De Rham's theorem for Orlicz cohomology in the case of Lie groups
aa r X i v : . [ m a t h . M G ] J un De Rham’s theorem for Orlicz cohomology in thecase of Lie groups
Emiliano Sequeira
Abstract
We prove the equivalence between the simplicial Orlicz cohomology and theOrlicz-de Rham cohomology in the case of Lie groups. Since the first one isa quasi-isometry invariant for uniformly contractible simplicial complexes withbounded geometry, we obtain the invariance of the second one in the case ofcontractible Lie groups. We also define the Orlicz cohomology of a Gromov-hyperbolic space relative to a point on its boundary at infinity, for which thesame results are true.
Orlicz cohomology has been studied in recent works ([KP15, Car16, Kop17, GK19,BFP20]) as a natural generalization of L p -cohomology. These kind of cohomologytheories can be defined in different contexts, which are related by some equivalencetheorems . A motivation to study them is that they provide quasi-isometry invari-ants, so they can be applied to quasi-isometry classification problems (see for example[Pan08, Car16]).For L p -cohomology we have a de Rham-type theorem, which establishes an equiv-alence between its simplicial version and its de Rham version (see [GKS88, Pan95,Gen14]). This equivalence is important to work with L p -cohomology because it allowsto use one or the other as appropiated. For example, one can prove the quasi-isometryinvariance of the simplicial L p -cohomology and then conclude that the de Rham L p -cohomology is also invariant under some hypothesis.In the case of Orlicz cohomology, it is proved in [Car16] the equivalence betweenboth versions only in degree 1. We present a proof in all degrees in the case of Liegroups equipped with left-invariant metrics. This proof has been obtained trying toimprove the results of Pansu and Carrasco on the large scale geometry of Heintzegroups, an special class of Lie groups that characterizes all connected homogeneousRiemannian manifolds with negative curvature.1inally, a relative version of the L p -cohomology can simplify the computations inthe case of Gromov-hyperbolic spaces (see [Seq19]), thus it can also be important toconsider a relative version of the Orlicz cohomology and prove the de Rham’s theoremin that context. We say that a real function φ : R → [0 , + ∞ ) is a Young function if: • it is even and convex; and • φ ( t ) = 0 if, and only if, t = 0.Observe that every Young function φ satisfieslim t → + ∞ φ ( t ) = + ∞ . Let (
Z, µ ) be a measure space and φ a Young function. The Luxembourg normassociated with φ of a measurable function f : Z → R = [ −∞ , + ∞ ] is defined by k f k L φ = inf (cid:26) γ > Z Z φ (cid:18) fγ (cid:19) dµ ≤ (cid:27) ∈ [0 , + ∞ ] . The
Orlicz space of ( Z, µ ) associated with φ is the Banach space L φ ( Z, µ ) = { f : Z → R measurable : k f k L φ < + ∞}{ f : Z → R measurable : k f k L φ = 0 } . It is not difficult to see that k f k L φ = 0 if, and only if, f = 0 almost everywhere.If µ is the counting measure on Z , we denote L φ ( Z, µ ) = ℓ φ ( Z ) and k k L φ = k k ℓ φ .Observe that if φ is the function t
7→ | t | p , then L φ ( Z, µ ) is the space L p ( Z, µ ). We referto [RR91] for a background about Orlicz spaces.
Remark 1.1. If K ≥ L Kφ ( Z, µ ) → L φ ( Z, µ )is clearly continuous and bijective, thus it is an isomorphism by the open mappingtheorem. This implies that the norms k k L Kφ and k k L φ are equivalent for all K > X equipped with a length distance has boundedgeometry if it has finite dimension and there exist a constant C > N : [0 , + ∞ ) → N such that1. the diameter of every simplex is bounded by C ;2. for every r ≥
0, the number of simplices contained in a ball of radius r is boundedby N ( r ). 2enote by X k the set of k -simplices in X and consider the cochain complex ℓ φ ( X ) δ → ℓ φ ( X ) δ → ℓ φ ( X ) δ → ℓ φ ( X ) δ → · · · where δ is the usual coboundary operator ( δθ ( σ ) = θ ( ∂σ )). The Orlicz cohomology of X associated with the Young function φ (or, more simply, the ℓ φ -cohomology of X ) isthe family of topological vector spaces ℓ φ H k ( X ) = Ker δ k Im δ k − . Since these spaces are not in general Banach spaces, it is something convenient to con-sider the reduced Orlicz cohomology of X associated with φ (or reduced ℓ φ -cohomology of X ) as the family of Banach spaces ℓ φ H k ( X ) = Ker δ k Im δ k − . If X is Gromov-hyperbolic and ξ is a point on its boundary at infinity ∂X , thenone can consider the relative Orlicz cohomology of the pair ( X, ξ ) associated with φ (or relative ℓ φ -cohomology of ( X, ξ )) as the family of topological vector spaces ℓ φ H k ( X, ξ ) = Ker δ | ℓ φ ( X k ,ξ ) Im δ | ℓ φ ( X k − ,ξ ) , where ℓ φ ( X k , ξ ) is the subspace of ℓ φ ( X k ) that consists of all k -cochains that are zeroon a neighborhood of ξ in X = X ∪ ∂X . We say that a k -cochain on X is zero orvanishes on a neighborhood of ξ if there exists an open set U in X which contains ξ ,such that θ ( ξ ) = 0 for every k -simplex σ contained in U . See for example [BHK01,Charpter 4] for a description of the topology in X .A map F : X → Y between two metric spaces (where the metric is denoted by | · − · | in both cases) is a quasi-isometry if there exist two constants λ ≥ ǫ ≥ x , x ∈ X , λ − | x − x | − ǫ ≤ | F ( x ) − F ( x ) | ≤ λ | x − x | + ǫ.
