aa r X i v : . [ m a t h . F A ] M a r Spectral Multipliers on 2-Step Stratified Groups, II
Mattia Calzi ∗ Abstract
Given a graded group G and commuting, formally self-adjoint, left-invariant, homogeneous differential operators L , . . . , L n on G , one of which is Rockland, we study the convolution operators m ( L , . . . , L n ) and their con-volution kernels, with particular reference to the case in which G is abelian and n = 1, and the case in which G is a 2-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and L , . . . , L n are either sub-Laplacians or central elements of the Lie algebra of G . Under suitable conditions, we prove that:i) if the convolution kernel of the operator m ( L , . . . , L n ) belongs to L , then m equals almost everywhere acontinuous function vanishing at ∞ (‘Riemann-Lebesgue lemma’); ii) if the convolution kernel of the operator m ( L , . . . , L n ) is a Schwartz function, then m equals almost everywhere a Schwartz function. Given a Rockland family ( L , . . . , L n ) on a homogeneous group G , following [24, 29] (see also [12]) wedefine a ‘kernel transform’ K which to every measurable function m : R n → C such that m ( L , . . . , L n )is defined on C ∞ c ( G ) associates a unique distribution K ( m ) such that m ( L , . . . , L n ) ϕ = ϕ ∗ K ( m )for every ϕ ∈ C ∞ c ( G ). The so-defined kernel transform K enjoys some relevant properties, which we listbelow; see [24, 29] for their proofs and further information. • there is a unique positive Radon measure β on R n such that K ( m ) ∈ L ( G ) if and only if m ∈ L ( β ),and K induces an isometry of L ( β ) into L ( G ); • there is a unique χ ∈ L ∞ ( R n × G, β ⊗ ν ), where ν denotes a Haar measure on G , such that forevery m ∈ L ( β ) K ( m )( g ) = Z R n m ( λ ) χ ( λ, g ) d β ( λ )for almost every g ∈ G ; • K maps S ( R n ) into S ( G ).We consider also some additional properties of particular interest, such as:( RL ) if K ( m ) ∈ L ( G ), then we can take m so as to belong to C ( R n );( S ) if K ( m ) ∈ S ( G ), then we can take m so as to belong to S ( R n ).In this paper, we shall investigate the validity of properties ( RL ) and ( S ) in two particular cases:that of a Rockland operator on an abelian group, and that of homogeneous sub-Laplacians and elementsof the centre on an M W + group (cf. Definition 4.1).Here is a plan of the following sections. In Section 2 we recall the basic definitions and notation,as well as some relevant results proved in [12]. In Section 3, we then consider abelian groups, andcharacterize the Rockland operators which satisfy property ( S ) thereon. In Section 4 we prepare themachinery for the study of homogeneous sub-Laplacians and elements of the centre on M W + groups, ∗ The author is partially supported by the grant PRIN 2015
Real and Complex Manifolds: Geometry, Topology andHarmonic Analysis , and is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). see Section 2 for precise definitions M W + groups will allow us to treat more than one homogeneoussub-Laplacian at a time. In Sections 5 and 6, then, we prove some sufficient conditions for properties( RL ) and ( S ) in this context.In Section 7 we present a particularly elegant result where all the good properties we consider areproved to be equivalent for the families which are invariant (in some sense) under the action of suitablegroups of isometries. In particular, this result covers the case of Heisenberg groups, thanks to theresults of Section 3. Finally, in Section 8 we consider products of Heisenberg groups and ‘decomposable’homogeneous sub-Laplacian thereon. In addition, we exhibit a Rockland family which is ‘functionallycomplete’ (cf. Definition 2.9) but does not satisfy property ( S ). In this section we recall some basic results and definitions from [12]. We shall then prove some usefulresults that were not considered therein.
As in [12], a Rockland family on a homogeneous group G (cf. [17]) is a jointly hypoelliptic, commutative,finite family L A = ( L α ) α ∈ A of formally self-adjoint, homogeneous, left-invariant differential operatorswithout constant terms. In this case, the L α are essentially self-adjoint on C ∞ c ( G ), and their closurescommute. In addition, L A is a weighted subcoercive system of operators (cf. [24, Proposition 3.6.3]), sothat the theory developed in [24] applies. Definition 2.1.
To every (Borel, say) measurable function m : R A → C such that m ( L A ) is defined (atleast) on C ∞ c ( G ), we associate a unique distribution K L A ( m ) (its ‘kernel’) on G such that m ( L A )( ϕ ) = ϕ ∗ K L A ( m )for every ϕ ∈ C ∞ c ( G ).We denote by E L A the space R A endowed with the dilations defined by r · ( λ α ) := ( r δ α λ α )for every r > λ α ) ∈ R A , where δ α is the homogeneous degree of L α . We shall oftenemploy the following short-hand notation: L L A ( G ) and S L A ( G ) will denote K L A ( L ∞ ( β )) ∩ L ( G ) and K L A ( L ∞ ( β )) ∩ S ( G ), respectively, while S ( G, L A ) will denote K L A ( S ( E L A )).Now, by [24, Theorem 3.2.7] there is a unique positive Radon measure β L A on E L A such that a Borelfunction m : E L A → C is square-integrable if and only if K L A ( m ) ∈ L ( G ) and such that, in this case, k m k L ( β L A ) = kK L A ( m ) k L ( G ) . The measure β L A is then equivalent to the spectral measure associated with L A . Using the existence of β L A and the fact that K L A maps S ( E L A ) in S ( G ), it is not hard to prove that a β L A -measurable functionadmits a kernel in the sense of Definition 2.1 if and only if there is a positive polynomial P on E L A suchthat m P ∈ L ( β L A ).Now, K L A can be extended to a continuous linear mapping from L ( β L A ) into C ( G ) (cf. [24, Propo-sition 3.2.12]), and there is a unique χ L A ∈ L ∞ ( β L A ⊗ ν G ), where ν G denotes a fixed Haar measure on G , such that K L A ( m )( g ) = Z E L A m ( λ ) χ L A ( λ, g ) d β L A ( λ )for every m ∈ L ( β L A ) and for almost every g ∈ G . Further, we denote by M L A : L ( G ) → L ∞ ( G ) the transpose of the mapping m
7→ K L A ( m )ˇ, so that M L A ( f )( λ ) = Z G f ( g ) χ L A ( λ, g ) d g for every f ∈ L ( G ) and for β L A -almost every λ ∈ E L A . Observing that M L A equals the adjoint of theisometry K L A : L ( β L A ) → L ( G ) on L ( G ) ∩ L ( G ), one may then prove that K L A ◦ M L A is the identityon L L A ( G ). This is basically a consequence of the Dunford–Pettis theorem, cf. [29]. .2 Products Assume that we have two Rockland families L A and L ′ A ′ on two homogeneous groups G and G ′ , respec-tively. Denote by L ′′ A ′′ the family whose elements are the operators on G × G ′ induced by the elementsof L A and L ′ A ′ , and observe that L ′′ A ′′ is a Rockland family. Theorem 2.2 ([12], Theorems 4.5 and 4.10) . The families L A and L ′ A ′ satisfy property ( RL ) (resp. ( S ) )if and only if L ′′ A ′′ does. Assume that we are given a Rockland family L A and a polynomial mapping P on E L A such that P ( L A ) is still a Rockland family (this is equivalent to saying that P is proper and has homogeneouscomponents with respect to the dilations of E L A ). Then, for every bounded measurable multiplier m wehave K P ( L A ) ( m ) = K L A ( m ◦ P ). As a consequence, if we want to establish properties ( RL ) or ( S ) for P ( L A ) on the base of our knowledge of L A , it is of importance to infer some properties of m from theproperties of m ◦ P . The results of this section address this problem.We begin with a definition. Definition 2.3.
Let X be a locally compact space, Y a set, µ a positive Radon measure on X , and π amapping from X into Y . We say that two points x, x ′ of Supp ( µ ) are ( µ, π )-connected if π ( x ) = π ( x ′ )and there are x = x , . . . , x k = x ′ ∈ π − ( π ( x )) ∩ Supp ( µ ) such that, for every j = 1 , . . . , k , for everyneighbourhood U j of x j in Supp ( µ ), and for every neighbourhood U j +1 of x j +1 in Supp ( µ ), the set π − ( π ( U j ) ∩ π ( U j +1 )) is not µ -negligible. We say that µ is π -connected if every pair of elements ofSupp ( µ ) having the same image under π are ( µ, π )-connected. Proposition 2.4.
Let E , E be two finite-dimensional affine spaces, L : E → E an affine mappingand µ a positive Radon measure on E . Assume that the support of µ is either a convex set and that L is proper on it, or that the support of µ is the boundary of a convex polyhedron on which L is properThen, µ is L -connected.Proof. The assertion is [12, Proposition 6.3] when the support of µ is convex. Then, assume that thesupport of µ is the boundary of a convex polyhedron C , on which L is proper. Consider first the case in which C is compact and has non-empty interior, E = R n , E = R n − and L ( x , . . . , x n ) = ( x , . . . , x n − ) for every ( x , . . . , x n ) ∈ E n . Define C ′ := L ( C ), so that C ′ is acompact convex polyhedron of E . Now, the functions f − : C ′ ∋ x ′ min { y ∈ R : ( x ′ , y ) ∈ C } and f + : C ′ ∋ x ′ max { y ∈ R : ( x ′ , y ) ∈ C } are well-defined; in addition, f − is convex while f + is concave. Therefore, f − and f + are continuouson ◦ C ′ by [10, Corollary to Proposition 21 of Chapter II, §
2, No. 10]. Now, observe that f − f + ; if f − ( x ′ ) = f + ( x ′ ) for some x ′ ∈ ◦ C ′ , then f − = f + on C ′ by convexity, and this contradicts the assumptionthat C has non-empty interior. Therefore, { ( x ′ , y ) : x ′ ∈ ◦ C ′ , f − ( x ′ ) < y < f + ( x ′ ) } is the interior of C ,so that ∂C ∩ ( ◦ C ′ × R ) is the union of the graphs Γ − and Γ + of the restrictions of f − and f + to ◦ C ′ .Since L induces homeomorphisms of Γ − and Γ + onto ◦ C ′ , it follows that ( x ′ , f − ( x ′ )) and ( x ′ , f + ( x ′ )) are( µ, L )-connected for every x ′ ∈ ◦ C ′ .Now, take x ′ ∈ ∂C ′ , and observe that { ( x ′ , y ) : y ∈ [ f − ( x ′ ) , f + ( x ′ )] } ⊆ ∂C . Take y ∈ [ f − ( x ′ ) , f + ( x ′ )]and an ( n − F of C which contains ( x ′ , y ). Observe that the support of χ F · µ is F .Indeed, clearly Supp ( χ F · µ ) ⊆ F . Conversely, take x in the relative interior of F . Then, every sufficientlysmall open neighbourhood of x intersects ∂C only on F , so that it is clear that x ∈ Supp ( χ F · µ ). Since F is the closure of its relative interior, the assertion follows. Then implies that ( x ′ , y ) , ( x ′ , y ′ ) are( µ, L )-connected for every y ′ ∈ R such that ( x ′ , y ′ ) ∈ F . Since ∂C is the (finite) union of its ( n − x, y ) , ( x, y ′ ) are ( µ, L )-connected for every y ′ ∈ [ f − ( x ′ ) , f + ( x ′ )]. Theassertion follows in this case. Now, consider the general case. Observe first that we may assume that C has non-empty interior.Then, take y ∈ L ( ∂C ) and a closed cube Q in E which contains y in its interior. Then, C ∩ L − ( Q ) isa compact polyhedron; in addition, ∂C ∩ L − ( ◦ Q ) = ∂ [ C ∩ L − ( Q )] ∩ L − ( ◦ Q ) . L − ( y ) ∩ ∂C are ( µ, L )-connected, we may assume that C is compact. Now, take x , x ∈ ∂C such that x = x and L ( x ) = L ( x ). Let L ′ be an affine mappingdefined on E such that L ′ ( x ) = L ′ ( x ) and such that the fibres of L ′ have dimension 1. Then, we mayapply above and deduce that x , x are ( µ, L ′ )-connected. It is then easily seen that x , x are also( µ, L )-connected, whence the result. Remark 2.5.
Notice that Proposition 2.4 is false when the support of µ is the boundary of a moregeneral convex set (on which L is proper). Indeed, choose E = R , E = R , L = pr , and C := { ( x, y, z ) ∈ E : 2 yz > x , z ∈ [0 , , y > } . Define C as the union of C and π ( C ), where π is the reflection along the plane pr − (1). Then, ∂C isthe union of C ′ := { ( x, y, z ) ∈ E : 2 yz = x , z ∈ [0 , , y > } and π ( C ′ ). Choose any continuous function m : C ′ → C , and define m : ∂C → C so that it equals m on C ′ and m ◦ π on π ( C ′ ). Then, m is clearly continuous. In addition, it is clear that C ′ intersects thefibres of L at one point at most, except for L − (0 , m can be chosen so that it is not constanton { (0 , } × [0 , χ ∂C · H cannot be L -connected. Proposition 2.6 ([12], Proposition 6.2) . Let
X, Y, Z be three locally compact spaces, π : X → Y a µ -measurable mapping, and µ a π -connected positive Radon measure on X . Assume that π ∗ ( µ ) is aRadon measure and that there is a disintegration ( λ y ) y ∈ Y of µ relative to π such that Supp ( λ y ) ⊇ Supp ( µ ) ∩ π − ( y ) for π ∗ ( µ ) -almost every y ∈ Y .Take a continuous mapping m : X → Z such that there is mapping m : Y → Z such that m ( x ) =( m ◦ π )( x ) for µ -almost every x ∈ X . Then, there is a π ∗ ( µ ) -measurable mapping m : Y → Z such that m = m ◦ π pointwise on Supp ( µ ) . Concerning the assumption on the disintegration, we shall often make use of a general result by Federer(cf. [16, Theorem 3.2.22]), which basically provides the disintegration of a wide family of measures. Weshall also derive Lemma 5.5 from it.For what concerns the composition of Schwartz functions, the techniques employed to prove [4,Theorem 6.1] can be effectively used to derive from [5, Theorem 0.2] and [7, Theorem 0.2.1] the followingresult:
Theorem 2.7 ([12], Theorem 7.2) . Let P : R n → R m be a polynomial mapping, and assume that R n and R m are endowed with dilations such that P ( r · x ) = r · P ( x ) for every r > and for every x ∈ R n .Let C be a dilation-invariant subanalytic closed subset of R n , and assume that P is proper on C and that P ( C ) is Nash subanalytic. Then, the canonical mapping Φ : S ( R m ) ∋ ϕ ϕ ◦ P ∈ S R n ( C ) has a closed range and admits a continuous linear section defined on Φ( S ( R m )) . In addition, ψ ∈ S R n ( C ) belongs to the image of Φ if and only if it is a ‘formal composite’ of P , that is, for every y ∈ R m thereis ϕ y ∈ E ( R m ) such that, for every x ∈ C ∩ P − ( y ) , the Taylor series of ϕ y ◦ P and ψ at x differ by theTaylor series of a function of class C ∞ which vanishes on C . In the statement, we denoted by S R n ( C ) the quotient of S ( C ) by the space of f ∈ S ( R n ) whichvanish on the closed set C . We refer the reader to [5, 6, 7] for the notion of (Nash) subanalytic sets;as a matter of fact, in the applications we shall only need to know that any convex subanalytic set isautomatically Nash subanalytic, since it is contained in an affine space of the same dimension, and thatsemianalytic sets are Nash subanalytic (cf. [5, Proposition 2.3]). Corollary 2.8 ([12], Corollary 7.3) . Let V and W be two finite-dimensional vector spaces, C a subana-lytic closed convex cone in V , and L a linear mapping of V into W which is proper on C . Take m ∈ S ( V ) and assume that there is m : W → C such that m = m ◦ L on C . Then, there is m ∈ S ( W ) suchthat m = m ◦ L on C . .4 Equivalence and Completeness Let us now add some definitions to those presented in [12].
