that [m, v] = 0. Therefore the stratification {fl!} is compatible with 6 and m. Hence g has a stratified nilshadow. The relations n-i (q) = q and 7r (n) = n are easy consequences of the above construction. Finally, if g is of type R, the radical q is of type R which, by Lemma 111.3. 10, is equivalent to the operators Sq(v) having purely imaginary eigenvalues for all v E n. Therefore, by construction, the operators i(v) have purely imaginary eigenvalues. Then, by (111.21), the operators S4(v) have purely imaginary eigenvalues for all v E 6. Hence 4 is of type R by another application of Lemma 111.3.10. But m m x 4 is of type R by is also of type R and hence the semidirect product 0 application of Proposition 11.4.11.

111.6

Twisted products

The nilshadow qN of the radical q was defined in Section 111.2 by the introduction of a new Lie bracket [ , ]N on q. If Q and QN are the connected, simply connected, Lie groups with Lie algebras q and qN, respectively, then it is clear

that QN should be obtained from Q by modification of the group product. In this section we analyze this new group structure together with various additional structural properties.

111.6 Twisted products

95

The Lie bracket on the nilshadow qN of the solvable Lie algebra q was obtained

by modifying the original Lie bracket with derivations {r(a) : a E q} of the Lie algebra. The corresponding simply connected groups QN and Q are related by an analogous modification of the group product. In particular QN is obtained by replacing the product on Q by a twisted multiplication defined with a group homomorphism which is determined in a straightforward manner from the derivations r(a). This is an invertible operation and one can reconstruct Q by modifying the product on QN. The procedure is a generalization of the construction of the semidirect product of two groups. The semidirect product is constructed from the direct product by a twisted multiplication but this latter procedure is not generally invertible. Combination of the techniques of passing from the nilshadow QN to the radical Q and from the direct product M x QN of a Levi subgroup M with the nilshadow QN to the semidirect product M x Q is of particular utility in the analysis of groups of polynomial growth. The construction of the nilshadow qN of the solvable Lie algebra q was a special case of a more general construction (111.6). We first analyze the modification of the group structure in this general setting. Thus we begin with a Lie algebra g with Lie bracket [ , ] and a representation r: g -* G(9) of g in itself by derivations satisfying (111.5), that is, r(r(a)b) = 0 for all a, b E g. Then one can introduce the new Lie bracket [ , ]T on g by (111.6), i.e.,

[a, b]T = [a, b] + r(a)b - r(b)a

(111.23)

for all a, b E g. Now we aim to compare the simply connected Lie groups G and GT with Lie algebras g = (g, [ , ]) and gT = (g, [ , ]T), respectively, and show that one can choose GT = G as topological spaces. First, given a representation r: g -+ G(g) of g in itself by derivations satisfying Condition (111.5), where g is the Lie algebra of a simply connected Lie group G,

r then lifts uniquely to a Lie group homomorphism T: G -+ Aut(G) from G into the automorphism group Aut(G) of G, i.e., there exists a unique Lie group homomorphism T: G - Aut(G) such that T (exp a) exp b = exp(et (0)b)

(111.24)

for all a, b E g. We call T the homomorphism associated with r. The condition c(r(a)b) = 0, for a, b E g, on the derivation r then leads to an invariance property of the group T, T (T (exp a) exp b) exp c = exp(eTW(°)b)c) = exp(eT(b)c) = T (exp b) exp c

for all a, b, c E g. Therefore T(T(g)h) = T(h) for all g, h E G. Secondly, given a Lie group homomorphism T: G -+ Aut(G) from G into the automorphism group Aut(G) one can define a twisted product T* on the set G by

g T* h = (T(h-')g) * h

96

111. Structure Theory

where g * h = gh denotes the group multiplication on G. A straightforward calculation, which we give in the proof of the following proposition, shows the product T* is associative if and only if T(T(g)h) = T(h)

for all g, h E G. If this condition is satisfied, then the corresponding product on the Lie algebra will be seen to be of the form [ , ]r with r satisfying the condition r(r(a)b) = 0, for all a, b E g. This converse is of a similar nature to the observation that the Lie algebra of the semidirect product of two groups is the semidirect product of the associated Lie algebras. For this converse we do not need G to be simply connected.

Let G be a (connected) Lie group with Lie algebra g, not necessarily simply connected, and not necessarily with polynomial growth. Further, let T: G -* Aut(G) be a Lie group homomorphism. Then one can define a homomorphism

T: G -> Aut(g), where Aut(g) denotes the group of automorphisms of g, by setting

T (g)b = dt T (g)(exp tb) I r_o.

(111.25)

We call T the homomorphism associated with T. Note that since T is a group homomorphism it follows that t H U(t) = T(g)(exp(-tb)) is a one-parameter group, i.e., U(s)U(t) = U(s + t) for all s, t E R, and U(0) = e. Therefore f(g) equals the infinitesimal generator and T(g)(exptb) = exp(tT(g)b)

(111.26)

for all g E G, b E g and t E R. Next, for each a E g one has a continuous one-parameter group of automor-

phisms t H T(exp(-ta)) of g. The infinitesimal generator r(a) of this group, defined by

d-

r(a)b = d f (exp ta)b

(111.27)

r-o for all b E g, gives a representation r: g -+ G(g) of g in itself by derivations. We call r the representation associated with T. Then, as in (111.24), one has for a general group

T(expa) expb = exp(T(expa)b) = exp(er(Q)b)

for alla,bE9. Now we can state the general structural result linking these groups. But first we recall that the Lie algebra of G is defined by 9 = TeG, the tangent space of G at the identity, as a vector space. Then if a E TeG and a is the unique left invariant vector field on G such that ale = a, one has (a cp)(g) = a(L(g-1).p). The Lie bracket on g is defined by [a, b] = [a, b]l J. The exponential map satisfies T,cp(exp(ta))Io = arp and one has the identities (dR(a)V)(g) = a(L(g-')(p) and

(dL(a)(p)(g) = -a(R(g)W).

111-6 Twisted products

97

Proposition 111.6.1 Let G be a Lie group and T : G -+ Aut(G) a Lie group homomorphism. Define f and r by (111.25) and (III.27), respectively. Suppose

T(T(g)h) = T(h)

(111.28)

for all g, h E G. Define the operation T* on the set G by

g T* h = (T

(h-t

)g) * h

(111.29)

where g * h = gh denotes the usual group multiplication on G. Then the following properties are valid. 1.

The operation T* is a group multiplication on G and GT = (G, T*) is a Lie group. The identity element of the Lie group GT = (G, T*) equals e and the inverse h(-t)T of h in GT is given by h(-')T = T(h)(h-t ). Moreover, g T* h(-t)T

= T(h)(gh-t) and

h T* g T* h(-' )T = T (h)((T (g-t)h)(gh-t ))

II.

for all g, h E G. The function T is a Lie group homonwrphism from GT into Aut(G). Hence

T(g T* h) = T(g)T(h) = T(gh) and

T(h(-')T) = T(h-t) = T(h)-t

for all g, h E G. III.

If a, b E g, then r(r(a)b) = 0. Moreover, if 9, = (g, [

,

twisted Lie algebra given by (111.23), then gT = (TeGT, [

,

]T) is the ]T) is the

Lie algebra of G. IV.

If g, h E G then T (T (g)h) = T (g). Moreover, the function T is a homomorphism from GT into Aut(g).

V.

The left regular representations on G and GT are related by

((dLG(T(g-t )a))(p)(g),

(111.30)

for all aE9,gEGandcpECOO(G). VI.

If a E g and r(a)a = 0, then exec a = expcT a where exec and expcT denote the exponential maps on G and GT, respectively.

VII.

If b E g is such that r(a)b = O for all a E 9, then T (g) exp b = exp b for

allgEG. VIII.

If ip is right Haar measure on G, then RGT (g)L i (G ; dµ) = L I (G; dµ) and

d p (h) (RG(g)(p)(h) = det T(g)I JG d(h) rp(h) JG

for all g E Gand cp E L1(G;dµ).

(I11.31)

98

III. Structure Theory

Proof The proof of Statement I is by straightforward computations using (111.28). One can verify the associative law as follows. First,

g T* (h T* k) = g T* (T (k-' )h * k)

= T(k-' * T(k-' )h-' )g * T (k-' )h * k = T (k-' )T (T (k-' )h-' )g * T (k-' )h * k for all g, h, k E G. Secondly, (gT*h)T*k=(T(h-')g*h)T*k

= T(k-')T(h-')g * T(k-')h *k for all g, h, k E G. Therefore the product T* on G is associative if and only if T (T (k-' )h-') = T (h-1) for all h, k E G, i.e., if and only if (111.28) is valid. The proofs of the other parts of Statement I are similar. Statement II now follows because

T(g T* h) = T(T(h-')g * h) = T(T(h-')g)T(h) = T(g)T(h) by use of (111.28).

Next we prove Statement V. Right translations RGT in GT are given by (111.32)

RGT (g)rp = (R(g)(p) o T (g-1)

for each g E G. Therefore

((dLGT(a))c)(g) = -a(RGT(g)rV) = -a((R(g)(p) o T(g-1)) dt

(R(g)co)(T (g-') exp(ta)) 11=0

_ -(T(g-')a)(R(g)rp) = ((dLG(T(g-1)a))(p)(g) for all a E g. Now consider Statement III. Let a, b E g = TeG. By differentiation of the relation T (T (exp sa) exp tb) exp c = T(exp tb) exp c,

it follows that c(r(a)b) = 0. So the Lie algebra gr is defined. Set A = dLG(a), A T = d LGT (a), B = d LG (b) and BT = d LGT (b). Then AT I e = A Ie = that (ATBrco)(e) = (ABTro)(e) d

= dt (BTro)(exp(-ta)) r-o

= d (dLG(T(expta)b)rp)(exp(-ta)) dt

r-0

-a so

III.6 Twisted products

99

for all rP E CO0(G) by Statement V. Setting

F(s, t) = (dLG(T(expsa)b)Q)(exp(-ta)), one has

(d F)(s, t)Is.r-0

ds

(dLG(T(expsa)b)co)(e)I

s_o

= (dLG(r(a)b)rp)(e)

and

(81F)(s,t)

s.1=0

= (AB(p)(e).

It follows that

(ArBrrp)(e) = (ABrp)(e) + (dLG(r(a)b)(p)(e).

(111.33)

Let [ , ]GT temporarily denote the Lie bracket of the Lie algebra TeGT of G. Combining (111.33) with a similar expression for (BTArV)(e) one obtains

(dLG([a, blr)rp)(e) = (dLG([a, b] + r(a)b - r(b)a)rp)(e)

= ((ABr - BrAr)(p)(e) = (dLGT([a, b]GT)v)(e) = (dLG([a, b]GT)(q)(e) where we used (111.30) in the last step. Comparing the left and right sides of this equation yields [ , lcr = [ . , lr So gr is the Lie algebra of GT. If a, b E 9, then T (T (exp a) exp b) = er("T (°)b) = er(b) = T (exp b). Hence

T(T(expa)h) = T(h) for all a E g and h E G and therefore T(T(g)h) = T(h) for all g, h E G. But then T(9 T* h) = T (T (h-I )g * h) = T (T (h-1)g) T (h) T(g) T(h) for all g, h E G and the proof of Statement IV is complete.

_

Next, consider Statement VI. If r(a)a = 0, define y: R -+ GT by y(t) _ expG(ta). Then y is a homomorphism since Y(s) T* Y(t) = (T(expG(-ta)) expG(sa)) * expG(ta)

= expG(se-rr(o)a) expG(ta) = expG((s + t)a) = y(s + t) for all s, t E R. Moreover, y (0) = a. Therefore y (t) = expGT (ta) for all t E R. If b E g is such that r(a)b = 0 for all a E g, then

T(expa)expb = exp(er(A)b) = expb for all a E g. Since G is connected Statement VII follows. Finally Statement VIII follows because for positive rp E Li (G ; dµ) one has

f

dµ(h) (RGT (g)(p)(h) = IG dµ(h) v(h T* g)

G

= IG

= Jd(h)?(T(g_I)h)

100

III. Structure Theory

by invariance under RG. If V has sufficiently small support one can then use the exponential map to make a change of coordinates. This gives the identity (III.31), with the determinant the Jacobian of the coordinate transformation. The identity then extends to all 'p E LI (G ; dµ) by linearity and density. The proof of the proposition is now complete.

One can associate with any Lie group homomorphism U: GI -+ Aut(G2) a representation U of GI in L2(G2) by setting

U(g)(p = 'p o U(g)-'

(III.34)

for all g E GI and all (p E L2(G2). In particular the representation T associated with the homomorphism T: G -* Aut(G) gives a representation of G in L2(G). The representation T is unitary if and only if I det f (g) i = 1 for all g E G by 11. 1.23. It follows from (III.26) that

dLG(a)T(g) = T(g)dLG(T(g-1)a)

(111.35)

for all a E g and g E G. Moreover, (111.32) can be rephrased as

RGr(g) = T(g)RG(9)

(111.36)

for all g E G. In particular an RG-invariant operator is RGT-invariant if and only if it commutes with T(g) for all g E G. The proof of Statement I of Proposition 111.6.1 established that the product T* is associative if and only if T satisfies (111.28). But associativity of T* is equivalent to the Jacobi identity for the bracket defined by (111.23). Moreover, the group property (111.28) is equivalent to the property (III.5) on the representation r. Therefore one has the following Lie algebraic conclusion.

Corollary 111.6.2 Let r: g -+ ,C(g) be a representation of g in itself by derivations. The following conditions are equivalent. 1.

II.

c(r(a)b) = O for all a, b E g. (a, b) i-+ [a, b] + r(a)b - r(b)a satisfies the Jacobi identity.

Subsequently we need to examine mappings between two groups with twisted products. It is an elementary exercise to prove the following lemma about the relation between a homomorphism between two groups and the related mapping between the twisted groups.

Lemma 111.6.3 For i E (1, 2) let G; be a Lie group with Lie algebra gi and let Ti: Gi_-- Aut(G;) be a Lie group homomorphism satisfying (111.28). Moreover let T; : G; -+ Aut(g;) and r; : gi -+ L(9i) be the homomorphism and representation associated with Ti. Let A: G -+ G2 be a Lie group homomorphism and let n: 91 --* 92 be the associated Lie algebra homomorphism. Suppose n rl (a)b = r2(7ra)7rb for all a, b E 91.

111.6 Twisted products

101

Then A is a Lie group homomorphism from (G )T, into (G2)T2 and n is a Lie algebra homomorphism from (gi)t, into (92)r2. Moreover,

ATt (g)h = T2(Ag)Ah

and

7rT1(g)a = T2(Ag)na

for allg,hEG1andaEgt. The procedure of passing from the group G to the group GT is not always invertible. Formally, one would expect the inversion to be effected by the operators T(g)-' but these do not necessarily give an automorphism of G or GT. For

example, T(g)-'T(h)-l = T(hg)-' and so the inverses have the automorphic property T (hg)-' = T(h) -' T (g)-' if and only if the T (g) commute. In fact this is a necessary and sufficient condition for invertibility of the transition from G to

G.

Proposition 111.6.4 Let G be a Lie group, T: G -+ Aut(G) a Lie group homomorphism satisfying (III.28) and GT = (G, TO the group constructed in Proposition 111.6.1. The following conditions are equivalent.

II.

There exists a Lie group homomorphism S: GT -+ Aut(GT) satisfying S(S(g)h) = S(h) for all g, h E G with the property that G = (Gr, s*). The T (g) commute for all g E G.

III.

T(g) E Aut(Gr)forall g E G.

1.

If these conditions are satisfied, then S(g) = T (g)-' for all g E G. Moreover, if T, S. r and a are the homomorphism and representation associated with T and S then r = -a and T(expG a) = S(expGT (-a)) for all a E 9.

Proof III. The first condition states that g * h = g s(T*) h for all g, h E G. Therefore

g * h(-t )r = S(h)g r*

=

h(-1)r

T((h(-')r)-')S(h)g *h(-I)r = T(h)S(h)g

*h(-I)r

for all g, h E G, where we have used Statement II of Proposition 111.6. 1. Then,

however, one must have T(h)S(h)g = g for all g, h E G. Therefore S(h) = T(h)-' for all h E G. Since S and T are homomorphisms, by assumption, it follows that

T(g)T(h) = S(g-')S(h-') = S(g-'h-') = S((hg)-') = T(hg) = T(h)T(g) for all g, h E G. Therefore the T(g) commute. II.

III. If g, h, k E G, then T(g)(h r* k) = T (g) T (k-' )h * T(g)k and

(T(g)h) r* (T(g)k) = T (k-1) T(g)h * T(g)k by (111.28). So Statements II and III are equivalent.

III. If the T(g) commute, then S: GT -* Aut(Gr) defined by S(g) _ T(g)-' is a Lie group homomorphism satisfying (111.28), again by Statement II of Proposition 111.6. 1. Then G = (GT, s*) by the foregoing calculation.

102

III. Structure Theory

Finally, if the conditions are valid, a, b E g and cp E C°O(G), then it follows from (111.33) that

(ArBr(p)(e) = (AB(p)(e) + (dLG(r(a)b)(p)(e)

where A = dLG(a), Ar = dLGT(a), B = dLG(b) and B. = dLGT(b). Similarly,

(ABrp)(e) = (A.Brcp)(e) + (dLGr(a(a)b)(p)(e) = (Ar Br(p)(e) + (d LG(a(a)b)co)(e)

by (111.30). Hence r(a)b = -a(a)b and r = -a. Then T (expG a) = er(a) _ (e°(0))-t = (S(expGT a))-1 = S(expGT(-a)) for all a E g. Again there is a Lie algebraic analogue.

Corollary 111.6.5 Let r: g -+ G(g) be a representation of g in itself by derivations satisfying (111.5). The following conditions are equivalent. I.

There exists a representation a: 9r -s £(9r) of 9r in itself by derivations satisfying (111.5) such that 9 = (gr )a

II.

The r(a) commute for all a E g.

III.

The r(a) are derivations of 9r for all a E 9.

Moreover, if these conditions are satisfied, then a = -r.

Composition of the maps G -* Gs, G -> GT of the type described in Proposition 111.6.1 is possible. But to define the iterated map G -- GsT it is necessary

that T be an automorphism of Gs. But T(g s* h) = T(S(h-')g)T(h). Hence T(g s* h) = T(g)T(h) for all g, h E G if and only if T(S(g)h) = T(h) for all g, h E G. In addition, one needs to be able to interpret the product map in a suitable fashion. If (ST)(g) is defined as a function over G by (ST)(g) = S(g)T(g), then (ST)(g) is a Lie group homomorphism if and only if S(g) and T(g) commute. These conditions are both necessary and sufficient for the composition.

Proposition 111.6.6 Let S, T : G - Aut(G) be Lie group homomorphisms satisfying the property (111.28). The following conditions are equivalent. 1.

T: Gs -* Aut(Gs), S: GT -+ Aut(GT) and ST: G -* Aut(G) are homomorphisms satisfying the property (111.28) and

GsT = (G, sT*) = (Gs, T*) = (GT. S*) = GTS II.

T(S(g)h) = T(h), S(T(g)h) = S(h) and [S(g), T(g)] = Ofor all g, h E G.

I11.6 Twisted products

103

Proof I=II. First, if the T (g) are automorphisms of Gs, then T(S(g)h) = T (h) for all g, h E G by the preceding discussion. A similar conclusion follows if S and T are interchanged. Moreover, if (ST)(g) is an automorphism of G, then [S(g), T (g)) = 0 for all g E G. III. Since S(g) and T(g) commute, (ST)(g) is clearly a Lie group homomorphism. But then

(ST)((ST)(g)h) = (ST)(S(g)T(g)h) = S(S(g)T(g)h)T(T(g)S(g)h) = S(T(g)h)T(S(g)h) = S(h)T(h)

for all g, h E G. Hence the product ST satisfies the property (111.28). Next, using the invariance property of S, one finds S(h(-i)r)g

g s(r*) h =

T* h = S(T(h)h-')g T* h = S(h-')g r* h.

Therefore,

g s(T*) h =

T(h-i)S(h-')g * h = (ST)(h-1)g

* h = g sr* h.

Hence equipping GT with the s* product gives GST. Similarly equipping Gs with

0

the r* product gives Gsr The corresponding Lie algebra statement is the following.

Corollary 111.6.7 Let a, r: g -- G(g) be representations of g in itself by derivations satisfying (111.5). The following conditions are equivalent. I.

a: gr --). G(gr) and r: ga -+ G(ga) are representations of 9r and ga, respectively, into themselves by derivations, a + r is a representation of g satisfying (111.5) and

(ga)r = (gt)a = ga+r II.

a(r(a)b) = 0 = r(a(a)b) and [a(a), r(a)] = 0 forall a, b E g.

The definition (111.29) of the twisted group product r* is one of two possible definitions associated with a given T. It is particularly appropriate for the analysis of left derivatives because of the simple relation (111.30). There is no comparable relation between right derivatives. The alternative definition

g *r h = g * T(g)h = g(T(g)h)

(111.37)

has similar group properties, for example, T(g *T h) = T(g)T(h) and g(-I)r = T (g-' )g-' , the corresponding Lie bracket is again given by (111.23) and (left invariant) Haar measure satisfies a relation analogous to (111.3 1) but with respect to left translations. Then, however, one has the relation ((d RGT (a))(o)(g) = ((d RG (T (g)a))co)(g),

104

III. Structure Theory

for the right derivatives and there is no comparable relation for the left derivatives.

Since our aim is to analyze subelliptic operators constructed from left derivatives, the definition (111.29) is the most convenient. Nevertheless, this convention does have implications for the identification of the semidirect product of two Lie groups. Let G 1, G2 be two Lie groups and let T: G t -+ Aut(G2) be a homomorphism. Further let G = G1 x G2 denote the direct product of G1 and G2. Define T: G -->

Aut(G) by T(gt, 82)(h1, h2) = (ht, T(gt)h2). Then T is a homomorphism and T (T (g)h) = T (h) for all g, h E G. Now we define the semidirect product GI Tx G2 of GI and G2 by GI Tx G2 = (Gi x G2, T*)One calculates straightforwardly that (gl, e) T* (e, 92) = (91, 92)

(111.38)

but

(e, 92) T* (gi, e) = (gt, T(gi 1)g2) = (gi, e) T* (e, T(g, 1)g2) Therefore

(91,92)(- OT = (g1 1, T(gi)g21)

111.39)

(hi, e) T* (e, 92) T* (hi, e)(-')T = (e, f(hl)92)-

(111.40)

and

More generally,

(gl, 92) T* (hi, h2) = (gtht, (T(hi 1)g2)h2) and

(h1, h2) T* (gi, 92) r* (ht, h2)(- 1)T

_ (higthi ',T(h1)(((T(gj t)h2)g2)h2t)).

(111.41)

In particular

(h1, h2) r* (e, 92) r* (h1,

h2)(-UT

= T(h1, h2)(e, h2g2h2 1).

(111.42)

It follows that Gi x {e} and {e} x G2 are subgroups of G1 Tx G2 and (111.41) establishes that (e) x G2 is a normal subgroup. The maps gi H (gt, e) from GI into GI Tx G2 and 82 H (e, 82) from G2 into G1 Fx G2 are homomorphisms. In what follows we identify gt with (gi, e) and $2 with (e, 82). Then 91 T* 92 = (g1,92)

and

hI

T*92T*h(-t)T

=T(ht)g2

g1, h i E G 1 and 92 E G2 by (111.38) and (111.42), respectively.

111.7 The nilshadow; analytic structure

105

Next, we consider the Liealgebras gl, 92 and g = gl ® 92 of G1, G2 and G. Define the homomorphism T: GI - Aut(82), by T(g)a = as T(g)(expta)I,=o and the representation f : gl -- G(92) by f (b)a = T,T (exp tb)a l r=o Then f (b) is a derivation for all b E 91. Let r be the representation associated with T. Then r(al, b2)(bl, b2) = (0, f(ai)b2) for all (al, b2), (bl, b2) E g. Hence Or = 91 tx 92 as Lie algebras. The verification was already given in Example 111.2.9. The Lie bracket [ , J. agrees with the definition (11.3) of the Lie bracket for the semidirect product adopted in Chapter II. Note that the alternative definition (111.37) of the twisted group product could be used to introduce a semidirect product G2 xT GI = (G2 x GI, *T) but this gives rather different ordering. For example, 92 *T gl = (91, 92)

and

h2 *T gl T* h(-I)r = T (h2)91

91 E GI and 82, h2 E G2. One can verify that with this convention the corresponding Lie bracket coincides with the commonly used definition (11.2). But, as mentioned above, this is not well suited to the discussion of right invariant differential operators although it would be appropriate for the analysis of left invariant operators. Exampk I11.6.8 Consider the three Lie algebras of Example 111.1.8. If or # 0, each of these is solvable but not nilpotent and each has an abelian nilradical and abelian nilshadow (see Example 111.2.1). The group actions can be explicitly calculated as twisted multiplication on the nilshadows. In each case the corresponding r is given by r (a) = S(a ,) (Example 111.2.8). (i) The analytic group with Lie algebra a2 is a semidirect product R T* R with multiplication given by

(xl.x2)(YI. Y2) = (XI +Y1,e-ayl x2+Y2) The second component corresponds to the nilradical. The twist T introduces a dilation.

(ii) The analytic group with Lie algebra e3 is a semidirect product R T* R2 with the second component corresponding to the nilradical. The multiplication is given by

(xt,x2)(Y1.Y2) = (x1 +YI.Ma(YI)x2+Y2). where

Ma (Y) = e-ay

cos y

s iny

-siny

cos y

The twist T gives a combined dilation and rotation. (iii) The analytic group with Lie algebra f is a semidirect product R2T*R2. The second component again corresponds to the nilradical and now the multiplication is given by (x1, x2) (YI. Y2) _ (xl + Y1. Ma (Y0 X2 + Y2),

where

cosy' siny' - sin y' cosy' with y = (y', y"). The twist T gives a dilation and a rotation which are independent. a M(y)=e"

111.7

The nilshadow; analytic structure

If G is simply connected, then Proposition 111.6.1 provides a framework to describe the passage from the solvable group Q to its nilshadow Q,v, or conversely,

106

III. Structure Theory

from QN to Q, by modification of the group product. Alternatively, one can pass

from a group G = M tx Q, expressed as the semidirect product of the Levi subgroup M acting on the radical Q, to the group GN = M K QN. Similarly, one can construct G = M x Q by modification of the product on the group GN = M X QN. The modified product which converts M x QN into the group G = M x Q can be expressed in terms of the homomorphism S associated with the representation or defined by (III.10). Then polynomial growth of G is characterized by unitarity

of the S(g) with respect to a suitably chosen inner product. Detailed analytic features of the group are directly related to spectral properties of the derivations a (a). For example, the unitarity allows for the construction of a common distance for the solvable group Q and its nilshadow QN.

Let G be a simply connected Lie group with radical Q and a Levi subgroup M. Let g, q and m denote the corresponding Lie algebras. First, if u is a subspace satisfying Properties I-III of Proposition Ill. 1.1, then gN was constructed in Section 111.2 by equipping g = m x q, the Lie algebra of the group G = M x Q, with the Lie bracket

[a, b]N = [a, b] + re(a)b - r9(b)a where rg(a) = -S(an) with an the component of a in D. The corresponding homomorphism TG: G - Aut(g) is given by TG(expa) = ero(a) and the homomorphism TG: G -- Aut(G) satisfies

TG(expa)expb = exp(e-S(a°)b)

(111.43)

for all a, b E g. We define the semidirect shadow GN as the topological space G

equipped with the group product g *G h = g TG* h = (TG(h)g)h. Secondly, since TG(g) IQ E Aut(Q) for all g E Q one can define TQ: Q Aut(Q) by TQ(g) = TG(g)IQ. Then we define the nilshadow QN of the group Q to be the space Q equipped with the product g *N It = g *QN h = (TQ(h-I )g)h = g *GN h. So the group QN is the restriction of the group GN to the set Q.

Thirdly, the Levi subgroup M of G is also a subgroup of GN and has Lie algebra m. Therefore GN is isomorphic to the semidirect product of QN and M since M and QN are simply connected by 11. 1. 17. The definition of GN as GN = (G, TG*), or QN = (Q, Tg*), can be inverted. Since the set S(v) is abelian, the TG(g) defined by (111.43) commute. Hence the map G H GN is invertible by Proposition 111.6.4. The inverse map is of course given by the homomorphisms associated with the derivations r(a) = S(ae). 0

Example 111.7.1 Let E3 be the covering group of the Euclidean motions group (see Examples 11.4.18 and 11.5.1) and at. a2. a3 a basis of the Lie algebra c3 with non-vanishing com-

mutators [at. a2] = a3 and [al, a3] = -a2. Choose n = spanal. The nilshadow of e3 is the three-dimensional abelian Lie algebra R3 and the nilshadow E3;N =_R3, the commutative group. If Ak = dLE3.N (ak) denote the infinitesimal generators, then Ak = -8/8xk = -ak on R3. Let SE3;N: R3 1 Aut(R3) be the homomorphism associated with the representation

at3;N: R -+ C(R3) given by oe3;N(a) = S(ad) = ade3ap. If Ak = dL93(ak), then it

111.7 The nilshadow; analytic structure

107

follows from Statement V of Proposition 111.6.1 with G = R3 and T = SE3;N that

At=-at

,

A2=-c,a2+s,a3 and A3=-s,a2-c,a3

where c, , s, : R3 -. Rare defined by c, (x) = cos x, and s, (x) = sin x, . Example 111.7.2 If Q is simply connected (and solvable) and b, , ..., bd is a basis for q

passing through n, t, 1)2, ... , br, where t, b2, ... , h, are as in Corollary 111.2.4, then by Statement 11.1.14 f o r all q E QN there exist g, , ... , d E R such that q = expQN , bI *N ... *N CXPQN dbd. But expQN kbk = exp a;kbk for all k e (1, ... , d) by Statement Ill of Proposition 111.6.1. So q = expttbt *N ... *N Since g *N h = (TQ(h-t)g)h and TQ (h - I) = I if h E n and TQ (h - t )g = g if g E exp u, by Statement V11 of Proposition 111.6.1, it follows that exp l;, b, *N ... *N exp gdbd = exp g, b1 ... exp Wd. Hence

expQN flb, *N ... *N expQN tdbd = expttb, ...exptdbd

for all l:,,.... td E R.

There is also a method for constructing the semidirect product G = M a Q from the direct product M x QN by modification of the group product. This will be of fundamental importance in the analysis of semigroup kernels in Chapter IV. We continue to assume that G is simply connected and adopt the standard iden-

tification of m E M with (m, e) E G = M x Q and of q E Q with (e, q) E G. Hence mq = (m, q) for all (m, q) E M X Q. Recall that the shadow ON of g is the direct product m x qN of m and the nilshadow qN of q. Let U: ON -+ L(ON) be the representation by derivations as in (III.10). So

a(b, a)(b', a') = (0, (adgb + S(an))a') for all b, b' E m and a, a' E U. Then (ON)a = g as Lie algebras by (111. 11). Let GN = M X QN and let S: GN -* Aut(GN) be the Lie group homomorphism associated with a. We call GN the shadow of G. Then S(expM b, expQN a) (expm b', expQN a') = (expM b', expQN (eadgbes(a,)a')) (111.44)

where we have distinguished between the exponential maps on M and QN. Since

G = GN = (GN)S as manifolds, (ON), = g equals the tangent space at the identity element of G and the Lie algebras (ON)a and g coincide, the Lie groups (GN)S and G must be equal. Hence g S* It = g * It for all g, It E G, where * is the multiplication on G. Alternatively stated, M a Q = (M x QN, S*). This can also be seen as follows. The homomorphism S is the composition of two homomorphisms TI, T2: GN -+ Aut(GN) defined by T1(m, q) = S(m, e) and T2(m, q) = S(e, q). It follows that these homomorphisms satisfy the commutation relations [TI (g), T2(g)) = 0, TI (T2(g)h) = TI (h) and T2(TI (g)h) = T2(h) for all g, h E GN. Therefore one can apply Proposition 111.6.6 to the composition. Modification of the group product by TI replaces the direct product by the semidirect product, i.e.,

MXQN=(MXQN,T,*)

and

MvQ=(MxQ,r,*)

108

III. Structure Theory

whilst modification with T2 replaces QN by Q, i.e.,

MxQ=(MxQN,T2*)

and

MxQ=(MxQN,T2*).

Then the composition of the homomorphisms gives

G = M x Q = (M N QN, r2*) = (M X QN, T,(T2*)) = (GN, S*) in conformity with Proposition 111.6.6. Note that T2 corresponds to the inverse, in the sense of Proposition 111.6.4, of the homomorphism (111.43) which replaces Q by QN. The homomorphism Ti is, however, not invertible in general since the Tt (g) do not commute, unless, for example, M = (e).

Moreover, Tt is the restriction of S to the Levi subgroup M and acts as automorphisms of QN, or of Q. Then it follows from the discussion at the end of Section 111.6, and in particular (111.39) and (111.40 ) that

S(m)q = mqm-t

(111.45)

for all m E M and q E Q. _ The effective action of the Lie algebra automorphisms S(g) associated with S is on the radical q, and even on the nilradical n, since (adm)u = S(u)u = (0). If Eli, ..., El to, tj are subspaces of q satisfying the properties of Corollary 111.2.4, then it follows from Corollary 111.2.4 that

S(g)m = m

,

S(g)n = tl

S(g)to = to

,

S(g)Et = tj

and

S(g)11 = Ilk

(III.46)

for all j E (1, ... , r) and g E G. Thus S(g) leaves the decomposition g = m ® n (D to (B ti (D 42 ®... ® h, invariant. This is of fundamental importance in Chapter V. It is subsequently useful to note that

S(expG a)) = S(exPG(am + au)) = S(exPGN (am + an)) = S(eXPGN a)

(111.47)

for all a E g, where am is the component of a in m. To establish this, first observe that if a E m® n and n E n are both close to 0 E g, then

expG (-a) expG (a + n) = expG c

with c E n by the Campbell-Baker-Hausdorff formula. Hence expG(a + n) _ expG a expG c and

S(expG(a + n)) = S(expG a)S(expG c) = S(expG a) by Statement IV of Proposition IV since S(expG n') = S(expGN n') = I for all n' E n. Similarly, S(expGN(a + n)) = S(expGN a).

111.7 The nilshadow; analytic structure

109

But a (a)a = 0 since a E m ® v. Therefore exPG a = expGN a by Statement VI of Proposition 111.6.1. Then (111.47) follows if a is close to 0 E g. The general statement is an immediate consequence of the group property and Statement IV of Proposition 111.6. 1.

Although a representation r: g -> G(g) of a Lie algebra g in itself by derivations with the property that r(r(a)b) = 0 for all a, b E g always lifts to a group homomorphism T satisfying the conditions of Proposition 111.6.1 on the (connected) simply connected, Lie group, it is useful to give conditions such that r lifts to a group homomorphism on a connected, not necessarily simply connected, Lie group.

Lemma III.7.3 Let G be a (connected) Lie group with Lie algebra 9 and let be the (connected) simply connected, covering group of G. Let A: G --> G be 0 the natural group homomorphism and set 1' = ker A. Let r: g -+ G(g) be a representation of g in itself by derivations with the property that r(r(a)b) = 0 for all a, b E 9. Finally let T: G -> Aut(G) be the homomorphism associated with r. Suppose the following conditions are valid. T(g)I' c T forall g E

1. 0

II.

T(g) = I for all g E r.

