Stratified Lie groups and potential theory for their sub-Laplacians

Springer Monographs in Mathematics A. Bonfiglioli • E. Lanconelli • F. Uguzzoni Stratified Lie Groups and Potential ...

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Springer Monographs in Mathematics

A. Bonfiglioli • E. Lanconelli • F. Uguzzoni

Stratified Lie Groups and Potential Theory for their Sub-Laplacians

A. Bonfiglioli Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]

F. Uguzzoni Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]

E. Lanconelli Università Bologna, Dip.to Matematica Piazza di Porta San Donato 5 40126 Bologna, Italy e-mail: [email protected]

Library of Congress Control Number: 2007929114

Mathematics Subject Classification (2000): 43A80, 35J70, 35H20, 35A08, 31C05, 31C15, 35B50, 22E60

ISSN 1439-7382 ISBN-10 3-540-71896-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71896-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author and VTEX using a Springer LATEX macro package Cover design: WMXDesign, Heidelberg, Germany Printed on acid-free paper

SPIN: 12029525

VA 41/3100/VTEX - 5 4 3 2 1 0

To Professor Bruno Pini and to our Families

Preface

With this book we aim to present an introduction to the stratified Lie groups and to their Lie algebras of the left-invariant vector fields, starting from basic and elementary facts from linear algebra and differential calculus for functions of several real variables. The second aim of this book is to perform a potential theory analysis of the sub-Laplacian operators m Xj2 , L= j =1

where the Xj ’s are vector fields, i.e. linear first order partial differential operators, generating the Lie algebra of a stratified Lie group. In recent years, these operators have received considerable attention in literature, mainly due to their basic rôle in the theory of subelliptic second order partial differential equations with semidefinite characteristic form.

1. Some Historical Overviews General second order partial differential equations with non-negative and degenerate characteristic form have appeared in literature since the early 1900s. They were first studied by M. Picone, who called them elliptic-parabolic equations and proved the celebrated weak maximum principle for their solutions [Pic13,Pic27]. The interest in this type of equations in application fields was originally found by A.D. Fokker, M. Planck and A.N. Kolmogorov. They discovered that partial differential equations with non-negative characteristic form arise in the mathematical modeling of theoretical physics and of diffusion processes [Fok14,Pla17,Kol34]. Since then, over the past half-century, this type of equations appeared in many other different research fields, both theoretical and applied, including geometric theory of several complex variables, Cauchy–Riemann geometry, partial differential equations, calculus of variations, quasiconformal mappings, minimal surfaces and convexity in sub-Riemannian settings, Brownian motion, kinetic theory of gases,

VIII

Preface

mathematical models in finance and in human vision. We report a short list of references for these topics at the end of this preface. A first systematic study of boundary value problems for wide classes of ellipticparabolic operators was performed by G. Fichera. In 1956 [Fic56a,Fic56b], he proved existence theorems of weak solutions of the “Dirichlet problem” and found the right subset of the boundary on which the data have to be prescribed. Some years later, several existence and regularity results for elliptic-parabolic operators were proved by O.A. Ole˘ınik and E.V. Radkeviˇc and by J.J. Kohn and L. Nirenberg (see the monograph [OR73] for a presentation and a wide survey on this subject). The methods used by these authors required particular assumptions on the Fichera boundary set and led to regularity results strongly depending on the regularity of the boundary data. 1.1. L. Hörmander’s Theorem The investigations of the local regularity properties of the solutions to ellipticparabolic equations, that is, regularity properties only depending on the given operator, have produced more interesting results. The most beautiful ones have been obtained for elliptic-parabolic equations with underlying algebraic-geometric structures of sub-Riemannian type. The milestone of these research field is a celebrated theorem of L. Hörmander proved in 1967. Theorem 1 (L. Hörmander, [Hor67]). Let X1 , . . . , Xm and Y be smooth vectors fields, i.e. linear first order partial differential operators with smooth coefficients in the open set Ω ⊆ RN . Suppose rank Lie{X1 , . . . , Xm , Y }(x) = N ∀ x ∈ Ω. (P.1) Then the operator L=

m

Xj2 + Y

(P.2)

j =1

is hypoelliptic in Ω, i.e. every distributional solution to Lu = f is of class C ∞ whenever f is of class C ∞ . Condition (P.1) simply means that at any point of Ω one can find N linearly independent differential operators among X1 , . . . , Xm , Y and all their commutators (the Lie algebra generated by {X1 , . . . , Xm , Y }). Hörmander’s work opened up a research field, the most remarkable contributions to which have been given by G.B. Folland, L.P. Rothschild and E.M. Stein. They developed and applied to (P.2) the singular integral theory in nilpotent Lie groups.1 1 The application of this theory also occurs in the developments started from the works by

¯ J.J. Kohn on the ∂-Neumann problem and the ∂¯b complex.

1. Some Historical Overviews

IX

By using these techniques, in 1975, G.B. Folland accomplished a functional analytic study of sub-Laplacians on stratified Lie groups [Fol75]. One year later L.P. Rothschild and E.M. Stein proved their celebrated lifting theorem (see [RS76]), enlightening the basic rôle played by the sub-Laplacians in the theory of second order partial differential equations which are sum of squares of vector fields. In force of this theorem, indeed, we can roughly say that: m 2 Every operator L = j =1 Xj satisfying the Hörmander rank condition “as close as we want” to a sub(P.1) can be lifted to an operator L Laplacian. 1.2. The Rank Condition The geometrical meaning of the rank condition (P.1) is clarified by the C. Carathéodory, W.L. Chow and P.K. Rashevsky theorem: If (P.1) is satisfied, then given two points x, y ∈ Ω, sufficiently close, there exists a piecewise smooth curve, contained in Ω and connecting x and y, which is the sum of integral trajectories of the vector fields ±X1 , . . . , ±Xm , ±Y . The appearance of (P.1) in Hörmander’s theorem seems to be suggested by some deep properties of the Kolomogorov operators (see also the Introduction in [Hor67]), which we now aim to discuss. In studying diffusion phenomena from a probabilistic point of view, A.N. Kolmogorov showed that the probability density of a system with 2n degrees of freedom satisfies an equation with non-negative characteristic form Ku = 0 in R2n × R, where R2n is the phase-space of the system. A prototype for K is the following operator K=

n j =1

∂x2j +

n

xj ∂yj − ∂t ,

(P.3)

j =1

where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) denote the velocity and the position vectors of the system, respectively. The operator K is “very degenerate”: its second order part only contains derivatives with respect to the variables x1 , . . . , xn . Nevertheless, as Kolmogorov showed, it has a fundamental solution Γ which is smooth out of its pole. This implies that K is hypoelliptic, that is, every distributional solution to Ku = f is of class C ∞ whenever f is of class C ∞ . The explicit expression of Γ is given by Γ (z, t; ζ, τ ) = γ ζ − E(t − τ )z, t − τ , z = (x, y), ζ = (ξ, η), (P.4) where γ (z, t) = 0 if t ≤ 0, and

(4π)n 1 −1 γ (z, t) = √ exp − C z, z if t > 0. 4 det C(t)

(P.5)

X

Preface

Here, ·, · stands for the usual inner product in R2n ; E(t) and C(t), respectively, denote the 2n × 2n matrices

t 0 0 , C(t) = E(s)A E(s)T ds. E(t) = exp −t In 0 0 Moreover, In denotes the identity matrix of order n and A = I0n 00 . We explicitly remark that C(t) > 0 for every t > 0. (P.6) This condition makes expression (P.5) meaningful and can be restated in geometrical– differential terms. Indeed, denoting Xj = ∂xj and Y = nk=1 xk ∂yk − ∂t , it can be proved that (P.6) is equivalent to the following rank condition: rank Lie{X1 , . . . , Xn , Y }(z, t) = 2n + 1 ∀ (z, t) ∈ R2n+1 .

(P.7)

It is also worthwhile to note that the Kolmogorov operator K can be written as K=

n

Xj2 + Y.

(P.8)

j =1

1.3. The Left Translation and Dilation Invariance The structure (P.4) of Kolmogorov’s fundamental solution suggests the relevance that a Lie group theoretical approach has in the analysis of Hörmander operators. Indeed, from the explicit expression of Γ one realizes that Γ (z, t; ζ, τ ) = γ (ζ, τ )−1 ◦ (z, t) , where ◦ is the following composition law making K := (R2n × R, ◦) a noncommutative Lie group (z, t) ◦ (z , t ) := z + E(t ) z, t + t , i.e. more explicitly, (x, y, t) ◦ (x , y , t ) = x + x , y + y + t x, t + t . In K one has (ζ, τ )−1 = (−E(−t) ξ, −τ ). It is easy to check that K is invariant w.r.t. the left translations on K and commutes with the following dilations: dλ (z, t) := (λx, λ3 y, λ2 t),

λ > 0.

For every λ > 0, dλ is an automorphism of K, so that (R2n × R, ◦, dλ ) is a homogeneous Lie group. It can be seen that its Lie algebra is the one generated by the vector fields Xj = ∂xj and Y = nk=1 xk ∂yk − ∂t appearing in (P.8).

1. Some Historical Overviews

XI

1.4. The Elliptic Counterpart: Stratified Groups and Sub-Laplacians For a proper comprehension and appreciation of this type of “parabolic”-type operators such as the above Kolmogorov operator K, it is crucial to possess a deep knowledge of their “elliptic” counterpart. This seems unavoidable, also bearing in mind that the underlying algebraic–geometric structures of these two different classes of operators are almost identical. Let us go back again, for a moment, to the Kolmogorov operator (P.3). If in that operator we square the term Y = nj=1 xj ∂xj +n − ∂t , we obtain the following “sum of square”-operator (which we may refer to as the “elliptic counterpart” of K): L :=

n j =1

∂x2j +

n

2 xj ∂xj +n − ∂t

(P.9)

.

j =1

The characteristic form of L is a non-negative quadratic form with non-trivial kernel. Then L has to be considered as a degenerate elliptic operator. However, it is hypoelliptic: the Hörmander rank condition (P.7) does not distinguish between L and K! Moreover, L is left-invariant on (R2n × R, ◦) (as we already know, so are the ∂xj ’s and Y ) but, this time, it commutes with the dilations δλ (z, t) = (λx, λ2 y, λ2 t),

λ > 0.

Also these dilations are automorphisms of (R2n × R, ◦), and G := (R2n × R, ◦, δλ ) becomes a stratified Lie group whose generators are the vector fields ∂xj ’s and Y . Then, according to our general agreement, L is a sub-Laplacian2 on G. 1.5. The Heisenberg Group In the lower-dimensional case n = 1, the operator (P.9) is ∂x2 + (x ∂y − ∂t )2 ,

(x, y, t) ∈ R3 .

(P.10)

Up to a change and a relabeling of the variables, this can be written as follows: (∂x2 + 2 y ∂t )2 + (∂y − 2 x ∂t )2 ,

(x, y, t) ∈ R3 ,

which, in turn, is the lower-dimensional version of the celebrated sub-Laplacian on the Heisenberg group. The Heisenberg group Hn is the stratified Lie group (R2n+1 , ◦) whose composition law is given by (z, t) ◦ (z , t ) = z + z , t + t + 2 Im z, z . (P.11) Here we identify R2n with Cn , and we use the notation 2 All these notions will be properly introduced in Chapter 1.

XII

Preface

(z, t) = (z1 , . . . , zn , t) = (x1 , y1 , . . . , xn , yn , t) for the points of Hn . In (P.11), ·, · stands for the usual Hermitian inner product in Cn . The dilation δλ (z, t) = (λz, λ2 t) is an automorphism of

Hn

(P.12)

and the vector fields

Xj = ∂xj + 2 yj ∂t ,

Yj = ∂yj − 2 xj ∂t

are left-invariant on (Hn , ◦). One readily recognizes that the following commutation relations hold: [Xj , Yj ] = −4 ∂t

(P.13a)

[Xj , Xk ] = [Yj , Yk ] = [Xj , Yk ] = 0 ∀ j = k.

(P.13b)

and

Identity (P.13a) is the canonical commutation relation between momentum and position in quantum mechanics. From (P.13a) it follows that rank Lie X1 , . . . , Xn , Y1 , . . . , Yn , ∂t = 2n + 1 at any point of R2n+1 . Then, by Hörmander’s Theorem 1, the sub-Laplacian on Hn Δ

Hn

:=

n

(Xj2 + Yj2 )

j =1

is hypoelliptic. The Heisenberg group and its Lie algebra originally arose in the mathematical formalizations of quantum mechanics (see H. Weyl [Weyl31]). Today, they appear in many research fields such as several complex variables, CR geometry, Fourier analysis and partial differential equations of subelliptic type. The Heisenberg sub-Laplacian is undoubtedly the most important prototype of the sub-Laplacians on non-commutative stratified Lie groups. 1.4. The Lifting Theorem Obviously, a generic Hörmander operator sum of squares of vector fields is not, in general, a sub-Laplacian on some stratified Lie group. Just consider, as an example, M = ∂x2 + (x ∂y )2

in R2 .

This operator satisfies the Hörmander rank condition, hence it is hypoelliptic. However, there is no Lie group structure in R2 making M left-invariant on it. Nevertheless, adding the new variable t, M can be lifted to the operator in (P.10) which is the sub-Laplacian on (a group isomorphic to) the Heisenberg group H1 . This is the source idea of the lifting theorem by L.P. Rothschild and E.M. Stein, which states, roughly speaking, that any Hörmander operator sum of squares of vector fields can be approximated by a sub-Laplacian on a stratified group. This result emphasizes the major rôle played by the sub-Laplacians in the theory of second order PDE’s with non-negative and degenerate characteristic form.

2. The Contents of the Book

XIII

1.5. Stratified Groups in Sub-Riemannian Geometry Stratified groups also appear naturally in sub-Riemannian geometry (frequently referred to as “Carnot” geometry). Roughly speaking, stratified groups play a rôle, for sub-Riemannian manifolds, analogous to that played by Euclidean vector spaces for Riemannian manifolds. More precisely, once it has been provided a suitable notion of tangent space at a point of a sub-Riemannian manifold, it turns out that (at a regular point) this tangent space is naturally endowed with a structure of nilpotent Lie group with dilations, a stratified Lie group (see J. Mitchell [Mit85] and A. Bellaïche in [BR96]). Furthermore, the analysis of a left invariant sub-Laplacian on a connected nilpotent Lie group (or more generally on a Lie group of polynomial growth) and the geometry at infinity of this group is described by a canonically associated dilationinvariant sub-Laplacian on a stratified Lie group. See G. Alexopoulos [Ale92,Ale02], S. Ishiwata [Ish03] and N.Th. Varopoulos [Varo00].

2. The Contents of the Book: An Overview A glance at the contents of the book and at our approach to the subjects is in order. The book is divided into three parts, and every part is, in its turn, subdivided in several chapters plus some appendices, if necessary. 2.1. Part I The first four chapters of Part I are devoted to an elementary and self-contained introduction to the stratified Lie groups in RN . Our presentation does not require a specialized knowledge neither in algebra nor in differential geometry. The approach is completely elementary, “constructive” whenever possible, abundant in examples and intended to be understandable by readers with basic backgrounds only in linear algebra and differential calculus in RN . Subsequently, we present the formal and abstract approach to the stratified Lie groups commonly used in literature, and we prove the equivalence of the abstract notion of stratified group to the “constructive” notion of homogeneous Carnot group. This equivalence is also provided in Part I. A very special emphasis is given to the examples. We introduce and discuss a wide range of explicit stratified Lie groups of arbitrarily large dimension and step. Some of them have been known in specialized literature for several years, such as the Heisenberg–Kaplan groups, the filiform groups and the Métivier groups. Many others have only appeared very recently, in particular what we shall call the Kolmogorov-type groups and the Bony-type groups. Other examples are completely new, some extracted from geometric control theory. Our long list of examples is also intended to be appreciated by readers working in geometry and analysis on Carnot groups. It provides a valuable benchmark set to

XIV

Preface

test new special properties of the groups, to exhibit explicit examples and counterexamples of the “pathologies” and the special features of Carnot groups. It is also payed a special attention to the Lie algebras of the groups by stressing their links with second order partial differential operators of Hörmander type (sum of squares of vector fields). In particular, given such an operator, we show necessary and sufficient conditions for it to be a sub-Laplacian on a suitable homogeneous stratified Lie group, and we explicitly show how to construct the related composition law. As a byproduct, this enables the reader to build up another plenty of examples of stratified groups and sub-Laplacians. Chapter 5 of Part I is dedicated to the analysis of the fundamental solution for the sub-Laplacians, a central topic of Part I. Here, the mainly used analytic tools are integration by parts and coarea-formulas. We start from the hypoellipticity of sub-Laplacians, easy consequence3 of the Hörmander Theorem 1. From this “assumption” on hypoellipticity, and with the aid of the strong maximum principle, whose proof is postponed to the Appendix of Chapter 5, we deduce the existence of a gauge function d for any given sub-Laplacian L, i.e. the existence of a positive non-constant homogeneous function d such that d 2−Q is L-harmonic away from the origin. Here Q stands for the homogeneous dimension of the group on which the sub-Laplacian lives (we always assume that Q ≥ 3). This property is one of the most striking analogies between L and the classical Laplace operator. We show that this leads to suitable mean value formulas on the d-balls, extending to this new setting the well-known Gauss theorem for classical harmonic functions. We then use these formulas (which will play a crucial rôle throughout the book) to prove Liouville-type theorems, Harnack-type inequalities, and a Sobolev–Stein embedding theorem. Furthermore, three sections are devoted to the following topics: some remarks on the analytic-hypoellipticity of sub-Laplacians, L-harmonic approximations and an integral representation formula for the fundamental solution. 2.2. Part II Part II of the book contains an exhaustive potential theory for the sub-Laplacians. Basically, our only starting point is the theorem by G.B. Folland asserting the existence for these operators of a homogeneous and smooth fundamental solution with pole at the origin. This key result allows us to perform a complete potential theory that parallel the one of the classical Laplace operator. The lack of explicit Poisson integral formulas forces us to follow an abstract approach to this theory. For this reason, in Chapter 6 we present some topics from abstract harmonic space theory, mainly inspired by the ones developed by H. Bauer [Bau66] and C. Costantinescu and A. Cornea [CC72]. This chapter mainly involves Perron–Wiener–Brelot method for the Dirichlet problem, harmonic minorants and majorants and balayage theory. 3 We do not go into the proof of this theorem, for it would require techniques very far from

the ones developed in the book.

2. The Contents of the Book

XV

Next, in Chapter 7 we show that every sub-Laplacian equips RN with a structure of harmonic space satisfying the axioms of the theory presented in Chapter 6. This is accomplished by using the Harnack-type theorem proved in Chapter 5, and then by showing the existence of a basis of the topology of RN formed by L-regular sets, i.e. by sets for which the Dirichlet problem for L is solvable in the usual classical sense (here we follow an idea by J.-M. Bony [Bon69]). In the subsequent chapters of Part II, we use the full strength of the abstract theory, together with the remarkable properties of the fundamental solution for L to deal with the arguments listed below: a) sub-mean characterizations of the L-subharmonic functions, and applications to the notion of convexity in Carnot groups; b) Green functions and Riesz representation theorems for L-subharmonic functions, with applications (among which Bôcher-type theorems); c) maximum principles on unbounded domains; d) L-capacity and L-polar sets, with applications: the Poisson–Jensen formula and the so-called fundamental convergence theorem; e) L-thinness and L-fine topology, with applications to the Dirichlet problem (and the derivation of Wiener’s criterion); f) the links between the Hausdorff measure naturally related to the gauge d and the capacity for L. In writing this part of the book we were also partially inspired by some monographs on potential theory for the classical Laplace operator—in particular the beautiful books by L.L Helms [Helm69], by W.K. Hayman and P.B. Kennedy [HK76] and by D.H. Armitage and S.J. Gardiner [AG01]. 2.3. Part III In Part III, we take up further topics on the algebraic and analytic theory of Carnot groups. In particular, this part of the book provides: a) the study of free Lie algebras; b) clear and complete proofs, in several contexts, of the fundamental and remarkable Campbell–Hausdorff formula4 ; c) the equivalence of sub-Laplacians under diffeomorphisms; d) the Rothschild–Stein lifting theorem and Folland’s lifting theorem (for stratified or homogeneous vector fields); e) the study of the algebraic structure of Heisenberg–Kaplan-type groups (also providing an explicit characterization of them) with a special emphasis to the remarkable form of their fundamental solutions, discovered by G.B. Folland and A. Kaplan (we also present the inversion and the Kelvin transform in the H-type groups of Iwasawa type); 4 In Chapter 15, we collect four theorems for the Campbell–Hausdorff formula: one for ho-

mogeneous vector fields, two for formal power series and one for general smooth vector fields.

XVI

Preface

f) the Carathéodory–Chow–Rashevsky connectivity theorem (for stratified vector fields) with applications; g) Taylor’s formula (with Lagrange and with integral remainder) on Carnot groups. The difficulty of finding “easy” and complete proofs of some of the above mentioned results in the existing literature is well known. By working with stratified vector fields we are able to overcome some of the lengthy steps of the proofs, while maintaining a good amount of generality in the final results.

3. How to Read this Book Besides Ph.D. students, the book is addressed to young and senior researchers. Indeed, one of the main efforts in presenting the material is to use an elementary approach and to reach, step by step, the level of current researches. Many parts of the book may be used for graduate courses and advanced lectures. The first four chapters of Part I are addressed to non-specialists in Lie group and Lie algebra theory. The first two chapters can be skipped by the readers having familiarity with the basics of differential geometry and Lie group theory. The reader already acquainted with Carnot group theory can pass directly to Chapter 5. In any case, beginners and specialists in the theory of stratified groups can exploit the first four chapters as a source for examples. Part II is the core of the monograph. The reader with some background in potential theory (and interested in the main case of sub-Laplacians) can pass directly to Chapter 7 and proceed throughout Part II, leaving Chapters 10 and 13 as a further reading. Part III is thought of as a more specialized lecture. Nonetheless, a deep understanding of, e.g. the Campbell–Hausdorff formula or of Heisenberg-type groups are amongst the main goals of this monograph. The book provides 21 illustrative figures, 250 exercises (each chapter has its own section of exercises) and an index of the basic notation. For the reading convenience, we furnish a synoptic diagram of the structure of the book on page XVII.

3. How to Read this Book

The synoptic diagram of the structure of the book.

XVII

XVIII

Preface

4. Some References on Theoretical and Applied Related Topics Here is a short list of references for related topics on analysis on stratified Lie groups and applications.5 Alexopoulos [Ale02], Altafini [Alt99], Bellaïche and Risler [BR96], Birindelli, Capuzzo Dolcetta and Cutrì [BCC97], Bahri [Bah04,Bah03], Barletta [Bar03], Barletta and Dragomir [BD04], Barletta, Dragomir and Urakawa [BDU01], Birindelli, Capuzzo Dolcetta and Cutrì [BCC98], Brandolini, Rigoli and Setti [BRS98], Capogna [Cap99], Capogna and Cowling [CC69], Capogna and Garofalo [CG98,CG03,CG06], Capogna, Garofalo and Nhieu [CGN00,CGN02], Capuzzo Dolcetta [CD98], Chandresekhar [Cha43], Citti [Cit98], Citti, Lanconelli and Montanari [CLM02], Citti, Manfredini and Sarti [CMS04], Citti and Montanari [CM00], Citti, Pascucci and Polidoro [CPP01], Citti and Sarti [CS06], Citti and Tomassini [CT04] Cowling, De Mari, Korányi and Reimann [CDKR02], Cowling and Reimann [CR03], Danielli, Garofalo, Nhieu and Tournier [DGNT04], Danielli, Garofalo and Salsa [DGS03], Dragomir [Dra01], Franchi, Gutiérrez and van Nguyen [FGvN05], Franchi, Serapioni and Serra Cassano [FSS03a,FSS03b,FSS01], Gamara [Gam01], Gamara and Yacoub [GY01], Garofalo and Lanconelli [GL92], Garofalo and Tournier [GT06], Garofalo and Vassilev [GV00], Golé and Karidi [GK95], Gutiérrez and Lanconelli [GL03], Gutiérrez and Montanari [GM04a,GM04b], Heinonen [Hei95b], Heinonen and Holopainen [HH97], Heinonen and Koskela [HK98], Huisken and Klingenberg [HK99], Jerison and Lee [JL87,JL88,JL89], Juutinen, Lu, Manfredi and Stroffolini [JLMS07], Korányi and Reimann [KR90,KR95], Lanconelli [Lan03], Lanconelli, Pascucci and Polidoro [LPP02], Lanconelli and Uguzzoni [LU00], Lu, Manfredi and Stroffolini [LMS04], Lu and Wei [LW97], Malchiodi and Uguzzoni [MU02], Manfredi and Stroffolini [MS02], Montanari [Mo01], Montanari and Lascialfari [ML01], Montgomery [Mon02], Montgomery, Shapiro and Stolin [MSS97], Monti and Morbidelli [MM05], Monti and Rickly [MR05], Monti and Serra Cassano [MSC01], Petitot and Tondut [PT99], Reimann [Rei01a,Rei01b], Slodkowski and Tomassini [ST91], Stein [Ste81], Uguzzoni [Ugu00], Varopoulos, Saloff-Coste and Coulhon [VSC92]. 5 The list is alphabetically ordered and the grouping of the references in different lines is

only meant for typographical readability.

5. Acknowledgments

XIX

5. Acknowledgments The authors would like to thank Chiara Cinti and Andrea Tommasoli for careful reading some chapters of the manuscript. We would also like to express our gratitude to Italo Capuzzo Dolcetta, Cristian E. Gutiérrez, Guozhen Lu and Juan J. Manfredi for the kind encouraging appreciation of our work. It is also a pleasure to thank the Springer-Verlag staff for the kind collaboration, in particular Dr. Catriona M. Byrne, Dr. Marina Reizakis and Dr. Susanne Denskus. Some topics presented in this book have partially appeared in the following papers by joint collaborations of the authors (and of one of us with C. Cinti) [Bon04, BC04,BC05,BL01,BL02,BL03,BL07,BLU02,BLU03,BU04a,BU04b,BU05a]. Bologna, Italy April 2007 Andrea Bonfiglioli Ermanno Lanconelli Francesco Uguzzoni

Contents

Part I Elements of Analysis of Stratified Groups 1

Stratified Groups and Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Vector Fields in RN : Exponential Maps and Lie Algebras . . . . . . . . . 1.1.1 Vector Fields in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Exponentials of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Lie Brackets of Vector Fields in RN . . . . . . . . . . . . . . . . . . . . . 1.2 Lie Groups on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Lie Algebra of a Lie Group on RN . . . . . . . . . . . . . . . . . . 1.2.2 The Jacobian Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The (Jacobian) Total Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The Exponential Map of a Lie Group on RN . . . . . . . . . . . . . 1.3 Homogeneous Lie Groups on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 δλ -homogeneous Functions and Differential Operators . . . . . 1.3.2 The Composition Law of a Homogeneous Lie Group . . . . . . 1.3.3 The Lie Algebra of a Homogeneous Lie Group on RN . . . . . 1.3.4 The Exponential Map of a Homogeneous Lie Group . . . . . . . 1.4 Homogeneous Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Sub-Laplacians on a Homogeneous Carnot Group . . . . . . . . . . . 1.5.1 The Horizontal L-gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Exercises of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 6 8 10 13 13 19 22 23 31 32 38 44 48 56 62 68 73

2

Abstract Lie Groups and Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1 Abstract Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.1.3 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.1.4 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.1.5 Commutators. ϕ-relatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.1.6 Abstract Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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2.1.7 Left Invariant Vector Fields and the Lie Algebra . . . . . . . . . . 2.1.8 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Some Properties of the Stratification of a Carnot Group . . . . 2.2.2 Some General Results on Nilpotent Lie Groups . . . . . . . . . . . 2.2.3 Abstract and Homogeneous Carnot Groups . . . . . . . . . . . . . . 2.2.4 More Properties of the Lie Algebra . . . . . . . . . . . . . . . . . . . . . 2.2.5 Sub-Laplacians of a Stratified Group . . . . . . . . . . . . . . . . . . . . 2.3 Exercises of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 112 116 121 125 128 130 138 144 147

3

Carnot Groups of Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Heisenberg–Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Homogeneous Carnot Groups of Step Two . . . . . . . . . . . . . . . . . . . . . 3.3 Free Step-two Homogeneous Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Exponential Map of a Step-two Homogeneous Group . . . . . . . . 3.6 Prototype Groups of Heisenberg Type . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 H-groups (in the Sense of Métivier) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Exercises of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 158 163 165 166 169 173 177

4

Examples of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Primer of Examples of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Euclidean Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Carnot Groups with Homogeneous Dimension Q ≤ 3 . . . . . 4.1.3 B-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 K-type Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Sum of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 From a Set of Vector Fields to a Stratified Group . . . . . . . . . . . . . . . . 4.3 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Vector Fields ∂1 , ∂2 + x1 ∂3 . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Classical and Kohn Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Bony-type Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Kolmogorov-type Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Sub-Laplacians Arising in Control Theory . . . . . . . . . . . . . . . 4.3.6 Filiform Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) . . . . . 4.4.1 Fields not Satisfying Hypothesis (H0) . . . . . . . . . . . . . . . . . . . 4.4.2 Fields not Satisfying Hypothesis (H1) . . . . . . . . . . . . . . . . . . . 4.4.3 Fields not Satisfying Hypothesis (H2) . . . . . . . . . . . . . . . . . . . 4.5 Exercises of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 183 184 184 186 190 191 198 198 200 202 204 205 207 210 210 212 215 215

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5

XXIII

The Fundamental Solution for a Sub-Laplacian and Applications . . . . 5.1 Homogeneous Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Control Distances or Carnot–Carathéodory Distances . . . . . . . . . . . . 5.3 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Fundamental Solution in the Abstract Setting . . . . . . . . . 5.4 L-gauges and L-radial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Gauge Functions and Surface Mean Value Theorem . . . . . . . . . . . . . . 5.6 Superposition of Average Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Harnack Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Liouville-type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Asymptotic Liouville-type Theorems . . . . . . . . . . . . . . . . . . . . 5.9 Sobolev–Stein Embedding Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Analytic Hypoellipticity of Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . 5.11 Harmonic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 An Integral Representation Formula for Γ . . . . . . . . . . . . . . . . . . . . . . 5.13 Appendix A. Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.1 A Decomposition Theorem for L-harmonic Functions . . . . . 5.14 Appendix B. The Improved Pseudo-triangle Inequality . . . . . . . . . . . 5.15 Appendix C. Existence of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Exercises of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 229 232 236 244 246 251 257 262 269 274 276 280 287 291 293 303 306 309 319

Part II Elements of Potential Theory for Sub-Laplacians 6

Abstract Harmonic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Sheafs of Functions. Harmonic Sheafs . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Harmonic Measures and Hyperharmonic Functions . . . . . . . . 6.2.2 Directed Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Harmonic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Directed Families of Harmonic and Hyperharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 B-hyperharmonic Functions. Minimum Principle . . . . . . . . . . . . . . . . 6.5 Subharmonic and Superharmonic Functions. Perron Families . . . . . . 6.6 Harmonic Majorants and Minorants . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 The Perron–Wiener–Brelot Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 S-harmonic Spaces: Wiener Resolutivity Theorem . . . . . . . . . . . . . . 6.8.1 Appendix: The Stone–Weierstrass Theorem . . . . . . . . . . . . . . 6.9 H-harmonic Measures for Relatively Compact Open Sets . . . . . . . . . 6.10 S∗ -harmonic Spaces: Bouligand’s Theorem . . . . . . . . . . . . . . . . . . . . 6.11 Reduced Functions and Balayage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Exercises of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 338 340 341 342 345 347 348 353 358 359 363 366 367 370 375 378

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7

The L-harmonic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The L-harmonic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Some Basic Definitions and Selecta of Properties . . . . . . . . . . . . . . . . 7.3 Exercises of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

381 381 388 392

8

L-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Sub-mean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Some Characterizations of L-subharmonic Functions . . . . . . . . . . . . . 8.3 Continuous Convex Functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Exercises of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397 397 401 411 422

9

Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 L-Green Function for L-regular Domains . . . . . . . . . . . . . . . . . . . . . . 9.2 L-Green Function for General Domains . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Potentials of Radon Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Potentials Related to the Average Operators . . . . . . . . . . . . . . 9.4 Riesz Representation Theorems for L-subharmonic Functions . . . . . 9.5 The Poisson–Jensen Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Bounded-above L-subharmonic Functions in G . . . . . . . . . . . . . . . . . 9.7 Smoothing of L-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 9.8 Isolated Singularities—Bôcher-type Theorems . . . . . . . . . . . . . . . . . . 9.8.1 An Application of Bôcher’s Theorem . . . . . . . . . . . . . . . . . . . 9.9 Exercises of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 425 427 432 435 441 445 451 455 458 462 463

10 Maximum Principle on Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . 10.1 MP Sets and L-thinness at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 q-sets and the Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Maximum Principle on Unbounded Domains . . . . . . . . . . . . . . . 10.4 The Proof of Lemma 10.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Exercises of Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

473 473 477 482 483 487

11 L-capacity, L-polar Sets and Applications . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Continuity Principle for L-potentials . . . . . . . . . . . . . . . . . . . . . . . 11.2 L-polar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Maria–Frostman Domination Principle . . . . . . . . . . . . . . . . . . . . . 11.4 L-energy and L-equilibrium Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 L-balayage and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 The Fundamental Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . 11.7 The Extended Poisson–Jensen Formula . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Further Results. A Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Further Reading and the Quasi-continuity Property . . . . . . . . . . . . . . 11.10 Exercises of Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

489 489 491 494 497 500 510 514 519 527 533

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12 L-thinness and L-fine Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The L-fine Topology: A More Intrinsic Tool . . . . . . . . . . . . . . . . . . . . 12.2 L-thinness at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 L-thinness and L-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Functions Peaking at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 L-thinness and L-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Wiener’s Criterion for Sub-Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 A Technical Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Wiener’s Criterion for L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Exercises of Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537 537 538 542 542 544 547 547 550 553

13 d-Hausdorff Measure and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 d-Hausdorff Measure and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 d-Hausdorff Measure and L-capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 New Phenomena Concerning the d-Hausdorff Dimension . . . . . . . . . 13.4 Exercises of Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

557 557 561 569 572

Part III Further Topics on Carnot Groups 14 Some Remarks on Free Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Free Lie Algebras and Free Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 14.2 A Canonical Way to Construct Free Carnot Groups . . . . . . . . . . . . . . 14.2.1 The Campbell–Hausdorff Composition . . . . . . . . . . . . . . . . 14.2.2 A Canonical Way to Construct Free Carnot Groups . . . . . . . . 14.3 Exercises of Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

577 577 584 584 586 589

15 More on the Campbell–Hausdorff Formula . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Campbell–Hausdorff Formula for Stratified Fields . . . . . . . . . . . 15.2 The Campbell–Hausdorff Formula for Formal Power Series–1 . . . . . 15.3 The Campbell–Hausdorff Formula for Formal Power Series–2 . . . . . 15.4 The Campbell–Hausdorff Formula for Smooth Vector Fields . . . . . . 15.5 Exercises of Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

593 593 599 605 610 616

16 Families of Diffeomorphic Sub-Laplacians ........................ 16.1 An Isomorphism Turning i,j ai,j Xi Xj into ΔG . . . . . . . . . . . . . . . 16.2 Examples and Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Canonical or Non-canonical? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Further Reading: An Application to PDE’s . . . . . . . . . . . . . . . . . . . . . 16.5 Exercises of Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

621 622 628 637 641 644 645

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17 Lifting of Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Lifting to Free Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 An Example of Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 An Example of Application to PDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Folland’s Lifting of Homogeneous Vector Fields . . . . . . . . . . . . . . . . 17.4.1 The Hypotheses on the Vector Fields . . . . . . . . . . . . . . . . . . . . 17.5 Exercises of Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

649 649 659 661 666 669 676

18 Groups of Heisenberg Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Heisenberg-type Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 A Direct Characterization of H-type Groups . . . . . . . . . . . . . . . . . . . . 18.3 The Fundamental Solution on H-type Groups . . . . . . . . . . . . . . . . . . . 18.4 H-type Groups of Iwasawa-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 The H-inversion and the H-Kelvin Transform . . . . . . . . . . . . . . . . . . . 18.6 Exercises of Chapter 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

681 681 686 695 702 704 709

19 The Carathéodory–Chow–Rashevsky Theorem . . . . . . . . . . . . . . . . . . . . 19.1 The Carathéodory–Chow–Rashevsky Theorem for Stratified Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 An Application of Carathéodory–Chow–Rashevsky Theorem . . . . . . 19.3 Exercises of Chapter 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

715

20 Taylor Formula on Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Polynomials and Derivatives on Homogeneous Carnot Groups . . . . . 20.1.1 Polynomial Functions on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.2 Derivatives on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Taylor Polynomials on Homogeneous Carnot Groups . . . . . . . . . . . . 20.3 Taylor Formula on Homogeneous Carnot Groups . . . . . . . . . . . . . . . . 20.3.1 Stratified Taylor Formula with Peano Remainder . . . . . . . . . . 20.3.2 Stratified Taylor Formula with Integral Remainder . . . . . . . . 20.4 Exercises of Chapter 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

733 734 734 736 741 746 751 754 766

715 727 730

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Index of the Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Part I

Elements of Analysis of Stratified Groups

1 Stratified Groups and Sub-Laplacians

In this first chapter, we introduce the main notation and the basic definitions concerning with vector fields in RN : algebras of vector fields, exponentials of smooth vector fields, Lie brackets. Then, we introduce the main geometric structure investigated throughout the book: the homogeneous Carnot groups. To this end, we first study Lie groups G on RN and the Lie algebra of their left-invariant vector fields. Subsequently, we equip G with a homogeneous structure by the datum of a well-behaved group of dilations {δλ }λ>0 on G. The composition of G thus takes a transparent form, allowing us to study homogeneous Lie groups on RN by very direct methods. In particular, it will be a simple exercise of calculus to verify that the relevant exponential and logarithmic maps are global polynomial diffeomorphisms, a result which will throughout account for a very useful tool. Finally, we introduce the notion of homogeneous Carnot group and of sub-Laplacian. A wide number of explicit examples of homogeneous Carnot groups will be given in Chapters 3 and 4, after (in Chapter 2) we have analyzed the due relationship between abstract Carnot groups and the homogeneous ones. Indeed, despite our notions of Lie group on RN , homogeneous and homogeneous-Carnot group are dependent on a fixed system of coordinates for the group (thus being non-intrinsic notions), every abstract Carnot group is, as we shall see, isomorphic via the exponential map to its Lie algebra, which is a homogeneous Carnot group. This basic fact provides another motivation for this introductory chapter.

1.1 Vector Fields in RN . Exponential Maps. Lie Algebras of Vector Fields 1.1.1 Vector Fields in RN Given N ∈ N, we set, as usual, RN = {(x1 , . . . , xN ) : x1 , . . . , xN ∈ R}. We use any of the notation ∂ ∂j , ∂xj , , ∂/∂ xj ∂xj

4


to indicate the partial derivative operator with respect to the j -th coordinate of RN . Let Ω ⊆ RN be an open (and non-empty) set. Given an N-tuple of scalar functions a1 , . . . , aN , aj : Ω → R, j ∈ {1, . . . , N }, the first order linear differential operator X=

N

(1.1)

aj ∂j

j =1

will be called a vector field on Ω with component functions (or simply, components) a1 , . . . , aN . If O ⊆ Ω is an open set and f : O → R is a differentiable function, we denote by Xf the function on O defined by Xf (x) =

N

aj (x) ∂j f (x),

x ∈ O.

j =1

Occasionally, we shall also use the notation Xf when f : O → Rm is a vector-valued function, to mean the component-wise action of X. More precisely,1 ⎛ ⎞ ⎛ ⎞ f1 (x) Xf1 (x) .. ⎠. if f (x) = ⎝ ... ⎠ , we set Xf (x) = ⎝ . fm (x)

Xfm (x)

1 We warn the reader that points in RN will be usually denoted as N -tuples x

=

(x1 , . . . , xN ). When this does not lead to confusion, the column-vector notation ⎛ ⎞ x1 ⎜ . ⎟ x = ⎝ .. ⎠ xN will also be allowed. For example, this last notation will be sometimes used (with no risk of misunderstanding) for vector-valued functions f (x) = (f1 (x), . . . , fN (x))T , e.g. I (x) = (x1 , . . . , xN )T will always denote (this time as a rule) the identity map in RN . However, there are cases in which we shall keep the notation well distinguished: namely, if f : RN → R is a differentiable function, its gradient ∇f = (∂1 f, . . . , ∂N f ) will always be written as a row-vector. On the other side, we shall always use the column-vector notation when vectors in RN appear in matrix calculation. For example, if x, y ∈ RN , x T y will denote the row×column product ⎞ ⎛ y1 N ⎜ . ⎟ xj yj . x T y = (x1 , . . . , xN ) ⎝ .. ⎠ = j =1 yN

1.1 Vector Fields in RN : Exponential Maps and Lie Algebras

5

(Typically, this notation will be used when f = I is the identity map of RN , i.e. I (x) = (x1 , . . . , xN )T ; see (1.2) below.) Furthermore, given a differentiable function f : O → Rm , we shall denote by Jf (x), x ∈ O the Jacobian matrix of f at x. Let C ∞ (O, R) (for brevity, C ∞ (O)) be the set of smooth (i.e. infinitely-differentiable) real-valued functions. If the components aj ’s of X are smooth, we shall call X a smooth vector field and we shall often consider X as an operator acting on smooth functions, f → Xf. X : C ∞ (O) → C ∞ (O), We shall prevalently deal with smooth vector fields. We shall denote by T (RN ) the set of all smooth vector fields in RN . Equipped with the natural operations, T (RN ) is a vector space over R. We adopt the following (non-conventional) notation: I will denote the identity map on RN and, if X is the vector field in (1.1), then ⎛ ⎞ a1 . XI := ⎝ .. ⎠ (1.2) aN will be the column vector of the components of X. This notation is obviously consistent with our definition of the action of X on a vector-valued function.2 Thus, XI may also be regarded as a smooth map from RN to itself (that is what some authors call a “vector field” on RN ). Often, many authors identify X and XI . Instead, in order to avoid any confusion between a smooth vector field as a function belonging to C ∞ (RN , RN ) and a smooth vector field as a differential operator from C ∞ (RN ) to itself,3 we prefer to use the different notation XI and X as described in (1.2) and (1.1), respectively. By consistency of notation, we may write Xf = (∇f ) · XI, where

∇ = (∂1 , . . . , ∂N )

is the gradient operator in RN , f is any real-valued smooth function on RN and · denotes the row×column product. For example, for the following two vector fields on R3 (whose points are denoted by x = (x1 , x2 , x3 )) X2 = ∂x2 − 2 x1 ∂x3 , (1.3a) X1 = ∂x1 + 2 x2 ∂x3 ,

we have X1 I (x) =

1 0 2 x2

,

X2 I (x) =

0 1 −2 x1

.

2 Indeed, since I = (I , . . . , I ) with I (x) = x , we have X(I ) = a . N j j j j 1 3 Or even from C ∞ (RN , RN ) to itself!

(1.3b)

6


1.1.2 Integral Curves A path γ : D → RN , D being an interval of R, will be said an integral curve of the smooth vector field X if γ˙ (t) = XI (γ (t)) for every t ∈ D. If X is a smooth vector field, then, for every x ∈ RN , the Cauchy problem γ˙ = XI (γ ), γ (0) = x

(1.4)

has a unique solution γX (·, x) : D(X, x) → RN . We agree to denote by D(X, x) the greatest open interval of R on which γX (·, x) exists. For example, if X1 is as in (1.3a), the solution to the relevant problem (1.4) is obtained by solving the following system of ODE’s (we write γ = (γ1 , γ2 , γ3 )) γ˙1 (t) = 1, γ˙2 (t) = 0, γ˙3 (t) = 2 γ2 (t), γ1 (0) = x1 ,

γ2 (0) = x2 ,

γ3 (0) = x3 ,

i.e. after simple computations, we obtain D(X1 , x) = R and γX1 (t, x) = (x1 + t, x2 , x3 + 2 x2 t).

(1.5)

Since X is smooth, t → γX (t, x) is a C ∞ function whose n-th Taylor expansion in a neighborhood of t = 0 is given by t2 tn γX (t, x) = x + t X (1) I (x) + X (2) I (x) + · · · + X (n) I (x) 2! n! t 1 (t − s)n X (n+1) I (γX (s, x)) ds. + n! 0

(1.6)

Hereafter, for k ∈ N, we denote by X (k) the vector field X (k) =

N (X k−1 aj )∂xj , j =1

being X 0 = I (the identity map) and X h , h ≥ 1, the h-th order iterated of X, i.e. X h := X · · ◦ X .

◦ · h times

In other words, we have


7

⎞ ⎞ ⎛ ⎛ ⎞ X(a1 ) X(X(a1 )) a1 . . .. ⎠,.... X (1) I = XI = ⎝ .. ⎠ , X (2) I = ⎝ .. ⎠ , X (3) I = ⎝ . aN X(aN ) X(X(aN )) ⎛

We remark that X h is a differential operator of order at most h, whereas X (h) is a differential operator of order at most 1. To check (1.6) we use (1.4). Writing γ (t) instead of γX (t, x), (1.4) gives (d/dt)|t=0 γ (t) = XI (x) and

γ (0) = x,

d d2 γ (t) = (XI )(γ (t)) = JXI (γ (0)) · γ˙ (0) = JXI (x) · XI (x) dt t=0 dt 2 t=0 ⎛ ⎞ ⎛ ⎞ ∇a1 (x) · XI (x) Xa1 (x) .. .. ⎠=⎝ ⎠ = X (2) I (x). =⎝ . . ∇aN (x) · XI (x)

XaN (x)

By iterating this argument, we obtain dk (k) γ (0) := k γ (t) = X (k) I (x), dt t=0

k ≥ 2.

Replacing this identity in the Taylor formula γ (t) = x +

n tk k=1

k!

γ

(k)

1 (0) + n!

t

(t − s)n γ (n+1) (s) ds,

0

we obtain (1.6). Since the identity map I is linear and since the first order part of X h coincides with X (h) , one has X (h) I ≡ X h I. Thus formula (1.6) can be rewritten as t2 tn γX (t, x) = x + t XI (x) + X 2 I (x) + · · · + X n I (x) 2! n! t 1 (t − s)n X n+1 I (γX (s, x)) ds. + n! 0 Example 1.1.1. For example, if X1 is as in (1.3a), since

1 0 (1) (2) X1 I = , X1 I = 0 = X1(k) I 0 2 x2 0 we have γX1 (t, x) = x + t X1 I (x) = as we directly found in (1.5).

x1 x2 x3

+t

1 0 2 x2

=

(1.7)

∀ k ≥ 3,

x1 + t x2 x3 + 2 x2 t

,

8


1.1.3 Exponentials of Vector Fields The expansion in (1.7) suggests to use an “exponential-type” notation. For this (more than notational) reason, we give the following definition. Definition 1.1.2 (Exponential of a vector field). Let X be a smooth vector field on RN . Following all the above notation, we set exp(tX)(x) := γX (t, x),

(1.8)

where γX (·, x) is the solution of (1.4). This definition makes sense4 for every X ∈ T (RN ), for every x ∈ RN and every t ∈ D(X, x). (See Fig. 1.1.)

Fig. 1.1. Figure of Definition 1.1.2

Then, being X smooth, for every n ∈ N, we have the expansion exp(tX)(x) =

n tk k=0

+

k!

1 n!

X k I (x)

t

(t − s)n X n+1 I exp(sX)(x) ds.

(1.9)

0

In particular, for n = 1, exp(tX)(x) = x + t XI (x) +

t

(t − s) X 2 I exp(sX)(x) ds.

(1.10)

0

If we define U := {(t, x) ∈ R × RN | x ∈ RN , t ∈ D(X, x)}, from the basic theory of ordinary differential equations (see, e.g. [Har82]) we know that U is open and the map 4 Definition 1.1.2 is well-posed also when X has only Lipschitz-continuous component func-

tions, but we shall prevalently deal, as already stated, with smooth vector fields.


9

U (t, x) → exp(tX)(x) ∈ RN is smooth. Moreover, from the unique solvability of the Cauchy problem related to smooth vector fields we get: t ∈ D(−X, x) iff −t ∈ D(X, x) and exp(−tX)(x) := exp((−t)X)(x) = exp(t (−X))(x), exp(−tX) exp(tX)(x) = x, exp((t + τ )X)(x) = exp(tX) exp(τ X)(x) , exp((tτ )X)(x) = exp(t (τ X))(x),

(1.11a) (1.11b) (1.11c) (1.11d)

when all the terms are defined. If D(X, x) = R, identities (1.11a)–(1.11d) hold true for every t, τ ∈ R. See also the note.5 Remark 1.1.3 (Pyramid-shaped vector fields). For our aims, the vector fields of the following type N X= aj (x1 , . . . , xj −1 ) ∂xj (1.12) j =1

will play a crucial rôle. In (1.12), the function aj only depends on the variables x1 , . . . , xj −1 , and we agree to let aj (x1 , . . . , xj −1 ) = constant when j = 1. Roughly speaking, such a remarkable kind of vector field is “pyramid”-shaped, ⎛ ⎞ a1 ⎜ ⎟ a2 (x1 ) ⎜ ⎟ ⎜ ⎟ (x , x ) a 3 1 2 ⎜ ⎟ X = ⎜ a4 (x1 , x2 , x3 ) ⎟ . ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . aN (x1 , . . . , xN −1 ) For example, the fields in (1.3b) have this form. For any smooth vector field X of the form (1.12), the map (x, t) → exp(tX)(x) is well defined for every x ∈ RN and t ∈ R. Indeed, if γ = (γ1 , . . . , γN ) is the solution to the Cauchy problem γ˙ = XI (γ ), γ (0) = x, x = (x1 , . . . , xN ), then γ˙1 = a1 and γ˙j = aj (γ1 , . . . , γj −1 ) for j = 2, . . . , N . As a consequence, t aj (γ1 (x, s), . . . , γj −1 (x, s)) ds, γ1 (x, t) = x1 + ta1 , γj (x, t) = xj + 0

5 Strictly speaking, according to the very definition of exp(tX)(x) := γ (t, x), one may X

observe that t and X should be kept separate in notation. Though, we explicitly remark that identity (1.11d) justifies the notation “tX” in exp(tX)(x).

10


and γj (x, t) is defined for every x ∈ RN and t ∈ R. Moreover, γ1 (·, t) only depends on x1 , whereas for j = 2, . . . , N , γj (·, t) only depends on x1 , . . . , xj . Let us put A1 (t) = ta1 and, for j = 2, . . . , N, t Aj (x, t) = Aj (x1 , . . . , xj −1 , t) := aj (γ1 (x, s), . . . , γj −1 (x, s)) ds. 0

Then, for every x ∈ RN , t ∈ R, ⎛

⎞

x1 + A1 (t) x2 + A2 (x1 , t) .. .

⎜ ⎜ exp(tX)(x) = ⎜ ⎝

⎟ ⎟ ⎟, ⎠

(1.13)

xN + AN (x1 , . . . , xN −1 , t) and the map x → exp(tX)(x) is a global diffeomorphism of RN onto RN for every fixed t ∈ R. Its inverse map y → L(y, t) is given by y → L(y, t) = exp(−tX)(y).

(1.14)

This last statement follows from identity (1.11b).

Remark 1.1.4. Let us consider a smooth function u : RN → R and the vector field in (1.1). Then Xu(x) = lim

t→0

u(exp(tX)(x)) − u(x) t

∀ x ∈ RN .

(1.15)

Indeed, since exp(tX)(x) = x + tXI (x) + O(t 2 ), the limit on the right-hand side of (1.15) is equal to the following one: lim

t→0

u(x + tXI (x)) − u(x) = ∇u(x) · XI (x) = Xu(x). t

1.1.4 Lie Brackets of Vector Fields in RN Given two smooth vector fields X and Y in RN , we define the Lie-bracket [X, Y ] as follows [X, Y ] := XY − Y X. This definition is only seemingly deceitful, for it writes [X, Y ] (which is a first order differential operator) as a difference Nof two second order differential operators. a ∂ and Y = Indeed, if X = N j j j =1 j =1 bj ∂j , a direct computation shows that the Lie bracket [X, Y ] is the vector field [X, Y ] =

N (Xbj − Y aj )∂j . j =1


11

As a consequence, ⎛

⎞ ⎛ ⎞ Xb1 Y a1 . . [X, Y ]I = ⎝ .. ⎠ − ⎝ .. ⎠ = JY I · XI − JXI · Y I. XbN Y aN

(1.16)

For example, if X1 , X2 are as in (1.3b) (page 5), we have [X1 , X2 ] = (X1 (−2x1 ) − X2 (2x2 )) ∂x3 = −4 ∂x3 . It is quite trivial to check that (X, Y ) → [X, Y ] is a bilinear map on the vector space T (RN ) satisfying the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 for every X, Y, Z ∈ T (RN ). We shall refer to T (RN ) (equipped with the above Lie-bracket) as the Lie algebra of the vector fields on RN . Any sub-algebra a of T (RN ) will be called a Lie algebra of vector fields. More explicitly, a is a Lie algebra of vector fields if a is a vector subspace of T (RN ) closed with respect to [ , ], i.e. [X, Y ] ∈ a for every X, Y ∈ a. We now introduce some other notation on the algebras of vector fields. Given a set of vector fields Z1 , . . . , Zm ∈ T (RN ) and a multi-index J = (j1 , . . . , jk ) ∈ {1, . . . , m}k , we set ZJ := [Zj1 , . . . [Zjk−1 , Zjk ] . . .]. We say that ZJ is a commutator of length (or height) k of Z1 , . . . , Zm . If J = j1 , we also say that ZJ := Zj1 is a commutator of length 1 of Z1 , . . . , Zm . A commutator of the form ZJ will also be called nested, in order to emphasize its difference from, e.g. a commutator of the form [[Z1 , Z2 ], [Z3 , Z4 ]]. What is striking is that this last commutator is a linear combination of nested ones,6 as we prove in Proposition 1.1.7. First, we give a definition. Definition 1.1.5 (The Lie algebra generated by a set). If U is any subset of T (RN ), we denote by Lie{U } the least sub-algebra of T (RN ) containing U , i.e. Lie{U } := h, where h is a sub-algebra of T (RN ) with U ⊆ h. We define

rank Lie{U }(x) := dimR ZI (x) | Z ∈ Lie{U } .

6 Namely,

[[Z1 , Z2 ], [Z3 , Z4 ]] = −[Z3 , [Z4 , [Z1 , Z2 ]]] + [Z4 , [Z3 , [Z1 , Z2 ]]].

12


Example 1.1.6. Let X1 and X2 be as in (1.3b) (page 5). Since [X1 , X2 ] = −4∂x3 and since any commutator involving X1 , X2 more than twice is identically zero, then Lie{X1 , X2 } = span{X1 , X2 , [X1 , X2 ]}, and rank(Lie{X1 , X2 }(x)) = 3

for every x ∈ R3

(the well-known Hörmander condition).

The following result holds. Proposition 1.1.7 (Nested commutators). Let U ⊆ T (RN ) be any set of smooth vector fields on RN . We set U1 := span{U },

Un := span{[u, v] | u ∈ U, v ∈ Un−1 },

n ≥ 2.

Then we have Lie{U } = span{Un | n ∈ N}. Moreover, [u, v] ∈ Ui+j

for every u ∈ Ui , v ∈ Uj .

We explicitly remark that, from the very definition of Un , the vector fields in Un are linear combination of nested brackets, i.e. brackets of the type [u1 [u2 [u3 [· · · [un−1 , un ] · · ·]]]] with u1 , . . . , un ∈ U . The above proposition then states that every element of Lie{U } is a linear combination of nested brackets. To show the idea behind the proof, let us take u1 , u2 , v1 , v2 ∈ U and prove that [[u1 , u2 ], [v1 , v2 ]] is a linear combination of nested brackets. By the Jacobi identity [X, [Y, Z]] = −[Y, [Z, X]] − [Z, [X, Y ]], one has [[u1 , u2 ], [ v1 , v2 ]] = −[v1 , [v2 , [u1 , u2 ]]] − [v2 , [[u1 , u2 ], v1 ]]

X

Y

Z

= −[v1 , [v2 , [u1 , u2 ]]] + [v2 , [v1 , [u1 , u2 ]]] ∈ U4 . Proof (of Proposition 1.1.7). We set U ∗ := span{Un | n ∈ N}. Obviously, U ∗ contains U and is contained in any algebra of vector fields which contains U . Hence, we are left to prove that U ∗ is closed under the bracket operation. Obviously, it is enough to show that, for any i, j ∈ N and for any u1 , . . . , ui , v1 , . . . , vj ∈ U , we have [u1 [u2 [· · · [ui−1 , ui ] · · ·]]]; [v1 [v2 [· · · [vj −1 , vj ] · · ·]]] ∈ Ui+j . We argue by induction on k := i + j ≥ 2. For k = 2 and 3, the assertion is obvious. Let us now suppose the thesis holds for every i + j ≤ k with k ≥ 4, and prove it also holds when i + j = k + 1. We can suppose, by skew-symmetry, j ≥ 3. Exploiting repeatedly the induction hypothesis and the Jacobi identity, we have

1.2 Lie Groups on RN

13

u; [v1 [v2 [· · · [vj −1 , vj ] · · ·]]] = −[v1 , [[v2 , [v3 , · · ·]], u]] − [[v2 , [v3 , · · ·]], [u, v1 ]]

length k

= {element of Uk+1 } − [[v1 , u], [v2 , [v3 , · · ·]]] = {element of Uk+1 } + [v2 , [[v3 , · · ·], [v1 , u]]] + [[v3 , · · ·], [[v1 , u]v2 ]]

length k

= {element of Uk+1 } + [[v2 , [v1 , u]], [v3 , · · ·]] (after finitely many steps) = {element of Uk+1 } + (−1)j −1 [[vj −i , [vj −2 , · · · [v1 , u]]], vj ] = {element of Uk+1 } + (−1)j [vj , [vj −i , [vj −2 , · · · [v1 , u]]]] ∈ Uk+1 . This ends the proof.

Corollary 1.1.8. Let Z1 , . . . , Zm ∈ T (RN ) be fixed. Then Lie{Z1 , . . . , Zm } = span ZJ | with J = (j1 , . . . , jk ) ∈ {1, . . . , m}k , k ∈ N . This (non-trivial) fact (direct consequence of Proposition 1.1.7) will be used throughout the next sections, often without mention. The following notation will be used when dealing with “stratified” (or “graded”) Lie algebras. If V1 , V2 are subsets of T (RN ), we denote [V1 , V2 ] := span [v1 , v2 ] | vi ∈ Vi , i = 1, , 2 . (1.17)

1.2 Lie Groups on RN 1.2.1 The Lie Algebra of a Lie Group on RN We first recall a well-known definition. Definition 1.2.1 (Lie group on RN ). Let ◦ be a given group law on RN , and suppose that the map RN × RN (x, y) → y −1 ◦ x ∈ RN is smooth. Then G := (RN , ◦) is called a Lie group on RN . Convention. For the simplicity of notation, we shall assume that the origin 0 of RN is the identity of G. This assumption is not restrictive. Indeed, if e ∈ G is the identity of G (a Lie group on RN ), we can consider new coordinates on RN given by the C ∞ -diffeomorphism defined by T (x) = x − e. Thus, we obtain a new Lie group on = (RN , ∗), where RN , G y ∗ y = (y + e) ◦ (y + e) − e,

14


are isomorphic Lie groups via the with identity T (e) = 0. Obviously, G and G diffeomorphism T . Furthermore, we shall see that every homogeneous Lie group on RN (see Section 1.3) has identity element equal to the origin of RN . Finally, the main subject of this book, i.e. Carnot groups, are naturally isomorphic to a homogeneous Lie group on RN . This justifies our convention and motivates the study of Lie groups on RN , i.e. roughly speaking, Lie groups with a global chart. Fixed α ∈ G, we denote by τα (x) := α ◦ x the left-translation by α on G. A (smooth) vector field X on RN is called left-invariant on G if X(ϕ ◦ τα ) = (Xϕ) ◦ τα

(1.18)

for every α ∈ G and for every smooth function ϕ : RN → R. We denote by g the set of the left-invariant vector fields on G. It is quite obvious to recognize that for every X, Y ∈ g and for every λ, μ ∈ R, we have λX + μY ∈ g and [X, Y ] ∈ g.

(1.19)

Then, g is a Lie algebra of vector fields, sub-algebra of T (RN ). It will be called the Lie algebra of G. Example 1.2.2. The map (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = x1 + y1 , x2 + y2 , x3 + y3 + 2 (x2 y1 − x1 y2 ) endows R3 with a structure of Lie group. In several next examples, we shall refer to H1 = (R3 , ◦) as the Heisenberg–Weyl group on R3 . It is a direct computation to show that the vector fields X1 = ∂x1 + 2 x2 ∂x3 ,

X2 = ∂x2 − 2 x1 ∂x3

(1.20)

are left invariant w.r.t. ◦. Consequently, X1 , X2 , [X1 , X2 ] ∈ h1 , say, the Lie algebra of H1 (this notation is not standard). We shall show that, precisely, h1 =

span{X1 , X2 , [X1 , X2 ]} = Lie{X1 , X2 }. From the theorem of differentiation of composite functions, we easily get the following characterization of left-invariant vector fields on G. Proposition 1.2.3 (Characterization of g. I). Let G be a Lie group on RN , and let g be the Lie algebra of G. The (smooth) vector field X belongs to g if and only if (XI )(α ◦ x) = Jτα (x) · (XI )(x)

∀ α, x ∈ G.

(1.21)

As usual, Jτα (x) denotes the Jacobian matrix at the point x of the map τα . Proof. For every smooth function ϕ on RN , we have (X(ϕ ◦ τα ))(x) = ∇(ϕ ◦ τα )(x) · XI (x) = (∇ϕ)(τα (x)) · Jτα (x) · XI (x) and


15

(Xϕ)(τα (x)) = (∇ϕ)(τα (x)) · XI (τα (x)). Then X ∈ g if and only if (∇ϕ)(τα (x)) · Jτα (x) · XI (x) = (∇ϕ)(τα (x)) · XI (τα (x))

(1.22)

N ∞ ∞ for Nevery α, x ∈ R and for every ϕ ∈ C (C , R). By choosing ϕ(x) = j =1 hj xj with hj ∈ R for 1 ≤ j ≤ N, (1.22) gives

hT · Jτα (x) · XI (x) = hT · XI (τα (x)) for every h ∈ RN , which obviously implies (1.21).

Interchanging α with x in (1.21), we obtain (XI )(x ◦ α) = Jτx (α) · (XI )(α) for all α, x ∈ G, so that, when α = 0, (XI )(x) = Jτx (0) · (XI )(0)

∀ x ∈ G.

(1.23)

This identity says that a left-invariant vector field on G is completely determined by its value at the origin (and by the Jacobian matrix at the origin of the left-translation). The following result shows that (1.23) characterizes the vector fields in g. Proposition 1.2.4. Let G be a Lie group on RN , and let g be the Lie algebra of G. Let η be a fixed vector of RN , and define the (component functions of the) vector field X as follows (1.24) XI (x) := Jτx (0) · η, x ∈ RN . Then X ∈ g. Proof. Definition (1.24) gives XI (α ◦ x) = Jτα◦x (0) · η,

α, x ∈ RN .

(1.25)

On the other hand, since the composition law on G is associative, we have τα◦x = τα ◦ τx , so that Jτα◦x (0) = Jτα (x) · Jτx (0). Replacing this identity in (1.25), we get XI (α ◦ x) = Jτα (x) · Jτx (0) · η, which implies, by (1.24), XI (α ◦ x) = Jτα (x) · XI (x). Then, by Proposition 1.2.3, X ∈ g. This ends the proof.

Corollary 1.2.5 (Characterization of g. II). Let G be a Lie group on RN , and let g be the Lie algebra of G. The vector field X belongs to g iff (XI )(x) = Jτx (0) · (XI )(0)

∀ x ∈ G.

(1.26)

16


Proof. If X satisfies (1.26), then, by Proposition 1.2.4 (setting η := XI (0)), we have X ∈ g. Vice versa, if X ∈ g, then we already showed that (1.26) follows from (1.21) of Proposition 1.2.3.

Example 1.2.6. If G = H1 (see Example 1.2.2), we have

1 0 0 Jτx (0) = 0 1 0 . 2x2 −2x1 1 For example, for X1 = ∂x1 + 2 x2 ∂x3 , we recognize that, for every x ∈ H1 ,

1 1 1 0 0 (X1 I )(x) = = 0 0 1 0 · 0 = Jτx (0) · (XI )(0). 2 x2 0 2x2 −2x1 1 The same obviously holds, e.g. for the fields X2 = ∂x2 − 2 x1 ∂x3 and [X1 , X2 ] = −4∂x3 . From Proposition 1.2.3 and identity (1.23) it follows that g is a vector space of dimension N. Indeed, the following proposition holds. Proposition 1.2.7 (Characterization of g RN . III). Let G be a Lie group on RN , and let g be the Lie algebra of G. The map J : RN → g,

η → J (η)

with J (η) defined by J (η)I (x) = Jτx (0) · η

(1.27)

is an isomorphism of vector spaces. In particular, dim g = N. Proof. We first observe that J is well defined since, by Proposition 1.2.4, J (η) ∈ g for every η ∈ RN . Moreover, by identity (1.23), J (RN ) = g. The linearity of J is obvious. Then it remains to prove that J is injective. Suppose J (η) = 0. This means that Jτx (0)·η = 0 for every x ∈ RN . In particular Jτ0 (0) · η = 0. On the other hand, since the left-translation τ0 is the identity map,

Jτ0 (0) · η = η. Then η = 0, and J is one-to-one. Example 1.2.8. The Lie algebra h1 of G = H1 (see Example 1.2.2 for the notation) is given by span{X1 , X2 , [X1 , X2 ]}. Indeed, X1 , X2 , [X1 , X2 ] are three linearly independent left-invariant vector fields and dim(h1 ) = 3, as stated in Proposition 1.2.7. Again using the same proposition, we could also argue as follows: X1 , X2 , [X1 , X2 ] are the vector fields obtained by multiplying Jτx (0) respectively times the basis of R3 (1, 0, 0)T ,

(0, 1, 0)T ,

(0, 0, −4)T .

In what follows, the next remarks will be very useful.


17

Remark 1.2.9. Let X ∈ g, and denote by η the value of XI at x = 0, i.e. η = XI (0). Then, by identity (1.23), XI (x) = Jτx (0) · η. As a consequence, for every smooth function ϕ on RN , d d ϕ(x ◦ (tη)) = ϕ(τx (tη)) dt t=0 dt t=0 = ∇ϕ(x) · Jτx (0) · η = ∇ϕ(x) · XI (x) = (Xϕ)(x). Then (Xϕ)(x) =

d ϕ(x ◦ (tη)), dt t=0

η = XI (0).

(1.28)

Identity (1.28) characterizes the left-invariant vector fields on G. This follows from the next remark. Remark 1.2.10. Let X be a vector field on RN . Assume that there exists η ∈ RN such that, for every ϕ ∈ C ∞ (RN , R), d (Xϕ)(x) = ϕ(x ◦ (tη)) ∀ x ∈ RN . (1.29) dt t=0 Then η = XI (0) and X ∈ g. Indeed, by taking ϕ(x) = xj = Ij (x) and x = 0 in (1.29), one gets d (XI )j (0) = (tη)j = ηj , dt t=0 i.e. XI (0) = η. Then (1.29) and the associativity of ◦ imply d d (Xϕ)(α ◦ x) = ϕ((α ◦ x) ◦ (tη)) = (ϕ ◦ τα )(x ◦ (tη)) dt t=0 dt t=0 = X(ϕ ◦ τα )(x) for every α, x ∈ G. Then X is left-invariant on G.

Collecting together the above remarks, we have proved the following result. Proposition 1.2.11 (Characterization of g. IV). Let G be a Lie group on RN , and let g be the Lie algebra of G. The vector field X belongs to g iff there exists η ∈ RN such that, for every ϕ ∈ C ∞ (RN , R), d (1.30) (Xϕ)(x) = ϕ(x ◦ (tη)) ∀ x ∈ RN . dt t=0 In this case η = XI (0).

18


Remark 1.2.12. For every x ∈ RN and X ∈ g, the following expansion holds exp(tX)(x) = x ◦ (tη) + o(t) as t → 0,

η = XI (0).

(1.31)

Indeed, since XI (x) = Jτx (0) · η, x ◦ tη = τx (tη) = τx (0) + tJτx (0) · η + o(t) = x + tXI (x) + o(t). Then (1.31) follows from (1.10), see page 8. We remark that two vector fields can be linearly independent in T (RN ) without being linearly independent at every point. Take, for example, ∂x1 and x1 ∂x2 in R2 . Moreover, two vector fields can be linearly dependent at every point without being linearly dependent in T (RN ). Take, for example, ∂x1 and x1 ∂x1 in R2 . The following result shows that neither of the previous situations can occur for left-invariant vector fields on a Lie group G on RN . Indeed, given a family of vector fields X1 , . . . , Xm ∈ g, the rank of the subset of RN spanned by {X1 I (x), . . . , Xm I (x)} is independent of x. More precisely, we have the following result. Proposition 1.2.13 (Constant rank). Let G be a Lie group on RN , and let g be the Lie algebra of G. Let X1 , . . . , Xm ∈ g. Then the following statements are equivalent: (i) (ii) (iii) (iv)

X1 , . . . , Xm are linearly independent (in g); X1 I (0), . . . , Xm I (0) are linearly independent (in RN ); ∃ x0 ∈ RN : X1 I (x0 ), . . . , Xm I (x0 ) are linearly independent (in RN ); X1 I (x), . . . , Xm I (x) are linearly independent (in RN ) for all x ∈ RN .

Proof. We first recall that, by identity (1.23), Xj I (x) = Jτx (0) · ηj , for every x ∈

RN .

with ηj = Xj I (0),

On the other hand, since τx −1 ◦ τx = I , Jτx −1 (x) · Jτx (0) = IN .

Hence Jτx (0) is non-singular for every x ∈ RN . Then (ii), (iii) and (iv) are equivalent. The equivalence between (i) and (ii) follows from Proposition 1.2.7. Indeed, with the notation of that proposition, for every j ∈ {1, . . . , m}, Xj = J (ηj ) with

ηj = Xj I (0), and J is an isomorphism of RN onto g. Example 1.2.14. The vector fields on R2 defined by X1 = ∂x1 ,

X2 = x1 ∂x2

do satisfy the so-called Hörmander condition rank(Lie{X1 , X2 }(x)) = 2

for every x ∈ R2 .

However, since X1 and X2 are independent as vector fields but X1 I (0), X2 I (0) are dependent as vectors of R2 , X1 and X2 are not left-invariant with respect to any group law on R2 .


19

1.2.2 The Jacobian Basis From Proposition 1.2.7 it follows that any basis of g is the image via J of a basis of RN . A natural definition is thus in order. Definition 1.2.15 (Jacobian basis). Let G be a Lie group on RN , and let g be the Lie algebra of G. If {e1 , . . . , eN } is the canonical basis7 of RN and J is the map defined in Proposition 1.2.7, we call {Z1 , . . . , ZN },

Zj := J (ej )

the Jacobian basis of g. (Note. We warn the reader that the notion of Jacobian basis is strictly related to the fact that, presently, G is a Lie group on RN , and we are making reference to the fixed Cartesian coordinates on RN . Hence, the Jacobian basis is not well-posed on general Lie groups. Moreover, if we perform a change of coordinates on RN , even a linear one, the Jacobian basis changes. Despite this “non-invariant”, non-coordinatefree nature of the Jacobian basis, the reader will soon recognize its usefulness.) From the very definition of J we obtain Zj I (x) = Jτx (0) · ej = j -th column of Jτx (0) so that, since Jτx (0) = IN ,

∀ x ∈ RN ,

(1.32)

Zj I (0) = ej .

From Remark 1.2.10 we also have d ∂ (Zj ϕ)(x) = ϕ(x ◦ tej ) = ϕ(x ◦ y) dt t=0 ∂yj y=0

(1.33)

for every ϕ ∈ C ∞ (RN ) and every x ∈ G. Consequently, the Jacobian basis {Z1 , . . . , ZN } of g is given by the N column of the Jacobian matrix Jτx (0) (whence the name). Moreover, Zj |0 = ∂/∂xj |0 and (Zj ϕ)(x) = (∂/∂yj )|y=0 ϕ(x ◦ y)

∀ ϕ ∈ C ∞ (RN ), x ∈ G.

(1.34)

Summing up the above results, we have the following equivalent characterizations of the Jacobian basis. 7 Id est, for every j ∈ {1, . . . , N },

1 , . . . , 0)T . ej = (0, . . . , j

20


Proposition 1.2.16 (Jacobian basis). Let G be a Lie group on RN , and let g be the Lie algebra of G. Let j ∈ {1, . . . , N } be fixed. Then there exists one and only one vector field in g, say Zj , characterized by any of the following equivalent conditions: 1. Zj |0 = (∂/∂xj )|0 , i.e. (Zj ϕ)(0) =

∂ϕ (0) ∂ xj

for every ϕ ∈ C ∞ (RN , R);

2. for every ϕ ∈ C ∞ (RN , R), it holds ∂ ϕ(x ◦ y) (Zj ϕ)(x) = ∂ yj y=0

for every x ∈ G;

3. if ej denotes the j -th vector of the canonical basis of RN , then Zj I (0) = ej ; 4. the column vector of the component functions of Zj is Zj I (x) = Jτx (0) · ej = j -th column of Jτx (0); 5. for every x ∈ G, we have d (Zj ϕ)(x) = ϕ(x ◦ (tej )) dt t=0

for every ϕ ∈ C ∞ (RN , R).

The system of vector fields Z := {Z1 , . . . , ZN } is a basis of g, the Jacobian basis. The coordinates of X ∈ g w.r.t. Z are, orderly, the entries of the column vector XI (0). (For the proof of the last statement, see Remark 1.2.20.) In the sequel, when we need to endow g with a differentiable structure, we shall consider the vector space structure of g, making g (in a natural way) a differentiable manifold. Although the choice of a basis for g is completely immaterial, we shall prevalently fix a system of coordinates on g by choosing the Jacobian basis, then identifying g with RN in a fixed way. Example 1.2.17 (The Jacobian basis of H1 ). The Jacobian basis for the Lie algebra of H1 (see Example 1.2.2) is given by Z1 = ∂x1 + 2 x2 ∂x3 ,

Z2 = ∂x2 − 2 x1 ∂x3 ,

Z3 = ∂x3 ,

since, in this case, the Jacobian matrix at 0 of the left-translation is

1 0 0 Jτx (0) =

0 1 0 . 2x2 −2x1 1


21

Example 1.2.18 (A non-polynomial nilpotent Lie group on R3 ). It is a simple exercise to verify that the following operation (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) := (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) endows R3 with a Lie group structure (G, ◦). The Lie algebra g of G is spanned by the vector fields, coefficient-vectors are given by the columns of the Jacobian matrix 1

0 cosh(x1 ) 0 Jτx (0) = 0 1 sinh(x1 ) 0 0 1 (the Jacobian basis), i.e. g = span{Z1 , Z2 , Z3 }, where 1 ∂x cosh(x1 ) 1 Z2 = ∂x2 Z3 = ∂x3 + sinh(x1 ) ∂x2 .

Z1 =

Note that G is nilpotent (of step two). Example 1.2.19 (A non-polynomial non-nilpotent Lie group on R2 ). It is a simple exercise to verify that the following operation on R2 (x1 , x2 ) ◦ (y1 , y2 ) = (x1 + y1 , y2 + x2 ey1 ) defines a Lie group structure, and the Jacobian basis is Z1 = ∂x1 + x2 ∂x2 , Z2 = ∂x2 . Hence, the relevant Lie algebra is not nilpotent, for [Z2 , Z1 ] = Z2 , so that, inductively, [· · · [[[Z2 , Z1 ], Z1 ], Z1 ] · · · Z1 ] = Z2

for all k ∈ N.

k times

Remark 1.2.20. Let us consider the map π : g → RN ,

X → π(X) := XI (0).

(1.35)

From the very definition of π, the following fact follows: if X ∈ g and we write η := π(X), then we have Jτx (0) · η = Jτx (0) · XI (0) = (XI )(x),

(1.36)

the last equality following from (1.23). Let now J be the map introduced in Proposition 1.2.7. Comparing (1.36) to (1.27), we recognize that J (π(X)) = J (η) = X,

π(J (η)) = η

∀ X ∈ g, η ∈ RN .

Thus π is the inverse map of J . We explicitly remark that π is the linear map which assigns, to every vector field X in g, the N-tuple η in RN of the coordinates of X with respect to the Jacobian basis. The coordinate of X with respect to this basis is simply given by XI (0) (see also Fig. 1.2).

22


Fig. 1.2. Figure of Remark 1.2.20

1.2.3 The (Jacobian) Total Gradient Let G = (RN , ◦) be a Lie group on RN , and let Z1 , . . . , ZN be the Jacobian basis8 of the Lie algebra g of G. For any differentiable function u defined on an open set Ω ⊆ RN , we consider a sort of “intrinsic” gradient of u given by (Z1 u, . . . , ZN u) (in the sequel, we shall call it (Jacobian) total gradient). Then it follows from (1.32) that (Z1 u(x), . . . , ZN u(x)) = ∇u(x) · Jτx (0)

∀ x ∈ Ω.

(1.37)

On the other hand, since Jτx (0) is non-singular and its inverse is given by Jτx −1 (0), we can write the Euclidean gradient of u in terms of its total gradient in the following way ∀ x ∈ Ω. (1.38) ∇u(x) = (Z1 u(x), . . . , ZN u(x)) · Jτx −1 (0) From (1.38) we immediately obtain the following result. We shall follow the notation of Remark 1.2.3. Proposition 1.2.21. Let G be a Lie group on RN , and let Z1 , . . . , ZN be the relevant Jacobian basis (or any basis for g). Let Ω ⊆ RN be an open and connected set. A function u ∈ C 1 (Ω, R) is constant in Ω if and only if its total gradient (Z1 u, . . . , ZN u) vanishes identically on Ω. (Note. A significant improvement of this result will be available in the stratified setting. See, e.g. Proposition 1.5.6). Proof. From (1.37) and (1.38) it follows that the total gradient of u vanishes at x ∈ Ω if and only if ∇u(x) = 0.

8 In this section, we consider the Jacobian basis for the sake of brevity. In fact, Z , . . . , Z N 1

can be replaced by any basis X1 , . . . , XN of g. Indeed, note that in this case there exists a N × N non-singular constant matrix M such that (X1 I (x) · · · XN I (x)) = M · (Z1 I (x) · · · ZN I (x)).


23

Example 1.2.22. When G = H1 , it indeed holds (Z1 u, Z2 u, Z3 u) = (∂x1 u + 2 x2 ∂x3 u, ∂x2 u − 2 x1 ∂x3 u, ∂x3 u)

1 0 0 = ∂x1 u, ∂x2 u, ∂x3 u · 0 1 0 = ∇u · Jτx (0), 2x2 −2x1 1 and, vice versa, (Z1 u, Z2 u, Z3 u) · Jτx −1 (0) = (Z1 u, Z2 u, Z3 u) ·

1 0 −2x2

0 1 2x1

0 0 1

= ∇u.

1.2.4 The Exponential Map of a Lie Group on RN The next lemma will be useful to define the notion of Exponential map from g to G, one of the most important tools in the Lie group theory. Lemma 1.2.23 (Completeness of g). Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let X ∈ g, and let γ : [t0 , t0 + T ] → RN be an integral curve of X. Then: (i) α ◦ γ is an integral curve of X for every α ∈ G. (ii) γ can be continued to an integral curve of X on the interval [t0 − T , t0 + 2T ]. Proof. (i) For every t ∈ [t0 , t0 + T ], we have (by (1.21)) d d (α ◦ γ (t)) = (τα (γ (t))) = Jτα (γ (t)) · γ˙ (t) dt dt = Jτα (γ (t)) · XI (γ (t)) = X(α ◦ γ (t)). (ii) Define Γ : [t0 − T , t0 + 2T ] → RN as follows: ⎧ −1 ⎪ ⎨ γ (t0 ) ◦ (γ (t0 + T )) ◦ γ (t + T ) if t0 − T ≤ t ≤ t0 , if t0 ≤ t ≤ t0 + T , Γ (t) := γ (t), ⎪ ⎩ −1 γ (t0 + T ) ◦ (γ (t0 )) ◦ γ (t − T ) if t0 + T ≤ t ≤ t0 + 2T . Then, by (i), Γ is an integral curve of X and, obviously, Γ |[t0 ,t0 +T ] ≡ γ .

From assertion (ii) of this Lemma we immediately obtain the following important statement: For every X ∈ g, the map (x, t) → exp(tX)(x) is well-defined for every x ∈ RN and every t ∈ R. The next corollary easily follows from the assertion (i) of Lemma 1.2.23.

24


Corollary 1.2.24. Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let X ∈ g and x, y ∈ G. Then x ◦ exp(tX)(y) = exp(tX)(x ◦ y)

(1.39)

for every t ∈ R. In particular, for y = 0, exp(tX)(x) = x ◦ exp(tX)(0). Proof. By Lemma 1.2.23-(i), t → x ◦ exp(tX)(y) is an integral curve of X. Moreover, (x ◦ exp(tX)(y))|t=0 = x ◦ y. Then (1.39) follows.

Definition 1.2.25 (Exponential map). Let G be a Lie group on RN , and let g be its Lie algebra. The exponential map of the Lie group G is defined by Exp : g → G,

Exp (X) = exp(1 · X)(0).

More explicitly, Exp (X) is the value at the time t = 1 of the path γ (t) solution to γ˙ (t) = XI (γ (t)), γ (0) = 0. (See Fig. 1.3.) From Corollary 1.2.24 and identity (1.11b) (with τ = −t) we get Exp (−X) ◦ Exp (X) = 0. Indeed, Exp (−X) ◦ Exp (X) = Exp (−X) ◦ exp(X)(0) = exp(X)(Exp (−X)) = exp(X)(exp(−X)(0)) = 0. Then we have

(Exp (X))−1 = Exp (−X).

Fig. 1.3. Figure of Definition 1.2.25

We give an explicit example of exponential map.

(1.40)


25

Example 1.2.26 (The Exp map on H1 ). Let us consider once again the Heisenberg– Weyl group H1 on R3 . In Example 1.2.8, we showed that a basis for its Lie algebra h1 is given by X1 , X2 , X3 , where X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 and X3 = [X1 , X2 ] = −4∂x3 . Let us construct the exponential map. We set, for ξ ∈ R3 , ξ · X := ξ1 X1 + ξ2 X2 + ξ3 X3

1 0 0 ξ1 = ξ1 + ξ2 + ξ3 0 = . ξ2 0 1 −4ξ3 + 2ξ1 x2 − 2ξ2 x1 2 x2 −2 x1 −4 By Definition 1.1.2, for fixed x ∈ H1 , we have exp(ξ · X)(x) = γ (1), where γ (s) = (γ1 (s), γ2 (s), γ3 (s)) is the solution to γ˙ (s) = (ξ · X)I (γ (s)) = (ξ1 , ξ2 , −4ξ3 + 2ξ1 γ2 (s) − 2 ξ2 γ1 (s)), γ (0) = x. Solving the above system of ODE’s, one gets ⎛

⎞ x1 + ξ1 ⎠. x 2 + ξ2 exp(ξ · X)(x) = ⎝ x 3 − 4 ξ3 + 2 ξ1 x 2 − 2 ξ2 x 1

As a consequence, by Definition 1.2.25, we obtain ⎛

⎞ ξ1 Exp (ξ · X) = exp(ξ · W )(0) = ⎝ ξ2 ⎠ , −4 ξ3 so that Exp is globally invertible and its inverse map is given by ⎞ ⎛ y1 Log (y) := (Exp )−1 (y) = ⎝ y2 ⎠ · X. − 14 y3 For example, we have ⎛

⎞ −ξ1 −1 Exp (−ξ · X) = ⎝ −ξ2 ⎠ = −Exp (ξ · X) = Exp (−ξ · X) , +4 ξ3 since the inverse of x in H1 coincides with −x. This fact tests, in this simple example, the validity of (1.40).

