ANNALES MATHÉMATIQUES
BLAISE PASCAL
Pierre Pansu & Francesca Tripaldi
Averages and the
`q,1cohomology of Heisenberg groups
Volume 26, no1 (2019), p. 81-100.<http://ambp.centre-mersenne.org/item?id=AMBP_2019__26_1_81_0>
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Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS
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Averages and the `
q,1cohomology of Heisenberg groups
Pierre PansuFrancesca Tripaldi
Abstract
Averages are invariants defined on the`1cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the
`1cohomology vanishes in these cases.
Moyennes et cohomologie`q,1des groupes d’Heisenberg
Résumé
Les moyennes sont des invariants definis sur la cohomologie`1des groupes de Lie. Nous démontrons que les moyennes dans les groupes abéliens et les groupes d’Heisenberg sont nulles. Ce résultat complète des travaux précédents et montre que la cohomologie`1est nulle pour les groupes de Lie étudiés.
1.
Introduction
1.1.
From isoperimetry to averages of
L1forms
The classical isoperimetric inequality implies that ifuis a compactly supported function onRn,kukn0 ≤Ckduk1, wheren0= n−n1. Equivalently, every compactly supported closed 1-formωadmits a primitiveusuch thatkukn0 ≤Ckωk1. More generally, ifωis a closed 1-form onRnwhich belongs toL1, does it have a primitive inLn0?
There is an obstruction. We observe that if each componentai ofω =Ín i=1aidxi
is again in L1, the integral∫
Rnaidx1. . .dxn is well defined and it is an obstruction for ωto be the differential of an Lq function (for every finiteq). Indeed, ifω =du, an =∂x∂un. For almost every(x1, . . . ,xn−1), the functiont7→ ∂x∂u
n(x1, . . . ,xn−1,t)belongs toL1andt 7→u(x1, . . . ,xn−1,t)belongs toLq. Sinceu(x1, . . . ,xn−1,t)tends to 0 along subsequences tending to+∞or−∞,
∫
R
∂u
∂xn(x1, . . . ,xn−1,t)dt=0, hence ∫
Rn
∂u
∂xn dx1. . .dxn=0.
A similar argument applies to other coordinates. Note thatandx1∧ · · · ∧dxn=(−1)n−1ω∧ (dx1∧ · · · ∧dxn−1).
P.P. is supported by Agence Nationale de la Recherche, ANR-15-CE40-0018 SRGI.
F.T. is supported by the Academy of Finland grant 288501 and the ERC Starting Grant 713998 GeoMeG.
Keywords:Heisenberg groups, Rumin complex,`pcohomology, parabolicity.
2010Mathematics Subject Classification:35R03, 58A10, 43A80.
More generally, ifGis a Lie group of dimensionn, there is a pairing, theaverage pairing, between closedL1k-formsωand closed left-invariant(n−k)-formsβ, defined by
(ω, β) 7→∫
G
ω∧β.
The integral vanishes if eitherω=dφwhereφ∈L1, orβ=dαwhereαis left-invariant.
Indeed, Stokes formula∫
Mdγ=0 holds for every complete Riemannian manifoldMand every L1formγsuch that dγ ∈ L1. Hence the pairing descends to quotients, the L1,1 cohomology
L1,1Hk(G)=closedL1k-forms/d(L1(k−1)-forms with differential inL1), and the Lie algebra cohomology
Hn−k(g)=closed, left-invariant(n−k)-forms/d(left-invariant(n−k−1)-forms).
1.2. `q,1
cohomology
It turns out that L1,1 cohomology has a topological content. By definition, the `q,p cohomology of a bounded geometry Riemannian manifold is the`q,pcohomology of every bounded geometry simplicial complex quasiisometric to it. For instance, of a bounded geometry triangulation. Contractible Lie groups are examples of bounded geometry Riemannian manifolds for which L1,1 cohomology is isomorphic to `1,1 cohomology.
We do not need define the `q,p cohomology of simplicial complexes here, since, according to Theorem 3.3 of [7], every`q,p cohomology class of a contractible Lie group can be represented by a formωwhich belongs toLp as well as an arbitrary finite number of its derivatives. If the class vanishes, then there exists a primitiveφofωwhich belongs toLqas well as an arbitrary finite number of its derivatives. This holds for all 1≤p≤q≤ ∞.
Although `p with p > 1, and especially `2 cohomology of Lie groups has been computed and used for large families of Lie groups, very little is known about `1 cohomology.
1.3.
From
`1,1to
`q,1cohomology
For instance, the averaging pairing is specific to`1cohomology and it has never been studied yet. The first question we want to address is whether the averaging pairing provides information on`q,1cohomology for certainq>1.
Question 1.1. Given a Lie groupG, for which exponentsqand which degreeskis the averaging pairing`q,1Hk(G) ⊗Hn−k(g) →Rwell-defined?
The question is whether there existsq >1 such that the pairing vanishes on allL1 forms which are differentials ofLqforms. We just saw that for abelian groupsRn, the pairing is well defined fork=1 and all finite exponentsq. Here is a more general result.
Theorem 1.2. LetGbe a Carnot group. In each degree1≤k≤n, there is an explicit exponentq(G,k)>1(see Definition 3.6) such that the averaging pairing is defined on
`q,1Hk(G)forq∈ [1,q(G,k)].