Definition 2.9.
We say that two Rockland families L A and L A are functionally equivalent if thereare two Borel functions m : E L A → E L A and m : E L A → E L A such that m ( L A ) = L A and m ( L A ) = L A .We shall say that a Rockland family L A is functionally complete if every β L A -measurable function m : E L A → C such that m ( L A ) is a differential operator equals a polynomial β L A -almost everywhere.Notice that there exist Rockland families which are not functionally complete; for example, if L is apositive Rockland operator, then ( L ) is a Rockland family which is not functionally complete. Further,observe that we cannot talk of a ‘completion’ of L A unless we know that the algebra of differentialoperators arising as functions of L A is (algebraically) finitely generated.The main point for considering functional completeness is the following result, which shows that prop-erty ( S ) implies functional completeness; nevertheless, the converse fails in general (cf. Proposition 8.6). Proposition 2.10.
Let L A be a Rockland family on a homogeneous group G . If L A satisfies property ( S ) , then it is functionally complete.Proof. Take a function of L A which is a left-invariant differential operator of degree δ , and let T be itsconvolution kernel; assume that L A satisfies property ( S ). Take τ ∈ S ( E L A ) such that τ ( λ ) = 0 for every λ ∈ E L A ; then K L A ( τ ) ∗ T ∈ S ( G ), so that it has a multiplier m ∈ S ( E L A ). If we define m := m τ , then m ∈ C ∞ ( E L A ) and K L A ( m ) = T . By means of [17, Theorem 1.37], we see that there are a family withfinite support ( P δ ′ ) δ of homogeneous polynomials, where P δ ′ has homogeneous degree δ ′ for every δ ′ ∈ [0 , δ ], and a function ω , such that m ( λ ) = X δ ′ δ P δ ′ ( λ ) + ω ( λ )for every λ ∈ E L A , and such that lim λ → ω ( λ ) | λ | δ = 0 . Now, clearly m ( r · λ ) = r δ m ( λ ) for every r > λ ∈ σ ( L A ); fix a non-zero λ ∈ σ ( L A ).Then, r δ m ( λ ) = m ( r · λ ) = X δ ′ δ P δ ′ ( r · λ ) + ω ( r · λ ) = X δ ′ δ r δ ′ P δ ′ ( λ ) + o (cid:0) r δ (cid:1) for r → + , so that P δ ′ ( λ ) = 0 for every δ ′ ∈ [0 , δ [ and P δ ( λ ) = m ( λ ). Therefore, m = P δ on σ ( L A ), sothat m = P δ µ L A -almost everywhere. By the arbitrariness of T , the assertion follows (cf. [27]). In this section, G denotes a homogeneous abelian group. In other words, G is the euclidean space R n endowed with dilations of the form r · x = ( r d x , . . . , r d n x n ) for r > x = ( x , . . . , x n ) ∈ R n , and somefixed d , . . . , d n >
0. Then, ∂ = ( ∂ , . . . , ∂ n ) is a homogeneous basis of the Lie algebra of G . We shallconsequently put a scalar product and the associated Hausdorff measures on G , and identify the Fouriertransform F with a mapping from S ′ ( G ) onto S ′ ( E − i∂ ). Proposition 3.1.
Let P be a polynomial mapping with homogeneous components from E − i∂ into R Γ for some finite set Γ . Then, L A = P ( − i∂ ) is a Rockland family if and only if P is proper. In this case,the following hold:1. σ ( L A ) = P ( E − i∂ ) ;2. a β L A -measurable function m admits a kernel in the sense of Definition 2.1 if and only if m ◦ P isa polynomial times an element of L ( E − i∂ ) ; in this case, K L A ( m ) = F − ( m ◦ P ) . Proof.
Since σ ( − i∂ ) = E − i∂ and − i∂ is Rockland, the assertions follow easily by the properties of theFourier transform. 5y means of [16, Theorem 3.2.22], one may obtain some relatively explicit formulae for β L A and χ L A .In the following result, we give complete answers to our main questions in the case of one operator. Theorem 3.2.
Let L be a positive Rockland operator on G . Then, χ L has a continuous representativewhich is of class C ∞ on R ∗ + × G ; in particular, property ( RL ) holds.In addition, take m ∈ C b ( β L ) , and let k be the greatest k ′ ∈ N ∗ such that P k ′ is a polynomial. Then,the following conditions are equivalent:1. K L ( m ) ∈ S ( G ) ;2. there are m , . . . , m k − ∈ S ( R ) such that m ( λ ) = P k − h =0 λ hk m h ( λ ) for every λ > .In particular, property ( S ) holds if and only if k = 1 . Before we pass to the proof of the preceding result, we need to establish a lemma.
Lemma 3.3.
Let A be a non-empty finite set and endow R A with a family of (not-necessarily isotropic)dilations. Take a positive, non-constant, homogeneous polynomial P in R [ A ] and assume that there is ahomogeneous element x of R A such that P ( x ) = 0 . Then, the following statements are equivalent:1. there are no positive homogeneous polynomials Q ∈ R [ A ] and no k ∈ N such that k > and P = Q k ;2. if m is a complex-valued function defined on R + such that m ◦ P is C ∞ on R A , then m may beextended to an element of C ∞ ( R ) .Proof. = ⇒ Take m : R + → C and assume that m ◦ P is C ∞ on R A . Notice that there is ahomogeneous polynomial P x ∈ R [ X ] such that P ( λx ) = P x ( λ ) for every λ ∈ R . In particular, m ◦ P x is of class C ∞ . In addition, P x ( X ) = a x X d x for some a x = 0 and d x ∈ N ∗ ; we may assume that a x = 1. It is then clear that m is C ∞ on R ∗ + ; further, m ◦ P x admits a Taylor series P j ∈ N e a j X j at 0.Therefore, m admits the asymptotic development P j ∈ N a x,j λ jdx for λ → + . Suppose that there aresome j ∈ N \ ( d x N ) such that a x,j = 0, and let j x be the least of them. Let q x , r x be the quotient andthe remainder, respectively, of the division of j x by d x .Define e m := m − P j x d x j =0 a x,j ( · ) jdx . Then, e m ◦ P x is C ∞ and ( e m ◦ P x )( λ ) = o (cid:16) | λ | j x d x (cid:17) . Hence, it isnot hard to see that e m may be extended to an element of C j x ( R ). Let us then prove that ∂ j x x P jxdx = ∂ j x x ( m ◦ P ) − q x X j =0 a x,d x j ∂ j x x P j − j x d x X j = j x +1 a x,j ∂ j x x P jdx − ∂ j x x ( e m ◦ P )extends to a continuous function on E := { x ′ ∈ R A : P ( x ′ ) = 0 } ∪ { } . Indeed, this is clear for the firsttwo terms, and follows from the above remarks for the fourth one. Let us then consider the third term.Notice that both ∂ x and P are homogeneous, and that ∂ j x x P jxdx is homogeneous of degree 0 on the x axis,hence on R A . Hence, ∂ j x x must be homogeneous of degree d j x d x , where d is the homogeneous degree of P .Then, ∂ j x x P jdx is homogeneous of degree d j − j x d x >
0, so that it may be extended by continuity at 0.Therefore, ∂ j x x P jxdx is a continuous function on E which is homogeneous of degree 0; hence, it isconstant, and its constant value must be j x ! = 0. Now, Fa`a di Bruno’s formula shows that P − rxdx = 1 j x ! P − rxdx ∂ j x x P jxdx = X P jxℓ =1 ℓβ ℓ = j x β ! (cid:18) j x d x (cid:19) | β | P q x −| β | j x Y ℓ =1 (cid:18) ∂ ℓx Pℓ ! (cid:19) β ℓ , where (cid:16) j x d x (cid:17) | β | := j x d x (cid:16) j x d x − (cid:17) . . . (cid:16) j x d x − | β | + 1 (cid:17) is the Pochhammer symbol. Then, P − rxdx is a rationalfunction, so that there are N, D ∈ R [ A ], with D = 0, such that P − rxdx = ND . Hence, P d x − r x = N dx D dx , so Notice that L = P ( − i∂ ) where P is a proper polynomial; unless G = R , in which case our analysis is trivial, P musthave a constant sign, so that we may assume that L is positive without loss of generality. Notice that λx denotes the scalar multiplication of x by λ , not the dilate λ · x of x by λ , which is meaningful only for λ > Here, | λ | denotes the usual absolute value of λ ∈ R . D d x divides N d x in R [ A ]. Since R [ A ] is factorial, it follows that D divides N , so that P − rxdx is a(positive) polynomial. Next, let g be the greatest common divisor of d x and d x − r x , and take d ′ , r ′ ∈ N ∗ so that d x = g d ′ and d x − r x = g r ′ . Then, (cid:16) P − rxdx (cid:17) d ′ = P r ′ . Since R [ A ] is factorial, this proves that there is a polynomial Q ∈ R [ A ] such that Q r ′ = P − rxdx and Q d ′ = P . Now, d ′ > d x does not divide d x − r x ; in addition, Q is positive since both P − rxdx and P are positive and d ′ , r ′ are coprime: contradiction. Therefore, a x,j = 0 for every j d x N , so that theconclusion follows easily. = ⇒ Suppose by contradiction that there are a positive homogeneous polynomial Q ∈ R [ A ]and k > P = Q k . Define m : λ λ k on R + . Then, m is not right-differentiable at 0;nevertheless, m ◦ P = Q since Q is positive, so that m ◦ P is C ∞ : contradiction. Proof of Theorem 3.2.
Notice that χ L ( λ, · ) is an eigenfunction of positive type and of class C ∞ of L ,with eigenvalue λ , and that χ L ( r · λ, g ) = χ L ( λ, r · g ) for every r > β L ⊗ ν G )-almost every( λ, g ) (cf. [29]). It is then easily seen that χ L has a continuous representative which is of class C ∞ on R ∗ + × G .Now, take m ∈ C b ( β L ) such that K L ( m ) ∈ S ( G ). Then, Proposition 3.1 implies that m ◦ P ∈ S ( E − i∂ ).Take a positive polynomial Q on E − i∂ so that P = Q k . Since [ m ◦ ( · ) k ] ◦ Q = m ◦ P , Lemma 3.3 impliesthat we may take e m ∈ E ( R ) so that m ◦ ( · ) k = e m on R + . In addition, it is clear that we may assumethat e m ∈ S ( R ). Now, let P ℓ ∈ N a ℓ λ ℓ be the Taylor development of e m at 0. Take, for h = 1 , . . . , k − m h ∈ C ∞ c ( R ) so that its Taylor development at 0 is P ℓ ∈ N a h + kℓ λ ℓ (cf. [20, Theorem 1.2.6]), and define m := m − P k − h =1 ( · ) hk m h on R + . Since clearly m has the asymptotic development P ℓ ∈ N a kℓ λ ℓ for λ → m ◦ ( · ) k is of class C ∞ , it is easily seen that m may be extended to an element of S ( R ). Therefore, m ( λ ) = P k − h =0 λ hk m h ( λ ) for every λ > m , . . . , m k − ∈ S ( R ) such that m ( λ ) = P k − h =0 λ hk m h ( λ ) for every λ >
0. Then, m ◦ P ∈ S ( E − i∂ ), so that K L ( m ) ∈ S ( G ) by Proposition 3.1. Corollary 3.4.
Let L : R n → R n ′ be a linear mapping which is proper on R n + . Then, L ( − ∂ , . . . , − ∂ n ) satisfies properties ( RL ) and ( S ) .Proof. This is a consequence of Theorems 2.2 and 3.2 when L is the identity. The general case thenfollows by means of Propositions 2.4 and 2.6, and Corollary 2.8. M W + Groups
Definition 4.1.
Let G be a 2-step stratified group, that is, a simply connected Lie group whose Liealgebra is decomposed as g = g ⊕ g with g = [ g , g ] and [ g , g ] = 0. For every ω ∈ g ∗ , define B ω : g × g ∋ ( X, Y )
7→ h ω, [ X, Y ] i . We say that G is an M W + group if B ω is non-degenerate for some ω = 0. We say that G is a M´etiviergroup if it is not abelian and B ω is non-degenerate for every ω = 0. A Heisenberg group is a M´etiviergroup with one-dimensional centre.Notice that a 2-step stratified group satisfies property M W + if and only if it satisfies the Moore-Wolfcondition (cf. [26]) and [ g , g ] is the centre of g .We shall endow a 2-step stratified group with the canonical dilations, so that r · ( X + Y ) = rX + r Y for every r >
0, for every X ∈ g and for every Y ∈ g . Since exp G : g → G is a diffeomorphism, thesedilations transfer to G .Now, to every symmetric bilinear form Q on g ∗ we associate a differential operator on G as follows: L := − X ℓ,ℓ ′ Q ( X ∗ ℓ , X ∗ ℓ ′ ) X ℓ X ℓ ′ , X ℓ ) is a basis of g with dual basis ( X ∗ ℓ ). As the reader my verify, L does not depend on thechoice of ( X ℓ ); actually, one may prove that −L is the symmetrization of the quadratic form induced by Q on g ∗ (cf. [19, Theorem 4.3]).By a ‘sum of squares’ we means a differential operator of the form L = − P kj =1 Y j , where Y , . . . , Y k are elements of g . If, in addition, Y , . . . , Y k generate g as a Lie algebra, then we say that L is asub-Laplacian. Thanks to [22], this is equivalent to saying that L is hypoelliptic. Lemma 4.2.
Let Q be a symmetric bilinear form on g ∗ , and let L be the associated operator. Then, L is formally self-adjoint if and only if Q is real. In addition, L is formally self-adjoint and hypoelliptic ifand only if Q is non-degenerate and either positive or negative. Definition 4.3.
Let V be a vector space and Φ a bilinear form on V . Then, we defined Φ : V ∋ v Φ( · , v ) ∈ V ∗ . Proposition 4.4.
