Sub-Riemannian limit of the differential form spectrum of contact manifolds

Michel Rumin

September 1998

1 Introduction

One natural approach to sub-Riemannian geometry lies in the study of the behavior of Riemannian objects in the sub-Riemannian limit. This consists in blowing up the metric transversely to the Carnot distribution. These metric spaces converge in the Gromov-Hausdorff sense to the sub-Riemannian ones [10].

However, very little is known about the convergence of some even basic and linear objects as the spectrum of the Laplacians on differential forms. We begin here this study in the contact case. We will see that the non-blowing parts of the spectrum of the Laplacians, d+δ and the signature operator ∗d−d∗, concentrate and are described by their counterparts coming from the contact complex studied in [16]. In particular, an interesting infinite dimensional collapsing eigenvalues phenomenon occurs on middle degree forms. It corresponds to the special second order differential D of the contact complex. These spectrum convergences of unbounded operators are first studied in Theorems 3.5 and 3.6 through the convergence of their resolvents.

The techniques are much inspired from adiabatic limits as developed for example in [2, 4, 6, 14]. Nevertheless, the algebraic and analytic situations here are quite different, in some sense opposite to the adiabatic case, where the unexploded directions need to be integrable and form a fibration. Anyway, this approach, pointed out by J.-M. Bismut, relies on some formal resemblances between the problems. Mainly, we will see in section 3, that the contact complex occurs as a natural spectral sequence in the sub-Riemannian blow up, just like the Leray spectral sequence of the fibration does in the adiabatic case.

These algebraic structures used to predict, in a formal power series sense at first, the different parts of the spectrum that blow up or collapse at different rates. The way to turn this into the actual convergence of the resolvents will rest here on the use of some L^{2} a priori estimates. They will come from a sub-Riemannian Bochner technique developed in section 5. This one has to bypass three facts. First, the Riemannian curvature diverges here, and makes a Riemannian like Bochner formula unusable. Also, the presence of an infinite collapsing spectrum phenomenon in middle degrees precludes a classical first order Sobolev control there. Lastly, again in these degrees, the second order differential D, coming with the spectral sequence structure, can’t directly come out and be dominated by the Riemannian Laplacian.

These basic resolvents’ convergences of Laplacians or signature operator are then taken as a starting point for the study of global and local convergences of heat kernels and eta functions, for non small times. These are shown to converge to their hypoelliptic contact counterparts in section 7.

The developed techniques, and most of the results, also apply to Galois coverings of compact contact manifolds. This can be used to study the asymptotic behavior of the heat kernel on forms on the Heisenberg groups, for large time. This problem is related to ours because, through the Heisenberg dilations, the large scale Riemannian geometry of this group looks like the local sub-Riemannian one. Here, the heat associated to the contact complex appears to be the self-similar limit of the Riemannian process, as stated in Theorem 7.14.

2 Notation and heuristic study

Let M be a contact manifold of dimension 2n + 1. We denote by H the contact field, that we’ll assume transversally oriented. Therefore, we can fix a contact form θ on M such that H = ker θ. The contact condition means that dθ restricted to H is a non-degenerate 2form. We choose an (almost) complex structure J on H such that dθ(X, JY) = g_{H}(X, Y) is symmetric positive definite on H. There is no obstruction for doing this since we don’t require J to be integrable : that is to induce an integrable CR structure. Also associated to θ is the transverse Reeb field T : it is the unique vector field satisfying θ(T) = 1 and L_{T} θ = 0. We can now consider the family of metrics

g_{ε} = g_{H} + \frac{g_{T}}{ε^{2}}

where g_{H} = dθ(·, J ·), g_{T} = θ^{2} and T is orthogonal to H. We plan to study the behavior of the Riemannian spectrum of (M, g_{ε}) in the sub-Riemannian limit ε → 0.

The exterior algebra of M splits in horizontal and vertical forms, which we’ll denote by

Ω^{∗}M = Ω^{∗}H ⊕ θ ∧ Ω^{∗}H

where Ω^{∗}H are forms vanishing on T. With respect to this decomposition, the exterior differential writes d(α_{H} + θ ∧ α_{T}) = (d_{H}α_{H} + Lα_{T}) + θ ∧ (L_{T} α_{H} − d_{H}α_{T}), that is :

d = (\begin{matrix}d_{H} & L \\ L_{T} & −d_{H} \end{matrix})

where d_{H} = Π_{Ω^{∗}H}d is the horizontal part of d, L_{T} the Lie derivative along T and L is the operator on Ω^{∗}H defined by Lα = dθ ∧ α. Observe that L_{T} and L both preserve horizontal and vertical forms. Before going ahead, we now work out a conjugation on the exterior algebra. This is the isometry from (Ω_{m}^{∗}M, g_{ε}) to (Ω_{m}^{∗}M, g_{1}) defined by

C_{ε}(α_{H} + θ ∧ α_{T}) = α_{H} + ε θ ∧ α_{T} .

We consider then d_{ε} = C_{ε}d C_{ε}^{−1} and its adjoint δ_{ε} = (d_{ε})^{∗_{g_{1}}} = C_{ε}δ^{g_{ε}}C_{ε}^{−1}, such that we have Δ_{ε} = d_{ε}δ_{ε} + δ_{ε}d_{ε} = C_{ε}Δ_{g_{ε}}C_{ε}^{−1}, and for the Hodge star operator : ∗_{g_{1}} = C_{ε} ∗_{g_{ε}} C_{ε}^{−1}. The advantage of working with these operators is that they act on a fixed metric space, the spectrum being unchanged under conjugation.

Now, we have

(1)  \{ \begin{matrix}d_{ε} & = (\begin{matrix}d_{H} & ε^{−1}L \\ ε L_{T} & −d_{H} \end{matrix})  δ_{ε} = (\begin{matrix}δ_{H} & ε L_{T}^{∗} \\ ε^{−1}Λ & −δ_{H} \end{matrix}) \\ ∗_{1} & = (\begin{matrix}0 & ∗_{H} \\ ∗_{H}(−1)^{p} & 0 \end{matrix}) \\ Δ_{ε} & = (\begin{matrix}Δ_{H} + ε^{2}L_{T}^{∗} L_{T} + ε^{−2}LΛ & ε^{−1}[δ_{H}, L] + ε[d_{H}, L_{T}^{∗}] \\ ε^{−1}[Λ, d_{H}] + ε[L_{T} , δ_{H}] & Δ_{H} + ε^{2}L_{T} L_{T}^{∗} + ε^{−2}ΛL \end{matrix}) \end{matrix}

where Λ is the adjoint of L.

From the structure of these matrices, we can see the differences between the behavior of the spectrum of Δ_{ε} on functions and forms. Indeed, we have on functions Δ_{ε} = Δ_{H} +ε^{2}L_{T}^{∗} L_{T}. This is a decreasing (ε → 0) family of positive elliptic operators dominating the hypoelliptic Kohn Laplacian Δ_{H}. The maxmin principle associated to subelliptic estimates shows then that the whole spectrum (eigenvalues and eigenfunctions) of Δ_{ε} converges towards the one of Δ_{H} (see [8]).

On the contrary, part of the spectrum on forms is divergent. For example, one has Δ_{ε}(θ) = \frac{1}{nε^{2}} θ and also Δ_{ε}(dθ) = d (Δ_{ε}θ) = \frac{1}{nε^{2}} (dθ). Of course, another part of the spectrum on 1-forms is still convergent, for we can take α_{ε} = d_{ε}f_{ε} where Δ_{ε}f_{ε} = λ_{ε}f_{ε} are convergent eigenfunctions. We observe also that these forms α_{ε} = d_{H}f_{ε} + εθ ∧ L_{T} f_{ε} are asymptotically horizontal (for g_{ε}).

The aim of this is to introduce the heuristic remark that despite the divergent terms in Δ_{ε}, part of its spectrum actually converges and the corresponding eigenforms should

concentrate on E = ker (\begin{matrix}LΛ & 0 \\ 0 & ΛL \end{matrix}) (the ε^{−2} term in Δ_{ε}). We note that since L is injective

on Ω^{p}H for p ≤ n − 1 (resp. surjective onto Ω^{p}H for p ≥ n + 1), one obtains that

for p ≤ n ,  E ∩ Ω^{p}M = {α ∈ Ω^{p}M, Λα = 0}

= Ω_{0}^{p}H (the primitive horizontal forms)

for p > n ,  E ∩ Ω^{p}M = θ ∧ (Ω_{L}^{p−1}H)

= {α ∈ θ ∧ (Ω^{p−1}H), Lα = 0} .

To complete these preliminary remarks, we now stress on the fact that a special phenomenon of collapsing eigenvalues can occur in the middle dimensions spaces Ω^{n}M and Ω^{n+1}M. We show this on some particular contact manifolds : we take M to be the boundary of a circular pseudoconvex domain of ℂ^{n+1}. Moreover, we suppose that the holomorphic S^{1} action z → e^{iφ}z which preserves the CR structure of M is transverse to H. We can then fix a contact form θ such that θ(\frac{∂}{∂φ}) = 1. Now, we consider the family of holomorphic (n + 1)-forms

α_{P} = P (z_{1}, ⋯, z_{n+1})dz_{1} ∧ ⋯ ∧ dz_{n+1} ,

where P is an homogeneous polynomial of degree k. Their restriction α̃_{P} to M are closed vertical forms (because α_{P} restricted to H is of bidegree (n + 1, 0)). So (1) gives d_{H}(i_{T} α̃_{P}) = 0. Moreover, we have

∗α_{P} = ∗_{H}(i_{T} α̃_{P})
= (−1)^{\frac{n(n+1)}{2}} J(i_{T} α̃_{P}) (see [18] I thm 2)
= (−1)^{\frac{n(n+1)}{2}} i^{n}(i_{T} α̃_{P})

so that finally

(d_{ε}∗)α_{P} = (−1)^{\frac{n(n+1)}{2}} i^{n}εL_{T} α̃_{P} by (1)
= (−1)^{\frac{n(n+1)}{2}} i^{n+1}ε(n + 1 + k)α̃_{P} .

That means the α̃_{P} generate an infinite dimensional space of collapsing eigenforms for the signature operator (and of course for the Laplacian since Δ_{ε}(α̃_{P}) = ε^{2}(n + 1 + k)^{2}α̃_{P}). This family of eigenvalues has been produced using an overabundance of strong geometric hypothesizes (embeddability in ℂ^{n} and transverse S^{1} CR action) although we a priori face a (sub)Riemannian problem. One goal of this article is to understand whether this phenomenon survives to non integrable J. There, holomorphic objects will disappear but small eigenvalues should remain.

3 The contact complex and the sub-Riemannian limit

The previous remarks lead us to think that the convergent part of the spectrum on forms

concentrates on the subspace E = ker (\begin{matrix}0 & L \\ Λ & 0 \end{matrix}) of Ω^{∗}M. It must therefore be described

by differential operators acting on E and invariant under the anisotropic change of metric g_{ε}. There is a natural candidate for this : the so-called contact complex in [16].

We briefly recall its construction. One considers the differential ideal I^{∗} generated by the contact forms i.e. I^{∗} = {θ ∧ α + dθ ∧ β} and its annulator J^{∗} = {α | θ ∧ α = dθ ∧ α = 0} = E ∩ Ω^{p>n}M. We have two induced complexes d_{Q} on Ω^{∗}M/I^{∗} and J^{∗} from the de Rham one, and a second order D from Ω^{n}M/I^{n} in J^{n+1} defined the following way. For α ∈ I^{n}, let Dα = dα̃ where α̃ is any lift of α in Ω^{n}M such that dα̃ is a vertical form. D is easily seen to be well-defined and independent of θ. However, a choice of θ and of a partial complex structure J on H allows us to identify the quotient space Ω^{∗}M/I^{∗} with the primitive horizontal forms Ω_{0}^{∗}H = E ∩ Ω^{p≤n}M. In this case one can take α̃ = α_{H}^{0} − θ ∧ L^{−1}d_{H}α_{H}^{0} where α_{H}^{0} is the representative of α in Ω_{0}^{∗}H, and L^{−1} the inverse of L from Ω^{n−1}H to Ω^{n+1}H. The formula for D reads then

(2)  Dα = θ ∧ (L_{T} + d_{H}L^{−1}d_{H})α_{H}^{0}

The main properties of this construction are :

Theorem 3.1 ([16]). The complex

0 → ℝ → C^{∞}M ⟶^{d_{Q}} Ω_{0}^{1}H ⟶^{d_{Q}} ⋯ Ω_{0}^{n}H ⟶^{D} J^{n+1} ⟶^{d_{Q}} ⋯ J^{2n+1} → 0

is a resolution and the associated Laplacians are hypoelliptic.

To enlighten both the construction and the role of this complex in our problem, we show that it can be derived from spectral sequence considerations (see P. Julg [12] for still another point of view on this complex). We have to find a filtration of the exterior algebra adapted to the contact sequence and the anisotropic blow-up of metric g_{ε}. We choose a basis θ_{H}^{1}, ⋯, θ_{H}^{2n} of Ω^{1}H and θ a contact form. We’ll say a (non zero) form α is of contact-weight N_{H}α = p if it is in the linear span of θ_{H}^{I_{1}} ∧θ^{I_{2}} with p = card(I_{1})+2 card(I_{2}), I_{1} ⊂ {1, ⋯, 2n}, I_{2} ⊂ {1}. This weight depends on the choice of the basis but induces a contact-intrinsic filtration of the exterior algebra by F^{p}Ω = {α | N_{H}α ≥ p}. Equivalently a q-form is in F^{p}Ω iff it vanishes on all q-uples of vectors in ⊗^{k_{1}}H⊗^{k_{2}}T M with k_{1}+2 k_{2} < p. (And finally F^{p}Ω = Ω^{p}M ⊕ vertical forms of degree p − 1). This filtration is stable under d, so that a spectral sequence can be associated to it (see [9]).

Remark 3.2. This construction has a natural generalization in (regular) Carnot-Caratheo-dory geometry [10]. If a manifold is given a r-step bracket generating distribution H, one can intrinsically filter the forms using the d non-decreasing weight N_{H_{1}} +2 N_{H_{2}} ⋯+r N_{H_{r}}, where H_{1} = H, H_{k} = [H, H_{k−1}]. (We hope to discuss some of the analytic properties of the associated spectral sequence elsewhere.)

It turns out that in our contact case this spectral sequence is merely equivalent to the contact complex as follows from the following straightforward proposition, stated in the standard notation of [9].

Proposition 3.3. 1. E_{0}^{p,q} (= \frac{F^{p}Ω^{p+q}}{F^{p+1}Ω^{p+q}} ) = 0 unless q = 0 or −1

E_{0}^{p,0} = horizontal (partial) p-forms E_{0}^{p,−1} = vertical (p − 1)-forms

d_{0} = (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) (induced by d on E_{0})

2. E_{1} (= ker d_{0}/ Im d_{0}) ≃ E = primitive horizontal forms ⊕ coprimitive vertical forms d_{1} = d_{Q} (including the fact that d_{1} = d_{Q} = 0 on E_{1}^{n,0})

3. E_{2} (= ker d_{Q}/ Im d_{Q}) = H_{Q} ≃ de Rham cohomology except in degree n and n + 1. d_{2} = 0 except in degree n where d_{2} = D.

4. E_{3} = E_{2} except E_{3}^{n,0} = ker D/ Im d_{Q} and E_{3}^{n+2,−1} = ker d_{Q}/ Im D

5. E_{3} = E_{4} = ⋯ ≃ de Rham cohomology.

Remark 3.4. Before going ahead, we insist on two particular features of this spectral sequence

• The first one is that D = d_{2} is only a priori defined from E_{1}^{n,0} ∩ ker d_{Q} into E_{1}^{n+2,−1}/ Im d_{Q}. But these spaces are actually E_{1}^{n,0} (= Ω_{0}^{n}H) and E_{1}^{n+2,−1} (= J^{n+1} by Theorem (3.1)), because

d_{Q} : E_{1}^{n,0} → E_{1}^{n+1,0} = 0
and d_{Q} : E_{1}^{n+1,−1} = 0 → E_{1}^{n+2,−1}

• The reason why this spectral sequence really computes the cohomology of M and not only its graded part (see [9])

Gr (H_{DR}) = ⊕_{p} \frac{F^{p}H_{DR}}{F^{p+1}H_{DR}}

is that they coincide! Indeed, this filtration degenerates the following way :

for q ≤ n, F^{p}H_{DR}^{q} = \begin{cases}H_{DR}^{q} & if p ≤ q \\ \{0\} & if p > q \end{cases}

for q > n, F^{p}H_{DR}^{q} = \begin{cases}\{0\} & if p < q + 1 \\ H_{DR}^{q} & if p ≥ q + 1 \end{cases}

This stems from the facts that there is no closed vertical form of degree p ≤ n (due to the injectivity of L : Ω^{p−1}H → Ω^{p+1}H), and that any form of degree p > n is cohomological to a vertical one (due to the the surjectivity of L : Ω^{p−2}H → Ω^{p}H).

We now come back to our formula for d_{ε}

d_{ε} = ε^{−1} (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) + (\begin{matrix}d_{H} & 0 \\ 0 & −d_{H} \end{matrix}) + ε (\begin{matrix}0 & 0 \\ L_{T} & 0 \end{matrix})

and observe that the components of this ε-splitting are also homogeneous with respect to

the above contact weight N_{H} and associate filtration. We mean that (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) preserves

N_{H}, (\begin{matrix}d_{H} & 0 \\ 0 & −d_{H} \end{matrix}) increases it by 1 and (\begin{matrix}0 & 0 \\ L_{T} & 0 \end{matrix}) by 2.

This possibly explains the appearance of the associated spectral sequence (and the contact complex) in our problem. In fact, there are strong algebraic similarities with the adiabatic case. Recall that one considers there a family of metrics g_{ε} = ε^{−2}g_{H} + g_{V} where, instead of our contact case, the unexploded V -directions need to form an integrable foliation with some global hypothesis on the leaves : they have to come from a compact fibration. One can then define the Leray spectral sequence associated with the d-stable filtration F^{p}Ω^{p+q} = linear span of θ_{H}^{I_{1}} ∧ θ_{V}^{I_{2}} with card(I_{1}) ≥ p, which again are homogeneous under the above change of metric^{1}.

A deep study of the adiabatic limit has been undertaken by many authors : Dai, Mazzeo-Melrose, Bismut and co-authors ... They have shown, for various geometric operators (Laplacians, signature, Dolbeault) that the asymptotic behavior when ε → 0 of the spectrum is encoded in the associated Leray spectral sequence. Roughly speaking, eigenforms associated to eigenvalues λ_{ε} of order less or equal ε^{r} converge in the nested spectral sequence’s spaces E_{r} (rather a Hodge theoretic version of them). Moreover, the magnified spectrum ε^{−r}λ_{ε} converges to the one of operators constructed from the successive differential d_{r} of the spectral sequence. This screening procedure converges to cohomology for a finite r with the spectral sequence degeneration.

This picture will apply in our sub-Riemannian problem. First of all we define the relevant Hodge versions of the quotient spectral spaces E_{r} of prop 3.3. Let

• F_{1} = ker (\begin{matrix}0 & L \\ Λ & 0 \end{matrix}) (= E) : it is the candidate limit space for the convergent part of

the spectrum.

