Geometric invariance of the semi-classical calculus on nilpotent graded Lie groups

In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. We analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. We interpret the geometric meaning of this invariance in the setting of filtered manifolds.


Introduction
Semi-classical analysis started to develop in the 70s and has proved a flexible tools for the analysis of PDEs [4]. It crucially relies on a microlocal viewpoint and the use of semi-classical pseudodifferential calculus. Such an approach has been recently developed on nilpotent graded Lie groups with the ambition of taking into account the non-commutativity of the group by using representation theory and the associated Fourier theory [2,9,6]. The symbols of the pseudodifferential operators are then fields of operators on the product G × G of the group G and its dual space G. Like in the Euclidean case [16], this semi-classical pseudodifferential calculus enjoys a non-commutative symbolic calculus which describes taking the adjoints and the composition as asymptotic sums of symbols [9,6]. These tools prove efficient for studying the propagation of oscillations or concentration effects [7] of Schrödinger equations, or related questions in PDEs such as the existence of observability inequalities [8].
In the Euclidean case, invariance to leading order by change of variables is an important property of the semiclassical pseudodifferential calculus. This invariance allows for the identification of the symbol as a function on the cotangent space, and is therefore the foundation of the semiclassical calculus on manifolds [16]. Here we investigate this property for symbols on graded Lie groups: what happens to the semi-classical pseudodifferential operators introduced in [6] when conjugated by the change of variables induced by a local smooth diffeomorphism that preserves the filtration of the group.
It turns out that a smooth function Φ between two graded Lie groups G and H that preserves the filtration of the groups is uniformly Pansu differentiable (see Theorem 1.5 below). After the publication of the present paper to Journal of Geometric Analysis, it was pointed out to us that this analysis had been performed in greater generality on filtered manifolds in [3,Section 7]. However, our analysis here is done from first principles on graded nilpotent Lie groups. Indeed, part of the article consists in revisiting the concept of Pansu differentiability in the context of graded nilpotent Lie groups and its link with being filtration preserving. This study allows us to define one-to-one maps between the phase spaces G× G and H × H associated with any smooth local diffeomorphisms Φ between two nilpotent groups G and H (see Theorem 1.8 below). We investigate the geometric interpretation of these results in the setting of filtered manifolds in the last Section 4.
1.1. Graded groups. A graded group G is a connected simply connected nilpotent Lie group whose (finite dimensional, real) Lie algebra g admits an N-gradation into linear subspaces, i.e.
With this way of describing g, all but a finite number of subspaces g j are trivial. We denote by j = n G the smallest integer such that all the subspaces g j , j > n G , are trivial. If the first stratum g 1 generates the whole Lie algebra, then g j+1 = [g 1 , g j ] for all j ∈ N 0 and n G is the step of the group; the group G is then said to be stratified, and also (after a choice of basis or inner product for g 1 ) Carnot.
The product law on G is derived from the exponential map exp G : g → G and the Dynkin formula for the Baker-Campbell-Hausdorff formula (see [ where the coefficients c r,s are known: (r j + s j ) Π ℓ i=1 r i !s i ! Above, the sum over ℓ is finite in the nilpotent case. In particular, the term for which s ℓ > 1 or s ℓ = 0 and r ℓ > 1 is zero, while the term adXad −1 Y (Y ) for s ℓ = 0, r ℓ = 1 is understood as X.
Here ln G denotes the inverse map to exp G ; we may drop the subscript G for exp and ln when the context is clear. The exponential mapping is a global diffeomorphism from g onto G. Once a basis X 1 , . . . , X n for g has been chosen, we may identify the points (x 1 , . . . , x n ) ∈ R n with the points x = exp(x 1 X 1 + · · · + x n X n ) in G. It allows us to define the (topological vector) spaces C ∞ (G) and S(G) of smooth and Schwartz functions on G identified with R n ; note that the resulting spaces are intrinsically defined as spaces of functions on G and do not depend on a choice of basis.
The exponential map induces a Haar measure dx on G which is invariant under left and right translations and defines Lebesgue spaces on G, together with a (non-commutative) convolution for functions f 1 , f 2 ∈ S(G) or in L 2 (G), We now construct a basis adapted to the gradation. Set n j = dim g j for 1 ≤ j ≤ n G . We choose a basis {X 1 , . . . , X n 1 } of g 1 (this basis is possibly reduced to ∅), then {X n 1 +1 , . . . , X n 1 +n 2 } a basis of g 2 (possibly {0}) and so on. Such a basis B = (X 1 , · · · , X n ) of g is said to be adapted to the gradation; here n = dim g = n 1 + . . . + n G .
The Lie algebra g is a homogeneous Lie algebra equipped with the family of dilations {δ r , r > 0}, δ r : g → g, defined by δ r X = r ℓ X for every X ∈ g ℓ , ℓ ∈ N [10,9]. We re-write the set of integers ℓ ∈ N such that g ℓ = {0} into the increasing sequence of positive integers υ 1 , . . . , υ n counted with multiplicity, the multiplicity of g ℓ being its dimension. In this way, the integers υ 1 , . . . , υ n become the weights of the dilations and we have δ r X j = r υ j X j , j = 1, . . . , n, on the chosen basis of g. The associated group dilations are defined by δ r (x) = rx := (r υ 1 x 1 , r υ 2 x 2 , . . . , r υn x n ), x = (x 1 , . . . , x n ) ∈ G, r > 0.
In a canonical way, this leads to the notions of homogeneity for functions and operators. It also motivates the definition of quasi-norms on G as continuous functions |·| : G → [0, +∞) homogeneous of degree 1 on G which vanishes only at 0. This often replaces the Euclidean norm in the analysis on homogeneous Lie groups. Any quasi-norm | · | on G satisfies a triangle inequality up to a constant: Any two homogeneous quasi-norms | · | 1 and | · | 2 are equivalent in the sense that For example, the Haar measure is Q-homogeneous where Q := ℓ∈N ℓn ℓ = υ 1 + . . . + υ n is called the homogeneous dimension of G.