Let Q and Q be two symmetric bilinear forms on g ∗ , and let L and L be theassociated operators. Then, L and L commute if and only if d Q ◦ d B ω ◦ d Q = d Q ◦ d B ω ◦ d Q for every ω ∈ g ∗ .Proof. Choose a basis ( X j ) j ∈ J of g and a basis ( T k ) k ∈ K of g . Let ( X ∗ j ) j ∈ J and ( T ∗ k ) k ∈ K be thecorresponding dual bases. Define a h,j ,j := Q h ( X ∗ j , X ∗ j ) for h = 1 , j , j ∈ J , so thatd Q h is identified with the matrix A h := ( a h,j ,j ) j ,j ∈ J for h = 1 ,
2. Analogously, define b k,j ,j := B T ∗ k ( X j , X j ) for every k ∈ K and for every j , j ∈ J , so that d B T ∗ k is identified with the matrix B k := ( b k,j ,j ) j ,j ∈ J for every k ∈ K . Now, define Y j ,j := ( X j X j + X j X j ) for every j , j ∈ J .Then, L h = X j ,j ∈ J a h,j ,j Y j ,j since Q h is symmetric. In addition, for every j , j , j , j ∈ J ,[ Y j ,j , Y j ,j ] = Y j ,j [ X j , X j ] + Y j ,j [ X j , X j ] + Y j ,j [ X j , X j ] + Y j ,j [ X j , X j ]since the elements of g = [ g , g ] lie in the centre of U ( g ). Next, observe that, for every j , j ∈ J ,[ X j , X j ] = X k ∈ K b k,j ,j T k . Therefore,[ L , L ] = X j ,j ,j ,j ∈ J X k ∈ K a ,j ,j a ,j ,j [ b k,j ,j Y j ,j + b k,j ,j Y j ,j + b k,j ,j Y j ,j + b k,j ,j Y j ,j ] T k = 2 X j ,j ∈ J X k ∈ K c k,j ,j Y j ,j T k , where c k,j ,j = X j ,j ∈ J ( a ,j ,j a ,j ,j + a ,j ,j a ,j ,j ) b k,j ,j for every k ∈ K and for every j , j ∈ J . Now, the distinct monomials in the family of the Y j ,j T k ,as j , j ∈ J and k ∈ K , are linearly independent (cf., for example, [11, Theorem 1 of Chapter I, § C k the matrix ( c k,j ,j ) j ,j ∈ J for every k ∈ K . Since A and A aresymmetric and since B k is skew-symmetric, we have C k = A B k A + A t B k A = A B k A − A B k A for every k ∈ K . The assertion follows easily.Now we shall present some results which will enable us to put our homogeneous sub-Laplacians in aparticularly convenient form. We state them in terms of the associated quadratic forms.8 roposition 4.5. Let ( V, σ ) be a finite-dimensional symplectic vector space over R . Let ( Q α ) α ∈ A bea family of positive, non-degenerate bilinear forms on V such that the d − Q α ◦ d σ , as α runs through A ,commute.Then, there is a finite family ( P γ ) γ ∈ Γ of projectors of V such that the following hold: • P γ is σ - and Q α -self-adjoint for every α ∈ A and for every γ ∈ Γ ; • I V = P γ ∈ Γ P γ and P γ P γ ′ = 0 for γ, γ ′ ∈ Γ , γ = γ ′ ; • the bilinear forms Q α ( P γ · , P γ · ) , as α ∈ A , are all multiples of one another for every γ ∈ Γ , γ = γ . For the proof, basically follow that of [22, Theorem 3.1 (c)] using commutativity in order to getsimultaneous diagonalizations. Applying [22, Theorem 3.1 (c)] (or simply [1, Corollary 5.6.3]) to therange of each P γ , we may find a symplectic basis of V which is Q α -orthogonal for every α ∈ A . Proposition 4.6.
Take a finite family ( L α ) α ∈ A of commuting homogeneous sub-Laplacians on an M W + group G , and let ( Q α ) α ∈ A be the corresponding family of non-degenerate positive bilinear forms on g ∗ .Then, there is a finite family ( P γ ) γ ∈ Γ of non-zero projectors of g such that the following hold:1. I g = P γ ∈ Γ P γ and P γ P γ = 0 for every γ , γ ∈ Γ such that γ = γ ;2. P γ is B ω - and b Q α -self-adjoint for every γ ∈ Γ , for every ω ∈ g ∗ , and for every α ∈ A ;3. for every γ ∈ Γ , the bilinear forms Q α (cid:0) t P γ · , t P γ · (cid:1) , as α runs through A , are mutually propor-tional.Proof. Fix ω ∈ g ∗ such that B ω is non-degenerate. Then, Proposition 4.5 and the remarks which followits proof imply that there is a basis X , . . . , X n of g such that d B ω and d Q α are represented by thematrices (cid:18) I − I (cid:19) and (cid:18) D α D α (cid:19) , respectively, for some diagonal matrix D α ( α ∈ A ). Denote by d α, , . . . , d α,n the diagonal elements of D α , and denote by ( a ω,j,k ) the matrix associated with d B ω , for every non-zero ω ∈ g ∗ .Now, assume that A has exactly two elements α , α . Then, defineΓ := (cid:26) d α ,j d α ,j : j ∈ { , . . . , n } (cid:27) and, for every γ ∈ Γ, let V γ be the vector subspace of g generated by the set (cid:26) X j , X n + j : d α ,j d α ,j = γ (cid:27) . Next, take j, k ∈ { , . . . , n } such that d α ,j d α ,j = d α ,k d α ,k . Apply Proposition 4.4, and observe that the ( j, k )-thcomponents of (the matrices representing) the equalityd Q α ◦ d B ω ◦ d Q α = d Q α ◦ d B ω ◦ d Q α give d α ,j a ω,j,k d α ,k = d α ,j a ω,j,k d α ,k , whence a ω,j,k = 0. Considering the components ( n + j, k ) , ( j, n + k ) and ( n + j, n + k ), we see that a ω,n + j,k = a ω,j,n + k = a ω,n + j,n + k = 0. Therefore, B ω ( V γ , V γ ) = { } for every non-zero ω ∈ g ∗ and forevery γ , γ ∈ Γ such that γ = γ . Then, define P γ as the projector of g onto V γ with kernel L γ ′ = γ V γ ′ .The general case follows easily.From now on, G will denote an M W + group, ( Q η ) η ∈ H a family of positive symmetric bilinear formson g ∗ , and ( T , . . . , T n ) a basis of g . We shall denote by L η the sum of squares induced by Q η , andwe shall assume that L A := ( L H , ( − iT k ) k =1 ,...,n ) is a Rockland family. Observe that this condition is9quivalent to the fact that the sum of the L η is hypoelliptic. We may therefore assume that Q η isnon-degenerate for every η ∈ H , in which case each L η is a (homogeneous) sub-Laplacian.We shall also endow g with a scalar product for which g and g are orthogonal, and which induces b Q η on g for some fixed η ∈ H . Up to a normalization, we may then assume that (exp G ) ∗ ( H n ) is thechosen Haar measure on G , where n is the dimension of G . We shall endow g ∗ with the scalar productinduced by that of g , and then with the corresponding Lebesgue measure, that is, H n .Define J Q η ,ω := d Q η ◦ d B ω : g → g for every η ∈ H and for every ω ∈ g ∗ .We shall denote by W the set of ω ∈ g ∗ such that B ω is degenerate, so that G is a M´etivier group ifand only if W = { } .We shall denote by Ω the set of ω ∈ g ∗ \ W where Card ( σ ( | J Q H ,ω | )) attains its maximum h . Bymeans of a straightforward generalization of the arguments of [23, § § Proposition 4.7.
The sets W and g ∗ \ Ω are algebraic varieties. In addition, there are three analyticmappings µ : Ω → (( R ∗ + ) h ) H P : Ω → L ( g ) h ρ : Ω → { , . . . , h } n such that the following hold: • the mapping Ω ∋ ω µ η,ρ k,ω ,ω ∈ R + extends to a continuous mapping ω e µ η,k,ω on g ∗ for every k = 1 , . . . , n and for every η ∈ H ; • for every h = 1 , . . . , h and for every ω ∈ Ω , P h,ω is a B ω - and b Q H -self-adjoint projector of g ; • if h = 1 , . . . , h and ω ∈ Ω , then Tr P h,ω = 2 Card( { k ∈ { , . . . , n } : ρ k,ω = h } ) ; • P hh =1 P h,ω = I g and P hh =1 µ η,h,ω P h,ω = | J Q η ,ω | for every ω ∈ Ω and for every η ∈ H . Definition 4.8.
Define n : Ω → ( N ∗ ) h so that n ,h,ω = Tr P h,ω for every ω ∈ Ω and for every h = 1 , . . . , h . By an abuse of notation, we shall denote by ( x, t ) the elements of G , where x ∈ g and t ∈ g , thus identifying ( x, t ) with exp G ( x, t ). For every x ∈ g , for every ω ∈ Ω and for every h = 1 , . . . , h , define x h,ω := √ µ η ,h,ω P h,ω ( x ) . By an abuse of notation, we shall write x ω instead of ( x h,ω ) h =1 ,...,h , and | x ω | instead of (cid:16)P hh =1 | x h,ω | (cid:17) / .The following two results are easy and their proof is omitted. Corollary 4.9.
The function ω µ η,ω ( n ,ω ) = e µ η,ω ( n ) = k J Q,ω k is a norm on g ∗ which is analyticon g ∗ \ W for every η ∈ H . Proposition 4.10.
The mapping g × Ω ∋ ( x, ω ) h X h =1 x h,ω = q − J Q,ω ( x ) extends uniquely to a continuous function on g × g ∗ which is analytic on g × ( g ∗ \ W ) . Indeed, if π is the projection of G onto its abelianization, then d π ( L A ) is a Rockland family, so that F (d π ( L A ))vanishes only at 0. Since F (d π ( L η )) > π ( T k ) = 0 for every η ∈ H and for every k = 1 , . . . , n , this implies that P η ∈ H F (d π ( L η )) vanishes only at 0, so that P η ∈ H Q η is non-degenerate and P η ∈ H L η is hypoelliptic. By an abuse of notation, we denote by | J Q H ,ω | the family ( | J Q η ,ω | ) η ∈ H . efinition 4.11. Define G ω , for every ω ∈ g ∗ , as the quotient of G by its normal subgroup exp G (ker ω ).Then, G is the abelianization of G , and we identify it with g . If ω = 0, then we shall identify G ω with g ⊕ R , endowed with the product( x , t )( x , t ) := (cid:18) x + x , t + t + 12 B ω ( x , x ) (cid:19) for every x , x ∈ g and for every t , t ∈ R . Hence, π ω ( x, t ) = ( x, ω ( t ))for every ( x, t ) ∈ G . Proposition 4.12.
Define e π : [ ω ∈ Ω { ω } × G ω ∋ ( ω, ( x, t )) ω ∈ Ω , and identify the domain of e π with Ω × ( g ⊕ R ) as an analytic manifold, so that ̟ becomes an analyticsubmersion.Then, e π defines a fibre bundle with base Ω and fibres isomorphic to G ′ := H n ⊕ R d . More precisely,for every ω ∈ Ω , there is an analytic trivialization ( U, ψ ) of e π such that the following hold: • U is an open neighbourhood of ω in Ω ; • ψ : e π − ( U ) → U × G ′ is an analytic diffeomorphism such that pr ◦ ψ = ̟ and such that ψ ω :=pr ◦ ψ : e π − ( ω ) → G ′ is a group isomorphism for every ω ∈ U ; • if ( X , . . . , X n , T, Y , . . . , Y d ) is a basis of left-invariant vector fields on G ′ which at the origininduce the partial derivatives along the coordinate axes, then d( ψ ω ◦ π ω )( L η ) = − d X k =1 Y k − n X k =1 e µ η,k,ω ( X k + X n + k ) and d( ψ ω ◦ π ω )( T ℓ ) = ω ( T ℓ ) T for every η ∈ H , for every ℓ = 1 , . . . , n , and for every ω ∈ U . The proof is omitted. It basically consists in using the projectors P h to propagate locally a givenbasis of eigenvectors and then in ‘symplectifying’ the new basis in order to meet the requirements.One may then give formulae for β L A and χ L A (see [24, 4.4.1] for the general procedure). Nevertheless,we shall (almost) only need to know that β L A is equivalent to χ Σ · H n , whereΣ := n ( µ ω ( n ,ω + 2 γ ) , ω ( T )) : ω ∈ g ∗ , γ ∈ N h o . ( RL ) In this section we shall present several sufficient conditions for the validity of property ( RL ). Unlike inthe cases considered in [12], we are able to prove continuity results for χ L A , even though under ratherstrong assumptions (cf. Theorem 5.2); we then deduce property ( RL ) under slightly weaker assumptions(cf. Theorem 5.3). Let us comment a little more on the assumptions of Theorem 5.3. Besides theconditions that Ω is g ∗ \ { } and that µ is constant on the unit sphere associated with the norm µ η ( n ),we need to add the condition that dim R µ ω ( R h ) = dim Q µ ω ( Q h ) for some, and then all, ω ∈ Ω. Eventhough this condition may appear peculiar, we cannot get rid of it without running into counterexamples,as Theorem 7.4 shows. Furthermore, observe that, even though Theorem 7.4 is the main applicationof Theorems 5.2 and 5.3, the latter result can be applied to more general homogeneous sub-Laplacianson H -type groups. For example, consider the complexified Heisenberg group H C , whose Lie algebrais endowed with an orthonormal basis X , X , X , X , T , T such that [ X , X ] = [ X , X ] = T and[ X , X ] = [ X , X ] = T , while the other commutator vanish. If L = − ( aX + bX + cX + dX ) with a, b, c, d > p ab , p cd ∈ Q , and either a = b or c = d , then Theorem 5.3 applies, but Theorem 7.4 doesnot unless a = b and c = d . 11he next results concern families of the form ( L , ( − iT , . . . , − iT n ′ )) for n ′ < n . Notice that, inthis case, we do not only reduce the number of elements of g , but we restrict to the case in whichCard( H ) = 1. In this case, indeed, the spectrum of ( L , ( − iT , . . . , − iT n ′ )) is no longer a countableunion of semianalytic sets, but a convex cone, so that things are somewhat easier and we can provemore general results than for the ‘full family’ L A . In Theorem 5.7, we show that property ( RL ) holdsif W = { } . With reference to the above example in the complexified Heisenberg group, this is the casewhen ac = bd and ad = bc .Our last result concerns the case of general M W + groups (cf. Theorem 5.8); even though its hy-potheses are more restrictive than in the preceding one, it nonetheless applies when G is a product ofHeisenberg groups and L is a sum of homogeneous sub-Laplacians on each factor (cf. Proposition 8.3). n ′ = n We begin with a technical lemma. Here, if V is a finite-dimensional vector space, then E ( V ) denotesthe space of continuous functions on V with the topology of locally uniform convergence, while E ′ c ( V )denotes its dual (that is, the space of Radon measures with compact support), endowed with the topologyof compact convergence. Lemma 5.1.