Then r lifts to a Lie group homomorphism T : G -+ Aut(G) such that T (T (g)h) = T (h) for all g, h E G. So T (exp a) exp b = exp(eT (a)b) for all a, b E 9. Moreover, T(A(g))A(h) = A(T(g)h) for all g, h E G and A is a homomorphism

from Gf onto G. Proof It follows from Condition I that A(T(g)h) = e for all g E G and h E I'. Hence there exists a unique Lie group homomorphism T: G -> Aut(G) such that T(g)A(h) = A(7 (g)h) for all g, h E G. Next T(g) = 16 for all g E r by Condition II. Then T(g) = IG for all g E r and there exists a unique Lie group

homomorphism T: G -+ Aut(G) such that T(A(g)) = T(g) for all g E G. The map T is the desired homomorphism.

By construction one has T(A(g))A(h) = A(7 (g)h) for all g, h E G. Then A 0

is a Lie group homomorphism from Gf onto GT by Lemma 111.6.3. 0

Example II1.7.4 If E3 and E3 are the Euclidean motions groups as in Example 11.4.18 and one realizes E3 by T x R2 and k3 by R3. then

/0 rq(alat +X2a2+A3a3) = f(

0 0

0 0

xt

0

-at

0

as a matrix with respect to the basis at, a2, a3. The lifting TQ of Tq on k3 is given by 1

Q(zt,x2,x3) =

0 0

0

cos

zt

-sinxi

0 sill xt

cosxt

/II

0

110

III. Structure Theory If A: E3 -* E3 is the natural map, then 1' = ker A = 2rrZ x ((0,0)). Since I'Q(1) _ {1}, one can factor out r and define TQ: E3 -* Aut(q). Similarly, YQ (x i , x2, x3) (yt , y2. y3) = (Y1, y2 Cos X1 - Y3 sin x j. Y2 sin x j + Y3 cos xt )

and one can factor out I' both in the x and y variables.

We now wish to define the shadow and nilshadow for a non-simply connected group G. The reasoning of Lemma 111.7.3 will later play a part in this, but first we need some results for the fundamental group F. Let G be a (connected) Lie group and G the covering group of G. If A: G G

is the natural map and r = ker A, then r is a discrete subgroup of G and G/ r is isomorphic to G (see II.1.18). Let m be a Levi subalgebra of g and M and Q the connected subgroups of G with Lie algebras m and q. Then M and Q are simply connected by 11. 1. 17. Moreover, M fl $ _ (e) and MQ = G. Since I- is 0 0 discrete, it is contained in the centre Z(G) of G by II.1.19. Let n be a subspace of q satisfying Properties I-III of Proposition 111. 1. 1. A basic result for the centre Z(G) is the following lemma. 0

0

0

0

Lemma 111.7.5 If m E M, q E Q and mq E Z(G), then m E Z(M) and there exists a unique a E ro(m, ti) fl q such that q = exp$ a. Moreover, s'(mq) = I, where SR: GN -+ Aut(GN) is the hontomorphism as in (111.44), and [II, a] = {0}.

Proof For simplicity we delete the circles on the groups and the homomorphism and write G, Q, M and S for G, Q, M and , . Let S and a be the homomorphism and representation associated with S. If th E M, then mq = = (mmm-1)(mgm-1). So m = inmm-1 and ritmgth-1

q = thgm-1.

(111.48)

In particular, m E Z(M). It is not a priori clear that there is an a E q such that q = expQ a, since the exponential map on a solvable group is not always surjective. But on nilpotent groups the exponential map is always surjective (see II.1.14). Therefore we circumvent this problem via a detour to the (nilpotent) nilshadow QN of the solvable group Q. If ( ) 1 , ... , il, are as in Corollary 111.2.4, k E (1, ... , r), a E 1)k fl n and

b=bm+bq =bm+bu+bn E g, then a(b)a = (S(b0) + adbm)a = (adb - K(bu) - adbn)a = [b, a] - [bu, a}N - [bn, a}N = [b, a] - [bq, a}N E [b, a] + qN;k+l So by (111.47) it follows that S(expb)a = S(expGN b)a = e0 (b)a E eadba + qN;k+1 = Ad(exp b)a + qN;k+l

111.7 The nilshadow; analytic structure

111

Since S(g) leaves the decomposition q = n ® (1) I n n) ® 42 ®... ® ry, invariant for

all g E G, it follows that S(g)a E Ad(g)a + qN;k+1 for all g E G. Alternatively, if a E u and b E g, then a (b)a = 0 and

S(expb)a = S(expGN b)a = e°(b)a = a E eadba + it = Ad(expb)a + it.

So S(g)a E Ad(g)a + it for all g E G. Since Ad(g) = I for all g E Z(G), it follows that S(g) = I for all g E Z(G). Hence S(g) = I for all g E Z(G). There exists a unique a E qN such that q = expGN a. Let b E M U u, t E R and

set h = expGN ib. Since a(m)(m ® u) = (0) it follows that S(m)h = h. Hence

S(q-1)h = S((mq)-1)S(m)h = h. If b E u, then mhm-1 = m exp(tb)m-1 = exp(tAd(m)b) = exp(tb) = h by Statement VI of Proposition 111.6.1 and Property I of Proposition 111. 1. 1. So q(mhm-1)q-1 = qhq-1

h = (tnq)h(mq)-1 =

by (111.48). Alternatively, if b E m then (111.48) states immediately that h = qhq-1. So in any case it follows from Statement I of Proposition 111.6.1 that

expGN a = q = hqh-1 =hs*qs*h(-1)s

= S(h)((S(q-1)h) *GN (q *GN h(-1)GN)) /l S(h)(h *GN q *GN

h(-I)GN)

= S(expGN tb) expGN (etad9Nba).

So if b E u, then expGN a =expGN

(erS(b) t WON ha)

a

=expGN (e

tadb

a)

and a = etadba for all t E R. Therefore [b, a] = 0 and S(b)a = 0 since S(b) is a polynomial in adb without constant term. Alternatively, if b E m then [b, a]9N = 0 and expGN a = S(expGN tb) expGN a = expGN (eadba). Therefore [b, a] = 0. So a E ro(m, u) and [a, u] = (0). But then a(a)a = 0 and Statement VI of Proposition 111.6.1 implies that q = expGN a = expa. Finally, if also a' E ro(m, u) n q and q = expa', then similarly 0 q = expGN a'. Therefore a = a' by the uniqueness of a. We now continue the description of the situation introduced before the statement of Lemma 111.7.5

Corollary 111.7.6 One has S(h) = 1, (h) = 1, and S(g)h = h for all h E r and g E G. Hence

h*BNg=hg=gh=g*8Nh

112

111. Structure Theory

for all h E r and g E G, where

*8N0

denotes the multiplication in

N. Therefore

r is a discrete central subgroup of GN.

Proof Since r C Z(G) the first two equalities follow from Lemma 111.7.5. If 0 0 m E M, q E Q and mq E r, then there exists an a E tu(m, v) fl q such that q = exp$ a = exp6N a by Lemma 111.7.5. Then a (b)a = 0 for all b E ON, where or is the representation associated to SR. So SR(g)q = q for all g E GN by Statement VII of Proposition 111.6.1. Then §(g)(mq) = mq for all g E 6N0 0 Next, recall that G 0 = (GN), . Therefore, if g E GN and h E I', then g *8N h

=

(§(h(-t)gN)g)

*&N h

= g * h = h *('

(-t)

where we used §(F) _ {1}. Corollary 111.7.6 allows us to define the shadow GN of the connected, possibly 0

non-simply connected Lie group G. Observe that the quotient groups G/ r and GN/ r are identical as manifolds, because g *6 r = g *6N F whenever g E

G.

Let %P: G/ r -+ G be the unique isomorphism such that M(gr) = Ag for all 0 g E G. We define GN to be the Lie group with underlying manifold G such 0 that %P: GN/ r -+ GN is a Lie group isomorphism. We call GN the shadow of G. Then the Lie algebra of GN is ON, the shadow of g. The group product of GN is denoted by *N and the inverse by (-1)N. Note that this definition of *N coincides with the previous definition on QN in the case that G is simply connected. Moreover, it is trivial that this definition of GN is the same as the previous definition in the case that G is simply connected. The following facts are straightforward consequences of the definition of GN. 0 0 The map A: GN -+ GN is a homomorphism and a covering map, with kernel r. Applying Lemma 111.7.3 and Corollary 111.7.6, it follows that ,S induces a homomorphism S: GN -* Aut(GN) such that

S(Ag)(Ah) = A(S¢(g)h)

(111.49)

for all g, h E G. Moreover, S(S(g)h) = S(h) for all g, h E G. Since gh = (h(-t)")g *,N h for all g, h E GN and A is a homomorphism both from G into G as from GN into GN, it follows that

gh =

(S(h(-t)N)g)

*N h

for all g, h E G. If S: GN -+ Aut GN denotes the homomorphism associated with S, then

S(Ag) = (g)

111.7 The nilshadow; analytic structure

113

for all g E 6. Hence it follows from the identities (111.46) and (111.47) that

S(g)m = m

,

S(g)v = v

S(g)to = to

,

S(g)t) = tj

and

S(g)l)1 = hj

(111.50)

for all jE{I,...,r)andgEGand S(expG a)) = S(expG (am + an)) = S(expGN (am + an)) = S(expGN a)

(111.51)

for all a E p, where am and av is the component of a in m and v. We emphasize that we do not assume that G is simply connected. We define the nilshadow QN of the radical Q of G to be the subgroup A($N) of GN. Since Q = A($), we see that Q and QN are identical as manifolds. Then the Lie algebra of QN equals qN, the nilshadow of q, and QN is nilpotent. Clearly 6N is the universal covering group of QN, and the covering map is A16N which

has kernel r n 6N0 The Levi subgroup M of G is also a subgroup of GN, since M = A(M). 0 Because the Lie algebra of M equals m it follows that M is also a Levi subgroup of GN. Moreover GN = M *N QN. Since m and qN commute in ON, it follows that M and QN are mutually commuting subgroups of GN. In general, the intersection

M n QN = M n Q may be nontrivial. Since mq = m *&N 9 for all m E M and q E $ one deduces that mq = m *N q for all in E M and q E Q.

Lemma 111.7.7 One has M n Q c Z(GN) n QN c Z(QN).

Proof Since the subgroups M and QN of GN are mutually commuting, it follows that m n QN c Z(QN) and m n QN c Z(M). But GN = M *N QN. Hence m n Q= M n QN c Z(GN). Lemma 111.7.8 If in E M and g E G, then mg = in *N g. In particular q' *N q = q'q for all q' E M n Q and q E Q. Moreover, M n Q is a subgroup of GN and

(q')(- ON = (q')-' for all q' E M n Q. Proof Since o (a)b = 0 for all a E g and b E m, it follows from Statement VII of Proposition VII that S(g)m = in for all g E G and m E M. Then

mg =

*N g = m *N g

whenever m E M and g E G. The other statements are easy.

In the simply connected case S(m) is an inner automorphism of Q for all In E

M. In particular S(m)q = niqm-1 for all in E M and q E Q by (111.45). A similar relation is true in the non-simply connected case.

Ill. Structure Theory

114

Lemma 111.7.9 Let GN be the shadow of G and S: GN -> Aut(GN) be given by (111.49). If m E M and q E Q, then S(m)q = m q m-t.

Proof For all m E M define the inner automorphism ym: G -+ G by ym (g) _ m

mg

E qAf, then

S(expM a) expQN b = S(expGN a) expGN b = expGN (e°(a)b) = expQN (eadaNab) = expGN a expGN b (expGN

a)(-t)N

= expM a expQN b(eXpM a)-t = yexpM a(expQN b).

Since expQN is surjective, by 11. 1. 14, it follows that S(m)q = ym(q) for all m E

expM m and q E Q. But both m ra S(m) and m -+ ym are homomorphisms. Therefore the lemma follows since M is connected.

0

The S(g) can be used to give a characterization of Lie groups with polynomial growth.

Proposition 111.7.10 Let GN be the shadow of G. Let S: GN -> Aut(GN) be given by (111.49) and let S: GN - Aut(gN) be the homomorphism associated with S. The following are equivalent. 1.

G is of polynomial growth.

H.

M is compact and Q is of polynomial growth.

III.

There exists an inner product (- , ) on g such that S(g) is orthogonal for all g E G N and ada is skew-symmetric for all a E M.

IV.

There exists an inner product ( - , ) on g, M > 0 and m E N such that (b, S(g)c) I :< M(1 + IgI)' IIb1I IIc1I

for all g E G N, b, c E g and ada is skew-symmetric a E M.

I f these conditions are valid and--1) 1, .. , 4r, to, eI are subspaces of q satisfying the properties of Corollary 111.2.4, then one can choose the inner product (- , ) on g such that the spaces m, u, to, et, h2, ... , hr are mutually orthogonal. Moreover, the Haar measure on G is a Haar measure on GN.

Proof I .II. The equivalence is a direct consequence of Proposition II.4.11. I=III. Suppose G has polynomial growth. Then g is of type R and there is an inner product (- , ) on g such that ada and S(v) are skew-symmetric for all a E m and v E u, by Proposition 111.3.12. Since q is an ideal the operator a(a) is skew-

symmetric for all a E g. Hence S(expa) = e°(a) is orthogonal and Statement III follows.

III=IV. This implication is evident.

IV=I. Let a E g. There exists a c > 0 such that I exp(ta)I < c ItI for all t E R. Let gC, S(g)c and a(a)c denote the complexification of g, S(g) and

111.7 The nilshadow; analytic structure

115

a (a), respectively, and extend (- , ) on g to a complex inner product (- , ) on 9C such that (a, b) = (a, b) for all a, b E g C 9C. Let k E C be an eigenvalue of a (a)C with eigenvector b E 9C. Then by assumption -

11b112etRex

=

I(b,eto(a)cb)I

= I(b,S(eXp(ta))Cb)I < M(1 + IeXp(ta)I)m11b112 < M(1 +cItl)'"IIb1I2

for all t E R. Therefore Re X = 0 and all eigenvalues of or (a) are purely imaginary. Applying this to a E u one concludes that the eigenvalues of S(v) are purely imaginary for all u E n. Then Statement I follows, as in the proof of the implication I=II in Proposition 111.3.12. Finally we prove that the Haar measure on GN is a Haar measure on C if G has polynomial growth. First notice that M is compact by Statement II. Let (- , ) be an inner product on g as in Statement III. Since S(g) is orthogonal one has I detS(g)I = 1 for all g E G. Let dg denote the Haar measure on GN. Since GN is the direct product of a compact and a nilpotent group, it is unimodular. So dg is right invariant. If rp E C,(G) is positive and h E G, then

f

dg (RG(h)t9)(g) = I detS(h)I

f

GN

GN

dg'(g) = J dg'p(g) GN

by Statement VIII of Proposition VIII. So dg is a right Haar measure on G. Finally, since G has polynomial growth, it is also unimodular and dg is left invariant D on G. Proposition 111.7. 10 establishes that if g is of type R, then one can arrange for the S(g) to be orthogonal. In particular,

IdetS(g)I = 1

(111.52)

for all g E G. This condition is of interest because it is equivalent to the S(g) preserving left and right Haar measure on the Lie group G with Lie algebra g. This follows from 11.1.23. This latter equivalence is a general property of Lie group theory which does not depend on the particular structure of the group or on the particular properties of the homomorphisms S. Although (111.52), is a con-

sequence of the type R property, the converse is not valid, as Example 11.4.7 demonstrates. Nevertheless one has the following result on subgroups. Lemma I1I.7.11 If G has polynomial growth, i.e., 9 is type R, and h is an ideal in g, then det Ad(g) I n = 1 for all g E G.

Proof Let a E g. Then S(a) leaves ll invariant. Therefore Tr S(a)Ib = 0 since S(a) has imaginary eigenvalues. So

detAd(expa)Ih = eTradalh = eTrs(a)Ih = 1

III. Structure Theory

116

0

Then det Ad(g) 14 = 1 for all g E G since G is connected.

Finally we establish that the distances on G and GN, or on the solvable group Q and its nilshadow QN, are the same with respect to an appropriate basis for the Lie algebra if G has polynomial growth. The next proposition gives an abstract version.

Proposition 111.7.12 Let G be a Lie group and T: G -+ Aut(G) a Lie group homomorphism. Suppose T (T (g)h) = T (h) for all g, h E G. Use the notation of Proposition III.6.1. Let bt, . . . , bd be an orthonormal basis with respect to an inner product ( - , ) on g and let dG ( - ; ). I IG, dG (. ; -) and I I Gr be the right invariant distances and moduli on the Lie groups G and GT with respect to the vector space basis bt , ... , bd of g =

If T(g) is orthogonal for all g E G, then dG(g ; h) = dGT(g ; h) and

1.

IS IG = IgIGT for all g, h E G.

If f (g) is bounded uniformly for all g E G, then the moduli

H.

IG and I I Gr

are equivalent and the distances dG( ; ) and dGT( ; ) are equivalent.

Proof

Suppose T (g) is orthogonal for all g E G. Define Ti,: G -+ R for

i, j E (1, ... ,d) by T(g)bi = Edj=1 T;j (g) bj. Then the matrix (Tij (g)) is orthogonal for all g E G. Let g E G and let y : [0, 1] -> G be an absolutely continuous path with y (0) _

e and y(l) = g, such that d

Y(t) _ >2Y1(t)dLG(bi)I

i-t

Y(r)

for almost every t E [0, 1]. Then Statement V of Proposition 111.6.1 gives

Y(t) _

;a

Yi(t)dLGT(T(Y(t))bi)I

r(r)

= EEYi(t)Tij(Y(t))dLGr(bi)I

j=i i-t

r(r)

for almost every t E [0, 1]. But then

18IGT _
2 y;(t)dLG(b;)I i=I

Y(r)

for almost every t E [0, 1]. Set co = T(h) o y. Since dLG(a)(rp o T(h)) _ (dLG(T(h)a)rp) o T(h) for all V E CO0(G), one has

w(t) _ 1=t

yt(t)dLG(T(h)bi)I.(r) = EEyt(t)T;j (h)dLG(bj)Iw(r) j=1 i=1

and the orthogonality of the matrix (T ij(h)) implies that I T (h)g I G 191G. The opposite inequality follows since T(h) is an automorphism. This completes the proof of Statement I. The proof of Statement II is similar.

An effective way to apply the above proposition is to use a basis for g adapted to the spectral decomposition of the S(v) with v E D. For the homogenization process in Chapter IV it is useful to have a basis which is in addition adapted to the lower central series {qN;j} of the nilshadow qN. This is simultaneously possible by Proposition 111.7. 10.

Corollary 111.7.13 Let G be a Lie group and T: G --> Aut(G) a Lie group homomorphism. Suppose T(T(g)h) = T(h) for all g, h E G. Use the notation of Proposition 111.6. 1. Suppose f (g) is bounded uniformly for all g E G with respect to some norm on g. Let I I' be the modulus on G with respect to an algebraic basis at , . . . , ad for g and let I I" be the modulus on GT with respect to an algebraic

basis b1, ... , bd for gT, where r is the representation associated with T. Then one has the following. 1.

For all S > 0 there exists a c > 0 such that T*h(-t)T1" < clgh_'I'

Ig

for all g, h E G with Ig T* h(- OT I" > S. II.

For all S > 0 there exists a c > 0 such that Igh-11'

for all g, h E G with Igh -'I' > S.

C 19 T*

h(- DTI"

III. Structure Theory

118

Proof First we extend the algebraic basis b1, . .. , bd" to a vector space basis b1..... bd", ... , bd of g. Let (- , ) be an inner product on g such that b1, ... , bd is an orthonormal basis. Let I IG and I IGr be the moduli on G and GT with respect to bl, ... , bd. Then by Statement II of Proposition 111.7.12 there exists a

cl > 1 such that c

IG 0 such that c2181" < 181Gr for all g E G with I81" > S. Similarly, there exists a c3 > 1 such that 1816 < c3181' for all g E G Igh-I

with 181G > c1 1 c23. Now let g, h E G and suppose that 19 T* h (- I )T I" > S. Then 18 T* h(- OT IGr ? c21g T* h(-1)r 1" > c28. Hence Igh-11G ? cj Ices. Therefore

I8h-1IG

c31gh-I I'. But then

h(-1)TIGr IS T* h(-1)T1" < c2 '19 T*

CIC2 I

I8h-I IG

C1C3C2

I Igh-1I'

as required. The proof of Statement II is similar.

O

The first principal application of Corollary 111.7.13 is with respect to G and GN if G has polynomial growth. This follows from Proposition 111.7.10. The second one is with respect to the radical Q.

Corollary 111.7.14 Let G be a Lie group with polynomial growth, radical Q and Levi subgroup M. Let n be a subspace of the Lie algebra g of G satisfying Properties I-III of Proposition 111. 1. 1 with respect to the Lie algebra in of M. Let I I' be the modulus on G associated with an algebraic basis a 1, ... , ad' for g. Moreover, let I I'QN be the modulus on the nilshadow QN with respect to an algebraic basis bl, ..., bd" for the nilshadow qN of the radical q of g. Then there exists a c > 0 such that qI)N1/QN Imq(mIgl)-11' < c (1 + Iq *N

for all m, nt 1 E M and q, qI E Q. In addition, for any neighbourhood n in Q of the identity element there exists a c > 0 such that c-11gl'QN < Iq1' 0 such that I8h-1I, < CO + Ig *N h(- ONIG N) 11

for all g, h E G. But if m, m 1 E M and q, q1 E Q, then I(mq) *N (?nlgl)(-1)NIGN = I(m *N q) *N ("11 *N ql)(-')N I"

= Im *N

m(,-1)N

*N q *N qj-1)NIGN

111.7 The nilshadow; analytic structure

119

q(-1)Nc < IM *N m(-t)NI"GN + Iq *N N 1

I

M * N mI

1

Gv + Iq *N

q(- ON It ,

QN'

Since M is compact, it is bounded and the first statement follows. The last statement can be proved similarly.

The modulus of an element of Q, written with the aid of suitable coordinates of the second kind, is easy to estimate, outside of a neighbourhood of the identity. Corollary 111.7.15 Let G be a Lie group with polynomial growth, radical Q and Levi subgroup M. Let u be a subspace of the Lie algebra g of G satisfying Properties I-III of Proposition 111. 1. 1 with respect to the Lie algebra m of M. Let I' be the modulus on G associated with an algebraic basis al .... , ads for g. Let b1, . . , bd be a basis for the radical q of g passing through n, t, h2, , hr, where t, 1 ) 1 , 112, ... , IIr are as in Corollary 111.2.4. Then there exists a c > 0 such I

.

that

c-' for all

a,El)k Proof

Iexpl;ibj ...exptdbd1'

E Rd with III II

1, where III II W = >

d IH, I'/W' and w; = k if

_ Define 4): Rd - Q by 4)(l;) = expl;1bi ...expi;'dbd. Then 4)(t;) _

expQN l;1b1 *N ... *N expQN tdbd for all 1; E Rd by Example 111.7.2. Hence I4) (H) I QN = I eX pQN 1b1 *N

*N expQN dbd I QN and by Statement II of 11.4.17

there exists a c > 0 such that c-' III II W _< I4 (f) I QN < c III II W for all t; E Rd with III II W > 1. By Corollary 111.7.14 the restriction to Q of the modulus I

I' is equivalent to the modulus I I QN on the complement of any neighbourhood of the identity element of Q. Then the corollary follows.

The modulus for exponential coordinates of the first kind is harder to determine.

Note that the nilradical N of G is a nilpotent Lie group and therefore it has a maximal compact subgroup by 11. 1.2 1. This maximal subgroup is in the centre of N.

Proposition 111.7.16 Let I I' be a modulus on a Lie group G with polynomial growth, associated with an algebraic basis for the Lie algebra g of G. Let c be the Lie algebra of the maximal compact subgroup of the nilradical N of G. Then one has the following. 1.

I f j E ( 1, ... , r) and a E qN; j, then there exists a c > 0 such that I exptal' < c ItI"j, forall t E R with ItI > 1.

H.

If to, tj are spaces as introduced in Corollary 111.2.4, ko E to, k1 E tl \(0) and n2 E q N; 2, then there exists a c > 0 such that I exp to I' > c t for all t E R with I t I > 1, where n ko + k i+ n2.

III.

If 11 E c, then suprER I exp tr:l' < oo.

120 IV.

III. Structure Theory IfnE

with nit cand j =max(I E {1,...,r) : n E qN;r+c}, then there

exists a c > 0 such that c- t I t l t / f 1. by II.4.3. So statement I is valid if j = 1. Since qN;2 c n, the other cases of Statement I follow from Statements III and IV.

Let n E n. Then exptn = expQN to for all t E R by Statement VI of PropoIQN be the modulus on QN with respect to the vector space basis bt, ... , bd for q. Then by Corollary 111.7.14 it suffices to prove the statesition 111.6. 1. Let I

ments with I exp to I' replaced by I exp to I QN

If n E c, then expQN to E expQN c. Since expQN c = expN c is compact Statement III follows. Next we prove Statement IV, starting with the lower bounds. There exist nt E

qN;j fl n and n2 E c such that n = nt + n2. Then expN to = expN tnt expN tn2 since c is central inn by Statement 11. 1.2 1. So by Statement III it suffices to show

that there exists a c > 0 such that c- t I t I I lj < I expN to I I QN _< c It 11 /j for all t E R with ItI > 1. Let n: n -+ n/c be the natural map. Then trnt E (n/c) j and 2rn I it (n/c)j+t by the maximality of j, where ((n/c)k ) is the lower central series of n/c. Moreover, let A: N -+ H be the natural map, where H = N/expN C.

Then H is simply connected by 11. 1.2 1. So by Statement I of 11.4.17 there exists

a c > 0 such that I expN t7rnt lH > c ltI" for all t E R with ItI > 1, where IH is a modulus on H. Since A is surjective there exists a c' > 0 such that c'IAgIH IgIQN forallg E N with IAgIH > c. Then I

I expN tnt IQ., > c' IA expN tnt Ill= Ic' I expN rtrnt IH > cc' ItI' Ij

for all t E R with ItI > 1. The upper bounds follow similarly by considering the surjective natural map from the covering group of N to N. This proves Statement IV. 0 0 0 Finally we prove Statement II. Let G be the covering group of G and A: G 0

0

G the natural map. Set r = ker A. By Lemma 111.7.5 for all n E N n I' there 0 0 exists a unique a E ro(m, v) fl q such that n = exp6 a,,, where Q and N are the radical and nilradical of O. Then a,, E n. Let c' = {a : n E l' fl 17} C n. Note that c' c m(m, v) fl n C to ® q.v,2. Then expo a = expk, a for all a E span c',

since c' C n. Hence c' = log ker(A I ,) because A;

is the natural Lie group

homomorphism from the covering group . / of N onto N. Therefore expv span c' is the maximal compact subgroup of N (see I1.1.21). But then c = span c'. Hence c+ q,v;2 c to ED q,y.2. Since kt # 0 it follows that n c+ qN.2. Then Statement II follows from Statement IV. Note that the last three statements of Proposition 111.7.16 are about I exp to I' with a E n, whilst the first one is for a E q. If c' is the Lie algebra of the maximal

compact subgroup of Q. then one can prove in a similar way as in the proof

Notes and remarks

121

of Statement IV that, for all a E qN with a ¢ c', there exists a c > 0 such that c-1 Itilli ckIAkAi

(IV.1)

k,1=1

with Ak = dLG(ak) where at, ... , ad' is an algebraic basis of the Lie algebra g of G and C = (ckl) is a complex-valued d' x d'-matrix whose real part is strictly positive-definite. In particular on L2(G) the operator H is the maximal accretive operator associated with the quadratic form d'

h(w) _

Ckl(Akcp, A1(p) k,1=I

d'

_

Ckl J dg (dLG(ak)W)(g) (dLG(al)(G)(g) k,1=1

(IV.2)

G

where 'p E D(h) = L2;1. Now we aim to reformulate H as an operator on the shadow GN.

It follows from the analysis in Section 111.7 that G = (GN, S*) where the twisted product is defined with the Lie group homomorphism S: GN -+ Aut(GN) given by (111.49) and GN is the shadow of G. Since G has polynomial growth

the Levi subgroup M is compact and the radical Q has polynomial growth by Proposition 11.4.11. Moreover, the Haar measure dg on G is a Haar measure on GN, by Proposition 111.7. 10. Note that GN = M X QN if G is simply connected, where QN is the nilshadow of Q. The Lie algebras g and ON of G and GN are equal as vector spaces and g = m ® u ® n as vector spaces, where m is the Lie algebra of M, u is a subspace satisfying Properties I-III of Proposition III.1.1 and n is the nilradical of g. If a E g, let am, an and an denote the components of a in the subspaces m, o and n, respectively. Then the representation a: ON -+ Aut(9N) associated with S is given by

a(a)b = (adgam + S(an))bn

for all a, b E 9 (see (Ill. 10)). Therefore S(expG a)b = b for b E m ® u and S(expG a)b = e(adaam+S(a,,))b for all b E q. One can now use the twisted product S* to re-express the operator H as an operator with variable coefficients on GN. It follows from Statement V of Proposition III.6.1 that

((dLG(a))'p)(g) = ((dLGN(S(g-1)a))'p)(g)

(IV.3)

for all a E g, g E G and 'p E C'O(G), where S is the homomorphism associated with S. Since the Haar measure dg on G is also Haar measure on GN it follows

IV. Homogenization and Kernel Bounds

126

that

h(W) = E ckl f dg (dLGN((g)ak)W)(g) (dLGN((g ')al)W)(g) N

k,1=1

This reformulation of (IV.2) establishes that the subelliptic operator H is a second-

order partial differential operator on L2(GN) with variable coefficients. In order to identify the coefficients explicitly we introduce the inner product (- , ) on g, constructed in Proposition 111.7.10, such that S(g) is orthogonal for all g E GN and the spaces in, v, to, Pt, ht , , 4, are mutually orthogonal, where to, ti, 41, ..., ll, are subspaces of q satisfying the properties of Corollary 111.2.4. In particular Co ®C1 = 41 n n. Since we are dealing with complex coefficients it is convenient to consider the complexification 9C of g and denote by (- , ) the unique (complex) inner product on gC such that (a, b) = (a, b) for all a, b E g. The operators S(g) extend to complex linear operators on gC, which we again denote by S(g), since there is no confusion. Let b_d,,, ..... bd be an orthonormal basis for g passing through the subspaces , h where d,,, = dim m - 1 and d = dim q. Set do = dim u m, b, to, e1 , 42, and d1 = dim 41 . Then m = span(b_dm, ... , bo), n = span(b1, . .., bdo), ht = span(b1, ... , bd,) and n = span(bdo+1, ... , bd). One can decompose each element S(g-1)ak with respect to the basis b j and obtain d

d

3(g-1)ak = L (bj,S(g-1)ak)bj = L' (S(g)bj,ak)bj j=-dd

j'=-dm

Therefore

(AkW)(g) = (dLGN(S(g-1)ak)W)(g) = E (S(g)bj, ak) (Bj(Mg) (IV.4)

j=-d, where Ak = d LG (ak) and we have set B j = d LGN (b j ). Hence d

f-

(IV.5)

cij(g) = L (S(g)bi, ak) ckl (S(g)bj, al)

(IV.6)

h(W)

r,i=-dm

N

with the coefficients ci j given by

k,1=1

forallgEG. The coefficients ci j are particularly simple for special choices of i, j or of g. If i < do, for example, bi E m ® n, and it follows from (111. 10) that a(a)bi = 0

for all a E 9. Hence S(g)bi = bi and the corresponding matrix elements are

IV.1 Subelliptic operators

127

constant. In particular, the ci j are constant if i, j < do. If, however, i > do + 1, then a(a)bi depends only on them and n components of a E g. Hence the matrix elements g s-- (S(g)bi,ak) only depend on these directions. This last property can be expressed in terms of the near-nilradical of Section 111.4.

Let GS denote the (connected) Lie group with Lie algebra n,.,,, the nearnilradical. Then GSn is the local direct product of the (connected) Lie group S with Lie algebra s and the nilradical N of G. But ns, = kera, by Proposition II1.4.2. Therefore S(g) = I for all g E GS,,. In particular, the coefficients ci j are functions over G/GS,,. They are constant over GS,,. The operator H is now represented through (IV.5) as a second-order operator d

H = - E Bi ci j Bj

(IV.7)

i.j=-dm

with coefficients ci j which may vary over the group if either one of the indices is strictly larger than do. If C = (ckl) is self-adjoint,_then it follows from (IV.6) that C = (cij) is also self-adjoint. More generally, 91C = R. Therefore the new representation of H does not change the symmetry property. But care has to be taken with the positivity properties, i.e., with ellipticity. Although the real part of the matrix C = (ci j) of coefficients is positive-definite, it is only strictly positivedefinite, uniformly over G, if al, ... , ad' is a vector space basis of g. This can be verified by observing that d

Re L i, j=-dm

d'

Re k,1=1

[mod with b = -i=-dm a;; bi where the 4, E C. But if a1, ... , ad' is a not a vector space basis, the right-hand side vanishes for g = e and any non-zero !;i such that (bt, ak) = 0 for all k E (1, ... , d'). Conversely, if al, ... , ad' is a vector space basis, then d'

d'

Re L (S(g)b4, ak) ckl k,1=1

al) ? u

I

ak)I2 >

k=1

with µ' > 0. The representation (IV.7) of the subelliptic operator H as an operator with variable coefficients depends on various factors, e.g., on the choice of the spaces m, n, to, ti, 41, ... , ll, and the inner product (- , ) on g. Different choices lead to different representations. In particular, the coefficients cij are determined by the homomorphism S, which in turn is determined by the representation a given by (Ill. 10). But the definition of a is dependent on a choice of the Levi subalgebra m and the subspace n. Hence the coefficients cij depend on the choice of m and n, satisfying Properties I-Ill of Proposition II1.1.1. An appropriate choice of m and u can lead to simplifications (see Example IV.3.7). Moreover, in the case of accidental symmetries the ci1 may be constant.

IV. Homogenization and Kernel Bounds

128

Example [V.1.1 The Lie algebra e3 of the three-dimensional group E3 of Euclidean motions (see Examples 111.3.2 and 111.3.13) has a basis at , a2, a3 with two non-vanishing commuta-

tors [a1, a2] = a3 and [a1, a3] = -a2. The nilradical n is the abelian subalgebra generated by a2, a3. Choose the complementary subspace and the inner product (- , ) as in Example 111.3.13, i.e., o is the one-dimensional span of a I and a 1, a2, a3 are orthonormal. The nilshadow is abelian. Let b1, b2, b3 be an orthonormal basis of the nilshadow passing through

o and n. Since n and n are orthogonal (bl, ai) = 0 = (bi, a1) for all i E (2, 3). Moreover, a (a 1 a 1 + a2a2 + a3a3) = x 1 ada 1 and one calculates (exa(al)bi,at)

= 0 (ea°(al)bl,ai)

(eaa(a1)bi, a2) = cos a (bi, a2) - sin )L (bi, a3)

,

= 0 and

(eaa(a 1)b1,

a3) = sin J. (bi, a2) + cos ,l (b1, a3)

for all i E (2, 3). Hence

H=-A2

-p(A2+A3)_-Bi-P(BZ+B3)

for all p > 0. Thus the subelliptic operator H has the same form and coefficients in both the Q and QN representations.

Although in this example the nilshadow is abelian and the operator strongly elliptic, neither of these conditions are essential. Example IV.1.2 Consider the five-dimensional solvable Lie algebra q of Example 111.2.12.

