Homogeneous Lie groups

Transcription

1 Chapter 3 Homogeneous Lie groups By definition a homogeneous Lie group is a Lie group equipped with a family of dilations compatible with the group law. The abelian group (R n, +) is the very first example of homogeneous Lie group. Homogeneous Lie groups have proved to be a natural setting to generalise many questions of Euclidean harmonic analysis. Indeed, having both the group and dilation structures allows one to introduce many notions coming from the Euclidean harmonic analysis. There are several important differences between the Euclidean setting and the one of homogeneous Lie groups. For instance the operators appearing in the latter setting are usually more singular than their Euclidean counterparts. However it is possible to adapt the technique in harmonic analysis to still treat many questions in this more abstract setting. As explained in the introduction (see also Chapter 4), we will in fact study operators on a subclass of the homogeneous Lie group, more precisely on graded Lie groups. A graded Lie group is a Lie group whose Lie algebra admits a (N)- gradation. raded Lie groups are homogeneous and in fact the relevant structure for the analysis of graded Lie groups is their natural homogeneous structure and this justifies presenting the general setting of homogeneous Lie groups. From the point of view of applications, the class of graded Lie groups contains many interesting examples, in fact all the ones given in the introduction. Indeed these groups appear naturally in the geometry of certain symmetric domains and in some subelliptic partial differential equations. Moreover, they serve as local models for contact manifolds and CR manifolds, or for more general Heisenberg manifolds, see the discussion in the Introduction. The references for this chapter of the monograph are [FS82, ch. I] and [oo76], as well as Fulvio Ricci s lecture notes [Ric]. However, our conventions and notation do not always follow the ones of these references. The treatment in this chapter is, overall, more general than that in the above literature since we also consider distributions and kernels of complex homogeneous degrees and adapt our analysis for subsequent applications to Sobolev spaces and to the op- The Editor(s) (if applicable) and The Author(s) 2016 V. Fischer, M. Ruzhansky, Quantization on Nilpotent Lie roups, Progress in Mathematics 314, DOI / _3 91

2 92 Chapter 3. Homogeneous Lie groups erator quantization developed in the following chapters. Especially, our study of complex homogeneities allows us to deal with complex powers of operators (e.g. in Section 4.3.2). 3.1 raded and homogeneous Lie groups In this section we present the definition and the first properties of graded Lie groups. Since many of their properties can be explained in the more general setting of homogeneous Lie groups, we will also present these groups Definition and examples of graded Lie groups We start with definitions and examples of graded and stratified Lie groups. Definition (i) A Lie algebra g is graded when it is endowed with a vector space decomposition (where all but finitely many of the V j s are {0}): g = V j such that [V i,v j ] V i+j. j=1 (ii) A Lie group is graded when it is a connected simply connected Lie group whose Lie algebra is graded. The condition that the group is connected and simply connected is technical but important to ensure that the exponential mapping is a global diffeomorphism between the group and its Lie algebra. The classical examples of graded Lie groups and algebras are the following. Example (Abelian case). The abelian group (R n, +) is graded: its Lie algebra R n is trivially graded, i.e. V 1 = R n. Example (Heisenberg group). The Heisenberg group H no giveninexample is graded: its Lie algebra h no can be decomposed as h no = V 1 V 2 where V 1 = no i=1 RX i RY i and V 2 = RT. (For the notation, see Example in Section 1.6.) Example (Upper triangular matrices). The group T no of n o n o matrices which are upper triangular with 1 on the diagonal is graded: its Lie algebra t no of n o n o upper triangular matrices with 0 on the diagonal is graded by t no = V 1... V no 1 where V j = no j i=1 RE i,i+j. (For the notation, see Example in Section 1.6.) The vector space V j is formed by the matrices with only non-zero coefficients on the j-th upper off-diagonal.

3 3.1. raded and homogeneous Lie groups 93 As we will show in Proposition , a graded Lie algebra (hence possessing a natural dilation structure) must be nilpotent. The converse is not true, see Remark 3.1.6, Part 2. Examples are stratified in the following sense: Definition (i) A Lie algebra g is stratified when g is graded, g = j=1 V j, and the first stratum V 1 generates g as an algebra. This means that every element of g can be written as a linear combination of iterated Lie brackets of various elements of V 1. (ii) A Lie group is stratified when it is a connected simply connected Lie group whose Lie algebra is stratified. Remark Let us make the following comments on existence and uniqueness of gradations. 1. A gradation over a Lie algebra is not unique: the same Lie algebra may admit different gradations. For example, any vector space decomposition of R n yields a graded structure on the group (R n, +). More convincingly, we can decompose the 3 dimensional Heisenberg Lie algebra h 1 as 3 h 1 = V j with V 1 = RX 1, V 2 = RY 1, V 3 = RT. j=1 This last example can be easily generalised to find several gradations on the Heisenberg groups H no, n o =2, 3,..., which are not the classical ones given in Example Another example would be 8 h 1 = V j with V 3 = RX 1, V 5 = RY 1, V 8 = RT, (3.1) j=1 and all the other V j = {0}. 2. A gradation may not even exist. The first obstruction is that the existence of a gradation implies nilpotency; in other words, a graded Lie group or a graded Lie algebra are nilpotent, as we shall see in the sequel (see Proposition ). Even then, a gradation of a nilpotent Lie algebra may not exist. As a curiosity, let us mention that the (dimensionally) lowest nilpotent Lie algebra which is not graded is the seven dimensional Lie algebra given by the following commutator relations: [X 1,X j ]=X j+1 for j =2,...,6, [X 2,X 3 ]=X 6, [X 2,X 4 ]=[X 5,X 2 ]=[X 3,X 4 ]=X 7. They define a seven dimensional nilpotent Lie algebra of step 6 (with basis {X 1,...,X 7 }). It is the (dimensionally) lowest nilpotent Lie algebra which is not graded. See, more generally, [oo76, ch.i 3.2].