Let V and e V be two finite-dimensional vector spaces over R , L a discrete subgroup of V , C the convex envelope of R + F for some finite subset F of L which generates V , and µ : V → e V a linearmapping which is proper on C . Assume that L ∩ ker µ generates ker µ , and take ξ ∈ µ ( C ) . Define V ξ := µ − ( ξ ) S ξ := V ξ ∩ Cn ξ := dim R S ξ ν ξ := 1 H n ξ ( S ξ ) χ S ξ · H n ξ . Take x ∈ C and define, for every λ ∈ R ∗ + and for every γ ∈ µ ( x + L ∩ C ) , ν λ,γ = 1 c γ X γ ′ ∈ L ∩ Cγ = µ ( x + γ ′ ) δ λ ( x + γ ′ ) , where c γ = Card (cid:0) µ − ( γ ) ∩ ( x + L ∩ C ) (cid:1) . Then, lim ( λγ,λ ) → ( ξ, γ ∈ µ ( x + L ∩ C ) ν λ,γ = ν ξ in E ′ c ( V ) .Proof. Define Σ := µ ( x + L ∩ C ), and define F ξ as the filter ‘( λ, γ ) ∈ R ∗ + × Σ , ( λγ, λ ) → ( ξ, ν λ,γ converges vaguely to ν ξ along F ξ . Indeed, the ν λ,γ areprobability measures supported in S λγ ⊆ C ∩ µ − ( K ) (1)eventually along F ξ , where K is a compact neighbourhood of ξ in e V . Since µ is proper on C , the assertionfollows.Now, let us prove that we may reduce to the case in which x = 0. Indeed, define ν λ,γ := 1 c γ X γ ′ ∈ L ∩ Cγ = µ ( x + γ ′ ) δ λγ ′ . It will then suffice to prove that ν λ,γ − ν λ,γ converges vaguely to 0 along F ξ . However, take ϕ ∈ C c ( V )and ε >
0. Then, there is a neighbourhood U of 0 in V such that | ϕ ( x ) − ϕ ( x ) | < ε for every x , x ∈ V such that x − x ∈ U . Therefore, (cid:12)(cid:12)(cid:12)D ν λ,γ − ν λ,γ , ϕ E(cid:12)(cid:12)(cid:12) < ε as long as λx ∈ U , hence eventually along F ξ .The assertion follows. Observe that C is a polyhedral convex cone. In addition, let n be the dimension of V , and let( F h ) h ∈ H be the (finite) family of ( n − C ; observe that F h is a convex cone forevery h ∈ H , so that 0 ∈ F h . Take, for every h ∈ H , some p h ∈ V ∗ such that F h = ker p h ∩ C and12 h ( C ) ⊆ R + . Then, C is the set of x ∈ V such that p h ( x ) > h ∈ H , and L ∩ ker p h generatesker p h for every h ∈ H .In addition, let H ξ be the set of h ∈ H such that p h ( S ξ ) = { } , and let H ′ ξ be its complement in H .We shall write p H ξ and p H ′ ξ instead of ( p h ) h ∈ H ξ and ( p h ) h ∈ H ′ ξ , respectively. Define V ′ ξ := V ξ ∩ ker p H ξ .Then, V ′ ξ ∩ p − H ′ ξ (cid:16) ( R ∗ + ) H ′ ξ (cid:17) is the interior of S ξ in V ′ ξ ; since by convexity V ′ ξ ∩ p − H ′ ξ (cid:16) ( R ∗ + ) H ′ ξ (cid:17) is not empty, V ′ ξ is the affine space generated by S ξ . Define W ξ := V ′ ξ − V ′ ξ , and observe that L ∩ W ξ generates W ξ . Indeed, the linear mapping( µ, p H ξ ) : V → e V × R H ξ maps L into the discrete subgroup µ ( L ) × Q h ∈ H ξ p h ( L ) of e V × R H ξ , and W ξ isthe kernel of ( µ, p H ξ ), whence the assertion.Therefore, there are two subspaces W ′ ξ and W ′′ ξ of V such that the following hold (cf. [9, Exercises 2and 3 of Chapter VII, § • W ξ ⊕ W ′ ξ = V and V ⊕ W ′′ ξ = V ; • L ∩ W ′ ξ and L ∩ W ′′ ξ generate W ′ ξ and W ′′ ξ , respectively, over R ; • ( L ∩ W ξ ) ⊕ ( L ∩ W ′ ξ ) ⊕ ( L ∩ W ′′ ξ ) = L as abelian groups.Therefore, we may endow V and e V with two scalar products such that W ξ , W ′ ξ , and W ′′ ξ are or-thogonal, and µ induces an isometry of W ′′ ξ into e V . We may further assume that k p h k h ∈ H . Define, for λ > γ ∈ Σ, r ξ,λ,γ := inf { r > S λγ ⊆ B ( S ξ , r ) } + λ, so that S λγ ⊆ B ( S ξ , r ξ,λ,γ ). Let us prove that r ξ,λ,γ converges to 0 along F ξ .Indeed, let U be an ultrafilter finer than F ξ . Denote by K the space of non-empty compact subsetsof V endowed with the Hausdorff distance d H . By (1), [8, Proposition 10 of Chapter I, §
6, No. 7]and [2, Theorem 6.1], it follows that S λγ has a (unique) limit S in K along U . Now, for every closedneighbourhood K of ξ in e V , S λγ ⊆ C ∩ µ − ( K )as long as λγ ∈ K , so that, by passing to the limit along U , S ⊆ C ∩ µ − ( K ) . By the arbitrariness of K , it follows that S ⊆ S ξ . Therefore, r ξ,λ,γ d H ( S, S λγ ) + λ, so that r ξ,λ,γ tends to 0 along U . Thanks to [8, Proposition 2 of Chapter I, §
7, No. 2], the arbitrarinessof U implies that r ξ,λ,γ converges to 0 along F ξ . Now, let π ξ be the affine projection of V onto V ′ ξ with fibres parallel to W ′ ξ ⊕ W ′′ ξ . Reasoningas in and taking into account, we see that ν λ,γ − ( π ξ ) ∗ ( ν λ,γ ) converges vaguely to 0 along F ξ , sothat it will suffice to prove that ( π ξ ) ∗ ( ν λ,γ ) converges vaguely to ν ξ along F ξ . Now, if n ξ = 0, then( π ξ ) ∗ ( ν λ,γ ) = δ ξ ′ = ν ξ , where ξ ′ is the unique element of S ξ . Therefore, we may assume that n ξ > ε > x, y ∈ S ξ,λ,γ := Supp (( π ξ ) ∗ ( ν λ,γ )). Assume that B ( x, ε ) ∩ p − H ξ (cid:16) R H ξ + (cid:17) ⊆ C , andthat r ξ,λ,γ < ε . Take y ′ ∈ Supp ( ν λ,γ ) such that π ξ ( y ′ ) = y , and let us prove that y ′ + x − y ∈ Supp ( ν λ,γ ).Indeed, it is clear that x − y ∈ λL ∩ W ξ , so that y ′ + x − y ∈ λL . Hence, it will suffice to prove that y ′ + x − y ∈ C . Now, since y ′ ∈ S λγ ⊆ B ( S ξ , ε ), it follows that there is x ′ ∈ S ξ such that | y ′ − x ′ | < ε ,so that ε > | y ′ − x ′ | = | y − x ′ | + | y ′ − y | since y − x ′ ∈ W ξ and y ′ − y ∈ W ′ ξ ⊕ W ′′ ξ . Therefore, | y ′ − y | < ε ; since, in addition, p h ( y ′ + x − y ) = p h ( y ′ ) > h ∈ H ξ , it follows that y ′ + x − y ∈ B ( x, ε ) ∩ p − H ξ (cid:16) R H ξ + (cid:17) ⊆ C . By the arguments of above, we see that there is a function c ξ,λ,γ on S ξ,λ,γ such that( π ξ ) ∗ ( ν λ,γ ) = X x ∈ S ξ,λ,γ c ξ,λ,γ ( x ) δ x , c ξ,λ,γ ( x ) > c ξ,λ,γ ( y ) eventually along F ξ whenever x, y ∈ S ξ,λ,γ and B ( x, ε ) ∩ p − H ξ (cid:16) R H ξ + (cid:17) ⊆ C for some fixed ε >
0. In particular, c ξ,λ,γ is constant on the set of x ∈ S ξ,λ,γ such that B ( x, ε ) ∩ p − H ξ (cid:16) R H ξ + (cid:17) ⊆ C .Now, let us prove that, if ε min h ∈ H ′ ξ min S ξ p h and if x ∈ V ′ ξ and B ( x, ε ) ∩ V ′ ξ ⊆ C , then B ( x, ε ) ∩ p − H ξ (cid:16) R H ξ + (cid:17) ⊆ C . Indeed, take x ′ ∈ B ( x, ε ), and assume that p h ( y ) > h ∈ H ξ . Take h ∈ H ′ ξ ,and observe that | p h ( y − x ) | | y − x | < ε , so that p h ( y ) = p h ( x ) + p h ( y − x ) > p h ( x ) − ε > ε . By the arbitrariness of h , it follows that y ∈ C . Finally, take a fundamental parallelotope P ξ of L ∩ W ξ , and extend c ξ,λ,γ to a function on V which is constant on x + λP ξ for every x ∈ π ξ ( λL ), and vanishes outside S ξ,λ,γ + λP ξ . Then, ν ξ,λ,γ := H nξ ( λP ξ ) c ξ,λ,γ · H n ′ ξ is a probability measure; in addition, as in we see that ( π ξ ) ∗ ( ν λ,γ ) − ν ξ,λ,γ converges vaguely to 0 along F ξ , so that it will suffice to show that ν ξ,λ,γ converges vaguely to ν ξ along F ξ . However, if S ′ ξ denotes the boundary of S ξ in V ′ ξ , then and imply that H nξ ( λP ξ ) c ξ,λ,γ is uniformlybounded eventually along F ξ , and converges on V \ S ′ ξ to a function g which is 0 on the complement of S ξ , and is constant on S ξ \ S ′ ξ . The assertion follows by dominated convergence. Theorem 5.2.
Assume that
Ω = g ∗ \ { } , that dim Q µ ω ( Q h ) = dim R µ ω ( R h ) for every ω ∈ Ω , and that P h is constant on Ω . Then, χ L A has a continuous representative.Proof. We shall simply write P h and n to denote the constant values of the functions ω P h,ω and ω n ,ω , respectively. In addition, we shall denote by | · | ′ the (homogeneous) norm µ η ( n ),and by S ′ the corresponding unit sphere. Choose ω ∈ S ′ and observe that µ η,h,ω = | ω | ′ µ η,h,ω and x h,ω = q | ω | ′ x h,ω for every ω ∈ Ω.For every ξ ∈ µ ω ( R h + ), let F ξ denote the filter ‘( λ, γ ) ∈ R ∗ + × Σ , ( λγ, λ ) → ( ξ, µ ( n + 2 N H ). In addition, define, for every λ ∈ R ∗ + and for every γ ∈ Σ, ν ′ λ,γ = X γ = µ ω ( n +2 γ ′ ) δ λ ( n +2 γ ′ ) , and ν λ,γ := ν ′ λ,γ ( R h ) · ν ′ λ,γ , so that ν λ,γ is a probability measure. Then, Lemma 5.1 implies that ν λ,γ converges to some probability measure ν ξ in E ′ c ( R h ) along F ξ . Denote by Λ mγ the γ -th Laguerre polynomial of order m , and by J m the Bessel function (of thefirst kind) of order m . Define, for every ( x, t ) ∈ R H × R , χ ( λ ( n + 2 γ ′ ) , λ, x, t ) = e − λ | x | + iλt (cid:0) n + γ ′ − h γ ′ (cid:1) h Y h =1 Λ n ,h − γ ′ h (cid:18) λ | x h | (cid:19) for every λ ∈ R ∗ + and for every γ ′ ∈ N h , and χ ( ξ ′ , , x, t ) := h Y h =1 n ,h − ( n ,h − (cid:0)p ξ ′ h | x h | (cid:1) n ,h − J n ,h − (cid:18)q ξ ′ h | x h | (cid:19) for every ξ ′ ∈ R h + . Then, χ extends to a continuous function on R h × R × R h × R .Next define, for every λ ∈ R + , f λ : R h ∋ x (2 λ ) | n − h | (cid:18) x λ + n − h n − h (cid:19) = h Y h =1 ( x h + λn ,h − λ ) · · · ( x h − λn ,h + 2 λ )( n ,h − , and observe that f λ converges locally uniformly to f as λ → + . Therefore, f λ · ν λ,γ converges to f · ν ξ in E ′ c ( R h ) along F ξ . If we define ν ′ λ,γ := ν λ,γ ( f λ ) f λ · ν λ,γ and ν ′ ξ := ν ξ ( f ) f · ν ξ , then ν ′ λ,γ converges to ν ′ ξ in E ′ c ( R h ) along F ξ .Define, for every ω ∈ Ω and for every γ ∈ Σ, χ (( | ω | ′ γ, ω ( T )) , ( x, t )) := D ν ′| ω | ′ ,γ , χ ( · , | ω | ′ , ( | x h,ω | ) hh =1 , ω ( t ) | ω | ′ ) E ,
14o that χ is a representative of χ L A (reason as in [24, 4.4.1], and take [3] into account). Now,lim ( λ,γ ) , F ξ χ (( λγ, λω ( T )) , ( x, t )) = D ν ′ ξ , χ ( · , , ( | x h,ω | ) hh =1 , ω ( t )) E uniformly as ξ runs through µ ( R h + ), as ω runs through S ′ , and as ( x, t ) runs through a compact subsetof G . Since D ν ′ ξ , χ ( · , , ( | x h,ω | ) hh =1 , ω ( t )) E does not depend on ω , it follows that χ is continuous on σ ( L A ) × G . The assertion follows from [9, Corollary to Theorem 2 of Chapter IX, §
4, No. 2].
Theorem 5.3.
Assume that
Ω = g ∗ \ { } , that dim Q µ ω ( Q h ) = dim R µ ω ( R h ) for every ω ∈ Ω , and that µ is constant where µ η ( n ) is constant. Then, L A satisfies property ( RL ) .Proof. Take ϕ ∈ L L A ( G ). Let S ′ be the unit sphere associated with the homogeneous norm | · | ′ : ω µ η ,ω ( n ). Now, observe that, arguing as in [25, Proposition 5.4], by means of the group Plancherelformula one may prove that, given any m ∈ L ∞ ( β L A ) such that K L A ( m ) ∈ L ( G ), we have π ∗ ( K L A ( m )) = m (d π ( L A )) for almost every (class of irreducible unitary representations) π in the dual of G . Then,comparing the irreducible representations of G and the G ω , we see that there is a negligible subset N of S ′ such that ( π ω ) ∗ ( ϕ ) ∈ L π ω ( L A ) ( G ω ) for every ω ∈ S ′ \ N . Observe, in addition, that the mapping ω ( π ω ) ∗ ( ϕ ) ∈ L ( g ⊕ R )is continuous on Ω, hence on S ′ . Now, fix ω ∈ S ′ , and take ( U, ψ ) as in Proposition 4.12. Then, it iseasily seen that the mapping U ∩ S ′ ∋ ω ( ψ ω ◦ π ω ) ∗ ( ϕ ) ∈ L ( G ′ )is continuous. Furthermore, observe that our assumptions imply that, with the notation of Proposi-tion 4.12, L ′ H := d( ψ ω ◦ π ω )( L H ) = − d X k =1 Y k − n X k =1 e µ η,k ( X k + X n + k ) ! η ∈ H does not depend on ω , while d( ψ ω ◦ π ω )( T ) = ω ( T ) T. Observe that ( L ′ H , − iT ) satisfies property ( RL ) by Theorem 5.2, and that ( ψ ω ◦ π ω ) ∗ ( ϕ ) ∈ L L ′ H , − iT ) ( G ′ )for every ω ∈ S ′ \ N , hence for every ω ∈ S ′ by continuity. Therefore, the mapping U ∩ S ′ ∋ ω
7→ M ( L ′ H , − iT ) (( ψ ω ◦ π ω ) ∗ ( ϕ )) ∈ C ( σ ( L ′ H , − iT ))is continuous. Now, take ω ∈ U ∩ S ′ \ N . Then, [24, Proposition 3.2.4], applied to the right quasi-regularrepresentation of G ′ in L ( G ), implies that M ( L ′ H , − iT ) (( ψ ω ◦ π ω ) ∗ ( ϕ ))( λ,
0) = M d π ( L H ) (( π ) ∗ ( ϕ ))( λ )for every λ ∈ R H such that ( λ, ∈ σ ( L ′ H , − iT ), that is, for every λ ∈ σ (d π ( L H )). By continuity,this proves that the mapping U ∩ S ′ ∋ ω
7→ M ( L ′ H , − iT ) (( ψ ω ◦ π ω ) ∗ ( ϕ ))( λ,
0) is constant for every λ ∈ σ (d π ( L H )). Taking into account the arbitrariness of U , we infer that there is a unique m ∈ C ( σ ( L A ))such that m ( λ, ω ( T )) = M ( L ′ H , − iT ) (( ψ U, ω / | ω |′ ◦ π ω / | ω |′ ) ∗ ( ϕ ))( λ, | ω | ′ )for every ( λ, ω ( T )) ∈ σ ( L A ) such that ω | ω | ′ ∈ U ∩ S ′ , where U runs through a finite covering of S ′ and ψ U is the associated local trivialization as above. Hence, ϕ = K L A ( m ) and the assertion follows.Here we prove a negative result. Proposition 5.4.