• F_{2} = ker(d_{Q} + δ_{Q}) ⊂ F_{1} with the particularity that d_{Q} = 0 on F_{1}^{n} = F_{1} ∩ Ω^{n}M and δ_{Q} = 0 on F_{1}^{n+1} (see prop 3.3) : the limit space for de Rham cohomology, except in middle degrees.

• F_{3} = F_{2} ∩ (Ω^{n}M + Ω^{n+1}M) : will be the limit space for the collapsing spectrum.

Now we define

• P_{ε} = d_{ε} + δ_{ε} with Δ_{ε} = P_{ε}^{2} = d_{ε}δ_{ε} + δ_{ε}d_{ε}

the signature operator, when dimM = 4l − 1 = 2n + 1, is

• S_{ε} = (∗d_{ε} − d_{ε}∗)w acting on Ω^{even}M with w = (−1)^{p} on Ω^{2p}M.

Their contact counterparts are • P_{Q} = d_{Q} + δ_{Q} and Δ_{Q} = P_{Q}^{2} = d_{Q}δ_{Q} + δ_{Q}d_{Q} acting on F_{1}.

1If the Reeb field induces a fibration one can take here V = T, and see that the adiabatic limit is the opposite of the sub-Riemannian one in this situation.

• P_{D} = D + D^{∗} acting on F_{2} (with the convention, motivated by prop 3.3, that D = 0 outside degree n)

• Δ_{D} = D^{∗}D + DD^{∗} on F_{2} (≠ 0 only on F_{3}) (a fourth order Laplacian)

the contact-signature operators :

• S_{Q} = (∗d_{Q} − d_{Q}∗)w on F_{1}^{even}

• S_{D} = (−1)^{l}D∗ on F_{2}^{even} (≠ 0 only on F_{2}^{n+1}).

We can formulate two first theorems, describing respectively the non-collapsing and collapsing spectrum on a compact contact manifold endowed with the family of metrics g_{ε}. All convergences are relative to the norm of bounded operators in L^{2}. Π denotes orthogonal projection.

Theorem 3.5. ∃λ ∈ ℂ such that

1. Π_{F_{3}^{⊥}}(λ − P^{ε})^{−1}Π_{F_{3}^{⊥}} ⟶_{ε→0} (λ − P^{Q})^{−1}Π_{F_{1}∩F_{3}^{⊥}};

2. the same holds with the signatures S_{ε} and S_{Q} instead;

3. Π_{F_{3}^{⊥}}(λ − Δ^{ε})^{−1}Π_{F_{3}^{⊥}} ⟶_{ε→0} (λ − Δ^{Q})^{−1}Π_{F_{1}∩F_{3}^{⊥}}.

Theorem 3.6. ∃λ ∈ ℂ such that

1. (λ − \frac{P_{ε}}{ε} )^{−1} ⟶_{ε→0} (λ − P_{D})^{−1}Π_{F_{2}};

2. idem with \frac{S_{ε}}{ε} and S_{D};

3. (λ − \frac{Δ_{ε}}{ε^{2}} )^{−1} ⟶_{ε→0} (λ − Δ_{D})^{−1}Π_{F_{2}}.

One classical consequence (see [15]) of convergence in the norm resolvent sense of self-adjoint operators is the convergence both of spectral projections (in particular of the eigenforms) and of the spectrum to the limit operator one’s (over bounded intervals of ℝ).

4 Miscellaneous formulas

To prove Theorems 3.5 and 3.6, we need first to relate the horizontal H-operators to the quotiented Q-ones. Also, in view of formula (1), we have to take care of the actions of

Δ_{H}, L_{T}, d_{H}, ⋯ with respect to the eigenspaces of (\begin{matrix}LΛ & 0 \\ 0 & ΛL \end{matrix}) (which is the ε^{−2} term of

Δ_{ε}). We recall these are the factors of the Lefschetz decomposition :

Ω^{∗}H = Ω_{0}^{∗}H ⊕ LΩ_{0}^{∗}H ⊕ ⋯ ⊕ L^{n}Ω_{0}^{∗}H.

where Ω_{0}^{∗}H denotes the primitive horizontal forms, and we have

(3)  \{ \begin{matrix}ΛL & = (k + 1)(n − p − k)^{+} Id & on L^{k}Ω_{0}^{p}H \\ [Λ, L] & = (n − p) Id & on Ω^{p}H \end{matrix}

These formulas are purely algebraic (order 0), relying only on the existence on H of the Hermitian structure induced by the relation g_{H} = dθ(·, J · ) with J almost complex structure (see [18]). We compute now the basic first and second order identities. We have

(4)  d_{H}^{2} = −LL_{T} and [d_{H}, L] = [d_{H}, L_{T}] = [L, L_{T}] = 0

Proof. Expansion of d^{2} = (\begin{matrix}d_{H} & L_{T} \\ L & −d_{H} \end{matrix})^{2} = 0.

Perhaps more surprising, the following key formula holds even for not integrable J. It can thus be called an almost-contact-Kähler identity

(5)  [Λ, d_{H}] = −δ_{H}^{J} (= −J^{−1}δ_{H}J).

Proof. Weil’s book ([18]) contains a proof of the Kähler identity [Λ, d] = −δ^{J} in the almost Kähler case : that is on symplectic manifolds with a compatible almost complex structure J and metric g = ω(·, J ·). This first order formula depends only on the following facts

• dω = 0 ⟺ [d, L] = 0,

• δ = − ∗ d∗,

• the above relation between g, ω and J.

In our situation, we have such a partial structure on H, taking

• ω = dθ,

• d_{H} and δ_{H} = − ∗_{H} d_{H}∗_{H} (see [16]).

So, a proof “à la Weil” (without any connection and framing) applies here.

Remark 4.1. The fact that (4) holds without zero order term will only be crucial in the study of the collapsing spectrum, where we’ll need a Bochner like formula with controllable

’error’ terms for \frac{Δ}{ε^{2}}.

We can express the action of d_{H} with respect to the Lefschetz decomposition. We have on Ω_{0}^{p}H

(6)  \{ \begin{matrix}δ_{H} & = δ_{Q} \\ d_{H} & = d_{Q} − \frac{L}{n − p + 1}δ_{Q}^{J} . \end{matrix}

Proof. The first formula comes from [Λ, δ_{H}] = 0 (by (4)).

The second one is a refinement of prop 4 in [16] thanks to the new (5)

We now come to second order relations^{2}

(7)  \{ \begin{matrix}L_{T} & = \frac{1}{n − p}δ_{Q}^{J}d_{Q} + \frac{1}{n − p + 1}d_{Q}δ_{Q}^{J} & on Ω_{0}^{p}H with p < n \\ L_{T} & = i_{T} D + d_{Q}δ_{Q}^{J} & on Ω_{0}^{n}H. \end{matrix}

2All order references are given with respect to the natural contact and hypoelliptic weight : 1 for an horizontal derivative, 2 for a transversal one.

Proof. Using (6) and [d_{H}, L] = 0, one can expand d_{H}^{2} = −LL_{T} (4). This leads to the first equation. Relation (2) : i_{T} D = L_{T} − d_{H}L^{−1}d_{H} and (6) immediately give the second one (d_{Q} = 0 on Ω_{0}^{n}H).

Using (7) and L = L^{J} (because dθ is a (1, 1) form), one can see that L_{T} preserves the Lefschetz decomposition and moreover, one has on the whole Ω^{∗}H :

(8)  \{ \begin{matrix}L_{T}^{∗} + L_{T}^{J} & = 0 \\ [Λ, L_{T}] & = 0 \end{matrix}

We study Δ_{H}. We have on Ω_{0}^{p}H with p ≤ n

(9)  Δ_{H} = Δ_{Q} + \frac{d_{Q}^{J}δ_{Q}^{J}}{n − p + 1} − \frac{L}{n − p + 2}(δ_{Q}δ_{Q}^{J} + δ_{Q}^{J}δ_{Q})

(where Δ_{Q} = d_{Q}δ_{Q} on Ω_{0}^{n}H).

Proof. This follows from Δ_{H} = d_{H}δ_{H} + δ_{H}d_{H} and (6).

One consequence of (9) is the fact that Δ_{H} almost preserves the decomposition Ω^{∗}H = Ω_{0}^{∗}H ⊕Im L. Indeed, δ_{Q}δ_{Q}^{J} +δ_{Q}^{J}δ_{Q} is a first order operator, vanishing when J is integrable. This can be deduced from the decomposition

(10)  d_{Q} = ∂_{Q} + ¯∂_{Q} + N + N̄, where N = Π_{Ω_{0}^{p+2,q−1}H}d_{Q}Π_{Ω_{0}^{p,q}H}

is an algebraic expression of the Nijenhuis tensor of J. So, we have shown the following, on Ω_{0}^{p}H = Ω_{0}^{p}H ⊕ (Im L)^{p}, p ≤ n,

(11)  Δ_{H} = (\begin{matrix}Δ_{Q} + \frac{d_{Q}^{J}δ_{Q}^{J}}{n − p + 1} & P_{N}^{(1)∗} \\ P_{N}^{(1)} & Π_{L}Δ_{H}Π_{L} \end{matrix}) ,

where P_{N}^{(1)} = −\frac{L}{n−p+2}(δ_{Q}δ_{Q}^{J} + δ_{Q}^{J}δ_{Q}).

In fact, Δ_{H} almost preserves (modulo first order differential operators) the full Lefschetz decomposition. Even more, [Δ_{H}, L] = first order. This comes easily from (5) and decomposition of (4) : d_{H}^{2} = −LL_{T} with respect to the bigraduation of Ω^{∗}H. This follows also immediately from the fact that Δ_{H} is an almost scalar operator on Ω^{p,q}H with principal part a Folland-Stein operator (see [16] prop 2):

(12)  Δ_{H} = Δ_{K} + i(p − q)L_{T} + first order,

where Δ_{K} = − ∑_{i=1}^{n}(X_{i}^{2} + Y_{i}^{2}) for a J-orthonormal base of H, is the Kohn Laplacian. Two other important properties of Δ_{H} also come from this.

• The first one is that Δ_{H} almost respects the complex bigrading of Ω^{∗}H. More precisely, the previous computations show that Δ_{H} = Δ_{H}^{J} + P_{J}^{(1)} where P_{J}^{(1)} is a first order operator vanishing when J is integrable and invariant under the transverse Reeb flow.

• Δ_{H} is hypoelliptic on Ω^{p,q}H when |p − q| < n.

In fact choosing a sub-Riemannian connection ∇ (like in [16], for example, but the horizontal part of the Levi-Cevita connection would also fit), one can obtain the following explicit a priori L^{2} estimates on Ω^{p}M (resp Ω_{0}^{p}H) for p < n

(13)  \{ \begin{matrix}(Δ_{H}α, α) + C∥α∥^{2} & ≥ \frac{1}{n}∥∇_{H}α∥^{2} \\ (Δ_{Q}α, α) + C∥α∥^{2} & ≥ \frac{n − p}{n − p + 2}(Δ_{H}α, α) \end{matrix}

where C is a constant depending only on the norm of the curvature of ∇ (including tensors like the Nijenhuis of J and its first horizontal covariant derivative, L_{T} J and horizontal curvature).

Proof. (see prop 6, 7 part I and lemma 10 part II of [16]). The first control is the basic L^{2} estimate for the Folland-Stein operators. This is obtained by writing

L_{T} = \frac{1}{n} ∑_{i=1}^{n}[∇_{Y_{i}}, ∇_{X_{i}}] + first order,

so that,

|(L_{T} α, α)| ≤ \frac{2}{n} ∑_{i=1}^{n} ∥∇_{Y_{i}}α∥∥∇_{X_{i}}α∥
≤ \frac{1}{n}(∇_{H}^{∗}∇_{H}α, α) + first order .

The second estimate relies on the use of auxiliary normalized Laplacians. Namely, (7) suggests to define, on Ω_{0}^{p}H for p < n, the following

(14)  Δ_{Q_{N}} = \frac{1}{n − p}δ_{Q}d_{Q} + \frac{1}{n − p + 1}d_{Q}δ_{Q} = δ_{Q_{N}} d_{Q_{N}} + d_{Q_{N}} δ_{Q_{N}} ,

with d_{Q_{N}} = \frac{1}{\sqrt{n − p}}d_{Q}. The basic property of these Laplacians is that they nearly preserve

J, because by (7),

(15)  Δ_{Q_{N}} − Δ_{Q_{N}}^{J} = (L_{T} J) + P_{Nijenhuis}^{(1)} = first order.

Now, (9) can be rewritten, on Ω_{0}^{p}H

(16)  Δ_{H} = (n − p)Δ_{Q_{N}} + d_{Q_{N}} δ_{Q_{N}} + d_{Q_{N}}^{J} δ_{Q_{N}}^{J} + P_{N}^{(1)} .

Together with d_{Q_{N}}^{J} δ_{Q_{N}}^{J} ≤ Δ_{Q_{N}}^{J} = Δ_{Q_{N}} + P_{N}^{(1)}, we obtain that

Δ_{H} + first order ≤ (n − p + 2)Δ_{Q_{N}} ≤ \frac{n − p + 2}{n − p}Δ_{Q} ,

which gives the second control, the first order part being absorbed by Cauchy-Schwartz and loosing an arbitrarily small amount of (Δ_{H}α, α).

Remark 4.2. These normalized Laplacians, that appear as a convenient tool here, have also the important feature that they nearly commute (modulo lower order terms) with all our algebra of operators : d_{Q} and δ_{Q} (of course), but also d_{Q}^{J} and δ_{Q}^{J} (because Δ_{Q_{N}} ≃ Δ_{Q_{N}}^{J}), except in middle degree.

Finally, we notice that the term [L_{T} , δ_{H}] arising in formula (1) for Δ_{ε} is a first order (horizontal) operator. More precisely,

[L_{T} , δ_{H}] = −[J^{−1}(L_{T} J), δ_{H}]

Proof. [L_{T} , d_{H}] = 0 (by (2)) implies [L_{T}^{∗} , δ_{H}] = 0. Then (8) gives

L_{T} + L_{T}^{∗} = L_{T} − L_{T}^{J} = L_{T} − J^{−1}L_{T} (J ·) = −J^{−1}(L_{T} J)

therefore,

[L_{T} , δ_{H}] = [L_{T} + L_{T}^{∗} , δ_{H}] = −[J^{−1}(L_{T} J), δ_{H}] .

Collecting (2) and the formulas of this section leads finally to the following expression of Δ_{ε} in the splitting, for p ≤ n,

Ω^{p}M = Ω_{0}^{p}H ⊕ θ ∧ (Ω_{0}^{p−1}H) ⊕ (Im L)^{p} ⊕ θ ∧ (Im L)^{p−1}
(17)  Δ_{ε} = D_{ε,H} + ε^{−2}D_{L}^{2} + ε^{2}D_{T}^{2} + Q_{N}^{(1)} + εQ_{J}^{(1)}

with

D_{ε,H} = (\begin{matrix}Δ_{Q} + \frac{1}{n − p + 1}d_{Q}^{J}δ_{Q}^{J} & −\frac{d_{Q}^{J}}{ε} & 0 & 0 \\ −\frac{δ_{Q}^{J}}{ε} & Δ_{Q} + \frac{1}{n − p + 2}d_{Q}^{J}δ_{Q}^{J} & −\frac{d_{Q}Π_{0}Λ}{ε(n − p + 2)} & 0 \\ 0 & −\frac{Lδ_{Q}}{ε(n − p + 2)} & Π_{L}Δ_{H}Π_{L} & −\frac{Π_{L}d_{H}^{J}Π_{L}}{ε} \\ 0 & 0 & −\frac{Π_{L}δ_{H}^{J}Π_{L}}{ε} & Π_{L}Δ_{H}Π_{L} \end{matrix} )

and

D_{L}^{2} = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & n − p + 1 & 0 & 0 \\ 0 & 0 & LΛ & 0 \\ 0 & 0 & 0 & ΛL \end{matrix}) = LΛΠ_{H} + ΛLΠ_{T}

D_{T}^{2} = (\begin{matrix}L_{T}^{∗} L_{T} & 0 & 0 & 0 \\ 0 & L_{T} L_{T}^{∗} & 0 & 0 \\ 0 & 0 & L_{T}^{∗} L_{T} & 0 \\ 0 & 0 & 0 & L_{T} L_{T}^{∗} \end{matrix} ) = L_{T}^{∗} L_{T} Π_{H} + L_{T} L_{T}^{∗} Π_{T}

Q_{N}^{(1)} = (\begin{matrix}0 & 0 & P_{N}^{(1)∗} & 0 \\ 0 & 0 & 0 & P_{N}^{(1)∗} \\ P_{N}^{(1)} & 0 & 0 & 0 \\ 0 & P_{N}^{(1)} & 0 & 0 \end{matrix}) = Π_{L}Δ_{H}Π_{0} + Π_{0}Δ_{H}Π_{L} (by (11))

Q_{J}^{(1)} = Π_{T} [L_{T} , δ_{H}]Π_{H} + Π_{H}[d_{H}, L_{T}^{∗}]Π_{T} ,

where Π_{H}, Π_{T}, Π_{0} and Π_{L} denote respectively the orthogonal projections on the horizontal, vertical, primitive forms and on Im L. The reason for this choice of splitting will become clear in next section.

5 A priori estimates

We now briefly describe the method we’ll follow to prove the resolvent convergence Theorems 3.5 and 3.6. As already mentioned it is inspired by adiabatic techniques as developed by Bismut and co-authors in [4] and [2]. This consists here of two main steps. First, ob-

tain a priori estimates of some Sobolev L^{2} norm of α from (Δ_{ε}α, α) or (\frac{Δ_{ε}}{ε^{2}} α, α). Then,

use them to justify some formal asymptotic inversion of λ − P_{ε} or λ − \frac{P_{ε}}{ε} in matrix form. By formal, we mean treating the differential parts of the expression as a priori bounded ones. The associated spectral sequence will naturally arise, as in the adiabatic case, while computing the successive asymptotics of the resolvents (λ − ε^{−r}P_{ε})^{−1}.

A well-known means to obtain a priori estimates from an expression (Δα, α) is to use a Bochner type formula Δ = ∇^{∗}∇ + curvature. The classical Riemannian one can’t apply here, because the curvature of the metrics g_{ε} diverges. In fact, one can show (see [16] II.5) that the sectional curvatures have the following behavior (for X unitary ∈ H):

K_{ε}(X, T) ∼ \frac{1}{4ε^{2}} and K_{ε}(X, JX) ∼ −\frac{3}{4ε^{2}} .

So that the Ricci curvature tensor is indefinite :

R_{ε}(T, T) ∼ \frac{n}{2ε^{2}} , R_{ε}(X, X) ∼ −\frac{1}{2ε^{2}} .