4 94 Chapter 3. Homogeneous Lie groups 3. To go back to the problem of uniqueness, different gradations may lead to morally equivalent decompositions. For instance, if a Lie algebra g is graded by g = j=1 V j then it is also graded by g = j=1 W j where W 2j +1 = {0} and W 2j = V j. This last example motivates the presentation of homogeneous Lie groups: indeed graded Lie groups are homogeneous and the natural homogeneous structure for the graded Lie algebra g = j=1v j = j=1w j is the same for the two gradations. Moreover, the relevant structure for the analysis of graded Lie groups is their natural homogeneous structure. 4. There are plenty of graded Lie groups which are not stratified, simply because the first vector subspace of the gradation may not generate the whole Lie algebra (it may be {0} for example). This can also be seen in terms of dilations defined in Section Moreover, a direct product of two stratified Lie groups is graded but may be not stratified as their stratification structures may not match. We refer to Remark for further comments on this topic Definition and examples of homogeneous Lie groups We now deal with a more general subclass of Lie groups, namely the class of homogeneous Lie groups. Definition (i) A family of dilations of a Lie algebra g is a family of linear mappings {D r,r>0} from g to itself which satisfies: the mappings are of the form D r =Exp(A ln r) = l=0 1 l! (ln(r)a)l, where A is a diagonalisable linear operator on g with positive eigenvalues, Exp denotes the exponential of matrices and ln(r) the natural logarithm of r>0, each D r is a morphism of the Lie algebra g, that is, a linear mapping from g to itself which respects the Lie bracket: X, Y g, r > 0 [D r X, D r Y ]=D r [X, Y ]. (ii) A homogeneous Lie group is a connected simply connected Lie group whose Lie algebra is equipped with dilations.

5 3.1. raded and homogeneous Lie groups 95 (iii) We call the eigenvalues of A the dilations weights or weights. The set of dilations weights, or in other worlds, the set of eigenvalues of A is denoted by W A. We can realise the mappings A and D r in a basis of A-eigenvectors as the diagonal matrices υ 1 r υ1 υ 2 A... and D r υ2 r.... υn r υn The dilations weights are υ 1,...,υ n. Remark Note that if {D r } is a family of dilations of the Lie algebra g, then D r := D r α := Exp(αA ln r) defines a new family of dilations { D r,r>0} for any α>0. By adjusting α if necessary, we may assume that the dilations weights satisfy certain properties in order to compare different families of dilations and in order to fix one of such families. For example in [FS82], it is assumed that the minimum eigenvalue is 1. raded Lie algebras are naturally equipped with dilations: if the Lie algebra g is graded by g = j=1v j, then we define the dilations D r := Exp(A ln r) where A is the operator defined by AX = jx for X V j. Theconverseistrue: Lemma If a Lie algebra g has a family of dilations such that the weights are all rational, then g has a natural gradation. Proof. By adjusting the weights (see Remark 3.1.8), we may assume that all the eigenvalues are positive integers. Then the decomposition in eigenspaces gives the the gradation of the Lie algebra. Before discussing the dilations in the examples given in Section and other examples of homogeneous Lie groups, let us state the following crucial property. Proposition The following holds: (i) A Lie algebra equipped with a family of dilations is nilpotent. (ii) A homogeneous Lie group is a nilpotent Lie group.

6 96 Chapter 3. Homogeneous Lie groups Proof of Proposition Let {D r =Exp(Aln r)} be the family of dilations. By Remark 3.1.8, we may assume that the smallest weight is 1. For υ W A let W υ g be the corresponding eigenspace of A. Ifυ R but υ/ W A then we set W υ := {0}. Thus D r X = r υ X for X W υ.moreover,ifx W υ and Y W υ then D r [X, Y ]=[D r X, D r Y ]=r υ+υ [X, Y ] and hence [W υ,w υ ] W υ+υ. In particular, since υ 1forυ W A, we see that the ideals in the lower series of g (see (1.18)) satisfy g (j) a j W a. Since the set W A is finite, it follows that g (j) = {0} for j sufficiently large. Consequently the Lie algebra g and its corresponding Lie group are nilpotent. Let be a homogeneous Lie group with Lie algebra g endowed with dilations {D r } r>0. By Proposition , the connected simply connected Lie group is nilpotent. We can transport the dilations to the group using the exponential mapping exp =expof (see Proposition (a)) in the following way: the maps exp D r exp 1, r > 0, are automorphisms of the group ; we shall denote them also by D r and call them dilations on. This explains why homogeneous Lie groups are often presented as Lie groups endowed with dilations. We may write rx := D r (x) for r>0andx. The dilations on the group or on the Lie algebra satisfy D rs = D r D s, r,s > 0. As explained above, Examples 3.1.2, and, are naturally homogeneous Lie groups: In Example 3.1.2: The abelian group (R n, +) is homogeneous when equipped with the usual dilations D r x = rx, r>0, x R n. In Example 3.1.3: The Heisenberg group H no is homogeneous when equipped with the dilations rh =(rx, ry, r 2 t), h =(x, y, t) R no R no R. The corresponding dilations on the Heisenberg Lie algebra h no are given by D r (X j )=rx j, D r (Y j )=ry j, j =1,...,n o, and D r (T )=r 2 T.

7 3.1. raded and homogeneous Lie groups 97 In Example 3.1.4: The group T no is homogeneous when equipped with the dilations defined by [D r (M)] i,j = r j i [M] i,j 1 i<j n o, M T no. The corresponding dilations on the Lie algebra t no are given by D r (E i,j )=r j i E i,j 1 i<j n o. As already seen for the graded Lie groups, the same homogeneous Lie group may admit various homogeneous structures, that is, a nilpotent Lie group or algebra may admit different families of dilations, even after renormalisation of the eigenvalues (see Remark 3.1.8). This can already be seen from the examples in the graded case (see Remark part 1). These examples can be generalised as follows. Example On R n we can define D r (x 1,...,x n )=(r υ1 x 1,...,r υn x n ), where 0 <υ 1... υ n,andonh no we can define D r (x 1,...,x no,y 1,...,y no,t)=(r υ1 x 1,...,r υno x no,r υ 1 y1,...,r υ no y no,r υ t), where υ j > 0, υ j > 0andυ j + υ j = υ for all j =1,...,n o. These families of dilations give graded structures whenever the weights υ j for R n and υ j,υ j,υ for H no are all rational or, more generally, all in αq + for afixedα R +. From this remark it is not difficult to construct a homogeneous non-graded structure: on R 3, consider the diagonal 3 3 matrix A with entries, e.g., 1 and π and 1 + π. Example Continuing the example above, choosing the υ j and υ j s rational in a certain way, it is also possible to find a homogeneous structure for H no such that the corresponding gradation of h no = j=1 V j does exist but is necessarily such that V 1 = {0}: we choose υ j,υ j positive integers different from 1 but with 1 as greatest common divisor (for instance for n o = 2, take υ 1 =3,υ 2 =2,υ 1 = 5,υ 2 = 6 and υ = 8). As an illustration for Corollary in the sequel, with this example, the homogeneous dimension is Q = =24 while the least common multiple is ν o =2 3 5 = 30, so we have here Q<ν o. If nothing is specified, we assume that the groups (R n, +) and H no are endowed with their classical structure of graded Lie groups as described in Examples and Remark We continue with several comments following those given in Remark