Assume that G is the product of k > M W + groups G , . . . , G k , and assume thateach G j is endowed with a homogeneous sub-Laplacian L ′ j . Assume that Card( H ) = 1 and that L = L ′ + · · · + L ′ k . Then, L A does not satisfy properties ( RL ) and ( S ) . It is easily proved that L L ′ H , − iT ) ( G ′ ) is closed in L ( G ′ ). roof. Take, for every j = 1 , . . . , k , a basis T j of the centre g j, of the Lie algebra of G j . Then, we mayassume that L A = ( L , − i T , . . . , − i T k ); define L ′ A ′ := ( L ′ , . . . , L ′ k , − i T , . . . , − i T k ). Then, there is aunique linear mapping L : E L ′ A ′ → E L A such that L A = L ( L ′ A ′ ). Now, take j ∈ { , . . . , k } and γ ∈ N h j ,and define C j,γ := { ( µ j,ω ( n ,j,ω + 2 γ ) , ω ( T j )) : ω ∈ Ω j } . Define C := [ γ ∈ Q kj =1 N hj k Y j =1 C j,γ j , and observe that β L ′ A ′ is equivalent to χ C · H n . Next, define N := R k × [ γ ∈ Q kj =1 Z hj γ =0 ω ( T ) : ω ∈ k Y j =1 Ω j , k X j =1 µ j,ω j ( γ j ) = 0 , and observe that L is one-to-one on C \ N . In addition, the µ j are analytic and homogeneous ofhomogeneous degree 1, and the components of the Ω j are unbounded; therefore, N is β L ′ A ′ -negligible.Hence, there is a unique m : E L A → E L ′ A ′ such that m ◦ L is the identity on C \ N , while m equals0 on the complement of L ( C \ N ). Then, m is β L A -measurable, and K L A ( m ) = L ′ A ′ δ e . Now, let usprove that m is not equal β L A -almost everywhere to a continuous function. Assume by contradictionthat L ′ A ′ δ e = K L A ( m ′ ) for some continuous function m ′ , and let π be the projection of G onto itsabelianization G ′ . Then, [24, Proposition 3.2.4], applied to the right quasi-regular representation of G in L ( G ′ ) implies that the operators d π ( L ′ ) , . . . , d π ( L ′ k ) belong to the functional calculus of π ( L ),which is absurd.To conclude, simply take τ ∈ S ( E L A ) such that τ ( λ ) = 0 for every λ ∈ E L A , and observe that K L A ( mτ ) = L ′ A ′ K L A ( τ ) is a family of elements of S ( G ), while mτ is not equal β L A -almost everywhereto any continuous functions. n ′ < n Before we state our main results, let us consider some technical lemmas.
Lemma 5.5.
Let M be a separable analytic manifold of dimension n endowed with a positive Radonmeasure µ which is equivalent to Lebesgue measure on every local chart. In addition, take k, h ∈ N anda analytic mapping P : M → R k with generic rank h such that π ∗ ( µ ) is a Radon measure. Then, thefollowing hold:1. P ( M ) is H h -measurable and countably H h -rectifiable;2. P ∗ ( µ ) is equivalent to χ P ( M ) · H h ;3. Supp ( P ∗ ( µ )) = P ( M ) ;4. if ( β y ) y ∈ R k is a disintegration of µ relative to P , then Supp ( β y ) = P − ( y ) for H h -almost every y ∈ P ( M ) . Notice that it is worthwhile for our analysis to consider the case in which M is possibly disconnected. Proof.
Observe first that M may be embedded as a closed submanifold of class C ∞ of R n +1 by Whitneyembedding theorem (cf. [15, Theorem 5 of Chapter 1]). We may therefore assume that µ = f · H n forsome f ∈ L ( χ M · H n ). Now, [28] implies that the set where P has rank < h , which is H n -negligible byanalyticity, has H h -negligible image under P . Since the image under P of the set where P has rank h isa countable union of analytic submanifolds of R k of dimension h , we see that P ( M ) is H h -measurableand countably H h -rectifiable. Therefore, we may make use of [16, Theorem 3.2.22], and infer that P ∗ ( µ )is equivalent to the restriction of H h to the set of y such that H n − h ( P − ( y )) >
0, and that we may find adisintegration ( β y ) of µ relative to P such that β y is equivalent to χ P − ( y ) · H n − h for P ∗ ( β )-almost every y ∈ R k . Now, the preceding arguments show that P − ( y ) is an analytic submanifold of dimension n − h of M for H h -almost every y ∈ P ( M ). As a consequence, Supp ( β y ) = Supp (cid:0) χ P − ( y ) · H n − h (cid:1) = P − ( y )for H h -almost every y ∈ P ( M ); for the same reason, we also see that P ∗ ( µ ) is equivalent to χ P ( M ) · H h .Finally, Supp ( P ∗ ( µ )) = P ( M ) since P is continuous and Supp ( µ ) = M .16 emma 5.6. Let E , E be two finite-dimensional vector spaces, C a convex subset of E with non-empty interior, and L : E → E a linear mapping which is proper on ∂C . Assume that for every x ∈ ∂C either L − ( L ( x )) ∩ ∂C = { x } or ∂C is an analytic hypersurface of E in a neighbourhood of x . Then, L induces an open mapping L ′ : ∂C → L ( ∂C ) .Proof. Take x ∈ ∂C , and assume that L − ( L ( x )) ∩ ∂C = { x } . Define U x,k := L − ( B ( L ( x ) , − k )) ∩ ∂C for every k ∈ N . Since L is proper on ∂C , U x,k is a compact neighbourhood of x for every k ∈ N . Inaddition, T k ∈ N U x,k = { x } ; hence, [8, Proposition 1 of Chapter 1, §
9, No. 3] implies that ( U x,k ) is afundamental system of neighbourhoods of x in ∂C , so that L ′ is open at x .Now, assume that L − ( L ( x )) ∩ ∂C = { x } . Then, there is a convex neighbourhood U of L − ( L ( x )) ∩ ∂C such that ∂C ∩ U is an analytic hypersurface of E . Assume by contradiction that ker L ⊆ T x ( ∂C ∩ U ),and take x ′ ∈ ∂C so that L ( x ′ ) = L ( x ) but x ′ = x . Since C is convex, we have [ x, x ′ ] ⊆ ∂C . Now, ∂C ∩ U is an analytic hypersurface and U is convex, so that the arbitrariness of x ′ implies that ℓ ∩ U ⊆ ∂C ,where ℓ is the line passing through x and x ′ . Since each x ′′ ∈ ℓ ∩ ∂C has a convex neighbourhood where ∂C is an analytic hypersurface, we see that the non-empty closed set ℓ ∩ ∂C is open in ℓ . It follows that ℓ ⊆ ∂C , which is absurd since then ℓ would be contained in the compact set L − ( L ( x )) ∩ ∂C . Therefore,ker L T x ( ∂C ∩ U ), so that L ′ is open at x . The assertion follows. Theorem 5.7.
Assume that
Card( H ) = 1 and that W = { } ; take a positive integer n ′ < n . Then,the family ( L , ( − iT j ) j =1 ,...,n ′ ) satisfies property ( RL ) .Proof. Consider the mapping L : E L A ∋ λ ( λ , ( λ ,j ) n ′ j =1 ) ∈ E ( L , ( − iT j ) j =1 ,...,n ′ ) ;observe that L ( L A ) = ( L , ( − iT j ) j =1 ,...,n ′ ). Define, in addition, L ′ in such a way that L = I R × L ′ , andidentify g ∗ with R n by means of the mapping ω ω ( T ). Define, for every γ ∈ N n , β γ : C c ( E L A ) ∋ ϕ Z g ∗ ϕ ( e µ ω ( n + 2 γ ) , ω ) | Pf( ω ) | d ω, so that β L A = π ) n n P γ ∈ N n β γ . Define ρ : R n ′ ∋ ω ′ min L ′ ( ω )= ω ′ µ ω ( n ) , and C := { ( ρ ( ω ′ ) + r, ω ′ ) : ω ′ ∈ R n ′ , r > } , so that ρ is a norm on R n ′ and C = L (Supp ( β )) = L ( σ ( L A )). Now, Corollary 4.9 implies that Supp ( β ) \ { } is an analytic submanifold of E L A . In addition,Corollary 5.5 implies that L ∗ ( β ) is equivalent to χ σ ( L , ( − iT j ) j =1 ,...,n ′ ) · H n ′ +1 and that, if ( β ,λ ) is adisintegration of β relative to L , then Supp ( β ,λ ) = L − ( λ ) ∩ Supp ( β ) for L ∗ ( β )-almost every λ ∈ C .In addition, Lemma 5.12 implies that the mapping L : Supp ( β ) → C is open. If we prove that L ∗ ( β γ )is absolutely continuous with respect to H n ′ +1 for every γ ∈ N n , the assertion will then follow fromProposition 2.6.Then, let us prove that L ∗ ( β γ ) is absolutely continuous with respect to H n ′ +1 for every γ ∈ N n .Notice that this will be the case if we prove that the analytic mapping Ω ∋ ω ( e µ ω ( n + 2 γ ) , L ′ ( ω ))is generically a submersion for every γ ∈ N n (cf. Lemma 5.5). Assume by contradiction that this is notthe case, so that there are γ ∈ N n and a component C of Ω such that e µ ′ ω ( n + 2 γ ) vanishes on ker L ′ for every ω ∈ C . As a consequence, there are ( r, ω ′ ) ∈ R × R n ′ such that L − ( r, ω ′ ) ∩ σ ( L A ) contains anopen segment. Then, there is a line ℓ in L ′− ( ω ′ ) such that ( { r } × ℓ ) ∩ σ ( L A ) contains an open segment;observe that 0 ℓ since the mapping ω e µ ω ( n + 2 γ ) is homogeneous and proper. Then, [23, Theorem6.1 of Chapter II] implies that there is an analytic function f : ℓ → R n such that f ( ω ) is a reordering of e µ ω for every ω ∈ ℓ . As a consequence, for every γ ′ ∈ N n the set of ω ∈ ℓ such that f ( ω )( n + 2 γ ′ ) = r is compact, hence discrete by analyticity. It is then easily seen that ( { r } × ℓ ) ∩ σ ( L A ) is countable, sothat it cannot contain any open segments: contradiction. The proof is therefore complete. Theorem 5.8.
Define C γ := { ( e µ ω ( n + 2 γ ) , ω ( T )) : ω ∈ g ∗ } for every γ ∈ N n . In addition, take n ′ < n and define L := id R × pr ,...,n ′ on E L A . Assume that the following hold: . Card( H ) = 1 ;2. χ C · β L A is L -connected3. for every f ∈ L L A ( G ) and for every γ ∈ N n , M L A ( f ) equals β L A -almost everywhere a continuousfunction on C γ .Then, L ( L A ) satisfies property ( RL ) . Observe that condition holds if C is the boundary of a polyhedron (cf. Corollary 2.4 below) andif Ω = g ∗ \ { } (cf. Lemma 5.12). With a little effort, one may prove that condition holds if n ′ = 1.We shall prepare the proof of Theorem 5.8 through several lemmas. Lemma 5.9.
Let V be a topological vector space, C a convex subset of V with non-empty interior, and W an affine subspace of V such that W ∩ ◦ C = ∅ . Then, W ∩ ∂C is the frontier of W ∩ C in W .Proof. Indeed, take x ∈ W ∩ ◦ C , and take x in the interior of W ∩ C in W . Then, there is y ∈ W ∩ C such that x ∈ [ x , y [, so that [10, Proposition 16 of Chapter II, §
2, No. 7] implies that x ∈ ◦ C . By thearbitrariness of x , this proves that W ∩ ◦ C is the interior of W ∩ C in W . Analogously, one proves that W ∩ C is the closure of W ∩ C in W , whence the result. Lemma 5.10.
Let f : R n → R be a convex function which is differentiable on an open subset U of R n .Let L be a linear mapping of R n onto R k for some k n , and assume that ( f, L ) has rank k on U .Then, for every y ∈ ( f, L )( U ) , the fibre ( f, L ) − ( y ) is a closed convex set which contains L − ( y ) ∩ U .Proof. Define π := ( f, L ) : R n → R × R k , and observe that ker L ⊆ ker f ′ ( x ) for every x ∈ U since π has rank k on U . Therefore, if y ∈ π ( U ), then f is locally constant on L − ( y ) ∩ U . Now, take twocomponents C and C of L − ( y ) ∩ U , and observe that they are open in L − ( y ). Take x ∈ C and x ∈ C . Then [ x , x ] ⊆ L − ( y ), so that there are x ′ , x ′ ∈ ] x , x [ such that f is constant on [ x , x ′ ]and on [ x ′ , x ]. By convexity, f must be constant on [ x , x ], hence on C ∪ C . By the arbitrariness of C and C , we infer that π − ( y ) ⊇ L − ( y ) ∩ U .Now, consider the closed convex set C := { ( λ, x ) : x ∈ R n , λ > f ( x ) } , and observe that ◦ C = { ( λ, x ) : x ∈ R n , λ > f ( x ) } since f is continuous, so that ∂C is the graph of f . Next, define W := (id R × L ) − ( y ) = { y } × L − ( y ),and observe that W ∩ ∂C = { y } × π − ( y ). Assume by contradiction that W ∩ ◦ C = ∅ . Then, Lemma 5.9implies that W ∩ ∂C is the frontier of W ∩ C in W , so that π − ( y ) has empty interior in L − ( y ). However, π − ( y ) contains L − ( y ) ∩ U , which is open in L − ( y ): contradiction. Therefore, { y }× π − ( y ) = W ∩ C is a closed convex set, whence the result. Lemma 5.11.