The non-vanishing commutators are [at, a21 = a3 + a4, [at, a3] = -a2 + a5, [al, a4] _ a5 and [a1,a5] = -a4. The nilradical n is the four-dimensional abelian Lie subalgebra generated bya2,a3,a4,a5.Choose e=spanat.Then an inner product as constructed in Proposition 111.3.12. In particular, ea(a) is orthogonal for all a E q and n and n are orthogonal. Let bl, ... , b5 be an orthonormal basis passing through n and n. Then one can calculate (eao (a 1)

bi , a j) = (b1, a -)'a (a 1) aj) = (b1, a-)ada l aj)

for atI i, j E (I ..... 5). Since n and n are orthogonal one has (ela(a1)b1,a,) = 0 = (eao(al)bi,at)ifi 54 1. Then AI =B1

,

A2=cosa(B3+B4)+sin A(B3+B5)

and

A3 =-sina(B3+B4)+cosa(B3+B5). Now a1, a2, a3 is an algebraic basis and

H = -Aj - p(AZ + A3) = -Bj - p((B3 + B4)2 + (B3 + B5)2) for all p > 0. Thus H is a subelliptic operator with constant coefficients in the QN -representation.

The coefficients cij of the subelliptic operator H acting on L2(GN) are finite sums of products of matrix elements of the unitary representation S of GN. Therefore they are almost periodic functions over GN by the discussion of Section II.11 and in particular 11. 11.6. The almost periodicity will play a key role in the sequel and it is convenient to introduce functions spaces suited to the description of the coefficients. Let H denote the Hilbert space given by equipping 9C with the scalar product ( , ). Then let E denote the vector space of complex-valued bounded functions over GN formed by the linear span of the complex functions l;b,c where b,c(g) =

IV. I Subelliptic operators

129

(S(g)b, c) with b, c E gC. Thus E is the linear span of the matrix elements of the unitary representation S of GN on gc More generally, we define ?{n for each n E N as the complex Hilbert space obtained by equipping the n-fold tensor product of 'H = gC with the inner product (- , - ) defined such that n

((C1 (9 ... ®C;,), (C1 (9 ... (9 C.)) = fl(Ci', Ci) i=1

for all ci, c' E 9C. The inner product is uniquely determined by this relation. Then En is defined as the complex linear span of products g H l r,,,,, (g) ... (g)

with the c', ci E gC. Since the unitary representation S on fi extends to a unitary representation Sn of GN on ?{n through the n-fold tensor product,

(Cl ®... (9 cn) H Sn(9)(C1 ®... (9 Cn) = S(g)c1 ®... ®S(g)cn

for all ci E 9C and g E GN, it follows that En is the linear span of the matrix elements b,c of Sn, defined by 6,,(g) = (Sn (g)b, c) with b, c E fn. We call En the corrector spaces. The coefficients cil of H are clearly elements of E2. They are linear combinations of products of matrix elements of 32 acting on H2. In fact cij = b;®b;,c with c = Ek,1=1 ckl(ak (9 al). This follows from (IV.6). It follows from these definitions that N1 = R, Si = S and El = E. Moreover, EI contains the constants and EI C E2 C E3 .... In addition, the elements of each En are smooth, and if >G E En and >[i' E En,, then En

E

is also invariant under LG and LGN since

LG(9) b,, = y0n(g)c = LGN (g) b.c

(IV.8)

for all b, c E Nn and g E G by Statement IV of Proposition III.6.1.

One can associate a version H8- of the subelliptic operator H with each of the unitary representations (H,, G,,, Sn) by the general method of Section 11.2. The representatives dSn(a) of a E g are the generators of the one-parameter subgroups t r-+ Sf(expGN (-ta)). But if n = 1, then S1 = S and d3(a) = a(a) by definition. Similarly, d3. (a) = an(a) where an(a) is the natural extension of a (a) to the tensor product space, e.g., if n = 2, then a2 (a) = a (a) ®I + I ®a (a). In particular, it follows from (IV.8) and (111.5 1) that

dLGN(a) b.r

for all b, c E Nn and a E g. Moreover, d'

IIS _ - L Ck/ an (ak )an (al) k.1=1

130

IV. Homogenization and Kernel Bounds

Note that H acts on En because E, is a subspace of L.(G), and the action of H is such that

H b,c = b,HH,c

for all b, c E fn. This relation is significant since it allows one to transfer information about the action of HH acting on 1-ln to information about H acting on En. It is, however, convenient Tor the discussion of transference to introduce another family of spaces which correspond to excision of the constant functions from En.

Let Pn (0) denote the orthogonal projection on the subspace of 7-1n which is pointwise invariant under Sn, i.e., the subspace {a E fn : Sn (g)a = a for all g E GN}. Then define 'HO = (1 - Pn(0))fn and En,o = span{ b.r : b, c E 7HO). Note that 6,(1-Pn(0))r = ci(l

-Pn(0))b,c = (1-P,,(0))b,(l-Pn(0))c E En,0

for all b, C E Nn. The subspaces En,o of En are particularly suited to the discussion of mean values of functions in En. Specifically, if f E En is given by = FT c1,ri with c', Cj E 7{n, then the mean M(VI) is given by

l=1

as a consequence of the identification 11. 11.7 of the mean of a unitary representation. Therefore m

*-

F l=(

c,.(I-Pn(0))rj E En.0

(IV.9)

SoEn,0={>(/EEn:M(V/)=0}. It follows from (IV.8) that M(LG(g)>!i) = M(>/') for all g E G and >L E En. Similarly, it follows from Statement IV of Proposition 111.6.1 that RG(g)1/r = RG,(g)>G for all g E G and >G E En. Hence one also has M(RG(g)>!i) = MW for all g E G and * E En. Since Pn(0) is the projection on the subspace of fn which is pointwise invariant under Sn, it follows that Sn also leaves R o = (1 - Pn(0))7Nn invariant. If S denotes the restriction of Sn to ho, then S is a unitary representation and the corresponding generator a°(a) oft r-* S (expGN(-ta)) is the restriction of an(a) to fo. One can associate a version HHo of H with the unitary representa-

tion (71O, GN, S) as above, and it follows that HS is the restriction of Hsn to R o. The important feature, which we next prove, is that HS has a strictly positive real part. Hence it is invertible on How

IV.2 Scaling

131

Lemma IV.1.3 If n E N, then ker H,-5 = P (0)?{,,. Hence for each c E ho there exists a unique x E NO such that x = c.

Proof One has HS P (0) = 0 since S (g) P (0) = P (0) for all g E G implies that a(a)P(O) = 6 for all a E g. Conversely, if x E N,, and HH,,x = 0, then x E Pn(0)'Hn. This follows by subellipticity since d'

Re(a, H3,. a) > {,c. L Ilan(ak)aII k_l

>_ 0

f o r all a E ln. Therefore H s. x = 0 implies a,, (ak)x = 0 for all k E {1, ... , d'}.

But al, ... , ad, is an algebraic basis and an is a representation of g on flu. Therefore an(a)x = 0 for all a E g. Hence Sn(g)x = x for all g E GN and x E Pn(0)'Hn. One concludes that Re H-o is strictly positive. Consequently, H3o S. S

is invertible and x = He c.

0

The latter result is crucial for the definition of Lie group extension of homogenization theory in Section IV.3. But before discussing this topic we turn to the definition of scaling.

IV.2

Scaling

The natural technique for exploiting the almost periodic nature of the subelliptic operator H expressed on GN is through homogenization theory. Much of this theory is, however, based on scaling arguments. But the group GN, and in particular its nilradical QN, do not automatically have any natural scale invariance. Therefore in this section we assume that g has a stratified nilshadow and that G is simply connected. Then M and Q are simply connected and GN = M X QN, and the nilshadow QN has a group of dilations. Then one can extend the dilations from QN to GN so that the compact component M is pointwise invariant. The Lie algebra ON of GN = M x QN is the direct product ON = m x qN of the Lie algebra m of the compact component M and the Lie algebra qN of the stratified nilshadow QN. Throughout this section we assume that qN has stratification (bj) and the stratification is compatible with v and m (see Section 111.5). In particular the lower central series qN;j of qN is given by qN;j = ®;_; bi and C b j and S(v)(lj c hj for all j E (1, ... , r). Then it follows from the definition (III.10) of the derivation a that a(a)lj c lei and a(a)m = (0) for all a E ON and j E { 1, ... , r}. For all k E {-d,,,, ... , d) define the weight wk = 0

if k < 0, and wk = j if bk E b j. Next for each u > 0 define the linear map d , ,..., , d). It follows Yu:ON - ON such that yu(bk) = uW4 bk for all k E ((-d,,,, that yu is a Lie algebra isomorphism, y,, y, = yuv and yu a(a) = a(a) yu for all u, v > 0 and a E ON. Moreover, ylgr, coincides with the group of dilations

132

IV. Homogenization and Kernel Bounds

of the stratified nilshadow qN. Let r,,: GN -> GN denote the Lie group isomorphism such that ru(expGN a) = expGN yu(a) for all u > 0 and a E ON. Then

ru(m) = m, rurU = r.U and ru S(g) = S(g) r for all u, v > 0, m E M and g E GN. Moreover, is the group of dilations defining the stratified group QN. Also ru. Bk = uwk Bk for all u > 0 and k E d). Next define the homomorphism S,,: GN --> Aut(GN) by SS(g) = S(ru(g))

for each u > 0. Then the associated representation au: ON -> G(ON) and the homomorphism S,,: GN - Aut(gN) are given by a) = e°(Yu(a))

Hence S,(g) = S(ru(g)). Introduce the group G = (GN)S,, = (M X QN, then the corresponding Lie algebra is given by gu = (ON)o,,. But [a,b)a. = [a,bJgN + a.(a)b - a.(b)a

= Yu ' ([Yua, Yub]9N) + Yu ' (a (Yua)Yub) - Yu ' (a(Yub)a) = Yu ' ([Yua, Yub19)

for all a, b E 9 and u > 0, where we have used yua(a) = a(a)Yu So yu: 9u -+ 9 is a Lie algebra isomorphism. Similarly, ru(Su(h(-t)N)g

ru(g *G. h) =

*N h)

(S((ru(h))(-l)N)ru(g))

=

*N ru(h) = ru(g) *G ru(h)

for all g, h E GN and u > 0. So ru: Gu -* G is a Lie group isomorphism. Next, fix a Haar measure dg on G and let din be the normalized Haar measure on M. If dq is a Haar measure on Q, then rp r-> fm dm f Q dq cp(mq) is a positive linear functional on C,(G) which is invariant under LG(m) and LG(q) for all m E M and q E Q, so it is a Haar integral on G. We choose the normalization of the Haar measure on Q such that IG dg (p (g) = fm dm fQ dq co(mq)

for all V E C,(G). It follows from Proposition 111.7.10 that dq is also a Haar measure on the group QN and dm dq is a Haar measure on the group GN. We choose dq to be the Haar measure on QN and dm dq to be the Haar measure on GN. Since QN is stratified it follows that

f

QN

dq ((p o ru)(q) =

u-')f QN

dq p(q)

IV.2 Scaling

133

for all W E LI(QN) and all u > 0 where D is the homogeneous dimension of QN. Since Fu is a Lie group isomorphism from G onto G it then follows that

fGUJMJQN

(IV.10)

is a Haar integral on G. The normalization of the Haar measure on G is the multiple u-D of the Haar integral (IV 10). So the Haar measure dg on G is such that fG=

u-D f dm M

J QN dq co(ru (m, q))

= fm dm fQ dq cp(m, q)

(IV. 11)

N

for all (p E C,(Gu). Our tactic is to use these dilation properties to extend the arguments of homogenization theory which are usually described in terms of dilations on Euclidean space. Therefore we need appropriate rescalings of the left derivatives and the subelliptic operators. First, for all k E 11, ... , d'), set Akul = d LG (uyu (ak)). Since a 1 , ... , ad, is an algebraic basis for g it follows that u yy 1(a 1), ... , u yu 1 (ad') is an algebraic basis for gu and

Aku'V = u(Ak(V o ru 1)) o r

(IV.12)

for all rp E CO0(Gu) because ru is a Lie group automorphism. Next, introduce the operators Hlul on Gu by d'

Hlul = -

ckt Akul Ajul

k./=l

Therefore H = Hp 1. Each of the H[u] is a subelliptic operator on G, with respect to the algebraic basis uyu 1(a 1), ... , uyu 1 (ad'), with coefficients independent of u. In particular the upper bound IICII and the ellipticity constant µ of the matrix of coefficients of Hlul are uniform for all u > 0. Secondly, the operator Hlul can be represented on GN by the same arguments

used for H in the previous section. Since Akul =

y(ak)) and Gu =

(GN)S. it follows from Statement V of Proposition 111.6.1 that

_

j=-d. d

(Su(g)bj, uyu 1(ak)) (Bj(P)(g)

_

E u" (3u(g)bj, ak) (Bjw)(g), 1=-dm

in analogy with (IV.4).

(IV.13)

134

IV. Homogenization and Kernel Bounds

Then H[ul on GN is given by d

H[.1 =- L Bi c;j 1 B j i.j=-dm

_

where

d'

jjl(g) =

u2-wi-W!

L (Su(g)bi, ak) Ckl (Su(g)bj, al). k,l=l

Thus on GN the replacement H -+ H[ul corresponds to the replacement cij -> c! j ] of the coefficients with the vector fields Bi and Bj unchanged.

Thirdly, each of the subelliptic operators H[ul generates a continuous holomorphic semigroup Slul on L2(Gu) with a bounded integrable kernel K[ul. In particular K,ul(h) (Siulgo)(g) = JGN dh

go(hg)

for all (p E CO0(Gu) and g E Gu. But then it follows from (IV.12) that H[ulco =

u2(H((p, rul)) o ru.

Therefore

S,"IW = (SU2,(co o I'ul)) o I'u

(IV.14)

and the semigroup S[ul generated by the rescaled operator H[ul is related to the rescaling of the semigroup S generated by H. Alternatively,

(Su2,((P o ru l))(ru(g)) =

J G

JGN

dh Ku2t(h) (go o r u

I)(h-t r

(g))

dh K2,(h) g((rI (h)g)

= uD f dh Ku2,(ru(h)) (p(h-I g) N

for all cp E CO0(Gu) and g E Gu. Hence Ku2,(I'u(h)) = u-DKlul (h) and

Kt(g) = u-D Kuulz,(ru I w)

(IV.15)

for all g E G and u, t > 0. This relationship will allow us to apply the general techniques of homogenization theory to obtain Gaussian bounds on the kernel K.

IV.3

Homogenization; correctors

Homogenization theory developed from the theory of strongly elliptic secondorder operators with periodic, or almost periodic coefficients, on Rd. Consider an

IV.3 Homogenization; correctors

135

operator d

H = - > ak ekl al k,1=1

with ak = a/axk, in divergence form. The coefficients of the rescaled operators d

Hlul

ak ckll al, k,1=1

where ck1l (x) = ckl(ux), oscillate more and more rapidly as u - oo. Then the key observation is that the H1 1 converge in the strong resolvent sense on L2(Rd ) to a second-order strongly elliptic operator H with constant coefficients, i.e.,

lim II((A1 + H1 1)-1 - (Al + uoo

0

(IV.16)

for all k E C with ReA > 0 and all rp E L2(Rd ). The coefficients ckl of H are 'averages' of the oscillating coefficients ckl but the averaging process is not straightforward. The 41 are non-linear functions of the ckl of the form d

ekl = M(ekl) - LM(ekjajXI) j=1

where M denotes the mean on Rd and the Xj are special functions called correctors. They are solutions of corrector equations d

HXj = - L akckj.

(IV.17)

k=1

The operator H is called the homogenization of H and the ekl the homogenized coefficients. The convergence property (IV.16) is usually expressed in an equivalent elliptic form which asserts the convergence of weak solutions of the Dirichlet problem 0 on bounded open subsets of Rd to solutions of the associated problem Hcp = 0 for the homogenized operator H. Let H = -d be a second-order operator on R where d = d/dx and r is a positive periodic function. The solution of the corrector equation NX = -dc satisfies Example IV.3.1

cdX = r - ro with ro a constant to be determined. 'Men dX = I - roc-1, and since M(dX) = O one deduces that ro = M(c) -1. Therefore M(c dX) = M(r-t)-l. and c = M(r) - M(cdX) =

M(r)-M(r-1)-1

Similar arguments are applicable to the subelliptic operators on GN since they are operators with almost periodic coefficients. The discussion separates naturally into three parts: the introduction of the correctors, the definition and characteri-

zation of the homogenization and the proof of convergence properties. In this section we discuss the definition of the correctors. Since scaling properties are

IV. Homogenization and Kernel Bounds

136

not relevant for the definition of the correctors it is not necessary to assume that G is simply connected, or that g has a stratified nilshadow. The coefficients ci j of H acting on L2(GN) are elements of the corrector space E2 defined at the end of Section IV.1 Hence it is natural to consider the corrector

equation d

HXj = -

Bicij

(IV.18)

i =-dm

on E2 analogous to the classical equation (IV 17) on Rd. In fact one can interpret

the equation on E = EI because the right-hand side of (IV.18) is in E1,0. To establish this, we first note that if i > d0, then a(bi) = 0 and (S(g)b, o(bi)t) = 0. Secondly, if i < do, then bi E m®u and S(g)bi = bi for all (S(g)bi, o(bi)t) = -(a(bi)bi, c) = 0. and g E GN. Therefore Consequently, d

do

do

L Bi bi.ak = > Bi bj.uk = - L a(b,)bi,ak = 0. i=-d,,,

i=-d,,,

i=-d,,,

Then it follows from (IV.4) that d

d

d'

E Bi Cij = L ckl E Sbl.ak Bibj.aj i=-d,,,

i=-d,,,

k.1=1

d'

d'

L cki Ak6j.u, =

L. ckl Sbj.a(ak)ag

k.1=1

k.1=1

Hence one has the identification d

_

Bi cij = bj.c,, E El

(IV.19)

i=-dm

with ca E gC given by d'

Ca = L ckl a(ak)a1.

(IV20)

k.1=1

But Ca E nC f1 (1 - P(0))nC. This follows by first observing that a(g)g c n. Hence ca E nc. Secondly, if b E P (0)g, one has (ca , b) = 0 because a (g) P (0) =

(0). Therefore ca E (1 - P(0))9E = (1 - P(0))nC since P(0)Im®d = 1. Hence bj.to E E1.0 It now follows from Lemma IV 1.3 that the corrector equation (IV 18) has a

solution Xj = b,.s E EI,o where x is the unique solution of the corresponding corrector equation

Hsx = -ca 7{oI

(IV21)

= (1 - P(0))7-1. In fact this solution of (IV.18) is the unique solution in on 61.0. This is a consequence of the following more general transference result.

IV.3 Homogenization; correctors

137

Proposition IV.3.2 Let H denote the subelliptic operator (IV.7) acting on E,,. Then for all n E N and W E Eo there exists a unique * E Eo satisfying H i/r = (p.

(IV.22)

Moreover * is the unique element of E,,,o such that

M(X(Hl - (p)) = 0 for all X E E,,,o. Explicitly, if (P = ETJ 11;c,,c1 with c'j, cj E fO = (I -

j

then >y =

with xj = (HS )-1 cj. n

Proof A general rp E Eo has a representation rp with cj E 7{O. But HS is invertible on 7-i by Lemma IV.1.3. Hence if xj = (HS)-1 c then n

H ETl=

1

/ x= l

jJ=

1

i

H30,i =

j This proves the existence l= , _J,gyp.

of the solution of (IV.22). To establish the uniqueness of the solution, it suffices to prove that if * E E,,,o

and M(X Hg') = 0 for all X E E0, then i/i = 0. To this end we convert E into a finite-dimensional Hilbert space 7-!M by the introduction of the inner product

4, *) M = with (p, >/i E E,,. This is well defined since ilp i/i E E2 and it clearly has the correct linearity properties. By Statement V of 11. 11.2 it is strictly positive-definite. The

group G acts by left translations on RM by unitaries and the derivatives Ak = d LG (ak), with k E (1, ... , d'), and the subelliptic operator H are defined on HM by restriction. Now if i/r E E,,,o and M(X 0 for all X E E o, then choosing X = ' one has d'

0=

H*)m ? AL(AkV, Ak*)M = M(IAk*12) k=1

by ellipticity. Hence Akl/r = 0 for all k E (1, ... , d') by faithfulness of the mean, Statement V of II. 11.2. Then d LG (a) i/r = 0 for all a E g since a 1, ... , ad' is an algebraic basis. Therefore i/r is constant. Hence M(i/r) = 0 implies >[i = 0 and 0 the solution is unique. It now follows from Proposition IV.3.2 that the corrector equation (IV. 18) has a unique solution Xj E E1,0 for all j E {-dm, ... , d}.

Corollary IV.3.3 Let H denote the subelliptic operator (IV.7) acting on E. Then

for all j E {-dm, ... , d) the corrector equation (IV.18) has a unique solution X j E E1,0

Explicitly, X j = tb/,x for all j E {-dm, ... , d} where x E 7{i is the unique solution of HS x = -ca with cv = ckl or (ak) al.

IV. Homogenization and Kernel Bounds

138

The functions X j introduced in Corollary IV.3.3 are called correctors or, more

specifically, first-order corrector. Note that Bickt = 0 if i > do because the coefficients cki only depend on the m and n directions, i.e., the directions corresponding to i < do. Hence the corrector equation (IV.18) simplifies to do

HXj = - L Bicij. i=-dm

Moreover, 2d

Bi ci j = 0 for all j E (-d,,,, ... , do) since ci j is constant for do}. Therefore X j = 0 for all j E {-d, ... , do}, i.e., the only correctors which are possibly non-zero are Xdo+l, , Xd Moreover, all the correctors are zero if ti = {0). all i , j E

m

Example IV.3.4 If tt = (01, then Xj = 0 for all j E (-dm...., d). Indeed, if tt = (0), then p is near-nilpotent by Corollary 111.4.3. Therefore Proposition 111.4.2 implies that a = 0

and ca = 0. Hence Xj = 0 for all j E {-dm, .... d).

Proposition IV.3.2 has two other useful corollaries. First, the solutions of the corrector equation respect spectral properties of S,,.

If E is an orthogonal projection on l,, then E commutes with for all g E GN. If these equivaa E g if and only if E commutes with as the linear span of the complex lent conditions are satisfied, then define functions l b,Ee with b, c E R, Then LG(g) b,Ec = b,Sn(g)Ec - 4b.E33n(g)c E En(E)

for all b, c E 9-L and g E G. Hence

C_ E (E) for all g E G. In

E commutes with particular, a E g, the projection E also commutes with H. In addition it follows from the c_

mean ergodic theorem, 11. 11.7, that

(a, EPn(0)b) = (Ea, Pn(0)b) = for all a, b E 9-1,,. So Similarly, define EO(E) = {>G E (IV.9) that E O(E) = E6((I -

(a, PR(0)Eb)

Note that &,,0 = E,, (I M(t') = 0). Then it follows from

ENO. Since E commutes with and with Hsn, the equation (IV.22) can be solved on E,,,O(E).

Corollary IV.3.5 Let n E N and let E be an orthogonal projection on 7{n such [E, Hsn] = 0 and E7-1O a E g. Then [E, that [E, c 7{O. Hence [EI,Ho. (HS )-1 ] = 0 and for all c E E7i° there exists a unique

x E E7-l° such that H3 ,x = c. In particular, if rp E En,o(E), then the unique of (IV.22) is also in E O(E).

solution

Proof

This follows from the explicit form of the solution given in ProposiO

tion IV.3.2.

Although the correctors X j vanish if j < do the correctors X j with b j E t are not zero in general. Corollary IV.3.5 gives the following characterization.

IV3 Homogenization; correctors

139

Lemma IV.3.6 If ca is as in (IV.20), then the following are equivalent. 1.

Xj = Ofor all j E (do + 1,...Id1).

II.

(bj,ca)=Oforall j E (do+1,...,d1).

Proof Let n: 7( -+ Ec be the natural projection. Then ir is an orthogonal projection which commutes with a(a) for all a E 9 by (111.50). Let x = -(HS)-l ca.

Then rrx, Jrca E 9-l? and nx = -(H3o)-l Jrca by Corollary IV.3.5. So lrca = 0 if and only if nx = 0. Then, by (111.50) and Corollary IV.3.3, the following

are equivalent: (b j , ca) = 0 for all j E (do + 1, ..., dl ); 7rca = 0; rrx = 0;

(S(g)bj, x) = 0 for all g E G and j E (do + 1, ... , dl); lbj,x = 0 for all

jE(do+1,...,dl};Xj=0foralljE(do+I....Id1).

D

Alternatively, it follows from Corollary IV.3.3 that the correctors X j are zero

if and only if ca = 0. But then by (IV.19) and (IV.20) this is the case if and only if the coefficients c; j are divergence-free, i.e., if 0 _dT B; c; j = 0. But the coefficients c; j and the element ca depend on the choice of m and u, satisfying Properties I-III of Proposition 111. 1. 1. An appropriate choice of m and n can lead to the correctors vanishing. Example IV.3.7 If the coefficients ckl of the subelliptic operator H are real, then there is a choice of m and u, satisfying Properties I-111 of Proposition 1 1 1 . 1 . 1 , such that the correc-

tors Xj = 0 f o r all j E (-d,,,, ... , d). The proof consists of a finite iterative procedure to choose m and o, satisfying Properties 1-111 of Proposition 111.1.1, such that the corresponding

derivation a = am,o gives ca = 0. First, let m and u denote an arbitrary choice and a the corresponding derivation. If u is an 3-invariant subspace of P, then for all c E (I - P(0))u the corrector equation Hg x = c has a unique solution in x E (I - P(O))u by Corollary IV.3.5. Since Ca E (I - P(0))n, the corresponding equation (IV.21) has a unique solution in x E (I - P(O))n. Secondly, let x be the solution of (IV.21) and introduce the corresponding automorphism 0 = eadx of g. Set ml = 4>(m) and ul = m(o). Then the pair ml and ul satisfy Properties 1Ill of Proposition 111.1.1 as an immediate corollary of Proposition 111.3.3. Let al = aml,t,1 denote the derivation corresponding to the new pair. Since al (m(a))m(b) = 4>(a(a)b) for all a, b E g, by Lemma 111.3.4, it follows that d' m-I

(ca,) = F_ ckla(m-lak)4-lal. k.1=I

Now we compare m-I (cai) and ca with the aid of the following lemma.

Lemma IV.3.8 If x E gNMj fl n for some j >I and m =

a(m-I(a))=a(a)

and

eadx. then

0-1(a)-a+a(a)xEgN;j+l

forallaE9 Proof

First, note that m-I

(a) =

e-adx(a)

= a - (adx)(a)+ 00E (-I')" (adx)"(a). n=2

n.

(IV.23)

IV. Homogenization and Kernel Bounds

140

Therefore 0-1 (a) - a E n and the first statement of the lemma follows immediately from the definition of a.

Secondly, X E n and hence a(x) = 0. Therefore (adx)(a) = (x,a]9N - a(a)x. Since x E qN;j. by assumption, and a(a)gN;j c qN;j n n, by Lemma 111.2.2, it fol-

lows that (adx)(a) E qN;j n n. But [nl,n2l = [nl,n2]N for all nl,n2 E n. Hence (adx)2(a) = [x. [x,a]]N E gN;j+l. Consequently (adx)^(a) E qN;j+l for all n > 2. Therefore m-1(a)-a+(adx)(a) E gN;j+l. But sincea(x) = Done has (adx)(a)+a(a)x = [x. a19N E qN; j+l The combination of these two inclusions gives the second statement of the lemma.

0

It now follows from the lemma and (IV.23) that

d'

rkl a((D-lak)(b-lal

m-1(ea1) = k,t=l d'

rkla(ak)(al-a(al)x)+y=ra+HSx+Y=y k,1=1

with y E qN;2, where we have used x E gN;I and the corrector equation (IV.21). Therefore

ral E qN;2 The vanishing of the correctors is established by an r-fold iteration of this argument where r is the rank of the nilshadow qN. Repeating the foregoing argument with m, u and a replaced by ml, of and aI and solving the corrector equation on the a-invariant subspace qN;2, one obtains a third triplet m2, 02 and a2 such that rat E qN;3. Then since gN;r+t = (0) an r-fold repetition eventually leads to a choice mr, nr and ar such that ca, = 0. Thus the corresponding solution of (IV.21) is zero.

Any special choice of m and v which leads to the correctors vanishing will depend on the subelliptic operator H. Therefore the subspaces then depend on H. Example IV.3.9 Let e3 be the solvable Lie algebra of the three-dimensional group E3 of Euclidean motions in the plane (see Example 11.4.18). Then e3 has a basis a,. a2, a3 with non-vanishing commutators [aI, a2l = a3 and [al, a3] = -a2. The nilradical is the span of (a2, a3) and one may choose n as the span of al. Then a(a) = S(av ). Hence S(al) = adal and S(a2) = 0 = S(a3). Therefore if H = - Fk 1=1 rklAkAl, with real coefficients rki,

then HS = -r11(adal )2 and ca = r12a3 - c13a2. The unique solution of HSx = -ra is

given byx=rlll(r13a2-r12a3)andol =ead`oisthespan of a, =r11 (cllal-cl2a2c13a3). If al is the corresponding derivation, r,,, = 0 since the nilshadow is abelian. The group is represented by vector fields on L2(R3) (see Example 11.5.1) and the replacement al -+ a corresponds to a change of coordinates xI -+ yl, x2 -+ Y2 = x2 - r12x1 /rl I and x3

.

Y3 = X3 - r13xl/rlI

The next corollary of Proposition IV.3.2 shows that each element of En with mean value zero, i.e., each element of 4,0, can be represented as a sum of subelliptic derivatives of functions in En,O

Corollary IV.3.10 Let H denote the subelliptic operator (IV.7) acting on En. Then f o r all n E N and t p E En,o there exists flit, * j ,. . d'

_

Ak>frk

k=l

hfd' E En,o satisfying

IV.4 Homogenized operators

141

Proof Let * E E,,,o denote the unique solution of (IV.22). For all k E { 1, ... , d'}

set ''k = -

IV.4

d=I cktAtlf. Then *k E En.o and _k=l Ak*k = H>1i = rp.

Homogenized operators

In this section we define the homogenization of the subelliptic operator H on the Lie group G by analogy with the definition of the classical theory on Euclidean space. The definition is not unique since it depends on a choice of the Levi subalgebra m and the complementary space n. Another complication arises if the Levi subalgebra m is non-trivial. Then there are two distinct candidates, a homogenized operator on L2(GN) and a second operator on L2(QN). If G is simply connected, the first operator is equal to the second tensored with the identity operator on L2(M), but for non-simply connected groups the situation is more complicated. Nevertheless, the homogenizations can be defined by following a procedure similar to the Rd-theory. The justification of this process relies on the subsequent proof, in Chapter VI, that the homogenization governs the asymptotic behaviour of the subelliptic operator and the associated semigroup. Throughout this section we do not assume that G is simply connected. First, let X j denote the unique solutions of the corrector equation (IV. 18), the functions given by Corollary IV.3.3. Secondly, introduce the functions d

cij = cij - L Cik BkXj k=-dm

for all i, j E (-d,,,, ... , d). The ci j are elements of E3 since cik E E2 and BkXj E El. In particular one can define the mean values d

_

cij = M(cij) = M(cij) - E M(cik BkXj) k=-dm

of the cij. Note that BkXj = 0 if k > do. Moreover, cij = c'ij is constant if i, j E (-d, ... . do) and ci j = ci j if j E ( -d,,,, ... , do). Also note that the ci j are divergence-free, d

d

L' Bicij= E Bicij+HXj=0 i=-dm

i=-dm

by the corrector equation (IV.18). The homogenization H of H is now defined on L2(QN) by dl

H = - L ci j Bl N) i,j=1

(IV.24)

142

IV. Homogenization and Kernel Bounds

with domain D(H) = n,,i=t D(B(i N )B(N)), where B`N) = dLQN(bi) for all i E {1, ... , d}. The second version of the homogenization is then defined on L2(GN) by d,

H0=- LcijBiB1

(IV.25)

i,j=l

with domain D(H0) = nd, =l D(Bi Bj). If G is simply connected, then Hp =

I®H. Next we examine some of the basic properties of the homogenization. In particular, we establish symmetry, positivity and invariance properties. To this end it is useful to introduce a second, equivalent, formulation of the homogenized coefficients.

For all i E { 1, ... , d}, define li: GN - R by a4i(m, expQN a) = -(a, bi). By the Campbell-Baker-Hausdorff formula it follows that expQN a expQN b = expQN c with c = a + b mod qN;2. Hence Sik for all i E { 1, ... , d1 } and d). Therefore if i, j E { 1, ... , dl), then kE d

ij =

_ (Bk W kl

(IV.26)

X j ).

k,l=-dm

It follows, however, from the corrector equation (IV.18) that d

U=-d,

M((Bk

Xj)) = 0

Ckl

for all >i E El. Subtracting the average of (IV.26) then gives

ij =

d

M(Bk(

Ek,

Xj))

(IV.27)

k,1=-d,,,

for all i , j E (1, ... , dl) and 1Li E E. Alternatively, if ri, ... , rd, E C, one can reformulate (IV27) as d

(r, Cr) = Y, M((rk - Bk*r)Ck, (rl - B1XT))

(IV28)

k,1=-d,,,

with r = d! l rifii, Xr =

l riXi and the convention that rk = 0 if k

{l....,d1}.

As a first application of (IV28) we show that the homogenization H commutes with the adjoint operation. The adjoint H* of the subelliptic operator H is a subelliptic operator and the coefficients of H* are given by the adjoint matrix C* = (ck1) with ck1 = Z. Although it is not at all obvious from the original definition, the homogenization process respects the adjoint operation, i.e., H* = H* with a similar relation for Ho and its adjoint. These relations follow from a similar property of the matrix of homogenized coefficients.

IV.4 Homogenized operators

143

Proposition IV.4.1 The adjoint of the homogenized matrix C is equal to the homogenization C* of the adjoins.

Proof Set >/ri = Xit, the correctors for the adjoint H*, in (IV.28). Then taking the complex conjugate gives dl

(T, Cr) _

M((Tk - BkXT)Clk (T1 - R1 X")) = (r, C* r) k.l=-d,,,

where the second identity also uses (IV.28). Therefore C* = C*.

Now we examine positivity and subellipticity properties. Note that bl^... , bd, is an algebraic basis of qN, by Lemma 111.2.5. Hence subellipticity of H on QN

corresponds to the dl x dl-matrix C of coefficients having a strictly positivedefinite real part. To establish this property we apply (IV.28) with *i = Xi. Thus

_

d1

(r, Cr) = E M((rk -

(Tl - BIXr))

(IV.29)

k.l=-dm

where one can restrict the sum to k, I < dl because BkXr = 0 and tk = 0 for k > d1. In particular if C = (ckl) is a (complex) self-adjoint matrix, then C = (cij) is self-adjoint and one has

(r, Cr) = min E WEE

M((Tk - BkW Ekl (r1 - Blrp)).

(IV.30)

k,l=-dm

This follows from (IV.29) and the corrector equation (IV. 18). If one replaces by Xr + eqr, the corrector equation ensures that the terms linear in E vanish.

Conversely, if V E E and (r, Cr) = k.l=-dm M((rk - k(P) Ckl (Tl - B10), then it follows from the second statement of Proposition IV.3.2 that p - M(') = Xr So the minimum in (IV.30) is unique if, in addition, M(V) = 0. This has several immediate implications.