Remark 1.2.27 (Local invertibility of Exp ). Let {X1 , . . . , XN } be a basis of g. Then, for every X ∈ g, X=

N j =1

ξj X j

for a suitable ξ = (ξ1 , . . . , ξN ) ∈ RN ,

26


Fig. 1.4. The relation between the group G, its algebra g and RN

so that

Exp (X) = exp

N

ξj Xj (0).

j =1

From the classical theory of ODE’s we know that the map N

ξj Xj (0) (ξ1 , . . . , ξN ) → exp j =1

is smooth. Then we can say that the map g X → Exp (X) ∈ G is smooth. From the Taylor expansion (1.10) (page 8) we get Exp (X) =

N

ξj ηj + O(|ξ |2 ),

as |ξ | → 0,

j =1

where ηj = Xj I (0). Denote by E the matrix whose column vectors are η1 , . . . , ηN . Then JExp (0) = E

(we fix on g coordinates related to the Xj ’s).

In particular, if {X1 , . . . , XN } = {Z1 , . . . , ZN } is the Jacobian basis of g, then JExp (0) = IN

(we fix on g coordinates related to the Zj ’s).

(1.41)


27

As a consequence, Exp is a diffeomorphism from a neighborhood of 0 ∈ g onto a neighborhood of 0 ∈ G. Where defined,9 we denote by Log the inverse map of Exp. Example 1.2.28. With the notation of Example 1.2.26, we recall that the Jacobian basis for h1 is given by (see Example 1.2.17) Z1 = X1 ,

Z2 = X2 ,

1 Z3 = − X3 . 4

Hence, by the computations in Example 1.2.26, we have 1 Exp (ξ1 Z1 + ξ2 Z2 + ξ3 Z3 ) = Exp ξ1 X1 + ξ2 X2 − ξ3 X3 4

! = (ξ1 , ξ2 , ξ3 ), (1.42)

and (1.41) is readily verified.

The next proposition is an easy consequence of Corollary 1.2.24 and shows an important link between the composition law in G and the exponential map. Proposition 1.2.29 (Exponentiation and composition). Let (G, ◦) be a Lie group on RN , and let g be its Lie algebra. Let x, y ∈ G. Assume Log (y) is defined. Then x ◦ y = exp(Log (y))(x).

(1.43)

Proof. Let X = Log (y). This means that y = Exp (X) = exp(X)(0). Then, by Corollary 1.2.24, we infer x ◦ y = x ◦ exp(X)(0) = exp(X)(x). This is precisely (1.43), and the proof is complete.

By writing y = Exp (X) in (1.43), we obtain x ◦ Exp (X) = exp(X)(x)

for every X ∈ g and every x ∈ G.

(1.44)

We give two examples of this proposition, the first is very simple, the second one a little more elaborated. Example 1.2.30 (of (1.43) for H1 ). Let us consider once again the Heisenberg–Weyl group H1 on R3 . Proceeding with the computations in Example 1.2.26, we have 9 We shall see in Chapter 2 that, in many important situations, such as for connected and

simply connected nilpotent Lie groups, Exp is globally invertible. Any Carnot group is a connected and simply connected nilpotent Lie group.

28


Fig. 1.5. Figure of Proposition 1.2.29

! ! 1 exp(Log (y))(x) = exp y1 , y2 , − y3 · X (x) 4 ⎛ ⎞ x1 + y1 ⎠ = x ◦ y, x2 + y2 =⎝ x3 + y3 + 2 y1 x2 − 2 y2 x1 which tests, in this simple example, the validity of (1.43).

Example 1.2.31. Let us consider once again the Lie group G introduced in Example 1.2.18. The relevant Jacobian basis is {Z1 , Z2 , Z3 }, where Z1 =

1 ∂x , cosh(x1 ) 1

Z2 = ∂x2 ,

Z3 = ∂x3 + sinh(x1 ) ∂x2 .

Given ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 and x ∈ G, we set ξ · Z = Z))(x) = γ (t) is the solution to

3

i=1 ξi

Zi , so that exp(t (ξ ·

γ˙ (t) = ξ1 Z1 I (γ (t)) + ξ2 Z2 I (γ (t)) + ξ3 Z3 I (γ (t)), i.e. more explicitly (writing γ (t) = (γ1 (t), γ2 (t), γ3 (t))), ⎧ −1 ⎪ ⎨ γ˙1 (t) = ξ1 (cosh(γ1 (t))) , γ1 (0) = x1 , γ˙2 (t) = ξ2 + ξ3 sinh(γ1 (t)), γ2 (0) = x2 , ⎪ ⎩ γ˙3 (t) = ξ3 , γ3 (0) = x3 . A direct computation gives ⎧ ⎪ ⎨ γ1 (t) = arcsinh(sinh(x1 ) + ξ1 t), 2 γ2 (t) = x2 + (ξ2 + ξ3 sinh(x1 ))t + ξ1 ξ3 t2 , ⎪ ⎩ γ3 (t) = x3 + ξ3 t.

γ (0) = x,


29

In particular, Exp (ξ · Z) = exp(1 (ξ · Z))(0), i.e. ! 1 Exp (ξ · Z) = arcsinh(ξ1 ), ξ2 + ξ1 ξ3 , ξ3 , 2 so that

! 1 Log (y1 , y2 , y3 ) = sinh(y1 ), y2 − sinh(y1 ) y3 , y3 · Z. 2

Collecting the above facts together, we get ! ! 1 sinh(y1 ) y3 , y3 · Z (x) 2 arcsinh(sinh(x1 ) + sinh(y1 )) = x2 + (y2 − 12 sinh(y1 ) y3 + y3 sinh(x1 )) + sinh(y1 ) y3 x3 + y3

arcsinh(sinh(x1 ) + sinh(y1 )) = = x ◦ y, x2 + y2 + y3 sinh(x1 ) x3 + y3

exp(Log (y))(x) = exp

sinh(y1 ), y2 −

1 2

as stated in (1.43) of Proposition 1.2.29.

Remark 1.2.32 (The composition of G induces a composition on g via Exp: The Campbell–Hausdorff operation). Suppose that Exp : g → G and Log : G → g are globally defined C ∞ maps, inverse to each other. We then define on g the operation X Y := Log (Exp (X) ◦ Exp (Y )), X, Y ∈ g. (1.45) It is immediately seen that defines a Lie group structure on g and Exp : (g, ) → (G, ◦) is a Lie-group isomorphism. Indeed, this last fact is obvious from the very definition of , whereas the associativity of on g follows immediately from the associativity of ◦ on G. One of the most striking facts about Lie algebras and Lie groups is that (under suitable hypotheses) the operation on g is well-posed and can be expressed in a somewhat “universal” way as a sum of iterated Lie-brackets of X and Y (see (2.43) and Theorem 2.2.13, page 129). For example, the first few terms are XY =X+Y +

1 1 1 [X, Y ] + [X, [X, Y ]] − [Y, [X, Y ]] + · · · . 2 12 12

(1.46)

We shall deal extensively on the composition law throughout the book. We give an example of when G is the Heisenberg–Weyl group on R3 .

30


Example 1.2.33 (The operation for H1 ). The reader is invited to compare this example to the diagram in Fig. 1.6. We use the notation and computations of Example 1.2.26. When G = H1 , we saw that Exp is globally invertible. Hence (1.45) makes sense. We fix X ∈ h1 . If Z1 , Z2 , Z3 is the Jacobian basis for h1 , and we set, for brevity, ξ := XI (0), we have (see Remark 1.2.20) X = ξ1 Z1 + ξ2 Z2 + ξ3 Z3 =: ξ · Z. Analogously, if Y ∈ h1 , we set η := Y I (0), so that Y = η · Z. Thus, we derive Log Exp (X) ◦ Exp (Y ) = Log Exp (ξ · Z) ◦ Exp (η · Z) = (see (1.42))

Log (ξ ◦ η)

= Log (ξ1 + η1 , ξ2 + η2 , ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) (again from (1.42)) = (ξ1 + η1 , ξ2 + η2 , ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) · Z = (ξ1 + η1 ) Z1 + (ξ2 + η2 ) Z2 + (ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) Z3 .

(1.47)

On the other hand, we consider (1.46), truncated to the commutators of length two (sine h1 is nilpotent of step two!), and we explicitly write down X Y in our case, thus obtaining 1 [ξ · Z, η · Z] 2 = ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + η1 Z1 + η2 Z2 + η3 Z3

(ξ · Z) (η · Z) = ξ · Z + η · Z +

1 [ξ1 Z1 + ξ2 Z2 + ξ3 Z3 , η1 Z1 + η2 Z2 + η3 Z3 ] 2 (here we use [Z1 , Z2 ] = −4 Z3 , [Z1 , Z3 ] = [Z2 , Z3 ] = 0) +

= (ξ1 + η) Z1 + (ξ2 + η2 ) Z2 + (ξ3 + η3 ) Z3 1 (−4 ξ1 η2 + 4 ξ2 η1 ) Z3 + 2 = (ξ1 + η) Z1 + (ξ2 + η2 ) Z2 + (ξ3 + η3 + 2 η1 ξ2 − 2 η2 ξ1 ) Z3 , which equals the last term in (1.47). As a consequence, we have proved that in this case it holds 1 Log Exp (X) ◦ Exp (Y ) = X + Y + [X, Y ]. 2 For the group considered in Examples 1.2.18 and 1.2.31, the reader is invited to test a similar formula.

1.3 Homogeneous Lie Groups on RN

31

1.3 Homogeneous Lie Groups on RN We begin by giving the definition of homogeneous Lie group (see also E.M. Stein [Ste81]). Definition 1.3.1 (Homogeneous Lie group (on RN )). Let G = (RN , ◦) be a Lie group on RN (according to Definition 1.2.1). We say that G is a homogeneous (Lie) group (on RN ) if the following property holds: (H.1)

There exists an N-tuple of real numbers σ = (σ1 , . . . , σN ), with 1 ≤ σ1 ≤ · · · ≤ σN , such that the “dilation” δλ : RN → RN ,

δλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN )

is an automorphism of the group G for every λ > 0. We shall denote by G = (RN , ◦, δλ ) the datum of a homogeneous Lie group on RN with composition law ◦ and dilation group {δλ }λ>0 . (Note. A note similar to that given after Definition 1.2.15 of Jacobian basis applies to the notion of homogeneous Lie group: the notion of homogeneous Lie group is not coordinate-free and strongly depends on the choice of a fixed system of coordinates on RN . Nonetheless, the reader will soon recognize the suitability of this notion.) The family of dilations {δλ }λ>0 forms a one-parameter group of automorphisms of G whose identity is δ1 = I, the identity map of RN . Indeed, we have δr s (x) = δr δs (x)

∀ x ∈ G, r, s > 0.

Moreover, (δλ )−1 = δλ−1 . In the sequel, {δλ }λ>0 will be referred to as the dilation group (or group of dilations) of G. From (H.1) it follows that δλ (x ◦ y) = (δλ x) ◦ (δλ y)

∀ x, y ∈ G

(1.48)

and, if e denotes the identity of G, δλ (e) = e for every λ > 0. This obviously implies that e = 0. This is consistent with our previous assumption that the origin is the identity of G. For example, the Heisenberg–Weyl group H1 (see Example 1.2.2, page 14) is a homogeneous Lie group if R3 is equipped with the dilations δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ). Remark 1.3.2. Suppose G = (RN , ◦) is a Lie group on RN such that there exists an N-tuple of positive real numbers σ = (σ1 , . . . , σN ) such that δλ : RN → RN ,

dλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN )

32


is an automorphism of the group G for every λ > 0. Then, modulo a permutation of the variables of RN , it is always not restrictive to suppose that σ1 ≤ · · · ≤ σN . Obviously, this permutation of the coordinates does not alter neither (the new permuted) G being a Lie group on RN nor the (relevant permuted) dilation δλ satisfying (1.48). Moreover, there exists a group of dilations δλ on G such that σ1 σN δλ (x1 , . . . , xN ) = (λ x1 , . . . , λ xN )

with 1 = σ1 ≤ · · · ≤ σN . Indeed, it suffices to take (once the σj ’s have been ordered increasingly) σj := σj /σ1 for every j = 1, . . . , N . Indeed, with this choice, we have δλ ≡ dλ1/σ1 , and δλ (x ◦ y) = δλ (x) ◦ δλ (y) follows from (1.48), restated for dλ , with λ replaced by λ1/σ1 . 1.3.1 δλ -homogeneous Functions and Differential Operators Before we continue the analysis of homogeneous Lie groups, we show some basic properties of homogeneous functions and homogeneous differential operators with respect to the family {δλ }λ . In this subsection, no group law is required on RN . Here, we only suppose that it is given on RN a family of maps δλ of the form δλ : RN → RN ,

δλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN ),

(1.49)

with fixed positive real numbers σ1 , . . . , σN . We set σ := (σ1 , . . . , σN ). A real function a defined on RN is called δλ -homogeneous of degree m ∈ R if a does not vanish identically and, for every x ∈ RN and λ > 0, it holds a(δλ (x)) = λm a(x). A non-identically-vanishing linear differential operator X is called δλ -homogeneous of degree m ∈ R if, for every ϕ ∈ C ∞ (RN ), x ∈ RN and λ > 0, it holds X(ϕ(δλ (x))) = λm (Xϕ)(δλ (x)). Let a be a smooth δλ -homogeneous function of degree m ∈ R and X be a linear differential operator δλ -homogeneous of degree n ∈ R. Then Xa is a δλ homogeneous function of degree m − n (unless Xa ≡ 0). Indeed, for every x ∈ RN and λ > 0, we have λn (Xa)(δλ (x)) = X(a(δλ (x))) = X(λm a(x)) = λm (Xa)(x). Given a multi-index α ∈ (N ∪ {0})N , α = (α1 , . . . , αN ), we define the δλ -length (or δλ -height) of α as


|α|σ = α, σ =

N

αi σi .

33

(1.50)

i=1

G-length of a multi-index. G -degree). When G = (RN , ◦, δλ ) is Definition 1.3.3 (G a homogeneous Lie group on RN with its given group of dilations {δλ }λ , we shall use the notation |α|G for the relevant δλ -length as defined in (1.50). In this case, we shall refer to |α|G as the G-length (or G-height) of α. Moreover, if p : G → R is a polynomial function (the sum below is intended to be finite) p(x) = cα x α , cα ∈ R, α

we say that degG (p) := max{|α|G : cα = 0} is the G-degree or δλ -(homogeneous) degree of p. Since x → xj and ∂/∂xj , j ∈ {1, . . . , N }, are obviously δλ -homogeneous of degree σj , the function x → x α and the differential operator D α are both δλ homogeneous of degree |α|σ . If a is a continuous function, δλ -homogeneous of degree m and a(x0 ) = 0 for some x0 ∈ RN , then m ≥ 0. Indeed, from a(δλ (x0 )) = λm a(x0 ) we get a(0) a(δλ (x0 )) = . λ→0 a(x0 ) a(x0 )

lim λm = lim

λ→0

Moreover, the continuous and δλ -homogeneous of degree 0 functions are precisely the constant (non-zero) functions. Indeed, a(x) = a(δλ (x)) = lim a(δλ (x)) = a(0). λ→0+

Let us now consider a smooth and δλ -homogeneous of degree m ∈ R function a and a multi-index α. Assume that D α a is not identically zero. Then, since D α a is smooth and δλ -homogeneous of degree m − |α|σ , it has to be m − |α|σ ≥ 0, i.e. |α|σ ≤ m. This result can be restated as follows: D α a ≡ 0 ∀ α such that |α|σ > m. Thus a is a polynomial function. Let a(x) = α∈A aα x α , where A is a finite set of multi-indices and aα ∈ R for every α ∈ A. Since a is δλ -homogeneous of degree m, we have λm aα x α = λm a(x) = a(δλ (x)) = aα λ|α|σ x α . α∈A

Hence

λm a

α

=

λ|α|σ

α∈A

aα for every λ > 0, so that |α|σ = m if aα = 0. Then

34


a(x) =

aα x α .

(1.51)

|α|σ =m

It is quite obvious that every polynomial function of the form (1.51) is δλ -homogeneous of degree m. Thus, we have proved the following proposition. Proposition 1.3.4 (Smooth δλ -homogeneous functions). Let δλ be as in (1.49). Suppose that a ∈ C ∞ (RN , R). Then a is δλ -homogeneous of degree m ∈ R if and only if a is a polynomial function of the form (1.51) with some aα = 0. As a consequence, the set of the degrees of the smooth δλ -homogeneous (non-vanishing) functions is precisely the set of the nonnegative real numbers A = {|α|σ : α ∈ (N ∪ {0})N }, with |α|σ = 0 if and only if a is constant. From the proposition above one easily obtains the following characterization of the smooth δλ -homogeneous vector fields. Proposition 1.3.5 (Smooth δλ -homogeneous vector fields). Let δλ be as in (1.49). Let X be a smooth non-vanishing vector field on RN , X=

N

aj (x) ∂xj .

j =1

Then X is δλ -homogeneous of degree n ∈ R if and only if aj is a polynomial function δλ -homogeneous of degree σj − n (unless aj ≡ 0). Hence, the degree of δλ homogeneity of X belongs to the set of real (possibly negative) numbers Aj = {σj − |α|σ : α ∈ (N ∪ {0})N }, whenever j is such that aj is not identically zero. Proof. A direct computation shows the “if” part of the proposition. Vice versa, if X(ϕ ◦ δλ ) = λn (X ϕ) ◦ δλ , the choice ϕ(x) = xj yields λσj aj (x) = λn aj (δλ (x)), whence aj is a (smooth) δλ -homogeneous function of degree σj − n. By Proposi tion 1.3.4, aj is a polynomial function. For example, the differential operators X1 = ∂x1 + 2 x2 ∂x3 ,

X2 = ∂x2 − 2 x1 ∂x3

(1.52)

on R3 are δλ -homogeneous of degree one with respect to the dilation δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ). Also, the vector fields x13 X1 = x13 ∂x1 + 2 x13 x2 ∂x3 and x2 X2 = x2 ∂x1 − 2 x1 x2 ∂x3 are respectively δλ -homogeneous of degrees −2 and 0 w.r.t. the same dilation.


35

Corollary 1.3.6. Let δλ be as in (1.49). Let X be a smooth non-vanishing vector field. Then X is δλ -homogeneous of degree n ∈ R iff δλ XI (x) = λn XI (δλ (x)). Proof. Let X = N j =1 aj ∂xj . By Proposition 1.3.5, X is δλ -homogeneous of degree n iff aj (δλ (x)) = λσj −n aj (x) for any j ∈ {1, . . . , N }. This is equivalent to T T δλ (XI (x)) = δλ a1 (x), . . . , aN (x) = λσ1 a1 (x), . . . , λσN aN (x) T = λn a1 (δλ (x)), . . . , aN (δλ (x)) = λn XI (δλ (x)). This ends the proof.

As a straightforward consequence, we have the following simple fact. Remark 1.3.7. Let δλ be as in (1.49). Let X = 0 be a smooth vector field on RN of the form N aj (x) ∂xj . X= j =1

If X is δλ -homogeneous of degree n ∈ R, then, for every aj non-identically zero, we must have n ≤ σj . As a consequence, it has to be n ≤ σN (i.e. the set of the δλ homogeneous degrees of the smooth vector fields is bounded above by the maximum exponent of the dilation). Hence, X has the form aj (x) ∂/∂xj . X= j ≤N, σj ≥n

Suppose now n > 0. Since aj is a polynomial function of degree σj − n and n > 0, then aj does not depend on xj , . . . , xN , aj (x) = aj (x1 , . . . , xj −1 ) (we agree to let aj (x1 , . . . , xj −1 ) = constant when j = 1). We already highlighted in the previous section the importance of these “pyramid”-shaped vector fields (see Remark 1.1.3). From this remark the next proposition straightforwardly follows. N Proposition 1.3.8. Let δλ be as in (1.49). Let X = j =1 aj (x) ∂xj be a smooth -homogeneous of positive degree. Then its adjoint operator X ∗ = vector field δ λ N ∗ − j =1 ∂j (aj ·) satisfies X = −X and X 2 = div(A · ∇ T ), where A is the square matrix (ai aj )i,j ≤N . Finally, X has null divergence.

(1.53)

36


Proof. By the previous remark, the coefficient aj does not depend on xj . Then, for every smooth function ϕ, X∗ ϕ = −

N

∂j (aj ϕ) = −

j =1

N

aj ∂j ϕ = −Xϕ.

j =1

Moreover, it holds X = 2

N

ai ∂i (aj ∂j ) =

i,j =1

N i=1

∂i

N

ai aj ∂j

= div(A · ∇ T ),

j =1

where A is as in the statement of the proposition. Finally, div(XI ) =

0, since aj is independent of xj .

N

j =1 ∂j (aj )

=

Vector fields with different degree of δλ -homogeneity are linearly independent if they do not vanish at the origin. Indeed, the following proposition holds. Proposition 1.3.9. Let δλ be as in (1.49). Let X1 , . . . , Xk ∈ T (RN ) be δλ -homogeneous vector fields of degree n1 , . . . , nk , respectively. If ni = nj for i = j and if Xj I (0) = 0 for every j ∈ {1, . . . , k}, then X1 , . . . , Xk are linearly independent. Proof. Let c1 , . . . , ck ∈ R be such that kj =1 cj Xj = 0. Then, for every smooth function ϕ, 0=

k

cj Xj (ϕ(δλ x)) =

j =1

cj λnj (Xj ϕ)(δλ x)

∀ x ∈ RN .

j =1

If we take ϕ(x) = h, x = 0=

k

k

N

j =1 hj

xj , this identity at x = 0 gives

cj λnj ηj , h

∀ h ∈ RN ,

∀ λ > 0,

j =1

where ηj = Xj I (0). Equivalently, " 0=

k

# cj λnj ηj , h .

j =1

Due to the arbitrariness of h ∈ RN , this gives that (since ni = nj if i = j )

k

j =1 cj

λnj ηj = 0 for all λ > 0, so

cj ηj = 0 for any j ∈ {1, . . . , k}. This implies cj = 0, since, for every j = 1, . . . , k, ηj = 0 by the hypothesis.


37

The following simple proposition will be useful in the sequel. Proposition 1.3.10. Let δλ be as in (1.49). Let X1 , X2 be δλ -homogeneous vector fields of degree n1 , n2 , respectively. Then [X1 , X2 ] is δλ -homogeneous of degree n1 + n2 (unless X1 and X2 commute). As a consequence, if n1 , n2 are both positive, then every commutator of X1 , X2 containing k1 times X1 and k2 times X2 vanish identically whenever k1 n1 + k2 n 2 > σ N . Proof. It suffices to note that, for every smooth function ϕ on RN , one has (X1 X2 )(ϕ(δλ (x))) = λn2 X1 ((X2 ϕ)(δλ (x))) = λn2 +n1 (X1 X2 )(ϕ(δλ (x))). This proves the first part of the assertion, since [X1 , X2 ] = X1 X2 − X2 X1 (and [X1 , X2 ] ≡ 0 iff X1 and X2 commute). Finally, let X be a commutator of X1 , X2 containing k1 times X1 and k2 times X2 . By the first part of this proof, it follows inductively that X is δλ -homogeneous of degree k1 n1 + k2 n2 (unless X ≡ 0). By Remark 1.3.7, we know that if a smooth vector field is δλ -homogeneous of degree n ∈ R, then n ≤ σN . This ends the proof.

For example, the differential operators X1 = ∂x1 + 2x2 ∂x3 , X2 = ∂x2 − 2x1 ∂x3 on the Heisenberg–Weyl group H1 are homogeneous of degree one with respect to the dilation δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ), and [X1 , X2 ] = −4∂x3 is indeed δλ homogeneous of degree two. Moreover, any commutator of X1 , X2 of length ≥ 3 vanish identically, as stated in the last part of Proposition 1.3.10. Corollary 1.3.11. Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN , and let g be the Lie algebra of G. Let X1 , . . . , Xk ∈ g be non-identically vanishing and δλ -homogeneous of degrees n1 , . . . , nk , respectively. If ni = nj for i = j , then X1 , . . . , Xk are linearly independent. Proof. Since Xj I (x) = Jτx (0)·Xj I (0) for every x ∈ RN , and Xj is non-identically vanishing, then Xj I (0) = 0 for any j ∈ {1, . . . , k}. Hence the assertion follows from the previous proposition.

Proposition 1.3.12 (Nilpotence of homogeneous Lie groups on RN ). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN , and let g be the Lie algebra of G. Then G is nilpotent of step ≤ σN , i.e. every commutator of vector fields in g containing more than σN terms vanishes identically. Moreover, if Zj is the j -th element of the Jacobian basis of g, Zj is δλ homogeneous of degree σj . Proof. Let Zj be the j -th element of the Jacobian basis of g. By Proposition 1.2.16-(5), for every ϕ ∈ C ∞ (RN , R), we have d d Zj (ϕ(δλ (x))) = (ϕ(δλ (x ◦ (tej )))) = (ϕ(δλ (x) ◦ δλ (tej ))) dt t=0 dt t=0 d = λσj (ϕ(δλ (x) ◦ (r ej ))) = λσj (Zj ϕ)(δλ (x)), dr r=0

38


i.e. Zj is δλ -homogeneous of degree σj . This proves the last part of the assertion. Since (Z1 , . . . , ZN ) is a linear basis for g, the above arguments show that every X ∈ g is a linear combination of δλ -homogeneous smooth vector fields of degrees at least σ1 ≥ 1. The first part of the assertion hence follows from Proposition 1.3.10.

1.3.2 The Composition Law of a Homogeneous Lie Group By using the elementary properties of the homogeneous functions showed in the previous section, we shall obtain a structure theorem for the composition law in a homogeneous Lie group (RN , ◦, δλ ). We first prove two lemmas. Lemma 1.3.13. Let δλ be as in (1.49). Let P : RN × RN → R be a smooth nonvanishing function such that P (δλ (x), δλ (y)) = λσj P (x, y)

∀ x, y ∈ RN , ∀ λ > 0,

for some j such that 1 ≤ j ≤ N . Assume also that P (x, 0) = xj ,

P (0, y) = yj .

(1.54)

Then P (x, y) = x1 + y1 if j = 1 and, if j ≥ 2, (x1 , . . . , xj −1 , y1 , . . . , yj −1 ), P (x, y) = xj + yj + P is a polynomial, the sum of mixed monomials in x1 , . . . , xj −1 , y1 , . . . , yj −1 . where P (δλ (x), δλ (y)) = λσj P (x, y). Finally, P (x, y) only depends on the xk ’s Moreover, P and yk ’s with σk < σj . Proof. By Proposition 1.3.4, P is a polynomial function of the following type: P (x, y) = cα,β x α y β , cα,β ∈ R. |α|σ +|β|σ =σj

On the other hand, by (1.54), xj = P (x, 0) =

cα,0 x α

|α|σ =σj

and yj = P (0, y) =

c0,β y α .

|β|σ =σj

Then P (x, y) = xj + yj +

cα,β x α y β .

(1.55)

|α|σ +|β|σ =σj , α,β=0

We can complete the proof by noticing that the condition |α|σ + |β|σ = σj , α, β = 0 is empty when j = 1, whereas it implies α = (α1 , . . . , αj −1 , 0, . . . , 0), β = (β1 , . . . , βj −1 , 0, . . . , 0) when j ≥ 2. As for the last assertion of the lemma, being α, β = 0 in the sum in the right-hand side of (1.55), the sum itself may depend only on the α’s and β’s with |α|σ , |β|σ
0,

∂Q (x, 0) ∂ yj

is δλ -homogeneous of degree m − σj (unless it vanishes identically). Proof. By Proposition 1.3.13, Q is a polynomial of the following type Q(x, y) = cα,β x α y β . |α|σ +|β|σ =m

Then, denoting by ej the j -th element of the canonical basis of RN , we have ∂Q (x, y) = ∂ yj

cα,β βj x α y β−ej ,

|α|σ +|β|σ =m

so that, since |ej |σ = σj , ∂Q (x, 0) = ∂ yj

cα,β x α .

|α|σ =m−σj ,β=ej

This completes the proof.

Now, we are in the position to prove the previously mentioned structure theorem for the composition law of a homogeneous Lie group on RN . Theorem 1.3.15 (Composition of a homogeneous Lie group on RN ). Let (RN , ◦, δλ ) be a homogeneous Lie group on RN . Then ◦ has polynomial component functions. Furthermore, we have (x ◦ y)1 = x1 + y1 ,

(x ◦ y)j = xj + yj + Qj (x, y),

2 ≤ j ≤ N,

and the following facts hold: 1. Qj only depends on x1 , . . . , xj −1 and y1 , . . . , yj −1 ; 2. Qj is a sum of mixed monomials in x, y; 3. Qj (δλ x, δλ y) = λσj Qj (x, y). More precisely, Qj (x, y) only depends on the xk ’s and yk ’s with σk < σj . Proof. Let j ∈ {1, . . . , N }, and define Pj : RN × RN → R, Since δλ is an automorphism of G, we have

Pj (x, y) = (x ◦ y)j .

40


Pj (δλ (x), δλ (y)) = (δλ (x ◦ y))j = λσj (x ◦ y)j = λσj Pj (x, y). Moreover, since x ◦ 0 = x, 0 ◦ y = y, we have Pj (x, 0) = xj ,

Pj (0, y) = yj .

Then the proof follows from Lemma 1.3.13.

For example, the Lie group G = (R3 , ◦) considered in Example 1.2.18 is not homogeneous with respect to any dilation on R3 , since its composition x ◦ y = (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) does not fulfill the requirements of Theorem 1.3.15. Corollary 1.3.16 (Inversion of a homogeneous Lie group on RN ). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN . Let j ∈ {1, . . . , N }. For every y ∈ G, we have (y −1 )j = −yj + qj (y), where qj (y) is a polynomial function in y, δλ -homogeneous of degree σj , only depending on the yk ’s with σk < σj . Proof. We let p : RN → R,

p(y) = y −1 .

For every y ∈ G, we have 0 = δλ (y −1 ◦ y) = δλ (y −1 ) ◦ δλ (y), whence δλ (y −1 ) = (δλ (y))−1 , i.e. δλ (p(y)) = p(δλ (y))

∀ y ∈ RN .

(1.56)

If j ∈ {1, . . . , N } is fixed and pj is the j -th component function of p, (1.56) means that pj is δλ -homogeneous of degree σj . As a consequence, since the inversion is a smooth map (by definition of Lie group), we can apply Proposition 1.3.4 and infer that pj is a polynomial function, δλ -homogeneous of degree σj . Now, we exploit the explicit form of the composition in Theorem 1.3.15. If x ◦ y = 0, then we have ()

xj = −yj

whenever σj = 1.

Hence, if σj = 2, we have xj = −yj + Qj (x, y), where Qj only depends on the xk ’s and yk ’s with σk = 1. As a consequence of (), we infer xj = −yj + qj (y)

whenever σj = 2,

where qj only depends on the yk ’s with σk = 1. An inductive argument now proves that xj = −yj + qj (y), where qj only depends on the yk ’s with σk < σj , and the proof is complete.


41

Example 1.3.17. Let us consider on R4 the composition law ⎞ ⎛ x 1 + y1 ⎟ ⎜ x2 + y2 ⎟. x◦y =⎜ ⎠ ⎝ x3 + y3 + y1 x2 x4 + y4 + y1 x22 + 2 x2 y3 Then ◦ equips R4 with a homogeneous Lie group structure with dilations δλ (x1 , x2 , x3 , x4 ) = (λx1 , λx2 , λ2 x3 , λ3 x4 ). Notice that the inversion on this group is given by ⎞ ⎛ −y1 ⎟ ⎜ −y2 ⎟. y −1 = ⎜ ⎠ ⎝ −y3 + y1 y2 −y4 − y1 y22 + 2 y2 y3 Corollary 1.3.18. Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN . Let j ∈ {1, . . . , N }. For every x, y ∈ G, we have (j ) (y −1 ◦ x)j = xj − yj + Pk (x, y)(xk − yk ), k:σk 1, y −1 ◦ x = (x1 − y1 , x2 − y2 , x3 − y3 − y1 x2 + y1 y2 ). The following result describes in a very explicit way the Jacobian matrix at 0 of the left-translation τx on a homogeneous Lie group on RN . Corollary 1.3.19 (The Jacobian basis of a homogeneous Lie group). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN . Then we have ⎛ ⎞ 1 0 ··· 0 .. ⎟ ⎜ (1) .. ⎜ a2 . 1 .⎟ ⎜ ⎟, (1.62) Jτx (0) = ⎜ . ⎟ . . .. .. ⎝ .. 0⎠ (1) (N −1) · · · aN 1 aN


43

(j )

where ai is a polynomial function δλ -homogeneous of degree σi − σj . As a consequence, if we let Zj = ∂xj +

N

(j )

ai

∂xi

for 1 ≤ j ≤ N − 1 and ZN = ∂xN ,

i=j +1

then Zj is a left-invariant vector field δλ -homogeneous of degree σj . Moreover, Jτx (0) = (Z1 I (x) · · · ZN I (x)). In other words, the Jacobian basis Z1 , . . . , ZN for the Lie algebra g of G is formed by δλ -homogeneous vector fields of degree σ1 , . . . , σN , respectively. Proof. By Theorem 1.3.15, the Jacobian matrix Jτx (0) takes the form (1.62) with (j )

ai (x) =

∂ Qi (x, 0). ∂ yj

(j )

Then, by Lemma 1.3.14, ai (x) is a polynomial function, δλ -homogeneous of degree σi − σj . This proves the first part of the corollary. The second one follows from Proposition 1.3.12.

Example 1.3.20. In Example 1.2.6, we showed that the Jacobian matrix of the left translation on H1 is

1 0 0 Jτx (0) = 0 1 0 . 2x2 −2x1 1 We recognize that the three columns of this matrix give raise to the Jacobian basis Z1 = ∂x1 + 2 x2 ∂x3 , Z2 = ∂x2 − 2 x1 ∂x3 and Z3 = ∂x3 and these vector fields are homogeneous of degree, respectively, 1, 1, 2 with respect to δλ (x1 , x2 , x3 ) =

(λx1 , λx2 , λ2 x3 ). The structure Theorem 1.3.15 of the composition law of (RN , ◦, δλ ) implies that the Lebesgue measure on RN is invariant under left and right translations on G. Indeed, by Theorem 1.3.15, the Jacobian matrices of the functions x → α ◦ x and x → x ◦ α have the following lower triangular form ⎛ ⎞ 1 0 ··· 0 ⎜ .⎟ ⎜ 1 . . . .. ⎟ ⎜ ⎟. ⎜ . . ⎟ . . . . ⎝ . . . 0⎠ ··· 1 Then, we have proved the following proposition. Proposition 1.3.21 (Haar measure on a homogeneous Lie group). Let G be a homogeneous Lie group on RN . Then the Lebesgue measure on RN is invariant with respect to the left and the right translations on G.

44


The above proposition is also restated as: the Lebesgue measure on RN is the Haar measure for G. If we denote by |E| the Lebesgue measure of a measurable set E ⊆ RN , we then have |α ◦ E| = |E| = |E ◦ α| ∀ α ∈ G. We also have that the Lebesgue measure is homogeneous with respect to the dilations {δλ }λ>0 . More precisely, as a trivial computation shows, |δλ (E)| = λQ |E|, where Q=

N

σj .

(1.63)

j =1

The positive number Q is called the homogeneous dimension of the group G = (RN , ◦, δλ ). For example, in the case of the Heisenberg–Weyl group H1 , where τα is given by τα (x) = (α1 + x1 , α2 + x2 , α3 + x3 + 2 (α2 x1 − α1 x2 )), and δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ), we have

1 0 0 λ 0 0 Jδλ (x) = 0 λ 0 , Jτα (x) = 0 1 0 , 2α2 −2α1 1 0 0 λ2 so that, for every α, x ∈ H1 and every λ > 0, we have det Jτα (x) = 1,

det Jδλ (x) = λ4 = λQ ,

since the homogeneous dimension of H1 is Q = 1 + 1 + 2 = 4. 1.3.3 The Lie Algebra of a Homogeneous Lie Group on RN The following remark holds. Remark 1.3.22. Let G be a homogeneous Lie group on RN with Lie algebra g. From Corollary 1.3.19 we easily obtain the splitting of g as a direct sum of linear spaces spanned by vector fields of constant degree of δλ -homogeneity. More precisely, let us recall that the exponents σj ’s in the dilation δλ of G (see (H.1) in Definition 1.3.1) satisfy σ1 ≤ · · · ≤ σN and can then be grouped together to produce real and natural numbers, respectively, say n1 , . . . , nr

and

N1 , . . . , Nr ,

such that n1 < n2 < · · · < nr ,

N1 + N2 + · · · + Nr = N,


defined by ⎧ n1 = σj ⎪ ⎪ ⎪ ⎨ n2 = σj .. ⎪ . ⎪ ⎪ ⎩ nr = σj

45

for 1 ≤ j ≤ N1 , for N1 < j ≤ N1 + N2 , for N1 + · · · + Nr−1 < j ≤ N1 + · · · + Nr−1 + Nr .

Let Z1 , . . . , ZN be the Jacobian basis of g. Define g1 = span{Zj | 1 ≤ j ≤ N1 }

and, for i = 2, . . . , r,

gi = span{Zj | N1 + · · · + Ni−1 < j ≤ N1 + · · · + Ni−1 + Ni }. By Corollary 1.3.19, the generators Zj ’s of gi are δλ -homogeneous vector fields of degree ni , 1 ≤ i ≤ r. Moreover, we obviously have g = g1 ⊕ · · · ⊕ gr . We also explicitly notice that, by Proposition 1.3.9, a vector field X ∈ g is δλ homogeneous of degree n iff, for a suitable i ∈ {1, . . . , r}, n = ni and X ∈ gi . In the next section, we shall deal with homogeneous groups in which ni = i for 1 ≤ i ≤ r, and the layer (or slice) gi , i ∈ {1, . . . , r}, is precisely generated by the commutators of length i of the vector fields in g1 . Example 1.3.23. The usual additive group (R3 , +) is a homogeneous Lie group if equipped with the dilation δλ (x1 , x2 , x3 ) = (λ2 x1 , λπ x2 , λ4 x3 ). The decomposition of the Lie algebra as in Remark 1.3.22 is span{∂x1 } ⊕ span{∂x2 } ⊕ span{∂x3 }. Moreover, R4 is a homogeneous Lie group if equipped with the group law ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟ x◦y =⎜ ⎝ x3 + y3 + 2 y1 x2 − 2 y2 x1 ⎠ x4 + y4 and the dilation δλ (x1 , x2 , x3 , x4 ) = (λx1 , λx2 , λ2 x3 , λ2 x4 ). The decomposition of the Lie algebra as in Remark 1.3.22 is g1 ⊕ g2 = span{X1 , X2 } ⊕ span{∂x3 , ∂x4 }, where X1 = ∂x1 + 2 x2 ∂x3 , X2 = ∂x2 − 2 x1 ∂x3 . Note that (see the notation in (1.17))

46


[g1 , g1 ] g2 . Observe that the above (R4 , ◦) is isomorphic to the homogeneous Lie group (R4 , ∗) with the composition law ⎞ ⎛ ξ1 + η1 ⎟ ⎜ ξ 2 + η2 ⎟ ξ ∗η =⎜ ⎠ ⎝ ξ 3 + η3 ξ 4 + η 4 + 2 η1 ξ 2 − 2 η 2 ξ 1 and the new group of dilations δλ (ξ1 , ξ2 , ξ3 , ξ4 ) = (λξ1 , λξ2 , λξ3 , λ2 x4 ). The decomposition of the Lie algebra as in Remark 1.3.22 is g1 ⊕ g2 = span{Z1 , Z2 , ∂ξ3 } ⊕ span{∂ξ4 }, where Z1 = ∂ξ1 + 2 ξ2 ∂ξ4 , Z2 = ∂ξ2 − 2 ξ1 ∂ξ4 . Note that this time we have [g1 , g1 ] = g2 . Definition 1.3.24 (Dilations on the Lie algebra of a homogeneous Lie group). Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN with Lie algebra g and dilation δλ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ). We define a group of dilations on g (which we still denote by δλ ) as follows: δλ is the (only) linear (auto)morphism of g mapping the j -th element Zj of the Jacobian basis for g into λσj Zj . In other words, if X ∈ g is written w.r.t. the Jacobian basis Z1 , . . . , ZN as X=

N

we then have δλ (X) =

cj Zj ,

j =1

N

cj λσj Zj .

j =1

We immediately recognize that, if π : g → RN is the map defined by π(X) = XI (0) (see also Remark 1.2.20), it holds π(δλ (X)) = δλ (π(X))

∀ X ∈ g.

(1.64)

Indeed, we have δλ (π(X)) = δλ π = δλ

N j =1

N

= δλ

cj Zj

N

cj π(Zj )

j =1

cj (Zj )I (0) = δλ (c1 , . . . , cN )

j =1

= (λσ1 c1 , . . . , λσN cN )


47

and, on the other hand, π(δλ (X)) = π δλ

N

=π

cj Zj

j =1

=

N

N

σj

cj λ Zj

j =1

cj λσj π(Zj ) = (λσ1 c1 , . . . , λσN cN ).

j =1

The following simple and very useful fact holds. Proposition 1.3.25. Let G be a homogeneous Lie group on RN with Lie algebra g. The dilation on g introduced in Definition 1.3.24 is a Lie algebra automorphism of g, i.e. (1.65) δλ ([X, Y ]) = [δλ (X), δλ (Y )] ∀ X, Y ∈ g. Proof. First we remark that, for every i, j ∈ {1, . . . , N }, δλ ([Zi , Zj ]) = λσi +σj [Zi , Zj ].

(1.66)

Indeed, since Zi and Zj are δλ -homogeneous of degrees σi and σj , respectively, then [Zi , Zj ] is a δλ -homogeneous vector field of degree σi + σj (see Proposition 1.3.10). This implies that, if we express [Zi , Zj ] w.r.t. the Jacobian basis [Zi , Zj ] =

N

ck Zk ,

k=1

then the sum runs over the k’s such that σk = σi + σj

(1.67)

(here we use, as a crucial tool, Corollary 1.3.11). Consequently, N

N δλ ([Zi , Zj ]) = δλ ck Zk = ck λσk Zk k=1

= λσi +σj

(by (1.67))

k=1

N

ck Zk = λσi +σj [Zi , Zj ].

k=1

This proves (1.66). N Let now X = N i=1 xi Zi and Y = j =1 yj Zj . Then we have $ δλ ([X, Y ]) = δλ = δλ

N i=1

N i,j =1

xi Zi ,

N j =1

%

yj Zj

xi yj [Zi , Zj ]

48

1 Stratified Groups and Sub-Laplacians N

=

xi yj δλ ([Zi , Zj ])

(see (1.66))

i,j =1 N

=

xi yj λσi +σj ([Zi , Zj ])

i,j =1

=

$N

σi

ci λ Zi ,

N

% σj

cj λ Zj

= [δλ (X), δλ (Y )].

j =1

i=1

This completes the proof.

1.3.4 The Exponential Map of a Homogeneous Lie Group Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN with Lie algebra g. The exponential map on g has some remarkable properties, due to the homogeneous structure of G. We prove such properties in what follows. Let Z1 , . . . , ZN be the Jacobian basis of g. By Corollary 1.3.19, Zj is δλ homogeneous of degree σj and takes the form Zj =

N

(j )

ak (x1 , . . . , xk−1 )∂xk ,

(1.68)

k=j (j )

(j )

where ak is a polynomial function δλ -homogeneous of degree σk −σj and aj ≡ 1. We now consider on g the dilation group introduced in Definition 1.3.24, i.e. with abuse of notation (soon justified) N

N δλ ξj Zj := λσj ξj Zj . (1.69) δλ : g −→ g, j =1

j =1

Remark 1.3.26 (Consistency of the dilations on g and G). The dilation (1.69) is consistent with the one in G. More precisely, if Z ∈ g then, for every λ > 0, it holds δλ (ZI (x)) = (δλ Z)I (δλ (x))

∀ x ∈ G.

(1.70)

We first check this identity in the case Z = Zj , j = 1, . . . , N. Since Zj is δλ -homogeneous of degree σj , by Corollary 1.3.6, we have δλ (Zj I (x)) = λσj (Zj I )(δλ (x)), so that (see (1.69)) Then, given Z =

δλ (Zj I (x)) = (δλ Zj )I (δλ (x)).

N

j =1 ξj

δλ (ZI (x)) =

Zj ∈ g, we have (since δλ is linear on g)

N

ξj δλ (Zj I (x)) =

j =1

=

N j =1

N

ξj (δλ Zj )I (δλ (x))

j =1

ξj (δλ Zj ) I (δλ (x)) = (δλ Z)I (δλ (x)).


49

From the previous remark, we easily obtain the following lemma. Lemma 1.3.27. Let G = (RN , ◦, δλ ) be a homogeneous Lie group on RN with Lie algebra g. Denote also by δλ the dilation (1.69) on g. Let γ : [0, T ] → RN be an integral curve of Z with Z ∈ g. Then Γ := δλ (γ ) is an integral curve of δλ (Z). Proof. Identity (1.70) gives Γ˙ = δλ (γ˙ ) = δλ (ZI (γ )) = (δλ Z)I (δλ (γ )) = (δλ Z)I (Γ ). This ends the proof.

We are now in the position to prove the following important theorem. Theorem 1.3.28 (Exponential map of a homogeneous Lie group). Let G = (RN , ◦, δλ ) be a homogeneous Lie group with Lie algebra g. Then Exp : g → G and Log : G → g are globally defined diffeomorphisms with polynomial component functions (provided g is equipped with its vector space structure and any fixed system of linear coordinates). Moreover, denote also by δλ the dilation on g defined in (1.69). Then, for every Z ∈ g and x ∈ G, it holds Exp δλ (Z) = δλ (Exp (Z)) and Log (δλ (x)) = δλ (Log (x)). (1.71) Proof. Let Z ∈ g, Z =

N

Z=

j =1 ξj

N k=1

Zj . From (1.68) we obtain

k

(j ) ξj ak (x1 , . . . , xk−1 )

∂xk .

(1.72)

j =1

Then the system of ODE’s defining Exp (Z) is “pyramid”-shaped, and the first part of the theorem follows from Remark 1.1.3. In order to prove the first identity in (1.71), we consider the solution γ to the Cauchy problem γ˙ = ZI (γ ),

γ (0) = 0.

By the very definition of Exp (Z), we have γ (1) = Exp (Z). Let us put Γ = δλ (γ ). By Lemma 1.3.27, Γ is an integral curve of δλ (Z). Moreover, Γ (0) = δλ (γ (0)) = δλ (0) = 0. Then Γ (1) = Exp (δλ (Z)), so that Exp (δλ (Z)) = Γ (1) = δλ (γ (1)) = δλ (Exp (Z)). This proves the first identity in (1.71). The second one is trivially equivalent to the first one.

50


The first part of Theorem 1.3.28 together with (1.40) and Proposition 1.2.29 (page 27) give the following corollary. Corollary 1.3.29. For every x, y ∈ G, we have and x −1 = Exp (−Log (x)).

x ◦ y = exp(Log (y))(x)

Remark 1.3.30 (Exp and Log preserve the mass). If Z is the vector field (1.72), then ZI (x) =

(1) ξ1 , ξ2 + ξ1 a2 (x1 ), . . . , ξN

+

N −1

(j ) aN (x1 , . . . , xN −1 )

.

j =1

This implies (see (1.13), page 10) Exp (Z) = exp(Z)(0) = ξ1 , ξ2 + B2 (ξ1 ), . . . , ξN + BN (ξ1 , . . . , ξN −1 ) , (1.73) where the Bj ’s are suitable polynomial functions. Then the Jacobian matrix of the map RN (ξ1 , . . . , ξN ) → Exp (ξ1 Z1 + · · · + ξN ZN ) ∈ RN takes the following form

⎛

1

0

⎜ 1 ⎜ ⎜ . . ⎝ .. .. ···

⎞ ··· 0 . .. . .. ⎟ ⎟ ⎟. .. . 0⎠ 1

(1.74)

Thus, with respect to the Jacobian basis of g and the canonical basis of G ≡ RN , Exp preserves the Lebesgue measure. The same property holds for the map Log , since Log = (Exp )−1 . NEquivalently, if Z1 , . . . , ZN is the Jacobian basis for g and, as usual, ξ · Z = j =1 ξj Zj , then Exp and Log have the following remarkable forms ⎛ ⎜ Exp (ξ · Z) = ⎜ ⎝

⎞

ξ1 ξ2 + B2 (ξ1 ) .. .

⎟ ⎟ ⎠

(1.75a)

⎟ ⎟ · Z, ⎠

(1.75b)

ξN + BN (ξ1 , . . . , ξN −1 ) and

⎛ ⎜ Log (x) = ⎜ ⎝

x1 x2 + C2 (x1 ) .. .

⎞

xN + BN (x1 , . . . , xN −1 ) where the Bi ’s and Ci ’s are polynomial functions (δλ -homogeneous of degree σi ) completely determined by the composition law on G.


51

For example, the above results are readily verified for H1 , since in that case (as we proved in Example 1.2.28, page 27) Exp is represented by the identity matrix (if the algebra of H1 is equipped with the Jacobian basis). A little more elaborated example is given below. Example 1.3.31. Let us consider on R4 the following composition law (we denote the points of R4 by x = (x1 , x2 , x3 , x4 )): ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟. x◦y =⎜ ⎝ ⎠ x3 + y3 + y1 x2 2 x4 + y4 + y1 x2 + 2 x2 y3 It is readily verified that ◦ equips R4 with a homogeneous Lie group structure, provided that the dilation group is given by δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ). Let us construct the exponential map. To begin with, the first two vector fields X1 , X2 of the related Jacobian basis can be found as follows (see (1.33)): for every ϕ ∈ C ∞ (R4 ) and every x ∈ R4 , we have ∂ ∂ (X1 ϕ)(x) = ϕ(x ◦ y), (X2 ϕ)(x) = ϕ(x ◦ y), ∂y1 y=0 ∂y2 y=0 so that the chain rule straightforwardly gives X1 = ∂x1 + x2 ∂x3 + x22 ∂x4 ,

X2 = ∂x2 .

A direct computation shows that [X1 , X2 ] = −∂3 − 2 x2 ∂4 ,

[X1 , [X1 , X2 ]] = 0,

[X2 , [X1 , X2 ]] = −2 ∂4 ,

whereas all commutators of length > 3 vanish. We now remark that X1 ,

X2 ,

[X1 , X2 ]

and [X2 , [X1 , X2 ]]

satisfy the following properties: – they are left-invariant w.r.t. ◦ (as iterated commutators of left-invariant vector fields, see also (1.19)); – they are linearly independent vector fields10 ; – they form a basis of g (since dim(g) = 4, see Proposition 1.2.7). 10 For example, we can notice that their respective evaluations at 0

⎛1⎞

⎛0⎞

⎜0⎟ ⎝ ⎠, 0 0

⎜1⎟ ⎝ ⎠, 0 0

⎛ 0 ⎞ ⎜ 0 ⎟ ⎝ ⎠, −1 0

⎛ 0 ⎞ ⎜ 0 ⎟ ⎝ ⎠ 0 −2

are linearly independent vectors of R4 (and use Proposition 1.2.13).

52


We now set W1 := X1 , W2 := X2 , W3 := [X1 , X2 ], W4 := [X2 , [X1 , X2 ]], and, for ξ ∈ R4 , we also let ξ · W := ξ1 W1 + ξ2 W2 + ξ3 W3 + ξ4 W4 . By the definition in (1.8), we have exp(ξ · W )(x) = γ (1), where γ (s) solves γ˙ (s) = (ξ · W )I (γ (s)) = (ξ1 , ξ2 , ξ1 γ2 − ξ3 , ξ1 γ22 − 2 ξ3 γ2 − 2ξ4 ), γ (0) = x. Solving the above system of ODE’s, one gets ⎛

⎞ x1 + ξ1 ⎟ ⎜ x 2 + ξ2 ⎟. exp(ξ · W )(x) = ⎜ 1 ⎠ ⎝ x 3 − ξ3 + 2 ξ1 ξ2 + ξ1 x 2 1 2 2 x4 − 2ξ4 + 3 ξ1 ξ2 − ξ2 ξ3 + ξ1 x2 + ξ1 ξ2 x2 − 2ξ3 x2

As a consequence, by Definition 1.2.25, we obtain ⎛

⎞ ξ1 ⎜ ⎟ ξ2 ⎟, Exp (ξ · W ) = exp(ξ · W )(0) = ⎜ 1 ⎝ ⎠ −ξ3 + 2 ξ1 ξ2 1 2 −2ξ4 + 3 ξ1 ξ2 − ξ2 ξ3

so that the inverse map of Exp is given by ⎛

⎞ x1 ⎜ ⎟ x2 ⎟ · W. Log (x) = ⎜ 1 ⎝ ⎠ −x3 + 2 x1 x2 1 1 1 2 − 2 x4 − 12 x1 x2 + 2 x2 x3

One can now directly check11 the validity of Theorem 1.3.28, Corollary 1.3.29 and Remark 1.3.30. We explicitly remark the vectors Wi ’s do not form the Jacobian basis for g, which, instead, is given by Z1 = W1 ,

Z2 = W2 ,

Z3 = −W3 ,

1 Z4 = − W4 . 2

Hence, if we write ξ · Z := ξ1 Z1 + ξ2 Z2 + ξ3 Z3 + ξ4 Z4 , we see that ! 1 (ξ, ξ2 , ξ3 , ξ4 ) · Z = ξ, ξ2 , −ξ3 , − ξ4 · Z, 2 11 For example, we see that the inverse of x ∈ G is given by

⎞ −x1 & ' ⎜ ⎟ −x2 ⎟. x −1 = Exp −Log (x) = ⎜ ⎠ ⎝ −x3 + x1 x2 −x4 + 2 x2 x3 − x1 x22 ⎛


53

whence, with respect to the Jacobian coordinates, the exponential and the logarithmic maps are respectively given by ⎞ ⎛ ξ1 ⎟ ⎜ ξ2 ⎟, Exp (ξ · Z) = ⎜ ⎠ ⎝ ξ3 + 12 ξ1 ξ2 ξ4 + 13 ξ1 ξ22 + ξ2 ξ3 ⎛ ⎞ x1 ⎜ ⎟ x2 ⎟ · Z. Log (x) = ⎜ 1 ⎝ ⎠ x3 − 2 x1 x2 1 2 x4 + 6 x1 x2 − x2 x3 Compare this to (1.74).

Theorem 1.3.28 has many important consequences. We collect some of them in the following remark. Remark 1.3.32. From Theorem 1.3.28 we infer, in particular, that Exp : g → G and Log : G → g are globally defined

C∞

maps. Hence, by Remark 1.2.32, the operation on g

X Y := Log (Exp (X) ◦ Exp (Y )),

X, Y ∈ g,

(1.76)

defines a Lie group structure isomorphic to (G, ◦). We consider on g the dilation (still denoted by δλ ) introduced in Definition 1.3.24. We claim that δλ is a Lie group automorphism of (g, ), i.e.

δλ (X Y ) = (δλ (X)) (δλ (Y ))

∀ X, Y ∈ g.

(1.77)

Roughly speaking, (g, , δλ ) is a homogeneous Lie group too. To prove the claim, we notice that δλ (X Y ) = δλ {Log (Exp (X) ◦ Exp (Y ))} (see (1.71)) = Log {δλ (Exp (X) ◦ Exp (Y ))} = Log {(δλ (Exp (X))) ◦ (δλ (Exp (Y )))} (see (1.71)) = Log {(Exp (δλ (X))) ◦ (Exp (δλ (Y )))} = (δλ (X)) (δλ (Y )). This proves our claim.12 12 We explicitly remark that, if we already knew that is defined by a “universal” composition

of iterated Lie brackets, we could derive (1.77) from (1.65) in Proposition 1.3.25. Indeed, ! 1 1 [X, [X, Y ]] + · · · δλ (X Y ) = δλ X + Y + [X, Y ] + 2 12 1 1 [δλ (X), [δλ (X), δλ (Y )]] + · · · = δλ (X) + δλ (Y ) + [δλ (X), δλ (Y )] + 2 12 = (δλ (X)) (δλ (Y )), the second identity following by a repeated application of (1.65).

54


We now identify g with RN taking coordinates with respect to the Jacobian basis. In other words, we consider the map (see Remark 1.2.20) π : g → RN ,

X → π(X) := XI (0).

Again, we transfer the Lie group structure of (g, ) into a Lie group (RN , ∗) in the natural way, by setting ξ ∗ η := π(π −1 (ξ ) π −1 (η)),

ξ, η ∈ RN .

(1.78)

As a consequence, (RN , ∗) is isomorphic to (g, ) and hence to (G, ◦). We finally consider on RN the same dilation δλ defined on G (this makes sense, since the underlying manifold for G is RN too). We claim that (RN , ∗, δλ ) is a homogeneous Lie group. In other words, we have to show that δλ is a Lie group automorphism of (RN , ∗). This follows from the argument below, δλ (ξ ∗ η) (see (1.64)) (see (1.77)) (see (1.64))

= δλ {π(π −1 (ξ ) π −1 (η))} = π{δλ (π −1 (ξ ) π −1 (η))} = π{(δλ (π −1 (ξ ))) (δλ (π −1 (η)))} = π{(π −1 (δλ (ξ ))) (π −1 (δλ (η)))} = (δλ (ξ )) ∗ (δλ (η)).

We can summarize the above remarked facts as follows. Given a homogeneous Lie group G = (RN , ◦, δλ ), we can consider a somewhat “more canonical” homogeneous Lie group on RN C-H(G) := (RN , ∗, δλ ) (which we may call “of Campbell–Hausdorff type”) obtained by the natural identification of the Lie algebra of G (equipped with the Campbell–Hausdorff composition law in (1.76)) to RN (via coordinates w.r.t. the Jacobian basis). (See also the Fig. 1.6.) Example 1.3.33. Let us consider the homogeneous Lie group on R3 with the dilation δλ (x) = δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ) and the composition law defined by ⎛

⎞ x1 + y1 ⎠. x2 + y2 x◦y =⎝ x3 + y3 + x1 y2

The Jacobian basis for g (the Lie algebra of G) is


55

Fig. 1.6. Figure related to Remark 1.3.32

Z1 = ∂1 ,

Z2 = ∂2 + x1 ∂3 ,

Z3 = [Z1 , Z2 ] = ∂3 . For ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 , we fix the notation ξ · Z := 3i=1 ξi Zi ∈ g. It is easy to show that the exponential and logarithmic maps are given by ⎞ ⎛ ⎞ ⎛ ξ1 x1 ⎠ , Log (x) = ⎝ ⎠ · Z. ξ2 x2 Exp (ξ · Z) = ⎝ x1 x2 ξ1 ξ2 x3 − 2 ξ3 + 2 Hence, the map ∗ considered in (1.78) is given by ⎛ ⎞ ξ 1 + η1 ⎠. ξ 2 + η2 ξ ∗η =⎝ 1 ξ3 + η3 + 2 (ξ1 η2 − ξ2 η1 ) The Jacobian basis related to C-H(G) = (R3 , ∗, δλ ) is (1 = ∂1 − ξ2 ∂3 , Z (2 = ∂2 + ξ1 ∂3 , Z (3 = [Z (1 , Z (2 ] = ∂3 . Z 2 2 Now, it is interesting to see what happens if we iterate this “C-G” process. It is easy to see that, if we consider once again the group obtained from C-H(G) in the same way (i.e. C-H(C-H(G))), we obtain nothing else than C-H(G) itself (a rigorous formulation of this fact will be given in Proposition 2.2.24). We remark that G and C-H(G) are isomorphic and the canonical sub-Laplacian of G ΔG = {(∂/∂x1 )}2 + {(∂/∂x2 ) + x1 (∂/∂x3 )}2 is “equivalent” to the canonical sub-Laplacian of C-H(G) )2 )2 ξ2 ξ1 ΔC-H(G) = (∂/∂ξ1 ) − (∂/∂x3 ) + (∂/∂ξ2 ) + (∂/∂x3 ) . 2 2

56


1.4 Homogeneous Carnot Groups We now enter into the core of the chapter by introducing the central definition of this book. Our definition here of Carnot group will be properly compared to the classical one in Section 2.2, page 121. Definition 1.4.1 (Homogeneous Carnot group). We say that a Lie group on RN , G = (RN , ◦), is a (homogeneous) Carnot group or a homogeneous stratified group, if the following properties hold: (C.1) RN can be split as RN = RN1 × · · · × RNr , and the dilation δλ : RN → RN δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , λ2 x (2) , . . . , λr x (r) ),

x (i) ∈ RNi ,

is an automorphism of the group G for every λ > 0. Then (RN , ◦, δλ ) is a homogeneous Lie group on RN , according to Definition 1.3.1. Moreover, the following condition holds: (C.2) If N1 is as above, let Z1 , . . . , ZN1 be the left invariant vector fields on G such that Zj (0) = ∂/∂xj |0 for j = 1, . . . , N1 . Then13 rank(Lie{Z1 , . . . , ZN1 }(x)) = N

for every x ∈ RN .

If (C.1) and (C.2) are satisfied, we shall say that the triple G = (RN , ◦, δλ ) is a homogeneous Carnot group. We also say that G has step r and N1 generators. The vector fields Z1 , . . . , ZN1 will be called the (Jacobian) generators of G, whereas any basis for span{Z1 , . . . , ZN1 } is called a system of generators of G. (Note. As already remarked for Lie groups on RN and homogeneous ones, the notion of homogeneous Carnot group is not coordinate-free. This fact will not distract us from recognizing its importance.) In the sequel, we use the following notation to denote the points of G x = (x1 , . . . , xN ) = (x (1) , . . . , x (r) ) with

(i) ) ∈ R Ni , x (i) = (x1(i) , . . . , xN i

i = 1, . . . , r.

(1.79a) (1.79b)

Furthermore, we shall denote by g the Lie algebra of G. Remark 1.4.2 (Equivalent definition of homogeneous Carnot group). An equivalent definition of homogeneous Carnot group can be given: Suppose G = (RN , ◦) is a Lie group on RN , and there exist positive real numbers τ1 ≤ · · · ≤ τN such that dλ (x) = (λτ1 x1 , . . . , λτN xN ) is a Lie group morphism of G for every λ > 0. Let g be the Lie algebra of G, and let g1 be the linear subspace 13 See the notation in Definition 1.1.5.

1.4 Homogeneous Carnot Groups

57

of g of the left-invariant vector fields which are dλ -homogeneous of degree τ1 . If g1 Lie-generates14 the whole g, then G is a homogeneous Carnot group according to Definition 1.4.1. Precisely, G has step r := τN /τ1 , it has m := dim(g1 ) generators, and it is a homogeneous Lie group with respect to the dilation δλ = dλ1/τ1 . Also, set σj := τj /τ1 , then {σ1 , σ2 , . . . , σN } are consecutive integers starting from 1 up to r. A sketch of the proof is in order. As we observed in Remark 1.3.2, δλ is a morphism of (G, ◦), i.e. G = (RN , ◦, δλ ) is a homogeneous Lie group on RN . Obviously, X ∈ g1 if and only if X is δλ homogeneous of degree 1. Let ν be the maximum of the integers k’s such that σk = 1. Let us denote by {Z1 , . . . , ZN } the Jacobian basis related to G and observe that (by Proposition 1.3.12), for every j ≤ N , Zj is δλ -homogeneous of degree σj . We claim that () ν = dim(g1 ) =: m, and {Z1 , . . . , Zm } is a basis for g1 . Indeed, let X ∈ g1 . Then X = ξ1 Z1 + · · · + ξN ZN for suitable scalars ξj ’s. Since X is δλ -homogeneous of degree 1, by Corollary 1.3.11 and the definition of ν, it holds ξj = 0 for every j > ν. Hence, g1 is spanned by {Z1 , . . . , Zν } whence (this system of vectors being linearly independent) the claimed () holds. By the assumption Lie(g1 ) = g and (), it follows ()

Lie(Z1 , . . . , Zm ) = g.

For every j ∈ N, j ≥ 2, let us set (see the notation in (1.17)) gj := [g1 , gj −1 ]. By Proposition 1.3.12, gj = {0} for every j > r := σN . Also, by Proposition 1.3.10, any X ∈ gj is δλ -homogeneous of degree j . Let now j ∈ {m + 1, . . . , N } be fixed. Then, by (), Zj is a linear combination of nested commutators of Z1 , . . . , Zm . But any such commutator is δλ -homogeneous of an integer degree in 1, . . . , r. This proves that σj (the δλ -homogeneous degree of Zj ) is integer and (again from Corollary 1.3.11) σj ∈ {1, . . . , r}. As a consequence, we have the splitting of RN , as requested in (C.1) of Definition 1.4.1, with N1 = m. Finally, let us prove that (C.2) holds too. This is obvious thanks to (), since (see the notation in Definition 1.1.5) rank(g(x)) ≥ rank Z1 I (x), . . . , ZN I (x) = rank Z1 I (0), . . . , ZN I (0) = N for every x ∈ G (see Proposition 1.2.13).

Example 1.4.3. The Heisenberg–Weyl group H1 is a Carnot group of step two and two generators. Indeed, it is a homogeneous Lie group (with dilations δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 )). Moreover (since the first two vector fields of the Jacobian basis are Z1 = ∂x1 + 2x2 ∂x3 and Z2 = ∂x2 − 2x1 ∂x3 ), we have rank(Lie{Z1 , Z2 }(x)) = 3 for every x ∈ R3 , 14 This means that Lie(g ) = g, see the notation in Proposition 1.1.7. 1

58


as we proved in Example 1.1.6. Thus, the above properties (C.1) and (C.2) are fulfilled. We now give an example of a homogeneous Lie group which is not a Carnot group. Let us consider the following composition law on R2 (x1 , x2 ) ◦ (y1 , y2 ) = (x1 + y1 , x2 + y2 + x1 y1 ). It can be readily verified that G = (R2 , ◦) is a Lie group (here (x1 , x2 )−1 = (−x1 , −x2 + x12 )). Moreover, G is a homogeneous group, if equipped with the dilation δλ (x1 , x2 ) := (λx1 , λ2 x2 ). Hence (C.1) is satisfied. However, (C.2) is not. Indeed, if Z1 = ∂x1 + x1 ∂x2 is the first vector field of the Jacobian basis, we have rank(Lie{Z1 }(x)) = 1 = 2 for every x ∈ R2 . Hence G is not a homogeneous Carnot group. Finally, let us remark that the triple (R2 , +, δλ ) is a homogeneous Carnot group if δλ (x1 , x2 ) = (λx1 , λx2 ), whereas if δλ (x1 , x2 ) = (λx1 , λ2 x2 ), (R2 , +, δλ ) is a homogeneous Lie group but not a Carnot one.

From properties (C.1) and (C.2) of Definition 1.4.1 and the results on the homogeneous Lie groups showed in Section 1.3 we immediately get the assertions contained in the following remarks. Remark 1.4.4. Let (RN , ◦, δλ ) be a homogeneous Carnot group. Then ◦ has polynomial component functions. Moreover, following the notation in (1.79a) and denoting x ◦ y by ((x ◦ y)(1) , . . . , (x ◦ y)(r) ), we have (x ◦ y)(1) = x (1) + y (1) ,

(x ◦ y)(i) = x (i) + y (i) + Q(i) (x, y),

2 ≤ i ≤ r,

where 1. Q(i) only depends on x (1) , . . . , x (i−1) and y (1) , . . . , y (i−1) ; 2. the component functions of Q(i) are sums of mixed monomials in x, y; 3. Q(i) (δλ x, δλ y) = λi Q(i) (x, y). Remark 1.4.5. Let (RN , ◦, δλ ) be a homogeneous Carnot group. Then we have ⎛ ⎞ IN1 0 ··· 0 .. ⎟ ⎜ (1) .. ⎜ J2 (x) IN2 . . ⎟ ⎜ ⎟, (1.80) Jτx (0) = ⎜ . ⎟ . . .. .. ⎝ .. 0 ⎠ (1) (r−1) (x) INr Jr (x) · · · Jr (i)

where In is the n × n identity matrix, whereas Jj (x) is a Nj × Ni matrix whose entries are δλ -homogeneous polynomials of degree j − i. In particular, if we let Jτx (0) = Z (1) (x) · · · Z (r) (x) , where Z (i) (x) is a N × Ni matrix, then the column vectors of Z (i) (x) define δλ homogeneous vector fields of degree i: those of the relevant Jacobian basis.


59

Remark 1.4.6. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group with Lie algebra g. Let Z1 , . . . , ZN be the Jacobian basis of g, i.e. Zj ∈ g and Zj (0) = ∂xj |0 ,

j = 1, . . . , N.

With a notation consistent with (1.79a) and (1.79b), we shall also denote the Jacobian basis by (1) (1) (r) (r) Z1 , . . . , ZN1 ; . . . ; Z1 , . . . , ZNr . Obviously, Zj(1) = Zj for 1 ≤ j ≤ N1 . By Corollary 1.3.19, Zj(i) is δλ -homogeneous of degree i and takes the form (i) Zj

=

(i) ∂/∂xj

Nh r (i,h) (h) + aj,k (x (1) , . . . , x (h−i) ) ∂/∂xk ,

(1.81)

h=i+1 k=1 (i,h)

where aj,k

is a δλ -homogeneous polynomial function of degree h − i. In par(1)

(1)

ticular, the Jacobian generators of G, i.e. the vector fields Z1 , . . . , ZN1 are δλ homogeneous of degree 1.

Remark 1.4.7. With the notation of the above remark, the Lie algebra g is generated by Z1 , . . . ZN1 , (1.82) g = Lie{Z1 , . . . ZN1 }. Indeed, the inclusion Lie{Z1 , . . . ZN1 } ⊆ g is obvious. Since dim(g) = N, in order to show the opposite inclusion, it is enough to prove that dim(Lie{Z1 , . . . ZN1 }) = N. By condition (C.2), there exists X1 , . . . , XN ∈ Lie{Z1 , . . . ZN1 } such that X1 I (0), . . . , XN I (0) are linearly independent vectors in RN . Then (by Proposition 1.2.13) X1 , . . . , XN are linearly independent in g. Hence N ≥ dim(Lie{Z1 , . . . ZN1 }) ≥ N, and this ends the proof.

Remark 1.4.8 (Stratification of the algebra of a homogeneous Carnot group). Let the notation of Remark 1.4.6 be employed. Let us denote by W (k) the vector space spanned by the commutators of length k of Z1 , . . . , ZN1 , W (k) := span ZJ | J ∈ {1, . . . , N1 }k . Obviously, W (k) ⊆ g, and every Z ∈ W (k) is δλ -homogeneous of degree k. Then, by Corollary 1.3.11 and Proposition 1.3.12, W (k) = {0} if k > r, while (k)

(k)

W (k) ⊆ span{Z1 , . . . , ZNk }

if 2 ≤ k ≤ r.

(1.83)

60


Then, if we agree to let (1)

(1)

W (1) = span{Z1 , . . . , ZN1 } = span{Z1 , . . . , ZN1 }, we have dim(W (k) ) ≤ Nk

for any k ∈ {1, . . . , r}.

(1.84)

On the other hand, by Proposition 1.1.7, (1)

(1)

span{W (1) , . . . , W (r) } = Lie{Z1 , . . . , ZN1 }. Thus, by Remark 1.4.7, g = span{W (1) , . . . , W (r) }, so that, since W (h) ∩ W (k) = {0} if h = k (see Corollary 1.3.11), we have g = W (1) ⊕ W (2) ⊕ · · · ⊕ W (r) . As a consequence, dim(g) = On the other hand, dim(g) = N =

r

r

dim(W (k) ).

k=1

k=1 Nk .

dim(W (k) ) = Nk

Then, by (1.84),

for any k ∈ {1, . . . , r},

and, by (1.83), (k)

(k)

W (k) = span{Z1 , . . . , ZNk }

if 1 ≤ k ≤ r.

We also have [W (1) , W (i−1) ] = W (i)

for 2 ≤ k ≤ r

(1.85a)

and [W (1) , W (r) ] = {0}. Indeed, let us put V1 :=

W (1)

(1.85b)

and

Vi := [V1 , Vi−1 ]

for i = 2, . . . , r.

By the definition of W (k) and Proposition 1.1.7, Vi ⊆ W (i) for i = 2, . . . , r. Then dim(Vi ) ≤ dim(W (i) ) = Ni . On the other hand, by Proposition 1.3.12, [V1 , Vr ] = {0}, and, by Proposition 1.1.7, (1)

(1)

g = Lie{Z1 , . . . , ZN1 } = span{V1 , V2 , . . . , Vr }.

Then N = ri=1 dim(Vi ) ≤ ri=1 Ni = N . This implies dim(Vi ) = Ni for every i ∈ {1, . . . , r}. As a consequence, Vi = W (i) for every i ∈ {1, . . . , r}, and (1.85a) and (1.85b) hold.


61

Summing up, we have proved the “stratification” of the Lie algebra g, i.e. the decomposition g = W (1) ⊕ W (2) ⊕ · · · ⊕ W (r) with [W (1) , W (i−1) ] = W (i) [W (1) , W (r) ] = {0}, where

(k)

for 2 ≤ k ≤ r,

(k)

W (k) = span{Z1 , . . . , ZNk }

if 1 ≤ k ≤ r.

Remark 1.4.9. Following all the notation and definitions in Remark 1.3.32, if G = (RN , ◦, δλ ) is a homogeneous Carnot group, then the Lie group C-H(G) := (RN , ∗, δλ ) obtained by the natural identification of the algebra of G to RN is a homogeneous Carnot group too. The proof of this fact is left to the reader as an exercise. Remark 1.4.10 (Stratified change of basis on a homogeneous Carnot group). Let (RN , ◦, δλ ) be a homogeneous Carnot group according to the Definition 1.4.1. As usual, we denote the points of G by x = (x (1) , . . . , x (r) )

with x (i) ∈ RNi

and the dilation group by δλ (x) = (λx (1) , . . . , λr x (r) ). Let C (1) , . . . , C (r) be r fixed non-singular matrices with C (i) of dimension Ni × Ni for every i = 1, . . . , r. We denote by C the N ×N matrix having C (1) , . . . , C (r) as diagonal blocks and 0’s elsewhere, i.e. ⎛ (1) ⎞ C ··· 0 . .. ⎠ .. C = ⎝ .. . . . (r) 0 ··· C Finally, we denote again by C the relevant linear change of basis on RN , i.e. the linear map C : RN → RN , C(x) = C · x. We define on RN a new composition law ∗ obtained by writing ◦ in the new coordinates defined by ξ = C(x). More precisely, we have ξ ∗ η := C (C −1 (ξ )) ◦ (C −1 (η)) ∀ ξ, η ∈ RN . (1.86) We claim that H = (RN , ∗, δλ ) is a homogeneous Carnot group isomorphic to G = (RN , ◦, δλ ). The proof of this (not obvious) assertion is left as an exercise (see also Section 16.3 of Chapter 16, page 637).

62


1.5 The Sub-Laplacians on a Homogeneous Carnot Group Definition 1.5.1 (Sub-Laplacian on a homogeneous Carnot group). If Z1 , . . . , ZN1 are the Jacobian generators of the homogeneous Carnot group G = (RN , ◦, δλ ), the second order differential operator ΔG =

N1

Zj2

(1.87)

j =1

is called the canonical sub-Laplacian on G. Any operator L=

N1

Yj2

(1.88)

j =1

where Y1 , . . . , YN1 is a basis of span{Z1 , . . . , ZN1 }, is simply called a sub-Laplacian on G. The vector valued operator ∇G = (Z1 , . . . , ZN1 )

(1.89)

will be called the canonical (or horizontal) G-gradient. Finally, if L is as in (1.88), the notation ∇L = (Y1 , . . . , YN1 ) will be used to denote the L-gradient (or horizontal L-gradient). Example 1.5.2. The canonical sub-Laplacian of the Heisenberg–Weyl group H1 is ΔH1 = {∂x1 + 2 x2 ∂x3 }2 + {∂x2 − 2 x1 ∂x3 }2 = (∂x1 )2 + (∂x2 )2 + 4(x12 + x22 ) (∂x3 )2 + 4 x2 ∂x1 ,x3 − 4 x1 ∂x2 ,x3 . A (non-canonical) sub-Laplacian on H1 is, for example, L = {(∂x1 + 2 x2 ∂x3 ) − (∂x2 − 2 x1 ∂x3 )}2 + {∂x2 − 2 x1 ∂x3 }2 = (∂x1 )2 + 2(∂x2 )2 + 4(x12 + (x1 + x2 )2 ) (∂x3 )2 − 2∂x1 ,x2 + 4(x1 + x2 ) ∂x1 ,x3 − 4(x1 + (x1 + x2 )) ∂x2 ,x3 . The following one is a (non-canonical!) sub-Laplacian on the classical additive group (R2 , +) L = {2 ∂x1 − 5 ∂x2 }2 + {−∂x1 + 3 ∂x2 }2 = 5(∂x1 )2 + 34(∂x2 )2 − 26 ∂x1 ,x2 . It is not difficult to verify that R4 equipped with the operation ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟ x◦y =⎜ 1 ⎝ ⎠ x3 + y3 + 2 (y2 x1 − y1 x2 ) 1 1 x4 + y4 + 2 (y3 x1 − y1 x3 ) + 12 (x1 − y1 )(y2 x1 − y1 x2 )

1.5 The Sub-Laplacians on a Homogeneous Carnot Group

63

and the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ) is a homogeneous Carnot group, say G, whose canonical sub-Laplacian is )2 1 1 1 ΔG = ∂1 − x2 ∂3 − x3 ∂4 − x1 x2 ∂4 2 2 12 )2 1 1 2 + ∂2 + x1 ∂3 + x1 ∂4 2 12 1 = ∂11 + ∂22 + (x12 + x22 )∂33 + x1 ∂23 − x2 ∂13 4 ! ! ! ! x2 x12 2 1 x1 x2 2 x1 x2 ∂44 + 1 ∂24 − x3 + ∂14 + + x3 + 4 6 6 6 6 ! !! x12 1 x1 x2 1 x1 + x2 x3 + ∂34 + x2 ∂4 . + 2 6 6 6 We notice that ΔG also contains a first order (underlined) partial differential opera tor.15 We would like to list some basic properties of the sub-Laplacians, straightforward (1) (1) consequences of the properties of the vector fields Z1 , . . . , ZN1 . In what follows 1 2 L= N j =1 Yj will denote any sub-Laplacian on G. (A0) L is hypoelliptic, i.e. every distributional solution to Lu = f is of class C ∞ whenever f is of class C ∞ (see Section 5.10, page 280, for further comments). This follows from the celebrated Hörmander hypoellipticity theorem [Hor67, Theorem 1.1], recalled in the Preface (Theorem 1, page VIII) and the fact that, 1 2 if L = N j =1 Yj , then the following rank-condition holds rank Lie {Y1 , . . . , YN1 }(x) = N

∀ x ∈ RN .

This is an obvious consequence of hypothesis (C.2) in Definition 1.4.1, page 56. (A1) L is invariant with respect to the left translations on G, i.e. for every fixed α ∈ G, L(u(α ◦ x)) = (Lu)(α ◦ x)

for every x ∈ G and every u ∈ C ∞ (RN ).

This holds since the Yj ’s are left-translation invariant on G. (A2) L is δλ -homogeneous of degree two, i.e. for every fixed λ > 0, L(u(δλ (x))) = λ2 (Lu)(δλ (x))

for every x ∈ G and every u ∈ C ∞ (RN ).

This holds since the Yj ’s are δλ -homogeneous of degree one, see Remark 1.4.6. 15 This cannot happen for the sub-Laplacians on groups of step two, provided the inverse

map on the group is −x, i.e. any sub-Laplacian on a step-two homogeneous Carnot group (whose inverse map is −x) contains only second order coordinate partial derivatives (see Section 3.2).

64


(A3) L can be written as

L = div(A(x)∇ T ),

where div denotes the divergence operator in N × N symmetric matrix

(1.90a)

RN ,

∇ = (∂1 , . . . , ∂N ), A is the

A(x) = σ (x) σ (x)T

(1.90b)

and σ (x) is the N × N1 matrix whose columns are Y1 I (x), . . . , YN1 I (x). In other words, A(x) is the Gram matrix of the system of vectors {Y1 I (x), . . . , YN1 I (x)}; hence, since these vectors are linearly independent for every x ∈ G, the rank of A(x) is N1 for every x ∈ G. Now, (1.90a) is a consequence of the following computation N

N1 N1 N Yk2 = (Yk I )i (x) ∂i (Yk I )j (x)∂j L= k=1

=

N

∂i

N1

*

(Yk I )i (x) (Yk I )j (x) ∂j ,

j =1

i=1

j =1

k=1 i=1 N

k=1

since (Yk I )i (x) does not depend on xi . Hence, this proves that L = div(A(x)∇ T ) with N

1 A(x) = (Yk I )i (x) (Yk I )j (x) = σ (x) σ (x)T . i,j =1,...,N

k=1

The matrix A takes the following block form A=

A1,1 A2,1

A1,2 A2,2

! (1.91)

,

where Ai,j stands for a mi ×mj matrix with polynomial entries, with m1 = N1 and m2 = N − N1 . Furthermore, A1,1 is constant and non-singular. Indeed, for 1 a suitable non-singular matrix B = (bj,k )N j,k=1 , we have Yj =

N1

bj,k Zk ,

j = 1, . . . , N1 .