Let f : R n → R be a convex function which is analytic on some open subset Ω of R n whose complement is H n -negligible. Let L be a linear mapping of R n onto R k for some k n , and let U be the union of the components of Ω where ( f, L ) has rank k . Then, ( f, L ) − ( y ) = L − ( y ) ∩ U for H k -almost every y ∈ ( f, L )( U ) .Proof. Define π := ( f, L ). Since the complement of Ω is H n -negligible, there is an H k -negligible subset N of R k such that L − ( y ) \ Ω is H n − k -negligible for every y ∈ R k \ N (cf. [16, Theorem 3.2.11]). Inaddition, observe that the set R k of x ∈ Ω \ U such that ker L ⊆ ker f ′ ( x ), that is, such that π ′ ( x ) hasrank k , is H n -negligible by the analyticity of f . Then, there is an H k -negligible subset N of R k suchthat L − ( y ) ∩ Ω \ U is H n − k -negligible for every y ∈ R k \ N ( loc. cit. ). Now, define N := R × ( N ∪ N ),and observe that χ π ( U ) · H k is equivalent to the (not necessarily Radon) measure π ∗ ( χ U · H n ) thanksto Corollary 5.5; since U \ π − ( N ) = U \ L − ( N ∪ N ) is H n -negligible, it follows that π ( U ) ∩ N is H k -negligible.Now, take y ∈ π ( U ) \ N . Then, Lemma 5.10 implies that π − ( y ) is a closed convex set which contains L − ( y ) ∩ U , so that its interior in L − ( y ) is not empty. Let U ′ be a component of Ω which is notcontained in U , and assume that π − ( y ) ∩ U ′ = ∅ . Since f is analytic on U ′ , and since π − ( y ) is a convex18et with non-empty interior in L − ( y ), we see that a component C of L − ( y ) ∩ U ′ is contained in R k .By the choice of N , this implies that C is H n − k -negligible; since C is non-empty and open in L − ( y ),this leads to a contradiction. Therefore, L − ( y ) ∩ U ⊆ π − ( y ) ⊆ L − ( y ) ∩ [ U ∪ ( R n \ Ω)] . By our choice of N , the set L − ( y ) \ Ω is H n − k -negligible; on the other hand, the support of χ π − ( y ) ·H n − k is π − ( y ) by convexity. Hence, L − ( y ) ∩ U is dense in π − ( y ), whence the result. Lemma 5.12.
Keep hypotheses and notation of Lemma 5.11. Assume, in addition, that lim x →∞ f ( x ) = + ∞ and that H n is ( f, L ) -connected. Then, for every m ∈ C ( R n ) such that m = m ′ ◦ L H n -almost everywherefor some m ′ : R × R k → C , there is m ′′ ∈ C ( R × R k ) such that m = m ′′ ◦ ( f, L ) pointwise.Proof. Define π := ( f, L ). Let ( β ,y ) y ∈ R × R k be a disintegration of χ U · H n relative to π and let( β ,y ) y ∈ R × R k be a disintegration of χ Ω \ U · H n relative to π . Then, Corollary 5.5 implies that: • π ∗ ( χ U · H n ) is equivalent to χ π ( U ) · H k ; • π ∗ ( χ Ω \ U · H n ) is equivalent to χ π (Ω \ U ) · H k +1 ; • Supp ( β ,y ) = π − ( y ) ∩ U for H k -almost every y ∈ π ( U ); • Supp ( β ,y ) = π − ( y ) ∩ Ω \ U for H k +1 -almost every y ∈ π (Ω \ U ).In addition, π ( U ) has Hausdorff dimension k , so that H k +1 ( π ( U )) = 0; in particular, π ∗ ( χ U · H n ) and π ∗ ( χ Ω \ U · H n ) are alien measures. If we define β y := β ,y for every y ∈ π ( U ) and β y := β ,y for every y ∈ ( R × R k ) \ π ( U ), then ( β y ) is a disintegration of H n relative to π .Now, Lemma 5.11 implies that π − ( y ) ∩ U = π − ( y ) for H k -almost every y ∈ π ( U ). Next, let usprove that π − ( y ) = π − ( y ) ∩ Ω \ U for H k +1 -almost every y π ( U ). Let us first prove that π − ( y )is the boundary of a compact convex set with non-empty interior in L − ( y ) for H k +1 -almost every y ∈ π (Ω \ U ).Indeed, by [28] there is an H k +1 -negligible subset N of π (Ω \ U ) such that π ′ ( x ) has rank k + 1 forevery x ∈ L − ( y ) ∩ Ω \ U and for every y ∈ π (Ω \ U ) \ N . Now, define C := { ( λ, x ) : x ∈ R n , λ > f ( x ) } , andobserve that id R × L is proper on C since lim x →∞ f ( x ) = + ∞ . Therefore, { y }× π − ( y ) = (id R × L ) − ( y ) ∩ ∂C is compact for every y ∈ R × R k . In addition, if y ∈ π (Ω \ U ) \ N , then (id R × L ) − ( y ) ∩ ◦ C = ∅ , sothat Lemma 5.9 implies that π − ( y ) is the boundary of a compact convex set with non-empty interiorin L − ( y ).Therefore, π − ( y ) is bi-Lipschitz homeomorphic to S n − k − , so that the support of χ π − ( y ) · H n − k − is π − ( y ) for such y . In addition, since R n \ Ω is H n -negligible, [16, Theorem 3.2.11] implies that π − ( y ) \ Ω is H n − k − -negligible for H k +1 -almost every y ∈ R × R k . Hence, π − ( y ) ∩ Ω \ U = π − ( y ) for H k +1 -almost every y π ( U ).Then, Proposition 2.6 implies that there is m ′′′ : π ( R n ) → C such that m = m ′′′ ◦ π ; since π is proper,this implies that m ′′′ is continuous on π ( R n ). Finally, since π is proper, π ( R n ) is closed, so that theassertion follows from [9, Corollary to Theorem 2 of Chapter IX, §
4, No. 2].
Proof of Theorem 5.8.
Until the end of this proof, we shall identify R n and g ∗ by means of the bijection ω ω ( T ); L ′ will denote pr ,...,n ′ , so that L = id R × L ′ . In addition, for every γ ∈ N n , define π γ : R n ∋ ω ( µ ω ( n + 2 γ ) , ω ), so that π γ is continuous and C γ is the graph of π γ .Take f ∈ L L ( L A ) ( G ) and let m be a representative of M L ( L A ) ( f ). Take, for every γ ∈ N n , acontinuous function m γ on C γ such that m γ = M L A ( f ) χ C γ · β L A -almost everywhere. Then, Lemma 5.12implies that there is a continuous function m ′ : E L ( L A ) → C such that m = m ′ ◦ L on C . Since β L ( L A ) need not be equivalent to L ∗ ( χ C · β L A ), though, this is not sufficient to conclude.For every γ ∈ N n , define β γ := χ C γ · β L A , and let U γ, be the union of the components C of Ω suchthat e µ ′ ω ( n + 2 γ ) does not vanish on ker L ′ for some ω ∈ C . Let U γ, be the complement of U γ, in Ω.Notice that β γ is equivalent to ( π γ ) ∗ ( H n ). In addition, Corollary 5.5 implies that the following hold: • L ∗ ( χ R × U γ, · β γ ) is equivalent to χ L ( π γ ( U γ, )) · H n ′ +1 ; • L ∗ ( χ R × U γ, · β γ ) is equivalent to χ L ( π γ ( U γ, )) · H n ′ ;19 χ R × U γ, · β γ has a disintegration ( β γ, ,λ ) λ ∈ E L ( L A ) relative to L such that L − ( λ ) ∩ π γ ( U γ, ) ⊆ Supp ( β γ, ,λ ) and β γ, ,λ is equivalent to the measure χ L − ( λ ) ∩ π γ ( U γ, ) · H n − n ′ for H n ′ -almostevery λ ∈ L ( π γ ( U γ, )).In particular, β L ( L A ) is equivalent to χ σ ( L ( L A )) · H n ′ +1 + µ , where µ is a measure absolutely continuouswith respect to H n ′ alien to H n ′ +1 . Now, observe that L ∗ ( χ R × U γ, · β γ ) is absolutely continuous withrespect to L ∗ ( β ); since ( m − m ′ ) ◦ L is β -negligible, there is an L ∗ ( β )-negligible subset N of E L ( L A ) suchthat m = m ′ on E L ( L A ) \ N . Since N is L ∗ ( χ R × U γ, · β γ )-negligible, this implies that ( m − m ′ ) ◦ L vanishes χ R × U γ, · β γ -almost everywhere. Since m ◦ L = m γ β γ -almost everywhere, it follows that m ′ ◦ L = m γ χ R × U γ, · β γ -almost everywhere, hence onSupp (cid:0) χ R × U γ, · β γ (cid:1) = Supp (cid:0) ( π γ ) ∗ ( χ U γ, · H n ) (cid:1) = π γ ( U γ, ) , since m ′ ◦ L and m γ are continuous, while π γ is proper.Next, consider χ R × U γ, · β γ . Tonelli’s theorem implies that L ′− ( λ ) \ Ω is H n − n ′ -negligible for H n ′ -almost every λ ∈ R n ′ . Now, if e N is an H n ′ -negligible subset of R × R n ′ , then pr ( e N ) is H n ′ -negligible since pr is Lipschitz. Therefore, there is an H n ′ -negligible subset N ′ of R n ′ such that, forevery λ ∈ L ( π γ ( U γ, )) \ ( R × N ′ ), • m ◦ L = m γ β γ, ,λ -almost everywhere; • L − ( λ ) ∩ π γ ( U γ, ) ⊆ Supp ( β γ, ,λ ); • L ′− ( λ ) \ Ω is H n − n ′ -negligible.Hence, if λ ∈ L ( π γ ( U γ, )) \ ( R × N ′ ), then m γ is constant on L − ( λ ) ∩ π γ ( U γ, ). In addition, fix λ ∈ L ( π γ ( U γ, )) \ ( R × N ′ ); then, L ′− ( λ ) = L ′− ( λ ) ∩ U γ, ∪ L ′− ( λ ) ∩ U γ, , so that L ′− ( λ ) ∩ U γ, ∩ L ′− ( λ ) ∩ U γ, = ∅ or L ′− ( λ ) ∩ U γ, = ∅ by connectedness.Now, let C be the set of components of L ′− ( λ ) ∩ U γ, ; observe that C is finite since L ′− ( λ ) ∩ Ω issemi-algebraic (cf. [13, Proposition 4.13]) and since L ′− ( λ ) ∩ U γ, is open and closed in L ′− ( λ ) ∩ Ω.In addition, observe that pr ◦ π γ is constant on each C ∈ C ; let λ ,C be its constant value. In particular,since pr ◦ π γ is proper and since C is finite, this implies that L ′− ( λ ) ∩ U γ, = ∅ . Further, m γ isconstant on π γ ( C ) ⊆ L − ( λ ,C , λ ) ∩ π γ ( U γ, ) for every C ∈ C . Now, there is C ∈ C such that L ′− ( λ ) ∩ U γ, ∩ C = ∅ ; since m γ ◦ π γ = m ′ ◦ L ◦ π γ on U γ, , and since m γ is continuous, it follows that m γ ◦ π γ = m ′ ◦ L ◦ π γ on C . Iterating this procedure, we eventually see that m γ ◦ π γ = m ′ ◦ L ◦ π γ on L ′− ( λ ). Therefore, m γ = m ′ ◦ L on L − ( λ ) ∩ C γ for every λ ∈ L ( π γ ( U γ, )) \ ( R × N ′ ).Now, observe that L − ( R × N ′ ) ∩ π γ ( U γ, ) is H n ′ -negligible since pr ◦ L ◦ π γ = L ′ and since H n ′′ isequivalent to the non-Radon measure L ′∗ ( H n ). Therefore, m γ = m ′ ◦ L β γ -almost everywhere, henceon C γ by continuity. By the arbitrariness of γ , this implies that m ′ ◦ L is a representative of M L A ( f ),so that m ′ is a continuous representative of M L ( L A ) ( f ). The assertion follows. ( S ) The results of this section are basically a generalization of the techniques employed in [3, 4]. The firstresult has very restrictive hypotheses, for the same reasons explained while discussing property ( RL ), buthold for the ‘full family’ L A (cf. Theorem 6.2); on the contrary, the second one holds under more generalassumptions, but only for families of the form ( L , ( − iT , . . . , − iT n ′ )) for n ′ < n (cf. Theorem 6.5).Notice that, even though Theorem 7.4 is the main application of Theorem 6.2, there are other familiesto which it applies as well. This happens for the family we considered while discussing property ( RL ) inthe case of Theorem 5.3.Notice that in all of our results we imposed the condition W = { } ; this is unavoidable (with ourmethods), since on W we cannot infer any kind of regularity from the ‘inversion formulae’ employed.Indeed, our auxiliary function | x ω | is not differentiable on W , in general. Nevertheless, this does notmean that property ( S ) cannot hold when W = { } , as Theorem 8.3 shows.Before stating our first result, let us recall a lemma based on some techniques developed in [18] andthen in [3]. 20 emma 6.1 ([12], Lemma 11.1) . Let L A be a Rockland family on a homogeneous group G ′ , and let T ′ , . . . , T ′ n be a free family of elements of the centre of the Lie algebra g ′ of G ′ . Let π be the canonicalprojection of G ′ onto its quotient by the normal subgroup exp( R T ′ ) , and assume that the following hold: • ( L A , iT ′ , . . . , iT ′ n ) satisfies property ( RL ) ; • d π ( L A , iT ′ , . . . , iT ′ n ) satisfies property ( S ) .Take ϕ ∈ S ( L A ,iT ′ ,...,iT ′ n ) ( G ′ ) . Then, there are two families ( e ϕ γ ) γ ∈ N n and ( ϕ γ ) γ ∈ N n of elements of S ( G ′ , L A ) and S ( L A ,iT ′ ,...,iT ′ n ) ( G ′ ) , respectively, such that ϕ = X | γ | Assume that Ω = g ∗ \ { } , that dim Q µ ω ( Q h ) = dim R µ ω ( R h ) for every ω ∈ Ω , and that µ is constant where µ η ( n ) is constant. Then, L A satisfies property ( S ) .Proof. We proceed by induction on n > Notice that the inductive hypothesis, Theorem 3.2, Theorem 5.3, and Lemma 6.1 imply that wemay find a family ( e ϕ γ ) of elements of S ( G, L H ), and a family ( ϕ γ ) of elements of S L A ( G ) such that ϕ = X | γ | Let U an open subset of R k × R n , and ϕ a mapping of class C ∞ of U into R . Assumethat ∂ ϕ ( x ) = 0 and that ∂ ϕ ( x ) is positive and non-degenerate for some x ∈ U .Then, there are an open neighbourhood V of in R k , an open neighbourhood V of x , in R n , anda C ∞ -diffeomorphism ψ from V × V onto an open subset of U such that ψ (0 , x , ) = x , ψ = pr , and ϕ ( ψ ( y )) = ϕ ( ψ (0 , y )) + k y k for every y ∈ V × V . orollary 6.4. Keep the hypotheses and the notation of Lemma 6.3. Take a function f ∈ C ∞ ( ψ ( V × V ) × R ) and a function g : V × R → C so that f ( x, ϕ ( x )) = g ( x , ϕ ( x )) for every x ∈ ψ ( V × V ) . Then, g can be modified so as to be of class C ∞ in a neighbourhood of ( x , , ϕ ( x )) .Proof. Indeed, the assumption means