Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry

1.1 Sub-Riemannian contact manifolds

A contact manifold is given by the couple ( M , H ) , where M is a smooth odd-dimensional (connected) manifold of dimension 2 n + 1 and H is the so-called contact structure on M , that is, H is a smooth distribution of hyperplanes which is maximally nonintegrable: given θ M a smooth 1-form defined on M such that H = ker ( θ M ) , then d θ M restricts to a nondegenerate 2-form on H . Roughly speaking, to be maximally nonintegrable means that the contact subbundle H is as far as possible from being integrable. Indeed, in general, for a subbundle defined by a 1-form η to be integrable, it is necessary and sufficient that η ∧ ( d η ) n ≡ 0 (see [31], Section 3.4). Therefore, a measure on a contact manifold ( M , H ) can be defined through the nondegenerate top degree form θ M ∧ ( d θ M ) n . There exists, in addition, a unique vector field ξ M transverse to ker θ M (the so-called Reeb vector field) such that θ M ( ξ M ) = 1 and ℒ ξ M = 0 .

Among stratified nilpotent Lie groups, the Heisenberg groups are the simplest example of a group endowed with a contact structure. We recall that the Heisenberg group H n is the Lie group with stratified nilpotent Lie algebra h of step 2

where the only nontrivial commutation rules are [ X j , Y j ] = T , j = 1 , … , n . We denote by θ H the 1-form on H n such that ker θ H = exp ( h 1 ) and θ H ( T ) = 1 . We recall also that H n can be identified with R 2 n + 1 through the exponential map. The stratification of the algebra induces a family of dilations δ λ in the group via the exponential map,

which are analogous to the Euclidean homotheties.

As already stressed earlier, the Heisenberg groups are the local model of all contact manifolds (see [31], p. 112).

Following Rumin (see [34], p. 288), we can assume that there is a metric g M , which is globally adapted to the symplectic form d θ M . Indeed, there exists an endomorphism J of H such that J 2 = − I d , d θ M ( X , J Y ) = − d θ M ( J X , Y ) for all X , Y ∈ H , and d θ M ( X , J X ) > 0 for all X ∈ H ⧹ { 0 } . Then, if X , Y ∈ H , we define g H ( X , Y ) ≔ d θ M ( X , J Y ) . Finally, we extend J to T M by setting J ξ M = ξ M and setting g M ( X , Y ) = θ M ( X ) θ M ( Y ) + d θ M ( X , J Y ) for all X , Y ∈ T M .

The couple ( M , H ) equipped with the Riemannian metric g H is called a sub-Riemannian contact manifold, and in the sequel, it will be denoted by ( M , H , g M ) , where g M is obtained as mentioned earlier. In any sub-Riemannian contact manifold ( M , H , g M ) , we can define a sub-Riemannian distance d M (see, e.g., [43]) inducing on M the same topology of M as a manifold. In particular, Heisenberg groups H n can be viewed as sub-Riemannian contact manifolds. If we choose on the contact sub-bundle of H n a left-invariant metric, it turns out that the associated sub-Riemannian metric is also left-invariant. It is customary to call this distance in H n a Carnot-Carathéodory distance (note that all left-invariant sub-Riemannian metrics on Heisenberg groups are bi-Lipschitz equivalent).

A natural setting when dealing with differential forms in Heisenberg groups is provided by Rumin’s complex ( E 0 • , d c H ) of differential forms in H n (see, e.g., [34]), since de Rham’s complex ( Ω • , d ) in R 2 n + 1 , endowed with the usual exterior differential d , does not fit the very structure of the group H n . Indeed, differential forms on h split into two eigenspaces under δ λ (see (2)); therefore, de Rham’s complex lacks scale invariance under the anisotropic dilations δ λ , basically since it mixes derivatives along all the layers of the stratification. Rumin’s substitute for de Rham’s exterior differential is a linear differential operator d c H from sections of E 0 h to sections of E 0 h + 1 ( 0 ≤ h ≤ 2 n + 1 ) such that ( d c H ) 2 = 0 .

We note explicitly that Rumin’s differential d c H may be a left-invariant differential operator of order higher than 1.

Rumin’s construction of the intrinsic complex makes sense for arbitrary contact manifolds ( M , H ) (see [34]). The main features of Rumin’s complex defined on M are the same as those already stated in H n . Assuming E 0 • = ⊕ h = 0 2 n + 1 E 0 h endowed with the exterior differential d c M , we have:

1. ( d c M ) 2 = 0 ;

2. the complex ( E 0 • , d c M ) is homotopically equivalent to de Rham’s complex ( Ω • , d ) ;

3. d c M : E 0 h → E 0 h + 1 is a homogeneous differential operator in the horizontal derivatives of order 1 if h ≠ n , whereas d c M : E 0 n → E 0 n + 1 is a homogeneous differential operator in the horizontal derivatives of order 2.

Given ( M , H , g M ) , the scalar product on H determines a norm on the line bundle T M / H . Therefore, for any h , the vector spaces E 0 h are endowed with a scalar product. By using θ M ∧ d θ M as a volume form, one obtains L p -norms on spaces of smooth Rumin’s differential forms on M (see Remark 3.4).