8 98 Chapter 3. Homogeneous Lie groups 1. The converse of Proposition does not hold, namely, not every nilpotent Lie algebra or group admits a family of dilations. An example of a nine dimensional nilpotent Lie algebra which does not admit any family of dilations is due to Dyer [Dye70]. 2. A direct product of two stratified Lie groups is graded but may be not stratified as their stratification structures may not match. This can be also seen on the level of dilations defined in Section Jumping ahead and using the notion of homogeneous operators, we see that this remark may be an advantage for example when considering the sub-laplacian L = X 2 + Y 2 on the Heisenberg group H 1. Then the operator L + k t for k N odd, becomes homogeneous on the direct product H 1 R when it is equipped with the dilation structure which is not the one of a stratified Lie group, see Lemma or, more generally, Remark In our definition of a homogeneous structure we started with dilations defined on the Lie algebra inducing dilations on the Lie group. If we start with a Lie group the situation may become slightly more involved. For example, R 3 with the group law xy = (arcsinh(sinh(x 1 ) + sinh(y 1 )),x 2 + y 2 + sinh(x 1 )y 3,x 3 + y 3 ) is a 2-step nilpotent stratified Lie group, the first stratum given by X =cosh(x 1 ) 1 x1, Y = sinh(x 1 ) x2 + x3, and their commutator is T =[X, Y ]= x2. It may seem like there is no obvious homogeneous structure on this group but we can see it going to its Lie algebra which is isomorphic to the Lie algebra h 1 of the Heisenberg group H 1. Consequently, the above group itself is isomorphic to H 1 with the corresponding dilation structure. 4. In fact, the same argument as above shows that if we defined a stratified Lie group by saying that there is a collection of vector fields on it stratified with respect to their commutation relations, then for every such stratified Lie group there always exists a homogeneous stratified Lie group isomorphic to it. Indeed, since the Lie algebra is stratified and has a natural dilation structure with integer weights, we obtain the required homogeneous Lie group by exponentiating this Lie algebra. We refer to e.g. [BLU07, Theorem ] for a detailed proof of this. Refining the proof of Proposition , we can obtain the following technical result which gives the existence of an adapted basis of eigenvectors for the dilations.

9 3.1. raded and homogeneous Lie groups 99 Lemma Let g be a Lie algebra endowed with a family of dilations {D r,r> 0}. Then there exists a basis {X 1,...,X n } of g, positive numbers υ 1,...,υ n > 0, and an integer n with 1 n n such that t >0 j =1,...,n D t (X j )=t υj X j, (3.2) and [g, g] RX n RX n. (3.3) Moreover, X 1,...,X n generate the algebra g, that is, any element of g can be written as a linear combination of these vectors together with all their iterated Lie brackets. This result and its proof are due to ter Elst and Robinson (see [ter97, Lemma 2.2]). Condition (3.2) says that {X j } n j=1 is a basis of eigenvectors for the mapping A given by D r =Exp(Aln r). Condition (3.3) says that this basis can be chosen so that the first n vectors of this basis generate the whole Lie algebra and the others span (linearly) the derived algebra [g, g]. Proof of Lemma WecontinuewiththenotationoftheproofofProposition For each weight υ W A,wechooseabasis {Y υ,1,...,y υ,d υ,y υ,d υ +1,...,Y υ,dυ } of W υ such that {Y υ,d υ +1,...,Y υ,dυ } is a basis of the subspace ( ) W υ Span [W υ,w υ ]. υ +υ =υ Since g = υ WA W υ,wehavebyconstructionthat [g, g] Span {Y υ,j : υ W A, d υ +1 j d υ }. Let h be the Lie algebra generated by {Y υ,j : υ W A, 1 j d υ}. (3.4) We now label and order the weights, that is, we write W A = {υ 1,...,υ m } with 1 υ 1 <...<υ m. It follows by induction on N =1, 2...,m that N j=1 W υ j is contained in h and hence h = g and the set (3.4) generate (algebraically) g. A basis with the required property is given by Y υ1,1,...,y υ1,d υ 1,...,Y υm,1,...,y υm,d for X 1,...,X υm n, and Y υ1,d υ 1 +1,...,Y υ1,d υ1,...,y υm,d υm +1,...,Y υm,d υm for X n +1,...,X n.

10 100 Chapter 3. Homogeneous Lie groups Homogeneous structure In this section, we shall be working on a fixed homogeneous Lie group of dimension n with dilations {D r =Exp(Aln r)}. We denote by υ 1,...,υ n theweights,listedinincreasingorderandwitheachvalue listed as many times as its multiplicity, and we assume without loss of generality (see Remark 3.1.8) that υ 1 1. Thus, 1 υ 1 υ 2... υ n. (3.5) If the group is graded, then the weights are also assumed to be integers with one as their greatest common divisor (again see Remark 3.1.8). By Proposition the Lie group is nilpotent connected simply connected. Thus it may be identified with R n equipped with a polynomial law, using the exponential mapping exp of the group (see Section 1.6). With this identification its unit element is 0 R n and it may also be denoted by 0 or simply by 0. We fix a basis {X 1,...,X n } of g such that AX j = υ j X j for each j. This yields a Lebesgue measure on g and a Haar measure on by Proposition If x or g denotes a point in the Haar measure is denoted by dx or dg. The Haar measure of a measurable subset S of is denoted by S. We easily check that D r (S) = r Q S, f(rx)dx = r Q f(x)dx, (3.6) where Q = υ υ n =TrA. (3.7) The number Q is larger (or equal) than the usual dimension of the group: n =dim Q, and may replace it for certain questions of analysis. For this reason the number Q is called the homogeneous dimension of. Homogeneity Any function defined on or on \{0} canbecomposedwiththedilationsd r. Using property (3.6) of the Haar measure and the dilations, we have for any measurable functions f and φ on, provided that the integrals exist, (f D r )(x) φ(x) dx = r Q f(x) (φ D 1 )(x) dx. (3.8) r

