cohomology of certain Carnot groups

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`q,pcohomology Carnot group case Local Poincar´e inequality

`

^q,p

cohomology of certain Carnot groups

Pierre Pansu, Univ. Paris-Sud. Joint with A. Baldi, B. Franchi, M. Rumin

February 22nd, 2018

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Motivation

Euclidean Poincar´e inequality Leray’s acyclic covering theorem revisited

Definition

X simplicial complex. Cochains are functions on simplices. When is an`^p cocycle the coboundary of an`^qcochain ? Set

`^q,pH^k(X) ={`^pk-cocycles}/d{`^qk−1-cochains}.

X finite : topological invariant.X infinite : quasi-isometry invariant. Well defined for discrete groups, gives rise to numerical invariants of discrete groups...

Example.X = subdivided line. Then all`^q,pH⁰(X) = 0,`^∞,1H¹(X) = 0, all other

`^q,pH¹(X)6= 0.

Example.X = tesselated plane. Then`^q,pH¹(X) = 0 if 1 p−1

q ≥1

2. Indeed, Sobolev inequality allows to handle the case of finitely supported cocycles. It states that, for a smooth compactly supported functionuon the plane, ifp<2,

kukq≤Ckdukp.

Questions. Handle infinitely supported cocycles ? Pass from discrete to continuous and backward ?

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Motivation

Definition

`^q,pH¹(X)6= 0.

q ≥1

kukq≤Ckdukp.

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Motivation

Definition

`^q,pH¹(X)6= 0.

q ≥1

kukq≤Ckdukp.

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Motivation

Definition

X Riemannian manifold. Set

L^q,pH^k(X) ={L^pclosedk-forms}/d{L^qk−1-formsωsuch thatdω∈L^p}.

Questions. Compute it. IfX is triangulated, doesL^q,pH^k(X) =`^q,pH^k(X) ?

Example.X =Rⁿ. ThenL^q,pH^k(X) = 0 if 1<p≤q<∞and 1 p−1

q =1 n. Proof. Let ∆ =d^∗d+dd^∗. Then ∆ has a pseudo-differential inverse which commutes withd.T=d^∗∆⁻¹has a homogeneous kernel of degree 1−n, hence is bounded L^p→L^qprovided ¹_p−_q¹=¹_n (Calderon-Zygmund 1952). Finally 1 =dT+Td.

Example.X = ball inRⁿ. ThenL^q,pH^k(X) = 0 if 1<p≤q<∞and 1 p−1

q≤ 1 n. Proof(Iwaniec-Lutoborsky 1993). Cartan’s homotopy formula provides a homotopy T, 1 =dT+Td, which has a homogeneous kernel of degree 1−n. It does not require forms to be globally defined. H¨older⇒qcan be lessened. Works for convex sets.

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Motivation

Definition

q =1 n. Proof. Let ∆ =d^∗d+dd^∗. Then ∆ has a pseudo-differential inverse which commutes withd.T=d^∗∆⁻¹has a homogeneous kernel of degree 1−n, hence is bounded L^p→L^qprovided ¹_p−¹

q =¹_n (Calderon-Zygmund 1952). Finally 1 =dT+Td.

q≤ 1 n. Proof(Iwaniec-Lutoborsky 1993). Cartan’s homotopy formula provides a homotopy T, 1 =dT+Td, which has a homogeneous kernel of degree 1−n. It does not require forms to be globally defined. H¨older⇒qcan be lessened. Works for convex sets.

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Motivation

Definition

q =1 n. Proof. Let ∆ =d^∗d+dd^∗. Then ∆ has a pseudo-differential inverse which commutes withd.T=d^∗∆⁻¹has a homogeneous kernel of degree 1−n, hence is bounded L^p→L^qprovided ¹_p−¹

q =¹_n (Calderon-Zygmund 1952). Finally 1 =dT+Td.

q≤1 n. Proof(Iwaniec-Lutoborsky 1993). Cartan’s homotopy formula provides a homotopy T, 1 =dT+Td, which has a homogeneous kernel of degree 1−n. It does not require forms to be globally defined. H¨older⇒qcan be lessened. Works for convex sets.

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Motivation

Theorem (Leray, circa 1946)

Vanishing of L^q,pH^·of all simplices suffices to prove that L^q,pH^·=`^q,pH^·for bounded geometry triangulated manifolds.

