Zolt Án

M Balogh

Urs Lang
email: urs.lang@math.ethz.ch

Pierre Pansu
email: pierre.pansu@u-psud.fr

LIPSCHITZ EXTENSIONS OF MAPS BETWEEN HEISENBERG GROUPS

Keywords:

Let H n be the Heisenberg group of topological dimension 2n + 1. We prove that if n is odd, the pair of metric spaces (H n , H n ) does not have the Lipschitz extension property.

Introduction

Since globally defined Lipschitz mappings have good Rademacher type differentiability properties, it is a matter of general interest to provide extension theorems for partially defined Lipschitz mappings. In this context, the following concept is useful: A pair of metric spaces (X, Y) is said to have the Lipschitz extension property if there exists a constant C ≥ 1 such that any Lipschitz mapping f : A → Y, A ⊂ X, has a Lipschitz extension F : X → Y with the property that Lip(F) ≤ C • Lip( f ). Here, we used the notation Lip(F) := inf{L ≥ 0 : d Y (F(x), F(y)) ≤ Ld X (x, y) for all x, y ∈ X} and similarly for Lip( f ).

There is a long history of results providing examples of metric pairs (X, Y) with or without the Lipschitz extension property. The first such result is due to McShane [START_REF] Mcshane | Extension of range of functions[END_REF] who proved that for an arbitrary metric space X, the pair (X, R) has the Lipschitz extension property, even with constant C = 1. This result was followed by the theorem of Kirszbraun [START_REF] Kirszbraun | Über die zusammenziehende und Lipschitzsche Transformationen[END_REF] who proved the Lipschitz extension property, also with constant C = 1, for pairs of Euclidean spaces (R m , R n ). Kirszbraun's theorem was generalized by Valentine [START_REF] Valentine | A Lipschitz condition preserving extension for a vector function[END_REF] to pairs of general Hilbert spaces (H 1 , H 2 ).

The Lipschitz extension problem becomes more challenging if the target space Y has no linear structure. An generalization of the Kirszbraun-Valentine theorem to metric spaces with a curvature bound in the sense of Alexandrov was established by Lang and Schroeder [START_REF] Lang | Kirszbraun's theorem and metric spaces of bounded curvature[END_REF], and the case of an arbitrary metric space X and a Hadamard manifold Y was discussed in [START_REF] Lang | Extensions of Lipschitz maps into Hadamard spaces[END_REF]. In the paper of Lang and Schlichenmaier [START_REF] Lang | Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions[END_REF], a general sufficient condition for the Lipschitz extension property of a metric pair (X, Y) is given.

The Lipschitz extension property for a pair of general Carnot groups (G 1 , G 2 ) with their canonical sub-Riemannian Carnot-Carathéodory metrics is studied for particular pairs of Carnot groups in [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF], [START_REF] Rigot | Lipschitz non-extension theorems into jet space Carnot groups[END_REF], [START_REF] Wenger | Lipschitz extensions into jet space Carnot groups[END_REF]. The situation is fully understood for the pair (R n , H m ), where G 1 = R n is the Euclidean space of dimension n and H m is the Heisenberg group of topological dimension 2m + 1. In this case, it is known that the pair (R n , H m ) has the Lipschitz extension property if and only if n ≤ m. The "only if" part of this equivalence is due to Balogh and Fässler [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF], whereas the "if" part is due to Wenger and Young [START_REF] Wenger | Lipschitz extensions into jet space Carnot groups[END_REF].

In this note, we shall study the Lipschitz extension property for pairs of Heisenberg groups (H n , H m ), a question that was posed in [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF]. Let us observe first that since H n contains isometric copies of R n , it follows from [START_REF] Balogh | Rectifiability and Lipschitz extensions into the Heisenberg group[END_REF] that if (H n , H m ) has the Lipschitz extension property, then we must necessarily have that n ≤ m.

The reason of this restriction on dimensions is as follows: It is a well-known fact of sub-Riemannian geometry that smooth Lipschitz mappings between sub-Riemannian spaces must satisfy the contact condition. This means that the horizontal distribution defining the sub-Riemannian metric must be preserved by the tangent map. This property gives an obstruction for the existence of smooth Lipschitz extensions. This principle can be transferred even for possibly non-smooth Lipschitz extensions, allowing us to prove the main result of this note: Theorem 1. If n ≥ 2k + 1, then the pair (H n , H 2k+1 ) does not have the Lipschitz extension property.

Since H 2k+1 is isometrically embedded in H n for 2k + 1 ≤ n it is enough to prove the statement for n = 2k+1. The idea of the proof is to exploit the consequences of the aforementioned contact condition. Indeed, this condition implies that a globally defined smooth contact map F : H 2k+1 → H 2k+1 is orientation preserving, which gives a major restriction on the nature of globally defined Lipschitz mappings. It allows us to define a map f : H → H 2k+1 from a subset H ⊂ H 2k+1 into H 2k+1 that does not have a globally defined Lipschitz extension. In order to prove the non-existence of such an extension we use results from mapping degree theory [START_REF] Lloyd | Degree Theory[END_REF], [START_REF] Fonseca | Degree Theory in Analysis and Applications[END_REF] and Pansu's result on almost everywhere differentiability [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisoméries des espaces symétriques de rang un[END_REF]. This argument does not work for the case of the pairs (H 2k , H 2k ) as smooth contact maps F : H 2k → H 2k are not necessarily orientation preserving. In this case the Lipschitz extension property is still open.

The paper is organized as follows: In Section 2 we recall the metric structure of the Heisenberg group and discuss regularity results of Lipschitz maps between Heisenberg groups to be used in the sequel. In Section 3 we prove a result on the computation of the mapping degree of maps between Heisenberg groups using the Pansu derivative. This result may be of independent interest. In Section 4 we give the proof of Theorem 1.

The Heisenberg metric and regularity of Lipschitz maps between Heisenberg groups

We recall background results on the geometry of the Heisenberg group and preliminary results on the differentiability properties of Lipschitz maps between Heisenberg groups.

The model of the Heisenberg group H n used in this paper is described as follows:

H n := R n × R n × R, endowed with the group operation (x, y, t) * (x ′ , y ′ , t ′ ) = (x + x ′ , y + y ′ , t + t ′ + 2(y • x ′ -x • y ′ )),
for (x, y, t), (x ′ , y ′ , t ′ ) ∈ H n . Here, x • y stands for the usual scalar product of vectors x, y ∈ R n .