(1.92)

k=1

On the other hand, see (1.81), Zk = ∂k +

N i=N1 +1

(k)

(k)

ai ∂i = ∂k +

N

(k)

∂i (ai ·),

(1.93)

i=N1 +1

where the ai ’s are suitable polynomial functions independent of xi . Replacing (1.93) in (1.92) and squaring, we obtain


L=

N1

Yj2 =

j =1

N1

ah,k ∂h,k +

65

∂h (ah,k ∂k ),

h,k≤N, h∨k>N1

h,k=1

where A1,1 = (ah,k )h,k≤N1 = B T B is a constant N1 × N1 matrix. Moreover, when h ∨ k := max{h, k} > N1 , then ah,k is a suitable polynomial function. We notice that if L = ΔG , then B = IN1 , so that A1,1 = IN1 . The above computations also give the expression of L with respect to the usual coordinate partial derivatives, L=

N1 k=1

Yk2 =

N

ai,j (x)∂i,j +

i,j =1

N

bj (x) ∂j ,

j =1

where ai,j (x) =

N1 (Yk I )i (x) (Yk I )j (x),

bj (x) =

k=1

N1

Yk ((Yk I )j (x)).

k=1

Analogous formulas hold for general sum of squares of vector fields (see Ex. 4 at the end of the chapter). (A4) If x ∈ G is fixed and A(x) is the matrix in (1.90a), then the quadratic form in ξ ∈ RN qL (x, ξ ) := A(x)ξ, ξ is called the characteristic form of L. We have qL (x, ξ ) =

N1 Yj I (x), ξ 2 , j =1

so that qL (x, ·) is obtained by formally replacing in L the coordinate derivatives ∂1 , . . . , ∂N by ξ1 , . . . , ξN . This can be easily seen from (1.90b), for we have qL (x, ξ ) = A(x) ξ, ξ = σ (x) σ (x)T ξ, ξ = σ (x)T ξ, σ (x)T ξ = |σ (x)T ξ |2 =

N1 Yj I (x), ξ 2 . j =1

Then qL (x, ξ ) ≥ 0 for every x ∈ G and every ξ ∈ RN , i.e. A(x) is positive semi-definite for every x ∈ G. Moreover, qL (x, ξ ) = 0 iff Yj I (x), ξ = 0 for every j ∈ {1, . . . , N1 }. Hence, for a fixed x ∈ G, the set N (x) of vectors ξ ’s which annihilate the quadratic form related to A(x) is a linear space given by ⊥ ⊥ N (x) := {ξ ∈ RN | qL (x, ξ ) = 0} = Y1 I (x) ∩ · · · ∩ YN1 I (x) . (1.94)

66


We recall that, by Proposition 1.2.13, since Y1 , . . . , YN1 are linearly independent in g, then Y1 I (x), . . . , YN1 I (x) are linearly independent in RN for every fixed x. Thus, if N1 < N , that is if r ≥ 2, for every x ∈ G there exists ξ ∈ RN \ {0} such that qL (x, ξ ) = 0. More precisely, the set of isotropic vectors for the quadratic form related to A(x) is a linear subspace of RN of dimension N − N1 , equal to the kernel16 of the matrix A(x). This means that if r ≥ 2, then L is not elliptic at any point of G. On the other hand, if N1 = N (that is the step r of G is 1) the block A1,1 in (1.91) has dimension N × N, while the other blocks disappear. Then L is a constant coefficient operator of the form L = N i,j =1 ai,j ∂i,j with A = (ai,j )i,j symmetric and strictly positive definite. Thus, we can summarize these results as follows: The sub-Laplacian L is a second order differential operator in divergence form with polynomial coefficients. The characteristic form of L is positive semi-definite. If the step of G is ≥ 2, then L is not elliptic at any point of G. If the step of G is 1, then L is an elliptic operator with constant coefficients. Example 1.5.3. The canonical sub-Laplacian of the Heisenberg–Weyl group H1 has been written in Example 1.5.2: in that case the characteristic form is q(x, ξ ) = (ξ1 )2 + (ξ2 )2 + 4(x12 + x22 )(ξ3 )2 + 4x2 ξ1 ξ3 − 4x1 ξ2 ξ3 = (ξ1 + 2x2 ξ3 )2 + (ξ2 − 2x1 ξ3 )2 . Hence q(x, ξ ) = 0 if and only if (1, 0, 2 x2 ), ξ = (0, 1, −2 x1 ), ξ = 0, 16 Indeed, given a N × N real symmetric matrix A, we denote by Isotr(A) the set of the

isotropic vectors w.r.t. the quadratic form related to A, i.e. Isotr(A) := {ξ ∈ RN | Aξ, ξ = 0}. Obviously, it holds Ker(A) ⊆ Isotr(A). In general, the reverse inclusion does not necessarily hold (as Isotr(A) is not necessarily a vector space!) as the following example shows: when ! 1 0 A := , 0 −1 we have Ker(A) = {(0, 0)}, whereas Isotr(A) = span{(1, 1)} ∪ span{(1, −1)}. However, if A is positive semi-definite, then Ker(A) = Isotr(A). Indeed, let R be a real, symmetric matrix such that A = R 2 . If ξ ∈ Isotr(A), it holds 0 = Aξ, ξ = R 2 ξ, ξ = R T Rξ, ξ = Rξ, Rξ = R ξ 2 . Hence R ξ = 0, so that A ξ = R 2 ξ = RRξ = 0. This gives ξ ∈ Ker(A).


67

so that, for any fixed x ∈ H1 , the set N (x) of vectors ξ ’s which annihilate the relevant quadratic form is (see (1.94)) N(x) = (1, 0, 2 x2 )⊥ ∩ (0, 1, −2 x1 )⊥ = span{(−2x2 , 2x1 , 1)}. Therefore, it is always one-dimensional. As we showed in (A4), N (x) can also be found as N (x) = Ker(A(x)), where

! 1 0 1 0 2x2 T A(x) = σ (x) σ (x) = · 0 1 0 1 −2x1 2x2 −2x1

1 0 2x2 = . 0 1 −2x1 2x2 −2x1 4(x12 + x22 ) This kernel is one-dimensional, as A(x) has rank 2 for every x ∈ H1 . (A5) The sub-Laplacian L is the second order partial differential operator related to the Dirichlet form u →

|∇L u|2 dx.

More precisely, let Ω ⊆ RN be an open set, and consider the functional 1 C (Ω, R) u → J (u) = 2 ∞

Ω

|∇L u|2 dx,

|∇L u|2 =

N1

(Yj u)2 .

j =1

Denoting by , the inner product in RN1 , we have J (u + h) − J (u) = ∇L u, ∇L h dx + J (h) Ω

C0∞ (Ω, R).

We call critical point of J any function u ∈ for every h ∈ C ∞ (Ω, R) such that ∇L u, ∇L h dx = 0 ∀ h ∈ C0∞ (Ω, R). Ω

Then, given u ∈ C ∞ (Ω, R), we have u is a critical point of J if and only if Lu = 0 in Ω. Indeed, since Yj∗ = −Yj , an integration by parts gives Ω

∇L u, ∇L h dx =

N1

Yj uYj h dx = −

j =1 Ω

=−

(Lu)h dx Ω

for every u ∈ C ∞ (Ω, R) and h ∈ C0∞ (Ω, R).

N1 j =1 Ω

(Yj2 u)h dx

68


1.5.1 The Horizontal L-gradient We end this section with some useful results on the horizontal L-gradient. N1 2 Proposition 1.5.4. Let L = j =1 Xj be a sub-Laplacian on the homogeneous ∞ Carnot group G. Let u ∈ C (G, R) be such that Xj u is a polynomial function of G-degree not exceeding m for every j = 1, . . . , N1 . Then u is a polynomial function of G-degree not exceeding m + 1. Proof. Let Z1 , . . . , ZN be the Jacobian basis of g, the Lie algebra of G. Since the Xj ’s have polynomial coefficients and g = Lie{X1 , . . . , XN1 }, pk := Zk u is a polynomial function, k = 1, . . . , N . Moreover, if we denote δλ (x) = δλ (x1 , . . . , xN ) := (λσ1 x1 , . . . , λσN xN ) with 1 = σ1 ≤ · · · ≤ σN = r, keeping in mind the stratification of g, one easily recognizes that degG pk ≤ m + 1 − σk , k = 1, . . . , N. Now, by using (1.38) and (1.62) of Section 1.1, we see that ∂xj u(x) =

N

pk (x) ak (x −1 ), (j )

j = 1, . . . , N,

k=j

where y → ak (y) is δλ -homogeneous of degree σk − σj . Then, since δλ (x −1 ) = (δλ (x))−1 and the map x → x −1 = Exp (−Log x) has polynomial components (see (j ) Theorem 1.3.28 and Corollary 1.3.29), the function x → ak (x −1 ) is a polynomial δλ -homogeneous of degree σk − σj . It follows that (j )

x → ∇u(x), x =

N

xj ∂xj u(x)

j =1

is a polynomial function of G-degree not exceeding m + 1. Therefore, also 1 1 d dt u(tx) dt = ∇u(tx), tx u(x) − u(0) = t 0 dt 0 is a polynomial function of G-degree not exceeding m + 1.

In order to state the next corollary, we introduce a new notation. Let β = (i1 , . . . , ik ) be a multi-index with components in the set {1, . . . , N1 }. We set X β := Xi1 ◦ · · · ◦ Xik and |β| = k. Corollary 1.5.5. Let the hypotheses of Proposition 1.5.4 hold. Let u ∈ C ∞ (G, R) be such that X β u = 0 ∀ β : |β| = m for a suitable integer m ≥ 1. Then u is a polynomial function on G of G-degree not exceeding m − 1.


69

Proof. We argue by induction on m. By Proposition 1.5.6, the assertion holds if m = 1. Suppose it holds for m = p, and let us prove that it holds for m = p + 1. Now, if X β u = 0 for any multi-index β with |β| = p + 1, then X γ (Xj u) = 0 ∀ γ : |γ | = p and for every j ∈ {1, . . . , N1 }. Hence, by the induction assumption, Xj u is a polynomial function of G-degree not exceeding p − 1 for every j ∈ {1, . . . , N1 }. From Proposition 1.5.4 it follows that u is a polynomial function of G-degree not exceeding p. This completes the proof.

Proposition 1.5.6. Let Ω be an open and connected subset of the homogeneous Carnot group G. Let L be any sub-Laplacian on G. Then a function u ∈ C 1 (Ω, R) is constant in Ω if and only if the relevant horizontal L-gradient ∇L u vanishes identically on Ω. Proof. It is obviously non-restrictive to suppose that L = ΔG . Suppose Z1 u, . . . , ZN1 u vanish identically on Ω. Since the Lie algebra of G is given by Lie{Z1 , . . . ZN1 } (see (1.82)), then for every vector field Zj of the Jacobian basis, we have Zj u ≡ 0. We end by applying Proposition 1.2.21.

Example 1.5.7. Our proof of Proposition 1.5.6, despite its simplicity, conceals a deep geometric argument, which can be applied in more general situations (namely, for vector fields satisfying Hörmander’s condition). We describe the underlying geometric idea by an explicit example, leaving to the reader the task to generalize it in more general cases. (See also Chapter 19, where we study the so-called Carathéodory– Chow–Rashevsky connectivity theorem.) Consider the Heisenberg–Weyl group H1 on R3 . The Jacobian generators of its Lie algebra are the vector fields X1 = ∂1 + 2x2 ∂3 ,

X2 = ∂2 − 2x1 ∂3 .

Proposition 1.5.6 then states that if a C 1 -function u satisfies X1 u = 0 and X2 u = 0 in a domain Ω ⊆ R3 ,

(1.95)

then u is constant in Ω. For the sake of brevity, rather than applying some (simple) connectedness argument, we may suppose Ω = R3 . The basic idea is the following one. Suppose that two points x, y ∈ R3 can be joined by an integral curve γ of one of the fields ±X1 or ±X2 , then u(x) = u(y). Indeed, suppose that γ : [0, T ] → R3 is such that γ (0) = x, γ (T ) = y and γ˙ (s) = XI (γ (s))

for all s ∈ [0, T ],

70


where X is one of the fields ±X1 or ±X2 . Then, taking into account (1.95), we have

T

u(y) − u(x) = 0

T

=

T d (u(γ (s))) ds = (∇u)(γ (s)), γ˙ (s) ds = ds 0 T + , (Xu)(γ (s)) ds = 0, (∇u)(γ (s)), XI (γ (s)) ds =

0

0

whence u(x) = u(y). As a consequence, we can prove that u is constant, if we show that any couple of points in H1 can be joined by a finite sequence of paths which are integral curves of ±X1 and ±X2 .

(1.96)

We shall refer to (1.96) by saying that H1 is (X1 , X2 )-connected. The (X1 , X2 )-connectedness has a deep motivation, namely the fact that X1 ,

X2 ,

[X1 , X2 ] are linearly independent.17

(1.97)

In its turn, the fact that (1.97) implies the (X1 , X2 )-connectedness has a profound motivation too, mainly based on the so-called Campbell–Hausdorff formula, as we shall explain at the end of this section. Before entering into the details of the Campbell–Hausdorff formula, we show how simple the argument is in the case of H1 . Indeed, let us fix a point P0 := (x1 , x2 , x3 ) ∈ H1 , and let us consider the path γ , integral curve of X1 starting from this point. As we showed in (1.5), we have γ (s) = (x1 + s, x2 , x3 + 2 x2 s).

(1.98)

We denote this point by P1 . We then proceed along the integral curve of X2 starting from P1 : at the time s, we arrive to the following point (as a simple calculation shows) P2 = (x1 + s, x2 + s, x3 + 2x2 s − 2(x1 + s)s). Moreover, we proceed along the integral curve of −X1 starting from P2 : at the time s, we arrive to the following point (it is enough to have in mind (1.98) and replace s with −s) P3 = (x1 , x2 + s, x3 − 2(x1 + s)s − 2s 2 ). Finally, we proceed along the integral curve of −X2 starting from P3 : at the time s, we arrive to the following point (again, it is enough to notice that the integral curve of −X2 at time s coincides with the integral curve of X2 at time −s) 17 When X , X are arbitrary smooth vector fields in R3 , the result 1 2

(X1 , X2 , [X1 , X2 ] linearly independent) ⇒ (R3 is [X1 , X2 ]-connected), is known as Carathéodory’s theorem. For a more general version of this result in RN (but under some further assumptions on the fields Xi ’s) we refer the reader to Chapter 19.


71

P4 = (x1 , x2 , x3 − 4s 2 ). An analogous calculation shows that, if we start from (x1 , x2 , x3 ) and proceed along the integral curves of, respectively, X2 , X1 , −X2 , −X1 , we arrive to 4 = (x1 , x2 , x3 + 4s 2 ). P 4 , this shows that we can join any two points having the Being s arbitrary in P4 and P same x1 , x2 -coordinates and the third one arbitrarily given. We now start from (x1 , x2 , x3 ) and, along an integral curve of X1 , at the time s we arrive to (x1 + s, x2 , x3 + 2x2 s). By the preceding argument, keeping fixed the first two coordinates, we can vary the third one, in order to arrive to the point (x1 + s, x2 , x3 ) (after finitely many integral curves of ±X1 , ±X2 ). Being s arbitrary, this shows that we can join any two points having the same x2 , x3 coordinates and the first one arbitrarily given. Finally, an obvious analogous argument shows that we can join any two points having the same x1 , x3 coordinates and the second one arbitrarily given. All these facts together prove (1.96). To end the section, we describe the reason why the validity of (1.97) implies the (X1 , X2 )-connectedness. For instance, let us denote by Γ (s) the point we obtain if (as we did above) we follow paths of X1 , X2 , −X1 , −X2 for a time s, this is the same as saying that we follow paths of sX1 , sX2 , −sX1 , −sX2 for unit time. More explicitly, Γ (s) = exp(−sX2 )(exp(−sX1 )(exp(sX2 )(exp(sX1 )(x)))). Very simple arguments (showed in the proof of Lemma 5.13.18, page 301) show that (see precisely (5.117), page 302) Γ (s) − x = [X1 , X2 ]I (x). s→0 s2 lim

(1.99)

Roughly speaking, (1.99) ensures that, by following suitable integral curves of ±X1 , ±X2 , we can arrive as close as we want to the endpoints of the integral curves of the commutator [X1 , X2 ]. So, if X1 , X2 , [X1 , X2 ] span R3 at every point, it is intuitively evident that we can connect any two points by suitable integral curves of ±X1 , ±X2 . This argument becomes completely apparent if we make use of the so-called Campbell–Hausdorff formula. For two vector fields X1 , X2 generating an algebra nilpotent of step two, the Campbell–Hausdorff formula states that18 ! 1 exp(X2 )(exp(X1 )(x)) = exp X1 + X2 + [X1 , X2 ] (x). (1.100) 2 18 Further details on the Campbell–Hausdorff formula can be found in Definition 2.2.11,

Theorem 2.2.13 (page 129), in Lemma 4.2.4 (page 194) and, mostly, in Theorem 15.1.1 (page 595).

72


We now apply three times formula (1.100), in order to simplify the above Γ (s). The following calculation then applies: Γ (s) = exp(−sX2 )(exp(−sX1 )(exp(sX2 )(exp(sX1 )(x)))) ! !! s2 = exp(−sX2 ) exp(−sX1 ) exp sX1 + sX2 + [X1 , X2 ] (x) 2 ! ! s2 s2 = exp(−sX2 ) exp sX1 + sX2 + [X1 , X2 ] − sX1 − [X2 , X1 ] (x) 2 2 = exp(−sX2 )(exp(sX2 + s 2 [X1 , X2 ])(x)) = exp(sX2 + s 2 [X1 , X2 ] − sX2 )(x) = exp(s 2 [X1 , X2 ])(x). This says that, following suitable integral curves of ±X1 , ±X2 , we can arrive wherever the integral curves of the commutator [X1 , X2 ] do arrive! So, again, if X1 , X2 , [X1 , X2 ] span R3 at every point, we can obviously connect any two points by suitable integral curves of ±X1 , ±X2 . We explicitly remark that the assumption that X1 , X2 generate an algebra nilpotent of step two has made the above calculation very transparent. For general smooth vector fields, the Campbell–Hausdorff formula leads to a formula with a remainder term, exp(−sX2 )(exp(−sX1 )(exp(sX2 )(exp(sX1 )(x)))) = exp(s 2 [X1 , X2 ] + Ox (s 3 ))(x).

Bibliographical Notes. “Carnot” groups seem to owe their name after an paper by C. Carathéodory [Car09] (related to a mathematical model of thermodynamics) dated 1909. The same denomination was then used in the school of M. Gromov [Gro96] and it is nowadays commonly used. The definition of stratified group given in this chapter is seemingly different from the classical one by G.B. Folland [Fol75] and by G.B. Folland & E.M. Stein [FS82] (see also M. Gromov [Gro96], P. Pansu [Pan89]). Indeed, here we focused on stratified groups having an underlying homogeneous structure. Nonetheless, we shall prove in Chapter 2 that any (abstract) stratified group is canonically isomorphic to a homogeneous one, so that our definition here is nonrestrictive and seems to be more operative and easier to deal with. A direct approach to homogeneous Carnot groups can also be found in E.M. Stein [Ste81] or in N.T. Varopoulos, L. Saloff-Coste, T. Coulhon [VSC92]; see also P. Hajlasz and P. Koskela [HK00].

1.6 Exercises of Chapter 1

73

1.6 Exercises of Chapter 1 Ex. 1) Verify that the following operation (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) := (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) endows R3 with a Lie group structure. Ex. 2) a) Verify that, for every fixed α ∈ R, the following operation ⎛ x1 + y1 , ⎜ x2 + y2 , ⎜ 1 ⎜ + y + x 3 3 2 (x1 y2 − x2 y1 ), ⎜ 1 ⎝ x4 + y4 + 2 (x1 y3 − x3 y1 ) + α2 (x2 y3 − x3 y2 ) 1 α (x1 − y1 ) (x1 y2 − x2 y1 ) + 12 (x2 − y2 ) (x1 y2 − x2 y1 ) + 12

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

defines on R4 a homogeneous Carnot group G. b) Verify that the Jacobian basis for the algebra g of the above G is ! 1 1 1 x3 + x2 (x1 + α x2 ) ∂4 , Z1 = ∂1 − x2 ∂3 − 2 2 12 ! 1 1 1 Z2 = ∂2 + x1 ∂3 + − α x3 + x1 (x1 + α x2 ) ∂4 , 2 2 12 ! 1 1 x1 + α x2 ∂4 , Z3 = [Z1 , Z2 ] = ∂3 + 2 2 Z4 = [Z1 , [Z1 , Z2 ]] = [Z1 , Z3 ] = ∂4 . Verify that the only other non-trivial commutator identity is [Z2 , [Z1 , Z2 ]] = α [Z1 , [Z1 , Z2 ]]. c) Verify that the exponential map for G (written w.r.t. the Jacobian basis) is the “identity map” in the following sense: if ξ = (ξ1 , ξ2 , ξ3 , ξ4 ) ∈ R4 , and ξ · Z ∈ g denotes the vector field 4i=1 ξi Zi , then Exp (ξ · Z) = ξ ∈ G. Ex. 3) With reference to what we proved in Example 1.2.33 (page 30), prove that, for the group considered in Examples 1.2.18 and 1.2.31 (pages 21 and 28) it holds 1 Log Exp (X) ◦ Exp (Y ) = X + Y + [X, Y ]. 2 Ex. 4) In this exercise, we provide some compact formulas for general sum of squares of vector fields. Suppose we are assigned m vector fields of class at least C 1 on RN ,

74


⎛α Xk =

N

⎞

⎜ α2,k ⎟ ⎟ Xk I = ⎜ ⎝ .. ⎠ , . αN,k

i.e. as usual,

αj,k ∂j ,

1,k

j =1

k ≤ m.

Let S be the N × m matrix whose k-th column vector is given by Xk I , ⎛α

1,1

S := ⎝ ... αN,1

···

α1,k .. .

α1,m ⎞ .. ⎠ = (α ) j,k j =1,...,N . .

···

αN,k

αN,m

k=1,...,m

With this notation, for every u ∈ C 1 (RN , R), we have ∇u · S = (X1 u, . . . , Xm u) =: ∇X u, where ∇ = (∂1 , . . . , ∂N ) is the usual gradient operator, and ∇X := (X1 , . . . , Xm ) is the “intrinsic gradient” related to the family {X1 , . . . , Xm }. We also define the N × N symmetric matrix m

T αi,k αj,k =: (ai,j )i,j ≤N . A := S · S = i,j =1,...,N

k=1

For every u ∈ C 1 (RN , R), it holds |∇X u|2 :=

m (Xk u)2 = ∇X u, ∇X u = ∇u · S, ∇u · S k=1

= (∇u · S) · (∇u · S)T = ∇u · (S · S T ) · (∇u)T = ∇u · A · (∇u)T , i.e. |∇X u|2 =

N

ai,j ∂i u ∂j u.

i,j =1

Show that, for every k = 1, . . . , m, it holds

N N N 2 αi,k αj,k ∂i,j + αi,k (∂i αj,k ) ∂j . Xk2 = i,j =1

j =1

i=1

Derive that, in the coordinate form, the sum of squares related to the vector 2 fields Xk ’s, L := m k=1 Xk , is given by


L=

N

2 ai,j ∂i,j +

i,j =1

N

75

bj ∂j ,

j =1

with ai,j =

m

αi,k αj,k ,

k=1

bj =

m N

αi,k (∂i αj,k ) =

k=1 i=1

m

Xk αj,k .

k=1

We note that bj is a sort of “X-divergence” of the j -th row of the matrix S. Besides, show that N

N

N

N 2 ∂i αi,k αj,k ∂j − ∂i αi,k · αj,k ∂j Xk = i=1

=

N i=1

∂i

j =1 N

αi,k αj,k ∂j

j =1

i=1

− div(Xk I ) · Xk .

j =1

Deduce that L can be written in the following equivalent ways:

N m

m N N N L= ∂i αi,k αj,k ∂j − αj,k (∂i αi,k ) ∂j i=1

j =1 k=1

i=1

j =1

j =1

k=1 i=1

N

m N N = ∂i ai,j ∂j − αj,k div(Xk I ) ∂j = div(A · ∇ T ) −

j =1 m k=1

k=1

div(Xk I ) · Xk

⎞ # div(X1 I ) . T T ⎠,∇ .. = div(A · ∇ ) − S · ⎝ div(Xm I ) T = div(A · ∇ ) − (div(X1 I ), . . . , div(Xm I )), ∇X . "

⎛

In particular, the sum of squares L is in divergence form if and only if ⎞ ⎛ div(X1 I ) m N .. ⎠ ≡ 0. ∀ j ≤ N, αj,k (∂i αi,k ) = 0, i.e. S · ⎝ . k=1 i=1 div(Xm I ) (1.101) For example, re-derive that any sub-Laplacian L on a homogeneous Carnot group N is in divergence form, because in this case it holds div(Xk I ) = i=1 ∂i αi,k = 0, for αi,k does not depend on xi . We explicitly remark that, in the case of non-homogeneous Carnot groups (which will be introduced in the

76


next chapter), this is not necessarily true, as the following example shows: R3 equipped with the composition x ◦ y = (arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 ) is a non-homogeneous Carnot group; the first two vector fields of the relevant Jacobian basis are X1 = (cosh(x1 ))−1 ∂1 ,

X2 = ∂3 + sinh(x1 ) ∂2 ,

so that the canonical sub-Laplacian is not a divergence form operator, for (1.101) is not satisfied, being, for j = 1, α1,1 div(X1 I ) = − sinh(x1 )/(cosh(x1 ))3 . Ex. 5) Prove the assertions made in Remark 1.4.9 (page 61). Ex. 6) Prove the following simple formulas of calculus for vector fields. Here, 2 N N u, f, g ∈ C 2 (RN , R), α ∈ C 2 (R, R), ϕ = (ϕ 1 , . . . , ϕ2 N ) ∈ C (R , R ), X is a sum of squares X = ∇(·) · XI is a vector field on RN , L = m j =1 j of vector fields on RN , ∇L = (X1 , . . . , Xm ) is the intrinsic gradient related to L. • X(f g) = (Xf ) g + f (Xg), • ∇L (f g) = (∇L f ) g + f (∇L g), • X 2 (f g) = (X 2 f ) g + 2 (Xf ) (Xg) + f (X 2 g), • L(f g) = (Lf ) g + 2 ∇L f, ∇L g + f (Lg), • X(α(u)) = α (u) Xu, • ∇L (α(u)) = α (u) ∇L u, • X 2 (α(u)) = α (u) (Xu)2 + α (u) X 2 u, |∇L u|2 + α (u) Lu, • L(α(u)) = α (u) N • X(u(ϕ(x))) = j =1 uj (ϕ(x)) Xϕj (x) = (∇u)(ϕ(x)), Xϕ(x), • ∇L (u(ϕ(x))) = N j =1 (∂j u)(ϕ(x)) ∇L ϕj (x); if we use the usual columnvector notation ⎞ ⎛ ϕ1 (x) . ϕ(x) = ⎝ .. ⎠ ϕN (x) for vectors in RN , the row-vector notation for the gradients ∇u = (∂1 u, . . . , ∂N u),

∇L u = (X1 u, . . . , Xm u),

and we introduce the X-Jacobian matrix ⎛ ⎞ ⎛ ⎞ X1 ϕ1 (x) · · · Xm ϕ1 (x) ∇L ϕ1 (x) ∂ϕ .. .. .. ⎠=⎝ ⎠ (x) := ⎝ . ··· . . ∂X X1 ϕN (x) · · · Xm ϕN (x) ∇L ϕN (x) (which is a N × m matrix) which can also be denoted by


77

∇L ϕ(x) = (X1 ϕ(x) · · · Xm ϕ(x)) ⎛ ⎛ ⎛ ⎞ ⎞⎞ ϕ1 (x) ϕ1 (x) = ⎝X1 ⎝ ... ⎠ · · · Xm ⎝ ... ⎠⎠ , ϕN (x)

ϕN (x)

then last formula can be rewritten as ∇L (u(ϕ(x))) = (∇u)(ϕ(x)) ·

∂ϕ (x), ∂X

or equivalently ∇L (u(ϕ(x))) = (∇u)(ϕ(x)) · ∇L ϕ(x), which is resemblant to the classical chain rule ∇(u(ϕ(x))) = (∇u)(ϕ(x)) · Jϕ (x), •

prove that X 2 (u(ϕ(x))) =

N

(∂i,j u)(ϕ(x))Xϕi (x) Xϕj (x)

i,j =1

+

N (∂j u)(ϕ(x))X 2 ϕj (x), j =1

or equivalently, following the previous notation (here Hessu (x) is the usual Hessian matrix of u at x), X 2 (u(ϕ(x))) = (Xϕ(x))T · Hessu (ϕ(x)) · (Xϕ(x)) + (∇u)(ϕ(x)) · X 2 ϕ(x), •

deduce from the previous formulas that L(u(ϕ(x))) =

N

+ , (∂i,j u)(ϕ(x)) ∇L ϕi (x), ∇L ϕj (x)

i,j =1

+

N (∂j u)(ϕ(x))Lϕj (x), j =1

which can also be rewritten as L(u(ϕ(x))) =

m (Xk ϕ(x))T · Hessu (ϕ(x)) · (Xk ϕ(x)) k=1

+ (∇u)(ϕ(x)) · Lϕ(x)

78


or, introducing the Gram matrix of the system of N vectors in Rm {∇L ϕ1 (x), . . . , ∇L ϕN (x)}, i.e. the N × N symmetric matrix !T ∂ϕ ∂ϕ Gϕ (x) := (x) · (x) ∂X ∂X ⎛ ⎞ ∇L ϕ1 (x) . ⎠ · ∇L ϕ1 (x) T · · · ∇L ϕN (x) T .. = ⎝ ∇L ϕN (x) ⎛ ⎞ ∇L ϕ1 (x), ∇L ϕ1 (x) · · · ∇L ϕ1 (x), ∇L ϕN (x) .. .. .. ⎠, = ⎝ . . . ∇L ϕN (x), ∇L ϕ1 (x)

···

∇L ϕN (x), ∇L ϕN (x)

the above formula can be rewritten as L(u(ϕ(x))) = trace(Hessu (ϕ(x)) · Gϕ (x)) + (∇u)(ϕ(x)) · Lϕ(x). Ex. 7) Let X be a smooth vector field on some open set Ω ⊆ RN , and let x ∈ Ω. Let γ be the solution to the system of ODE’s γ˙ (t) = XI (γ (t)),

γ (0) = x,

defined on the open (maximal) interval D(X, x) ⊆ R. Denote (momentarily) γ (t) by E(X, x, t). Let λ ∈ R be arbitrary. Prove the homogeneity relation E(X, x, λ t) = E(λ X, x, t)

for all t such that λ t ∈ D(X, x). (1.102)

Let now t0 ∈ D(X, x) be fixed. Take λ = t0 and t = 1 in (1.102) and derive that (for the arbitrariness of t0 ) E(X, x, t) = E(t X, x, 1)

for all t ∈ D(X, x).

(1.103)

Throughout the following exercise, we shall adopt our usual notation exp(tX)(x) to denote E(X, x, t). Equation (1.103) says that the notation is not improper. Ex. 8) In this exercise, we give a generalization of formula (1.7) (page 7). Let Ω ⊆ RN be an open set, and let X be a smooth vector field on Ω. Let also f be a smooth function on Ω. Finally, we fix x ∈ Ω. Then the function t → f (exp(tX)(x)) is C ∞ near t = 0, and its Taylor expansion at 0 is given by f (exp(tX)(x)) =

n tk

(X k f )(x) k! k=0 1 t (t − s)n X n+1 f (exp(sX)(x)) ds. + n! 0

(1.104)


79

Indeed, prove by induction that it holds dk (f (exp(tX)(x))) = (X k f )(exp(tX)(x)). dt k Derive from it the (very useful) formula dk (f (exp(tX)(x))) = (X k f )(x). dt k t=0 Derive from (1.44) that if G = (RN , ◦) is a Lie group on RN and X is leftinvariant on G, then dk (f (x ◦ Exp (tX))) = (X k f )(x). dt k t=0 Ex. 9) Let Ω ⊆ RN be an open set, and let X be a smooth vector field on Ω. Let us consider the (autonomous) equation γ˙ = XI (γ ). From general results on the existence and uniqueness of solution of ODE’s (see, e.g. [Har82]) we know that, for every fixed compact set K ⊂ Ω, there exists δ = δ(X, K) > 0 such that the solution γ (t, x) to γ˙ = XI (γ ), γ (0) = x exists for every t ∈ ]−δ, δ[ and every initial value x ∈ K. From the uniqueness of the solution (and the autonomous nature of the equation) derive that γ (t, γ (s, x)) = γ (t + s, x),

|t| + |s| < δ,

x ∈ K.

(1.105)

Deduce from this fact that, for every fixed t0 ∈ ]−δ, δ[, the mapping K x → γ (t0 , x) is injective. (Hint: If γ (t0 , x) = γ (t0 , y), then γ (t +t0 , x) = γ (t, γ (t0 , x)) = γ (t, γ (t0 , y)) = γ (t + t0 , y), and evaluate at t = −t0 .) If O is an open set such that O ⊂ Ω is compact, prove that (whenever |t0 | < δ(X, O)) the map O x → γ (t0 , x) is a C ∞ -diffeomorphism onto its image. (Hint: Again thanks to (1.105), the inverse map is given by y → γ (−t0 , x).) Ex. 10) Let Y1 , . . . , Ym be smooth vector fields on an open set Ω ⊆ RN . Let ξ = (ξ1 , . . . , ξm ) ∈ Rm . Since the dependence on ξ of ξ1 Y1 I + · · · + ξm Ym I is smooth, by general results on ODE’s (see, e.g. [Har82]), given a compact set K ⊂ Ω, we infer the existence of ε > 0 such that the solution γ to γ˙ =

m

ξj Yj I (γ ),

γ (0) = x

j =1

exists for every x ∈ K, every ξ satisfying |ξ | < ε and every t ∈ ]−ε, ε[ (here |ξ | denotes any fixed norm on Rm ). Moreover, the dependence of γ on (x, ξ, t) is smooth. Using (1.103) (and the notation therein), observe that

80


Fig. 1.7. The flow of the vector field X

E

! ξj Yj , x, t

=E

j

! for all t ∈ ]−ε, ε[

tξj Yj , x, 1

j

and derive that the map (u1 , . . . , um ) → Θx (u) := exp

uj Yj (x)

j

is well-defined (and smooth) on the open ball B = {u ∈ Rm : |u| < ε 2 } (for any x ∈ K). Prove that the Jacobian matrix of Θx at u = 0 is the matrix whose j -th column vector is Yj I (x) or, equivalently, that the differential d0 Θx sends the tangent vector (∂/∂uj )|0 ∈ T0 (B) to Yj |x ∈ Tx (Ω). Hint: Use the following fact ! ! ∂ ∂ 2 I (x) + exp uj Yj (x) = uj Yj I (x) + Ou→0 (|u| ) ∂uj 0 ∂uj 0 j

j

= Yj I (x). As a corollary, deduce that if m = N and Y1 I (x), . . . , YN I (x) are linearly independent, then Θx is a C ∞ -diffeomorphism of a neighborhood of 0 onto a neighborhood of x. Finally, generalize (1.104) of Exercise 8 (we follow the notation therein) and obtain the following Taylor expansion at u = 0 f

exp

m j =1

uj Yj

k m ∞ 1 uj Yj f (x). (x) ∼ k! k=0

j =1

(1.106)


Hint: Set

F (u) := f

exp

m

81

and G(t, u) := F (tu).

uj Yj (x)

j =1

Hence G(t, u) = f (exp(tX)(x)), where X = m j =1 uj Yj . Use Ex. 8 to derive that

k m ∂ k G(t, u) = uj Yj f (x), ∂t k t=0 j =1

and the Taylor expansion of F at u = 0 follows, as usual, from that of G at t = 0. Ex. 11) (Rellich–Pohozaev identities). We introduce the following notation. If Ω ⊂ RN is a domain with boundary regular enough, we denote by ν the outer unit normal to ∂Ω and by dσ the Hausdorff (N − 1)-dimensional measure on ∂Ω. Provide the details for the following proposition. Proposition Let G be an arbitrary homogeneous Carnot group, and 1.6.1. 2 be a sub-Laplacian on G. Let Ω be a bounded domain X let L = m i=1 i in G, regular for the divergence theorem. Finally, let Z be a vector field of class C 1 (G). Then, for every ϕ ∈ C 2 (Ω), we have m 2 Xi ϕ Xi I, ν Zϕ dσ − ZI, ν |∇L ϕ|2 dσ ∂Ω i=1

∂Ω

m =2 Xi ϕ [Xi , Z]ϕ + 2 Lϕ Zϕ − |∇L ϕ|2 div(ZI ). (1.107) Ω i=1

Ω

Ω

Proof. By applying two times the divergence theorem (and recalling that Xi∗ = −Xi ), we obtain 2 |∇L ϕ| ZI, ν dσ = div(|∇L ϕ|2 ZI ) ∂Ω Ω |∇L ϕ|2 div(ZI ) + Z(|∇L ϕ|2 ) = Ω Ω = |∇L ϕ|2 div(ZI ) + 2 Z(∇L ϕ), ∇L ϕ Ω Ω m 2 |∇L ϕ| div(ZI ) + 2 [ZI, Xi ]ϕ Xi ϕ = Ω

+2

Ω

Ω i=1

Xi (Zϕ) Xi ϕ

Ω i=1

=

m

|∇L ϕ|2 div(ZI ) + 2

m

+2

∂Ω i=1

This ends the proof.

m [ZI, Xi ]ϕ Xi ϕ Ω i=1

Zϕ Xi ϕ Xi I, ν dσ − 2

Zϕ Lϕ. Ω

82


We now specify the integral identity (1.107) when Z is given by the socalled generator of the translations. Let ◦ be the group law on the Carnot group G, and fix z0 ∈ G. We denote by Z z0 the following vector field on G d z0 Z I (z) = ((hz0 ) ◦ z). (1.108) dh h=0 Recalling that, for every i = 1, . . . , m, Xi (z) = (d/dh)h=0 (z ◦ (hei )) (here ei is the i-th versor of the canonical basis of RN ), derive that the bracket [Xi , Z z0 ] vanishes identically. Hint: ◦ is associative and d d f ((sz0 ) ◦ (z ◦ (hei ))) [Xi , Z z0 ]f (z) = dh h=0 ds s=0 d d − f (((sz0 ) ◦ z) ◦ (hei )). ds s=0 dh h=0 Furthermore, the divergence of the vector field Z z0 vanishes identically. Indeed, we recall that ◦ has the form z ◦ ζ = ((z ◦ ζ )(1) , . . . , (z ◦ ζ )(r) ), where (z ◦ζ )(1) = z(1) +ζ (1) ,

(z ◦ζ )(j ) = z(j ) +ζ (j ) +Q(j ) (z, ζ ),

2 ≤ j ≤ r,

Q(j ) being a function with values in RNj and whose components are mixed polynomials in z and ζ such that Q(j ) (δλ z, δλ ζ ) = λj Q(j ) (z, ζ ). We then recognize that the components of Z z0 (z) in the j -th layer have the following form (j )

(1)

(j −1)

(j )

(Z z0 (z))(j ) = z0 + z0 , q1 (z) + · · · + z0

(j )

, qj −1 (z),

(j )

where qi is a RNi -valued function whose components are polynomials δλ -homogeneous of degree j − i. In particular, (Z z0 (z))(j ) does not depend on zj , whence div(Z z0 ) = 0. From Proposition 1.6.1 and the above remarks, the next result immediately follows. Proposition 1.6.2. Let G be a homogeneous Carnot group, and let L = m 2 be a sub-Laplacian on G. Let Ω be a bounded domain in G, regX i=1 i ular for the divergence theorem. Finally, for a fixed z0 ∈ G, let Z z0 be the vector field defined in (1.108). Then, for every ϕ ∈ C 2 (Ω), we have 2

m z0 Xi ϕ Xi I, ν Z ϕ dσ −

∂Ω i=1

∂Ω

Lϕ Z z0 ϕ.

=2 Ω

Z z0 I, ν |∇L ϕ|2 dσ


83

Ex. 12) (Maps commuting with a sum of squares). a) Consider a second order differential operator N

L :=

2 ai,j (x) ∂i,j +

i,j =1

N

bi (x) ∂i

i=1

with coefficients ai,j and bi in C 2 (RN ) such that ai,j = aj,i for all i and j . Consider ψ ∈ C 2 (RN , RN ). Prove that the necessary and sufficient conditions in order to have L(u ◦ ψ) = (L u) ◦ ψ

∀ u ∈ C 2 (RN )

are the following ones: ⎧ bk (ψ) = Lψk ∀ k = 1, . . . , N, ⎪ ⎪ ⎨ N ⎪ ai,j (ψ) = ar,s ∂r ψi ∂s ψj ∀ i, j = 1, . . . , N. ⎪ ⎩

(1.109)

(1.110)

r,s=1

(Hint: Take u(x) = xk and then u(x) := xi xj .) If Jψ denotes the usual Jacobian matrix of ψ and ⎞ ⎛ b1 (x) ⎟ ⎜ A(x) := ai,j (x) 1≤i,j ≤N , B(x) := ⎝ ... ⎠ , bN (x) so that

L = trace A(x) · Hess + ∇, B(x),

then system (1.110) can be rewritten as B(ψ(x)) = Lψ(x) A(ψ(x)) = Jψ (x) · A(x) · (Jψ (x))T .

(1.111)

p b) Suppose furthermore that L is a sum of squares L = k=1 Xk2 with (k) Xk = N i=1 σi ∂i , so that p

p

N N (k) (k) (k) 2 σi σj ∂i,j + Xk σi L= ∂i . i,j =1

k=1

i=1

k=1

Prove that the necessary and sufficient conditions for (1.109) to hold are ⎧ p ⎪ (k) ⎪ ⎪ (Xk σi )(ψ) = Lψi ∀ i = 1, . . . , N, ⎪ ⎨ k=1

p ⎪ ⎪ (k) (k) ⎪ ⎪ σi (ψ) σj (ψ) = ∇L ψi , ∇L ψj ⎩

∀ i, j = 1, . . . , N,

k=1

(1.112)

84


where ∇L u := (X1 u, . . . , Xp u). Set, as usual, ⎞ σ1(k) ⎟ ⎜ Xk I := ⎝ ... ⎠ . ⎛

(k)

σN Considering the N × p matrix (j )

S(x) := (σi (x))i≤N, j ≤p = (X1 I (x) · · · Xp I (x)), we have

A(x) =

p

(k) (k) σi σj

= S(x) · (S(x))T , i,j ≤N

k=1

so that Jψ (x) · A(x) · (Jψ (x))T = (Jψ (x) · S(x)) · (Jψ (x) · S(x))T . Moreover, ⎞ ∇L ψ 1 ⎟ ⎜ = ⎝ ... ⎠ . ∇L ψ N ⎛

Jψ (x) · S(x) = (Xj ψi (x))i≤N, j ≤p As a consequence,

(Jψ (x) · S(x)) · (Jψ (x) · S(x))T = (∇L ψi (x), ∇L ψj (x))i,j ≤N . Moreover, B(x) =

p k=1

(k)

Xk σi

i≤N

⎛ p ⎜ =⎝

k=1

p k=1

(k) ⎞ Xk σ1 ⎟ .. ⎠. .

Xk σN(k)

Finally, (1.112) becomes B(ψ(x)) = Lψ(x), A(ψ(x)) = (∇L ψi (x), ∇L ψj (x))i,j ≤N . Ex. 13) Consider the Lie group (and the notation) in Example 1.2.19 (page 21). Show that, for every ξ1 , ξ2 ∈ R, the integral curve of ξ1 Z1 + ξ2 Z2 starting at (0, 0) is (ξ1 t, (ξ2 eξ1 t − ξ2 )/ξ1 ) if ξ1 = 0, γ (t) = (0, ξ2 t) if ξ1 = 0.


85

Equivalently, considering the function f : R → R,

f (z) :=

ez − 1 if z = 0, z

f (0) := 1,

we have γ (t) = (ξ1 t, ξ2 t f (ξ1 t)). Derive that Exp : g → G,

exp(ξ1 Z1 + ξ2 Z2 ) = (ξ1 , ξ2 f (ξ1 )).

Prove that Exp is smooth (indeed, real analytic!) and globally invertible. Find the Log function. (Hint: The function f is analytic and invertible.)

2 Abstract Lie Groups and Carnot Groups

The aim of this chapter is to prove that, up to a canonical isomorphism, the classical definition of stratified group (or Carnot group) (see Definition 2.2.3) coincides with our definition of homogeneous Carnot group, as given in Chapter 1. To this aim, we begin by recalling some basic facts about abstract Lie groups, providing all the terminology and the main results about manifolds, tangent vectors, left-invariant vector fields, Lie algebras, homomorphisms, the exponential map. Our exposition in Section 2.1 is self-contained and is intended to provide the topics from differential geometry and Lie group theory, which are strictly necessary to read this book. In Section 2.2, we provide the cited equivalence between the two notions of Carnot group. This is accomplished by showing that the Lie algebra g of the abstract Carnot group G possesses a natural structure of homogeneous Carnot group. Indeed, the group operation on g is the one induced by that of G via the exponential map, and the dilations on g are modeled on its stratification. A central rôle here will be played by the Campbell–Hausdorff formula, that we assume in this chapter by recalling (without proofs) some of its abstract and very general properties. Nonetheless, we shall devote Chapter 15 to provide a self-contained investigation of such a remarkable formula in the significant case of homogeneous stratified groups.

2.1 Abstract Lie Groups 2.1.1 Differentiable Manifolds Let N ∈ N, and let us define, for i = 1, . . . , N , the coordinate projections on RN (whose points will be denoted by ξ = (ξ1 , . . . , ξN ) ∈ RN with ξ1 , . . . , ξN ∈ R) πi : RN −→ R,

πi (ξ ) := ξi .

Definition 2.1.1 (N -dimensional locally Euclidean space). An N -dimensional locally Euclidean space M is a Hausdorff topological space such that every point of M

88


has a neighborhood in M homeomorphic to an open subset of RN . If ϕ is a homeomorphism between a connected open set U ⊆ M and an open subset of RN , we say that ϕ : U → RN is a coordinate map, xi := πi ◦ ϕ : U → R is a coordinate function, and the pair (U, ϕ) (sometimes also denoted by (U, x1 , . . . , xN )) is a coordinate system or a chart. If m ∈ U and ϕ(m) = 0, we say that the coordinate system is centered at m. Definition 2.1.2 (Differentiable manifold). A C ∞ differentiable structure F on a locally Euclidean space M is a collection of coordinate systems {(Uα , ϕα ) : α ∈ A } with the following properties: • α∈A Uα = M; • ϕα ◦ ϕβ−1 is C ∞ for every α, β ∈ A (whenever it is defined); • F is maximal w.r.t. the second property in the sense that if (U, ϕ) is a coordinate system such that ϕ ◦ ϕα−1 and ϕα ◦ ϕ −1 are C ∞ for every α ∈ A, then (U, ϕ) ∈ F. An N -dimensional C ∞ differentiable manifold is a couple (M, F), where M is a second countable N -dimensional locally Euclidean space and F is a C ∞ differentiable structure. As usual, when we say “M is an N-dimensional C ∞ differentiable manifold”, we leave unsaid that M is equipped with the fixed datum of a C ∞ differentiable structure F on M. Let M and M be differentiable manifolds of dimension N and N , respectively, and let f : M −→ M . Then we say that f is C ∞ in m ∈ M if, for every (or, equivalently, for at least one) coordinate system (U, ϕ) of M and for every (or, equivalently, for at least one) coordinate system (U , ϕ ) of M such that m ∈ U and f (m) ∈ U , the function ϕ ◦ f ◦ ϕ −1 : ϕ(U ) ⊆ RN −→ ϕ (U ) ⊆ RN is C ∞ in a neighborhood of ϕ(m). Let now μ(t) be a C ∞ function defined on a real interval and with values in M. We say that μ(t) is a curve passing through m at the time t0 if μ(t0 ) = m. We say that a C ∞ real valued function f defined in a neighborhood of m ∈ M is horizontal in m if d f (μ(t)) =0 dt t=0 for every curve passing through m at the time t = 0. Remark 2.1.3. The following characterizations hold: (I). It is immediate to see that a real-valued function f is C ∞ in a neighborhood of m ∈ M if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a

2.1 Abstract Lie Groups

89

(ordinary) C ∞ real-valued function f on (the open subset of RN ) ϕ(U ) such that f = f◦ ϕ on U . (For example, any coordinate map xi = πi ◦ ϕ is C ∞ on U for πi is smooth.) (II). Analogously, μ is a curve passing through m ∈ M at t0 if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a (ordinary) C ∞ curve ν : ]a, b[ ⊆ R −→ ϕ(U ) ⊆ RN with t0 ∈]a, b[, ν(t0 ) = ϕ(m) and μ = ϕ −1 ◦ ν on ]a, b[. (III). Furthermore, we show that f is horizontal in m ∈ M if and only if there exist a coordinate system (U, ϕ) with m ∈ U and a C ∞ real-valued function f on ϕ(U ) such that f = f◦ ϕ on U and the ordinary gradient (∇ f)(ϕ(m)) is null. Indeed, suppose this fact holds. Let μ be any curve passing through m at t = 0. By (II), there exist a coordinate system (V , ψ) with m ∈ V and a C ∞ curve ν : ]a, b[ −→ ψ(U ) with 0 ∈ ]a, b[, ν(0) = ψ(m) and μ = ψ −1 ◦ ν on ]a, b[. Then, for ε > 0 sufficiently small (so that ν(t) ∈ ψ(U ∩ V ) for every t ∈ ]−ε, ε[), the composition f ◦ μ = f◦ ϕ ◦ ψ −1 ◦ ν is well defined and smooth on ]−ε, ε[ and, by the ordinary chain rule, it holds d f (μ(t)) = (∇ f) ϕ(ψ −1 (ν(0))) · Jϕ◦ψ −1 (ν(0)) · ν˙ (0) = 0, dt t=0 for (∇ f)(ϕ(ψ −1 (ν(0)))) = (∇ f)(ϕ(m)) = 0. This proves that f is horizontal in m. Vice versa, suppose f is horizontal in m. By (I), there exist a coordinate system (U, ϕ) with m ∈ U and a C ∞ real-valued function f on ϕ(U ) such that f = f◦ ϕ on U . We aim to prove that (∇ f)(ϕ(m)) = 0. Let h ∈ RN be any fixed vector. Let ε > 0 be small enough, so that, for every |t| < ε, we have ν(t) := ϕ(m) + t h ∈ ϕ(U ). Then, by (II), μ := ϕ −1 ◦ ν is a C ∞ curve passing through m at t = 0. Hence, being f horizontal in m, we have d d f (μ(t)) f ◦ ϕ ◦ ϕ −1 ◦ ν (t) = 0= dt d t t=0 t=0 = (∇ f)(ν(0)) · ν˙ (0) = (∇ f)(ϕ(m)), h . Consequently, due to the arbitrariness of h, this proves that (∇ f)(ϕ(m)) = 0. (IV). Finally, we prove that f is horizontal in m ∈ M if and only if for every coordinate system (V , ψ) with m ∈ V , the C ∞ real-valued function f = f ◦ ψ −1 on ψ(V ) satisfies

90


(∇ f)(ψ(m)) = 0. The “if” part follows by (III). Vice versa, suppose f is horizontal in m ∈ M. By (III), there exist a coordinate system (U, ϕ) with m ∈ U and a C ∞ real-valued function g on ϕ(U ) such that f = g ◦ ϕ on U and (∇g)(ϕ(m)) = 0. As a consequence, (∇ f)(ψ(m)) = (∇(f ◦ ψ −1 ))(ψ(m)) = (∇(g ◦ ϕ ◦ ψ −1 ))(ψ(m)) = (∇g)(ϕ ◦ ψ −1 ◦ ψ(m)) · Jϕ◦ψ −1 (ψ(m)) = (∇g)(ϕ(m)) · Jϕ◦ψ −1 (ψ(m)) = 0, for (∇g)(ϕ(m)) = 0.

Example 2.1.4. (i). Any constant function on M is trivially horizontal at every point of M. The same is true for a function vanishing near m. (ii). Let m ∈ M, and fix any coordinate system (U, ϕ) with m ∈ U . Let i, j ∈ {1, . . . , N } be chosen. Consider the relevant coordinate functions xi , xj , i.e. xi = πi ◦ ϕ and xj = πj ◦ ϕ on U . Denote x 0 := ϕ(m). We show that f := (xi − xi0 ) (xj − xj0 ) is horizontal at m. By (III) of Remark 2.1.3, in order to prove that f is horizontal at m, it suffices to show that ∇(f ◦ ϕ −1 )(ϕ(m)) = 0. But this is obvious since, for every u ∈ ϕ(U ), (f ◦ ϕ −1 )(u) = (πi ◦ ϕ − xi0 ) (πj ◦ ϕ − xj0 ) (ϕ −1 (u)) = (ui − xi0 ) (uj − xj0 ), and trivially

∂ (ui − xi0 )(uj − xj0 ) = 0 ∂ uk x 0

for every k = 1, . . . , N . (iii). If f is horizontal at m ∈ M, f (m) = 0 and g is any smooth function in a neighborhood of m, then f g is horizontal at m ∈ M. Indeed, let (U, ϕ) be a coordinate system with m ∈ M. By (IV) of Remark 2.1.3, we need to show that ∇((f g) ◦ (ϕ −1 ))(ϕ(m)) = 0. This is obvious since ∇ (f g) ◦ (ϕ −1 ) = (g ◦ ϕ −1 ) ∇ f ◦ (ϕ −1 ) + (f ◦ ϕ −1 ) ∇ g ◦ (ϕ −1 ) and recalling that ∇(f ◦ ϕ −1 )(ϕ(m)) = 0 (for f is horizontal in m) and f (m) = 0 by hypothesis.

As we shall see in the proof of Proposition 2.1.7, the functions in (i), (ii), (iii) of Example 2.1.4 are “infinitesimally” the only functions worth considering.


91

2.1.2 Tangent Vectors Definition 2.1.5 (Tangent vector, space and bundle). Let M be an N -dimensional C ∞ differentiable manifold. A tangent vector v at m ∈ M is a linear functional, defined on the collection of the real-valued functions C ∞ in some neighborhood of m, such that v(f ) = 0 whenever f is horizontal in m. We denote by Mm the set of the tangent vectors at m ∈ M, and we say that Mm is the tangent space to M at m. We finally set

T (M) := (2.1) {m} × Mm = (m, v) : m ∈ M, v ∈ Mm . m∈M

T (M) is called the tangent bundle to M. Remark 2.1.6. Suppose f is a real-valued C ∞ function defined in a neighborhood of m ∈ M. Hence, there exists a coordinate system (U, ϕ) such that m ∈ U , and f is defined on U . Consider the open set ϕ(U ) ⊆ RN . For n ∈ {1, 2, 3, 4}, denote by Bn the Euclidean ball centered at ϕ(m) with radius n ε, where ε > 0 is small enough, so that B4 is contained in ϕ(U ). Let χ be a smooth cut-off function on RN such that χ ≡ 1 on B1 and χ ≡ 0 on RN \ B2 . Then, consider the function on M χ(ϕ(n)) f (n) if n ∈ ϕ −1 (B3 ), f (n) := n ∈ M. 0 otherwise, It is easy to see that f ≡ f on the set ϕ −1 (B1 ), open neighborhood of n in M, and that ϕ is defined and smooth on M (note that χ ◦ ϕ is C ∞ on U by Remark 2.1.3-(I)). Since f − f vanishes in a neighborhood of m, by Example 2.1.4-(i), we have v(f − f ) = 0 for every tangent vector v at m. Hence, by linearity, v(f ) = v(f ). This shows that in the definition of vector field it is not restrictive to suppose that a tangent vector at a point is a linear functional defined on the collection of the real-valued functions C ∞ in M. We remark that Mm is a vector space with the usual operations of sum of functionals and multiplication of a functional times a scalar factor. Denoting by dim M the dimension N of the differentiable manifold M, we have the following result. (Note. The reader will recognize in the proof below that a tangent vector at a point is indeed a linear “differential” operator of the first order defined on the C ∞ functions on M.) Proposition 2.1.7. Let M be an N-dimensional C ∞ differentiable manifold. Then dim(Mm ) = N = dim M. Proof. Let v ∈ Mm . Let (U, ϕ) be a coordinate system such that m ∈ U . Let f be an arbitrary C ∞ function defined in a neighborhood of ϕ(m). We set

92


v(f) := v(f ),

where f := f◦ ϕ.

By Remark 2.1.3-(I), v is meaningful, and it is easily seen that v is a linear functional defined on the functions C ∞ in a neighborhood of x 0 := ϕ(m). In particular, v (precisely, a suitable restriction of v) is linear on the vector space V of the real-valued linear functions a(x) defined on RN (where N := dim M; note that V = (RN )∗ , the usual dual space of RN ). Now, any linear functional on V is uniquely represented by a vector λ ∈ RN , i.e. v(a(u)) = a( λ). Let now f = f(u) be any function C ∞ in a neighborhood of x 0 . By Taylor expansion with the integral remainder, we have f(u) = f(x 0 ) + ∇ f(x 0 ), u − x 0 1 N

∂ 2 f 0 + x + t (u − x 0 ) dt (ui − xi0 )(uj − xj0 ) (1 − t) ∂ui ∂uj 0 i,j =1

=: f(x 0 ) − a(x 0 ) + a(u) + R(u) for every u near x 0 . By Example 2.1.4, we see that f(x 0 ) − a(x 0 ) and R ◦ ϕ are horizontal in m. Indeed, the summand in the (i, j )-sum defining R ◦ ϕ is the product of (ui − xi0 )(uj − xj0 ) ◦ ϕ = (xi − xi0 )(xj − xj0 ) (where xi , xj are the relevant coordinate functions), which is horizontal and vanishing in m, times the smooth function near m

1

U n → g(n) := 0

(1 − t)

∂ 2 f 0 x + t (ϕ(n) − x 0 ) dt. ∂ui ∂uj

We now set f = f◦ ϕ, so that f = f(x 0 ) − a(x 0 ) + a ◦ ϕ + R ◦ ϕ = a ◦ ϕ + {a horizontal function in m}. As a consequence of the definition of tangent vector, we infer v(f ) = v(a ◦ ϕ) = v (a ◦ ϕ) ◦ ϕ −1 λ. = v(a(u)) = ∇(f ◦ ϕ −1 )(x 0 ), Now, it is easy (and is left as an exercise) to prove that the map Mm v → λ ∈ RN is linear, injective and surjective. This completes the proof.

(2.2)


93

Remark 2.1.8. It is easy to see that v ∈ Mm if and only if for every coordinate map (U, ϕ) with m ∈ U (or, equivalently, for at least one such coordinate map) there exists λ = λ(ϕ, m) ∈ RN such that v(f ) = ∇(f ◦ ϕ −1 )(ϕ(m)), λ for every f , real valued and C ∞ near m. The “only if” part follows from the proof of Proposition 2.1.7 (see in particular (2.2)). Suppose now that there exists (U, ϕ) and λ as above. Let f be horizontal in m ∈ M. We need to show that ∇(f ◦ ϕ −1 )(ϕ(m)), λ = 0. By Remark 2.1.3, the function f = f ◦ ϕ −1 satisfies (∇ f)(ϕ(m)) = 0. This definitely suffices for what we needed to prove.

Remark 2.1.9. Remark 2.1.8 furnishes another very useful characterization of tangent vectors (actually, an alternative definition frequently used in literature). If m ∈ M, then v ∈ Mm if and only if there exists a C ∞ curve on M passing through m at t0 such that d v(f ) = (f (μ(t))) for every f , real valued and C ∞ near m. d t t0 Indeed, if v is defined in this way, we have, for any given coordinate map (U, ϕ), m ∈ U, d d (f ◦ ϕ −1 ) ◦ (ϕ ◦ μ) (t) (f (μ(t))) = v(f ) = dt t d t t0 0 d = ∇(f ◦ ϕ −1 )(ϕ(m)), (ϕ ◦ μ)(t0 ) , dt whence v satisfies the condition in Remark 2.1.8 with λ=

d (ϕ ◦ μ)(t0 ). dt

Vice versa, suppose v ∈ Mm . Fix any coordinate map (U, ϕ) with m ∈ U . Then, again by Remark 2.1.8, there exists λ ∈ RN such that ( ) v(f ) = ∇(f ◦ ϕ −1 )(ϕ(m)), λ for every f , real valued and C ∞ near m. Consider the curve on M given by (see (II) in Remark 2.1.3) μ(t) := ϕ −1 (ϕ(m) + t λ), |t| < ε, with ε > 0 small enough. Obviously, μ passes through m at t = 0. Moreover,

94


d d (f ◦ ϕ −1 ) ◦ (ϕ ◦ μ) (t) (f (μ(t))) = dt 0 dt 0 d = ∇(f ◦ ϕ −1 )(ϕ(m)), ϕ(m) + t λ (0) dt 0 −1 = ∇(f ◦ ϕ )(ϕ(m)), λ . Comparing to ( ), we have proved our assertion.

Definition 2.1.10 (Partial derivatives on M). Let M be an N -dimensional C ∞ differentiable manifold. Let (U, ϕ) be a coordinate system with coordinate functions x1 , . . . , xN (xi := πi ◦ ϕ), and let m ∈ U . For every i ∈ {1, . . . , N } we define a tangent vector, denoted ∂ ∈ Mm , ∂x i m

by setting

∂ ∂ (f ) := (f ◦ ϕ −1 )(ξ ) ∂ x i m ∂ ξi ϕ(m)

(2.3)

for every C ∞ function f defined in a neighborhood of m. We remark that f → ((∂/∂ xi )|m )(f ) actually defines an element of Mm (see Remark 2.1.8 with λ = the i-th element of the canonical basis of RN ). Note that the definition of (∂/∂ xi )|m is not coordinate-free: despite the notation, forgetful of ϕ, it depends on the coordinate map ϕ, as xi itself. Example 2.1.11. With the notation of the above definition, ∂ (xj ) = δi,j (of Kronecker). ∂ xi m Indeed, xj ◦ ϕ −1 = πj , so that (2.3) gives ∂ ∂ (x ) = (ξj ) = δi,j . j ∂ x i m ∂ ξi ϕ(m) Remark 2.1.12. (i). If v ∈ Mm , then (by collecting together Remark 2.1.8 and Example 2.1.11) we have the following suggesting formula: v=

N

i=1

∂ v(xi ) · , ∂ x i m

xi := πi ◦ ϕ,

and then (∂/∂ x1 )|m , . . . , (∂/∂ xN )|m is a basis for Mm . (ii). In particular, two tangent vectors v, w ∈ Mm coincide if and only if

(2.4)


v(πi ◦ ϕ) = w(πi ◦ ϕ)

95

∀ i = 1, . . . , N,

for every (or, equivalently, for at least one) coordinate system (U, ϕ) such that m ∈ U. (iii). The above formula (2.4) allows to represent a tangent vector in a very explicit way, once a coordinate system has been fixed. Indeed, if ϕ : U −→ RN is a coordinate map, ϕi is the i-th component of ϕ, (ξ1 , . . . , ξN ) are the coordinates on RN , m ∈ M, f is a C ∞ function defined in U and v ∈ Mm , then we have v(f ) =

N

i=1

v(ϕi ) ·

∂(f ◦ ϕ −1 ) (ϕ(m)). ∂ ξi

(2.5)

2.1.3 Differentials Definition 2.1.13 (Differential at a point). Let ψ : M −→ M be a C ∞ map between two differentiable manifolds, and let m ∈ M. The differential of ψ at m is the linear map dm ψ : Mm −→ Mψ(m) defined as follows: if v ∈ Mm , dm ψ(v) is the tangent vector in Mψ(m) acting in the following way: if f is a C ∞ function in a neighborhood of ψ(m), we set dm ψ(v) (f ) := v(f ◦ ψ). (2.6)

Even if many authors use to write dψ instead of dm ψ, we shall keep the notation well distinguished, preserving “dψ” for a suitable further notion. Remark 2.1.14. We verify that (2.6) actually defines a tangent vector at ψ(m). Let f be a C ∞ function in a neighborhood of ψ(m), and let (U, ϕ), (U , ϕ ) be coordinate systems in M and M , respectively, with m ∈ U , ψ(m) ∈ U . By Remark 2.1.8, there exists λ ∈ RN such that v(g) = ∇(g ◦ ϕ −1 )(ϕ(m)), λ for every g, C ∞ near m. Hence, we have (we denote λ as a column vector) dm ψ(v) (f ) = v(f ◦ ψ) = v f ◦ (ϕ )−1 ◦ ϕ ◦ ψ ◦ ϕ −1 ◦ ϕ = ∇ f ◦ (ϕ )−1 ◦ ϕ ◦ ψ ◦ ϕ −1 (ϕ(m)), λ = ∇ f ◦ (ϕ )−1 ϕ (ψ(m)) · Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · λ = ∇ f ◦ (ϕ )−1 ϕ (ψ(m)) , Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · λ . This shows that dm ψ(v) (f ) = ∇ f ◦ (ϕ )−1 ϕ (ψ(m)) , λ ,

(2.7a)

96


where

λ = Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · λ.

(2.7b)

Again thanks to Remark 2.1.8, this proves that dm ψ(v) is a tangent vector at ψ(m).

Remark 2.1.15. We remark that, by (2.4), (2.7a) and (2.7b), if (U, x1 , . . . , xN ) and (U , y1 , . . . , yN ) are coordinate systems at m ∈ M and ψ(m) ∈ M , respectively, we have N ∂ ∂ ∂ = (y ◦ ψ) · . (2.8) dm ψ j ∂ x i m ∂ x i m ∂ yj ψ(m) j =1

Indeed, (2.8) follows from (2.7b) by taking f = πj ◦ ϕ , λ = the i-th element of the standard basis of RN and by recalling that yj ◦ ψ = πj ◦ ϕ ◦ ψ and ∂ (yj ◦ ψ) = ∂i (πj ◦ ϕ ◦ ψ ◦ ϕ −1 ) (ϕ(m)) = Jϕ ◦ψ◦ϕ −1 (ϕ(m)) . j,i ∂ xi m Remark 2.1.16 (Transformation of tangent vectors via a differential). In general, a vector field v at m transforms under the differential of ψ as follows: N

∂ dm ψ v(πi ◦ ϕ) · ∂ x i m i=1 N N

∂ −1 = v(πi ◦ ϕ) · ∂i πj ◦ ϕ ◦ ψ ◦ ϕ ∂ yj ψ(m) j =1

=

N

j =1

where

i=1

Jϕ ◦ψ◦ϕ −1 (ϕ(m)) · v(ϕ)

j

∂ , ∂ yj ψ(m)

(2.9)

⎛

⎞ v(π1 ◦ ϕ) ⎜ ⎟ .. v(ϕ) = ⎝ ⎠ . v(πN ◦ ϕ)

and (U, ϕ), (U , ϕ ) are, respectively, coordinate systems at m ∈ M, ψ(m) ∈ M (M, M are differentiable manifolds of dimensions N, N , respectively). Remark 2.1.17 (Differential of the composition and of the inverse). Let ψ : M −→ M , φ : M −→ M be C ∞ maps between differentiable manifolds M, M , M . Then it is easily seen that φ ◦ ψ is C ∞ , and we have dm (φ ◦ ψ) = dψ(m) φ ◦ dm ψ

∀ m ∈ M.

(2.10)

Furthermore, let ψ : M −→ M be a C ∞ map between two differentiable manifolds. Suppose ψ −1 : M −→ M is a C ∞ map too. In this case, we say that ψ is a diffeomorphism.


97

It is immediate to observe that if ψ is a diffeomorphism and m ∈ M, then dm ψ is also invertible (for every m ∈ M) and dψ(m) (ψ −1 ) : Mψ(m) −→ Mm

is the inverse function of

. dm ψ : Mm −→ Mψ(m)

Definition 2.1.18 (dψ as a map on the tangent bundle). Let ψ : M → M be a C ∞ map between two differentiable manifolds M, M . We set dψ : T (M) → T (M ),

dψ(m, v) := (ψ(m), dm ψ(v)).

(2.11)

Note that, whereas dm ψ is a map from Mm to Mψ(m) (for any fixed m ∈ M), dψ is a map from T (M) to T (M ).

2.1.4 Vector Fields The following definition is one of the most important in differential geometry. Definition 2.1.19 (Vector field). Let Ω ⊆ M be an open subset of a differentiable manifold M. A vector field X on Ω is an application X : Ω −→ T (M) such that, X(m) = (m, v(m)) ∈ T (M)

∀ m ∈ Ω.

Equivalently, we have X(m) = (m, v(m)),

where v(m) ∈ Mm for every m ∈ Ω.

In order to avoid any confusion, we make explicit the following conventional identification of a vector field with its projection onto the second argument. This identification is frequently tacitly employed in literature. Convention–Notation. If T (M) is the tangent bundle of a differentiable manifold M, and, for every m ∈ M, v ∈ Mm , we set π(m, v) := v, then the following map is well posed on T (M): π : T (M) → Mm , (m, v) → v. m∈M

In the sequel, if X is a vector field on an open set Ω ⊆ M, we shall use the notation X(m) for the map X : Ω → T (M), m → X(m), whereas Xm will denote the map

98


Ω→

Mm ,

m → Xm := (π ◦ X)(m).

m∈M

So, the above positions can be summarized as X(m) = (m, Xm )

for every m ∈ M.

(2.12)

Finally, if f is a C ∞ function on Ω and X is a vector field on Ω, we shall denote (with an abuse of notation) by X(f ) or shortly Xf the function on Ω whose value at m is Xm (f ), i.e. (2.13) Xf : Ω → R, (Xf )(m) := Xm (f ). Definition 2.1.20 (Smooth vector field). Let X be a vector field defined on a manifold M. We say that X is C ∞ (or smooth) if, for every open set Ω ⊆ M and every smooth real-valued function f on Ω, the function Xf as defined in (2.13) is smooth on Ω. Remark 2.1.21. It is straightforward to verify that X is a smooth vector field on M if and only if, for every coordinate system (U, x1 , . . . , xN ), the functions a1 , . . . , aN defined on U by N

∂ Xm = ai (m) · ∂ x i m i=1

C∞

(see (2.12) and (2.4)), are functions on U . Following the above notation, we have ai (m) = Xm (xi ),

where xi = πi ◦ ϕ.

Recalling (2.5), we see that, if (U, ϕ) is a coordinate system, a smooth vector field acts on a function f ∈ C ∞ (U ) in the following way: Xm (f ) =

N

ai (m) · ∂i (f ◦ ϕ −1 ) (ϕ(m))

i=1

=

N

Xm (πi ◦ ϕ) · ∂i (f ◦ ϕ −1 ) (ϕ(m)),

(2.14)

i=1

where a1 , . . . , aN ∈ C ∞ (U ) are fixed functions depending on X and on the coordinate map. This shows that a vector field X on M is smooth iff, for every coordinate system (U, ϕ), the functions m → Xm (π1 ◦ ϕ), . . . , Xm (πN ◦ ϕ) are C ∞ on U .

Remark 2.1.22 (Smooth vector fields as operators on C ∞ (M, R)). Let X be a smooth vector field on a differentiable manifold M. Besides a map from M to T (M), it is possible to identify X with the map


X : C ∞ (M, R) → C ∞ (M, R),

99

f → Xf,

where (see (2.13)) Xf : M → R,

m → (Xf )(m) = Xm f.

We denote by X (M) the set of the smooth vector fields considered as linear operators (i.e. endomorphisms) on C ∞ (M, R). Note that X (M) is a vector space over R. The correspondence between X as a vector field on M and as an endomorphism on C ∞ (M, R) is faithful in the sense that, if X, Y are vector fields such that Xf ≡ Yf for every f ∈ C ∞ (M, R), then X = Y . (Indeed, take any m ∈ M, any coordinate system (U, ϕ) around m, any i ∈ {1, . . . , N } and choose f = πi ◦ ϕ. Then apply Remark 2.1.12-(ii).) Remark 2.1.23. Definition 2.1.13 defines dm ψ as a map from Mm to Mψ(m) . Moreover, in Definition 2.1.18, we introduced a natural map denoted by dψ between the tangent bundles T (M) and T (M ). It may be thought that a third natural map can be defined between X (M) and X (M ) by mapping X ∈ X (M) into the vector field Y such that Yψ(m) = dm ψ(Xm ). Unfortunately, in general, this defines a “vector field” only on the points of ψ(M) and not on the whole M . However, if ψ is a diffeomorphism, this can be done as we describe below.

Definition 2.1.24 (dψ as a map on X (M)). Suppose ψ : M → M is a C ∞ diffeomorphism of differentiable manifolds M, M . We set dψ : X (M) −→ X (M ),

X → dψ(X),

where, for every f ∈ C ∞ (M , R), {dψ(X)}m (f ) = dψ −1 (m ) ψ(Xψ −1 (m ) )(f )

∀ m ∈ M .

(2.15a)

Since ψ is onto, (2.15a) is equivalent to set {dψ(X)}ψ(m) = dm ψ(Xm ) for every m ∈ M.

(2.15b)

We say that X ∈ X (M) and the above defined dψ(X) ∈ X (M ) are ψ-related (see also Definition 2.1.34). We leave to the reader the verification that dψ(X) is indeed a smooth vector field on M according to the definition of X (M ) in Remark 2.1.22. The following example shows that the above mapping X → dψ(X) appears naturally when a “change of variable” occurs. Example 2.1.25 (Related vector fields via a diffeomorphism). We consider a simple but significant example. Let T : RN → RN be a C ∞ -diffeomorphism (i.e. an invertible map, smooth together with its inverse function) from RN onto itself. We consider on the domain of T a fixed system of Cartesian coordinates (x1 , . . . , xN ), and we equip the image of T (which still coincides with RN = T (RN )) with the new system of coordinates y defined by y = T (x). We use the following notation

100


T : (RN , x) → (RN , y). on To every smooth vector field X on (RN , x), there corresponds a vector field X (RN , y) in a natural way, roughly speaking, by expressing X in the new coordinates y. Namely, given any f = f (y), f ∈ C ∞ ((RN , y), R), we set y (f ) := XT −1 (y) (f ◦ T ), X also written as or equivalently,

(y) := X(f ◦ T )(T −1 (y)), Xf )(T (x)) = X(f ◦ T )(x). (Xf

(2.16)

or that X and X are T -related. With reference to (2.16), we say that T turns X into X, is the representation1 of X in the new system of coordinates Roughly speaking, X defined by y = T (x). 1 A simple example is in order. Let R2 be equipped with coordinates x = (x , x ) and 1 2

consider the linear change of coordinates given by y = (y1 , y2 ) = T (x1 , x2 ) := (2x1 − x2 , −5x1 + 3x2 ). Following the above definition, the ordinary partial derivatives X1 := ∂x1 and X2 := ∂x2 1 and X 2 , respectively, where are turned into the operators X 1 f (y) = ∂ f (2x1 − x2 , −5x1 + 3x2 ) X ∂x1 −1 x=T

(y)

= 2∂y1 f (y1 , y2 ) − 5 ∂y2 f (y1 , y2 ), ∂ f (2x1 − x2 , −5x1 + 3x2 ) X2 f (y) = ∂x2 −1 x=T

(y)

= −∂y1 f (y1 , y2 ) + 3 ∂y2 f (y1 , y2 ). Since the Jacobian matrix of T equals JT (x1 , x2 ) =

2 −1 −5 3

,

2 are the vector fields whose component functions at y are respectively given 1 and X then X by 2 −1 1 · , JT (T −1 (y)) · X1 I (T −1 (y)) = −5 3 0 2 −1 0 · . JT (T −1 (y)) · X2 I (T −1 (y)) = −5 3 1 Hence ∂x1 is turned by T into 2 ∂y1 − 5 ∂y2 and ∂x2 is turned by T into − ∂y1 + 3 ∂y2 . Consequently, the ordinary Laplace operator Δ = (∂x1 )2 + (∂x2 )2 is turned by T into the following second order constant coefficient differential operator of elliptic type (which is a sub-Laplacian on (R2 , +)!)


101

If we write, as usual, X = ∇ · XI , the chain rule then gives (y) = ∇(f ◦ T )(T −1 (y)) · XI (T −1 (y)) Xf = (∇y f )(y) · JT (T −1 (y)) · XI (T −1 (y)). In other words, for every y ∈ (RN , y), one has )(y) y = ∇y · (XI X

)(y) = JT (T −1 (y)) · XI (T −1 (y)), with (XI

(2.17a)

or equivalently, for every x ∈ (RN , x), )(T (x)) T (x) = ∇T (x) · (XI X

)(T (x)) = JT (x) · XI (x). with (XI

(2.17b)

Keeping in mind (2.9) of Remark 2.1.16, (2.17b) shows that T (x) = dx T (Xx ), X

(2.17c)

which gives a significant interpretation of the differential map (at least in the present case when T is a diffeomorphism): Given a vector field X, dx T (Xx ) is the tangent vector at T (x) which represents Xx with respect to the change of variable y = T (x).

In what follows, we introduce an important definition. The adjectives “regular” and “smooth” will always mean “of class C ∞ ”. Definition 2.1.26 (Tangent vector to a curve). Let μ : [a, b] → M be a regular curve. The tangent vector to the curve μ at time t is defined by d ∈ Mμ(t) . μ(t) ˙ := dt μ (2.18) d r r=t Hence, fixed t ∈ [a, b], if f is C ∞ near μ(t), we have d (f (μ(r))). μ(t)(f ˙ )= d r r=t Remark 2.1.27. Note that, as Remark 2.1.9 shows, any tangent vector at a point of M can be represented as the tangent vector to a certain curve at a suitable time. Definition 2.1.28 (Integral curve). Let X be a smooth vector field on the differentiable manifold M. A regular curve μ : [a, b] −→ M is called an integral curve of X if μ(t) ˙ = Xμ(t) for every t ∈ [a, b]. (2.19) = 5(∂y1 )2 + 34(∂y2 )2 − 26 ∂y1 ∂y2 . Δ is simply the ordinary Laplace operator expressed in a new system of In other words, Δ coordinates y in R2 . In Chapter 16, we show that every second order constant coefficient differential operator of elliptic type in RN is simply the ordinary Laplace operator expressed in a suitable new system of coordinates via a linear change of basis.

102


More explicitly, (2.19) means that d (f (μ(r))) = X(f )(μ(t)) d r r=t for every smooth function f on M and every t ∈ [a, b]. We remark that if Xm = N i=1 ai (m)(∂/∂ xi )|m and (U, ϕ) is a coordinate system such that μ(t) ∈ U , then (2.19) is also equivalent to d

ϕi (μ(r)) = ai (μ(t)), d r r=t which becomes (setting γ (t) := ϕ(μ(t)) ∈ RN ) γ˙i (t) = (ai ◦ ϕ −1 )(γ (t)),

i = 1, . . . , N,

(2.20)

which is an ODE on RN . Equivalently, making more explicit the rôle of X, μ : [a, b] → M is an integral curve of X if and only if, whenever (U, ϕ) is a coordinate system and (t0 , t1 ) ⊂ [a, b] is such that μ(t) ∈ U for every t ∈ (t0 , t1 ), it holds μ(t) = ϕ −1 (γ (t)), where γ˙ (t) = Xϕ −1 (γ (t)) (ϕ)

∀ t ∈ (t0 , t1 )

and Xm (ϕ) = (Xm (π1 ◦ ϕ), . . . , Xm (πN ◦ ϕ)). Remark 2.1.29. By the above observation on the coordinate form of Definition 2.1.28 and by recalling the existence “in small” for smooth systems of ODE’s, we infer that, given any smooth vector field X on M and fixed any m ∈ M, there exists one and only one integral curve of X passing through m at time 0. Hence the following definition is well-posed. Definition 2.1.30 (Complete vector field). Let X be a smooth vector field on the differentiable manifold M. We say that X is complete if, for every m ∈ M, the integral curve μ of X such that μ(0) = m is defined on the whole R (i.e. its maximal interval of definition is R). 2.1.5 Commutators. ϕ-relatedness In the sequel, we denote by C ∞ (M, R) or, shortly, C ∞ (M) the set of the smooth real-valued functions defined on a differentiable manifold M. It is immediate to observe that if X is a smooth vector field on M and f ∈ C ∞ (M, R), we have Xf ∈ C ∞ (M, R). We explicitly recall that, here and in the sequel, we use the notation in (2.13): Xf : M → R, (Xf )(m) = Xm (f ). As a consequence, the following definition is well posed.


103

Definition 2.1.31 (Commutators). Let X and Y be smooth vector fields on a differentiable manifold M. We define a vector field on M (called the commutator of X and Y ) in the following way: [X, Y ] : M → T (M),

[X, Y ](m) := (m, [X, Y ]m ),

where [X, Y ]m (f ) := Xm (Yf ) − Ym (Xf ) for every m ∈ M and every f ∈

(2.21)

C ∞ (M, R).

Definition 2.1.31 is well posed as it follows from (i) in the proposition below. Proposition 2.1.32. If X, Y and Z are smooth vector fields on M, we have: (i) [X, Y ] is a smooth vector field on M; (ii) [X, Y ]m = −[Y, X]m for every m ∈ M; (iii) [[X, Y ], Z]m + [[Y, Z], X]m + [[Z, X], Y ]m = 0 for every m ∈ M. Proof (Sketch). The verification that these facts hold true, can be performed via a computation in a coordinate map. This reduces the above identities (ii) and (iii) to identities between usual differential operators in RN . For instance, let us prove (i). Let m ∈ M be fixed and choose any coordinate map (U, ϕ) such that m ∈ M. According to Definition 2.1.20, we have Xm (f ) =

N

Xm (πi ◦ ϕ) ∂i (f ◦ ϕ −1 ) (ϕ(m)).

i=1

As a consequence, Xm (Yf ) =

N

i,j =1

=

N

∂ Yϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ ) (ϕ(m)) Xm (πi ◦ ϕ) ∂ ξi Xm (πi ◦ ϕ) · ∂ξi Yϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ )

i,j =1

+ Yϕ −1 (ξ ) (πj ◦ ϕ)

∂2 (f ◦ ϕ −1 )(ξ ) . ∂ ξi ξj ξ =ϕ(m)

By reversing the rôles of X and Y and then subtracting, we obtain (note, before the second equality sign, the cancellation of the second order terms) Xm (Yf ) − Ym (Xf ) =

N

i,j =1

Xm (πi ◦ ϕ) ∂ξi Yϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ ) ξ =ϕ(m)

104


−

N

i,j =1

+

N

i,j =1

−

N

i,j =1

=

N

j =1

Ym (πi ◦ ϕ) ∂ξi Xϕ −1 (ξ ) (πj ◦ ϕ) ∂ξj (f ◦ ϕ −1 )(ξ ) ξ =ϕ(m) 2 ∂ −1 Xm (πi ◦ ϕ) Ym (πj ◦ ϕ) (f ◦ ϕ ) (ϕ(m)) ∂ ξi ξj 2 ∂ Ym (πi ◦ ϕ) Xm (πj ◦ ϕ) (f ◦ ϕ −1 ) (ϕ(m)) ∂ ξi ξj

N

Xm (πi ◦ ϕ) ∂ξi Yϕ −1 (ξ ) (πj ◦ ϕ) (ϕ(m))

i=1

− Ym (πi ◦ ϕ) ∂ξi Xϕ −1 (ξ ) (πj ◦ ϕ) (ϕ(m)) ∂ξj (f ◦ ϕ −1 )(ϕ(m)). Thus, [X, Y ]m is actually a tangent vector at m (see, for instance, Remark 2.1.8), and, following the notation in Definition 2.1.20, we have proved that, in a coordinate system (U, ϕ) around m, [X, Y ]m (f ) =

N

cj (m) ∂j (f ◦ ϕ −1 )(ϕ(m))

with

j =1

cj (m) = [X, Y ]m (πj ◦ ϕ) = Xm Y (πj ◦ ϕ) − Ym X(πj ◦ ϕ)

∀ j ≤ N. (2.22)

We see that the maps cj ’s are smooth on U , so that, by Remark 2.1.21, [X, Y ] is a smooth vector field on M. Now, (iii) in the assertion can be proved with an analogous coordinate-computation, whereas (ii) is trivial.

Remark 2.1.33 (Commutators in X (M)). Consider the alternative definition of smooth vector field as an element of X (M), see Remark 2.1.33. The commutator operation rewrites as an operation on X (M) in the following way: Given X, Y ∈ X (M), we consider the operator on C ∞ (M, R) defined by [X, Y ] : C ∞ (M, R) → C ∞ (M, R),

f → [X, Y ]f,

where ([X, Y ]f )(m) := [X, Y ]m f = Xm (Yf ) − Ym (Xf ). Then, obviously, [X, Y ] ∈ X (M) is the operator on C ∞ (M, R) related to the (usual) vector field [X, Y ]. With this meaning of the commutation, Proposition 2.1.32 rewrites as: If X, Y and Z belong to X (M), we have: (i) [X, Y ] ∈ X (M); (ii) [X, Y ] = −[Y, X]; (iii) [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.


105

be a C ∞ function between two Definition 2.1.34 (ϕ-relatedness). Let ϕ : M → M on M are called ϕ-related differentiable manifolds. The vector fields X on M and X if we have ◦ ϕ. dϕ ◦ X = X (2.23a) (see Definition 2.1.18) and, Here, dϕ is intended as a map from T (M) to T (M) as usual, the vector fields are maps from the underlying differentiable manifolds to the relevant tangent bundles. Hence, (2.23a) is intended as an equality of functions Hence, ϕ-relatedness is equivalent to saying that the following from M to T (M). diagram is commutative ϕ

M

M X

X dϕ

T (M)

T (M).