Even the scalar curvature goes to −∞ as −\frac{n}{2ε^{2}} .^{3}

We remark also that trying to use a more suited sub-Riemannian connection (preserving horizontality, like the one in [16] II for example) instead of Levi Civita’s could possibly be fruitful for estimates based on (Δ_{ε}α, α) but will introduce quite uncontrollable terms

like (\frac{curvature}{ε^{2}}α, α) while studying (\frac{Δ_{ε}}{ε^{2}} α, α) (for the collapsing spectrum). It’s why we turn ourself towards a formula like (17) comparing the two geometric Laplacians Δ_{ε} and Δ_{Q} (instead of ∇^{∗}∇). Observe (17) has the characteristic that the curvature like “error terms” Q^{(1)} are only of non diagonal type (and we hope that the non-primitive and horizontal components collapse on the bounded spectrum).

Although formula (17) has a lot of positive diagonal terms, it still contains a divergent one of order 1. We will absorb it in a positive term, even if weakening the diagonal. This is the aim of the following Bochner type formula. We use the decomposition of Ω^{p}M described in (17) and note Π_{H,0}, Π_{T,0}, Π_{H,L}, Π_{T,L} the projections on the different factors.

Lemma 5.1. (Sub-Riemannian Bochner formula) One has on Ω^{p}M, for p ≤ n,

Δ_{ε} = Δ_{Q}Π_{H,0} + \frac{K_{ε}^{∗}K_{ε}}{n − p + 1} + (Δ_{Q} + \frac{1}{n − p + 2}(d_{Q}^{J}δ_{Q}^{J} − d_{Q}δ_{Q}))Π_{T,0} + L_{ε}^{∗}L_{ε}

+ Π_{H,L}(Δ_{H} − \frac{1}{2}d_{H}^{J}Π_{L}δ_{H}^{J})Π_{H,L} + \frac{M_{ε}^{∗}M_{ε}}{2} + Π_{T,L}Δ_{H}Π_{T,L}

+ \frac{1}{ε^{2}} P_{L}^{2} + ε^{2}D_{T}^{2} + Q_{N}^{(1)} + εQ_{J}^{(1)}.

3Notice that the “universal” constants ruling the asymptotics are just the curvatures of the Heisenberg group. This is because our contact manifold viewed in the magnified metrics \frac{g_{ε}}{ε^{2}} = \frac{g_{H}}{ε^{2}} + \frac{g_{T}}{ε^{4}} converges to its tangent cone, the Heisenberg group.

where

K_{ε} = (δ_{Q}^{J} −\frac{n − p + 1}{ε} 0 0)
L_{ε} = (0 \frac{Lδ_{Q}}{n − p + 2} −\frac{1}{ε} 0)
M_{ε} = (0 0 Π_{L}δ_{H}^{J}Π_{L} −\frac{2}{ε})

and

P_{L}^{2} = (LΛ − 1)Π_{H,L} + (ΛL − 2)Π_{T,L} = (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & LΛ − 1 & 0 \\ 0 & 0 & 0 & ΛL − 2 \end{matrix}) .

Proof. The principle of this decomposition is the absorption of the divergent non diagonal terms of (17) by repeated applications of Cauchy-Schwartz, starting from the first (less positive diagonal term) to the fourth (more positive) factor in Ω^{p}M. We observe that except for the two first terms (that will have strong geometric and analytic senses), the proposed decomposition of Δ_{ε} is not so canonical. We mean the non-diagonal terms could have been balanced by other combinations of diagonal ones, changing so the L_{ε} and M_{ε}. This doesn’t matter as the goal of lemma 5.1 is to obtain a priori estimates.

We come to this. Recall that (section 3)

F_{1} = primitive horizontal forms ⊕ coprimitive (∈ ker L) vertical forms .

Definition 5.2. Let ∥ ∥_{1,ε} be the following Sobolev L^{2} norm

• For α ∈ Ω^{p}M, p ≠ n, n + 1,

∥α∥_{1,ε}^{2} = ∥α∥_{1,H}^{2} + ε^{2}∥α∥_{1,T}^{2} + \frac{1}{ε^{2}} ∥Π_{F_{1}^{⊥}}α∥_{0}^{2} + ∥α∥_{0}^{2}

• For α ∈ Ω^{n}M,

∥α∥_{1,ε}^{2} = ∥Π_{F_{1}^{⊥}}α∥_{1,H}^{2} + ∥δ_{Q}α_{H,0}∥_{0}^{2} + ε^{2}∥α∥_{1,T}^{2} + \frac{1}{ε^{2}} ∥Π_{(ker Λ)^{⊥}}α∥_{0}^{2}

+ ∥δ_{Q}^{J}α_{H,0} − \frac{α_{T,0}}{ε} ∥_{0}^{2} + ∥α∥_{0}^{2}

• For α ∈ Ω^{n+1}M,

∥α∥_{1,ε}^{2} = ∥ ∗ α∥_{1,ε}^{2} = ∥Π_{F_{1}^{⊥}}α∥_{1,H}^{2} + ∥d_{Q}α_{T,ker L}∥_{0}^{2} + ε^{2}∥α∥_{1,T}^{2} + \frac{1}{ε^{2}} ∥Π_{(ker L)^{⊥}}α∥_{0}^{2}

+ ∥δ_{Q}i_{T} α_{T,ker L} − \frac{Λ}{ε}α_{H,ker L}∥_{0}^{2} + ∥α∥_{0}^{2}

where the lower indices of the norms indicates the number of horizontal or transversal derivatives controlled in L^{2}.

These norms are adapted to our problem for we have Proposition 5.3. ∃C_{1}, C_{2} > 0 such that, ∀ε ∈]0, 1]

C_{1}∥α∥_{1,ε}^{2} ≤ (Δ_{ε}α, α)_{0} + ∥α∥_{0}^{2} ≤ C_{2}∥α∥_{1,ε}^{2} .

Proof. The right inequality amounts to the continuity of P_{ε} = d_{ε} + δ_{ε} from the Hilbert space H_{1,ε} = (Ω^{∗}M, ∥ ∥_{1,ε}) into H_{0} = (Ω^{∗}M, ∥ ∥_{0}) = L^{2}(Ω^{∗}M). This is clear outside the middle degrees from (1). The only point to check on Ω^{n}M is the continuity of

(α_{H,0}^{n}, α_{T,0}^{n}) → (d_{H}α_{H,0}^{n} + \frac{1}{ε}Lα_{T,0}^{n}) ,

the upper matrix line of d_{ε} (see (1)), but this is precisely, by (6) : −L(δ_{Q}^{J}α_{H,0}^{n} − \frac{1}{ε}α_{T,0}^{n}), included expression in definition of ∥α^{n}∥_{1,ε} (this strange group is also nothing else but the K_{ε}α^{n} term of lemma 5.1).

The left inequality comes of course from the Bochner type formula lemma 5.1.

First, the so-called P_{L}^{2} is actually positive due to LΛ ≥ 2Id on Ω^{p}H ∩ Im L, and ΛL ≥ 4Id on θ ∧ (Im L)^{p−1} (p ≤ n) as come from (3). Therefore

\frac{(P_{L}^{2}α, α)_{0}}{ε^{2}} ≥ \frac{∥Π_{Im L}α∥_{0}^{2}}{ε^{2}} .

We now look at the four Laplacian like diagonal terms of lemma 5.1. We have to check that each of them controls its respective part of ∥α∥_{1,H}^{2}.

• As concerns Δ_{Q} on Ω_{0}^{p}H, we know by (13) that, for p < n,

(Δ_{Q}α_{H,0}, α_{H,0})_{0} + ∥α_{H,0}∥_{0}^{2} ≥ C∥α_{H,0}∥_{1,H}^{2}

Of course, Δ_{Q} = d_{Q}δ_{Q} is not hypoelliptic on Ω^{n}H, so that (d_{Q}δ_{Q}α_{H,0}^{n}, α_{H,0}^{n})_{0} only controls ∥δ_{Q}α∥_{0}^{2}.

• On θ ∧ Ω^{p−1}H, we have, for p ≤ n,

Δ_{Q} + \frac{1}{n − p + 2}(d_{Q}^{J}δ_{Q}^{J} − d_{Q}δ_{Q}) ≥ \frac{n − p + 1}{n − p + 2}Δ_{Q}

controls ∥α_{T,0}∥_{1,H}^{2} (modulo C∥α∥_{0}^{2}) by (13).

• We have on Ω_{Im L}^{p}H, for p ≤ n,

Π_{L}(Δ_{H} − \frac{1}{2}d_{H}^{J}Π_{L}δ_{H}^{J})Π_{L} ≥ Π_{L}(Δ_{H}^{J} − \frac{1}{2}d_{H}^{J}δ_{H}^{J} + 1st order)Π_{L},

for Δ_{H} = Δ_{H}^{J} + 1st order (section 4),

≥ Π_{L}(δ_{H}^{J}d_{H}^{J} + \frac{1}{2}d_{H}^{J}δ_{H}^{J} + 1st order)Π_{L}

≥ \frac{1}{2}Π_{L}(Δ_{H}^{J} + 1st order)Π_{L} = \frac{1}{2}Π_{L}(Δ_{H} + 1st order)Π_{L},

controls ∥α_{H,L}∥_{1,H}^{2} by (13) (extends for Δ_{H} on Ω_{L}^{n}H because [Δ_{H}, L] is first order (section 4)).

• On θ ∧ Ω_{L}^{p−1}H with p ≤ n, Δ_{H} controls ∥α_{T,L}∥_{1,H}^{2} by (13).

We now look at the “curvature terms” Q_{N}^{(1)} and εQ_{J}^{(1)}. They have no diagonal component from Ω_{0}^{p}H in itself by (17). One can therefore obtain, using Cauchy-Schwartz, the following control : ∃K such that ∀ε ∈]0, 1],

|(Q_{N}^{(1)}α, α)| + ε|(Q_{J}^{(1)}α, α)| ≤ 2K∥α∥_{0}∥Π_{F_{1}^{⊥}}α∥_{1,H} ≤ K(c^{−1}∥α∥_{0}^{2} + c∥α∥_{1,H}^{2})

for any arbitrarily small c.

Lastly, we still have to understand the domination of \frac{1}{ε^{2}} ∥α_{T,0}∥_{0}^{2} for p < n. Thanks to previous work and lemma 5.1, we now know that (Δ_{ε}α, α)_{0} + ∥α∥_{0}^{2} controls C∥α∥_{1,H}^{2} and

∥K_{ε}α∥_{0}^{2} = ∥δ_{Q}^{J}α_{H,0} − \frac{n − p + 1}{ε}α_{T,0}∥_{0}^{2} ≥ \frac{1}{ε^{2}} ∥α_{T,0}∥_{0}^{2} − C′∥α_{H,0}∥_{1,H}^{2} .

Before going on, we remark that the deficiency of control of ∥Π_{F_{3}}α∥_{1,H}^{2} by (Δ_{ε}α, α)_{0} in middle degree is unavoidable. The contrary would contradict, by the compact Sobolev embedding of H_{1,H} space in L^{2}, the infinite dimensional collapsing spectrum phenomenon we already observed in section 2.

This also precludes the possibility of independent controls of ∥δ_{Q}^{J}α_{H,0}∥_{0} and \frac{1}{ε}∥α_{T,0}∥_{0}

instead of ∥δ_{Q}^{J}α_{H,0} − \frac{1}{ε}α_{T,0}∥_{0} = ∥K_{ε}α∥_{0} in ∥α∥_{1,ε} for α ∈ Ω^{n}M. Or else, the control

of ∥δ_{Q}^{J}α_{H,0}∥_{0} added to the already obtained one of ∥δ_{Q}α_{H,0}∥_{0} would give a domination of ∥Π_{(Ω^{n,0}H⊕Ω^{0,n}H)^{⊥}}α_{H,0}∥_{1,H}, because the system (δ_{Q}, δ_{Q}^{J}) is hypoelliptic on (Ω^{n,0}H ⊕ Ω^{0,n}H)^{⊥} ∩ Ω_{0}^{n}H as comes from (11) : Δ_{H} = d_{Q}δ_{Q} + d_{Q}^{J}δ_{Q}^{J} + 1st order on Ω_{0}^{n}H, and (12). This is still impossible since the collapsing spectrum should be described by the one of D^{∗}D, not concentrated on Ω^{n,0}H ⊕ Ω^{0,n}H. The conclusion is that the term

∥δ_{Q}^{J}α_{H,0} − \frac{n − p + 1}{ε}α_{T,0}∥_{0} = ∥K_{ε}α∥_{0} arising in lemma 5.1 and ∥ ∥_{1,ε} is analytically relevant (at least in middle degree for the moment).

It has also a geometric meaning. Namely, we observe there exist lifting maps r (implicit in [16]) from Ω_{0}^{p}H in Ω^{p}M for p ≤ n, defined by

(18)

r(α_{H,0}) = α_{H,0} − θ ∧ L^{−1}d_{H}α_{H,0}  with L^{−1} left inverse of L,
= α_{H,0} + θ ∧ \frac{1}{n − p + 1}δ_{Q}^{J}α_{H,0}  by (6) and L^{−1} = (ΛL)^{−1}Λ.

r(α_{H,0}) is the unique vertical extension of α_{H,0} such that d r(α_{H,0}) restricted to H is primitive.

A consequence of this definition is that dr = rd_{Q} on Ω_{0}^{p}H, p < n (dr(α_{H,0}) is a closed extension of d_{Q}α_{H,0} and so is r(d_{Q}α_{H,0})) and by definition (section 3) D = dr on Ω_{0}^{n}H. This means that r, together with the projection Π_{H,0} on Ω_{0}^{∗}H, realize an homotopy between the de Rham and contact complexes.

Now, we observe that

K_{ε}α = 0 = δ_{Q}^{J}α_{H,0} − \frac{n − p + 1}{ε}α_{T,0} and P_{L}α = 0 = Π_{L}α,

are precisely the differential defining equations of Im r (through the conjugation C_{ε}(α_{H} + θ ∧ α_{T}) = α_{H} + εθ ∧ α_{T} we have done since section 2).

Remark 5.4. This space Im r has stronger geometric invariance it seems to possess at first sight. It is in fact a (C^{1}) contact-invariant space. This comes from its definition

Im r = {α ∈ Ω^{p}M, α and dα restricted to H are primitive}.

Indeed, following A. Weil ([18] corollary of thm I.3), ker Λ = ker L^{n−p+1} with L = dθ ∧ · acting on H-partial p-forms. So this space doesn’t depend on the chosen adapted J on H, neither of the choice of the contact form because L → fL on partial forms when θ → fθ.

Coming back to prop 5.3, we now know we can’t hope to control \frac{1}{ε^{2}} ∥α_{T,0}∥_{0}^{2} with (Δ_{ε}α, α)_{0} in degree n, although we are aiming to prove that, even there, the bounded spectrum (ε → 0) of Δ_{ε} concentrates on F_{1}^{n} = Ω_{0}^{n}H. Even if loosing some speed of convergence, the component α_{T,0} is actually collapsing to 0.

Proposition 5.5. ∃C > 0 such that ∀ε ∈]0, 1], ∀α ∈ Ω^{n}M,

(Δ_{ε}α, α)_{0} + ∥α∥_{0}^{2} ≥ \frac{C}{ε} ∥α_{T,0}∥_{0}^{2} .

In order to prove this, we fix some notation

H_{0} = (Ω^{∗}M, ∥ ∥_{0}) = L^{2}(Ω^{∗}M)
H_{1,H} = (Ω^{∗}M, ∥ ∥_{1,H}) = L^{2} forms with 1 horizontal derivative in L^{2}

H_{0} will be our pivot space, that is we identify it with its dual H_{0}^{∗}. Let H_{1,H}^{∗} = H_{−1,H} denotes the dual of H_{1,H}. We have the dense inclusions

H_{−1,H} ⊃ H_{0} = H_{0}^{∗} ⊃ H_{1,H}
with ∥ ∥_{−1,H} ≤ ∥ ∥_{0} ≤ ∥ ∥_{1,H}.

Recall that for α ∈ H_{0}, ∥α∥_{−1,H} = sup_{∥β∥_{1,H}≤1}(α, β)_{0}.

Now, prop 5.5 immediately stems from prop 5.3 and the following

Lemma 5.6. ∃C > 0, such that ∀ε ∈]0, 1], ∀α ∈ Ω^{n}M,

C∥α∥_{1,ε}^{2} ≥ \frac{1}{ε^{2}} ∥α_{T,0}∥_{−1,H}^{2} + ∥α_{T,0}∥_{1,H}^{2} ≥ \frac{1}{ε}∥α_{T,0}∥_{0}^{2} .

Proof. The first inequality amounts to the observation that

∥α∥_{1,ε}^{2} ≥ ∥δ_{Q}^{J}α_{H,0} − \frac{α_{T,0}}{ε} ∥_{0}^{2} ≥ ∥δ_{Q}^{J}α_{H,0} − \frac{α_{T,0}}{ε} ∥_{−1,H}^{2}

≥ \frac{1}{ε^{2}} ∥α_{T,0}∥_{−1,H}^{2} − ∥δ_{Q}^{J}α_{H,0}∥_{−1,H}^{2}

with δ_{Q}^{J} continuous from H_{0} in H_{−1,H} since d_{Q}^{J} is from H_{1,H} in H_{0}.

The second inequality is of well known type in elliptic context (Peetre’s inequality). Here is a possible “elementary” proof in this hypoelliptic case. We choose a partially defined connection ∇_{H} and put, for α_{T,0} ∈ θ ∧ Ω_{0}^{n−1}H,

∥α∥_{1,H}^{2} = ∥α∥_{0}^{2} + ∥∇_{H}α∥_{0}^{2} .

Denote by Π_{[0,K]} (resp Π_{]K,+∞[}) the spectral projections associated to the spectral measures of ∇_{H}^{∗}∇_{H} in [0, K] and ]K, +∞[ for some K ≥ 0.

• Obviously,

∥Π_{[0,K]}α∥_{1,H}^{2} = (∇_{H}^{∗}∇_{H}Π_{[0,K]}α, Π_{[0,K]}α)_{0} + ∥Π_{[0,K]}α∥_{0}^{2} ≤ (K + 1)∥α∥_{0}^{2} and by duality, (K + 1)∥α∥_{−1,H}^{2} ≥ ∥Π_{[0,K]}α∥_{0}^{2}.

• We also have,

∥α∥_{1,H}^{2} ≥ (∇_{H}^{∗}∇_{H}Π_{]K,+∞[}α, Π_{]K,+∞[}α)_{0} + ∥Π_{]K,+∞[}α∥_{0}^{2} ≥ (K + 1)∥Π_{]K,+∞[}α∥_{0}^{2} . Adding these two inequalities with K + 1 = ε^{−1} gives the result.