It is easy to check that the vector fields A Lipschitz curve γ :

X i = ∂ ∂x i + 2y i ∂ ∂t , Y i = ∂ ∂y i -2x i ∂ ∂t are left invariant for i = 1, . . . , n. For p ∈ H n the space H p = span{X i , Y i : i = 1, . . . ,
[0, 1] → H n is said to be horizontal if γ ′ (t) ∈ H γ(t) for a.e. t ∈ [0, 1]
. For p, q ∈ H n we denote by d cc (p, q) the sub-Riemannian Carnot-Carathéodory distance between p and q defined as

d cc (p, q) := inf 1 0 |γ ′ (t)|dt,
where the inf is taken over all horizontal curves connecting p and q. Such horizontal curves exist between any pair of points p, q ∈ H n . This follows from the fact that the horizontal distribution is bracket-generating. For an explicit computation of the geodesics and the exact formulae of the metric we refer to [START_REF] Monti | Some properties of Carnot-Carathéodory balls in the Heisenberg group[END_REF] (see also [START_REF] Capogna | An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem[END_REF] or [START_REF] Ambrosio | Optimal mass transportation in the Heisenberg group[END_REF]).

It is easy to check that d cc is invariant under left translation. Moreover, it is compatible with the homogeneous dilations δ s : H n → H n defined by δ s (x, y, t) = (sx, sy, s 2 t), i.e. d cc (δ s (p), δ s (q)) = sd cc (p, q) for all s > 0 and p, q ∈ H n .

In this paper the metric on H n is understood to be the above defined Carnot-Carathéodory metric d cc . However the main result of the paper remains valid for any metric d on H n that is bi-Lipschitz equivalent to d cc .

It is easy to see that a smooth Lipschitz map f : H n → H n has the property that its tangent map preserves the horizontal bundle H . Since the horizontal bundle is the kernel of the one-form

θ = dt -2 2k+1 i=1 (y i dx i -x i dy i ),
we obtain that there exists a real valued smooth function λ such that f * θ = λ • θ.

If f is merely Lipschitz with no additional smoothness assumptions, then λ is only an L ∞ function and the equality f * θ = λ • θ holds almost everywhere. This follows from the following differentiability result:

A Lipschitz map f : H n → H n is called P-differentiable at a point p ∈ H n if there exists a group homomorphism P( f )(p) : H n → H n such that (1) lim s→0 δ 1/s ( f (p)) -1 * f (p * δ s q) = P( f )(p)(q)
for all q ∈ H n . Here * stands for the group operation, δ s is the homogeneous group dilation, and p -1 is the inverse to p in H n . Pansu proved in [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisoméries des espaces symétriques de rang un[END_REF] that any Lipschitz map is Pdifferentiable almost everywhere, moreover in a point of P-differentiability the limit ( 1) is uniform in q if q is taking its values in a compact set.

Let us define the singular set of a Lipschitz map f :

H n → H n as S = S 1 ∪ S 2 , S 1 := {p ∈ H n : f is not P-differentiable at p}, S 2 := {p ∈ H n : f is P-differentiable at x but P( f )(p) is not injective},
and the regular set as

R = H n \ S.
For d > 0 we denote by H d the d-dimensional Hausdorff measure with respect to the Heisenberg metric. By Pansu's theorem we have H 2n+2 (S 1 ) = 0, which implies by the Lipschitz continuity of f that also H 2n+2 ( f (S 1 )) = 0. Furthermore, we observe that if p ∈ S 2 , the metric Jacobian J f (p) of f at p vanishes, where

J f (p) := lim inf r→0+ H 2n+2 ( f (B(p, r))) H 2n+2 (B(p, r)) = 0 and B(p, r)
is the (open) ball with center p and radius r in the Heisenberg metric. For any measurable set A ⊂ H n we have

H 2n+2 ( f (A)) ≤ A J f (p) dH 2n+2 (p)
(see e.g. Chapter 6 of [START_REF] Capogna | An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem[END_REF]), so H 2n+2 ( f (S 2 )) = 0 and

H 2n+2 ( f (S)) = 0.

The mapping degree of Lipschitz maps in Heisenberg groups

In this section we will use the P-derivative in order to estimate the mapping degree of Lipschitz maps between Heisenberg groups. For background results on degree theory we refer to [START_REF] Lloyd | Degree Theory[END_REF], [START_REF] Fonseca | Degree Theory in Analysis and Applications[END_REF].

The considerations of this section are based on the fact that smooth contact maps f : H 2k+1 → H 2k+1 have a positive Jacobi determinant, and so they preserve orientation. To see this, let

θ = dt -2 2k+1 i=1 (y i dx i -x i dy i )
be the contact form in H 2k+1 . Observe that if we denote by v the volume form in H 2k+1 , then there exists a dimensional constant c(k) > 0 such that v = c(k) • θ ∧ (dθ) 2k+1 . If f is a smooth contact mapping then f * θ = λθ for some real valued smooth function λ. The relation between the Jacobi determinant of f and λ can be computed as follows:

f * v = c(k) • f * θ ∧ ( f * dθ) 2k+1 = c(k) • λθ ∧ (d(λθ)) 2k+1 = c(k) • λθ ∧ (dλ ∧ θ + λ dθ) 2k+1 = c(k) • λ 2k+2 θ ∧ (dθ) 2k+1 = λ 2k+2 v,
which implies that det D f = λ 2k+2 ≥ 0. In particular, it follows that group homomorphisms f : H 2k+1 → H 2k+1 are orientation preserving.

We want to extend this observation from the class of smooth Lipschitz maps to possibly non-smooth Lipschitz mappings. This is possible by a version of the Rademacher differentiability theorem due to Pansu [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisoméries des espaces symétriques de rang un[END_REF] according to which Lipschitz maps between Carnot groups are differentiable almost everywhere; and their tangent map satisfies the contact condition in the points of differentiability. Note that differentiability is understood here in terms of the underlying group operation and compatible homogeneous dilations as indicated in the previous section. The classical Rademacher differentiability theorem that is formulated in terms of vector addition and scalar multiplication as in the Euclidean space fails for Lipschitz maps between Carnot groups. An example showing this fact is due to Magnani [START_REF] Magnani | The coarea formula for real-valued Lipschitz maps on stratified groups[END_REF]. Nevertheless, Pansu's differentiability theorem can be used to estimate the mapping degree for maps between Carnot groups. The main result of this section is the following statement:

∈ f (U) \ [ f (∂U) ∪ f (S)].
In this case we shall prove that deg( f, U, p) > 0.