First, choosing rp = 0 gives e < M(C). Secondly, if C1 and C2 are two strictlypositive d' x d'-matrices and Cl > C2, then Cl > C2 by (IV.6). Therefore Ci > C2 by (IV.30). Thirdly, one can apply (IV.30) to obtain an estimate for the self-adjoint real part 91C of a general C. Explicitly, one has

(r, 91Cr) = Re(r, Cr) = L ReM((rk - BkXr) ckl (Ti - BIXr)) k,l=-dm d,

L M((rk - k XT) (91C)kl (TI - BIXr))k,l=-dm

But it follows from (IV.6) that 91C = 91C. Hence one deduces from (IV.30) that

(r, 91Cr) > (r, 9I-CT)

144

IV. Homogenization and Kernel Bounds

or, equivalently, 91C > 91C. Since subellipticity of H is equivalent to 9tC > Al and the order is preserved by homogenization one deduces that 91C > Al. Thus the proof of subellipticity is reduced to proving subellipticity of the homogenization of the sublaplacian associated with the algebraic basis al, ... , ad'. In fact one can prove somewhat more. Proposition IV.4.2 Let H denote the subelliptic operator (IV. 1) with coefficients

ckl and ellipticity constant A. The homogenizationis subelliptic on QN. In particular the real part 91C of the dl x dl-matrix C = (cij) is strictly positivedefinite. Moreover, there exists a µo > 0, independent of the coefficients ckl, such that £RC > µo A 1. The best possible Ao is the ellipticity constant of the homogenization L of the sublaplacian L = - k=1 A. Proof Let L = - Fk-1 Ak be the sublaplacian corresponding to the algebraic basis on GN. Then the d' _x d'-matrix of coefficients is the identity matrix I and the corresponding matrix T has coefficients which can be expressed as

Iij(g) _ L (S(g)bi, ak) (ak, S(g)bj),

(IV.31)

k=1

for all i, j E

d). Then using (IV.29) the homogenized coefficients are

given by d

lij =

Xi) lkl

Xj))

(IV.32)

k,l=-d,,,

for all i, j E { I, ... , d1 } where the Xi are now the correctors for L. Therefore inserting (IV.31) into (IV.32) and appealing to (IV.4) one finds that d'

lij =

M(Ak(i - Xi)

Xi))-

k=1

Hence d'

(r, IT) _

M(IAkwrI2) k=1

Xi). But Akcpr E E1 by Corollary IV.3.3 and (IV.4). with Wr = ! I Since the mean M is faithful, by 11, 11.2, the lower bound is zero if and only if

Akwr = 0 for all k E (1, ... , d'). Since al, ... , ad' is an algebraic basis for g this implies that Bi cpr = 0 for all i E { -d,,,, ... , d) and cpr must be constant. As F11 1 riXi is bounded, it would follow that d! 1 rii;i is bounded which implies

rl = ... = rdl = 0. Hence the real part 91C of C = (ci j) is strictly positivedefinite and L is a subelliptic operator on QN. Let j2 be the ellipticity constant of the matrix 1, i.e., the largest real number such that I > µ 1. Then µ is clearly independent of the coefficients ckl .Moreover, it follows from the argument preceding the proposition that 91C > µ1 > µ p 1.

IV.4 Homogenized operators

145

Finally we show that the ellipticity constant µ equals the ellipticity constant µE of the subelliptic operator L on QN. Obviously dl

d,

i.j=1

f(Bw)IiJ(Brco)>_/2IIBcoII2 j=1

for al l w E CO0(QN), so µZ > µ. Next. fix i E C°0(QN) with II*II2 = 1. Let r1, ... , rd, E R and for all u > 0 set w = V1, where r.l; = Yd' I r j t;j . e'r.g

Then IIwu1I2 = 1 and d1

di

u L Iij (B;N)wu,

B;N)(Pu)

i.j=l

=

u-2(wu, Lwu) >- µL u-2

IIB(N)(P.II2

j=1

for all u > 0. But (IV.33) it follows that

e"4

(IV.33)

iurj rp,,. Taking the limit u -+ oo in

(r,Ir)IIvII2

and lc > t4 E. Therefore µ = µi. In the classical Rd-theory of homogenization the geometric factor p. is equal to one. The inherent reason for this simplification is that at, ... , ad, is a vector space basis which is orthonormal with respect to the natural inner product on Rd. In the Lie group setting the value of µo depends on the choice of the inner product (- , ) on g. But since the only directions involved in the homogenization are those of the subspace 4 1, the dependency is only on the choice of the inner product on 41. This choice is governed by the requirement that the restriction of the S(g) to fI should be orthogonal. In the subelliptic setting it is not evident that there is a natural choice of basis or of inner product. One can also deduce some basic properties of the homogenization from symmetry. First we consider the invariance properties of the matrix of coefficients C. -

The matrix C can be viewed as the matrix of a linear operator Cn, on l i defined by invariant its restriction to h 1, >d! I ci j b. Moreover, since S leaves Chi b, = f,1

which we denote by Sn1, is a unitary representation of GN in 41 by Statement IV of Proposition 111.6. 1.

Lemma IV.4.3 The linear operator Chi commutes with the unitary group representation Sh, on 41, i.e., [Ch,, Sh1(g)] = Ofor all g E G.

Proof It follows from Lemma IV4.10 that the homogenized coefficients cij = M(ai j) with aij (g) = (S2(g)(bi 0 bj), a) for some a E 7-12. Moreover, the unitary operator Shi (h) has a matrix representation Sij (h) = (S(h )bi , bj ). Therefore d,

L aik(g) Skj(h) = (S2(g)(I k=1

®S1(h-1)(bi

(9 bj),a)

146

IV. Homogenization and Kernel Bounds (S2(gh-)(31(h)

_

(9 1)(bi ®bj), a)

dl

_ L Sik(h)

akj(gh-1).

k=1

Taking means over g, and using the invariance of the mean, one deduces that Cht commutes with each Sh, (h). 0

Since the linear operator eh, commutes with Shl (g) for all g E G one can use representation theory to deduce some general features of the structure of the matrix. The unitary representation (Shy, hl) can be decomposed as a direct sum of irreducible unitary representations (Sa, ht,a) such that Sh,

(D S,, a

and

h1 =®41,a a

If the decomposition is multiplicity-free, i.e., if the Sa are mutually inecluivalent, then it follows from the commutation property of Lemma IV.4.3 that C takes a diagonal form. Explicitly, C =

a P.

where ca E C and Pa is the orthogonal projection with range l]t,a. If some of the irreducible components in the decomposition occur with multiplicity, then the form of C is slightly more complicated as there can be off-diagonal components mapping between the subspaces corresponding to the unitarily equivalent components. Example IV.4.4 Let c3 be the three-dimensional Lie algebra of the Euclidean motions. The nilradical n is two-dimensional and the complementary subspace n is one-dimensional. Hence

S decomposes into a trivial one-dimensional representation on o and a non-trivial twodimensional representation on n. Therefore the homogenized matrix e is diagonal regardless of the choice of subelliptic operator.

The simplest examples with non-diagonal homogenization require the Lie algebra to be at least five-dimensional. One can then construct examples starting with a sublaplacian L for which the matrix of coefficients is diagonal with respect to the basis at, ... , ad, but the homogenization L is not diagonal with respect to

the basis bl,...,bd,. Example IV.4.5 Let q be the five-dimensional solvable Lie algebra with basis bl.... , b5

and commutation relations (bl, b2] = b3, [bl, b3] = -b2, [bl, b4] = b5 and [bl. b5] = -b4. Then n = span(b2, ... , b5); one can choose o = span bl and the nilshadow is abelian. Let Q be the simply connected Lie groups with Lie algebra q and identify QN with R5 in the natural manner. If Bk = dLQ(bk) and Bk = dLQN (bk), then 8k = -8k and it follows from Statement V of Proposition 111.6.1 that Bt = -a,. Moreover

B2=-r182+s1 a3

B4=-C1a4+s1a5

B3=-S1-2 -rla3

B5=-sla4-rta5

IV.4 Homogenized operators

147

for all x e R5 = QN where cl (x) = cos x land sl (x) = sin x I. Set a t = bl , a2 = bN + b4,

a3 = b3, a4 = b4 and as = b5. Then a,, ..., as is a basis for q. Let H = - Ek=t Ak be the corresponding Laplacian. Then with the notation of Section IV.3 one calculates that

I = 1, c = 0, c14 = 0 and j24 (x) = cost XI _+ sing x1 = I for all x E Rs. Then HX4 = -BIc16 = 0 and X4 = 0. Moreover, c21BIX4 = 0. Hence M(c21 B1 X4)= 1 # 0. So the matrix C is not diagonal. In this example all correctors vanish and C = M(C) but the matrix M(C) of averaged coefficients is not diagonal. Even if the matrix of averaged coefficients is diagonal, then off-diagonal contributions from the correctors are possible. Set al = bl, a2 = bl + a3 b3, a4 = bl + b4 2 and a3 = b5. Then al, - a5 is a basis for q. Let H = - k=t Ak be the corresponding 0 for all x E RS. Laplacian. Then cl I = 3.121(x) = cosx1, 114(x) = cosxl and One easily verifies that M(c;j) = 0 for all i, j. Next, (HX4)(x) = -(BIC14)(x) _ - sinxl and x4(x) = 3-1 sin xl. Moreover, V21 BI X4)(x) = -3-1 cost x1. Hence i'24 = M(c24)M(c21 it X4) = 6-1 # 0. Again the matrix C is not diagonal. -

- -

The invariance property of Lemma IV.4.3 can also be expressed as an invariance of the homogenization H, or of Ho. Let S. GN --)' G(L2(GN)) denote the unitary representation of GN defined in terms of S by (111.34). Explicitly, S(grp = rp o S(g-1). Since Sleaves QN invariant one can also define an operator T: GN -4 G(L2(QN)) by T (g)rp = (p o S(g-1).

Proposition IV.4.6 The homogenization HO commutes with Son L2(GN) and H commutes with T on L2(QN).

Proof It follows from (111.35) that dl

BiS(g)rp = ESij(g-1)Bjco j=l

for all i E (1, ... , dl), g E G and rp E C°O(GN), where we have used the invariance of hl under the action of S(g). Again Sij(h) = (S(h)b1, bj) denotes the matrix elements of S(h). Then dt

(S(g)t, HoS(g)(p) = L eij (B, S(g)*, BjS(g)w) i.j=l _

dl

L eij Sik(g-1)Sjl(g-1) (S(g) Bk/, S(g) Birp) i,j,k,l=l

for all gyp, r/i E CO0(GN) and g E G. But S(g) is unitary since detS(g) = 1. Moreover, di

L cij Sik(g-I) Sjl(g-I) _ (Sbt (g-1)bl, ell, Sbl (g-1)bk) i,j=l

_ (Sh1(g-1)bl,Sb,(g-I )Cb,bk) = (bl, Cb,bk) = Ck1

148

IV. Homogenization and Kernel Bounds

for all g E G and k, l E (1..... dl ), where we have used Lemma IV.4.3 in the third step. Therefore d,

(S(g)*, HoS(g)(V) = L ckl (Bkf, B11P)

Hod)

k,1=1

for all Y, }li E CO0(G). So bx density it follows that S(g)D(Ho) = D(Ho) and S(g)*H0S(g) = Ho. Since S(g) is unitary the first statement of the proposition is established. The proof of the second statement is essentially identical.

0

There is an immediate corollary which follows from the relation (111.36).

Corollary IV.4.7 The homogenizations H and H0 are right invariant operators, with respect to the groups Q and G, on L2(Q) and L2(G), respectively. Proof The operator H0 is RGN-invariant by definition. Moreover, it commutes

with S(g) for all g E G by Proposition IV.4.6. Since RG(g) = S(g) RGN(g) by (111.36), it is then RG-invariant on L2(G). The RQ-invariance of H on L2(Q) follows similarly. 0 The homogenization also commutes with the adjoint action of M on Q. Define

ym: Q -+ Q, for each m E M, by y,,,(q) = mqm-1 and then define y,,,

E

G(L2(Q))by ymco=(p oym1 Corollary IV.4.8 The homogenization H commutes with ym on L2(QN) for all

mEM. Proof It follows from Lemma 111.7.9 that );m = T(m). Therefore the corollary is a special case of Proposition IV.4.6.

0

The invariance has a direct implication for the semigroup kernel.

Proposition IV.4.9_If K is the kernel of the semigroup generated by H on QN, then K,(q qi 1) = K,(q *N q(-i)") and K,(q) = K,(mgn:-1) for all q, qi E Q,

mEMandt>0.

Proof Let S be the semigroup generated by H on L2(QN) and define K, by K, (q ; qi) = K,(q *N q(-')'). Since the Haar measures on the Lie groups Q and QN are equal one has

(S, )(q)= f dgiK,(q;gI) 0. But S, is RG-invariant since H is RGinvariant. Therefore K,(q ; qi) = k, (q q e) for all q, qi E Q and the first statement of the proposition follows. The second statement follows similarly. 0

IV.4 Homogenized operators

149

One can give a more direct proof of the symmetry properties of the homogenized operators using a coordinate-free characterization. First note that the coefficients ci j can be expressed as d

_

cij =M(c;j)+ E M((Bkcik)Xj)k=-d,,,

The equivalence with the original definitions follows from the left invariance of the mean on GN. This implies that M(Bk(ctk Xi)) = 0 for all k E {-dm, ... , d}. Hence M(cik BkXj) = -M((Bkctk) Xj). Next note that Ek=-dm Bkcik

with co =

k,l=l clk a(ak)at by (IV.19). Moreover, Xj = bi,x with x = -%o) co and co = Ek,t=t ck1 a(ak) at by Corollary IV.3.3. In addition c;j = d' ckl(ak (9 at). Therefore one has ci j = M(ai j) where bi®bt,c with c = Ek,1=1

a;j(g) = (S2(g)(b; ®bj),c+c`o (9 x)

(IV.34)

for all i, j E {-dm, ... , d}. This formulation allows one to characterize the homogenized coefficients in terms of the projection P2(0) and the algebraic basis.

Lemma IV.4.10 If cQ = _k,1=1 ckl a(ak) at E (I - P(0))n and X E NO is the unique solution of HS1x = -ca then

c;j = (bi ®bj, P2(0)(c + co (9 x))

(IV.35)

for all i, j E (-dm, ..., d) with c = _k 1=1 ckl (ak 0 at) E E2 and co = L.k,l=l clk a(ak)a,.

Proof Since c;j = M(a;j) with aij given by (IV.34) the representation of the lemma follows by the mean ergodic theorem, Statement 11. 11.7.

0

The identity (IV.35) now allows a coordinate-free description of the homog-

enized operators H and Hp. For all n E N and p E [1, oo] let 8LG:7{ -> G(LP;oo(G)) be the linear map such that

aLG(cl ®...(9

(IV.36)

for all cl, ... , c E g. Then Lemma IV.4.10 implies the following identities for the homogenizations.

Lemma IV.4.11 If c, co and x are as in Lemma IV.4.10 and E is the natural projection from 'H2 onto 41 ® 4 1, then

HILp;.(QN) = aLQN(EP2(0)(c+c`Q (9 x)) and

HOILp.c, (GN) = 8LGN(EP2(0)(c+c` (9 x))

for all p E [1, oo].

150

IV. Homogenization and Kernel Bounds

One can use the coordinate-free description to give an alternative proof of the symmetries in Proposition IV.4.6 and Corollary IV.4.8. It follows as in (111.35) that

dLQ, (a)T (g)co = T(g)dLQN(S(g-')a)W for all a E q, g E G and rp E C°O(QN). Hence 8LQN (a)T (g)w = T (g) aLQN (S2(g-' )a)co

for all a E q ® q, g E G and cp E CO0(QN) where aLQN is defined by (IV.36). If c, c`o and x are as in Lemma IV.4.10 and E is the natural projection from x2 onto 41 ® 41, then it follows from Lemma IV.4.11 that Hrp = aLQN, (EP2(0)(c + c`o (9 x))rp

for all (p E C°O(QN). Obviously E commutes with S2(g) for all g E G. But then

HT(g)rp = 8LQN(EP2(0)(c+c`o (9 x))T(g),p

= T (g) aLQN (S2(g-')EP2(0)(c + c , (& x))(G = T(g) 8LQN(EP2(0)(c +co (9 x))w = T(g) H(P Since CO0(QN) is dense in ndj=i

it follows that H commutes

with T.

IV.5

Homogenization; convergence

In the standard theory of periodic operators on Rd the homogenization j7 of a strongly elliptic operator H is identified as the dilation invariant limit of the rescaled operators as u - oo. This identification is usually expressed by the convergence of weak solutions of the Dirichlet problems HJ,p = 0 on bounded open subsets of Rd to solutions of the associated problem Hcp = 0 for H. Alternatively, it can be formulated globally on Rd by the strong resolvent convergence (IV. 16). In the Lie group setting there are analogous identifications of the homogenization, both local and global. The local estimates are subsequently used in the derivation of Gaussian bounds on the kernels of the subelliptic semigroups and the global estimates are exploited to obtain bounds on derivatives of the kernels. We next discuss these convergence results starting with the global theory because it is technically simpler. In the Euclidean theory the key global property is the strong resolvent convergence (IV. 16). Initially we prove a version of this result for the Lie group operators using a formula which compares the subelliptic operator H and its homogenization. It is useful for the subsequent scaling arguments to trace the derivatives in-

volved. Therefore we introduce the following weights. Define wk = 0 if k < 0

IV5 Homogenization; convergence

151

and wk = j if bk E ll j for k E (1, ... , d). (This definition coincides with the definition given in Section IV.2 for groups with a stratified nilshadow.) Now the

multi-index a = (i1, ... ,

with ik E {-d,,,, ... , d}, has length lal = n and

weighted length Ila II = wi, + ... + wi.. Moreover, set P = fu dm LGN (m), to be the projection on the space of functions which are constant in the M-directions.

Proposition IV.5.1 For all of E J (d) and k E { 1, ... , d' } there are ra,k E E3 and ra E E4 such that d (E

(Ho - H)(V = -H

_

d'

L

x Bjcp) +

Ial=2 k=I

j=1

E raBarp lal cikXJBiBkBjrP. i, j,k=-dm

One may again discard all terms with i, j, k < 0 by another application of (IV.39), except for i < 0 in the first term. Together one obtains d

E

XjBj(o)+

(Ho-H)co=-H(>

j=1

cijBiB,co

I k=-d,,, BkckiXJ) = 0 for all i, j E (1, ... , d), and apply Corollary IV.3.10 to deduce that for all i, j E ( 1 , ... , d) and k E (1, ... , d'} there are functions ri'jk E E3 such that d'

d

Cij - 6ij +

BkOkiXi = k=-dm

Akr' k=1

Therefore the last sum in (IV.41) can be written as d

d

i,j=l

k=1

- E (Cij -Cij +1: d

d'

rd

L:

i, j=1 k=1 d

d'

d'

> rhkAkBiBjco

i, j=1 k=1

_

d

d'

d

Ak(rijkBiBjco)+ i,j=1 k=1

i,j=1 k=I l=-dm

_ rijkrkl BIBiBjW

IV.5 Homogenization; convergence

where we expanded Ak = F

_

153

rk1Bi, with rki E £1, using (IV.4). But by

(IV.39) one may discard all terms where I < 0. Then it is easy to see that all terms contribute to the right side of (IV.37).

The comparison formula (IV.37 ) can be used to obtain a strong convergence result for the rescaled operators defined in Section IV.2. But to use the scaling formalism of Section IV.2 we must again assume that G is simply connected

and that its Lie algebra has a stratified nilshadow with stratification 14j). We adopt these assumptions throughout the remainder of the section. Then Proposition IV.5.1 has an immediate corollary corresponding to the rescaled operators. If

t/': GN - C is a function, we set 0(u) = >G o r. for all u > 0. Corollary IV.5.2 If ra,k E £3 and ra E £4 are the coefficients of Proposition IV.5.1, then d

d'

(Ho - Hl. 1)(p = Hlul(E uX Bjp) + j=1

u1-IlatlAkyl (ra(k BaV) lal=2 k=l

+ E u2-Manr," Bate

(IV.42)

1a153Slall

for all u > 0, p E [1, ooj and tp E PLP,oo(GN).

Proof The statement of the corollary follows from (IV.37) by scaling, with the help of the identities AM = u(F 1).Ai, u-" B, = (fu 1).Bi and Hl. j =

u2(r 1).H. The relation (IV.37), or its scaled version (IV.42), can be viewed as the firstorder in an expansion of the difference Ho - H, or of Ho - Hlul. The first-order term is explicitly identified in terms of the first-order correctors Xj. In Proposition VI.1.11 we will give an improvement of the expansion in which the secondorder term is identified in terms of second-order correctors Xij which are solutions of an analogue of (IV. 18). The expansion (IV.42) immediately allows one to deduce strong resolvent con-

vergence of the rescaled operators to the operator Ho. Recall from Section 11.2 that the L2-contraction semigroup generated by the subelliptic opera-

tor H is holomorphic in a sector A(8) = (z e C : I arg zI < 0) with 0 > 9c = arctan(t The holomorphic extension of the semigroup is contrac11C11-1).

tive throughout the subsector 0(9c). In particular the resolvent X r-. (XI + H)-1 is defined and satisfies bounds II(XI + H)-1112-.2 < IXI-1 for all X E L(9c). The Similarly, the semisame is true for the resolvents of the rescaled operators group generated by the operator Ho is holomorphic and contractive in a sector A(9e) where 0 is completely determined by the matrix of coefficients C = (cij ). Hence one has bounds II(XI + Ho)-1 II2-.2 < IXI-1 for all ,1 E 0(00). Let I I' denote the subelliptic modulus on G with respect to the algebraic basis a1, ... , ad', and for u > 0 let I I;, denote the corresponding modulus on G with -

154

IV. Homogenization and Kernel Bounds

respect to the re-scaled basis uy,,-' (al ), ..., uyu 1(ad'). It follows from (IV.12) that Iru(g)I' = ulg1, for all g E Gu. Proposition IV.5.3 The operators Hlu) converge in the strong resolvent sense in

L2(GN) as u - oo. Moreover lim II((1A/ + Hlul)-' - (Al + Ho)-' P)w112 = 0

U- 00

for all k E 0(90) and tp E L2(GN), where Bo = 9c A Ac-

Proof The proof is in two steps. The first step establishes convergence on the subspace PL2(GN). This is basically a corollary of the expansion (IV.42). The second step establishes that (Al + H[u))-1(I - P) converges strongly to zero. Step 1 If rp E PL2(GN), then

((Al +

Hl,,))-1

- (Al + Ho)-1)(p = (XI + H[u))-' (Ho - Hlul)(A1 + go)-'(p

and (Al + Ho)-1 PL2(GN) C PL2(GN). Therefore using (IV.42) one deduces that 11((a/ +

Hlul)-'

- (Al +

Ho)-')w112

< clu-1IIHlui(kl + Hl.))-' 112-.2 max 1<j 1 the required

convergence follows immediately for (p E PL2;oo(GN). It follows for all rp E PL2(GN) by continuity. Step 2 First, note that L2(GN) = L2(Gu) for all u > 0 and that II*112 = for all >r E L2(GN) since Haar measure on G is a Haar II*IIL2(GN) = III measure on GN by Proposition 111.7.10. Secondly, for all u > 0 and g E GN, one has II (1 -

Hlu))-1 d'

181u1

\k=1

l/ 1

IIA[u] (AI +

< (plµ)-'/21g1;, 14 112 = (IAI/L)-'/2u-'

2

I

I lu (9)I' 11 w112

IV.5 Homogenization; convergence

155

by II.2.6. In particular,

11(1- LGN(m))0,1 +

11(1- LG.(m))(AI + (I,IIi)-'12u-' Iru(m)I' IIWII2

= (I),Ip)-'"2u-' Iml' Ilwll2 for all m E M. Therefore

11(1- P)(,tl + Hlul)-'VIIL2(GN)
/in weakly. Then subtraction of these two equations gives XI

n:i' (Pn) =

(IV.49)

V"n)

But to;i converges strongly to 4i in L2(S2) and i/in converges weakly to 1/i in L2(S2). Therefore d, n

lim

qj)

00

(IV.50)

j=I

where the second equality follows from (IV.48). Now we evaluate the limit of the left-hand side of (IV.49). Since d'

((Akn]X)ckl AM +

(Hlul, x) k,1=I

Aknlckl

(Allu x))

IV.5 Homogenization; convergence

161

one can write ([H1

1,

MO +Mn2

X{

with d

Mn;i = I=t and

d'

Mn;i

L ((A'u"]X) cki

Aku(p.)

d

((BIX)Emuj1 1,m=-dm

It follows immediately from the strong convergence of n;i and the weak convergence of ?7,,;j that d,

lim Mnj =

noo

(IV.51)

j=I

Now we compute the limit of Mn2; as n -* oo.

(-dm,

First, Bm n;i = 0 unless m E dl }. Then (aim un x)("")). Secondly, BIX = 0 for all 1 E {-din, ... , 0}. Moreover, cmu 1 = Therefore d dd 1

Mn2i

= - L L un-wl ((BIX) Cml) (aim - (BmXi )(u")), *On) 1=1 m=-d,"

cpn) is bounded uniformly inn. So in the limit

But ((BIX) c, i ) (aim -(BmXif

n --* oo all terms with wl > 1 vanish. Moreover, it follows from Lemma IV.5.7 that, by passing to a subsequence if necessary, we may assume that Vn converges weakly to rp in LZ(supp X) as n -+ oo. Hence, using (IV.47) to identify the weak scaling limit as a mean value and the relation

(c*), = L M(cml (aim - BmXi )) m=-dm

for the coefficients of the homogenization of the adjoint, one obtains d,

"lim Mn2i

am (c*) l=1

(IV.52)

162

IV. Homogenization and Kernel Bounds

Then combining (IV.49), (IV.50), (IV51) and (IV.52) gives d,

d,

d

i7j) - Fc (Bix, (p) = j=1

r!j) j=1

1=1

where we have used the identity (c'),i = cii which follows from Proposition IV.4.1. Then, by rearrangement, one has d,

(IV.53) ,=1

for all i E { 1, ... , d1 }. Therefore (IV.48) gives d,

(X,

d,

>2(BiX, iii)

L c (B,BiX, (G) = ((H')oX, w)

i=1

i.,=1

Hence

(x','Gb) = (H'X','V) for all X' E CO0(S2').

Similarly, it follows from (IV.53) that d,

(X" 71

) = C (B,'1 X', (cb) ,=I

for all x' E CO0(Q') and i E (1, ... , d1). The matrix e acting on Cdr must,

C) = 0 and however, be injective because if 4 E Cdr and Ct4 = 0, then = 0 by Proposition IV.4.2. Therefore e is invertible. Let (bij) be the inverse of C. Then d,

bki ,j) _ -(Bk'V)x', w')

(X i=I

for all x' E C°O(Q') and k E

{ 1, ... , d1 }. Hence one concludes that Wb E (S2) and HVb = ,1b weakly on S2'. HZ(N)

IV.6

Kernel bounds; stratified nilshadow

In this and the next section we establish global Gaussian upper bounds on the kernel K of the semigroup S generated by the complex subelliptic operator H on a group G with polynomial growth. The previous properties of homogenization play an essential role in the proof. The Gaussian bounds are subsequently used in Chapter VI to identify the asymptotic behaviour of K and S. If H is real, these bounds are relatively easy to derive (see Section 11.5) and homogenization theory

IV.6 Kernel bounds; stratified nilshadow

163

is irrelevant. The proof for complex operators is, however, considerably more complicated and it appears that homogenization theory is essential. The proof of the bounds is in two steps. First, we establish the results on a special class of groups, notably those with a stratified nilshadow. In this step dilation theory and the convergence results of the previous section play an important role. Secondly, the result is lifted to general groups by extension and embedding arguments of the type used earlier for nilpotent groups (see Section 11.9). This two-step process is not essential but it is conceptually simpler than immediately dealing with the general case. In this section we assume the Lie algebra g of G has a stratified nilshadow compatible with n and m. Moreover, we suppose that the local dimension D' of G with respect to the algebraic basis at , ... , ad, equals the dimension D of G at infinity and that D' > 2. In addition we restrict attention to simply connected groups. We use the notation and conventions of Sections IV.2 and IV.5. The kernel bounds are then derived under these three restrictions. The first restriction allows the use of the scaling arguments of Section IV.2. The proof consists of verifying the hypotheses of Theorem 11.10.5 for the rescaled derivatives A IM, ... , AM and the subelliptic operators H[u] uniformly for all u > 1. In particular, we verify that the various parameters in the hypotheses of the theorem can be chosen independently of u. The first two hypotheses are relatively simple to verify since they only involve the subelliptic geometry. The third, however, the De Giorgi estimates, is more difficult and requires non-commutative extensions of various estimates used in the standard Euclidean setting. The proof exploits the almost periodic nature of the rescaled operators H[u] represented on GN and relies on the homogenization estimates of the previous section. Theorem 11. 10.5 applied to the kernel KIUI of the semigroup Slut generated by HIu) then gives local Gaussian bounds which are

uniform u > 1. Next, one can exploit the rescaling relation (IV.15) between the kernel Klul and the rescaled K to obtain global Gaussian bounds on the kernel K from the local bounds on the kernel KM. Since K1(g) = u-p Ku" Z,(r t (g)) for all g E G and u, t > 0 the choice u = tt/2 gives

K,(g) = t-o12 K,`'nl

(IV.54)

Therefore, Gaussian bounds for K,ul for t < 1 which are uniform for all large u give Gaussian bounds on K for all large t. In Section IV.7 we now remove the three special restrictions, namely the assumptions that the nilshadow is stratified, the local and global dimensions are equal, and the group is simply connected. This second part of the proof is fairly straightforward. It is largely based on arguments used earlier to prove kernel bounds for special classes of groups.

The initial aim is to verify the three hypotheses of Theorem 11.10.5 for the operators HIul on G uniformly for all large u. Consider the growth properties. For u > 0 let B'(u)(p) = (g E G : gI' < p), with p > 0, denote the balls associated with the modulus I I' . Since I r(g)I' _

164

IV. Homogenization and Kernel Bounds

u Jgj' for all g E G one has

and

B'(ru) = ru(B'(u)(r))

(IV.55)

l,(B'(us)(r)) = B'(u)(rs)

(IV.56)

for all r, s, u > 0. The large balls B'(u)(r) satisfy a uniformity property. Let ( , ) again denote an inner product on g such that S(g) is orthogonal for all g E GN and the spaces m, b, t, 1)2, , 4, are mutually orthogonal, where ((lk) is the stratification of qN compatible with m and u, and f = h, fl n. Further, let b-dn, ..... bd be an orthonormal basis for g passing through m, u, t, (2, ..., h,. Let B'(N)(p) denote the ball of radius p on QN associated with the algebraic basis b, , ... , bd, Moreover, let B(M)(p) be the ball of radius p on M with respect to the vector space basis b-dn , ... , bo.

Lemma IV.6.1 There exists a c > 1 such that B(M)(ru) x B'(N)(r) C B'(u)(cr) and B'(u)(r) C B(M)(cru) x B'(N)(cr) uniformly forall r, u > 0 with ru > 1.

Proof Since the S(g) with g E GN are orthogonal with respect to the inner product (- , ) on 9, there exists a c > 1 such that the inclusions are valid for u = 1, by Corollary 111.7.13, applied with h = e. Now let r, u > 0 with ru > 1. Then it follows from (IV.55) that

B(M)(ru) x B'(N)(r) = ru 1(B(M)(ru) x B'(N)(ru))

C ru'(B'(cru)) = B(u)(cr) and similarly that B'(u)(r) = ru I (B'(ru)) C ru I (B(M)(cru) x B'(N)(cru)) _ B(M)(cru) x B'(N)(ru). D Next we use (IV 11) to estimate the volume V'(u)(r) of the ball B'(u)(r) in Gu. By definition

V'(u)(r) = fG

= JM dm Jdq 1BO )(r)(m,q)

I

/'

=

dm J IM

dq IB'()(r(m, q)) = u-DV'(ur)

QN

where we have used (IV.55). Since we assume that the local dimension D' and the dimension D at infinity of G are equal there exists a c >_ 1 such that 0-1 rD' < V'(r) < Or D' uniformly for all r E (0, oo). But then

c-' rD < V'(u)(r) < crD uniformly for all u > 0 and r > 0. This implies Condition I of Theorem 11. 10.5 uniformly on Gu.

IV6 Kernel bounds; stratified nilshadow

165

We proceed next to estimate the parameters of the Neumann-type Poincar6 inequality occurring in Condition II of Theorem 11. 10.5. Let H2(;)(s2) denote the local Sobolev space defined by (IV.43) equipped with the norm IIwII'2;,,u,,. Then,

for 'p E L1,1()c(Gu), we denote the average of tp over 0 by ('),,,n, and if 0 =

B't' (r), we set IIWII2,u,r = II47II2,B'(u>(r) etc. Proposition IV.6.2 There exists a CN > 0 such that Iiw - (W)u,r 112,ur < cN r2II V gO112,u.r

uniformlyfor all u > 0, r > Oandg E HZ"j)(B'(u)(r)). Proof There exists a CN > 0 such that (IV.57)

11W - (w)r 112,r < cN r2II V'w J12,r

uniformly for all r > 0 and go E Cb (G) (see Notes and Remarks). Then (IV.57) is valid for all r > 0 and go E H2;1 (B(r)) by approximation from Theorem II.10.1.

Now let u > 0, r > 0 and V E H2;, (B'lu)(r)). Set 1' = W o ru 1. Then it follows from (IV.12) that * E H2;1(B'(ru)). Moreover, A; iJt = u-1(A;ulrp) o I'u as distribution. So I

Iw

- (w)u,r

11

112

2 . u,r = u-DING - (%' )ru 2,ru

< u-o cN (ru)2Il V'*II2 r = CN

r2II VugojI2 u,r

0

and the proposition follows.

This proposition clearly implies Condition II of Theorem IL 10.5 uniformly in u. It remains to establish a uniform version of Condition III of the theorem. The proof is based on two Caccioppoli inequalities. Let H Z ' , , (Q') denote the local Sobolev space defined by (IV.44) on the ho-

mogeneous nilpotent group QN, with respect the algebraic basis b1, ... , bd, equipped with the norm Then, for W E L1,1a(QN), we denote the average of (p over Q' by (cp)N,n', and if 0' = B'1N)(r), we set II'112,N,r = IIWI12,BI(N)(r), etc.

Lemma IV.6.3 There exist cl > I and Q E (0, 1) such that I.

I1ousv112,u,or < ct r-211w - (co)u.r112,u.r

uniformly for all u > 0, r E (0, 11 and g E H2;1)(B'"u)(r)) satisfying Hiulrp = 0 weakly on BM (r), II.

II o'(N)V 112,N,or < ci r-21Iw - ((p)N,rI12,N,r

uniformlyforall r E (0, 11 and rp E weakly on B'(N)(r).

satisfying Hip = 0

166

IV. Homogenization and Kernel Bounds

Proof Statement II has been proved in Appendix A.1, Lemma A.1.3. The proof of Statement I is more delicate, since it requires the constant to be uniform in u.

