This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of ΨHDO's, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of Folland-Stein and providing us with sharp regularity estimates for hypoelliptic operators on Heisenberg manifolds; (ii) Criterions on the principal symbol of P to invert the heat operator P + ∂t and to derive the small time heat kernel asymptotics for P ; (iii) Weyl asymptotics for hypoelliptic operators which can be reformulated geometrically for the main geometric operators on CR and contact manifolds, that is, the Kohn Laplacian, the horizontal sublaplacian and its conformal powers, as well as the contact Laplacian. For dealing we cannot make use of the standard approach of Seeley, so we rely on a new approach based on the pseudodifferential approach representation of the heat kernel. This is especially suitable for dealing with positive hypoelliptic operators. We will deal with more general operator in a forthcoming paper using another new approach. The results of this paper will be used in another forthcoming paper dealing with an analogue for the Heisenberg calculus of the noncommutative geometry which, in particular, will allow us to make use in the Heisenberg setting of Connes' noncommutative geometry, including the operator theoretic framework for the local index formula of Connes-Moscovici.

This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes' pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a detailed description is still missing. In this paper, we focus on constructing an oscillating integral for noncommutative tori and laying down the main functional analysis ground for understanding Connes' pseudodifferential calculus. In particular, this allows us to give a precise explanation of the definition of pseudodifferential operators on noncommutative tori. More generally, this paper introduces the main technical tools that are used in the 2nd part of the series to derive the main properties of these operators.2010 Mathematics Subject Classification. 58B34, 58J40.

In this paper we attempt to give a systematic account on privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel [20] by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group G satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by G.(2.1)We further say that g is Carnot algebra when g w+1 = [g 1 , g w ] for w = 1, . . . , r − 1.Remark 2.2. The conditions (2.1) automatically imply that g is nilpotent of step r.Remark 2.3. Carnot algebras are also called stratified nilpotent Lie algebras (see [30]).Remark 2.4. Any commutative real Lie algebra g (i.e., any real vector space) is a step 1 Carnot algebra with g 1 = g. Remark 2.5. A classification of n-dimensional Carnot algebras of step n − 1 was obtained by Vergne [60]. A classification of rigid Carnot algebras was obtained by Agrachev-Marigo [3].In what follows, by the Lie algebra of a Lie group we shall mean the tangent space at the unit element equipped with the induced Lie bracket.3 Definition 2.6. A graded nilpotent Lie group (resp., Carnot group) is a connected simply connected nilpotent real Lie group whose Lie algebra is a graded nilpotent Lie algebra (resp., Carnot algebra).Remark 2.7. We refer to the monographs [14,25,29] detailed accounts on nilpotent Lie groups and Carnot groups.Let G be a step r graded nilpotent Lie group with unit e. Then its Lie algebra g = T G(e) is canonically identified with the Lie algebra of left-invariant vector fields on G. More precisely, with any ξ ∈ g is associated the unique left-invariant vector field X ξ on G such that X ξ (e) = ξ. In addition, as G is a connected simply connected nilpotent Lie group, its exponential map is a global diffeomorphism exp : g → G (see, e.g., [25,32]). For any ξ ∈ G, the flow exp(tX ξ ) exists for all times t ∈ R. We then havewhere exp(X ξ ) := exp(tX ξ )(e) |t=1 .In addition, the flow R ∋ t → exp(tX ξ ) is a one-parameter subgroup of G. Conversely, any one-parameter subgroup of G is generated by a (unique) left-invariant vector field. By assumption g comes equipped with a grading g = g 1 ⊕ g 2 ⊕ · · · ⊕ g r which is compatible with its Lie algebra bracket. This grading then gives rise to a family of anisotropic dilations ξ → t · ξ, t ∈ R, which are linear maps given by (2.2) t · (ξ 1 + ξ 2 + · · · + ξ r ) = tξ 1 + t 2 ξ 2 + · · · + t r ξ r , ξ j ∈ g j . 7

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold pM, Hq we get a smooth groupoid that encodes the smooth deformation of the pair MˆM to the tangent group bundle GM . This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].