that f (cid:16) y, ϕ ( ψ (0 , y )) + k y k (cid:17) = g (cid:16) y , ϕ ( ψ (0 , y )) + k y k (cid:17) for every y ∈ V × V . Define, for every y ∈ V , e f y : V ∋ y f (( y , y ) , ϕ (cid:16) ψ (0 , y )) + k y k (cid:17) and e g y : R ∋ t g ( y , ϕ ( ψ (0 , y )) + t ) . Then, the mapping V ∋ y e f y belongs to E ( V ; E ( V )), and e f y ( y ) = e g y (cid:16) k y k (cid:17) for every y ∈ V and for every y ∈ V .Now, [30] easily implies that the mapping Φ : E R ( R + ) ∋ h h ◦ k · k ∈ E ( R k )is an isomorphism onto the set of radial functions of class C ∞ . Since there is a continuous linearextension operator E R ( R + ) → E ( R ) (cf., for instance, [5, Corollary 0.3]), we find a continuous linearmapping Φ : Φ ( E R ( R + )) → E ( R ) such thatΦ ( h ) ◦ k · k = h for every even function h ∈ E ( R ). Then, take τ ∈ D ( V ) so that τ equals 1 on a neighbourhood V ′ of 0in V , and define e G y : V ∋ t Φ ( τ e f y )( t ) . Then, e G y (cid:16) k y k (cid:17) = e g y (cid:16) k y k (cid:17) for every y ∈ V ′ and for every y ∈ V . In addition, the mapping y e G y belongs to E ( V ; E ( R )), so that there is e G ∈ E ( V × R ) such that e G ( y , t ) = e G y ( t ) for every y ∈ V and for every t ∈ R . Then, g (cid:16) y , ϕ ( ψ (0 , y )) + k y k (cid:17) = e G (cid:16) y , k y k (cid:17) for every y ∈ V and for every y ∈ V ′ . Define G : V × R ∋ ( y , t ) e G ( y , t − ϕ ( ψ (0 , y ))) , so that G ∈ E ( V × R ) and f ( x, ϕ ( x )) = G ( x , ϕ ( x ))for every x ∈ ψ ( V ′ × V ), whence the result. Theorem 6.5. Assume that Card( H ) = 1 and that W = { } , and let S be the analytic hypersurface { ω ∈ g ∗ : µ ω ( n ,ω ) = 1 } . Assume that, for every ω ∈ S such that (cid:10) T , . . . , T n ′ (cid:11) ◦ ⊆ T ω ( S ) , the Gaussiancurvature of S at ω is non-zero. Take n ′ ∈ { , . . . , n − } and define L A ′ := ( L , ( − iT , . . . , − iT n ′ )) .Then, L A ′ satisfies property ( S ) . The condition on S is satisfied, for example, if ω µ ω ( n ,ω ) is a hilbertian norm. Observe, in addi-tion, that the Gaussian curvature of S vanishes on a negligible set in virtue of the strict convexity of thenorm ω µ ω ( n ,ω ). Therefore, for almost every ( T ′ , . . . , T ′ n ′ ) ∈ g n ′ the family ( L , ( − iT ′ , . . . , − iT ′ n ′ ))satisfies property ( S ). We denote by E R ( R + ) the quotient of E ( R ) by the set of ϕ ∈ E ( R ) which vanish on R + . roof. We proceed by induction on n ′ . Observe first that the assertion follows from Theorem 3.2when n ′ = 0. Then, assume that n ′ > 0. By the inductive assumption, it is easily seen that d π ( L A ′ )satisfies property ( S ), where π is the canonical projection of G onto G / exp G ( R T ).Take ϕ ∈ S L A ′ ( G ). Then, Theorem 5.7 and Lemma 6.1 imply that we may find a family ( e ϕ γ ) γ ∈ N n ′ of elements of S ( G, L ), and a family ( ϕ γ ) γ ∈ N n ′ of elements of S L A ′ ( G ) such that ϕ = X | γ | 0. Now, define f M ( ω ) := Z G ϕ ( x, t ) e − | x ω | + iω ( t ) d( x, t )for every ω ∈ g ∗ . Arguing as in the proof of Theorem 6.2 and taking into account Proposition 4.10, wesee that f M ∈ S ( g ∗ ) and that f M vanishes of order ∞ at 0. Now, observe that m ( µ ω ( n ,ω ) , L ′ ( ω ( T ))) = f M ( ω )for every ω ∈ g ∗ . In addition, Σ := R + ( { } × S ) is a closed semianalytic subset of E L A since it is theclosure of the graph of an analytic function (defined on g ∗ \ { } ); in addition, L is proper on Σ and L (Σ) = σ ( L A ′ ) is a subanalytic closed convex cone, hence Nash subanalytic. By Theorem 2.7, in orderto prove that m ∈ S E L A ′ ( σ ( L A ′ )) it suffices to show that f M is a formal composite of L ′ . Now, theassertion is clear at 0 since f M vanishes of order ∞ at 0. Then, take ω ∈ S . If ker L T ω ( T ) ( S ( T )), then L ′ is a submersion at ω , so that the assertion follows in this case. Otherwise, as in the proof of Lemma 5.6we see that L ′− ( L ′ ( ω )) = { ω } , so that the assertion follows from Corollary 6.4. By homogeneity, theassertion follows for every ω = 0. Therefore, m ∈ S E L A ′ ( σ ( L A ′ )), whence the result. H -Type Groups In this section we shall deal with the following situation: G is an H -type group and there is a finitefamily ( v η ) η ∈ H of subspaces of g such that v η ⊕ g , with the induced structure, is an H -type Lie algebrafor every η ∈ H and such that v η and v η commute and are orthogonal for every η , η ∈ H such that η = η . We shall define n := (cid:0) dim v η (cid:1) η ∈ H .We shall then consider, for every η ∈ H , the group of linear isometries O ( v η ) of v η , and define acanonical action of O := Q η ∈ H O ( v η ) on the vector space subjacent to g as follows: ( L η )(( v η ) , t ) :=(( L η · v η ) , t ) for every ( L η ) ∈ O and for every (( v η ) , t ) ∈ g ⊕ g .A projector of D ′ ( G ) is then canonically defined as follows: π ∗ ( T ) := Z O ( L · ) ∗ ( T ) d ν O ( L )for every T ∈ D ′ ( G ); here, ν O denotes the normalized Haar measure on O . Proposition 7.1. The following hold: . π induces a continuous projection on D ′ r ( G ) , S ′ ( G ) , E ′ r ( G ) , E r ( G ) , S ( G ) , D r ( G ) and L p ( G ) forevery r ∈ N ∪ {∞} and for every p ∈ [1 , ∞ ] ;2. if ϕ , ϕ ∈ D ( G ) , then h π ∗ ( ϕ ) , ϕ i = h ϕ , π ∗ ( ϕ ) i and h π ∗ ( ϕ ) | ϕ i = h ϕ | π ∗ ( ϕ ) i ; 3. if µ is a positive measure on G , then also π ∗ ( µ ) is a positive measure; in addition, π ∗ ( ν G ) = ν G ;4. if T ∈ D ′ ( G ) is O -invariant, then also ˇ T is O -invariant;5. if T is supported at e , then π ∗ ( T ) is supported at e ;6. if ϕ , ϕ ∈ D ( G ) are O -invariant, then also ϕ ∗ ϕ is O -invariant and ϕ ∗ ϕ = ϕ ∗ ϕ . The proof is based on [14] and is omitted.Now, let L η be the differential operator corresponding to the restriction of the scalar product to v ∗ η ;in other words, L η is minus the sum of the squares of the elements of any orthonormal basis of v η . Let T , . . . , T n be an orthonormal basis of g , and define L A := (( L η ) η ∈ H , ( − iT , . . . , − iT n )).Recall that a left-invariant differential operator X is π -radial if and only if π ∗ ( X e ) = X e , that is, ifand only if X e is O -invariant. Nevertheless, this does not imply that X is O -invariant. Proposition 7.2. L A is a Rockland family and generates (algebraically) the unital algebra of left-invariant differential operators which are π -radial.Proof. Since P η ∈ H L η is the operator associated with the scalar product of g ∗ , it is clear that L A is aRockland family.Now, take an O -invariant distribution S on G which is supported at e . Let p : G → G / [ G, G ] bethe canonical projection. Then, p ( S ) is O -invariant and supported at p ( e ). By means of the Fouriertransform, we see that there is a unique polynomial P ∈ R [ H ] such that p ( S ) = P ( p ( L H,e )). Therefore,there are S , . . . , S n ∈ D ′ ( G ) such that Supp ( S k ) ⊆ { e } for every k = 1 , . . . , n , and such that S = P ( L H,e ) + n X k =1 T k,e S k . Reasoning by induction, it follows that S belongs to the unital algebra (algebraically) generated by L A,e .Conversely, it is clear that T , . . . , T n are π -radial. On the other hand, a direct computation shows that L η,e = − P v ∈ B ∂ v , where B is any orthonormal basis of v η . Hence, L η,e is O -invariant.Now, we shall consider some image families of L A . More precisely, we shall fix µ ∈ ( R H ) H ′ so thatthe induced mapping from R H into R H ′ is proper on R H + . Then, we shall define L : E L A ∋ ( λ , λ ) ( µ ( λ ) , λ ) ∈ R H ′ × g ∗ and consider the family L ( L A ). Then, L ( L A ) is a Rockland family since L isproper on σ ( L A ) by construction. Proposition 7.3. Set d := dim Q µ ( Q H ) . Then, there are a β L ( L A ) -measurable function m : E L ( L A ) → C d and a linear mapping L ′ : R d → C H ′ such that the following hold: • there is µ ′ ∈ (( Q ∗ + ) H ) d such that m ( L A ) = µ ′ ( L H ) ; • ( L ′ ( m ( L A )) , ( − iT j ) n j =1 ) = L ( L A ) ; • m equals β L ( L A ) -almost everywhere a continuous function if and only if d = dim R µ ( R H ) .Proof. Indeed, we may find d linearly independent Q -linear functionals p , . . . , p d on µ ( Q H ). Let µ ′ , . . . , µ ′ d be the elements of Q H associated with p ◦ µ, . . . , p d ◦ µ . Then, µ ′ , . . . , µ ′ d are linearly in-dependent over Q , hence over C by tensorization. Now, define L ′′ h := P η ∈ H µ ′ h,η L η , so that the family( L ′′ , . . . , L ′′ d ) is linearly independent over C . Next, take h ∈ { , . . . , d } , and observe that, if λ ∈ R n \ { } and γ , γ ∈ N H are such that ( | λ | µ ( n + 2 γ ) , λ ) = ( | λ | µ ( n + 2 γ ) , λ ) , then µ ( γ − γ ) = 0, so that ( | λ | µ ′ ( n + 2 γ ) , λ ) = ( | λ | µ ′ ( n + 2 γ ) , λ ) . β L A -measurable function m : E L ( L A ) → C d such that m h ( L ( λ ′ )) = µ ′ h ( λ ′ )for every λ ′ ∈ σ ( L A ) ∩ ( R H × ( R n \ { } )); hence, L ′′ h δ e = K L A ( m h ) for every h = 1 , . . . , d . Next, observethat, for every η ′ ∈ H ′ there is ( L ′ η ′ , , . . . , L ′ η ′ ,d ) ∈ Q d such that d X h =1 L ′ η ′ ,h ( p h ◦ µ ) = µ η ′ on Q H . Therefore, P kh =1 L ′ η ′ ,h µ ′ h = µ η ′ , whence ( L ′ ( m ( L A )) , ( − iT j ) n j =1 ) = L ( L A ).If d = dim R µ ( R H ), then m × id R n is a homeomorphism of σ ( L ( L A )) onto σ ( L ′′ , . . . , L ′′ d , ( − iT h ) n h =1 ).Conversely, assume that m can be taken so as to be continuous. Then, m × id R n and L ′ × id R n areinverse of one another between σ ( L ( L A )) and σ ( L ′′ , . . . , L ′′ d , ( − iT h ) n h =1 ). In particular, L ′ induces ahomeomorphism of µ ′ ( R H + ) onto µ ( R H + ), so that these two cones must have the same dimension. Hence, d = dim R ( µ ( R H )). Theorem 7.4. The following conditions are equivalent:(i) χ L ( L A ) has a continuous representative;(ii) L ( L A ) satisfies property ( RL ) ;(iii) every element of S L ( L A ) ( G ) has a continuous multiplier;(iv) L ( L A ) satisfies property ( S ) ;(v) L ( L A ) is functionally complete;(vi) dim Q µ ( Q H ) = dim R µ ( R H ) .If, in addition, L ( L A ) is not functionally complete, then there is some L ′ , corresponding to some µ ′ ∈ ( R H ) H ′′ , such that L ′ ( L A ) is functionally complete and functionally equivalent to L ( L A ) .Proof. (i) = ⇒ (ii). Obvious. (ii) = ⇒ (iii). Obvious. (iii) = ⇒ (vi). Assume, on the contrary, that dim Q µ ( Q H ) > dim R µ ( R H ), and keep the notationof Proposition 7.3. Then, m h cannot be taken so as to be continuous for some h ∈ { , . . . , d } . Take ϕ ∈ S ( E L ( L A ) ) so that ϕ ( λ ) = 0 for every λ ∈ E L A . Then, K L ( L A ) ( m h ϕ ) = ( µ ′ h ( L H )) K L ( L A ) ( ϕ ) ∈ S ( G ) , but m h ϕ is not equal β L ( L A ) -almost everywhere to any continuous functions, whence the result. (vi) = ⇒ (iv). This follows from Theorem 6.2. (iv) = ⇒ (v). This follows from Proposition 2.10. (v) = ⇒ (vi). This follows from Proposition 7.3. (vi) = ⇒ (i). This follows from Theorem 5.2. In this section, ( G α ) α ∈ A will be a family of Heisenberg groups each of which is endowed with a homoge-neous sub-Laplacian L α . Define L := P α ∈ A L α , and denote by T a finite family of elements of g , whichis the centre of the Lie algebra of G := Q α ∈ A G α .Before we proceed to the main results of these section, let us introduce some more notation. Forevery α ∈ A , we shall denote by T α a basis of the centre of the Lie algebra of G α , so that we may identifycanonically g with L α ∈ A R T α . Then, there is a basis ( X α, , . . . , X α, n ,α , T α ) of the Lie algebra of G α such that [ X α,k , X α,n ,α + k ] = T α for every k = 1 , . . . , n ,α , while the other commutators vanish, andsuch that there is µ α ∈ ( R ∗ + ) n ,α such that L α = − n ,α X k =1 µ α,k ( X α,k + X α,n ,α + k ) . We shall denote by g ,α the vector space generated by X α, , . . . , X α, n ,α , and we shall set n :=( n ,α ) α ∈ A . 