The proof is based on the fact that for group homomorphisms this property holds true as indicated above. This fact will be combined with a localization argument using linearization by the P-differential of f at regular points. The first step is to show that

if p ∈ f (U) \ [ f (∂U) ∪ f (S)], the pre-image f -1 (p) ⊂ R ∩ U is a finite set.
To prove this, assume by contradiction that there exists an infinite sequence (p n ) n of distinct points in R ∩ U such that f (p n ) = p for each n ∈ N. By passing to a subsequence we can assume that p n → p 0 for some p 0 ∈ U. By continuity of f we obtain that f (p 0 ) = p. Since p f (∂U) ∪ f (S), this implies that p 0 ∈ R ∩ U. We shall obtain the desired contradiction by showing that P( f )(p 0 ) is non-injective.

To see this, let us write p n = p 0 * δ t n q n for some q n ∈ ∂B(0, 1) and t n > 0, t n → 0. By passing to a further subsequence if necessary we can assume that q n → q for some q ∈ ∂B(0, 1). Since

f (p 0 ) = p = f (p n ) = f (p 0 * δ t n q n ), we have P( f )(p 0 )(q) = lim t n →0 δ 1/t n ( f (p 0 )) -1 * f (p 0 * δ t n q) = lim t n →0 δ 1/t n ( f (p 0 * δ t n q n )) -1 * f (p 0 * δ t n q) ,
but this last limit equals 0 by the Lipschitz continuity of f and the fact that q n → q. So P( f )(p 0 )(q) = 0, in contradiction to p 0 ∈ R.

By the first step we can assume that

f -1 (p) = {p 1 , . . . , p n } ⊂ R ∩ U.
By general properties of the mapping degree ( [START_REF] Lloyd | Degree Theory[END_REF], [START_REF] Fonseca | Degree Theory in Analysis and Applications[END_REF]) we have that

deg( f, U, p) = n i=1 deg( f, U i , p),
where U i are arbitrary small, non-intersecting open sets containing p i for i = 1, . . . , n.

In the following we shall prove that deg( f, U i , p) = 1 for every i. The idea is to construct a homotopy from f to its linearization P( f )(p i ) in U i with the property that the image of ∂U i under the homotopy does not meet the point p.

Let us fix an index i ∈ {1, . . . , n}. To simplify the notation, we can assume without loss of generality that p = p i = 0, U i = B(0, r) for some small r > 0 (to be chosen later), and P( f )(p i ) = P( f )(0) = A, where A : H 2k+1 → H 2k+1 is an injective homogeneous group homomorphism (i.e. commuting with dilations).

The P-differentiability of f at 0 now reads as

lim s→0 δ 1/s f (δ s q) = A(q),
and the convergence is uniform in q ∈ ∂B(0, 1). Equivalently, for every ǫ > 0 there exists an η > 0 such that (2) d H ( f (δ s q), A(δ s q)) < ǫ • s for all q ∈ ∂B(0, 1) and s < η. Furthermore, by the injectivity and homogeneity of A there exists an a > 0 such that

(3) d H (A(δ s q), 0) ≥ a • s
for all q ∈ ∂B(0, 1) and s > 0. Let us choose ǫ := a and U i = B(0, r) for some r < η.

Consider the homotopy H : [0, 1] × B(0, r) → H 2k+1 given by

H(s, q) = A(q) * δ s [(A(q)) -1 * f (q)].
Clearly H(0, •) = A and H(1, •) = f . For all s ∈ [0, 1] and q ∈ ∂B(0, r) we have

d H (H(s, q), 0) ≥ d H (A(q), 0) -d H (H(s, q), A(q)),
where d H (A(q), 0) ≥ a • r by ( 3) and d H (H(s, q), A(q)) = s d H ( f (q), A(q)) < ǫ • r by ( 2), so that

d H (H(s, q), 0) > 0.
By the homotopy invariance property of the mapping degree ( [START_REF] Lloyd | Degree Theory[END_REF], [START_REF] Fonseca | Degree Theory in Analysis and Applications[END_REF]) we obtain that

deg( f, U i , p) = deg( f, B(0, r), 0) = deg(A, B(0, r), 0) = 1,
finishing the proof.

Remark 3.

Observe that the statement of Proposition 2 no longer holds for maps f :

H 2k → H 2k .
Indeed, the mapping f :

H 2k → H 2k given by f (x, y, t) = (-x, y, -t) is a contact map since f * θ = -θ,
and it is orientation reversing: detD f ≡ -1. More generally, it is easy to see that every integer is the degree of some contact map between domains in H 2k .

A partially defined Lipschitz map in the Heisenberg group

In this section we define a Lipschitz mapping on a subset H of H n to itself which does not have a Lipschitz extension to H n in the case when n = 2k + 1.

The set H will be a topological 2n-sphere foliated by geodesics connecting the origin p 0 to a point p 1 on the vertical axis. For simplicity of notation we choose

p 1 = (0, 0, 4π) ∈ R n × R n × R.
To define H we use a parametrization Φ : S 2n-1 × [0, 2π] → H n , where

S 2n-1 := {(a, b) ∈ R n × R n : |a| 2 + |b| 2 = 1}
is the unit sphere in R 2n . The map Φ : S 2n-1 × [0, 2π] → H n will be given by

Φ((a, b), s) := sin(s) • a + (1 -cos(s)) • b, -(1 -cos(s)) • a + sin(s) • b, 2(s -sin(s)) for ((a, b), s) ∈ S 2n-1 × [0, 2π]. The hypersurface H := Φ(S 2n-1 × [0, 2π]
) is the boundary of a bounded domain Ω ⊂ H n and is foliated by the unit speed geodesics s → Φ((a, b), s) connecting p 0 and p 1 . For details we refer to the papers of Ambrosio-Rigot [START_REF] Ambrosio | Optimal mass transportation in the Heisenberg group[END_REF] and Monti [START_REF] Monti | Some properties of Carnot-Carathéodory balls in the Heisenberg group[END_REF].