Since, for every m ∈ M, we have (dϕ ◦ X)(m) = dϕ(m, Xm ) = (ϕ(m), dm ϕ(Xm )), whereas

◦ ϕ)(m) = X(ϕ(m)) ϕ(m) ), (X = (ϕ(m), X

then (2.23a) can be rewritten as the collection of identities ϕ(m) ∀ m ∈ M dm ϕ(Xm ) = X between tangent vectors at ϕ(m), i.e. between two functionals on Finally, condition (2.23a) is equivalent to the following ones: ϕ(m) (f ), Xm (f ◦ ϕ) = X

(2.23b) R). C ∞ (M,

)◦ϕ X(f ◦ ϕ) = (Xf

(2.23c) (2.23d)

for every m ∈ M and for every f smooth in a neighborhood of ϕ(m) in M. Remark 2.1.35. When ϕ is a diffeomorphism, for every X ∈ X (M), there always ∈ X (M) which is ϕ-related to X (see Remark 2.1.23), namely, exists a (unique) X = dϕ(X). with the notation of Definition 2.1.24, X With reference to (2.17c), we have an interpretation of ϕ-relatedness when ϕ : are ϕ-related if and only if X is the RN → RN is a diffeomorphism: X and X expression of X in the new coordinates defined by the new Cartesian coordinates y = ϕ(x).

Remark 2.1.36 (ϕ-relatedness in X (M)). Consider the alternative definition of vector field as in Remark 2.1.22. Then, the notion of ϕ-relatedness of vector fields rewrites R), ∈ X (M) are ϕ-related if, for every f ∈ C ∞ (M, as follows: X ∈ X (M) and X the functions X(f ◦ ϕ), (Xf ) ◦ ϕ on M do coincide, i.e. ϕ(m) f for every m ∈ M, Xm (f ◦ ϕ) = X or, equivalently, (2.23b) holds. The following simple result will be soon of a crucial importance.

106


be a C ∞ Proposition 2.1.37 (ϕ-relatedness and commutators). Let ϕ : M → M function between two differentiable manifolds. Let X, Y be smooth vector fields on Y be smooth vector fields on M. If X is ϕ-related to X and Y is ϕ-related M and X, to Y , then [X, Y ] is ϕ-related to [X, Y ]. Proof. We have to prove Y ] ◦ ϕ, dϕ ◦ [X, Y ] = [X, and dϕ◦Y = Y ◦ϕ. To this end, let m ∈ M and f ∈ C ∞ (M). provided dϕ◦X = X◦ϕ By the equivalent restatement (2.23c) of ϕ-relatedness, we have to show that Y ]ϕ(m) (f ). ( ) dm ϕ [X, Y ]m (f ) = [X, By definition of dm ϕ, we have dm ϕ [X, Y ]m (f ) = [X, Y ]m (f ◦ ϕ) (see (2.21)) = Xm Y (f ◦ ϕ) − Ym X(f ◦ ϕ) f ) ◦ ϕ − Ym (Xf )◦ϕ (see (2.23d)) = Xm (Y (f ) − dm ϕ(Ym ) X(f ) = dm ϕ(Xm ) Y ϕ(m) Y (f ) − Y ϕ(m) X(f ) (see (2.23b)) = X Y ]ϕ(m) (f ). (see (2.21)) = [X, This gives ( ) thus ending the proof.

2.1.6 Abstract Lie Groups Definition 2.1.38 (Lie group). A Lie group G is a differentiable manifold G along with a group law ∗ : G × G −→ G such that the applications G × G (x, y) → x ∗ y ∈ G,

G x → x −1 ∈ G

are smooth.2 In the following, we shall always denote by e the identity of (G, ∗). Moreover, fixed σ ∈ G, we denote by τσ the left translation on G by σ , i.e. the map G x → τσ (x) := σ ∗ x ∈ G. In case, when more than only one composition law is involved, we may also write τσ∗ instead of τσ . 2 The notion of “smoothness” of a map from G × G should be properly defined. The prod-

uct of two differentiable manifolds is indeed endowed with a differentiable structure in a natural way. Here, we just remark that a function f : G × G → R is smooth if, for every couple of coordinate systems (U, ϕ) and (V , φ) on G, the function RN × RN ⊇ ϕ(U ) × φ(V ) (u, v) → f ϕ −1 (u) ∗ φ −1 (v) ∈ R is smooth. For more details see, e.g. [War83].


107

Definition 2.1.39 (Lie algebra). A (real) Lie algebra is a real vector space g with a bilinear operation [·, ·] : g × g −→ g (called (Lie) bracket) such that, for every X, Y , Z ∈ g, we have: 1. (anti-commutativity) [X, Y ] = −[Y, X]; 2. (Jacobi identity) [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0. A very remarkable fact is that, given any Lie group, there exists a certain finitedimensional Lie algebra such that the group properties are reflected into properties of the algebra. For instance (as we shall see later on), any connected and simply connected Lie group is completely determined (up to isomorphism) by its Lie algebra. Therefore, the study of a Lie group is often reduced to the study of its Lie algebra. Remark 2.1.40. Straightforwardly adapting the proof of Proposition 1.1.7 in Section 1.1 (see page 12), we have: if X1 , . . . , Xm are elements of an (abstract) Lie algebra, then a system of generators of Lie{X1 , . . . , Xm } is given by the commutators XI := [Xi1 , [Xi2 , [Xi3 , . . . [Xik−1 , Xik ] . . .]]], where {i1 , i2 , . . . , ik } ⊆ {1, . . . , m} and I = (i1 , i2 , . . . , ik ), k ∈ N. Indeed, the proof of Proposition 1.1.7 is only based on anti-commutativity and the Jacobi identity. 2.1.7 Left Invariant Vector Fields and the Lie Algebra Definition 2.1.41 (Left invariant vector fields). Let G be a Lie group. A smooth vector field X on G is called left invariant if, for every σ ∈ G, X is τσ -related to itself, i.e. (2.24) dτσ ◦ X = X ◦ τσ . Here dτσ is intended as a map from T (G) to itself (as in Definition 2.1.18). As shown by the remarks after Definition 2.1.34, condition (2.24) is equivalent to the following one: ∀ x, σ ∈ G. (2.25a) (dx τσ )(Xx ) = Xσ ∗x Applying (2.25a) at the identity e, it follows immediately that if X is a left invariant vector field, we have ∀ σ ∈ G, (2.25b) de τσ (Xe ) = Xσ which proves that a left invariant vector field is determined by its action at the origin. Equality (2.25b) is actually equivalent3 to (2.25a). Moreover, (2.25a) can also be written as 3 Indeed, by applying (2.25b) with σ replaced by σ ∗ x, subsequently (2.10) and finally using

again (2.25b) with σ replaced by x, we have Xσ ∗x = de τσ ∗x (Xe ) = dx τσ de τx (Xe ) = dx τσ (Xx ).

108


Xx (f ◦ τσ ) = Xσ ∗x (f ) for every x, σ ∈ G and every f ∈ C ∞ (G, R), or again as (the most commonly used) Xx y → f (σ ∗ y) = (Xf )(σ ∗ x),

(2.25c)

(2.25d)

i.e. comparing it to (1.18) (page 14), the very analogue of the usual left-invariance. Before giving the following central Definition 2.1.42, we pause a moment in order to recall the multiple ways a smooth vector field can be thought of. A smooth vector field on G is a map X : G → T (G) such that, for every x ∈ G, it holds X(x) = (x, Xx ), where Xx ∈ Gx for every x ∈ G and such that, for every f ∈ C ∞ (G, R), the function x → Xx (f ) is smooth on G. A smooth vector field can be identified to the operator X : C ∞ (G, R) −→ C ∞ (G, R), f → Xf : G → R, x → Xx f. The set of the vector fields, as the above described operators, is denoted by X (G). Obviously, the set of the left invariant operators on G gives rise to a relevant subset in X (G), following the above identification. We are ready to give the following central definition. Definition 2.1.42 (Algebra of a Lie group). Let G be a Lie group. Then the subset of X (G) of the smooth left invariant vector fields on G is called the (Lie) algebra of G. It will be denoted by g. More precisely, following Remark 2.1.22, we henceforth identify a left invariant vector field X on G with the following operator X : C ∞ (G, R) → C ∞ (G, R) such that, for every f ∈ C ∞ (G, R), the function Xf on G is defined by (Xf )(x) := Xx f

∀ x ∈ G.

Hence, g is a (linear) set of endomorphisms on C ∞ (G, R), g ⊆ X (G). Note that, from the left invariance of X ∈ g, we have (Xf )(x) = X(f ◦ τx )(e)

∀ x ∈ G ∀ f ∈ C ∞ (G, R).

Along with the above definition of the algebra of a Lie group, there is a wide commonly used identification of g with Ge described in the following theorem. Theorem 2.1.43 (The Lie algebra of a Lie group). Let G be a Lie group and g be its algebra. Then we have:


109

(i) g is a vector space, and the map α : g −→ Ge , X → α(X) := Xe is an isomorphism between g and the tangent space Ge to G at the identity e of G. As a consequence, dim g = dim Ge = dim G; (ii) The commutator of smooth left invariant vector fields (see also Remark 2.1.33) is a smooth left invariant vector field; (iii) g with the commutation operation is a Lie algebra. Proof. (i). It is evident that g (thought of as a subset of X (G)) is a vector space and that α is linear. Let us prove that α is injective. If α(X) = α(Y ), we have (see (2.25b)) Xσ = dτσ (Xe ) = dτσ (α(X)) = dτσ (α(Y )) = dτσ (Ye ) = Yσ , i.e. X = Y . Let us prove that α is surjective. If v ∈ Ge , we set Xσ := (de τσ )(v)

for every σ ∈ G.

Here, de τσ is the differential of the map τσ at the identity e of G. By definition, we have Xσ ∈ Gσ , i.e. σ → (σ, Xσ ) is a vector field on G. Moreover, it is not difficult4 to prove that X is smooth. On the other hand, X is left invariant since Xσ ∗x = de τσ ∗x (v) = (dx τσ ◦ de τx )(v) = dx τσ (Xx ), which gives (2.25a). Finally, we have 4 Indeed, let us prove that, for every f ∈ C ∞ (G, R), G σ → X f is smooth. Fix σ

σ0 ∈ G, and let (U, ϕ) be a coordinate system around σ0 . We have to prove that the map ϕ(U ) u → (Xf )(ϕ −1 (u)) is smooth. By definition, it holds (Xf )(ϕ −1 (u)) = Xϕ −1 (u) (f ) = (de τϕ −1 (u) )(v)(f ) = v(f ◦ τϕ −1 (u) ) = ().

By Remark 2.1.8, fixed a coordinate system (E, χ ) around e, there exists λ = λ(χ , e, v) ∈ RN such that ! " ! " () = λ, ∇ f ◦ τϕ −1 (u) ◦ χ −1 (χ (e)) = λ, ∇v f (ϕ −1 (u) ∗ χ −1 (v)) (χ (e)) . By the very definition of Lie group, ϕ(U ) × χ (E) (u, v) → f (ϕ −1 (u) ∗ χ −1 (v)) is smooth, so that ϕ(U ) u → ∇v f (ϕ −1 (u) ∗ χ −1 (v)) (χ (e)) is smooth, and this ends the proof that X is smooth.

110


α(X) = Xe = de τe (v) = v, i.e. α is surjective. Since α is an isomorphism of vector spaces, we have dim g = dim Ge = dim G (see Proposition 2.1.7). (ii). Let σ ∈ G be arbitrary. Let X and Y ∈ g, i.e. X is τσ -related to itself and Y is τσ -related to itself. From Proposition 2.1.37 it follows that [X, Y ] is τσ -related to itself, i.e. [X, Y ] ∈ g. (iii). The last statement of the theorem follows from Proposition 2.1.32 (see also Remark 2.1.33).

Incidentally, in the above proof, we have explicitly written the inverse map of the natural identification α : g −→ Ge , X → α(X) := Xe . Indeed, the inverse of α is given by α −1 : Ge −→ g, v → X, where Xσ = (de τσ )(v) for every σ ∈ G.

(2.26)

Example 2.1.44 (The Lie algebra of (R, +)). It is obvious that the Lie algebra r of the usual Euclidean Lie group (R, +) is d span , dr where

d : C ∞ (R, R) → C ∞ (R, R), f → f . dr With the usual formalism Xx for vector fields, this rewrites as d f = f (t) for every t ∈ R. d r t

Remark 2.1.45 (Left invariance in coordinates). We now make explicit the left invariance condition in terms of the coefficients ai of the coordinate form of a smooth vector field N

∂ ai (σ ) · . Xσ = ∂x i=1

i σ

We know that a smooth vector field X is left invariant on G if and only if (see (2.25b)) ( )

de τσ (Xe ) = Xσ

∀ σ ∈ G.

In turn (see Remark 2.1.21), fixed a coordinate system (E, χ) around the identity e, ( ) is equivalent to state that, for every σ ∈ G and for every (or, equivalently, for at least one) coordinate system (U, ϕ) around σ , it holds

2.1 Abstract Lie Groups N

111

∂ ∂ (f ◦ ϕ −1 )(ϕ(σ )) = Xe (χi ) · (f ◦ τσ ◦ χ −1 )(χ(e)) ∂ ui ∂ vi N

Xσ (ϕi ) ·

i=1

i=1

for every smooth function f around σ (or, equivalently, for the N smooth functions ϕ1 , . . . , ϕN ). Writing f = f ◦ ϕ −1 ◦ ϕ, on the right-hand side the above becomes N

Xσ (ϕi ) ·

i=1

=

N

j =1

=

N

j =1

=

N

Xe (χi )

i=1 N

i=1

N

j =1

∂ (f ◦ ϕ −1 )(ϕ(σ )) ∂ ui ∂ ∂ (ϕj ◦ τσ ◦ χ −1 )(χ(e)) · (f ◦ ϕ −1 )(ϕ(σ )) ∂ vi ∂ uj

∂ Jϕ◦τσ ◦χ −1 (χ(e)) Xe (χi ) · (f ◦ ϕ −1 )(ϕ(σ )) j,i ∂ uj

∂ Jϕ◦τσ ◦χ −1 (χ(e)) · Xe (χ) · (f ◦ ϕ −1 )(ϕ(σ )). j ∂ uj

Here, Xe (χ) = (Xe (χ1 ), . . . , Xe (χN ))T . Comparing the far left-hand and right-hand sides, we rewrite all as the matrix identity Jϕ◦τσ ◦χ −1 (χ(e)) · Xe (χ) = Xσ (ϕ).

(2.27)

Here, Xe (ϕ) = (Xe (ϕ1 ), . . . , Xe (ϕN ))T . As a consequence, we have proved that a smooth vector field X on the Lie group G is left invariant if and only if, fixed a coordinate system (E, χ) around the identity e, for every σ ∈ G, and for every (or, equivalently, for at least one) coordinate system (U, ϕ) around σ , it holds ∂ ai (σ ) , Xσ = ∂ x i σ i=1 ⎛ ⎞ ⎞ ⎛ a1 (σ ) Xe (χ1 ) ⎜ ⎟ ⎟ ⎜ .. where ⎝ ... ⎠ = Jϕ◦τσ ◦χ −1 (χ(e)) · ⎝ ⎠. . N

aN (σ )

(2.28)

Xe (χN )

Formula (2.28) is particularly useful for those Lie groups admitting a single global coordinate system defined on the whole group (such as Carnot groups or GLN (R), to give some example). Remark 2.1.46 (Bracket in the identity). Since the Lie algebra can be identified to Ge , it is interesting to analyze what form takes the bracket as seen as an operation on Ge . Let (E, χ) be a fixed coordinate system around the identity e ∈ G. We know that (see (2.14) and (2.22))

112


Xe f =

N

λj ∂j |χ(e) (f ◦ χ −1 ),

j =1

Ye f =

N

μj ∂j |χ(e) (f ◦ χ −1 ),

j =1

[X, Y ]e f =

N

νj ∂j |χ(e) (f ◦ χ −1 ),

j =1

where λj = Xe (χj ), μj = Ye (χj ), and νj = Xe (Y χj ) − Ye (Xχj )

λi ∂ui |χ(e) Yχ −1 (u) χj − μi ∂ui |χ(e) Xχ −1 (u) χj = i

i

(after a lengthy computation using also (2.28)) N

$ ∂ # −1 ∂ = χj χ (u) ∗ χ −1 (v) . (λi μk − μi λk ) ∂u ∂v i χ(e)

i,k=1

k χ(e)

2.1.8 Homomorphisms Definition 2.1.47 (Homomorphisms). Let (G, •) and (H, ∗) be Lie groups. A map ϕ : G −→ H is a homomorphism of Lie groups if it is C ∞ and if ϕ(x • y) = ϕ(x) ∗ ϕ(y)

∀ x, y ∈ G.

A map ϕ is an isomorphism of Lie groups if it is a homomorphism of Lie groups and a diffeomorphism of differentiable manifolds. An isomorphism of G onto itself is called an automorphism of G. Let (g, [·, ·]1 ) and (h, [·, ·]2 ) be Lie algebras. A map ϕ : g −→ h is a homomorphism of Lie algebras if it is linear and if ∀ X, Y ∈ g. ϕ [X, Y ]1 = [ϕ(X), ϕ(Y )]2 A map ϕ is an isomorphism of Lie algebras if it is a bijective homomorphism of Lie algebras. An isomorphism of g onto itself is called an automorphism of g. Example 2.1.48. Suppose (G, •) is a Lie group and M is a differentiable manifold. Suppose T : G → M is a C ∞ -diffeomorphism. We can consider on M a Lie group structure naturally induced by • via T . Precisely, we equip M with the following composition law (2.29) M × M (x, y) → x ∗ y := T T −1 (x) • T −1 (y) ∈ M. It is an easy exercise to verify that ∗ defines on M a Lie group structure, T : (G, •) → (M, ∗) is a Lie-group isomorphism, the identity of (M, ∗) is T (eG ) (eG being the identity of (G, •)) and, finally, the inverse of x in M with respect to ∗ is given by


x −1 = T

113

−1 . T −1 (x)

Now, let X be a •-left invariant vector field on G. We showed in Example 2.1.25 how on M simply setting (see, for instance (2.17c)) to relate to X a vector field X y = dT −1 (y) T (XT −1 (y) ) X

for every y ∈ M.

(2.30)

are T -related according to Definition 2.1.34. This obviously ensures that X and X Moreover, we claim that is ∗-left invariant. X (2.31) This is equivalent to σ ∗y (f ) y (f (σ ∗ ·)) = X X

∀ y, σ ∈ M

∀ f ∈ C ∞ (M, R).

Now, this last equality follows from the following calculation y (f (σ ∗ ·)) = X

(see (2.30)) = dT −1 (y) T (XT −1 (y) ) (f (σ ∗ ·)) = (XT −1 (y) ) f (σ ∗ T (·))

(see (2.29)) = (XT −1 (y) ) f T (T −1 (σ ) • (·))

(X is •-left inv.) = (XT −1 (σ )•T −1 (y) ) f T (·)

(see (2.29)) = (XT −1 (σ ∗y) ) f T (·) = dT −1 (σ ∗y) T (XT −1 (σ ∗y) ) (f ) σ ∗y (f ). (see (2.30)) = X Roughly speaking, this gives the following fact. Let T : RN → RN be a C ∞ -diffeomorphism defining on RN a change of variable y = T (x). Suppose the domain (RN , x) is equipped with a group law •, and define on the image (RN , y) the induced group law ∗ as in (2.29). Then, if X is a which expresses X left-invariant vector field w.r.t. • on (RN , x), the vector field X in the y-coordinates is a left-invariant vector field w.r.t. ∗ on (RN , y).

Let ϕ : G −→ H be a homomorphism of Lie groups. Since ϕ sends the identity of G to the identity of H, the differential de ϕ of ϕ is a linear map from Ge to He (for brevity we denote by e the identity element both in G and in H). Now, by means of the natural identification between Ge and the algebra g of G and between He and the algebra h of H (see Theorem 2.1.43), de ϕ thus induces a natural linear map between g and h, which we denote by dϕ. In other words, the following definition naturally arises. Definition 2.1.49 (Differential of a homomorphism). If ϕ : G −→ H is a homomorphism of Lie groups, we define dϕ : g −→ h

114


as the linear map such that, for every X ∈ g, dϕ(X) is the only element of h satisfying {dϕ(X)}eH = deG ϕ(XeG ) (∈ HeH ).

(2.32)

At this point, some confusion in the notation may occur. Indeed, Definition 2.1.49 defines the differential of a Lie-group homomorphism ϕ : G → H as a map from the Lie-algebra g to the Lie-algebra h; but we already defined (see Definition 2.1.18) dϕ as a map between the relevant tangent bundles, when ϕ is smooth, and dϕ (see Definition 2.1.24) as a map between the smooth vector fields as operators, when ϕ is a diffeomorphism. The following theorem shows that the three definitions are consistent (see also Remark 2.1.51). Theorem 2.1.50. Let G and H be Lie groups with associated algebras g and h, respectively. Let ϕ : G −→ H be a homomorphism of Lie groups. Then: (i) for every X ∈ g, we have that X and dϕ(X) are ϕ-related, i.e.

∀ x ∈ G; dϕ(X) ϕ(x) = dx ϕ(Xx )

(2.33)

Condition (2.33) characterizes dϕ(X), i.e. dϕ(X) is the only left invariant vector field Y ∈ h such that Yϕ(x) = dx ϕ(Xx ) for every x ∈ G; (ii) dϕ : g −→ h is a homomorphism of Lie algebras. := dϕ(X). We have to prove that (see ReProof. Let X ∈ g be arbitrary. Let X mark 2.1.36 and (2.23b)) ( )

ϕ(σ ) = dσ ϕ(Xσ ) X

∀ σ ∈ G.

H Let σ ∈ G. We shall use the notation τσG and τϕ(σ ) for the left translations on G and on H and the notation eG , eH for the identity elements in G and H, respectively. Moreover, the group laws on G and H will be respectively denoted by ◦G and ◦H . Since ϕ is a Lie group homomorphism, we have H H G G (τϕ(σ ) ◦ ϕ)(·) = ϕ(σ ) ◦ ϕ(·) = ϕ(σ ◦ ·) = (ϕ ◦ τσ )(·),

i.e. the following identity of maps on G holds H G τϕ(σ ) ◦ ϕ ≡ ϕ ◦ τσ .

and X are left invariant) As a consequence, we have (also recall that X ϕ(σ ) = X H (see (2.25b)) = deH τϕ(σ ) (XeH ) H H (see (2.32)) = deH τϕ(σ ) (deG ϕ(XeG )) = deG (τϕ(σ ) ◦ ϕ)(XeG )

(see (2.34)) = deG (ϕ ◦ τσG )(XeG ) = dσ ϕ(deG τσG (XeG )) (see (2.25b)) = dσ ϕ(Xσ ). This is precisely ( ), and (i) is proved.

(2.34)


115

Finally, we have to prove dϕ([X, Y ]) = [dϕ(X), dϕ(Y )]

∀ X, Y ∈ g.

By means of the first part (i) of the assertion, we have that X (resp. Y ) is ϕrelated to dϕ(X) (resp. dϕ(Y )). Thus, by Proposition 2.1.37, [X, Y ] is ϕ-related to [dϕ(X), dϕ(Y )], i.e. dx ϕ([X, Y ]x ) = [dϕ(X), dϕ(Y )]ϕ(x)

∀ x ∈ G.

In particular, for x = eG , we have deG ϕ([X, Y ]eG ) = [dϕ(X), dϕ(Y )]eH . Now, by definition, dϕ([X, Y ]) is the unique vector field in h whose value in eH is deG ϕ([X, Y ]eG ), then dϕ([X, Y ]) = [dϕ(X), dϕ(Y )], and the theorem is completely proved.

Remark 2.1.51 (Consistency of the notation “dϕ”). Let G and H be Lie groups with associated algebras g and h, respectively. Let ϕ : G → H be a homomorphism of Lie groups. In Definition 2.1.18, dϕ was defined as the map dϕ : T (G) → T (H),

dϕ(x, Xx ) = (ϕ(x), Yϕ(x) ),

where Yϕ(x) = dx ϕ(Xx ), Hence, by Theorem 2.1.50, dϕ, as defined in Definition 2.1.49, is just the “projection” of the previously defined dϕ on the second component. In case ϕ is also an isomorphism, we have defined dϕ in Definition 2.1.24. A comparison of (2.15b) and (2.33) shows that dϕ and dϕ of Definition 2.1.49 actually coincide.

Remark 2.1.52. (i) From Theorem 2.1.50 it immediately follows that if ϕ : G −→ H is a Lie group isomorphism then dϕ : g −→ h is a Lie algebra isomorphism. (ii) Let us now suppose we are given two Lie groups on RN as defined in Chapter 1: (G, ◦) and (H, ∗). Let the associated Lie-algebras be g and h, respectively. Suppose it is given a Lie group isomorphism ϕ from G to H. As usual, the identity elements in G and H are supposed to be the origin 0 of RN . Then dϕ sends the ◦Jacobian basis of g in a basis of h that in 0 coincides with the column vectors of the matrix Jϕ (0). Indeed, if Z1 , . . . , ZN is the Jacobian basis related to (G, ◦) and if f ∈ C ∞ (H, R), we have {dϕ(Zk )}0 (f ) = {d0 ϕ((Zk )0 )}(f ) = {Zk }0 (f ◦ ϕ) = ∂xk |0 (f ◦ ϕ) =

N

(∂ξj f )(0) · (∂xk ϕj )(0). j =1

In particular, dϕ sends the ◦-Jacobian basis of g in the ∗-Jacobian basis of h if and only if Jϕ (0) is the identity matrix.

116


(iii) Let F, G, H be three Lie groups with Lie algebras f, g, h. Let ϕ : F → G,

ψ :G→H

be Lie-group homomorphisms. Then d(ψ ◦ ϕ) = dψ ◦ dϕ. Indeed, for every X ∈ f, d(ψ ◦ ϕ)(X) is the only element of h such that ( ) d(ψ ◦ ϕ)(X) e = deF (ψ ◦ ϕ)(XeF ). H

But also (dψ ◦ dϕ)(X) when applied on eH coincides with the right-hand side of ( ). Indeed, (dψ ◦ dϕ)(X) e = dψ dϕ(X) e = deG ψ dϕ(X) e H H G = deG ψ deF ϕ(XeF ) = deF (ψ ◦ ϕ)(XeF ).

2.1.9 The Exponential Map We begin with a simple but crucial result on Lie groups, following from an ODE’s result. We recall that, according to Definition 2.1.30, a smooth vector field X on a Lie group G is complete if, for every x ∈ G, the integral curve μ of X such that μ(0) = x is defined on the whole R. Proposition 2.1.53 (Completeness of the left invariant vector fields). The left invariant vector fields on a Lie group G are complete. Proof. Let X ∈ g be fixed, g being the Lie algebra of G. By simple prolongation results for ODE’s, it is enough to prove that there exists ε = ε(X) > 0 such that, for every x ∈ G, the integral curve μx of X such that μx (0) = x is defined on [−ε, ε]. Let e be the identity of G. Let ε > 0 be such that the integral curve μe of X with μe (0) = e is defined on [−ε, ε]. Then, it is an immediate consequence of the left invariance of X to derive that the map [−ε, ε] t → (τx ◦ μe )(t) =: ν(t) ∈ G coincides with the above μx . Indeed, ν(0) = τx (μe (0)) = τx (e) = x and (see the Definition 2.1.26 for ν˙ (t)) d d = dt (τx ◦ μe ) ν˙ (t) = dt ν d r r=t d r r=t d = dμe (t) τx ◦ dt μe d r r=t = dμe (t) τx (μ˙e (t)) = dμe (t) τx (Xμe (t) ) (see (2.24)) = X(τx ◦μe )(t) = Xν(t) . This ends the proof.


117

Definition 2.1.54 (The exponential curve expX (t)). Let G be a Lie group with Lie algebra g. Let X ∈ g be fixed. By Proposition 2.1.53, the integral curve μ(t) of X passing through the identity of G when t = 0 is defined on the whole R. We set expX (t) := μ(t). By the very Definition 2.1.28 of integral curve, we have ⎧ ⎨ expX (0)= eG , d expX : R → G with = XexpX (t) ⎩ dt expX d r r=t

∀ t ∈ R.

(2.35)

In terms of functionals on C ∞ (G, R), (2.35) can be written more explicitly as

d f (expX (r)) = XexpX (t) (f ) ∀ f ∈ C ∞ (G, R). (2.36a) dr r=t In particular, when t = 0,

d f (expX (r)) = Xe (f ) dr r=0

∀ f ∈ C ∞ (G, R).

Again from (2.35) with t = 0 we infer d = Xe . d0 expX dt 0

(2.36b)

(2.36c)

Remark 2.1.55 (The exponential curve as a homomorphism). It is a simple exercise on ODE’s (see also Ex. 9 at the end of Chapter 1) to verify that expX : (R, +) −→ (G, ∗) is a Lie group homomorphism. In other words, it holds expX (r + s) = expX (r) ∗ expX (s)

for every r, s ∈ R.

Hence, Definition 2.1.49 can be applied. The differential of expX is the following map (here r is the Lie algebra of (R, +)) d expX : r −→ g

is such that λ

d → λ X dt

∀ λ ∈ R,

d d expX = X. (2.37) dt Indeed, by definition of the differential of a homomorphism, d expX (d/dt) is the unique vector field of g such that d d d expX . = d0 expX dt dt e 0 i.e. it holds

The above right-hand side equals Xe thanks to (2.36c), whence (2.37).

118


For future reference, we collect some other useful formulas for expX (t), immediate consequence of the facts proved above. Theorem 2.1.56. Let (G, ∗) be a Lie group with algebra g. Let X ∈ g. Then: (i) expX (r + s) = expX (r) ∗ expX (s) for every r, s ∈ R; (ii) expX (−t) = (expX (t))−1 for every t ∈ R; (iii) expX (0) = e; (iv) R t → expX (t) ∈ G is a smooth curve; (v) expX (t) is the unique integral curve of X passing through the identity at time zero, so that, for every x ∈ G, t → x ∗ (expX (t)) is the unique integral curve of X passing through x at time zero. Note that (i) in the above theorem jointly with the left invariance of X and (2.36c) gives back5 (2.36b). We are ready to give the fundamental definition. Definition 2.1.57 (Exponential map). Let (G, ∗) be a Lie group with Lie algebra g. Following the notation in Definition 2.1.54, we set Exp : g −→ G, X → Exp(X) := expX (1). Exp is called the exponential map (related to the Lie group G). The following results hold. Proposition 2.1.58. Let (G, ∗) be a Lie group with Lie algebra g. For every X ∈ g, we have (i) Exp(t X) = expX (t) for every t ∈ R; (ii) Exp((r + s)X) = Exp(r X) ∗ Exp(sX) for every r, s ∈ R; (iii) Exp(−tX) = (Exp(tX))−1 , for every t ∈ R. 5 Indeed, a direct computation gives

d dr

# $ f (expX (r)) =

r=t

=

d dr

d dr

r=0

# $ f (expX (r + t))

# $ f (expX (t) ∗ expX (r)) =

r=0

= d0 expX

d dr

d dr

# $ (f ◦ τexpX (t) )(expX (r))

r=0

(f ◦ τexpX (t) ) = Xe (f ◦ τexpX (t) ) = XexpX (t) (f ). r=0

Here we used (2.36c) and (2.25c).


119

Proof. Fix t ∈ R and consider the curve s → μ(s) := expX (s t). We have μ(0) = expX (0) = e. By Theorem 2.1.56, we have μ(s) ˙ = t XexpX (s t) = t Xμ(s) . Thus μ is the integral curve of tX, and, again by Theorem 2.1.56, we have μ(s) = exptX (s), i.e. expX (s t) = exptX (s). For s = 1, we have expX (t) = exptX (1) = Exp (t X). Therefore, this yields Exp ((r + s) X) = expX (r + s) = expX (r) ∗ expX (s) = Exp (r X) ∗ Exp (s X), and moreover, −1 −1 = Exp (t X) . Exp (−t X) = expX (−t) = expX (t) The proposition is thus completely proved.

Theorem 2.1.59. Let G and H be Lie groups with associated algebras g and h. We denote by ExpG and ExpH the exponential maps related to G and to H, respectively. Finally, let ϕ : G −→ H be a Lie group homomorphism. Then the following diagram is commutative: G

ϕ

ExpG

g

H ExpH

dϕ

h.

Proof. Let X ∈ g. We have to prove that Exp H (dϕ(X)) = ϕ Exp G (X) .

(2.38)

We set for brevity x := Exp H (dϕ(X)). By definition, we have x = expH dϕ(X) (1), where expH dϕ(X) (t) is the unique integral curve of dϕ(X) on H passing through the identity eH of H at time zero (see Theorem 2.1.56). We set y := ϕ(Exp G (X)), i.e. y = ϕ(expG X (1)), where expG X (t) is the unique integral curve of X on G passing through the identity eG of G at time zero. We want to show that G expH dϕ(X) (t) = ϕ(expX (t))

∀ t ∈ R.

When t = 1, this gives x = y, which is the claimed (2.38).

(2.39)

120


To this end, we show that μ(t) := ϕ(expG X (t)) is an integral curve of dϕ(X) on H passing through eH at time zero. By uniqueness, this will yield μ(t) = expH dϕ(X) (t), i.e. (2.39). Since ϕ is a homomorphism, we have μ(0) = ϕ(expG X (0)) = ϕ(eG ) = eH . Finally, since ϕ and expG X are Lie-group homomorphisms, μ is a homomorphism. Thus, we have d d d G = dμ = d ϕ ◦ exp μ(t) ˙ = dt μ X d r r=t dt dt μ(t) μ(t) d G = dϕ ◦ dexpX = dϕ(X) μ(t) . dt μ(t) In the first equality, we used (2.18); in the second, (2.33); in the third, we exploited the very definition of μ; the fourth follows from Remark 2.1.52-(iii); in the fifth equality, we used (2.37). This ends the proof.

Remark 2.1.60 (Some notable commutative diagrams). Theorem 2.1.59 has some important consequences: suppose that Exp G and Exp H are diffeomorphisms with inverse functions Log G and Log H , respectively. Let us analyze the following commutative diagram: G

ϕ

Exp G

g

H Log H

dϕ

h.

We deduce that, given the Lie group homomorphism ϕ, the map Log H ◦ ϕ ◦ Exp G : g −→ h coincides with the differential of ϕ, which is a Lie algebra homomorphism (in particular, a linear map!). Under the same hypotheses on Exp G and Exp H , suppose we are given a Lie algebra homomorphism ψ : g −→ h. Let us consider the map Exp H ◦ ψ ◦ Log G : G −→ H. If such a map is a Lie group homomorphism,6 then the differential of this map coincides with ψ. Indeed, the following diagram is commutative G

Exp H ◦ψ◦Log G

Log G

g

H Exp H

ψ

h.

6 We shall see that Exp ◦ ψ ◦ Log is always a Lie group homomorphism whenever G and H G

H are connected and simply connected nilpotent Lie groups.

2.2 Carnot Groups

121

Finally, given a Lie group homomorphism ϕ : G −→ H, by the commutative diagram G

ϕ

H

Log G

g

Exp H dϕ

h

the map Exp H ◦ dϕ ◦ Log G : G −→ H coincides with ϕ and is thus a Lie group homomorphism. (For other related topics7 not employed in this book, see, e.g. [War83].)

2.2 Carnot Groups We give two definitions of Carnot groups: the first one (which we already introduced in Chapter 1, Section 1.4, page 56) is the most convenient for our purposes and it seems very natural in an Analysis context; the second one is the classical one from Lie group theory. Then, we compare the two definitions showing that, up to isomorphism, they are equivalent. For reader’s convenience, we recall the definition of homogeneous Carnot group. Definition 2.2.1 (Homogeneous Carnot group). Let RN be equipped with a Lie group structure by the composition law ◦. Let also RN be equipped with a homogeneous group structure by a family {δλ }λ>0 of automorphisms of (RN , ◦) (called dilations) of the following form δλ (x (1) , x (2) , . . . , x (r) ) = (λx (1) , λ2 x (2) , . . . , λr x (r) ).

(2.40)

Here x (i) ∈ RNi for i = 1, . . . , r and N1 + N2 + · · · + Nr = N. Let g be the algebra of the group (RN , ◦). For i = 1, . . . , N1 , let Zi be the (unique) vector field of g agreeing with ∂/∂ xi at the origin. If the following assumption holds (H1) then G =

the Lie algebra generated by Z1 , . . . , ZN1 is the whole g,

(RN , ◦, δλ )

is called a homogeneous Carnot group.

7 For example, the following results hold.

Theorem 2.1.61. Let G and H be Lie groups with related algebras g and h. 1. Suppose that G is connected. Let ϕ, ψ : G −→ H be Lie group homomorphisms. Then ϕ and ψ coincide if and only if the Lie algebra homomorphisms dϕ and dψ from g to h coincide. 2. Suppose that G is simply connected. Let ψ : g −→ h be a Lie algebra homomorphism. Then there exists a unique Lie group homomorphism ϕ : G −→ H such that dϕ = ψ. 3. Suppose that G and H are simply connected. If g and h are isomorphic Lie algebras, G and H are isomorphic Lie groups.

122


Given a Lie algebra h, for any two subsets V , W of h, we set

[V , W ] = span [v, w] v ∈ V , w ∈ W . Remark 2.2.2. Let m ∈ N, and let X1 , . . . , Xm ∈ h. Then, for every k ∈ N, any Lie bracket of height k of X1 , . . . , Xm is a linear combination of nested brackets of the form (see Proposition 1.1.7, page 12) [Xi1 , [Xi2 , . . . [Xik−1 , Xik ] . . .]],

i1 , i2 , . . . , ik ∈ {1, . . . , m}.

Definition 2.2.3 (Stratified Lie group). A stratified group (or Carnot group) H is a connected and simply connected Lie group whose Lie algebra h admits a stratification, i.e. a direct sum decomposition [V1 , Vi−1 ] = Vi if 2 ≤ i ≤ r, (2.41) h = V1 ⊕ V2 ⊕ · · · ⊕ Vr such that [V1 , Vr ] = {0}. Convention. In the rest of the chapter, we shall use the term “stratified group” to refer to a Carnot group according to the above abstract, classical Definition 2.2.3. We shall instead use the prefix “homogeneous” for Carnot groups as in Definition 2.2.1. From Chapter 3 onwards, after we have proved that (see Theorem 2.2.18) every stratified group is isomorphic to a homogeneous Carnot group, we shall use the term “Carnot” group almost indifferently. Yet, whenever the homogeneous structure will have to be stressed, or whenever it will be important to distinguish stratified versus homogeneous Carnot groups, we shall re-invoke the prefix homogeneous. Remark 2.2.4. Our Definition 2.2.1 is very natural to deal with in an analytic context. However, its only inconvenient property is that it is not invariant under isomorphism of Lie groups, since the homogeneity property heavily depends on the choice of coordinates on the underlying manifold RN . For example, consider the following group law in R3 : x ◦ y = arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 . It is an easy exercise to prove that (R3 , ◦) is a homogeneous Carnot group8 according to Definition 2.2.3: indeed, the stratification is given by span{(cosh(x1 ))−1 ∂x1 , ∂x3 + sinh(x1 )∂x2 } ⊕ span{∂x2 }. It is also evident that (R3 , ◦) is not a homogeneous group (if it were so, the composition law should have polynomial component functions!, see Theorem 1.3.15). However, (R3 , ◦) is isomorphic to the homogeneous Carnot group H1 = (R3 , ∗, δλ ), being 8 We remark that

2 2 (cosh(x1 ))−1 ∂1 + ∂3 + sinh(x1 )∂2

is the canonical sub-Laplacian on this group.

2.2 Carnot Groups

123

x ∗ y = (x1 + y1 , x2 + y2 , x3 + y3 + 2x2 y1 − 2x1 y2 ), and δλ (x) := (λx1 , λx2 , λ2 x3 ). The main goal of this section is to prove that in every equivalence class of isomorphic stratified groups there is one group which is homogeneous, according to Definition 2.2.1. Example 2.2.5. Consider the open subset of R3 π π × R. Ω := (0, ∞) × − , 2 2 Then Ω is equipped with a structure of (non-homogeneous) Carnot group by the composition (ξ = (ξ1 , ξ2 , ξ3 ) ∈ Ω and analogously for η ∈ Ω) ⎛ ⎞ ξ1 η1 ⎠. arctan(ξ1 + η1 + tan ξ2 + tan η2 − ξ1 η1 ) ξ ∗η =⎝ ξ3 + η3 + 2(ξ1 ln η1 + tan ξ2 ln η1 − η1 ln ξ1 − tan η2 ln ξ1 ) This can be seen, for example, by remarking that G = (Ω, ∗) is isomorphic to the Heisenberg–Weyl group H1 (see Example 1.2.2) via the isomorphism ϕ : Ω → H1 ,

ϕ(ξ1 , ξ2 , ξ3 ) = (ln ξ1 , ξ1 + tan ξ2 , ξ3 ).

Indeed, recalling that the composition law on H1 is given by (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = x1 + y1 , x2 + y2 , x3 + y3 + 2 (x2 y1 − x1 y2 ) , and the inverse map of ϕ is ϕ −1 : H1 → Ω, we notice that

ϕ −1 (x1 , x2 , x3 ) = exp(x1 ), arctan(x2 − exp(x1 )), x3 , ξ ∗ η = ϕ −1 (ϕ(ξ ) ◦ ϕ(η))

∀ ξ, η ∈ G.

This suffices to prove that G is a Carnot group. Equivalently, we show the stratification condition for g, the algebra of G: first we find the ∗-left-invariant vector fields which coincide with the partial derivatives ∂ξ1 , ∂ξ2 , ∂ξ3 at the identity of G, say π −1 eG = ϕ (0, 0, 0) = 1, − , 0 . 4 These vector fields Z1 , Z2 , Z3 can be found via the following formula (here f ∈ C ∞ (Ω, R)) ∂ (Zi f )(ξ ) = (f (ξ ∗ η)) ∀ ξ ∈ G. ∂ ηi η=eG A direct computation then shows that

124


Z1 = ξ1 ∂ξ1 +

1 − ξ1 ∂ξ + (2ξ1 + 2 tan ξ2 − 2 ln ξ1 ) ∂ξ3 , 1 + tan2 (ξ2 ) 2

2 ∂ξ − 4 ln ξ1 ∂ξ3 , 1 + tan2 (ξ2 ) 2 Z3 = ∂ξ3 . Z2 =

Equivalently, the Zi ’s are the vector fields whose column-vector of the coefficients is given by the columns of the Jacobian matrix of the map τξ (η) = ξ ∗ η at eG , ⎛ Jτξ (eG ) = ⎝

0

ξ1 1−ξ1 1+tan2 (ξ2 ) 2ξ1 + 2 tan ξ2 − 2 ln ξ1

2 1+tan2 (ξ2 ) −4 ln ξ1

⎞ 0 0⎠. 1

As a consequence, the stratification of g is given by g = span{Z1 , Z2 } ⊕ span{[Z1 , Z2 ]}, and all commutators of Z1 , Z2 with length > 2 vanish identically. Indeed, another direct computation gives [Z1 , Z2 ] = −8 ∂ξ3 . We can now directly check the validity of Theorem 2.1.50-(2), i.e. that dϕ is an algebra-isomorphism from g to h1 , the algebra of H1 . To this end, we remark that dϕ is the map that to a vector field Z ∈ g assigns the vector field in h1 whose vector of the coefficients (at x ∈ H1 ) is given by (dϕ(Z))I (x) = Jϕ (ϕ −1 (x)) · (ZI )(ϕ −1 (x)). Now, we have ⎛

⎞ ξ −1 0 0 Jϕ (ξ ) = ⎝ 1 1 + tan2 (ξ2 ) 0 ⎠ , 0 0 1 ⎛ −x ⎞ e 1 0 0 1 + (x2 − ex1 )2 0 ⎠ . Jϕ (ϕ −1 (x)) = ⎝ 1 0 0 1 Consequently, it holds ⎛

e−x1 ⎝ 1 (dϕ(Z1 ))I (x) = 0

0 1 + (x2 − ex1 )2

0

⎞ ⎛ ⎞ ⎞⎛ ex1 1 0 x1 1−e ⎠. 1 0 ⎠ ⎝ 1+(x −ex1 )2 ⎠ = ⎝ 2 2x2 − 2x1 1 2x2 − 2x1

This means that dϕ(Z1 ) = ∂x1 + ∂x2 + (2x2 − 2x1 ) ∂x3 = X1 + X2 ,

2.2 Carnot Groups

125

where, as usual, X1 = ∂x1 + 2x2 ∂x3 , X2 = ∂x2 − 2x1 ∂x3 are the generators of the algebra of H1 . Analogously, we have ⎞ ⎛ ⎞ ⎛ −x ⎞⎛ 0 e 1 0 0 0 (dϕ(Z2 ))I (x) = ⎝ 1 1 + (x2 − ex1 )2 0 ⎠ ⎝ 1+(x22−ex1 )2 ⎠ = ⎝ 2 ⎠ , −4x1 0 0 1 −4x1 i.e. dϕ(Z2 ) = 2∂x2 − 4x1 ∂x3 = 2X2 . Moreover, ⎛

e−x1 (dϕ([Z1 , Z2 ]))I (x) = ⎝ 1 0

0 1 + (x2 − ex1 )2 0

⎞⎛ ⎞ ⎛ ⎞ 0 0 0 0⎠⎝ 0 ⎠ = ⎝ 0 ⎠, −8 −8 1

i.e. dϕ([Z1 , Z2 ]) = −8 ∂x3 = 2[X1 , X2 ]. Finally, we can straightforwardly check the algebra-isomorphism condition for dϕ (see Theorem 2.1.50-(2)), namely dϕ([Z1 , Z2 ]) = 2[X1 , X2 ] = [X1 + X2 , 2X2 ] = [dϕ(Z1 ), dϕ(Z2 )]. This completes our example.

2.2.1 Some Properties of the Stratification of a Carnot Group If H is a Carnot group, its Lie algebra admits at least a stratification, but it can have more than one. For example, if H = H1 is the Heisenberg–Weyl group on R3 (see Example 1.2.2 and its notation), its Lie algebra admits the stratifications span{X1 , X2 } ⊕ span{[X1 , X2 ]}, span{X1 − 3 [X1 , X2 ], X2 } ⊕ span{[X1 , X2 ]}, span{X1 + X2 , 3X1 + [X1 , X2 ]} ⊕ span{[X1 , X2 ]}. Definition 2.2.6 (Basis adapted to the stratification). Let H be a Carnot group. Let V = (V1 , . . . , Vr ) be a fixed stratification of the Lie algebra h of H as in (2.41). We say that a basis B of h is adapted to V if (1) (1) (r) (r) (2.42) B = E 1 , . . . , E N1 ; . . . ; E 1 , . . . , E Nr , where, for i = 1, . . . , r, we have Ni := dim Vi , and (i) (i) E1 , . . . , ENi is a basis for Vi . Obviously, every Carnot group admits an adapted basis to any of its stratifications.

126


Definition 2.2.7 (Step and number of generators). Let H be a stratified group. Let (V1 , . . . , Vr ) be any stratification of the algebra of H, as in Definition 2.2.3. We say that H as step (of nilpotency) r and has m generators, where m := dim(V1 ). The following proposition shows that the above definitions are well-posed, i.e. they do not depend on the particular stratification of H. Proposition 2.2.8. Let H be a stratified group. Suppose that (V1 , . . . , Vr ) and r ) be any two stratifications of the algebra of H, as in Definition 2.2.3. 1 , . . . , V (V i ) for every i = 1, . . . , r. Moreover, the algebra of Then r = r and dim(Vi ) = dim(V H is a nilpotent Lie algebra of step r. Hence, the natural number Q :=

r

idim(Vi )

i=1

depends only on the stratified nature of H and not on the particular stratification. Q is called the homogeneous dimension of H. Proof. From the very Definition 2.2.3, we see that if (V1 , . . . , Vr ) is a stratification of h, the algebra of H, then h is a nilpotent9 Lie algebra of step r. Hence r depends only on h and not on the stratification. i ) for every i = 1, . . . , r is not trivial and follows The fact that dim(Vi ) = dim(V from Lemma 2.2.9 below.

Lemma 2.2.9 (The two-stratification lemma). Let H be a stratified group with Lie algebra h. Suppose V := (V1 , . . . , Vr ) and W := (W1 , . . . , Wr ) are two stratifications of h. Then, for every couple of bases V and W of h respectively adapted to the stratifications V and W , the transition matrix between the two bases is non-singular and has the block-triangular form ⎞ ⎛ (1) M 0 ··· 0 ⎜ .. ⎟ . ⎜ M (2) . . . ⎟ ⎟, ⎜ ⎟ ⎜ .. .. .. ⎝ . . . 0 ⎠ ··· M (r) 9 We recall that, given an abstract Lie algebra (g, [·, ·]), g is called nilpotent of step r, r ∈ N,

if for every X1 , . . . , Xr+1 ∈ g,

[X1 , [X2 , . . . [Xr , Xr+1 ] . . . ]] = 0, and there exist Y1 , . . . , Yr ∈ g such that [Y1 , [Y2 , . . . [Yr−1 , Yr ] . . . ]] = 0.

2.2 Carnot Groups

127

where, for every i = 1, . . . , r, the block M (i) is a Ni × Ni non-singular matrix (Ni being the common value of dim(Vi ) = dim(Wi )). Proof. Let the notation in the assertion be fixed. We also set, for every i = 1, . . . , r, Ni = dim(Vi ) and Mi = dim(Wi ). From V1 ⊆ W1 ⊕ · · · ⊕ Wr and from the stratification condition for V and for W , we infer that Vi = [V1 , · · · [V1 , V1 ] · · ·] )* + ( i times / , . ⊆ {W1 ⊕ · · ·}, · · · {W1 ⊕ · · ·}, {W1 ⊕ · · ·} · · · ⊆ Wi ⊕ Wi+1 ⊕ · · · ⊕ Wr . Hence (the second column obtained by reversing the rôles of V and W ), ⎧ V1 ⊆ W1 ⊕ W2 ⊕ · · · ⊕ Wr , W1 ⊆ V1 ⊕ V2 ⊕ · · · ⊕ Vr , ⎪ ⎪ ⎪ ⎪ .. ⎨ .. . . ⎪ ⎪ Wr−1 ⊆ Vr−1 ⊕ Vr , Vr−1 ⊆ Wr−1 ⊕ Wr , ⎪ ⎪ ⎩ Vr ⊆ Wr , Wr ⊆ Vr . All this proves that

( )

⎧ W1 ⊕ W2 ⊕ · · · ⊕ Wr = V1 ⊕ V2 ⊕ · · · ⊕ Vr , ⎪ ⎪ ⎪ ⎪ . ⎨. . ⎪ ⎪ Wr−1 ⊕ Wr = Vr−1 ⊕ Vr , ⎪ ⎪ ⎩ Wr = Vr .

In particular (recalling that dim(A ⊕ B) = dim(A) + dim(B)), this yields ⎧ M 1 + M2 + · · · + Mr = N 1 + N 2 + · · · + N r , ⎪ ⎪ ⎪ ⎪ ⎨ .. . ⎪ ⎪ Mr−1 + Mr = Nr−1 + Nr , ⎪ ⎪ ⎩ Mr = N r , whence, Ni = Mi , i.e. dim(Vi ) = dim(Wi ) for every i = 1, . . . , r. From ( ) it immediately follows the block-form for the matrix of the change of basis between V and W.

The following proposition shows that “to be a Carnot group” is an invariant under isomorphism of Lie groups. Proposition 2.2.10. Let H be a stratified group. Suppose G is a Lie group isomorphic to H. Then G is a stratified group too. Moreover, H and G have the same step, the same number of generators and even the dimensions of the layers of the relevant

128


stratifications are preserved. Also, H and G have the same homogeneous dimension Q. More precisely, suppose ϕ : H → G is a Lie group isomorphism and that (V1 , . . . , Vr ) is a stratification of h, the algebra of H, as in Definition 2.2.3. Then, if g is the algebra of G, a stratification for g is given by (dϕ(V1 ), . . . , dϕ(Vr )), where dϕ is the differential of ϕ (see Definition 2.1.49) which is an isomorphism of Lie algebras (and of vector spaces). Proof. We follow the notation in the assertion. Set Wi := dϕ(Vi ) for every i = 1, . . . , r. The proposition will be proved if we demonstrate that g = W1 ⊕ W2 ⊕ · · · ⊕ Wr

and

[W1 , Wi−1 ] = Wi [W1 , Wr ] = {0}.

if 2 ≤ i ≤ r,

Now, from the linearity and the invertibility of dϕ, it holds g = dϕ(h) = dϕ(V1 ⊕ · · · ⊕ Vr ) = dϕ(V1 ) ⊕ · · · ⊕ dϕ(Vr ) = W1 ⊕ · · · ⊕ Wr . Finally, dϕ being a Lie algebra homomorphism (see Theorem 2.1.50-(ii)) it is easy to see that [W1 , Wi−1 ] equals dϕ(Vi ) = Wi , 2 ≤ i ≤ r, [dϕ(V1 ), dϕ(Vi−1 )] = d [V1 , Vi−1 ] = dϕ({0}) = {0}. This ends the proof.

2.2.2 The Campbell–Hausdorff Formula and Some General Results on Nilpotent Lie Groups Before proving that, up to isomorphism, Carnot groups and homogeneous Carnot groups provide equivalent notions, we recall some results about the so-called Campbell–Hausdorff formula. Definition 2.2.11. Let h be a nilpotent Lie algebra. For X, Y ∈ h, we set10 X Y :=

(−1)n+1 n≥1

n

pi +qi ≥1

(ad X)p1 (ad Y )q1 · · · (ad X)pn (ad Y )qn −1 Y ( nj=1 (pj + qj )) p1 ! q1 ! · · · pn ! qn !

1≤i≤n

= X+Y +

1 [X, Y ] + 2

10 We use the notation (ad A)B = [A, B]. Moreover, if q = 0, the term in the sum (2.43) is n by convention · · · (ad X)pn−1 (ad Y )qn−1 (ad X)pn −1 X. Clearly, if qn > 1, or qn = 0 and pn > 1, the term is zero.

2.2 Carnot Groups

1 1 [X, [X, Y ]] − [Y, [X, Y ]] 12 12 1 1 [Y, [X, [X, Y ]]] − [X, [Y, [X, Y ]]] − 48 48 + {brackets of height ≥ 5}.

129

+

(2.43)

The sum over n actually runs on {1, . . . , r}, where r < ∞ is the step of nilpotency of h. The same is true for the sum over the pi ’s and qi ’s, for which it is left unsaid that ni=1 (pi + qi ) ≤ r. We shall refer to the operation defined in (2.43) as the Campbell–Hausdorff operation on h. The inner sum in (2.43) can be reordered so that the brackets appear with increasing height (here N0 = N ∪ {0}), XY =

r

(−1)n+1 n=1

×

n

r

h=1

(p1 ,q1 ),...,(pn ,qn )∈ N0 ×N0 (p1 ,q1 ),...,(pn ,qn )=(0,0) (p1 +q1 )+··· +(pn +qn )=h

×

(adX)p1 (ad Y )q1 · · · (adX)pn (ad Y )qn −1 Y . ( nj=1 (pj + qj )) p1 ! q1 ! · · · pn ! qn !

Remark 2.2.12. Since h is nilpotent, (2.43) is a finite sum, and defines a binary operation in h. A striking fact is that this operation is actually associative! Indeed, much more holds: (h, ) is a Lie group. We explicitly remark that the inverse of X ∈ h w.r.t. is simply −X. We shall prove this fact in Corollary 2.2.15 below, by making use of two abstract remarkable results whose proofs are out of our scope here (we refer the reader to the monographs [CG90] and [Var84] for more references). First of all we recall the following result (see [CG90, p. 13]), which also gives the well known Campbell–Hausdorff formula (2.44). Theorem 2.2.13 (Corwin and Greenleaf [CG90, Theorem 1.2.1]). Let (H, ∗) be a connected and simply connected Lie group. Suppose that the Lie algebra h of H is nilpotent. Then defines a Lie group structure on h (h being equipped with the manifold structure resulting from its finite-dimensional vector space structure) and Exp : (h, ) → (H, ∗) is a group-isomorphism. In particular, we have Exp (X) ∗ Exp (Y ) = Exp (X Y )

∀ X, Y ∈ h.

(2.44)

We now recall the third fundamental theorem of Lie (see [Var84, Theorem 3.15.1]). This is a very deep result in Lie group theory.

130


Theorem 2.2.14 (The third fundamental theorem of Lie). Let h be a finite-dimensional Lie algebra. Then there exists a connected and simply connected Lie group whose Lie algebra is isomorphic to h. Collecting the above two theorems, we obtain the following result. Corollary 2.2.15. Let h be a finite-dimensional nilpotent Lie algebra. Then defines a Lie group structure on h. Moreover, the Lie algebra associated to the Lie group (h, ) is isomorphic to the algebra h. Proof. Since h is a finite-dimensional Lie algebra, by Theorem 2.2.14 there exists a connected and simply connected Lie group (say, G) whose Lie algebra g is isomorphic to h. Let ϕ : g → h be a Lie algebra isomorphism. Since h is nilpotent by hypothesis, so is g (since g is isomorphic to h). Hence G satisfies the hypotheses of Theorem 2.2.13. As a consequence, g equipped with the operation defined in (2.43) is a Lie group. We denote by g such a composition law on the Lie algebra g. Analogously, we denote by h a similar operation on h. Now, the isomorphism ϕ between the two Lie algebras g and h transfers the operation g into h . More precisely, for every X, Y ∈ g, we have ϕ(X g Y ) 1 1 [X, [X, Y ]g ]g + · · · = ϕ X + Y + [X, Y ]g + 2 12 (since ϕ is a Lie algebra homomorphism) 1 1 [ϕ(X), [ϕ(X), ϕ(Y )]h ]h + · · · = ϕ(X) + ϕ(Y ) + [ϕ(X), ϕ(Y )]h + 2 12 = ϕ(X) h ϕ(Y ). (2.45) We explicitly remark that in the last equality we used, as a crucial tool, the “universal” way in which is defined on a Lie algebra. In particular, the associativity of g on g directly implies the associativity of h on h. Thus, the first part of the assertion of the corollary is proved. The second part follows from Ex. 2 at the end of the chapter.

Remark 2.2.16. If h and g are finite-dimensional nilpotent Lie algebras and ϕ : h → g is an algebra-homomorphism, then ϕ is also a group-homomorphism between (h, ) and (g, ). This directly follows from (2.45). 2.2.3 Abstract and Homogeneous Carnot Groups We now aim to prove that, up to isomorphism, the definitions of classical and homogeneous Carnot group are equivalent. To begin with, we prove the following simple fact:

2.2 Carnot Groups

131

Proposition 2.2.17 ((Homogeneous ⇒ stratified) Carnot). A homogeneous Carnot group is a stratified group. Proof. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group. Clearly, G is connected and simply connected. Let g be the algebra of G. (i) For i = 1, . . . , r and j = 1, . . . , Ni , let Zj be the vector field of g agreeing (i)

with ∂/∂ xj at the origin. We set (i)

(i)

Vi := span{Z1 , . . . , ZNi }. Remark 1.4.8 proves that (V1 , . . . , Vr ) is a stratification of g, as in Definition 2.2.3. This ends the proof.

We are now in the position to prove the main result of this section. The proof is a detailed argument of what is commonly used (without comments) in literature, i.e. the identification of the group with its algebra. isom.

Theorem 2.2.18 ((Stratified ⇒ homogeneous) Carnot). Let H be a stratified group, according to Definition 2.2.3. Then there exists a homogeneous Carnot group H∗ (according to our Definition 2.2.1) which is isomorphic to H. We can choose as H∗ the Lie algebra h of H (identified to RN by a suitable choice of an adapted basis of h) equipped with the composition law defined by the Campbell–Hausdorff operation (2.43) in Definition 2.2.11. In this case, a groupisomorphism from H∗ to H is the exponential map Exp : (h, ) → (H, ∗). Proof. Let (H, ∗) be as in Definition 2.2.3. Let h be the algebra of H. Let h = V1 ⊕ · · · ⊕ Vr be a fixed stratification of h as in (2.41). By Proposition 2.2.8, h is nilpotent of step r. Then Theorem 2.2.13 yields that Exp : (h, ) → (H, ∗) is a Lie-group isomorphism,

(2.46)

where is as in (2.43). We now prove that (a coordinate version of) (h, ) is a homogeneous Carnot group according to Definition 2.2.1. We fix a basis for h adapted to its stratification (see Definition 2.2.6): for i = (i) ) be a basis for Vi . Then consider 1, . . . , r, set Ni := dim Vi , and let (E1(i) , . . . , EN i the basis for h given by (1) (1) (r) (r) E = E 1 , . . . , E N1 ; . . . ; E 1 , . . . , E Nr . By means of this basis, we fix a coordinate system on h, and we identify h with RN , where N := N1 + · · · + Nr . More precisely, we consider the map11 11 See also the map introduced in (1.35) of Remark 1.2.20, page 21. Incidentally, we notice

that (h, πE ) is a coordinate system for the whole h, determining its differentiable structure. Obviously, along with πE , we have other coordinate maps (for example those arising by a different choice of a linear basis of h, adapted or not) and the differentiable structure of h does not depend on E.

132


Fig. 2.1. Turning a stratified Carnot group into a homogeneous one

πE : h → RN ,

E · ξ :=

Ni r

(i)

(i)

ξj Ej → (ξ (1) , . . . , ξ (r) ),

i=1 j =1 (i)

(i)

where ξ (i) = (ξ1 , . . . , ξNi ) ∈ RNi for every i = 1, . . . , r. Next, we set Ψ := Exp ◦ (πE )−1 : RN → H,

Ψ (ξ ) = (Exp (E · ξ )).

(See also Fig. 2.1.) Notice that, more explicitly, r N i

(i) (i) Ψ (ξ ) = Exp ξj E j

∀ ξ ∈ RN .

(2.47)

with the composition law E defined by ξ E η := Ψ −1 Ψ (ξ ) ∗ Ψ (η) , ξ, η ∈ RN ,

(2.48)

i=1 j =1

Finally, we equip

RN

We define a family of dilations {Δλ }λ>0 on the Lie algebra h as follows: r r

Δλ Xi := λi Xi , where Xi ∈ Vi . Δλ : h → h, i=1

(2.49a)

i=1

Obviously, Δλ is a vector-space automorphism of h.

(2.49b)

(Note that N, the Ni ’s and the form of Δλ do not depend neither on the choice of the stratification of h nor on the adapted basis; see, for instance, Lemma 2.2.9.) Obviously, Δλ turns into a family of dilations {δλ }λ>0 on RN via Ψ by setting δλ := πE ◦ Δλ ◦ πE−1 . We claim that H∗ := (RN , E , δλ ) is a homogeneous Carnot group (of step r and N1 generators) isomorphic to (H, ∗) via the Lie group isomorphism Ψ . To prove the claim, we split the proof in steps.

(2.49c)

2.2 Carnot Groups

133

(I). By the very definition of E and Ψ , we have Ψ (ξ E η) = Ψ (ξ ) ∗ Ψ (η)

∀ ξ, η ∈ RN ,

(2.50a)

which, in turn, is equivalent to (exploit (2.46) and recall that Ψ = Exp ◦ πE−1 ) πE−1 (ξ E η) = πE−1 (ξ ) πE−1 (η)

∀ ξ, η ∈ RN ,

(2.50b)

∀ X, Y ∈ h.

(2.50c)

or, equivalently, πE (X Y ) = πE (X) E πE (Y ) Now, we recognize that (2.50a) and (2.50b) mean that (RN , E ),

(h, ),

(H, ∗)

are isomorphic Lie groups via the Lie-group isomorphisms πE−1

Exp

(RN , E ) −→ (h, ) −→ (H, ∗). In particular, Ψ = Exp ◦ πE−1 : (RN , E ) → (H, ∗) is a Lie-group isomorphism.

(2.51)

(II). We now investigate the dilation δλ . The stratified notation hE·ξ =

Ni r

(i)

(i)

ξj E j

i=1 j =1

for an arbitrary vector of h and the fact that πE (E · ξ ) = ξ

(2.52)

suggests the notation RN ξ = (ξ (1) , . . . , ξ (r) ) for the points in RN . We claim that, with the above notation, δλ introduced in (2.49c) has the form in (2.40), i.e. δλ (ξ (1) , ξ (2) , . . . , ξ (r) ) = (λξ (1) , λ2 ξ (2) , . . . , λr ξ (r) ). Indeed, δλ (ξ ) = (see (2.49c)) πE ◦ Δλ ◦ πE−1 (ξ ) (see (2.52)) = πE Δλ (E · ξ ) r N i

(i) (i) ξj E j = πE Δλ i=1 j =1

(2.53)

134


(see (2.49b))

= πE

(see (2.49a))

= πE

Ni r

(i) (i) ξj Δλ (Ej )

i=1 j =1 Ni r

(i) (i) ξj λi E j

i=1 j =1

(see (2.52))

= πE E · λξ (1) , . . . , λr ξ (r) = λξ (1) , . . . , λr ξ (r) .

Next, we proceed by showing that Δλ is an automorphism of the Lie-group (h, ), i.e. (2.54) Δλ (X Y ) = Δλ (X) Δλ (Y ) ∀ X, Y ∈ h, ∀ λ > 0. Recalling Remark 2.2.16, it is enough to prove that Δλ [X, Y ] = [Δλ (X), Δλ (Y )] for every X, Y ∈ h. If X = ri=1 Xi and Y = ri=1 Yi , where Xi , Yi ∈ Vi , we have [Xi , Yj ] ∈ Vi+j (by the stratification condition), whence r r

Δλ [Xi , Yj ] = λi+j [Xi , Yj ] Δλ [X, Y ] =

=

i,j =1 r

[λi Xi , λj Yj ] =

i,j =1

i,j =1 r

[Δλ (Xi ), Δλ (Yj )] = [Δλ (X), Δλ (Y )].

i,j =1

Now, a joint application of (2.50b), (2.50c) and (2.54) prove that δλ is a Lie-group automorphism of (RN , E ), i.e. δλ (ξ E η) = δλ (ξ ) E δλ (η)

∀ ξ, η ∈ RN ,

∀ λ > 0.

(III). Thus, H∗ := (RN , E , δλ ) is a homogeneous Lie group on RN , as defined in Definition 1.3.1, page 31. Let now h∗ be the Lie algebra of H∗ . Dealing with a Lie group on RN (and the fixed Cartesian coordinates ξ ’s on RN ), the Jacobian basis related to the composition E is well-posed. We denote by (1) (1) (r) (r) Z = Z1 , . . . , ZN1 ; . . . ; Z1 , . . . , ZNr this Jacobian basis, i.e. Zk is the vector field in h∗ agreeing at the origin with (i) ∂/∂ ξk . The proof is complete if we show that the Lie algebra generated by Z1 , . . . , ZN1 coincides with the whole h∗ . (i)

To this end, we first observe that, thanks to (2.51), d Ψ : h∗ → h is an algebraisomorphism (see Theorem 2.1.50). Furthermore, from Remark (2.1.52)-(iii), we have

2.2 Carnot Groups

135

dΨ = (d Exp ) ◦ d(πE−1 ) . (1)

(1)

Moreover, since E1 , . . . , EN1 is a system of Lie-generators for h (by the very definition of stratification!), it is enough to prove that (i)

(i)

d Ψ (Zk ) = Ek

for every i = 1, . . . , r and every k = 1, . . . , Ni .

(2.55)

In order to prove (2.55), we recall that a left-invariant vector field is determined by its value at the identity. Hence, (2.55) will follow if we show that (i) (i) d Ψ (Zk ) e = (Ek )e . For every f ∈ C ∞ (H, R), we have (i) (i) (i) d Ψ (Zk ) e (f ) = d0 Ψ (Zk )0 (f ) = (Zk )0 (f ◦ Ψ ) r N i

(i) (i) (i) ξj E j = (∂/∂ ξk )|ξ =0 f Exp d (i) = f Exp (t Ek ) dt t=0 d = f expE (i) (t) k dt =

i=1 j =1

t=0 (i) (Ek )e (f ).

In the first equality, we used the very definition of the differential of a homomorphism (see Definition 2.1.49); in the second one, we used the definition of the dif(i) ferential at a point; in the third equality, we exploited the very definition of Zk and that of Ψ (see (2.47)); the fourth equality is a triviality from calculus; the fifth and the sixth equalities are the core of the computation: they follow, respectively, from Proposition 2.1.58-(1) and from (2.36b). The theorem is thus completely proved.

Remark 2.2.19. From Theorem 2.2.18 and Remark 2.2.12, it follows that any stratified group is isomorphic to a homogeneous Carnot group in which the group inversion law is simply given by x −1 = −x. This is commonly assumed in great part of the literature involving Carnot groups, without further comments. See Proposition 2.2.22 for more details. Remark 2.2.20 (Change of adapted basis). We give some more information on Theorem 2.2.18. Let H be a fixed stratified group with Lie algebra h. Let h = V1 ⊕· · ·⊕Vr be a given stratification of h, as in (2.41). We know that r and every Ni := dim(Vi ) (for i = 1, . . . , r) are invariants of the stratified group. So is N := N1 + · · · + Nr (= dim(h)). We fix any arbitrary basis E for h adapted to its stratification. We use the notation in the proof of Theorem 2.2.18. We therein demonstrated that RN is a homogeneous Carnot group isomorphic to (H, ∗) if RN is equipped with the composition law E defined by (2.48) and the dilation (2.53).

136


The map Ψ is a Lie-group isomorphism from (RN , E ) to (H, ∗). Another way to write (2.48) is obviously Exp E · (ξ E η) = (Exp (E · ξ )) ∗ (Exp (E · η)). (2.56a) Suppose now that we choose another basis E (adapted to the same stratification), with analogous notation as for E. RN can be equipped with an another composition E characterized by · ( · · Exp E ξ E η) = (Exp (E ξ )) ∗ (Exp (E η)). (2.56b) (i)

For every i ∈ {1, . . . , r} and every j ∈ {1, . . . , Ni }, there exist scalars ch,j ’s such that (i) (i) (i) (i) E 1 + · · · + cN E Ni . Ej(i) = c1,j i ,j Let us introduce the notation (i) C (i) := ch,j 1≤h≤N , 1≤j ≤N i

i

and C to denote the N × N matrix whose diagonal blocks are C (1) , . . . , C (r) , ⎛ (1) ⎞ C ··· 0 . . . .. .. ⎠ , C = ⎝ .. 0 · · · C (r) and, finally, let us use once again the notation C to denote the linear map C : RN → RN ,

x → Cx.

We note that if a vector field in h has coordinates ξ ∈ RN with respect to the basis i.e. E, then12 it has coordinates Cξ with respect to the basis E, · (Cξ ) E·ξ =E

for every ξ ∈ RN .

12 Indeed, it holds

E·ξ =

Ni r

(i) (i) ξj Ej i=1 j =1

⎛ ⎞ Ni Ni Ni Ni r r

(i) (i) (i) (i) (i) (i) ⎝ = ξj ch,j Eh = ch,j ξj ⎠ E h i=1 j =1

=

h=1

i=1 h=1

j =1

Ni Ni r r

(i) (i) (i) = C (i) · ξ (i) E (C · ξ )h E h h i=1 h=1

· (Cξ ). =E

h

i=1 h=1

(2.57)

2.2 Carnot Groups

137

We now claim that the same happens for the relevant group structures G := (RN , E ),

:= (RN , ), G E

Roughly i.e. the linear map C : RN → RN is a group isomorphism from G to G. speaking, the same change of basis in h from the basis E to the basis E acts as a Lie-group isomorphism from (RN , E ) to (RN , E). Indeed, from (2.56a), (2.56b) and (2.57) we have Exp E · (ξ E η) = (Exp (E · ξ )) ∗ (Exp (E · η)) · (Cξ ) ∗ Exp E · (Cη) = Exp E · ((Cξ ) (Cη)) = Exp E E

= Exp E · C −1 (Cξ ) E (Cη) , which implies

ξ E η = C −1 (Cξ ) E (Cη) ,

(2.58a)

i.e. C(ξ E η) = (Cξ ) E (Cη)

∀ ξ, η ∈ RN .

(2.58b)

Now, (2.58b) is equivalent to say that C : (RN , E ) → (RN , E),

x → Cξ

is a Lie-group isomorphism.

Example 2.2.21 (From “stratified” to “homogeneous”). We consider once again the Lie group introduced in Examples 1.2.18 and 1.2.31. To be consistent with the notation in Theorem 2.2.18, we denote this group by H. We explicitly remark that H is a Carnot group which is not homogeneous (w.r.t. the coordinates initially assigned to it). We find an explicit isomorphism of Lie groups turning H into a homogeneous Carnot group: the existence of such an isomorphism is indeed ensured by Theorem 2.2.18, whereas an explicit way to construct it comes from the very proof of Theorem 2.2.18 (see also Remark 2.2.20). We have H = (R3 , ∗), where x ∗ y = arcsinh(sinh(x1 ) + sinh(y1 )), x2 + y2 + sinh(x1 )y3 , x3 + y3 . (1)

(1)

A stratification of the algebra h of H is V1 ⊕ V2 , where V1 = span{E1 , E2 }, (2) V2 = span{E1 }, being (1)

E1 =

1 ∂x , cosh(x1 ) 1

(1)

E2 = ∂x3 + sinh(x1 ) ∂x2 ;

The relevant dilation on h is given by

(2)

E1 = ∂x2 .

138


δλ : h −→ h, δλ (ξ1 E1(1)

+ ξ2 E2(1)

+ ξ3 E1(2) ) := ξ1 λ E1(1) + ξ2 λ E2(1) + ξ3 λ2 E1(2) .

Since h is a nilpotent algebra of step two, the Campbell–Hausdorff operation on h is 1 X Y = X + Y + [X, Y ]. 2 (1)

(1)

(2)

Now, since [E1 , E2 ] = E1 , a direct computation gives (ξ1 E1(1) + ξ2 E2(1) + ξ3 E1(2) ) (η1 E1(1) + η2 E2(1) + η3 E1(2) ) 1 (1) (1) (2) = (ξ1 + η1 ) E1 + (ξ2 + η2 ) E2 + ξ3 + η3 + (ξ1 η2 − ξ2 η1 ) E1 . 2 (1)

(1)

(2)

Via the choice of the basis E = {E1 , E2 , E1 }, the Lie group (h, ) can be identified to H∗ = (R3 , E ), where 1 ξ E η = ξ1 + η1 , ξ2 + η2 , ξ3 + η3 + (ξ1 η2 − ξ2 η1 ) . 2 The Lie group isomorphism between (H∗ , E ) and (H, ∗) is the “exponential-type map” H∗ (ξ1 , ξ2 , ξ3 ) → Exp (ξ1 E1 + ξ2 E2 + ξ3 E1 ) ∈ H. (1)

(1)

(2)

Now, from the computation13 in Example 1.2.31 (page 28) we derive that this map is given by 1 (ξ1 , ξ2 , ξ3 ) → Ψ (ξ1 , ξ2 , ξ3 ) = arcsinh(ξ1 ), ξ3 + ξ1 ξ2 , ξ2 . 2 The map Ψ is the isomorphism of Lie groups turning the homogeneous Carnot group H∗ = (R3 , E , δλ ) (where δλ (ξ1 , ξ2 , ξ3 ) = (λξ1 , λξ2 , λ2 ξ3 )) into the nonhomogeneous Carnot group H. We recognize that x ∗ y = Ψ (Ψ −1 (x) E Ψ −1 (y))

∀ x, y ∈ H.

2.2.4 More Properties of the Lie Algebra of a Stratified Group In the proof of Theorem 2.2.18, we described how to identify a stratified group with a homogeneous Carnot group on RN . More precisely, given a stratified group H, the equivalence class of the Lie groups which are isomorphic to H contains at least one (in fact, infinite) homogeneous Carnot group H∗ on RN , according to Definition 2.2.1. Namely, H∗ is a “coordinate-copy” of (h, ), the Lie algebra of H equipped with the Campbell–Hausdorff operation. 13 We warn the reader that a slight change of notation here is needed, if compared to the

notation in Example 1.2.31, interchanging ξ2 and ξ3 .

2.2 Carnot Groups

139

Since, in the specialized literature, one often meets with phrases such as “it is not restrictive to suppose that. . . ” H has some distinguished properties, it is advisable to look more closely to the properties of (h, ). We furnish some of these properties in the following proposition which collect several already proved facts. Proposition 2.2.22. Let H be a stratified group with Lie algebra h and exponential map Exp H : h → H. Let also be the Campbell–Hausdorff operation on h defined in (2.43). Let V1 ⊕ · · · ⊕ Vr be a stratification of h, as in (2.41). Let E be any basis for h adapted to the stratification, as in Definition (2.2.6). Set N := dim(h), consider the map πE : h → RN , where, for every X ∈ h, πE (X) is the N -tuple of the coordinates of X w.r.t. E. Then the binary operation on RN defined by ∀ x, y ∈ RN x E y = πE πE−1 (x) πE−1 (y) has the following properties: (1) G := (RN , E ) is a Lie group on RN ; G is isomorphic to H via the map Ψ = Exp H ◦πE−1 and to (h, ) via πE , whence (G, E ) and (h, ) are stratified groups. (2) Let Z = {Z1 , . . . , ZN } be the Jacobian basis related to G; then, denoting the adapted basis by E = {E1 , . . . , EN }, we have dΨ (Zi ) = Ei

for every i = 1, . . . , N,

or, equivalently, Zi (f ◦ Ψ ) ≡ Ei (f ) ◦ Ψ on G ∞ C (H, R). Moreover, if g is the algebra of G, the exponential map

for every f ∈ Exp G : g → G is a linear map and it sends Zi in the i-th element of the standard basis of G ≡ RN , whence Exp G (x1 , . . . , xN )Z = (x1 , . . . , xN ), being (x1 , . . . , xN )Z = x1 Z1 + · · · + xN ZN . (3) The inversion on G is the Euclidean inversion −x. (4) For every i ∈ {1, . . . , N }, we have (x E y)i = xi + yi + Ri (x, y), where Ri (x, y) is a polynomial function depending on the xk ’s and yk ’s with k < i, and Ri (x, y) can be written as a sum of polynomials each containing a factor of the following type xh yk − xk yh

with h = k and h, k < i.

(5) Let Δλ be the linear map on h such that, for every i = 1, . . . , r, Δλ (X) = λi X

whenever X ∈ Vi .

Let δλ := πE ◦ Δλ ◦ πE−1 . Then (RN , E , δλ ) is a homogeneous Carnot group of the same step and number of generators as H.

140


Proof. (1). The proof of (1) is contained in the proof of Theorem 2.2.18. (2). In the proof of Theorem 2.2.18, we demonstrated that (see (2.55)) (i)

(i)

d Ψ (Zk ) = Ek

∀ i ∈ {1, . . . , r}, k ∈ {1, . . . , Ni } (1) (1) (r) (r) if E = E1 , . . . , EN1 ; . . . ; E1 , . . . , ENr (1) (1) (r) (r) and Z = Z1 , . . . , ZN1 ; . . . ; Z1 , . . . , ZNr

(2.59)

are, respectively, the usual notation for the adapted basis E and the notation for the Jacobian basis resulting from the coordinates induced by πE on RN . This proves the first part of (2). As to the first part, consider the following diagram (h∗ denotes the algebra of the Lie group (h, ) and Exp h : h∗ → h the related exponential map) (G, E ) Exp G

g

πE−1

(h, )

Exp H

(H, ∗)

dExp h d πE−1

h∗

Exp H d Exp H

h.

From the commutativity result in Theorem 2.1.59 it holds

Exp G = πE ◦ Exp h ◦ d πE−1 = πE ◦ (Exp H )−1 ◦ Exp H ◦ d Exp H ◦ d πE−1 = πE ◦ {d Exp H } ◦ d πE−1 ,

whence

Exp G = πE ◦ d Exp H ◦ d πE−1 .

(2.60)

Hence, Exp G is a linear map since πE , d Exp H and d πE−1 are linear. We now prove that (i) (i) Exp G (Zk ) = 0(1) , . . . , ek , . . . , 0(r) ∀ i ≤ r, k ≤ Ni ,

(2.61)

(i)

where ek denotes the k-th element of the standard basis of RNi (for k ∈ {1, . . . , Ni }). To this end, first notice that (i) (i) πE (Ek ) = 0(1) , . . . , ek , . . . , 0(r) ∀ i ≤ r, k ≤ Ni . (2.62) Then, by applying (2.60), we infer (i) (i) (i) Exp G (Zk ) = πE ◦ d Exp H ◦ d πE−1 (Zk ) = πE d Ψ (Zk ) (i) (i) = πE (Ek ) = 0(1) , . . . , ek , . . . , 0(r) . In the second equality, we invoked the definition of Ψ and Remark (2.1.52)-(iii), whereas in the third we exploited (2.59); finally, the last equality is (2.62).

2.2 Carnot Groups

141

Now, the linearity of Exp G and (2.61) produce Exp G (x1 , . . . , xN )Z r N i

(i) (i) xj Zj = Exp G i=1 j =1

=

Ni r

(i) (i) xj Exp G (Zj )

i=1 j =1

=

r

(1)

0 ,...,

xj(i) ej(i) , . . . , 0(r)

j =1

i=1

= (x

Ni

,...,x

(r)

(i) (1)

xj

i=1 j =1

(1)

=

Ni r

(i)

0 , . . . , ej , . . . , 0(r)

=

r

(1) 0 , . . . , x (i) , . . . , 0(r) i=1

).

(3). Since in (1) we proved that (G, E ) is isomorphic to (h, ) and πE provides a Lie-group isomorphism between them, which is also a linear map of vector spaces, we infer that, for any x ∈ G, −1 = πE − πE−1 (x) = −πE πE−1 (x) = −x. (x)−1 = πE πE−1 (x) Here we used the fact that the inversion in (h, ) is X → −X, for, by the Campbell– Hausdorff operation, we have X (−X) = X − X +

1 1 [X, −X] + [X, [X, −X]] + · · · = 0. 2 12

(4). To begin with, we know that, by definition, x E y = πE πE−1 (x) πE−1 (y) ∀ x, y ∈ RN . N Writing x = (x1 , . . . , xN ), we have X := πE−1 (x) = i=1 xi Ei . Analogously, N −1 Y := πE (y) = j =1 yj Ej . Let us consider the Campbell–Hausdorff operation X Y . Considering the form of the operation in (2.43), we see that, besides the summands N N N

xi E i + yj E j = (xi + yi )Ei , X+Y = i=1

j =1

i=1

any other summand has the following form: it is given by a rational number times ( ) (ad X)α1 (ad Y )β1 · · · (ad X)αn (ad Y )βn [X, Y ] , or an analogous term with [X, Y ] replaced by [Y, X]. Let us analyze the case [X, being analogous). Replacing X and Y (but in the final [X, Y ]) by NY ] (the other N x E and i i i=1 j =1 yj Ej , respectively, ( ) is a finite sum of summands of the following type:

142


(

)

p(x, y) [Ek1 , · · · [EkM , [X, Y ]]],

where p(x, y) is a polynomial, M ∈ N and i1 , . . . , iM ∈ {1, . . . , N }. Now, (

) is equal to p(x, y)[Ek1 , · · · [EkM , [X, Y ]]] =

N

p(x, y) xi yj [Ek1 , · · · [EkM , [Ei , Ej ]]]

i,j =1

=

{ · · ·} = { · · ·} + { · · ·} i=j

=

i<j

i>j

p(x, y) (xh yk − xk yh ) [Ek1 , · · · [EkM , [Eh , Ek ]]].