We shall see that q(Rn,k) = n0 = n−n1 in all degrees. For Heisenberg groups, q(H2m+1,k)=2m+22m+1 ifk,m+1, andq(H2m+1,m+1)= 2m+22m .
1.4.
Vanishing of the averaging pairing
The second question we want to address is whether the averaging pairing is trivial or not.
Question 1.3. Given a Lie groupG, for which exponentsqand which degreeskdoes the averaging pairing`q,1Hk(G) ⊗Hn−k(g) →Rvanish?
The pairing is always nonzero in top degreek =n. Indeed, there existL1 n-forms (even compactly supported ones) with nonvanishing integral. However, this seems not to be the case in lower degrees.
Theorem 1.4. LetGbe an abelian group or a Heisenberg group of dimensionn. In each degree1≤k<n, the averaging pairing vanishes on`q,1Hk(G)forq∈ [1,q(G,k)].
In combination with results of [2], Theorem 1.4 implies a vanishing theorem for`q,1 cohomology. In fact, [2] shows how in the Heisenberg case (and more trivially in the abelian case) it is possible to constructLq(G,k)primitives for dc-closedL1forms with vanishing averages, for all degrees k except top degree. It should be stressed that in proving this result, the vanishing of averages is a necessary extra assumption.
Corollary 1.5. LetGbe an abelian group or a Heisenberg group of dimensionn. In each degree0≤k<n,`q,1Hk(G)=0forq≥q(G,k).
This is sharp. It is shown in [7] that`q,1Hk(G),0 ifq<q(G,k). Also, in top degree, not only is`q(G,n),1Hn(G),0, but the kernel of the averaging map`q(G,n),1Hn(G) → R=H0(g)∗does not vanish. This is in contrast with the results of [1] in the case where p > 1, where it is proved that `q(G,k),pHk(G) = 0 for all degrees k, including top degree. The results of [2] rely in an essential manner on analysis of the Laplacian on L1, inaugurated by J. Bourgain and H. Brezis, [4], adapted to homogeneous groups by S. Chanillo and J. van Schaftingen, [5].
1.5.
Methods
The Euclidean spaceRn is n-parabolic, meaning that there exist smooth compactly supported functionsξonRntaking value 1 on arbitrarily large balls, and whose gradient has an arbitrarily smallLnnorm. Ifωis a closedL1form andβa constant coefficient form, and ifω=dψ,ψ ∈Ln0, Stokes theorem gives
∫
ξω∧β =
∫
ψ∧dξ∧β
≤ kψkn0kdξknkβk∞
which can be made arbitrarily small.
This argument extends to Carnot groups of homogeneous dimensionQ, which areQ- parabolic. For this, one uses Rumin’s complex, which has better homogeneity properties under Carnot dilations than de Rham’s complex. When Rumin’s complex is exactly homogeneous (e.g. for Heisenberg groups in all degrees, only for certain degrees in general), one gets a sharp exponentq(G,k). This leads to Theorem 1.2.
In Euclidean space, every constant coefficient form βhas a primitiveαwith linear coefficients (for instance, dx1∧ · · · ∧dxk =d(x1dx2∧ · · · ∧dxk)). On the other hand, there exist cut-offs which decay like the inverse of the distance to the origin. Therefore
∫
ξω∧β =
∫
ω∧dξ∧α
≤ kωkL1(supp(dξ))
which tends to 0. This argument extends to Heisenberg groups in all but one degree. To complete the proof of Theorem 1.4, one performs the symmetric integration by parts, integratingωinstead ofβ. For this, one produces primitives ofωon annuli, of linear growth.
1.6.
Organization of the paper
In Section 2, the needed cut-offs are constructed. Theorem 1.2 is proven in Section 3. In order to integrateω∧βby parts, one needs understand the behaviour of wedge products in Rumin’s complex on Heisenberg groups, this is performed in Section 4. Section 5 exploits the linear growth primitives of left-invariant forms. In Section 6, controlled primitives of L1forms are designed, completing the proof of Theorem 1.4.
2.
Cut-offs on Carnot groups
2.1.Annuli
We first construct cut-offs with anL∞control on derivatives. In a Carnot groupG, we fix a subRiemannian metric and denote byB(R)the ball with centereand radiusR. We
fix an orthonormal basis of horizontal left-invariant vector fieldsW1, . . . ,Wn1. Given a smooth functionuonG, and an integerm∈N, we denote by∇muthe collection of order mhorizontal derivativesWi1. . .Wim,(i1, . . . ,im) ∈ {1, . . . ,n1}m, and by|∇mu|2the sum of their squares.
Lemma 2.1. LetGbe a Carnot group. Letλ >1. There existsC=C(λ)such that for all R>0, there exists a smooth functionξRsuch that
(1) ξR=1onB(R).
(2) ξR=0outsideB(λR).
(3) For allm∈N,|∇mξR| ≤C/Rm.
Proof. We achieve this first whenR=1, and then setξR=ξ1◦δ1/R. Lemma 2.2. Given f a (vector valued) function which is homogeneous of degreed∈N under dilations, then∇f is homogeneous of degreed−1.