We claim that to each point p ∈ H \ {p 0 , p 1 } there is a unique value ((a, b), s) ∈ S 2n-1 × (0, 2π) such that Φ((a, b), s) = p, whereas Φ -1 (p 0 ) = S 2n-1 × {0} and Φ -1 (p 1 ) = S 2n-1 × {2π}. To verify this, let (x, y, z) ∈ H \ {p 0 , p 1 }. Then z ∈ (0, 4π), and since the function φ : (0, 2π) → (0, 4π), φ(s) = 2(ssin(s)) is strictly increasing and onto, we get a unique value s ∈ (0, 2π) such that φ(s) = z. Furthermore, the system (4) sin(s)

• a + (1 -cos(s)) • b = x -(1 -cos(s)) • a + sin(s) • b = y
has a unique solution (a, b), proving the claim. Consider now the symplectic reflection R : R 2n → R 2n given by R(a, b) = (-a, b). Observe that R * ω = -ω for the standard symplectic form ω = n i=1 dx i ∧ dy i . We define the mapping f : H → H using R as follows:

If p ∈ {p 0 , p 1 }, set f (p) = p. If p ∈ H \ {p 0 , p 1 }, let ((a, b), s) = Φ -1 (p) and put f (p) = Φ(R(a, b), s).
Observe that the mapping interchanges the geodesics according to the reflection (a, b) → R(a, b) while it preserves the height s of points on H. Lemma 4. The mapping f : H → H defined above is a Lipschitz map with respect to the restriction of the sub-Riemannian Carnot-Carathéodory metric d cc on H in both the source and target space.

Proof. We have to find a constant C > 0 such that

d cc ( f (p), f (p ′ )) ≤ C d cc (p, p ′ ) for all p, p ′ ∈ H.
Let p, p ′ ∈ H. We can assume that p, p ′ ∈ H \ {p 0 , p 1 }, so that there are unique preimages ((a, b), s) = Φ -1 (p) and ((a ′ , b ′ ), s ′ ) = Φ -1 (p ′ ). Consider a third point q ∈ H lying on the same geodesic as p ′ but having the height of p, that is, q = Φ((a ′ , b ′ ), s). By the triangle inequality we have

d cc ( f (p), f (p ′ )) ≤ d cc ( f (p), f (q)) + d cc ( f (q), f (p ′ )).
Since q and p ′ are on the same geodesic, the same is true for f (q) and f (p ′ ). It follows that

d cc ( f (q), f (p ′ )) = |s -s ′ | = |d cc (p 0 , p) -d cc (p 0 , p ′ )| ≤ d cc (p, p ′ ).
Hence we obtain [START_REF] Kirszbraun | Über die zusammenziehende und Lipschitzsche Transformationen[END_REF]

d cc ( f (p), f (p ′ )) ≤ d cc ( f (p), f (q)) + d cc (p, p ′ ).
We claim that there is a constant

C ′ > 0 such that (6) d cc ( f (p), f (q)) ≤ C ′ d cc (p, q).
Here we will use in an essential way that p and q have the same heights. In order to verify the above estimate we shall consider another metric d K that is bi-Lipschitz equivalent to d cc and prove that

(7) d K ( f (p), f (q)) = d K (p, q).
The estimate [START_REF] Lang | Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions[END_REF] follows from [START_REF] Lang | Extensions of Lipschitz maps into Hadamard spaces[END_REF] and the bi-Lipschitz equivalence of d cc and d K .

For two points (x, y, t), (x ′ , y ′ , t ′ ) ∈ H n the Korányi gauge metric d K is given by the expression

d K ((x, y, t), (x ′ , y ′ , t ′ )) = (|x -x ′ | 2 + |y -y ′ | 2 ) 2 + |t -t ′ + 2(x • y ′ -y • x ′ )| 2 1/4
, where | • | and • denote the Euclidean norm and scalar product in R n . It can be checked that this defines a metric indeed. This metric is no longer given as a sub-Riemannian metric but is bi-Lipschitz equivalent to d cc . We refer to [START_REF] Capogna | An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem[END_REF] for details.

To verify equality [START_REF] Lang | Extensions of Lipschitz maps into Hadamard spaces[END_REF] we shall use the explicit forms of p and q:

p = sin s • a + (1 -cos s) • b, -(1 -cos s) • a + sin s • b, 2(s -sin s) , q = sin s • a ′ + (1 -cos s) • b ′ , -(1 -cos s) • a ′ + sin s • b ′ , 2(s -sin s) ,
where s ∈ (0, 2π) and (a, b), (a ′ , b ′ ) ∈ S 2n-1 . The respective expressions for f (p) and f (q) are obtained by replacing a by -a as well as a ′ by -a ′ and leaving b and b ′ unchanged. Now a direct computation gives

(8) d K (p, q) = (2 -2 cos s) 1/2 (|a -a ′ | 2 + |b -b ′ | 2 ) 2 + 4|a • b ′ -b • a ′ | 2 1/4 .
To calculate d K ( f (p), f (q)), all we need to do is to replace on the right side of (8) a by -a and a ′ by -a ′ . After doing that, we see that its value will not change. This gives ( 7) and hence [START_REF] Lang | Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions[END_REF]. Notice that in this computation it was crucial that the two points p and q have the same heights. Otherwise, the second part of the right side of (8) will be different. Combining ( 5) and ( 6) we obtain

d cc ( f (p), f (p ′ )) ≤ C ′ d cc (p, q) + d cc (p, p ′ ).
As d cc (p, q) ≤ d cc (p, p ′ ) + d cc (p ′ , q) ≤ 2d cc (p, p ′ ), this gives

d cc ( f (p), f (p ′ )) ≤ (2C ′ + 1) d cc (p, p ′ ),
concluding the proof of the Lemma.

We are now ready to prove our main result about the existence of partially defined Lipschitz maps on H 2k+1 with no Lipschitz extension to the whole Heisenberg group H 2k+1 . This follows from the following: Theorem 5. If n = 2k + 1, then the mapping f : H → H as in Lemma 4 has no Lipschitz extension to the whole Heisenberg group, i.e. there is no Lipschitz map F : H n → H n such that F| H = f . Proof. By contradiction assume that there is a Lipschitz extension F : H n → H n such that F| H = f . Proposition 2 implies that if n = 2k + 1, then deg(U, F, p) ≥ 0 for any p F(∂U).