Diffeomorphisms between graded groups.
1.2.1. Filtration preserving maps. In this paragraph, we explain the notion of a (smooth or at least C 1 ) map preserving the natural filtrations between two graded nilpotent groups G and H. We first need to set some notation. For each x ∈ G, we denote the differential of the left-translation L x : G → G by x ∈ G at the identity by For y ∈ H, we denote by τ H y the analogue map on H; we may omit the superscript G or H if the context is clear. These maps allow us to view derivatives between Lie groups as acting on the Lie algebras of the groups: if Φ is a smooth map from an open set U of G to H, then we define for x . By definition, the following diagram commutes : Note that an equivalent definition for d x Φ : g → h is given by In this article, we will use the shorthand: and for instance (1.2) may be rewritten as Definition 1.1. Let Φ be a smooth function from an open set U of G to H. We say that Φ preserves the filtration at x ∈ U when we have

1.2.2.
Pansu differentiability. For smooth diffeomorphisms (or at least sufficiently differentiable), the property of preserving the filtration is related to the notion of Pansu differentiability, which we define following [11]: Let Φ be a map from an open set U of G to H, and let x ∈ U . The function Φ is Pansu differentiable at the point x when for any z ∈ G, the following limit exists: The limit is denoted by P-D x Φ(z); the map z → P-D x Φ(z) is called the Pansu derivative of Φ at x. Part (2) of this definition means that for every point (x, z) ∈ U × G, there exists a compact neighborhood of (x, z) in U × G where the limit in (1.4) holds uniformly. Note that this implies that the limit in (1.4) holds uniformly on any compact subset of U × G. A similar assumption is made in Section 4 of [15] with a subtle difference: the limit in (1.4) is supposed to be locally uniform only in z ∈ G; however, the Pansu derivative at every point is required to be an automorphism of the group G. We will see in Section 2 that condition (2) in Definition 1.3 implies that the Pansu derivative is a group morphism, and an automorphism when Φ is a diffeomorphism.
Example 1.4. Let Φ : H → R 3 ; (p, q, t) → (p, q, t) be the identity map from the Heisenberg group to (R 3 , +). Equip both groups with the gradation corresponding to the non-isotropic dilations δ ε (p, q, t) = (εp, εq, ε 2 t). One computes, for x = (p 1 , q 1 , t 1 ) and y = (p 2 , q 2 , t 2 ), ) which converges as ε → 0 if and only if p 1 q 2 = p 2 q 1 . In particular, the identity map between H and (R 3 , +) is Pansu differentiable at the origin but not in a neighborhood of the origin.
The following property is crucial for our analysis and will be proved in Section 2 below. It states that for smooth functions, the notion of uniform Pansu differentiability on an open set coincides with preserving filtration above this set: This result has been proved in [15] in the case of Carnot groups. The filtration preserving maps then are contact maps. The Heisenberg group is a prime example of Warhust's result [15]. One contribution of this paper is to extend the result to the case of graded groups with a proof 'from first principle'. After the publication of the present paper to Journal of Geometric Analysis, it was pointed out to us that this analysis had been performed in greater generality on filtered manifolds in [3,Section 7].
Our proof of Theorem 1.5 will also yields the following properties: Corollary 1.6. We continue with the notation and setting of Theorem 1.5. Let Φ be a smooth function from an open set U of G to H which is uniformly Pansu differentiable on U .
yields continuous morphisms of topological vector spaces from C ∞ (H) to C ∞ (U, C ∞ (G)) and also from S(H) to C ∞ (U, S(G)). • If Φ is a smooth diffeomorphism from U onto its image, then G and H are isomorphic.
1.3. The dual set and the semi-classical pseudodifferential calculus.
1.3.1. The dual set. Recall that a representation (H π , π) of G is a pair consisting of a Hilbert space H π and a group morphism π from G to the set of unitary transforms on H π . In this paper, the representations will always be assumed (unitary) strongly continuous, and their Hilbert spaces separable. A representation is said to be irreducible if the only closed subspaces of H π that are stable under π are {0} and H π itself. Two representations π 1 and π 2 are equivalent if there exists a unitary transform U called an intertwining map that sends H π 1 on H π 2 with The dual set G is obtained by taking the quotient of the set of irreducible representations by this equivalence relation. We may still denote by π the elements of G and we keep in mind that different representations of the class are equivalent through intertwining operators. The following properties are straightforwards: Lemma 1.7 (Pullback of Irreps). Let θ : G → H be a continuous group homomorphism, and (π, H π ) be a representation of H. Then • π • θ is a representation of G, • π → π • θ preserves unitarity and the unitary equivalence class of π, • π → π • θ preserves irreducibility if θ is surjective.
Hence if θ is surjective, we have a map In particular, any automorphism of G induces an automorphism of G, and hence any subgroup of Aut(G) acts on G. For instance, the dilations δ r , r > 0, on a graded Lie group G provide an action of R + on G via (1.5) δ r π(x) = π(rx), x ∈ G, π ∈ G, r > 0.