11 3.1. raded and homogeneous Lie groups 101 Therefore, we can extend the map f f D r to distributions via f D r,φ := r Q f,φ D 1 r, f D (), φ D(). (3.9) We can now define the homogeneity of a function or a distribution in the same way: Definition Let ν C. (i) A function f on \{0} or a distribution f D () ishomogeneous of degree ν C (or ν-homogeneous) when f D r = r ν f for any r>0. (ii) A linear operator T : D() D () ishomogeneous of degree ν C (or ν-homogeneous) when T (φ D r )=r ν (Tφ) D r for any φ D(), r>0. Remark We will also say that a linear operator T : E F,whereE is a Fréchet space containing D() as a dense subset, and F is a Fréchet space included in D (), is homogeneous of degree ν C when its restriction as an operator from D() tod () is. For example, it will apply to the situation when T is a linear operator from L p () to some L q (). Example (Coordinate function). The coordinate function x j =[x] j given by x =(x 1,...,x n ) x j =[x] j, (3.10) is homogeneous of degree υ j. Example (Koranyi norm). The function defined on the Heisenberg group H no by H no (x, y, t) ( ( x 2 + y 2) 2 + t 2 ) 1/4, where x and y denote the canonical norms of x and y in R no, is homogeneous of degree 1. It is sometimes called the Koranyi norm. Example (Haar measure). Equality (3.8) shows that the Haar measure, viewed as a tempered distribution, is a homogeneous distribution of degree Q (see (3.7)). We can write this informally as see (3.6). d(rx) =r Q dx,

12 102 Chapter 3. Homogeneous Lie groups Example (Dirac measure at 0). TheDiracmeasureat0istheprobability measure δ 0 given by fdδ 0 = f(0). It is homogeneous of degree Q since for any φ D() andr>0, we have δ 0 D r,φ = r Q δ 0,φ D 1 r = r Q φ( 1 r 0) = r Q φ(0) = r Q δ 0,φ. Example (Invariant vector fields). Let X g be viewed as a left-invariant vector field X or a right-invariant vector field X (cf. Section 1.3). We assume that X is in the υ j -eigenspace of A. Then the left and right-invariant differential operators X and X are homogeneous of degree υ j. Indeed, X(f D r )(x) = t=0 {f D r (x exp (tx))} = t=0 {f (rx exp (r υj tx))} = r υj t =0 {f (rx exp (t X))} = r υj (Xf)(rx), and similarly for X. The following properties are very easy to check: Lemma (i) Whenever it makes sense, the product of two functions, distributions or operators of degrees ν 1 and ν 2 is homogeneous of degree ν 1 ν 2. (ii) Let T : D() D () be a ν-homogeneous operator. Then its formal adjoint and transpose T and T t, given by (Tf)g = f(t g), (Tf)g = f(t t g), f,g D(), are also homogeneous with degree ν and ν respectively. Consequently for any non-zero multi-index α =(α 1,...,α n ) N n 0 \{0}, the function x α := x α1 1...xαn n, (3.11) and the operators ( ) α ( ) α1 ( ) αn :=...,X α := X α1 1...Xn αn and x x 1 x X α α1 αn := X 1... X n, n are homogeneous of degree [α] :=υ 1 α υ n α n. (3.12) Formula (3.12) defines the homogeneous degree of the multi-index α. It is usually different from the length of α given by α := α α n.

13 3.1. raded and homogeneous Lie groups 103 For α = 0, the function x α and the operators ( x )α, X α, Xα are defined to be equal, respectively, to the constant function 1 and the identity operator I, which are of degree [α] :=0. With this convention for each α N n 0, the differential operators ( x )α, X α and X α are of order α but of homogeneous degree [α]. One easily checks for α 1,α 2 N n 0 that [α 1 ]+[α 2 ]=[α 1 + α 2 ], α 1 + α 2 = α 1 + α 2. Proposition Let the operator T be homogeneous of degree ν T and let f be a function or a distribution homogeneous of degree ν f. Then, whenever Tf makes sense, the distribution Tf is homogeneous of degree ν f ν T. In particular, if f D () is homogeneous of degree ν, then X α f, X α f, α f are homogeneous of degree ν [α]. Proof. The first claim follows from the formal calculation (Tf) D r = r ν T T (f D r )=r ν T T (r ν f f)=r ν T +ν f Tf. The second claim follows from the first one since X α, Xα f and α f are well defined on distributions and are homogeneous of the same degree [α] givenby (3.12) Polynomials By Propositions and we already know that the group law is polynomial. This means that each [xy] j is a polynomial in the coordinates of x and of y. The homogeneous structure implies certain additional properties of this polynomial. Proposition For any j =1,...,n, we have [xy] j = x j + y j + c j,α,β x α y β. α,β N n 0 \{0} [α]+[β]=υ j In particular, this sum over [α] and [β] can involve only coordinates in x or y with degrees of homogeneity strictly less than υ j. For example, for υ 1 : [xy] 1 = x 1 + y 1, for υ 2 : [xy] 2 = x 2 + y 2 + c 2,α,β x α y β, [α]=[β]=υ 1 for υ 3 : [xy] 3 = x 3 + y 3 + c 3,α,β x α y β, [α]=υ 1, [β]=υ 2 or [α]=υ 2, [β]=υ 1

14 104 Chapter 3. Homogeneous Lie groups and so on. Proof. Let j =1,...,n. From the Baker-Campbell-Hausdorff formula (see Theorem 1.3.2) applied to the two vectors X = x 1 X x n X n and Y = y 1 X y n X n of g, we have with our notation that [xy] j = x j + y j + R j (x, y) where R j (x, y) is a polynomial in x 1,y 1,...,x n,y n.moreover,r j must be a finite linear combination of monomials x α y β with α + β 2: R j (x, y) = c j,α,β x α y β. α,β N n 0 α + β 2 We now use the dilations. Since the function x j is homogeneous of degree υ j, we easily check R j (rx, ry) =r υj R j (x, y) for any r>0and this forces all the coefficients c j,α,β with [α] +[β] υ j to be zero. The formula follows. Recursively using Proposition , we obtain for any α N n 0 \{0}: with (xy) α =[xy] α1 1...[xy]αn n = β 1,β 2 N n 0 [β 1]+[β 2]=[α] c β1,β 2 (α)x β1 y β2, (3.13) c β1,0(α) = { 0 ifβ1 α 1 ifβ 1 = α and c 0,β2 (α) = { 0 ifβ2 α 1 ifβ 2 = α. (3.14) Definition A function P on is a polynomial if P exp is a polynomial on g. For example the coordinate functions x 1,...,x n defined in (3.10) or, more generally, the monomials x α defined in (3.11) are (homogeneous) polynomials on. It is clear that every polynomial P on can be written as a unique finite linear combination of the monomials x α,thatis, P = c α x α, (3.15) α N n 0 where all but finitely many of the coefficients c α C vanish. The homogeneous degree of a polynomial P written as (3.15) is D P := max{[α] :α N n 0 with c α 0},