By Lemma A.1.2 there exist c, o > 0, and for all R > 0 a function qR E CO0(B'(R)) such that qR(g) = I for all g E B'(a R), 0 < qR < 1 and IIAi qR Iloo
0 and i E { 1, ... , d'}. For u > 0 and R > 0, define qR°) = qR o r,,. If u > 0, R > 0, g E Gu and qR)(g) ,E 0, then r'u(g) E suppgRu c B'(Ru). So ulgI', = Iru(g)I' < Ru and g E B'(u)(R). Similarly, qR)(g) = 1 for all g E B'(u)(aR). Obviously, 0 < ?I (RU) < Finally, it follows from (IV.12) that IIA;uIgR) Iloo = u lI (Ai i Ru) o ru lloo =

R-' uniformly for all i E (1, ... , d'}, u > 0 and R > 0.

u II Ai qRu Iloo < c

Now Statement I follows from Statement II of Lemma A. 1.3. By the Poincard inequality of Proposition IV.6.2 there exists a cN > 1 such that (IV.58)

Ilw - ((v)u,rll2,u,r < CN and

II2,N,r 0 such that for alI R E (0, 1] and Or E HH(N)(B'(N)(R)) which satisfy RV, = 0 weakly on BIM (R) one has llV'(N)* II22 ,N,r < CDG

(r/R)D-2+2v,

for all 0 < r < R. Hence in combination with (IV.59) and Statement II of Lemma IV.6.3 it follows that

III// -

2,N,r < CNr2 IIo'(N)>LII22,N,r Q-D+2-2v,r2(r/R)D-2+2v,

< CDG CN Ct CDG cN

a-D+2-2v, (r/R)D+2v,

II* -

II2,N,R

whenever 0 < r < QR, R < 1 and * E HZ(N)(B'(N)(R)) satisfying RV/ = 0 clCDGCNa-D+2-2v,(r/R)D+2v, _ weakly on B'(^')(R). But one then factors (c1CDGCN(aR)-D+2-2vir2(v,-vo))rD+2vo So take R = c-1 and r sufficiently

small.

1

IV.6 Kernel bounds; stratified nilshadow

167

Proposition IV.6.5 For all v E (0, 1) there exist ro E (0, 1) and uo > 1 such that for all u > up and W E H2(1)(B'(u)(1)) satisfying H(UJSP = 0 weakly on BO) (1) one has i

2

D-2+2v

u ro - r0

2

1

11VuW112,u,1

Proof Let vo E (v, 1). Let ro be as in Lemma IV.6.4. Suppose there is no such up. Then for all n E N there exist un > c V ar0 1 V c-2par0 I V p v n and 'pn E H2(,")(B'(u")(1)) such that Hlu"lcpn = 0 weakly on B'(u")(1) and i

' D-2+2v 2 110u"Wn112,un,I' r0

2

II

where p > 0 is such that B(M)(pc-1) = M. We may assume that ((Pn)u",I = 0 cN for all n E N by and IIou"wn112,u",I = 1 for all n E N. Then IIwn112,u",I the Poincar6 inequality (IV.58). By Lemma IV.6.1 one has M x B'(N)(c-1) C B'(u")(1) for all n E N since un > c v p. Applying Proposition IV.5.6 to the set M x B'(N)(c-1) and the restrictions of the functions 'p to the set M x B'(N) (c-1), it follows that there exists a subsequence of rps, 'P2, ..., which we also denote by 'pl, (p2, ..., such that rpn converges

weakly on M x B'(N)(c-1) to a 'p satisfying V = Prp and H* = 0 weakly on B'(N)(c-1), where * = cpb. Moreover, since caro < c-1 one may assume by Lemma IV.5.7 that tpn converges to 1 ®>/i strongly in L2 (M x B'(N)(ca-1 ro)). Then D-2+2v = _ liminfroD-2+2v 2 IIDu"(Pn112,u",I

ro

1

n-+oo

< lim inf II Vu" 'pn 112,u",ro n-+oo

< lim infcl a2 r0 21Icon - ((Pn)u",a )r0 2,ua-)ro

by the Caccioppoli inequality of Lemma IV.6.3. Next, note that v F-1 fn Iw - v12 has its minimum for v = (v)n, the average of

'p over Q. Moreover, B'(u)(a-1 ro) C B(M)(ca-irou) x B'(N)(ca-1 ro) = M x B'(N)(ca-fro) by Lemma IV.6.1 whenever a-lrou > 1 and ca-Irou > PC-I. Therefore with rI = cc -I ro one has liminf IIwn - (Wn) u".o ro112 2,u",a-)ro n-.oo )

< lim inf J Icon - (*)N,r) I2 n-'0MxB'(N)(Ca-)ro) MxB'(N)(ca-)r0) 2

111 ®>fr - W N,r) 12

= II G - (*)N,r) II2,N,rl.

168

IV. Homogenization and Kernel Bounds

It follows from Lemma IV.6.4 and the normalization I M I = 1 that

r0D-z+2v < C1

a2 rp 2

rp

ro -2+2vp lim inf

12

I

fMXB'(N)(c-1)

< r0D-2+2vo lim roof II'Pn II2,u,,, I = r0D-2+2vo 2

Then one has a contradiction since v < vp and r0 < 1. These local estimates extend to global estimates by various applications of scal-

ing. Subsequently, we frequently use the following. If 0 < r < R, s, u > 0, (p E HZ(i)(B'(u)(R)) satisfying HIu1(p = 0 weakly on B'(u)(r) and >/i = (p o I's, then by (IV.56) and HIu.cll/i = 0 weakly on B'(us)(rs-1). The last relation is established by noting that if X E CO0(B'(us)(rs-1)), then X' = X 0 rs 1 E C°O(B'(u)(r)) and E

d

(x,H(usl*)_

s2-Wr-Wi

i,1dm

B'(Yd)(rs )) d

= S2-D

f

dg(Bix)(g)j;;J(rs(g))(Bj*)(g)

dg (Bi x') (g) ojj l (g) (Bj(p)(g)

= S2-D(X', Hlul(p) = 0 where the second step follows by (IV.56). Moreover, 11 V,' W 112,us,p = s2-D II V, (P 2,u.n.s

for all p E (0, Rs-1].

Lemma IV.6.6 For all v E (0, 1) there exist rp E (0, 1) and up > 1 such that for

all r E (0, r0), u > r

u0 and (p E H2'(U)(B'(u)(1)) satisfying HIuJ(p = 0 weakly

on B'(u) (1) one has 2

-D+2-2v D-2+2v

r

Ilouwll2,u,r ro (k+l)uoand lp E H. )(B'(")(l))satisfyingH1"lrp = 0 weakly on B'(") (1). Set s = ro and >G = w o r, Then, by the previous remark, tilr E HZ(us)(B1(u,)(y.-1)) C HZ(us)(B'("s)(1)), us > uo and Hl",llk = 0 weakly on B'("s)(s-I) and hence on B'("s)(1). Therefore Proposition IV.6.5 implies that I1VUs*ll2,us.ro < ro

Hence IIV wll`ur4`1

rp-2+2vII0.,wll2.u.rt

0 such that for all u > 0 and w E H!") (B'(") (1)) satisfying H(u l w = 0 weakly on B'(") (1) one has IIV,,VIIj.u.r < crD

2+2vIIV

wIlj.u.l

(IV.61)

for all r E (0, ro].

Proof Let ro, uo be as in Lemma IV.6.6. Let u > 0, r E (0, ro] and w E H. r(B'(")(1)) satisfying Hlulw = 0 weakly on B'(")(1). If u > r-luo, then (IV.61) is valid with c = ra D+2-2v by Lemma IV.6.6. So we may assume that u < r- Iuo. Let CDG be the De Giorgi constant for the operator Hl"ol associated with the order v (see Proposition II. 10.2). Set o r_10. Then Hluol* = 0 weakly on B'("o)(uuo 1) and ,uo.p

(p1R)D-2+2v11ouoJr112.uo.tt

CDG

for all 0 < p < R < (uuo I A 1). Therefore Ilo;,wll2 u,._'uap < CDG

(p/R)D-2+2vIIV

Iiz.u.u-,uoR

for all 0 0, R E (0, 1], g E G. and rp E I(B'(u)(g; R)) satisfying Hlulcp = 0 weakly on B'(") (g ; R) one has 1IoMW112.u.B'" (g;r)

- CDG

(r1R)D-2+2v IIouwII2

u.a (6) (g; R)

for all 0 < r < R. Proof

Let ro, c be as in Lemma IV.6.7. Let R E (0, 1], u > 0 and (P E

H2(1)(B'(u)(R)) satisfying HIuIV = 0 weakly on B'(") (R). Set >y = W o TR. Then

E H^(uR)(B'(uR)(I)) and HIURI* = 0 weakly on B'(u)(1). Therefore one has cpD-2+2vIIVVR*02,uR.1 for all p E (0, ro], by Lemma IV.6.7. Then p IIVLR*I12,UR,p
0 and g E G, where I I' is the modulus associated with the algebraic basis and V' is the corresponding volume. Proof Proposition IV.6.9 gives the result under three additional restrictions, simply connectedness, D = D', and a stratified nilshadow. The general result follows from the special case by an extension of the techniques described in Section 11.9 for nilpotent groups. The aim is to embed G in a larger, simply connected, Lie

group Go with stratified nilshadow and with D = D' and also to extend H to a subelliptic operator H on Go. The construction is arranged such that Gaussian bounds on the kernel can be deduced by projection of the Gaussian bounds for the kernel K of H on Go. The latter follows from Proposition IV.6.9. By Proposition 111.5.1 there exists a Lie algebra b of type R with a stratified nilshadow together with a surjective homomorphism ir: g -+ g. Then there exist a Levi subalgebra m of p, a vector space 6 satisfying Properties I-III of Proposition 111. 1. 1, and a stratification {f)k) for the nilshadow qN of the radical q of g such that the stratification (6k) is compatible with m and 6. Moreover, there exist d" > d' and 61, ... , ad"" E g such that iii, ad" is an algebraic basis for g, d"). Next rr(ai) =ai for all i E {1,...,d'}andir(ai) =0foralli E fd'+ 1, we consider the dimensions. Let G be the connected, simply connected, Lie group with Lie algebra g. Then G has polynomial growth since g is of type R. Let D be its dimension at infinity and D' the local dimension corresponding to the algebraic basis ai 5 1 ,.- ads-. Since in general D i4 D' we introduce a connected, simply connected. Lie group

172

IV. Homogenization and Kernel Bounds

Go with similar properties to G, and an algebraic basis, such that on Go the local dimension equals the dimension at infinity. The basic idea is to define Go = G x G' where G' is a Lie group chosen to balance the dimensions. The balance can be achieved for any pair of D, D' by constructing G' as a multiple of a group with corresponding dimensions which differ by 1, -1 or 0.

First, if D' > D, set Go = G X

(H3)D'-D, where H3 is the connected simply

connected 3-dimensional Heisenberg group (see Example 11.4.16). Then Go is simply connected and has polynomial growth. We identify the Lie algebras g and 4D'-D with subspaces of the Lie algebra go of Go as usual, where h is the Lie algebra of H3. If bt , b2, b3 is a vector space basis for 4. then the local dimension is

3 and the4D,-D, dimension at infinity is 4. Therefore if ad"+i, ... , adis a vector space then a 1 , ... , adis an algebraic basis for go with local dimension basis f o r

and dimension at infinity equal to D' + 3(D' - D). The radical of go equals q x !)D'-D and m is a Levi subalgebra for go. The subspace u satisfies Properties IIII of Proposition 111.1.1 with respect to m. Then the corresponding nilshadow of the radical of go equals qN x 1D'-D The latter Lie algebra is stratified in

the natural way since both qN and h are stratified. Moreover, the stratification is compatible with nt and n. Finally, define no: go --* g by Jro(a, b) = ir(a) for all (a, b) E x 11D'-D. Then no is a surjective homomorphism, iro(a;) = a; for all

i E {1,...,d')and no(a;)=0for all i E {d'+1,...,d"'}. Secondly, if D' < D, set Go = G x (E3 )D-D" where E3 is the connected simply connected 3-dimensional covering group of the Euclidean motion group (see Example 11.4.18). Then Go is simply connected and has polynomial growth. Again we identify the Lie algebras g and (e3)D-D' with subspaces of the Lie algebra go of Go as usual, where e3 is the Lie algebra of E3. If b1, b2, b3 is a basis for e3 satisfying the commutation relations [b1, b2] = b3, [b1, b3] = -b2 and [b2, b3] = 0, then b1, b2 is an algebraic basis for e3 with local dimension 4. The dimension at infinity of E3 equals 3. Moreover, span(bi) is a subspace of e3 satisfying Properties I-III of Proposition 111.1.1. In each of the D - D' copies of e3 one has such a basis, of which we denote the k-th copy by bik), b2 ), b3k) Now define ad"+2k-I = and ad""+2k = b2k) for all k E (1, ... , D - D') and

set d"' = d" + 2(D - D'). It follows that a1, ... , adis an algebraic basis for go with local dimension and dimension at infinity equal to D' + 4(D - D'). The (e3)D_D,

and m is a Levi subalgebra for go. The subspace radical of go equals q x b(D-D'l) satisfies Properties I-III of Proposition 111. 1.1 no = n ® span(b('), .. , with respect to m. Then the corresponding nilshadow of the radical of go equals qN x (R3)D-D' since (e3)N = R3. The nilshadow is again stratified in the natural way and the stratification is compatible with m and n. Finally, define the surjective

homomorphism Jro: go -- g by no(a, b) = n(a) for all (a, b) E g x Again ,ro(a;) = ai for all i E {1, ... , d'} and ,ro(a;) = 0 for all i E (d' + (e3)D-D'

1,,d"'

Thirdly, if D' = D, set Go=G. go=B.Jro=irandd"'=d". So in any case there exists a connected, simply connected Lie group Go of

polynomial growth, an algebraic basis ai 5 1 ,-

ad,,, of go and a surjective homo-

IV.7 Kernel bounds; general case

173

morphism,ro: go -+ g such that go has a stratified nilshadow. the local dimension

and the dimension at infinity are equal, :ro(ai) = ai for all i E (1, ... , d') and

tro(ai) = 0 for all i E (d' + 1....,d"'). Let A: Go --), G be the natural homomorphism. For all i E { 1, ... , d"'} let Ai = d LGo (ai) denote the infinitesimal generators on Go. Set

Aj. Then N is a subelliptic operator. Moreover.

H(tpoA)=(H(p)cA for all

E C.,(G). Therefore

Kr(A(8)) = J

dlt K,(glt)

ker A

for all g E Go by (11.40). By Proposition IV.6.9, the kernel k satisfies good Gaussian upper bounds. Hence by transference, Corollary 11.8.6, the complex kernel K on G also satisfies good Gaussian bounds, i.e., there exist b. c > 0 such that

I Kt(g)I < c

V'(t)-1122

e-b(1g1')2t--'

uniformly for all t > 0 and g E G. As a corollary of Theorem IV.7.1 we deduce that the semigroup S is uniformly bounded on all the LP-spaces. For real symmetric operators this is an almost obvious result that follows from the Beurling-Deny criteria, but for complex operators it requires much more detailed argument. The uniform boundedness property is

a key element of the theory of complex elliptic equations or of a more general theory of systems of equations. There appears to be no general reasoning which yields the boundedness in the complex case and avoids detailed analysis.

Corollary IV.7.2 Let S be the semigroup generated by the closure of a pure second-order complex subelliptic operator with constant coefficients on a Lie group with polynomial growth. Then S is uniformly bounded on L. uniformly for all p E [ 1, oc J. Hence there exists a c > 0 such that

IIStlip-p 0.

Proof This follow by integrating the kernel bounds of Theorem IV.7. 1.

In the special case of a real symmetric operator, i.e., an H with the matrix of coefficients real symmetric, the semigroup kernel K satisfies Gaussian lower

174

IV. Homogenization and Kernel Bounds

bounds (11.25). Such bounds cannot be expected for complex operators since the kernel will also be complex. If, however, the operator is self-adjoint, but not necessarily real, then one can deduce that the real part of the kernel is positive within a cone {g E G : IgI' < Kt112}. This is established by a two step argument. The first step is a corollary of the following proposition and the second step will be given in Corollary V.2.13. V'(t)-1/2 Proposition IV.7.3 There exists a c > 0 such that (Kr * Kt)(e) > c for all t > 0, where Kt is the kernel of the semigroup generated by the adjoint H` of H.

Proof First, define Lr = Kr * Kl K. Since Kl (g) = Kr(g-') one has

Lt(g) = JG dh K,(h)K1(gh)

JG dh I K(h)I2 = Lr(e)

f or all g E G and t > 0. Secondly, since H is a pure second-order operator

fdg K,(g) = 1 = f Kr (g). Therefore f dg L1(g) = 1. Thirdly, K and Kt both satisfy the Gaussian bounds of Theorem IV7.1. and it follows from the next lemma that L is also bounded by a Gaussian.

Lemma IV.7.4 For each b > 0 there exists b', c > 0 such that Gb,r * Gb,s . c Gb',s+r for all t, s > 0 where Gb,r(g) = V'(t)-1/2e-b(I5I')2r-1

Proof Let K/ denote the kernel of the sublaplacian - _k=1 A. Then there are Ch c2, bi, b2 such that c1 Gb, r < Kr < c2 Gb2,1 for all t > 0 by (11.25). The result follows immediately since KL is a convolution semigroup. The proof of the proposition now follows by combination of these observations. One concludes that

L1(e) > V'(Kt 1/2)-1 (1

-f

dg L,(g))

g :lgl'>-Kr'I2)

V'(Kt 1/2)-1 (1 - j

dg c

V'(t)-1/2e-b(Isl')2r-')

g :lgl'>Kt'/2)

for a suitable choice of b, c > 0 uniformly for all K, t > 0. But the last integral tends to zero as K -+ oo. Therefore, there is K such that L, (e) > 2-' V'(Kt1/2)-' for all t > 0. Finally, since the group has polynomial growth one must have a bound V'(Kt 1/2) < k VI(t)112 for all t > 0. Corollary IV.7.5 If the subelliptic operator H is self-adjoins, then there is a c >

0 such that K,(e) > c V'(t)-'/'for all t > 0. This follows immediately since, by assumption, K = Kt and hence Lr = K2,.

Notes and remarks

175

Notes and Remarks The Rd-theory of homogenization is described at length in the book by Bensoussan, Lions and Papanicolaou [BLP] or, alternatively, in the book by Zhikov, Kozlov and Oleinik [ZKO]. Alexopoulos introduced homogenization theory to the asymptotic analysis of semigroup kernels on Lie groups of polynomial growth. In [Ale2] he considered the asymptotic behaviour of the semigroup kernel associated with a sublaplacian on a solvable Lie group of polynomial growth. Subsequently, [Alel] he generalized his results to general Lie groups of polynomial growth. An extended description of these results is given in [Ale3]. The last reference also examines sublaplacians with a class of drift terms which do not affect the asymptotic behaviour. Our description and application of homogenization theory resembles that of Alexopoulos in broad outline but differs from his approach in much of the detail.

Section IV,1 The reformulation of the subelliptic operator H as a second-order operator corresponding to a basis of g passing through the spectral subspaces m, u, to, ti, i 1, ... , 11, is similar to that of Alexopoulos [Ale 1 ]. But Alexopoulos identifies QN with a direct product X of copies of R and T by use of exponential coordinates of the second kind. Explicitly, he introduces a unitary map

T: L2(QN) - L2(X) by (T*)(x) = >G((P(x)) with

4(x)

=expQN(xibi)...expQN(xdbd).

Therefore H is viewed as an operator on M x X and the coefficients c;j are bounded functions which are trigonometric polynomials in the first do-variables. (In fact Alexopoulos considers left invariant operators as opposed to our right invariant operators and so there is some change of ordering that is necessary in any comparisons with the present description.)

Section IV.2 It is not essential to assume that the nilshadow qN of the radical q is stratified. The detailed arguments are independent of this assumption [EIR10]. But in the general situation the linear maps yu: ON -+ ON are not automorphisms of ON.

Section IV.3 The corrector equation (IV. 18) and homogenized operator (IV.24) are the direct analogues of the corresponding quantities occurring for strongly elliptic, periodic, operators in the standard Rd-theory (see [BLP], Chapter 1, Section 2, or [ZKO], Chapter 1). This analogy can be pursued by the choice of coordinates as above. The situation is particularly simple in the case of a solvable Lie group of polynomial growth [Ale2]. Then the coefficients c;j of H only depend on the first do-variables. Therefore the right-hand side of the corrector equation (IV.18), and the correctors, are functions of these variables. But H, restricted to functions of x1 , ... , xdo, is an operator with constant coefficients. Hence (IV.18) is an elliptic differential equation soluble by elementary considerations. This strategy is less straightforward in the general case since the coefficients then depend on the Levi subgroup M. Thus the correctors depend on M and the coordinates xj, ... , xdo. Moreover, the restriction of H now contains derivatives in the m-

176

IV. Homogenization and Kernel Bounds

directions. Therefore the existence of solutions to (IV. 18) requires more sophisticated arguments. The identification of the correctors with elements of the Lie algebra g is new.

The tactic of transference of properties of the representation (G, S, g) resolves the problems of existence, uniqueness, etc. of the correctors without reference to arguments of partial differential equations. The representation (IV.28) of the homogenized matrix is a direct extension of (2.24) in [BLP] and (IV.30) is a generalization of (1.33) in [ZKO]. Propositions IV.4.1 and IV.4.2 are then extensions of the comparable Rd results (see, for example, [BLP], Chapter 1, Section 3). The observation, Corollary IV4.7, that the homogenized operator H is right invariant with respect to G appears to be new. The comparison (IV37) of Proposition IV.5.1 originates with the idea of correction terms in the estimation of convergence described in Chapter 1, Sections 2.4 and 5.1, of [BLP]. Similar considerations appear in [ZKO], Chapter 1, Section 1.4.

The basic tactic to prove that the solution uE of a scaled equation HEUr; = f converges to the solution u of the homo enized equation Hu = f is to consider the difference zE = uE - u + e Ei=l B ji. But formally one has HEz, _

Hu - HEi + EHE 8ju. Therefore, with H = Hi and z = zi, one has (Hu - H)u = -H Ed_t 8j4 + Hz and Proposition IV.5.1 is seen to give a quantative expression for the remainder Hz. The use of correction terms, both with first-order and higher-order correctors, is a common procedure. In particular it is a basic technique for Avellaneda and Lin [AvL 11. Alexopoulos has formulated this method in a manner similar to ours, e.g., in Lemma 16.5.1 of [Ale3].

Section IV.5 The equivalence of strong resolvent convergence and strong convergence of the corresponding semigroups is called the Trotter-Kato theorem. It was first proved by Trotter [Tro] and the proof was subsequently clarified by Kato [Katl]. The theorem depends on the assumption that the limit of the resolvents is the resolvent of a semigroup generator or that the limit of the semigroups is a continuous semigroup. Proposition IV.5.3 and Corollary IV.5.4 give an example where these conditions are only satisfied on a subspace, the subspace PL2(GN). A general version of the Trotter-Kato theorem which covers semigroups on Banach space having only weak continuity properties can be found in [BrR], Theorem 3.1.26. The convergence result Proposition IV.5.6 is a version of the standard statement of homogenization theory, [BLP], Chapter 1, Theorem 3.1. or [ZKO], Chapter 1, Theorem 1.4. The proof is modeled on the adjoint argument of [BLP], Chapter 1, Section 3.2. The main technical differences in the Lie group version are the mixed Dirichlet-Neumann conditions explained after the proposition and the domain complications caused by subellipticity. Since H is subelliptic, the form domain H;(")(S2) of Hl,,) can vary with u. In the standard theory of strongly elliptic operators this problem does not arise since the corresponding form domain H2; I (S) is basis-independent. Section IV,6 Most of this section comes from [EIR10], where also the large time Gaussian kernel bounds of Theorem IV.7.1 have been proved. The Neumann-type

Notes and remarks

177

Poincar6 inequality (IV.57) which is the basis of the proof of Proposition IV.6.2 was established by Saloff-Coste and Stroock, [SaS], (P.1). The general strategy that we have followed to prove De Giorgi estimates is based on the ideas of Giaquinta [Gia2]. Avellaneda and Lin [AvL1] [AvL2] combined these methods with homogenization theory to obtain De Giorgi estimates uniform in a scaling parameter. The proof of Proposition IV.6.8 follows their arguments. The equivalence of De Giorgi estimates, formulated as a Dirichlet condition, and Gaussian bounds was proved by Auscher [Aus] for divergence form operators on Rd with uniformly continuous coefficients. These ideas were then extended to complex subelliptic operators on Lie groups in [EIR8] to prove Gaussian bounds. Auscher's theorem gives a precise statement for the equivalence of elliptic and parabolic estimates which has been a perennial part of the folklore of partial differential equations. Section IV.7 The idea of augmenting the dimension by tensoring has been used to analyze low dimensional problems on Rd (see, for example, [Varo2], Section 15.4 and [Dav2], page 121). The proof of local lower bounds follows an argument of Varopoulos [Varo2] which was developed more fully in [EIR7].

V Global Derivatives

In the previous chapter we applied techniques of homogenization theory to derive global Gaussian bounds on the subelliptic semigroup kernel K. Once these bounds are established, one can use quite different techniques based on L2-estimates to obtain global bounds on the derivatives of K. The nature of these bounds is sensitive to the direction of the derivatives and, in particular, the local and global singularities are usually quite different. The differences reflect the global geometry of the group. The kernel K is infinitely-often differentiable and its subelliptic derivatives satisfy canonical Gaussian bounds for small t. Specifically, one has bounds

I(A'K,)(g)I

ct-Ia1/2V'(t)-I/2ewte-b(IrI')2t-1

(V.1)

for all g E G, t > 0 and multi-indices ct by Proposition 11.6.2. These bounds have the optimal singularity as t -i 0. Moreover, since these derivatives are in all directions of the algebraic basis al, ... , ads one can estimate the behaviour of multiple derivatives in general directions. Although the bounds (V.1) give the correct behaviour for small t, they give no essential information about the global behaviour. The dominant factor for large t is the exponential ewt which corresponds to the semigroup property. The global behaviour of derivatives is much more involved than the local behaviour even for real symmetric operators. If the kernel K of the subelliptic operator H satisfies global Gaussian bounds, then the global behaviour of derivatives in the subelliptic directions is given by I(AkKt)(g)I lal. For all a = (k1, ... , k,) E J(g), define 8a, = 1 if there exists an i E (1, ... , n) with ki E {-d ..... 0}; otherwise

set Sa, = 0. So 8a, = 1 if and only if a has an index in the s-direction.

Proposition V.1.1 If G is simply connected and its Lie algebra has a stratified nilshadow with stratification then there exists an m > 0 and, for all a E J (g), a c > 0 such that II B° Sr 112-.2 1. The moral of this proposition is that any derivative in an s-direction automatically gives an exponential decay. The next proposition gives a detailed estimate if all the derivatives are in the direction of the nilradical n.

182

V. Global Derivatives

Proposition V.1.2 If G is simply connected and its Lie algebra has a stratified nilshadow with stratification {1), }, then for all a E J(n) there exists a c > 0 such that IIB°`St112-.2 _< ct-II«II/2

(V.3)

for all t > 1. The propositions demonstrate that the asymptotic behaviour of the derivatives of the semigroup is dependent on the direction of the derivatives. The exponential decrease for derivatives in the s-directions is directly related to the compactness of the group with Lie algebra s. It reflects a spectral property, a spectral gap, and the value of the exponent w is related to the gap. The asymptotic behaviour is analogous to that of compact groups described by Proposition 11.7. 1. Secondly, each derivative in an n-direction gives at least a t-112-decrease. Multiple derivatives B° with a E J(n) generally have a faster decrease property. The rate of decrease is determined by the weighted length Ilarll of a and not simply the number lal of derivatives. It is also possible to establish L2-bounds on derivatives in other directions. If R is a general right invariant differential operator without constant term, i.e., if R E span{BY : y E J+(g)}, then one has bounds IIRS1112-.2 1. Moreover, there are R for which these estimates are optimal, i.e., there are multi-derivatives which have the t-1/2-decrease. These estimates, together with versions of (V.2) and (V.3) for general groups of polynomial growth, will be a consequence of the Gaussian bounds we derive in Section V.2. The optimality of the bounds will be established in Section V.3. Example V.1.3 Let H be a complex, second-order, subelliptic operator on a compact Lie group G. The spectrum of H on each of the Lp-spaces L p(G) is a countable discrete subset of the open right half-plane with a possible accumulation point at infinity. Each point in the spectrum corresponds to an eigenvalue with finite multiplicity. Moreover, zero is a simple eigenvalue and the corresponding eigenfunctions are constant. The spectrum and the eigenspaces are independent of the value of p E [1, 00). If S is the semigroup generated by H, then each S, is a Hilbert-Schmidt operator on L2(G) and the Hilbert-Schmidt norm I l IS, l s , 1112 satisfies IIISt1112 = TIL2(St St) = IIKt112

for all t > 0. Moreover, if P is the eigenprojection corresponding to the zero eigenvalue,

ct,° = inf(Re(rp, Hcp) : rp E D(H), II(1 - P)tpll2 = 1)

and s,(°)=(I-P)St=S,(1-P), then II4°)1122' c-"0t for all t > 0. Therefore IIISt°)1112 5 11S(0-)E)1 112-.2111 S,, 1112

5 IIS(l 6),112-2111Set1112 5 e-410(t-6)t11Kft112

VI L2-bounds

183

for all e E (0, 1) and i > 0. Since IIKet 112 5 a V'(er)-1 /4 for some a > 0 it follows that there is a b > 0 such that I 5 111St 1112 5 1 + b

e-D'/4e-ap(1-e)t

for all e E (0, 11 and t > 0. Similarly, 11 1 A*St 1112 5 IIA'S1112-2111St-11112 for all t >I and

the Hilbert-Schmidt norms of the derivatives are also bounded by a multiple of

e-",0(1-e)r

ast - 0o. Propositions V.1.1 and V.1.2 give bounds for multiple left derivatives Bi = d LG (bi) but one can also estimate the asymptotic behaviour of left derivatives Bi = dLGN(bi) with respect to the product group GN = M X QN. In fact the family of estimates (V.3) is equivalent to a similar family for the Bi. The equivalence is of practical use in establishing the validity of the estimates and is

also useful for discussing the behaviour of multiple derivatives in a general order. Let S: GN -+ Aut(gN) be the homomorphism associated with the homomorphism S as in (111.44). The passage from the G-derivatives to the GN-derivatives is given by (IV.3). First, it follows that d

Bi = E Sij BJ

(V.5)

j=-dm

for all i E {-d. ... . d), where Sij denotes the operator of multiplication by the matrix element Sij E E given by Sij(g) = (S(g)bj, bi). Conversely, d

Bi = L Sj1Bj. j =-dm

But S(g)b j = b j if j < do and Sij = Sij if i < do or j < do. Therefore Bi = Bi if i < do, i.e., the G-derivatives and the GN-derivatives coincide in the m and 0 directions. In particular, if a E J (s), then B" = Ba. _ Secondly, if i > do+ 1, then Sij = 0 for j < do since S(g)n = n. Therefore the G-derivatives in the n-directions are linear combinations of the GN-derivatives in the n-directions and conversely. But one also has S(g)hk = hk by (111.46). Hence the transformation between the derivatives respects the weighting. Thus

Bi =

S,, R j

and

Bi

=

jE(do+l.....d)

S ji B, . jE(do+1.....d) Wj=Wj

WJ=Wj

Next, the functions Sij are constant in the n-directions and BkSij = 0 = BkSij for all k > do + 1. Therefore the relations for the mufti-derivatives in the ndirections simplify. If a E J(n), then Ba =

Sa,y By yEJ(n)

and

Ba =

Sy,a By yEJ(n)

IY1=1a1

IYI=la1

IIYII=IIaII

IIY11=11aII

(V.6)

184

V. Global Derivatives

with Say E eIal. In particular the family of estimates II B'S, 1122 < C. t-I1all/2 for a E J(n) and t > 1 is equivalent to the family of estimates II k' St 112.2 cat - Ila II /2 for a E J (n) and t > 1. In the proof of Proposition V.1.2 we will use these various equivalences to simplify the arguments. The rest of this section is devoted to the proof of the propositions. The proof will be divided into the two distinct components, s-derivatives and n-derivatives.

V1.1

Compact derivatives

We first argue that one can separate the discussion of the compact derivatives, i.e., the derivatives in the s-directions from the other derivatives in Proposition V.1.1.

Lemma V.1.4 If a E J (s) and 0 E J (g), then there exists a c > 0 such that IIBrBflS1II22 < c esup JIB'S,/21122 IIBOS,/2II22 OE.t(S)

Ial=lal

forallt>0. Proof

Since s is semisimple and of type R it follows from Statements I and III of Proposition 11.4.8 that there exists a positive-definite invariant symmetric bilinear form (. , ) on s. Let c_d, . ... . co E s be an orthonormal basis for s with Ck, then the invariance respect to the inner product ( , ). If [c;, c/ F_kO=_ds

implies that cv = for all i, j, k E {-d ..... 0). Set Cj = dLG(CJ) for all j E (-d..... 0) and introduce the Casimir operator 0

C"

A

j=-ds

Then [C;, A] = 0 for all i E (-d..... 0) since c = -c. Hence one has [dLG(a), 0] = 0 for all a E s. But also [dLG(a), 0] = 0 for all a E q since 81

[s, q] = 0. Moreover, by II.1.3 there exists an ideal sl of m such that m = s ®s-L

and [s, s-L] = (0). Then, in addition, [dLG(a), 0] = 0 for all a E sl. Since g = s ®sl ® q, it follows that 0 commutes with all right-invariant vector fields

on G. In particular 0 commutes with B' for all P E J(n), and with S, for all t > 0. But IIo'11(pll2 = >j=-d, IICjw112. Hence there exists a c > 0 such that IIBjV 112 < c IIO1/2W112 for all rp E L2;00 and j E {-d ..... 0). Since 0 commutes with each of the B j this estimate may be iterated to give IlBawll2 O and ro E 1)(W). III.

There exist N E N and v > 0 such that IIHN(P11

> vN IIw11

for all rP E D(HN).

Proof I=II. It follows by integration of S that H r(,)-1

II H-z 11 0 such that 11H N(P11 < en lI HN+n(PI1 -i- CnE-N Ilwll

(V.9)

for all V E D(HN+") and E > 0 as a consequence of 11.2.1. I fence it follows from Condition III that IIHN+npll

(VN -Cne-N)E-nlI(Pll

for all V E D(HN+") and all e > 0. Therefore there is a K > 0 such that IIHN+ncPll

> KN11VII

for all n E {0, ... , N - 1) and ' E D(H N+" ). Another straightforward application of (V.9) leads to the further conclusion that there are a, r > 0 such that

((Al -

aNll,pll

V.1 L2-bounds

187

forallnE{0,...,N-1),,pED(HN+n)and,AECwith l,?I 4-1a sin 9. It follows from the integral representation that one has bounds II S, II < M e-` for alt t > 1 with w = 4-1 a sin 9. As S is uniformly

bounded, these bounds extend to all t > 0 with an enlarged value for M, i.e.. Condition I is satisfied. This completes the proof of Lemma V.1.7.

0

We continue with the proof of Lemma V.1.5.

In order to apply Lemma V.1.7 we note that if a E J+(s), then B°S, = B°(I -

Ps)Sr = B°S1(l - Ps)S,-1 and IIBG'Slll2-.211(1- PP)S,-11122

for all t > 1. Therefore the bounds (V.7) follow if t H 11(l - Pa)S,Il2. ,2 = IIS,(l - Ps)112.2 is exponentially decreasing. But this is a direct corollary of Lemma V.1.7 if there are N E N and v > 0 such that

-Ps*112?,N11(/-Pa)cV112 for all W E D(H N ). This can, however, be established by the following argument.