Proof. Given f: G → R homogeneous of degree d under dilations, we have that f(δλp)=λdf(p).
By applying a horizontal derivative to the left hand side of the equation, namely
∇=Wjwithj ∈ {1, . . . ,n1}, we get
∇[f(δλp)]=Wj[f(δλp)]=df ◦dδλ(Wj)p =df λ(Wj)δλp =λ·df Wj
δλp. If we now apply∇to the right hand side, we get
∇[λdf(p)]=Wj[λdf(p)]=λd·df(Wj)p. We have therefore proved that df
δ
λp =λd−1·df
p when restricted to horizontal derivatives, so we finally get our result
∇f(δλp)=λd−1∇f(p).
2.2.
Parabolicity
Second, we construct cut-offs with a sharperLQcontrol on derivatives.
Letrbe a smooth, positive function onG\ {e}that is homogeneous of degree 1 under dilations (one could think of a CC-distance from the origin, but smooth) and let us define the following function
χ(r)= log(λR/r)
log(λR/R) =log(λR/r)
log(λ) . (2.1)
One should notice that χ(λR)=0,χ(R)=1, and thatχis smooth.
Definition 2.3. Using the smooth functionχintroduced in (2.1), we can then define the cut-off functionξas follows
ξ(r)=
1, onB(R)
χ(r), onB(λR) \B(R) 0, outsideB(λR).
Lemma 2.4. The cut-off functionξdefined above has the following property: for every integerm∈N,k∇mξkQ/m→0asλ→ ∞.
Proof. We compute
∇ξ= ( 1
logλ∇r
r ifR<r< λR,
0 otherwise.
Let f be the vector valued function f = ∇rr onG\ {e}. Then f is homogeneous of degree
−1. According to Lemma 2.2,∇m−1f is homogeneous of degreem. It follows that
∫
B(λR)\B(R)
∇m−1f
Q/m
≤C
∫ λR
R
1 rm
Q/m
rQ−1dr
=C
∫ λR
R
rQ−1
rQ dr=Clog(λ).
Therefore
k∇mξkQ/m ≤C(logλ)−1+(m/Q)
tends to 0 asλtends to infinity, providedm<Q. Let us stress that, in general, for values ofmgreater than or equal toQ, the final limit will not be zero. However, the values ofm which we will be considering are very specific.
This estimate will in fact be used in the proof of Proposition 3.3, and in that setting the degreesmthat can arise will be all the possible degrees (or equivalently weights) of the differential dcon an arbitrary Rumink-formφof weightw. If we denote byMthe maximalmthat could arise in this situation, then one can show thatM <Q.
Let us first assume that the maximal orderM for the dcon k-forms (which is non trivial for 0≤k <n) is greater or equal toQ. Then, givenφa Rumink-form of weight w, then dcφ =ÍM
i=1βw+i, where each βw+i is a Rumin(k+1)-form of weightw+i. If we consider the Hodge of the βw+i withQ ≤i ≤ M, then these forms are Rumin (n−k−1)-forms of weightQ− (w+i)=Q−w−i≤Q−w−Q<−w≤0, which is impossible.
Therefore M <Q, which means that indeed in all the cases that we will take into consideration, theLQ/mnorm of∇mξwill always go to zero asλ→ ∞.
Remark 2.5. One says that a Riemannian manifoldMisp-parabolic(see [8]) if for every compact setK, there exist smooth compactly supported functions onMtaking value 1 on Kwhose gradient has an arbitrarily smallLpnorm. The definition obviously extends to subRiemannian manifolds.
Lemma 2.4 implies that a Carnot group of homogeneous dimensionQisQ-parabolic.
3.
The averaging map in general Carnot groups descends to cohomology
Definition 3.1. LetGbe a Carnot group of dimensionnand homogeneous dimension Q. Fork=1, . . . ,n, letW(k)denote the set of weights arising in Rumin’s complex in degreek. For a Rumink-formω, letω= Õ
w∈W(k)
ωw
be its decomposition into components of weightw. Let dc=Í
jdc,j be the decomposition of dcinto weights/orders. LetJ (k,w)denote the set of weights/orders jsuch that dc,j onk-forms of weightwis nonzero, in other words,
J (k,w):={j ∈N|dc,jωw ,0 for someωof degreek}. We will denote byJ (k)the set of all the possible weights/orders, that is
J (k)= Ø
w∈W(k)
J (k,w).
Let us defineLχ(k)as follows Lχ(k)=
(
φ= Õ
w∈W(k−1)
φw ∈E0k−1
∀j ∈ J (k−1,w), φw ∈ LQ/Q−j )
and ifJ (k−1,w)ˆ =∅for some ˆw, then we don’t require anything onφwˆ.
Lemma 3.2. LetGbe a Carnot group. Fix a left-invariant subRiemannian metric making the direct sumg=É
giorthogonal. TheL2-adjointd∗cofdcis a differential operator.
Fix a degreek. Let
dc= Õ
j∈ J(k)
dc,j
be the decomposition ofdcinto weights (dc,j increases weights of Rumin forms by j, hence it has horizontal order j). Then the decomposition ofd∗con a(k+1)-form into weights/orders is
d∗c= Õ
j∈ J∗(k)
d∗c,j.