15 3.1. raded and homogeneous Lie groups 105 which is often different from its isotropic degree: d P := max{ α : α N n 0 with c α 0}. For example on H no,1+tis a polynomial of homogeneous degree 2 but isotropic degree 1. Definition We denote by P() the set of all polynomials on. For any M 0 we denote by P M the set of polynomials P on such that D P M and by P M iso the set of polynomials on such that d P M. We also define in the same way P <M, P =M, P M and so on, and similarly for P iso. It is clear that P() is an algebra, for pointwise multiplication, which is generated by the x j s. It is not difficult to see: Lemma The subspaces P M and P M iso of P are finite dimensional with bases {x α : α N n 0, [α] M} and {x α : α N n 0, α M}, respectively. Furthermore, M 0 P M P M iso P υnm. Proof. The first part of the lemma is clear. For the second, because of (3.5), we have α N n 0 α [α] υ n α. (3.16) Therefore, P P d P D P υ n d P, and the inclusions follow. By Proposition , [xy] j is in P υj as a function of x for each y, andalso as a function of y for each x. Hence each subspace P M is invariant under left and right translation. This is not the case for P M iso (unless Piso M C or =(Rn, +)); consequently, it will not be of much use to us Invariant differential operators on homogeneous Lie groups We now investigate expressions for left- and right-invariant operators on homogeneous Lie groups. Proposition The left and right-invariant vector fields X j and X j, for any j =1,...,n, can be written as X j = X j = x j + x j + P j,k 1 k n υ j<υ k Q j,k 1 k n υ j<υ k = + x k x j = + x k x j 1 k n υ j<υ k 1 k n υ j<υ k x k P j,k x k Q j,k,

16 106 Chapter 3. Homogeneous Lie groups where P j,k and Q j,k are homogeneous polynomials on of homogeneous degree υ k υ j > 0. Proof. For any x, we denote by L x : the left-translation, i.e. L x (y) = xy. Letj =1,...,n. Recall that X j is the differential operator invariant under left-translation which agrees with x j at 0, that is, for any f C ()andx o, we have (X j f) L xo (0) = X j (f L xo )(0) and X j (f)(0) = f (0). x j Thus (X j f)(x o ) = (X j f) L xo (0) = X j (f L xo )(0) = (f L xo )(0) x j n f = (x o ) [x ox] k (0), x k x j k=1 by the chain rule. But by Proposition , [x o x] k (0) = [x o ] k + x k + x j x j = δ j,k + β=e j,α N n 0 \{0} [α]+[β]=υ k α,β N n 0 \{0} [α]+[β]=υ k c k,α,β x α o, c k,α,β x α o x β (0) where e j is the multi-index with 1 in the j-th place and zeros elsewhere, and δ j,k is the Kronecker delta. The assertion for X j now follows immediately, and the assertion for X j is proved in the same way using right translations. Proposition gives, in particular, so that for υ n : X n = for υ n 1 : X n 1 = for υ n 2 : X n 2 =, x n + P n 1,n, x n 1 x n + P n 2,n 1 + P n 2,n, x n 2 x n 1 x n x n = X n, x n 1 = X n 1 P n 1,n X n, x n 2 = X n 2 P n 2,n 1 (X n 1 P n 1,n X n ) P n 2,n X n,

17 3.1. raded and homogeneous Lie groups 107 and so forth, with similar formulae for the right-invariant vector fields. This shows that there are formulas for the x j s of the same sort as for the X j s and X j s, that is, = X j + p j,kx k = Xj + q j,k X k, (3.17) x j 1 k n 1 k n υ j<υ k υ j<υ k where p j,k and q j,k are homogeneous polynomials on of homogeneous degree υ k υ j > 0. Remark iven the formulae above and the condition on the degree, it is not difficult to see that the P j,k and Q j,k in Proposition and the p j,k and q j,k in (3.17), with υ k >υ j, are polynomials in (x 1,...,x k 1 )and commute with X k, Xk and respectively. x k 2. The first part of Proposition and its proof are valid for any nilpotent Lie group (see Remark 1.6.7, part (1)). In our setting here, the homogeneous structure implies the additional property that the P j,k and Q j,k are homogeneous. Corollary For any α N n 0 \{0}, X α = X α = β N n 0, β α [β] [α] β N n 0, β α [β] [α] P α,β Xβ = Q α,β X β = β N n 0, β α [β] [α] β N n 0, β α [β] [α] X β p α,β, X β q α,β, where P α,β,p α,β,q α,β,q α,β are homogeneous polynomials of homogeneous degree [β] [α]. Proof. By Proposition we obtain recursively for any α N n 0 \{0} that X α = β N n 0, β α [β] [α] ( ) β P α,β, (3.18) x with P α,β homogeneous polynomial of degree [β] [α]. Similar formulae yield X α in terms of the ( β s. x) Recursively from (3.17), we also obtain similar formulae for ( α x) in terms of the X β or X β. The assertion comes form combining these formulae, with a similar argument for p α,β and q α,β.

18 108 Chapter 3. Homogeneous Lie groups Corollary For any M 0, the maps {( ) α (i) P P (0)}, x α N n 0, [α] M (ii) P {X α P (0)} α N n 0, [α] M { }, (iii) P Xα P (0) α N n 0, [α] M, are linear isomorphisms from P M to C dim P M. Also, the maps {( ) α (i) P P (0)}, x α N n 0, [α]=m (ii) P {X α P (0)} α N n 0, [α]=m { }, (iii) P Xα P (0) are linear isomorphisms from P =M to C dim P =M. α N n 0, [α]=m, Proof. By Lemma , the vector subspace P M of P is finite dimensional, with basis {x α : α N n 0, [α] M}. Hence case (i) is a simple consequence of Taylor s Theorem on R n. Note that in the formula (3.18), P α,β is a constant function when [α] =[β] and P α,β (0) = 0 when [α] > [β]. Hence X α 0 = β N n 0, β α [β]=[α] P α,β ( x) β 0. We have similar result from the other formulae relating X α, Xα and ( α. x) Cases (ii) and (iii) follow from these observations together with case (i). The case of the homogeneous polynomials of order M is similar. We may use the following property without referring to it. Corollary Let α, β N n 0. The differential operator X α X β is a linear combination of X γ with [γ] N n 0, [γ] =[α]+[β]: X α X β = γ N n 0, γ α + β [γ]=[α]+[β] c α,β,γx γ. (3.19) The differential operator X α Xβ is a linear combination of Xγ with [γ] N n 0, γ α + β and [γ] =[α]+[β].