188

V. Global Derivatives

First, consider 0 = - 1:0=_ds B acting on L2(G). Further consider the opwith dLG1(Bj), acting on L2(G5). Since erator 05 = G5 is compact and connected, the operator A. has a compact resolvent, and there is a ), > 0 such that 05 > Al on the orthogonal complement of the constant func-

tions in L2(G5). Let W E (I - P5)C°O(G) c C°0(G) and define Vg E C°°(G5) (Bjcp)(sg), by cog(s) = w(sg) for all g E G, S E G5. Observe that and V. is orthogonal to the constants because PSrp = 0. Since G/G5 is a group it follows from the Weil formula (11.38) that there exists a normalization of the Haar

measure dh on G/G5 such that

f dg Vi(g) = J for all f E

dh Vb°(h)

GIGe

G

where *b

E

C, (G/G5) is defined by vrb(gG5) _

fG, ds >/r(sg). Then 0

0

E (IBjcI2)b(gG5) = E f ds I(B;w)(Sg)I2 s

l=-ds

1=-ds

0

f ds I(BP5)w8)(S)I2 ,

1=-ds

f.

ds IV,(S)12 = A (1W12)b(gG5)

for all g E G. Therefore

E IIB;w112 = I =-ds

1=-ds

f

/G.

dh (IB;w 12)b(hh)

f /G. dh(I412)b(h)IIVI12 for all rp E (I - P5)C°O(G). Next, since al, ... , ad' is an algebraic basis, each Bj can be expressed as a polynomial in the Ak. The lowest order term in these polynomials is at least one and the highest order term at most r,,, the rank of the algebraic basis a I, ... , ad'. Therefore there exists a ,l1 > 0 such that sup IIAawll2 aEJ(d')

- Al IIwI12

(V.11)

lal n, there is a c > 0 such that sup IIA(VI12 _< CE-2N+1 1IHN12WII2 +E I4112 aEJ(d') lal=n

(V.12)

for all cp E D(HN12) and all E E (0, 1].

Proof Set

Mn,, = sup

IIAaSIVI12

aEJ(d') Ia l 0, for a suitable c > 0. But one also has M2._1,,

H)(2n-1)12Sr(VII2

cl 11(1 +

< C2 (IISrWI12 + II H(2i-I)/2SrwII2) < C3 (1 + t-n+I/2)IIOII2

(V.14)

for all t > 0. The first inequality follows from the second statement of Proposition 11.6.2, the second by (II.10), and the third follows from the holomorphy estimates 11.2.3 since S is contractive on L2. Then combining (V.13) and (V.14) gives

+ t-nl2)II0I2

C4(t-1/4

sup II AaS, p112 < dal=n

for all t > 0. Next, by the usual Laplace transform algorithm for the resolvent, for all N > n, there exists a cs > 0 such that 00

d t t-I a-£4` tN/2II AaS,V/112

IIAa(H + E41)-N/2*112 < I'(N/2)-1 fo 0 such that II Ba Sr 112-.2 < II BQS1112-2 for all r > 2. Hence if co > 0 is as in IIB°`S1112-.21ISr-1112-.2 Lemma V.1.5 then it follows from Lemma V.1.4 that for all a E J (s) and .8 E g there exists a c > 0 such that 1. But s is an ideal in g. Therefore Proposition V.I.1 follows.

V1.2

Nilpotent derivatives

Next we establish the bounds of Proposition V. 1.2 for derivatives of the semigroup in the directions of the nilradical n. At this point we explicitly assume that the nilshadow Q;y of the radical Q is stratified. This assumption was of no consequence in the earlier arguments, but it now allows us to use scaling arguments.

It follows from (V.6) that (V.3) is equivalent to proving that for all at E J(n) there exists a c > 0 such that II

c1-tnu%2

(V.15)

V.1 L,-bounds

191

for all t > 1, for the G,v-derivatives B;. Since the nilshadow Q.v is stratified we can use scaling arguments. These arguments are particularly simple on L2 because the dilations are implemented by operators r,,: L2 --> L2 defined by

rurP= 0 and gP E L2(G) = L2(G,v). Then one has

ruAkru = dLG.(y. lak) = 1

u-lA[ku]

for all k E (1, ... , d') by (IV.12). Hence I uS,1'u 1 = Sl°1_, for all r > 0. Moreover, r B J I'u = u -Wr Bj for j E (1, ..., d). Therefore 1

IIBaSrI12--2 = I1ru(BaSr)ru l 112-.2 = u-fiailIIB°Su`-lz,ll2.+2

for all u > 0. Hence setting u = r 1 /2 one has

L IIBaAkSrli2-.2 d'

IIBaSddl2-.2 +r1/2

k=1

d'

= r-laC/2(IIBaS"

118'A1,1111S,,.121112-.2)

k=1

for all t > 0. Consequently, we have reduced the proof of the bounds (V.15), for all t > 1, to establishing that, for all a E J(n), there exists a c{a > 0 such that IIBaS;ul112-.2 1. The proof requires some preparation. Consider the unitary representation U of the Lie group N in L2(GN) defined by U(n) = LGN (n). Form E No define the space

Xm = n D(Ba) aEJm(n)

with norm 111(pHlm = max

aEJm(n)

where 11

112

IIBa(P112,

is the L2-norm on L2(GN). So X. is the Banach space of m-

times differential vectors with respect to the representation U with the usual norm with respect to the vector space basis bd,+1 + I ,.. bd for n. Define the seminorm Nm: Xm -+ R by

Nm(rp) = max IlBawll2. aEJ(n) 'a!=m

Next we need bounds on Nm((XI +

Nlul)-1(o).

192

V. Global Derivatives

Lemma V.1.10 For all in E N there exist X10 > 0 and c > 0 such that Nm((AI + Hlu1)-1(p) < ck-1 Nm(cp)

forallA>A0,u> 1andrpE L2;oo (GN). Proof First Akul = Ed=-dm U I - W; rkJ 1 Bj with rkl) = rkj o 1'u and rkj (g) _ (S(g)bj, ak). But Birkj = 0 for all i > do since the rkj do not depend on the nilpotent directions. Therefore one deduces that there exists a c > 0 such that II[Ba, Ak"]]wll2 1, k > 0, V E L2;W(GN) and write >G = (Al + Hlul)-'gyp. For all a E J(n) with IaI = in one has IIB°`k112 G)

< Nm(i(/) N.(,p) < 2-'(N. (*)2 + N. (V)' and

I Re( B">//, [H1"1, B"}*), d'

d'

< L Icijl . 1(9'*, [At"l,

+ L Icijl . l(B"k,

i.j=l

< c2 Nm(*)

max

jE(1.....d')

+c2

Aiu1[A'u1

i,j=l

Nm(A("1

)

J

max

iE(1.... ,d'I

Nm(A!"Itfi)Nm(1f/)+cc2Nm(>//)2

by anti symmetry of AM and an estimate on the commutator [Ai"1, B"[. Hence max

kE(t,....d')

Nm(A1k"1*)2 < µ-1 NmM2+4c2µ l Nm(*)

max

JE(1....,d')

+2(µ-' +c2+µ-tcc2)Nm(*)2. Therefore max kell.....d')

µ-112 Nm(W) + C3 Nm(iG)

where c3 that Nm(%(') < A-1(1 +c2 µ-I/2)Nm(cP) + C2(C3

So the lemma follows with Ap = 1 + 4c2(cl +C3 )2`. Now we are able to prove Proposition V.1.2.

with (V. 19) it follows

),-1

+ct A-112) N.('//). 11

V. Global Derivatives

194

Proof of Proposition V.1.2 Let M E N, m > 2. It follows from Corollary A.4.2 in Appendix A.4 and (11. 15) that there exist c > 0 and v' > 0 such that d'

11(1- LGNg)w112 -c(IgIGN)" ( Iw112 + L IIAku1w112) k=1

uniformly for all u > 1, w E L2;oo and g E GN, where I IGN is the modulus on GN with respect to the basis b_dm ..... bd. Therefore

II (1 - LGN(g))()1 + Hlul)-'WII2 :5 c (I expGN (-tbj)I" )'1` d'

(II(A/ + Hlul)-'(P112 + L

Hlu)) 'wI 2)

k=l

Hence

II(1-

Hlul)-1wII2 < C(1

uniformly for all w E L2(GN), n E N, ,k > 1 and u > 1, where I IN is the norm on N with respect to the basis bdo+1, ... , bd. So by Statement I of 11.12.2 and Statement III of 11. there exists a co > 0 such that II(Xl +

C0A-1/214112

(V.20)

for all u > 1, A > 1 and w E Xo, where we have introduced the real interpolation spaces with respect to the K-method of Peetre. By Lemma V.1.10 there exist cl > and Ao > 0 such that III(A1 + Hlu))_'wIIIm < cl )-' IIIwIIIm

for all A > A0, u > 1 and cp E L2;,O(GN). But L2;c (GN) is dense in X by Corollary 11.8.3. Hence if A = 1 v A0, then the map (Al + Hlu))-' is continuous from Xo into X0 with norm bounded by I and from into Xwith norm bounded by cl. Therefore, by interpolation (see Statement 11.12.1), for all y E (0, 1) the map (Al + H[u))-1 is continuous from (X0, Xm)y,2;K into (Xo, XX,)y,2;K with norm bounded by cl. But by (V.20) the map (Al + Hlu))-1 is also continuous from X0 into (X0, X,,,),,,2,K with norm bounded by co. Hence,

by interpolation, for all y e (0, 1) the map (Al + Hlu))-1 is continuous from (Xo, into (Xo, Xm)y+(I_y),,.2:K with norm bounded by co + cl. Using interpolation once more, it follows that there exists an N E N such that the map (Al + Hlu))-N is continuous from X() into (X0, X,,1)1_t,,,,> l ,',K with norm

bounded by (1 + co + cI)N. By Statement II of 11.12.2 one has the continuous embedding (X0, Hence there exists a c2 > 0 such that IIBa(pII2 < IIIWIIIm-I < C2 IIPII(xO,.vm),

,2m)-1.2:K

V.2 Gaussian bounds

195

for all ry E L2;oo(GN) and a E J(n) with Ial = m - 1. Then IIB°rP112 _5 C2(1 +CO+CI)NII().I + for all rp E

In particular,

IIB°Sju1112-2 1 such that II H Sr 112-.2 < c3 t -1 for all t > 0. Then IIH[u)S;j;,1I2-+2 = u211HSn-t 2112.2 -< C311 uniformly for all u > 0. Hence 118"Sj")II2_p2

< C4

for all u > 1, where C4 = c2C3(2(l + CO + c1),k)NN. Thus (V.16) has been

0

established and the proof of Proposition V.1.2 is complete.

It is not generally true that one has bounds IIB°'Sr112_2 -< ct-lal/2

(V.21)

for all t > 1 and for derivatives in arbitrary directions. These bounds can fail for two derivatives in the n-directions. Example V.1.11 The Lie algebra c3 of the three-dimensional group E3 of Euclidean motions has a basis (b1. b2, b3) with non-zero commutators [b1, b2l = b3, [bl. b3l = -b2. Choose

a second basis at = bl - b2, a2 = b2 and a3 = b3. If S is the semigroup generated by S1II2-2 = ('te)-I12, i.e., the bounds (V.21) fail for two derivatives in the u-directions. This is a reformulation of Example 11.5.2.

The bounds (V.21) can also fail for two derivatives in the m-directions (see Example V.3.5) or one derivative in an m-direction and one derivative in a 0direction (see Example V.3.6).

V.2

Gaussian bounds

In this section we turn to the derivation of large time Gaussian bounds on multiple

derivatives of the semigroup kernel. The main ingredients in the derivation of these bounds are the Gaussian bounds on the kernel given by Theorem IV.7.1,

196

V. Global Derivatives

and the large time L2-bounds of Propositions V.1.1 and V.1.2. Although the latter propositions only dealt with derivatives in the directions of the near-nilradical, we will establish bounds for general right-invariant differential operators. In order to formulate the bounds it is convenient to introduce a variation of the d) set wi = 0 if i < do and wi = wi if i > do. weighting. For all i E

Moreover, for all a = 01, ... , E J(9), set I la I l = w., + ... + wi.. It is also convenient to define the Gaussian function Gb,, on G, for all b, t > 0, by V'(t)-1/2 e-b(19 Gb.t(g) = 1

I

The principal result which covers all derivatives is given by the following.

Theorem V.2.1 There exist b, w > 0 such that for all multi-indices a E J (9) and i E {-d,,,, ( - d ,. .. , , do} there is a c > 0 such that I BaKtl
1. The theorem establishes Gaussian bounds with a characteristic asymptotic behaviour of the derivatives. The large-time behaviour reflects the L2-bounds given by (V.1.1), (V.1.2) and (V.4), but now we no longer assume that the nilshadow

is stratified. Any s-derivative gives an exponential decrease with large t and a derivative in the direction bi E n decreases as t-W!/2. Moreover, a general multiderivative leads to at least a t-I/2-decay. Estimates on compact groups and nilpotent groups, e.g., on Td and Re, show that the first bounds cannot be improved, and in the next section we will discuss the t-1/2-behaviour of general derivatives in greater detail. The Gaussian bounds can of course be used to strengthen the preliminary results of the previous section. One has, for example, IIBaK,111

1 and of E J (g) one has IIBaK3t11oo = IIBaS3:IUI-+°o

IIBaS21112-.oollS,III_2 = II Bu s, II2-.2II St II2-.oo Il St II2-oo

IIBaStll2-.2IIK,II211K, II2,

where Kt is the kernel of S. Then the corollary follows from the bounds of Propositions V.1.1 and V.1.2, the Gaussian bounds on K and Kt, together with the volume doubling property. The next lemma is the key to turning Lw-bounds on derivatives of the kernel and Gaussian bounds on the kernel itself into Gaussian bounds on the derivatives.

Lemma V.2.3 Suppose a E 9, t > 1, b, co, c2, c, S > 0, 4 E C°O(G), I4'(g)I

coGb,,(9)

for all g E G, V'(t)-I/2

IIdL(a)2(bI1o0 < c2

and Iexpua1' wi + wr, and one can argue as before. II

Lemma V.2.6 Let G be a (not necessarily simply connected) Lie group. Then for

all i, j E {-d,,,, ... , d} there exists a c > 0 such that

j C(l+(Igl')u,-'

W;)

if lLi ?_W'

I (bj, Ad(g-))br)I < 0

otherwise

for all g E G.

Proof Suppose that G is simply connected. Since G = M Q and M is compact it follows that Ad(M)b, is a bounded subset of m ® o or lbw, fl n if i < do or i > do, respectively. So it suffices to consider the case g E Q. By Example 111.7.2 there exists a E Rd such that g = exp 1;) b) ... exp dbd. Then e-a4do+Ibdal i o

Ad(g-') = o

Since S(v) is an orthogonal transformation, leaving m ® n, t, 1)2,..invariant, it suffices by Corollary 111.7.15 to show that there exists a c > 0 such that I(bj,e-a4dbd...e-agdo+I bdo4

c II

IIwf

if wj > W. otherwise

0

uniformly for all E Rd with 1. But this follows from Lemma V.2.5 by expanding the (terminating) power series of the exponentials of the nilpotent endomorphisms. 0 Finally, we drop the assumption that G is simply connected. If G is the covering group of G and A: G -* G is the natural map, then Ad(Ag) = Ad(g) for all g E G, where Ad denotes the Adjoint map on Therefore the bounds transfer El to G by Statement 11.4.4. G.

The next step is to establish simple Gaussian bounds on the multi-derivatives of the kernel, i.e., bounds with no additional decrease corresponding to the derivatives.

Lemma V.2.7 If G is simply connected and its Lie algebra has a stratified nilshadow with stratification (fj ), then for all a E J(g) there exists b, c > 0 such that

BK,l 1.

(V.26)

V. Global Derivatives

200

Proof First, IIB°Ktlloo = IIBa`Stllt_oo 1. Then there exist b', c' > 0 such that I Bi R Kt I < c' t

-a

t -_w /2 Gb',, for all t

> 1.

Proof For all j E {-dm, ... , d} define the function >//ij: G -+ R by >Gij(g) = LG(g)*ij(g)Bj for all g E G. (bj, Ad(g-1)bi). Then BiLG(g) Hence

d

BiE (1*ij)*Bjtp j=-dm

for all i E {-d,, ... , d) and tp, >/i E L 1;oo(G). In particular, d

('ij RKt) * BjK,

BiRK2t = Bi((RK,) * Kr) =

(V.27)

j=-dm

for all t > 0. By Proposition V.2.4 and Lemma V.2.7 there exist bi, ci > 0 such that I Bj Kr I < ct t-Wj/2Gb,,, for all t > 1 and j E {-dm, ... , d}. Moreover, by Lemma V.2.6, one can restrict the sum in (V.27) to j E (-dm, . . , d) with wj > wi. Suppose wj > wi. Then there exists a C2 > 0 such that IvIij(g)l c2(1 + (Igl')wj W') for alI g E G. Therefore, by the assumption on RK, one has .

I Vlij (g) (R Kt ) (g) 1

+(Igl't-1/2)wj-jyjt(wj-. )/2)t-6Gb,t(g)

cc2(1

< c c2 (1 + C3) t-3 tLj-w')/2 G2-lb,t(g)

e-r'nxz. Then for all g E G and t > 1, where C3 = supx>oxu'j-w' I (V,ij RK,) * Bj K1I < CC1 C2 (1 + C3) t-b t-_l2 G2-1b,t * Gb,.,


0 such that (V.28)

I Aa K1 I < c t-1/2 Gb,t

for all t > 1. But if these bounds are valid for one fixed a E J+(d'), then they are also valid for AkAa K, for all k E { 1, ... , d'). This follows from Lemma V.2.8 by writing Ak as a linear combination of B_d,...... Bd. Hence the bounds (V.28) are valid for all a E J1 (d') once one can prove they are valid for all a with IaI = 1. The existence of Gaussian bounds on subelliptic derivatives of the kernel is a classical result for real symmetric subelliptic operators. It was discussed in Section 11.5. The proof is by combination of the Gaussian bounds on the kernel, L2bounds on the subelliptic derivatives of the semigroup and a Davies perturbation argument. Specifically, one obtains bounds on the subelliptic derivative using exponentially weighted norms.

Lemma V.2.10 There exist b, c > 0 such that

I(AkKt)(g)I 5 ct-112Gb.t(9)

for all k E (1,...,d'), t > Oandg E G. Proof

For all p > 0 let UP denote the multiplication operator with action

(Up(p)(g) = ePlxl'(p(g). We first prove bounds on II UPAkK1 112. Let d'

W_ {,/r E CO0(G) : ,/i real, +/i(e) = 0 and sup) I(Ak>G)(g)I2 < 1). k=I

So D" = (* E Di

i/'(e) = 0), where D, is the set defined by (11. 16). Then let UP denote multiplication by ePL, where p > 0 and ' E D','. It follows from :

(11. 18) that

IIUPAkKr1I2 = sup IIUP AkKt112 VIEDI

V. Global Derivatives

202

Therefore in order to bound II Up Ak K, 112 it suffices to obtain an appropriate bound on IIUP Ak Kt 112 uniform in >G.

Secondly, IIUP AkK,II2 = II(Ak - (Akl)!)UPK,112 0 such that

Re(h - hp)(cp) < eRehp (,p) uniformly for all cp E D(h), P E R, >li E D, and E E (0, 1). Combining these estimates one deduces that there is a c > 0 such that d'

IIUPAkK,112 G E D". Thirdly, by the Gaussian bounds on K there exist c, co > 0 such that II UP K,112 < IIUpK,112 < c V'(t)-1,4

eVp2,

(V.29)

for all t > 0. But the Gaussian bounds on t H K, extend to the subsector A(0 _ {z : I argzl < 9} with 0 < 9c of the sector of holomorphy. Moreover, Up K, satisfies a Cauchy representation UPK,=(2ni)-I

1

Jr

dzUpKZ

z-t

where r is a circle of radius rt, with r < sin 0, centred at t. Using the identity HP UP K, _ -d (UP K,)/dt one then obtains bounds II H11 UP K,112 < rt

sup 11 Up Kz 112 < c t-I ZEr

V'(t)-I"aeu p2,

203

V.2 Gaussian bounds

for some c, w > 0, uniformly for all p, t > 0 and * E D". Finally, the combination of these bounds gives II Up A, Kt I12
0, uniformly for all p, t > 0 and * E D". Hence (V.30)

IIUpAkKr112 0. Next, dheplhl'l(AkK,12)(h)lep(Ixl'-Ihl')IK,12(h-'g)I

IIUpAkKtlloo < sup gEGJG < IIUpAkK,121121IUpKr/2112

< 21/2C2 t-1/2

for all p, t > 0, where the second step uses the triangle inequality for the modulus and the last step the bounds (V.29) and (V.30). Then, by the volume doubling property, V'(t) < c V'(t/2) for all t > 0. Hence there exists a c' > 0 such that

II UpAkKrll.
0 and p E [ 1, oo].

Corollary V.2.12 If G is simply connected and its Lie algebra has a stratified nilshadow with stratification (I)/}, then for all a E J(9) and i E (-dm, ... , do) there exist b, c > 0 such that I Ba Bi Kt I < c l-(IIIaI

for all t > 1.

II+1)/2 Gb,t

204

V. Global Derivatives

Proof This follows from Lemma V.2.10 and the discussion preceding it.

O

At this point we are prepared to prove Theorem V.2.1. Initially we continue to assume that G is simply connected and the nilshadow is stratified. Finally these assumptions are removed by transference.

Proof of Theorem V.2.1 Suppose that G is simply connected and its Lie algebra has a stratified nilshadow with stratification {l)k). Our first aim is to prove the Gaussian bounds (V.22) and (V.23) with a uniform parameter b. It follows from the Gaussian bounds of Theorem IV.7.1 that there exist co, cuo > 0 such that II Up K, 111 < coe'

''

for all p, t > 0. Then for each a E J (g) there exist, by Corollary V.2.9, b, c > 0 such that I Ba Kt I c c t-Illalll/2 Gb,r

for all t > 1. Hence there is a v > 0 such that II Up Ba Kt II oo < c t-Illalll/2 V'(t)-112 evn2t

for all p > 0 and t > 1. Alternatively, it follows from the local kernel bounds (11.29) that there are c', v', N > 0 such that c't-N V'(t)-112ev'(1+p2)t

II UpBaKtlloo
0. Set e = coo(v V v')-1. Using the inequalities II Up Ba Kt 11 oo < II Up Ba KE1 Il o0 II Up K(l -E)t 111 < co e(1-`)"° 2' II Up Ba KEt 1100

one deduces that II UpB"Kt 11 0o

V'(et)-1/2e("+(V-")E)p2t

cco(et)-Illalll/2 0 such that

I(BaK,)(g)I : c' e-2-ldtvr r-Illalll/2 G2_lb.,(g) for all t > I and g E G. Similarly one can prove that for all a E J(g) and i E (-dm ..... do) there exists a c' > 0 such that e_2-'Sara,, t-(Illalll)/2

J(BaBi Kr)(g)I < C

G2-ib.r(g)

for all t >_ 1 and g E G. This completes the proof of Theorem V.2.1 if G is simply connected and its Lie algebra has a stratified nilshadow with stratification (hk ). Finally, if G is merely connected and j, etc., are as in Proposition 111.5. 1, then

hi c_ qN;j = trgN; j by Corollary III.3.I7. Then the general case follows from

0

Corollary 11.8.6 as at the end of the proof of Theorem IV.7. 1.

The bounds on the subelliptic derivatives of the kernel allow us to complete the discussion of the lower bounds on kernels of self-adjoint operators started in Section IV.7. If His self-adjoint, then Corollary IV.7.5 establishes that the value of the corresponding kernel Kr is real at the identity and there is a c > 0 such that

K,(e) > c V'(t)-1 /2 for all t > 0. But

IKr(g) - K,(e)I : c IgI'

sup

ke(1.....d')

IIAkK,Iloo 5 c

Igl't-1/2

for all g E G and t > 0. Therefore one has the following. Corollary V.2.13 If the subelliptic operator is self-adjoins, then there are c, K >

0 such that Re K( (g) ? c V'(t)-1 /2 for all g E G and t > 0 such that Ig!' < Kt 1/2.

Proof The estimate preceding the corollary implies that there is a K > 0 such that Re K,(g) > 2-1 K,(e) if IgI' < K11/2. Then the corollary follows from Corollary 1V.7.5.

0

The corollary applies if the coefficients of the subelliptic operator H are real and symmetric because then the operator is automatically self-adjoint. In this case Kr is real and satisfies Gaussian lower bounds (see Section 11.5).

V.3

Anomalous behaviour

In Section V.2 we established that for each right invariant differential operator R E span(BY : y E J+(g)) one has Gaussian bounds I(RK,)(g)I < ct-1 /2Gb.,(g)

V. Global Derivatives

206

for all g E G and t > 1. Next we prove that the t-1/2-decrease is optimal in general. It can only be improved if the group is near-nilpotent. For example, if g 0 nsn, then there are second-order derivatives which have an asymptotic t-1/2behaviour. Thus derivatives in the directions outside the near-nilradical can have a seemingly anomalous slow decrease for large t. There are two complementary results of this nature. The first involves a derivative in a general direction which is not in the near-nilradical. The second involves second-order subelliptic derivatives and is based on the fact that directions outside the near-nilradical automatically intervene.

Theorem V.3.1 If a E g but a V n, then there exists a b E g and, for each n E No, a c> 0 such that

c-I t-1/2 < IIdL(a)ndL(b)St112,2 < ct-1/2

(V.31)

and

c-1 t- 1/2 Ve(t)-1 /2 < IIdL(a)° dL(b) Ktlloo

ct-112

(V.32)

V'(t)-112

for all t > 1. The upper bounds follow from Theorem V.2.1 and the onus of the proof is to establish the lower bounds. The proof is based on the following crucial observation.

Proposition V.3.2 Let a E g and suppose that there exists a cl > 0 such that I exp(ta) I' > cI t

for all t > 1. Then there exists a c2 > 0 such that IIdL(a)St112-2 ? c2t-1/2

and

IIdL(a)K,Iloo > c2t-1/2

V'(t)-1/2

for all t > 1.

Set L, = Kt * Kt, where Kt is the kernel of the semigroup generated by the dual H' of H. Then it follows from Proposition IV.7.3 that there exists a c2 > 0 such that L,(e) > c2 V'(t)-1 /2 for all t > 0. Moreover, since K and Kt satisfy Gaussian bounds, one deduces from Lemma IV.7.4 that there are b, c3 > 0 for all t > 0 and g E G. Hence such that I L,(g)I < C3 V'(t)-1 /2 there exists a K > 0 such that I L, (e) - L,(g)I > 2-1C2 V'(t)-1 /2 for all t > 0 and g E G with (Igl')2 > K t. In particular,

Proof

e-b(I81')2t-'

2-Ic2 V'(t)-1 /2 < I L,(e) - Lt(expsa)l < s IIdL(a)L: IIoo = s II (dL(a)Kt) * K,'1100 < s IIdL(a)Ktlloo II K' III

V.3 Anomalous behaviour

for all s > (c- IK1/2t1/2) V 1. But there exists a c4 > 0 such that IIK, 111

207 C4

f o r all t > 0. Choosing s = c i K 1 /2 t 1 /2 gives the second estimate of the proposition if t is large enough. But then IIdL(a)K,+t'II < IIdL(a)K,II IIKt'111 1

c4 II d L (a) K, II oo and the bounds extend to all t > 1. Finally, II d L(a)K, II oo < IIdL(a)St13112,2II K,13112 by the semigroup property

and there exists a c5 > 0 such that IIKt13112 < c5 V'(t)-t/2 uniformly for all t > 0. Hence the first estimate of the proposition follows.

The following implication of the proposition will be used in the proof of the theorem.

Corollary V.3.3 If to, CI are as in Corollary 111.2.4, ko E to, k1 E fI\{0} and n2 E qN;2, then there exists a c > 0 such that IIdL(ko+k1 +n2)StII2.y2 >

ct-I/2

and

IIdL(ko+k1 +n2)Kt112-.2 >-

ct-1/2

V'(t)-1/2

for all t > 1.

Proof

This follows immediately from Proposition V.3.2 and the second state-

0

ment of Proposition 111.7.16.

Proof of Theorem V.3.1 We only prove (V.31) since the proof of (V.32) is similar.

Since a it n,n there exists a b E p1 such that (ada)nb ¢ qN;2 for all n E No by Proposition 111.4.2, where C1 is a subspace as in Corollary 111.2.4. Let n E No. Since [g, {!1 } c t1 ® qN;2, it follows from Corollary V.3.3 that there exists a c > 0 such that

IIdL((ada)nb)S:II2-.2 > ct-1/2 for all t > 1. Next we use the identity

n-I/\ dL(a)n dL(b) = dL((ada)nb) + E I n)

dL((ada)kb)dL(a)n-k

k=0\ /

But (ada)kb E 4 1 ® qN;2 c n for all k E (0, ... , n). Hence by Theorem V.2.1 there exists a c > 0 such that IIdL((ada)kb)dL(a)n--kStII2-.2 < ct-1 for all t > 1 and k E (0, ..., n - 1). Then the lower bounds in (V.31) follow easily. An alternative way of expressing the key technical estimate used in the proof of the theorem is the following.

Corollary V.3.4 If a E g, b E e1 and (ada)(b) ¢ qN;2, then there is a c > 0 such that IIdL(a)dL(b) 51112-2 > ct-1/2 for all t > 1.

208

V. Global Derivatives

One can use these estimates to construct further examples for which one has a slow decrease of multiple derivatives. We have already shown in Example V.1.11 that the canonical estimates IIB1 St112-2 < ct-la1/2 can fail fort > 1 for two derivatives in the u-directions. The next example shows that they can also fail for two derivatives in the m-directions. Example V3.5 Let g be the six-dimensional Lie algebra with a basis al, ._a6 a6 and commutators

(at,a2l=a3

[a2,a3]=at

[at, a4] = a5

[a2 a5l = a6

(a3 a61 = a4

[a2 a6] = -a5

[a3 a41 = -a6

-a4

The corresponding simply connected group is a semidirect product of the covering group of SO (3) acting on R3. The radical q and the nilradical n are equal and are given by the abelian algebra span(a4, a5, a6). Therefore qN = q is abelian and u = {0). One can choose the Levi suhalgebra m as the subalgebra span(a I. a2. a3) and a = (0).

Let S be the semigroup generated by the strongly elliptic operator H = - F6-1 A. 2 < e t-1 /2 for all i E (1, ..., 6)

Then it follows from (11.12) that one has bounds II A; St 112

and t > 0. But (ll,Ajl = 0 and A,AjSt = A,Stj2AjStj2 for all j E (1,2,3), i it. ..6) and t > 0. Therefore under these conditions one has bounds 11 Ai A j St II2-2

E

.

ct -Alternatively, one has bounds 11Aj A1St112-2 < r't - I for all i, j E {4,5,6) and t > I by Proposition V.1.2. But since q = n is abelian it follows that qN;2 = (0) and tl = span(a4, a5, a6). Then (adal)(a4) = a5 V qN;2, and it follows from Corollary V.3.4 that IIA I A4S, 112-.2 > r t-1/2 for all t > 1. Similarly, one deduces that the G(L2)-norms of AI A5$, A2A5St, A2A6Sr, A3A4St and A3A6St have lower bounds rt-1 j2 for all t > 1. Finally let m' be the subalgebra generated by (a 1 +a4, a2 - a6, a3 ). Then m' = eada5 M. Hence m' is a Levi subalgebra. If, however, a, = at + a4 E m'. then

I(A1A,S,112-2 ? IIAIA4St112-2 - IIAIAISt112-2 - IIA4A4S1 112-2 - IIA4AISt112-2 and it follows from the foregoing bounds that there exists a r > O such that 1I A, A St 112-.2

rt-1/2forallt> I. The next example combines Examples V.1.11 and V.3.5 to give an example in which there are c > 0, io E {-dm, ... , 0} and jo E { 1, ... , do}, i.e., bio E m and bjo E u, such that II Bi Bj St 112_.2 > c t-1/2 for all i, j E (io, jo) and t > 1. So in general two (even mixed) derivatives on the semigroup in the m or u-directions fail to have the canonical decay. Example V.3.6 Let g be the 10-dimensional Lie algebra spanned by b,, i E (I.... , 4) and bj,k, j E (1, 2, 3), k E (1.2) and commutators

[bl

[b2

[b3 b3,k] = bl,k

[bl b2,kl = -bl.k

[b2 b3,k1 = -b2,k

[b3 bl,k) = -b3,k

[b4 bj,I ] = bj,2

[b4 bj,21 = -bj, I

for all j E {1.2.3) and k e (1, 2). The corresponding simply connected group G is the covering group of the semidirect product of (SO(3))° x R acting on R6, where the action on

R6 - R3 ® R2 is the tensor product of the action of (SO(3))° on R3 with the action of R

V.3 Anomalous behaviour

209

on R2 by rotations as in the covering group of the Euclidean motions group. Here (SO(3))° is the covering group of SO(3). The nilradical n is spanned by the elements bj,k, and the radical is q = span(b4) ® n.

As a Levi subalgebra one can choose m = span(bl, b2, b3), and then o = span(b4). Then qN;2 = (0) and t1 = n. Set ai = bi for all i E ( 1 , ..., 4), a4+k = bk, l and a7+k = bk, 2 for all k E (1, 2, 3 ). Consider the strongly elliptic operator H = - 1:10 A. One easily verifies that [H. Aj ] = 0 for all j E (1, ... , 4). Then repetition of the reasoning of the previous example yields bounds ct-

IIA1AjSt112-2 5 f o r all t > 1 , whenever (i, j) E 1 1 , .... 10) x ( 1 , ... , 4} or (i, j) E 15, ..., 10) x (5, .... 10). Moreover, by the previous reasoning one has bounds

IIA1AjSih12-2 > c'1_

1/2

for all t > 1, whenever (i, j) E (1, ..., 4) x (5, ..., 10) and, in addition,

( i , j) V ((l, 7). (1, 10), (2, 5), (2, 8), (3, 6), (3, 9)). In the latter cases the elements commute, for example [al , a71 = 0.

Consider the Levi subalgebra m' = ea5 m and the subspace o' = e 5 n. Set bk = eaa5 (bk) E m' U o' and Bk = dLG (bk) for all k E 11, ... , 4). Noting that bk = bk [bk, 61.11 one calculates that b = b1 - b2,1, b' = b2, b'3 = b3 + b3,1 = a3 + a7 and b` = b4 - b1,2 = a4 - as. Reasoning from the above bounds on Ai Aj St, it is then routine to verify that one has bounds 1IBk1 B', S, II2-2 >

1/2

whenever t > l and k1, k2 E (3, 4).

The second result on slow decrease of multiple derivatives only involves the subelliptic directions. In particular it establishes that one has canonical asymptotic behaviour for all second-order subelliptic derivatives if and only if the Lie group G is near-nilpotent, i.e., it is a local direct product of a compact semisimple Lie group and a nilpotent Lie group. Thus it is a generalization of Proposition 11.5.3.

Theorem V.3.7 The following conditions are equivalent. I.

G is near-nilpotent, i.e., G is a local direct product of a compact semisimple Lie group and a nilpotent Lie group.

II.

g is near-nilpotent, i.e., g = ns.n.

III.

For each a E J (d') there exists a c > 0 such that II Aa St 1122 < c

for all t > 1. IV.

For all a E J(d') with lal = 2 one has lim 11/2 IIAaS1 II2-.2 = 0f-.00

V.