1.3.2.
The Fourier transform. The Fourier transform of an integrable function f ∈ L 1 (G) at a representation π of G is the operator acting on H π via Note that F G (f )(π) ∈ L(H π ); in this paper, we denote by L(H) the Banach space of bounded operator on the Hilbert space H. Note also that if π 1 , π 2 are two equivalent representations of G with π 1 = U −1 • π 2 • U for some intertwining operator U, then Hence, this defines the measurable field of operators {F G (f )(π), π ∈ G} modulo equivalence. Here, the unitary dual G is equipped with its natural Borel structure, and the equivalence comes from quotienting the set of irreducible representations of G together with understanding the resulting fields of operators modulo intertwiners. The Plancherel measure is the unique positive Borel measure µ on G such that for any f ∈ C c (G), we have the Plancherel formula Here · HS(Hπ ) denotes the Hilbert-Schmidt norm on H π . This implies that the group Fourier transform extends unitarily from L 1 (G) ∩ L 2 (G) to L 2 (G) onto L 2 ( G) := G H π ⊗ H * π dµ(π) which we identify with the space of µ-square integrable fields on G.
The Plancherel formula also yields an inversion formula for any f ∈ S(G) and x ∈ G: where Tr Hπ denotes the trace of operators in L(H π ), the set of bounded linear operators on H π . This formula makes sense since, for f ∈ S(G), the operators Ff (π) are trace-class and G Tr Hπ Ff (π) dµ(π) is finite.

Semiclassical analysis.
For an open set U ⊂ G, we denote by A 0 (U × G) the space of symbols σ = {σ(x, π) : (x, π) ∈ U × G} of the form where κ ∈ C ∞ c (U, S(G)); that is, the map x → κ x is compactly supported on U and valued in the set of Schwartz class functions S(G) with Schwartz norms depending smoothly on x ∈ U . We call x → κ x the convolution kernel of σ.
As the Fourier transform is injective, it yields a one-to-one correspondence between A 0 (U × G) and C ∞ c (U, S(G)). In this way, A 0 (U × G) inherits from C ∞ c (U, S(G)) the structures of topological vector space and smooth compactly supported section of the Schwartz bundle over U . These structures are easier to grasp than being a set of measurable fields of operators on U × G modulo intertwiners as described in Section 1.3.2. The results of this paper develop a deeper geometric interpretation of the space of symbols, here A 0 (U × G), in terms of densities; this will be discussed in Section 4.
With the symbol σ ∈ A 0 (U × G), we associate the (family of) semi-classical pseudodifferential operators Op ε (σ), ε ∈ (0, 1], defined via or equivalently, in terms of the convolution kernel κ x is the following rescaling of the convolution kernel: . This second expression for Op ε (σ) explains the choice of vocabulary for the convolution kernel of a symbol.
The rescaled convolution kernel is different from the integral kernel of Op ε (σ), which is given by: . We emphasize the convolution kernel over the integral kernel, although the latter is used in the pioneer work [12] on pseudodifferential theory on filtered manifolds (see also [13,14]). The reason is that our approach here is different and aims to be symbolic. It is inspired from semi-classical analysis in Euclidean setting and based on the fact that the symbols obtained via the Fourier transforms of convolution kernels proved a flexible tool for dealing with applications (see [8,7]).
Our semi-classical pseudodifferential theory may be used to analyse the oscillations of families (u ε ) ε>0 that are bounded in L 2 (G). One considers the functionals ℓ ε defined on The limit points of ℓ ε as ε goes to 0 have some structures. When the family (u ε ) ε>0 is L 2 -normalized and after possibly further extraction of subsequences, the limit points of ℓ ε define states of the C *algebra A(U × G) obtained by completion of A 0 (U × G) for the norm For describing the structure of these limit points, we consider the set of pairs (γ, Γ) where γ is a positive Radon measure on U × G and is a positive γ-measurable field of trace-class operators satisfying This set is then equipped with the equivalence relation: The equivalence class of (γ, Γ) is denoted by Γdγ, it is called a positive vector-valued measure. We denote by M + ov (U × G) the set of these equivalence classes. The positive continuous linear functionals of the Given the sequence (ε k ) k∈N , one has The positive vector-valued measure Γdγ is called a semi-classical measure of the family (u ε ) for the sequence ε k . Our aim is to analyze how these notions can be transferred from (an open subset of) a graded group G to another one H via a smooth local diffeomorphism that preserves the filtration of the group.
1.4. Main result. Let us consider two graded groups, G and H, and a smooth diffeomorphism, Φ, from an open set U of G to H that is filtration preserving and thus uniformly Pansu differentiable by Theorem 1.5.
We denote by J Φ the Jacobian of Φ with respect to Haar measures chosen on G and H, and we consider the operator U Φ which associates to a function f defined on Φ(U ) ⊂ H the function Conjugation by U Φ pulls back any S ∈ L(L 2 (Φ(U ))) to an operator The map Φ also induces a transformation I Φ between convolution kernels. Indeed, for a function κ : The geometric meaning of this map is discussed in Section 4. By Lemma 1.7, the map is well-defined. Moreover, it induces an isomorphism of topological vector spaces defined in the following way: for σ ∈ A 0 (Φ(U ) × H), the symbol ( GΦ) * σ given by Theorem 1.8. Let G and H be two graded groups, and Φ a smooth diffeomorphism from an open set U of G to H. Assume that Φ is filtration preserving on U (and thus uniformly Pansu differentiable on U ). Let σ ∈ A 0 (Φ(U ) × H), then in L(L 2 (U )), Note that when we assume that the smooth diffeomorphism Φ is uniformly Pansu differentiable on U , this implies that the graded groups G and H are isomorphic (see Corollary 1.6).
Theorem 1.8 is proved in Section 3 below. It crucially relies on the Pansu differentiability of filtration preserving maps (see Section 2). It also has straightforward consequences on semi-classical measures that we now present. We associate with Φ a map from Then, the result of Theorem 1.8 implies the next corollary.
Corollary 1.9. Let G and H be two graded groups, and let Φ a smooth diffeomorphism from an open set U of G to H. Assume that Φ is filtration preserving on U and thus uniformly Pansu differentiable on U . Let (f ε ) ε>0 be a bounded family in L 2 (U ) and Γdγ be a semi-classical measure of (f ε ) ε>0 for the sequence ε k . Then, Γ Φ dγ Φ is a semi-classical measure of the family (U * Φ f ε ) ε>0 for the sequence ε k .
The proof of Theorem 1.8 is developed in Section 3 and heavily uses the results of Section 2 via Theorem 1.5 and its consequences in Corollary 1.6. Similar results should hold in the microlocal case where no specific scale ε is chosen, as developed in [5]. Finally, we develop the geometric interpretation of the convolution kernel in Section 4.
Acknowledgments. The authors thank Antoine Julia and Pierre Pansu for inspiring discussions.