Let us choose U = Ω such that ∂U = H and let p be any point in the interior of U, for instance, take p = (0, 0, 2π). Since F is a reflection on the boundary we have deg(U, F, p) = -1, which gives a contradiction, proving the statement.

To check that deg(U, F, p) = -1 we use the fact that deg(U, F, p) = deg(U, F, p) for any continuous map F : U → R 2n+1 with the property that F∂U = f . We will use the relation (4) first to find an explicit formula for the mapping f : H → H in cartesian coordinates as [START_REF] Lloyd | Degree Theory[END_REF] f (x, y, t) =

          cos ϕ(t) • x + sin ϕ(t) • y sin ϕ(t) • x -cos ϕ(t) • y t          
, where ϕ : [0, 4π] → [0, 2π] is the inverse function of φ : [0, 2π] → [0, 4π], φ(s) = 2(ssin s).

Observe now that ϕ : (0, 4π) → (0, 2π) is a smooth function and formula [START_REF] Lloyd | Degree Theory[END_REF] itself in fact gives a continuous extension F : U → U that is smooth in U. The above formula indicates that F preserves horizontal planes. In horizontal planes the map looks like a reflection followed by a rotation where the rotation angle depends on the height of the horizontal plane.

Since deg(U, F, p) is independent on the choice of the extension we have deg(U, F, p) = deg(U, F, p). Because the mapping F is smooth in U we can calculate the degree by an explicit computation as deg(U, F, p) = sign det D F( F-1 (p)) = sign det D F(0, 0, 2π) = (-1) n .

In the case when n = 2k + 1 we obtain that deg(U, F, p) = -1, finishing the proof.

In conclusion, let us come back to the general problem of determining exactly the pairs of Heisenberg groups (H n , H m ) with the Lipschitz extension property. Consider the case n = 1. In [START_REF] Wenger | Lipschitz extensions into jet space Carnot groups[END_REF] it is shown that the pair (H 1 , H 3 ) does have the Lipschitz extension property. On the other hand, by Theorem 1 this is not case for (H 1 , H 1 ). Therefore it only remains to study the extension problem for the pair (H 1 , H 2 ). We believe that understanding the Lipschitz extension property in this particular situation will shed light also on the general case of pairs (H n , H m ).

Proposition 2 .

 2 Let k ≥ 0 be an integer, f : H 2k+1 → H 2k+1 a Lipschitz map, U ⊂ H 2k+1 an open set, and p ∈ H 2k+1 \ f (∂U). If p f (S), then deg( f, U, p) ≥ 0. Proof. Let us consider a (possibly) non-smooth Lipschitz map f : H 2k+1 → H 2k+1 as in the statement. There are two cases to consider: If p f (U), then deg( f, U, p) = 0 by the definition of the degree ([9], [4]) and we are done. Let us therefore assume that p

  n} is the horizontal plane at p. Varying p ∈ H n gives rise to a vector bundle H , the horizontal plane distribution. Let us choose the Riemannian metric on H which makes {X i , Y i : i = 1, . . . , n} orthonormal at each point p ∈ H n and denote the length of horizontal vectors v ∈ H p in this metric by |v|.

Acknowledgements:

The authors thank Jeremy T. Tyson, Stefan Wenger, Piotr Hajlasz and Enrico Le Donne for many stimulating conversations on the subject of the paper.

Z.M.B. is supported by MAnET Marie Curie Initial Training Network and the Swiss National Science Foundation, U.L. is supported by the Swiss National Science Foundation, P.P. is supported by MAnET Marie Curie Initial Training Network and Agence Nationale de la Recherche, ANR-10-BLAN 0116.


Emmanuel Wiernik
email: emmanuel.wiernik@ens.uvsq.fr

Hermann Nabi

Bruno Pannier

Sébastien Czernichow

Olivier Hanon

Tabassome Simon

Jean-Marc Simon

Frédérique Thomas

Cyril Ducolombier

Nicolas Danchin

Frédéric Limosin

Silla M Consoli

Cédric Lemogne

Perceived stress, sex and occupational status interact to increase the risk of future high blood pressure: the IPC cohort study

Keywords: Epidemiology, gender, hypertension, interaction, occupational status, prospective study, socioeconomic position, stress

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Although hypertension has several known risk factors such as obesity, smoking, excessive alcohol or salt intake, most patients with hypertension lend great importance to psychological stress in the regulation of blood pressure (BP) and in the need for taking antihypertensive drugs [START_REF] Marshall | Lay perspectives on hypertension and drug adherence: systematic review of qualitative research[END_REF]. The association between acute psychological stress and a transient BP elevation is well established [START_REF] Chida | Chronic psychosocial factors and acute physiological responses to laboratory-induced stress in healthy populations: a quantitative review of 30 years of investigations[END_REF]. However, epidemiological studies do not consistently show psychological stress to be associated with BP in the long-term [START_REF] Sparrenberger | Does psychosocial stress cause hypertension? A systematic review of observational studies[END_REF]. One possible reason for these conflicting results is that the moderating role of some factors has been overlooked. Here we aimed to test the hypothesis that the association between perceived stress and high BP might depend on gender or occupational status.