There exists a v E (0, 1) such that

lim t1/2 sup (Igl')-III(/ - L(g))AkSt112,2 = 0 too gEG\(e)

for all k E {1,...,d'}.

t-jaj/2

V. Global Derivatives

210

The equivalence I. II is by definition, the implication II=III follows from Theorem V.2.1, the implication III=IV is evident, and the implication IV=>V

Proof

follows by interpolation. Therefore it suffices to prove the implications V=IV and IV=II. V=IV. First Condition V implies that there is a function f : [ 1, oo) -+ R such that lim, . t 1/2 f (t) = 0 and 11(/ - L(g))AkSr112.+2 1. Secondly, let i E { 1, ... , d'}. Then it follows from spectral theory that there exists a c > 0 such that

IIA11= c

fds s' (' II D(s)3

(V.33)

II2)

for all rP E D(A1), where Di (s) = (I - L(exp sai )) for all s E R. But

IIDi(S)3Aks,II2-.2 1. Then it follows from a combination of (V.33), (V.34) and (V.35) that there exists a c3 > 0 such that

IIAiAksr112" 1. Condition IV follows immediately.

Finally, we prove the implication IV=II. First, for all y E J(d') and k, I E (1,

... , d') there exists a c1 > 0 such that IIAY'AYAkA1S11I2.+2 < Cl t-1/2

for all t > 1 by Theorem V.2.1. Since IIAYAkA,Sr112.2
0 it follows that limt.,, tl/2 IIAYAkAIS:112-.2 = 0. Thus lim t1/2 IIAYSrll2_+2 = 0 t-.oo

for ally E J(d') with IYI > 2. Secondly, m ® to ® qN;2 is a subalgebra since qN;2 is an ideal in g and [m®Co,m®Eo] C m®[to,to] C mED gN;2.Hence if a[al E m®to ED gN;2for all a E J(d) with IaI > 2 then [g, g] C m ®eo ® qN;2 Thirdly, o(g)fil c [g, g] since S(v) is a polynomial in adv without constant term for all v E U.

Now suppose that g is not near-nilpotent. Then by Proposition 111.4.2 there

exist a E g and b E E1 such that cr(a)b i4 0. But then cr(a)b E tl fl [g, g] by Corollary 111.2.4. Therefore there exists a Of E J(d') with Ial > 2 such that

' m (D to ® qN;2 Write a[a] = m + ko + kl + n2 with m E m, ko E to, kl E tl and n2 E qN;2. Then k1 0. Let s be the rank of the algebraic

a[a]

basis al, ... , ad'. Since m = [m, m] by 11.1.4 there exist cp E R such that m = F-8EJ(d'); 2 0 such that I and x, y e R3. Choosing y = 0 gives 11831, Iloo sc" t-t j2 t-c for all t > 1. Therefore by scaling 11a3Kt IIoo < c"t-E for all t > I and 83K1 = 0. This is a contradiction. Hence the estimate IKt(gh-t) - K, (g +N ht-UN)l $ ct-112Gb,t(gh-t)

for all r > I cannot be improved in general.

The asymptotic estimate (VI. 1) does simplify in special cases. If H is real symmetric, then it can be arranged that first-order correctors X j and X j are zero (see Example IV.3.7). Nevertheless, the first-order corrections do not vanish, in general. Example VI.1.4 If all the first-order correctors Xj and X vanish (or if all the correctors with j E { 1. ... , dt } vanish), then the unweighted third-order coefficients all vanish (or the unweighted coefficients with a E J(dl) and lal = 3 vanish). This is established by the following argument. Since the correctors Xk vanish one has e;jk = EI _dm JVt(c;I BI X jk) Hence if is the inner product on E2 defined in the proof of Lemma IV.3.2, then it follows from (IV. 19) that d

d

Cijk = L (Cil BIXjk)M = - L (Blcil Xjk)M = where co =

(fib1ct0. Xjk)M.

Ed',;=1 clk a (ak )aj . But by Lemma IV.3.6 one has X, = 0 for all i E (I , ... , d1 )

if and only if (b;,co) =0foralli E (I,....dI), and this is the case if and only if g t b;,co for all i E {1.....d1}.

=0

Despite the implication in Example VI. 1.4 the weighted third-order coefficients

ca with ltrl = 2 and Ilctll = 3 do not vanish even if the correctors and adjoint correctors are zero, i.e., X j = Xt = 0 for all j E (-dm. ... . d). Example VI.1.5 Consider the five-dimensional solvable Lie algebra gas in Example 111.2.12

with basis bi, ... , b5 and commutation relations [b1, b2) = b3 + b4, [bi, b3) = -b2 + b5, [b1, b4) = b5 and [b1, b5) = -b4. Then one can choose o = spanbt, tt = span(b2, b3) and qN;2 = span(b4, b5}. If G is the simply connected Lie group with Lie algebra g and c l, s1: G -i R are defined by ct(expl;tbt ...exprsbs) = cosgt st (exp (:t bt ... exp gsbs) = sin t1

(see Example 111.7.2), then Bt = Bt and

B2=elB2-s1B3

B4=c1B4-stB5

=st82+11B3

B4 =s1B4+c1B5

B3

Let H = -(B1 + ... + BS) - (B2 + B4)2. Then H is a strongly elliptic operator and

H=-(B +...+BS)-(c1B2-s1B3+c1B4-s1B5)2.So 0

((ij) =

1

-ctst

r1

-ctsi

0 0

-rts1

I

-CISI

SI

e

-e1st

I

-Ctst

0

-c1S1

S

-1'1st

222

VI. Asymptotics

Since B1elj = Blcjt = 0 for all j one has Xj = Xt = 0 for all j E (1, ..., 5). Then c;j = M (cjj) and

2(c;j) =

2

0

0

0

0

0

2

0

1

0

0

0

2

0

1

0

1

0

2

0

0

0

1

0

2

In particular, if a = (2, 4). then tall = 3 and is = 2-1 # 0.

One can obtain a conclusion analogous to Theorem VI. 1.2 for the semigroup S, but for this we need to define an extension Et to G of an operator E on QN similar

to the extension of the functions from QN to G. Specifically, if 1 < p < r < 00 and E is a bounded operator from Lp(QN) to Lr(QN), we define the bounded operator Ell from Lp(G) to Lr(G) by

Jdml((J®E)%fr)(miq'q)

(EOv)(mq) = IM n Q1-1 E

y'EMnQ M

((I (&LQN(q'))(PM

= IM n QI-1

0 E)*)(m, q),

q'EMnQ

where t(i: M x Q - C is given by i/r(m, q) = rp(mq) and PM =

Jdm LM(m) E L(Lp(M), Lr(M))

is the projection onto the constant functions in Lr(M). Note that ElLp(GN) e PLr(GN), where

P=

dmLG(m) JM

is the projection onto the functions on GN which are constant along M. Moreover,

if LQN (q') E = E LQN (q') for all q' E M n Q, then (E')a = (El)*. In addition,

if 1 < p < q < r < oo and El: Lr(QN) -p Lq(QN) and E2: Lr(QN) Lr(QN) are continuous operators such that LQN(q') E2 = E2 LQ, (q') for all q' E M n Q, then (E2E1)a = EZ E. If G is simply connected, thenEO = PM ® E on GN = M x QN. In particular the extension of the semigroup S is given by -V = PM ®S.

Vl. I Asymptotics of semigroups

223

Theorem VI.1.6 Let G be a (not necessarily simply connected) Lie group. Then there exists a c > 0 such that d1

IISr-3" +>2(Xj( j=t

- 1: aEJ(d)

fo

_

_

r/2

C. (

ds SJ B

lC

2
for all t > I and p, r E (1,00] with p < r. Theorem VI.1.6 is a corollary of Theorem VI.1.2, the Minkowski inequality. and interpolation theory together with the following lemma which relates E"" and his where E is the operator of convolution with >i.

Lemma VI.1.7 Let E: Lp(QN) - Lp(QN) be the bounded convolution operator Ecp = >y * So where >G E LI(QN) fl C(QN) and the convolution is with respect to left translations on QN. Then EU: L p (GN) -+ L p (GN) is the bounded convolution operator on GN with kernel i/rp.

Proof The statement follows by observing that

(Ekp)(mq) = IM n QI-I

=J M

f

qM

>2 J dml J dqt *((q'q) *N

qj(-')x)so(mtq

)

Q

dqt VfG(mq *N

Q

forallcoELp(g),mEMandgEQ.

0

The main aim of the remainder of this section is to prove Theorem VI.1.2. The proof is indirect as there are many technical problems. In principal there are five main steps which can be outlined as follows. First, we assume G is simply connected and has a stratified nilshadow (see Section III.5). We then establish

appropriate bounds on IIS, - PM 0 Sr Ilper

for 1 < p < r < oo with the

exceptions of (p, r) = (1, 1) and (p, r) = (oo, 00). Secondly. we prove the bounds of Theorem VI. 1.6 on the second-order asymptotic expansion of S, with the exception of the L t -+ L) and LOO - La, bounds. Thirdly, these bounds give

224

VI. Asymptotics

Lw-bounds on the second-order asymptotic expansion of K and by interpolation one deduces bounds K,(gh-t) I

- (1 0 K,)(g *N h(-1)N)I < ct-1 /2

Gb,t(gh-1)

uniformly for all g, h E G and t > 1. Fourthly, these bounds allow transference to a general (connected) group from the simply connected group with stratified nilshadow. Fifthly, we prove the second-order expansion for the kernel on the general group. Note that the second-order expansion for the semigroup in Theorem VI. 1.6 follows, by the above argument, including the cases p = r E (1, oo). The proof of the theorem is separated in several lemmas and propositions. In order to avoid repetition of the details of the proof for the second-order expansion, in the second and fifth steps, we combine the proofs in Lemma VI.1.12. The fact that M fl Q {e} creates many complications. We first establish the first-order asymptotic approximation of the semigroup for a simply connected group with a stratified nilshadow.

Proposition VI.1.8 Suppose G is simply connected and the nilshadow qN of the radical q has a stratification compatible with v and m for some subspace v satisfying Properties I-III of Proposition 111. 1. 1. Then for all p, r E [ 1, 00] with p < r and [p, r] fl (1, oo) # 0 there exists a c > 0 such that II St - PM ® StIIp-.r

1, and constant ct > 0. We shall prove (VI.3) for t = 2.

Proof of Proposition VI.1.8 The proof is based on the inequalities IISZ°1- OPII p-r < IIS2'"1(1- P)Il p_r + 11S1'"1(1 - P)(SI"1

- S,°)PIIp-.r

+IISI"1P(sI"1-s,°)PIIp-.r+II(SI"1-)S^°1PIIp.r < IIS2'"1(1 - P)II p-+r + IISI"1(1- P)I1p-.r(IISI"1IIp- p +

+ for all u > 1.

IISI"1P(SIu1

II?lIp-p)

- s,°)llp-.r + IISI"1- s,° )s, °Pll p-.r

(VI.4)

VI.1 Asymptotics of semigroups

225

First, it follows from the Gaussian bounds of Theorem IV.7.1 that there exists a c > 0 such that II Sr II p-. p < c uniformly for all t > 0. Hence by the scaling relation (IV.14) one has Ils;u)Ilp-.p 0. Secondly, it follows from the Gaussian bounds (V.2.1) for the subelliptic derivatives Ak K, that there exists a c > 0 such that 1, where c' = cd' fM dm ImI'. The same argument also applies to the dual H* of H. Hence there exists a c > 0 such that IISiu)(1

- P)IIp +r 1. Thirdly, we estimate II(Siu) - Sb)PSi")Ilp-+r. which is the dual of the third term and the most difficult to handle in (VI.4). Let q E [1, 00], rp E PLq;oo(GN),

0 E (0,9G A00),X E A(ir/2+0)andu > 1. Set

_ (G),,u = (Al + Hiu))-1 (p

- (,t1 +

d

Ho)-1 cp +

u-U'i X(u) Bj (AI + Ho)-1 J=1

where B! = d LG,,, (b1), as in Chapter IV. Then

(X1 + H[u))(pa.u = (Ho - Hiu))(AI + Ho)-lip

+

_

d

u- `J(A1 + H1 1)Xju) i=1

_ H)-1 v

226

Vl. Asymptotics d

_ ,l>2u-WlXjIu) Bj(,kl + Ho)-I cp j=1

_

d'

+

k

a,k

aEJ(d) k=1 Ia1=2

+

u2-Ila11r(au)Ba(AI

L:

+ Ho)-I V

aEJ(d) Ia10

BaS2t-.c

0

_ ds Ss BaS2-s

0

d

+ 1: 1: e, j Xa Bi Bj haS2r-s i,j=1 !a!=3

-

f ds S(HX)B' 3111-s*

(VI.17)

But d'

d'

d'

Xa4P = L Ak (Va,krP) - 1, E Va.k rk/ Al o k=1 k=1

k=1

for all rp E PL p;x(G). Therefore SXa Bast = aEJ3 (d)

r

__ Bast

ac-J3(d) k=l d'

d'

Sr *..k rki BI aEJ3 (d) k=1 k=1

So the first term on the right-hand side of (VI. 17) contributes to the first two terms on the right-hand side of (VI. 15). Similarly, the third term on the right-hand side of (VI. 17) can be expanded. This proves Statement 11. 0

VI. Asymptotics

236

Corollary VI.1.13 Suppose G is simply connected and the nilshadow qN of the radical q has a stratification compatible with u and m for some subspace u satisfying Properties I-III of Proposition 111. 1. 1. Then there exist b, c > 0 such that IK1(gh-1)

- (1 (9 k1)(9 *N h(-I)N)I < ct-1 /2Gb,r(gh-1)

uniformly for all g, h E G and t > 1.

Proof Let p, r E [1, oo] with p < r and [p, r] fl (1, oo) 0 0. We first prove that there exists a c > 0 such that

I

-

f

t/ 2

cPM ®dS 2 0 such that IISS - S°PIIq-4r < CS-(q -r 1)D/2s-1/2 uniformly for all s > 1. Moreover, c's-(p-'-q-')D/2s-3/2 for all there exists a c' > 0 such that II BaS°s PII p-q
1,ifllall=3.Then J

`dsll(S.-gSP)Ba`2t_SPIIp-r < CC' 1 ds s-(q-'-r-')D/2s-1 /2t-(q-'-r-')D/2t-3/2 1

`

< C 1 t-(p-'-r-')D/2t-I

VI.I Asymptotics of semigroups

237

uniformly for all t > 1. Taking p = 1 and r = oc it follows that there exists a c > 0 such that

`Kr(gh-I)

_ - (1 ® K1)(9 *,v h+

di

> Xj(g) (1 ®B(N)Kr)(g *N

h(-))w)

j=1 dl

- L(1

*N h(-I)N) Xj (h)

j=1

r/2

-

ca(1 ®ds Kc *N B'K,-)(g *N aEJ(d)

J

2 f d4 V1;.4(Xrh)

J:fq(m) Q

for all m E M. Then applying the Weil formula twice gives

fdco() = fdg cpb(8)

ffq

= fm dm fQ d9 cp°(m9)

c1

f

=c'1

fMdm

=c'1

fMdm J d4

(m)

d4cp(mm r9)

c'kl f di(P(B)

Soc' = (k1)-1. Now let Mfr E Li (QN) fl C(QN). Then it follows from Statement I that

i°(A(9)) = J A*(4h) mQ for all 4 E QN, where the multiplication qh is on Q. If in E M and 4 E Q, then

(1 (9 *)b(A(m4)) =

f d(1 (9 f)m4h) j r

= (k1)-'

_

;.j

dh (1 0

jm;q;h)

nQ

_ (k1)-' r jdi; nQ

*((m'4(mm)qh)

VI. Asymptotics

244

r` L *b(A((mjm;)-i4(mjm;)4;)) ;.iJ.

So

ib(qq) = fba(mq)

ltb(mq) = (1 (9 f)b(mq) = IM n QI-1 q'EMnQ

for all m E M and q E Q and the proof of Statement II is complete. This completes the proof of the lemma. We now transfer the first-order approximation for the kernel on the special Lie group of Corollary VI.1.13 to a general Lie group (with polynomial growth).

Proposition VI.1.16 Let G be a (not necessarily simply connected) Lie group. Then there exist b, c > 0 such that K,(gh-t) I

- I (g *N h(-1)N)I < ct-112 Gb,,(gh-t )

uniformly for all g, h E G and t > 1.

Proof

Let K be the kernel on GN of the semigroup generated by H. Similarly let K be the homogenized kernel on QN. It follows from Corollary VI.1.13 that there exist b, c > 0 such that

a

ap

IKr

..,

- Krl 0.

Proof Let S be the semigroup generated by the second version of the homoga enized operator H0 of h on G (see (IV.25)). Then HoA*ry = A*HOrp for all tP E Cb;2(G), where A*rp = (p o A. Hence Sr A*(p = A*30rp for all (P E CbrG) L5 (in) is the projection onto the functions on GN and t > 0. But if P = fey

which are constant along M then X* P = PA*. Moreover, St P = S, for all t > 0 since a is simply connected. Therefore for all rP E Cb(G) and t > 0 one has

A*)P(P = Sr PA*w = Sr A*(P. Using the Well formula (11.38) one deduces that Kad

A

St A*rP)(i) =

r (h) (A*(P)(h-1

G

=f

dh

Kp(h) rp((Ah)-1 Ai)

c

= J dh Kp(h) rp(h - Ag) c

=fA 0, (h) ry(h-1 Ag) _ ( (p)(Ai) _ (A* w)(i) for all i E G. Since G is surjective the lemma follows.

O

Proposition VI.1.16 can be reformulated in the following simple form.

Corollary VI.1.18 Let G be a (not necessarily simply connected) Lie group. Then there exist b, c > 0 such that

IKr(mq) - Kr(q)I

ct-112 Gb,t(mq)

uniformly for all m E M, q E Q and t> 1. Proof There exist b", c" > 0 such that I (B; N)Kt)(q)I
0 and q E QN, where I IN is the modulus on QN with respect to the algebraic basis bl, ... , bd,. Let q' E Q and y: [0, 1] -+ QN be an absolutely continuous path such that di

y(s) i-1

yi (s) Bl N)I r(s)

for almost all s E [0, 1 ] and Iyi (s)1 < 21gIN for all i E (1, ... , d1 ) ands E [0, 1]. Then for all t > 0 and q E QN one has di

1

I Kt(q'q) - Kr(q)I

fo

ds

< 2c"d1IgI'

lyi(s)I I(B,N)I:)(y(s)q)I

J0

I

ds t-I/2Vi(t)-1/2e-b"(ly(s)gIN)2t-.

< where we used the inequalities

(IgIN)2 < 2(Iy(s)IN)2 + 2(Iy(s)g1')2 _ 8(Iq'IN)2 + 2(I y(s)gI N)2. So for all q' E Q there exist b, c > 0 such that

I Kr(q'q) - K1(q)I
1. Then by Corollary 111.7.14 there exist b', c' > 0 such that e-b(IgIN)21-1 < c' e-b'(Imgl')21-'

(VI.20)

uniformly for all m E M, q E Q and t > 1. Then the corollary follows by a combination of Proposition VI.1.16, (VI.19) and (VI.20). Finally, before we can prove Theorem VI.1.2, we need one more lemma.

Lemma VI.1.19 For all t > I let 1r E LI (QN) fl C(QN). Suppose there exist b, c > 0 and N E No such that

I*t(q)I

1. Then there exist b', c' > 0 such that *G(g *N h(-1)N)
1.

C't-N/2Gb'.t(gh-1)

VI.1 Asymptotics of semigroups

247

Proof This follows again from Corollary 111.7.14. Proof of Theorem VI.1.2 Since by Lemma V1.1.7 the left-hand side of (VIA) is the kernel of the left-hand side of (VI.14), with r replaced by 2t, we have to prove that the kernel of DM DO) + D(2) + (D`2)tyr

can be bounded by a multiple of (g, h) H t-1 Gb,,(gh-1) uniformly for all g, h E G and t > 1 for a suitable b > 0. The estimate for the first term follows from Lemma VI. 1.7, Proposition VI.1.16 and Lemma IV.7.4. Since the third

term can be handled by duality of the second term, it remains to consider the second term. The second term we rewrite as in (VI.15). Then the kernel of each term, except the sixth, in (VI.17) can be estimated by a multiple of (g, h) r-+ t-1 Gb.,(gh-1)

uniformly for all g, h E G and t > I for a suitable b > 0, using Lemmas IV.7.4 and VI.1.19. Hence it remains to estimate the kernel of the sixth term

C(Ss -) (B1-s)t fds 2 1, where again I IN is the modulus on QN with respect to the algebraic basis bi , ... , bd, . Fix ql' E M n Q. Since m n Q is a group one has

IM n QI-'

E,

(Kr(q'q)

- Kt(q))

q'EMnQ

= 2-' IM n QI-1

(K:(q'q) + K1((q')-'q)

- 2Kr(q)).

q'EMnQ

Therefore by Lemma 111.7.8 it remains to show that there exist b, c > 0 such that Kt((q')(-1)N

Kt(q' *N q) +

*N q) - 2Kt(q)I

ct-1

V'(t)-1/2e-b(IeIN)21-'

uniformly for all q E Q and t > 1. There exists an a E q such that q' = expN a. Applying the equality

f(1)+ f(-1) -2f(0)

f dv] dw f"(w) v

which is valid for all f E C2(R) to the function f (u) = Kg((expN ua) *N q) it follows that K,((q')(-I)N

I Kt(q' *N q) +


0 with u2t < 1 and by Statement 11.4.15 IIS°Pllp-.r < C t-SD/2

for all t > 0. Then if SD < 1 and BD' < 2, one calculates that there is a c > 0 such that 00

11W +

(Al + Ho)-' Pll p-r < fo d t e-at Ilsiul

cu-

I

(1

-

+),-(I -SD)/2)

for all A > 0 and u > 1. This gives a strong quantitative version of the resolvent convergence of Proposition IV.5.3 for real X. If A E A(6c A BE) is complex one finds a similar estimate.

250

VI. Asymptotics

VI.2

Asymptotics of derivatives

In Theorem VI. 1.2 we derived the first- and second-order terms in the asymptotic

expansion of the semigroup kernel in terms of the kernel of the homogenization. These estimates demonstrate that the asymptotic evolution is governed by the homogenized system on the shadow group. The details of the asymptotic approximation can be further analyzed by considering the derivatives of the kernel, and our next aim is to establish the asymptotic form of multiple derivatives of the kernel and the first-order correction terms. The derivative estimates then give information about the boundedness of the

Riesz transforms. In particular, the first-order transforms, which are formally given by AkH-112, can be shown to be bounded on all Lp-spaces with p E (1, oo). Some care has to be taken in interpreting the Riesz transforms since H is not injective in general. Therefore, the operator H-112 is not densely-defined. Clearly this is the situation if the group is compact. Nevertheless, we shall prove

that if p E (1, oo), then D(H'12) = L'p;1(G), and there exists a c > 0 such for all , E D(H1/2) and k E The nathat IIAkWIIp c II ture of the bounds on the higher-order transforms is more complicated as there is usually a difference between the local and global behaviour of the derivatives. This is directly related to the different singularity structure of the derivatives of the semigroup for large and small t. If Y E J(g), set ey = 1 if lyI > 0 and ey = 0 if IyI = 0.

Theorem VI.2.1 There exist b, w > 0 such that for all a E J(5), P E J(n) and y E J(g) there exists a c > 0 such that

(BaBf

ByK,)(gh-1)

- (BaBt3ByKx(2))(g; h) d,

+ )(BYXj)(g)(BaB1'(BjN)Kt)0(2))(g; h)I j=1

< c t-1 /2e-da(ott-(11011+ty)/2Gb.t(gh-1)

uniformly forall g, h E G and t > 1. Here *0(2)(g ; h) = *O(g *N h(-1)")for all E LI(QN) fl C(QN) and the derivatives on Ka(2) and are with respect to the first variable.

Proof Notice that the theorem is evident if lal ¢ 0 because the first term on the left-hand side has an exponential decrease in t by Theorem V.2.1 and the other two terms on the left-hand side vanish. Therefore in the rest of the proof we may assume that la I = 0.

For all t > 0 set

U, = St -

so

+

Xs (B(' 6 EJ4 (d)

S,)p

VI.2 Asymptotics of derivatives

251

and

Vr = -

ca (B('

X8 (B(N)8HS1)' +

S,)'

6EJ(d)

BE J2- (d)

2 0 and for all fl' E J(n) and y' E J(g) a *p,y' E COO(G) such that B' B" S,cp

= E E (ts'y' K,) * B' BY ,p '5'EJ(n) y'EJ(g) t'51?Il$4 1y'1=1y1

and I*'5'y'(g)I < c'(1 + (Igl')1# II-tdll for all g E G. Then (VI.23) follows by differentiation.

El

Proof of Theorem VI.2.1 (g, h) F-- (BO

Note that ByK,)(gh-t)

- (BO By ,)(g *N h(- UN)

+ E (BBBY(X6 (B(N)aK,)"))(g *N

h(-t)N)

t_ 3. By (V.2.1) there exists a b > 0 such that the kernel BO BY K,12 of B'5 BY S,12 can be bounded by a multiple of the function (g, h) r-> t-(11,61`0/2 Gb,,(gh-1)

uniformly for all t >_ 2. Moreover, by Theorem VI-1.2 there exists a b > 0 such that the kernel K,/2 - (K,a)' + F-1 0, uniformly for all t > 2 and s E [t/2, t - 1]. Hence the kernel of the second term

VI.2 Asymptotics of derivatives

253

on the right-hand side of (VI.23) can be bounded by a multiple of the function (g, h) r-> t-(11611+ey)/2 Gb.t(gh-1) uniformly for all t > 2. Finally, it follows straightforwardly from Leibniz's rule, (VI.22) and the bounds

of Lemma VI.1.19 that there exists a b' > 0 such that the kernel of the operator B 6' BY' V, can be bounded by a multiple of the function

(g, h) -

s-3/2 s-116'11/2 Gb's(gh-1)

for all 0' E J(n) and Y' E J(g). Then it follows from Lemma IV.7.4 that the kernel of the third term on the right-hand side of (VI.23) can be bounded by a multiple of the function (g, h) r- t-1t-(11611+ey)/2Gb,t(gh-1) uniformly for all t > 2, for a suitable b > 0. This completes the proof of the Theorem VI.2. 1. 0

Corollary VI.2.3 There exists an w > 0 such that for all a E J(S), P E J(n) and Y E J (g) there exists a c> 0 such that d1

IIBaBOByS1 - B°B6 BY S, + L By XjrB6(BjN)St))tII per l=1

< c t-1 /2e-dQwtt-(11611+ey)/2t-(p-1-r-')D/2

uniformly for all t > 1 and p, r E [ 1, oo) with p < r. It is now not difficult to prove that the first-order Riesz transforms are bounded on Lp(G) if p E (1, oo). Moreover, if G is near-nilpotent, then the Riesz transforms of all orders are bounded on Lp(G) again for p E (1, oo). Both statements follow from the next proposition.

Proposition VI.2.4 Let p E ( 1 , oo), n E No, (k1, ..., kn+I) E { 1, ... , d'} and suppose that ak...... ako E n,". Then D(H(n+I)12) C D(A'5) in the L p-sense, where a = (k1, ... , kn+I ), and there exists a c > 0 such that IIA'

for all tp E

Proof

II

0 such that d

IIAGSt - dLG(nl (a))0'90 - E(Akn+1 Xj)dLG(nl (a))'(Bj')Sg)'llp_.p j=1

_ d IIAaS1 - Aa(St)0 - 1:(Ak,+1 xj)A'(Bj')St)'Ilp-.p

j=l + II(Aa - dLG(n1(a))a)

II p +p

d

+ E 11(Ak,,+1 xj)(A' - dLG(,r1(a))16)(B(N)St)UII j=1

255

V1.2 Asymptotics of derivatives 0. These follow for t < 1 from the bounds (VI.25) and they follow for t > 1 from Corollary VI.2.3 and Theorem V.2.1. One deduces that

c r((n + 1)/2)-1

0 00

J

dt e`(1 00

+ Ilr((n + 1)/2)-1

e-r)mt-(2m+1)/2(1

+

)(2m-n-20

dt a-e'(1 - e-')m t(n-1)/2

J0 di

/

I dLG(n1(a))"S, + L(Ak,+, xj)dLG (71

p-pp

j=1

But the first term on the right-hand side has a finite bound, uniform for all e > 0. The second term requires more work. If Y E J(d1) and I i = n + 1, then sup,>0 J I B (N)y(el + H)-(n+l)/2II Lp(QN)-+LP(QN) < oo. Hence

sup II (B(N)y(e1 +

H)-(n+1)/2)allp-.p < 00.

e>0

In particular, the Lp-norm of the operator 00

r((n + 1)/2)-1

J0

dt e-" (I - e-)mt(n-1)/2(B(N)yst)

_

00

sup 11r((n

+ 1)/2)-1 fo

(-1)1(7)(BNl + e)1 +

dt a-E'(1 - e-t)mt(n-1)/2.

a>0 d1 !

l=1

/,, (El + d'

L Ekl (AkV, A1(Ei +

H)-'I2(p)

k.1=1

+ E (u', (E/ + H) -1/2(p)

for all cp E Lp(G) and' E D(H*) C Lq(G), where q is the exponent dual to p. Hence by the above there exists a c > 0 such that d'

1((E1 + H')'/'*,(p)I 0, (p E Lp(G) and >/i E D(H') C Lq(G). Therefore d'

II(El + H')1/2*IIq

cE11ZII*IIq +C

IIAk*IIq k=1

and the upper bound follows by density and taking the limit E -+ 0. Similarly, one proves the next theorem.

Theorem VI.2.6 If G is near-nilpotent and p E (1, oo), then for all n E N there exists a c > 0 such that D(Hnl2) = Lp;n and

c-1 max IIAa4llp < IIHn124Ilp < c aEJ(d') lal=n

max IlAawlip

,EJ(d') laI=n

for all cp E L'p;n(G).

Again the lower bounds follow from Proposition VI.2.4 since G is near-nilpotent. The upper bounds can then be deduced by a slight variation of the above duality argument. One can combine this last result with Theorem V.3.7 to deduce that boundedness of the second-order subelliptic Riesz transforms is characteristic of nearnilpotency of the group. Moreover, one can extend the L2-statements of Theorem V.3.7 to Lp-statements for all p E (1, 00).

V1.2 Asymptotics of derivatives

257

Corollary VI.2.7 The following conditions are equivalent. I.

G is near-nilpotent.

U.

There exist p E (I. oc) and c > 0 such that IIAkA/SOllp -< c IIHvllp

forallcp E D(H)andk,I E (1,...,d'). III.

For all p E (1, oc) and a E J(d') there exists a c > 0 such that IIAaS4IIp-.p

1. IV.

There exists a p E (1, oo) such that

lint t1/2 IIAaS,Itp-p = 0 for all a E J (d') with Ia I = 2-

V

There exist p E (1, oo) and v E (0, 1) such that

lim t1/2 sup (Igl')-°II(l - L(g))AtSrll p.p = 0

r-.00

(VI.26)

gEGllel

for all k E (1,...,d'). Proof

Since H has a bounded Hoc-holomorphic functional calculus for each p E (1, oo) (see Notes and Remarks), for all n E N and p E (1, oc) there exists a c > 0 such that II H"12 Sr II p-. p < c r"/2 for all t > 0. Then the implication I=III follows from Theorem VI.2.6. The implication IlI=1V is trivial. Similarly, the implications 1=:;II=::IV follow. The implication IV=V follows by interpolation with the bounds of Corollary V.2.11. Finally, assume Condition V is valid. Since one has bounds

II(/ - L(g))AkS,llq.q < 2ilAkS,Ilq-.q -$

ct-112

(VI.27)

uniformly f o r all g E G. k E (1, ... , d'), t > 0 and q E (1, oo) it follows by interpolation of the bounds (V1.26) with the bounds (VI.27) for q = oo if p < 2

and with q = 1 if p > 2 that the L2-condition V of Theorem V.3.7 is valid. Therefore G is near-nilpotent by Theorem V.3.7.

It is evident from the last corollary that one cannot hope to bound the higherorder Riesz transforms A° H - la1/2 for a general group of polynomial growth. The difficulty arises because the derivatives have a different local and global behaviour. Formally 00

Aa H-IaI/2 = c-1 1* I0 dt tk '2-I AaS'

V1. Asymptotics

258

where c = fo dt tlal/2-1e-'. If lal = 1, then IIA°S,llp-.p =

O(t-I/2) both as

t - 0 and t -+ oo. But if lal > 1, then the local and global singularities can be quite different. Hence the estimation of the derivatives in terms of the subelliptic operator separates into two different problems, local and global. One is forced to use different variants of the Riesz transforms to describe the local and global bounds. One way to separate the problem is to consider transforms

R= f

1

Rat) = f dt tll/2AaS,

,

0

Rat)

=

J0

dt

tal/2-le-w1AaS,

,

dt tI hI/2-1 AaSt.

RaR) =

with co > 0. But the latter can be expressed directly in terms of H. One has Rat) = claiAa(W1 + and if H is injective

H)-I«I/2

h(g) = claiAaH-Iall2e-H

In the first case the extra term wl takes care of the global singularity and in th second case the factor a-11 takes care of the local singularity. The boundedness of the local Riesz transforms Rat) follows from Statement II of Proposition 11.6.2 and we next discuss the global transforms. We consider the global transforms corresponding to multiple B-derivatives. H)-6e-H is bounded in LP for all y E J(g), Obviously the operator BY (El +

3 >OandpE[1,00). Theorem VI.2.8 Let a E J(s), P E J(n), y E J(g), v > 0 and p E [1, 001. Suppose 1.

a E J+(s), or

if.

V < (11011 +Ey)/2, or,

III.

V

01$11+Ey)/2andp¢(1,oo).

Then

sup IIBaB-6BY(EI + H)-"e-HIIP-.P < 00. E>0

Hence, if in addition G is not compact, the operator BaB,6 B1 H-"e-H extends to a bounded operator on L p.

Proof It follows from Theorem V.2.1 that there exist c, co > 0 such that II Ba Bf BY(E1 + H)-"e-H ll p-.p 00

< r(v)-l

d

fo

tt"-l e-6'ljB°B1BYSr+1llp-+P

259

V1.2 Asymptotics of derivatives 00

< c 1'(v)-1 fo dt tv-I e-Ere-daw(r+I) (t +

I)-(0,61+L,),)/2

for all e > 0. Hence if a E J+(s) or v < (II0II + ey)/2, then 00

I

dt t"-1 e-Ere-5.w(r+I) (t + 00 _
0

If y ¢ J(q), then BOBy(e! + H)-De-H)t = 0 and (VI.30) is obviously valid. If y E J(q), then II BOBy((e! + H)-"e-R )III p_.p = II (B(N).B(N)Y(EI + H)-"e-H I iip_p < II B(N)0 B(N)Y (el +

H)-De-H

II p- p

and the theorem has been reduced from G to the nilpotent group QN. Next write B(N)AB(N)Y =

Ca B(N)a. aEJ(d1) Ial?11011+IIYII

If a E J(d1) and Jul > 2v then it follows as before that supe>0 II B(N)a(el + H)-9e-H II p-. p < oo. Alternatively, if a E J(d,) and Ial = 2v, then < Ile-H IIp.p sup J I B (N)a(EI + H)-°II sup J I B (N)a(EI + H)-De-fl

e>0

Ilp-.p

a>0

< 00

by Proposition VI.2.4. Therefore the second term in (VI.28) is bounded, uniformly

for all e > 0. The third term in (VI.28) can be bounded similarly, uniformly for all e > 0 and the proof of the theorem is complete. 0

Corollary VI.2.9 If b E g, then sup6>0 IIdLG(b)(el + H)-I/2e-H Il pip < 00 for all p E (1, oo). Hence the operator dLG(b)H-112e-H extends to a bounded operator on L p, if in addition G is not compact.