2. For every y ∈ Y there exists x ∈ X such that | F ( x ) − y | ≤ ǫ .Every quasi-isometry F : X → Y admits a quasi-inverse. That is, quasi-isometry F : Y → X such that F ◦ F and F ◦ F are at bounded uniform distance to the identity.If X and Y are Gromov-hyperbolic, then the quasi-isometry F induces a homeomor-phism between their boundaries ∂F : ∂X → ∂Y (see for example [GdlH90, Charpter7]). We will use also the notation F ( ξ ) = ∂F ( ξ ) if ξ ∈ ∂X .We say that a metric space X is uniformly contractible if it is contractible and thereis a function ψ : [0 , + ∞ ) → [0 , + ∞ ) such that every ball B ( x, r ) = { x ′ ∈ X : | x ′ − x | 7→ | ω | x = sup {| ω x ( v , . . . , v k ) | : v i ∈ T x M for i = 1 , . . . , k, with k v i k x = 1 } in the measure space ( M, dV ), where k k x is the Riemannian norm on T x M and dV isthe Riemannian volume on M . We denote by L φ C k ( M ) the completion of L φ Ω k ( M )with respect to | | L φ . Observe that the derivative of differential forms induces a con-tinuous map d = d k : L φ C k ( M ) → L φ C k +1 ( M ).We can consider the Orlicz-de Rham cohomology of M associated with φ (or L φ -cohomology of M ) as the family of topological vector spaces L φ H k ( M ) = Ker d k Im d k − , and the reduced Orlicz-de Rham cohomology of M (or reduced L φ -cohomology of M )as the family of Banach spaces L φ H k ( M ) = Ker d k Im d k − . Remark 1.3. A measurable k -form on M is a function x ω x , where ω x is analternating k -linear form on the tangent space T x M , such that the coefficients of ω for4very parmetrization of M are all measurable. We consider L φ ( M, Λ k ) the space of L φ -integrable measurable k -forms up to almost everywhere zero forms. It is a Banachspace equipped with the Luxemburg norm k k L φ .Since L φ Ω k ( M ) ⊂ L φ ( M, Λ k ) and the inclusion is continuous, one can prove usingH¨older’s inequality ( k f g k L ≤ k f k L φ k g k L φ ∗ , where φ ∗ is the convex conjugate of φ )that L φ C k ( M ) can be seen as a space of k -measurable forms in L φ ( M, Λ k ) which weakderivatives are defined and belong to L φ ( M, Λ k +1 ).We say that ̟ is the weak derivative of ω ∈ L φ ( M, Λ k ) if for every differential( n − k )-form with compact support α one has Z M ̟ ∧ α = ( − k − Z M ω ∧ dα. If M is Gromov-hyperbolic and ξ ∈ ∂M , we consider the subspace L φ C k ( X, ξ ) ⊂ L φ C k ( X ) consisting of all k -forms that are zero in almost every point of a neighborhoodof ξ . Then we can define the relative Orlicz-de Rham cohomology of the pair ( M, ξ )associated with φ (or relative L φ -cohomology of ( M, ξ )) as the family of topologicalvector spaces L φ H k ( M, ξ ) = Ker d | L φ C k ( X,ξ ) Im d | L φ C k − ( X,ξ ) . Consider a Lie group G equipped with a left-invariant Riemannian metric. We denoteby dx the volume on G and by L x and R x the left and right translation by x ∈ G respectively.Suppose that there exists a uniformly contractible simplicial complex X with boundedgeometry that is quasi-isometric to G . Then we can define the simplicial Orlicz coho-mology and the reduced simplicial Orlicz cohomology of G as the families of spaces ℓ φ H k ( G ) = ℓ φ H k ( X ) and ℓ φ H k ( G ) = ℓ φ H k ( X )Observe that, because of Theorem 1.2, it is well-defined up to isomorphisms.If G is Gromov-hyperbolic and ξ ∈ ∂G , we can consider the relative simplicialOrlicz cohomology of the pair ( G, ξ ) as the family of spaces ℓ φ H k ( G, ξ ) = ℓ φ H k ( X, ξ ) , where ξ is the image of ξ by a quasi-isometry F : G → X . Remark 1.4. Let M be a complete Riemannian manifold with bounded geometry.This means that it has positive injectivity radius and its sectional curvature is uniformlybounded from above and below. Assume that dim( M ) = n .One can consider X M a triangulation of M with bounded geometry such that every n -simplex is bi-Lipschitz diffeomorphic to the standard Euclidean simplex of the same5imension (see [Att94]). For every vertex v of X M we define U ( v ) as the interior ofthe union of all simplices containing v . Observe that X M is the nerve of the covering U = { U ( v ) : v ∈ X M } , and that every non empty intersection U ∩ . . . ∩ U k of elementsof U is bi-Lipschitz equivalent to the unit ball in R n with uniform Lipschitz constant.In general, we can consider X M as the nerve of any open covering satisfying theabove properties and equip it with a length metric such that every simplex is isometricto the standard Euclidean simplex of the same dimension. It is clear that X M isquasi-isometric to M in this case. Moreover, there is a family of quasi-isometries F : X M → M verifying F ( U ) ∈ U for all vertex U ∈ U , we call them canonicalquasi-ismometries .If M is Gromov-hyperbolic and ξ is a point in ∂M , observe that all canonical quasi-isometries are at bounded uniform distance from each other and as a consequence theyinduce the same map on the boundary. Denote by ξ ∈ X M the point corresponding to ξ by a canonical quasi-isometry. We say that ( X M , ξ ) is a simplicial pair corresponding to ( M, ξ ). As we saw with the first construction, if M is uniformly contractible we cansuppose that X M is also uniformly contractible.Since a Lie group G equipped with a left-invariant metric is always complete andhas bounded geometry, then one can consider the simplicial complex X G . If G isin addition contractible, then it is uniformly contractible and its (relative/reduced)simplicial Orlicz cohomology is well-defined.Our main result is the following: Theorem 1.5. Let G be a Lie group equipped with a left-invariant metric and X G asimplicial complex as in Remark 1.4, then1. The (reduced) L φ -cohomology of G and the (reduced) ℓ φ -cohomology of X G areisomorphic.2. If G is Gromov-hyperbolic and ξ ∈ ∂G , the relative L φ -cohomology of the pair ( G, ξ ) and the relative ℓ φ -cohomology of the pair ( X G , ξ ) are isomorphic. As a consequence of the proof of Theorem 1.5 we will obtain: Theorem 1.6. If G is a Lie group equipped with a left-invariant metric, then thecochain complexes ( L φ Ω k ( G ) , d ) and ( L φ C k ( G ) , d ) are homotopically equivalent. Thesame result is true for the relative complexes in the Gromov-hyperbolic case. A more general version of Theorem 1.6 is proved in [KP15].Theorem 1.5 implies that if G is contractible and φ is a Young function, then ℓ φ H k ( G ) is isomorphic to L φ H k ( G ) and ℓ φ H k ( G ) is isomorphic to L φ H k ( G ). If G is inaddition Gromov-hyperbolic and ξ is a point in ∂G , then ℓ φ H k ( G, ξ ) is isomorphic to L φ H k ( G, ξ ).Combining this with Theorem 1.2 we get:6 orollary 1.7. If F : G → G is a quasi-isometry between two contractible Lie groupsequipped with left-invariant metrics and φ is a Young function, then for every k ∈ N • the topological vector spaces L φ H k ( G ) and L φ H k ( G ) are isomorphic; and • the Banach spaces L φ H k ( G ) and L φ H k ( G ) are isomorphic.Furthermore, if G and G are Gromov-hyperbolic and ξ is a point in ∂G , then thespaces L φ H k ( G , ξ ) and L φ H k ( G , F ( ξ )) are isomorphic for every k . Let X be a simplicial complex with bounded geometry and fix a Young function φ .Observe that every element θ ∈ ℓ φ ( X k ) has a