19 3.1. raded and homogeneous Lie groups 109 Proof. The differential operator X α X β is a left-invariant differential operator of order α + β by (3.18), and it is a linear combination of X γ, γ α + β (see Section 1.3), X α X β = c α,β,γx γ. γ N n 0, γ α + β By homogeneity, for any r>0 and any function f C (), we have on one hand, X α X β (f D r )=r [α]+[β] (X α X β f) D r, and on the other hand, X α X β (f D r ) = = c α,β,γx γ (f D r ) γ N n 0, γ α + β c α,β,γr [γ] (X γ f) D r. γ N n 0, γ α + β Choosing f suitably (for example f being polynomials of homogeneous degree at most [α] +[β], see Corollary ), this implies that if [α] +[β] [γ] then c α,β,γ = 0, showing (3.19). The property for the right-invariant vector fields is similar Homogeneous quasi-norms We can define an Euclidean norm E on g by declaring the X j s to be orthonormal. We may also regard this norm as a function on via the exponential mapping, that is, x E = exp 1 x E. However, this norm is of limited use for our purposes, since it does not interact in a simple fashion with dilations. We therefore define: Definition A homogeneous quasi-norm is a continuous non-negative function x x [0, ), satisfying (i) (symmetric) x 1 = x for all x, (ii) (1-homogeneous) rx = r x for all x and r>0, (iii) (definite) x = 0 if and only if x =0. The -ball centred at x with radius R>0 is defined by B(x, R) :={y : x 1 y <R}.

20 110 Chapter 3. Homogeneous Lie groups Remark With such definition, we have for any x, x o, R>0, since z x o B(x, R) x 1 o In particular, we see that It is also easy to check that x o B(x, R) =B(x o x, R), (3.20) z B(x, R) x 1 x 1 o z <R z B(x o x, R). B(x, r) =xb(0,r). B(0,r)=D r (B(0, 1)). Note that in our definition of quasi-balls, we choose to privilege the left translations. Indeed, the set {y : yx 1 < R} may also be defined as a quasi-ball but one would have to use the right translation instead of the left x o -translation to have a similar property to (3.20). An important example of a quasi-norm is given by Example on the Heisenberg group H no. More generally, on any homogeneous Lie group, the following functions are homogeneous quasi-norms: for 0 <p<, and for p = : n (x 1,...,x n ) p = x j p j=1 υ j 1 p, (3.21) (x 1,...,x n ) = max x j 1 υ j. (3.22) 1 j n In Definition we do not require a homogeneous quasi-norm to be smooth away from the origin but some authors do. Quasi-norms with added regularity always exist as well but, in fact, a distinction between different quasi-norms is usually irrelevant for many questions of analysis because of the following property: Proposition (i) Every homogeneous Lie group admits a homogeneous quasi-norm that is smooth away from the unit element. (ii) Any two homogeneous quasi-norms and on are mutually equivalent: in the sense that a, b > 0 x a x x b x.

21 3.1. raded and homogeneous Lie groups 111 Proof. Let us consider the function Ψ(r, x) = D r x 2 E = n r 2υj x 2 j. Let us fix x 0. The function Ψ(r, x) is continuous, strictly increasing in r and satisfies Ψ(r, x) 0 and Ψ(r, x) +. r 0 r + Therefore, there is a unique r>0 such that D r x E = 1. We set x o := r 1. Hence we have defined a map j=1 \{0} x x 1 o (0, ) which is the implicit function for Ψ(r, x) = 1. This map is smooth since the function Ψ(r, x) is smooth from (0, + ) \{0} to (0, ) and r Ψ(r, x) isalways different from zero. Setting 0 o := 0, the map o clearly satisfies the properties of Definition This shows part (i). For Part (ii), it is sufficient to prove that any homogeneous quasi-norm is equivalent to o constructed above. Before doing so, we observe that the unit spheres in the Euclidean norm and the homogeneous quasi-norm o coincide, that is, S := {x : x E =1} = {x : x o =1}. Let be any other homogeneous norm. Since it is a definite function (see (iii) of Definition ) its restriction to S is never zero. By compactness of S and continuity of, there are constants a, b > 0 such that x S a x b. For any x \{0}, lett>0begivenbyt 1 = x o.wehaved t x S, and thus a D t x b and a x o = t 1 a x t 1 b = b x o. The conclusion of Part (ii) follows. Remark If is graded, the formula (3.21) for p =2υ 1...υ n gives another concrete example of a homogeneous quasi-norm smooth away from the origin since x x p p is then a polynomial in the coordinate functions {x j }. Proposition and our examples of homogeneous quasi-norms show that the usual Euclidean topology coincides with the topology associated with any homogeneous quasi-norm:

22 112 Chapter 3. Homogeneous Lie groups Proposition If is a homogeneous quasi-norm on R n,thetopology induced by the -balls B(x, R) :={y : x 1 y <R}, x and R>0, coincides with the Euclidean topology of R n. Any closed ball or sphere for any homogeneous quasi-norm is compact. It is also bounded with respect to any norm of the vector space R n or any other homogeneous quasi-norm on. Proof of Proposition It is a routine exercise of topology to check that the equivalence of norm given in Proposition implies that the topology induced by the balls of two different homogeneous quasi-norms coincide. Hence we can choose the norm given by (3.22) and the corresponding balls B (x, R) :={y : x 1 y <R}. We also consider the supremum Euclidean norm given by and its corresponding balls (x 1,...,x n ) E, = max 1 j n x j, B E, (x, R) :={y : x + y E, <R}. That the topologies induced by the two families of balls {B (x, R)} x,r>0 and {B E, (x, R)} x,r>0 must coincide follows from the following two observations. Firstly it is easy to check for any R (0, 1) B (0,R 1 υ 1 ) B E, (0,R) B (0,R 1 υn ). Secondly for each x, the mappings Ψ x : y x 1 y and Ψ E,x : y x + y are two smooth diffeomorphisms of R n. Hence these mappings are continuous with continuous inverses (with respect to the Euclidean topology). Furthermore, by Remark , we have Ψ x (B (x, R)) = B (0,R) and Ψ E,x (B E, (x, R)) = B E, (0,R). The second part of the statement follows from the first and from the continuity of homogeneous quasi-norms. The next proposition justifies the terminology of quasi-norm by stating that every homogeneous quasi-norm satisfies the triangle inequality up to a constant, the other properties of a norm being already satisfied.