We denote by ∗ the Hodge duality associated with the inner product in E 0 • and the volume form (see also Section 4), and by δ c M , the formal adjoint in L 2 ( M , E 0 • ) of the operator d c M ; we have δ c M = ( − 1 ) h ∗ d c M ∗ on E 0 h (see [34]).

1.2 Bounded geometry

In the sequel, we will assume that M is a noncompact manifold with bounded geometry, without boundary. But, of course, our approach covers also the case of a compact manifold without boundary.

Examples of noncompact sub-Riemannian contact manifolds with bounded geometry are given in [6] (see, in particular, Remark 4.10 therein and Section 7).

The formulation of a Gaffney-type inequality in this setting requires different statements, depending on the degree of the differential forms we are considering. Indeed, the fact that the operator d c M has order 1 or 2, depending on the degree of the form on which it acts, will be a major issue in the proofs of our results and will change the class of Sobolev spaces in our inequalities.

To introduce the notion of Sobolev space in M , we will make use of the analogous notion in Heisenberg groups H n . This notion is associated with the stratification of their algebra and nowadays is quite classical: we refer, e.g., to Section 4 of [14] and Section 2.1.1 of this article Roughly speaking, if ℓ ∈ N and p ≥ 1 , the Sobolev space W ℓ , p ( H n ) (the so-called Folland-Stein Sobolev space) can be defined as follows. Fix an orthonormal basis { W i , i = 1 , … , 2 n + 1 } of h such that W i ∈ h 1 for i = 1 , … , 2 n and W 2 n + 1 ∈ h 2 . We call homogeneous order of a monomial in { W i } its degree of homogeneity with respect to the group dilations δ λ , λ > 0 . We say that a differential form on H n belongs to W ℓ , p ( H n ) if all its derivatives of homogeneous order ≤ ℓ along { W i } belong to L p ( H n ) . Using C k -bounded charts, this notion extends to C k -bounded geometry sub-Riemannian contact manifolds M (we refer to Section 3 for precise definitions).

Our main result reads as follows.

This article is organized as follows. Section 2 contains some definitions and properties of Heisenberg groups and a very short review of Rumin’s complex. Section 3 contains a precise definition of manifold with bounded geometry and the definition of its associated Sobolev spaces. Section 3 also contains a fine result concerning the existence of a map that associates an orthonormal symplectic basis of ker θ M with the canonical orthonormal symplectic basis of Ker θ e H , where θ e H denotes the form θ H evaluated at the point e ∈ H n . It should be noted that throughout the article, we will use both symbols α p and α ( p ) to indicate when the form α is evaluated at the point p . In our setting, if ψ : H n → M is a contact diffeomorphism, we have ψ ♯ d c M = d c H ψ ♯ , where ψ ♯ is the pullback of the map ψ . This is not the case when we replace the differential with the codifferential; in Section 4, we show that if M has bounded geometry we can locally control the difference ψ ♯ δ c M − δ c H ψ ♯ with constants depending only on the geometry of M . In Section 5, we prove at first a local Gaffney inequality, and then we pass from the local case to the global one using an ad hoc covering for M with balls of suitably small radius obtained in Section 3.

We now list a few difficulties we have to deal with in this note. To prove the Gaffney inequality on M , we pass from the corresponding result for the “flat” model H n proved in [4]. A delicate point is to show that we can replace the contactomorphism ϕ x appearing in Definition 1.1 with another contactomorphism ψ x , which “sends” an orthonormal symplectic basis of ker θ x M into the canonical orthonormal symplectic basis of ker θ e H and that still depends only on the bounded geometric constants and not on the point x . This is accomplished in Theorem 3.10, and it is a convenient step if we want to write the difference ψ ♯ δ c M − δ c H ψ ♯ in local coordinates. Another subtle issue comes from the fact that the order of the differentials d c M and d c H can be one or two depending on the degree of the form, and this problem is reflected in several proofs (e.g., in Section 4, when we estimate ψ ♯ δ c M − δ c H ψ ♯ , or in Theorem 5.4 when we have to estimate the commutators between differentials and functions).

2.1 Heisenberg groups

We denote by H n the ( 2 n + 1 )-dimensional Heisenberg group, identified with R 2 n + 1 through exponential coordinates. A point p ∈ H n is denoted by p = ( x , y , t ) , with both x , y ∈ R n and t ∈ R . If we consider two points in H n , p = ( x , y , t ) and q = ( x ˜ , y ˜ , t ˜ ) , the (noncommutative) group operation is denoted by p ⋅ q ≔ ( x + x ˜ , y + y ˜ , t + t ˜ + 1 2 ∑ j = 1 n ( x j y ˜ j − y j x ˜ j ) ) . In this system of coordinates, the unit element of H n , which will be denote by e , is the zero of the vector space R 2 n + 1 , and p − 1 = − p .

For any q ∈ H n , the (left) translation τ q : H n → H n is defined as follows:

The Lebesgue measure in R 2 n + 1 is a Haar measure in H n .