natural linear extension θ : C k ( X ) → R ,where C k ( X ) = ( m X i =1 t i σ i : t , . . . , t m ∈ R , σ , . . . , σ m ∈ X k ) . The support of a chain c = P mi =1 t i σ i in C k ( X ) (with t i = 0 for all i = 1 , . . . , m ) is | c | = { σ , . . . , σ m } . We also define the uniform norm and the length of c by k c k ∞ = max {| t | , . . . , | t m |} , and ℓ ( c ) = m. Proposition 2.1. The usual coboundary operator δ = δ k : ℓ φ ( X k ) → ℓ φ ( X k +1 ) iscontinuous.Proof. Let θ be a cochain in ℓ φ ( X k ), then k δθ k ℓ φ = inf γ > X σ ∈ X k +1 φ (cid:18) θ ( ∂σ ) γ (cid:19) ≤ . The bounded geometry implies that there is a constant N (1) such that every k -simplex τ in X is on the boundary of at most N (1) ( k + 1)-simplices. Then X σ ∈ X k +1 φ (cid:18) θ ( ∂σ ) γ (cid:19) ≤ X τ ∈ X k N (1) φ (cid:18) θ ( τ ) γ (cid:19) , which implies k δθ k ℓ φ ≤ inf ( γ > X τ ∈ X k N (1) φ (cid:18) θ ( τ ) γ (cid:19) ≤ ) = k θ k ℓ N (1) φ . The proof ends using the equivalence between k k ℓ N (1) φ and k k ℓ φ (Remark 1.1).7o prove Theorem 1.2 we need the following lemmas. Lemma 2.2 ([BP03]) . Let X and Y be two uniformly contractible simplicial complexeswith bounded geometry. Then any quasi-isometry F : X → Y induces a family of maps c F : C k ( X ) → C k ( Y ) which verify:(i) ∂c F ( σ ) = c F ( ∂σ ) for every σ ∈ X k .(ii) For every k ∈ N there exist constants N k and L k (depending only on k and thegeometric data of X, Y and F ) such that k c F ( σ ) k ∞ ≤ N k and ℓ ( c F ( σ )) ≤ L k forevery σ ∈ X k Furthermore, the Hausdorff distance between c F ( σ ) and F ( σ ) is uniformly bounded. Lemma 2.3 ([BP03]) . Consider F, G : X → Y two quasi-isometries between uniformlycontractible simplicial complexes with bounded geometry. If F and G are at boundeduniform distance, then there exists an homotopy h : C k ( X ) → C k +1 ( Y ) between c F and c G . This means that(i) ∂h ( v ) = c F ( v ) − c G ( v ) if v ∈ X , and(ii) ∂h ( σ ) + h ( ∂σ ) = c F ( σ ) − c G ( σ ) if σ ∈ X k , k ≥ .Moreover, k h ( σ ) k ∞ and ℓ ( h ( σ )) are uniformly bounded by constants N ′ k and L ′ k thatdepend only on k and the geometric data of X, Y, F and G .Proof of Theorem 1.2 (part 2). We define the pull-back of a cochain θ ∈ ℓ φ ( Y k , F ( ξ ))as the composition F ∗ θ = θ ◦ c F . The map c F given by Lemma 2.2 is not unique, then F ∗ depends on the choice of it.Let us prove that F ∗ : ℓ φ ( Y k , F ( ξ )) → ℓ φ ( X k , ξ ) is well-defined and continuous: k F ∗ θ k ℓ φ = inf ( γ > X σ ∈ X k φ (cid:18) θ ( c F ( σ )) γ (cid:19) ≤ ) ≤ inf γ > X σ ∈ X k φ N k γ X τ ∈| c F ( θ ) | | θ ( τ ) | ≤ ≤ inf γ > X σ ∈ X k X τ ∈| c F ( θ ) | ℓ ( c F ( σ )) φ (cid:18) N k L k γ | θ ( τ ) | (cid:19) ≤ , where N k and L k are the constants given by Lemma 2.2.Since F is a quasi-isometry and the Hausdorff distance between c F ( v ) and F ( v ) isuniformly bounded for all v ∈ X , we can find a constant C k such that if dist ( σ , σ ) > k , then c F ( σ ) ∩ c F ( σ ) = ∅ . Using the bounded geometry of X we have that every τ ∈ Y k satisfies τ ∈ | c F ( σ ) | for at most D = N ( C + C k ) simplices σ ∈ X k . This implies k F ∗ θ k ℓ φ ≤ inf ( γ > X σ ∈ Y k Dφ (cid:18) N k L k γ | θ ( τ ) | (cid:19) ≤ ) = N k L k k θ k ℓ Cφ (cid:22) k θ k ℓ φ . Hence F ∗ θ ∈ ℓ φ ( X k ). We write f (cid:22) g for a pair of non-negative functions f and g ifthere exists a constant K such that f ≤ Kg .Now we prove that for every θ in ℓ φ ( Y k , F ( ξ )), the cochain F ∗ θ is zero on someneighborhood of ξ . Assume that θ is zero on V ⊂ Y , F ( ξ ) ∈ V . If σ ∈ X k and v ∈ X is a vertex of σ , d H ( c F ( σ ) , F ( v )) ≤ d H ( c F ( σ ) , c F ( v )) + d H ( c F ( v ) , F ( v )) , (1)where d H denotes the Hausdorff distance. By the properties of c F the distance (1) isuniformly bounded by a constant ˜ C k . We define ˜ V = { y ∈ Y : dist ( y, V c ∩ Y ) > ˜ C k } .Since F is a quasi-isometry, there exists U ⊂ X a neighbourhood of ξ such that F ( U ∩ X ) ⊂ ˜ V . For every k -simplex σ ⊂ U , we have c F ( σ ) ⊂ V and then F ∗ θ ( σ ) = 0.We conclude that F ∗ θ vanishes on U .Since c F commutes with the boundary, we have δF ∗ = F ∗ δ , which implies that F ∗ defines a continuous map in cohomology, denoted by F : ℓ φ H k ( Y, F ( ξ )) → ℓ φ H k ( X, ξ ). We have to prove that F is an isomorphism.Claim: If F, G : X → Y are two quasi-isometries at bounded uniform distance, then F = G .We have to construct a family of continuous linear maps H k : ℓ φ ( Y k , F ( ξ )) → ℓ φ ( X k − , ξ ) such that:(i) F ∗ θ − G ∗ θ = H δθ for every θ ∈ ℓ φ ( Y , F ( ξ )), and1. F ∗ θ − G ∗ θ = H k +1 δθ + δH k θ for every θ ∈ ℓ φ ( Y k , F ( ξ )), k ≥ H k θ : X k → R , H k θ ( σ ) = θ ( h ( σ )), where h is the map given by Lemma2.3. Using the same argument as for F ∗ , one can show that H k is well-defined andcontinuous from ℓ φ ( Y k , F ( ξ )) to ℓ φ ( X k ). To see that H k θ vanishes on some neigh-borhood of ξ observe that h ( σ ) have uniformly bounded length, which implies that d H ( c F ( σ ) , h ( σ )) is uniformly bounded.Using the definition of H k one can easily verify ( i ) and ( ii ), which proves the claim.As a consequence of the claim we have that F does not depend on the choice of c F . Moreover, if T : Y → Z is another quasi-isometry, a possible choice of the function c T ◦ F is the composition c T ◦ c F . In this case ( T ◦ F ) ∗ = F ∗ ◦ T ∗ and as a consequence( T ◦ F ) = F ◦ T . 