1≤h r,

(2.70)

and V1 generates (by iterated commutators) all g. In fact, in (2.70), it holds [Vi , Vj ] = Vi+j if i + j ≤ r. This is yet another equivalent definition of Carnot group, frequently adopted in literature. Hint: Use the following facts: If G is a stratified group according to Definition 2.2.3, then (setting Vi := {0} if i > r) it holds Vi = [V1 , · · · [V1 , V1 ]] )* + (

for every i ∈ N.

i times

In particular, V1 Lie-generates all the Vi ’s (whence it generates also g = V1 ⊕ · · · ⊕ Vr ). This also gives (using Proposition 1.1.7, page 12)


149

/ , [Vi , Vj ] = [V1 , · · · [V1 , V1 ]], [V1 , · · · [V1 , V1 ]] )* + ( )* + ( i times

j times

⊆ [V1 , · · · [V1 , V1 ]] = Vi+j . )* + ( i + j times

In particular, (2.70) holds. Vice versa, let G satisfy the above hypothesis (H∗). Set W1 := V1 and Wi := [W1 , Wi−1 ] = [V1 , · · · [V1 , V1 ]] )* + (

for i ≥ 2.

i times

Prove that condition (2.70) implies that Wi ⊆ Vi for every 1 ≤ i ≤ r and Wi = {0} for every i > r. Moreover, the second hypothesis in (H∗) (i.e. V1 Lie-generates g) ensures that g = W1 + · · · + Wr . Now, a simple linear algebra argument shows that the following conditions W1 + · · · + Wr = g = V1 ⊕ · · · ⊕ Vr ,

Wi ⊆ Vi

∀i ≤r

are sufficient to derive that Wi = Vi for every 1 ≤ i ≤ r. As a consequence, we have [V1 , Vj ] = [W1 , Wj ] = Wj +1 = Vj +1 whenever 1 + j ≤ r, and [V1 , Vj ] = [W1 , Wj ] = {0} whenever 1+j > r, so that G is a Carnot group according to Definition 2.2.3. Ex. 5) Let g be a Lie algebra. Consider the so-called lower central series of g, i.e. the sequence of subspaces defined by g(1) := g,

g(j ) := [g, g(j −1) ],

j ≥ 1.

In other words, for every j ≥ 2, we have g(j ) = [g, [g, · · · [g, g]]] . ( )* + j times

Prove the following facts: • It holds g(1) ⊇ g(2) ⊇ · · · ⊇ g(j ) ⊇ g(j +1) ⊇ · · · for every j ∈ N; • if there exists r ≥ 1 such that g(r) = g(r+1) , then g(r) = g(j ) for every j ≥ r; • g is nilpotent of step r iff g(r+1) = {0}, but g(r) = {0}, • g is nilpotent of step r iff g(1) g(2) · · · g(r) g(r+1) = {0}. Ex. 6) Prove the following fact: A (finite dimensional) nilpotent Lie algebra g of step two is necessarily stratified.

150


Indeed, let us set V2 = [g, g] and choose any V1 such that g = V1 ⊕ V2 : then it also holds [V1 , V1 ] = V2 and [V1 , V2 ] = {0}. Hint: for every X, Y ∈ g, write X = X1 + X2 and Y = Y1 + Y2 , where Xi , Yi ∈ Vi , and observe that [g, g] [X, Y ] = [X1 , Y1 ] ∈ [V1 , V1 ]. Ex. 7) Consider the Carnot group on R4 (whose points are denoted by (x, y) with x ∈ R, y = (y1 , y2 , y3 ) ∈ R3 ) with the composition law ⎛

⎞ x+ξ y 1 + η1 ⎜ ⎟ (x, y) ◦ (ξ, η) = ⎝ ⎠. y2 + η2 + 12 (xη1 − ξy1 ) 1 y3 + η3 + 12 (xη2 − ξy2 ) + 12 (x − ξ )(xη1 − ξy1 ) Its Lie algebra g is spanned by 1 1 1 y1 ∂y2 − y2 ∂y3 − xy1 ∂y3 , 2 2 12 1 1 2 x ∂y3 , Y1 = ∂y1 + x ∂y2 + 2 12 1 Y2 = ∂y2 + x ∂y3 , 2 Y3 = ∂y3 , X = ∂x −

and the following commutator relations hold: [X, Y1 ] = Y2 ,

[X, Y2 ] = Y3 ,

[X, Y3 ] = 0,

[Yi , Yj ] = 0,

i, j ∈ {1, 2, 3}. Prove that the following are three different stratifications of g: g = span{X, Y1 } ⊕ span{Y2 } ⊕ span{Y3 }, g = span{X, Y1 + Y2 } ⊕ span{Y2 + Y3 } ⊕ span{Y3 }, g = span{X, Y1 + Y2 + Y3 } ⊕ span{Y2 + Y3 } ⊕ span{Y3 }. Ex. 8) Let g be a (finite dimensional) stratified Lie algebra, i.e. suppose we have g = V (1) ⊕ V (2) ⊕ · · · ⊕ V (r) with (set V (i) := {0} whenever i > r) [V (1) , V (i) ] = V (i+1) for every i ∈ N. Prove the following facts: • •

•

g is nilpotent of step r; V (r) = g(r) , the r-th element of the lower central series for g (see Ex. 5); (1) (r) (Hint: for every X1 , · · · , Xr ∈ g, write Xi = Xi + · · · + Xi , where (1) (r) Xi ∈ V (1) , · · · , Xi ∈ V (r) , and observe that [X1 , · · · [Xr−1 , Xr ]] = (1) (1) (1) [X1 , · · · [Xr−1 , Xr ]].) (1) Suppose V ⊕ · · · ⊕ V (r) and W (1) ⊕ · · · ⊕ W (r) are two stratifications of g. Prove that


W (r) = V (r) , (r) W (r−1) = V (r−1) ⊕ Ur−1 (r−1)

151

(r)

(where Ur−1 is a subspace of V (r) ), (r)

W (r−2) = V (r−2) ⊕ Ur−2 ⊕ Ur−2 (r−1)

(r)

(where Ur−2 , Ur−2 are subspaces of V (r−1) , V (r) , respectively). In other words, we have W (r) = V (r) , W (r−1) = V (r−1) (modulo V (r) ), W (r−2) = V (r−2) (modulo V (r−1) + V (r) ), .. . (1) W = V (1) (modulo V (2) + · · · + V (r−1) + V (r) ). Precisely, prove the following facts: (i) dim(V (i) ) = dim(W (i) ) =: Ni for every i = 1, . . . , r. Choose two bases V and W of g adapted respectively to the stratifications with the V (i) ’s and with the W (i) ’s. Set V = {V1 , . . . , VN } and W = {W1 , . . . , WN } and prove the existence of non-singular matrices M (i) ’s of order Ni × Ni such that (with clear meaning of the notation) ⎛ (1) ⎞ M ··· ⎛ ⎛ ⎞ ⎞ W1 .. ⎟ V1 .. ⎜ (2) . . ⎟ ⎝ .. ⎠ 0 M ⎝ ... ⎠ = ⎜ . ⎜ . ⎟ . .. .. ⎝ .. ⎠ . . WN VN 0 ··· 0 M (r) (Hint: By the preceding part of the exercise, we have W (r) = V (r) = g(r) . Now consider the quotient g/g(r) . It holds (1)

(r−1)

(1)

(r−1)

W/W (r) ⊕ · · · ⊕ W/W (r) = g/g(r) = V/V (r) ⊕ · · · ⊕ V/V (r) , and these are stratifications of the Lie algebra g/g(r) too. Consequently, (r−1)

(r−1)

W/W (r) = V/V (r) . Proceed inductively. Alternatively, argue as in the two-stratification Lemma 2.2.9.) Ex. 9) Let g be a stratified Lie algebra, i.e. g = V (1) ⊕ V (2) ⊕ · · · ⊕ V (r) with (set V (i) := {0} whenever i > r) [V (1) , V (i) ] = V (i+1) for every i ∈ N. Prove that the lower central series of g (see Ex. 5 above) is given by g(k) =

r 2

V (i)

∀ k = 1, . . . , r,

g (k) = {0} ∀ k > r.

i=k

3r (i) arguing inductively as follows: (Hint: First prove that g(k) ⊆ i=k V 3r (2) (1) (r) (i) = [g, g] = [V + · · · + V , V (1) + · · · + V (r) ] ⊆ g i=2 V ;

152


3 3 3 g(3) = [g, [g, g]] ⊆ [ ri=1 V (i) , ri=2 V (i) ] ⊆ ri=3 V (i) , etc. Vice versa, it holds V (i) = span{[X1 , · · · [Xi−1 , Xi ] · · ·] : X1 , . . . , Xi ∈ V (1) } ⊆ [[g, · · · [g, g] · · ·]] = g(i) , ( )* + i times

then derive

3r i=k

V (i) = V (k) ⊕ · · · ⊕ V (r) ⊆ g(k) ⊕ · · · ⊕ g(r) = g(k) .)

Ex. 10) The natural identification between the Lie algebra g of Lie group G with the tangent space Ge to G at the identity must be “handled with care”, as the following example shows. Suppose G and H are Lie groups with Lie algebras g, h, respectively. We denote by e both the identity of G and that of H. Suppose F : G → H is a C ∞ -map such that F (e) = e. Obviously, we have de F : Ge → He . But Ge is identified with g and He is identified with h. Consequently, F induces a natural map, say f , between g and h. The following question arises: as in the case when F is a group homomorphism, does f represent pointwise the differential of F ? Id est (see (2.33)), does it hold (for every X ∈ g and every x ∈ G) ?

(f (X))F (x) = dx F (Xx ). The answer is in general negative. Indeed, with the above notation (see also (2.26)) the map f : g → h is defined in the following way: for every X ∈ g and every y ∈ H, we have (f (X))y = de τy (de F (Xe )), i.e. f (X) ∈ h is the vector field on H defined by H y → de (τy ◦ F )(Xe ). We ask whether ?

dx F (Xx ) = de (τF (x) ◦ F )(Xe )

(∗1)

∀ X ∈ g, x ∈ G.

Since X is left-invariant, it holds Xx = de τx (Xe ), so that dx F (Xx ) = dx F (de τx (Xe )) = de (F ◦ τx )Xe . As a consequence, (∗1) holds iff (recall that Ge = {Xe : X ∈ g} by Theorem 2.1.43-(1)) (∗2)

?

de (F ◦ τx ) = de (τF (x) ◦ F )

∀ x ∈ G.

(Note that this certainly holds if F is a group homomorphism!) In turn, this is equivalent to (∗3)

?

dx F = dx (τF (x) ◦ F ◦ τx −1 )

∀ x ∈ G.


Note that (∗3) is a “differential equation” dx F = de τF (x) ◦ de F ◦ dx τx −1 F (e) = e.

153

for every x ∈ G,

For example, prove that the above system holds for G = H = (RN , +) iff F is a linear map, i.e. F is an automorphism of the Lie group (RN , +). Consequently, the map F : (R, +) → (R, +), F (x) = x 2 furnishes a counterexample. Ex. 11) Suppose that G = (RN , ◦) and H = (Rn , ∗) are two Lie groups on RN and Rn , respectively. Let g and h denote the relevant Lie algebras. Finally, let ϕ : G → H be a homomorphism of Lie groups. If dϕ : g → h is the map defined in Definition 2.1.49, prove that it operates in the following way: for every X ∈ g, dϕ(X) is the left invariant vector field on H whose column vector of the coefficients at the point y ∈ Rn is given by dϕ(X) I (y) = Jτy ◦ϕ (0) · XI (0), or, equivalently, dϕ(X) I (y) = Jτy (0) · Jϕ (0) · XI (0). Observe that (since Jτ0 (0) is the identity matrix, and since the coordinates of a vector field Y w.r.t. the Jacobian basis are given by Y I (0)) the matrix representing the linear map dϕ with respect to the relevant Jacobian bases on g and h, respectively, is simply Jϕ (0). Ex. 12) Provide the ODE’s details for what is stated in the first paragraph of the proof of Proposition 2.1.53, page 116. Ex. 13) Suppose G = (RN , ◦) and H = (Rn , ∗) are two Lie groups on RN and Rn , respectively (according to the definition given in Chapter 1). Let g and h be the relevant Lie algebras. Let ϕ : G → H be a homomorphism of Lie groups. Show that the differential of ϕ, as defined in Definition 2.1.49, is the map dϕ : g → h such that, for every X ∈ g, dϕ(X) is the only vector field in h such that (2.71) dϕ(X)I (ϕ(x)) = Jϕ (x) (XI )(x). Ex. 14) The following is an example to the “change of adapted basis”, described in Remark 2.2.20. We follow the therein notation. Example 2.3.1 (Stratified groups of step two). Let (H, ∗) be a homogeneous Carnot group of step two on RN . We shall prove in Section 3.2, page 158, that if Z denotes the Jacobian basis related to H, then it holds

154


(ξ, τ ) Z (ξ , τ ) =

ξj + ξj (j = 1, . . . , m) τi + τi + 12 B (i) ξ, ξ (i = 1, . . . , n)

,

where m is the number of generators of H, n = N − m and if {X1 , . . . , Xm ; T1 , . . . , Tn } re-denotes the Jacobian basis, then B (i) is the m × m matrix (i) B (i) = bh,k 1≤h,k≤m defined by [Xk , Xh ] =

n

(i) k=1 bh,k Ti .

is another basis of the algebra of g adapted to its stratification, say If Z = {X 1 , . . . , X m ; T1 , . . . , Tn }, Z there exist two non-singular matrices U (of order m × m) and V (of order n × n) such that 1 I · · · X m I = X1 I · · · Xm I · U, X T1 I · · · Tn I = T1 I · · · Tn I · V . A simple computation shows that the new composition law Z is given by (using, for example, (2.58a)) ξj + ξj (j = 1, . . . , m) (ξ, τ ) Z (ξ , τ ) = , (i) ξ, ξ (i = 1, . . . , n) τi + τi + 12 B where, if V −1 = (wi,j )i,j ≤n , (i)

B

=U · T

n

wi,j B

j =1

The test of this fact is left to the reader.

(j )

· U.

3 Carnot Groups of Step Two

The aim of this chapter is to collect some results and many explicit examples of Carnot groups of step two. Some examples are well known in literature, some are new. To begin with, we present the most studied (and by far one of the most important) among Carnot groups, the Heisenberg–Weyl group. Then, we turn our attention to general homogeneous Carnot groups of step two and m generators, m ≥ 2. In particular, we show that they are naturally given with the data on Rm+n of n skewsymmetric matrices of order m. The set of examples that we provide here contains the free step-two homogeneous groups, the prototype groups of Heisenberg-type (which will be widely studied in Chapter 18) and the H-groups in the sense of Métivier.

3.1 The Heisenberg–Weyl Group Let us consider in Cn × R (whose points we denote by (z, t) with t ∈ R and z = (z1 , . . . , zn ) ∈ Cn ) the following composition law (z, t) ◦ (z , t ) = (z + z , t + t + 2 Im(z · z )).

(3.1)

In (3.1), we have set (i obviously denotes the imaginary unit) Im(x + iy) = y (x, y ∈ R), whereas z · z denotes the usual Hermitian inner product in Cn , z · z =

n (xj + iyj )(xj − iyj ). j =1

Hereafter we agree to identify Cn with R2n and to use the following notation1 to denote the points of Cn × R ≡ R2n+1 : 1 Someone may say that the notation (z, t) ≡ (x , y , . . . , x , y , t) would be more appron n 1 1

priate, but the other notation is so deeply entrenched that we have no choice but to go along with it.

156


(z, t) ≡ (x, y, t) = (x1 , . . . , xn , y1 , . . . , yn , t) with z = (z1 , . . . , zn ), zj = xj + iyj and xj , yj , t ∈ R. Then, the composition law ◦ can be explicitly written as (3.2) (x, y, t) ◦ (x , y , t ) = x + x , y + y , t + t + 2y, x − 2x, y , where ·, · denotes the usual inner product in Rn . It is quite easy to verify that (R2n+1 , ◦) is a Lie group whose identity is the origin and where the inverse is given by (z, t)−1 = (−z, −t). Let us now consider the dilations δλ : R2n+1 → R2n+1 ,

δλ (z, t) = (λz, λ2 t).

A trivial computation shows that δλ is an automorphism of (R2n+1 , ◦) for every λ > 0. Then Hn = (R2n+1 , ◦, δλ ) is a homogeneous group. It is called the Heisenberg–Weyl group in R2n+1 . For example, when n = 1, the Heisenberg–Weyl group H1 in R3 is equipped with the composition law (x, y, t) ◦ (x , y , t ) = (x + x , y + y , t + t + 2 (yx − xy )), while, when n = 2, the Heisenberg–Weyl group H2 in R5 is equipped with the composition law (x1 , x2 , y1 , y2 , t) ◦ (x1 , x2 , y1 , y2 , t )

= (x1 + x1 , x2 + x2 , y1 + y1 , y2 + y2 , t + t + 2 (y1 x1 + y2 x2 − x1 y1 − x2 y2 )).

The Jacobian matrix at the origin of the left translation τ(z,t) is the following block matrix ⎞ ⎛ In 0 0 0 ⎠, In Jτ(z,t) (0, 0) = ⎝ 0 2 y T −2 x T 1 where In denotes the n × n identity matrix, while 2 y T and −2 x T stand for the 1 × n matrices (2y1 · · · 2yn ) and (−2x1 · · · − 2xn ), respectively. Then, the Jacobian basis of hn , the Lie algebra of Hn , is given by Xj = ∂xj + 2yj ∂t ,

Yj = ∂yj − 2xj ∂t ,

j = 1, . . . , n,

T = ∂t .

Since [Xj , Yj ] = −4 ∂t , we have rank Lie{X1 , . . . , Xn , Y1 , . . . , Yn }(0, 0) = dim span{∂x1 , . . . , ∂xn , ∂y1 , . . . , ∂yn , −4∂t } = 2n + 1. This shows that Hn is a Carnot group with the following stratification2 hn = span{X1 , . . . , Xn , Y1 , . . . , Yn } ⊕ span{∂t }.

(3.3)

2 Obviously, there exist other possible stratifications, but the above one is the most frequently

adopted and the one we shall refer to.

3.1 The Heisenberg–Weyl Group

157

The step of (Hn , ◦) is r = 2 and its Jacobian generators are the vector fields Xj , Yj (j = 1, . . . , n). The canonical sub-Laplacian on Hn (also referred to as Kohn Laplacian) is then given by ΔHn =

n 2 Xj + Yj2 . j =1

An explicit formula for ΔHn can be found in Ex. 1, at the end of this chapter. Finally, we exhibit the explicit form of the exponential map for Hn . It is given by3 Exp ((ξ, η, τ ) · Z) = (ξ, η, τ ). (3.4) n Here we have set (ξ, η, τ ) · Z = j =1 (ξj Xj + ηj Yj ) + τ T . We now want to perform a change of variables in Hn , inspired by the change of basis in hn , which turns the Jacobian basis into the new basis Xj∗ := Xj ,

Yj∗ := Yj

(j = 1, . . . , n), T ∗ := [X1 , Y1 ] = −4 T . As above, we set (ξ, η, τ ) · Z ∗ = nj=1 (ξj Xj∗ + ηj Yj∗ ) + τ T ∗ . In other words, we have chosen another basis of hn adapted to the stratification (3.3). Then, what we aim to do (see the proof of Theorem 2.2.18, page 131) is to equip hn with a Lie-group structure isomorphic (via Exp ) to that of (Hn , ◦); we identify hn with R2n+1 via the cited basis Z ∗ : this amounts to equip R2n+1 with the composition ∗ such that Log (Exp ((ξ, η, τ ) · Z ∗ )) ◦ (Exp ((ξ , η , τ ) · Z ∗ )) = (ξ, η, τ ) ∗ (ξ , η , τ ) · Z ∗ . The explicit expression for ∗ is easily found,4 (ξ, η, τ ) ∗ (ξ , η , τ )

1 1 = ξ + ξ , η + η , τ + τ − η, ξ + ξ, η , 2 2

(3.5a)

or, equivalently, 1 (ζ, τ ) ∗ (ζ , τ ) = ζ + ζ , τ + τ + Bζ, ζ 2

with B=

0 In

−In 0

(3.5b)

.

The natural isomorphism ϕ : (R2n+1 , ∗) → (Hn , ◦) is given by ϕ(ξ, η, τ ) = Exp ((ξ, η, τ ) · Z ∗ ) = Exp ((ξ, η, −4τ ) · Z) = (ξ, η, −4τ ). 3 See (3.11a) in Section 3.5, page 167, and Remark 3.2.4, page 163. 4 For all the details, see Section 4.3.2, page 200.

158


3.2 Homogeneous Carnot Groups of Step Two Let m, n ∈ N. Set RN := Rm × Rn and denote its points by z = (x, t) with x ∈ Rm and t ∈ Rn . Given an n-tuple B (1) , . . . , B (n) of m × m matrices with real entries, let

1 (3.6) (x, t) ◦ (ξ, τ ) = x + ξ, t + τ + Bx, ξ . 2 Here Bx, ξ denotes the n-tuple (1) B x, ξ , . . . , B (n) x, ξ

also written as

m

Bi,j xj ξi

i,j =1

and ·, · stands for the inner product in Rm . One can easily verify that (RN , ◦) is a Lie group whose identity is the origin and where the inverse is given by (x, t)−1 = −x, −t + Bx, x . We highlight that the inverse map is the usual −(x, t) if and only if, for every k = 1, . . . , n, it holds B (k) x, x = 0 ∀ x ∈ Rm , i.e. iff the matrices B (k) ’s are skew-symmetric. It is also quite easy to recognize that the dilation (3.7) δλ : RN → RN , δλ (x, t) = (λx, λ2 t) is an automorphism of (RN , ◦) for any λ > 0. Then G = (RN , ◦, δλ ) is a homogeneous Lie group. We explicitly remark that the composition law of any Lie group in Rm × Rn , homogeneous w.r.t. the dilations {δλ }λ as in (3.7), takes the form (3.6) (see Theorem 1.3.15, page 39). The Jacobian matrix at (0, 0) of the left translation τ(x,t) takes the following block form ⎛ ⎞ Im 0 ⎠, Jτ(x,t) (0, 0) = ⎝ 1 B x I n 2 (k)

where, if B (k) = (bi,j )i,j ≤m for k = 1, . . . , n, Bx denotes the matrix m

(k) bi,j xj . j =1

More explicitly, we have

k≤n, i≤m

⎛

⎜ ⎜ ⎜ ⎜ Jτ(x,t) (0, 0) = ⎜ ⎜ ⎜ ⎜ ⎝

Im

1 2

1 2

m

(1) j =1 b1,j

m

xj · · ·

.. .

(n) j =1 b1,j xj

··· ···

0m×n

1 2

1 2

m

(1) j =1 bm,j

m

xj

.. .

(n) j =1 bm,j

In xj

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

3.2 Homogeneous Carnot Groups of Step Two

159

Then the Jacobian basis of g, the Lie algebra of G, is given by m

n 1 (k) bi,l xl (∂/∂tk ) Xi = (∂/∂xi ) + 2 k=1

l=1

1 = (∂/∂xi ) + (Bx)i , ∇t , 2 Tk = ∂/∂tk , k = 1, . . . , n.

i = 1, . . . , m,

(3.8)

Here, we briefly denoted by (Bx)i the vector of Rn (1) (B x)i , . . . , (B (n) x)i , where (B (k) x)i is the i-th component of B (k) x. An easy computation shows that [Xj , Xi ] =

n n 1 (k) (k) (k) bi,j − bj,i ∂tk =: ci,j ∂tk . 2 k=1

k=1

(k)

We have denoted by C (k) = (ci,j )i,j ≤m the skew-symmetric part of B (k) , i.e. C (k) =

1 (k) B − (B (k) )T . 2

Let us now assume that C (1) , . . . , C (n) are linearly independent. This implies that the m2 × n matrix ⎛ ⎞ (1) (n) C1,1 ··· C1,1 ⎜ (n) ⎟ ⎜ C (1) ··· C1,2 ⎟ ⎜ ⎟ 1,2 ⎜ .. .. ⎟ ⎜ . ··· . ⎟ ⎜ ⎟ ⎜ (n) ⎟ ⎜ C (1) ··· C1,m ⎟ 1,m ⎜ ⎟ ⎜ (1) (n) ⎟ ⎜ C2,1 ⎟ ··· C2,1 ⎜ ⎟ ⎜ .. .. ⎟ ⎜ . ··· . ⎟ ⎜ ⎟ ⎜ (proceed analogously up to) ⎟ ⎜ ⎟ ⎜ .. .. ⎟ ⎜ . ··· . ⎟ ⎝ ⎠ (1) (n) ··· Cm,m Cm,m has rank equal to n. As a consequence, span{[Xj , Xi ] | i, j = 1, . . . , m} = span{∂t1 , . . . , ∂tn }. Therefore, rank Lie{X1 , . . . , Xm }(0, 0) = dim span{∂x1 , . . . , ∂xm , ∂t1 , . . . , ∂tn } = m + n. This shows that G is a Carnot group of step two and Jacobian generators X1 , . . . , Xm .

160


We explicitly remark that the linear independence of the matrices C (1) , . . . , C (n) is also necessary for G to be a Carnot group. Then, we have proved the following proposition. Proposition 3.2.1 (Characterization. I). Every homogeneous Lie group G on RN , homogeneous with respect to the dilation δλ : RN → RN ,

δλ (x, t) = (λx, λ2 t)

(where x ∈ Rm , t ∈ Rn and N = m + n), is equipped with the composition law

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ 2 2 for n suitable m × m matrices B (1) , . . . , B (n) . Moreover, a characterization of homogeneous Carnot groups of step two and m generators is given by the above G = (Rm+n , ◦, δλ ), where the skew-symmetric parts of the B (k) ’s are linearly independent. We remark that the above arguments show that there exist Carnot groups of any dimension m ∈ N of the first layer and any dimension n ≤ m(m − 1)/2 of the second layer: it suffices to choose n linearly independent matrices B (1) , . . . , B (n) in the vector space of the skew-symmetric m × m matrices (which has dimension m(m − 1)/2) and then define the composition law as in (3.6). Finally, by means of the general results on stratified groups in Chapter 2, we obtain the following theorem. Theorem 3.2.2 (Characterization. II). Every N -dimensional (not necessarily homogeneous) stratified group of step two and m generators is naturally isomorphic to a homogeneous Carnot group (RN , ◦, δλ ) with Lie group law as in (3.6) for some m × m linearly independent skew-symmetric matrices B (k) ’s. The group of dilations is given by (3.7), and the inverse of x is −x. By (3.8), we can write explicitly the canonical sub-Laplacian of the Lie group G = (RN , ◦) with ◦ as in (3.6). It is given by ΔG = Δx +

n 1 (h) B x, B (k) x ∂th tk 4 h,k=1

+

m

B (k) x, ∇x ∂tk +

k=1

n 1 trace(B (k) ) ∂tk . 2 k=1

(3.9)

3.2 Homogeneous Carnot Groups of Step Two

161

Here, we denoted Δx =

m

∂xi ,xi

and ∇x = (∂x1 , . . . , ∂xm ).

i=1

We recognize that ΔG contains partial differential terms of second order only if trace(B (k) ) = 0 for every k = 1, . . . , n. This happens, for example, if the B (k) ’s are skew-symmetric, i.e. if the inverse map on G is x → −x. Example 3.2.3. Following all the above notation, let us take m = 3, n = 2 and ⎛ ⎞ ⎛ ⎞ 1 1 0 0 0 −1 B (2) = ⎝ 0 1 0 ⎠ . B (1) = ⎝ −1 0 0 ⎠ , 0 0 0 1 0 0 Then the composition law on R5 = R3 × R2 as in (3.6) becomes (denoting (x, t) = (x1 , x2 , x3 , t1 , t2 ) and analogously for (ξ, τ )) ⎛ ⎞ x 1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟ ⎜ ⎟, x + ξ 3 3 (x, t) ◦ (ξ, τ ) = ⎜ ⎟ 1 ⎝ t1 + τ1 + 2 (x1 ξ1 + ξ1 x2 − ξ2 x1 ) ⎠ t2 + τ 2 +

1 2

(x2 ξ2 − ξ1 x3 − ξ3 x1 )

and the dilation is δλ (x1 , x2 , x3 , t1 , t2 ) = (λx1 , λx2 , λx3 , λ2 t1 , λ2 t2 ). Then G = (R5 , ◦, δλ ) is a homogeneous Carnot group, for the skew-symmetric parts of B (1) and B (2) are linearly independent, ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 −1 1 (1) 1 B − (B (1) )T = ⎝ −1 0 0 ⎠ , B (2) − (B (2) )T = ⎝ 0 0 0 ⎠ . 2 2 0 0 0 1 0 0 In fact, we can compute the first three vector fields of the Jacobian basis and verify that they are Lie-generators for the whole Lie algebra, 1 1 X1 = ∂x1 + (x1 + x2 )∂t1 − x3 ∂t2 , 2 2 1 1 X2 = ∂x2 − x1 ∂t1 + x2 ∂t2 , 2 2 1 X3 = ∂x3 + x1 ∂t2 , 2 [X1 , X2 ] = −∂t1 , [X1 , X3 ] = ∂t2 , 1 [X2 , X3 ] = ∂t2 . 2

162


The related canonical sub-Laplacian is ΔG = ∂x1 ,x1 + ∂x2 ,x2 + ∂x3 ,x3 1 (x1 + x2 )2 + (−x1 )2 ∂t1 ,t1 + (−x3 )2 + (x2 )2 + (x1 )2 ∂t2 ,t2 + 4 + 2 (x1 + x2 )(−x3 ) + (−x1 )(x2 ) ∂t1 ,t2 + {(x1 + x2 ) ∂x1 − x1 ∂x2 } ∂t1 + {−x3 ∂x1 + x2 ∂x2 + x1 ∂x3 } ∂t2 1 1 + ∂t1 + ∂t2 . 2 2 ΔG contains first order terms, for trace(B (1) ) = 0 = trace(B (2) ). On the contrary, if ⎛ ⎞ ⎛ ⎞ 1 1 0 0 −2 0 B (1) = ⎝ −1 0 0 ⎠ , B (2) = ⎝ 2 1 0 ⎠ , 0 0 0 0 0 0 then the composition law on R5 given by ⎛

⎞ x 1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟ ⎜ ⎟ x + ξ (x, t) ◦ (ξ, τ ) = ⎜ 3 3 ⎟ ⎝ t1 + τ1 + 1 (x1 ξ1 + ξ1 x2 − ξ2 x1 ) ⎠ 2 t2 + τ2 + 12 (x2 ξ2 − 2ξ1 x2 + 2ξ2 x1 )

does not define a homogeneous Carnot group, because the skew-symmetric parts of B (1) and B (2) are linearly dependent, ⎛ ⎞ ⎛ ⎞ 0 1 0 0 −2 0 1 (1) 1 B − (B (1) )T = ⎝ −1 0 0 ⎠ , B (2) − (B (2) )T = ⎝ 2 0 0 ⎠ . 2 2 0 0 0 0 0 0 In fact, the only admissible dilation would be δλ (x, t) = (λ x1 , λ x2 , λ x3 , λ2 t1 , λ2 t2 ), but the first three vector fields of the related Jacobian basis are not Lie-generators for the whole Lie algebra, since 1 X1 = ∂x1 + (x1 + x2 )∂t1 − x2 ∂t2 , 2

1 1 x2 + x1 ∂t2 , X2 = ∂x2 − x1 ∂t1 + 2 2 X3 = ∂x3 , [X1 , X2 ] = −∂t1 + 2∂t2 , [X1 , X3 ] = [X2 , X3 ] = 0.

3.3 Free Step-two Homogeneous Groups

163

Remark 3.2.4. The composition law on the Heisenberg–Weyl group Hk on R2k+1 , which is given by (x, y, t) ◦ (x , y , t ) = x + x , y + y , t + t + 2 (y, x − x, y ) , is of the form (3.6) with m = 2k, n = 1 and

0 B (1) = 4 −Ik

Ik 0

.

3.3 Free Step-two Homogeneous Groups In this section, following the notation of the previous one, we fix a particular set of matrices B (k) ’s and consider the relevant homogeneous Carnot group (Fm,2 , ), which will serve as prototype for what we shall call free Carnot group of step two and m generators. Throughout the section, m ≥ 2 is a fixed integer. Let i, j ∈ {1, . . . , m} be fixed with i > j , and let S (i,j ) be the m × m skewsymmetric matrix whose entries are −1 in the position (i, j ), +1 in the position (j, i) and 0 elsewhere. For example, if m = 3, we have ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 1 S (3,1) = ⎝ 0 0 0 ⎠ , S (2,1) = ⎝ −1 0 0 ⎠ , 0 0 0 −1 0 0 ⎛ ⎞ 0 0 0 S (3,2) = ⎝ 0 0 1 ⎠ . 0 −1 0 Then, we agree to denote by (Fm,2 , ) the Carnot group on RN associated to these m(m − 1)/2 matrices according to (3.6) of the previous section. We set n := m(m − 1)/2, N = m + n = m(m + 1)/2, I := {(i, j ) | 1 ≤ j < i ≤ m}. We observe that the set I has exactly n elements. In the sequel of this section, we shall use the following notation, different from the one used in the previous section: instead of using the notation t for the coordinate in the “second layer” of the group, we denote the points of Fm,2 by (x, γ ), where x = (x1 , . . . , xm ) ∈ Rm , γ ∈ Rn , and the coordinates of γ are denoted by γi,j

where (i, j ) ∈ I.

Here we have ordered I in an arbitrary (henceforth) fixed way. Then, the composition law is given by

xh + xh , h = 1, . . . , m . (x, γ ) (x , γ ) = + 1 (x x − x x ), (i, j ) ∈ I γi,j + γi,j j i 2 i j

164


For example, when m = 3, we have ⎛

⎞ x1 + x1 ⎜ ⎟ x2 + x2 ⎜ ⎟ ⎜ ⎟ x3 + x3 ⎜ ⎟ (x, γ ) ◦ (x , γ ) = ⎜ γ2,1 + γ + 1 (x2 x − x1 x ) ⎟ . 2,1 1 2 ⎟ 2 ⎜ ⎜ + 1 (x x − x x ) ⎟ ⎝ γ3,1 + γ3,1 1 3 ⎠ 2 3 1 + 1 (x x − x x ) γ3,2 + γ3,2 2 3 2 3 2 By (3.8), we can compute the Jacobian basis Xh ,

h = 1, . . . , m,

Γi,j ,

(i, j ) ∈ I,

of fm,2 , the Lie algebra of Fm,2 : it holds m

(i,j ) 1 Xh = (∂/∂xh ) + Sh,l xl (∂/∂γi,j ) 2 1≤j 0, the dilation δλ is an automorphism of B. To prove this claim we need the following lemma. Lemma 4.1.1. For every t ∈ R and λ > 0, we have E(λt) Dλ = Dλ E(t)

(4.2c)

where E(t) = exp(tB), B is as in (4.2a) and Dλ is the dilation in (4.2b). Proof. Since B k = 0 for every k ≥ r + 1, one has E(t) =

r

t k B k /k!,

k=0

and (4.2c) holds for every t ∈ R and λ > 0 iff λ k B k Dλ = Dλ B k

∀ k ≥ 0, ∀ λ > 0.

(4.2d)

This identity holds true when k = 0. An easy direct computation shows that it also holds true for k = 1. As a consequence, λ2 B 2 Dλ = λB(λBDλ ) = λB(Dλ B) = (λBDλ )B = (Dλ B)B = Dλ B 2 . Then (4.2d) holds true for k = 2. An iteration of this argument shows (4.2d) for k ≥ 2.

4.1 A Primer of Examples of Carnot Groups

187

From this lemma the Claim 1 easily follows. Indeed, for every z = (t, x), z = ∈ R1+N , we have

(t , x )

(δλ z) ◦ (δλ z ) = = (by Lemma 4.1.1) = = =

(λt, Dλ x) ◦ (λt , Dλ x ) (λt + λt , Dλ x + E(λt )Dλ x) (λ(t + t ), Dλ x + Dλ E(t )x) δλ (t + t , x + E(t )x) δλ (z ◦ z ).

Thus, B = (R1+N , ◦, δλ ) is a homogeneous group whose first layer is R × Rp0 = {(t, x (0) ) | t ∈ R, x (0) ∈ Rp0 }. Moreover, the vector fields in the Jacobian basis related to this first layer are given by (4.2e) Y = ∂t + Bx, ∇x , ∂x1 , . . . , ∂xp0 . Claim 2. We have rank(Lie{Y, ∂x1 , . . . , ∂xp0 }(0, 0)) = 1 + N. Once this claim is proved, it will follow that (R1+N , ◦, δλ ) is a Carnot group of step r + 1 with 1 + p0 generators, which are the vector fields in (4.2e). Thus the related canonical sub-Laplacian is given by ΔB = Y 2 + ΔRp0 ,

where ΔRp0 =

p0

∂x2j .

(4.2f)

j =1

This sub-Laplacian will be said of Kolmogorov type. To prove Claim 2, the following lemma will be useful. Lemma 4.1.2. In Rp × Rq , let us consider the vector field Z = Ay · (∇z )T , where A is a q × p matrix, y ∈ Rp and z ∈ Rq . Suppose rank(A) = q ≤ p. Then span [∂yi , Z] | i = 1, . . . , p = span{∂z1 , . . . , ∂zq }. Proof. Let A = (ai,j )i≤q, j ≤p . Then [∂yi , Z] =

q

aj,i ∂zj ,

i = 1, . . . , p,

j =1

so that, since rank(A) = q,

dim span [∂yi , Z] | i = 1, . . . , p = q. This implies (4.2g).

(4.2g)

188

4 Examples of Carnot Groups

We now prove Claim 2. Since B has the form (4.2a), we can write Y = ∂t +

r

Bi x (i−1) · (∇x (i) )T .

i=1

Then, by applying Lemma 4.1.2, we get span [∂xi , Y ] | i = 1, . . . , p0 = span [∂xi , B1 x (0) · (∇x (1) )T ] | i = 1, . . . , p0 = span ∂x (1) | i = 1, . . . , p1 . i

Another application of Lemma 4.1.2 gives span [∂x (1) , Y ] | i = 1, . . . , p1 = span ∂x (2) | i = 1, . . . , p2 . i

i

Iterating this argument, we get Lie{Y, ∂x1 , . . . , ∂xp0 } = Lie{Y, ∂x1 , . . . , ∂xN }. This obviously proves the claim.

Note. The groups of Kolmogorov type were introduced by E. Lanconelli and S. Polidoro [LP94] in studying a class of hypoelliptic ultraparabolic operators including the classical prototype operators of Kolmogorov–Fokker–Planck. The composition law in [LP94] was suggested by the structure of the fundamental solution of the operator ∂x21 + x1 ∂x2 − ∂x3 in R3 given by A.N. Kolmogorov in [Kol34]. Example 4.1.3. With the notation of Section 4.1.4, an example of K-type group is given by the choice p0 = p1 = 1, B1 = (1), whence 0 0 1 0 B= , N = p0 + p1 = 2, exp(s B) = . (1) 0 s 1 Hence, our K-type group B is R3 (whose points are denoted by (t, x1 , x2 )) equipped with the operation (t, x1 , x2 ) ◦ (s, y1 , y2 ) = (t + s, x1 + y1 , x2 + y2 + s x1 ) and the dilation δλ (t, x1 , x2 ) = (λt, λx1 , λ2 x2 ). This is naturally isomorphic to the Heisenberg–Weyl group H1 . Again following the notation of Section 4.1.4, another example of K-type group is given by the choice p0 = p1 = p2 = 1, B1 = B2 = (1), whence ⎛ ⎞ ⎛ ⎞ 1 0 0 0 0 0 B = ⎝ (1) 0 0 ⎠ , N = p0 + p1 + p2 = 3, exp(s B) = ⎝ s 1 0 ⎠ . s2 0 (1) 0 s 1 2

4.1 A Primer of Examples of Carnot Groups

189

Hence, our K-type group B is R4 (whose points are denoted by (t, x1 , x2 , x3 )) equipped with the operation (t, x1 , x2 , x3 ) ◦ (s, y1 , y2 , y3 ) s2 = t + s, y1 + x1 , y2 + x2 + s x1 , y3 + x3 + sx2 + x1 , 2 and the dilation δλ (t, x1 , x2 , x3 ) = (λt, λx1 , λ2 x2 , λ3 x3 ). Alternatively, the same non-trivial block (1) 0

0 (1)

in the latter matrix B can also be realized by the choice 1 0 B1 = , p0 = p1 = 2, 0 1 whence we have N = p0 + p1 = 4 and 0 0 , B = 1 0 0 0 1

where 0 =

0 0 . 0 0

Hence, our K-type group B is R5 (whose points are denoted by (t, x) = (t, x1 , x2 , x3 , x4 )) equipped with the operation (t, x) ◦ (s, y) = (t + s, y1 + x1 , y2 + x2 , y3 + x3 + s x1 , y4 + x4 + s x2 ), and the dilation δλ (t, x1 , x2 , x3 , x4 ) = (λt, λx1 , λx2 , λ2 x3 , λ2 x4 ). Remark 4.1.4. Assume that the matrix B is as in (4.2a). If we define dλ : R1+N → R1+N , dλ (t, x (0) , . . . , x (r) ) = (λ2 t, λx (0) , λ3 x (1) , . . . , λ2r+1 x (r) ), then {dλ }λ>0 is a group of automorphisms of B. For a proof of this statement, we directly refer to [LP94]. This remark shows that (R1+N , ◦, dλ ) is a homogeneous Lie group. It can be also easily proved that the ultraparabolic operator L = Δp0 + Y

(4.3)

is left-invariant (w.r.t. ◦) and homogeneous of degree two with respect to {dλ }λ>0 . Operator (4.3) generalizes the prototypes of the ones introduced by Kolmogorov in [Kol34].

190


4.1.5 Sum of Carnot Groups Suppose we are given two homogeneous Carnot groups G(1) = (RN , ◦(1) ), G(2) = (RM , ◦(2) ) with dilations (1)

x ∈ G(1) ,

(2)

y ∈ G(2) ,

δλ (x) = (λ x (1) , . . . , λr x (r) ), δλ (y) = (λ y (1) , . . . , λs y (s) ), where x (i) ∈ RNi , i ≤ r,

N1 + · · · + Nr = N

and y (i) ∈ RMi , i ≤ s, M1 + · · · + Ms = M. Let ΔG(1) =

N1 j =1

Xj2

and ΔG(2) =

M1

Yj2

j =1

be the canonical sub-Laplacians on G(1) and G(2) , respectively. We define a homogeneous Carnot group G on RN +M as follows. Suppose r ≤ s. If (x, y) ∈ RN × RM , we consider the following permutation of the coordinates R(x, y) = (x (1) , y (1) , . . . , x (r) , y (r) , y (r+1) , . . . , y (s) ). We then denote the points of G ≡ RN +M by z = R(x, y). We finally define the group law ◦ and the dilation δλ on G as one can expect: for every z = R(x, y), ζ = R(ξ, η) ∈ G, we set z ◦ ζ = R(x ◦(1) ξ, y ◦(2) η),

δλ z = R(δλ(1) x, δλ(2) y).

It is then easily checked that (G, ◦, δλ ) is a homogeneous stratified group of step s and N1 + M1 generators. Moreover, the canonical sub-Laplacian on G is the sum of the sub-Laplacians on G(1) and G(2) : ΔG = ΔG(1) + ΔG(2)

N1 M1 2 = Xj + Yj2 . j =1

j =1

For example, if G(1) is the ordinary Euclidean group on R2 and G(2) is the Heisenberg–Weyl group on R3 , then the “sum” of G(1) and G(2) is the Carnot group on R5 (whose points are denoted z = (x, y) = (x1 , x2 , y1 , y2 , y3 )) with the composition law x1 + x1 , x2 + x2 , , (x, y) ◦ (x , y ) = y1 + y1 , y2 + y2 , y3 + y3 + 2(y2 y1 − y1 y2 ) and dilation δλ (x, y) = (λx1 , λx2 , λy1 , λy2 , λ2 y3 ).

4.2 From a Set of Vector Fields to a Stratified Group

191

4.2 From a Set of Vector Fields to a Stratified Group In the analysis of PDE’s, problem naturally arises: given a linear second the following 2 , where the X ’s are smooth vector fields on RN , does X order operator L = m j j =1 j N there exist a Lie group on R with respect to which L is a sub-Laplacian? And, if the answer is affirmative, is the group law explicitly expressible? The aim of this section is to answer these questions. First of all, we recall some notation. For every k ∈ N, denote W (k) = span XJ | J ∈ {1, . . . , m}k , where, if J = (j1 , . . . , jk ), X(j1 ,...,jk ) = [Xj1 , · · · [Xjk−1 , Xjk ] · · ·]. Assume the vector fields Xj ’s satisfy the following conditions: (H0) X1 , . . . , Xm are linearly independent and δλ -homogeneous of degree one with respect to a suitable family of dilations {δλ }λ>0 of the following type δλ : RN → RN ,

δλ (x) = δλ (x (1) , . . . , x (r) ) := (λx (1) , . . . , λr x (r) ),

where r ≥ 1 is an integer, x (i) ∈ RNi for i = 1, . . . , r, N1 = m and N1 + · · · + Nr = N; (H1) dim(W (k) ) = dim{XI (0) : X ∈ W (k) } for every k = 1, . . . , r; (H2) dim(Lie{X1 , . . . , Xm }I (0)) = N. By the results2 of the previous sections, conditions (H0)–(H1)–(H2) are necessary for m j =1 Xj to be a sub-Laplacian on a suitable homogeneous Carnot group (see Proposition 1.2.13, Remark 1.4.8 and the very definitions of Carnot group and sub-Laplacian). Moreover, the hypotheses (H0)–(H1)–(H2) are independent, as the following examples show: • The vector fields ∂x1 , ∂x2 on R3 satisfy (H0) with respect to the dilation (λx1 , λx2 , λ2 x3 ). Moreover, they satisfy (H1) but not (H2); • The vector fields X1 = ∂x1 + x2 ∂x4 , X2 = ∂x2 , X3 = ∂x3 + x2 ∂x4 + x22 ∂x5 in R5 satisfy (H0) with respect to the dilations (λx1 , λx2 , λx3 , λ2 x4 , λ3 x5 ). Moreover, since [X1 , X2 ] = −∂x4 , [X1 , X3 ] = 0, [X2 , X3 ] = ∂x4 + 2x2 ∂x5 and [X2 , [X2 , X3 ]] = 2∂x5 , the given vector fields satisfy (H2) but not (H1); • the vector fields ∂x1 + x1 ∂x2 , ∂x2 on R2 satisfy (H1) and (H2) but do not satisfy (H0) with respect to any dilation (λσ1 x1 , λσ2 x2 ). (See also Section 4.4.) We are going to show that conditions (H0)–(H1)–(H2) are sufficient for the solvability of our problem. To begin with, we notice that the vector fields in W (k) are δλ -homogeneous of degree k (see Proposition 1.3.10). Moreover, by Proposition 1.3.9 and hypothesis (H1)

192


W (i) ∩ W (j ) = {0} if i = j . By Remark 1.3.7, we also have W (k) = {0} for every k ≥ r + 1. Then, by Proposition 1.1.7, (4.4) Lie{X1 , . . . , Xm } = W (1) ⊕ · · · ⊕ W (r) . Moreover, by hypotheses (H0) and (H2), dim(W (1) ) = dim(span{X1 , . . . , Xm }) = m and rk=1 dimW (k) = N. The following proposition shows a crucial link between the dimension of the W (k) ’s and the Nk ’s in (H0). Proposition 4.2.1. If X1 , . . . , Xm satisfy dim(W (k) ) = Nk for any k ∈ {1, . . . , r}.

hypotheses

(H0)–(H1)–(H2),

(k)

then

(k)

Proof. Let us set Mk := dim(W (k) ) and fix a basis {Z1 , . . . , ZMk } of W (k) , 1 ≤ k ≤ r. Then (k) (k) {Z1 I (0), . . . , ZMk I (0)} span W (k) I (0), so that, by (H1), it is a basis of W (k) I (0) := {XI (0) : X ∈ W (k) }. By (4.4), the set of vector fields (1) (r) Z1(1) , . . . , ZM , . . . , Z1(r) , . . . , ZM r 1

(4.5)

is a basis of Lie{X1 , . . . , Xm }. Then, by (H2), the column vectors of the matrix

(1) (1) (r) (r) A := Z1 I (0) · · · ZM1 I (0) · · · Z1 I (0) · · · ZMr I (0) span RN . We shall show that they are also linearly independent. By Proposition 1.3.5 (k) and Remark 1.3.7, the vector fields Zj can be written as (k)

Zj =

Ns r

(k,j )

as,i

(s)

(∂/∂xi ),

s=k i=1 (k,j )

where as,i (k,j ) as,i (0)

is a polynomial function δλ -homogeneous of degree s − k. In particular,

= 0 for every k < s ≤ r. As a consequence, the matrix A takes the form

⎛

A(1) ⎜ .. ⎝ . 0

··· .. . ···

⎞ 0 .. ⎟ , . ⎠ (r) A

(k,j ) where A(k) = ak,i (0) 1≤i≤N

The block A(k) has dimension Nk × Mk and rank Mk since

k,

1≤j ≤Mk

.

4.2 From a Set of Vector Fields to a Stratified Group (k)

193

(k)

{Z1 I (0), . . . , ZMk I (0)} is a basis for W (k) I (0) and dim(W (k) I (0)) = dim(W (k) ) = Mk . It follows that Nk ≥ Mk for 1 ≤ k ≤ r. On the other hand, since the column vectors of A span RN , r

r

Mk ≥ N =

k=1

Nk .

k=1

Then Mk = Nk for any k ∈ 1, . . . , r.

Following the previous proof, we infer that the matrix

(1) (1) (r) (r) Z1 I (x) · · · ZN1 I (x) · · · Z1 I (x) · · · ZNr I (x) ,

x ∈ RN ,

takes the following form ⎛

A(1) ⎜ ⎜ ⎜ .. ⎝ .

0 A(2) .. .

··· ··· .. .

0 0 .. .

···

A(r)

⎞ ⎟ ⎟ ⎟, ⎠

where A(1) , . . . , A(r) are square constant non-singular matrices. As a consequence, if the vector fields X1 , . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), then they also satisfy

(H1)∗ dim W (k) I (x) = dim W (k) ∀ k ≤ r, ∀ x ∈ RN .

(H2)∗ dim Lie{X1 , . . . , Xm }I (x) = N ∀ x ∈ RN . Condition (H2)∗ is the well-known Hörmander’s hypoellipticity condition for the 2. X partial differential operator m j =1 j Throughout the remaining part of this section, X1 , . . . , Xm will be a given set of smooth vector fields satisfying hypotheses (H0)–(H1)–(H2). The family {δλ }λ>0 will denote the family of dilations in (H0). We let a = Lie{X1 , . . . , Xm }. (k)

(k)

(1)

(1)

(4.6)

Finally, for every k = 1, . . . , r, Z1 , . . . , ZNk will be a fixed basis for W (k) . We know that (r)

(r)

{Z1 , . . . , ZN } := {Z1 , . . . , ZN1 , . . . , Z1 , . . . , ZNr } is a basis of a. In particular, dim(a) = N. For every ξ = (ξ1 , . . . , ξN ) = (ξ (1) , . . . , ξ (r) ), we set

(4.7)

194


ξ ·Z =

N

ξj Zj =

j =1

Nk r

(k)

(k)

ξj Zj .

k=1 j =1 (k)

Obviously, a = {ξ · Z : ξ ∈ RN }. Moreover, from the structure of Zj matrix (4.6)) we get r N (k) ξj aj (x1 , . . . , xk−1 ) ∂xk , ξ ·Z = k=1

(see the

j =1

(k)

where aj is a suitable polynomial function independent of xk , . . . , xN . Then, by Remark 1.1.3, the map (x, t) → exp(t ξ · Z)(x) is well defined for every x ∈ RN and t ∈ R. Furthermore,1 Exp : RN −→ RN ,

Exp(ξ ) := exp(ξ · Z)(0)

is a global diffeomorphism with polynomial component functions. Its inverse function, which we shall denote by Log, has polynomial components too. We are now ready to define a composition law on RN , suggested by Corollary 1.3.29. The notation “exp” is introduced in Definition 1.1.2, page 8. Definition 4.2.2. If X1 , . . . , Xm satisfy hypotheses (H0)–(H1)–(H2), we set x, y ∈ RN ,

x ◦ y := exp(Log(y) · Z)(x).

(4.8)

Remark 4.2.3. We shall show that G := (RN , ◦, δλ ) is a Carnot group whose Lie algebra g is a in (4.7). Since the group operation depends only on the Lie algebra itself, it will follow, in particular, that the definition of ◦ is independent of the choice of the basis (k) {Zj | 1 ≤ k ≤ r, 1 ≤ j ≤ Nk } of a. The main task of the proof is to show that ◦ is associative. To this end, we shall use the following result, which is a consequence of the Campbell–Hausdorff–Dynkin formula. Lemma 4.2.4 (Particular case of Campbell–Hausdorff formula). With the hypotheses (H0)–(H1)–(H2) on X1 , . . . , Xm and set a = Lie{X1 , . . . , Xm }, the following result holds: for every X, Y ∈ a, there exists a unique V ∈ a such that

exp(Y ) exp(X)(x) = exp(V )(x) for every x ∈ RN . (4.9) 1 We explicitly note that we are using the notation Exp to denote a map on RN instead of on

an algebra of vector fields.


195

We postpone the proof of this deep result to Part III of the book (see Chapter 15). We would like to stress that V in (4.9) depends only on X and Y , in particular, it is independent of x ∈ RN . Note that no mention of the word “associativity” is made in the above lemma, yet it will be the turning point in proving the associativity of ◦. The statement of Lemma 4.2.4 can be rewritten as follows: for every ξ , η ∈ RN , there exists a unique ζ ∈ RN , which we shall denote by ξ ∗ η, such that

exp(η · Z) exp(ξ · Z)(x) = exp (ξ ∗ η) · Z (x) (4.10) for every x ∈ RN . Then, by Definition 4.2.2, for every x, y ∈ RN , we have

x ◦ y = exp(Log(y) · Z)(x) = exp(Log(y) · Z) exp(Log(x) · Z)(0)

= exp (Log(x) ∗ Log(y)) · Z (0) = Exp Log(x) ∗ Log(y) . Then we have the identity

x ◦ y = Exp Log(x) ∗ Log(y)

∀ x, y ∈ RN ,

which is equivalent to the following one Log(x ◦ y) = Log(x) ∗ Log(y),

∀ x, y ∈ RN .

(4.11)

With this identity at hand, we easily get the proof of the following theorem. Theorem 4.2.5. Let ◦ be the composition law introduced in Definition 4.2.2. Then (RN , ◦) is a Lie group. Proof. 1) I DENTITY ELEMENT. Since Exp(0) = 0 = Log(0), for every x ∈ RN , we have x ◦ 0 = exp(Log(0) · Z)(x) = x, 0 ◦ x = exp(Log(x) · Z)(0) = Exp(Logx) = x. Then 0 is the identity of (RN , ◦). 2) I NVERSE ELEMENT. For any x ∈ RN , we have x ◦ Exp(−Log(x)) = exp(−Log(x) · Z)(x)

= exp(−Log(x) · Z) exp(Log(x) · Z)(0) = 0. An analogous argument shows that Exp(−Log(x)) ◦ x = 0. Then, for every x ∈ RN ,

x −1 = Exp(−Log(x)).

3) A SSOCIATIVITY. For every x, y, z ∈ RN , we have

196


(x ◦ y) ◦ z = exp(Log(z) · Z)(x ◦ y)

= exp(Log(z) · Z) exp(Log(y) · Z)(x)

(by (4.10)) = exp (Log(y) ∗ Log(z)) · Z (x). On the other hand, by (4.11),

x ◦ (y ◦ z) = exp(Log(y ◦ z) · Z)(x) = exp (Log(y) ∗ Log(z)) · Z (x). This, together with the previous identity, shows the associativity of ◦. To complete the proof of the theorem, we only have to note that the maps (x, y) → x ◦ y = exp(Log(y) · Z)(x),

x → x −1 = Exp(−Log(x))

are smooth.

In order to prove that {δλ }λ>0 is a family of automorphisms of (RN , ◦), we need the following lemma. Lemma 4.2.6. For every x, ξ ∈ RN , we have

δλ exp(ξ · Z)(x) = exp((δλ ξ ) · Z)(δλ (x))

∀ λ > 0.

(4.12)

Proof. The path R t → γ (t) = exp(tξ · Z)(x) is the solution to the Cauchy problem N γ˙ (t) = ξj Zj I (γ (t)), γ (0) = x. j =1

We have N

d (δλ (γ (t))) = δλ (γ˙ (t)) = ξj δλ Zj I (γ (t)) dt j =1

(by Corollary 1.3.6, page 35) =

N

ξj λσj (Zj I )(δλ (γ (t)))

j =1

= ((δλ ξ ) · Z)(δλ (γ (t))). Moreover, δλ (γ (0)) = δλ (x). This shows that μ := δλ (γ ) solves the Cauchy problem μ(t) ˙ = (δλ (ξ ) · Z)I (μ(t)), μ(0) = δλ (x). Thus δλ (γ (t)) = exp(t (δλ ξ ) · Z)(δλ (x)). By replacing t = 1 in this identity, we obtain (4.12).

Theorem 4.2.7. G := (RN , ◦, δλ ) is a homogeneous group.


197

Proof. By Theorem 4.2.5, we only have to prove that δλ is an automorphism of (RN , ◦). By Lemma 4.2.6, for every ξ ∈ RN , we have δλ (Exp(ξ )) = δλ (exp(ξ · Z)(0)) = exp((δλ ξ ) · Z)(0) = Exp(δλ (ξ )), so that, since Log = Exp−1 , Log(δλ (ξ )) = δλ (Log(ξ )). Then, for every x, y ∈ RN , δλ (x ◦ y) = δλ (exp(Log(y) · Z)(x)) (by Lemma 4.2.6) = exp(δλ (Log(y)) · Z)(δλ (x)) = exp(Log(δλ (y)) · Z)(δλ (x)) = (δλ (x)) ◦ (δλ (y)). This completes the proof.

By using the associativity property of the composition law ◦, it is easy to show that the vector fields Z1 , . . . , ZN are invariant with respect to the left translations on (RN , ◦, δλ ). Theorem 4.2.8. The vector fields Z1 , . . . , ZN are left-invariant on G. Proof. If ej = (0, . . . , 1, . . . , 0) (1 being the j -th component), then ej · Z = Zj and x ◦ Exp(t ej · Z) = exp(tZj )(x) for every x ∈ RN and t ∈ R. For any fixed α ∈ RN and u ∈ C ∞ (RN ), let us denote uα (x) := u(α ◦ x). From (1.15) (page 10) and the associativity of ◦, we obtain d uα x ◦ Exp(tej · Z) Zj (u(α ◦ x)) = Zj (uα (x)) = d t t=0

d u (α ◦ x) ◦ Exp(tej · Z) = (Zj u)(α ◦ x), = d t t=0 for every x ∈ RN . This completes the proof.

Corollary 4.2.9. Let g be the Lie algebra of G. Then g = a, where a = Lie{X1 , . . . , Xm }. Proof. By Theorem 4.2.8, we have a ⊆ g. On the other hand, by (4.7) and Proposition 1.2.7, dim(a) = N = dim(g). Thus a = g.

Let us now consider the vector fields Y1 , . . . , YN1 ∈ g such that Yj (0) = ∂xj , 1 ≤ j ≤ N1 . We know that N1 = m. Since Yj is δλ -homogeneous of degree one (see Corollary 1.3.19), by Corollaries 4.2.9 and 1.3.11 (the latter on page 37) and identity (4.4), we have Yj ∈ span{X1 , . . . , Xm }, 1 ≤ j ≤ m. Then, since Y1 , . . . , Ym are linearly independent,

198


span{Y1 , . . . , Ym } = span{X1 , . . . , Xm }. This identity obviously implies Lie{Y1 , . . . , Ym } = Lie{X1 , . . . , Xm } = a = g. Thus Y1 , . . . , Ym are the of g. Then G = (RN , ◦, δλ ) is a homogeneous mgenerators 2 2 Carnot group, ΔG = j =1 Yj is its canonical sub-Laplacian and L = m j =1 Xj is a sub-Laplacian on G.

We can summarize the results of this section by stating the following theorem. Theorem 4.2.10. Let X1 , . . . , Xm be smooth vector fields in RN satisfying hypotheses (H0)–(H1)–(H2). Let {δλ }λ>0 be the family of dilations defined in (H0). Finally, let ◦ be the composition law on RN introduced in Definition 4.2.2. Then G = (RN , ◦, δλ ) is a homogeneous Carnot group of step r and with m generators whose Lie algebra g is Lie-generated by X1 , . . . , Xm , i.e. g = Lie{X1 , . . . , Xm }. Moreover, the second order partial differential operator L = Laplacian on G.

m

2 j =1 Xj

is a sub-

4.3 Further Examples In this section, we exhibit some non-trivial examples of homogeneous Carnot groups constructed starting from a set of vector fields satisfying hypotheses (H0)–(H1)–(H2) of the previous section. We introduce the notation: given n ∈ N, Bn will denote the following n × n (nilpotent of step n) matrix ⎛ ⎞ 0 0 ··· 0 ⎜1 0 ··· 0 ⎟ ⎜ ⎟ . (4.13) Bn := ⎜ . . . . . . . ... ⎟ ⎝ .. ⎠ 0 ···

1

0

4.3.1 The Vector Fields ∂1 , ∂2 + x1 ∂3 Let us consider in R3 the vector fields X1 = ∂x1 ,

X2 = ∂x2 + x1 ∂x3 .

We denote by x = (x1 , x2 , x3 ) the points of R3 . It is straightforwardly verified that {X1 , X2 } satisfy conditions (H0)–(H1)–(H2) of the previous section with respect to the dilations

4.3 Further Examples

δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ),

199

λ > 0.

Then, by Theorem 4.2.10, the operator L = ∂x21 + (∂x2 + x1 ∂x3 )2 is a sub-Laplacian (indeed, the canonical one) on a suitable homogeneous Carnot group G = (R3 , ◦, δλ ). We now construct the composition law ◦ by using Definition 4.2.2. With the notation of the previous section, we have W (1) = span{X1 , X2 },

W (2) = span{X3 },

where X3 = [X1 , X2 ] = ∂x3 . For every ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 , we have ξ ·X =

3

ξj Xj = ξ1 ∂x1 + ξ2 ∂x2 + (ξ2 x1 + ξ3 )∂x3 ,

j =1

so that exp(ξ · X)(x) = γ (1), where γ = (γ1 , γ2 , γ3 ) and ⎧ γ1 (0) = x1 , ⎪ ⎨ γ˙1 (t) = ξ1 , γ˙2 (t) = ξ2 , γ2 (0) = x2 , ⎪ ⎩ γ˙3 (t) = ξ2 γ1 (t) + ξ3 , γ3 (0) = x3 . An easy computation shows that 1 exp(ξ · X)(x) = x1 + ξ1 , x2 + ξ2 , x3 + ξ3 + ξ2 x1 + ξ1 ξ2 . 2 As a consequence, 1 Exp(ξ ) = exp(ξ · X)(0) = ξ1 , ξ2 , ξ3 + ξ1 ξ2 , 2 1 Log(η) = Exp−1 (η) = η1 , η2 , η3 − η1 η2 . 2 Then, by Definition 4.2.2, the composition law determined by X1 , X2 , X3 is given by x ◦ y = exp(Log(y) · X)(x) 1 = exp y1 X1 + y2 X2 + y3 − y1 y2 X3 (x1 , x2 , x3 ) 2 1 1 = x1 + y1 , x2 + y2 , x3 + y 3 − y1 y2 + y2 x1 + y1 y2 , 2 2 i.e. x ◦ y = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ).

200


4.3.2 Classical and Kohn Laplacians In this section, we re-derive the classical Laplace operator and the Kohn Laplacian (already introduced in Sections 4.1.1 and 3.1, respectively) starting from the relevant vector fields. • The only example of homogeneous Carnot group of step 1 is the usual additive group (RN , +). If δλ denotes the usual dilation on RN δλ (x1 , . . . , xN ) = (λ x1 , . . . , λ xN ), a family of fields satisfying hypotheses (H0)–(H1)–(H2) is necessarily of the form {Xj }j ≤N , where N Xj = ai,j ∂i i=1

2 and A = (ai,j )i,j is a non-singular N ×N matrix so that N j =1 Xj is a strictly-elliptic N constant-coefficient operator. Given ξ, x ∈ R , it holds N ξj X j , exp(ξ · X)(x) = γ (1) set ξ · X := j =1

where γ˙ (r) = A · ξ and γ (0) = x. Hence exp(ξ · X)(x) = x + A · ξ,

Exp(ξ ) = A · ξ,

Log(y) = A−1 · y.

We find out x ◦ y = exp(Log(y) · X)(x) = x + A · Log(y) = x + A · A−1 · y = x + y, the usual additive structure of RN . The canonical sub-Laplacian related to this group is the ordinary Laplace operator N (∂xj )2 . Δ= j =1

• Consider now on R2N +1 (whose points are denoted by z = (x, y, t), x, y ∈ t ∈ R) the 2N vector fields

RN ,

Xj := ∂xj + 2yj ∂t ,

Yj := ∂yj − 2xj ∂t ,

j = 1, . . . , N .

If R2N +1 is equipped with the dilation δλ (z) = (λx, λy, λ2 t), the former 2N vector fields satisfy hypotheses (H0)–(H1)–(H2). We set T := [Xj , Yj ] = −4 ∂t . Given ζ = (ξ, η, τ ), z = (x, y, t) ∈ R2N +1 , one has N exp (ξj Xj + ηj Yj ) + τ T (z) = (μ(1), ν(1), ρ(1)), j =1


201

where

(μ, ˙ ν, ˙ ρ)(r) ˙ = ξ, η, −4τ + 2 ν(r), ξ − 2 μ(r), η , This gives (set ζ · Z :=

N

j =1 (ξj Xj

(μ, ν, ρ)(0) = (x, y, t).

+ ηj Yj ) + τ T )

exp(ζ · Z)(z) = x + ξ, y + η, t − 4τ + 2 y, ξ − 2 x, η , whence Exp(ζ · Z) = (ξ, η, −4τ ),

Log(z ) = (x , y , −t /4) · Z.

(4.14)

Fixed z, z ∈ R2N +1 , we have

z ◦ z = exp Log(z ) · Z (z) = x + x , y + y , t + t + 2 y, x − 2 x, y . (4.15) We recognize the well-known group (HN , ◦) on R2N +1 . Its canonHeisenberg–Weyl N 2 2 ical sub-Laplacian ΔHN = j =1 (Xj + Yj ) is the Kohn Laplacian on HN . As we know from Chapter 1, ◦ induces a Lie group structure ∗ (of “Campbell–Hausdorfftype”) on the Lie algebra hN of HN in the way re-described hereafter. For (ξ, η, τ ) ∈ RN × RN × R, we agree to set (ξ, η, τ )Z := (ξ, η, τ ) · Z =

N

(ξj Xj + ηj Yj ) + τ T ∈ hN .

j =1

Then the group law ∗ on hN is defined by

(ξ1 , η1 , τ1 )Z ∗ (ξ2 , η2 , τ2 )Z = Log Exp(ξ1 , η1 , τ1 )Z ◦ Exp(ξ2 , η2 , τ2 )Z

= Log (ξ1 , η1 , −4 τ1 ) ◦ (ξ2 , η2 , −4 τ2 )

= Log ξ1 + ξ2 , η1 + η2 , −4τ1 − 4τ2 + 2 η1 , ξ2 − 2 ξ1 , η2 1 1 = ξ1 + ξ2 , η1 + η2 , τ1 + τ2 − η1 , ξ2 + ξ1 , η2 . 2 2 Z If we drop the notation (·)Z and identify hN with R2N +1 via the basis (X1 , . . . , XN , Y1 , . . . , YN , T ), we have somewhat “more intrinsic” group (R2N +1 , ∗) canonically related to HN , where (ξ1 , η1 , τ1 ) ∗ (ξ2 , η2 , τ2 ) 1 1 = ξ1 + ξ2 , η1 + η2 , τ1 + τ2 − η1 , ξ2 + ξ1 , η2 . 2 2

(4.16)

This group is canonically related to HN in the sense that we make precise below: if we consider the change of coordinate system on HN induced by the exponential-type coordinates in (4.14), i.e.

(x, y, t) = Exp (ξ, η, τ )Z = (ξ, η, −4τ ),

202


then, with respect to this new coordinates, the vector fields Xj and Yj are respectively turned into2 j := ∂ηj + 1 ξj ∂τ . j := ∂ξj − 1 ηj ∂τ , Y X 2 2 j ’s, we turn to construct a related j ’s and Y Starting from these new vector fields X Carnot group (as we did above): after simple computations we obtain the very same group law ∗ as in (4.16). 4.3.3 Bony-type Sub-Laplacians Let us consider in R1+N the operator 2 2 ∂ ∂ 2 ∂ N ∂ L= + t +t + ··· + t , ∂t ∂x1 ∂x2 ∂xN

(t, x1 , . . . , xN ) ∈ R1+N ,

quoted by J.-M. Bony in [Bon69, Rémarque 3.1] as an example of a sum of squares satisfying Hörmander condition but nevertheless with a “very degenerate” characteristic form. L is not a sub-Laplacian on any Carnot group since the vector Clearly, j ∂/∂x vanishes on the hyperplane t = 0. It is however sufficient to field N t j j =1 add a new coordinate in order to lift L to a sub-Laplacian. Indeed, consider on R2+N , whose points are denoted by (t, s, x), t, s ∈ R, x ∈ RN , the following operator L := T 2 + S 2 , where

t2 tN ∂x2 + · · · + ∂x . 2! N! N It is readily verified that the pair T , S satisfies hypothesis (H0) with respect to the family of dilations defined by T := ∂t ,

S := ∂s + t ∂x1 +

δλ (t, s, x) := (λ t, λ s, λ2 x1 , λ3 x2 , . . . , λN +1 xN ). For every k = 1, . . . , N, we then consider the vector field t N −k Xk := [T , [T , · · · [T , S] · · ·]] = ∂xk + t ∂xk+1 + · · · + ∂x .

(N − k)! N k times

With the notation of Section 4.2, we have W (1) = span{T , S} and, for k = 1, . . . , N, W (k+1) = span{Xk }. It is easy to recognize that the hypothesis (H1) is satisfied. Finally, we have 2 Indeed, for every u ∈ C ∞ (HN , R), u = u(x, y, t), we set v := u ◦ Exp, i.e. v =

v(ξ, η, τ ) = u(ξ, η, −4τ ), so that it holds ∂ξj v = (∂xj u) ◦ Exp, ∂ηj v = (∂yj u) ◦ Exp, ∂τ v = −4 (∂t u) ◦ Exp. Consequently, (Xj u) ◦ Exp = ∂ξj v −

1 ηj ∂τ v, 2

(Yj u) ◦ Exp = ∂ηj v +

1 ξj ∂τ v. 2


203

dim Lie{T , S}I (0) = 2 + N, whence also the hypothesis (H2) holds. As a consequence, L is a sub-Laplacian on a suitable homogeneous Carnot group (G, ◦) on R2+N with step 1 + N and with 2 generators. We now turn to construct the group multiplication ◦ on G by using Definition 4.2.2. Let α, β ∈ R and ξ ∈ RN be fixed. We have j N j j −k ξk Xk = α, β, β t /j + ξk t /(j − k)! αT +βS + k=1

. j =1,...,N

k=1

This yields exp[α, β, ξ ](t, s, x)

N := exp αT + βS + ξk Xk (t, s, x) = (τ, σ, γ )(1), k=1

where (here j runs from 1 to N) τ˙ (r) = α, τ (0) = t, σ˙ (r) = β, σ (0) = s, j γ˙j (r) = β τ j (r)/j ! + k=1 ξk τ j −k (r)/(j − k)!,

γj (0) = xj .

From a direct integration it follows that exp[α, β, ξ ](t, s, x) is given by

j (α + t)j +1 − t j +1 (α + t)j −k+1 − t j −k+1 + α + t, β + s, xj + β ξk . (j + 1)α (j − k + 1)α k=1

We agree to put ((α + t)j +1 − t j +1 )/α = (j + 1)t j when α = 0. We now define the following matrices ⎞ ⎛ 1 0 0 ··· 0 ⎜ α 1 0 ··· 0⎟ ⎟ ⎜ 2! ⎜ 2 .. ⎟ .. α ⎜ α . 1 .⎟ F (α) := ⎜ 3! ⎟, 2! ⎟ ⎜ . . .. .. ⎜ .. .. . . 0⎟ ⎠ ⎝ α N−2 α N−1 α · · · 1 N! (N −1)! 2! ⎛ α ⎞ ⎛ ⎞ α 2! ⎜ α2 ⎟ ⎜ α2 ⎟ ⎜ 3! ⎟ ⎜ 2! ⎟ ⎜ ⎜ 3 ⎟ ⎟ ⎜ α3 ⎟ ⎜ ⎟ V (α) := ⎜ 4! ⎟ , U (α) := ⎜ α3! ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ ⎝ . ⎠ αN (N +1)!

and the functions

αN N!

204


(α, t) := α −1 ((α + t)F (α + t) − t F (t)), F (α, t) := α −1 ((α + t)V (α + t) − t V (t)). V It then holds

(α, t) · ξ + β V (α, t) , exp[α, β, ξ ](t, s, x) = α + t, β + s, x + F

Exp(α, β, ξ ) = α, β, F (α) · ξ + β V (α) ,

Log(τ, σ, y) = τ, σ, F −1 (τ ) · (y − σ V (τ ) . Let (t, s, x) and (τ, σ, y) ∈ R2+N be given. Then we have (t, s, x) ◦ (τ, σ, y)

(τ, t) · F −1 (τ ) · (y − σ V (τ )) + σ V (τ, t) . = τ + t, σ + s, x + F If we now prove that the following identities hold (see also (4.13)) (τ, t) = exp(t BN ) · F (τ ), F

(τ, t) − exp(t BN ) · V (τ ) = U (t), V

(4.17)

then the explicit form of the multiplication ◦ turns out to be (t, s, x) ◦ (τ, σ, y) = (τ + t, σ + s, x + exp(t BN ) · y + σ U (t)) N 1 1 = τ + t, σ + s, x1 + y1 + σ t, . . . , xN + yk t N −k + σ tN . (N − k)! N! k=1

The first identity in (4.17) follows by proving that, for every i, j ∈ {1, . . . , N } with i ≥ j , one has t i−k τ k−j (τ + t)i−j +1 − t i−j +1 = · , (i − j + 1)! τ (i − k)! (k − j + 1)! i

k=j

which readily follows by applying Newton’s binomial formula to the left-hand side. The second identity is equivalent to ti (τ + t)i+1 − t i+1 τ k t i−k = − · , i! (i + 1)! τ (k + 1)! (i − k)! i

i = 1, . . . , N,

k=1

which can be proved analogously. 4.3.4 Kolmogorov-type Sub-Laplacians We now reconsider the examples in Sections 4.1.3 and 4.1.4, and we show how to obtain the composition law of the groups of Kolmogorov type by using the results of Section 4.2. We consider in R1+N (whose points are denoted by (t, x (0) , . . . , x (r) ))


205

the vector fields introduced in (4.2e). Then, by the results proved in Sections 4.1.3– 4.1.4, they satisfy conditions (H0)–(H1)–(H2) in Section 4.2 with respect to the following family of dilations δλ (t, x (0) , . . . , x (r) ) = (λ t, λ x (0) , . . . , λr+1 x (r) ) (we refer the reader directly to the notation in Sections 4.1.3–4.1.4). For k = 0, . . . , r (k) (k) and j = 1, . . . , pk , we set Zj := ∂/∂ xj . If t, τ ∈ R and x, ξ ∈ RN are fixed, we have pk r+1 (k) (k) exp[τ, ξ ](t, x) := exp τ Y + ξj Zj (t, x) = (μ(1), γ (1)), k=1 j =1

where μ(r) ˙ = τ,

μ(0) = t;

γ˙ (r) = ξ + τ B · γ (r),

γ (0) = x.

This yields exp[τ, ξ ](t, x) = τ + t, exp(τ B) · x + Exp(τ, ξ ) = τ,

1

exp(τ (1 − r)B) · ξ dr ,

0 1

exp(τ (1 − r)B) · ξ dr ,

0

Log(s, y) = s,

1

exp(s(1 − r)B) dr

−1

·y .

0

As a consequence,

(t, x) ◦ (s, y) = t + s, y + exp(sB)x . We explicitly remark that this is the same group multiplication treated in Section 4.1.3. 4.3.5 Sub-Laplacians Arising in Control Theory We here discuss an example of homogeneous Carnot group arising from control theory, and we refer to [Alt99] for a description of the relevance of this example in that context. In RN , we consider the following vector fields X1 := ∂1 + x2 ∂3 + x3 ∂4 + · · · + xN −1 ∂N ,

X2 := ∂2 .

For every k = 3, . . . , N , we have Xk := [Xk−1 , X1 ] = ∂k , whence it is readily verified that X1 and X2 fulfill hypotheses (H0)–(H1)–(H2) with respect to the family of dilations δλ (x1 , x2 , x3 , . . . , xN ) := (λ x1 , λ x2 , λ2 x3 , . . . , λN −1 xN ).

206


As a consequence, L = X12 + X22 is a sub-Laplacian on a suitable homogeneous Carnot group (G, ◦) on RN with step N − 1 and with 2 generators. In [Alt99] it is given a representation of G by means of matrices of the following form ⎛ ⎞ 1 x2 x3 x4 · · · xN N−2 2 ⎜ ⎟ x x ⎜ 0 1 x1 2!1 · · · (N1−2)! ⎟ ⎜ ⎟ ⎜ .. ⎟ .. ⎜0 0 ⎟ . . 1 x 1 ⎜ ⎟ ≡ (x1 , x2 , . . . , xN ) ∈ G, ⎜ ⎟ 2 . x1 . ⎜0 0 ⎟ . 0 1 ⎜ ⎟ 2! ⎜. . . ⎟ .. ... ... ⎝ .. .. x1 ⎠ 0 0 ··· 0 0 1 whereas the Lie group law is given by the matrix product. We hereafter show how to obtain the composition law following the lines described in Section 4.2. Let ξ ∈ RN be fixed. We have N ξk Xk = (ξ1 , ξ2 , ξ3 + ξ1 x2 , . . . , ξN + ξ1 xN −1 ) = ξ + ξ1 H x, k=1

where H is the following N × N matrix (see also (4.13)) 0 0 . H := 0 BN −2 This gives exp[ξ ](x) := exp( N k=1 ξk Xk )(x) = γ (1), where γ˙ (r) = ξ + ξ1 H γ (r), γ (0) = x, whence r exp(ξ1 (r − t) H )ξ dt. γ (r) = exp(ξ1 r H )x + 0

In particular,

1

exp[ξ ](x) = exp(ξ1 H )x + Exp(ξ ) =

exp(ξ1 (1 − t) H )ξ dt,

0 1

exp(ξ1 (1 − t) H )ξ dt.

0

It is straightforward to recognize that, for every ρ ∈ R, we have 1 0 exp(ρ H ) = . 0 exp(ρ BN −1 ) Given y = (y1 , y ) ∈ RN , the equation y = Exp(ξ ) is equivalent to the following ξ ) ∈ RN ) system (setting ξ = (ξ1 ,


y 1 = ξ1 ,

y= 0

1

207

ξ dt. exp ξ1 (1 − t)BN −1

As a consequence,

1

Log(y) = y1 , 0

exp(y1 (1 − t)BN −1 ) dt

−1

· y .

For any fixed x, y ∈ RN , this gives x ◦ y = exp(Log(y))(x) = y + exp(y1 H )x = (y1 + x1 , y + exp(y1 BN −1 ) x) N 1 N −j = y1 + x1 , y2 + x2 , y3 + x3 + y1 x2 , . . . , y N + y xj . (N − j )! 1 j =2

4.3.6 Filiform Carnot Groups In this section, we give the definition of filiform Carnot group. To this end, we recall the definition of filiform Lie algebra (see, e.g. [OV94, page 61]). Definition 4.3.1 (Filiform Lie algebra). Let h be a Lie algebra of finite dimension n ≥ 3. For every k ∈ N, we recall that the terms of the lower (or descending) central series for h are h1 := h,

h2 := [h, h],

...,

hk := [h, hk−1 ] = [h[h[· · · [h, h] · · ·]]],

k ∈ N.

k times

Then, the Lie algebra h is called filiform if codimh (hk ) = k

for every k such that 3 ≤ k ≤ n.

In other words, h is filiform if and only if dim(hk ) = n − k

for every k: 3 ≤ k ≤ n (where n = dim(h)).

(4.18)

We explicitly remark that for the lower central series we have h1 ⊇ h2 ⊇ h3 ⊇ · · · ⊇ hk−1 ⊇ hk

∀ k ∈ N.

Remark 4.3.2. If dim(h) = 3, (4.18) says that h is filiform iff dim(h3 ) = 0. Hence, simple arguments show that the only filiform Lie algebras of dimension three are: 1) Lie{X1 , X2 , X3 } with {X1 , X2 , X3 } linearly independent and [X2 , X1 ] = [X3 , X1 ] = [X3 , X2 ] = 0. Any Lie group with such a Lie algebra is isomorphic to the classical (R3 , +).

208


2) Lie{X1 , X2 } with {X1 , X2 , [X2 , X1 ]} linearly independent and [X2 , [X2 , X1 ]] = [X1 , [X2 , X1 ]] = 0. Any Lie group with such a Lie algebra is isomorphic to the Heisenberg–Weyl group H1 on R3 . Let us now suppose that n := dim(h) ≥ 4. From (4.18) we have dim(hn−1 ) = 1 and dim(hn ) = 0, whence every filiform algebra is nilpotent, and, precisely, (if n ≥ 4) an n-dimensional filiform algebra is nilpotent of step n−1 (i.e. any commutator of length ≥ n vanishes, and there exists at least one non-vanishing commutator of length = n − 1). A very useful fact is that the converse is also true, as the following proposition states. Proposition 4.3.3 (Characterization). Let h be a Lie algebra of finite dimension n ≥ 3. • If n ≥ 4, then h is filiform if and only if it is nilpotent of step n − 1. • If n = 3, the same is true as in the previous case, except the case of the commutative R3 . Note 4.3.4. Since the linear dimension, the step of nilpotency and being isomorphic to R3 are all invariants of isomorphic Lie algebras, by Proposition 4.3.3, we derive: if g is a Lie algebra isomorphic to a filiform one, then g is filiform too. Proof (of Proposition 4.3.3). The “only if” part follows from Remark 4.3.2. We turn to the “if” part. It can be proved by an inductive argument. For example, we give the proof when n = 4. Suppose that h is nilpotent of step 3. We have to prove that h fulfills (4.18), i.e. dim(h3 ) = 1

and dim(h4 ) = 0.

The second equality follows from the step-3-nilpotence of h. By contradiction, suppose that dim(h3 ) ≥ 2. Since h2 ⊇ h3 and h2 must contain a commutator of length 2 which is not of length 3, then dim(h2 ) ≥ 3. Finally, since there must exist at least two elements in h1 \ h2 , then the inequality dim(h2 ) ≥ 3 implies dim(h1 ) ≥ 5, which contradicts the assumption dim(h1 ) = dim(h) = 4. This completes the proof for n = 4. The proof in the general case follows these ideas and is left to the reader.