This completes the proof of the a priori estimates we’ll need for thm 3.5. We continue this section with the study of \frac{Δ_{ε}}{ε^{2}} . We define the relevant norm here. Definition 5.7. • For α ∈ Ω^{p}M, p ≤ n, let

∥α∥_{1′,ε}^{2} = \frac{1}{ε^{2}} (Δ_{Q}α_{H,0}, α_{H,0})_{0}^{2} + \frac{1}{ε^{2}} ∥Π_{F_{1}^{⊥}}α∥_{1,H}^{2} + ∥α∥_{1,T}^{2} + \frac{1}{ε^{2}} ∥K_{ε}α∥_{0}^{2}

+ \frac{1}{ε^{4}} ∥Π_{(ker Λ})^{⊥}α∥_{0}^{2} + ∥α∥_{0}^{2}

where K_{ε}α = δ_{Q}^{J}α_{H,0} − \frac{(n−p+1)}{ε}α_{T,0} (lemma 5.1)

• ∥α∥_{1′,ε} = ∥ ∗ α∥_{1′,ε} if α ∈ Ω^{p}M, p ≥ n + 1. We’ll prove Proposition 5.8. ∃C_{1}, C_{2} > 0, such that ∀α ∈ Ω^{p}M,

C_{1}∥α∥_{1′,ε}^{2} ≤ \frac{1}{ε^{2}} (Δ_{ε}α, α)_{0} + ∥α∥_{0}^{2} ≤ C_{2}∥α∥_{1′,ε}^{2} .

Proof. Along the lines of proof of prop 5.3. We refer to it for some details.

The right inequality comes from the continuity of \frac{P_{ε}}{ε} = \frac{d_{ε}+δ_{ε}}{ε} from H_{1′,ε} = (Ω^{∗}M, ∥ ∥_{1′,ε}) into H_{0} = L^{2}. This follows from (2) and (6).

The left inequality is a use of the Bochner type formula lemma 5.1. We still have

\frac{1}{ε^{4}} (P_{L}^{2}α, α)_{0}^{2} ≥ \frac{1}{ε^{4}} ∥Π_{(ker Λ)^{⊥}}α∥_{0}^{2} .

This, associated to the diagonal terms in lemma 5.1 gives the control of \frac{1}{ε^{2}} ∥Π_{(ker Λ)^{⊥}}α∥_{1,H}^{2}. About \frac{α_{T,0}}{ε} ∈ θ ∧ Ω_{0}^{p−1}H we have two estimations.

• The first one is that the (T, 0) diagonal term of lemma 5.1 satisfies

\begin{aligned}\frac{1}{ε^{2}} (Δ_{T,0}α, α)_{0} & = \frac{1}{ε^{2}} ((Δ_{Q} + \frac{1}{n − p + 2}(d_{Q}^{J}δ_{Q}^{J} − d_{Q}δ_{Q}))α_{T,0}, α_{T,0})_{0} \\ & ≥ \frac{1}{2ε^{2}} (Δ_{Q}α_{T,0}, α_{T,0})_{0} . \end{aligned}

• And also, following lemma 5.6, ∃C > 0 such that ∀ε ≤ 1,

\begin{aligned}\frac{1}{ε^{2}} ∥α_{T,0}∥_{−1,H}^{2} & ≤ ∥δ_{Q}^{J}α_{H,0} − \frac{α_{T,0}}{ε} ∥_{0}^{2} + C∥α∥_{0}^{2} \\ & ≤ ∥\frac{K_{ε}α}{ε}∥_{0}^{2} + C∥α∥_{0}^{2} . \end{aligned}

Now we have seen in lemma 5.6, that ∃K > 0, ∀α ∈ θ ∧ Ω_{0}^{p−1}H,

∥α∥_{0}^{2} ≤ K^{−1}∥α∥_{−1,H}^{2} + K∥α∥_{1,H}^{2}

where one can take ∥α∥_{1,H}^{2} = ∥α∥_{0} + (Δ_{Q}α, α)_{0} (due to positivity of Δ_{Q} and a priori estimate (13)). Using this with K = \frac{1}{2} and α = \frac{α_{T,0}}{ε} gives

\frac{1}{2ε^{2}} ∥α_{T,0}∥_{0}^{2} ≤ \frac{2}{ε^{2}} ∥α_{T,0}∥_{−1,H}^{2} + \frac{1}{2ε^{2}} (Δ_{Q}α_{T,0}, α_{T,0})_{0}^{2} ,

and finally the control, for a C > 0,

\frac{1}{4ε^{2}} ∥α_{T,0}∥_{1,H}^{2} ≤ (\frac{Δ_{T,0}}{ε^{2}} α, α)_{0} + ∥\frac{K_{ε}α}{ε}∥_{0}^{2} + C∥α∥_{0}^{2} .

We are left with the domination of the (now not so small) curvature terms \frac{1}{ε}(Q_{J}^{(1)}α, α)_{0} and \frac{1}{ε^{2}} (Q_{N}^{(1)}α, α)_{0}. In fact, collecting our estimations, we still know that there exists a constant C_{1}, depending only of the dimension of M, and C_{2}, depending of θ, J and M (by the norms of the sub-Riemannian curvatures) such that

(19)  (\frac{Δ_{ε}}{ε^{2}} α, α)_{0} ≥ C_{1}∥α∥_{1′,ε}^{2} − C_{2}∥α∥_{0}^{2} − \frac{1}{ε}|(Q_{J}^{(1)}α, α)_{0}| − \frac{1}{ε^{2}} |(Q_{N}^{(1)}α, α)_{0}| .

\frac{1}{ε}(Q_{J}^{(1)}α, α)_{0} is easily handled like in proof of prop 5.3 : ∃K > 0 such that

\frac{1}{ε}|(Q_{J}^{(1)}α, α)_{0}| ≤ 2K∥α∥_{0}∥\frac{Π_{F_{1}^{⊥}}α}{ε}∥_{1,H}
≤ K(c^{−1}∥α∥_{0}^{2} + c∥α∥_{1′,ε}^{2})

for any arbitrary small constant c.

By (17), Q_{N}^{(1)} is a first order operator exchanging ker Λ and Im L, and preserving horizontality and verticality of forms. Thus, we can decompose

\frac{1}{ε^{2}} (Q_{N}^{(1)}α, α)_{0} = \frac{2}{ε^{2}} (Q_{N}^{(1)}α_{H,0}, α_{H,L})_{0} + \frac{2}{ε^{2}} (Q_{N}^{(1)}α_{T,0}, α_{T,L})_{0} .

- The second term is controlled by

Kε∥\frac{α_{T,0}}{ε} ∥_{1,H} ∥\frac{α_{T,L}}{ε^{2}} ∥_{0} ≤ Kε∥α∥_{1′,ε}^{2} .

- The remaining one is more delicate because α_{H,0} doesn’t collapse. One obtains,

\frac{1}{ε^{2}} |(Q_{N}^{(1)}α_{H,0}, α_{H,L})| ≤ ∥Q_{N}^{(1)}α_{H,0}∥_{0}∥\frac{α_{H,L}}{ε^{2}} ∥_{0}

≤ K_{N}∥α_{H,0}∥_{1,H}∥α∥_{1′,ε} (20) where

K_{N} = sup_{∥α_{H,0}∥_{1,H}≤1} ∥Q_{N}^{(1)}α_{H,0}∥ = |||Q_{N}^{(1)}Π_{H,0}|||_{(1,H),0}

ie the norm of Q_{N}^{(1)} as an operator from H_{1,H} ∩ Ω_{0}^{p}H in H_{0} ∩ Ω_{L}^{p}H.

We need to control ∥α_{H,0}∥_{1,H}. The discussion breaks in two cases.

For p < n, one can put on Ω_{0}^{p}H (see (13))

∥α_{H,0}∥_{1,H}^{2} = (Δ_{Q}α_{H,0}, α_{H,0})_{0}^{2} + ∥α∥_{0}^{2} .

So that, by definition of ∥ ∥_{1′,ε}, one has

∥α_{H,0}∥_{1,H} ≤ ε∥α∥_{1′,ε} + ∥α∥_{0} .

This, added in (20), concludes the proof of prop 5.8 in these degrees.

For p = n, Δ_{Q} = d_{Q}δ_{Q} isn’t hypoelliptic. We put then ∥α∥_{1,H}^{2} = ∥∇_{H}α∥_{0}^{2} + ∥α∥_{0}^{2}, where ∇_{H} is a partially defined connection. We’ll need the following control

Lemma 5.9. ∃C > 0 such that ∀α ∈ Ω^{n}M, ∀ε < 1,

∥α_{H,0}∥_{1,H} ≤ 2∥α∥_{1′,ε} + C∥α∥_{0} .

Proof. Recall that on Ω^{n}M,

∥α∥_{1′,ε} ≥ \frac{1}{ε^{2}} ∥δ_{Q}α_{H,0}∥_{0}^{2} + \frac{1}{ε^{2}} ∥δ_{Q}^{J}α_{H,0} + \frac{α_{T,0}}{ε} ∥_{0}^{2} + ∥α∥_{1,T}^{2}
≥ ∥δ_{Q}α_{H,0}∥_{0}^{2} + ∥δ_{Q}^{J}α_{H,0}∥_{0}^{2} + ∥α_{H,0}∥_{1,T}^{2} .

Now we observe that this control is elliptic. More precisely, by (11) and (12), one has on Ω_{0}^{p,q}H with p + q = n,

d_{Q}δ_{Q} + d_{Q}^{J}δ_{Q}^{J} = Δ_{H} + first order
= ∇_{H}^{∗}∇_{H} + i(p − q)L_{T} + first order ,

so that,

∥δ_{Q}α_{H,0}∥_{0}^{2} + ∥δ_{Q}^{J}α_{H,0}∥_{0}^{2} + ∥L_{T} α∥_{0}^{2} ≥ ∥∇_{H}α∥_{0}^{2} − n|(L_{T} α, α)_{0}|

+ ∥L_{T} α∥_{0}^{2} + (P_{H}^{(1)}α, α)_{0}

≥ \frac{1}{2}∥∇_{H}α∥_{0}^{2} + \frac{1}{2}∥L_{T} α∥_{0}^{2} − C∥α∥_{0}^{2}

for some constant C.

This lemma associated with (20) gives us that

\frac{1}{ε^{2}} |(Q_{N}^{(1)}α_{H,0}, α_{H,L})_{0}| ≤ K_{N}(2∥α∥_{1′,ε} + C∥α∥_{0})∥α∥_{1′,ε}

≤ 2K_{N}∥α∥_{1′,ε}^{2} + \frac{K_{N}C}{2} (c∥α∥_{1′,ε}^{2} + \frac{1}{c}∥α∥_{0}^{2})

for any arbitrary small c.

This estimation, together with (19) is seen to complete the proof of prop 5.8 in the case 2K_{N} < C_{1}, but seems to be critical otherwise. The following rescaling argument shows we can always reduce to this case 2K_{N} < C_{1}.

Indeed, consider the transformation θ → kθ for some k > 0 in the metrics g_{ε,θ} = dθ(·, J ·) + \frac{θ^{2}}{ε^{2}} . We have then • ∥α∥_{0}^{2} → k^{n+1−p}∥α∥_{0}^{2} for α ∈ Ω^{p}H

• ∥α∥_{1,H}^{2} = ∥∇_{H}α∥_{0}^{2} + ∥α∥_{0}^{2} → k^{n−p}∥∇_{H}α∥_{0}^{2} + k^{n−p+1}∥α∥_{0}^{2}

• Q_{N}^{(1)} = Π_{L}Δ_{H}Π_{0} + Π_{0}Δ_{H}Π_{L} → k^{−1}Q_{N}^{(1)} because δ_{H} → k^{−1}δ_{H} and d_{H} is unchanged.

This shows that for k ≥ 1,

K_{N,kθ} = sup \frac{∥Q_{N,kθ}^{(1)}α∥_{0,kθ}}{∥α∥_{(1,H),kθ}} ≤ \frac{1}{\sqrt{k}} K_{N,θ} ,

and therefore 2K_{N,kθ} < C_{1} for k large enough (recall that following (19), C_{1} is independent of θ). Proposition 5.8 is then proved for the family of metrics g_{ε,kθ} with ε → 0. We finally observe that g_{ε,kθ} = kg_{\frac{ε}{\sqrt{k}} ,θ} so that Δ_{ε,kθ} = \frac{1}{k}Δ_{\frac{ε}{\sqrt{k}} ,θ}, giving thus prop 5.8 for our original family g_{ε′,θ} with ε′ = \frac{ε}{\sqrt{k}} → 0, at least for ε′ small enough. Of course, the controls extend up to any positive constant, as can been seen with the help of the classical Bochner formula. Indeed, the Riemannian curvature stay bounded there, giving uniform a priori L^{2} estimates.

Before leaving this section, we emphasize the geometric meaning of the controls of prop 5.8. We already knew from prop 5.3 and 5.5 that the bounded eigenforms of Δ_{ε} (ε → 0) should concentrate on F_{1}, which geometrically interprets as the E_{1} term of the contact spectral sequence of prop 3.3. Now, prop 5.8 tells us that bounded eigenforms for \frac{Δ_{ε}}{ε^{2}} (as harmonic forms for example) have to collapse (at least in weak-limit sense for the moment) on F_{2} = F_{1} ∩ ker Δ_{Q} : the Hodge version of the E_{2} term of the spectral sequence. Moreover, the controlled term \frac{1}{ε}∥K_{ε}α∥_{0} in ∥α∥_{1′,ε} shows that the weak limit of C_{ε}^{−1}α has to be in ker K_{ε}C_{ε} = ker K_{1}, that is, as already mentioned, in the image of the homotopy operator r from the contact to the de Rham complex. Now, we remember that d_{ε}C_{ε}r = εD on Ω_{0}^{n}H, where D is the second order operator of the contact complex. Of course, this D wasn’t directly controllable (in L^{2} norm) by the first order \frac{d_{ε}+δ_{ε}}{ε}, but nevertheless, indirectly appears here through the control of \frac{1}{ε}∥K_{ε}α∥_{0} and this homotopical formula.

6 Proofs of Theorems 3.5 and 3.6

We will do it here for P_{ε} = d_{ε} + δ_{ε}. As will become clear, the proof would equally work for the signature operator S_{ε} = (∗d_{ε} − d_{ε}∗)w. The general method, widely used in the adiabatic setting (see [2, 4]) will rely on the previous a priori L^{2} estimates and trying to solve, in a matrix form, the equations for λ ∉ ℝ,

(λ − P_{ε})β_{ε} = α with α ∈ F_{3}^{⊥}
and (λ − \frac{P_{ε}}{ε})β_{ε} = α .

Recall that we have to work orthogonally to

F_{3} = (ker δ_{Q} ∩ F_{1}^{n}) ⊕ (ker d_{Q} ∩ F_{1}^{n+1})

in the first case because it is the expected limit space for the collapsing spectrum of P_{ε}. (F_{3} is a closed space in L^{2} if δ_{Q} and d_{Q} are taken in distributional sense as continuous operators from L^{2} = H_{0} in (H_{1,H})^{∗} = H_{−1,H}).

First, we need to decompose P_{ε} with respect to the splitting of Ω^{∗}M in F_{1} and F_{1}^{⊥}, paying some particular attention to the components in F_{3} ⊂ F_{1}^{n} ⊕ F_{1}^{n+1}, and even to θ ∧ (Ω_{0}^{n−1}H) ⊕ Ω_{ker L}^{n+1}H, due to the unsplitable controlled combinations

K_{ε}^{n} = δ_{Q}^{J}Π_{F_{1}^{n}} − \frac{i_{T}}{ε} Π_{θ∧Ω_{0}^{n−1}H} and K_{ε}^{n+1} = δ_{Q}i_{T} Π_{F_{1}^{n+1}} − \frac{Λ}{ε}Π_{Ω_{ker L}^{n+1}H} .

We’ll therefore use the decompositions,

F_{1} = F_{1}^{tame} ⊕ F_{1}^{tough} and F_{1}^{⊥} = F_{1}^{⊥,tame} ⊕ F_{1}^{⊥,tough} ,

with

F_{1}^{tough} = F_{1}^{n} ⊕ F_{1}^{n+1} , F_{1}^{tame} = F_{1} ∩ (F_{1}^{tough})^{⊥} = ⊕_{p≠n,n+1}F_{1}^{p}

and

F_{1}^{⊥,tough} = (θ ∧ Ω_{0}^{n−1}H) ⊕ Ω_{ker L}^{n+1}H , F_{1}^{⊥,tame} = F_{1}^{⊥} ∩ (F_{1}^{⊥,tough})^{⊥} .

Note that tame and tough parts are respectively in ∗ (and ∧) duality.

We start the decomposition of d_{ε}. We already know by (1) that we have in Ω^{∗}M = Ω^{∗}H ⊕ θ ∧ Ω^{∗}H,

d_{ε} = (\begin{matrix}d_{H} & ε^{−1}L \\ εL_{T} & −d_{H} \end{matrix}) = D_{H} + εD_{T} + ε^{−1}D_{L} .

By (8), we have [L, L_{T}] = [Λ, L_{T}] = 0, so that

D_{T} ((F_{1}^{n})^{⊥}) ⊂ F_{1}^{⊥,tame} , D_{T} (F_{1}^{n+1}) = 0 and D_{T} (F_{1}^{n}) ⊂ F_{1}^{n+1}

where, thanks to (7),

D_{T} α = θ ∧ L_{T} α = Dα + θ ∧ d_{Q}δ_{Q}^{J}α for α ∈ F_{1}^{n} = Ω_{0}^{n}H .

From (6), d_{H}(L^{k}α) = L^{k}(d_{Q}α − \frac{L}{n−p+1}δ_{Q}^{J}α) for α ∈ Ω_{0}^{p}H = F_{1}^{p} (p ≤ n), and by definition d_{Q}(θ ∧ α) = d_{ε}(θ ∧ α) = D_{H}(θ ∧ α) for θ ∧ α ∈ θ ∧ ker L = F_{1}^{p} (p > n). So, we see that

Π_{F_{1}}D_{H}Π_{F_{1}} = d_{Q} , Π_{F_{1}^{n}}D_{H}Π_{F_{1}^{⊥}} = 0 and Π_{F_{1}^{n+1}}D_{H}Π_{F_{1}^{⊥}} = −θ ∧ d_{Q}i_{T} Π_{θ∧Ω_{0}^{n−1}H}.

Bringing all this together, we have found that in F_{1} ⊕ F_{1}^{⊥},

d_{ε} = (\begin{matrix}d_{Q} + εD & Π_{F_{1}^{tame}}D_{H}Π_{F_{1}^{⊥,tame}} \\ Π_{F_{1}^{⊥}}D_{H}Π_{F_{1}} & Π_{F_{1}^{⊥}}D_{H}Π_{F_{1}^{⊥}} + ε^{−1}D_{L} \end{matrix}) + εθ ∧ d_{Q}K_{ε}^{n} + ε (\begin{matrix}0 & 0 \\ Π_{F_{1}^{⊥,tame}}D_{T} Π_{F_{1}^{tame}} & D_{T} \end{matrix} ) .

Before taking the adjoint, we can put it in a slightly more convenient form, if we observe that,

εθ ∧ d_{Q}K_{ε}^{n} = ε(K_{ε}^{n+1})^{∗}K_{ε}^{n} + LK_{ε}^{n}

= ε(K_{ε}^{n+1})^{∗}K_{ε}^{n} + L(δ_{Q}^{J}Π_{F_{1}^{n}} − \frac{i_{T}}{ε} Π_{θ∧Ω_{0}^{n−1}H})

= ε(K_{ε}^{n+1})^{∗}K_{ε}^{n} + Π_{F_{1}^{⊥}}D_{H}Π_{F_{1}^{n}} − ε^{−1}D_{L}Π_{F_{1}^{⊥,tough}} .