9inally, if F : Y → X is a quasi-inverse of F , then by the claim ( F ◦ F ) and ( F ◦ F ) are the identity in cohomology. Since ( F ◦ F ) = F ◦ F and ( F ◦ F ) = F ◦ F ,the statement follows. Suppose that M is a smooth manifold of dimension n and ( Z, µ ) is a measure space.We say that Φ = { Φ ( x,z ) : x ∈ M, z ∈ Z } is a family of measurable k -forms on M if forevery ( x, z ) ∈ M × Z , Φ ( x,z ) is an alternating k -form on the tangent space T x M andall coefficients of Φ with respect to every parametrization (depending on x ∈ M and z ∈ Z ) are measurable. It is a smooth family of k -forms if its coefficients are smooth.We say that Φ is integrable on Z if for every x ∈ M , the function z 7→ | Φ | ( x,z ) = sup {| Φ ( x,z ) ( v , . . . , v k ) | : v i ∈ T x M for i = 1 , . . . , k, with k v i k x = 1 } belongs to L ( Z, µ ). In this case we can consider the k -form ω x ( v , . . . , v k ) = (cid:18)Z Z Φ ( x,z ) dµ ( z ) (cid:19) ( v , . . . , v k ) = Z Z Φ ( x,z ) ( v , . . . , v k ) dµ ( z ) . (2)Observe that for all x ∈ M , | ω | x ≤ Z Z | Φ | ( x,z ) dµ ( z ) = k Φ ( x, · ) k L . Lemma 3.1. Let { Φ ( x,z ) : x ∈ M, z ∈ Z } be a measurable family of k -forms such that: • It is integrable on Z , then we can define ω as in (2). • For every fixed z ∈ Z the k -form x Φ ( x,z ) is locally integrable and has weakderivative d Φ ( x,z ) . • The function z 7→ | d Φ | ( x,z ) belongs to L ( Z, µ ) for every x ∈ M .Then ω is locally integrable and has weak derivative dω x = Z Z d Φ ( x,z ) dµ ( z ) . (3)The previous lemma follows directly from definition of weak derivative.To prove that a measurable k -form ω on M is smooth it is enough to verify thatfor every set of k vector fields { X , . . . , X k } the function f ( x ) = ω x ( X ( x ) , . . . , X k ( x ))10s smooth on M . A sufficient condition for f to be smooth is that for every set ofvector fields { Y , . . . , Y m } there exists L Y m · · · L Y f ( x )for all x ∈ M . The Lie derivative with respect to the field Y is defined by L Y f ( x ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( ϕ t ( x )) , where ϕ t is the flow associated with Y .From the above observation and the classical Leibniz Integral Rule one can concludethe following lemma: Lemma 3.2. Let M and N be two Riemannian manifolds and { Φ ( x,z ) : x ∈ M, z ∈ Z } a smooth family k -forms on M . If Φ ( x,y ) has compact support for every y ∈ N , thenthe k -form on M defined by ω x = Z N Φ ( x,y ) dV N ( y ) belongs to Ω k ( M ) and its derivative is dω = R N d Φ( · , y ) dV N ( y ) . Now consider a Lie group G . By a kernel on G we mean a smooth function κ : G → [0 , 1] such that: • supp( κ ) is a compact neighborhood of e ∈ G , and • R G κ ( x ) dx = 1.If ω is a locally integrable k -form on G we consider its convolution with κ as the k -form ( ω ∗ κ ) x = Z G ( R ∗ z ω ) x κ ( z ) dz. Lemma 3.3. There exists a constant C > such that for every locally integrable k -form ω on G and x ∈ G we have | ω ∗ κ | x ≤ C | ω | ∗ κ ( x ) , where | ω | ∗ κ is the convolution of the function x 7→ | ω | x with the kernel κ .Proof. Let v , . . . , v k be vectors in T x G , then | ( ω ∗ κ ) x ( v , . . . , v k ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z G ( R ∗ z ω ) x ( v , . . . , v k ) κ ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z G | ( R ∗ z ω ) x ( v , . . . , v k ) | κ ( z ) dz = Z G | ω x · z ( d x R z ( v ) , . . . , d x R z ( v k )) | κ ( z ) dz. R z ◦ L x = L x ◦ R z , we have | d e ( R z ◦ L x ) | = | d e ( L x ◦ R z ) | (here | | is the usualoperator norm) and therefore | d x R z ◦ d e L x | = | d z L x ◦ d e R z | . Using that L x is an isometrywe obtain | d x R z | = | d e R z | for every x ∈ G . The function z 7→ | d e R z | is continuous,then it has a maximum M in supp( κ ). If k v k = . . . = k v k k = 1, | ω x · z ( d x R z ( v ) , . . . , d x R z ( v k )) | ≤ M k | ω | x · z , which implies | ω ∗ κ | x ≤ C | ω | ∗ κ ( x ) with C = M k .A consequence of Lemma 3.3 is that the convolution of a locally integrable form isalso locally integrable. Proposition 3.4. Let ω be a locally integrable k -form on G , then:(i) If ω has weak derivative dω , then the convolution ω ∗ κ has weak derivative and d ( ω ∗ κ ) = dω ∗ κ. (ii) The convolution ω ∗ κ is a differential form.Proof. (i) For every z we have d ( R ∗ z ω ) = R ∗ z dω in a weak sense. To see this take β ∈ Ω n − k − ( G ) with compact support, then Z G ( R ∗ z dω ) ∧ β = Z G R ∗ z ( dω ∧ R ∗ z − β )= Z G dω ∧ R ∗ z − β = ( − k +1 Z G ω ∧ dR ∗ z − β = ( − k +1 Z G ( R ∗ z ω ) ∧ dβ. Therefore the weak derivative with respect to x ∈ G of the k -form Φ ( x,z ) =( R ∗ z dω ) x κ ( z ) is d Φ( x, z ) = ( R ∗ z dω ) x κ ( z ) . Since z d Φ( x, z ) has compact support for all x ∈ G , by Lemma 3.1 we conclude( dω ∗ κ ) = Z G ( R ∗ z dω ) κ ( z ) dz is the weak derivative of the convolution ω ∗ κ .(ii) Suppose first that ω = f is a 0-form, which is equivalent to say that it is a locallyintegrable function on G . Consider Y a vector field on G with flow ϕ t . Firstobserve that f ∗ κ ( x ) = Z G f ( x · z ) κ ( z ) dz = Z G f ( y ) κ ( x − · y ) dy. L Y ( f ∗ κ )( x ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f ∗ κ ( ϕ t ( x )))= ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Z G f ( y ) κ ( ϕ t ( x ) − · y ) dy. Since ϕ is smooth and κ is smooth with compact support, the classical Leibnizintegral Rule implies that this derivative exists and L X ( f ∗ κ )( x ) = Z G f ( y ) ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 κ ( ϕ t ( x ) − · y ) dy. Using this argument we can prove by induction that L Y m . . . L Y m ( f ∗ κ )( x ) existsfor all x ∈ G and every family of vector fields Y , . . . , Y m , which implies that f ∗ κ is smooth.Now consider { e , . . . , e n } a basis of T e G and X , . . . , X n the right-invariant fieldsverifying X i ( e ) = e i . Let ϕ it be the flow associated with X i for every i = 1 , . . . , n .If ω is a k -form with k ≥ f i ,...,i k ( x ) = ( ω ∗ κ ) x ( X i ( x ) , . . . , X i k ( x )) . To prove that ω ∗ κ is smooth it is enough to prove that all these functions aresmooth. Observe that if g i ,...,i k ( x ) = ω ( x )( X i ( x ) , . . . , X i k ( x )) , then f i ,...,i k = g i ,...,i k ∗ κ . This reduces the general case to the case k = 0 andfinishes the proof.The last lemma of the section relates the L φ -norm with the L -norm in the case offinite measure. Lemma 3.5. If µ is finite, then L φ ( Z, µ ) ⊂ L ( Z, µ ) and the inclusion is continuous,with norm bounded depending only on µ ( Z ) and φ .Proof. Let f ∈ L φ ( Z, µ ), then k f k L φ = inf (cid:26) γ > Z Z φ (cid:18) fγ (cid:19) dµ ≤ (cid:27) ≥ inf (cid:26) γ > µ ( Z ) φ (cid:18) µ ( Z ) Z Z fγ dµ (cid:19) ≤ (cid:27) From this we obtain k f k L ≤ µ ( Z ) φ − (1 /µ ( Z )) k f k L φ .13 Proof of main theorem Let U be an uniformly locally finite open covering on the Lie group G such that everynon-empty intersection U ∩ · · · ∩ U k is uniformly bi-Lipschitz to the unit Euclideanball. By uniformly locally finite we mean that there exists a uniform constant C suchthat every point in G belongs