23 3.1. raded and homogeneous Lie groups 113 Proposition If is a homogeneous quasi-norm on, there is a constant C>0 such that xy C ( x + y ) x, y. Proof. Let be a quasi-norm on.let B := {x : x 1} be its associated closed unit ball. By Proposition , B is compact. As the product law is continuous (even polynomial), the set {xy : x, y B} is also compact. Therefore, there is a constant C>0such that x, y B xy C. Let x, y. If both of them are 0, there is nothing to prove. If not, let t>0be given by t 1 = x + y > 0. Then D t (x) andd t (y) arein B, sothat and this concludes the proof. t xy = D t (xy) = D t (x)d t (y) C, Note that the constant C in Proposition satisfies necessarily C 1 since 0 = 0 implies x C x for all x. It is natural to ask whether a homogeneous Lie group may admit a homogeneous quasi-norm which is actually a norm or, equivalently, which satisfies the triangle inequality with constant C = 1. For instance, on the Heisenberg group H no, the homogeneous quasi-norm given in Example turns out to be a norm (cf. [Cyg81]). In the stratified case, the norm built from the control distance of the sub-laplacian, often called the Carnot-Caratheodory distance, is also 1-homogeneous (see, e.g., [Pan89] or [BLU07, Section 5.2]). This can be generalised to all homogeneous Lie groups. Theorem Let be a homogeneous Lie group. Then there exist a homogeneous quasi-norm on which is a norm, that is, a homogeneous quasi-norm which satisfies the triangle inequality xy x + y x, y. A proof of Theorem by Hebisch and Sikora uses the correspondence between homogeneous norms and convex sets, see [HS90]. Here we sketch a different proof. Its idea may be viewed as an adaptation of a part of the proof that the control distance in the stratified case is a distance. Our proof may be simpler than the stratified case though, since we define a distance without using horizontal curves. Sketch of the proof of Theorem If γ :[0,T] is a smooth curve, its tangent vector γ (t o )atγ(t o ) is usually defined as the element of the tangent space T γ(to) at γ(t o ) such that γ (t o )(f) = d dt f(γ(t)) t=to, f C ().

24 114 Chapter 3. Homogeneous Lie groups It is more convenient for us to identify the tangent vector of γ at γ(t o )withan element of the Lie algebra g = T 0. We therefore define γ (t o ) g via γ (t o )(f) := d dt f(γ(t o) 1 γ(t)), f C (). t=to We now fix a basis {X j } n j=1 of g such that D rx j = r υj X j. We also define the map : g [0, ) by X := max x n j 1 υ j, X = x j X j g. j=1,...,n iven a piecewise smooth curve γ :[0,T], we define its length adapted to the group structure by l(γ) := T 0 j=1 γ (t) dt. If x and y are in, wedenotebyd(x, y) the infimum of the lengths l(γ) of the piecewise smooth curves γ joining x and y. Since two points x and y can always be joined by a smooth compact curve, e.g. γ(t) = ((1 t)x) ty, thequantity d(x, y) is always finite. Hence we have obtained a map d : [0, ). It is a routine exercise to check that d is symmetric and satisfies the triangle inequality in the sense that we have for all x, y, z, that d(x, y) =d(y, x) and d(x, y) d(x, z)+d(z,y). Moreover, one can check easily that l(d r (γ)) = r l(γ) and l(zγ) = l(γ), thus we also have for all x, y, z and r>0, that d(zx,zy) =d(x, y) and d(rx, ry) =rd(x, y). (3.23) Let us show that d is non-degenerate, that is, d(x, y) =0 x = y. First let E be the Euclidean norm on g R n such that the basis {X j } n j=1 is orthonormal. We endow each tangent space T x with the Euclidean norm obtained by left translation of the Euclidean norm E. Hence we have for any smooth curve γ at any point t o γ (t o ) Tγ(to) = γ (t o ) E. Now we see that if X = n j=1 x jx j g is such that X E, := max x j 1, j=1,...,n then X E X E, X.

25 3.1. raded and homogeneous Lie groups 115 This implies that if γ :[0,T] is a smooth curve satisfying t [0,T] γ (t) Tγ(t) < 1, (3.24) then where l is the usual length l(γ) C l(γ), (3.25) l(γ) := T 0 γ (t) Tγ(t) dt, and C>0apositive constant independent of γ. Let d be the Riemaniann distance induced by our choice of metric on the manifold, that is, the infimum of the lengths l(γ) of the piecewise smooth curves γ joining x and y. Very well known results in Riemaniann geometry imply that d induces the same topology as the Euclidean topology. Moreover, there exists a small open set Ω containing 0 such that any point in Ω may be joined to 0 by a smooth curve satisfying (3.24) at any point. Then (3.25) yields that we have d (0,x) Cd(0,x) for any x Ω. This implies that d is non-degenerate since d is invariant under left-translation and is 1-homogeneous in the sense of (3.23), Checking that the associated map x x = d(0,x) is a quasi-norm concludes the sketch of the proof of Theorem Even if homogeneous norms do exist, it is often preferable to use homogeneous quasi-norms. Because the triangle inequality is up to a constant in this case, we do not necessarily have the inequality xy x C y. However, the following lemma may help: Proposition We fix a homogeneous quasi-norm on. For any f C 1 (\{0}) homogeneous of degree ν C, for any b (0, 1) there is a constant C = C b > 0 such that f(xy) f(x) C y x Re ν 1 whenever y b x. Indeed, applying it to a C 1 (\{0}) homogeneous quasi-norm, we obtain b (0, 1) C = C b > 0 x, y y b x = xy x C y. (3.26) Proof of Proposition Let f C 1 (\{0}). Both sides of the desired inequality are homogeneous of degree Re ν so it suffices to assume that x =1 and y b. By Proposition and the continuity of multiplication, the set {xy : x = 1 and y b} is a compact which does not contain 0. So by the (Euclidean) mean value theorem on R n,weget f(xy) f(x) C y E. We conclude using the next lemma.