Example 4.3.5. Let h be a Lie algebra of finite dimension n satisfying the following commutator identities: if {X1 , . . . , Xn } is a basis (in the sense of vector spaces) forh, we have X3 = [X1 , X2 ],

X4 = [X1 , X3 ],

X5 = [X1 , X4 ],

...,

Xn = [X1 , Xn−1 ],

whereas all other commutators of the Xi ’s vanish identically. In other words, it holds


⎧ ⎪ ⎨ [X1 , Xi ] = Xi+1 [X1 , Xn ] = 0, ⎪ ⎩ [Xi , Xj ] = 0

209

for every i = 2, . . . , n − 1, for every 1 < i < j ≤ n.

Since h is evidently nilpotent of step n−1, then, by Proposition 4.3.3, h is filiform. In Sections 4.3.3 and 4.3.5 we have furnished two explicit models for a similar algebra. Vice versa, the following remarkable fact holds. Theorem 4.3.6 (Bratzlavsky [Bra74]). Let h be a filiform Lie algebra of finite dimension n. Then there exists a basis {Xi }i for h (in the sense of vector spaces) such that Xi+1 = [X1 , Xi ] ( for every i = 2, . . . , n − 1) and [X1 , Xn ] = 0. Hence any filiform Lie algebra has a basis of the following form: n − 2 times

X1 , X2 , [X1 , X2 ], [X1 , [X1 , X2 ]], [X1 , [X1 , [X1 , X2 ]]], . . . , [X1 , · · · [X1 , X2 ] · · ·] .

=:X3

=:X4

=:X5

=:Xn

In general, nothing is said about the remaining commutators [Xi , Xj ] for 1 < i < j ≤ n. Definition 4.3.7 (Filiform Carnot group). A Carnot group is said filiform if its Lie algebra is a filiform Lie algebra. (Note. If G and H are isomorphic Carnot groups and G is filiform, then H is filiform too. This follows from Note 4.3.4.) By Proposition 4.3.3 and Remark 4.3.2, we have the following result. Proposition 4.3.8. A Carnot group on RN (with N ≥ 3) is filiform if and only if it is nilpotent of step N − 1 (except the trivial case of the usual Euclidean (R3 , +)). Remark 4.3.9. For example, among the Heisenberg–Weyl groups Hn the only filiform one is H1 ; analogously, the only filiform Carnot group of step two (hence nilpotent of step two) has necessarily dimension 3: up to isomorphism, this is H1 . Moreover, the Carnot groups considered in Sections 4.3.3 and 4.3.5 are filiform. Again from Proposition 4.3.3 it follows that a filiform Carnot group G on RN (non-Euclidean) is characterized by a stratification of its algebra g of the following type g = V1 ⊕ · · · ⊕ VN −1

with dim(V1 ) = 2, dim(Vi ) = 1 ∀ i = 2, . . . , N − 1.

In particular, a non-Euclidean filiform Carnot group has necessarily two generators.

210


4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2) In this section, we exhibit some examples of vector fields not satisfying one of the hypotheses (H0), (H1) or (H2) on page 191: we formally try to construct a group for each situation and we show that what we obtain is not a Carnot group! 4.4.1 Fields not Satisfying Hypothesis (H0): The Group is not Well Posed We consider on R2 the following two polynomial vector fields X := (1 + x 2 ) ∂x ,

Y := (1 + y 2 ) ∂y .

It is easily seen that they satisfy hypotheses (H1) and (H2) on page 191, but not (H0). Indeed, we have

dim W (1) = 2 = dim span {(1 + x 2 , 0), (0, 1 + y 2 )} ∀ (x, y) ∈ R2 , and all the vector spaces W (k) ’s for k ≥ 2 reduce to {0}, for X and Y commute. However, X and Y are not δλ -homogeneous with respect to any dilation on R2 (for their component functions contain zero degree terms), hence (H0) is not fulfilled. This invalidates our construction of a Carnot group canonically related to X and Y , the construction being heavily dependent on the well-behaved properties of the fields and, in particular, dependent on the delicate Campbell–Hausdorff-type Lemma 4.2.4. It is nonetheless important to remark that a Campbell–Hausdorff formula holds in a more general setting than the one we presented here. For instance, it holds3 for vector fields satisfying the Hörmander condition (i.e. our hypothesis (H2)) and hence in the present situation too. However, we remark that the present case is complicatedby the lack of homogeneity of X and Y , which implies that the exponential series k≥0 X k I /k! may fail to converge everywhere. We formally try to construct a group in the present situation. As done in previous sections, we fix ζ := (ξ, η) ∈ R2 and consider the vector field ζ · Z := ξ X + η Y . We formally let, as usual, Exp(ζ · Z) := (x(1), y(1)), where (x(s), y(s)) solves

(x(s), ˙ y(s)) ˙ = (ζ · Z)I (x(s), y(s)) = ξ (1 + x 2 (s)), η (1 + y 2 (s)) , (x(0), y(0)) = (0, 0). After a simple computation, we get Exp(ζ · Z) = (tan ξ, tan η),

Log(x, y) = (arctan x, arctan y) · Z.

In particular, we explicitly remark that Exp is only defined for 3 The reader is referred to Nagel–Stein–Wainger [NSW85].

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2)

211

π π π π (ξ, η) ∈ − , + × − ,+ . 2 2 2 2 We now fix (x1 , y1 ), (x2 , y2 ) ∈ R2 , and, again, we formally set (x1 , y1 ) ◦ (x2 , y2 ) := (x(1), y(1)), where (x(s), y(s)) solves

(x(s), ˙ y(s)) ˙ = arctan x2 (1 + x 2 (s)), arctan y2 (1 + y 2 (s)) (x(0), y(0)) = (x1 , y1 ). After a simple computation, we derive (x1 , y1 ) ◦ (x2 , y2 ) = (tan(arctan x1 + arctan x2 ), tan(arctan y1 + arctan y2 )) x1 + x2 y1 + y2 . , = 1 − x1 x2 1 − y1 y2 Where defined, this operation is commutative, associative, (0, 0) is the neutral element and has the inverse (x, y)−1 = (−x, −y). However, the operation is defined only away from the subset of R2 × R2 (x1 , y1 ), (x2 , y2 ) : x1 x2 = 1, or y1 y2 = 1 . Moreover, there does not exist any ε > 0 such that |x| < ε, |y| < ε implies |(x + y)/(1 − xy)| < ε. This makes it impossible to define a group from ◦. Another formal argument is more successful in this case. Indeed, we formally compute the Campbell–Hausdorff-type operation on the “algebra”: we identify (x, y) ∈ R2 with x X + y Y and we set

(ξ1 , η1 ) ∗ (ξ2 , η2 ) ≡ Log Exp(((ξ1 , η1 ) · Z) ◦ (Exp(ξ2 , η2 ) · Z))

= Log (tan ξ1 , tan η1 ) ◦ (tan ξ2 , tan η2 ) tan ξ1 + tan ξ2 tan η1 + tan η2 = Log , 1 − tan ξ1 tan ξ2 1 − tan η1 tan η2

= Log tan(ξ1 + ξ2 ), tan(η1 + η2 ) = (ξ1 + ξ2 , η1 + η2 ) · Z ≡ (ξ1 + ξ2 , η1 + η2 ) ∈ R2 . Thus, the operation on the “algebra” of the formal group related to ◦ is indeed a group law (the usual Euclidean structure on R2 ! reflecting the commutative nature of the algebra generated by the commuting vector fields {X, Y }). Analogously, we consider the change of coordinate system induced by the Log map, i.e. we consider new coordinates defined by (ξ, η) := (arctan x, arctan y).

212


With respect to these new coordinates, the vector fields X and Y are respectively turned into4 = ∂η . = ∂ξ , Y X These fields do fulfill hypotheses (H0), (H1) and (H2) of page 191, and the relevant homogeneous Carnot group is obviously the usual additive structure on R2 , exactly as for the operation ∗ above. For another example of polynomial vector fields satisfying hypotheses (H1) and (H2), but not (H0), see Ex. 18 at the end of this chapter. 4.4.2 Fields not Satisfying Hypothesis (H1): The Operation is not Associative We consider on R3 (whose points are denoted by (x, y, t)) the vector fields X := ∂x + y 2 ∂t ,

Y := ∂y .

It holds [X, Y ] = −2y ∂t ,

[X, [X, Y ]] = 0,

[Y, [X, Y ]] = −2∂t ,

and the commutators of length > 3 vanish identically. It is immediately seen that hypothesis (H2) is fulfilled, whereas (H1) is not. Indeed, W (2) = span{−2y ∂t } is one-dimensional, whereas {ZI (0) : Z ∈ W (2) } = {(0, 0, 0)} is zero dimensional. We remark that X and Y are δλ -homogeneous of degree 1 with respect to the (unusual) “dilation”: δλ : R3 −→ R3 ,

δλ (x, y, t) := (λx, λy, λ3 t).

We try formally to consider the relevant “exponential” map: we fix ζ := (ξ, η, τ ) ∈ R3 and consider the vector field ζ · Z := ξ X + η Y + τ [X, Y ]. We formally let, as usual, Exp(ζ · Z) := (x(1), y(1), t (1)), where (x(s), y(s), t (s)) solves

(x(s), ˙ y(s), ˙ t˙(s)) = ξ, η, ξ y 2 (s) − 2τ y(s) (x(0), y(0), t (0)) = (0, 0, 0). 4 Indeed, let v = v(ξ, η) ∈ C ∞ (R2 , R) and set u = u(x, y) := v(arctan x, arctan y). We

also set Log(x, y) := (arctan x, arctan y), so that u = v ◦ Log. We have ∂x u(x, y) =

i.e. ∂x u(x, y) =

1 (∂ξ v) ◦ Log, 1 + x2

1 (∂ξ v) ◦ Log, 1 + tan2 (ξ )

∂y u(x, y) =

1 (∂η v) ◦ Log, 1 + y2

∂y u(x, y) =

1 (∂η v) ◦ Log. 1 + tan2 (η)

This immediately gives Xu(x, y) = (∂ξ v) ◦ Log, Y u(x, y) = (∂η v) ◦ Log.

4.4 Fields not Satisfying One of the Hypotheses (H0), (H1), (H2)

213

After a simple computation, we get

1 2 Exp((ξ, η, τ ) · Z) = ξ, η, ξ η − τ η , 3 and this map is not globally invertible since Exp((0, 0, 0) · Z) = Exp((0, 0, 1) · Z). The singularity of this map reflects the fact that the system of vector fields X, Y, [X, Y ] is pointwise non-everywhere linearly independent! On the other hand, the everywhere well-posedness of the map is a consequence of the δλ -homogeneity of the vector fields X, Y . Since the pointwise dependence of X, Y, [X, Y ] plays a negative rôle, a natural question arises. What if we considered the pointwise everywhere linearly independent vector fields X, Y , [Y, [X, Y ]]? We proceed once again formally: let (ξ, η, τ ), (x, y, t) ∈ R3 be fixed, and let us consider

exp ξ X + ηY + τ [Y, [X, Y ]] (x, y, t) = (γ1 (1), γ2 (1), γ3 (1)), where

(γ˙1 (s), γ˙2 (s), γ˙3 (s)) = (ξ, η, ξ γ22 (s) − 2 τ ), (γ1 (0), γ2 (0), γ3 (0)) = (x, y, t).

A simple computation now gives ξ η2 + ξ y2 + ξ η y − 2 τ + t . (γ1 (1), γ2 (1), γ3 (1)) = x + ξ, y + η, 3 As a consequence, when (x, y, t) = (0, 0, 0), it holds (by the slight abuse of notation Exp(ξ, η, τ ) := Exp(ξ X + η Y + τ [Y, [X, Y ]])) ξ η2 − 2τ . Exp(ξ, η, τ ) = ξ, η, 3 This map is now everywhere invertible, and its inverse function is (by an analogous abuse of notation) x y2 t Log(x, y, t) = x, y, − . 6 2 Finally, given (x1 , y1 , t1 ), (x2 , y2 , t2 ) ∈ R3 , we set (formally following the ideas in Definition 4.2.2) (x1 , y1 , t1 ) ◦ (x2 , y2 , t2 )

= exp x2 X + y2 Y + (x2 y22 /6 − t2 /2)[Y, [X, Y ]] (x1 , y1 , t1 )

= x1 + x2 , y1 + y2 , t1 + t2 + x2 y12 + x2 y2 y1 . This binary operation on R3 is not associative, for

(4.19)

214


((0, 1, 0) ◦ (0, 1, 0)) ◦ (1, 0, 0) = (1, 2, 4) = (1, 2, 3) = (0, 1, 0) ◦ ((0, 1, 0) ◦ (1, 0, 0)). Moreover, X and Y are not invariant under left “translations” with respect to ◦ (as we know, the left-invariance is closely linked to the associativity; see the proof of Theorem 4.2.8). We explain more closely the fact that ◦ lacks to be associative by directly studying the Campbell–Hausdorff formula in the present situation. First, we fix x = (x1 , x2 , x3 ), (α1 , α2 , α3 ) and (β1 , β2 , β3 ) in R3 . Then, starting from x, we proceed along the integral path of the vector field A := α1 X + α2 Y + α3 [Y, [X, Y ]], so that, at unit time, we arrive (by definition) in exp(A)(x). Then, starting from exp(A)(x) and following the integral curve of the field B := β1 X + β2 Y + β3 [Y, [X, Y ]], we arrive (at unit time) in exp(B)(exp(A)(x)). Now, the Campbell–Hausdorff formula says that5 we get (at unit time) to this same final point even if we start from x and proceed along the integral path of the vector field C =AB =A+B +

1 1 1 [A, B] + [A, [A, B]] − [B, [A, B]]. 2 12 12

(4.20)

A direct computation shows that it holds 1 C = (α1 + β1 )X + (α2 + β2 )Y + (α1 β2 − α2 β1 )[X, Y ] 2 1 + α3 + β3 + (α2 − β2 )(α1 β2 − α2 β1 ) [Y, [X, Y ]]. 12 As we explained after the statement of Lemma 4.2.4 (page 194), we could certainly construct the desired associative operation on R3 , if C could be expressed as a constant-coefficient linear combination of X, Y, [Y, [X, Y ]]. But here we have C = (α1 + β1 )X + (α2 + β2 )Y 1 1 + α3 + β3 + (α1 β2 − α2 β1 )x2 + (α2 − β2 )(α1 β2 − α2 β1 ) [Y, [X, Y ]], 2 12 which shows that there do not exist three constants γ1 , γ2 , γ3 such that C = γ1 X + γ2 Y + γ3 [Y, [X, Y ]]. This can be seen as the very motivation for the lack of associativity of ◦ as defined in (4.19). 5 We explicitly remark that the Campbell–Hausdorff formula holds in the present case too,

see Chapter 15.


215

Remark 4.4.1. Nonetheless, we will be able to “lift” the above vector fields X, Y to suitable vector fields (on a larger vector space) satisfying hypotheses (H0)–(H1)– (H2), only exploiting the homogeneity property of X and Y and the fact that they fulfil Hörmander’s condition: this will be properly explained in Chapter 17 (see, in particular, Section 17.4, page 666), thus showing that the hypotheses of the present chapter can be somewhat weakened to produce, in case, “lifted” groups. 4.4.3 Fields not Satisfying Hypothesis (H2): The Group is Undefined We consider on R3 the following two vector fields X := ∂x1 ,

X2 := ∂x2 .

It is easily seen that they satisfy hypothesis (H0) with dilation δλ (x1 , x2 , x3 ) = (λx1 , λx2 , λ2 x3 ) and hypothesis (H1), but they do not satisfy hypothesis (H2) on page 191. Obviously, the exponential map (ξ1 , ξ2 , ξ3 ) → exp(ξ1 X1 + ξ2 X2 ) = (ξ1 , ξ2 , 0) does not even define a bijection from R3 onto R3 . Formally, the related “composition” would be (x1 , x2 , x3 ) ◦ (y1 , y2 , y3 ) = (x1 + y1 , x2 + y2 , 0) which possess no inverse map!

Bibliographical Notes. For some topics on filiform Lie algebras, we followed A.L. Onishchik and E.B. Vinberg [OV94, page 61]. Some of the topics presented in this chapter also appear in [Bon04].

4.5 Exercises of Chapter 4 Ex. 1) Consider on R4 the vector fields 1 x2 ∂3 − 2 1 X2 := ∂2 + x1 ∂3 + 2

X1 := ∂1 −

1 1 x3 ∂4 − x1 x2 ∂4 , 2 12 1 2 x ∂4 . 12 1

216


Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ). Set (1)

Z1 := X1 ,

(1)

Z2 := X2 ,

(2)

Z1 := [X1 , X2 ],

(3)

Z1 := [X1 , [X1 , X2 ]].

Following the notation of Section 4.2, verify that exp(ξ · Z)(x) equals ⎛ ⎞ x1 + ξ1 ⎜ ⎟ x 2 + ξ2 ⎜ ⎟, 1 ⎝ ⎠ x3 + ξ3 + 2 (ξ2 x1 − ξ1 x2 ) 1 1 x4 + ξ4 + 2 (ξ3 x1 − ξ1 x3 ) + 12 (x1 − ξ1 )(ξ2 x1 − ξ1 x2 ) whence Exp(ξ ) = ξ and Log(x) = x (i.e. precisely, Exp(ξ · Z) = ξ and Log(x) = x · Z). Finally, prove that the relevant composition law is ⎛ ⎞ x1 + y1 ⎜ ⎟ x2 + y2 ⎜ ⎟. 1 ⎝ ⎠ x3 + y3 + 2 (y2 x1 − y1 x2 ) 1 (x1 − y1 )(y2 x1 − y1 x2 ) x4 + y4 + 12 (y3 x1 − y1 x3 ) + 12 Ex. 2) Write down the canonical sub-Laplacian ΔG of the group obtained in Ex. 1 and observe that ΔG contains the first order differential term 16 x2 ∂4 . Ex. 3) Consider on R4 the vector fields X1 := ∂1 + x2 ∂3 + x22 ∂4 ,

X2 := ∂2 .

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the same dilation as in Ex. 1. Set (1)

Z1 := X1 ,

(1)

Z2 := X2 ,

(2)

Z1 := [X1 , X2 ],

(3)

Z1 := [X2 , [X1 , X2 ]].

Following the notation of Section 4.2, verify that ⎞ ⎛ x 1 + ξ1 ⎟ ⎜ x 2 + ξ2 ⎟. exp(ξ · Z)(x) = ⎜ ⎠ ⎝ x3 − ξ3 + 12 ξ1 ξ2 + ξ1 x2 1 2 2 x4 − 2ξ4 + 3 ξ1 ξ2 − ξ2 ξ3 + ξ1 x2 + ξ1 ξ2 x2 − 2ξ3 x2 Deduce that

and

⎛

⎞ ξ1 ⎜ ⎟ ξ2 ⎟ Exp(ξ · Z) = ⎜ ⎝ ⎠ −ξ3 + 12 ξ1 ξ2 −2ξ4 + 13 ξ1 ξ22 − ξ2 ξ3 ⎛

⎞ x1 ⎜ ⎟ x2 ⎟ · Z. Log(x) = ⎜ ⎝ ⎠ −x3 + 12 x1 x2 1 − 12 x4 − 12 x1 x22 + 12 x2 x3


217

Finally, prove that the relevant composition law is ⎞ ⎛ x 1 + y1 ⎟ ⎜ x2 + y2 ⎟. x◦y =⎜ ⎠ ⎝ x3 + y3 + y1 x2 x4 + y4 + y1 x22 + 2 x2 y3 Ex. 4) Show that the inverse map for the Lie group in Ex. 3 is given by ⎞ ⎛ −x1 ⎟ ⎜ −x2 ⎟. x −1 = ⎜ ⎠ ⎝ −x3 + x1 x2 −x4 + 2 x2 x3 − x1 x22 Ex. 5) a) Consider the Lie group and the notation in Ex. 3. For every ξ, η ∈ RN , prove that Log(Exp(ξ · Z) ◦ Exp(η · Z)) equals ⎛ ⎞ ξ1 + η1 ⎜ ⎟ ξ 2 + η2 ⎜ ⎟. 1 ⎝ ⎠ ξ3 + η3 + 2 (ξ1 η2 − ξ2 η1 ) 1 1 ξ4 + η4 + 2 (ξ2 η3 − ξ3 η2 ) + 12 (ξ2 − η2 ) (ξ1 η2 − ξ2 η1 ) Denote the above composition law in R4 by ξ ∗ η. Deduce that F = (R4 , ∗) is a Carnot group isomorphic to G = (R4 , ◦). Find the Liegroup isomorphism turning F into G. Considering the natural identification R4 ξ ←→ ξ · Z ∈ g (g being the Lie-algebra of G) define a composition law on g dual to ∗. Compare this operation to the Campbell–Hausdorff composition law in (2.43), motivating your remarks. b) Consider on R4 the change of coordinates modeled on the Log map, i.e. the new coordinates ξ on R4 defined by ⎛ ⎞ x1 ⎜ ⎟ x2 ⎟. ξ = L(x) := ⎜ 1 ⎝ ⎠ −x3 + 2 x1 x2 1 − 12 x4 − 12 x1 x22 + 12 x2 x3 Prove that X1 and X2 are respectively turned6 by L into the following vector fields: 1 := ∂1 − 1 ξ2 ∂3 − 1 ξ22 ∂4 , X 2 12 1 1 1 X2 := ∂2 + ξ1 ∂3 + ξ1 ξ2 − ξ3 ∂4 . 2 12 2 6 I.e. if v = v(ξ ) and u(x) = v(L(x)), then X u(x) = (X i v)(L(x)). i

218


1 and c) Carry out an exercise similar to Ex. 3 for the above vector fields X 4 X2 on R . Verify that the relevant composition law is the same as ∗ in Ex. 5-(a). Remark that the linear change of coordinates in R4 given by (ξ1 , ξ2 , ξ3 , ξ4 ) → (x1 , x2 , x3 , x4 ) := (ξ2 , ξ1 , −ξ3 , −ξ4 ) 1 and X 2 into X2 and X1 , respectively, of Ex. 1. (Why?) turns X Ex. 6) Let α ∈ R be fixed. Consider on R4 the vector fields 1 1 1 Z1 = ∂1 − x2 ∂3 − x3 + x2 (x1 + α x2 ) ∂4 , 2 2 12 1 1 1 Z2 = ∂2 + x1 ∂3 + − α x3 + x1 (x1 + α x2 ) ∂4 . 2 2 12 Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with a suitable dilation. Verify that the relevant composition law is the same as ◦ in Ex. 1-(a) of Chapter 1. Ex. 7) For (x1 , y1 ), (x2 , y2 ) ∈ R2 \ {(x, y) ∈ R2 : xy = 1}, set x1 + x2 y1 + y2 . , (x1 , y1 ) ◦ (x2 , y2 ) := 1 − x1 x2 1 − y1 y2 Prove that ◦ is commutative, associative, has (0, 0) as neutral element and has the inverse (x, y)−1 = (−x, −y). However, prove that there does not exist any ε > 0 such that |x| < ε, |y| < ε implies |(x + y)/(1 − xy)| < ε. This makes it impossible to define a group from ◦. Ex. 8) Consider on R4 the vector fields 1 1 x2 ∂3 + x22 ∂4 , 2 6 1 1 X2 := ∂2 + x1 ∂3 + x3 − x1 x2 ∂4 . 2 6

X1 := ∂1 −

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λ2 x3 , λ3 x4 ). Verify that the relevant composition law is ⎛ ⎞ x 1 + y1 ⎜ ⎟ x2 + y2 ⎟. x◦y =⎜ ⎝ ⎠ x3 + y3 + 12 (x1 y2 − x2 y1 ) 1 x4 + y4 + (x3 y2 − x2 y3 ) + 6 (y2 − x2 ) (x1 y2 − x2 y1 ) Verify that the related Jacobian basis is X1 ,

X2 ,

[X1 , X2 ],

1 − [X2 , [X1 , X2 ]]. 2


219

Ex. 9) Write the following second order constant-coefficient strictly elliptic operator L on R2 as a sum of squares of vector fields L = 10 (∂x1 )2 + 10 ∂x1 ,x2 + 5 (∂x2 )2 . Is L a sub-Laplacian on a suitable homogeneous Carnot group? Which one? Ex. 10) Following the notation of Section 4.1.4, prove that an example of K-type group is given by the choice p0 = p1 = p2 = · · · = pr = 1, B1 = B2 = · · · = Br = (1), whence ⎞ ⎛ 0 0 ··· 0 0 ⎜1 0 ··· 0 0⎟ ⎟ ⎜ ⎜ .. .. ⎟ .. ⎜ . . .⎟ N = p0 + p1 + · · · + pr = 1 + r, B = ⎜0 1 ⎟, ⎟ ⎜ .. .. . . ⎝. . . 0 0⎠ 0 0 ··· 1 0 so that the relevant K-type group B is R2+r (whose points are denoted by (t, x1 , x2 , x3 , . . . , xr+1 )) equipped with the operation ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ s+t s t ⎟ y1 + x1 ⎜ x1 ⎟ ⎜ y 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ y2 + x2 + s x1 ⎜ x2 ⎟ ⎜ y2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ s ⎜ x3 ⎟ ◦ ⎜ y3 ⎟ = ⎜ y + x + s x + x ⎟ 3 3 2 2 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎝ . ⎠ ⎝ . ⎠ ⎝ . ⎠ 2 r xr+1 yr+1 yr+1 + xr+1 + s xr + s2 xr−1 + · · · + sr! x1 and the dilation δλ (t, x1 , x2 , x3 , . . . , xr+1 ) = (λt, λx1 , λ2 x2 , λ3 x3 , . . . , λr+1 xr+1 ). Ex. 11) Following the notation of Section 4.1.4, find an explicit general expression for the composition law on the K-type group such that r = 1 and B1 = Ip0 . Ex. 12) Consider on R4 the vector fields Z1 = ∂x1 + x1 ∂x4 ,

Z1 = ∂x2 ,

Z1 = ∂x3 + x2 ∂x4 .

Prove that they fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x1 , x2 , x3 , x4 ) := (λx1 , λx2 , λx3 , λ2 x4 ). Moreover, [Z1 , Z2 ] = 0 = [Z1 , Z3 ], [Z2 , Z3 ] = ∂x4 . Set Z4 := [Z2 , Z3 ]. Following the notation of Section 4.2, verify that ⎞ ⎛ x 1 + ξ1 ⎟ ⎜ x 2 + ξ2 ⎟, exp(ξ · Z)(x) = ⎜ ⎠ ⎝ x 3 + ξ3 x4 + ξ4 + x1 ξ1 + 12 ξ12 + ξ3 x2 + 12 ξ3 ξ2

220


whence

⎞ ξ1 ⎟ ⎜ ξ2 ⎟, Exp(ξ · Z) = ⎜ ⎠ ⎝ ξ3 ξ4 + 12 ξ12 + 12 ξ3 ξ2 ⎛ ⎞ y1 ⎜ ⎟ y2 ⎟ · Z. Log(y) = ⎜ ⎝ ⎠ y3 y4 − 12 y12 − 12 y3 y2 ⎛

We explicitly remark that Exp(Z1 ) = (1, 0, 0, 1/2) = (1, 0, 0, 0), which shows that the Jacobian basis of g do not necessarily correspond to the canonical basis of G ≡ RN via the exponential map. (But see also Proposition 2.2.22, page 139.) Finally, prove that the relevant composition law is ⎞ ⎛ x1 + y1 ⎟ ⎜ x2 + y2 ⎟. ⎜ ⎠ ⎝ x3 + y3 x4 + y4 + y3 x2 + y1 x1 Instead, consider the basis {X1 , . . . , X4 } of g corresponding to the canonical basis {e1 , . . . , e4 } of R4 via the exponential map, i.e. Xi := Log(ei ) · Z. Verify that X1 = Z1 −

1 Z2 , 2

X2 = Z2 ,

X3 = Z3 ,

X4 = Z4 .

Now, equip R4 with the following composition law: given ξ, η ∈ R4 , we define ξ ∗ η ∈ R4 as the only vector such that Log(Exp(ξ · X) ◦ Exp(η · X)) = (ξ ∗ η) · X. Obviously, (R4 , ∗) is a Lie group isomorphic to (R4 , ◦). However, check out that ⎞ ⎛ ξ1 + η1 ⎟ ⎜ ξ 2 + η2 ⎟, ξ ∗η =⎜ ⎠ ⎝ ξ 3 + η3 ξ4 + η4 − 12 ξ1 − 12 η1 − 12 ξ3 η2 + 12 ξ2 η3 so that (R4 , ∗) is not a homogeneous Carnot group, for the only “dilations” on R4 which are homomorphisms of (R4 , ∗) have the form δλ (x1 , x2 , x3 , x4 ) = (λα1 x1 , λα2 x2 , λα3 x3 , λα4 x4 ) with α4 = α1 = α2 + α3 .


221

Ex. 13) According to the definition given in Section 4.1.5, page 190, write the composition law of the sum of the Heisenberg–Weyl groups H1 and H2 . Do the same for H1 and H1 . The groups obtained are prototype H-type groups according to the definition given in Section 3.6, page 169? Why? Ex. 14) Consider the fields on R2 X1 := ∂1 ,

X2 := x1 ∂2 .

Verify that [X1 , X2 ] = ∂2 , whereas all commutators of length > 2 vanish. It holds W (1) I (x) = span{(1, 0), (0, x1 )},

W (2) I (x) = span{(0, 1)}.

Whence dim(W (1) I (x)) = 2 only when x1 = 0, whereas

dim W (2) I (x) = 1 for every x ∈ R2 , whence hypothesis (H1) is not satisfied, but (H2) is. Moreover, X1 and X2 are homogeneous of degree 1 w.r.t. the “dilation” δλ (x1 , x2 ) := (λx1 , λ2 x2 ). Try to construct formally the Exp, Log and ◦ maps as in Section 4.4.2, page 212, and verify that 1 Exp(ξ1 , ξ2 ) = ξ1 , ξ1 ξ2 , 2 2 x2 2 x1 y2 , x ◦ y = x 1 + y1 , x2 + y2 + . Log(x1 , x2 ) = x1 , x1 y1 Observe that these maps are not everywhere defined, ◦ is not associative and X1 , X2 are not invariant w.r.t. ◦. Ex. 15) Let us consider on R2 the following polynomial vector fields X1 := ∂x1 ,

X2 := ∂x2 + x1 ∂x1 .

It is easily seen that they satisfy hypotheses (H1) and (H2) on page 191, but not7 (H0). We explicitly remark that W (k) = span{X1 } for every k ∈ N, k ≥ 2, whence Lie{X1 , X2 } is not a nilpotent algebra. Prove that the same holds for the group formally related to this case, according to the construction of Section 4.2. Following our usual notation, verify that the following facts hold.8 etξ2 − 1 tξ2 γ (t) = x1 e + ξ1 , x2 + tξ2 ξ2 7 Precisely, the only “dilation” map for which X and X are homogeneous is δ (x , x ) = λ 1 2 1 2 (λα x1 , x2 ) (for every α; X1 is δλ -homogeneous of degree α and X2 of degree 0). 8 We agree to denote by (eξ − 1)/ξ the (analytic) function equal to 1 when ξ = 0 and (eξ − 1)/ξ when ξ = 0.

222


γ˙ = (ξ1 X1 + ξ2 X2 )I (γ ), γ (0) = x, ξ2 e −1 exp(ξ1 X1 + ξ2 X2 )(x1 , x2 ) = ξ1 , ξ2 , ξ2 y2 Log(y1 , y2 ) = y1 y X 1 + y2 X 2 , 2 e −1 x ◦ y = (x1 ey2 + y1 , x2 + y2 ). solves

Moreover, verify that (R2 , ◦) is a Lie group (not nilpotent) and that X1 and X2 Lie-generate the algebra of this group. Ex. 16) Give a complete proof of Proposition 4.3.3. Ex. 17) Let n ∈ N be fixed. Let us consider the Lie algebra kn with a basis {X, Y1 , Y2 , . . . , Yn } with commutator relations [Yi , Yj ] = 0,

1 ≤ i, j ≤ n,

[X, Yj ] = Yj +1 , [X, Yn ] = 0.

1 ≤ j ≤ n − 1,

Then kn is an (n + 1)-dimensional Lie-algebra nilpotent of step n; also, kn is stratified with stratification kn = span{X, Y1 } ⊕ span{Y2 } ⊕ span{Y2 } ⊕ · · · ⊕ span{Yn }. Prove that a model for kn is given by the algebra of vector fields on R1+n (the points are denoted by (x, y) with x ∈ R, y = (y1 , . . . , yn ) ∈ Rn ) spanned by X = ∂x ,

Yj =

n x k−j ∂y , (k − j )! k

j = 1, . . . , n.

k=j

Show that X, Y1 fulfill hypotheses (H0)–(H1)–(H2) of page 191 with the dilation δλ (x, y1 , y2 , . . . , yn ) = (λx, λy1 , λ2 y2 , . . . , λn yn ). Prove that the relevant composition law is (x, y) ◦ (ξ, η) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

x+ξ y 1 + η1 y 2 + η2 + η 1 x 2 y3 + η3 + η2 x + η1 x2! 2 3 y4 + η4 + η3 x + η2 x2! + η1 x3! .. . 2

n−2

n−1

x x + η1 (n−1)! yn + ηn + ηn−1 x + ηn−2 x2! + · · · + η2 (n−2)!

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠


223

Compare to the Bony-type sub-Laplacians in Section 4.3.3, page 202. Another realization of kn as a matrix algebra can be described as follows. Denote by Kn the set of the (n + 1) × (n + 1) matrices of the form ⎛ ⎞ 0 x 0 ··· 0 yn ⎜ ⎟ ⎜ 0 0 x ... 0 yn−1 ⎟ ⎜ ⎟ ⎜ .. .. .. ⎟ .. ⎜. . 0 . . ⎟ ⎜ ⎟ M(x, y) := ⎜ ⎟. . .. x ⎜ 0 y3 ⎟ ⎜ ⎟ ⎜ 0 x y2 ⎟ ⎜ ⎟ ⎝ 0 y1 ⎠ 0 ··· 0 0 Consider the map ϕ : kn → Kn ,

ϕ xX +

n

yj Yj

= M(x, y).

j =1

Then ϕ is a Lie-algebra isomorphism. Now, consider the set Kn := {exp(M) : M ∈ Kn }, where exp denotes the exponential of a square matrix. It is remarkable to observe that (Kn , ·) (where · denotes the usual product of matrices) is a Lie group with the Lie algebra isomorphic to kn . In particular, Kn is nilpotent of step n and a Carnot group, for kn is stratified. Unfortunately, Kn is not a homogeneous group (but, obviously, it is isomorphic to (kn , ), being the Campbell–Hausdorff multiplication, which is a homogeneous Carnot group if we identify kn to Rn+1 in the obvious way). We sketch the verification that Kn is closed under the operation ·, for this involves the Campbell–Hausdorff formula. We aim to prove that if A, A ∈ Kn , then A·A ∈ Kn . To this end, let M, M ∈ Kn be such that A = exp(M), A = exp(M ). Moreover, since ϕ is an isomorphism between kn and Kn , there exist X, X ∈ kn such that M = ϕ(X), M = ϕ(X ). Then A · A = exp(ϕ(X)) · exp(ϕ(X )) 1 = exp ϕ(X) + ϕ(X ) + [ϕ(X), ϕ(X )] + · · · 2 1 = exp ϕ(X + X + [X, X ] + · · ·) 2

since X X ∈ kn . = exp ϕ(X X ) ∈ Kn , In the second equality, we have used the Campbell–Hausdorff formula for the exponential of matrices, which here involves a finite sum, since in Kn

224


we have strictly upper triangular matrices. In the third one, we used the fact that ϕ is a Lie-algebra morphism. Incidentally, we have also proved that exp ◦ϕ : (kn , ) → (Kn , ◦) is a Lie-group isomorphism. Ex. 18) With reference to the preceding exercise (we adopt the therein notation), we study the Lie algebra k3 . We have k3 = span{X, Y1 , Y2 , Y3 } with [X, Y1 ] = Y2 , [X, Y2 ] = Y3 , i, j ∈ {1, 2, 3}. Let us also set ⎧ ⎛ 0 x ⎪ ⎪ ⎨ ⎜0 0 K3 = M(x, y) := ⎜ ⎝0 0 ⎪ ⎪ ⎩ 0 0

0 x 0 0

[X, Y3 ] = 0,

[Yi , Yj ] = 0,

⎫ ⎞ y3 ⎪ ⎪ ⎬ y2 ⎟ 4 ⎟ : (x, y) = (x, y1 , y2 , y3 ) ∈ R . y1 ⎠ ⎪ ⎪ ⎭ 0

Verify that K3 (equipped with the usual bracket of matrices) is a Lie algebra and that the map 3 ϕ : k3 → K3 , ϕ xX + yj Yj = M(x, y) j =1

is a Lie-algebra isomorphism. Now, consider the set K3 := {exp(M) : M ∈ K3 }, where exp denotes the exponential of a square matrix. Verify that ⎛ ⎞ 2 1 x x2! y3 + 2!1 xy2 + 3!1 x 2 y1 ⎜0 1 x ⎟ y2 + 2!1 xy1 ⎟. exp(M(x, y)) = ⎜ ⎝0 0 1 ⎠ y1 0 0 0 1 Verify that (K3 , ·) is a Lie group (here, · is the usual product of matrices), by first checking that

exp(M(x, y)) · exp(M(ξ, η)) = exp M (x, y) ◦ (ξ, η) , where

⎞ x+ξ ⎟ ⎜ y1 + η1 ⎟. (x, y) ◦ (ξ, η) = ⎜ 1 ⎠ ⎝ y2 + η2 + 2 (xη1 − ξy1 ) 1 (x − ξ )(xη1 − ξy1 ) y3 + η3 + 12 (xη2 − ξy2 ) + 12 ⎛

Deduce that (K3 , ·) is isomorphic to the Lie group (k3 , ), where is the Campbell–Hausdorff multiplication.


225

Ex. 19) Analogously to what we did in the preceding two exercises, we study the Lie algebra h1 (related to the Heisenberg group H1 ). We have h1 = span{X, Y, Z} with [X, Y ] = Z, Let us also set

⎧ ⎨

[X, Z] = [Y, Z] = 0. ⎛

0 x H1 = M(x, y, z) := ⎝ 0 0 ⎩ 0 0

⎫ ⎞ z ⎬ y ⎠ : (x, y, z) ∈ R3 . ⎭ 0

Verify that H1 (equipped with the usual bracket of matrices) is a Lie algebra and that the map ϕ : h1 → H1 ,

ϕ(xX + yY + zZ) = M(x, y, z)

is a Lie-algebra isomorphism. Now, consider the set H1 := {exp(M) : M ∈ H1 }, where exp denotes the exponential of a square matrix. Verify that ⎛ ⎞ 1 x z + 12 xy ⎠. exp(M(x, y, z)) = ⎝ 0 1 y 0 0 1 Verify that (H1 , ·) is a Lie group (here, · is the usual product of matrices), by first checking that

exp(M(x, y, z)) · exp(M(ξ, η, ζ )) exp M (x, y, z) ◦ (ξ, η, ζ ) , where

⎛

⎞ x+ξ ⎠. y+η (x, y, z) ◦ (ξ, η, ζ ) = ⎝ z + ζ2 + 12 (xη − ξy)

Deduce that (H1 , ·) is isomorphic to the Lie group (h1 , ), where is the Campbell–Hausdorff multiplication, which, in turn, is isomorphic to the usual Heisenberg–Weyl group H1 on R3 . Ex. 20) We here consider the Lie algebra n4 of the strictly upper triangular matrices of dimension 4. We use the notation ⎫ ⎧ ⎛ ⎞ 0 x 1 x4 x6 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎜ 0 0 x2 x5 ⎟ 6 ⎟ : x = (x1 , . . . , x6 ) ∈ R . n4 = M(x) := ⎜ ⎝ ⎠ 0 0 0 x3 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 Clearly, n4 is a Lie algebra of dimension 6, nilpotent of step 3. Now, consider the set

226


N4 := {exp(M) : M ∈ N4 }, where exp denotes the exponential of a square matrix. Verify that exp(M(x)) ⎛ 1 x1 ⎜0 1 =⎜ ⎝0 0 0 0

x4 + 12 x1 x2 x2 1 0

x6 +

1 2!

(x1 x5 + x3 x4 ) + x5 + 12 x2 x3 x3 1

1 3!

x1 x2 x3

⎞ ⎟ ⎟. ⎠

Verify that (N4 , ·) is a Lie group (here, · is the usual product of matrices), by first checking that exp(M(x)) · exp(M(y)) = exp(M(x ◦ y)), where

⎛

x1 + y1 x2 + y2 x3 + y3 x4 + y4 + 12 (x1 y2 − x2 y1 )

⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟. 1 x◦y =⎜ ⎜ ⎟ x5 + y5 + 2 (x2 y3 − x3 y2 ) ⎜ ⎟ ⎜ x + y + 1 (x y − x y ) + 1 (x y − x y ) ⎟ 6 5 1 3 4 ⎟ ⎜ 6 2 1 5 2 4 3 ⎜ ⎟ 1 (x3 − y3 )(x2 y1 − x1 y2 ) + 12 ⎝ ⎠ 1 + 12 (x1 − y1 )(x2 y3 − x3 y2 ) Deduce that (N4 , ·) is isomorphic to the Lie group (n4 , ), where is the Campbell–Hausdorff multiplication. Verify that the above ◦ defines on R6 a homogeneous Carnot group of step 3 with 3 generators and dilations δλ (x) = (λx1 , λx2 , λx3 , λ2 x4 , λ2 x5 , λ3 x6 ). The first three vector fields of the Jacobian basis are 1 1 1 X1 = ∂x1 − x2 ∂x4 + − x5 + x2 x3 ∂x6 , 2 2 12 1 1 1 X2 = ∂x2 + x1 ∂x4 − x3 ∂x5 − x1 x3 ∂x6 , 2 2 6 1 1 1 x4 + x1 x2 ∂x6 . X3 = ∂x3 + x2 ∂x5 + 2 2 12 The commutator relations are 1 1 x3 ∂x6 , [X2 , X3 ] = ∂x5 + x1 ∂x6 , 2 2 [X1 , [X2 , X3 ]] = ∂x6 = −[X3 , [X1 , X2 ]],

[X1 , X2 ] = ∂x4 −

whereas all other commutators are zero.

5 The Fundamental Solution for a Sub-Laplacian and Applications

In this chapter, we enter the core of the study of the sub-Laplacians L on the homogeneous Carnot groups (and hence on the stratified Lie groups) of homogeneous dimension Q ≥ 3, by showing that they possess a fundamental solution Γ resemblant to the fundamental solution cN |x|2−N of the usual Laplace operator Δ on RN , N ≥ 3. This property is one of the most striking analogies between L and the classical Laplace operator. Indeed, we shall see that it holds Γ = d 2−Q , where Q is the homogeneous dimension of G and d is a symmetric homogeneous norm on G, smooth out of the origin (the relevant definitions will be given in Section 5.1). We shall also call d an L-gauge. To do this, we first fix some results on homogeneous norms and the Carnot– Carathéodory distance. Then, in Section 5.3, we define the fundamental solution Γ , whose existence follows from the hypoellipticity and the homogeneity properties of L. We then collect many of its remarkable properties. As a first application, we provide mean value formulas for L, generalizing to the sub-Laplacian setting the Gauss theorem for classical harmonic functions. These formulas will play a central rôle throughout the book and are proved by only using integration by parts and the coarea theorem (see Theorems 5.5.4 and 5.6.1). From the mean value formulas, we derive Harnack-type inequalities for L and the Brelot convergence property for monotone sequences of L-harmonic functions (see Theorem 5.7.10). Furthermore, as an application of the Harnack theorem, in Section 5.8 we derive several Liouville-type theorems for L. As another application of the properties of the fundamental solution of L, we prove the Sobolev–Stein embedding theorem in the stratified group setting (see Section 5.9). To end with the applications of Γ , we provide three sections devoted to the following topics: some remarks on the analytic-hypoelliptic sub-Laplacians, Lharmonic approximations, and finally an integral representation formula for the fundamental solution by R. Beals, B. Gaveau and P. Greiner [BGG96]. Finally, three appendices close the chapter. The first is devoted to the weak and the strong maximum principles for L. The second one provides an improved ver-

228


sion of the pseudo-triangle inequality. In the third appendix, we prove in details the existence of geodesics on Carnot groups. As a direct application of the maximum principles, we give a decomposition theorem for L-harmonic functions, resemblant to the decomposition of a holomorphic function on an annulus of C into the sum of the regular and singular parts from its Laurent expansion. Convention. Throughout this chapter, we fix a stratified group H of step r and m generators. Q denotes the homogeneous dimension of H. Together with H, a stratification V = (V1 , . . . , Vr ) of the algebra of H will be fixed. Moreover, L will be any sub-Laplacian on H related to the given stratification. We recall that any stratification of the algebra of H brings along a homogeneous Carnot group on RN isomorphic to H. Hence, together with the couple (H, V ), we fix G = (RN , ◦, δλ ), a homogeneous Carnot group isomorphic to H, as described in Proposition 2.2.22. We let Ψ : G → H be the Lie-group isomorphism, as in the cited proposition. We still denote by L the sub-Laplacian on G which is Ψ -related to the sub-Laplacian L on H (see (2.68), page 147). Obviously, the “homogeneous version” G of H depends upon the stratification V but not on the sub-Laplacian L. Thus, any definition and result given henceforward for homogeneous Carnot groups has its counterpart (and is actually intended) for any couple (H, V ), where H is an abstract stratified group and V is a stratification for G. Notation. We introduce the notation for the homogeneous Carnot group G = (RN , ◦, δλ ). Its dilations {δλ }λ>0 are denoted by δλ (x) = δλ (x (1) , . . . , x (r) ) = (λx (1) , . . . , λr x (r) ),

x (i) ∈ RNi ,

1 ≤ i ≤ r.

We denote by m := N1 the number of generators of G and assume that the homogeneous dimension Q = N1 + 2 N2 + · · · + r Nr ≥ 3. As we showed in Chapter 1, Section 1.4 (page 56), the sub-Laplacian L on G can be written as follows (see (1.90a)) L=

m

Xj2 = div(A(x)∇ T ),

j =1

where {X1 , . . . , Xm } is a family of vector fields that form a linear basis of the first layer of g, the Lie algebra of G. The matrix A is given by ⎛ ⎞ (X1 I (x))T . ⎠ .. A(x) = X1 I (x) · · · Xm I (x) · ⎝ T (Xm I (x))

5.1 Homogeneous Norms

229

and takes the following block form (see (1.91))

A1,1 A1,2 A= , A2,1 A2,2 where A1,1 is a strictly positive definite constant m × m matrix. The characteristic form of L N1 qL (x, ξ ) := A(x)ξ, ξ = Xj I (x), ξ 2 ,

x, ξ ∈ RN ,

(5.1a)

j =1

is non-negative definite and, for every fixed x ∈ RN , the set {ξ ∈ RN | qL (x, ξ ) = 0} is a linear space of dimension N − m. The vector-valued operator ∇L := (X1 , . . . , Xm )

(5.1b)

is called the L-gradient operator in G. Due to identity (5.1a), we have |∇L u| = 2

m

|Xj u|2 = A∇ T u, ∇ T u ,

u ∈ C 1 (RN , RN ).

(5.1c)

j =1

5.1 Homogeneous Norms Definition 5.1.1. We call homogeneous norm on (the homogeneous Carnot group) G, every continuous1 function d : G → [0, ∞) such that: 1. d(δλ (x)) = λ d(x) for every λ > 0 and x ∈ G; 2. d(x) > 0 iff x = 0. Moreover, we say that d is symmetric if 3. d(x −1 ) = d(x) for every x ∈ G. Example 5.1.2. Define |x|G :=

r

|x

(j )

|

2r! j

1 2r!

,

x = (x (1) , . . . , x (r) ) ∈ G,

(5.2)

j =1

where |x (j ) | denotes the Euclidean norm on RNj . Then | · |G is a homogeneous norm on G smooth out of the origin. It is symmetric if x −1 = −x for any x ∈ G. In general, if G is any Carnot group (with inverse x −1 not necessarily equal to −x) the map x → |Log (x)|G is a symmetric homogeneous norm on G smooth out of the origin. This follows from the facts that Log (δλ (x)) = δλ (Log (x)) and Log (x −1 ) = −Log (x). 1 With respect to the Euclidean topology.

230


Example 5.1.3 (Control norm). Let d be the control distance related to a system of generators of G (see Section 5.2). Define d0 (x) := d(x, 0),

x ∈ G.

Then (see Theorem 5.2.8 in Section 5.2) we shall see that d0 is a symmetric homogeneous norm on G. From the next (elementary) proposition it will follow that the homogeneous norms on G are all equivalent. Proposition 5.1.4 (Equivalence of the homogeneous norms). Let d be a homogeneous norm on G. Then there exists a constant c > 0 such that c−1 |x|G ≤ d(x) ≤ c |x|G

∀ x ∈ G,

(5.3)

where | · |G has been defined in (5.2). Proof. Due to the δλ -homogeneity of d and | · |G , inequalities (5.3) hold taking c := max{H, 1/ h}, where H := sup{d(x) : |x|G = 1},

h := inf{d(x) : |x|G = 1}.

We explicitly remark that H < ∞ and h > 0, since the set {x : |x|G = 1} is a compact subset of G not containing the origin and d is a continuous function strictly positive in G \ {0}. Corollary 5.1.5. For every fixed (non-necessarily symmetric) homogeneous norm d on G, there exists a constant c > 0 such that c−1 d(x) ≤ d(x −1 ) ≤ c d(x) ∀ x ∈ G.

(5.4)

Proof. The function x → d(x −1 ) is a homogeneous norm on G. Indeed, recall that δλ is an automorphism of G, whence δλ (x −1 ) = (δλ (x))−1 . Then the assertion follows from Proposition 5.1.4. Any homogeneous norm turns out to be locally Hölder continuous with respect to the Euclidean metric in the following sense. Proposition 5.1.6. Let d be a homogeneous norm on G. Then, for every compact set K ⊂ RN , there exists a constant cK > 0 such that d(y −1 ◦ x) ≤ cK |x − y|1/r where r is the step of G.

∀ x, y ∈ K,

(5.5)

5.1 Homogeneous Norms

231

(See also Proposition 5.15.1 (page 309) in Appendix C for an estimate from below of d(y −1 ◦ x).) Proof. Let K ⊂ RN be a compact set. It is easy to see that there exists a constant c = c(K) > 0 such that |x|G ≤ c |x|1/r for every x ∈ K, where | · |G has been defined in (5.2). We now use Proposition 5.1.4, and we obtain that there exists a constant c = c(K) > 0 such that d(x) ≤ c |x|1/r for every x ∈ K. Hence, (5.5) will follow if we prove that there exists a constant c = c(K) > 0 such that |y −1 ◦ x| ≤ c |x − y| for every x, y ∈ K. If we apply the mean value theorem to the function F (x, y) := y −1 ◦ x, we obtain |y −1 ◦ x| = |F (x, y) − F (x, x)| ≤ max JF (x, t x + (1 − t)y) |x − y| ≤ c |x − y|, t∈[0,1]

and the assertion is proved. Any homogeneous norm satisfies a kind of pseudo-triangle inequality. Proposition 5.1.7 (Pseudo-triangle inequalities. I). Let d be a homogeneous norm on G. Then there exists a constant c > 0 such that: 1) d(x ◦ y) ≤ c(d(x) + d(y)), 2) d(x ◦ y) ≥ 1c d(x) − d(y −1 ), 3) d(x ◦ y) ≥ 1c d(x) − c d(y) for every x, y ∈ G. Proof. Due to the δλ -homogeneity of d, inequality 1) is equivalent to the following one d(x ◦ y) ≤ c if d(x) + d(y) = 1. This inequality holds true taking c := max{d(x ◦ y) : d(x) + d(y) = 1}. Obviously, 1 ≤ c < ∞. From inequality 1) we now obtain d(x) = d((x ◦ y) ◦ y −1 ) ≤ c(d(x ◦ y) + d(y −1 )), whence 2). Now, 3) follows from 2) and Corollary 5.1.5 with a suitable change of the constant c. Given a homogeneous norm d0 on G, the function G × G (x, y) → d(x, y) := d0 (y −1 ◦ x) is a pseudometric on G. Indeed, we have the following proposition. Proposition 5.1.8 (Pseudo-triangle inequalities. II). With the above notation, there exists a positive constant c > 0 such that:

232


1) d(x, y) ≤ c d(y, x) for every x, y ∈ G (here c can be taken = 1 iff d0 is also symmetric), 2) d(x, y) ≤ c(d(x, z) + d(z, y)) for every x, y, z ∈ G (the pseudo-triangle inequality for d), 3) d(x, y) = 0 iff x = y. Proof. It immediately follows from Corollary 5.1.5 and Proposition 5.1.7.

5.2 Control Distances or Carnot–Carathéodory Distances We begin with some important definitions. Definition 5.2.1 (X-subunit path). Let X = {X1 , . . . , Xm } be any family of vector fields in RN . A piece-wise regular path γ : [0, T ] → RN is said to be X-subunit if γ˙ (t), ξ 2 ≤

m Xj I (γ (t)), ξ 2

∀ ξ ∈ RN ,

j =1

almost everywhere in [0, T ]. We shall denote by S(X) the set of all X-subunit paths, and we put l(γ ) = T if [0, T ] is the domain of γ ∈ S(X). We explicitly remark that every integral curve of ±Xj (j ∈ {1, . . . , m}) is X-subunit. Convention. We assume RN is X-connected in the following sense (a proof of this fact in the case of stratified vector fields will be given in Theorem 19.1.3 on page 716): For every x, y ∈ RN , there exists γ ∈ S(X),

γ : [0, T ] → RN

such that γ (0) = x and γ (T ) = y.

Then the following definition makes sense. Definition 5.2.2 (X-Carnot–Carathéodory distance). Suppose RN is X-connected. Then, for every x, y ∈ RN , we set

dX (x, y) := inf l(γ ) : γ ∈ S(X), γ (0) = x, γ (T ) = y . (5.6) Under suitable hypotheses, the above inf is actually a minimum. For example, in Appendix C, we show that this occurs if {X1 , . . . , Xm } are generators of the first layer of the stratified algebra of a homogeneous Carnot group (see also [HK00]). Proposition 5.2.3 (dX is a metric). If RN is X-connected, then the function (x, y) → dX (x, y) is a metric on RN , called the X-control distance or the Carnot–Carathéodory distance related to X.

5.2 Control Distances or Carnot–Carathéodory Distances

233

In what follows, when there is no risk of confusion, we shall simply write d instead of dX . Proof. It is quite easy to see that d is non-negative, symmetric and satisfies the triangle inequality. To prove positivity, i.e. d(x, y) = 0 ⇒ (x = y), (5.7) we compare d with the Euclidean metric (which has an interest in its own). Given x ∈ RN and r > 0, define n |Xj I (z)| : |z − x| ≤ r , (5.8a) M(x, r) := sup j =1

where | · | denotes the Euclidean norm. We next show the following inequality M(x, |x − y|) d(x, y) ≥ |x − y|

∀ x, y ∈ RN .

(5.8b)

This will obviously imply (5.7). By contradiction, assume (5.8b) is false for some x and y in RN . Then there exists γ ∈ S(X), γ : [0, T ] → RN , such that γ (0) = x, γ (T ) = y and M(x, |x − y|) T < |x − y|. As a consequence, if we put

t ∗ := sup t ∈ [0, T ] : |γ (s) − x| < |x − y| for 0 ≤ s ≤ t , we have |γ (t ∗ ) − x| =

0

t∗

m γ˙ (s) ds ≤

t∗

|Xj I (γ (s))| ds

j =1 0

≤ M(x, |x − y|) t ∗ ≤ M(x, |x − y|) T < |x − y|. It follows that t ∗ = T and |y − x| = |γ (T ) − x| < |x − y|. This contradiction proves (5.8b). If the vector fields X1 , . . . , Xm are left invariant w.r.t. the translations on a Lie group G = (RN , ◦), then the X = {X1 , . . . , Xm }-control distance has the same property. Indeed, we have the following proposition. Proposition 5.2.4 (Control distance of a left-invariant family). Let G = (RN , ◦) be a Lie group on RN , and let d be the control distance related to a family of left invariant vector fields X = {X1 , . . . , Xm } on G. Then d(x, y) = d(y −1 ◦ x, 0) and

d(x −1 , 0) = d(x, 0)

∀ x, y ∈ G,

(5.9a)

∀ x ∈ G.

(5.9b)

The proof of this proposition will easily follow from the next lemma.

234


Lemma 5.2.5. In the hypotheses of Proposition 5.2.4, let γ : [0, T ] → RN be a X-subunit curve. Then α ◦ γ is X-subunit for every α ∈ G. Proof. If we denote by Γ the path α ◦ γ , we have Γ˙ (s) = Jτα (γ (s)) · γ˙ (s) almost everywhere in [0, T ]. Then, for every ξ ∈ RN , T 2 Γ˙ (s), ξ 2 = γ˙ (s), Jτα (γ (s)) ξ ≤ = =

m j =1 m j =1 m

(since γ is X-subunit)

T 2 Xj I (γ (s)), Jτα (γ (s)) ξ 2 Jτα (γ (s)) · Xj I (γ (s)), ξ

(Proposition 1.2.3, page 14)

m 2 2 Xj I (α ◦ γ (s)), ξ = Xj I (Γ (s)), ξ .

j =1

j =1

This proves that Γ is X-subunit. Proof (of Proposition 5.2.4). Let x, y ∈ G, and let γ be a X-subunit path connecting x and y. For every α ∈ G, by the previous lemma, α ◦ γ is a X-subunit path connecting α ◦ x and α ◦ y. Then d(α ◦ x, α ◦ y) ≤ d(x, y). Since x, y, α are arbitrary, this inequality obviously also implies d(x, y) ≤ d(α ◦ x, α ◦ y). Thus, we have proved d(α ◦ x, α ◦ y) = d(x, y) ∀ x, y, α ∈ G.

(5.10)

Choosing in (5.10) α = y −1 , we obtain (5.9a). Putting x = 0 in (5.9a), we obtain d(y −1 , 0) = d(0, y) = d(y, 0), which is (5.9b). This completes the proof. The control distance related to homogeneous vector fields is homogeneous. More precisely, the following assertion holds. Proposition 5.2.6 (Control distance of a homogeneous family). Let d be the control distance related to a family X = {X1 , . . . , Xm } of smooth vector fields in RN . Assume the Xj ’s are λ -homogeneous of degree one with respect to the “dilations” λ : R N → R N ,

λ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ),

where σ1 , . . . , σN are positive real numbers. Then d(λ (x), λ (y)) = λ d(x, y) ∀ x, y ∈ RN ∀ λ > 0. For the proof of this proposition we need the following lemma.

(5.11)

5.2 Control Distances or Carnot–Carathéodory Distances

235

Lemma 5.2.7. In the hypotheses of Proposition 5.2.6, let γ : [0, T ] → RN be a X-subunit curve. Then, for every λ > 0, the curve Γ : [0, λ T ] → RN ,

Γ (s) := λ (γ (s/λ))

is a X-subunit path. Proof. For every ξ ∈ RN , we have (note that λ is a linear and symmetric map for it is represented by a diagonal matrix) 2 2 Γ˙ (s), ξ 2 = λ−2 λ (γ˙ (s/λ)), ξ = λ−2 γ˙ (s/λ), λ (ξ ) (since γ is X-subunit) ≤ λ−2

m 2 Xj I (γ (s/λ)), λ (ξ ) j =1

= (Corollary 1.3.6, page 35) =

m j =1 m

−1 2 λ λ (Xj I (γ (s/λ))), ξ 2 Xj I (λ (γ (s/λ))), ξ .

j =1

Since λ (γ (s/λ)) = Γ (s), this proves the lemma. Proof (of Proposition 5.2.6). Let x, y ∈ RN , and let γ : [0, T ] → RN be a Xsubunit curve connecting x and y. By the previous lemma, Γ (s) = λ (γ (s/λ)) (0 ≤ s ≤ λ t) is a X-subunit curve, so that, since Γ connects λ (x) and λ (y), d(λ (x), λ (y)) ≤ l(Γ ) = λ T = λ l(γ ). Then, being γ an arbitrary X-subunit curve connecting x and y, d(λ (x), λ (y)) ≤ λ d(x, y) ∀ x, y ∈ RN ∀ λ > 0.

(5.12)

x ) and 1/λ ( y ), respecThis inequality obviously implies (replace x and y with 1/λ ( tively, and then λ with 1/ λ; then remove “∼”) d(x, y) ≤

1 d(λ (x), λ (y)) λ

∀ x, y ∈ RN ∀ λ > 0.

Then (5.12) holds with the equality sign and the proposition is proved. Theorem 5.2.8 (Control distance of a homogeneous Carnot group). Let G be a homogeneous Carnot group on RN , and let d be the control distance related to any family of generators for G. Then G x → d0 (x) := d(x, 0) is a symmetric homogeneous norm on G.

(5.13)

236


Proof. By means of Propositions 5.2.3, 5.2.4 and 5.2.6, we are only left to prove that d0 is continuous. For the proof of this fact, we refer to Theorem 19.1.3 on page 716. Remark 5.2.9. The homogeneous norm (5.13), in general, is not smooth. Corollary 5.2.10. Let G be a Carnot group. Denote by d the control distance related to a family of generators for G. Then, for every compact subset K of G, there exists a positive constant C(K) such that d(x, y) ≤ C(K) |x − y|1/r

∀ x, y ∈ K,

where r denotes the step of G. (See also Proposition 5.15.1 (page 309) in Appendix C for an estimate from below of d(x, y).) Proof. It follows from Propositions 5.1.6, 5.2.4 and Theorem 5.2.8.

5.3 The Fundamental Solution Throughout the sequel, we shall make use of some maximum principles for subLaplacians, which (for the reader’s convenience) we postpone to Appendix A of the present chapter (see Section 5.13). For our purposes, it is convenient to give the definition of fundamental solution of a sub-Laplacian L on a homogeneous Carnot group as follows. Definition 5.3.1 (Fundamental solution). Let G be a homogeneous Carnot group on RN . Let L be a sub-Laplacian on G. A function Γ : RN \ {0} → R is a fundamental solution for L if: (i) Γ ∈ C ∞ (RN \ {0}); (ii) Γ ∈ L1loc (RN ) and Γ (x) −→ 0 when x tends to infinity; (iii) LΓ = −Dirac0 , being Dirac0 the Dirac measure supported at {0}. More explicitly (recall that L∗ = L, being L∗ the formal adjoint of L), Γ Lϕ dx = −ϕ(0) ∀ ϕ ∈ C0∞ (RN ). (5.14) RN

Theorem 5.3.2 (Existence of the fundamental solution). Let L be a sub-Laplacian on a homogeneous Carnot group G (whose homogeneous dimension Q is > 2). Then there exists a fundamental solution Γ for L. (Note. Such a fundamental solution is indeed unique, as it will be proved in Proposition 5.3.10.)

5.3 The Fundamental Solution

237

Proof. The existence of such a fundamental solution may be proved by means of very general arguments from the theory of distributions, based on the hypoellipticity of L and of its formal adjoint L∗ (= L), jointly with the well-behaved homogeneity properties of L. Indeed, from the hypoellipticity of L (see property (A0), page 63) we infer the existence of a “local” fundamental solution satisfying LΓ = −Dirac0 on a neighborhood of the origin (see F. Trèves [Tre67, Theorems 52.1, 52.2]). Moreover, by using the homogeneity properties of L, a “local-to-global” argument can be performed. It is out of our scopes here to give the details. The complete proof is due to G.B. Folland and can be found in [Fol75, Theorem 2.1] (see also L. Gallardo [Gal82] for some further properties of Γ obtained via probabilistic techniques). An alternative proof can be found in [BLU02, Theorem 3.9]. From the integral identity (5.14) and condition (i) in the above Definition 5.3.1 we immediately get the L-harmonicity of Γ out of the origin. Indeed, if we replace in (5.14) a test function ϕ with support in RN \ {0}, by the smoothness of Γ out of the origin, we can integrate by parts obtaining (LΓ ) ϕ dx = 0 ∀ ϕ ∈ C0∞ (RN \ {0}). RN

This obviously implies LΓ = 0 in RN \ {0}.

(5.15)

A simple change of variable and the left-invariance of L w.r.t. the translations on G give the following theorem. Theorem 5.3.3 (Γ left-inverts L). Let L be a sub-Laplacian on a homogeneous Carnot group G. If Γ is a fundamental solution for L, then Γ (y −1 ◦ x) Lϕ(x) dx = −ϕ(y) ∀ ϕ ∈ C0∞ (RN ) (5.16) RN

and every y ∈ RN . Proof. The change of variable z = y −1 ◦ x gives −1 Γ (y ◦ x) Lϕ(x) dx = Γ (z) (Lϕ)(y ◦ z) dz. RN

RN

(5.17)

On the other hand, since L is left-invariant on G, then (Lϕ)(y ◦ z) = L(ϕ(y ◦ z)). Replacing this identity in (5.17) and using (5.14) with ϕ(·) replaced by ϕ(y ◦ ·), one gets the thesis.

238


Remark 5.3.4. The integral identity (5.16) means that L(Γ (y −1 ◦ ·)) = −Diracy in the weak sense of distributions. Here Diracy denotes the Dirac measure supported at {y}. Due to identity (5.16), we can say that Γ is a left inverse of L. We next prove that Γ is a right inverse too. Theorem 5.3.5 (Γ right-inverts L). Let L be a sub-Laplacian on a homogeneous Carnot group G. If Γ is a fundamental solution for L, then, for every ϕ ∈ C0∞ (RN ), the function RN y → u(y) :=

RN

Γ (y −1 ◦ x) ϕ(x) dx

(5.18a)

is smooth and satisfies the equation Lu = −ϕ.

(5.18b)

The proof of this theorem requires some prerequisites. First of all, we note that a change of variable in the integral at the right-hand side of (5.18a) gives Γ (z) ϕ(y ◦ z) dz. u(y) = RN

Then we can differentiate under the integral sign and get the smoothness of u. Moreover, if supp(ϕ) ⊆ {x : d(x) ≤ R} (here d denotes a fixed homogeneous norm on G), then |ϕ(z)| dz |u(y)| ≤ sup{|Γ (z)| : d(y ◦ z) ≤ R} RN =: C(y) |ϕ(z)| dz. (5.19) RN

On the other hand, by Corollary 5.1.5 and Proposition 5.1.7, d(z) ≥

1 d(y) − cd(y ◦ z), c

for a suitable positive constant c independent of x, y, z. As a consequence, 1 C(y) ≤ sup |Γ (z)| : d(z) ≥ d(y) − cR , c so that, since Γ (z) vanishes as z goes to infinity, inequality (5.19) implies lim u(y) = 0.

y→∞

(5.20)

We then show a crucial property of the (ε, G)-mollifiers. The relevant definition is the following one.


239

Definition 5.3.6 (Mollifiers). Let G = (RN , ◦, δλ ) be a homogeneous Carnot group on RN . Let O be a fixed non-empty open neighborhood of the origin 0. Let also be given a function J ∈ C0∞ (RN , R), J ≥ 0, such that supp(J ) ⊂ O and J = 1. RN

For any ε > 0, we set Jε (x) := ε −Q J (δ1/ε (x)). Let u ∈ L1loc (RN ). We set, for every x ∈ RN , uε (x) := (u ∗G Jε )(x) := u(y) Jε (x ◦ y −1 ) dy RN = u(z−1 ◦ x) Jε (z) dz.

(5.21)

δε (O)

We call uε a mollifier of u (or (ε, G)-mollifier) related to the kernel J . Note that this mollifier depends only on G = (RN , ◦, δλ ) and J . For the use of mollifiers in a context of subelliptic PDE’s, see also [CDG97]. Example 5.3.7. Let G = (RN , ◦, δλ ) be a homogeneous Carnot group on RN . Let be a fixed homogeneous symmetric norm on G. We set a notation which will be used throughout the book. For every x ∈ G and every r > 0, we set

B (x, r) := y ∈ G : (x −1 ◦ y) < r . We say that B (x, r) is the -ball with center x and radius r. Also, fixed a point x ∈ G and a set A ⊂ G, we let

-dist(x, A) := inf (x −1 ◦ a) : a ∈ A . We call -dist(x, A) the -distance of x from A. The notation dist (x, A) will also be available. Let now a function J ∈ C0∞ (RN ), J ≥ 0 be given such that J = 1. supp(J ) ⊂ B (0, 1) and RN

For any ε > 0, we set Jε (x) := ε −Q J (δ1/ε (x)). Let u ∈ L1loc (Ω), Ω ⊆ RN open. For the open set Ωε := {x ∈ Ω : -dist(x, ∂Ω) > ε}, we define

uε (x) := (u ∗G Jε )(x) := = B (0,ε)

for every x ∈ Ωε .

B (x, ε)

u(y) Jε (x ◦ y −1 ) dy

u(y −1 ◦ x) Jε (y) dy

(5.22)

240


We call uε a mollifier of u (or (ε, G)-mollifier) related to the homogeneous norm . Note that this mollifier depends only on G = (RN , ◦, δλ ), J and . Remark 5.3.8. Let the notation in Definition 5.3.6 be fixed. Let u ∈ L1loc (RN ). Then the following fact holds: ()

uε → u as ε → 0 in L1loc (RN ).

Indeed, we have (perform the change of variable y = δ1/ε (z)) u(z−1 ◦ x) ε −Q J (δ1/ε (z)) dz = u(δε (y −1 ) ◦ x)J (y) dy. uε (x) = δε (O)

O

J = 1), −1 dx |uε (x) − u(x)| dx = u(δ (y ) ◦ x) J (y) dy − u(x) ε RN O −1 dx = u(δ (y ) ◦ x) − u(x) J (y) dy ε RN O u(δε (y −1 ) ◦ x) − u(x) dx J (y) dy. ≤

As a consequence (recall that RN

O

RN

O

Given σ > 0, by well-known results, there exists ε = ε(σ, G, O) > 0 such that if 0 < ε < ε, then the integral in braces is ≤ σ for every fixed y ∈ O. This proves that |uε (x) − u(x)| dx ≤ σ J (y) dy = σ ∀ 0 < ε < ε, RN

O

i.e. () holds. Proposition 5.3.9 (L-harmonicity of the mollifier). Let G = (RN , ◦, δλ ) be a homogeneous Carnot group on RN . Let L be a sub-Laplacian on G. Let u ∈ L1loc (RN ) be a weak solution to Lu = 0 in RN , i.e. u Lϕ dy = 0 ∀ ϕ ∈ C0∞ (RN ). (5.23) RN

Then, if uε denotes the mollification on G w.r.t. any kernel J (as in Definition 5.3.6), we have (5.24) Luε = 0 in G for every ε > 0. Proof. First, note that, being supp(Jε ) ⊂ δε (O), u(y −1 ◦ x) Jε (y) dy. uε (x) = RN

For every test function ϕ, we thus have (Fubini–Tonelli’s theorem certainly applies)


RN

uε (x) Lϕ(x) dx =

RN

=

RN

=

RN

=

RN

241

Lϕ(x) u(y −1 ◦ x) Jε (y) dy dx N R

Jε (y) (Lϕ)(x)u(y −1 ◦ x) dx dy N R

Jε (y) (Lϕ)(y ◦ z) u(z) dz dy N R

Jε (y) L z → ϕ(y ◦ z) u(z) dz dy. RN

The inner integral in the far right-hand side is equal to zero by the hypothesis (5.23). Then uε (x) Lϕ(x) dx = 0 ∀ ϕ ∈ C0∞ (RN ), RN

so that the claimed (5.24) follows from the hypoellipticity of L and the fact that L∗ = L. With Proposition 5.3.9 at hand, it is easy to prove the uniqueness of the fundamental solution. Proposition 5.3.10 (Uniqueness of the fundamental solution). Let L be a subLaplacian on a homogeneous Carnot group G. The fundamental solution of L (whose existence is granted by Theorem 5.3.2) is unique. Proof. Let Γ and Γ be fundamental solutions for L. Then the function u = Γ − Γ has the following properties: u ∈ L1loc (RN ), u(x) → 0 as x → ∞ and u Lϕ dy = 0 ∀ ϕ ∈ C0∞ (RN ). RN

As a consequence, by Proposition 5.3.9, Luε = 0 in RN for every ε > 0. Thus, since uε (x) → 0 as x → ∞ (argue as in (5.20)), the maximum principle in Section 5.13 implies uε = 0 in RN . On the other hand, uε → u as ε → 0 in L1loc (RN ) (see Remark 5.3.8). Then u = 0 almost everywhere in RN , so that Γ = Γ in RN \ {0}. We now prove that Γ is “G-symmetric” with respect to the origin. More precisely, the following assertion holds. Proposition 5.3.11 (Symmetry of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ (x −1 ) = Γ (x)

∀ x ∈ G \ {0}.

Proof. Given ϕ ∈ C0∞ (RN ), define Γ (y −1 ◦ x) Lϕ(y) dy, u(x) := − RN

x ∈ G.

242


The function u is smooth and vanishes at infinity (see (5.20)). Moreover, for every ψ ∈ C0∞ (RN ), Lu(x) ψ(x) dx = u(x) Lψ(x) dx RN RN

−1 Lϕ(y) Γ (y ◦ x) Lψ(x) dx dy =− N RN R = Lϕ(x) ψ(x) dx (by Theorem 5.3.3). RN

This proves that L(u − ϕ) = 0 in G. Thus, since u − ϕ vanishes at infinity, by the maximum principle, u = ϕ in RN . In particular, ϕ(0) = u(0) = − Γ (y −1 ◦ x)Lϕ(y) dy ∀ ϕ ∈ C0∞ (RN ), RN

so that x → Γ (x −1 ) is a fundamental solution of L (see Definition 5.3.1). The uniqueness of the fundamental solution (Proposition 5.3.10) implies Γ (x −1 ) = Γ (x) for every x ∈ G \ {0}. Finally, we are in the position to prove Theorem 5.3.5. Proof (of Theorem 5.3.5). Let u be the function defined in (5.18a). Then u ∈ C ∞ (RN ) and, for any test function ψ ∈ C0∞ (RN ), one has (Lu)(y) ψ(y) dy = u(y) Lψ(y) dy RN RN

Lψ(y) Γ (y −1 ◦ x) ϕ(x) dx dy = N RN R

ϕ(x) Γ (y −1 ◦ x)Lψ(y) dy dx = N RN R

ϕ(x) Γ (x −1 ◦ y) Lψ(y) dy dx. = RN

RN

Here we used Proposition 5.3.11 ensuring that Γ (y −1 ◦ x) = Γ ((x −1 ◦ y)−1 ) = Γ (x −1 ◦ y). Now, by identity (5.16), the inner integral at the far right-hand side is equal to −ψ(x). Then (Lu)(y) ψ(y) dy = − ϕ(x) ψ(x) dx ∀ ψ ∈ C0∞ (RN ). RN

RN

This gives identity (5.18b).

From the uniqueness of Γ we easily obtain its δλ -homogeneity. Proposition 5.3.12 (δλ -homogeneity of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ is δλ -homogeneous of degree 2 − Q, i.e. Γ (δλ (x)) = λ2−Q Γ (x)

∀ x ∈ G \ {0} ∀ λ > 0.


243

Proof. For any fixed λ > 0, define Γ (x) := λQ−2 Γ (δλ (x))

∀ x ∈ G \ {0}.

It is quite obvious that Γ ∈ C ∞ (RN \ {0}) ∩ L1loc (RN ) and Γ (x) → 0 as x → ∞. Moreover, for every test function ϕ ∈ C0∞ (RN ), Γ (x) Lϕ(x) dx = λQ−2 Γ (δλ (x)) Lϕ(x) dx RN

RN

(by using the change of variable y = δλ (x)) Γ (y) (Lϕ)(δ1/λ (y)) dy = λ−2 RN

(since L is δλ -homogeneous of degree 2) Γ (y)L ϕ(δ1/λ (y)) dy = −ϕ(δ1/λ (0)) = −ϕ(0). = RN

This proves that Γ is a fundamental solution of L. Then, by Proposition 5.3.10, Γ = Γ and the assertion is proved. From the strong maximum principle in Theorem 5.13.8 we obtain another important property of Γ . Proposition 5.3.13 (Positivity of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then Γ (x) > 0 ∀ x ∈ G \ {0}. Proof. Let ϕ ∈ C0∞ (RN ), ϕ ≥ 0. Define Γ (y −1 ◦ x)ϕ(x) dx, u(y) := RN

y ∈ G.

The function u is smooth, vanishes at infinity and satisfies the equation Lu = −ϕ (see (5.20) and Theorem 5.3.5). Then Lu ≤ 0 in G and lim u(y) = 0. y→∞

By the maximum principle in Corollary 5.13.6, it follows that u ≥ 0 in G. Hence Γ (y −1 ◦ x) ϕ(x) dx ≥ 0 ∀ ϕ ∈ C0∞ (RN ), ϕ ≥ 0. RN

Thus, Γ ≥ 0, so that, since it is L-harmonic in G\{0}, the strong maximum principle in Theorem 5.13.8 implies Γ ≡ 0 or Γ (x) > 0 for any x = 0. The first case would contradict identity (5.14). Then Γ > 0 at any point of G \ {0}. This ends the proof.

244


Corollary 5.3.14 (Pole of Γ ). Let L be a sub-Laplacian on a homogeneous Carnot group G. The fundamental solution Γ of L has a pole at 0, i.e. lim Γ (x) = ∞.

x→0

(5.25)

Proof. Since Γ is smooth and strictly positive out the origin, we have h := min{Γ (x) : d(x) = 1} > 0. Here d denotes any fixed homogeneous norm on G. Then, by Proposition 5.3.12, Γ (x) = d(x)2−Q Γ (δ1/d(x) (x)) ≥ h d 2−Q (x). From this inequality (5.25) immediately follows. 5.3.1 The Fundamental Solution in the Abstract Setting The aim of this brief section is to show how to derive a “fundamental solution” for a sub-Laplacian L on an abstract stratified group H, starting from the fundamental on a homogeneous-Carnot-group copy G solution of the related sub-Laplacian L of H. Many other alternative “more intrinsic” definitions may be certainly provided, for example, by making use of the integration on an abstract Lie group. We considered more “in the spirit” of our exposition to pass through the homogeneous group G. Besides, this also makes unnecessary to furnish the (lengthy) theory of integration on manifolds. m 2 Let H be a stratified group with an algebra h. Let L = j =1 Xj be a subLaplacian on H, and let V = (V1 , . . . , Vr ) be the stratification of h related to L, according to Definition 2.2.25, page 144. Let also E be a basis for h adapted to the stratification V . By Proposition 2.2.22 on page 139 (whose notation we presently follow), there exists a homogeneous Carnot group G = (RN , E ) which is isomorphic to H. Let Ψ : G → H be the isomorphism as in Proposition 2.2.22-(1). With the therein notation, we have Ψ = Exp ◦ πE−1 , where, for every ξ ∈ RN , πE−1 (ξ ) is the vector field in h having ξ as N-tuple of the coordinates w.r.t. the basis E. m be the vector fields in g (the algebra of G) which are Ψ -related 1 , . . . , X Let X to X1 , . . . , Xm , respectively, i.e. j ) = Xj dΨ (X

for every j = 1, . . . , m.

We set := L

m j )2 , (X j =1


245

is a sub-Laplacian on the homogeneous Carnot group G) we let Γ and (since L denote its (unique) fundamental solution. A possible definition of a fundamental solution for L is Γ := Γ ◦ Ψ −1 .

(5.26)

Our task here is to show how Γ depends on the (arbitrary) choice of the above basis E. We show that, roughly speaking “up to a multiplicative factor”, Γ in (5.26) is intrinsic (see below for the precise statement). To this aim, let E1 and E2 be two bases of h adapted to the stratification V . With the above notation, let (for i = 1, 2) Gi = (RN , Ei ) and gi denotes the algebra of Gi . Moreover, we set Ψi := Exp ◦ πE−1 , i

i = 1, 2.

(5.27)

Again for i = 1, 2, we also let (i) m (i) , . . . , X X 1

be the vector fields in gi which are Ψi -related to X1 , . . . , Xm , respectively, i.e. (i) ) = Xj dΨi (X j We set i := L

for every j = 1, . . . , m.

m (i) )2 , (X j

(5.28)

i = 1, 2.

j =1

i . We claim that Finally, Γi denotes the fundamental solution of L . Γ2 = c1,2 · Γ1 ◦ (Ψ1−1 ◦ Ψ2 ), where c1,2 := det πE1 ◦ πE−1 2 This will give

Γ2 ◦ Ψ2−1 = c1,2 · Γ1 ◦ Ψ1−1 ,

(5.29) (5.30)

which proves the “almost” intrinsic nature of the definition (5.26) of Γ . We explicitly remark that πE1 ◦πE−1 is a linear automorphism of RN , so that its determinant is well2 posed. (Note. We explicitly remark that the uniqueness of the fundamental solution of L up to a multiplicative factor is a matter of fact. Indeed, even in the homogeneous Carnot setting, the fundamental solution depends on the measure within the integral in (5.14) of Definition 5.3.1. In order to have a unique fundamental solution, the choice of the Lebesgue measure on RN was quite natural (though arbitrary). Instead, in the context of an abstract Lie group H, the Haar measure is unique only up to a multiplicative factor, and there is no effective way to prefer a Haar measure instead of another. These remarks show that the constant in (5.30) is perfectly justified.) We now turn to the claimed (5.29). By invoking Proposition 5.3.10, set Γ1,2 := c1,2 · Γ1 ◦ (Ψ1−1 ◦ Ψ2 ),

246


then (5.29) will follow if we show that Γ1,2 satisfies (i), (ii) and (iii) of Defini2 . tion 5.3.1 w.r.t. the sub-Laplacian L To begin with, observe that (thanks to (5.27)) −1 −1 = πE1 ◦ πE−1 Ψ1−1 ◦ Ψ2 = Exp ◦ πE−1 ◦ Exp ◦ π , (5.31) E 1 2 2 and this last map is a linear isomorphism of RN . As a consequence, since Γ1 ∈ C ∞ (RN \ {0}), we immediately infer that the same holds for Γ1,2 . This proves (i). Moreover, since Γ1 ∈ L1loc (RN ) and Γ1 (x) −→ 0 when x tends to infinity, the same holds for Γ1,2 , again thanks to (5.31). This proves (ii). Finally, we prove (iii). First, note that from (5.28) one gets (1) ) = Xj = dΨ2 (X (2) ) dΨ1 (X j j

for every j = 1, . . . , m,

i.e. for every j = 1, . . . , m, (1) = d(Ψ −1 ◦ Ψ2 )(X (2) ). X j j 1 Set Ψ1,2 := Ψ1−1 ◦ Ψ2 . This gives 1 = dΨ1,2 (L 2 ), L i.e. it holds 1 f ) ◦ Ψ1,2 = L 2 (f ◦ Ψ1,2 ) (L

∀ f ∈ C ∞ (RN , R).

(5.32)

Let now ϕ ∈ C0∞ (RN ). Then we have 2 ϕ) = c1,2 · 2 ϕ)(x) dx Γ1,2 (L Γ1 (Ψ1,2 (x))(L RN

RN

, see (5.31)) (by the linear change of variable y = Ψ1,2 (x) = πE1 ◦ πE−1 2 −1 dy 2 ϕ) Ψ (y) Γ1 (y)(L (see (5.32)) = c1,2 · 1,2 c1,2 RN 1 ϕ ◦ Ψ −1 (y) dy Γ1 (y)L = 1,2 RN

1 and ϕ ◦ Ψ −1 ∈ C ∞ (RN )) (Γ1 is the fundamental solution of L 1,2 0 −1 (0) = −ϕ(0). = − ϕ ◦ Ψ1,2 This proves (iii), and the proof is complete.

5.4 L-gauges and L-radial Functions On every homogeneous Carnot group, there exist distinguished smooth symmetric homogeneous norms playing a fundamental rôle for the sub-Laplacians. We call these norms gauges, according to the following definition.