Filtration preserving diffeomorphisms
Our aim in this section is to prove the equivalence between the uniform Pansu differentiability on an open set and the preservation of the filtration above this set, see Theorem 1.5. We will start with introducing some notations and concepts in order to give a precise meaning to these properties. ( (2) If Φ is uniformly Pansu differentiable on U , then the Pansu derivative at every point x ∈ U is a group morphism from G to H : Proof. (1) This comes from the definition and the observation that for r > 0, Passing to the limit as ε goes to 0, we obtain Then, fixing a homogeneous quasi-norm on G and using that z = δ |z| z with |z| = 1, one writes (2) We write and use the uniformity of the convergence.
The hypothesis of uniformity for the Pansu differentiability is needed to show that the Pansu derivative is a group morphism (see Part (2) above) but also for the following composition property: Proof. Writez for (w, z) ranging in a compact subset of U ′ × F and ε ∈ (0, 1]. Observe that By continuity of Φ, and uniform Pansu differentiability of Ψ, the pair (Ψ(w),z ε ) ranges in a compact subset of Ψ(U ′ ) × G on which the map converges uniformly as ε → 0 (by uniform Pansu differentiability of Φ). Since lim ε→0zε = P-D w Ψ(z), the limit of (2.1) as ε → 0 is P-D Ψ(w) Φ(P-D w Ψ(z)).
Remark 2.3. Let us consider a map Φ from an open set U of a graded group G to a graded group H such that Φ is a bijection from U onto its image Φ(U ) which is open. If Φ and its inverse Φ −1 are uniformly Pansu differentiable on U and Φ(U ) respectively, then we may apply Lemmata 2.1 and 2.2 and use P-D x Id = Id to obtain In particular, the groups G and H are isomorphic.

Filtration preserving smooth maps.
In this section, we study how to characterize smooth maps that are filtration preserving.
A matrix-valued viewpoint will be helpful for a deeper understanding of Definition 1.

For a smooth function Φ from an open set
j=1 g j and h = ⊕ ∞ j=1 h j adapted to the respective gradations (see Section 1.1). This matrix can be written by blocks M Φ,i,j (x) associated with the gradation. As the map d x Φ is linear, in order to identify the blocks M Φ,i,j (x), it is enough to let V vary in g j and calculate the projection on h j of d x Φ(V ). For this, we denote by pr h,j the projection onto h j along ⊕ j ′ =j h j ′ . We may allow ourselves to remove the subscript h (and write pr j instead of pr h,j ) when the context is clear.
Let us illustrate this point with the following equivalences: We obtain easily the following implication between preserving the filtration and uniform Pansu differentiability: Lemma 2.5. Let Φ be a smooth function from an open set U of G to H, and let x ∈ U . If Φ is Pansu differentiable at x, then Φ preserves the filtration at x.
Proof. Let V ∈ g j . We have by taking t = ε j in (1.2), while the properties of dilations yield As Φ is Pansu differentiable at x, the argument inside pr h,i • ln H above has a limit. Hence, if i > j, we have pr h,i (d x Φ(V )) = 0 and we conclude with Lemma 2.4.
In the next sections, we will analyze the reverse implication to the one in Lemma 2.5, using this matrix-valued point of view; this will give Theorem 1.5.