In other words, the adjointd∗c,j ofdc,j decreases weights by j and has horizontal order j. In fact, if we denote byJ∗(k+1,w)e the set of weights/ordersjsuch thatd∗c,j on (k+1)-forms of weightewis non-zero, that is
J∗(k+1,w)e ={j∈N|d∗c,jαw˜ ,0for someαof degreek+1},
then there is a clear relationship between the sets of indicesJ (k,w)andJ∗(k+1,w),e namely
J∗(k+1,w)e = Ø
w∈W(k)
{j∈ J (k,w) |w+j =w}e .
And from this relationship, we get directly the following identity:
J∗(k+1)= Ø
˜ w∈W(k+1)
J∗(k+1,w)e =J (k).
Moreover, since the formulad∗c=(−1)n(k+1)+1∗dc∗applies to any Rumink-form, we also have the equalityJ∗(n−k,Q−w)=J (k,w).
Let us stress that this also impliesJ (k)=J∗(n−k)=J (n−k−1).
Proposition 3.3. Ifω,φ,βare Rumin forms withω∈L1of degreek,βleft-invariant of complementary degreen−k,φ∈Lχ(k)anddcφ=ω, then
∫
G
ω∧β=0.
Proof. Without loss of generality, one can assume thatβhas pure weightQ−wfor some w∈ W(k). Then its Hodge-star∗βhas pure weightw. Letξbe a smooth cut-off. By definition of theL2-adjoint, we have
∫
G
(dcφ) ∧ξ β=∫
G
hdcφ,∗ξ βidvol=∫
G
hφ,d∗c(∗ξ β)idvol
= Õ
j∈ J∗(k,w)
∫
G
hφw−j,d∗c,j(∗ξ β)idvol.
Sinceφ ∈ Lχ(k), for any j ∈ J∗(w−j)we haveφw−j ∈ LQ/Q−j, by definition of Lχ(k). Hence, applying Hölder’s inequality, we obtain
∫
G
ω∧ξ β
≤ Õ
j∈ J∗(k,w)
kφw−jkQ/(Q−j)k∇jξkQ/jkβk∞.
It is therefore sufficient to takeξas the cut-off function introduced in Definition 2.3, so that by Lemma 2.4 we get that∫
Gω∧β=0.
Example 3.4. Euclidean spaceRn. ThenW(k)={k}andJ (k)={1}in all degrees.
Proposition 3.3 states that the averaging map descends to Lq,1 cohomology, where q=n−n1.
Example 3.5. Heisenberg groupsH2m+1. We have thatW(k) = {k} for k ≤ m, and W(k) = {k+1} when k ≥ m+1. J (k) = {1} in all degrees but k = m, where J (m) = {2}, so that Proposition 3.3 states that the averaging map descends to Lq,1 cohomology, whereq= Q−Q2 in degreem+1 andq= Q−Q1 in all other degrees.
3.1.
Link with
`q,1cohomology
Let 1≤p≤q ≤ ∞. According to Theorem 3.3 of [7], every`q,pcohomology class of a Carnot group contains a formωwhich belongs toLpas well as an arbitrary finite number of its derivatives. If the class vanishes, then there exists a primitiveφofωwhich belongs toLq as well as an arbitrary finite number of its derivatives. There exists a homotopy between de Rham and Rumin’s complex given by differential operators, therefore the same statement applies to Rumin’s complex. In particular, Rumin forms can be used to compute`q,pcohomology.
Let ω be a Rumink-form which belongs to L1 as well as a large number of its horizontal derivatives. Assume thatωrepresents the trivial cohomology class. Then there exists a Rumin(k−1)-formφwhich belongs toLq as well as its horizontal derivatives up to orderQ, and such that dcφ=ω. By Sobolev’s embedding theorem,φbelongs to L∞, hence toLq0for allq0≥q. This suggests the following notation.
Definition 3.6. LetGbe a Carnot group of dimensionnand homogeneous dimensionQ. Let 1≤k≤n. Define
j(k)=min Ø
w∈W(k)
J (k−1,w).
and
q(G,k):= Q Q−j(k).
Proof of Theorem 1.2. LetGbe a Carnot group. Letωbe a Rumink-form onG, which belongs toL1 as well as a large number of its derivatives. Assume thatω=dcφwith φ∈Lq(G,k). Then
∀w∈ W(k),∀ j∈ J (k−1,w), φw−j ∈LQ/(Q−j), thereforeφ∈ Lχ(k). Proposition 3.3 implies that averages∫
ω∧βvanish. This completes
the proof of Theorem 1.2.
Example 3.7. Euclidean space. Thenj(k)=1 in all degrees.
Example 3.8. Heisenberg groupsH2m+1. Thenj(k)=1 in all degrees but k=m+1, wherej(m+1)=2.
For these examples, as we saw before, one need not invoke [7] sinceJ (k)has only one element in each degree.
Example 3.9. Engel groupE4. Thenj(k)=1 in degrees 1 and 4, j(k)=2 in degrees 2 and 3. One concludes that the averaging map is well-defined in`q,1 cohomology for q≤ Q−Q1 in degrees 1 and 4, and forq≤ Q−Q2 in degrees 2 and 3. Here,Q=7.
3.2.