to at most C elements of U . Take X G as in the Remark1.4, that is, X ℓ = U ℓ = { U ∩ · · · ∩ U k = ∅ : U , . . . , U ℓ ∈ U } , and every simplex is isometric to the standard one of the same dimension.We consider, for a fixed Young function φ , the following cochain complexes: • L φ Ω k ( G, U ) is the space of all differential forms ω ∈ Ω k ( G ) such that ω | U and dω | U are in L φ Ω k ( U ) for every U ∈ U , and the functions U 7→ k ω | U k L φ and U 7→ k dω | U k L φ are in ℓ φ ( U ). The norm of ω ∈ L φ Ω k ( G, U ) is defined by | ω | L φ = k θ k ℓ φ + k θ ′ k ℓ φ , where θ ( U ) = k ω | U k L φ and θ ′ ( U ) = k dω | U k L φ . Naturally, the map defining thecochain complex is the usual derivative. • I φ Ω k ( G, U ) = L φ Ω k ( G ) ∩ L φ Ω k ( G, U ) with the norm | | I φ = | | L φ + | | L φ . • If G is Gromov-hyperbolic and ξ is a point in ∂G , we consider L φ Ω k ( G, U , ξ ) thesubcomplex consisting of all forms ω ∈ L φ Ω k ( G, U ) that vanish on a neighborhoodof ξ . In this case we also define I φ Ω k ( G, U , ξ ) = L φ Ω k ( G, ξ ) ∩ L φ Ω k ( G, U , ξ ).If φ ( t ) = | t | p for every t ∈ R , then L φ Ω k ( G ) = L φ Ω k ( G, U ) = I φ Ω k ( G, U ) andthe norms on these spaces are equivalent. However, it is not true for general Youngfunctions, as one can see in the following example: Example 4.1. We take the Young function φ : R → [0 , + ∞ ), φ ( t ) = φ p,κ ( t ) = | t | p log( e + | t | − ) κ , with p > κ > 0. This is a doubling Young function , which means that there is aconstant D such that φ (2 t ) ≤ Dφ ( t ) for every t ∈ R . This condition implies some niceproperties of the corresponding Orlicz space.We want to construct a 1-form ω in L φ Ω ( R , U ) and out of L φ Ω ( R ), where U = { U n = ( n − ǫ, n + 1 + ǫ ) : n ∈ Z } with ǫ > { a n } n ∈ Z be a sequence of positive numbers such that: • P a pn = + ∞ , and This example was given to me by Marc Bourdon. P φ ( a n ) < + ∞ .Take for every n ∈ Z an interval A n in R such that A n ⊂ ( n + 2 ǫ, n + 1 − ǫ ) and long ( A n ) = a pn (we can suppose that a n is small enough for every n ∈ Z ). Consider thefunction f : R → R defined by f = X n ∈ Z A n . On the one hand, if γ > Z R φ (cid:18) f ( t ) γ (cid:19) dt = X n ∈ Z Z n +1 n φ (cid:18) A n ( t ) γ (cid:19) dt = X n ∈ Z a pn φ (cid:18) γ (cid:19) = + ∞ . But on the other hand Z U n φ (cid:18) f ( t ) γ (cid:19) dt = a pn φ (cid:18) γ (cid:19) = a pn γ p log( e + γ ) κ ≤ (cid:18) a n γ (cid:19) p , which implies k f | U n k L φ ≤ a n and then X n ∈ Z φ ( k f | U n k L φ ) ≤ X n ∈ Z φ ( a n ) < + ∞ . (4)It is not difficult to see, using the doubling condition, that (4) implies that {k f | U n k} n ∈ Z belongs to ℓ φ ( Z ).We can find a smooth function g close enough from f such that g − f ∈ L φ ( R ) and {k ( g − f ) | U n k L φ } n ∈ Z ∈ ℓ φ ( Z ) and consider the 1-form ω = g dt . Since | ω | t = | g ( t ) | and dω = 0 we can see that ω ∈ L φ Ω ( R , U ) and ω / ∈ L φ Ω ( R ).In this case the other inclusion is true. One can prove that L φ Ω k ( R ) ⊂ L φ Ω k ( R , U )for k = 0 , φ ( s ) φ ( t ) ≤ κ φ ( st ). In fact, this inclusion can beproved for every Riemannian manifold with bounded geometry and every doublingYoung function satisfying an inequality φ ( t ) φ ( s ) ≤ Cφ ( st ) for all s, t ∈ R and someconstant C .We will prove Theorem 1.5 in three steps. Proposition 4.2 (First step) . The cochain complexes ( ℓ φ C ∗ ( X G ) , δ ) and ( L φ Ω ∗ ( G, U ) , d ) are homotopically equivalent. So are the relative cochain complexes ( ℓ φ C ∗ ( X G , ¯ ξ ) , δ ) and ( L φ Ω ∗ ( G, U , ξ ) , d ) . To prove the proposition we need some lemmas. The first one is a L φ -version ofLemma 8 in [Pan95]. Lemma 4.3. Let B be the unit ball in the Euclidean space R n . The cochain complex ( L φ Ω ∗ ( B ) , d ) retracts to the complex ( R → → → . . . ) . roof. Fix x ∈ B . Suppose that χ : Ω k ( B ) → Ω k − ( B ) is defined for all k ≥ k − τ ⊂ B , we have Z τ χ ( ω ) = Z C τ ω for every differential k -form ω . The cone C τ is defined as follows: If τ = ( x , . . . , x k − ),then C τ = ( x, x , . . . , x k − ). The function χ will depend on x , we write χ x = χ ifnecessary.Claim: χd + dχ = Id . (5)Take σ a k -simplex in B and ω ∈ Ω k ( B ), then Z σ χ ( dω ) = Z C σ dω = Z ∂C σ ω, where the last equality comes from Stokes’ theorem. If ∂σ = τ + . . . + τ k , we have Z σ χ ( dω ) = Z σ ω − k X i =0 Z C τi ω = Z σ ω − Z ∂σ χ ( ω ) = Z σ ω − Z σ dχ ( ω ) . Since the equality holds for every k -simplex we conclude (5) (see for example [Whi57,Chapter IV]).For x ∈ B we consider ϕ = ϕ x : [0 , × B → B , ϕ x ( t, y ) = ty + (1 − t ) x and η t : B → [0 , × B , η t ( y ) = ( t, y ). We look for an explicit expression for χ ( ω ): Z σ χ ( ω ) = Z C σ ω = Z ϕ ([0 , × σ ) ω = Z [0 , × σ ϕ ∗ ω = Z σ Z η ∗ s ( ι ∂∂t ϕ ∗ ω ) ds. Where ∂∂t is the vector field on [0 , × B defined by ∂∂t ( s, y ) = (1 , k -form ̟ with respect to a vector field V is the ( k − ι V ̟ x ( v , . . . , v k − ) = ̟ x ( V ( x ) , v , . . . , v k − ) . We conclude that χ ( ω ) = Z η ∗ t ( ι ∂∂t ϕ ∗ ω ) dt. Observe that the family of k -forms { η ∗ t ( ι ∂∂t ϕ ∗ ω ) } t ∈ [0 , satisfies the hypothesis ofLemma 3.2, then χ ( ω ) is smooth. By definition and the claim it satisfies equality (5).Observe that if ω is closed, then χ ( ω ) is a primitive of ω , so it is enough to prove theclassic Poincar´e’s lemma. However, in our case we need a primitive in L φ , so we takea convenient average. 