2.3.
Characterization of Pansu differentiability for smooth maps. If Φ is Pansu differentiable at x ∈ U , we set p-d x Φ := ln H • P-D x Φ • exp G , so that we have the following diagram : This defines the map p-d x Φ : g → h. An equivalent definition for this map is given by for V ∈ g, having used the shorthand (1.3). Clearly, Φ is Pansu differentiable at x ∈ U if and only if the limit in (2.2) exists for all V ∈ g, and it is uniformly Pansu differentiable on U if and only if these limits hold locally uniformly on U × g.
The map p-d x Φ may not be linear in general but it will be under mild hypotheses. Indeed, when z → P-D x Φ(z) is a continuous group morphism, for instance when Φ is uniformly Pansu differentiable on an open neighbourhood of x, then p-d x Φ is linear with As above, we can adopt a matrix-valued point of view and define P-M Φ (x) to be the matrix whose columns are the images of B by p-d x Φ, that is, the vectors p-d x Φ(X j ), j = 1, . . . , dim G, expressed in the basis C. The (rectangular) matrix P-M Φ (x) is of the same size as M Φ (x). It makes sense to look at P-M Φ (x) and its blocks P-M Φ,i,j (x) associated with the gradation, or rather to the quantities pr h,i (p-d x Φ(V )), V ∈ g j . As a consequence of the definitions of the objects involved and of homogeneous properties, for each V ∈ g j with i, j = 1, 2, . . . , Actually, very few of the quantities pr h,i (p-d x Φ(V )), V ∈ g j , are non-zero: Lemma 2.6. Let Φ be a smooth function from an open set U of G to H, and let x ∈ U . We always have (regardless of whether Φ is uniformly Pansu differentiable or preserves the filtration) Proof of Lemma 2.6. For each x ∈ U and V ∈ g j , we have and the last argument of pr h,i tends to d x Φ(V ) as ε goes to 0.
(1) By Lemma 2.6 and equation (2.3), if Φ is Pansu differentiable at x then for each V ∈ g j with i, j = 1, 2, . . . , (2) Let us give a matrix interpretation of Part (1). The matrix P-M Φ (x) defined above is blocklower-diagonal in the sense that all the blocks P-M Φ,i,j (x), i < j, strictly above the diagonal are 0. Furthermore, the diagonal blocks coincide with those of M Φ (x). We will see later that one can say more (see Theorem 2.9). This matrix interpretation does not depend on the bases B and C chosen to describe the matrix as long as they are adapted to the gradations.
The existence of the limits on the left-hand side of (2.3) turns out to also be a sufficient condition for a smooth map to be Pansu differentiable, and thus gives a characterization of smooth Pansu differentiable maps (or at least for C k -maps for some k large enough).
Proposition 2.8. Let Φ be a smooth function from an open set U of G to H. The function Φ is uniformly Pansu diffferentiable on U if and only if for each x ∈ U , V ∈ g j , with i, j = 1, 2, . . . , the limit of exists as ε goes to 0 and, all these limits hold locally uniformly on U × g j for each i = 1, 2, . . .

Proof of Proposition 2.8.
We drop the indices of the groups and Lie algebras in the notation for the logarithmic and exponential maps and the projections. In view of Remark 2.7 (1), it remains to show the reverse implication. Hence, we assume that the limits in (2.4) exist and we aim at proving that for any x ∈ U , V ∈ g, i = 1, 2, . . ., the limit of pr i • ln H δ ε −1 (Φ x (exp G (δ ε V ))) exists and that this holds locally uniformly with respect to (x, V ) in U × g. We will prove this recursively on i = 1, 2, . . . First, we need to set some conventions.
Let us recall that the map Θ : R dim G → R dim G given by the following exponential coordinates of the second kind on g is a global diffeomorphism of R dim G , and that Θ(δ ε V ) = (εV 1 , . . . , ε n G V n G ).
We will use the equality We will use the star product (1.1) from the Dynkin formula and its properties coming from the gradation property for H via the properties we now describe. Here m ≥ 2 is an integer, which will be equal to n G below. Consider the star product of m elements W 1 , . . . , W m in h projected onto h i . This product yields a polynomial map h m → h whose linear part is the sum pr i (W 1 )+. . .+pr i (W m ). The map P i = P i,m : h m → h defined by their difference Furthermore, P i depends only on the projections of the vectors onto h j ′ , j ′ < i. In order to express this technically, for each j ∈ N, we denote by pr <j = pr 1 + . . . + pr j−1 the projection onto ⊕ j ′ <j h j ′ along ⊕ j ′ ≥j h j ′ , with the convention pr <1 := 0. We have: . . , W m ) = P i (pr <i W 1 , . . . , pr <i W m ).
Let us come back to (2.5) and set m = n G . We use a recursive argument and start with i = 1; in this case, P 1 = P 1,n G (defined above) is identically zero. Therefore, composing (2.5) with ε −1 pr 1 •ln yields the expression The other steps of the recursion are proved in the following manner. At a general recursive step i = 2, 3, . . . , we have where Q x,i (V ) := P i (ln •Φ x (exp(V 1 )), ln •Φ x expV 1 (expV 2 ), . . .) , having used the polynomial P i = P i,n G defined above. The properties of P i imply Applying the recursive assumption to each term involving pr <i , the limit of ε −i Q x,i (δ ε V ) holds locally uniform as ε goes to 0. Therefore, in the expression the first term on the right-hand side has a locally uniform limit and the other ones too by the hypotheses in (2.4). This shows the i th step and terminates the proof.