Results for Heisenberg groups
H2m+1In [2], it is proven that every closedL1k-form,k ≤2m, whose averages∫
ω∧βvanish, is the differential of a form inLq, whereq=q(k)=Q−Q1 unless whenk=m+1, where q(m+1)= Q−Q2. In other words,
Theorem 3.10 ([2]). Let G = H2m+1 and let k = 1, . . . ,2m. The averaging map Lq(k),1Hk(G) →H2m+1−k(g)∗is injective.
The goal of subsequent sections is to prove that the image of averaging map is 0 in all degreesk≤2m. This will prove thatLq(k),1Hk(G)=0. Note that fork=2m+1, both facts fail: the averaging map is not zero (one can check with compactly supported forms) and it is not injective either (see [2]).
4.
Wedge products between Rumin forms in Heisenberg groups
We shall rely on Stokes formula on Heisenberg groupsH2m+1. We need a formula of the form d(φ∧β)=(dcφ) ∧β±φ∧dcβ. This does not always hold in general for Carnot groups. In fact, the complex of Rumin formsE0•equals the Lie algebra cohomologyH•(g), and therefore carries a natural cup product induced by the wedge product, but which in general differs from the wedge product. We shall see that in the case of Heisenberg groups, the difference does not show up in many degrees. This has already been investigated in [3].
Let us take into consideration the original construction of the Rumin complex in the (2m+1)-dimensional Heisenberg groupH2m+1as appears in [9].
GivenΩ•the algebra of smooth differential forms, one can define the following two differential ideals:
• I• :={α=γ1∧τ+γ2∧dτ}, the differential ideal generated by the contact formτ, and
• J•:={β∈Ω• |β∧τ=β∧dτ=0}.
Remark 4.1. By construction, the idealJ•is in fact the annihilator ofI•. In other words, given two arbitrary formsα∈ J•andβ∈ I•, we haveα∧β=0.
One can quickly check that the subspacesJh=J•∩Ωhare non-trivial forh≥m+1, whereas the quotientsΩh/Ihare non-trivial forh≤m, whereIh=I•∩Ωh.
Moreover, the usual exterior differential descends to the quotientsΩ•/I•and restricts to the subspacesJ•as first order differential operators:
dc:Ω•/I•→Ω•/I• and dc:J•→ J•. In [9] Rumin then defines a second order linear differential operator
dc:Ωm/Im→ Jm+1
which connects the non-trivial quotientsΩ•/I•with the non-trivial subspacesJ•into a complex, that is dc◦dc=0,
Ω0/I0 d−−→c Ω1/I1 d−−→c . . .−d−→c Ωm/Im−−d→ Jc m+1 d−−→ Jc m+2 d−−→c . . .−d−→ Jc 2m+1. Proposition 4.2. InH2m+1, the wedge product of Rumin forms is well-defined and satisfies the Leibniz rule
dc(α∧β)=dcα∧β+(−1)hα∧dcβ
if eitherh≥m+1ork≥m+1orh+k<m, whereh=deg(α)andk=deg(β).
Proof. In order to study whether the wedge product between Rumin forms is well-defined, we will consider this differential operator dcin the following two cases:
(i) dc:Ωh/Ih→Ωh+1/Ih+1whereh<m, (ii) dc:Jh→ Jh+1whereh>m.
Let us first stress that in the first case, givenα∈Ωh/Ih, we have that dcα=dα mod Ih+1 for h <m. SinceIis an ideal, ifh+k≤m,α∧β∈Ωh+k/Ih+kis well defined.
Ifh+k<m, the identity d(α∧β)=(dα) ∧β+(−1)hα∧dβpasses to the quotient.
It is important to notice that, however, ifh+k=m,h >0,k>0, dcα∧β+(−1)hα∧dcβ involves only first derivatives ofαandβ, and thus cannot be equal to dc(α∧β). Ifh=0 andk=m, dc(α∧β)involves second derivatives ofα, and therefore cannot be expressed in terms of dcα.
On the other hand, in the second case, givenβ∈ Jh, the Rumin differential coincides with the usual exterior differential,
dcβ=dβ for h>m.
Therefore, givenα∈Ωh/Ihandβ∈ Jkwithh <mandk >m, the wedge product α∧βis well-defined and belongs toJh+k, and the usual Leibniz rule also applies:
dc(α∧β)=d(α∧β)=(dα) ∧β+(−1)hα∧ (dβ)=dcα∧β+(−1)hα∧dcβ . If h =mandk ≥m+1, h+k ≥2m+1, so the identity between differentials holds trivially.
To conclude, the wedge product of Rumin forms is well defined and satisfies the Leibniz rule dc(α∧β)=dcα∧β+(−1)hα∧dcβif eitherh ≥m+1, ork≥m+1, or
h+k<m.
5.
Averages on Heisenberg group: generic case
5.1.Primitives of linear growth
Lemma 5.1. Letβbe a left-invariant Ruminh-form in the Heisenberg groupH2m+1. If h,m+1,βadmits a primitiveαof linear growth, i.e. at Carnot–Carathéodory distance rfrom the origin,|α| ≤C r.
Proof. Let β∈ E0h be a left-invariant form. Then dcβ= 0, and βhas weightw = h (ifh ≤m) orh+1 (ifh >m+1). We know that the Rumin complex is locally exact, that is∃α∈E0h−1such that dcα=β.