16efine h ( ω ) = 1Vol (cid:0) B (cid:1) Z B χ x ( ω ) dx, where B = B (cid:0) , (cid:1) .Since ( x, y ) χ x ( ω ) y is smooth in both variables we can use again Lemma 3.2 toshow that h ( ω ) is in Ω k ( B ). Note that this works because we take the integral on aball with closure included in B . Moreover, the derivative of h is dh ( ω ) = 1Vol (cid:0) B (cid:1) Z B dχ x ( ω ) dx. Then using (5) we have dh ( ω ) + h ( dω ) = ω (6)for all ω ∈ Ω k ( B ) with k ≥ h is well-defined and continuous from L φ Ω k ( B ) to L φ Ω k − ( B ).To this end we first bound | χ x ( ω ) | y for y ∈ B and ω ∈ Ω k ( B ). Since ι ∂∂t ϕ ∗ ω is a formon [0 , × B that is zero in the direction of ∂∂t , we have | η ∗ t ( ι ∂∂t ϕ ∗ ω ) | y = | ι ∂∂t ϕ ∗ ω | ( t,y ) for every t ∈ (0 , 1) and y ∈ B . After a direct calculation we get the estimation | ι ∂∂t ϕ ∗ ω | ( t,y ) ≤ t k − | y − x || ω | ϕ ( t,y ) . Using the assumption that t ∈ (0 , | χ ( ω ) | y ≤ Z | y − x || ω | ϕ ( t,y ) dt. (7)Consider the function u : R n → R defined by u ( z ) = | ω | z if z ∈ B and u ( z ) = 0 inthe other case. Using (7) and the change of variables z = ty + (1 − t ) x we haveVol (cid:0) B (cid:1) | h ( ω ) | y ≤ Z B ( ty, − t ) Z | z − y | u ( z )(1 − t ) − n − dtdz = Z B ( y, | z − y | u ( z ) (cid:18)Z B ( ty, − t )( z )(1 − t ) − n − dt (cid:19) dz. Observe that B ( ty, − t )( z ) = 1 implies that | z − y | ≤ − t ). Then we have Z B ( ty, − t )( z )(1 − t ) − n − dt ≤ Z − | z − y | (1 − t ) − n − dt = Z | z − y | r − n − dr (cid:22) | z − y | n . This implies V ol ( B ) | h ( ω ) | y (cid:22) Z B ( y, | z − y | − n u ( z ) dz. Using this estimate we have k h ( ω ) k L φ = inf (cid:26) γ > Z B φ (cid:18) | h ( ω ) | y γ (cid:19) dy ≤ (cid:27) (cid:22) inf (cid:26) γ > Z B φ (cid:18)Z B ( y, | z − y | − n u ( z ) γ dz (cid:19) dy ≤ (cid:27) . R B ( y, | z − y | − n dz < + ∞ , we can use Jensen’s inequality and write k h ( ω ) k L φ (cid:22) inf (cid:26) γ > B (0 , Z B Z B (0 , φ (cid:18) u ( z )Vol( B (0 , γ (cid:19) dz | z − y | n − dy ≤ (cid:27) = inf (cid:26) γ > B (0 , Z B (0 , φ (cid:18) u ( z )Vol( B (0 , γ (cid:19) (cid:18)Z B dy | z − y | n − (cid:19) dz ≤ (cid:27) . We have that there exists a constant K > R B dy | z − y | n − ≤ K for all z ∈ B (0 , k h ( ω ) k L φ (cid:22) Vol( B (0 , k ω k L ˜ Kφ (cid:22) k ω k L φ , where ˜ K = K Vol( B (0 , dh ( ω ) = ω − h ( dω ) we also have k dh ( ω ) k φ ≤ k ω k φ + k h ( dω ) k φ (cid:22) k ω k φ + k dω k φ . We conclude that h is well-defined and bounded for the norm | | L φ in all degrees k ≥ ω = df for certain function f we observe that η ∗ t ( ι ∂∂t ϕ ∗ x df )( y ) = df ϕ x ( t,y ) ( y − x ) = ( f ◦ α ) ′ ( t ) , where α is the curve α ( t ) = ϕ x ( t, y ). Then χ x ( df )( y ) = f ( y ) − f ( x ), from which we get h ( df ) = f − V ol ( B ) Z B f. We define h : L p Ω ( B ) → L p Ω − ( B ) = R by h ( f ) = V ol ( B ) R B f , which is clearlycontinuous because B has finite Lebesgue measure. Then the identity (6) is true forall ω ∈ L φ Ω k ( B ) and h is continuous in all degrees.The following lemma can be proved by a direct application of the Lipschitz condi-tion. Lemma 4.4. Let M and N be two Riemannian manifolds and f : M → N a bi-Lipschitz diffeomorphism with constant L . Then for every k ∈ N the pull-back f ∗ : L φ Ω k ( N ) → L φ Ω k ( M ) is continuous and its operator norm is bounded depending on L , k , φ and n = dim( M ) . A bicomplex is a family of topological vector spaces { C k,ℓ } k,ℓ ∈ N together with con-tinuous linear maps d ′ : C k,ℓ → C k +1 ,ℓ and d ′′ : C k,ℓ → C k,ℓ +1 . We denote it by( C ∗ , ∗ , d ′ , d ′′ ). Lemma 4.5 (Lemma 5,[Pan95]) . Let ( C ∗ , ∗ , d ′ , d ′′ ) be a bicomplex with d ′ ◦ d ′′ + d ′′ ◦ d ′ = 0 .Suppose that for every ℓ ∈ N the complex ( C ∗ ,ℓ , d ′ ) retracts to the subcomplex ( E ℓ :=Ker d ′ | C ,ℓ → → → · · · ) . Then the complex ( D ∗ , δ ) , defined by D m = M k + ℓ = m C k,ℓ and δ = d ′ + d ′′ , is homotopically equivalent to ( E ∗ , d ′′ ) . 18 complete proof of Lemma 4.5 can be found in [Seq19]. Proof of Proposition 4.2. Let us define the bicomplex C k,ℓ = ( ω ∈ Y U ∈U ℓ L φ Ω( U ) : {k ω U k L φ } U ∈U ℓ , {k dω U k L φ } U ∈U ℓ ∈ ℓ φ ( U ℓ ) ) , equipped with the norm k ω k = k θ k ℓ φ + k θ ′ k ℓ φ , where θ ( U ) = k ω U k L φ and θ ′ ( U ) = k dω U k L φ . The derivatives are defined by • ( d ′ ω ) U = ( − ℓ dω U for every ω ∈ C k,ℓ , • If ω ∈ C k,ℓ and W ∈ U ℓ +1 , W = U ∩ . . . ∩ U ℓ +1 , then( d ′′ ω ) W = ℓ +1 X i =0 ( − i ( ω U ∩ ...U i − ∩ U i +1 ∩ ... ∩ U ℓ +1 ) | W . It is easy to see that d ′ and d ′′ are well-defined and continuous, and that d ′ ◦ d ′′ + d ′′ ◦ d ′ =0. Observe that the elements of Ker d ′ | C ,ℓ are the functions g ∈ Q U ∈U ℓ L φ Ω ( U )satisfying the following conditions: • dg U = 0 for all U ∈ U ℓ , then g U is constant. • {k g U k L φ } U ∈U ℓ ∈ ℓ φ ( U ℓ ) = ℓ φ ( X ℓG ).Using the construction of X G and the fact that U is bi-Lipschitz (with uniform Lipschitzconstant) to the Euclidean unit ball we have that Ker d ′ | C ,ℓ is isomorphic to ℓ φ ( X ℓG )and d ′′ coincides with the derivative on this space.On the other hand, the elements of Ker d ′′ | C k, are of the form ω = { ω U } U ∈U with ω U | U ∩ U ′ = ω U ′ | U ∩ U ′ if U ∩ U ′ = ∅ . We can take a k -form ˜ ω in L φ Ω k ( M ) such that ˜ ω | U = ω U for all U ∈ U , then thereis an isomorphism between Ker d ′′ | C k, and L φ Ω k ( G ) for which d ′ coincides with thederivative on the second space.Claim 1: The cochain complex ( C ∗ ,ℓ , d ′ ) retracts to (Ker d ′ | C ,ℓ → → . . . ) for all ℓ ∈ N .For every U ∈ U ℓ consider f U : U → B an L -bi-Lipschitz diffeomorphism ( L doesnot depend on U and B is the unit ball in the corresponding Euclidean space). Wedefine H : C k,ℓ → C k − ,ℓ by ( Hω ) U = ( − ℓ f ∗ U h ( f − U ) ∗ ω U , h : L φ Ω k ( B ) → L φ Ω k − ( B ) is the map given by Lemma 4.3. Here we consider C − ,ℓ = Ker d ′ | C ,ℓ and d ′ : C − ,ℓ → C ,ℓ the inclusion. One can easily verify that Hd ′ + d ′ H = Id. Using Lemma 4.4 we have that H is continuous.Claim 2: The cochain complex ( C k, ∗ , d ′′ ) retracts to (Ker d ′′ | C k, → → . . . ) for all k ∈ N .We have to construct a family of bounded linear maps P : C k,ℓ → C k,ℓ − ( ℓ ≥ P ◦ d ′′ + d ′′ ◦ P = Id, where C k, − = Ker d ′′ | C k, and d ′′ : C k, − → C k, is theinclusion.Consider { η U } U ∈U a partition of unity with respect to U . If ℓ ≥ ω ∈ C k,ℓ ,then we define ( P ω ) V = X U ∈U η U ω U ∩ V , for all V ∈ U ℓ − . For ω ∈ C k, and V ∈ U we put( P ω ) V = X U ∈U η U ω U | V . A direct calculation shows that P is as we wanted.Applying Lemma 4.5 we obtain that a complex ( D ∗ , δ ) is homotopically equivalentto (Ker d ′ | C , ∗ , d ′′ ) and (Ker d ′′ | C ∗ , , d ′ ). The proof in the non-relative case ends usingthe above identifications.To prove the relative case we have to consider the bicomplex ( C ∗ , ∗ ξ , d ′ , d ′′ ), where C k,ℓξ is the subspace consisting of the elements ω of C k,ℓ for which there exists V ⊂ G ,a neighborhood of ξ , such that ω U ≡ U ⊂ V . The above argument works in thiscase because all