2.4.
A refinement on Theorem 1.5. This section is devoted to the statement of the theorem below which implies Theorem 1.5 and Corollary 1.6 but is more technical. It will be shown in Section 2.5. We shall use the following notation: with a subspace v of g and an open subset U of G, we associate the open set (ii) Besides, for such a map Φ: (1) The Pansu derivative yields a smooth function (x, z) → P-D x Φ(z) on U × G and the map x → P-D x Φ is smooth from U to Hom(G, H). (2) For any x ∈ U, V ∈ g j , Consequently, the Jacobian J Φ (x) equals the Jacobian of the map z → P-D x (z).
(3) For any (x, V ) ∈ U × g and any ε > 0 small enough: where the function ρ is valued in h, continuous on R g,U and smooth on the open subset Ω g,U .
Theorem 1.5 follows from Part (i) while the second part of Corollary 1.6 follows from Part (i) and Remark 2.3. The first Part of Corollary 1.6 is a consequence of Theorem 2.9 (ii) (1).
Before entering into the proof Theorem 2.9, let us give a technical corollary that will be useful in Section 3: Corollary 2.10. Here we fix a homogeneous a quasi-norm | · | G on G and we keep the same notation for the function on the underlying Lie algebra g, and similarly for H. We denote byB r the corresponding closed balls about 0 of radius r > 0.
We continue with the assumptions of Theorem 2.9. We fix a compact subset K of U . Let r 0 > 0 be so that KB r 0 ⊂ U . Then there exists a constant C > 0 such that for any (x, V, ε) ∈ K × g × (0, 1] satisfying ε|V | G ≤ r 0 , we have: |ρ(x, W, ε)| H .

2.5.1.
A technical lemma. The proof of Theorem 2.9 relies on the following property which is of interest on its own, all the more that it holds for any Lie group, not necessarily nilpotent. (1) For every x ∈ U and V ∈ g, we have for t in a small neighbourhood of 0, where the coefficients c p ∈ R comes from the Dynkin formula in (1.1). In particular c 0 = 1.
The first two relations in Part (2) come readily from the definitions of the objects involved. This together with the consequence of Part (1) implies the rest of Part (2).

2.5.2.
Proof of Theorem 2.9. Let us start with the equivalence (i).
Proof of Part (i) in Theorem 2.9. By Lemma 2.5, it suffices to prove the reverse implication: we assume that Φ preserves the filtration at every x ∈ U and we want to show that it is uniformly Pansu differentiable on U . Let V ∈ g j . By Proposition 2.8, it suffices to show that the limit in (2.4) exists and holds locally uniformly. Here, we will show the existence of the limits as the local uniformity will be a natural consequence of the existence. We only need considering j < i by Lemma 2.6.
As Φ preserves the filtration at every x ∈ U , by Lemma 2.4, we have in a neighborhood of t = 0 Therefore, differentiating in t, we obtain that in the same neighborhood of t = 0 and equation (2.6) may be generalised into The latter relation implies that we have for any p ∈ N and ℓ = 1, 2, . . .: The previous fact together with Lemma 2.11 (2) implies for any k = 2, 3, . . .
We now combine these observations with the Taylor expansion of t → pr h,i • ln H (Φ x (exp G (tV )) for t = ε j . We push the expansion at a higher order depending on how far j is from i. Indeed, for j < i, we can find an integer k ≥ 1 such that i k+1 < j < i k−1 (with the convention that i k−1 = +∞ if k = 1) and we push the Taylor expansion up to order k: By the first part of Lemma 2.11 (2), we have by (2.6). Therefore, by (2.9), as long as (k − 1)j < i, we have This shows the existence of the limits in (2.4) for j < i/(k − 1) such that j > i/(k + 1), and thus recursively for all j < i. Furthermore, one checks easily that they hold locally uniformly, so Part (i) of Theorem 2.9 is proved.
The next points of Theorem 2.9 come from the preceding analysis: Proof of Part (ii) in Theorem 2.9. Additionally to the limits in (2.4), we have obtained that for any V ∈ g j and pr h,i ∈ h i , we have where the functions r i,j are valued in h i , smooth on the open subset Ω g j ,U and continuous on R g j ,U . This implies Point (2) and that (x, V ) → P-D x Φ(exp G V ) is smooth on U × g j for any j = 1, 2, . . . Then, the smoothness of P-D x stated in Point (1) follows from the Baker-Campbell-Hausdorff formula and the fact that it is a group morphism. Finally, the existence of R in Point (3) follows for general V ∈ g, using (2.5) and the Baker-Campbell-Hausdorff formula.