Let us consider the Taylor expansion ofαat the origin in exponential coordinates, and let us group terms according to their homogeneity under dilationsδt:
α=α0+· · ·+αw−1+αw+αw+1+. . .
where we denote byαdthe term with homogeneous degree d, i.e.δ∗tαd =tdαd. Since dccommutes with the dilationsδt, the expansion of dcαis therefore
dcα=dcα0+· · ·+dcαw−1+dcαw+dcαw+1+. . . .
The expansion ofβis given instead byβ=β, since it is a left-invariant form, hence homogeneous of degreew, so thatβ=dcαw.
Let us notice thatαwhas degreeh−1 andh,m+1, so it has weightw−1. Since it is homogeneous of degreewunderδλ, its coefficients are homogeneous of degree 1, that is they are linear in horizontal coordinates, henceαwhas linear growth, that is|α| ≤C r.
Proposition 5.2. Givenω∈E0kanL1,dc-closed Rumin form inH2m+1, then the integral
∫
H2m+1
ω∧β
vanishes for all left-invariant Rumin formsβof complementary degree,β∈ E02m+1−k, providedk,m.
Proof. Letωbe anL1, dc-closed Rumink-form,k,m. Letβbe a left-invariant Rumin h-form, withh=2m+1−k,m+1. Letαbe a linear growth primitive ofβ,|α| ≤C R.
Letξbe a smooth cut-off such thatξ=1 onB(R),ξ=0 outsideB(λR)and|dcξ| ≤C0/R. Since, according to Proposition 4.2,
d(ξω∧α)=dc(ξω∧α)=dc(ξω) ∧α+(−1)kξω∧dcα
=dcξ∧ω∧α+(−1)kξω∧β, Stokes formula gives
∫
H2m+1
ξω∧β =
∫
B(λR)\B(R)dcξ∧ω∧α
≤
∫
B(λR)\B(R)
|dcξ||α||ω|
≤λCC0kωkL1(H2m+1\B(R)). On the other hand,
∫
H2m+1
(1−ξ)ω∧β
≤ kβk∞kωkL1(H2m+1\B(R)). Both terms tend to 0 asRtends to infinity, thus∫
H2m+1ξω∧β=0.
This proves Theorem 1.4 except in degreek=m. The argument collapses in this case, since primitives of left-invariant(m+1)-forms have at least quadratic growth.
6.
Averages on Heisenberg group: special case
We now describe a symmetric argument: produce a primitive of theL1formωwith linear growth. It applies for all degrees butm+1, and so covers the special casek=m.
Sinceωis not inL∞but isL1, linear growth needs be taken in theL1sense: theL1 norm of the primitive in a shell of radiusRisO(R). It is not necessary to produce a global primitive with this property. It is sufficient to produce such a primitiveφRin theR-shell B(λR) \B(R). Indeed, Stokes formula leads to an integral
∫
H2n+1
ξω∧β=±
∫
B(λR)\B(R)dcξ∧φ∧β
which does not depend on the choice of primitiveφ.
6.1. `q,1
cohomology of bounded geometry Riemannian and subRiemannian manifolds
By definition, the`q,pcohomology of a bounded geometry Riemannian manifold is the
`q,pcohomology of every bounded geometry simplicial complex quasiisometric to it. For instance, of a bounded geometry triangulation.
Combining results of [2] and Leray’s acyclic covering theorem (in the form described in [6]), one gets that forq=nn−1, the`q,1cohomology of a bounded geometry Riemannian n-manifoldMis isomorphic to the quotient
Lq,1H·(M)=L1(M) ∩ker(d)/(L1∩dLq(M))
of closed forms inL1by differentials of forms inLq. In particular, ifMis compact, for allp≤ n−n1,Lp,1H·(M)is isomorphic to the usual (topological) cohomology ofM.
Similarly, ifM is a bounded geometry contact subRiemannian manifold of dimen- sion 2m+1 (hence Hausdorff dimensionQ=2m+2), forq=Q/(Q−1)(respectively q=Q/(Q−2)in degreem+1), the`q,1cohomology ofM is isomorphic to the quotient
Lq,1c H·(M)=L1(M) ∩ker(dc)/(L1∩dcLq(M)) of dc-closed Rumin forms by dc’s of Rumin forms inLq.
This applies in particular to Heisenberg groupsH2m+1, and also to shells in Heisenberg groups, but with a loss on the width of the shell.
6.2. L1
Poincaré inequality in shell
B(λ) \B(1)Lemma 6.1. There exist radii0< µ <1< λ < µ0such that everydc-exactL1Rumin k-formωonB(µ0) \B(µ)admits a primitiveφonB(λ) \B(1)such that
kφkL1(B(λ)\B(1)) ≤C· kωkL1(B(µ0)\B(µ)).
In Euclidean space, the analogous statement can be proved as follows. Up to a biLipschitz change of coordinates, one replaces shells with products[0,1] ×Sn−1. On such a product, a differential form writesω=at+dt∧btwhereatandbtare differential forms onSn−1.ωis closed if and only if eachatis closed and
∂at
∂t =bt. Givenr∈ [0,1], define
φr =et+dt∧ ft where et=∫ t r
bsds, ft=0.