25 roposition 8.1. Assume that Card( A ) > . If T generates g , then the families ( L , − i T ) and ( L A , − i T ) are functionally equivalent. In addition, ( L , − i T ) does not satisfy properties ( RL ) and ( S ) .Proof. See Theorem 5.2 and its proof. Lemma 8.2. Let µ be a linear mapping of R n onto R m such that ker µ ∩ R n + = { } . Define Σ := µ ( R n + ) × { } , and Σ := { ( λµ ( n + 2 γ ) , λ ) : λ > , γ ∈ N n } ∪ Σ . If ϕ ∈ C ∞ ( R m × R ) vanishes on Σ , then ϕ vanishes of order ∞ on Σ .Proof. Take x = ( λµ ( n + 2 γ ) , 0) for some λ > γ ∈ N n . Then, for every k ∈ N , (cid:18) x , λ k + 1 (cid:19) = (cid:18) λ k + 1 µ ( n + 2((2 k + 1) γ + k n )) , λ k + 1 (cid:19) ∈ Σ . Therefore, it is easily seen that ∂ h ϕ ( x ) = 0 for every h ∈ N . Since the set { ( λµ ( n + 2 γ ) , 0) : λ > , γ ∈ N n } is dense in Σ , it follows that ∂ h ϕ vanishes on Σ for every h ∈ N . Then, observe that, since we assumedthat µ ( R n ) = R m , the closed convex cone Σ generates R m × { } , so that Σ is the closure of its interiorin R m × { } . The assertion follows easily. Theorem 8.3. Assume that Card( A ) > . If T does not generate g , then the family ( L , − i T ) satisfiesproperties ( RL ) and ( S ) .Proof. Let us prove that ( L , − i T ) satisfies property ( RL ). Consider the Rockland family ( L , − iT A )and take α ∈ A ; take ω ∈ R A . Define C γ := ( X α ∈ A | ω α | µ α ( n ,α + 2 γ α ) , ω ! : ω ∈ R A ) for every γ ∈ N n , so that C is the boundary of a convex polyhedron. If L : E ( L , − iT A ) → E ( L , − i T ) is theunique continuous linear mapping such that L ( L , − iT A ) = ( L , − i T ), then χ C · β ( L , − iT A ) is L -connectedby Proposition 2.4. Now, define L ′ A ′ := (( − X α,k − X α,n ,α + k ) k =1 ,...,n ,α , − iT α ) α ∈ A , so that L ′ A ′ satisfiesproperties ( RL ) and ( S ) by Theorems 2.2 and 7.4. Take f ∈ L L , − i T ) ( G ), and let e m be its continuousmultiplier relative to L ′ A ′ (cf. Theorem 7.4). Then, m γ : C γ ∋ X α ∈ A | ω α | µ α ( n ,α + 2 γ α ) , ω ! e m (( | ω α | ( n ,α + 2 γ α ) , ω α ) α ∈ A )is a continuous function on C γ which equals M ( L , − iT A ) ( f ) χ C γ · β ( L , − iT A ) -almost everywhere. Therefore,the assertion follows from Theorem 5.8. Assume that T generates a hyperplane of g , and let us prove that ( L , − i T ) satisfies property( S ). Take m ∈ C ( E ( L , − i T ) ) such that K ( L , − i T ) ( m ) ∈ S ( G ), and consider the (unique) linear mapping L ′ : E L ′ A ′ → E ( L , − i T ) such that L ′ ( L ′ A ′ ) = ( L , − i T ). Then, there is m ∈ S ( E L ′ A ′ ) such that m ◦ L ′ = m on σ ( L ′ A ′ ). Next,define, for every ε ∈ {− , + } A and for every γ ∈ N n , S ε,γ := ((cid:0) | ω α | ( n ,α + 2 γ α ) , ω α (cid:1) α ∈ A : ω ∈ Y α ∈ A R ε α ) , so that S ε,γ is a closed convex semi-algebraic set of dimension Card( A ). Assume that L ′ is not one-to-one on S ε,γ . Since L ′ is proper on σ ( L ′ A ′ ), for every λ ∈ L ′ ( S ε,γ ) the fibre L ′− ( λ ) intersects S ε,γ on a closed segment whose end-points lie in the relative boundary of S ε,γ . Therefore, L ′ ( S ε,γ ) givesno contribution to S ε ′ ∈{− , + } A L ′ ( S ε ′ ,γ ); in particular, we may find a subset E of {− , + } A such that S ε ∈ E L ′ ( S ε, ) = L ′ ( σ ( L ′ A ′ )) and such that L ′ is one-to-one on S ε, for every ε ∈ E .26ow, Corollary 2.8 implies that for every ε ∈ E there is m ′ ε ∈ S ( E ( L , − i T ) ) such that m ′ ε ◦ L ′ = m on S ε, . Nevertheless, we must prove that these functions m ′ ε can be patched together to form a Schwartzmultiplier of K ( L , − i T ) ( m ). Then, take λ ∈ σ ( L , − i T ). We shall distinguish some cases.Assume that there are ε , ε ∈ E such that L ′ ( S ε , ) ∩ L ′ ( S ε , ) has non-empty interior and suchthat λ ∈ L ′ ( S ε , ) ∩ L ′ ( S ε , ). Then, m ′ ε = m ′ ε on L ′ ( S ε , ) ∩ L ′ ( S ε , ), so that m ′ ε − m ′ ε vanishesof order infinity on the closure of the interior of L ′ ( S ε , ) ∩ L ′ ( S ε , ), which is L ′ ( S ε , ) ∩ L ′ ( S ε , ) byconvexity. In particular, m ′ ε − m ′ ε vanishes of order infinity at λ .Next, assume that there are ε , ε ∈ E and λ ′ ∈ S ε , ∩ S ε , such that L ′ ( λ ′ ) = λ . Then, λ ′ ∈ S ε k ,γ for every γ ∈ N n such that γ α = 0 if ε ,α = ε ,α , and for k = 1 , 2; let Γ ε ,ε be the set of such γ .Now, clearly m ′ ε k ◦ L ′ = m on S ε k ,γ for every γ ∈ Γ ε ,ε . Taking into account Lemma 8.2, we see thatthe restriction of ( m ′ ε − m ′ ε ) ◦ L ′ to Q α ∈ A V α vanishes of order ∞ at λ ′ , where V α is R ( n ,α , ε ,α ) if ε ,α = ε ,α while V α = R n ,α +1 otherwise. Since either ε = ε or L ′ : Q α ∈ A V α → E ( L , − i T ) is onto, itfollows that m ′ ε − m ′ ε vanishes of order ∞ at λ .Then, assume that there are ε , ε ∈ E such that λ ∈ L ′ ( S ε , ) ∩ L ′ ( S ε , ), but that L ′ ( S ε , ) ∩ L ′ ( S ε , ) has empty interior and λ L ′ ( S ε , ∩ S ε , ). Let us prove that there is ε ∈ E such that λ ∈ L ′ ( S ε , ∩ S ε , ) and such that L ′ ( S ε , ) ∩ L ′ ( S ε , ) has non-empty interior. Indeed, observe thatthere is a unique liner mapping L ′′ such that L ′′ ( L ′ A ′ ) = ( L , − iT A ). In addition, if S = S ε ∈ N A S ε, , then L ′′ induces a homeomorphism of S onto S ′ = L ′′ ( S ). Furthermore, S ′ is the boundary of the convexenvelope C ′ of σ ( L , − iT A ), which is a convex polyhedron, and ker L has dimension 1. Next, observe that L ( S ′ ) = L ( C ′ ) = σ ( L , − i T ) and that L is proper on C ′ ; put an orientation on ker L , fix a linear section ℓ of L , and define g + : σ ( L , − i T ) ∋ λ max { t ∈ ker L : ℓ ( λ ) + t ∈ S ′ } and g − : σ ( L , − i T ) ∋ λ min { t ∈ ker L : ℓ ( λ ) + t ∈ S ′ } . Then, g − and g + are convex and concave, respectively, hence continuous on the interior of σ ( L , − i T ).Observe that the union of the graphs of g − and g + is S ε ∈ E L ′′ ( S ε, ). Now, let E , ± be the set of ε ∈ E such that L ′′ ( S ε, ) is contained in the graph of g ± . Observe that E is the disjoint union of E , − and E , + , since g − ( λ ) = g + ( λ ) for every λ in the interior of σ ( L A ) (cf. the proof of Proposition 2.4).Therefore, σ ( L A ) = S ε ∈ E , ± L ′ ( S ε, ); since L ′ ( S ε, ) is closed for every ε ∈ E and since E is finite,this proves that the union of the L ′ ( S ε, ) such that ε ∈ E , ± and λ ∈ L ′ ( S ε, ) is a neighbourhood of λ in σ ( L A ). Next, since λ L ′ ( S ε , ∩ S ε , ), we may assume that ε ∈ E , + and ε ∈ E , − . Then,there is ε ∈ E , + such that λ ∈ L ′ ( S ε , ) and L ′ ( S ε , ) ∩ L ′ ( S ε , ) has non-empty interior, so that λ ∈ L ′ ( S ε , ∩ S ε , ). Therefore, the preceding arguments show that m ′ ε − m ′ ε vanishes of order ∞ at λ . Hence, by means of Theorem 2.7 we see that there is m ′ ∈ S ( E ( L , − i T ) ) such that m ′ ◦ L = m on σ ( L ′ A ′ ), so that m ′ = m on σ ( L , − i T ), whence the result in this case. Now, consider the general case, and take m ∈ C ( E ( L , − i T ) ) such that K ( L , − i T ) ( m ) ∈ S ( G ). Take afinite subset T ′ of g which contains T and generates a hyperplane of g , so that implies that ( L , − i T ′ )satisfies property ( S ). Observe that σ ( L , − i T ′ ) is a convex semi-algebraic set. Therefore, the assertionfollows easily from Corollary 2.8. Lemma 8.4. Let G ′ and G ′′ be two non-trivial homogeneous groups, L ′ and L ′′ two positive Rocklandoperators on G ′ and G ′′ , respectively. Then, the operator L ′ + L ′′ on G ′ × G ′′ satisfies property ( S ) .Proof. Let π be the canonical projection of G ′ × G ′′ onto its abelianization G ′′′ , that is, onto its quotientby the normal subgroup [ G ′ × G ′′ , G ′ × G ′′ ]. Then, Theorem 3.2 implies that d π ( L ′ + L ′′ ) satisfies property( S ). Now, take ϕ ∈ S L ′ + L ′′ ( G ′ × G ′′ ). Then, Theorem 3.2 and [24, Theorem 3.2.4], applied to the rightquasi-regular representation of G ′ × G ′′ in L ( G ′′′ ), imply that π ∗ ( ϕ ) ∈ S d π ( L ′ + L ′′ ) ( G ′′′ ), so that thereis m ∈ S ( R ) such that K d π ( L ′ + L ′′ ) ( m ) = π ∗ ( ϕ ). Since σ ( L ′ + L ′′ ) = R + = σ (d π ( L ′ + L ′′ )), we see that ϕ = K L ′ + L ′′ ( m ), whence the result. Theorem 8.5. Let G ′ be a homogeneous group endowed with a positive Rockland operator L ′ which ishomogeneous of degree . Then, the following hold:1. ( L + L ′ , − i T ) satisfies property ( RL ) ;2. if L ′ satisfies property ( S ) , then also ( L + L ′ , − i T ) satisfies property ( S ) . not require that G ′ is graded, so that the requirement that L ′ has homogeneousdegree 2 can be met up to rescaling the dilations of G ′ . In addition, if L ′ is not positive, then ( L + L ′ , − i T )is not a Rockland family, since the mapping σ ( L , − i T , L ′ ) ∋ ( λ , λ , λ ) ( λ + λ , λ ) is not proper. Proof. Let us prove that L A satisfies property ( RL ). Observe that, if the assertion holds when T generates g , then the assertion follows by means of Propositions 2.4 and 2.6. Therefore, we may assumethat T is a basis of g .Define L ′ A ′ := ((( − X − X n ,α , . . . , − X n ,α − X n ,α ) , − iT α ) α ∈ A , L ′ ), and observe that L ′ A ′ satisfiesproperty ( RL ) by Theorems 2.2 and 7.4. Define S := n(cid:0) | ω α | n ,α , ω α (cid:1) α ∈ A : ω ∈ R A o , so that S is a closed semi-algebraic set of dimension Card( A ). Then, apply Proposition 2.6 with β = χ S × R + β L ′ A ′ , observing that L : S × R + → σ ( L + L ′ , − i T ) is a proper bijective mapping, hencea homeomorphism. Since L ∗ ( β ( L + L ′ , − i T ) ) is equivalent to L ∗ ( β ) thanks to [16, Theorem 3.2.22], theassertion follows. Now, assume that L ′ satisfies property ( S ), and let us prove that ( L + L ′ , − i T ) satisfies property( S ). Observe that, if we prove that the assertion holds when T generates g , then the general case willfollow by means of Corollary 2.8. Therefore, we shall assume that T = ( T α ) α ∈ A .Observe first that L ′ A ′ satisfies property ( S ) by Theorem 2.2. Then, take m ∈ C ( σ ( L A )) such that K L A ( m ) ∈ S ( G × G ′ ). It follows that there is m ∈ S ( E L ′ A ′ ) such that m ◦ L = m on σ ( L ′ A ′ ). Since S × R + is a closed semi-algebraic set, by Theorem 2.7 it will suffice to show thatthe class of m in S ( S × R + ) is a formal composite of L . Now, this is clear at the points of the form (cid:0)P α ∈ A | ω α | µ α ( n ,α ) + r, ω (cid:1) , where ω ∈ ( R ∗ ) A and r > 0. Arguing by induction on Card( A ) and takingLemma 8.4 into account, the assertion follows by means of Lemma 6.1.As a complement to Theorem 8.5, we present the following pathological case. Proposition 8.6. Let ( X, Y, T ) be a standard basis of H , and let L ′ be a positive Rockland operator ona homogeneous group G . Assume that ( L ′ ) satisfies property ( S ) and that L ′ h is homogeneous of degree for some h > . Then, the Rockland family ( − X − Y + L ′ h , − iT ) is functionally complete and satisfiesproperty ( RL ) , but does not satisfy property ( S ) .Proof. Define L := − X − Y . Then, Theorem 8.5 implies that ( L + L ′ h , − iT ) is a Rockland familywhich satisfies the property ( RL ). Next, take some ϕ ∈ D ( E ( L , − iT, L ′ ) ) supported in { ( λ ′ , λ ′ , λ ′ ) : λ ′ < | λ ′ | − λ ′ h } and equal to pr on a neighbourhood of (1 , , m : ( λ , λ ) ϕ (cid:16) | λ | , λ , h p λ − | λ | (cid:17) is not equal to any elements of S ( E L A ) on σ ( L A ). On the other hand, K L A ( m ) = K ( L , − iT, L ′ ) ( ϕ ) ∈S ( H × R ). Hence, L A does not satisfy property ( S ). Now, let us prove that L A in functionally complete. Take m ∈ C ( E L A ) such that K L A ( m ) issupported in { e } . Notice that we may assume that m is continuous since L A satisfies property ( RL ).Projecting onto the quotient by { } × R , we see that there is a unique polynomial P on E L A whichcoincides with m on σ ( L , − iT ). On the other hand, the family ( L , − iT, L ′ ) is functionally complete sinceit satisfies property ( S ) (cf. Theorem 2.2 and Proposition 2.10). Hence, there is a unique polynomial Q on E ( L , − iT, L ′ ) such that m ( λ + λ h , λ ) = Q ( λ , λ , λ )for every ( λ , λ , λ ) ∈ σ ( L , − iT, L ′ ). Hence, P ( λ + λ h , λ ) = Q ( λ , λ , λ )for every ( λ , λ , λ ) ∈ { (cid:16) k | r | , r, h p k | r | (cid:17) : r ∈ R , k ∈ N + 1 , k ∈ N } . Now, the closure of this latterset in the Zariski topology is E ( L , − iT, L ′ ) , so that m = P on σ ( L A ). The assertion follows.28 cknowledgements I would like to thank professor F. 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