5.4 L-gauges and L-radial Functions

247

Definition 5.4.1 (L-gauge). Let L be a sub-Laplacian on a homogeneous Carnot group G. We call L-gauge on G a homogeneous symmetric norm d smooth out of the origin and satisfying (5.33) L(d 2−Q ) = 0 in G \ {0}. An L-radial function on G is a function u : G \ {0} → R such that u(x) = f (d(x))

∀ x ∈ G \ {0}

for a suitable f : (0, ∞) → R and a given L-gauge d on G. The L-gauges are deeply related to the fundamental solution of L. Proposition 5.4.2. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let Γ be the fundamental solution of L. Then (Γ (x))1/(2−Q) if x ∈ G \ {0}, d(x) := 0 if x = 0 is an L-gauge on G. Proof. The assertion follows from condition (i) in Definition 5.3.1, (5.15), Propositions 5.3.11, 5.3.12, 5.3.13 and Corollary 5.3.14. In the next section, we shall show the reverse part of Proposition 5.4.2 (see Theorem 5.5.6): if d is an L-gauge on G, then there exists a positive constant βd such that Γ = βd d 2−Q is the fundamental solution of L. As a consequence, by Proposition 5.3.10, the L-gauge is unique up to a multiplicative constant. In Section 9.8, we shall also prove the following fact: if d is a homogeneous norm on G, smooth out of the origin and such that L(d α ) = 0 in G \ {0} for a suitable α ∈ R, α = 0, then α = 2 − Q and d is an L-gauge on G (see Corollary 9.8.8, page 461, for the precise statement). The sub-Laplacian of an L-radial function takes a noteworthy form. Proposition 5.4.3. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let f (d) be a smooth L-radial function on G \ {0}. Then

Q−1 L(f (d)) = |∇L d|2 f (d) + f (d) , (5.34) d 2 where, if L = m j =1 Xj , we set ∇L = (X1 , . . . , Xm ).

248


Proof. An easy computation gives (see also Ex. 6, Chapter 1) L(f (d)) =

m

Xj2 (f (d))

m m 2 = f (d) (Xj d) + f (d) Xj2 (d),

j =1

so that

j =1

j =1

L(f (d)) = f (d) |∇L d|2 + f (d) L(d).

(5.35)

(Note that (5.35) holds with the only assumption that f and d are smooth functions on some open subsets of G and R, respectively, and f ◦ d is defined.) Applying this formula to the function f (s) = s 2−Q and keeping in mind (5.33), we obtain 0 = (1 − Q) d −Q |∇L d|2 + d 1−Q L(d), hence

Q−1 . d Identity (5.34) follows by replacing this last identity in (5.35). L(d) = |∇L d|2

If is any (sufficiently smooth) homogeneous norm on G, the integration of a -radial function over a “-radially-symmetric” domain reduces to an integration of a single variable function. Proposition 5.4.4. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let be any homogeneous norm on G smooth on G \ {0}. Let f () be a function defined on the -ball B (0, r) of radius r centered at the origin, B (0, r) := {x ∈ G : (x) < r}. Then, if f () ∈ L1 (B (0, r)), it holds f ((x)) dx = Q ω

r

s Q−1 f (s) ds,

(5.36)

0

B (0,r)

where ω denotes the Lebesgue measure of B (0, 1), ω := |B (0, 1)|. Proof. The coarea formula gives f ((x)) dx = B (0,r)

0

r

f (s)

{=s}

1 N −1 ds. dH |∇ |

(5.37)

On the other hand, by using the δλ -homogeneity of , we have

r 1 dH N −1 ds = |B (0, r)| = ω r Q 0 {=s} |∇| for every r > 0. Differentiating this last identity with respect to r, we obtain 1 dH N −1 = Qω r Q−1 . (5.38) {=r} |∇| By using this identity in (5.37), we immediately get (5.36).

5.4 L-gauges and L-radial Functions

249

Corollary 5.4.5. Let L be a sub-Laplacian on a homogeneous Carnot group G. Let be any homogeneous norm on G. The function α is locally integrable in RN if and only if α > −Q. Proof. If is also smooth on G \ {0}, by Proposition 5.4.4, r α (x) dx = Qω s α+Q−1 ds, 0

B (0,r)

and the assertion trivially follows. If is only continuous, we argue as follows. If α ≥ 0, the assertion is trivial. If α < 0, we have N α (x) dx = α (x) dx B (0,r)

k+1 k k=0 {r/2 ≤ 0, t ∈ Rn . Assume v is smooth. Then B (k) x, ∇x u(x, t) = B (k) x, x/|x| ∂r v(|x|, t) = 0, since B (k) x, x = 0 for every x ∈ Rm . As a consequence, keeping in mind (3.14) and (3.16), we obtain the following form of ΔH and |∇H | for cylindrically-symmetric functions u(x, t) = v(|x|, t): m−1 1 vr + r 2 Δt , r 4 1 |∇H u|2 = vr2 + r 2 |∇t v|2 . 4 ΔH u = vrr +

(5.39g)

Here, r = |x| and vr = δr v, vrr = δrr v.

5.5 Gauge Functions and Surface Mean Value Theorem Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an Lgauge. The aim of this section is to prove a mean value theorem on the boundary of the d-balls for the L-harmonic functions. When L is the classical Laplace operator, our result will give back the Gauss theorem for classical harmonic functions. We shall obtain the mean value theorem as a byproduct of a representation formula for general

252


C 2 functions. These representation formulas will play a major rôle throughout the book. For any x ∈ G and r > 0, we recall that we have already defined the d-ball of center x and radius r as follows: Bd (x, r) := {y ∈ G : d(x −1 ◦ y) < r}.

(5.40)

Then Bd (x, r) = x ◦ Bd (0, r). By using the translation invariance of the Lebesgue measure and the δλ -homogeneity of d, one also easily recognizes that |Bd (x, r)| = r Q |Bd (0, 1)| =: ωd r Q .

(5.41)

We explicitly remark that ∂Bd (x, r) := {y ∈ G : d(x −1 ◦ y) = r} is a smooth manifold of dimension N − 1. Indeed, by Sard’s lemma, this holds true for almost every r > 0. The assertion then follows for every r > 0, since ∂Bd (x, r) is diffeomorphic to ∂Bd (x, 1) via the dilation δr . (Note that, so far, d may be any homogeneous norm smooth out of the origin.) Definition 5.5.1 (The kernels of the mean value formulas). Let L be a subLaplacian on the homogeneous Carnot group G, and let d be an L-gauge. We set, for x ∈ G \ {0}, ΨL (x) := |∇L d|2 (x). Moreover, for every x, y ∈ G with x = y, we define the functions ΨL (x, y) := ΨL (x −1 ◦ y)

and KL (x, y) :=

|∇L d|2 (x −1 ◦ y) . |∇(d(x −1 ◦ ·))|(y)

(5.42)

Remark 5.5.2. We explicitly remark that ΨL is δλ -homogeneous of degree zero, a fact which will be used repeatedly. We would like to recall that ΨL appears in the “radial” form of L, see (5.34). We also explicitly remark that, while ΨL is translation-invariant (i.e. ΨL (α ◦ x, α ◦ y) = ΨL (x, y)), the function KL does not necessarily share the same property. Moreover, when L = Δ is the classical Laplace operator, then ΨL = KL = 1. For a general sub-Laplacian L, we shall prove the following fact (see Proposition 9.8.9, page 462): The function ΨL is constant if and only if G is the Euclidean group.

5.5 Gauge Functions and Surface Mean Value Theorem

253

For instance, consider the following example. Example 5.5.3. Let H = (Rm+n , ◦, δλ ) be a group of Heisenberg type. Denoting by (x, t) the points of H, x ∈ Rm , t ∈ Rn , we proved in Example 5.4.7 that the “Folland function” d(x, t) := (|x|4 + 16 |t|2 )1/4 is a ΔH -gauge. Using (5.39g), we obtain ΨH (x, t) = |∇H d(x, t)|2

6 2 2 4 r r rs r s2 = + 16 3 = + 16 4 , d d d d4 d where r = |x| and s = |t|. Then ΨH (x, t) =

|x|2 |x|4 + 16 |t|2

,

(x, t) = (0, 0).

Let Ω ⊆ G be an open set and u, v ∈ C 2 (Ω). By using the divergence form of the sub-Laplacian L (see (1.90a) on page 64), L = div(A(x) · ∇ T ), we easily get v Lu − u Lv = div(v A · ∇ T u) − div(u A · ∇ T v).

(5.43a)

Let us now assume that Ω is bounded with boundary ∂Ω of class C 1 and exterior normal ν = ν(y) at any point y ∈ ∂Ω. Then, if u, v are of class C 2 in a neighborhood of Ω, integrating (5.43a) on Ω and using the divergence theorem, we obtain the Green identity2 v A · ∇ T u, ν − u A · ∇ T v, ν dH N −1 . (5.43b) (v Lu − u Lv) dH N = Ω

∂Ω

Hereafter, dH N (respectively, dH N −1 ) stands for the N -dimensional (respectively, (N − 1)-dimensional) Hausdorff measure in RN . If we choose v ≡ 1 in (5.43b), we 2 Another way to write the Green identity is the following one: if L = m X 2 , we have j =1 j

Ω

(v Lu − u Lv) dH N =

∂Ω

⎛ ⎝v

m j =1

Xj u Xj I, ν − u

m

⎞ Xj v Xj I, ν ⎠dH N −1 .

j =1

This follows from (5.43b), recalling that (see (1.90b), page 64) A is the N × N symmetric matrix A(x) = σ (x) σ (x)T , where σ (x) is the N × m matrix whose columns are X1 I (x), . . . , Xm I (x).

254


obtain3

Lu dH

N

=

Ω

A · ∇ T u, ν dH N −1 ,

(5.43c)

∀ u ∈ C0∞ (Ω).

(5.43d)

∂Ω

so that

Lu dH N = 0 Ω

Let us now consider an arbitrary open set O ⊆ RN such that Bd (x, r) ⊂ O for a suitable r > 0. For 0 < ε < r, we define the “d-ring” Dε,r := Bd (x, r) \ Bd (x, ε) = {y ∈ RN : ε < d(x −1 ◦ y) < r}. Given u ∈ C 2 (O), we apply the Green identity (5.43b) to the functions u and v := d 2−Q (x −1 ◦ ·) on the open set Dε,r . Since v is L-harmonic in G \ {0} (note that here we apply, for the first time and with crucial consequences, the fact d is an L-gauge), we obtain v Lu = Sr (u) − Sε (u) + Tε (u) − Tr (u), (5.43e) Dε,r

where we have used the following notation v A · ∇ T u, ν dH N −1 , Sρ (u) := ∂Bd (x,ρ) u A · ∇ T v, ν dH N −1 . Tρ (u) := ∂Bd (x,ρ)

Since v is constant on ∂Bd (x, ρ), keeping in mind (5.43c), we have 2−Q T N −1 2−Q A · ∇ u, ν dH =ρ Lu dH N , (5.43f) Sρ (u) = ρ ∂Bd (x,ρ)

Bd (x,ρ)

so that, by means of (5.41), Sε (u) = ε 2 O(|Lu|) −→ 0,

as ε → 0.

(5.43g)

To evaluate Tρ (u), we first remark that on ∂Bd (x, ρ) one has ν=

∇(d(x −1 ◦ ·)) , |∇(d(x −1 ◦ ·))|

and A · ∇ T (d(x −1 ◦ ·)), ∇ T (d(x −1 ◦ ·)) |∇(d(x −1 ◦ ·))| −1 ΨL (x ◦ ·) . (see (5.1c) and (5.42)) = (2 − Q) ρ 1−Q |∇(d(x −1 ◦ ·))|

A · ∇ T v, ν = (2 − Q) d 1−Q (x −1 ◦ ·)

3 Or, equivalently (see the previous note),

Ω

Lu dH N =

m

∂Ω j =1

Xj u Xj I, ν dH N −1 .

5.5 Gauge Functions and Surface Mean Value Theorem

Therefore, keeping in mind the second definition in (5.42), 1−Q Tρ (u) = (2 − Q) ρ u(y) KL (x, y) dH N −1 (y),

255

(5.43h)

∂Bd (x,ρ)

so that Tε (u) = (u(x) + o(1)) Tε (1),

as ε → 0.

(5.43i)

To compute Tε (1), we observe that (5.43e) with u = 1 gives Tε (1) = Tr (1) for 0 < ε < r < ∞. Hence, for every ε > 0,

Tε (1) = T1 (1) = (2 − Q) ∂Bd (x,1)

KL (x, ·) dH N −1 .

(5.43j)

(5.43k)

Finally, since v = d 2−Q (x, ·) ∈ L1 (Bd (x, r)) (see Corollary 5.4.5) lim v Lu dH N = v Lu dH N . ε→0 Dε,r

D0,r

Therefore, as ε → 0, identity (5.43e) together with (5.43f)-(5.43k) give v Lu dH N = r 2−Q Lu dH N + T1 (1) u(x) Bd (x,r) Bd (x,r) − (2 − Q) r 1−Q u KL (x, ·) dH N −1 .

(5.43l)

∂Bd (x,r)

We now observe that

T1 (1) = (2 − Q) ∂Bd (x,1)

does not depend on x, i.e.

(Q − 2) ∂Bd (x,1)

KL (x, ·) dH N −1

KL (x, ·) dH N −1

= (Q − 2) ∂Bd (0,1)

Indeed, from (5.43j) we have 1 1 = T1 (1) Tr (1) r Q−1 dr Q 0 1 = (2 − Q)

KL (0, ·) dH N −1 =: (βd )−1 .

(5.43m)

KL (x, y) dH N −1 (y) dr (by the coarea formula) = (2 − Q) ΨL (x, y) dH N −1 (y) Bd (x,1) = (2 − Q) ΨL dH N . 0

∂Bd (x,r)

Bd (0,1)

256


Here we used the very definition (5.42) of the functions ΨL and KL and the left invariance of ΨL . Incidentally, we have also proved that −1 (βd ) = Q(Q − 2) ΨL dH N . (5.43n) Bd (0,1)

From identity (5.43l), moving terms around, we obtain (Q − 2) βd u(x) = u KL (x, ·) dH N −1 r Q−1 ∂Bd (x,r) − βd (d 2−Q (x −1 ◦ ·) − r 2−Q ) Lu dH N .

(5.44)

Bd (x,r)

We have thus proved the following fundamental result. Theorem 5.5.4 (Surface mean value theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge on G. Let O be an open subset of G, and let u ∈ C 2 (O, R). Then, for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O, we have u(x) = Mr (u)(x) − Nr (Lu)(x), where

(5.45)

(Q − 2) βd Mr (u)(x) = KL (x, z) u(z) dH N −1 (z), r Q−1 ∂Bd (x,r) Nr (w)(x) = βd (d 2−Q (x −1 ◦ z) − r 2−Q ) w(z) dH N (z),

(5.46)

Bd (x,r)

and βd and KL are defined, respectively, in (5.43m) and (5.42). In particular, if Lu = 0, i.e. u is L-harmonic in O, we have u(x) = Mr (u)(x).

(5.47) RN ,

Remark 5.5.5. When L = Δ is the classical Laplace operator in N ≥ 3, the kernel KL is constant (KL ≡ 1). Then, in this case, 1 N −1 Mr (u)(x) = u(y) dH (y) =: − − u(y) dH N −1 y σN r N −1 |x−y|=r |x−y|=r (see (5.43m) and (5.46)) and (5.47) gives back the Gauss theorem for classical harmonic functions. From Theorem 5.5.4, we straightforwardly also obtain the following result (see also Corollary 9.8.8 on page 461 for yet another improvement). Theorem 5.5.6 (“Uniqueness” of the L-gauges. I). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let d be an L-gauge on G, and let βd be the positive constant defined in (5.43m). Then Γ = βd d 2−Q is the fundamental solution of L.

(5.48)

5.6 Superposition of Average Operators

257

Proof. Let ϕ ∈ C0∞ (Ω) and choose r > 0 such that supp(ϕ) ⊂ Bd (0, r). Then, by the mean value formula (5.45) (being u ≡ 0 on ∂Bd (0, r)), ϕ(0) = −βd (d 2−Q (z) − r 2−Q ) Lϕ(z) dH N (z). Bd (0,r)

On the other hand, by identity (5.43d), r 2−Q Lϕ dH N = 0. Bd (0,r)

Then, if Γ is the function defined in (5.48), Γ ∈ L1loc (RN ) thanks to Corollary 5.4.5 and −ϕ(0) = Γ (z) Lϕ(z) dH N (z) RN

C0∞ (Ω).

Moreover, Γ is smooth in G \ {0} and Γ (z) → 0 as z → ∞, for every ϕ ∈ since Q − 2 > 0. Thus, by Definition 5.3.1, Γ is the fundamental solution of L. From Theorem 5.5.4 we obtain the following asymptotic formula for L. Theorem 5.5.7 (Asymptotic surface formula for L). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open, and let u ∈ C 2 (Ω, R). Then, for every x ∈ Ω, we have lim

r→0+

Mr (u)(x) − u(x) = ad Lu(x), r2

(5.49a)

where ad := βd

(d 2−Q (y) − 1) dy.

(5.49b)

Bd (0,1)

Proof. The change of variable z = x ◦ δr (y) in the integral defining Nr gives Nr (1)(x) = ad r 2

for every x ∈ G.

As a consequence, if u ∈ C 2 (Ω, R), from (5.45) we obtain Mr (u)(x) − u(x) = Nr (Lu)(x) = Nr (Lu − Lu(x))(x) + Nr (1)(x) Lu(x) = ad r 2 (Lu(x) + o(1)), as r → 0+ . This proves (5.49a).

5.6 Superposition of Average Operators. Solid Mean Value Theorems. Koebe-type Theorems As in the previous section, L and d will respectively denote a sub-Laplacian and an L-gauge on the homogeneous Carnot group G. We shall denote by Bd (x, r) the

258


d-ball with center x ∈ G and radius r ≥ 0 and by Q the homogeneous dimension of G. We shall also assume, as usual, Q ≥ 3. Given an open set O ⊆ RN and a real number r > 0, we let Or := {x ∈ O | d-dist(x, ∂O) > r}, where

d-dist(x, ∂O) := inf d(y −1 ◦ x). y∈∂O

Since the d-balls are connected, one easily recognizes that Bd (x, ρ) ⊆ O for every x ∈ O and 0 < ρ < d-dist(x, ∂O). It follows that Bd (x, ρ) ⊆ O

∀ x ∈ Or and 0 < ρ ≤ r.

1 Let ϕ : R → R be an L -function vanishing out of the interval ]0, 1[ and such that R ϕ = 1. For r > 0, define

1 t ϕ , r r

ϕr (t) :=

t ∈ R.

Let us now consider a function u ∈ C 2 (O). Then, if x ∈ Or , from (5.45) we obtain u(x) = Mρ (u)(x) − Nρ (Lu)(x)

for 0 < ρ ≤ r.

We now multiply both sides of this identity times ϕr (ρ) and integrate with respect to ρ. We thus get u(x) = Φr (u)(x) − Φr∗ (Lu)(x),

where

∞

Φr (u)(x) :=

x ∈ Or ,

(5.50a)

ϕr (ρ) Mρ (u)(x) dρ

(5.50b)

ϕr (ρ) Nρ (w)(x) dρ.

(5.50c)

0

and Φr∗ (w)(x) :=

∞

0

By using the coarea formula, the average operator Φr can be written as follows (we agree to let u = 0 out of O) Φr (u)(x) = u(z) φr (x −1 ◦ z) dz (5.50d) RN

with

φr (z) := r −Q φ δ1/r (z)

and φ(z) := (Q − 2)βd ΨL (z) We note that φ vanishes out of Bd (0, 1) and

ϕ(d(z)) . d(z)Q−1

(5.50e)


RN

∞

φ(z) dz = (Q − 2)βd 0

∞

=

ϕ(ρ) ρ Q−1

d(z)=ρ

259

|∇L d(z)|2 dH N −1 (y) dρ |∇d(z)|

ϕ(ρ) dρ = 1.

0

Here we used (5.42), (5.43h), (5.43k) and (5.43m). If the function ϕ is smooth and its support is contained in ]0, 1[, then φ ∈ C0∞ (RN ), supp(φ) ⊆ Bd (0, 1) and x → Φr (u)(x) is a smooth map in Or , whenever u is just an L1loc (O)-function. We would like to explicitly remark that in the integral (5.50d) the function z → φr (x −1 ◦ z) vanishes out of Bd (x, r). As a consequence, if x ∈ Or , that integral is performed on a compact set contained in O, since Bd (x, r) ⊆ O. If we choose Q t Q−1 if 0 < t < 1, ϕ(t) = 0 otherwise, then

Φr (u) = Mr (u) and Φr∗ (w) = Nr (w),

where Mr (u)(x) := and nd Nr (w)(x) := Q r

Bd (x,r)

r

ρ 0

md rQ

Q−1

ΨL (x −1 ◦ y) u(y) dy,

(5.50f)

2−Q −1 2−Q d w(y) dy dρ, (x ◦ y) − ρ

Bd (x,ρ)

(5.50g)

being md := Q(Q − 2)βd and nd := Q βd .

(5.50h)

Hence, from (5.50a), we obtain the following theorem. Theorem 5.6.1 (Solid mean value theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge on G. Let O be an open subset of G, and let u ∈ C 2 (O, R). Then, for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O, we have u(x) = Mr (u)(x) − Nr (Lu)(x), where

(5.51)

260


md Mr (u)(x) = Q ΨL (x −1 ◦ y) u(y) dH N (y), r Bd (x,r)

2−Q −1 nd r Q−1 d ρ (x ◦ y) − ρ 2−Q w(y) dy dρ, Nr (w)(x) = Q r Bd (x,ρ) 0 and md , nd and ΨL are defined in (5.50h) (see also (5.43m)) and (5.42). In particular, if Lu = 0, i.e. u is L-harmonic in O, we have u(x) = Mr (u)(x).

(5.52)

Remark 5.6.2. When L = Δ is the classical Laplace operator in RN , N ≥ 3, one has ΨL ≡ 1. Then, in this case, 1 Mr (u)(x) = u(y) dy =: −− u(y) dH N (y) ωN r N |x−y| 0 such that Bd (x, r) ⊂ O. Then u ∈ C ∞ (O) and

Lu = 0 in O.

Proof. Assume condition (i) is satisfied. Then u(x) = Mρ (u)(x) for 0 < ρ ≤ r. (5.53) Let ϕ ∈ C0∞ (]0, 1[, R) be such that R ϕ = 1. Multiply both sides of (5.53) times ϕr (ρ) = ϕ(ρ/r)/r. An integration with respect to ρ gives u(x) = Φr (u)(x)

∀ x ∈ Or ,

where Φr (u) is the integral operator (5.50b). From (5.50b) we get u(x) = u(z) φr (x −1 ◦ z) dz, O

where φr (z) = r −Q φ(δ1/r (z)) and φ is the smooth function defined by (5.50e). It follows that u ∈ C ∞ (O). Condition (i) and identity (5.45) now give Nr (Lu)(x) = 0


261

for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O. Since the kernel appearing in the integral operator Nr is strictly positive, this implies Lu = 0 in O. Thus, the theorem is proved if condition (i) is fulfilled. Let us now assume (ii). Since ρ → Mρ (u)(x) is continuous on ]0, r] and r Q Mr (u)(x) = Q ρ Q−1 Mρ (u)(x) dρ, r 0 from (ii) we get Qr Q−1 u(x) =

d Q d (r u(x)) = Q dr dr

r

ρ Q−1 Mρ (u)(x) dρ

0

= Qr Q−1 Mr (u)(x). Hence u(x) = Mr (u)(x) for every x ∈ O and r > 0 such that Bd (x, r) ⊂ O. Then u satisfies condition (i), so that u ∈ C ∞ (O) and Lu = 0 in O. Remark 5.6.4 (Another Koebe-type result). In the hypotheses of Theorem 5.6.3, if u : O → R is continuous and satisfies u(x) = Φr (u)(x)

∀ r > 0 ∀ x ∈ Or ,

where Φr is the integral operator in (5.50d) related to a smooth function ϕ ∈ C0∞ (]0, 1[, R) (see also (5.50b)), then u ∈ C ∞ (O). As a consequence, by identity (5.50a), Φr∗ (Lu) = 0 in Or for every r > 0. It follows that Lu = 0 in O.

From Theorem 5.6.1 we obtain another asymptotic formula for the sub-Laplacians. Theorem 5.6.5 (Asymptotic solid formula for L). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open, and let u ∈ C 2 (Ω, R). Then, for every x ∈ Ω, we have lim

r→0+

Mr (u)(x) − u(x) = ad Lu(x), r2

where ad = Q ad /(Q + 2) (and ad is as in (5.49b)). Proof. From (5.50g) we obtain (by recalling the definition (5.49b) of ad )

2−Q Q βd r Q+1 Nr (1)(x) = Q ρ (y) − 1 dy dρ = r 2 ad . d r 0 Bd (0,1) Then, by Theorem 5.6.1 (arguing as in the proof of Theorem 5.5.7), we get Mr (u)(x) − u(x) = ad r 2 (Lu(x) + o(1)), and (5.54) follows.

as r → 0+ ,

(5.54)

262


5.7 Harnack Inequalities for Sub-Laplacians In this section, we shall prove some type of Harnack inequalities for sub-Laplacians. Our main tool will be the solid average operator Mr in (5.50f). Let d be an L-gauge on G, let cd be the positive constant of the pseudo-triangle inequality (see Proposition 5.1.8) d(x −1 ◦ y) ≤ cd d(x −1 ◦ z) + d(z−1 ◦ y) ∀ x, y, z ∈ G, (5.55) and let ΨL = |∇L d|2 be the kernel appearing in the average operator Mr in (5.50f). The following lemma will play a fundamental rôle in this section: it will allow us to compare the average on different d-balls of a given non-negative function. Lemma 5.7.1. Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. For every r > 0, there exists a point z0 = z0 (r) ∈ G satisfying the following conditions: (i) d(z0 ) = λ r, (ii) ΨL (x −1 ◦ y) ≥ μ and ΨL (y −1 ◦ x) ≥ μ for every x ∈ Bd (z0 , r) and every y ∈ Bd (0, 2 cd r). Here λ ≥ 1 and μ > 0 are real constants independent of r (depending only on G, d and L). Proof. We split the proof into three steps. (I) The set {y ∈ G \ {0} | ΨL (y) = 0} has empty interior. Indeed, suppose by contradiction ΨL (y) = 0 for every y in a neighborhood U of a suitable y0 = 0. Then, since ΨL = |∇L d|2 , this gives ∇L ΨL ≡ 0 on U , so that, by Proposition 1.5.6 (page 69), d is constant in an open set containing y0 . As a consequence, the function r → d(δr (y0 )) = r d(y0 ) is constant near r = 1. This implies d(y0 ) = 0, which is a contradiction with the assumption y0 = 0. (II) From step (I) and the homogeneity of ΨL we infer the existence of a point y0 ∈ G, d(y0 ) = 1, such that ΨL (y0 ) > 0 and ΨL (y0−1 ) > 0. Then, by the continuity of ΨL out of the origin, there exist two positive constants σ and μ (with σ ≤ 1) such that ΨL (ξ ◦ y0 ◦ η) ≥ μ,

ΨL (ξ ◦ y0−1 ◦ η) ≥ μ

(5.56)

for every ξ , η ∈ G satisfying the inequalities d(ξ ), d(η) ≤ 2 cd σ . (III) For any fixed r > 0, we let z0 := δλ r (y0 ) with λ = 1/σ . Let x ∈ Bd (z0 , r) and y ∈ Bd (0, 2 cd r). From the second inequality in (5.56) we obtain ΨL (x −1 ◦ y) = ΨL (x −1 ◦ z0 ) ◦ z0−1 ◦ y (by the homogeneity of ΨL ) = ΨL δ1/(λ r) (x −1 ◦ z0 ) ◦ y0−1 ◦ δ1/(λ r) (y) ≥ μ,

5.7 Harnack Inequalities

263

since d(δ1/(λ r) (x −1 ◦ z0 )) ≤ σ and d(δ1/(λ r) (y)) ≤ 2 cd σ . Analogously, from the first inequality in (5.56) we obtain ΨL (y −1 ◦ x) = ΨL y −1 ◦ z0 ◦ (z0−1 ◦ x) = ΨL δ1/(λ r) (y −1 ) ◦ y0 ◦ δ1/(λ r) (z0−1 ◦ x) ≥ μ, since d(δ1/(λ r) (z0−1 ◦ x)) ≤ σ and d(δ1/(λ r) (y −1 )) ≤ 2 cd σ . To state the next theorem, it is convenient to introduce the following constants θ0 := cd (1 + cd (1 + λ)),

θ := cd (λ + θ0 ).

(5.57)

Theorem 5.7.2 (Non-homogeneous Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open. Finally, let x0 ∈ Ω and r > 0 be such that Bd (x0 , θ r) ⊂ Ω (here θ is as in (5.57), see also Lemma 5.7.1 and (5.55)). Then, for any p ∈ (Q/2, ∞], we have (5.58) sup u ≤ c inf u + r 2−Q/p Lu Lp (B (x , θ r)) d 0 B (x , r) Bd (x0 ,r)

d

0

for every non-negative smooth function u : Ω → R. Here c is a positive constant depending only on G, d, L and p and not depending on u, r, x0 and Ω. Proof. Since L is left invariant, we may assume x0 = 0. We split the proof in several steps. Mr will be the average operator in (5.50f) and z0 = z0 (r) the point of G given by Lemma 5.7.1. (I) There exists an absolute constant4 c > 0 such that Mr (u)(x) ≤ c Mθ0 r (u)(z0 )

∀ x ∈ Bd (0, r).

(5.59)

Indeed, for every x ∈ Bd (0, r), we have Bd (x, r) ⊆ Bd (0, 2 cd r) ⊆ Bd (0, θ0 r) ⊂ Ω, whence (being u non-negative) Mr (u)(x) ≤

sup ΨL

G\{0}

m d

rQ

u(z) dz

Bd (x,r)

(by the first inequality in Lemma 5.7.1-(ii)) c1 ΨL (z0−1 ◦ z) u(z) dz, ≤ Q r Bd (x,r) where c1 = md /μ supG\{0} ΨL . We remark that c1 < ∞, since ΨL is smooth out of the origin and δλ -homogeneous of degree zero. Then, since we also have 4 Hereafter, we call absolute constant any positive real constant independent of u, r, x 0

and Ω.

264


Bd (x, r) ⊆ Bd (z0 , θ0 r) ⊆ Bd (0, θ r) ⊂ Ω, we get (again by the non-negativity of u) c1 c1 Mr (u)(x) ≤ Q ΨL (z0−1 ◦ z) u(z) dz = Mθ0 r (u)(z0 ). r md Bd (z0 ,θ0 r) This proves (5.59) with c = c1 /md . (II) There exists an absolute constant c > 0 such that Mr (u)(z0 ) ≤ c Mθ0 r (u)(y)

∀ y ∈ Bd (0, r).

(5.60)

This inequality can be proved just by proceeding as in the previous step, by using the second inequality in Lemma 5.7.1-(ii) and the inclusions Bd (z0 , r) ⊆ Bd (y, θ0 r) ⊆ Bd (0, θ r) ⊂ Ω. (III) Let finally x, y ∈ Bd (0, r). Then, by repeatedly using the solid mean value theorem 5.6.1, we have u(x) = Mr (u)(x) − Nr (Lu)(x) ≤ = =

(by (5.59))

cMθ0 r (u)(z0 ) − Nr (Lu)(x) c u(z0 ) + Nθ0 r (Lu)(z0 ) − Nr (Lu)(x) c Mr (u)(z0 ) − Nr (Lu)(z0 ) + Nθ0 r (Lu)(z0 ) − Nr (Lu)(x).

On the other hand, from (5.60), Mr (u)(z0 ) ≤ c Mθ0 r (u)(y) = c u(y) − Nθ0 r (Lu)(y) . By using this last estimate in the previous one, we infer that u(x) is bounded from above by a suitable absolute constant c times u(y) + |Nθ0 r (Lu)(y)| + |Nr (Lu)(z0 )| + |Nθ0 r (Lu)(z0 )| + |Nr (Lu)(x)| for all x, y ∈ Bd (0, r). An elementary computation based on Hölder’s inequality shows that |Nr (w)(x)| ≤ cp r 2−Q/p w

Lp (Bd (x, r))

(5.61)

for Q/2 < p ≤ ∞, p = p/(p − 1) and

1/p cp := nd (d(z)2−Q − 1) dz . d(z) 0 such that Bd (x0 , θ r) ⊂ Ω. The constant c depends only on G, L and d and does not depend on u, r, x0 and Ω. By using a covering argument, from the non-homogeneous Harnack inequality of Theorem 5.7.2 one obtains the following theorem. Theorem 5.7.4 (Non-homogeneous, non-invariant Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω be an open subset of G, and let K and K0 be compact and connected subsets of Ω such that K ⊂ int(K0 ). Then, for every p ∈ (Q/2, ∞], there exists a positive constant c = c(K, K0 , Ω, L, d, p, Q) such that sup u ≤ c inf u+ Lu Lp (K ) (5.63) 0 K K

for every u ∈ C ∞ (Ω, R), u ≥ 0. Proof. Let {Dj | j = 1, . . . , q} be a finite family of d-balls Dj = Bd (xj , rj ) such that (here θ is as in (5.57)): q (i) K ⊂ j =1 Dj , (ii) θ Dj := Bd (xj , θ rj ) ⊂ K0 for any j = 1, . . . , q, (iii) Dj ∩ Dj +1 = ∅ for every j ∈ {1, . . . , q − 1}. We explicitly remark that such a covering exists, since K is compact, connected and contained in the interior of K0 . By Theorem 5.7.2, we have sup u ≤ c inf u+ Lu Lp (θ Dj ) , j = 1, . . . , q, Dj

Dj

for every u ∈ C ∞ (Ω, R), u ≥ 0. The constant c is independent of u. Then, inequality (5.63) will follow by a repeated application of the following elementary lemma. Lemma 5.7.5. Let A1 and A2 be arbitrary sets such that A1 ∩ A2 = ∅. Suppose u : A1 ∩ A2 → R is a non-negative function satisfying (5.64) sup u ≤ c inf u + Li , i = 1, 2, Ai

Ai

266


for suitable constants c≥1 and Li ≥ 0, i = 1, 2. Then sup u ≤ c2 inf u + L1 + L2 . A1 ∪A2

A1 ∪A2

Proof. We have to show that u(x) ≤ c{u(y) + L1 + L2 }

(5.65)

for every x, y ∈ A1 ∪ A2 . Now, if x, y ∈ A1 or x, y ∈ A2 , then inequality (5.65) directly follows from (5.64). Suppose x ∈ A1 and y ∈ A2 and choose a point z ∈ A1 ∩ A2 . By hypothesis (5.64), u(x) ≤ c {u(z) + L1 }

and u(z) ≤ c {u(y) + L2 }.

Hence u(x) ≤ c {c (u(y) + L2 ) + L1 }, and (5.65) follows. From Theorem 5.7.4 one obtains the following improvement of Theorem 5.7.2. Corollary 5.7.6 (Non-homogeneous invariant Harnack inequality). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let Ω ⊆ G be open, and let r, R and R0 be real constants such that 0 < r < R < R0 . Assume r R , ≤ ρ and Bd (x0 , R0 ) ⊂ Ω R R0 for suitable ρ < 1 and x0 ∈ Ω. Then, for every p ∈ (Q/2, ∞], there exists a constant c > 0 such that sup u ≤ c inf u + R 2−Q/p Lu Lp (B (x , R)) d 0 B (x , r) Bd (x0 ,r)

d

(5.66)

0

for every u ∈ C ∞ (Ω, R), u ≥ 0. The constant c depends only on G, L, d, p and ρ and does not depend on u, r, R, R0 , Ω and x0 . Proof. Since L is left invariant, we may assume x0 = 0. Let us put uR (x) := u(δR (x)),

x ∈ δ1/R (Ω).

Then, by applying Theorem 5.7.4 to the function uR , the compact sets K = Bd (0, ρ) and K0 = Bd (0, 1) and the open set Bd (0, 1/ρ) (which is contained in Bd (0, R0 /R) ⊆ δ1/R (Ω)), we obtain sup uR ≤ c inf uR + LuR Lp (Bd (0,1)) Bd (0,ρ)

Bd (0,ρ)

=c

inf uR + R 2−Q/p Lu Lp (Bd (0,R)) ,

Bd (0,ρ)

5.7 Harnack Inequalities

267

with c > 0 depending only on the parameters in the assertion of the corollary. Note that we have also used the δλ -homogeneity of L (of degree 2). From this inequality (5.66) follows, since sup uR = Bd (0,ρ)

sup Bd (0,R ρ)

u ≥ sup u and Bd (0,r)

inf uR ≤ inf u.

Bd (0,ρ)

Bd (0,r)

This ends the proof. Theorem 5.7.4 and the δλ -homogeneity of L easily imply the following Harnack inequality on rings. Corollary 5.7.7 (Harnack inequality on rings). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let u be a smooth nonnegative function on the ring

AR,c := x ∈ G | cR < d(x) < R/c , where R > 0 and 0 < c < 1. Let 0 < a < b < c. Then, for every p ∈ (Q/2, ∞], there exists a constant c > 0 such that sup u ≤ c inf u + R 2−Q/p Lu Lp (A ) . (5.67) R,b AR,a AR,a The constant c depends only on G, L, p, d, a, b and c and does not depend on u and R. Proof. The change of variable x → δR (x) reduces (5.67) to the analogous inequality with R = 1. This last one follows from Theorem 5.7.4. Remark 5.7.8 (The Harnack inequality in the abstract setting). According the convention in the incipit of the chapter, given an abstract stratified group H, via the isomorphism Ψ between H and a homogeneous Carnot group G, all the Harnack inequalities of the present section do possess a counterpart in H. For example, it suffices to consider the results in Remark 2.2.28 in order to derive the following result from Theorem 5.7.4. Theorem 5.7.9 (A Harnack inequality in the abstract setting). Let H be an abstract stratified group, and let L be a sub-Laplacian on H. Let Ω be an open subset of H, and let K be a compact and connected subset of Ω. Then there exists a positive constant c = c(H, L, Ω, K) such that sup u ≤ c inf u, K

K

for every non-negative function u ∈ C ∞ (Ω, R) satisfying Lu = 0 in Ω. We close this section by giving the following “monotone convergence” theorem.

268


Theorem 5.7.10 (The Brelot convergence property). Let H be an abstract stratified group, and let L be a sub-Laplacian on H. Let Ω ⊆ H be open and connected. Let {un }n∈N be a sequence of L-harmonic functions in Ω, i.e. un ∈ C ∞ (Ω, R) and Lun = 0 in Ω

for every n ∈ N.

Assume the sequence {un }n∈N is monotone increasing and sup{un (x0 )} < ∞

(5.68)

n∈N

at some point x0 ∈ Ω. Then there exists an L-harmonic function u in Ω such that {un }n∈N is uniformly convergent on every compact subset of Ω to u. Proof. By the results in Remark 2.2.28, it suffices5 to consider the case when L is a sub-Laplacian on a homogeneous Carnot group G. Let K be a compact subset of Ω. Since Ω is connected, there exists a compact and connected set K ∗ such that K ⊆ K ∗ ⊂ Ω and x0 ∈ K0 . Then, by Theorem 5.7.4, sup(un − um ) ≤ sup(un − um ) ≤ cinfK ∗ (un − um ) K

K∗

≤ c(un (x0 ) − um (x0 ))

for every n ≥ m ≥ 1.

The constant c is independent of n and m. Then, by condition (5.68), {un }n is uniformly convergent on K. Since K is an arbitrary compact subset of Ω, {un }n∈N is locally uniformly convergent to a continuous function u : Ω → R. On the other hand, by the solid mean value Theorem 5.6.1, for every x ∈ Ω and r > 0 such that Bd (x, r) ⊂ Ω, we have un (x) = Mr (un )(x)

∀ n ∈ N.

Letting n tend to infinity (by the uniform convergence un → u), we get u(x) = Mr (u)(x)

∀ x ∈ Ω, r > 0 : Bd (x, r) ⊂ Ω,

and now the Koebe-type Theorem 5.6.3 implies u ∈ C ∞ (Ω, R) and Lu = 0 in Ω.

5 Indeed, following the notation in Remark 2.2.28 and in the assertion of the above theorem,

the following facts hold: the (abstract) sub-Laplacian L is Ψ -related to the sub-Laplacian on G; set := Ψ −1 (Ω), is open and connected L un := un ◦ Ψ , Ω x0 := Ψ −1 (x0 ), then Ω in G (recall that Ψ is a homeomorphism), the sequence { un }n∈N is monotone increasing un = 0 on Ω. Finally, if K is a compact subset of H, then bounded in on Ω, x0 and L := Ψ −1 (K) is a compact subset of G, and K un − u|. sup |un − u| = sup | K

K

5.8 Liouville-type Theorems

269

The above Brelot convergence property implies the following strong minimum principle (see also Section 5.13 for a more exhaustive investigation of maximum– minimum principles). Corollary 5.7.11 (Strong minimum principle). Let L be a sub-Laplacian on an abstract stratified group H. A non-negative solution to Lu = 0 on an open connected set Ω ⊆ H vanishes identically iff it vanishes at a point. Proof. Apply the result of Theorem 5.7.10 to the sequence {n · u | n ∈ N}.

5.8 Liouville-type Theorems The classical Liouville theorem for entire harmonic functions also holds in the subLaplacian setting. Indeed, the Harnack inequality of Corollary 5.7.3 implies the following theorem: Theorem 5.8.1 (Liouville theorem for sub-Laplacians). Let H be an abstract stratified group. Let L be a sub-Laplacian on H. Let u ∈ C ∞ (H, R) be a function satisfying u ≥ 0 and Lu = 0 in H. Then u is constant. Proof. By the results in Remark 2.2.28, it suffices6 to consider the case when L is a sub-Laplacian on a homogeneous Carnot group G. Define m := inf u and v := u − m. G

Then v ≥ 0 and Lv = 0 in G. From Harnack inequality (5.62) we obtain sup v ≤ c Bd (0,r)

inf v,

Bd (0,r)

c independent of r.

From this inequality, letting r tend to infinity, we obtain 0 ≤ sup v ≤ c inf v = 0, G

which implies v ≡ 0 and u ≡ m.

G

Theorem 5.8.1 can be viewed as a particular case of the following stronger version of the Liouville property for L. 6 Indeed, the (abstract) sub-Laplacian L is Ψ -related to the sub-Laplacian L on G and, set

u = 0 on G. Once we have proved that u := u ◦ Ψ , we have u ≥ 0 and L u is constant on G, the same follows for u = u ◦ Ψ −1 .

270


Theorem 5.8.2 (Liouville-type theorem-polynomial lower bound). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be an entire solution to Lu = 0, i.e. a function u ∈ C ∞ (G, R) such that Lu = 0 in G. Assume there exists a polynomial function p on G such that u≥p

in G.

Then7 u is a polynomial function and degG u ≤ degG p. The inequality degG u ≤ degG p can be strict: take, for example, u ≡ 0, p = −x12 , so that degG u = 0 < 2 = degG p. Remark 5.8.3. All the results in this section involving polynomial functions are obviously related to the fixed coordinates on a homogeneous Carnot group. Abstract counterparts of these results are available in the obvious way. A polynomial function on the abstract stratified group H is a function P : H → R such that p := P ◦ Exp is a polynomial function on the vector space h (the Lie algebra of H), i.e. p is a polynomial function when expressed in coordinates w.r.t. any (or equivalently, w.r.t. at least one) basis for h. All the details are left to the reader. Theorem 5.8.2 is an easy consequence of the following one. Theorem 5.8.4 (Liouville-type theorem-polynomial sub-Laplacian). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be a smooth function on G satisfying u ≥ 0 and Lu = w in G, where w is a polynomial function. Then u is a polynomial function and degG u ≤ 2 + degG w. More precisely,

degG u =

degG w 2 + degG w

if degG u = 0, if degG u ≥ 2.

(5.69)

The case degG u = 1 cannot occur, since a polynomial of G-degree 1 cannot be a non-negative function. The proof of this theorem will follow from a representation formula that we shall deduce from identity (5.50a) in Section 5.6. We first show how Theorem 5.8.2 can be obtained from Theorem 5.8.4. Proof (of Theorem 5.8.2). We first recall that L is a differential operator, δλ -homogeneous of degree two, and its coefficients are polynomial functions. Then, if u ≥ p and Lu = 0, we have u − p ≥ 0 and L(u − p) = w, 7 See Definition 1.3.3 on page 33 for the definition of the G-degree deg (p) of a polynomial G

p and the G-length |α|G of the multi-index α.


271

where w := −Lp is a polynomial whose G-degree does not exceed max{0, degG p − 2}. From Theorem 5.8.4 it follows that u − p is a polynomial and that degG (u − p) ≤ 2 + degG (w) ≤ 2 + max{0, degG p − 2} = max{2, degG p}. Hence, u = (u − p) + p is a polynomial function and its G-degree does not exceed max{2, degG p}. If degG p ≥ 2, this gives the assertion of Theorem 5.8.2. It remains to consider the cases degG p = 0 and degG p = 1 (indeed, in these cases the above argument only proves that degG u ≤ 2, which is weaker than degG u ≤ degG p). In both cases, Lp ≡ 0, so that the hypotheses of Theorem 5.8.2 rewrites as u − p ≥ 0,

L(u − p) = 0

in G.

Hence, by Liouville Theorem 5.8.1, u − p is constant, whence u = (u − p) + p = p + constant, so that u is a polynomial function of the same G-degree as p. We next prove a representation formula having its own interest and useful for the proof of Theorem 5.8.4. Proposition 5.8.5. Let L be a sub-Laplacian on the homogeneous Carnot group (G, ∗), and let d be an L-gauge on G. Let u ∈ C ∞ (G, R) be such that Lu = w

in G,

(5.70a)

(α)

(5.70b)

where w is a polynomial function. Then u(x) = Φr (u)(x) −

CQ w (α) (x) r 2+|α|G

|α|G ≤m

for any x ∈ G and r > 0. Φr denotes the integral operator (5.50b) related to a smooth function ϕ ∈ C0∞ ((0, 1), R). Moreover, m is the G-degree of w, w (α) (x) := Dyα |y=0 w(x ∗ y) (α)

and, for any α with |α|G ≤ m, CQ is a positive constant depending only on α, Q and d. In particular, w (α) is a polynomial function with G-degree not exceeding m − |α|G . Proof. Identity (5.50a) in Section 5.6 and hypothesis (5.70a) give u(x) = Φr (u)(x) − Φr∗ (w)(x),

272


where Φr∗ (w)(x) =

∞

ϕr (ρ) Nρ (w)(x) dρ, and (d 2−Q (x −1 ∗ y) − ρ 2−Q ) w(y) dH N (y) Nρ (w)(x) = βd Bd (x,ρ) (d 2−Q (y) − ρ 2−Q ) w(x ∗ y) dH N (y). = βd 0

Bd (0,ρ)

We now claim that w(x ∗ y) =

w (α) (x) yα , α!

(5.71)

|α|G ≤m

where w (α) = Dyα |y=0 w(x ∗ y) is a polynomial function of G-degree ≤ m − |α|G . Taking this claim for granted for a moment, we have Φr∗ (w)(x)

∞ w (α) (x) ρ dρ βd = ϕ (d 2−Q (y) − ρ 2−Q ) y α dH N (y) α! r r 0 Bd (x,ρ) |α|G ≤m

(by using the change of variables y = δρ (z) and ρ = r σ ) (α) = CQ w (α) (x) r 2+|α|G , |α|G ≤m

where (α) CQ

βd = α!

∞

0

ϕ(σ )

(d

2−Q

(z) − 1) z dH (z) dσ. α

N

Bd (0,1)

Then, we are left with the proof of (5.71). Since w is a polynomial function and (x, y) → x ∗ y has polynomial components too, one has w(x ∗ y) = cα,β x α y β |α|G +|β|G ≤n

for a suitable positive integer n and real constants cα,β . We have to prove only that n ≤ m. Now, since degG w ≤ m, w(z) = cγ zγ , cγ ∈ R for any γ . |γ |G ≤m

Then, since δλ is an automorphism of the group G, cα,β λ|α|G +|β|G x α y β = w(δλ (x) ∗ δλ (y)) |α|G +|β|G ≤n

= w(δλ (x ∗ y)) =

cγ λ|γ |G (x ∗ y)γ

|γ |G ≤m

for every x, y ∈ G and λ > 0. As a consequence,


273

cα,β x α y β = 0 ∀ x, y ∈ G,

m m, we have X β u(x) = X β Φr (u)(x) ∀ x ∈ G. Then, since the Xj ’s are left-invariant on (G, ∗) and δλ -homogeneous of degree one, we have (see also (5.50d) and (5.50e)) X β u(x) = u(y) X β φr (x −1 ∗ y) dy G −|β| (y −1 ∗ x) dy, =r u(y) X β φ r RN

(z) := where φ

φ(z−1 )

and

β = r −Q X β φ ∗ δ1/r . X φ r

Hence, X β u(x) = r −|β|

(z) dz. u(x ∗ δr (z−1 )) X β φ

(5.73)

Bd (0,1)

Using inequality (5.72) in (5.73), we obtain |X β u(x)| ≤ c r −|β| (1 + |x| + r)m+2 for every x ∈ G and r > 0. The constant c depends on u(0), but it is independent of x and r. Letting r tend to infinity, we obtain X β u(x) = 0

∀x ∈ G

and for every β with |β| > m + 2. By the cited Corollary 1.5.5, this implies that u is a polynomial function of G-degree ≤ m + 2.

274


Using the same argument as above, we can prove the following improvement of Theorem 5.8.4. Theorem 5.8.6 (Liouville: polynomial sub-Laplacian and bound). Let L be a subLaplacian on the homogeneous Carnot group G. Let u : G → R be a smooth function satisfying u ≥ p and Lu = w in G, where p and w are polynomial functions. Then u is a polynomial and degG u ≤ max{degG p, 2 + degG w}. Proof. Set v = u − p. We have v ≥ 0,

Lv = w − Lp

in G.

Since L has polynomial coefficients, we can apply Theorem 5.8.4 to derive degG (u − p) ≤ 2 + degG (w − Lp) ≤ 2 + max{degG w, degG Lp} max{degG w, degG p − 2} if degG p ≥ 2, ≤ 2+ max{degG w, 0} if degG p ≤ 1 max{2 + degG w, degG p} if degG p ≥ 2, = 2 + degG w if degG p ≤ 1 = max{degG p, 2 + degG w}. Summing up all the above statements, we obtain the following Liouville theorem “of polynomial type”. Theorem 5.8.7 (Liouville-type theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u : G → R be smooth and such that u≥p

and Lu = w

in G,

where p and w are polynomial functions. Then u is a polynomial, and if w ≡ 0, degG p degG u ≤ max{degG p, 2 + degG w} otherwise. 5.8.1 Asymptotic Liouville-type Theorems We close this section by giving some more “asymptotic” Liouville-type theorems, easy consequences of Theorem 5.8.2. Theorem 5.8.8 (Asymptotic Liouville. I). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be an entire solution to Lu = 0. Assume there exists a real number m ≥ 0 such that u(x) = O(m (x)),

as x → ∞.

(5.74)


275

Then u is a polynomial function and degG u ≤ [m].

(5.75)

Here is (any fixed) homogeneous norm on G and [m] denotes the integer part of m, i.e. [m] ∈ Z and [m] ≤ m < [m] + 1. Proof. By condition (5.74) and Proposition 5.1.4, we get u(x) ≥ p(x) where

p(x) = −c 1 +

r

∀ x ∈ G, [m]+1 |x

|

(j ) 2 r!/j

,

j =1

and c is a suitable positive constant. Then, by Theorem 5.8.2, u is a polynomial function. Let n := degG u. Assume, by contradiction, n ≥ [m] + 1. Writing u(x) =

n

uk (x),

k=0

where uk is δλ -homogeneous of degree k, from condition (5.74) we get n k=0

u(x) (x)k−n uk δ1/(x) (x) = −→ 0, (x)n

as (x) → ∞.

Hence, un (y) = 0 for every y ∈ G such that (y) = 1, which implies degG u ≤ n−1, a contradiction. Then n ≤ [m] and the proof is complete. Theorem 5.8.8 together with Theorem 5.6.1 give the following corollary. Corollary 5.8.9 (Asymptotic Liouville. II). Let L be a sub-Laplacian on the homogeneous Carnot group G, and let d be an L-gauge. Let u be an entire solution to Lu = 0. Assume there exists x0 ∈ G and a real number m ≥ 0 such that |u(y)| dy = O(r m ), as r → ∞. (5.76) − Bd (x0 ,r)

Then u is a polynomial function and degG u ≤ [m]. In (5.76) we have set

1 − := . |D| D D

276


Proof. Let x ∈ G \ {0} and put r = max{d(x), d(x0 )}. By the pseudo-triangle inequality (5.55), Bd (x, r) ⊆ Bd (x0 , 2 cd r). Then, from the solid mean value Theorem 5.6.1, we obtain |u(x)| = |Mr (u)(x)| ≤ Mr (|u|)(x) md |u(y)| dy, ≤ Q sup ΨL r Bd (x0 ,2cd r) G\{0} |u(x)| ≤ c −

so that

|u(y)| dy,

Bd (x0 ,2cd r)

being c > 0 independent of x and r. This inequality and (5.76) give u(x) = O(r m ) = O(d(x)m ),

as x → ∞.

Then, by Theorem 5.8.8, u is a polynomial function with the G-degree ≤ [m].

Remark 5.8.10. By using Proposition 5.1.4, one easily recognizes that condition (5.76) in the previous corollary is equivalent to the following one |u(y)| dy = O(r m+Q ), as r → ∞, (5.77) Br

where Br := {y ∈ G : (y) < r} is the ball of radius r centered at the origin, with respect to a homogeneous norm on G. Corollary 5.8.11 (Asymptotic Liouville. III). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let u be an entire solution to Lu = 0, and 1 ≤ p ≤ ∞. Assume (with the notation of Remark 5.8.10) u Lp (Br ) = O(r m+Q/p ),

as r → ∞.

(5.78)

Then u is a polynomial function of the G-degree not exceeding [m]. Proof. The Hölder inequality gives |u(y)| dy ≤ c r Q(1−1/p) u Lp (Br ) . Br

Then (5.78) implies (5.77), and the assertion follows from Remark 5.8.10 and Corollary 5.8.9.

5.9 Some Results on G -fractional Integrals and the Sobolev–Stein Embedding Inequality Let L be a sub-Laplacian on the homogeneous Carnot group G of homogeneous dimension Q. Let 0 < α < Q. Given a function f : G → R, we formally define

5.9 Sobolev–Stein Embedding Inequality

Iα (f )(x) :=

G

277

f (y) dy, (d(x, y))Q−α

where d(x, y) stands for d(y −1 ◦ x) and z → d(z) is an L-gauge function on G. By analogy with the Euclidean setting, we shall call Iα the G-fractional integral of order α. The following theorem, when G is the usual Euclidean group (RN , +), gives back a celebrated theorem by Hardy, Littlewood and Sobolev. Theorem 5.9.1 (Hardy–Littlewood–Sobolev for sub-Laplacians). Let L be a subLaplacian on the homogeneous Carnot group G, and let d be an L-gauge. Suppose 1 < α < Q and 1 < p < Q α . Let q > p be defined by 1/q = 1/p − α/Q. Then there exists a positive constant C = C(α, p, G, L, d) such that Iα (f )q ≤ C f p

for every f ∈ Lp (G).

Here, we use the notation · r to denote the Lr norm in G ≡ RN with respect to the Lebesgue measure. In the classical Euclidean setting, several proofs of this theorem are known. The simplest proof seems to be due to L.I. Hedberg [Hed72] and it makes use of the maximal function theorem by Hardy–Littlewood–Wiener. This last theorem holds in general metric spaces equipped with a doubling measure.8 In particular, it holds in our context. Fixed a sub-Laplacian L and an L-gauge d, we define the L-maximal function ML (f ) of a function f ∈ Lp (G, C), 1 < p < ∞, as follows ML (f )(x) := sup − |f (y)| dy, x ∈ G, r>0

Bd (x,r)

−

1 = . |D| D D The function x → ML (f )(x) is lower semicontinuous. Indeed, if ML (f )(x) > α, there exists r > 0 such that −Bd (x,r) |f (y)|dy > α. Then, since |f (y)| dy x → − where we used the notation

Bd (x,r)

is a continuous function (see Ex. 5 at the end of this chapter), there exists δ > 0 such that 8 A Radon measure μ on a quasi-metric space (X, d) is doubling if there exists a positive

constant Cd such that

0 < μ(B(x, 2r)) ≤ Cd μ(B(x, r))

for every d-ball with center at x and radius r.

278


ML (f )(z) ≥ −

if d(x −1 ◦ z) < δ.

|f (y)| dy > α

Bd (z,r)

The L-maximal function theorem is the following one. L-Maximal Function Theorem. Let 1 < p < ∞. With the above notation, there exists a positive constant C = C(p, G, L, d) such that ML (f )p ≤ C f p

for every f ∈ Lp (G, C).

A proof of this theorem in general doubling metric spaces (which is out of our scopes here) can be found in the monograph [Ste81, Chapter 2], by E.M. Stein. Starting from this result, we now prove Theorem 5.9.1 by using the idea in L.I. Hedberg’s paper [Hed72]. Proof (of Theorem 5.9.1). For every fixed t > 0 and x ∈ G, we have

+ |f (y)| (d(x, y))α−Q dy Iα (f )(x) = =:

d(x,y)≤t d(x,y)≥t (t) (t) Iα (f )(x) + Eα (f )(x).

The Hölder inequality gives (t) Eα (f )(x) ≤ f p

1−1/p (α−Q)p/(p−1)

(d(x, y))

d(x,y)≤t

= (see (5.36)) C f p

∞

dy

1−1/p

τ Q−1+(α−Q)p/(p−1) dτ

t

= C f p t α−Q/p . On the other hand, one has Iα(t) (f )(x)

=

∞ −k−1 0. We now choose

x ∈ G,

5.9 Sobolev–Stein Embedding Inequality

t=

f p ML (f )(x)

so that

(p α)/Q

Iα (f )(x) ≤ C f p

Hence, being q (1 − pα/Q) = p, we have q Iα (f )q

≤

(pqα)/Q Cf p

279

p/Q ,

(ML (f )(x))1−(p α)/Q . G

(ML (f )(x))p dx

(by the L-maximal function theorem) (p q α)/Q

≤ Cf p

p

q

f p = Cf p ,

since (p q α)/Q + p = q. The theorem is proved. From this theorem and the representation formula (5.16), one easily obtains an inequality extending the classical Sobolev embedding theorem to the homogeneous Carnot groups. Theorem 5.9.2 (Sobolev–Stein embedding). Let L be a sub-Laplacian on the homogeneous Carnot group G of homogeneous dimension Q. Suppose 1 < p < Q. Then there exists a positive constant C = C(p, G, L) such that for every u ∈ C0∞ (RN , R), uq ≤ C ∇L up where 1/q = 1/p − 1/Q

i.e. q =

Qp . Q−p

Proof. Let u ∈ C0∞ (RN , R). Using the representation formula (5.16), we have u(x) = − Γ (x −1 ◦ y) Lu(y) dy. Keeping in mind that L = right-hand side, we obtain

G

m

2 j =1 Xj

u(x) =

RN

and Xj∗ = −Xj , by integrating by parts at the

(∇L Γ )(x −1 ◦ y) ∇L u(y) dy.

(5.79)

On the other hand, out of the origin, we have ∇L Γ = βd ∇L (d 2−Q ) = (2 − Q) βd d 1−Q ∇L d, so that, since ∇L d is smooth in G \ {0} and δλ -homogeneous of degree zero, |∇L Γ | ≤ C d 1−Q , for a suitable constant C > 0 depending only on L. Using this inequality in (5.79), we get

280


|u(x)| ≤ C

G

|∇L u(y)| d(x, y)1−Q dy = C I1 (|∇L u|)(x).

Then, by the Hardy–Littlewood–Sobolev Theorem 5.9.1, uq ≤ C I1 (|∇L u|)q ≤ C ∇L up , where 1/q = 1/p − 1/Q This ends the proof.

⇔q=

Qp . Q−p

5.10 Some Remarks on the Analytic Hypoellipticity of Sub-Laplacians In this section, we collect some results on the hypoellipticity (especially in the analytic sense) of sub-Laplacians. It is far from our scopes here to give proofs of the results of this section, the interested reader will be properly referred to the existing literature. First of all, we recall the relevant definition. Definition 5.10.1 ((Analytic) hypoellipticity). We say that a differential operator L defined on an open set Ω ⊆ RN is hypoelliptic (respectively, analytic hypoelliptic) in Ω if, for every open set Ω ⊆ Ω and every f ∈ C ∞ (Ω , R) (respectively, f real analytic in Ω ), any solution u to the equation Lu = f on Ω (in the weak sense of distributions) is of class C ∞ (Ω , R) (respectively, is real analytic on Ω ). In the sequel, we shall write u ∈ C ω (Ω) to mean that u is real analytic on Ω. Moreover, we may also use the notation C ∞ -hypoelliptic and C ω -hypoelliptic to mean, respectively, hypoelliptic and analytic hypoelliptic. In the very special case of a homogeneous differential operator L with constant coefficients in RN , the problem of hypoellipticity is completely solved by the following result (see, e.g. [Hor69]). Let L be a homogeneous differential operator with constant coefficients in RN . Then the following statements are equivalent: 1) L is C ∞ -hypoelliptic in RN , 2) L is C ω -hypoelliptic in RN , 3) L is elliptic in RN . Moreover, if L has constant coefficients (but it is not necessarily homogeneous), then the equivalence of (2) and (3) still holds true. As a consequence of the above result, all sub-Laplacians on the Euclidean group E = (RN , +) (see Section 4.1.1) are C ∞ and C ω -hypoelliptic.

5.10 Analytic Hypoellipticity of Sub-Laplacians

281

We next focus our attention to the C ∞ -hypoellipticity. Let {X1 , . . . , Xm } be C ∞ vector fields on RN . We recall the well-known rank (or bracket) condition (also referred to as Hörmander’s hypoellipticity condition): dim Lie{X1 , . . . , Xm }I (x) = N ∀ x ∈ RN , i.e. for every x ∈ RN , there exists a set of N iterated brackets of the Xi ’s which are linearly independent at x. The following celebrated result holds (see Hörmander [Hor67]). the rank condition is suffiIf {X1 , . . . , Xm } are C ∞ -vector fields on RN , then n 2 cient for the C ∞ -hypoellipticity of the operator L = j =1 Xj . Moreover, if the coefficients of the Xj ’s are analytic, then the rank condition is also necessary for the C ∞ -hypoellipticity of L. (See also Derridj [Der71], Helffer–Nourrigat [HN79], Kohn [Koh73], Ole˘ınik– Radkeviˇc [OR73], Rothschild–Stein [RS76].) See also Bony [Bon69,Bon70] for a partial converse of the rank condition, namely If the sum of squares L of smooth vector fields is C ∞ -hypoelliptic, then the rank condition holds on an open set, dense in RN . (See also Fedi˘ı [Fed70], Kusuoka–Stroock [KuSt85], Bell–Mohammed [BM95] for examples of sums of squares which are C ∞ -hypoelliptic but do not satisfy the rank condition everywhere.) The problem of C ∞ -hypoellipticity is thus completely solved for any subLaplacian L on any homogeneous Carnot group, since L is a sum of squares of polynomial vector fields satisfying the rank condition. It is also interesting to remark that, for any homogeneous left-invariant differential operator L on a stratified Lie group (hence in particular for our sub-Laplacians), the C ∞ -hypoellipticity of L is equivalent to a Liouville-type property for L, namely the property stating that the only bounded functions u on G such that Lu = 0 are the constant functions. (See [Rot83]; see also [HN79,Gel83].) We finally turn our attention to the C ω -hypoellipticity. The problem of analytic hypoellipticity is more involved and only partial results are known. To begin with, we consider the rank condition, which played a central rôle for C ∞ -hypoellipticity. Unfortunately, if L is a sum of squares of analytic vector fields, then the rankcondition is not sufficient for analytic hypoellipticity. (See, for instance, Trèves [Tre78], Tartakoff [Tar80], Grigis–Sjöstrand [GS85]; see also explicit counterexamples in [BG72,Hel82,PR80,HH91,Chr91].) In the sequel of the section, we collect some of these results (in particular, several explicit negative ones) for our subLaplacians on Carnot groups. The first one is encouraging: The canonical sub-Laplacian on the Heisenberg–Weyl group Hn is analytic hypoelliptic. It is not difficult to prove this result making use of the real analyticity (out of the origin) of the fundamental solution Γ for the canonical sub-Laplacian on Hn , for instance (Q = 2n + 2 denotes the homogeneous dimension of Hn ) Γ (x, t) = cQ

(|x|4

1 . + |t|2 )(Q−2)/4

(5.80)

282


Since (see Kaplan [Kap80]; see also Example 5.4.7, page 250, and Chapter 18) the fundamental solution Γ for the canonical sub-Laplacian on every H-type group has exactly the same form as in (5.80), then the canonical sub-Laplacian on any H-type group is analytic-hypoelliptic. This is only a partial result of what is true on (a class of operators containing the) Heisenberg-type groups, as we shall see below; but, unfortunately (as we shall also see in a moment), it must be soon realized that C ω -hypoellipticity rarely occurs within the non-Euclidean setting of Carnot groups. To this end, we cite a first “negative” result (see Helffer [Hel82]): If G is a Carnot group of step two, and L is a sub-Laplacian on G, then a necessary condition for L to be C ω -hypoelliptic is that G is a HM-group. (See Definition 3.7.3, page 174, for the definition of HM-group.) Hence, at least within the setting of step-two Carnot groups, in order to find a C ω -hypoelliptic sub-Laplacian, we must restrict our attention to the sub-class of HM-groups. Fortunately, a complete answer on the C ω -hypoellipticity for subLaplacians is available on HM-groups. This is given by the following result (see Métivier [Met81]). If G is a HM-group and L ∈ Um (g) (i.e. L is a homogeneous operator of degree m in the relevant enveloping algebra), then L is C ω -hypoelliptic if and only if L is C ∞ -hypoelliptic. Consequently, since any sub-Laplacian L on G belongs to U2 (g) and L is C ∞ hypoelliptic by the rank condition, then L is also C ω -hypoelliptic. This result covers quite large classes of remarkable cases: for example, since the Heisenberg–Weyl groups, the Iwasawa-groups, and (more generally) the H-type groups all belong to the HM-group class, we now know that all their sub-Laplacians are C ω -hypoelliptic. On the converse, it is easy to exhibit a step-two group where no sub-Laplacian is real analytic: take any Carnot group where the first layer has odd dimension (for, in that case, it cannot be a HM-group; see Example 5.10.2 below). The problem finally rest on the investigation of Carnot groups of step strictly greater than two. Unfortunately, a complete answer to the analytic-hypoellipticity in that case is still an open problem. Quoting Rothschild [Rot84], it is reasonable to conjecture that if G is not a HM-group, then there is no L ∈ Um (g) (hence, no subLaplacian) which is analytic-hypoelliptic. For example, from a result by M. Christ [Chr93, Theorem 1.5] we infer that if G is a filiform Carnot group of dimension ≥ 4, then no sub-Laplacian on G is analytic-hypoelliptic. In Examples 5.10.4, 5.10.5, 5.10.6 below, we exhibit some other explicit negative results. To begin, we give two examples: the first (respectively, the second) is an example of a sub-Laplacian on a homogeneous Carnot group of step two which is not (respectively, which is) analytic-hypoelliptic; in the latter case, we explicitly write the (analytic) fundamental solution. Example 5.10.2. Let R4 (the points are denoted by (x, y, z, t) with x, y, z, t ∈ R) be equipped with the dilation δλ (x, y, z, t) = (λx, λy, λz, λ2 t) and the composition law


⎛ ⎞ ⎛ ⎞ ⎛ ξ x ⎜y ⎟ ⎜η⎟ ⎜ ⎝ ⎠◦⎝ ⎠=⎝ ζ z t τ t

283

⎞ x+ξ y+η ⎟ ⎠. z+ζ +τ +xη

Then G = (R4 , ◦, δλ ) is a homogeneous Carnot group of step two, and ΔG = (∂x )2 + (∂y + x ∂t )2 + (∂z )2 is its canonical sub-Laplacian. It is easily seen that G is isomorphic to the sum (in the sense of Section 4.1.5) of the Heisenberg-Weyl group H1 on R3 and the usual Euclidean group (R, +) (note that the canonical sub-Laplacians on both these groups are analytic hypoelliptic!). It can be proved that ΔG is not analytic hypoelliptic! Indeed, by the cited result of Helffer [Hel82], this follows from the fact that G is not a HM-group, since the first layer of the stratification has odd dimension. This example is a particular case of the family of not analytic hypoelliptic subLaplacians (see Rothschild [Rot84]) L=

n (∂xj )2 + (∂yj + xj ∂t )2 + (∂z )2 ,

(5.81)

j =1

which, in turn, are inspired by a famous counterexample by Baouendi–Goulaouic [BG72]. Indeed, in [BG72] it is proved that the operator L :=

n (∂xj )2 + (xj ∂t )2 + (∂z )2 j =1

on Rn+2 (the points are denoted by (x, z, t), x = (x1 , . . . , xn ) ∈ Rn , z ∈ R, t ∈ R) is not C ω -hypoelliptic on Rn+2 . Now, we notice that the operator L in (5.81) is a “lifted” version of L, so that it is easy to prove that L is not C ω -hypoelliptic on R2n+2 if L is not C ω -hypoelliptic on Rn+2 . Indeed, suppose to the contrary that L is C ω -hypoelliptic on R2n+2 . Then take a function f = f (x, z, t) real analytic on an open set Ω ⊆ Rn+2 and a solution u = u(x, z, t) to Lu = f on Ω. If we set f(x, y, z, t) := f (x, z, t) and u(x, y, z, t) := u(x, z, t), then we notice that L u(x, y, z, t) = Lu(x, z, t) = f (x, z, t) = f(x, y, z, t)

(5.82)

:= {(x, y, z, t) : (x, z, t) ∈ Ω, y ∈ R}. Since f is clearly on the open set Ω then (5.82) and the supposed C ω -hypoellipticity of L imply real analytic on Ω, u∈ This obviously means that u is real analytic on Ω. Thus we have shown that C ω (Ω). L is C ω -hypoelliptic on Rn+2 , contrarily to what is proved in [BG72]. Example 5.10.3. This example is taken from Balogh–Tyson [BT02]. Let us consider the group G on R5 (the points are denoted by (x1 , x2 , x3 , x4 , t) ∈ G, (x1 , x2 , x3 , x4 ) ∈ R4 corresponds to the first layer of the stratification, t ∈ R to the second

284


one) with dilation δλ (x1 , x2 , x3 , x4 , t) = (λx1 , λx2 , λx3 , λx4 , λ2 t) and the composition law ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ξ1 x 1 + ξ1 x1 x 2 + ξ2 ⎜ x2 ⎟ ⎜ ξ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x 3 + ξ3 ⎜ x3 ⎟ ◦ ⎜ ξ3 ⎟ = ⎜ ⎟. ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ x4 ξ4 x 4 + ξ4 1 t τ t + τ + 2 (x2 ξ1 − x1 ξ2 + 2x4 ξ3 − 2x3 ξ4 ) Following our conventional notation, G is the homogeneous Carnot group of step two (with m = 4 generators and n = 1) with relevant matrix ⎛ ⎞ 0 1 0 0 ⎜ −1 0 0 0 ⎟ B (1) = ⎝ ⎠. 0 0 0 2 0 0 −2 0 Then G is obviously a HM-group, for B is a non-singular skew-symmetric matrix. In particular, by the cited result of Métivier [Met81], the canonical sub-Laplacian L is C ω -hypoelliptic. We can see this directly, for the relevant fundamental solution Γ has been explicitly written by Balogh–Tyson in [BT02] (making use of a remarkable formula by Beals–Gaveau–Greiner, see [BGG96]; we describe this formula closely in Section 5.12, page 291): it is apparent that Γ is analytic out of the origin! Indeed, it holds Γ (x1 , x2 , x3 , x4 , t) = c d 2−Q (x1 , x2 , x3 , x4 , t), where c is a suitable positive constant, Q = 6 is the homogeneous dimension of G, and d is the homogeneous norm defined by d(x1 , x2 , x3 , x4 , t)

2

1/8 1 2 1 2 x1 + x2 + x32 + x42 + t 2 = 2 2 " 3/8

2 1 2 1 2 1 2 1 2 x + x + x32 + x42 + t 2 · x1 + x2 + 2 2 2 1 2 2 " −1/8

2 1 2 1 2 1 2 1 2 2 2 2 2 2 x + x + x3 + x4 + t · x1 + x2 + x3 + x4 + . (5.83) 2 2 2 1 2 2 This formula gives the remarkable example of an explicit fundamental solution of a group which is not a H-type group! We then turn our attention to groups of step greater than two. Following the idea in the argument at the end of Example 5.10.2, we can give infinite examples of groups of arbitrarily high step with a sub-Laplacian which is not C ω -hypoelliptic: it suffices to take the sum (in the sense of Section 4.1.5) of the group G of Example 5.10.2 with another Carnot group. Let us now quote some other examples taken or inspired by the existing literature.


285

Example 5.10.4. This example is taken from a paper by Christ [Chr95] (see also [Hel82,PR80]). Consider the Carnot group on R4 with the composition 9 low ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ξ1 x 1 + ξ1 x1 x 2 + ξ2 ⎟ ⎜ x2 ⎟ ⎜ ξ2 ⎟ ⎜ ⎠. ⎝ ⎠◦⎝ ⎠=⎝ x3 ξ3 x 3 + ξ3 − ξ2 x 1 x4 ξ4 x4 + ξ4 + 2ξ3 x1 − ξ2 x12 It is easily seen that G is a filiform homogeneous Carnot group of step three and two generators, with dilations δλ (x1 , x2 , x3 , x4 ) = (λx1 , λx2 , λ2 x3 , λ3 x4 ), and such that the first two vector fields of the relevant Jacobian basis are X1 = ∂x1 ,

X2 = ∂x2 − x1 ∂x3 − x12 ∂x4 .

Then, it can be proved that the canonical sub-Laplacian on G 2 ΔG = X12 + X22 = (∂x1 )2 + ∂x2 − x1 ∂x3 − x12 ∂x4 is not C ω -hypoelliptic. (Compare the group in this example to the Bony-type subLaplacian with N = 2 in Section 4.3.3, page 202.) Example 5.10.5. It is known (see [Chr91,Chr93,HH91,PR80]) that in R3 (with coordinates (t, s, x)) the operator L = (∂t )2 + (∂s − t m ∂x )2

(5.84)

is not C ω -hypoelliptic for any m ∈ N, m ≥ 2. In this example, fixed m as above, we give a suitable sub-Laplacian L “lifting” L: as a consequence (arguing as at the end of Example 5.10.2), L cannot be C ω -hypoelliptic, since L does not possess this property. Take N ∈ N, N ≥ m ≥ 2, and consider the following Bony-type sub-Laplacian (see Section 4.3.3, page 202): we equip R2+N (whose points are denoted by (t, s, x), t, s ∈ R, x ∈ RN ) by the composition law (t, s, x1 , x2 , x3 , . . . , xN ) ◦ (τ, σ, ξ1 , ξ2 , ξ3 , . . . , ξN ) ⎛ t +τ ⎜ s+σ ⎜ ⎜ x1 + ξ1 + σ t ⎜ 2 ⎜ x2 + ξ2 + ξ1 t + σ t2! =⎜ 2 3 ⎜ x3 + ξ3 + ξ2 t + ξ1 t2! + σ t3! ⎜ ⎜ .. ⎜ . ⎝ xN + ξN + ξN −1 t + ξN −2 9 Compare to Ex. 3, Chapter 4, page 216.

t2 2!

+ · · · + ξ1

t N−1 (N −1)!

⎞

+σ

tN N!

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

286


and the group of dilations defined by δλ (t, s, x1 , x2 , x3 , . . . , xN ) = (λt, λs, λ2 x1 , λ3 x2 , λ4 x3 , . . . , λN +1 xN ). Then, G = (R2+N , ◦, δλ ) is a filiform homogeneous Carnot group of step N + 1 (note that, since N ≥ m ≥ 2, then the step N + 1 is ≥ 3) and two generators, and

2 t2 tN ΔG = (∂t )2 + ∂s + t ∂x1 + ∂x2 + · · · + ∂xN 2! N! is its canonical sub-Laplacian. We now prove that ΔG is not C ω -hypoelliptic (starting from the fact that L in (5.84) is not). Indeed, since L is not C ω -hypoelliptic, there exists an open set Ω ⊆ R3 and a function u = u(t, s, x) on Ω such that Lu ∈ C ω (Ω) but u ∈ / C ω (Ω). Let us now consider the function (recall that m ≤ N ) u = u(t, s, x1 , . . . , xN ) := u(t, s, −m!xm ) defined on the open subset of R2+N

:= (t, s, x1 , . . . , xN ) | (t, s, −m! xm ) ∈ Ω . Ω As a consequence, it holds u(t, s, x1 , . . . , xN ) = (Lu)(t, s, −m! xm ) on Ω, ΔG (for Lu ∈ C ω (Ω)) and (for u ∈ whence ΔG u ∈ C ω (Ω) u∈ / C ω (Ω) / C ω (Ω)). This proves that ΔG is not analytic-hypoelliptic. Example 5.10.6. Let m, k ∈ N be such that 0 ≤ m ≤ k. Consider the operator on R3 (whose points are denoted by (x1 , x2 , x3 )) L = (∂x1 )2 + (x1m ∂x2 )2 + (x1k ∂x3 )2 .

(5.85)

O.A. Ole˘ınik and E.V. Radkeviˇc [OR72] (see also [Him98]) proved that L is C ω hypoelliptic if and only if m = k. Our aim in this example is to “lift” (in a suitable sense) the vector fields on R3 appearing in (5.85) ∂x1 ,

x1m ∂x2 ,

x1k ∂x3

(5.86)

to three vector fields (in a larger space, namely R3+m+k ) generating a homogeneous Carnot group. It will easily follow (arguing as in the last paragraph of Example 5.10.5) that the relevant sub-Laplacian is not C ω -hypoelliptic if L is not. When m = k = 0, then L = ΔR3 is the ordinary Laplace operator on R3 ; when m = 0, k = 1, we obtain a non-analytic-hypoelliptic operator already considered in Example 5.10.2 (see also Baouendi–Goulaouic [BG72]). Let us now suppose that k > m ≥ 1. We equip R3+m+k with the following coordinates (the semicolon will denote different layers in a suitable homogeneous Carnot group structure)

5.11 Harmonic Approximation

287

P = (x1 , y1 , z1 ; y2 , z2 ; y3 , z3 ; . . . , ym , zm ; x2 , zm+1 ; zm+2 ; zm+3 ; . . . ; zk ; x3 ). We define a group of dilations by setting (recall that k ≥ m + 1) δλ (P ) = λx1 , λy1 , λz1 ; λ2 y2 , λ2 z2 ; λ3 y3 , λ3 z3 ; . . . ; λm ym , λm zm ; λm+1 x2 , λm+1 zm+1 ; λm+2 zm+2 ; λm+3 zm+3 ; . . . ; λk zk ; λk+1 x3 . Let us also consider on R3+m+k the following vector fields X = ∂x1 , Y = ∂y1 + x1 ∂y2 + x12 ∂y3 + · · · + x1m−1 ∂ym + x1m ∂x2 , Z = ∂z1 + x1 ∂z2 + x12 ∂z3 + · · · + x1k−1 ∂zk + x1k ∂x3 . It is then not difficult to see that X, Y, Z are δλ -homogeneous of degree one and they fulfill hypotheses (H0)–(H1)–(H2) of page 191. Hence, by the results of Section 4.2 (page 191), we can define a suitable homogeneous Carnot group structure on R3+m+k such that ΔG = X 2 + Y 2 + Z 2 is the relevant canonical sub-Laplacian. Now, since ΔG (u(x1 , x2 , x3 )) = (Lu)(x1 , x2 , x3 ) for any smooth function u on R3+m+k depending only on x1 , x2 , x3 , we can argue as in the last paragraph of Example 5.10.5 to infer that ΔG is not C ω -hypoelliptic (since L is not).

5.11 Harmonic Approximation Let L be a sub-Laplacian on the homogeneous Carnot group G. In this section, we give some conditions ensuring that a L-harmonic function defined in a neighborhood of a compact set K contained in an open set Ω can be uniformly approximated on K by a sequence of L-harmonic functions in Ω. In some of the results of this section, we assume that L is analytic-hypoelliptic (see Section 5.10). Γ = d 2−Q , will denote its fundamental solution. To begin with, we prove the following lemma. Lemma 5.11.1. Let L be a sub-Laplacian on the homogeneous Carnot group G = (RN , ◦, δλ ), and let Γ = d 2−Q be its fundamental solution. Suppose also that L is analytic-hypoelliptic.

288


Let K ⊆ G be compact. Then there exists R = R(K, G, L) > 0 such that, for every z ∈ G with d(z) > R, it holds 1 α D Cα (z) · y α , Cα (z) = Γ (z−1 ◦ y) , Γ (z−1 ◦ y) = α! y=0 N α∈(N∪{0})

the series converging uniformly on the d-disc {y ∈ G : d(y) < d(K)}, where d(K) := supz∈K d(z). Proof. Let ζ ∈ G be such that d(ζ ) = 1. Since η → Γ (ζ −1 ◦ η) is analytic close to η = 0 and ∂Bd (0, 1) = {ζ ∈ G : d(ζ ) = 1} is compact, there exists ρ > 0, independent of ζ , such that Γ (ζ −1 ◦ η) = aα (ζ ) · ηα , uniformly on Bd (0, ρ), α∈(N∪{0})N

1 α aα (ζ ) = D Γ (ζ −1 ◦ η) . α! η=0

where

Let us now choose R > 0 such that R > d(K)/ρ. If z ∈ G, d(z) > R, then Γ (z−1 ◦ y) = d 2−Q (z) Γ δd −1 (z) z−1 ◦ δd −1 (z) (y) α aα δd −1 (z) z−1 · δd −1 (z) (y) = d 2−Q (z)

=:

α∈(N∪{0})N

Cα (z) · y α .

α∈(N∪{0})N

The series converges uniformly for y ∈ G such that d(y) d δd −1 (z) (y) = < ρ. d(z) In particular, this holds for y ∈ Bd (0, R ρ) ⊇ Bd (0, d(K)).

The next lemma does not require the analytic-hypoellipticity of L. Lemma 5.11.2. Let L be a sub-Laplacian on the homogeneous Carnot group G = (RN , ◦, δλ ), and let Γ = d 2−Q be its fundamental solution. Let u be an L-harmonic function on the ball Bd (0, r). Suppose u(x) =

∞

uk (x),

x ∈ Bd (0, r),

k=1

where uk is a continuous δλ -homogeneous function in the whole G of δλ -degree mk , k ∈ N. If the series is uniformly convergent on the compact subsets of Bd (0, r) and mk = mh for every k = h, then every uk is L-harmonic in G.