So that, using previous study, we get

d_{ε} = (\begin{matrix}d_{Q} + εD & Π_{F_{1}^{tame}}(D_{H} + εD_{T})Π_{F_{1}^{⊥,tame}} \\ Π_{F_{1}^{⊥,tame}}(D_{H} + εD_{T})Π_{F_{1}^{tame}} & Π_{F_{1}^{⊥}}(D_{H} + εD_{T})Π_{F_{1}^{⊥}} + ε^{−1}D_{L}Π_{F_{1}^{⊥,tame}} \end{matrix})
+ ε(K_{ε}^{n+1})^{∗}K_{ε}^{n}

and finally,

P_{ε} = d_{ε} + δ_{ε}

= (\begin{matrix}d_{Q} + δ_{Q} + ε(D + D^{∗}) & Π_{F_{1}^{tame}}(P_{H} + εP_{T})Π_{F_{1}^{⊥,tame}} \\ Π_{F_{1}^{⊥,tame}}(P_{H} + εP_{T})Π_{F_{1}^{tame}} & Π_{F_{1}^{⊥}}(P_{H} + εP_{T})Π_{F_{1}^{⊥}} + ε^{−1}Π_{F_{1}^{⊥,tame}}P_{L}Π_{F_{1}^{⊥,tame}} \end{matrix})

+ ε((K_{ε}^{n+1})^{∗}K_{ε}^{n} + (K_{ε}^{n})^{∗}K_{ε}^{n+1})

(21)

where

P_{H} = D_{H} + D_{H}^{∗} , P_{T} = D_{T} + D_{T}^{∗} and P_{L} = D_{L} + D_{L}^{∗} .

Remarks 6.1. 1. We deliberately don’t break the K_{ε} terms in matrix form because we still know we won’t be able to control each component individually whereas the entire combinations are controllable.

2. We mention here a technical difference with the adiabatic methods as developed by Bismut and others. Namely, the operator P_{ε} isn’t a perturbation of a diagonal operator, nor converges to such.

We are now in position to prove thm 3.5 and 3.6. Let λ ∈ ℂ \ ℝ and α ∈ F_{3}^{⊥}. By self-adjointness of P_{ε}, β_{ε} = (λ − P_{ε})^{−1}α exists, and one has by prop 5.3, for some C > 0,

∥α∥_{0}^{2} = ∥(λ − P_{ε})β_{ε}∥_{0}^{2} ≥ | Im λ|^{2}∥β_{ε}∥_{0}^{2} + ∥P_{ε}β_{ε}∥_{0}^{2}
≥ C∥β_{ε}∥_{1,ε}^{2} .

Now, by definition 5.2 and lemma 5.6, ∥β_{ε}∥_{1,ε}^{2} controls \frac{1}{ε^{2}} ∥Π_{F_{1}^{⊥,tame}}β_{ε}∥_{0}^{2} and \frac{1}{ε}∥Π_{F_{1}^{⊥,tough}}∥_{0}^{2}. This gives us the uniform control

∥Π_{F_{1}^{⊥}}β_{ε}∥_{0}^{2} = ∥Π_{F_{1}^{⊥}}(λ − P_{ε})^{−1}α∥_{0}^{2} ≤ εC^{−1}∥α∥_{0}^{2} ,

so we can focus now on Π_{F_{1}}β_{ε}. For this, we use (21) to obtain the Π_{F_{1}∩F_{3}^{⊥}} projection of the equation

(λ − P_{ε})β_{ε} = α .

It is

(λ − (d_{Q} + δ_{Q}))Π_{F_{1}∩F_{3}^{⊥}}β_{ε} − Π_{F_{1}^{tame}}(P_{H} + εP_{T})Π_{F_{1}^{⊥,tame}}β_{ε}

− εΠ_{F_{1}∩F_{3}^{⊥}}((K_{ε}^{n+1})^{∗}K_{ε}^{n} + (K_{ε}^{n})^{∗}K_{ε}^{n+1})β_{ε} = Π_{F_{1}}α ,

or equivalently,

(λ − P_{Q})Π_{F_{1}∩F_{3}^{⊥}}β_{ε} − Π_{F_{1}}α = Π_{F_{1}^{tame}}(P_{H} + εP_{T})Π_{F_{1}^{⊥,tame}}β_{ε}

+ εΠ_{F_{1}∩F_{3}^{⊥}}(θ ∧ d_{Q}K_{ε}^{n} + d_{Q}^{J}K_{ε}^{n+1})βε . (22)

We are led to compose this with the inverse of (λ − P_{Q}) on F_{1} ∩ F_{3}^{⊥}. We gather here some analytic facts we’ll need about it and the inverse of (λ − P_{D}) on F_{3}.

Proposition 6.2. 1. For λ ∈ ℂ \ ℝ, (λ − P_{Q}) (resp (λ − P_{D})) induces a bicontinuous bijection between the Hilbert spaces F_{1}∩F_{3}^{⊥}∩H_{1,H} and F_{1}∩F_{3}^{⊥} in L^{2} (resp F_{3}∩H_{2,H} and F_{3}).

2. P_{Q} (resp P_{D}) extends to a self-adjoint operator on F_{1} ∩ F_{3}^{⊥} (resp F_{3}) in L^{2} with domain F_{1} ∩F_{3}^{⊥} ∩H_{1,H} (resp F_{3} ∩H_{2,H}) and discrete spectrum without accumulation points.

3. Π_{F_{2}∩F_{1}^{tame}} is continuous from H_{0} (= L^{2}) to any H_{k,H}, k ∈ ℕ.

4. Π_{F_{1}∩F_{3}^{⊥}} extends continuously from H_{1,H} and H_{2,H} to themselves.

Proof. We define P = Δ_{Q} + P_{D} on F_{1}, that is P = d_{Q}δ_{Q} + δ_{Q}d_{Q} on F_{1}^{tame} and P =

(\begin{matrix}d_{Q}δ_{Q} & D^{∗} \\ D & δ_{Q}d_{Q} \end{matrix}) on F_{1}^{tough} = F_{1}^{n} ⊕ F_{1}^{n+1}. By Theorem 3.1, it is a symmetric second order

hypoelliptic operator. So, it follows classically that on a compact manifold, P extends to a self-adjoint operator on L^{2} with domain H_{2,H}. The assertions on P_{D} come then from the restrictions of P and its resolvent to the stable space F_{3} = (F_{1}^{n} ∩ker δ_{Q})⊕(F_{1}^{n+1} ∩ker d_{Q}).

For the properties of P_{Q}, we consider

Q_{λ} = (λ + P_{Q})(λ^{2} − P)^{−1} ,

which induces a continuous map from F_{1} ∩ F_{3}^{⊥} (in L^{2}) to F_{1} ∩ F_{3}^{⊥} ∩ H_{1,H}. It is a right inverse of (λ − P_{Q}) on F_{1} ∩ F_{3}^{⊥}. We also observe that Q_{λ} = (λ^{2} − P)^{−1}(λ + P_{Q}) on F_{1} ∩ F_{3}^{⊥} ∩ H_{1,H}. This shows that Q_{λ}(λ − P_{Q}) = Id holds on F_{1} ∩ F_{3}^{⊥} ∩ H_{2,H} and therefore extends continuously on all F_{1} ∩ F_{3}^{⊥} ∩ H_{1,H}. So we get the first point on P_{Q}, from which its self-adjointness is a classical consequence (see [15] VIII).

The third point comes from the fact that F_{2} ∩ F_{1}^{tame} is the kernel of the hypoelliptic operator Δ_{Q} on F_{1}^{tame}. The statements about Π_{F_{1}∩F_{3}^{⊥}} result from its L^{2} boundedness and its commutation with P_{Q} and P_{D}.

We apply this proposition to the expression (22), observing that

• P_{H} is a first order (horizontal) operator, so that (λ − P_{Q})^{−1}Π_{F_{1}^{tame}}P_{H}Π_{F_{1}^{⊥,tame}} is bounded from H_{0} = L^{2} to itself.

• P_{T} must be viewed by (8) as a second order operator, therefore continuous from H_{1,H} to H_{−1,H}.

We get finally, ∃C > 0 such that for α ∈ F_{3}^{⊥} and β_{ε} = (λ − P_{ε})^{−1}α,

∥Π_{F_{1}∩F_{3}^{⊥}}β_{ε} − (λ − P_{Q})^{−1}Π_{F_{1}}α∥_{0}

≤ C(∥Π_{F_{1}^{⊥,tame}}β_{ε}∥_{0} + ε∥K_{ε}β_{ε}∥_{0} + ε∥Π_{F_{1}^{⊥,tame}}β_{ε}∥_{1,H})

≤ Cε∥α∥_{0} by definition 5.2 and proposition 5.3.

This completes the proof of the uniform (L^{2}) convergence,

Π_{F_{3}^{⊥}}(λ − P_{ε})^{−1}Π_{F_{3}^{⊥}} ⟶_{ε→0} (λ − P_{Q})^{−1}Π_{F_{1}∩F_{3}^{⊥}} .

Writing, by spectral theory,

(λ^{2} − Δ_{ε})^{−1} = \frac{1}{2λ}((λ − P_{ε})^{−1} + (λ + P_{ε})^{−1}) ,

we obtain also the uniform convergence of the resolvents of Laplacians,

(λ^{2} − Δ_{ε})^{−1} ⟶_{ε→0} (λ^{2} − Δ_{Q})^{−1}Π_{F_{1}} on Ω^{p}M, p ≠ n, n + 1,

and

Π_{F_{3}^{⊥}}(λ^{2} − Δ_{ε})^{−1}Π_{F_{3}^{⊥}} ⟶_{ε→0} (λ^{2} − Δ_{Q})^{−1}Π_{F_{1}∩F_{3}^{⊥}} on Ω^{n}M ⊕ Ω^{n+1}M .

We now proceed with Theorem 3.6. For λ ∈ ℂ \ ℝ and α ∈ Ω^{∗}M, we consider the equation (λ − \frac{P_{ε}}{ε})β_{ε} = α. First of all, by definition 5.7 and proposition 5.8, we know that, for some C > 0,

∥α∥_{0}^{2} = ∥(λ − \frac{P_{ε}}{ε})β_{ε}∥_{0}^{2} ≥ | Im λ|^{2}∥β_{ε}∥_{0}^{2} + ∥\frac{P_{ε}}{ε} β_{ε}∥_{0}^{2}

≥ C∥β_{ε}∥_{1′,ε}^{2}

≥ C(\frac{1}{ε^{2}} (Δ_{Q}Π_{F_{1}}β_{ε}, Π_{F_{1}}β_{ε})_{0} + \frac{1}{ε^{4}} ∥Π_{F_{1}^{⊥,tame}}β_{ε}∥_{0}^{2}

+ \frac{1}{ε^{2}} ∥Π_{F_{1}^{⊥,tough}}β_{ε}∥_{0}^{2}) .

Also, from prop 6.2, 0 is isolated in the spectrum of P_{Q} on F_{1} ∩ F_{3}^{⊥}. Therefore, ∃C > 0 such that

\frac{1}{ε^{2}} (Δ_{Q}Π_{F_{1}}β_{ε}, Π_{F_{1}}β_{ε})_{0} ≥ \frac{C}{ε^{2}} ∥Π_{F_{2}^{⊥}}β_{ε}∥_{0}^{2} ,

where we recall that F_{2} = ker Δ_{Q}. Thus,

Π_{F_{2}^{⊥}}(λ − \frac{P_{ε}}{ε})^{−1} ⟶_{ε→0} 0 uniformly in L^{2} .

We use again (21) to obtain the F_{2} component of the equation (λ − \frac{P_{ε}}{ε})β_{ε} = α. It is

(λ − (D + D^{∗}))Π_{F_{2}}β_{ε} − Π_{F_{2}∩F_{1}^{tame}}(\frac{P_{H}}{ε} + P_{T})Π_{F_{1}^{⊥,tame}}β_{ε}

− Π_{F_{3}}(θ ∧ d_{Q}K_{ε}^{n} + d_{Q}^{J}K_{ε}^{n+1})β_{ε} = Π_{F_{2}}α ,

which in turn can be decomposed in F_{2} ∩ F_{1}^{tame} and F_{3} = F_{2} ∩ F_{1}^{tough} components, giving respectively,

(23)  \{ \begin{matrix}λΠ_{F_{2}∩F_{1}^{tame}}β_{ε} − Π_{F_{2}∩F_{1}^{tame}}(\frac{P_{H}}{ε} + P_{T})Π_{F_{1}^{⊥,tame}}β_{ε} & = Π_{F_{2}∩F_{1}^{tame}}α \\ (λ − P_{D})Π_{F_{3}}β_{ε} − Π_{F_{3}}(θ ∧ d_{Q}K_{ε}^{n} + d_{Q}^{J}K_{ε}^{n+1})β_{ε} & = Π_{F_{3}}α . \end{matrix}

Using prop 6.2, we obtain that, for some C > 0,

∥λΠ_{F_{2}∩F_{1}^{tame}}β_{ε} − Π_{F_{2}∩F_{1}^{tame}}α∥_{0} ≤ \frac{1}{ε}∥Π_{F_{1}^{⊥,tame}}β_{ε}∥_{0} ≤ Cε∥β_{ε}∥_{1,ε}
≤ Cε∥α∥_{0} by prop 5.8 and definition 5.7,

and also

∥Π_{F_{3}^{⊥}}β_{ε} − (λ − P_{D})Π_{F_{3}}α∥_{0} ≤ ∥Π_{F_{3}}(θ ∧ d_{Q}K_{ε}^{n} + d_{Q}^{J}K_{ε}^{n+1})β_{ε}∥_{−2,H}

≤ C∥K_{ε}β_{ε}∥_{0} ≤ Cε∥β_{ε}∥_{1′,ε} ≤ Cε∥α∥_{0} by proposition 5.8.

So that finally, observing that P_{D} = 0 on F_{3}^{⊥}, we have got

(λ − \frac{P_{ε}}{ε} )^{−1} ⟶_{ε→0} (λ − P_{D})^{−1}Π_{F_{2}} ,

uniformly in L^{2}, and the corresponding result on Laplacians, by the same trick as above,

(λ^{2} − \frac{Δ_{ε}}{ε^{2}} )^{−1} = \frac{1}{2λ}((λ − \frac{P_{ε}}{ε})^{−1} + (λ + \frac{P_{ε}}{ε})^{−1}) ⟶_{ε→0} (λ^{2} − Δ_{D})^{−1}Π_{F_{2}} .

7 On the convergence of heat kernels

The heat operators e^{−tΔ} are of particular importance in Riemannian geometry. For instance, their global and local traces play a key role in the index theorem and analytic torsion theory. Also, eta invariants involve global and local traces like Tr(P e^{−tP^{2}}) for P Dirac operators.

We plan here to begin the study of the behavior of some of such traces in the sub-Riemannian limit of the metric, that is with respect to g_{ε} = dθ(·, J ·) + \frac{θ^{2}}{ε^{2}} when ε → 0.

7.1 The global trace convergence

For P a self-adjoint operator on a compact manifold M with smooth kernel K_{P} (x, y), the trace of P will be denote

Tr_{M}(P) = ∫_{M} Tr(K_{P} (x, x))dvol = ∑_{Spectrum(P)} λ_{i} ,

where (λ_{i})_{i∈ℕ} is the pure point spectrum of P.

Theorem 7.1. Let M be a compact contact manifold endowed with the family of metrics g_{ε}. One has, uniformly for t ≥ t_{0} > 0,

1. on Ω^{p}M with p ≠ n, n + 1,

Tr_{M}(e^{−tΔ_{ε}}) ⟶_{ε→0} Tr_{M}(e^{−tΔ_{Q}}) (Δ_{Q} acting on F_{1}^{p})

2. on Ω^{n}M ⊕ Ω^{n+1}M,

Tr_{M}(e^{−t \frac{Δε}{ε^{2}}} ) ⟶_{ε→0} Tr_{M}(e^{−tΔ_{D}})

where Δ_{D} = D^{∗}D + DD^{∗} acts on F_{3} = (F_{1}^{n} ∩ ker δ_{Q}) ⊕ (F_{1}^{n+1} ∩ ker d_{Q}).

3. Similarly, for the signature operator S_{ε} acting on Ω^{even}M, one has

Tr_{M}(\frac{S_{ε}}{ε} e^{−t \frac{Δε}{ε^{2}}} ) ⟶_{ε→0} Tr_{M}(S_{D}e^{−tΔ_{D}})

where (section 3) S_{D} = (−1)^{l}D∗ on F_{3}^{n+1} for n = 2l + 1.

Proof. This follows easily from the resolvent convergence results (thm 3.5 and 3.6) and the developed Bochner technique. We consider the spectrum, arranged in increasing order, (λ_{i}(ε))_{i∈ℕ} of Δ_{ε} (resp \frac{Δ_{ε}}{ε^{2}} in middle degree). A classical consequence of norm resolvent convergence (see [15]), is the convergence of the spectrum over bounded open intervals of ℝ. So that, for each A > 0 and not in Spectrum(Δ_{Q}),

Spectrum(Δ_{ε}) ∩ [0, A[⟶_{ε→0} Spectrum(Δ_{Q}) ∩ [0, A[

(resp with \frac{Δ_{ε}}{ε^{2}} and Δ_{D}). This implies, for the ith eigenvalue, λ_{i}(ε) ⟶_{ε→0} λ_{i}(Δ_{Q}) (resp Δ_{D}).

We will be able to turn this into the convergence of the sum ∑_{i∈ℕ} e^{−tλ_{i}(ε)} if we can find a fixed summable domination of it. But this comes from the Bochner technique. Namely, for p ≠ n, n + 1, prop 5.3 tell us that ∀α ∈ Ω^{p}M,

((Δ_{ε} + Id)α, α) ≥ C∥α∥_{1,H}^{2} = (Δ′_{H}α, α)

for an hypoelliptic Laplacian Δ′_{H}. So the maxmin principle gives that

λ_{i}(ε) ≥ λ_{i}(Δ′_{H}) − 1 and e^{−tλ_{i}(ε)} ≤ e^{−t(λ_{i}(Δ′_{H})−1)} ,

whose sum is convergent by hypoellipticity of Δ′_{H}.

The corresponding domination in middle degree is elliptic as comes from definition 5.7, prop 5.8 and lemma 5.9,

((\frac{Δ_{ε}}{ε^{2}} + Id)α, α)_{0} ≥ ∥α∥_{1,(H+T)}^{2} .

Lastly, the eta trace in 3 is controlled with

|λ_{i}(\frac{S_{ε}}{ε})e^{−tλ_{i}^{2}( \frac{Sε}{ε})}| ≤((1 + λ_{i}^{2}(\frac{S_{ε}}{ε}))e^{−tλ_{i}^{2}( \frac{Sε}{ε})} =((1 + λ_{i}(\frac{Δ_{ε}}{ε^{2}} ))e^{−tλ_{i}( \frac{Δε}{ε^{2}} )}
≤ Ce^{− \frac{t}{2} λ_{i}( \frac{Δε}{ε^{2}} )} .