For a general review on Heisenberg groups and their properties, we refer to [24,39,42]. We limit ourselves to fix some notation, following [6].

First, we notice that Heisenberg groups are smooth manifolds (and therefore are Lie groups). In particular, the pullback of differential forms is well defined as follows (see, e.g., [21], Proposition 1.106).

The Heisenberg group H n can be endowed with the homogeneous norm (known as Korányi norm)

and we define the gauge distance (a true distance, see [39], p. 638) as follows:

The metric d behaves well with respect to left-translations, that is,

for all q , p , p ′ ∈ H n . Finally, the balls for the metric d are the so-called Korányi balls

We denote by h the Lie algebra of the left-invariant vector fields of H n . The standard basis of h is given, for i = 1 , … , n , by

The only nontrivial bracket relations are [ X j , Y j ] = T , for j = 1 , … , n . The horizontal subspace h 1 is the subspace of h spanned by X 1 , … , X n and Y 1 , … , Y n . Coherently, from now on, we refer to X 1 , … , X n , Y 1 , … , Y n (identified with first-order differential operators) as the horizontal derivatives. Denoting by h 2 the linear span of T , the two-step stratification of h is expressed by

The stratification of the Lie algebra h induces a family of nonisotropic dilations { δ λ } , λ > 0 , in H n so that for any p = ( x , y , t ) ∈ H n ,

Notice that the gauge norm is positively δ λ -homogenous (i.e., d ( δ λ ( p ) , λ ( p ′ ) ) = λ d ( p , p ′ ) for all q , p , p ′ ∈ H n and λ > 0 ) so that the Lebesgue measure of the ball B ( x , r ) is r 2 n + 2 up to a geometric constant (the Lebesgue measure of B ( e , 1 ) ). Thus, the homogeneous dimension of H n with respect to δ λ , λ > 0 , equals

It is well known that the topological dimension of H n is 2 n + 1 , since as a smooth manifold it coincides with R 2 n + 1 , whereas the Hausdorff dimension of ( H n , d ) is Q .

The vector space h can be endowed with an inner product, indicated by ⟨ ⋅ , ⋅ ⟩ , making X 1 , … , X n , Y 1 , … , Y n and T orthonormal.

Throughout this article, we write also

As in [15], we also adopt the following multi-index notation for higher-order derivatives. If I = ( i 1 , … , i 2 n + 1 ) is a multi-index, we set

By the Poincaré-Birkhoff-Witt theorem, the differential operators W H , I form a basis for the algebra of left invariant differential operators in H n . Furthermore, we set

to denote the order of the differential operator W H , I , and

to denote its degree of homogeneity with respect to group dilations.

2.2 Multilinear algebra and Rumin’s complex in Heisenberg groups

The dual space of h is denoted by ⋀ 1 h . The basis of ⋀ 1 h , dual to the basis { X 1 , … , Y n , T } , is the family of covectors { d x 1 , … , d x n , d y 1 , … , d y n , θ H } , where

is called the contact form in H n .

We indicate again by ⟨ ⋅ , ⋅ ⟩ the inner product in ⋀ 1 h that makes ( d x 1 , … , d y n , θ H ) an orthonormal basis. Coherently with the previous notation (6), we set

The volume ( 2 n + 1 ) -form ω 1 H ∧ ⋯ ∧ ω 2 n + 1 H will be also written as d V .

We denote by ⋀ 0 h ≔ ⋀ 0 h = R and, for 1 ≤ h ≤ 2 n + 1 , we can define

In the sequel, we shall denote by Θ h the basis of ⋀ h h defined by

The inner product ⟨ ⋅ , ⋅ ⟩ on ⋀ 1 h yields naturally an inner product ⟨ ⋅ , ⋅ ⟩ on ⋀ h h making Θ h an orthonormal basis.

If 1 ≤ h ≤ 2 n + 1 , the Hodge isomorphisms

are defined by setting

If v ∈ ⋀ h h , we define its dual v ♮ ∈ ⋀ h h by the identity ⟨ v ♮ ∣ w ⟩ ≔ ⟨ v , w ⟩ , and analogously we define φ ♮ ∈ ⋀ h h for φ ∈ ⋀ h h .

Throughout this article, the elements of ⋀ h h are identified with left invariant differential forms of degree h on H n .

The same construction given earlier can be performed starting from the vector subspace h 1 ⊂ h , obtaining horizontal h-covectors and horizontal h-vectors

Moreover,

provides an orthonormal basis of ⋀ h h 1 .

The symplectic 2-form d θ H ∈ ⋀ 2 h 1 is defined by

Keeping in mind that the Lie algebra h can be identified with the tangent space to H n at x = e (see, e.g., [21], Proposition 1.72), starting from ⋀ h h we can define by left translation a fiber bundle over H n , which is still denoted by ⋀ h h . We can think of h -forms as sections of ⋀ h h . We denote by Ω h the vector space of all smooth h -forms. In addition, the symplectic 2-form − d θ H induces on h 1 a symplectic structure. Notice that { W 1 H , … , W 2 n H } is a symplectic basis of ker θ H .