Then dφr=ω−ar. Set φ=∫ 1
0 φrdr, so that dφ=ω−ω¯ where ω¯ =∫ 1 0 ardr.
Note that eachar, and hence ¯ω, is an exact form onSn−1. Since kωk¯ L1(Sn−1)≤ kωkL1([0,1]×Sn−1),
according to Subsection 6.1, there exists a form ¯φonSn−1such that d ¯φ=ω¯ and kφk¯ L1(Sn−1)≤Ckω¯kL1(Sn−1).
Henceφ−φ¯is the required primitive.
The Heisenberg group case reduces to the Euclidean case thanks to a smoothing homotopy constructed in [2]. In fact, sinceφmerely needs be estimated inL1norm (and not in the sharpLqnorm), only the first, elementary, steps of [2] are required, resulting in the following result.
Lemma 6.2. For every radiiµ <1< λ < µ0, there exists a constantCwith the following property. For everydc-exactL1Rumin formωon the large shellB(µ0) \B(µ)ofH2m+1, there exist L1 Rumin forms Tω and Sω on the smaller shell B(λ) \ B(1) such that ω=dcTω+Sωon the smaller shell,
kTωkL1(B(λ)\B(1))+kSωkW1,1(B(λ)\B(1)) ≤CkωkL1(B(µ0)\B(µ)). Here, theW1,1norm refers to theL1norms of the first horizontal derivatives.
Proof. Pick a smooth function χ1with compact support in the large shellA. According to Lemma 6.2 of [2], there exists a left-invariant pseudodifferential operatorKsuch that the identity
χ1 =dcKχ1+Kdcχ1 holds on the space of Rumin forms
L1∩d−c1L1 :={α∈L1(A) : dcα∈L1(A)}.
Kis the operator of convolution with a kernelkof type 1 (resp. 2 in degreen+1). Using a cut-off, writek=k1+k2wherek1has support in an-ball andk2 is smooth. Since k1 =O(r1−Q)orO(r2−Q) ∈ L1, the operatorK1of convolution withk1is bounded onL1. HenceT =K1χ1is bounded onL1forms defined onA. WhereasS=dcK2χ1is bounded fromL1toWs,1for every integers. Ifµ0> λ+2andµ <1−2, the multiplication of ωbyχ1has no effect on the restriction of dcK1ωorK1dcωto the smaller shell, hence, in restriction to the smaller shell,
dcTω=dcK1χ1ω=(dcK1+K1dc)ω.
It follows that
Sω=dcK2χ1ω=(dcK2+K2dc)ω, and finally, in restriction to the smaller shell,
ω=dcTω+Sω.
Proof of Lemma 6.1. Using the exponential map, one can use simultaneously Heisenberg and Euclidean tools. Pickλ, µ, µ0such that the medium Heisenberg shellB(µ0−2)\B(µ+ 2)contains the Euclidean shellAeucl =Beucl(2) \Beucl(1), which in turn contains the smaller Heisenberg shellB(λ) \B(1). Apply Lemma 6.2 to a dc-closedL1formωdefined on the larger Heisenberg shell. Up to dcTω, and up to restricting to the medium shell, one can replaceωwithSωwhich has its first horizontal derivatives inL1. Apply Rumin’s homotopyΠE =1−dd−01−d−01d to get a usuald-closed differential formβ=ΠESω belonging toL1. Use the Euclidean version of Lemma 6.1 to get anL1primitiveγ, dγ=β, on the Euclidean shell Aeucl. Apply the order zero homotopyΠE0 =1−d0d−01−d−01d0
to get a Rumin formφ=ΠE0γ. Its restriction to the smaller Heisenberg shell satisfies
dcφ=ωand itsL1norm is controlled bykωk1.
6.3. L1
Poincaré inequality in scaled shell
B(λR) \B(R)Let 0< µ <1< λ < µ0. Letωbe a Rumink-form on the scaled annulusB(µ0R) \B(µR). Assume that there exists a Rumin(k−1)-formφon the thinner shell onB(λR) \B(R) such thatω=dcφon that shell.
Let’s denote the dilation byRas
δR:B(λ) \B(1) →B(λR) \B(R) then we can consider the pull-back of both forms:
• ωR:=δ∗R(ω)onB(λ) \B(1), and
• φR:=δ∗R(φ)onB(λ) \B(1).
Sinceδ∗Rcommutes with the Rumin differential dc, we have ωR=δ∗R(ω)=δ∗R(dcφ)=dc(δ∗Rφ)=dcφR.
Then, forωRwe have kδ∗RωkL1(B(λ)\B(1))
=∫
B(λ)\B(1)
|ω(δR(x))| ·Rwdx =
|{z}
y=δR(x)
Rw
∫
B(λR)\B(R)|ω|(y) · 1 RQ−1dy
=Rw−(Q−1)
∫
B(λR)\B(R)|ω|(y)dy=Rw−(Q−1)· kωkL1(B(λR)\B(R), so that
kωRkL1(B(λ)\B(1))=Rw−(Q−1)kωkL1(B(λR)\B(R)) (6.1) wherewis the weight of thek-formω.