maps preserve the subspaces C k,ℓξ .Since L p Ω k ( G, U ) = L p Ω k ( G ), the previous proposition finishes the proof of the L p -case. Proposition 4.6 (Second step) . The complexes ( L φ Ω ∗ ( G, U ) , d ) and ( I φ Ω ∗ ( G, U ) , d ) are homotopically equivalent. The same is true for complexes ( L φ Ω ∗ ( G, U , ξ ) , d ) and ( I φ Ω ∗ ( G, U , ξ ) , d ) . Combining Propositions 4.2 and 4.6 we have the following diagram: ℓ φ C ( X G ) δ / / ∼ (cid:20) (cid:20) ℓ φ C ( X G ) δ / / ∼ (cid:20) (cid:20) ℓ φ C ( X G ) δ / / ∼ (cid:20) (cid:20) · · ·L φ Ω ( G, U ) d / / T T ∼ (cid:20) (cid:20) L φ Ω ( G, U ) d / / T T ∼ (cid:20) (cid:20) L φ Ω ( G, U ) d / / T T ∼ (cid:20) (cid:20) · · ·I φ Ω ( G, U ) d / / T T I φ Ω ( G, U ) d / / T T I φ Ω ( G, U ) d / / T T · · · roof. Consider the family of maps ∗ κ : L φ Ω k ( G, U ) → I φ Ω k ( G, U ) given by the con-volution with a smooth kernel κ .Claim 1: For a fixed k = 0 , . . . , dim( G ) the map ∗ κ : L φ Ω k ( G, U ) → L φ Ω k ( G, U ) iswell-defined and continuous.Let γ > U ∈ U , using Lemma 3.3 we have Z U φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ Z U φ (cid:18)Z G C | ω | x · z γ κ ( z ) dz (cid:19) dx ≤ Z U φ (cid:18)Z x · supp( κ ) C | ω | y γ dy (cid:19) dx ≤ Z U φ X U ′ ∈N U (cid:13)(cid:13)(cid:13)(cid:13) Cω | U ′ γ (cid:13)(cid:13)(cid:13)(cid:13) L ! dx, where N U = { U ′ ∈ U : U ′ ∩ ( x · supp( κ )) = ∅ for some x ∈ U } . The bounded geometryimplies that there exists N a uniform bound of N U . Using this bound and Jensen’sinequality we have Z U φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ Vol( U ) N U X U ′ ∈N U φ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) N Cω | U ′ γ (cid:13)(cid:13)(cid:13)(cid:13) L (cid:19) . Let V be a uniform bound for Vol( U ). Because of Lemma 3.5, there exists a constant D such that if β is in L φ Ω k ( U ) with U ∈ U , then k β k L ≤ D k β k L φ . Therefore Z U φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ V N U X U ′ ∈N U φ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) DN Cω | U ′ γ (cid:13)(cid:13)(cid:13)(cid:13) L φ (cid:19) . If γ ≥ DN C k ω | U ′ k L φ for all U ′ ∈ N U , then Z U φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ V. By Remark 1.1 there exists a constant C ( V ) such that k ω ∗ κ | U k L φ ≤ C ( V ) k ω ∗ κ | U k L φV ≤ C ( V ) DN C X U ′ ∈N U k ω | U ′ k L φ . Denote L = C ( V ) DN C and take γ > X U ∈U φ (cid:18) k ω ∗ κ | U k L φ γ (cid:19) ≤ X U ∈U φ Lγ X U ′ ∈N U k ω | U ′ k L φ ! ≤ X U ∈U N U X U ′ ∈N U φ (cid:18) N Lγ k ω | U ′ k L φ (cid:19) . R > U ′ ∈ U , U ′ ∈ N U for at most R open sets U ∈ U .Then X U ∈U φ (cid:18) k ω ∗ κ | U k L φ γ (cid:19) ≤ X U ∈U Rφ (cid:18) N Lγ k ω | U ′ k L φ (cid:19) . This means that, if θ ( U ) = k ω | U k L φ and ϑ ( U ) = k ω ∗ κ | U k L φ , then k ϑ k ℓ φ ≤ N L k θ k ℓ Rφ (cid:22) k θ k ℓ φ . Using the same argument with d ( ω ∗ κ ) = dω ∗ κ we can conclude that | ω ∗ κ | L φ (cid:22) | ω | L φ ,which finishes the proof of Claim 1.Claim 2: Let k = 0 , . . . , dim( G ). The map ∗ κ : L φ Ω k ( G, U ) → L φ Ω k ( G ) is well-defined and continuous.As above, if γ > Z G φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ X U ∈U Z U φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ X U ∈U Vol( U ) N U X U ′ ∈N U φ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) DN Cω | U ′ γ (cid:13)(cid:13)(cid:13)(cid:13) L p hi (cid:19) ≤ X U ∈U V Rφ (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) Lω | U γ (cid:13)(cid:13)(cid:13)(cid:13) L φ (cid:19) . If we use again the notation θ ( U ) = k ω | U k L φ , we have k ω k L φ ≤ L k θ k ℓ V Rφ (cid:22) k θ k ℓ φ .Doing the same with the derivative we conclude the Claim 2.Claims 1 and 2 imply that ∗ κ : L φ Ω k ( G, U ) → I φ Ω k ( G ) is well-defined and contin-uous. Furthermore, by Proposition 3.4 we know that ∗ κ commutes with the derivative.We will define a family of continuous maps h : L φ Ω k ( G, U ) → L φ Ω k − ( G, U ) suchthat (cid:26) h ( df ) = f − i ( f ∗ κ ) if f ∈ L φ Ω ( G, U ) h ( dω ) + dh ( ω ) = ω − i ( ω ∗ κ ) if ω ∈ L φ Ω k ( G, U ) , k ≥ , (8)where i denotes the inclusion (which is clearly continuous). If h maps continuously I φ Ω k ( G, U ) on I φ Ω k − ( G, U ) for every k ≥ 1, the complexes ( L φ Ω ∗ ( G, U ) , d ) and( I φ Ω ∗ ( G, U ) , d ) are homotopically equivalent. L φ Ω ( G, U ) d / / ∗ κ (cid:20) (cid:20) L φ Ω ( G, U ) d / / ∗ κ (cid:20) (cid:20) h s s L φ Ω ( G, U ) / / ∗ κ (cid:20) (cid:20) h s s · · ·I φ Ω ( G, U ) d / / i T T I φ Ω ( G, U ) d / / i T T h k k I φ Ω ( G, U ) / / i T T h k k · · · z ∈ supp( κ ) we consider Z ∈ Lie( G ) a left-invariant vector field such thatexp( Z ) = z . Let ϕ Zt be the flow associated with Z . Observe that ϕ Zt ( x ) = L x ϕ Zt ( e ) = x · exp( tZ ) . Given ω ∈ L φ Ω k ( G, U ) with k = 1 , . . . , dim( G ), we define h ( ω ) x = − Z G (cid:18)Z ( ϕ Zt ) ∗ ι Z ω x dt (cid:19) κ ( z ) dz. By Lemma 3.2 it is smooth and its derivative is dh ( ω ) x = − Z G (cid:18)Z ( ϕ Zt ) ∗ dι Z ω x dt (cid:19) κ ( z ) dz. Claim 3: h and ∗ κ verify (8).Take ω a k -form in L φ Ω k ( G, U ) for k ≥ 1, thus h ( dω ) + dh ( ω ) = − Z G (cid:18)Z ( ϕ Zt ) ∗ ( ι Z d + dι Z ) ω dt (cid:19) κ ( z ) dz. Recall the Cartan formula L Z ω = ι Z dω + dι Z ω (see for example [GHL04]). Usingthe identity ∂∂t ( ϕ Zt ) ∗ ω = ( ϕ Zt ) ∗ L Z ω we obtain h ( dω ) + dh ( ω ) = − Z G (cid:18)Z ( ϕ Zt ) ∗ L Z ω dt (cid:19) κ ( z ) dz = − Z G (cid:18)Z ∂∂t ( ϕ Zt ) ∗ ω dt (cid:19) κ ( z ) dz = − Z G (( ϕ Zt ) ∗ ω − ω ) κ ( z ) dz = ω − ω ∗ κ. Now consider f ∈ L φ Ω ( G, U ). We have h ( df ) x = − Z G (cid:18)Z df x · exp( tZ ) ( Z ( x )) dt (cid:19) κ ( z ) dz. Let α : [0 , → G be the curve α ( t ) = x · exp( tZ ). We have α ′ ( t ) = Z ( α ( t )), then( f ◦ α ) ′ ( t ) = d α ( t ) f ( α ′ ( t )) = df x · exp( tZ ) ( Z ( α ( t ))) . Therefore h ( df ) x = − Z G (cid:18)Z ( f ◦ α ) ′ ( t ) dt (cid:19) κ ( z ) dz = Z G ( f ( α (1)) − f ( α (0))) κ ( z ) dz = f ( x ) − f ∗ κ ( x ) . h : L φ Ω k ( G, U ) → L φ Ω k − ( G, U ) is well-defined and continuous for every k = 1 , . . . , dim( G ).First we estimate the operator norm of hω at a point x ∈ G . Consider v , . . . , v k − ∈ T x G , then | h ( ω ) x ( v , . . . , v k − ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z