Thus Iwaniec-Lutoborsky =⇒ `^q,pH^·=L^q,pH^·if 1 p−1

q≤ 1 n.

Proposition (Rumin)

Proposition persists in the form`^q,pH^·=L^q,p∞H^·, for all1≤p,q≤ ∞, where L^p∞={forms all of whose derivatives are in L^p}.

”Loss on differentiability is allowed”. Useful since no restriction on exponentq. Proposition (Pansu)

Proposition merely requires even weaker analytic information : it suffices that a closed form on ball of radius 2 have a primitive on unit ball with controlled norms.

”Loss on domain is allowed”. Useful to prove quasiisometry invariance.

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Motivation

q≤ 1 n. Proposition (Rumin)

”Loss on differentiability is allowed”. Useful since no restriction on exponentq.

Proposition (Pansu)

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Motivation

q≤ 1 n. Proposition (Rumin)

”Loss on differentiability is allowed”. Useful since no restriction on exponentq.

Proposition (Pansu)

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Vanishing result

Role of anisotropic pseudodifferential calculus Rumin’s complex

Need homogeneous Laplacian. Use homogeneous groups. Special case : Carnot groups G,g=g1⊕ · · · ⊕gs,δt=tⁱ ongi.

Kohn’s Laplacian. For a functionu, letdRu=du|g₁ and let ∆u=d_R^∗dR. This homogeneous of degree 2 under Carnot dilationsδt.

In general, there is no differential homogeneous Laplacian on forms, since left-invariant forms split underδt into several weight spaces.

Example. On the Heisenberg group, two weights onk-forms,kandk+ 1, since Λ^kg^∗= Λ^kg^∗₁⊕Λ^k−1g^∗₁⊗g^∗₂.

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Vanishing result

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Vanishing result

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Vanishing result

A pseudodifferential homogeneous Laplacian. Let|∇|= ∆^1/2. Let|∇|^Nbe the operator acting componentwise which is|∇|^won forms of weightw. Then d^∇:=|∇|^−Nd|∇|^N is pseudodifferential of order 0, so is its Laplacian

∆^∇= (d^∇)^∗d^∇+d^∇(d^∇)^∗. Both are homogeneous.

∆^∇admits a pseudodifferential inverse (Helffer-Nourrigat 1979, Christ-Geller-Glowacki-Polin 1992), henced^∇admits a homotopy

K^∇:= (d^∇)^∗(∆^∇)⁻¹, 1 =d^∇K^∇+K^∇d^∇, hence a homogeneous homotopy K:=|∇|^NK^∇|∇|^−Nford.

K^∇is bounded onL^p(Folland 1975), thusK is bounded on L^N,p:={α;|∇|^−Nα∈L^p}, and onL^N−m,p for every constantm.

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Vanishing result

K^∇is bounded onL^p(Folland 1975), thusK is bounded on L^N,p:={α;|∇|^−Nα∈L^p}, and onL^N−m,p for every constantm.

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Vanishing result

K^∇is bounded onL^p(Folland 1975), thusK is bounded on L^N,p:={α;|∇|^−Nα∈L^p}, and onL^N−m,pfor every constantm.

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Vanishing result

On functions,

L^p∞=

∞

\

m=0

L^N−m,p.

On the space of forms whose weight lies betweenaandb, Ω^b_a, Ω^b_a∩

∞

\

m=a

L^N−m,p⊂Ω^b_a∩L^p∞⊂

∞

\

m=b

L^N−m,p.

|∇|^−µis bounded fromL^ptoL^qif _p¹−¹_q= ^µ_Q,Q=P

idim(gi) (Folland 1975). Whence the graded Poincar´e inequality

L^N−m,p⊂L^N−m+µ,q.

Assume thatK(Ω^b_a)⊂Ω^b_a0⁰. Ifµ=b−a⁰, then

K(L^p∞∩Ω^b_a)⊂Ω^b_a0⁰∩

∞

\

m=b

L^N−m,p⊂Ω^b_a0⁰∩

∞

\

m=b

L^N−m+µ,q= Ω^b_a0⁰∩

∞

\

m=a⁰

L^N−m,q⊂L^q∞,

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Vanishing result

On functions,

L^p∞=

∞

\

m=0

L^N−m,p.

∞

\

m=a

∞

\

m=b

L^N−m,p.

idim(gi) (Folland 1975).