Invariance of the semi-classical calculus on nilpotent Lie group
This section is devoted to the proof of Theorem 1.8. We will first recall (see [6,7]) some properties of the semi-classical calculus that will be useful in the proof.
3.1. L 2 -boundedness in the semi-classical calculus. If σ ∈ A 0 (U × G), then and the right-hand side defines the semi-norm · A 0 (U × G) on A 0 (U × G). Consequently, the operators Op ε (σ) are uniformly bounded on L 2 (U ) with bound: The next lemma shows that the behaviour of the integral kernel near the diagonal {(x, x) : x ∈ U } in U contains all the information about Op ε (σ) at leading order.
Definition 3.1. A cut-off along the diagonal of U is a function χ ∈ C ∞ (U × U ) such that for each x ∈ U , the map z → χ(x, xz −1 ) extends trivially to a smooth function on G that is identically equal to 1 on a neighborhood U 1 of 0 and vanish outside a bounded neighborhood U 0 of 0; moreover, the neighborhoods U 1 and U 0 are assumed to be independent of x ∈ U . We may call the smallest U 0 the diagonal support of χ.
For example, if χ 1 ∈ C ∞ c (G) is a function equal to 1 close to 0 and with support small enough, then (x, y) → χ 1 (xy −1 ) is a cut-off along the diagonal of U whose diagonal support is the support of χ.
3.2. Proof of Theorem 1.8. Let σ ∈ A 0 (Φ(U ) × H) and denote by κ : x → κ x its convolution kernel. Set σ := ( GΦ) * σ ∈ A 0 (U × G) and denote by κ = I Φ κ its convolution kernel. Note that the x-supports are transported via Φ: Let us first proceed to a reduction of the problem by using Lemma 3.2. Let χ ∈ C ∞ (Φ(U ) × Φ(U )) be a cut-off along the diagonal of Φ(U ). Consider the function χ defined on U × U by Then χ is a cut-off along the diagonal of U .
With the two cut-off functions χ and χ in hands, by Lemma 3.2, we can restrict to proving that the operator R ε whose integral kernel is We observe that the operator R ε may be written in the form ε,x (y) = ε −Q r ε,x (δ −1 ε y) and r ε,x is the function in C ∞ c (U, S(G)) given by We now aim to prove that (3.2) ∃c 0 > 0 ∀ε ∈ (0, 1] as, by (3.1), this implies R ε L(L 2 (U )) ≤ c 0 ε. We observe that the x-support of r ε,x is included in K 0 . We may assume that the diagonal support of χ is as small as we need below and therefore that the diagonal support of χ is included in a small ballB r 0 of G with a radius r 0 as small as we need; here, we have fixed a homogeneous quasi-norm | · | G . We can decompose I ε ≤ I 1,ε + I 2,ε with For I 1,ε , we use the Taylor estimates due to Folland and Stein (see [10,Section 1.41] or [9, Section 3.1.8]): we have for any x ∈ K 0 and z ∈ δ −1 , this implies that I 1 ≤ c 1 ε for some constant c 1 > 0.

Geometric meaning of the convolution kernel
As for the semi-classical calculus in the Euclidean setting, it is important to keep in mind the geometric aspects of the elements that one considers. Indeed, the objects defined using the group Fourier Transform depend on a choice of Haar measure on the graded Lie group G. This makes little difference in the group context, where the Haar measure is determined up to a constant, but will matter more if one wants to extend this non-commutative semi-classical approach on manifolds. In this section, we explain how to avoid choosing a normalization for the Haar measure when defining semi-classical quantization on graded Lie groups. We achieve this by giving a geometrically intrinsic definition of the convolution kernel and thus of the semiclassical symbols.
We will start with well-known geometric considerations and set out their usual conventions. The manifolds (often denoted by M ) are assumed to be smooth. If F = ∪ x∈M F x is a fiber bundle over M , π F : F → M will denote its canonical projection. Here, all the fiber bundles are smooth, and we denote by Γ(F ) the space of its smooth sections.
4.1. Generalities on bundles. Let E be a real vector bundle over a manifold M of rank r.
For s ∈ R * , the s-density bundle associated to E is the set |Λ| s (E) of all maps µ x : (E x ) r := E x × · · · × E x → R, for which When E = T M , we simply write |Λ| s M for the s-density bundle over M . An s-density µ on M is a smooth section of |Λ| s M . When s = 1 or s = 1 2 , we say that µ is a density or half-density respectively.
We say that a density µ is a positive density when µ x (v 1 , . . . , v r ) > 0 for all x ∈ M, v i ∈ E x . A nonvanishing n-form ω on an open set U ⊂ M determines a positive density |ω|, and identifies s-densities on U with functions via µ = f |ω| s . In particular, an s-density µ on M , is written µ(y) = f (y)|dy| s in local coordinates (y i ).
The canonical L 2 -space. Note that two half-densities determine a 1-density; indeed, given u = f µ where U Φ is defined as in (1.8) and U µ , U ν in (4.1). Vertical bundles. Consider a fiber bundle F = ∪ x∈M F x over a manifold M . With f ∈ F , we associate x = π F (f ) and f x the corresponding point in the fiber F  . Suppose F = E is a vector bundle. Then the vertical space V e (E) = T ex E x above e ∈ E (here x = π E (e)) is naturally identified with E x . Hence, the vertical bundle V(E) is isomorphic to the pull-back bundle (π E ) * (E) = ∪ e∈E E π E (e) of E by the isomorphism of vector bundles vert E : (π E ) * (E) −→ V(E): This example may be generalised to the case of a group bundle: Example 4.2 (The vertical bundle of a Lie group bundle). Suppose F = G = ∪ x∈M G x is a Lie group bundle. The vertical space V g (G) = T gx (G x ) above g ∈ G (here x = π G (g)) is naturally isomorphic to the Lie algebra g x of the Lie group G x . Therefore, the vertical bundle V(G) is naturally isomorphic to the pull-back bundle ∪ g∈G g π G (g) = (π G ) * (g) of the corresponding Lie algebra bundle g = ∪ x∈M g x . More precisely, the natural isomorphism of vector bundles vert G :