Notes and Remarks

261

Notes and Remarks Section VI.1 The first term in the asymptotic expansion of K in Theorem VI. 1.2, i.e., Corollary VI.1.13, has been derived for sublaplacians by Alexopoulos [Ale3], 1/1 by using a Corollary 1.14.7, but with the factor treplaced by Berry-Esseen estimate. The proof of Proposition VI. 1.8 follows the usual proof in homogenization theory for convergence of the resolvents. See [AvL3], Lemma 1. An injective and maximal accretive operator on a Hilbert space has a bounded Hoc-holomorphic functional calculus by [ADMI. Theorem G. On a Lie group,

if the semigroup kernel satisfies Gaussian bounds, then H has a bounded Hxholomorphic functional calculus on Lp(G) for all p E (1, oo) by [DuR], Theorem 3.4. This is used in the proofs of Lemma VI.1.10, Proposition VI.2.4 and Corollary VI.2.7.

Section VI.2 The identity (VI.24) comes from [Ale3], page 95, where it was used to prove asymptotics for the first-order derivative of the kernel in a general direction. Unfortunately a term like the third term in (VI.23) is missing in [Ale3]. In the case of the sublaplacian the boundedness of the first-order Riesz transforms in Theorem V1.2.5 was proved by Saloff-Coste [Sall], Theoreme 6, for p E (1.2] and by [Ale fl, Theorem 2, for p E (1, oo). The boundedness of all higher-order Riesz transforms on near-nilpotent Lie groups, Theorem VI.2.6, has been proved for the sublaplacian in [ERS2], Proposition 4.1 and for general H in [DER], Theorem 4.1. The operator B«H-1°II2S1 considered in Theorem VI.2.8 and Corollary VI.2.9 is equivalent to a Riesz transform at infinity introduced by Alexopoulos [Ale3], page 20. In [Ale3], Theorem 1.15.3, Alexopoulos proved that for a sublaplacian the first-order Riesz transforms at infinity are bounded on Lt, for all p E (1, oc).

Appendices

A. 1

De Giorgi estimates

The application of De Giorgi estimates is based in part on a Poincard inequality. A strong form of the Neumann-type Poincare inequality was established by Jerison

[Jer] for vector fields on Rd satisfying the H6rmander condition and the corresponding subelliptic balls. We need a weak Lie group version of Jerison's result. Throughout this section we assume that G has polynomial growth. Proposition A.1.1 There exist CN > 0 and RN E (0, 1] such that II

- (W)g"112.2 g,r -< CN r2 IIo'wII2,g,r

). for all g E G, r E (0, RN ] and cP E H?: I (Br '(g)).

Proof It follows from [Jer] that there exist cN, RN > 0 such that IIw - (OrII2.r 0 such that IIV'w112.g.ar < cr-111w - (4)g.r112.g.r

for all r E (0, 1], g E G and rp E H;.1(B'(g ; r)) satisfying Hip = 0 weakly on B'(g; r). 11.

If a E (0, 1) and for all r E (0, 1) there is a cut-off function rir E CO0(B'(r)) such that 0 < rir < 1 and hr = 1 on B'(ar), and in addition sup

max

rE(0.00) kE[i....,d')

r IIAkhrIloc < oo,

then

IIV'W112.g.ar < cr-t 11c' - (W)g.r112.g,r

f o r all r E (0, 1 ], g E G and rp E H2,1(B'(g ; r)) satisfying Hrp = 0 weakly on B'(g ; r), where d'

c = 2µc1 IICII L sup r II Akt1r Il,O k=1 rE(0,1)

and µc is the ellipticity constant.

Proof We follow the proof of the usual Caccioppoli inequality (see [Gia2], page 20). By translation invariance we need only consider g = e.

266

Appendix

Choose or and 17r as in Lemma A.1.2. Then AC11?7ro'w112 < Re(1]rO'co, Ct)rO'[o)

V'(w - ((p),), CV'w)

= d'

= Re L ([17 , A1](w - ((p)r), c,j Aj w) t.i=1

where the last step uses Hrp = 0 on B'(r). But 1172r, Ail = -2gr(A;r7r). Hence

AChrO'wll2 < -2 Re((O'gr)(V - MA Cr/rO'w) < 211CI1 II(V'17r)(49 - ((P)r)112 Ilnro'2112

Therefore IIV'wI12,ar 2. Fix n > 1 + D'/2. Then L2;n C L100;1 and one has Sobolev inequalities IIw1100 < n-D'12 NZ;n(co)

where N2;n denotes the

+cE-D'/'2IIwI12 ,

(A.3)

L2;n-seminorm, and

II'pI 00 < e'

-D'/2-'

N2,n(w) + C

E-D'/2-111W112

(A.4)

for some c > 0, uniformly for all e E (0, 1] and tV E L'2;n (see, for example, [Rob2] page 368). Moreover, 11(1 - L(g))wIloo < 21g1'

for all g E G. Next consider the (subelliptic) Holder space C"(G) of continuous functions over G for which sup O 0, uniformly for E E (0, 1] and tp E L2;n.

Now if (p is defined on B'(g; R), then we extend it to a function on G by setting it to zero outside B'(g ; R). Let a and 17R E CO0(G), for R E (0, 1], be as in Lemma A. 1.2. It follows from (A.5) that there exists a c > 0 such that (A.6)

IIInRWIIIC-

for all E E (0, 1], R E (0, 1] and W E L2;n. Note that (A.5) gives an estimate on L2 but this can be replaced by an estimate on L2 because the modular function is bounded on the ball B'(R) c_ B'(1). Now we are prepared to prove our first elliptic regularity result.

Lemma A.1.4 Let H = -

c,F, Ak At be a pure second-order subelliptic operator with complex coefficients ck,. I.

k:t=,

If R > 0 and tp E H2;I(B'(R)) satisfies Hcp = 0 weakly on the ball B'(R), then ,W E CO0(B'(R)) for all >G E C°0(B'(R)).

II.

For all a, 0 E J (d') there exist ca,a > 0 such that II Aa(Aa?]R)(pll2 0 such that IIA°V,112 fB,(g) IV - ((P)g,r12 is increasing. Moreover, l

(A.8)

0((P)g,r = W(g)

for almost every g E G, if W E L1,10c.

The Morrey and Campanato spaces are usually defined for G = Rd but their basic properties extend to the general Lie group situation. For example, one has the identities L2 n M2,y = L2 n M2,y if y E [0, D'), M2,D, = LOO and L2 n M2,y = L2 n C(y-D')/2' if y E (D', D'+ 2). Moreover, the norms on the spaces are equivalent. In the proof of Theorem 11. 10.5 we explicitly need the following inclusions and control of the norms of the embeddings. Lemma A.2.1

If 0 < y < D', then L2 n M2,y C_ M2,y, and there exists an a > 0,

1.

depending only on y, D' and the constant c in (11.46), such that y

< a (Ill 'I ll1,,t2 y + IIwll2)

for all ca E L2 n M2,y. IfO < y < D', V E L'2;1, is such that AkW E M2, y for all k E (1, ... , d'), then

II.

a (e2 1IIV'c llM2,y + E-(Y+') II(PII2)

for all 8 E (0, 2] and E E (0, 1], where a = 1 + c,, 2 +

RN-

(y+6)/2

with

RN and cN the constants occurring in the Poincare inequality (11.45). III.

If D' < y < D'+ 2, then L2 n M2,y a L,,, and there exists a c > 0, depending only on y, D' and the constant c in (11.46), such that IIwlloo -< E(y-D,)/2IIWIIM2.y +cE-D'12II(PI12

for all tP E L2 n M2,y and E E (0, 1]. In the second statement we have used the obvious notation IIV'wPIIM2.y = sup sup

( r-'

5EG rE(0,11 \

d'

J B;(g)

1/2

dh >

for all c E M2,y+s n L2;1, such that V'V E M2,y.

k=1

I1

272

Appendix

Proof If y E [0, D' + 2), w E L2 n M2,y g E G and 0 < r < R < 1, then r° I (w)g,r

- ((P)g.R12 < 20 fB;

Iw - (w)g,r12 + 2c

f

Iw - (w)g,R12

B: (g)

(g)

< 4E R' II IwIIIM2

Y'

so

(A.9)

I (w)g.r - (w)g,RI < 201 /2 r-D /2 Ry'2 II IwIIIM2,y

for all 0 < r < R < 1. Now we prove Statement I. If y < D', R E (0, 1] and k E No, then it follows from (A.9) that

k-l I((P)g.2-kR - ((P)g,Rl
0 with a > P there exist £ > 0 and C > 0 (depending only on A, a and P) such that for all B > 0, all Ro > 0 and all increasing functions 4: (0, R0] -* [0, oo) with the property that

(P (r) < A((r/R)°f +£)O(R) + BRP for all r, R E R with 0 < r < R < R0, one has the following estimates:

ot (r) < C((r/R)'4(R) + Br's)

forall0G)(g)I2 < 1. gEG k=1

Next, for each }G E D, define the family of bounded multiplication operators U,,, with p E R, by Upcp = e-P* p. Then SP is the strongly continuous semigroup on L2 given by

Sp=UPS,Up1 where we suppress the dependence on frin the notation. The generator Hp of SP is the sectorial operator associated with the form

hp((p) = h(w) + ph(1)(w) +

p2h121((P)

where D(hp) = D(h) = L';1, h(1)(w)

d'

d'

_ L Ckl (Akw, (A,*)(p) - L Ckl ((Ak*)w, Alw) k,1=1

k,1=1

and d'

h(2)((P)

L Ckl((AkVl)(P, (AI*)w) k,1=1

The starting point for the iterative argument to bound the L2 - LC'O crossnorm of SP is bounds on the L2 - L2 norms of SP and its derivatives. These can be expressed in terms of µ and IICII uniformly for ' E Di.

Lemma A.3.3 There exist a, co > 0, whose values depend on the coefficients C = (ckl) of H only through the ellipticity constant Et and the norm IICII, such that IISpII22 (14 cos 0 - 11C 11 1 sin 01)IIo'X II2 for all X E L'2; I . Moreover, since Vf E D11 , one has IReei°h(')(x)I 0, p E R, ' E Di and 9 E (-90, 90). Integrating this differential inequality gives the bounds ae°'(1+p2)Rez

(A.12)

11 SP II2-.2 5

for all p E R and Z E C\(0) with arg z E (-90,90). This includes the first bound of the lemma. One may now obtain the bounds on HpSp = -dSP/dt by using the Cauchy integral representation

SP = (2ni)-'

dz (z - t)-' SP

f,(1)

where Cr(1) is the circle of radius r centred at t and r = t sin(290). One readily obtains 11-2 II HpSPII2-.2 0 and w > 0, depending only on the parameters of Theorem 11. 10.5, such that Sf L2 C Loo for all t > 0 and p E R and (A.13)

11SPw11 oo

uniformlyforwEL2,t>0,pERand*EDi. Proof First, SP L2 C L;.1 and there exist a, w > 0, depending only on µ and IICII, such that at-1/2em(1+P2)`Bwll2

11SPw112 0, p E R and k E (1, ... , d'}. Moreover, there exist by, wy > 0, depending only on A. IICII, V- CDG and y such that IISPQUM2,,. G. Therefore HSP(p = SPHp(p - c )SPtp

d'

d'

k=1=1

k=1

- L C'P)AkSP(p - > Akck(P)SPV

weakly. Now if t < 1 we apply Proposition A.3.1 to SPCA with r identified as the first three terms on the right, rk = ck (P) SP cp and e = t 1 14.

First, one has E-(Y+a)IISfpIIz;I < a t-(y+6)14t-112ea(I+P2)1IIw112

by the L2-estimates of Lemma A.3.3. Secondly, SPHpcp E SP(DO0(Hp)) C M2,y and E2-a11SPHpcGII

E2-a11

M2.' =
0, p E R and tG E D,. Hence /2aa't-D'l2e(wva)(I+p2)re-P(*(g)-L(e))

IK,(g)I < 2D

forallgEG,r>0,pERand4rE Di. But IgI' = inff J k(g) - t(e)1 : rJr E Dj } for all g E G by (11.17). Minimizing with respect to 1G and p gives

IKt(g)I
0

If K64"' denotes the kernel of the semigroup S°"'u', then for all w E GZ Y) (Gu ) one has 11(1- S°lU')wIIL2(Gu) = fGY dg Kl"1YI(g)11(1- LGY(g)wIIL2(GY)

IIwIIC2 yl(Gu)

dg K(g) (181u b) t-1/2)y

tr/2( fG"

for all t > 0. So it suffices to show that the quantity between brackets is bounded above uniformly in t and u. Now KA"u'(g) = uDKu ""(I'u(g)) by (IV.15). Also there exist bl, b2, c2 > 0 such that c21W(b)(t))-1/2e-b)(Igjjb))2j-)

< Ke"''(g) '2e-b2(Ix11(0)2t-)

c2(V'(b)(t))-

uniformly for all t > 0 and g E G, where V'(b) (t) = 1(g E G : Ig 1 i b) < 1)1 (see (11.25)). Since G has polynomial growth there exists a c4 > 0 such that V'(b)(wt) < C4 V'(b)(t) uniformly for all t > 0, where w = 2b, b2 1. Using the identity ulglub) = Iru(g)I'(b) it follows that K°Rul(g)

(Igl;;b) t-1/2)y (V'(b)(u2t))-1/2e-bz(IglRb))2j-)

< C2 uD


0 and rp E C, b, S.u

bedded in the space L2

;

is continuously em-

So the space G2 Y)

with norm bounded by 1 + c2 c3 c4/`

y

Thirdly, for all y E (0, 1) let (L2(G.).

be the interpolation

space by the K-method of Peetre, where L26)(G.) is the space with respect to the

algebraic basis uyu 1(b-d ), ... , uy 1(bd, ). Then {rp E L2(Gu) : suet-YK,)t) < 001, r>0

where K(u)(t) = inf{lIw - (PI IIL2(G.) + tIIw1IIL26, : W1 E L26 )(G.)} for all

E

L2(Gu). The norm on the space (L2(G.). L2b))y,a;K equals = IItPII L2(G.) + Supt-YK(u)(t) t>0

(b

II(PII

Then

-y11(I -

t-YKQ(t)

(P11L2b)(G

I1WIIL2, + tt

2I

PIIL2(G.) +

11S

for all i > 0. But S°I''

00

=

ds SA1y1

J

f``

Also he-'h < s-le-Jh(esh - 1) = s-1(1 - e-ft') for all s > 0 and h > 0. So by spectral theory

t1-r II

..r

AINI

t1-YI ids r

2'

*IIL2(G.)

r2

< 2t1-Y

j

= 21-r/2(1

00

-

ds s-3/2 (s/2)Y1IIWlILs.. Y)-1

IIrPIILS-

A.4 Rellich lemma

285

for all W E LZ.Y and t > 0. So

1-'K,,(t) < (1

+21-y/2(1

- Y)-1) IIwI1Ls"' +

for all t E (0, 1] and the space Ls*u is continuously embedded in the interpolation

space (L2(Gu), L2.bj)(Gu))y,oo;K with norm bounded by 1 + 21-y/2(1 - y)-1. Fourthly, it follows from Statement V of Proposition III.6.1 that ((dLGN(a))(p)(g) = d

I =-dm

for all a E g, W E CO0(Gu) and g E G. Since the T(q) are orthogonal transformations which leave the spaces m and 1)1 invariant, it follows that there exists a c > 0 such that IIBA0112 < C

uniformly for all i E IIBrcv112 0 and W E CO0(Gu). Hence max JE( dm..-d1)

1(bj))wI12

(A.16)

uniformly for all i E {-dm, ... , d1), u > 1 and V E C°O(Gu). Then by density (A.16) is valid for all rp E LZd)(G,,). Since b_d...... bd, is an algebraic basis for ON, one can define the space L2;1(GN) with respect to this algebraic basis. Then LZd)(Gu) is continuously embedded in L2;1(GN) and the norm of

the inclusion map is bounded by 1 + c by (A.16), if u > 1. Hence, by interpolation, the space (L2(Gu), LZd)(Gu))y,oo;K is continuously embedded in (L2(GN), L2;1(GN))y,oo;K and the norm of the inclusion map is bounded by

1+c. Fifthly, if G2;y(GN) denotes the Lipschitz space on GN with respect to the algebraic basis b_d,,,, ... , bd,, then the space (L2(GN), L2;I (GN))y,oo;K is continuously embedded in GZ;y(GN) and the norm of the inclusion map is bounded

by2+p+dl. Sixthly, if G2;y(GN) denotes the Lipschitz space on GN with respect to the basis b_d,,, ... , bd and r is the rank of the algebraic basis b1 , ... , bd, in qN, then the space GZ;y(GN) is continuously embedded in L2;yl,(GN) by (11.15).

Combining the six maps one deduces that for all u > 1 the space L2;1(Gu) is continuously embedded in G2;1/(r,)(GN) and the norm of the inclusion map is bounded by a constant M', independent of u.

286

Appendix

At this stage we can prove the first part of Condition II in Theorem A.4.1 For all n E N and g E B(1) one has II(/ - LGN(g))(VfhIL2(QXQ1) s IgIGNrs) IIwnhIL2:1/(,s)(GN)

M'

IBIGNrs)IIwnhILz:1(G.)

M'(1 +d')C

IgIGNrs)

uniformly for all n E N, where I IGN is the modulus on GN with respect to the basis b_dm, ... , bd. So the functions (pn are equicontinuous with respect to left translations on GN, which is the first part of Condition II in Theorem A.4.1. Finally we also have to show that the functions 0n are equicontinuous with respect to right translations on GN. It follows, as in the proof of the second and third step, that the space G2:I/(rs)(GN) is continuously embedded in the interpolation space (L2(GN), L2;1(GN))1/(rs),oo;K Let T E C°O(GN) be such that 0 < r 1 the space L'2.1(Gu) is continuously embedded in C' norm of the inclusion map is bounded by a constant M, independent of u. Then 11(1 - LGN(g))w112 1, cp E L'211(Gu) and g E GN.

Notes and Remarks Appendix A.1 The main reference for Neumann-type Poincarb inequalities for vector fields satisfying the Hormander condition is the paper of Jerison [Jer]. This paper is used in the proof of Proposition A.1.1. Since Iw(g) -w(hg)12 < (1 +e)2(Ih1')2

f

1

d'

dt (IAkw>(Y(t)g)12

0

k=1

where y: [0, 1 ] -> G is an absolutely continuous path from e to h with length less than (1 + e)IhI' one can average over the product B' x B' to obtain an estimate 2

/

2

IIw - ((P)g.r II2 .g,r < 5r2I B2r I I Br l -' II V'w I12.g,3r'

Then the doubling property of the volume gives a Poincare-type inequality but with a ball of triple the radius in the upper bound. It is not too difficult to deduce a comparable inequality with a ball of radius rr, with r > 1 on the right, but to show that one can take r = 1 requires some delicate reasoning using a Whitney-type covering. A simplified version of the proof of Jerison is in SaloffCoste [Sal2], Theorem 5.6.1, together with additional references to related material.

The other results in this section are extracted from [EIR8], Sections 2 and 3. The proof of the Caccioppoli inequalities in Lemma A. 1.3 comes from Giaquinta [Gia2], page 20. Appendix A.2 A good source for Morrey and Campanato spaces on domains in R" are the books [Gial] and [Gia2] of Giaquinta. Lemma A.2.1 comes from the appendix in [EIR8] and the proof of Proposition 3.2 in [Gia2].

288

Appendix

Appendix A.3 This section is extracted from Section 4 in [EIRE], which is inspired by the paper [Aus] of Auscher, which is in turn inspired by Chapter III in [Gial]. In particular, Lemma A.3.2 equals Lemma 111.2.1 in [Gial]. The main onus in the Lie group setting is the lack of scaling and this has been replaced by interpolation at many places. Appendix A.4 The Fre chet-Kolmogorov theorem, Theorem A.4.1 has been proved in [Rob2], Appendix D.1.3. On a general Lie group one has to take the LP-spaces with respect to the right Haar measure. The proof of Statement II of Lemma IV.5.7 in the appendix uses various embeddings between Lipschitz spaces and all the parts are in the paper [EIR2] for one fixed group. But in the current setting we had to obtain a constant uniformly in the scaling parameter. This statement has been proved first in [EIR10], Lemma 2.2, In addition, [EIR101 gives a version which does not assume that g has a stratified nilshadow.

References

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Index of Notation

ai.... ,ad, algebraic basis

a[a]

[air, [a;2, ... [a;,,-,

14 15

Al (= dLG(ai))

14

A' (= A;, ... A,,,)

15

Ak[°l (= ad adjoint representation A(G)

133 8

52

Aut(G) group of automorphisms of G Aut(g) group of automorphisms of g

b_d,,,,...,bd

12.5

B; (=dLGN,(bi))

1216

B1N) (=dLQN,(b1))

142.

Bp = B'(p)

24

B(p) = B'

25

B'(g)

49

B'(°)(p) ball on G

L63

300

Index of Notation

B(M)(p) ball on M

B'(N)(p) ball on QN C matrix of coefficients IICII norm coefficients Cn,

Cki d' x d' matrix of constants Cjj

CN

C(G) continuous functions Cb(G) bounded continuous functions Cb,,, (G)

C,(G) continuous functions with compact support CO0(G) infinitely-differentiable functions

CO0(G) infinitely-differentiable functions with compact support

C"'(G)

d (=dim G) d,,, (= dim m - 1)

ds (=dims- 1) do (=dim n)

dt (=dimh1) d(g ; h) distance d'(g ; h) subelliptic distance D dimension at infinity

D' local dimension

D't dg Haar measure dL(a) left representative

Index of Notation

301

dR(a) right representative d U (a) generator

dU(a)° e identity element Ea E En

En,o

E3 Euclidean motions group O

E3 covering group, Euclidean motions group exp exponential map

g(-I)" inverse of g in GN

L12

G Lie group

6 covering group

12

G,

15

Gsn near-nilradical

Gb,, (Gb,,(g) =

127

V'(t)-1/2 e-b(IRI )2 )

G(d', r) free Lie group GN shadow

19 11

107, 112

GN semidirect shadow GT

G.

97

(GN)S.)

132.

H operator on G HY fractional power

H* dual operator on G H homogenization, operator on QN HS

HS Hp homogenization, operator on GN

142

Hluj operator on G.

133

21 (= 9C)

128

Index of Notation

302

xn

(/-P.(0))xn)

'H, H2-,j

(n)

H2 )(S2) f'2; I HZ(i)(c2) HH(N)(c2')

(d')

J+(d') Jn (d')

J (d') J (g)

J+(9) J (n)

J+(n) J (a)

J+(s) K kernel of S K kernel on QN Of 'S

Kt kernel of S' KI") kernel of SI")

K(a) nilpotent part Jordan decomposition

L = LG ((LG(h)(p)(g) = LP = Lp(G)

Lp;n = Lp;n(G) Lp;oo = Lp;oo(G) L'p;n = Lp,n(G) M Levi subgroup of G

M mean on A(G) M2,y Morrey space

W(h-ig))

Index of Notation

303

M2y Campanato space

271

Nm seminorm

191

P projection

38

P (= fey dm LGN (m))

151

PM (= & dm LM (m))

222

JG. dsLG(s))

L85

P(0) mean ergodic projection

54

Pa

P.(0)

L3Q

Q radical of G

106,113

QN r rank of qN

70

R = RG ((RG(h)(G)(g) = cp(gh))

13

91C (= 2-1(C + C'))

if

91H (= 2-1(H + H')) RN

5Q

S semigroup on G

17

S semigroup generated by H on QN

semigroup on G

217 134

30 semigroup on GN generated by ho

224

S ( G -> Aut(G))

146

S (S,,: GN -+ AUt(GN))

132

J

110

(SR: Go

N-

AUt(GN))

Sn

129

Sn

130

S (S,,: GN -> Aut(9N))

L32 L45

S(a) semisimple part Jordan decomposition

14

T

95

Index of Notation

304

T (= {zEC:Izl=1}) T

TeG tangent space at e

U representation Up

V'(P)=IB,I) v'(u)(P) (= IB't"t(P)I) Wk

W.

X representation space

X;, = X'(U) space of C"-vectors X00 = Xoo(U) space of C°O-vectors

X,,.p(U) (Yo, YI)y,p;K

Z(G) centre of G Yu Yu : 9

0

Yu Yu(bk) = uu'k bk 0

I' I' = ker A

i'u Iu:G-+G ru ru : G u -> G

FU ?uco=cooru Sa

A modular function

A(0) (= {z E C\{0} : I argzI < 9}) sY

rf, cut-off function

Oc (= arctan(i,t IICII-')) KY 0

0

0

A A: G -> G µ ellipticity constant b,c

(6,c(8) = (S(g)b, c))

Index of Notation

305 77

oa

L29

au

132

Orq,

74 76

r TB

Tq

(pb (?(q) = fu dm co(r, q)) Xi first-order corrector Xa WI'

p

g Lie algebra G 9

ON (= m x qN) gN semidirect shadow or 0. (_ (9N)ou)

g(d', r) free Lie algebra hk subspace of qN

132

9 28, 71

C subspace of qN

71

Co subspace of qN

7-1

Ei subspace of qN

71

m Levi subalgebra m

9 21

n nilradical n,,, near-nilradical

q radical 9

qk lower central series qN nilradical

a

69

306

Index of Notation

s semisimple part near-nilradical o complementary subspace V

ro(m, o)

m(o) (= {b E q : S(o)b = {0}}) 3(g) a1

Hall

Ilalll

[, IN Lie bracket on qN [

]N Lie bracket on ON

,

]N Lie bracket on gN

[

Jr

I

I

I modulus

I subelliptic modulus

I,, modulus on G 11

III

norm on X;,

II

IIIC''

111.111.

III

IIIAt2,Y

II

Ilp norm on Lp

II

Ilp;n norm on Lp;,,

II

III

;n

norm on Lp n

II' Ilp-*r II

11

11

II

112,n

112,g,r (= 11 112. N,,

112,u,r

= II

( = II

112,B'(g;r)) '

' 112,u.B'(ld)(r))

165

165

Index of Notation

307

* convolution ((w * Vi) (g) = fG dh ip(h) * (h -g))

* = *G multiplication on G *N multiplication on GN

112

T* multiplication on GT

97

1 (1(g) = 1) x semidirect product >a semidirect product

Index

accretive operator, 125, 261 adjoint form, 12 adjoint operator, 17 adjoint representation, 8

canonical behaviour, 209 211 Caratheodory, 22

adjoint semigroup, 17 20 affine transformations, 68, 70 algebraic basis, 14 almost periodic function, 52 62 128 almost periodic representation, 54 62 anomalous behaviour, 205 asymptotic expansion, 219 223, 247

Casimir operator, 1$4_

ball, 24 Berry-Esseen estimate, 261 Beurling-Deny criteria, 58, 173 boundary conditions Dirichlet, 50 157 Neumann, 50 L56 bounded continuous function, 14 bounded representation, 54

closed subgroup, 12 C"-element, 15, 40

Caccioppoli inequality, 165, 2287

compact resolvent, 38 60 compatible stratification, 91 conjugacy theorem of Mal'cev-Har-

Cartan subalgebra, 9 56 64 65 67 80, 81, 121, 122 Cauchy integral, 277 Cauchy representation, 202 centre of Lie algebra, 12 66 centre of Lie group, 12 110 L19 characteristic ideal, 85, 88 C°O-element, 15 C'-space, 15, 40 42

C"-space, 15 40 42 43, 55 coefficients, 217 221 230 homogenized, 135 142 143 145 149 239

real, 21, 31 5$

compact Lie group, 26 27 38 60 114, 182

Campanato space, 62 271, 287 Campbell-Baker-Hausdorff formula, 8

ish-Chandra, 56, 8

L22

310

Index

ergodic theorem, 5,4 62 138, 11

connected Lie group, 8 continuous function, 1.4 continuous representation, 15 contraction, 35

239

Euclidean motions group, Q. 33 59

68 79 85 106,

contraction semigroup, 17, 21 58, 59 convolution, 20 corrector, 138. 139. 175, 217, 220. 222 230

009

110, 113, 241 covering space, 59

cut-off function, 213, 263-265 Davies perturbation, 32, 201, 276 Davies perturbation method, 60 De Giorgi constant, 49 269 De Giorgi estimates, 49 51,6_2 166 170, 177, 269, 273 derivation, 10, 1 L 65 70 72-74, 96

100 102 143 derivative bounds, 196 dilation, 8 11, 47, 131 dimension

at infinity, 26 28 29 homogeneous, 35 local, 25 Dirichlet boundary conditions, 50 151

discrete subgroup, 2 34 distance, 22-24, L16

subelliptic, 2, 59 270 dual exponents, 14 dual representation, 14 15 dual space, 15 C°O-element, 15

C"-element, 15, 40 ellipticity constant, 16 equivalent modulus, 22 23 equivalent representations, 54

12 L

exponential growth, 26 exponential map, 8

faithful, 53 11 L44

first-order, 138, 221, 239 second-order, 212 corrector equation, 136 175 corrector space, 129 covering group, 12, 23, 46, 59,

0099

128, 140, 146, 195, 209, 220

form, 16 adjoint, 17 invariant bilinear, 27 symmetric bilinear, 27 Fourier theory, 21 36 171 fractional power, 17 56 5.8 free Lie group, 11 free nilpotent Lie algebra, 9 Fr6chet-Kolmogorov theorem, 281, 288

gap, spectral, 40 190, 212 Gaussian, 31 Gaussian lower bounds, 31 Gaussian upper bounds, 31 171, 1966 generating basis, 14 generating subspace, 72 generator, 9 14, 15 graded, 9 grading, _9 35

Haar measure, 2 44 57 132 211 Heisenberg group, 29, 112 Heisenberg Lie algebra, 36 75 Hilbert-Schmidt operator, 1$2 HOO-holomorphic functional calculus, 22 8, 254, 257, 261 homogeneous dimension, 29 35 133 homogeneous Lie group, 8 1 L 60 homogenization, 141, 149 175 homogenized coefficients, 135, 142 143 145, 149, 239 homogenized operator, 141,149,175

homomorphism associated with T, 96

Index

homomorphism associated with r, 95 infinitely-differentiable function, 14 infinitesimal generator, 14 interpolation space, 55, 194, 212, 283, 284 interpolation theory, 20, 55, 57, 212, 223, 270

invariant bilinear form, 27 invariant subspace, 10, 67 isoperimetric inequality, 32 Jacobi identity, 3, 46

Jordan decomposition, 10, 67, 78, 79

K-method of Peetre, 55, 194, 284 kernel, 20 Laplace transformation, 19, 189, 227, 229 left regular representation, 13 length, 15, 151 Levi subalgebra, 9, 10, 12, 56 Levi subgroup, 12 lift, 11

lifting, 59 Lipschitz space, 56, 282 local dimension, 25, 29 local direct product, 34, 20,9 211 local Gaussian bounds, 36 lower bounds, 31, 174, 205

lower central series, 8, 28, 29, 70, 85, 91, 131

maximal compact subgroup, 12, 28, 119, 121

mean, 52, 53, 62 faithful, 53, 137, 144 mean ergodic theorem, 54, 62, 138, 149, 239

311

Nash inequality, 32, 58 near-nilpotent Lie algebra, 88, 90, 209 Lie group, 34, 60, 209 211, 256, 257

near-nilpotent ideal, 87 near-nilradical, 88, 89, 127, 206 Neumann boundary conditions, 50, 156

nilpotent endomorphism, 9 nilpotent Lie algebra, 8 nilpotent Lie group, 11-13, 28, 46 nilpotent set, 10, 65 nilradical, 9, 10, 12, 72, 86, 119 nilshadow, 69, 73, 106, 113, 121 stratified, 91, 131, 171, 238 uniqueness, 78, 82

normalizer, 9

operator accretive, 125, 261 adjoint, 17 Hilbert-Schmidt, 182 homogenized, 141, 149, 175 real, 21, 31 strongly elliptic, 16 subelliptic, 16

parametrix method, 35-37, 48, 59 Peetre, K-method, 55 Poincar6 inequality Dirichlet-type, 49, 274 Neumann-type, 50, 51, 166, 177,

263.271.287 polynomial growth, 26, 114 positive kernel, 21, 31 positive semigroup, 21, 58 predual space, 15 projection, 38, 54, 72, 130, 138, 151, 185, 222, 239

modular function, 12, 13, 21, 26, 57 modulus, 22-24, 116-119 equivalent, 22, 23

quadratic form, 16

Morrey space, 62, 270 287 multi-index, 15 multi-index notation, 14

radical, 8, 12 rank, 8, 14, 70

312

Index

real coefficients, 21, 31, 58 real operator, 21, 31 Rellich lemma, 159, 281 representation, 8, 15 adjoint, 8

almost periodic, 54

bounded,54 equivalent, 54 left regular, 13 representation associated with T, 96 representative, 15 left, 14 representative, right, 14 resolvent, 19 Riesz transform, 37, 39, 48, 60, 61, 253, 255, 261 Riesz transform at infinity, 261 scaling, 131 scaling identity, 36 second-order corrector, 217 semidirect product, 11, 28, 57 semidirect shadow, 76, 106 semigroup, 17 adjoint, 17, 20 contraction, 17, 21, 58, 59 positive, 21, 58

semisimple endomorphism, 10 semisimple Lie group, 8, 11, 27 semisimple representation, 10 shadow, 77, 107, 112 simply connected subgroup, 12 Sobolev inequality, 32, 266 Sobolev space, 15, 48, 156, 157, 165 solvable Lie algebra, 8 solvable Lie group, 1 I SO(3), 208, 209 C°O-space, 15, 40, 42

C"-space, 15, 40, 42, 43, 55

spectral gap, 40, 190, 212 spectral property of holomorphic semigroup, 186 spectral theory, 38 spectrum, 38, 182, 190 stratification, 9, 91 compatible, 91 stratified Lie algebra, 9, 91 stratified Lie group, 11, 29, 46, 54 stratified nilshadow, 91, 131, 171,238 strongly elliptic operator, 16 subelliptic derivative, 201 subelliptic distance, 23, 59, 270 subelliptic Gaussian, 31 subelliptic modulus, 23 subelliptic operator, 16 symmetric bilinear form, 27

tangent space, 8 tensor product, 129, 177, 208

transference, 40, 48, 61, 136, 173, 204, 212, 238 Trotter-Kato theorem, 176 twisted Lie bracket, 73, 122 twisted product, 95 type R, 26-28, 31, 59, 68, 77, 83, 84, 89, 91, 114, 121

unimodular Lie group, 13, 26 upper bounds, 31 volume, 25 volume growth, 59 weakly (on S2), 49 weight, 131, 150, 181, 196 weighted length, 151, 181, 196 Weil formula, 44, 61, 188, 241

Nick Dungey, A. F. M. ter Elst, and Derek W. Robinson Analysis on Lie Groups with Polynomial Growth

on Lie Groups with Polrnontiul 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 ssith the theory of second-order. right invariant. elliptic operators on a large class of manifolds: Lie groups sith polynomial grussth. In s%stematically developing the analytic and algebraic background on Lie groups with polynomial growth, it is possible to describe the large time behas for for the semigroup generated by a complex second-order operator ssith the aid of homogenization theon and to present an asi mptolic expansion. Further, the text goes beyond the classical homogenization theon by converting an analtical problem into an algebraic one. .-tnuhvsis

her Jeature%:

Completely self-contained ssork. including a review of sell-established local theory for elliptic operators and a summan of the essential aspects

or Lie group theon Numerous illustrative examples :Appendices covering technical subtleties

:1n exhaustive bibliographN and index This work is aimed at graduate students as sell as researchers in the above areas. Prerequisites include knowledge of basic results from semigroup the-

on and Lie group theory.

Birkhditser ISBN 0-8176-3225-5 sssss.hirkhaw,(:r.Cont