As regards gender, differences exist in levels of perceived stress [START_REF] Cohen | Who's Stressed? Distributions of Psychological Stress in the United States in Probability Samples from 1983, 2006, and 2009[END_REF], type of stressors [START_REF] Matud | Gender differences in stress and coping styles[END_REF], the way to report and to cope with stress [START_REF] Nolen-Hoeksema | Emotion regulation and psychopathology: the role of gender[END_REF], but also in the prevalence [START_REF] Godet-Mardirossian | Patterns of hypertension management in France (ENNS 2006-2007)[END_REF], pathophysiology [START_REF] Gudmundsdottir | Hypertension in women: latest findings and clinical implications[END_REF] (e.g. the contribution of the renin-angiotensin system [START_REF] Maric-Bilkan | Sex differences in hypertension: contribution of the renin-angiotensin system[END_REF]) and risk factors of hypertension (e.g. menopause [START_REF] Yanes | Postmenopausal hypertension[END_REF] or the use of contraceptives [START_REF] Pimenta | Hypertension in women[END_REF]). In addition, there is some evidence suggesting that psychological variables might be related to several cardiovascular outcomes to a different extent among men and women [START_REF] Mendes De Leon | Depression and risk of coronary heart disease in elderly men and women: New Haven EPESE, 1982-1991. Established Populations for the Epidemiologic Studies of the Elderly[END_REF][START_REF] Joukamaa | Mental disorders and cause-specific mortality[END_REF][START_REF] Williams | Depression and risk of heart failure among the elderly: a prospective communitybased study[END_REF][START_REF] Lemogne | Gender differences in the association between depressive mood and mortality: a 12year follow-up population-based study[END_REF][START_REF] Iso | Perceived mental stress and mortality from cardiovascular disease among Japanese men and women: the Japan Collaborative Cohort Study for Evaluation of Cancer Risk Sponsored by Monbusho (JACC Study)[END_REF][START_REF] André-Petersson | Social support at work and the risk of myocardial infarction and stroke in women and men[END_REF][START_REF] Toivanen | Is the impact of job control on stroke independent from socioeconomic status?: a large-scale study of the Swedish working population[END_REF][START_REF] Tsutsumi | Prospective Study on Occupational Stress and Risk of Stroke[END_REF][START_REF] Gilbert-Ouimet | Repeated exposure to effort-reward imbalance, increased blood pressure, and hypertension incidence among white-collar workers: effort-reward imbalance and blood pressure[END_REF]. Likewise, perceived stress might be associated with BP differently in men and women.

With regard to occupational status, a recent study conducted by our group showed that there was a significant interaction between perceived stress and occupational status in relation to BP. Precisely, we found that perceived stress was negatively associated with high BP among individuals of high occupational status but positively associated among those of low status or unemployed [START_REF] Wiernik | Occupational status moderates the association between current perceived stress and high blood pressure: evidence from the IPC cohort study[END_REF]. This finding is in line with previous evidence showing that job strain may relate to BP at work site [START_REF] Landsbergis | Lower socioeconomic status among men in relation to the association between job strain and blood pressure[END_REF] or to the risk of stroke [START_REF] Tsutsumi | Impact of occupational stress on stroke across occupational classes and genders[END_REF] differently across occupational categories. However, this previous study, as several others, was cross-sectional. Hence, it remains unknown whether this association could be replicated in studies with a prospective design.

The main aim of this study was to examine the longitudinal association between perceived stress and the subsequent occurrence of high BP, and to explore the potential moderating role of gender or occupational status on this association.

Methods

Participants

According to the longitudinal design of the study, our target population was composed of all subjects who had at least two health checkups at the "Investigations Préventives et Cliniques" (IPC) Center (Paris, France) from January 1996 to December 2011. This medical Center, which is subsidized by the French national health care system, offers all working and retired individuals and their families a free medical examination with a minimum interval ranging from one to five years. It carries out approximately 25,000 examinations per year for people living in the Paris area. Our target population was composed of all subjects who had at least two health checkups at the IPC Center in the period from January 1996 to December 2011, with a time interval between visits of 1 year or more. All clinical and biological parameters were evaluated on the same day at the examination. In the case of participants who benefited from more than two examinations, only data from the first and second examinations were considered. Eligibility criteria were: 30 years of age or more at the first visit (owing to the low prevalence of hypertension in younger individuals), able to fill out the study questionnaires, no missing data for selected variables (see below) and a normal BP at the first visit: a systolic BP <140 mmHg, a diastolic BP <90 mmHg [START_REF] Chobanian | Seventh Report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure[END_REF] and not using antihypertensive drugs. To minimize potential biases, individuals with a history of cardiovascular or renal disease, which may confound the association between perceived stress and BP, and those who reported using psychotropic drugs, which may blur this association [START_REF] Licht | Depression is associated with decreased blood pressure, but antidepressant use increases the risk for hypertension[END_REF], were not included. The IPC Center received authorization from a local ethics committee and from the "Comité National d'Informatique et des Libertés" to conduct these analyses. All subjects gave their informed consent at the time of each examination. The data were rendered anonymous before analysis.

Blood pressure and outcome

After a 10-minute rest period, supine brachial systolic and diastolic BP were measured 3 times by trained nurses in the right arm using an automated sphygmomanometer. A standard cuff size was used, but a large cuff was utilized if necessary. The mean of the last 2 measurements was considered as the BP value. The primary outcome of the present study was a high BP at the second visit, defined as having a systolic BP ≥140 mmHg or a diastolic BP ≥90 mmHg or using antihypertensive drugs (see below).

Psychological variables

Perceived stress was measured with the French version of the 4-item Perceived Stress Scale (PSS-4) [START_REF] Cohen | A global measure of perceived stress[END_REF][START_REF] Lesage | Psychometric properties of the French versions of the Perceived Stress Scale[END_REF]. Each item is rated on a 0 to 4 scale (see Text S1, Supplemental Digital Content 1). The PSS-4 total score ranges from 0 to 16 and has a one-factor structure and a satisfactory internal consistency (α=0.73). It measures the degree to which situations in one's life over the past month were appraised as stressful (e.g. "In the past month, how often have you felt it was difficult to control the important things in your life?"). In order to obtain meaningful odds ratios, the variable was rescaled using the difference between the 25 th and the 75 th percentile as the unit.

Since stress is linked to depressive mood that may in turn be associated with a lower BP [START_REF] Nabi | Trajectories of depressive episodes and hypertension over 24 years: the Whitehall II prospective cohort study[END_REF], depressive mood was included as a covariate. It was measured with a French 13-item questionnaire (QD2A, Questionnaire of Depression 2 nd version, Abridged) [START_REF] Pichot | Un questionnaire d'autoévaluation de la symptomatologie dépressive, le Questionnaire QD2: I. Construction, structure factorielle et propriétés métrologiques[END_REF][START_REF] Pichot | Un questionnaire d'autoévaluation de la symptomatologie dépressive, le questionnaire QD2. II: Forme abrégée QD2A[END_REF]. Building on previous questionnaires, this [START_REF] Joukamaa | Mental disorders and cause-specific mortality[END_REF]-item questionnaire was specifically designed for depression screening in community studies and has a high internal consistency (α=0.91). Participants had to give a yes/no answer to each item as regards their current emotional state (e.g. "I am disappointed and disgusted with myself," "I'm sad these days," "I feel hopeless about the future"). The number of "yes" answers is summed, a total score ≥7 indicating a high probability of major depression.