Whence the graded Poincar´e inequality

∞

\

m=b

∞

\

m=b

∞

\

m=a⁰

L^N−m,q⊂L^q∞,

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Vanishing result

On functions,

L^p∞=

∞

\

m=0

L^N−m,p.

∞

\

m=a

∞

\

m=b

L^N−m,p.

idim(gi) (Folland 1975).

Whence the graded Poincar´e inequality

∞

\

m=b

∞

\

m=b

∞

\

m=a⁰

L^N−m,q⊂L^q∞,

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Vanishing result

Proposition

`^q,pH^k(G) = 0if1<p,q<∞,¹_p−¹_q≥^b−a_Q⁰, where b is the maximal weight in degree k, a⁰is the minimal weight in degree k−1.

Example. For Heisenberg group,b=k+ 1,a⁰=k−1,b−a⁰= 2. Unsharp.

Strategy. Replacedwith a subcomplex that uses less weights : Rumin’s complex (1994, 1999).

Forms onG split into several weights under dilations. Letd0be the weight 0 (algebraic) part ofd. Pick complements ofker(d0) andim(d0) and defined₀⁻¹. Then powers of 1−d₀⁻¹d−dd₀⁻¹stabilize to a projector ΠEonto a subcomplexE. Π0= 1−d₀⁻¹d0−d0d₀⁻¹projects to a subspaceERof forms where less weights occur. Set

dR= Π0◦d◦ΠE.

ThenRumin’s complex(ER,dR) is homotopic to de Rham’s complex.

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Vanishing result

Proposition

`^q,pH^k(G) = 0if1<p,q<∞,¹_p−¹_q≥^b−a_Q⁰, where b is the maximal weight in degree k, a⁰is the minimal weight in degree k−1.

Example. For Heisenberg group,b=k+ 1,a⁰=k−1,b−a⁰= 2. Unsharp.

Strategy. Replacedwith a subcomplex that uses less weights : Rumin’s complex (1994, 1999).

Forms onG split into several weights under dilations. Letd0be the weight 0 (algebraic) part ofd. Pick complements ofker(d0) andim(d0) and defined₀⁻¹. Then powers of 1−d₀⁻¹d−dd₀⁻¹stabilize to a projector ΠEonto a subcomplexE.

Π0= 1−d₀⁻¹d0−d0d₀⁻¹projects to a subspaceERof forms where less weights occur.

Set

dR= Π0◦d◦ΠE.

ThenRumin’s complex(ER,dR) is homotopic to de Rham’s complex.

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Vanishing result

Example. For the 3-dimensional Heisenberg group,

E_R¹consists of horizontal 1-forms,dR:E_R⁰→E_R¹is the horizontal gradient.

E_R²consists of vertical 2-forms. Π_E extends a horizontal 1-formαin such a way thatdΠEαis vertical (unique choice). HencedRα=dΠEαinvolves second derivatives.

In general, for the 2m+ 1-dimensional Heisenberg group,E_R has one weight in each degree,w=k ifk≤m,w=k+ 1 ifk≥m+ 1, anddR:E_R^m→E_R^m+1has order 2.

Theorem (Pansu-Rumin)

Let G be a Carnot group of homogeneous dimension Q. Let[a,b]be the scope of weights in Rumin k-forms, let[a⁰,b⁰]be the scope of weights in Rumin k−1-forms.

1 `^q,pH^k(G) = 0provided1<p,q<∞and 1 p−1

q≥ b−a⁰ Q .

2 `^q,pH^k(G)6= 0if1≤p,q≤ ∞, ¹_p−¹_q <^max{1,b_Q⁰^−a}.

Example. For Heisenberg groups,b−a⁰=b⁰−a= 1, except in middle dimension whereb−a⁰=b⁰−a= 2.

Example. For all Carnot groups,b−a⁰=b⁰−a= 1 in degrees 1 andn=dim(G).

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Vanishing result

Example. For the 3-dimensional Heisenberg group,

E_R¹consists of horizontal 1-forms,dR:E_R⁰→E_R¹is the horizontal gradient.

E_R²consists of vertical 2-forms. Π_E extends a horizontal 1-formαin such a way thatdΠEαis vertical (unique choice). HencedRα=dΠEαinvolves second derivatives.