Filtered manifolds.
A filtered manifold is a manifold M equipped with a filtration of the tangent bundle T M by vector bundles here we are using the convention that H i = T M for i > n M . For each x ∈ G, the quotient x 20 is naturally equipped with a structure of graded Lie algebra, and we denote by G x M the corresponding graded Lie group. As explained in [13], the unions are naturally equipped with a bundle structure that are called the osculating group and Lie algebra bundles over M .  An approach to a pseudodifferential calculus on the non-equiregular case is given in [1].
In what follows, it will be helpful to describe a smooth function f on GM by denoting the associated function on the fiber G x M via for each x ∈ M.
N the corresponding group morphism. This induces [13] morphisms between the osculating bundles GΦ : GM → GN and GΦ : GM → GN that we may call the group and (resp.) Lie algebra osculating maps.
Open sets of graded Lie groups are naturally equipped with a structure of filtered manifolds. The next statement summarises the main results in Section 2: Moreover, in this case, the Pansu derivative and the osculating map coincide at every x ∈ U : A Haar measure µ x on the group fiber G x M naturally identifies with a left-invariant density, which we also call µ x , on G This isomorphism allows us to lift densities x → µ x in |Λ|GM to vertical densities z →μ z viǎ µ z (vert z (V 1 ), . . . , vert z (V n )) = µ x (V 1 , . . . , V n ); V 1 , . . . , V n ∈ G x M, x = π GM (z).
This lift yields a map of sections: Γ (|Λ|GM ) → Γ (|Λ|V(GM )). We shall subsequently blur the distinction between a Haar density on M , its lift to a vertical density on GM , and a smooth Haar system on GM . Conversely, as dim(|Λ|G x M ) = 1 at every x ∈ M , any vertical density κ on GM may be written as in (4.2) for a unique smooth function κ on GM . This defines an isomorphism of topological vector spaces I µ : C ∞ (GM ) −→ C ∞ (GM, |Λ|V);κ −→ κ =κµ.
In the next paragraph, we will use the (smooth Fréchet) bundle S(GM, |Λ|V) of fiberwise Schwartz densities; that is, the collection {κ x : z x → κ x (z x )} x∈M of densities on the fibers G x M for which the function z x →κ(z x ) is Schwartz.

Schwartz vertical densities.
A vertical density κ is said to be a Schwartz vertical density when the corresponding function κ ∈ C ∞ (GM ) from (4.2) is in Γ c (S(GM )). Although this definition requires the choice of a smooth Haar system µ = {µ x } x∈M on M , the resulting property is independent of µ as the x-support of x → κ x is assumed to be compact. We denote by Γ c (S(GM, |Λ|V)) the space of vertical Schwartz densities (compactly supported over M ). Although these spaces are independent of the choice of Haar system, fixing such a Haar system µ establishes an identification between Γ c (S(GM )) and Γ c (S(GM, |Λ|V)) via the isomorphism of topological vector spaces: 4.4.1. Convolution kernels as vertical Schwartz densities. We start with the following general observation: Lemma 4.6. Let F : GM → GN be a smooth map. Assume that F x : G x M → G y N (with y = π GN (F(z x )) is a group morphism for each x ∈ M . Then, the differential of F preserves the vertical bundles: dF (V(GM )) ⊆ V(GN ).
Proof. For each z ∈ GM and V ∈ G x M with x = π GM (z), we have: where F x (V ) := d dt t=0 F x exp GxM (tV ) . Given a map F as in Lemma 4.6, we may now define the pullback of a vertical density ν on GN to a vertical density F * ν on GM via (F * ν) z (V 1 , . . . , V n ) := ν F(z) (d zx F x (V 1 ), . . . , d zx F x (V n )); V 1 , . . . , V n ∈ T zx (G x M ) , x = π GM (z).
If in addition F is a diffeomorphism (and hence an isomorphism of filtered manifold), the pushforward is defined by F * := (F −1 ) * .
Given a morphism of filtered manifolds Φ : M → N , we may apply the above to F = GΦ the osculating map, and define the pullback (GΦ) * ν of a vertical density ν on GN to GM . If Φ is in addition a diffeomorphism, the pushforward yields the following isomorphism of topological vector spaces: GΦ * : Γ c (S (GN, |Λ|V)) → Γ c (S (GM, |Λ|V)) .
If µ and ν are vertical densities on M and N respectively, we define the operator  This result shows that convolution kernels have the geometric structure of vertical Schwartz densities.
Lemma 4.7 motivates Γ c (S (GM, |Λ|V)) as the geometric space of convolution kernels for future study of semi-classical pseudodifferential operators on filtered manifolds. It is also interesting to notice that setting κ (ε) := (δ ε ) * κ, we have for any smooth Haar system, by homogeneity of Haar measures on graded Lie groups, where Q is the (constant) homogeneous dimension of the group fibers of GM .
Proof of Lemma 4.7. Let G and H be graded Lie groups. We may assume that both groups are modeled on R n , in which case the Lebesgue measure |dz| is a Haar measure for both. If U is an open subset of G, then V(GU ) ∼ = (U × G) × g whence |Λ|V(GU ) ∼ = (U × G) × |Λ|g,