G (cid:18)Z ω ϕ Zt ( x ) ( Z ( ϕ Zt ( x )) , d x ϕ Zt ( v ) , . . . , d x ϕ Zt ( v k − )) dt (cid:19) κ ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z G (cid:18)Z | ω ϕ Zt ( x ) ( Z ( ϕ Zt ( x )) , d x R exp( tZ ) ( v ) , . . . , d x R exp( tZ ) ( v k − )) | dt (cid:19) κ ( z ) dz As in the proof of Lemma 3.3 we have a uniform bound | d x R exp( tZ ) | ≤ M for all z ∈ supp( κ ). Moreover, since left-invariant fields have constant norm we can write k Z ( y ) k y = C for every y ∈ G . Hence, if k v k x = . . . = k v k − k x = 1, | h ( ω ) x ( v , . . . , v k − ) | ≤ Z G (cid:18)Z CM k − | ω | ϕ Zt ( x ) dt (cid:19) κ ( z ) dz, which implies | h ( ω ) | x ≤ Z G (cid:18)Z CM k − | ω | ϕ Zt ( x ) dt (cid:19) κ ( z ) dz. (9)Using (9) and Jensen’s inequality we obtain φ (cid:18) | h ( ω ) | x γ (cid:19) ≤ Z G (cid:18)Z φ (cid:18) CM k − γ | ω | ϕ Zt ( x ) (cid:19) dt (cid:19) κ ( z ) dz. (10)For U ∈ U denote θ ( U ) = k ω | U k L φ and ϑ ( U ) = k hω | U k L φ . If γ > Z U φ (cid:18) | h ( ω ) | x γ (cid:19) dx ≤ Z U Z G (cid:18)Z φ (cid:18) CM k − γ | ω | ϕ Zt ( x ) dt (cid:19)(cid:19) κ ( z ) dzdx = Z G (cid:18)Z Z U φ (cid:18) CM k − γ | ω | ϕ Zt ( x ) (cid:19) dxdt (cid:19) κ ( z ) dz. The identity d x ϕ Zt = d x R exp( tZ ) allows us to find m > m < | J ac x ( ϕ Zt ) | forall z ∈ supp( κ ). Then Z U φ (cid:18) | h ( ω ) | x γ (cid:19) dx ≤ Z G (cid:18)Z Z E ( U ) m φ (cid:18) CM k − γ | ω | y (cid:19) dydt (cid:19) κ ( z ) dz = 1 m Z E ( U ) φ (cid:18) CM k − γ | ω | y (cid:19) dy, where E ( U ) is a neighborhood of U with uniform radius (independent of U ) such that ϕ Zt ( x ) ∈ E ( U ) for all z ∈ supp( κ ) and x ∈ U . Consider V U = { V ∈ U : V ∩ E ( U ) = ∅} , γ ≥ CM k − max {k ω | V k L Sm φ : V ∈ V U } , where S ≥ V U for all U ∈ U , Z U φ (cid:18) | h ( ω ) | x γ (cid:19) dx ≤ , which implies k h ( ω ) | U k L φ ≤ CM k − max {k ω | V k L Sm φ : V ∈ V U } ≤ M X V ∈V U k ω | V k L φ for some constant M that does not depend on U . Therefore X U ∈U φ (cid:18) k h ( ω ) | U k L φ γ (cid:19) ≤ X U ∈U φ M X V ∈V U k ω | V k L φ γ ! ≤ X U ∈U X V ∈V U V U φ (cid:18) S Mk ω | V k L φ γ (cid:19) ≤ X U ∈U N φ (cid:18) S Mk ω | V k L φ γ (cid:19) , where N ≥ { U ∈ U : V ∈ V U } for all V ∈ U . From here we obtain k ϑ k ℓ φ ≤ S Mk θ k ℓ N φ (cid:22) k θ k ℓ φ . Using the identity dh ( ω ) = ω − i ( ω ∗ κ ) and the above estimate we obtain | h ( ω ) | L φ (cid:22) | ω | L φ . Claim 5: The map h : L φ Ω k ( G ) → L φ Ω k − ( G ) is well-defined and continuous forevery k = 1 , . . . , dim( G ).Using (10) we have Z G φ (cid:18) | h ( ω ) | x γ (cid:19) dx ≤ Z G Z G Z φ CM k − | ω | ϕ Zt ( x ) α ! dt ! κ ( z ) dzdx ≤ Z G (cid:18)Z Z G m φ (cid:18) CM k − | ω | y γ (cid:19) dydt (cid:19) κ ( z ) dz = Z G m φ (cid:18) CM k − | ω | y γ (cid:19) dy. From this we obtain k h ( ω ) k L φ (cid:22) k ω k L φ ; and using again the equality (8) we have | h ( ω ) | L φ (cid:22) | ω | L φ .By Claims 4 and 5 we conclude that h is well-defined and continuous from I φ Ω k ( G, U )to I φ Ω k − ( G, U ), which finishes the poof in the non-relative case.The same argument works in the relative case. The only thing we have to verifyis that the maps ∗ κ and h preserve the relative subcomplexes, which is easy using thecompactness of supp( κ ). 25he proof of Theorem 1.5 finishes with the following proposition. Proposition 4.7. The complexes ( I φ Ω ∗ ( G, U ) , d ) , ( L φ Ω ∗ ( G ) , d ) , and ( L φ C ∗ ( G ) , d ) arehomotopically equivalent. The same result is true for the corresponding relative com-plexes.Proof. In this case we consider ∗ κ and h defined as in Proposition 4.6. We have toprove that they are well-defined and continuous. Identities as (8) are clearly satisfied. L φ C ( G ) d / / ∗ κ (cid:20) (cid:20) L φ C ( G ) d / / ∗ κ (cid:20) (cid:20) h s s L φ C ( G ) d / / ∗ κ (cid:20) (cid:20) h s s · · ·I φ Ω ( G, U ) d / / i T T i (cid:20) (cid:20) I φ Ω ( G, U ) d / / i T T i (cid:20) (cid:20) h s s I φ Ω ( G, U ) d / / i T T i (cid:20) (cid:20) h s s · · · L φ Ω ( G ) d / / ∗ κ T T L φ Ω ( G ) d / / ∗ κ T T h s s L φ Ω ( G ) d / / ∗ κ T T h s s · · · The map h : L φ C k ( G ) → L φ C k − ( G ) is the continuous extension of h : L φ Ω k ( G ) → L φ Ω k − ( G ), then we have that all maps h in the diagram are continuous.We have to prove that the map ∗ κ : L φ C k ( G ) → I φ Ω k ( G, U ) is well-defined andcontinuous for every k = 0 , . . . , dim( G ). Then so is ∗ κ : L φ Ω k ( G ) → I φ Ω k ( G, U ) . Tothis end, first observe that if ω ∈ L φ C k ( G ) then ω ∗ κ ∈ Ω k ( G ) by Lemma 3.4.Let γ > 0. Using the estimate given in Lemma 3.3 we have Z G φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ Z G φ (cid:18) C | ω | ∗ κ ( x ) γ (cid:19) dx = Z G φ (cid:18)Z G C | ω | xz γ κ ( z ) dz (cid:19) dx ≤ Z G Z G φ (cid:18) C | ω | xz γ (cid:19) κ ( z ) dzdx. In the last line we use Jensen’s inequality. As before we take m > m < | J ac x ( R z ) | for all x ∈ G and z ∈ supp( κ ), then Z G φ (cid:18) | ω ∗ κ | x γ (cid:19) dx = Z G (cid:18)Z G φ (cid:18) C | ω | xz γ (cid:19) | J ac x ( R z ) || J ac x ( R z ) | dx (cid:19) κ ( z ) dz ≤ Z G (cid:18)Z G m φ (cid:18) C | ω | y γ (cid:19) dy (cid:19) κ ( z ) dz = Z G m φ (cid:18) C | ω | y γ (cid:19) dy. γ ≥ C k ω k L φm we have Z G φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ , which implies k ω ∗ κ k L φ ≤ C k ω k L φm (cid:22) k ω k L φ . In the same way we have k d ( ω ∗ κ ) k L φ = k dω ∗ κ k L φ (cid:22) k dω k L φ and as a conclusion | ω ∗ κ | L φ (cid:22) | ω | L φ .On the other hand we denote ϑ ( U ) = k ω ∗ κ | U k L φ and estimate Z U φ (cid:18) | ω ∗ κ | x γ (cid:19) dx ≤ Z U φ (cid:18)Z G C | ω | xz γ κ ( z ) dz (cid:19) dx ≤ Dφ (cid:18)Z E ( U ) C | ω | y γ dy (cid:19) , where E ( U ) is a neighborhood of U with radius independent of U , and D is a constant(also independent of U ). We can deduce from this that k ω ∗ κ | U k L φ ≤ Cφ − (1 /D ) Z E ( U ) | ω | y dy. In order to simplify the notation we write C = Cφ − (1 /D ) . Then X U ∈U φ (cid:18) k ω ∗ κ | U k L φ γ (cid:19) ≤ X U ∈U φ (cid:18) C γ Z E ( U ) | ω | y dy (cid:19) ≤ X U ∈U E ( U )) Z E ( U ) φ (cid:18) C Vol( E ( U )) | ω | y γ (cid:19) dy. 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