In general, for the 2m+ 1-dimensional Heisenberg group,E_R has one weight in each degree,w=k ifk≤m,w=k+ 1 ifk≥m+ 1, anddR:E_R^m→E_R^m+1has order 2.

Theorem (Pansu-Rumin)

Let G be a Carnot group of homogeneous dimension Q. Let[a,b]be the scope of weights in Rumin k-forms, let[a⁰,b⁰]be the scope of weights in Rumin k−1-forms.

1 `^q,pH^k(G) = 0provided1<p,q<∞and 1 p−1

q≥ b−a⁰ Q .

2 `^q,pH^k(G)6= 0if1≤p,q≤ ∞, ¹_p−¹_q <^max{1,b_Q⁰^−a}.

Example. For Heisenberg groups,b−a⁰=b⁰−a= 1, except in middle dimension whereb−a⁰=b⁰−a= 2.

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Smoothing homotopy Result

Theorem (Baldi-Franchi-Pansu)

Closed L^pforms defined on the Heisenberg 2-ball have dR-primitives on the unit ball which are L^q, provided1<p,q<∞and_p¹−_q¹≤_2m+2¹ (resp. _2m+2² in degree m+ 1).

Proof. No subRiemannian Cartan homotopy formula.

Instead, one uses Rumin’s homotopy ΠE followed by Iwaniec-Lutoborsky’s Euclidian homotopy. Since ΠEis differential, a preliminary smoothing homotopy is needed. Fix a subRiemannian metric in order to define adjoints. ∆R:=d_R^∗dR+dRd_R^∗is replaced with ∆R:= (d_R^∗dR)²+dRd_R^∗oud_R^∗dR+ (dRd_R^∗)²in degreesmandm+ 1. Then ∆R is maximally hypoelliptic, thusTR=d_R^∗∆⁻¹_R has a smooth and

homogeneous kernel.

LetKRbe the kernel ofTR. WriteKR=K¹+K²whereK1has small support andK2

is smooth. ThenT_R=T¹+T²,

1 =d_RT¹+T¹d_R+S

whereSis smoothing.T¹and thereforeS map forms defined on ball of radius 2 to forms defined on unit ball.T¹mapsL^p toL^qlikeTR.Swins the derivatives that ΠE

looses.

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Instead, one uses Rumin’s homotopy ΠEfollowed by Iwaniec-Lutoborsky’s Euclidian homotopy. Since ΠEis differential, a preliminary smoothing homotopy is needed.

Fix a subRiemannian metric in order to define adjoints. ∆R:=d_R^∗dR+dRd_R^∗is replaced with ∆R:= (d_R^∗dR)²+dRd_R^∗oud_R^∗dR+ (dRd_R^∗)²in degreesmandm+ 1. Then ∆R is maximally hypoelliptic, thusTR=d_R^∗∆⁻¹_R has a smooth and

homogeneous kernel.

1 =d_RT¹+T¹d_R+S

looses.

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Fix a subRiemannian metric in order to define adjoints. ∆R:=d_R^∗dR+dRd_R^∗is replaced with ∆R:= (d_R^∗dR)²+dRd_R^∗oud_R^∗dR+ (dRd_R^∗)²in degreesmandm+ 1.

Then ∆R is maximally hypoelliptic, thusTR=d_R^∗∆⁻¹_R has a smooth and homogeneous kernel.

1 =d_RT¹+T¹d_R+S

looses.

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Fix a subRiemannian metric in order to define adjoints. ∆R:=d_R^∗dR+dRd_R^∗is replaced with ∆R:= (d_R^∗dR)²+dRd_R^∗oud_R^∗dR+ (dRd_R^∗)²in degreesmandm+ 1.

Then ∆R is maximally hypoelliptic, thusTR=d_R^∗∆⁻¹_R has a smooth and homogeneous kernel.

1 =dRT¹+T¹dR+S

looses.

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Corollary (Baldi-Franchi-Pansu)

Rumin’s complex can be used to compute`^q,p-cohomology of contact 2m+ 1-manifolds with bounded geometry, provided1<p,q<∞and ¹_p−¹

q ≤ ¹

2m+2

(resp._2m+2² in degree m+ 1) :`^q,pH^·=L^q,pH^·(dc). Also, the complex(ER,dR) admits a smoothing homotopy.

Question. Cases whenp= 1 orq=∞? Work in progress with Baldi and Franchi.