Likewise, for the(k−1)-formφwe get
kδ∗RφkL1(B(λ)\B(1))=Rw˜−(Q−1)kφkL1(B(λR)\B(R)), (6.2) where in this caseewis the weight of the formφ.
Since we are working onH2m+1andω=dcφ(and likewiseωR=dcφR), we have
• we=w−1, ifk,m+1, and
• we=w−2, ifk=m+1.
According to Lemma 6.1, one can find a(k−1)-formφRonB(λ) \B(1)such that kφRkL1(B(λ)\B(1))≤C· kωRkL1(B(µ0)\B(µ))
so, using the equalities (6.1) and (6.2) we get the following inequality:
kφkL1(B(λR)\B(R)) ≤C·Rw−w˜kωkL1(B(µ0R)\B(µR))
which divides into the following two cases
• kφkL1(B(λR)\B(R))≤C·R· kωkL1(B(µ0R)\B(µR))ifk,m+1, and
• kφkL1(B(λR)\B(R))≤C·R2· kωkL1(B(µ0R)\B(µR))ifk=m+1.
Only the first case is useful for our purpose.
6.4.
Independence on the choice of primitive
Let us consider the exact Rumin formω∈E0kand letφ, ψ∈E0k−1be two primitives ofω onB(λR) \B(R), i.e. dcψ =dcφ=ω. Letβbe an arbitrary left-invariant Rumin formβ of complementary degree 2m+1−k.
If k , 2m, then Hk(B(λR) \B(R)) = 0, which means that there exists a Rumin (k−2)-formαsuch that dcα=ψ−φ.
Ifk,m+1, the degree ofβis 2m+1−k,m, so dc(ξ β)=(dcξ)∧β+ξdcβ=(dcξ)∧β, thus γ =(dcξ) ∧βis a well-defined dc-closed Rumin form (this is a special case of Proposition 4.2).
If on top we also assumek,m+2, givenγ=dc(ξ β)=dcξ∧β, we have that the form α∧γhas degree 2m ≥m+1, so we can apply Proposition 4.2 and obtain the following equality
d(α∧γ)=dc(α∧γ) ±α∧dcγ=(dcα) ∧γ=(ψ−φ) ∧ (dcξ) ∧β.
Since by construction dcξhas compact support inB(λR) \B(R),
∫
H2m+1
dcξ∧ (ψ−φ) ∧β=0.
Therefore, whenk,m+1,m+2,2m, we can replace a given primitiveψwith any other arbitrary primitiveφofωon the scaled shellB(λR) \B(R).
6.5.
Vanishing of averages
Proposition 6.3. Givenω∈E0kanL1,dc-closed Rumin form inH2m+1, then the integral
∫
H2m+1
ω∧β
vanishes for all left-invariant Rumin formsβof complementary degree,β∈ E02m+1−k, providedk,m+1,m+2,2m.
Proof. We assume thatk,m+1,m+2,2m.
Letψbe a global primitive ofωonH2m+1. Let us first analyse the following identities ξω∧β=ξ(dcψ) ∧β=−dcξ∧ψ∧β+d(ξψ∧β).
Let φbe the primitive ofω onB(λR) \B(R)introduced in Section 6.3. We can then replace∫
H2m+1dcξ∧ψ∧βwith∫
H2m+1dcξ∧φ∧β, and by applying Stokes’ theorem, we
get
∫
H2m+1
ξω∧β =
∫
B(λR)\B(R)dcξ∧ψ∧β
=
∫
B(λR)\B(R)dcξ∧φ∧β
≤ kdcξk∞kβk∞kφkL1(B(λR)\B(R)).
Finally, knowing thatkdcξk∞ ≤C0/Rand applying the Poincaré inequality on the shell
kφkL1(B(λR)\B(R))≤C·R· kωkL1(B(µ0R)\B(µR))
found in Subsection 6.3, we finally get
∫
H2n+1
ξω∧β
≤CC0kωkL1(B(µ0R)\B(µR)). Using the cut-off functionξintroduced in Definition 2.3, we have
∫
H2m+1
ξω∧β=∫
B(R)
ω∧β+∫
B(λR)\B(R)ξω∧β . Hence
∫
B(R)
ω∧β
≤
∫
H2n+1
ξω∧β +
∫
B(λR)\B(R)ξω∧β
≤CC0kωkB(µ0R)\B(µR)+∫
B(λR)\B(R)kβk∞· |ω|
≤C00kωkL1(B(µ0R)\B(µR)). SincekωkL1(B(µ0R)\B(µR))→0 asR→ ∞, we get our result
∫
H2m+1
ω∧β=0.
This completes the proof of Theorem 1.4.
Remark 6.4. Let us notice that this method would not work in the case wherek=m+1, since we would obtain the following inequality
∫
B(R)
ω∧β≤C·RkωkL1(B(λR)\B(R))
which is not conclusive.
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Pierre Pansu
Laboratoire de Mathématiques d’Orsay Université Paris-Sud, CNRS
Université Paris-Saclay 91405 Orsay FRANCE
pierre.pansu@math.u-psud.fr
Francesca Tripaldi Department of Mathematics and Statistics
University of Jyväskylä 40014, Jyväskylä FINLAND
francesca.f.tripaldi@jyu.fi