The QD2A has been found to predict suicide in the IPC Cohort Study [START_REF] Lemogne | Heart rate and completed suicide: evidence from the IPC cohort study[END_REF].

Occupational status

Occupational status was categorized in 5 classes: [START_REF] Marshall | Lay perspectives on hypertension and drug adherence: systematic review of qualitative research[END_REF] high (e.g. managers); [START_REF] Chida | Chronic psychosocial factors and acute physiological responses to laboratory-induced stress in healthy populations: a quantitative review of 30 years of investigations[END_REF] medium (e.g. clerks or first line supervisors); (3) low (e.g. blue collar workers); [START_REF] Cohen | Who's Stressed? Distributions of Psychological Stress in the United States in Probability Samples from 1983, 2006, and 2009[END_REF] unemployed participants (i.e. seeking employment); (5) participants without a paid occupation (e.g. housewives). Retired participants were assigned to their last occupational category. The distinction of three categories among working participants is standard among occupational cohorts examining the relationships between psychosocial variables and physical health outcomes [START_REF] Kivimäki | Job strain as a risk factor for coronary heart disease: a collaborative meta-analysis of individual participant data[END_REF].

Other variables

Other variables included time interval between visits, age, gender, living status (living alone or not), smoking status (non-smoker, ex-smoker, current smoker of 1-10 cigarettes/day, 11-20 cigarettes/day, >20 cigarettes/day), alcohol intake (0 glasses/week, 1-6 glasses/week for women or 1-13 glasses/week for men, 7-20 glasses/week for women or 14-27 glasses/week for men, >21 glasses/week for women or >28 glasses/week for men), and regular physical activity (i.e. estimated equivalent to at least one hour/day of walking). Personal history of cardiovascular or renal disease, and family history of hypertension were self-reported (yes, no), as well as current medications including diuretics, antihypertensive drugs (other than diuretics), medications "to sleep" or "for anxiety or depression." Among participants reporting taking diuretics, only those that reported doing this "to lower BP" were considered as taking an antihypertensive drug. Perceived health status was collected with a 10-point scale (with 10 considered to be "excellent health"). Body mass index (BMI) was calculated and categorized in 4 classes (<18.5; 18.5-24.9; 25-29.9; ≥30 kg/m 2 ). Resting heart rate (HR) was measured in beats per minute with a 10-cycle electrocardiogram (HR = 60 / RR interval in seconds) and fasting glycemia in mmol/L. Menopausal status was self-reported for women and categorized in 3 classes (not menopausal at second visit, menopausal at second visit only, already menopausal at first visit).

Statistical analysis

The dependent variable was high BP at follow-up (i.e. the second visit) and, according to the main aim of this study, our independent variables were those measured at baseline (i.e. the first visit). The relationship between perceived stress at baseline and the likelihood of having high BP at follow-up was examined using univariate and multivariate binary logistic regression models with odds ratio (OR) and 95% confidence interval (CI). Except for BP and BMI, all variables were analyzed as continuous when available as such. First, we tested the main effect of stress, as well as interactions between stress and gender and between stress and occupational status in relation to the risk of high BP at follow-up. The interactions were tested by including in the same model the two variables of interest (e.g. gender and perceived stress, separately) as well as their interaction term (i.e. gender by perceived stress). In the case of statistically significant interactions, analyses were stratified by gender or occupational status, including all other covariates. Statistical analysis was carried out with the PASW Statistics software (version 18.0, SPSS Inc., Chicago). S1 (see Supplemental Digital Content 3) compared the baseline characteristics of individuals who had only one health check-up at the IPC center to the characteristics of those who came twice. The final study population consisted of 19,766 participants (13,652 men and 6,114 women) with a mean age of 46.8±9.3 years. The mean perceived stress score was 3.6±2.9 with a 5-point difference between the 25 th and the 75 th percentile.

Results

Figure S1 (see Supplemental Digital Content 2) presents the flow chart of the study population selection and Table

After a mean follow-up of 5.8±2.1 years, 3,774 participants (19.1%) had high BP, including 417 with an antihypertensive drug (mean systolic / diastolic BP: 136±17 / 81±10 mmHg) and 3,357 without drug (mean systolic / diastolic BP: 149±11 / 88±9 mmHg). When men and women were included together, baseline perceived stress was not associated with greater risk of high BP at follow-up (OR [CI]: 1.00 [0.93-1.06]; p=0.86). However, there was a significant interaction between baseline perceived stress and gender (p=0.02) in relation to high BP at follow-up, but not between baseline perceived stress and occupational status (p=0.60). Thus, the association between baseline perceived stress and high BP at follow-up has been examined separately in men and women. Table 1 shows the characteristics of study participants.

Univariate analyses in men and women separately are displayed in Table S2 (see Supplemental Digital Content 4). Fully adjusted models in men and women are displayed in Table 2. Adjusting for all variables, the association between baseline perceived stress and high BP at follow-up was significant in women (OR [CI]: 1.20 In addition, we found a significant interaction between perceived stress and occupational status among women (p=0.02), suggesting that the association between baseline perceived stress and future high BP might be different across occupational categories in women. Adjusting for all other variables, baseline perceived stress was positively associated with future high BP among women of low or medium occupational status, but not among other occupational categories (Table 3 and Figure 1). The OR values among women of high, medium and low status suggested a possible linear increase of the association across occupational status. To examine whether this linear trend was significant, we restricted our analyses to women of high, medium or low occupational status and found a significant interaction between perceived stress and occupational status taken as a linear variable (p=0.005). Taking the use of antihypertensive drugs as an alternative endpoint, baseline perceived stress remained positively associated with future high BP among women of medium occupational status (OR [CI]: 1.74 [1.06-2.88]; p=0.03). For women of low occupational status, only 5 women were taking antihypertensive drugs at follow-up, thus preventing further statistical analyses.

Discussion

Summary of results

This prospective study aimed to examine the association between baseline perceived stress and the incidence of high BP, and to explore a potential moderating effect of gender or occupational status. After adjustment for all variables, current perceived stress was associated with future high BP in women only. In addition, the interaction between occupational status and perceived stress was significant in women.