7.2 The local convergence of heat kernels outside middle degrees

Let K_{ε}(x, y) and K_{Q}(x, y) be the respective kernels of e^{−tΔ_{ε}} and Π_{F_{1}}e^{−tΔ_{Q}}Π_{F_{1}}. We will be interested now in the problem of the local convergence of K_{ε} when ε → 0.

Theorem 7.2. Let M be a compact contact manifold and p ≠ n, n + 1, then

K_{ε}(x, y) ⟶_{ε→0} K_{Q}(x, y)

uniformly on M in C^{k} norm for any k ∈ ℕ.

For this, we have to improve the uniform L^{2} convergence of e^{−tΔ_{ε}} towards Π_{F_{1}}e^{−tΔ_{Q}}Π_{F_{1}}, which comes with Theorem 3.5 and functional calculus, to a uniform convergence at the kernel level. To perform this, we’ll follow a general method used in Bismut-Lebeau’s work on Quillen’s metrics (see [4], part XI, p165–179).

Here is a sketch of it. At first, we’ll extend the regularizing properties of the resolvent (λ − Δ_{ε})^{−1} to higher order Sobolev spaces. This will be done by a commutator technique (adapted to the hypoelliptic context here). Then we’ll turn this into uniform regularizing properties of e^{−tΔ_{ε}} by using a contour integral such as ∫_{Γ} e^{−tλ}(λ−Δ_{ε})^{−k}dλ, with k integer. This will give for k large enough a uniform L^{2} control of K_{ε}, and for k even larger a sufficient Sobolev control to deduce the equicontinuity and uniform convergence of the kernels.

We begin this. The a priori estimate prop 5.3 and Lax-Milgram assure that (Id +Δ_{ε}) induces a bicontinuous bijection from H_{1,ε} into H_{−1,ε} together with the following uniform norm control, for ε ∈]0, 1],

(24)  ∥(Id +Δ_{ε})^{−1}α∥_{1,ε} ≤ C_{1}^{−1}∥α∥_{−1,ε} .

To extend this to higher order regularizing properties, we introduce some families of Sobolev spaces modeled on ∥ ∥_{1,ε}.

Definition 7.3. For k ∈ ℕ, ε > 0 and α ∈ Ω^{p}M, let

∥α∥_{k,ε} = ∥α∥_{k,H} + \frac{1}{ε}∥Π_{F_{1}^{⊥}}α∥_{k−1,H} + ε∥T α∥_{k−1,H} + ∥α∥_{0} ,

and H_{k,ε} be the corresponding Sobolev space (the (k, H) indices in norms denote the number of horizontal derivatives controlled in L^{2}).

The estimate (24) will be generalized the following way.

Lemma 7.4. For k ∈ ℕ, ∃C_{k} such that ∀ε ∈]0, 1], (Id +Δ_{ε})^{−1} extends continuously from H_{2k−1,H} into H_{2k+1,ε}, with the uniform operator’s norms controls,

∥(Id +Δ_{ε})^{−1}∥_{(2k−1,H),(2k+1,ε)} ≤ C_{k} and
∥(Id +Δ_{ε})^{−1} \frac{Π_{F_{1}^{⊥}}}{ε} ∥_{(2k,H),(2k+1,ε)} ≤ C_{k} .

We observe that all this is clear for a fixed ε by elliptic regularity of Δ_{ε}. The problem is to handle its regularity degeneracy and divergences when ε → 0. The commutator technique allows a quite explicit approach of this.

We cover the manifold M with small enough open sets such that on each of them we can choose orthonormal vector fields (for g_{1}) : X_{1}, X_{2} = JX_{1}, ⋯, X_{2n−1}, X_{2n} = JX_{2n−1} in H, and T (the Reeb field). Let θ^{1}, ⋯, θ^{2n}, θ^{0} = θ be the dual base. For X ∈ T M, we note ∂_{X} the differential operator acting on forms defined by

α = ∑_{I⊂[0,2n]} α_{I}θ^{I} ⟼ ∂_{X}α = ∑_{I}(X.α_{I})θ_{I} .

We gather some useful facts about them. P^{(k)} will denote linear differential operators of order k (in horizontal derivatives).

• [∂_{X}, L] = [∂_{X}, Λ] = 0 (because L = ∑_{i=1}^{n} θ^{2i−1}∧θ^{2i}∧· and Λ are constant coefficients algebraic operators)

• [∂_{X}, ∂_{Y}] = ∂_{[X,Y]}

• [∂_{X}, d_{H}] = P^{(1)} + P^{(0)}∂_{T} for X in H (by previous point and the local Cartan’s formula d_{H} = ∑_{i=1}^{2n} ∂_{X_{i}}(θ_{i} ∧ ·) + P^{(0)})

• [∂_{T} , d_{H}] = P^{(1)} (here, [T, X] ∈ H for all X ∈ H because the Reeb field T preserves H)

• [∂_{X}, L_{T}] = P^{(1)} for X ∈ H (because L_{T} = ∂_{T} + P^{(0)})

• [∂_{T} , L_{T}] = P^{(0)}

• similar formulas hold with adjoint and J-conjugate (δ_{H}, L_{T}^{∗}, d_{H}^{J}) instead.

All this and (17) give easily the commutators we’ll need.

Proposition 7.5. For X, Y ∈ H, one has

[∂_{XY}^{2} , Δ_{ε}] = P^{(3)} + P^{(2)}∂_{T} + ε(P^{(2)} + P^{(1)}∂_{T}) + ε^{2}P^{(2)}∂_{T}
+ ε^{−1}Π_{F_{1}^{⊥}}(P^{(2)} + P^{(1)}∂_{T}) + ε^{−1}(P^{(2)} + P^{(1)}∂_{T})Π_{F_{1}^{⊥}} ,

and,

[∂_{T} , Δ_{ε}] = P^{(2)} + εP^{(1)} + ε^{2}P^{(0)}∂_{T} + ε^{−1}(Π_{F_{1}^{⊥}}P^{(1)} + P^{(1)}Π_{F_{1}^{⊥}}) .

The first commutator doesn’t seem so ”good” because it’s still of hypoelliptic order four. Even so, its high order part factorizes through ∂_{T}, which itself has good commutation properties (on the Heisenberg group, our local model, T is even in the center of the group).

Proof of lemma 7.4. We begin now the proof of lemma 7.4.The case k = 0 follows from (24). We suppose it’s true for the rank k and proceed by recurrence. We use proposition 7.5 to write for X, Y ∈ H,

∂_{XY}^{2} (Id +Δ_{ε})^{−1} = (Id +Δ_{ε})^{−1}∂_{XY}^{2} − (Id +Δ_{ε})^{−1}[∂_{XY}^{2} , Δ_{ε}](Id +Δ_{ε})^{−1}

= (Id +Δ_{ε})^{−1}∂_{XY}^{2}

− (Id +Δ_{ε})^{−1} \begin{matrix}( & P^{(3)} + ε(P^{(2)} + P^{(1)}∂_{T}) + ε^{2}P^{(2)}∂_{T} \\ + ε^{−1}(Π_{F_{1}^{⊥}}P^{(2)} + P^{(2)}Π_{F_{1}^{⊥}}))(Id +Δ_{ε})^{−1} \end{matrix}

− (Id +Δ_{ε})^{−1}(P^{(2)} + ε^{−1}(Π_{F_{1}^{⊥}}P^{(1)} + P^{(1)}Π_{F_{1}^{⊥}}))∂_{T} (Id +Δ_{ε})^{−1} (25)

with,

∂_{T} (Id +Δ_{ε})^{−1} = (Id +Δ_{ε})^{−1}∂_{T} − (Id +Δ_{ε})^{−1}[∂_{T} , Δ_{ε}](Id +Δ_{ε})^{−1}

= (Id +Δ_{ε})^{−1}∂_{T}

− (Id +Δ_{ε})^{−1} \begin{matrix}( & P^{(2)} + εP^{(1)} + ε^{2}P^{(0)}∂_{T} \\ + ε^{−1}(Π_{F_{1}^{⊥}}P^{(1)} + P^{(1)}Π_{F_{1}^{⊥}}))(Id +Δ_{ε})^{−1} . \end{matrix} (26)

We use this to estimate the norm at rank k + 1,

(27) ∥(Id +Δ_{ε})^{−1}α∥_{(2k+3,ε)} = ∑_{X,Y ∈{X_{1},··· ,X_{2n}}} ∥∂_{XY}^{2} (Id +Δ_{ε})^{−1}α∥_{(2k+1,ε)}

+ ∥(Id +Δ_{ε})^{−1}α∥_{(2k+1,ε)} .

We have to check the continuity properties of each term of (25) (fixed multiplicative constants will be overviewed) :

•  ∥(Id +Δ_{ε})^{−1}∂_{XY}^{2} α∥_{(2k+1,ε)} ≤ ∥∂_{XY}^{2} α∥_{(2k−1,H)} by rank k hypothesis
≤ ∥α∥_{(2k+1,H)} ,

•  ∥(Id +Δ_{ε})^{−1}P^{(3)}(Id +Δ_{ε})^{−1}α∥_{(2k+1,ε)} ≤ ∥P^{(3)}(Id +Δ_{ε})^{−1}α∥_{(2k−1,H)}
≤ ∥(Id +Δ_{ε})^{−1}α∥_{(2k+2,H)} ,

and we use that ∥ ∥_{(2k+2,H)} ≤ c∥ ∥_{(2k+3,H)} + c^{−1}∥ ∥_{(2k+1,H)} for any arbitrarily small c (which can be proved elementary, as in elliptic theory, by integration by parts and Cauchy-Schwartz :

∥α∥_{(2k+2,H)}^{2} = (∇_{H}^{2k+2}α, ∇_{H}^{2k+2})_{0} = (∇_{H}^{2k+3}α, ∇_{H}^{2k+1}α)_{0}
≤ ∥α∥_{(2k+3,H)}∥α∥_{(2k+1,H)}
≤ c^{2}∥α∥_{(2k+3,H)}^{2} + c^{−2}∥α∥_{(2k−1,H)}^{2} ).

So that finally c∥(Id +Δ_{ε})^{−1}α∥_{(2k+3,H)} ≤ c∥(Id +Δ_{ε})^{−1}α∥_{(2k+3,ε)} can be absorbed in the left side of (27).

• ∥(Id +Δ_{ε})^{−1}(ε(P^{(2)} + P^{(1)}∂_{T}) + ε^{2}P^{(2)}∂_{T} )(Id +Δ_{ε})^{−1}α∥_{(2k+1,ε)}
≤ ∥(εP^{(2)} + P^{(1)}ε∂_{T} + ε^{2}P^{(2)}∂_{T})(Id +Δ_{ε})^{−1}α∥_{(2k−1,H)}
≤ ∥α∥_{(2k−1,H)} + ε^{2}∥(Id +Δ_{ε})^{−1}α∥_{(2k+3,H)} ,

this last term being again absorbed in the left side of (27),

• ∥(Id +Δ_{ε})^{−1}(\frac{Π_{F_{1}^{⊥}}}{ε} P^{(1)} + P^{(1)} \frac{Π_{F_{1}^{⊥}}}{ε})(Id +Δ_{ε})^{−1}α∥_{(2k+1,ε)}

≤ ∥P^{(1)}(Id +Δ_{ε})^{−1}α∥_{(2k,H)} + ∥\frac{Π_{F_{1}^{⊥}}}{ε} (Id +Δ_{ε})^{−1}α∥_{(2k,H)}

≤ ∥(Id +Δ_{ε})^{−1}α∥_{(2k+1,ε)} ≤ ∥α∥_{(2k−1,H)}

• Using same methods, we easily get that

(Id +Δ_{ε})^{−1}(P^{(2)} + ε^{−1}Π_{F_{1}^{⊥}}P^{(1)} + ε^{−1}P^{(1)}Π_{F_{1}^{⊥}})

is continuous from H_{2k+1,ε} into itself, and that, thanks to (26), ∂_{T} (Id +Δ_{ε})^{−1} is from H_{2k+1,H} to H_{2k+1,ε}. They can therefore be composed as in (25), so that we obtain finally the required boundedness of ∥(Id +Δ_{ε})^{−1}∥_{(2k+1,H),(2k+3,ε)}.

We are left with the control of ∥(Id +Δ_{ε})^{−1} \frac{Π_{F_{1}^{⊥}}}{ε} ∥_{(2k+2,H),(2k+3,ε)}, which is the second part of our recurrence hypothesis at rank k + 1. This can be handled in the same way as

our previous estimation. We just have to compose the equations (25) and (26) by \frac{Π_{F_{1}^{⊥}}}{ε} by the right, and commute it with ∂_{XY}^{2} in (25) and ∂_{T} in (26). All terms can then be controlled using the rank k hypothesis.

In order to relate the heat operator e^{−tΔ_{ε}} to (λ − Δ_{ε})^{−1} by a contour integral formula, we have to extend our resolvent estimates to λ’s surrounding ℝ^{+}. Let therefore U be the domain of ℂ defined by

U = {λ ∈ ℂ, ℜλ ≤ −1 or |ℑλ| ≥ 1} .

We will need

Lemma 7.6. For k ∈ ℕ, ∃C_{k} such that ∀ε ∈]0, 1] and λ ∈ U, the following operator’s norm control holds,

∥(λ − Δ_{ε})^{−1}∥_{(2k−1,H),(2k+1,H)} ≤ C_{k}(1 + |λ|)^{2k+2} .

Proof. The case λ = −1 comes from lemma 7.4. Moreover, we know that (λ − Δ_{ε})^{−1} is related to (Id +Δ_{ε})^{−1} by the first resolvent equation (see [15])

(i)  (λ − Δ_{ε})^{−1} = −(Id +Δ_{ε})^{−1} + (λ + 1)(Id +Δ_{ε})^{−1}(λ − Δ_{ε})^{−1}

(ii) = −(Id +Δ_{ε})^{−1} + (λ + 1)(λ − Δ_{ε})^{−1}(Id +Δ_{ε})^{−1} .

To start the proof for k = 0, we also observe that, Δ_{ε} being a positive self-adjoint operator, it satisfies for λ ∈ U,

∥(λ − Δ_{ε})^{−1}α∥_{0} ≤ \frac{∥α∥_{0}}{dist(λ, SpecΔ_{ε})} ≤ ∥α∥_{0} .

A first use of the resolvent equation (i) gives therefore,

∥(λ − Δ_{ε})^{−1}α∥_{1,H} ≤ ∥(λ − Δ_{ε})^{−1}α∥_{1,ε}

≤ C∥α∥_{−1,H} + (|λ| + 1)∥(λ − Δ_{ε})^{−1}α∥_{−1,H} by lemma 7.4 or (24)

≤ (C + |λ| + 1)∥α∥_{0} ,

which can be used again in (ii) to obtain

∥(λ − Δ_{ε})^{−1}α∥_{1,H} ≤ C(1 + |λ|)^{2}∥α∥_{−1,H} .

Lemma 7.6 is then proved by recurrence using (i) and lemma 7.4.

We can now complete the proof of Theorem 7.2. Setting Γ = ∂U, we have by the functional calculus

e^{−tΔ_{ε}} = \frac{1}{2iπ} ∫_{Γ} e^{−tλ}(λ − Δ_{ε})^{−1}dλ ,

but also, ∀k ∈ ℕ^{∗},

= \frac{(−t)^{k−1}}{2iπ(k − 1)!} ∫_{Γ} e^{−tλ}(λ − Δ_{ε})^{−k}dλ .

This, together with lemma 7.6, implies that e^{−tΔ_{ε}} is a uniformly bounded operator, when ε → 0, from H_{0} into H_{k,H} for any k ∈ ℕ. The same is true with ordinary L^{2} elliptic Sobolev spaces H_{k}, because a transverse derivative is recovered by two horizontal ones. Therefore, if Δ = Id +∇^{∗}∇ denotes an invertible elliptic Laplacian and ∥A∥_{HS} = Tr(A^{∗}A)^{\frac{1}{2}} the Hilbert-Schmidt’s norm (see [15]), one has ∀l, m ∈ ℕ,

∥Δ_{x}^{l}Δ_{y}^{m}K_{ε}(x, y)∥_{L^{2}} = ∥Δ^{l}e^{−tΔ_{ε}}Δ^{m}∥_{HS}
= ∥Δ^{l}e^{−tΔ_{ε}}Δ^{m+k}Δ^{−k}∥_{HS} for any k ∈ ℕ,
≤ ∥Δ^{l}e^{−tΔ_{ε}}Δ^{m+k}∥_{L^{2}}∥Δ^{−k}∥_{HS} ,

with ∥Δ^{−k}∥_{HS} < +∞ for k large enough, and Δ^{l}e^{−tΔ_{ε}}Δ^{m+k} uniformly bounded from L^{2} in itself when ε → 0 by lemma 7.6 and duality.

The Sobolev inequalities give then a uniform control when ε → 0 of K_{ε} and its derivatives in sup norm. This allows to improve the week convergence of K_{ε} towards K_{Q} (coming with the L^{2} uniform convergence of e^{−tΔ_{ε}} → Π_{F_{1}}e^{−tΔ_{Q}}Π_{F_{1}}) to a strong uniform one.

7.3 The local convergence of heat kernels in middle degrees

We now come to the the problem of the local convergence of the kernel K_{ε}(x, y) of e^{−t \frac{Δε}{ε^{2}}} on Ω^{n}M ⊕Ω^{n+1}M. We already know that it weakly converges towards the kernel K_{D}(x, y) of Π_{F_{3}}e^{−tΔ_{D}}Π_{F_{3}}, where Δ_{D} = D^{∗}D + DD^{∗} and F_{3} = (F_{1}^{n} ∩ ker δ_{Q}) ⊕ (F_{1}^{n+1} ∩ ker d_{Q}).

We want to improve this convergence using the same general methods as above. Although, thanks to proposition 5.8 and lemma 5.9, Id +\frac{Δ_{ε}}{ε^{2}} now dominates an elliptic Laplacian, it contains more divergent terms. This makes finally the commutator technique more delicate to apply than previously. In particular, one important point in the proof of the

regularizing properties of (Id +Δ_{ε})^{−1} was the uniform boundedness of \frac{Π_{F^{⊥}_{1}}}{ε} (Id +Δ_{ε})^{−1} and

(Id +Δ_{ε})^{−1} \frac{Π_{F^{⊥}_{1}}}{ε} on higher order Sobolev spaces (lemma 7.4). This was obtained with the use of basic operators, the ”scalar” derivatives ∂_{X}, that had to preserve the components F_{1} and F_{1}^{⊥}. The problem in middle degree will be similarly to find operators that respect the relevant splitting of forms, which is no longer of algebraic nature here.