5.11 Harmonic Approximation

289

Proof. Let ϕ ∈ C0∞ (RN , R) with supp ϕ ⊆ Bd (0, 1). Since u is L-harmonic in Bd (0, r), we have 0= u(x) L ϕ(δλ−1 (x)) dx ∀ λ < r. Bd (0,r)

The homogeneity of L and the change of variable y = δλ−1 (x) give 0 = λQ−2 u(δλ (y)) Lϕ(y) dy Bd (0,r/λ)

= λQ−2 =λ =λ

∞

uk (δλ (y)) Lϕ(y) dy

k=1 Bd (0,r/λ) ∞ Q−2 mk

uk (y) Lϕ(y) dy

λ

Q−2

k=1 ∞

Bd (0,r/λ)

λ

uk (y) Lϕ(y) dy.

mk

k=1

Bd (0,1)

Note that, in the last equality, we were able to replace Bd (0, r/λ) by Bd (0, 1), since λ < r and ϕ is supported in a compact set in Bd (0, 1). Then uk (y) Lϕ(y) dy = 0 ∀ ϕ ∈ C0∞ (Bd (0, 1)) ∀ k ∈ N. Bd (0,1)

This means that Luk = 0 in Bd (0, 1) in the weak sense of distributions. Since L is hypoelliptic, this implies the L-harmonicity of uk in Bd (0, 1), and so in G, due to the δλ -homogeneity of uk , k ∈ N. We are now ready to prove the announced approximation theorem. Theorem 5.11.3 (An approximation theorem). Let L be a sub-Laplacian on the homogeneous Carnot group G. Suppose that L is analytic-hypoelliptic. Let Ω ⊆ G be an open set such that ∂Ω = ∂Ω. Let K ⊆ Ω be compact and satisfy the following condition: if ω is a bounded connected component of Ω \ K, then ∂ω ⊆ ∂Ω.

(5.87)

Then, for every function h which is L-harmonic in a neighborhood of K, there exists a sequence (hn )n∈N of L-harmonic functions in Ω such that lim hn = h,

n→∞

uniformly on K.

Proof. By general results from functional analysis, it is well known that it suffices to prove the following statement.10 10 This is also known as “Caccioppoli’s completeness method”.

290


Let μ be a signed Radon measure supported on K and satisfying h dμ = 0

(5.88)

K

for every L-harmonic function h in Ω. Then (5.88) holds for every function h which is L-harmonic just in a neighborhood of K. The crucial part of the proof is the following assertion. Claim. If μ satisfies the previous hypotheses, then u(x) := Γ (y −1 ◦ x) dμ(y) = Γ (x −1 ◦ y) dμ(y) (5.89) K

K

is identically zero in Ω \ K. We first show how to complete the proof of the theorem by using this claim. Let h be an L-harmonic function in an open set Ω0 ⊇ K, Ω0 ⊆ Ω. Choose a function φ ∈ C0∞ (Ω0 ) such that φ = 1 in an open set Ω1 ⊇ K, Ω 1 ⊆ Ω0 . Then φ h ∈ C0∞ (Ω0 ) and φ h = h in Ω1 . By the representation formula (5.16), we have Γ (x −1 ◦ y)L(φ h)(x) dx (φ h)(y) = − Ω0 =− Γ (x −1 ◦ y) L(φ h)(x) dx, y ∈ Ω0 . Ω0 \Ω 1

It follows that h(y) dμ(y) = (φ h)(y) dμ(y) K K

−1 Γ (x ◦ y) L(φ h)(x) dx dμ(y) =: (). =− K

Ω0 \Ω 1

Thus, by interchanging the integrals and keeping in mind (5.89), we infer L(φ h)(x)u(x) dx = 0, () = − Ω0 \Ω 1

since u = 0 in Ω \ K ⊇ Ω0 \ Ω 1 . Thus, we are left with the proof of the Claim. Let ω be a connected component of Ω \ K. We have to prove that u ≡ 0 in ω. We first suppose that ω is bounded. Then ∅ = ∂ω ⊆ ∂Ω. In particular, this implies that ∂Ω = ∅, hence Ω = G for ∂Ω = ∂Ω. For every x0 ∈ G \ Ω, the function y → Γ (x0−1 ◦ y) is L-harmonic in Ω. Therefore, by the assumption (5.88), Γ (y −1 ◦ x0 ) dμ(y) = 0. u(x0 ) = K

This proves that u ≡ 0 in G \ Ω. As a consequence,

5.12 An Integral Representation Formula for Γ

D α u(x) = 0

for every multi-index α

291

(5.90)

and for every x ∈ G \ Ω. Since u ∈ C ∞ (G \ K), it follows that (5.90) also holds at any point x ∈ ∂Ω = ∂Ω. In particular, since ∂ω ⊆ ∂Ω, (5.90) holds at some point x ∈ ∂ω. Then ω ≡ 0 in a neighborhood of x, since u is L-harmonic, hence real analytic, close to x. The connectedness of ω and again the analyticity of u imply that u ≡ 0 in ω. Let us now assume that ω is unbounded. By Lemma 5.11.1, for every x ∈ ω with d(x) sufficiently large, we have Cα (x) · y α , uniformly in Bd (0, d(K)). Γ (x −1 ◦ y) = α∈(N∪{0})N

Then

u(x) =

Γ (x

−1

◦ y) dμ(y) =

K

∞

um (x, y) dμ(y),

m=0 K

where

um (x, y) =

Cα (x) · y α .

|α|G =k

The function um is δλ -homogeneous of degree m and the series ∞

um (x, ·)

m=0

is uniformly convergent on Bd (0, d(K)) to the L-harmonic function y → Γ (x −1 ◦ y). Then, by Lemma 5.11.2, um (x, ·) is L-harmonic in G, so that, by the assumption (5.88), um (x, y) dμ(y) = 0 ∀ m ≥ 0. K

Thus we have proved that u(x) = 0 for every x ∈ ω, with d(x) sufficiently large. Since ω is connected and u is analytic in ω, this implies u ≡ 0 in ω and completes the proof of the theorem.

5.12 An Integral Representation Formula for the Fundamental Solution on Step-two Carnot Groups The aim of this section is to state a remarkable result in the paper [BGG96] by R. Beals, B. Gaveau and P. Greiner. This result provides a somewhat explicit integral representation formula of the fundamental solution of the canonical sub-Laplacian on a general Carnot group of step two. We shall see in Section 16.3 (page 637 in Part III) that, given a Carnot group G1 and an arbitrary sub-Laplacian L on G1 , there exists a Carnot group G2 isomorphic

292


to G1 such that L corresponds (via the related isomorphism in the relevant Lie algebras) to the canonical sub-Laplacian ΔG2 on G2 . Moreover, if G2 (whence G1 ) has step two, we saw in Proposition 3.5.1 (page 168) that we can perform another Liegroup isomorphism sending G2 into the homogeneous Carnot group G3 such that the composition law on G3 is given by (we follow our usual notation)

1 (1) 1 (n) (x, t) ◦ (ξ, τ ) = x + ξ, t1 + τ1 + B x, ξ , . . . , tn + τn + B x, ξ (5.91) 2 2 and the matrices B (i) ’s are skew-symmetric. This last isomorphism also sends the canonical sub-Laplacian of G2 into the canonical sub-Laplacian of G3 . Now, the cited result in [BGG96] furnishes an integral formula for the canonical subLaplacian on a homogeneous Carnot group of step two whose composition law ◦ has precisely the above form and the matrices B (i) ’s are skew-symmetric. Our argument above shows that (and how) we can obtain a representation formula for the fundamental solution of any (not necessarily canonical) sub-Laplacian on any homogeneous Carnot group of step two. We now state the remarkable result in [BGG96]. We explicitly remark that in [BGG96] the formalism of complex Hamiltonian mechanics is followed: we slightly change the therein notation. Theorem 5.12.1 (Beals–Gaveau–Greiner, [BGG96]). Let G = Rm+n (whose points are denoted by (x, t), x ∈ Rm , t ∈ Rn ) be equipped with a homogeneous Carnot group structure by the dilation δλ (x, t) = (λx, λ2 t) and the composition law matrices of in (5.91), where the B (k) ’s are n skew-symmetric linearly independent order m × m. Consider the canonical sub-Laplacian ΔG = ni=1 Xi2 , where m n 1 (k) Xi = ∂/∂xi + bi,l xl ∂/∂tk , i = 1, . . . , m 2 k=1

l=1

(k) (here bi,l denotes the entry of position (i, l) of B (k) ). Then, for every (x, t) with x = 0, the fundamental solution Γ of ΔG is given by √ det(V(B(τ ))) Γ (x, t) = cQ dτ, (5.92) 1 Rn ( 2 W(B(τ )) · x, x − ι t, τ )Q/2−1

where ι is the imaginary unit of C and cQ is the dimensional constant cQ =

( Q 2 − 1) . 2 (2 π)Q/2

(5.93)

Here we used the following notation: Q = m + 2 n is the homogeneous dimension of G, in (5.93) is Euler’s Gamma function, τ = (τ1 , . . . , τn ) ∈ Rn , B(τ ) =

1 (τ1 B (1) + · · · + τn B (n) ), 2

5.13 Appendix A. Maximum Principles

293

V and W are the real-analytic functions prolonging z/ sin(z) and z/ tan(z), respectively, at z = 0, i.e. V(z) =

∞ (−1)j z2j , (2j + 1)!

W(z) =

j =0

∞ (−1)j 22j B2j 2j z (2j )! j =0

(here the B2j ’s are the Bernoulli numbers). Moreover, for every t ∈ Rn \{0}, we have Γ (0, t) = lim Γ (x, t). 0=x→0

We remark that a general integral formula for Γ is provided in [BGG96] comprising the case x = 0 too, by shifting the contour Rn into the complex domain Cn (see [BGG96, Theorem 3, page 315]). The δλ -homogeneity of Γ in (5.92) (of degree 2 − Q) should be noted. As we saw in Example 5.10.3 (page 283), formula (5.92) can, in some cases, give explicit fundamental solutions. This can be done by using the fact that, if λ1 (τ ), . . . , λm (τ ) and v1 (τ ), . . . , vm (τ ) denote the eigenvalues and corresponding eigenvectors of the matrix B(τ ) (over the complex field), normalized in such a way that |vj (τ )| = 1 for j = 1, . . . , m, we have m # det V(B(τ )) =

λj (τ ) , sin(λj (τ ))

j =1 m

W(B(τ )) · x, x =

j =1

2 λj (τ ) x, vj (τ ) . tan(λj (τ ))

(In the last formula, the inner product is, of course, that of Cm .)

5.13 Appendix A. Maximum Principles In this section, we shall prove some weak and strong maximum principles for L, an arbitrary sub-Laplacian on a homogeneous Carnot group G. To begin with, we prove some elementary lemmas. Lemma 5.13.1. Let Ω ⊂ RN be a bounded open set and let u : Ω → R be an arbitrary function. Then there exists a point x0 ∈ Ω such that lim sup u(x) = sup u. x→x0

(5.94)

Ω

Proof. We argue by contradiction and assume that (5.94) is false. Then, for every x ∈ Ω, there exists an open neighborhood Vx of x such that sup u < sup u. Ω∩Vx

Ω

(5.95)

294


The family {Vx : x ∈ Ω} is an open covering of Ω, so that, since Ω is compact, we have p $ Ω⊆ Vxj , p ∈ N, j =1

for suitable x1 , . . . , xp ∈ Ω. Then sup u = max Ω

sup u : j = 1, . . . , p .

(5.96)

Ω∩Vxj

On the other hand, by (5.95), the right-hand side of (5.96) is strictly less than supΩ u. This contradiction proves the lemma. Lemma 5.13.2. Let A and B be N ×N symmetric matrices with constant real entries. Assume A ≥ 0 and B ≤ 0. Then trace(A · B) ≤ 0. Proof. Let R := A1/2 be a symmetric square root of A. Then trace(A·B) = trace(R· R · B) = trace(R · B · R) = trace(R T · B · R) ≤ 0, since B ≤ 0. Lemma 5.13.3. Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω ⊆ G be an arbitrary open set, and let u : Ω → R be a C 2 real function. Assume that u has a local maximum at x0 ∈ Ω. Then Lu(x0 ) ≤ 0.

(5.97)

Proof. We know that L = div(A · ∇ T ), where A is a N × N symmetric matrix with polynomial entries and A(x) ≥ 0 at any point x ∈ RN . Then L = trace(A · D 2 u) + b, ∇u ,

(5.98)

where D 2 u = (∂xi xj )i,j ≤N is the Hessian matrix of u and b is the vector-valued function whose j -th component is given by bj =

N

∂xi ai,j .

(5.99)

i=1

Since u has a local maximum at x0 , we have ∇u(x0 ) = 0 and D 2 u(x0 ) ≤ 0. Then, by Lemma 5.13.2, Lu(x0 ) = trace(A(x0 ) · D 2 u(x0 )) ≤ 0. This ends the proof.

We are now able to give a simple proof of the following weak maximum principle.


295

Theorem 5.13.4 (Weak maximum principle). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω be a bounded open subset of G. Let u : Ω → R be a C 2 function such that Lu ≥ 0 in Ω, (5.100) lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω. Then u ≤ 0 in Ω. Proof. We know that the matrix A in (5.98) has the following block form (see also (1.91), page 64)

A1,1 A1,2 A= , A2,1 A2,2 where A1,1 = (ai,j )i,j ≤m is a constant m × m symmetric matrix strictly positive definite. Then a1,1 > 0. Let b1 be given by (5.99) with j = 1. Define λ := 2 sup x∈Ω

b1 (x) , a1,1

M :=

sup

exp(λ x1 ),

(x1 ,...,xN )∈Ω

and h(x) = h(x1 , . . . , xN ) := M − exp(λ x1 ). A trivial computation shows that h(x) ≥ 0 and

Lh(x) < 0 for every x ∈ Ω.

(5.101)

For an arbitrary ε > 0, let us now consider the function uε := u − ε h. Due to inequalities (5.101) and condition (5.100), we have Luε > 0 in Ω

and lim sup uε (x) ≤ 0 for every y ∈ ∂Ω.

(5.102)

x→y

By Lemma 5.13.1, there exists a point x0 ∈ Ω such that lim sup uε (x) = sup uε . x→x0

(5.103)

Ω

We want to show that x0 ∈ ∂Ω. Arguing by contradiction, we assume x0 ∈ Ω. Then, by the continuity of u in Ω, uε (x0 ) = lim sup uε (x), x→x0

so that, by (5.103), uε (x0 ) = maxΩ uε . As a consequence, by Lemma 5.13.3, Luε (x0 ) ≤ 0. This contradicts the first inequality in (5.102). Thus x0 ∈ ∂Ω. Then, by (5.103) and the second condition in (5.102), sup uε = lim sup uε (x) ≤ 0. Ω

x→x0

Hence, u − ε h = uε ≤ 0 in Ω for every ε > 0. Letting ε tend to zero, we obtain u ≤ 0 in Ω. The theorem is thus completely proved.

296


Note 5.13.5. The previous proof can be applied also to continuous functions satisfying the inequality Lu ≥ 0 in the asymptotic sense of Exercises 8 and 9 at the end of the chapter (see also Ex. 10). Corollary 5.13.6. Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω be an unbounded open subset of G. Let u : Ω → R be a C 2 function such that ⎧ in Ω, ⎪ ⎨ Lu ≥ 0 lim supx→y u(x) ≤ 0 for every y ∈ ∂Ω, (5.104) ⎪ ⎩ lim sup |x|→∞ u(x) ≤ 0. Then u ≤ 0 in Ω. Proof. Let ε > 0 be arbitrary but fixed. The third condition in (5.104) implies the existence of a real positive constant R such that u(x) − ε < 0 in Ω \ ΩR ,

(5.105)

where ΩR := {x ∈ Ω : |x| < R}. It follows that L(u − ε) = Lu ≥ 0 in ΩR , lim supx→y u(x) ≤ 0 for every y ∈ ∂ΩR . Then, by Theorem 5.13.4, u − ε ≤ 0 in ΩR . This inequality, together with (5.105), gives u ≤ ε in Ω for every ε > 0. Hence u ≤ 0 in Ω. A particular case of Corollary 5.13.6 is the following one. Corollary 5.13.7. If L is as in Corollary 5.13.6, the only entire L-harmonic function vanishing at infinity is the null function. Proof. Let u : G → R be an entire L-harmonic function vanishing at infinity, i.e. u ∈ C ∞ (G, R) satisfies Lu = 0 in G, lim|x|→∞ u(x) = 0. Then, by applying Corollary 5.13.6 both to u and −u, we get u ≡ 0.

The rest of this section is devoted to the proof of the following strong maximum principle. Theorem 5.13.8 (Strong maximum principle). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω be a connected open subset of G. Let u : Ω → R be a C 2 function such that u ≤ 0 and Lu ≥ 0 in Ω.

(5.106)

Suppose there exists a point x0 ∈ Ω such that u(x0 ) = 0. Then u(x) = 0 for every x ∈ Ω.


297

The proof of this theorem requires several preliminary results. In what follows, we shall denote by | · | the standard Euclidean norm and by D(z, r) the ball D(z, r) := {x ∈ RN : |x − z| < r}. Definition 5.13.9. Let F be a relatively closed subset of Ω. We say that a vector ν ∈ RN \ {0} is orthogonal to F at a point y ∈ Ω ∩ ∂F if D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y}.

(5.107)

If this inclusion holds, we shall write ν⊥F at y. We also put

F ∗ := y ∈ Ω ∩ ∂F | there exists ν: ν⊥F at y . With the above notation, we explicitly remark that F ∗ = ∅ if F ⊂ Ω, F = Ω. Indeed, since Ω is connected, Ω ∩ ∂F is not empty. Take a point z ∈ Ω ∩ ∂F , a ball D(z, R) ⊆ Ω and a point x0 ∈ D(z, R/2). Let y ∈ Ω ∩ ∂F be such that r := |x0 − y| = dist(x0 , ∂F ). Then y ∈ F ∗ and ν := 2r (x0 − y)⊥F at y. The following Hopf-type lemma will be crucial for the proof of the strong maximum principle in Theorem 5.13.8. Lemma 5.13.10 (A Hopf-type lemma for sub-Laplacians). Let L be a sub-Laplacian on the homogeneous Carnot group G. Let Ω ⊆ G be open, and let u : Ω → R be a C 2 function satisfying the inequalities in (5.106). Let F := {x ∈ Ω : u(x) = 0}.

(5.108)

Assume ∅ = F = Ω. Then, for every y ∈ F ∗ and ν⊥F at y, we have qL (y, ν) = 0,

(5.109)

where qL (x, ξ ) := A(x) · ξ, ξ is the characteristic form of L defined in (5.1a). Proof. Let y ∈ F ∗ and ν⊥F at y. Then D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y}. Since F ∗ ⊆ F ∩ Ω, y is a maximum point for u (see (5.108)). Then ∇u(y) = 0. We now argue by contradiction assuming that (5.109) is false. Hence qL (y, ν) > 0. Let us now consider the function h(x) := exp(−λ |x − z|2 ) − exp(−λ r 2 ),

298


where z = y + ν and r = |ν|. The positive constant λ will be fixed later on. A direct and easy computation shows that Lh(y) = 4λ2 exp(−λ r 2 ) · qL (y, ν) + O(1/λ) , as λ → ∞. Then, we can choose and fix λ > 0 in such a way that Lh > 0 in a suitable neighborhood V of y. Obviously, we may assume V ⊂ Ω. Let us now consider the bounded open set U := V ∩ D(z, r). Note that ∂U = Γ1 ∪ Γ2 , where Γ1 = V ∩ ∂D(z, r) and Γ2 = D(z, r) ∩ ∂V . Since Γ2 is a compact subset of Ω \F and u < 0 in Ω \F , there exists ε > 0 such that u + ε h < 0 in Γ2 . On the other hand, being h = 0 on ∂D(z, r) and u ≤ 0 in Ω, we have u+ε h ≤ 0 on Γ1 . Then, since L(u+ε h) ≥ ε Lh ≥ 0 in U , from the maximum principle of Theorem 5.13.4, we obtain u + ε h ≤ 0 in U . Since u(y) = h(y) = 0, this implies h(y + t ν) − h(y) u(y + t ν) − u(y) ≤ −ε t t

for 0 < t < 1.

(5.110)

Letting t tend to zero in this inequality, we get ∇u(y), ν ≤ −ε ∇h(y), ν = −2ε exp(−λ r 2 ) r 2 . This contradicts the condition ∇u(y) = 0 and completes the proof.

Corollary 5.13.11. Let the hypotheses and notation of the previous lemma hold. Let 2 also L = m j =1 Xj . Then we have Xj I (y), ν = 0

∀ y ∈ F ∗ ∀ ν⊥F at y

and for every j = 1, . . . , m. Proof. It follows from the previous lemma, by just noticing that qL (x, ξ ) =

m Xj I (y), ξ 2 . j =1

Another crucial definition is the following one. Definition 5.13.12 ((Positively) invariant set w.r.t. a vector field). Let X ∈ T (RN ) be a smooth vector field in RN , and let F be a relatively closed subset of Ω. We say that F is positively X-invariant if, for any integral curve γ of X, γ : [0, T ] → Ω such that γ (0) ∈ F , we have γ (t) ∈ F for every t ∈ [0, T ]. We say that F is X-invariant if it is positively X-invariant with respect to both X and −X.


299

It is easy to verify that the condition XI (y), ν ≤ 0

∀ y ∈ F ∗ ∀ ν⊥F at y

(5.111)

is necessary for the positive X-invariance of F . Indeed, let y ∈ F ∗ , ν⊥F at y and γ : [0, T ] → Ω be an integral curve of X such that γ (0) = y. Let F be positively X-invariant. Since D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y}, we have |γ (t) − (y + ν)|2 ≥ |ν|2 and |γ (0) − (y + ν)|2 = |ν|2 for every t ∈ [0, T ]. This means that the real function t → |γ (t) − (y + ν)|2 has a minimum at t = 0. As a consequence, d 0≤ |γ (t) − (y + ν)|2 = γ˙ (0), γ (0) − (y + ν) = XI (y), −ν . d t t=0 Hence (5.111) holds. We will show that this condition is also sufficient for F to be positively X-invariant. To prove this statement, we need the following elementary lemma. Lemma 5.13.13. Let g : [0, T ] → R be a continuous function such that g(t + h) − g(t) ≤M h

lim sup h→0−

∀ t ∈ (0, T ],

(5.112)

for a suitable M ∈ R. Then g(t) ≤ g(0) + M t

∀ t ∈ [0, T ].

Proof. Let ε > 0 be fixed. Condition (5.112) implies that the real function t → g(t) − g(0) − (M + ε)t has a maximum at t = 0. Indeed, suppose to the contrary that there exist ε0 > 0 and t0 ∈ (0, T ] such that g(t) − g(0) − (M + ε) t ≤ g(t0 ) − g(0) − (M + ε0 ) t0

∀ t ∈ [0, T ].

In particular, for t = t0 + h and h < 0 small enough, this gives g(t0 + h) − g(t0 ) ≥ (M + ε0 ), h which contradicts the hypothesis. Then, g(t) − g(0) − (M + ε) t ≤ 0 for every t ∈ [0, T ]. Letting ε tend to zero, we obtain the assertion. Proposition 5.13.14 (Nagumo–Bony). Let X ∈ T (RN ) be a smooth vector field in RN , and let F be a relatively closed subset of Ω. Then F is positively X-invariant if and only if (5.113) XI (y), ν ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

300


Proof. We only need to show the “if” part. Let γ : [0, T ] → Ω be an integral curve of X such that x0 := γ (0) ∈ F . Define δ(t) := dist(γ (t), F ),

0 ≤ t ≤ T.

We have to prove that δ(t) = 0 for every t ∈ [0, T ]. Let V be a bounded neighborhood of x0 containing γ ([0, T ]), and let L :=

|XI (x) − XI (z)| |x − z| x,z∈V , x=z sup

(5.114)

be the Lipschitz constant of X on V . We may suppose L T < 1/2 and V = D(x0 , r) with D(x0 , 2r) ⊆ Ω. We claim that L(t) := lim sup h→0−

δ(t + h) − δ(t) ≤ L δ(t) h

∀ t ∈ (0, T ].

(5.115)

If δ(t) = 0, inequality (5.115) is trivial, since h < 0 and δ(t + h) ≥ 0. Suppose δ(t) > 0 and choose a sequence hn ↑ 0 such that L(t) = lim

n→∞

δ(t + hn ) − δ(t) . hn

Let us now denote x := γ (t) and xn := γ (t + hn ). Since γ ([0, T ]) ⊂ D(x0 , r) and D(x0 , 2r) ⊆ Ω, for every n ∈ N there exists a point zn ∈ F ∩ Ω such that |xn − zn | = dist(xn , F ) = δ(t + hn ). Obviously, we may suppose that zn → z ∈ F ∩ D(x0 , r), so that, since xn → x, |x − z| = lim |xn − z| = lim dist(xn , F ) n→∞

n→∞

= dist(x, F ) = δ(t). Moreover, ν :=

1 (x − z)⊥F at z. 2

Then δ(t + hn ) − δ(t) = |xn − zn | − |x − z| ≥ |xn − zn | − |x − zn | xn − x, zn − x . ≥ |xn − x| ≥ − |x − zn | Hence

) L(t) ≤ lim

n→∞

* ) * x − zn xn − x x−z = γ˙ (t), , |x − zn | hn |x − z|

2 XI (x), ν |x − z| 2 = XI (z) − XI (x), ν + XI (z), ν . |x − z| =

(5.116)


301

From (5.116) and (5.113), together with (5.114), we finally get L(t) ≤ L |x − z| = L δ(t). This completes the proof of (5.115). This inequality, combined with Lemma 5.13.13, gives δ(t) ≤ δ(t) − δ(0) ≤ L T sup δ, [0,T ]

so that sup[0,T ] δ ≤ 1/2 · sup[0,T ] δ. Hence δ ≡ 0, and the proof is complete. Corollary 5.13.15. The closed set F is X-invariant if and only if XI (y), ν = 0,

∀y ∈ F∗

∀ ν⊥F at y.

Proof. It straightforwardly follows from Proposition 5.13.14 and Definition 5.13.12. This corollary, together with Corollary 5.13.11, immediately gives the following result. m 2 Corollary 5.13.16. Let L = j =1 Xj be a sub-Laplacian on the homogeneous Carnot group G. Let u : Ω → R be a C 2 function satisfying the inequalities (5.106). Let F := {x ∈ Ω : u(x) = 0}. Assume ∅ = F = Ω. Then F is invariant with respect to X1 , . . . , Xm . Proposition 5.13.17. Let F be a relatively closed subset of the open set Ω ⊆ RN . Assume ∅ = F = Ω. Then a := {X ∈ T (RN ) : F is X-invariant} is a Lie algebra of vector fields. Proof. Let X, Y ∈ a, and let λ, μ be real constants. By Corollary 5.13.15, we have XI (y), ν = Y I (y), ν = 0 for every y ∈ F ∗ and for every ν⊥F at y. Then λ XI (y) + μ Y I (y), ν = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y. Hence a is a linear space. The next lemma will complete the proof of the proposition. Lemma 5.13.18. In the notation of Proposition 5.13.17, if X, Y [X, Y ] ∈ a.

∈ a, then

Proof. Let y ∈ F ∗ and ν⊥F at y. For every t > 0 define √ √ √ √ Γ (t) := exp(− tY ) ◦ exp(− tX) ◦ exp( tY ) ◦ exp(− tX) (y).

302


Here ◦ denotes the composition of maps, whereas “exp” denotes the exponential of a vector field as introduced in Definition 1.1.2, page 8. Let T > 0 be such that Γ (t) ∈ Ω for 0 ≤ t ≤ T . By using the Taylor expansion (1.7) of exp (on page 7) with n = 2, a direct computation gives Γ (t) = y + t JY I (y) · XI (y) − JXI (y) · Y I (y) + o(t) = y + t[X, Y ]I (y) + o(t), as t ↓ 0. Then lim t↓0

Γ (t) − y = [X, Y ]I (y). t

(5.117)

On the other hand, since F is X and Y invariant, Γ (t) ∈ F for every t ∈ [0, T ]. As a consequence, since D(y + ν, |ν|) ⊆ (Ω \ F ) ∪ {y} and Γ (0) = y, |Γ (t) − (y + ν)|2 ≥ |ν|2 = |Γ (0) − (y + ν)|2 . Then, by using also (5.117), we have d 0≥ |Γ (t) − (y + ν)|2 = 2[X, Y ]I (y), ν . d t t=0 Hence

[X, Y ]I (y), ν ≤ 0 ∀ y ∈ F ∗ ∀ ν⊥F at y.

By swapping X with Y , we also get [Y, X]I (y), ν ≤ 0, so that [X, Y ]I (y), ν = 0 ∀ y ∈ F ∗ ∀ ν⊥F at y. Then, by Corollary 5.13.15, [X, Y ] ∈ a.

Finally, we are in the position to give the proof of Theorem 5.13.8. Proof (of Theorem 5.13.8). Let us define F := {x ∈ Ω : u(x) = 0 }. By hypothesis, x0 ∈ F . Then F is a non-empty relatively closed subset of Ω. We have to prove that F = Ω. By contradiction, assume F = Ω. Then, since Ω is connected, F ∗ = ∅. Let y ∈ F ∗ , and let ν⊥F at y. By Proposition 5.13.17, ZI (y), ν = 0 ∀ Z ∈ g. Since dim(g) = N, this obviously implies ν = 0. On the other hand, by the very definition of a vector orthogonal to F , we have ν ∈ RN \ {0}. This contradiction completes the proof.


303

5.13.1 A Decomposition Theorem for L-harmonic Functions In this section, we give a decomposition theorem for L-harmonic functions, resemblant to the decomposition of a holomorphic function on an annulus of C into the sum of the regular and singular parts from its Laurent expansion (for the classical case of the Laplace operator, see also S. Axler, P. Bourdon, W. Ramey [ABR92]). Our main tool is the maximum principle from the previous section (precisely, we use Corollary 5.13.7, page 296). In the sequel, we assume Q ≥ 3. Moreover, in the proof of Theorem 5.13.20, we adopt the following notation: G = (RN , ◦, δλ ) is a homogeneous Carnot group, L is a sub-Laplacian on G, Γ = d 2−Q is the fundamental solution for L (see Proposition 5.4.2). If d is the above L-gauge, A is any subset of G and λ > 0, we agree to set Aλ := {x ∈ G | d-dist(x, A) < λ}, where

d-dist(x, A) := inf{d(x −1 ◦ a) | a ∈ A}.

Moreover, we use the following simple lemma. Lemma 5.13.19. Let K be a compact subset of G, and let f be bounded on K. Then the function F : G → R,

Γ (y −1 ◦ x) f (y) dy

F (x) := K

is L-harmonic on G \ K and vanishes at infinity. Moreover, ifμ is a Radon measure on RN with compact support K, the same is true for G(x) := RN Γ (y −1 ◦x) dμ(y). Proof. It easily follows by differentiation under the integral sign (recall also that Γ is locally integrable and vanishes at infinity). We are now ready to state and prove the following assertion. Theorem 5.13.20 (The decomposition theorem). Let the hypotheses in the incipit of this section hold. Let K ⊂ Ω ⊆ G, with K compact and Ω open. If u is Lharmonic in Ω \ K, then u has a decomposition of the form u = r + s, where r is L-harmonic in Ω and s is L-harmonic in G \ K. Furthermore, it can be assumed that s vanishes at infinity, and in this case the above decomposition is unique. Proof. Suppose the theorem holds true whenever Ω is bounded. We show that it holds true even for an unbounded Ω. Indeed, let u ∈ H(Ω \ K), where K is compact and Ω is an (unbounded) open set. Let R > 0 be such that K ⊂ Bd (0, R). Set := Ω ∩ Bd (0, R). Then K ⊂ Ω and, by our assumption, u can be decomposed as Ω s ∈ H(G \ K) and s → 0 at infinity. We consider the u = r + s, where r ∈ H(Ω), function r := u−s on Ω. Then r is L-harmonic in Ω \K and extends L-harmonically

304


(which is an open neighborhood of K). This across K, since r coincides with r on Ω ends the proof, since u = r + s, with r ∈ H(Ω), s ∈ H(G \ K) and s → 0 at infinity. We can then suppose that Ω is bounded. We fix the following notation (see also Fig. 5.1): B(λ) := closure of (∂Ω)λ , C(λ) := closure of Kλ , A(λ) := Ω \ (B(λ) ∪ C(λ)).

Fig. 5.1. Figure for the proof of Theorem 5.13.20

Since K is compact and ∂Ω is closed, we can choose λ > 0 small enough so that B(λ) ∩ C(λ) = ∅. Since Ω is bounded, we can choose a cut-off function ψλ ∈ C0∞ (RN ) such that supp(ψλ ) ⊂ Ω \ K,

ψλ ≡ 1 on A(λ).

We consider the function u ψλ , and we agree to consider this function trivially prolonged on RN to be zero. Hence this trivial prolongation belongs to C0∞ (RN ). By (5.16) in Theorem 5.3.3 (page 237) (being ψλ ≡ 1 on A(λ)), Γ (x −1 ◦ y) L(u ψλ )(y) dy u(x) = (u ψλ )(x) = − RN

+ + + =− A(λ) B(λ) C(λ) RN \(A(λ)∪B(λ)∪C(λ))

Γ (y −1 ◦ x) L(u ψλ )(y) dy ∀ x ∈ A(λ). (5.118) + =− B(λ)

C(λ)

In the last equality we used the fact that ψλ ≡ 1 on A(λ) jointly with Lu = 0 on Ω, and the fact that ψλ is supported in Ω. We now set Γ (y −1 ◦ x) L(u ψλ )(y) dy for x ∈ Ω \ B(λ), rλ (x) := − B(λ) sλ (x) := − Γ (y −1 ◦ x) L(u ψλ )(y) dy for x ∈ G \ C(λ). C(λ)


305

Hence, (5.118) gives the decomposition u(x) = rλ (x) + sλ (x)

∀ x ∈ A(λ).

(5.119)

From Lemma 5.13.19, we infer that rλ and sλ are L-harmonic on the respective sets of definition and that sλ vanishes at infinity. Let now 0 < μ < λ. Then obviously A(λ) ⊂ A(μ). From the decomposition in (5.119), we get an analogous decomposition u(x) = rμ (x) + sμ (x)

∀ x ∈ A(μ).

(5.120)

We claim that the decompositions (5.119) and (5.120) are compatible, i.e. rλ (x) = rμ (x) sλ (x) = sμ (x)

∀ x ∈ Ω \ B(λ), ∀ x ∈ G \ C(λ).

(5.121)

To prove the claim, we first remark that from (5.119) and (5.120) we obtain rλ (x) − rμ (x) = sμ (x) − sλ (x) Now, let us consider the following function sμ (x) − sλ (x) S : G → R, S(x) := rλ (x) − rμ (x)

∀ x ∈ A(λ).

(5.122)

for every x ∈ G \ C(λ), for every x ∈ Ω \ B(λ).

We claim that S has the following properties: i) S is well-posed: indeed, thanks to (5.122), sμ − sλ coincides with rλ − rμ on the set {G \ C(λ)} ∩ {Ω \ B(λ)} = A(λ); ii) S vanishes at infinity: indeed, this is true for sμ and sλ ; iii) S is L-harmonic on G: indeed, this is true for sμ − sλ on the open set G \ C(λ), and this is true for rλ − rμ on the open set Ω \ B(λ) (recall that C(μ) ⊂ C(λ) and B(μ) ⊂ B(λ)). Now, by the maximum principle (precisely, see Corollary 5.13.7, page 296) we infer that S ≡ 0, which is equivalent to the claimed (5.121). Now, let us fix a decreasing sequence of positive λn ’s such that λn → 0 as n → ∞. We set r(x) := rλn (x) s(x) := sλn (x)

∀ x ∈ Ω (where n ∈ N is such that x ∈ Ω \ B(λn )), ∀ x ∈ G \ K (where n ∈ N is such that x ∈ G \ C(λn )).

Thanks to (5.121), the definition of r(x) and s(x) are unambiguous.11 Now, the decomposition u(x) = r(x) + s(x) ∀x ∈Ω \K 11 We are also using the trivial fact that Ω \B(λ ) ↑ Ω and RN \C(λ ) ↓ RN \K as n → ∞. n n

306


follows from the analogous decomposition (5.119) (and the fact that A(λ) ↑ Ω \ K as n → ∞). Moreover, it is easily seen that r ∈ H(Ω), s ∈ H(G \ K) and s vanishes at infinity. This gives the desired decomposition of u as in the assertion. Finally, the uniqueness of the decomposition in the assertion (under the assumption that s vanishes at infinity) is another easy consequence of the same maximum principle quoted above. Indeed, suppose we are given two decompositions r2 + s2 = u = r1 + s1

on Ω \ K,

where ri ∈ H(Ω),

si ∈ H(G \ K),

lim si (x) = 0,

x→∞

i = 1, 2.

Then setting S : G → R,

S(x) :=

s1 (x) − s2 (x) r2 (x) − r1 (x)

for every x ∈ G \ K, for every x ∈ Ω,

we see that S ∈ H(G), S vanishes at infinity, and we end the proof following the same arguments as in the previous paragraph.

5.14 Appendix B. The Improved Pseudo-triangle Inequality Let G = (RN , δλ , ◦) be a homogeneous Carnot group, and let d be a symmetric homogeneous norm on G, smooth out of the origin.12 For example, d could be an L-gauge on G for some sub-Laplacian L on G. We know from Proposition 5.1.7 (page 231) that (even without assumptions on smoothness or symmetry of d) d satisfies the pseudo-triangle inequality d(a ◦ b) ≤ c (d(a) + d(b))

for every a, b ∈ G.

Here c≥1 is a constant depending on d and G. The aim of this brief appendix is to prove the following improvement of the pseudo-triangle inequality. For the following result, see also [DFGL05, (2.6)]. Proposition 5.14.1 (The improved pseudo-triangle inequality). Let d be a symmetric homogeneous norm on the homogeneous Carnot group G. Furthermore, suppose d is smooth out of the origin. Then there exists a constant β ≥ 1 depending only on d and G such that d(y ◦ x) ≤ β d(y) + d(x)

for every x, y ∈ G.

(5.123)

12 In other words (see the definition at the beginning of Section 5.1, page 229), the present d

has the following properties: d ∈ C(G, [0, ∞)) ∩ C ∞ (G \ {0}); d(δλ (x)) = λ d(x) for every λ > 0 and x ∈ G; d(x) = 0 iff x = 0; d(x −1 ) = d(x) for every x ∈ G.

5.14 Appendix B. The Improved Pseudo-triangle Inequality

307

Proof. Since (5.123) holds when y = 0, we can suppose y = 0. Since d is δλ homogeneous of degree one, (5.123) is equivalent to d(δ1/d(y) (y) ◦ δ1/d(y) (x)) ≤ β + d(δ1/d(y) (x)).

(5.124a)

By using the symmetry of d and setting ξ −1 = δ1/d(y) (y),

η−1 = δ1/d(y) (x),

(5.124a) is equivalent to (note that d(δ1/d(y) (y)) = 1) d(η ◦ ξ ) − d(η) ≤ β

for every ξ, η ∈ G: d(ξ ) = 1.

(5.124b)

By the usual13 pseudo-triangle inequality for d, (5.124b) holds when η ∈ Bd (0, M), i.e. d(η) ≤ M (where M = M(d, G) % 1 will be chosen in the sequel). Indeed, if η ∈ Bd (0, M), we have d(η ◦ ξ ) − d(η) ≤ c (d(η) + d(ξ )) − d(η) = (c − 1)d(η) + c≤(c − 1)M + c =: β. We can hence suppose η ∈ / Bd (0, M). Roughly speaking, we will show that we can drop η from (5.124b), by an argument of left-translation along curves which are supported away from zero, when M is large enough. Then, (5.124b) will follow from the classical mean value theorem. We now make this precise. Set Z := Log (ξ ) ∈ g, where Log is the logarithmic map related to G and g is the Lie algebra of G. Consider the integral curve γ of Z starting from η, i.e. with our usual notation γ (t) = exp(tZ)(η) = η ◦ exp(tZ)(0) = η ◦ Exp (tZ). Here we used Corollary 1.2.24 (page 24) and the definition of exponential map (see page 24). Obviously, we have γ (0) = η and γ (1) = η ◦ Exp (Z) = η ◦ Exp (Log (ξ )) = η ◦ ξ . If we show that we can choose M % 1 such that d(γ (t)) ≥ 1

for every t ∈ [0, 1]

(5.124c)

(recall that γ depends on η, besides ξ ), then [0, 1] t → u(t) := d(γ (t)) is smooth (for d is smooth out of the origin by hypothesis) so that the classical Lagrange mean value theorem applies and gives ˙ = sup (∇d)(γ (t)), γ˙ (t) d(η ◦ ξ ) − d(η) = u(1) − u(0) ≤ sup |u(t)| t∈[0,1] t∈[0,1] = sup (∇d)(γ (t)), (ZI )(γ (t)) t∈[0,1] = sup (Zd)(γ (t)). (5.124d) t∈[0,1]

Let X1 , . . . , XN be the Jacobian basis for the Lie algebra g of G. Hence, 13 See Proposition 5.1.7-1, page 231.

308

5 The Fundamental Solution for a Sub-Laplacian and Applications N

Z = Log (ξ ) =

(5.124e)

pj (ξ ) Zj ,

j =1

where the pj ’s are polynomials, so that there exists a constant C such that sup |pj (ξ )| ≤ C1

for all j = 1, . . . , N ,

(5.124f)

d(ξ )=1

since {ξ ∈ G : d(ξ ) = 1} is a compact set (see, e.g. Proposition 5.1.4, page 230). Moreover, Zj is δλ -homogeneous of degree σj ≥ 1 (see Corollary 1.3.19, page 42), so that Zj d is δλ -homogeneous of degree 1 − σj ≤ 0. Consequently, it is bounded on G \ Bd (0, 1), say sup |(Zj d)(ζ )| ≤ C2

for all j = 1, . . . , N.

(5.124g)

d(ζ )≥1

We now use again the claimed (5.124c) and, by collecting together (5.124f) to (5.124g), we derive that (5.124d) yields d(η ◦ ξ ) − d(η) ≤ sup (Zd)(ζ ) d(ζ )≥1

N = sup pj (ξ ) (Zj d)(ζ ) ≤ N C1 C2 . d(ζ )≥1 j =1

This proves (5.124b). We are then left to prove (5.124c). From the pseudo-triangle inequality for d (see Proposition 5.1.7-2, page 231) we have d(γ (t)) = d(η ◦ Exp (tZ)) ≥ ≥

1 d(η) − d(Exp (tZ)) c

1 1 M− sup d(Exp (t Log (ξ ))) =: M − m(d, G), c c t∈[0,1], d(ξ )=1

whence (5.124c) follows by choosing M = c(1 + m(d, G)). The finiteness of m(d, G) follows from N d(Exp (t Log (ξ ))) = d Exp tpj (ξ ) Zj j =1

≤ sup d Exp |ζj |≤C1

N

ζj Zj

< ∞,

j =1

uniformly for t ∈ [0, 1], d(ξ ) = 1. Here we used (5.124f) and the fact that N q(ζ ) := Exp ζj Zj j =1

has polynomial coefficient functions (see (1.75a), page 50). This completes the proof.

5.15 Appendix C. Existence of Geodesics

309

Note that, β being the constant in Proposition 5.14.1, if we replace x, y in (5.123) by, respectively, y −1 ◦ z and z−1 ◦ x, we get d(y −1 ◦ x) ≤ β d(y −1 ◦ z) + d(z−1 ◦ x)

for every x, y, z ∈ G.

(5.125a)

Moreover, by interchanging y and z in the above inequality (and using the symmetry of d) one gets |d(y −1 ◦ x) − d(z−1 ◦ x)| ≤ β d(y −1 ◦ z)

for every x, y, z ∈ G.

(5.125b)

5.15 Appendix C. Existence of Geodesics Let G = (RN , δλ , ◦) be a homogeneous Carnot group. Let g = W (1) ⊕ W (2) ⊕ · · · ⊕ W (r) be a stratification of the Lie algebra of G, as in Remark 1.4.8 (page 59). Let X = {X1 , . . . , Xm } be any basis of W (1) . We consider the related Carnot–Carathéodory distance dX . The aim of this section is to prove that the “inf” defining dX in (5.6) on page 232 is actually a minimum. In other words, fixed any x, y ∈ G, we show the existence of a X-subunit curve γ : [0, T ] → G connecting x and y (i.e. γ (0) = x, γ (T ) = y) such that T = dX (x, y). We shall call any such curve a X-geodesic (for x and y). The existence of geodesics can be proved in many general cases (see [Bus55] and [HK00]). Our argument here will make crucial use of the δλ -homogeneity and left-invariant properties of the system X. The resulting proof will be quite simple (for a more general proof, see the note after Theorem 5.15.5). First, we recall that, by Propositions 5.2.4, 5.2.6 and Theorem 5.2.8, dX (x, y) = d0 (y −1 ◦ x)

for every x, y ∈ G,

(5.126)

where d0 (z) := dX (z, 0),

z ∈ G,

(5.127)

and d0 is a homogeneous (symmetric) norm on G. As usual, we denote the dilation of G by δλ (x) = δλ (x1 , . . . , xN ) = (λσ1 x1 , . . . , λσN xN ),

λ > 0, x ∈ G,

where 1 = σ1 ≤ · · · ≤ σN = r are consecutive integers and r is the step of nilpotency of G. We are ready to prove the following result. Proposition 5.15.1. Let G be a homogeneous Carnot group, and let d be any homogeneous norm on G. For every compact set K ⊂ G, there exists a constant cK > 0 such that (cK )−1 |x − y| ≤ d(y −1 ◦ x) ≤ cK |x − y|1/r

∀ x, y ∈ K,

where r is the step of G and | · | is the Euclidean norm on G ≡ RN .

(5.128)

310


In particular, (5.128) holds true if d(y −1 ◦ x) is replaced by dX (x, y), where dX is the control distance related to any basis of generators (of the first layer of the stratification of the Lie algebra) of G. (Note. More generally, the estimates in (5.128) also hold when d = dX (with a suitable r), where X is a system of smooth vector fields satisfying the Hörmander condition, see [Lan83,NSW85,VSC92,Gro96]. The first equality in (5.128) holds for a general Carnot–Carathéodory distance d = dX , see [HK00] and Ex. 25 at the end of the chapter.) Proof. Once (5.128) has been proved, the last assertion of the proposition follows from (5.126), by taking d = d0 (d0 as in (5.127)). We then turn to prove (5.128). If | · | denotes the absolute value on R, set : G → [0, ∞),

(x) =

N

|xj |1/σj .

(5.129)

j =1

Obviously, is a homogeneous (symmetric) norm on G. By the equivalence of all homogeneous norms on G (see Proposition 5.1.4), there exists a constant c = c(, d, G) ≥ 1 such that c−1 (x) ≤ d(x) ≤ c (x)

∀ x ∈ G.

(5.130)

Thus (5.128) will follow if we show that, given a compact set K ⊂ G, (cK )−1 |x − y| ≤ (y −1 ◦ x) ≤ cK |x − y|1/r

∀ x, y ∈ K,

(5.131)

for a suitable constant cK > 0. Now, we recall the result in Corollary 1.3.18 (page 41): for every j ∈ {1, . . . , N } and every x, y ∈ G, we have (j ) (y −1 ◦ x)j = xj − yj + Pk (x, y) (xk − yk ), k: σk 1

where ⎛

1

⎜ ⎜ a2,1 (x) aj,h j,h=1,...,N = ⎜ ⎜ .. ⎝ . aN,1 (x)

1 .. . ···

⎞ 0 .. ⎟ .⎟ ⎟ = Jτ (0) x ⎟ 0⎠ 1

··· .. . .. .

0

aN,N −1 (x)

is the Jacobian matrix of the left translation by x on G (i.e. τx (y) = x ∗ y) and the aj,h ’s are polynomial functions, δλ -homogeneous of degree σj − σh . (1) (1) As a consequence, Zh Zk coincides with ∂2 (1) (1)

∂ xh xk

+

∂ aj,k j : σj =2

(0) ∂xj = (1)

∂ xh

∂2 (1) (1)

∂ xh xk

+

(2) N2

∂ aj,k j =1

(1)

∂ xh

(0) ∂x (2)

at 0. Here, we used a “stratified” notation for Jτx (0), namely ⎛

IN1

0

⎜ (2) ⎜ A1 (x) Jτx (0) = ⎜ .. ⎜ ⎝ .

IN2 .. . ···

(r)

A1 (x) Notice that

··· .. . .. . (r) Ar−1 (x)

⎞ 0 .. ⎟ . ⎟ ⎟. ⎟ 0 ⎠ INr

(2) (aj,k (x))1≤j ≤N2 ,1≤k≤N1 = A(2) 1 (x).

This shows that (iv) is equivalent to (2)

2 ∂ a

1 (1) (1) j,k (1) (1) (2) {Zh Zk f (0) + Zk Zh f (0)} + (0) Zj f (0) (1) 2 ∂x

N

j =1

?

=

∂ 2 f (0) ∂

(1) (1) xh xk

+

N2

∂

j =1

∂

(2) aj,k (0) ∂x (2) f (0). (1) j xh

Arguing as above, this is in turn equivalent to ∂ 2 f (0) (1) (1)

∂ xh xk

+

(2) N2

∂ aj,k j =1

(1)

∂ xh

(0) ∂x (2) f (0) j

h

j

770

20 Taylor Formula on Carnot Groups

+

∂ 2 f (0)

?

=

(2) (2) N2 ∂ aj,k ∂ aj,h 1 (0) + (0) ∂x (2) f (0) (1) (1) j 2 ∂ xk j =1 ∂ xh (1) (1)

∂ xh xk

+

(2) N2

∂ aj,k j =1

(1)

∂ xh

(0) ∂x (2) f (0). j

Thus, we are left to prove that, for all j ≤ N2 and all h, k ≤ N1 , it holds (2)

(2)

∂ aj,k

(0) + (1)

∂ aj,h (1) xk

?

(0) = 0.

∂ xh ∂ (2) (2) Now, the matrix aj,k = A1 involves the (derivatives of the) components of the left translation along the second layer. We know that the left translation is given by τx (y) = x ∗ y = x (1) + y (1) , x (2) + y (2) + Q(2) (x, y), . . . where Q(2) (x, y) is a polynomial in x (1) , y (1) of (ordinary) degree 2, mixed in x, y. Hence, for every j ∈ {1, . . . , N2 }, there exists a square matrix B j of order N1 such that (2)

(2)

(2)

(τx (y))j = xj + yj + B j · x (1) , y (1) . We now use the hypothesis x −1 = −x to derive that B j is skew-symmetric. Consequently, this gives N1 (2) ∂ j (2) τ (x) = (y) = xi(1) Bi,k . aj,k x j (1) ∂ yk 0 i=1 Hence, (2) ∂ aj,k (1)

∂ xh

(0) +

(2) ∂ aj,h (1)

∂ xk

j

j

(0) = Bh,k + Bk,h = 0,

for B j is skew-symmetric. The assertion is proved. Ex. 7) Suppose f, g ∈ C n+1 (G, R) are such that (Z1 , . . . , Zm )α f ≡ (Z1 , . . . , Zm )α g

∀ α ∈ (N ∪ {0})m : |α| = n + 1.

Show that there exists a polynomial function p, δλ -homogeneous of degree at most n, such that f = g + p on G. (Hint: Apply Theorem 20.3.3 to f − g and derive that f (h) − g(h) = Pn (f − g, 0)(h) for every h ∈ G.) Derive that (Zi1 , . . . , Zim )α f ≡ (Zi1 , . . . , Zim )α g for every α ∈ (N ∪ {0})m such that |α| = n + 1 and for every i1 , . . . , im ∈ {1, . . . , m}. This proves that the (Z1 , . . . , Zm )α f ’s with |α| = n + 1 determine all the (Zi1 , . . . , Zim )α f ’s with |α| = n + 1.


771

Ex. 8) Provide a detailed proof of Theorem 20.3.3. Ex. 9) Prove the following improvement of (20.55) and (20.65). For every x ∈ G, we have |Rn (x, x0 )| ≤

n+1 k

C k=1

k! ×

d(x0−1 ◦ x)σi1 +···+σik

i1 ,...,ik ≤N, σi1 +···+σik ≥n+1

×

sup d(z)≤C d(x0−1 ◦x)

|Zi1 · · · Zik u(x0 ◦ z)|,

where C is a constant depending only on G, d and the basis X . Here, the σi ’s are the same as in (20.2).

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[SY06] [Tai88] [Tar80] [Tha98] [Tha94] [Tho82] [Tre67] [Tre78]

[TW97] [TW99] [TW02a] [TW02b] [Ugu99] [Ugu00] [UL97] [UL02]

[Var84] [Varo87] [Varo00] [VSC92]

Strichartz, R.: Self-similarity on nilpotent Lie groups. Contemp. Math., 140, 123–157 (1992) Stroock, D.W., Varadhan, S.R.S.: On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math., 25, 651– 713 (1972) Sun, M., Yang, X.: Lipschitz continuity for H -convex functions in Carnot groups. Commun. Contemp. Math., 8, 1–8 (2006) Taira, K.: Diffusion Processes and Partial Differential Equations. Academic Press Inc., Boston, MA (1988) Tartakoff, D.S.: On the local real analyticity of solutions to b and ∂-Neumann problem. Acta Math., 145, 117–204 (1980) Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Progress in Mathematics, 159, Birkhäuser (1998) Thangavelu, S.: A Paley–Wiener theorem for step two nilpotent Lie groups. Rev. Mat. Iberoamericana, 10, 177–187 (1994) Thompson, R.C.: Cyclic relations and the Goldberg coefficients in the Campbell– Baker–Hausdorff formula. Proc. Am. Math. Soc., 86, 12–14 (1982) Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York, London (1967) Trèves, F.: Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ∂-Neumann problem. Comm. Partial Differential Equations, 3, 475–642 (1978) Trudinger, N.S., Wang, X.-J.: Hessian measures. I. Topol. Methods Nonlinear Anal., 10, 225–239 (1997) Trudinger, N.S., Wang, X.-J.: Hessian measures. II. Ann. of Math., 150, 579–604 (1999) Trudinger, N.S., Wang, X.-J.: Hessian measures. III. J. Funct. Anal., 193, 1–23 (2002) Trudinger, N.S., Wang, X.-J.: On the weak continuity of elliptic operators and applications to potential theory. Amer. J. Math., 124, 369–410 (2002) Uguzzoni, F.: A Liouville-type Theorem on halfspaces for the Kohn Laplacian. Proc. Amer. Math. Soc., 127, 117–123 (1999) Uguzzoni, F.: A note on Yamabe-type equations on the Heisenberg group. Hiroshima Math. J., 30, 179–189 (2000) Uguzzoni, F., Lanconelli, E.: On the Poisson kernel for the Kohn Laplacian. Rend. Mat. Appl., 17, 659–677 (1997) Uguzzoni, F., Lanconelli, E.: Degree theory for VMO maps on metric spaces and applications to Hörmander operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (V), 1, 569–601 (2002) Varadarajan, V.S.: Lie Groups, Lie Algebras and Their Representations. Graduate Texts in Mathematics, Springer-Verlag, New York, (1984) Varopoulos, N.T.: Fonctions harmoniques sur les groups de Lie. C. R. Acad. Sci. Paris, 304, 519–521 (1987) Varopoulos, N.T.: Geometric and potential theoretic results on Lie groups. Canad. J. Math., 52, 412–437 (2000) Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge (1992)

References [VM96]

[Wan05] [War83] [Weyl31]

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Vodopýanov, S.K., Markina, I.G.: Foundations of the nonlinear potential theory of subelliptic equations. Reshetnyak, Y.G., et al. (ed.), Sobolev Spaces and Adjacent Problems of Analysis. Tr. Inst. Mat. Im. S. L. Soboleva SO RAN, 31, Izdatel’stvo Instituta Matematiki SO RAN, Novosibirsk, 100–160 (1996) Wang, C.: Viscosity convex functions on Carnot groups. Proc. Amer. Math. Soc., 133, 1247–1253 (2005) Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, 94, Springer-Verlag, New York (1983) Weyl, H.: The Theory of Groups and Quantum Mechanics. Methuen, London (1931)

Index of the Basic Notation1

∂j , ∂xj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 X= N j =1 aj ∂j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Xf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 T (RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 X ≡ XI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ∇ .........................................................................5 γX (t, x), D(X, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 X (k) , X h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 exp(tX)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 exp(X)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 [X, Y ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 103 [Zj1 , · · · [Zjk−1 , Zjk ] · · ·] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ZJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Lie{U } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 rank(Lie{U }(x)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 [V , W ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 G := (RN , ◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 τα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 107 Jτα (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 η → J (η) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 {e1 , . . . , eN } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 π : g → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 (Z1 u, . . . , ZN u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 X Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 δλ : RN → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1 The page number refers to the first time the symbol is introduced within the text. In case of

multiple numbering, this means that the symbol is employed in multiple contexts.

790

Index of the Basic Notation

σ = (σ1 , . . . , σN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 G = (RN , ◦, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31, 56 |α|σ , |α|G , degG (p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 X ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 div(A · ∇ T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Jτx (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 |E| (E ⊆ RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Q= N j =1 σj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 126 g = g1 ⊕ · · · ⊕ gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 δλ : g → g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ξ · Z, E · ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50, 133 C-H(G) := (RN , ∗, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 x (1) , . . . , x (r) , x (i) ∈ RNi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 r and m = N1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 126 W (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 g = W (1) ⊕ · · · ⊕ W (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ΔG , L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62, 144 ∇G , ∇L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 L = div(A(x)∇ T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A(x) = σ (x) σ (x)T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 qL (x, ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Isotr(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 u → J (u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 X β := Xi1 ◦ · · · ◦ Xik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 (X1 , X2 )-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 πi : RN −→ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ϕ : U → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 (Uα , ϕα ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 v(f ), Mm , T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 M, N, M , N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 ∂/∂ xi |m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 dm ψ : Mm → Mψ(m) dψ : T (M) → T (M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 X : Ω −→ T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 π(m, v) := v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 X, Y , Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Xm , X(f )(m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 f → Xf , Xf : M → R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 X (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 dψ : X (M) → X (M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 X and X μ(t) ˙ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 [X, Y ] : M → T (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 G, H, F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 [·, ·] : g × g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107


791

g, h, f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 107 ◦, ∗, • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 τα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 x −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 α : g → Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 ϕ : G → H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 dϕ : g → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 expX (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Exp : g → G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24, 49, 118 Log : G → g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 49, 120 h = V1 ⊕ · · · ⊕ Vr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 (1) (r) B = (E1 , . . . , ENr ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 ad X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 πE : h → RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131, 139 Ψ := Exp ◦ (πE )−1 : RN → H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131, 139 Δλ : h → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 δλ := πE ◦ Δλ ◦ πE−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 H∗ := (RN , E , δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Exp h : h∗ → h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 HL (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Im(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 Hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 hn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 ΔHn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Bx, ξ , B (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158 (Fm,2 , ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 γi,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 H = (Rm+n , ◦, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169, 681 ρ : N → N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 HM-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Bη , g∗2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 E = (RN , +, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 (B, ◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 B = (R1+N , ◦, δλ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 h1 , . . . , hk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 d, |x|G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 d0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 dX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 S(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 l(γ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

792


X-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Dirac0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 uε , u ∗G Jε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Jε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 B (x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Bd (x, r), B˙d (x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252, 459 D(x, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 -dist(x, A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 ωd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 KL , ΨL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Mr , Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Mr , Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Φr , Φr∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 βd , md , nd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255, 259 a d , ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257, 261 −D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Iα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 ML (f )(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 F ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 ν⊥F at y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 AL(u)(x), AL(u)(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 lim infy→x , lim supy→x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 l.s.c., u.s.c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338 u, uˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 (E, T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 F : V → F(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 HϕV , HfV , HfΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 361, 388 μVx , μΩ x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341, 367, 388 H∗ (Ω), H∗ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 F ↑, F ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 (E, H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 H(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346, 388 B-H∗ (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 K ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 B(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 S(Ω), S(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353, 389 uV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 (H-D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Ω Uf , UΩ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Ω

Hf , HΩ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 R(∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361


793

R∞ (∂Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 + S c (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 sx0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 sxΩ0 := HfΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 u0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 f f RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 RA , L H(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Lε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 (G, H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 X 2 u(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 V := Exp (V1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 [x ◦ h−1 , x ◦ h] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Vx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 V -Convv (Ω), V -ConvH (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 GΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425, 427 Γ ∗ μ, GΩ ∗ μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 k, K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 mr [u], Mr [u] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 λx,r (y), Λx,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 μ, μ[u], μu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 S b (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 + S (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 I (μ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 V (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 UE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 RuE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 ΦEu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 RuE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 WK , VK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 μK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 C(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502, 508 C∗ (E), C ∗ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 GΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 g Ω (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 M, M0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 M(E), M0 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 A B, μ|A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 νE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 μ, ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 E + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 (ρ) Mφ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 mφ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

794 (ρ)


M(α) , m(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 α(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 n(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 I (φ, Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 fm,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 H (m, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 eX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 F = {F1 , . . . , FH } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .650 z, z⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Jz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 U (1) , . . . , U (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 v : G → b, z : G → z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 σ (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 u∗ (x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 Pk [x1 , . . . , xq ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 r k=s Pk [x1 , . . . , xq ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 jq , c(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 Pn = span{x α : α ∈ In } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 α Z α := Z1α1 · · · ZNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 Pn (f, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 πn : P → Pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Pn (f, x0 )(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 ζ1 (h), . . . , ζN (h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 Hesssym u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764

Index

B-hyperharmonic function, 348–350, 352 minimum principle, 351 B-invariant set, 349–351 δλ -degree (of a polynomial), 33 δλ -homogeneous degree (of a polynomial), 33 G-cone, 480 G-degree (of a polynomial), 33 G-polynomial, 734 S-harmonic space, 363, 367 positivity axiom (in a), 363 S∗ -harmonic space, 370 on a Carnot group, 382 L-maximal function, 277 q-set, 477 associative algebra, 600 asymptotic solid sub-Laplacian, 261 asymptotic surface sub-Laplacian, 257 balayage, 376 balayage (w.r.t. a sub-Laplacian), 501 barrier function (in a S∗ -harmonic space), 371 Bôcher-type theorem, 460, 461 Bouligand theorem, 371, 391, 544 bounded above L-subharmonic functions, 474 Brelot convergence property, 268 Caccioppoli–Weyl’s lemma for subLaplacians, 408 Campbell–Hausdorff formula, 128, 129, 194, 585–587, 600, 659

formula for formal power series, 600 formula for homogeneous vector fields, 194, 593, 594 operation, 29, 128, 130, 131, 142, 143, 585, 586, 595, 626, 652 canonical sub-Laplacian, 621, 622 of a Heisenberg-type group, 171, 692 capacitable set (w.r.t. a sub-Laplacian), 509 capacitary distribution (w.r.t. a subLaplacian), 502, 505 capacitary potential (w.r.t. a sub-Laplacian), 502, 508 capacity of a compact set (w.r.t. a sub-Laplacian), 502 and polarity, 507 characterization of the-, 503, 505 monotonicity, 506 right continuity, 506 strong subadditivity, 506 capacity (w.r.t. a sub-Laplacian), 509 and polarity, 509 exterior-, 509 interior-, 508 of Borel sets, 509 Carnot group, 56, 122, 131, 198 B-groups, 184 canonical sub-Laplacian of a-, 62 classical definition, 122 Euclidean group, 183, 280 filiform-, 207

796

Index

free-, 578, 584, 587, 588, 625, 658, 661, 662 harmonic space, 381 Heisenberg–Weyl group, 155 HM-groups, 174 homogeneous-, 121 K-type groups, 186 of Heisenberg type, 169, 681 of Iwasawa type, 702 of step r, 56, 198 of step two, 158, 163, 661, 666 of type HM, 174 stratified change of basis on a-, 61, 165 sum of-, 190 with homogeneous dimension Q ≤ 3, 184 central series (lower-), 149, 207 characteristic set (of a smooth open set), 710 Choquet lemma, 339 commutator, 103 nested-, 11 of homogeneous fields, 37 of length k, 11 continuity principle for potentials, 489 convergence axiom, 345, 347 in a Carnot group, 382 convex function equivalence between H- and v-, 417 H-, 416 horizontally-, 416 v-, 411 Cornea theorem, 344, 347 covering lemma, 566 decomposition theorem, 303 derivatives on G, 740 differentiable manifold, 88 differential, 95 of a homomorphism, 113, 114, 121 of a smooth map, 95 of the exponential map, 119 dilation, 31, 48, 56, 121, 132, 191, 593, 627, 638, 649, 656, 669 -invariance of a sub-Laplacian, 63 differential operator homogeneous with respect to-, 32, 34 function homogeneous with respect to-, 32, 34 invariance of the Lebesgue measure with respect to-, 44 on the Lie algebra g, 46

Dini–Cartan theorem, 343 Dirichlet problem (in a harmonic space), 359, 361, 371 doubling measure, 277 down directed (family of functions), 342, 343, 349, 356, 357 down directed (family of sets), 349 eikonal equation, 466 energy w.r.t. a sub-Laplacian, 528 equilibrium distribution (w.r.t. a subLaplacian), 497, 505 equilibrium potential (w.r.t. a sub-Laplacian), 497 as barrier function, 514 fundamental theorem on-, 498 uniqueness of the-, 508 equilibrium value (w.r.t. a sub-Laplacian), 497 characterization of the-, 505 exponential function generated by a system of vector fields, 194 map, 24, 49, 118, 129, 131, 626 inverse function of-, 27, 49, 626 of a homogeneous group of step 2, 166 of a vector field, 8, 23, 117 extended L-Green function, 516, 518 extended maximum principle for Lsubharmonic functions, 493 extended Poisson–Jensen’s formula, 518 exterior L-capacity, 509 filiform Carnot group, 207, 209, 285 fine topology (w.r.t. a sub-Laplacian), 537 fractional integral (in a Carnot group), 277 free Carnot group, 578 free homogeneous Carnot group, 584, 586–588, 625, 629, 656, 658, 659, 661, 662 free nilpotent Lie algebra, 577, 579, 583, 586, 624, 650, 656 Hall basis for the-, 579 fundamental solution, 236, 425, 456, 477, 649, 661–663, 665 H-type group, 696 fundamental theorem on L-equilibrium potentials, 498

Index gauge function (w.r.t. a sub-Laplacian), 247 generalized solution (in the sense of Perron–Wiener–Brelot), 359, 361, 390 generators of a homogeneous Carnot group, 56 geodesics (for a homogeneous Carnot group), 309, 314 gradient (canonical G-), 62 Grayson–Grossman theorem, 582 Green function extended-, 516, 518 of a general domain, 427 approximation of the-, 429 symmetry of the-, 431 of an L-regular set, 425, 426, 445, 446, 448 potential of a measure related to a-, 432 H-groups (in the sense of Métivier), 174 H-inversion map, 705 H-Kelvin transform, 705 H-type algebra, 681 H-type group, 681 canonical sub-Laplacian, 692 fundamental solution, 696 H-inversion map, 705 H-Kelvin transform, 705 Iwasawa group, 702 prototype-, 169, 687 Hall basis, 579, 581, 583, 586, 587 Hardy–Littlewood–Sobolev theorem, 277 harmonic function (w.r.t. a harmonic sheaf), 340, 346, 347, 353, 357 harmonic function (w.r.t. a sub-Laplacian), 146, 381, 388–390, 408, 433, 441–445, 458–461 Brelot convergence property for a-, 268 decomposition theorem for a-, 303 removable (or isolated) singularity, 458 harmonic majorant (least-), 358 harmonic measure (w.r.t. a harmonic sheaf), 341, 367 harmonic measure w.r.t. a sub-Laplacian, 388, 391, 426, 445, 518 of a polar set, 515 harmonic minorant (greatest-), 358, 427 harmonic sheaf, 340 harmonic space, 345, 347, 348, 353 in a Carnot group, 381

797

Harnack inequality, 265–267 on rings, 267, 460 Hausdorff dimension w.r.t. d, 558, 568 Hausdorff measure w.r.t. d, 557 height δλ -height of a multi-index, 32 G-height of a multi-index, 33 Heisenberg group polarized, 180 Heisenberg–Weyl group, 155, 156, 172, 580, 625, 630, 632, 659, 702 Heisenberg-type algebra, 681 Heisenberg-type group, 681 canonical sub-Laplacian, 692 fundamental solution, 696 H-inversion map, 705 H-Kelvin transform, 705 Iwasawa group, 702 prototype-, 687 Hessian (horizontal-), 412 higher-order derivatives on G, 738 HM-groups, 174 homogeneous Carnot group, 56, 121, 131, 198, 206, 586, 630, 637, 649, 656, 659 generators of a-, 56 homogeneous dimension, 44, 126, 128, 184, 303, 477, 559, 663 homogeneous Lie group on G homogeneous dimension of a-, 477 homogeneous Lie group on RN , 31, 196 homogeneous dimension of a-, 44 of step two, 158, 163 homogeneous norm, 229 pseudo-triangle inequality for a-, 231, 484 homomorphism, 114, 130 of Lie algebras, 112 of Lie groups, 112, 121 Hopf-type lemma, 297 horizontal Hessian, 412 horizontal segment, 415 horizontal Taylor formula, 763 horizontally convex function, 416 equivalence with v-convex function, 417 Hörmander condition, 12, 69, 185, 193, 202, 210, 281 hyperharmonic function (w.r.t. a harmonic sheaf), 341, 348, 352, 353, 355, 356

798

Index

hypoelliptic-(ity), 188, 193, 280, 434, 441, 633 analytic, 280 hypoharmonic function (w.r.t. a harmonic sheaf), 342 interior L-capacity, 508 isolated singularity, 458–461 Iwasawa-type group, 702 Jacobi identity, 11, 103, 107 Jacobian basis, 19–22, 26, 45, 50, 58, 59, 69, 115, 143, 144, 156, 159, 185, 187, 588, 622, 625, 627–629, 638, 639, 643, 649, 653, 656, 658–662, 666 of a homogeneous Lie group, 42, 43 Kelvin transformation, 704 l.s.c. function, 338 Lagrange mean-value theorem (on a Carnot group), 746 Laplace operator, 100, 183, 445, 621, 622 left-translation, 106 length δλ -length of a multi-index, 32 G-length of a multi-index, 33 Levi–Cartan theorem, 343, 347, 348 Lie algebra, 11, 14, 107, 197 filiform-, 207 free nilpotent-, 577 generated by a set, 11 Jacobian basis, 19 of a Carnot group, 59 of a Lie group, 108 Lie group, 106, 195 Carnot group, 56 composition law of a homogeneous-, 39, 41, 50, 58 homogeneous on RN , 31 on RN , 13 structure on a Lie algebra, 130 Lie polynomial, 600 Lifting, 653–656, 659, 661, 665, 666 Liouville-type theorems, 269, 270, 274, 461, 633 asymptotic-, 274–276 lower functions, 359 lower regularization, 339, 501

lower solution, 359 Lusin-type theorem, 495 Mac Laurin polynomial (on a Carnot group), 742 Maria–Frostman domination principle, 495, 499, 503 maximum principle, 474 -set, 474 for L-subharmonic functions, 409 extended-, 493 on unbounded open sets, 474 strong version, 426 weak version, 388, 389, 474 mean value formula, 391, 447 mean value operator, 397, 456 solid-, 399, 404, 405, 432, 441, 447, 458 superposition formula, 457, 458 surface-, 401, 404, 410, 447, 456 measure function, 557 minimum principle for B-hyperharmonic functions, 351, 360 mollifier, 239, 240, 401, 455 MP set, 474 mutual L-energy, 528 non-characteristic exterior ball condition, 384, 385, 387 peaking function at a point, 542 Perron family, 356–358, 362 Perron–Wiener–Brelot operator, 359 Perron–Wiener–Brelot solution (related to a sub-Laplacian), 390 Perron-regularization, 355 Poisson–Jensen’s formula, 445, 448 extended-, 518 polar set (w.r.t. a sub-Laplacian), 491, 493, 495, 508, 544, 559, 568 and harmonic measure, 515 characterization in terms of capacity, 507, 509 polar∗ set (w.r.t. a sub-Laplacian), 498, 499, 508 the L-irregular points are a-, 514 polarized Heisenberg group, 180 polynomial functions on G, 734 positivity axiom, 345, 351, 354 in a S-harmonic space, 363 in a Carnot group, 382

Index potential of a measure (related to the fundamental solution Γ ), 445, 451, 458 continuity principle (for the), 489 potential of a measure (related to the L-Green function GΩ ), 432, 433, 441, 443, 444 prototype H-type group, 169, 687 pseudo-triangle inequality, 231, 400, 484 improved-, 306 PWB function, 361, 428 quasi-continuity of L-superharmonic functions, 528 reduced function, 376 reduced function (w.r.t. a sub-Laplacian), 501 regular point, 371 regular point w.r.t. a sub-Laplacian, 518, 542, 544 and L-polarity∗ , 514 regular set (w.r.t. a harmonic sheaf), 341, 345, 346, 348, 353, 355 regular set w.r.t. a sub-Laplacian, 385, 387, 388, 391 approximation by-, 430 Green function of a-, 425 regularity axiom, 345, 346 in a Carnot group, 383 removable singularity, 458–461 resolutive function, 361 characterization of the-, 367 Riesz measure (of an L-subharmonic function), 441–443, 445, 447, 451, 502, 518 Riesz-type representation theorem, 441, 443–445, 451, 458 Riesz-type representation theorem, 484 segment (horizontal-), 415 separation axiom, 345, 352 in a Carnot group, 382 sheaf (of functions), 340, 353 harmonic-, 340 Sobolev–Stein embedding theorem, 279 Stone–Weierstrass theorem, 366 stratification, 45, 122, 131, 583 of a Carnot group, 60, 309, 314

799

stratified change of basis, 61, 165 stratified group, 122, 131 harmonic function, 146 sub-Laplacian of a-, 144 stratified Lagrange mean-value theorem, 746 stratified Taylor formula, 750 stratified Taylor inequality, 749 strong maximum principle, 296 sub-Laplacian, 62, 66, 198, 623, 625, 637, 641, 650 arising in control theory, 205 Caccioppoli–Weyl’s lemma (for-), 408 canonical-, 62, 198, 625, 637, 641, 650, 663 on a Heisenberg-type group, 692 characteristic form of a-, 65 degenerate-ellipticity of a-, 66 harmonic measure related to a-, 388 harmonic space related to a-, 381 invariance with respect to the dilation, 63 of a stratified group, 144 of Bony-type, 202, 223, 285 of Kolmogorov-type, 204 Perron–Wiener–Brelot solution related to a-, 390 q-set w.r.t. a-, 477 regular set w.r.t a-, 388 Riesz measure of a function subharmonic w.r.t. a-, 441 subharmonic function w.r.t. a-, 389 superharmonic function w.r.t. a-, 389 thin set w.r.t. a-, 474 sub-mean function, 397–399, 401 sub-mean properties, 399 subharmonic function (w.r.t. a harmonic sheaf), 353 criterion (for subharmonicity), 354 subharmonic function (w.r.t. a subLaplacian), 389, 401, 402, 404, 405, 411, 441, 442, 445, 447, 451, 456, 494, 516, 518 bounded above in G, 451 bounded above-, 474 extended maximum principle for-, 493 Riesz-type representation theorem for a-, 441, 443 subharmonic smoothing for a-, 456 subharmonic minorant, 356, 358

800

Index

superharmonic function (w.r.t. a harmonic sheaf), 353 characterization of the-, 353 superharmonic function (w.r.t. a subLaplacian), 389, 410, 432, 457, 458, 491 superharmonic majorant, 358 surface mean value theorem, 391, 404 symmetrized horizontal Hessian, 764 tangent bundle, 91 tangent space, 91, 109 tangent vector, 91 Taylor formula (on a Carnot group), 750 Taylor inequality (on a Carnot group), 749 Taylor polynomial (on a Carnot group), 745 thin set w.r.t. a sub-Laplacian, 474 thinness of a set at a point (w.r.t. a sub-Laplacian), 538, 550, 553 total gradient, 22 total set in C0+ (G), 529 u.s.c. function, 338 up directed (family of functions), 342–344, 347, 348, 354

upper functions, 359 upper regularization, 339 upper solution, 359 v-convex function, 411 equivalence with H-convex function, 417 vector field, 4, 97 complete-, 102, 116 generating a Carnot group, 191 integral curve (of a), 6, 101 left-invariant, 14, 17, 107, 197 completeness of the-, 116 Lie-bracket, 10 related-, 99, 105, 106, 114 smooth-, 98 weak maximum principle, 295 Wiener resolutivity theorem, 364, 390 Wiener’s criterion for sub-Laplacians, 547, 550 Wiener’s regularity test for sub-Laplacians, 553 Zorn’s lemma, 350

Springer Monographs in Mathematics This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should also describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Abhyankar, S.S. Resolution of Singularities of Embedded Algebraic Surfaces 2nd enlarged ed. 1998 Alexandrov, A.D. Convex Polyhedra 2005 Andrievskii, V.V.; Blatt, H.-P. Discrepancy of Signed Measures and Polynomial Approximation 2002 Angell, T.S.; Kirsch, A. Optimization Methods in Electromagnetic Radiation 2004 Ara, P.; Mathieu, M. Local Multipliers of C∗ -Algebras 2003 Armitage, D.H.; Gardiner, S.J. Classical Potential Theory 2001 Arnold, L. Random Dynamical Systems corr. 2nd printing 2003 (1st ed. 1998) Arveson, W. Noncommutative Dynamics and E-Semigroups 2003 Aubin, T. Some Nonlinear Problems in Riemannian Geometry 1998 Auslender, A.; Teboulle, M. Asymptotic Cones and Functions in Optimization and Variational Inequalities 2003 Banagl, M. Topological Invariants of Stratified Spaces 2006 Banasiak, J.; Arlotti, L. Perturbations of Positive Semigroups with Applications 2006 Bang-Jensen, J.; Gutin, G. Digraphs 2001 Baues, H.-J. Combinatorial Foundation of Homology and Homotopy 1999 Bonfiglioli, A., Lanconelli, E., Uguzzoni, F. Stratified Lie Groups and Potential Theory for their SubLaplacians 2007 Böttcher, A.; Silbermann, B. Analysis of Toeplitz Operators 2006 Brown, K.S. Buildings 3rd printing 2000 (1st ed. 1998) Chang, K.-C. Methods in Nonlinear Analysis 2005 Cherry, W.; Ye, Z. Nevanlinna’s Theory of Value Distribution 2001 Ching, W.K. Iterative Methods for Queuing and Manufacturing Systems 2001 Chudinovich, I. Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes 2005 Coates, J.; Sujatha, R. Cyclotomic Fields and Zeta Values 2006 Crabb, M.C.; James, I.M. Fibrewise Homotopy Theory 1998 Dineen, S. Complex Analysis on Infinite Dimensional Spaces 1999 Dugundji, J.; Granas, A. Fixed Point Theory 2003 Ebbinghaus, H.-D.; Flum, J. Finite Model Theory 2006 Edmunds, D.E.; Evans, W.D. Hardy Operators, Function Spaces and Embeddings 2004 Elstrodt, J.; Grunewald, F.; Mennicke, J. Groups Acting on Hyperbolic Space 1998 Engler, A.J.; Prestel, A. Valued Fields 2005 Fadell, E.R.; Husseini, S.Y. Geometry and Topology of Configuration Spaces 2001 Fedorov, Yu. N.; Kozlov, V.V. A Memoir on Integrable Systems 2001 Flenner, H.; O’Carroll, L.; Vogel, W. Joins and Intersections 1999 Gelfand, S.I.; Manin, Y.I. Methods of Homological Algebra 2nd ed. 2003 (1st ed. 1996) Griess, R.L. Jr. Twelve Sporadic Groups 1998 Gras, G. Class Field Theory corr. 2nd printing 2005 Greuel, G.-M.; Lossen, C.; Shustin, E. Introduction to Singularities and Deformations 2007 Hida, H. p-Adic Automorphic Forms on Shimura Varieties 2004 Ischebeck, E; Rao, R.A. Ideals and Reality 2005 Ivrii, V. Microlocal Analysis and Precise Spectral Asymptotics 1998 Jakimovski, A.; Sharma, A.; Szabados, J. Walsh Equiconvergence of Complex Interpolating Polynomials 2006 Jech, T. Set Theory (3rd revised edition 2002) Jorgenson, J.; Lang, S. Spherical Inversion on SLn (R) 2001

Kanamori, A. The Higher Infinite corr. 2nd printing 2005 (2nd ed. 2003) Kanovei, V. Nonstandard Analysis, Axiomatically 2005 Khoshnevisan, D. Multiparameter Processes 2002 Koch, H. Galois Theory of p-Extensions 2002 Komornik, V. Fourier Series in Control Theory 2005 Kozlov, V.; Maz’ya, V. Differential Equations with Operator Coefficients 1999 Lam, T.Y. Serre’s Problem on Projective Modules 2006 Landsman, N.P. Mathematical Topics between Classical & Quantum Mechanics 1998 Leach, J.A.; Needham, D.J. Matched Asymptotic Expansions in Reaction-Diffusion Theory 2004 Lebedev, L.P.; Vorovich, I.I. Functional Analysis in Mechanics 2002 Lemmermeyer, F. Reciprocity Laws: From Euler to Eisenstein 2000 Malle, G.; Matzat, B.H. Inverse Galois Theory 1999 Mardesic, S. Strong Shape and Homology 2000 Margulis, G.A. On Some Aspects of the Theory of Anosov Systems 2004 Miyake, T. Modular Forms 2006 Murdock, J. Normal Forms and Unfoldings for Local Dynamical Systems 2002 Narkiewicz, W. Elementary and Analytic Theory of Algebraic Numbers 3rd ed. 2004 Narkiewicz, W. The Development of Prime Number Theory 2000 Neittaanmaki, P.; Sprekels, J.; Tiba, D. Optimization of Elliptic Systems. Theory and Applications 2006 Onishchik, A.L. Projective and Cayley–Klein Geometries 2006 Parker, C.; Rowley, P. Symplectic Amalgams 2002 Peller, V. (Ed.) Hankel Operators and Their Applications 2003 Prestel, A.; Delzell, C.N. Positive Polynomials 2001 Puig, L. Blocks of Finite Groups 2002 Ranicki, A. High-dimensional Knot Theory 1998 Ribenboim, P. The Theory of Classical Valuations 1999 Rowe, E.G.P. Geometrical Physics in Minkowski Spacetime 2001 Rudyak, Y.B. On Thorn Spectra, Orientability and Cobordism 1998 Ryan, R.A. Introduction to Tensor Products of Banach Spaces 2002 Saranen, J.; Vainikko, G. Periodic Integral and Pseudodifferential Equations with Numerical Approximation 2002 Schneider, P. Nonarchimedean Functional Analysis 2002 Serre, J-P. Complex Semisimple Lie Algebras 2001 (reprint of first ed. 1987) Serre, J-P. Galois Cohomology corr. 2nd printing 2002 (1st ed. 1997) Serre, J-P. Local Algebra 2000 Serre, J-P. Trees corr. 2nd printing 2003 (1st ed. 1980) Smirnov, E. Hausdorff Spectra in Functional Analysis 2002 Springer, T.A.; Veldkamp, F.D. Octonions, Jordan Algebras, and Exceptional Groups 2000 Székelyhidi, L. Discrete Spectral Synthesis and Its Applications 2006 Sznitman, A.-S. Brownian Motion, Obstacles and Random Media 1998 Taira, K. Semigroups, Boundary Value Problems and Markov Processes 2003 Talagrand, M. The Generic Chaining 2005 Tauvel, P.; Yu, R.W.T. Lie Algebras and Algebraic Groups 2005 Tits, J.; Weiss, R.M. Moufang Polygons 2002 Uchiyama, A. Hardy Spaces on the Euclidean Space 2001 Üstünel, A.-S.; Zakai, M. Transformation of Measure on Wiener Space 2000 Vasconcelos, W. Integral Closure. Rees Algebras, Multiplicities, Algorithms 2005 Yang, Y. Solitons in Field Theory and Nonlinear Analysis 2001 Zieschang, P.-H. Theory of Association Schemes 2005

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