26 116 Chapter 3. Homogeneous Lie groups The next lemma shows that locally a homogeneous quasi-norm and the Euclidean norm are comparable: Lemma We fix a homogeneous quasi-norm on. Then there exist C 1,C 2 > 0 such that C 1 x E x C 2 x 1 υn E whenever x 1. Proof of Lemma By Proposition , the unit sphere {y : y =1} is compact and does not contain 0. Hence the Euclidean norm assumes a positive maximum C1 1 and a positive minimum C υn 2 on it, for some C 1,C 2 > 0. Let x. We may assume x 0.Thenwecanwriteitasx = ry with y = 1 and r = x. We observe that since ry 2 E = n yj 2 r 2υj, j=1 we have if r 1 r υn y E ry E r y E. Hence for r = x 1, we get x E = ry E r y E x C1 1 and x E = ry E r υn y E x υn C υn 2, implying the statement Polar coordinates There is an analogue of polar coordinates on homogeneous Lie groups. Proposition Let be a homogeneous Lie group equipped with a homogeneous quasi-norm. Then there is a (unique) positive Borel measure σ on the unit sphere S := {x : x =1}, such that for all f L 1 (), we have f(x)dx = f(ry)r Q 1 dσ(y)dr. (3.27) 0 S In order to prove this claim, we start with the following averaging property: Lemma Let be a homogeneous Lie group equipped with a homogeneous quasi-norm.iff is a locally integrable function on \{0}, homogeneous of degree Q, then there exists a constant m f C (the average value of f) such that for all u L 1 ((0, ),r 1 dr), we have f(x)u( x )dx = m f u(r)r 1 dr. (3.28) 0

27 3.1. raded and homogeneous Lie groups 117 The proof of Lemma yields the formula for m f in terms of the homogeneous quasi-norm, m f = f(x)dx. (3.29) 1 x e However, in Lemma we will give an invariant meaning to this value. Proof of Lemma Let f be locally integrable function on \{0}, homogeneous of degree Q. We set for any r>0, { f(x)dx if r 1, 1 x r ϕ(r) := r x 1 f(x)dx if r<1. The mapping ϕ :(0, ) C is continuous and one easily checks that ϕ(rs) =ϕ(r)+ϕ(s) for all r, s > 0, by making the change of variable x sx and using the homogeneity of f. It follows that ϕ(r) =ϕ(e)lnr and we set m f := ϕ(e). Then the equation (3.28) is easily satisfied when u is the characteristic function of an interval. By taking the linear combinations and limits of such functions, the equation (3.28) is also satisfied when u L 1 ((0, ),r 1 dr). Proof of Proposition For any continuous function f on the unit sphere S, we define the homogeneous function f on \{0} by f(x) := x Q f( x 1 x). Then f satisfies the hypotheses of Lemma The map f m f is clearly a positive functional on the space of continuous functions on S. Hence it is given by integration against a regular positive measure σ (see, e.g. [Rud87, ch.vi]). For u L 1 ((0, ),r 1 dr), we have f( x 1 x)u( x )dx = f(x) x Q u( x )dx = m f r Q 1 u(r)dr r=0 = f(y)u(r)r Q 1 dσ(y)dr. 0 S Since linear combinations of functions of the form f( x 1 x)u( x ) are dense in L 1 (), the proposition follows. We view the formula (3.27) as a change in polar coordinates.

28 118 Chapter 3. Homogeneous Lie groups Example For 0 <a<b< and α C, wehave { α x α Q dx = C 1 (b α a α ) if α 0 a< x <b ln ( ) b with C = σ(s). a if α =0 And if α R and f is a measurable function on such that f(x) =O( x α Q ) then f is integrable either near if α<0, or near 0 if α>0. The measure σ in the polar coordinates decomposition actually has a smooth density. We will not need this fact and will not prove it here, but refer to [FR66] and [oo80]. Now, the polar change of coordinates depends on the choice of a homogeneous quasi-norm to fix the unit sphere. But it turns out that the average value of the ( Q)-homogeneous function considered in Lemma does not. Let us prove this fact for the sake of completeness. Lemma Let be a homogeneous Lie group and let f be a locally integrable function on \{0}, homogeneous of degree Q. iven a homogeneous quasi-norm, let σ be the Radon measure on the unit sphere S giving the polar change of coordinate (3.27). Then the average value of f defined in (3.28) is given by m f = fdσ. (3.30) This average value m f is independent of the choice of the homogeneous quasinorm. Proof of Lemma For any homogeneous quasi-norm, using the polar change of coordinates (3.27), we obtain b f(x)dx = f(rx)dσ(x)r Q 1 dr a< x <b a S b b ( = f(x)dσ(x)r 1 dr = r 1 dr f(x)dσ(x) = ln b ) m f. a S a S a This shows (3.30), taking a = 1 and b = e, see (3.29) and the proof of Lemma Let and be two homogeneous quasi-norms on. We denote by B r := {x : x r} and B r := {x : x r}, the closed balls around 0 of radius r for and, respectively. By Proposition , Part (ii), there exists a constant a>0 such that B a B 1.Wealso have B a B 2a B 2 and, with the usual sign convention for integration, we have = +. B 2\ B 1 = B 2\ B a B 1\ B a S B 2\ B 2a B 2a \ B a B 1\ B a

29 3.1. raded and homogeneous Lie groups 119 Using the homogeneities of f and of the Haar measure, we see, after the changes of variables x =2y and x = az, that f(x)dx = f(y)dy and f(x)dx = f(z)dz. B 2\ B 2a B 1\ B a B 2a \ B a B 2 \ B 1 Hence f = f. B 2\ B 1 B 2 \ B 1 Using the first computations of this proof, the left and right hand sides are equal to (ln b/a)m f and (ln b/a)m f, respectively, where m f and m f are the average values for and.thusm f = m f Mean value theorem and Taylor expansion Here we prove the mean value theorem and describe the Taylor series on homogeneous Lie groups. Naturally, the space C 1 () here is the space of functions f such that X j f are continuous on for all j, etc. The following mean value theorem can be partly viewed as a refinement of Proposition Proposition We fix a homogeneous quasi-norm on. There exist group constants C 0 > 0 and η>1 such that for all f C 1 () and all x, y, we have f(xy) f(x) C 0 n j=1 y υj sup (X j f)(xz). z η y In order to prove this proposition, we first prove the following property. Lemma The map φ : R n defined by φ(t 1,...,t n )=exp (t 1 X 1 )exp (t 2 X 2 )...exp (t n X n ), is a global diffeomorphism. Moreover, fixing a homogeneous quasi-norm on, there is a constant C 1 > 0 such that (t 1,...,t n ) R n,j=1,...,n, t j 1 υ j C1 φ(t 1,...,t n ). The first part of the lemma is true for any nilpotent Lie group (see Remark Part (ii)). But we will not use this fact here. Proof. Clearly the map φ is smooth. By the Baker-Campbell-Hausdorff formula (see Theorem 1.3.2), the differential dφ(0) : R n T 0 is the isomorphism dφ(0)(t 1,...,t n )= n t j X j 0, j=1