Recall that we know there that F_{1}^{n} has to be decomposed further in F_{3}^{n} = F_{1}^{n} ∩ ker δ_{Q}

and F_{1}^{n} ∩ Im d_{Q} because ∥\frac{Π_{F^{⊥}_{3}}}{ε} α∥_{1,H} is controlled whereas Π_{F_{3}}α doesn’t collapse in ∥ ∥_{1′,ε}. This splitting of F_{1}^{n} implies in turn an anisotropic convergence of the component in F_{1}^{⊥,tough} = θ ∧ Ω_{0}^{n−1}H. Indeed, we know that

\frac{K_{ε}α}{ε} = \frac{1}{ε}(δ_{Q}^{J}α_{H,0} − \frac{α_{T,0}}{ε})

= \frac{1}{ε}δ_{Q}^{J}Π_{F_{3}^{⊥}}α + \frac{1}{ε}δ_{Q}^{J}Π_{F_{3}}α − \frac{α_{T,0}}{ε^{2}}

is controlled, showing that Π_{(δ_{Q}^{J}(F_{3}))^{⊥}}α_{T,0} should a priori collapse at speed ε^{2} whereas its orthogonal part Π\overline{δ_{Q}^{J}(F_{3})}α like ε only.

With this in mind, we now look for the possible operators that we can use in the commutator technique. The basic second order operator that preserves the decomposition F_{3}⊕F_{1}^{n}∩F_{3}^{⊥} is d_{Q}δ_{Q}. It controls two derivatives of the F_{1}^{n}∩F_{3}^{⊥} component by hypoelliptic regularity. We have to complete it on F_{1}^{⊥,tough} = θ ∧Ω_{0}^{n−1}H in order that the whole nearly

commutes with \frac{K_{ε}}{ε}. We have by (7)

\frac{1}{ε}δ_{Q}^{J}(d_{Q}δ_{Q}Π_{F_{1}^{n}}) = \frac{1}{ε}(L_{T} − \frac{1}{2}d_{Q}δ_{Q}^{J})δ_{Q}Π_{F_{1}^{n}}

= \frac{1}{ε}(L_{T} δ_{Q} + \frac{1}{2}d_{Q}δ_{Q}δ_{Q}^{J} + d_{Q}P_{N(J)}^{(1)})Π_{F_{1}^{n}} ,

where P_{N(J)}^{(1)} is a first order operator function of the Nijenhuis tensor of J. Its non vanishing

seems to be an obstruction in this method because the term ^{\frac{d_{Q}P_{N(J)}^{(1)}}{ε}} is uncontrollable on F_{3}. So, we will suppose in all the following that the complex structure J is integrable (gives a CR structure). In this case, we have found that on F_{1}^{n} ⊕ θ ∧ Ω_{0}^{n−1}H,

(28)  \frac{K_{ε}}{ε} (\begin{matrix}d_{Q}δ_{Q} & 0 \\ 0 & \frac{1}{2}(d_{Q}δ_{Q} + d_{Q}^{J}δ_{Q}^{J}) \end{matrix}) − \frac{1}{2}(d_{Q}δ_{Q} + d_{Q}^{J}δ_{Q}^{J})\frac{K_{ε}}{ε} = L_{T} \frac{δ_{Q}}{ε} Π_{F_{1}^{n}} .

We will see that this d_{Q}δ_{Q} + d_{Q}^{J}δ_{Q}^{J} controls the θ ∧ (ker δ_{Q} ∩ ker δ_{Q}^{J})^{⊥} component, which is contained in the above fast collapsing space θ ∧ (δ_{Q}^{J}(ker δ_{Q}))^{⊥} when J is integrable. We therefore introduce

F_{3,T} = θ ∧ (ker δ_{Q} ∩ ker δ_{Q}^{J})

the remaining part of θ ∧ Ω_{0}^{n−1}H to be controlled. Before coming to this, we still have to complete our first regularizing operator to the rest F_{1}^{⊥,tame} = Ω^{n}M ⊖ (F_{1}^{n} ⊕ θ ∧ Ω_{0}^{n−1}H) of the algebra, so that the extension nearly commutes with the full \frac{Δ_{ε}}{ε^{2}} . For this, we can use (16) and Remark 4.2, to write on Ω_{0}^{n−1}H,

(29)  \frac{1}{2}(d_{Q}δ_{Q} + d_{Q}^{J}δ_{Q}^{J}) = d_{Q_{N}} δ_{Q_{N}} + d_{Q_{N}}^{J} δ_{Q_{N}}^{J} = Δ_{H} − Δ_{Q_{N}} .

Now Δ_{H} − Δ_{Q_{N}} makes sense on all ⊕_{p<n−1}Ω_{0}^{p}H and, by (16), is hypoelliptic there. Then, we extend it to the full Lefschetz decomposition of F_{1}^{⊥,tame}, by requiring that it commutes with L. This choice has been induced by the fact that, thanks to (6), (15), (16) and Remark 4.2, Δ_{H} −Δ_{Q_{N}} ”nearly” commutes with all our algebra of operators : d_{Q}, δ_{Q}, d_{Q}^{J}, δ_{Q}^{J}, outside F_{1}^{n}. Namely, one has on (F_{1}^{n})^{⊥}, if J is integrable, the following commutation relations

(30)  \{ \begin{matrix}[Δ_{Q_{N}} , δ_{Q}] & = 0 \\ [Δ_{Q_{N}} , δ_{Q}^{J}] & = [Δ_{Q_{N}} − Δ_{Q_{N}}^{J} , δ_{Q}^{J}] = P_{L_{T} J}^{(1)} \\ [Δ_{H}, δ_{Q}] & = [d_{Q_{N}}^{J} δ_{Q_{N}}^{J} , δ_{Q}] = L_{T}^{J} δ_{Q}^{J} \\ [Δ_{H}, δ_{Q}^{J}] & = [Δ_{H} − Δ_{H}^{J}, δ_{Q}^{J}] + [Δ_{H}^{J}, δ_{Q}^{J}] = P_{L_{T} J}^{(1)} − L_{T} δ_{Q} . \end{matrix}

Anyway, in order to be able to use this, and in view of (28), we still have to find a vertical derivative ∂_{T} which nicely commutes with \frac{Δ_{ε}}{ε^{2}} . Again, in the previous section, this was easily done because our underlying splitting of forms in anisotropic components F_{1} and F_{1}^{⊥} was algebraic (define by a 0 order operator). Here, this ∂_{T} has to preserve both F_{3}^{n} = F_{1}^{n} ∩ ker δ_{Q}, and commutes in a controllable way with \frac{K_{ε}α}{ε} = \frac{1}{ε}(δ_{Q}^{J}α_{H,0} − \frac{α_{T,0}}{ε}). A natural, although very strong hypothesis, that allows this, is to suppose that the complex structure is invariant under the Reeb flow. In this case, one has by (8), L_{T}^{∗} = −L_{T}^{J} = −L_{T}, so that

(31)  [L_{T} , δ_{Q}] = 0 = [L_{T} , δ_{Q}^{J}] = [L_{T} , d_{Q}] = [L_{T} , d_{Q}^{J}] ,

and finally [L_{T} , \frac{Δ_{ε}}{ε^{2}} ] = 0, which can be seen directly because the Reeb flow preserves the

metrics g_{ε} = dθ(·, J ·) + \frac{θ^{2}}{ε^{2}} .

Lasly, we have to find a third operator in order to control the remaining components

F_{3}^{n} = F_{1}^{n} ∩ ker δ_{Q} and F_{3,T}^{n} = θ ∧ (ker δ_{Q} ∩ ker δ_{Q}^{J}) .

A second order one to try on F_{3}^{n} is ∗D. Again, we have to find an admissible conjugate of it through K_{ε}. One has on F_{1}^{n}, by (1) (7) and [18],

∗D = ∗θ ∧ (L_{T} − d_{Q}δ_{Q}^{J})
= (−1)^{\frac{n(n+1)}{2}} J(L_{T} − d_{Q}δ_{Q}^{J}) ,

so that,

δ_{Q}^{J}((−1)^{\frac{n(n+1)}{2}} ∗ D) = δ_{Q}^{J}((−1)^{\frac{n(n+1)}{2}} ∗ D)Π_{F_{3}}
= (δ_{Q}^{J}JL_{T} − δ_{Q}^{J}Jd_{Q}δ_{Q}^{J})Π_{F_{3}}
= −J[δ_{Q}, L_{T}]Π_{F_{3}} + (Jδ_{Q}d_{Q})Π_{F_{3,T}} δ_{Q}^{J}Π_{F_{3}} ,

where [δ_{Q}, L_{T}] = 0 by (31), and on F_{3,T}

Jδ_{Q}d_{Q}Π_{F_{3,T}} = JΔ_{Q_{N}} Π_{F_{3,T}} = Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}} = Π_{F_{3,T}} Jδ_{Q}d_{Q}

because Δ_{Q_{N}} = Δ_{Q_{N}}^{J} (by (15)) preserves F_{3,T} = θ ∧ (ker δ_{Q} ∩ ker δ_{Q}^{J}) (that we will often confuse with i_{T} F_{3,T}). So, we have found that on F_{1}^{n}

δ_{Q}^{J}((−1)^{\frac{n(n+1)}{2}} ∗ D) = (Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}})δ_{Q}^{J}Π_{F_{3}}
= (Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}})δ_{Q}^{J} − Π_{F_{3,T}} Jδ_{Q}d_{Q}δ_{Q}^{J}Π_{F_{3}^{⊥}} ,

with by (7), d_{Q}δ_{Q}^{J} = L_{T} on ker D ⊃ F_{3}^{⊥}. Thus, we finally get the commutation relation

(32)  δ_{Q}^{J}((−1)^{\frac{n(n+1)}{2}} ∗ D) = (Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}})δ_{Q}^{J} − Π_{F_{3,T}} Jδ_{Q}L_{T} Π_{F_{3}^{⊥}} ,

and we have found an admissible counterpart of ∗D on θ ∧ Ω_{0}^{n−1}H. Indeed, we have

\frac{K_{ε}}{ε} (\begin{matrix}(−1)^{\frac{n(n+1)}{2}} ∗ D & 0 \\ 0 & Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}} \end{matrix} ) =(Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}})\frac{K_{ε}}{ε}
− Π_{F_{3,T}} Jδ_{Q_{N}} L_{T} Π_{F_{3}^{⊥}} .

(33)

We have completed our search of basic regular operators to apply in the commutator technique. Before coming to this, we need to collect some analytic information on Π_{F_{3,T}}. We first observe that Π_{ker δ_{Q}∩ker δ_{Q}^{J}} = Π_{ker ∂_{Q}^{∗}∩ker \overline{∂}_{Q}^{∗}}. This is convenient because the projections Π_{ker ∂_{Q}^{∗}} and Π_{ker} \overline{∂}_{Q}^{∗} commute. To see this, we define

Δ_{∂_{Q_{N}}} = ∂_{Q_{N}}^{∗} ∂_{Q_{N}} + ∂_{Q_{N}} ∂_{Q_{N}}^{∗} and Δ_{\overline{∂}_{Q_{N}}} = \overline{∂}_{Q_{N}}^{∗} \overline{∂}_{Q_{N}} + \overline{∂}_{Q_{N}} \overline{∂}_{Q_{N}}^{∗} .

It follows from (7), (10) and (14) that when J is integrable and invariant under T, one has on Ω_{0}^{p}H for p < n,

(34)  \begin{cases}∂_{Q_{N}}^{∗} \overline{∂}_{Q_{N}} + \overline{∂}_{Q_{N}} ∂_{Q_{N}}^{∗} = ∂_{Q_{N}}^{2} = ∂_{Q_{N}} \overline{∂}_{Q_{N}} + \overline{∂}_{Q_{N}} ∂_{Q_{N}} = 0 \\ Δ_{∂_{Q_{N}}} + Δ_{\overline{∂}_{Q_{N}}} = Δ_{Q_{N}} and Δ_{\overline{∂}_{Q_{N}}} − Δ_{∂_{Q_{N}}} = iL_{T} \end{cases}

Now we will need the following facts.

Lemma 7.7. 1. Δ_{∂_{Q_{N}}} (resp Δ_{\overline{∂}_{Q_{N}}}) is hypoelliptic on Ω_{0}^{p,q}H for 1 ≤ p ≤ n − 1 (resp 1 ≤ q ≤ n − 1).

2. Π_{ker ∂_{Q_{N}}^{∗}} and its conjugate Π_{ker} \overline{∂}_{Q_{N}}^{∗} are continuous from any H_{k,H} to themselves.

3. One has Π_{ker δ_{Q}∩ker δ_{Q}^{J}} = Π_{ker ∂_{Q_{N}}^{∗}} Π_{ker} \overline{∂}_{Q_{N}}^{∗} = Π_{ker} \overline{∂}_{Q_{N}}^{∗} Π_{ker ∂_{Q_{N}}^{∗}} .

Proof. To see the first point, we use (16) and above (34) to write on Ω_{0}^{p,q}H with p+q < n,

Δ_{H} = (n − p − q)Δ_{Q_{N}} + d_{Q_{N}} δ_{Q_{N}} + d_{Q_{N}}^{J} δ_{Q_{N}}^{J}
≤ (n − p − q + 2)Δ_{Q_{N}} = (n − p − q + 2)(2Δ_{∂_{Q_{N}}} + iL_{T}) ,

so that by (12)

2(n − p − q + 2)Δ_{∂_{Q_{N}}} ≥ Δ_{H} − i(n − p − q − 2)L_{T}
= Δ_{K} + i(2p − n − 2)L_{T} + first order ,

which is an hypoelliptic Folland-Stein operator for 1 < p < n. This estimation is too crude for p = 1 but still gives anyway that Δ_{∂_{Q_{N}}} controls the H^{1,0} derivatives because

Δ_{K} − inL_{T} ≃ 2Δ_{∂_{b}} ≃ 2∇_{H}^{1,0}^{∗}∇_{H}^{1,0} .

This in turn gives that ∃C such that \overline{∂}_{Q_{N}} \overline{∂}_{Q_{N}}^{∗} ≤ CΔ_{∂_{Q_{N}}} because \overline{∂}_{Q}^{∗} = \overline{∂}_{H}^{∗} is an expression of these H^{1,0} derivatives. Lastly we can now refine the use of (16) to obtain

Δ_{H} = (n − 1 − q)Δ_{Q_{N}} + 2∂_{Q_{N}} ∂_{Q_{N}}^{∗} + 2\overline{∂}_{Q_{N}} \overline{∂}_{Q_{N}}^{∗}
≤ (n − 1 − q)(2Δ_{∂_{Q_{N}}} + iL_{T}) + 2(1 + C)Δ_{∂_{Q_{N}}} ,

which this time leads to an hypoelliptic control.

The other statements of lemma 7.7 are then direct consequences of this first point and (34).

Remark 7.8. The operators Δ_{∂_{Q_{N}}} are not hypoelliptic on the remaining spaces Ω_{0}^{0,p}H since their principal part vanishes (at least) on forms with anti-CR components (and in fact more since these are quotiented versions of the more classical ∂_{H} ones).

We can now come to the commutator technique.

Proposition 7.9. Let P_{F_{3}} and P_{F_{3}^{⊥}} be defined on Ω^{n}M = F_{1}^{n} ⊕ θ ∧ Ω_{0}^{n−1}H ⊕ F_{1}^{⊥,tame} by (the power of i is for the symmetry)

P_{F_{3}} = i^{n−1} (\begin{matrix}(−1)^{\frac{n(n+1)}{2}} ∗ D & 0 & 0 \\ 0 & Π_{F_{3,T}} JΔ_{Q_{N}} Π_{F_{3,T}} & 0 \\ 0 & 0 & 0 \end{matrix})

and

P_{F_{3}^{⊥}} = (\begin{matrix}d_{Q}δ_{Q} & 0 & 0 \\ 0 & Δ_{H} − Δ_{Q_{N}} & 0 \\ 0 & 0 & Δ_{H} − Δ_{Q_{N}} \end{matrix} ) .

1. One has, if J is integrable and invariant under the Reeb flow,

[P_{F_{3}}, \frac{Δ_{ε}}{ε^{2}} ] = −\frac{K_{ε}^{∗}}{ε} P_{H}^{(1)}L_{T} \frac{Π_{Im d_{Q}}}{ε} + \frac{Π_{Im d_{Q}}}{ε} P_{H}^{(1)}L_{T} \frac{K_{ε}^{∗}}{ε} ,

and

[P_{F_{3}^{⊥}}, \frac{Δ_{ε}}{ε^{2}} ] = −\frac{K_{ε}^{∗}}{ε} L_{T} \frac{δ_{Q}}{ε} Π_{F_{1}^{n}} + \frac{Π_{F_{1}^{⊥,tame}}}{ε^{2}} P_{H}^{(1)}L_{T} \frac{Π_{F_{1}^{⊥}}}{ε} + \frac{Π_{F_{1}^{⊥}}}{ε} P_{H}^{(2)}L_{T} \frac{Π_{F_{1}^{⊥}}}{ε} − adjoints.

2. The system (P_{F_{3}}, P_{F_{3}^{⊥}}) is hypoelliptic of order 2, which means that ∀k ∈ ℕ and

α ∈ Ω^{n}M

∥P_{F_{3}}α∥_{k,H} + ∥P_{F_{3}^{⊥}}α∥_{k,H} + ∥α∥_{k,H} ≥ ∥α∥_{k+2,H} .

Proof. The first point stems from formulas for Δ_{ε} as given in lemma 5.1 and (17). The case of P_{F_{3}} follows then from (33). About P_{F_{3}^{⊥}}, we have first of all

[P_{F_{3}^{⊥}}, \frac{d_{Q}δ_{Q}}{ε^{2}} Π_{F_{1}^{n}}] = 0 = [P_{F_{3}^{⊥}}, D_{T}^{2} ] ,

and by (28) and (29)

[P_{F_{3}^{⊥}}, \frac{K_{ε}^{∗}K_{ε}}{ε^{2}} ] = [P_{F_{3}^{⊥}}, K_{ε}^{∗}]\frac{K_{ε}}{ε^{2}} + \frac{K_{ε}}{ε^{2}} [P_{F_{3}^{⊥}}, K_{ε}]

= −\frac{K_{ε}^{∗}}{ε} L_{T} \frac{δ_{Q}}{ε} Π_{F_{1}^{n}} − adjoint.

Also, thanks to (28), we get that on F_{1}^{⊥}

[P_{F_{3}^{⊥}}, δ_{Q}] = P_{H}^{(1)}L_{T} = [P_{F_{3}^{⊥}}, δ_{Q}^{J}] ,

and finally

[P_{F_{3}^{⊥}}, non diagonal terms of \frac{Δ_{ε}}{ε^{2}} on F_{1}^{⊥}] = \frac{F_{1}^{⊥,tame}}{ε^{2}} P_{H}^{(1)}L_{T} \frac{Π_{F_{1}^{⊥}}}{ε} − adjoint,

together with

[P_{F_{3}^{⊥}}, second order diagonal terms of \frac{Δ_{ε}}{ε^{2}} on F_{1}^{⊥}] = \frac{Π_{F_{1}^{⊥}}}{ε} P_{H}^{(2)}L_{T} \frac{Π_{F_{1}^{⊥}}}{ε} − adjoint.

The second point is clear outside θ ∧ Ω_{0}^{n−1}H because (d_{Q}δ_{Q}, ∗D) is hypoelliptic on F_{1}^{n}, and Δ_{H} − Δ_{Q} is on Ω_{0}^{p}H for p < n − 1 by (16). On θ ∧ Ω_{0}^{n−1}H = F_{3,T} ⊕ F_{3,T}^{⊥}, one has

∥P_{F_{3}}α∥_{k,H} = ∥JΔ_{Q_{N}} Π_{F_{3,T}} α∥_{k,H} ≥ ∥Π_{F_{3,T}} α∥_{k+2,H} − ∥α∥_{k,H} ,

by hypoellipticity of Δ_{Q_{N}} and continuity of Π_{F_{3,T}} (lemma 7.12). About the remaining F_{3,T}^{⊥}, it can be orthogonally decomposed, thanks to lemma 7.7 and (34) in the following way

F_{3,T}^{⊥} = ∂_{Q}(ker ∂_{Q}^{∗} ∩ ker \overline{∂}_{Q}^{∗}) ⊕ \overline{∂}_{Q}(ker ∂_{Q}^{∗} ∩ ker \overline{∂}_{Q}^{∗}) ⊕ (Im ∂_{Q} ∩ Im \overline{∂}_{Q}) ,

on which P_{F_{3,T}^{⊥}} = d_{Q_{N}} δ_{Q_{N}} + d_{Q_{N}}^{J} δ_{Q_{N}}^{J} = 2(∂_{Q_{N}} ∂_{Q_{N}}^{∗} + \overline{∂}_{Q_{N}} \overline{∂}_{Q_{N}}^{∗}) acts as

P_{F_{3,T}^{⊥}} = (\begin{matrix}2Δ_{∂_{Q_{N}}} & 0 & 0 \\ 0 & 2Δ_{\overline{∂}_{Q_{N}}} & 0 \\ 0 & 0 & 2Δ_{Q_{N}} \end{matrix} )

completing the hypoelliptic control.

We can now state the refinement of Theorem 7.1 under the geometric assumptions which we have been leaded to.

Theorem 7.10. Let M be a compact contact manifold endowed with the family of metrics g_{ε}, and such that the complex structure is integrable and invariant under the Reeb flow. Then one has uniformly for t ≥ t_{0} > 0,

1. on Ω^{p}M with p = n or n + 1,

e^{−t \frac{Δε}{ε^{2}}} ⟶_{ε→0} e^{−tΔ_{D}}Π_{F_{3}}

2. for the signature operator S_{ε} acting on Ω^{even}M,

\frac{S_{ε}}{ε} e^{−t \frac{Δε}{ε^{2}}} ⟶_{ε→0} S_{D}e^{−tΔ_{D}} ,

smoothly at the kernel level.

Remark 7.11. It seems unlikely such strong geometric hypothesizes are really necessary to pass from the global statement, we already obtained in Theorem 7.1, to this local one. In index theory, some geometric price has sometimes to be paid to do this, but only for small time problems, which is not the case here!

The proof will follow the lines of the previous section and we refer to it for more details. We introduce higher order L^{2} Sobolev spaces and extend the regularizing properties of the resolvent on them.

Lemma 7.12. For k ∈ ℕ, ε > 0 and α ∈ Ω^{n}M, let

∥α∥_{k,ε} = ∥α∥_{k,H} + \frac{1}{ε}∥Π_{F_{3}^{⊥}}α∥_{k,H} + \frac{1}{ε}∥K_{ε}α∥_{k−1,H} + \frac{1}{ε^{2}} ∥Π_{F_{1}^{⊥,tame}}α∥_{k−1,H} .

Then (Id +\frac{Δ_{ε}}{ε^{2}} )^{−1} extends continuously from H_{2k−1,H} to H_{2k+1,ε} with the uniform boundedness, when ε → 0, of the following operators

(Id +\frac{Δ_{ε}}{ε^{2}} )^{−1}  : H_{2k−1,H} → H_{2k+1,ε}
(Id +\frac{Δ_{ε}}{ε^{2}} )^{−1} \frac{Π_{F_{3}^{⊥}}}{ε} : H_{2k−1,H} → H_{2k+1,ε}
(Id +\frac{Δ_{ε}}{ε^{2}} )^{−1} \frac{K_{ε}^{∗}}{ε} : H_{2k,H} → H_{2k+1,ε}
(Id +\frac{Δ_{ε}}{ε^{2}} )^{−1} \frac{Π_{F_{1}^{⊥,tame}}}{ε^{2}} : H_{2k,H} → H_{2k+1,ε}

Proof. We can proceed by recurrence as in lemma 7.4. The case k = 0 follows from the basic L^{2} a priori control of proposition 5.8. The recurrence then uses the commutator method applied with P_{F_{3}}, P_{F_{3}^{⊥}} and L_{T}, as proposition 7.9 allows it.

The proof of the first point of Theorem 7.10 is then completed in the same manner as in previous section. Lastly, the statement for the signature operator comes from the obtained uniform smoothing properties of e^{−t \frac{Δε}{ε^{2}}} and the continuity of \frac{S_{ε}}{ε} from L^{2} to H_{−1′,ε} as seen in prop 5.8.

7.4 On the large scale behavior of the heat kernels on forms on the Heisenberg groups

Although we have been concerned so far by compact contact manifolds, we now observe that the developed techniques can also been used in the following non-compact situation, namely, the study of the asymptotic behavior of the heat operators on forms on the Heisenberg groups for large times. The first thing to remark for this is that most of the convergence theorems obtained here for compact contact manifolds M also holds on Galois coverings Γ → M̃ → M of them (like their universal covering spaces with Γ = π_{1}M).

Proposition 7.13. 1. The resolvent convergence Theorems 3.5 and 3.6 hold on M̃ except that the collapsing of the F_{2} component in 3.6, instead of being uniform in L^{2}, is only weak on M̃ and strong on compact sets of M̃.

2. The local convergence theorems of heat kernels 7.2 and 7.10 hold on compact sets of M̃ × M̃.

Proof. The basic L^{2} estimates of section 5 rely on the Bochner technique and still apply here. The only changes in the proofs in section 6 thus depend on proposition 6.2. It is still true on M̃, at the exception of the statement on the discreteness of the spectra.

Indeed, the only point to check here is the essential self-adjointness of the operator P = Δ_{Q} + P_{D} considered in the proof of proposition 6.2. It can be obtained using the same general methods as described for elliptic operators by Atiyah in [1]. Namely, P being a maximally hypoelliptic operator of order 2, it satisfies basic estimates

∥P α∥_{0} + ∥α∥_{0} ≥ C∥α∥_{2,H}

for compactly supported forms in a fixed compact set of M̃. The fact is that, like in the elliptic case of [1], these estimates give a global one by Γ-translating and adding them using a Γ-invariant partition of unity.

Now, in the proofs of section 6, the discreteness of the spectrum of Δ_{Q} was only used to obtain the uniform control

∥β_{ε}∥_{1′,ε}^{2} ≥ \frac{1}{ε^{2}} (Δ_{Q}β_{ε}, β_{ε})_{0} ≥ \frac{C}{ε^{2}} ∥Π_{F_{2}^{⊥}}β_{ε}∥_{0}^{2} ,

where β_{ε} = (λ− \frac{P_{ε}}{ε})^{−1}α. That gave the uniform collapsing of Π_{F_{2}^{⊥}} component in Theorem 3.6. Instead of this, we only get here that

(Δ_{Q}β_{ε}, β_{ε})_{0} ≤ ε^{2}∥β_{ε}∥_{1′,ε}^{2} ≤ ε^{2}∥α∥_{0}^{2} .

Therefore, if we decompose Π_{F_{2}^{⊥}}β_{ε} with respect to the spectral spaces E(]0, ε]) and E(]ε, ∞[) associated to Δ_{Q}, we obtain that,

∥Π_{E(]ε,∞[)}β_{ε}∥_{0}^{2} ≤ \frac{1}{ε}(Δ_{Q}β_{ε}, β_{ε})_{0} ≤ ε∥α∥_{0}^{2} .

So, this component is still uniformly collapsing, whereas we only have that

(Π_{E(]0,ε])}β_{ε}, γ)_{0} = (β_{ε}, Π_{E(]0,ε])}γ)_{0} → 0 ,

for all γ ∈ L^{2} by the spectral theorem. Anyway, we conserve a uniform first order Sobolev control of β_{ε} through its (1′, ε) norm, and this still implies its strong convergence to 0, on compact sets. This completes the proof of the first point of the proposition. The remaining one is clear since the methods of section 7.2 and 7.3 still apply here, again in restriction to compact sets, to take profit of the Sobolev controls of the kernels.

We now come to the Heisenberg group H^{2n+1}. We recall that its Lie algebra 𝔥^{2n+1} is generated by X_{i}, Y_{i}, 1 ≤ i ≤ n and T in the center, with the commutation relations [Y_{i}, X_{j}] = δ_{ij}T. We endowed H^{2n+1} with a left invariant adapted metric g_{1} = dθ(·, J ·)+θ^{2} with θ dual to T and JX_{i} = Y_{i}. It is well known that these groups admit compact quotients M = H^{2n+1}/Γ (which are non trivial circle bundles over a torus), thus the previous proposition will apply to them. The link with our large scale riemannian problem is provided by the parabolic dilations h_{ε}. These are automorphisms of H^{2n+1} acting on 𝔥^{2n+1} by h_{ε} = ε Id on H =Vect (X_{i}, Y_{i}) and h_{ε}T = ε^{2}T. Moreover, h_{ε} induces an isometry between the spaces (H^{2n+1}, g_{1}) and (H^{2n+1}, \frac{g_{ε}}{ε^{2}} ), so that

h_{ε}^{∗}Δ_{g_{1}} = Δ_{\frac{gε}{ε^{2}}} = ε^{2}Δ_{g_{ε}} .

We recall also that, since section 2, we work with the conjugate operator Δ_{ε} = C_{ε}Δ_{g_{ε}}C_{ε}^{−1}, where C_{ε} is the point-wise map defined on Ω^{∗}M by C_{ε}(α_{H} + θ ∧ α_{T}) = α_{H} + εθ ∧ α_{T}. Lastly, if P is an operator on Ω^{∗}H^{2n+1}, the kernel of h_{ε}^{∗}P transforms like

K_{h_{ε}^{∗}P} (x, y) = ε^{−2n−2}C_{ε}^{−1}K_{P} (h_{ε}^{−1}x, h_{ε}^{−1}y)C_{ε} ,

when expressed in a fixed volume form, and with tangent spaces identified through translations. Thus, by proposition 7.13, we finally obtain the asymptotic behavior at large time of the heat kernels on forms on H^{2n+1}.

Theorem 7.14. One has on Ω^{p}H^{2n+1} for p ≠ n, n + 1

ε^{−2n−2}K_{e^{− \frac{t}{ε^{2}} Δ}}(h_{ε}^{−1}x, h_{ε}^{−1}y) ⟶_{ε→0} K_{e^{−tΔ_{Q}}}(x, y) ,

and on Ω^{n}H^{2n+1} ⊕ Ω^{n+1}H^{2n+1}

ε^{−2n−2}K_{e^{− \frac{t}{ε^{4}} Δ}}(h_{ε}^{−1}x, h_{ε}^{−1}y) ⟶_{ε→0} K_{e^{−tΔ_{D}}} (x, y) ,

uniformly on compact sets of H^{2n+1} × H^{2n+1} and non small t.

Thus, heat on forms ”at infinity” of the Heisenberg groups converges to the contact complex ones, its higher order degeneracy as a spectral sequence, in middle degrees, being directly reflected in the lower speed of diffusion there. This is of course consistent with a time-dilation rescaling, the contact Laplacians being homogeneous under h_{λ}. Namely, one has

h_{λ}^{∗}Δ_{Q,g_{1}} = Δ_{Q, \frac{g_{λ}}{λ^{2}}} = λ^{2}Δ_{Q,g_{1}} and h_{λ}^{∗}Δ_{D,g_{1}} = λ^{4}Δ_{D,g_{1}} ,

so that,

h_{λ}^{∗}(e^{−tΔ_{Q}}) = e^{−tλ^{2}Δ_{Q}} and h_{λ}^{∗}(e^{−tΔ_{D}}) = e^{−tλ^{4}Δ_{D}}.

We now mention that this result gives, using von Neumann’s Γ-dimension, some numerical information on the spectrum of Δ near 0. Recall (see [1]) that associated to the Γ-cocompact action are the notions of Γ-trace of operators and the related Γdimension of closed Γ-invariant spaces. Briefly, if P is a positive Γ-equivariant Hermitian operator acting on some Hilbert space H which is a free Γ-module, its Γ-trace is defined by Tr_{Γ}(P) = ∑_{i∈I}(P e_{i}, e_{i}) where one has chosen one vector e_{i} by Γ-orbit of a Γequivariant Hilbert base of H. If V is a closed Γ-invariant subspace in H, its Γ-dimension is dim_{Γ}(V) = Tr_{Γ}(Π_{V}), where Π_{V} is the orthogonal projection on V. Lastly, in the case we consider where P acts on Ω^{∗}M with smooth kernel one has Tr_{Γ}(P) = ∫_{F} Tr(K_{P} (x, x))dvol, F being a fundamental domain of the Γ-action. Moreover, since our operators are left invariant on H^{2n+1}, this formula simplifies further to Tr_{Γ}(P) = Tr(K(e, e))vol(F). Thus, the previous theorem leads to the following

Corollary 7.15. For Δ acting on Ω^{p}H^{2n+1}, let θ_{Δ}(t) = Tr_{Γ}(e^{−tΔ}) (this is also the Laplace transform of the spectrum distribution function of Δ : θ_{Δ}(t) = ∫ e^{−tλ}dN_{Γ}(λ) with N_{Γ}(λ) = dim_{Γ}(E_{Δ}([0, λ])) (see [11])). One has, when ε → 0

θ_{Δ}(ε^{−2}) ∼ \begin{cases}ε^{2n+2}θ_{Δ_{Q}}(1) & if p ≠ n, n + 1 \\ ε^{n+1}θ_{Δ_{D}}(1) & otherwise, \end{cases}

Therefore, the decay exponent of heat on p-forms is α_{p} = n + 1 if p ≠ n, n + 1 and \frac{n+1}{2} otherwise.

These numbers α_{p} (or sometimes twice these numbers) are called the Novikov-Shubin numbers, of M = H^{2n+1}/Γ here, and have some geometric interest since Gromov and Shubin proved in [11] their homotopical invariance (see also [5]). Lott in [13], computed them on H^{3} using representation theory, and also gave (sharp) bounds on the higher dimensional Heisenberg groups. Recently, and independently of this work, Schubert announced in [17] he found them in all dimensions, using again representation theory and algebraic methods (however, it seems there is a gap in this proof). The proof we gave here has some geometric flavour. But still, as homotopical invariants, they deserve an ”homotopical” approach. A direct one is indeed possible. (I am grateful to P. Pansu for his unyielding and finally communicative conviction about this).

another computation of α_{p}. Recall that in sections 2 and 3 we observed that the construction of the contact complex, and in fact the proof of its local exactness in [16], come with explicit homotopy operators, the lifting maps r in (18), towards the de Rham complex. They are (first order) operators from Ω_{0}^{p}H to Ω^{p}M satisfying dr = rd_{Q} for p < n and dr = D for p = n. Above degree n, the contact complex is naturally embedded as a sub-complex of de Rham’s one, so we can extend r by the inclusion map here. Reversely, we have a natural projection Π from Ω^{∗}M to Ω^{∗}M/I ≃ Ω_{0}^{∗}M until degree n and satisfying (by definition) Πd = d_{Q}Π. Again, this map Π has an extension from Ω^{∗}M to J^{∗} in higher degrees. To define this, let L_{R}^{−1} = Λ(LΛ)^{−1} be the right inverse of L (exists since L is surjective in the degrees we consider), and define h = θ ∧ L_{R}^{−1}Π_{H} (that is h = ( \begin{matrix}0 & 0 \\ L_{R}^{−1} & 0 \end{matrix} ) in Ω^{∗}M = Ω^{∗}H ⊕ θ ∧ Ω^{∗}H. Note also that in some sense h is d_{0}^{−1}, the inverse of the zero order term of d by prop 3.3). It is easy to check that Π = Id −hd − dh maps Ω^{∗}M to J^{∗} = θ ∧ ker L. In fact Π and r induce an homotopy equivalence between the de Rham and contact complexes since they are seen to satisfy Πr = Id and rΠ = Id −hd − dh (where one uses the left, instead of right, inverse of L in degree ≤ n).

Now, one would like to use the fundamental result of Gromov and Shubin in [11] that boundedly homotopy equivalent Hilbert complexes have the same Novikov-Shubin numbers since, as we already observed, the ones of the contact complex are clear on the Heisenberg group by its homogeneity through the parabolic dilations. Our problem here is that the first order maps Π and r are unbounded in L^{2}, but it is bypassed if instead of working with the full de Rham complex, one restricts or projects it to some spectral space E([0, λ]) associated to Δ (and the same on the contact complex). Indeed, the projection Π_{E([0,λ])} is a regularizing map and induces a bounded homotopy equivalence between the de Rham and cut-off de Rham complexes, because Id −Π_{E([0,λ])} = Π_{E(]λ,∞[)} = dH + Hd where H = δΔ^{−1}Π_{E(]λ,∞[)} is bounded.

Remark 7.16. We would like to mention, still about these α_{p}, that this work also suggests a third natural approach, relying only on the Bochner formula and related basic estimates of section 4, together with the following variational principle to estimate the distribution function of Laplacians. Namely, one has (see [11]),

N_{Γ}(λ) (= dim_{Γ}(E_{Δ}([0, λ]))) = sup_{L∈L_{λ}} dim_{Γ}(L) ,

where L_{λ} is the set of all Γ-invariant closed spaces such that (Δα, α) ≤ λ∥α∥^{2} for all α ∈ L. One can check (left to the tireless reader) that the injection in the Bochner formula of projection and lifting of basic relevant spectral spaces leads rather easily to another computation of the α_{p}, and even, outside middle degrees, to the full equivalent

N_{Γ,Δ}(λ) ∼_{λ→0} N_{Γ,Δ_{Q}}(λ) = λ^{n+1}N_{Γ,Δ_{Q}}(1) .

As a conclusion of this work, we think that it can suggest two directions of investigations. The first question raised, mimicking a famous one, is : ”What do we hear at the infinite of the nilpotent groups?” We mean by this to study the Laplacians and the limit of heat and wave propagation on forms at large scale on these groups. As we have just seen here, the answer to this question for the Heisenberg group is more or less ”the contact complex”. A clue for more general groups is certainly provided by the existence, we mentioned in section 3, of a spectral sequence naturally generalizing the contact complex in Carnot-Caratheodory geometry.

The other direction is to study further the behavior, in the contact case, of global spectral Riemannian invariants (like eta invariant or analytic torsion) in the limit we considered. Although the short time problems are out of scope of this work, we have seen that some hypoelliptic eta and zeta functions naturally come out that should be related to the finite part of their Riemannian counterparts in the sub-Riemannian limit. What we could really expect in this problem is still quite unclear to us, but an efficient analytic tool to investigate it could probably be the extended Heisenberg pseudo-differential calculus, recently developed by Epstein, Mendoza and Melrose in [7], and containing both the elliptic and hypoelliptic Heisenberg calculi (see [3]).

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———————————— Mathématique, Bât. 425, Université de Paris-Sud, 91405 Orsay, France

e-mail : rumin@topo.math.u-psud.fr