In analyses stratified by occupational categories, perceived stress was positively associated with high BP among women of medium or low occupational status, with OR suggesting a linear increase of the risk.

Explanatory hypotheses

Many hypotheses could explain gender differences in the association of baseline perceived stress with the occurrence of high BP. First, men and women are not exposed to the same stressors [START_REF] Matud | Gender differences in stress and coping styles[END_REF]. Therefore, perceived stress in women may relate to stressors associated with high BP to a greater extent than in men. Second, women might have reported their emotional state, and thus perceived stress, more accurately than men for at least two reasons: a better ability to recognize this emotional state [START_REF] Barrett | Sex differences in emotional awareness[END_REF] and a higher tendency to disclose emotional state once recognized [START_REF] Tamres | Sex differences in coping behavior: a metaanalytic review and an examination of relative coping[END_REF]. A better accuracy in the estimation of stress might thus have led to a better accuracy when looking for an association with high BP. Third, there is strong evidence for gender differences in emotion regulation strategies: in particular, women exhibit higher levels of rumination, defined as a perseverative focus on one's negative emotions and the causes and consequences of them, leading to mentally reliving stressful events [START_REF] Nolen-Hoeksema | Emotion regulation and psychopathology: the role of gender[END_REF]. Rumination is associated with poor BP recovery after acute stress [START_REF] Radstaak | The role of affect and rumination in cardiovascular recovery from stress[END_REF] and people who tend toward greater rumination exhibit more BP reactivity to repeated mental stress tasks [START_REF] Johnson | The effect of trait rumination on adaptation to repeated stress[END_REF].

Therefore, a greater proneness to ruminate among women might explain why perceived stress may lead to higher levels of BP in the long run. In addition, women might also cope with stress with behaviors that were not adjusted for in the current analysis. For example, perceived stress is associated with more frequent consumption of sweets/fast foods in women especially [START_REF] Mikolajczyk | Food consumption frequency and perceived stress and depressive symptoms among students in three European countries[END_REF] and might be a barrier to engage in health-promoting behaviors like physical exercise or refraining from snacking [START_REF] Soffer | The role of stress in the relationships between gender and healthpromoting behaviours[END_REF].

Finally, at a neural level, the impact of acute stress on blood pressure depends on the interplay of inhibitory control from prefrontal regions with excitatory input from limbic regions [START_REF] Gianaros | A review of neuroimaging studies of stressor-evoked blood pressure reactivity: Emerging evidence for a brain-body pathway to coronary heart disease risk[END_REF]. During acute stress, men might respond with a more favorable cortical-limbic balance than women that may result in lower effects of stress on BP levels [START_REF] Wang | Gender difference in neural response to psychological stress[END_REF][START_REF] Domes | The neural correlates of sex differences in emotional reactivity and emotion regulation[END_REF].

Differences in the association of baseline perceived stress with the occurrence of high BP according to occupational status among women mirror the results of our previous cross-sectional study among men and women [START_REF] Wiernik | Occupational status moderates the association between current perceived stress and high blood pressure: evidence from the IPC cohort study[END_REF] as well as those from stress at work studies [START_REF] Landsbergis | Lower socioeconomic status among men in relation to the association between job strain and blood pressure[END_REF]. Occupational categories may differ in terms of exposure to workrelated stress and particularly to job strain, which combines high job demands with low control at work [START_REF] Kivimäki | Job strain as a risk factor for coronary heart disease: a collaborative meta-analysis of individual participant data[END_REF] and tends to be associated with both high BP and lower occupational category [START_REF] Landsbergis | Lower socioeconomic status among men in relation to the association between job strain and blood pressure[END_REF][START_REF] Rosenthal | Occupational stress and hypertension[END_REF]. Beyond job strain, perceived stress may also relate to exposure to occupational stressors that are specific to certain occupational categories (e.g., noise, cardiotoxic chemicals) and differently associated with the risk of hypertension [START_REF] Tomei | Occupational exposure to noise and the cardiovascular system: a meta-analysis[END_REF][START_REF] Poreba | Ambulatory blood pressure monitoring and structural changes in carotid arteries in normotensive workers occupationally exposed to lead[END_REF]. Participants with lower occupational status might also have been less likely to deal with stress with adaptive health behaviors (e.g., physical activity) and more with detrimental ones (e.g., alcohol consumption) [START_REF] Siegrist | Work stress and health risk behavior[END_REF]. Finally, lower socioeconomic status may be associated with lower probability to be diagnosed with hypertension or to take an antihypertensive drug once diagnosed [START_REF] Grotto | Hypertension and socioeconomic status[END_REF], even in countries with an equal access to care [START_REF] De Gaudemaris | Socioeconomic inequalities in hypertension prevalence and care: the IHPAF Study[END_REF][START_REF] Paulsen | Socioeconomic status influences blood pressure control despite equal access to care[END_REF]. This is consistent with the small number of women of low occupational status taking antihypertensive drugs at follow-up. However, it is not clear why the above-mentioned mechanisms should apply for women only. This is unlikely to be explained in terms of statistical power due to the greater number of men among each occupational categories in which the association was significant. First, according to the Karasek model, differences in the ratio between job demands and latitude decision across occupational categories might be greater in women than men [START_REF] Niedhammer | Exposition aux facteurs psychosociaux au travail du modèle de Karasek en France : étude méthodologique à l'aide de l'enquête nationale Sumer[END_REF]. Second, according to the Siegrist model, differences in the ratio between reward and efforts at work might also be greater in women [START_REF] Niedhammer | Effort-reward imbalance model and self-reported health: cross-sectional and prospective findings from the GAZEL cohort[END_REF]. Interestingly, effort-reward imbalance has recently been found to predict future high BP in women only [START_REF] Gilbert-Ouimet | Repeated exposure to effort-reward imbalance, increased blood pressure, and hypertension incidence among white-collar workers: effort-reward imbalance and blood pressure[END_REF]. According to these two well-defined, internationally recognized models, such discrepancies may explain why perceived stress might have a particular impact among women of lower status. Third, in addition to occupational stressors, women are more likely than men to be exposed to non-occupational stressors such as household tasks, child care, care of sick or elderly relatives [START_REF] Lundberg | The total workload of male and female white collar workers as related to age, occupational level, and number of children[END_REF]. Perceived stress in women of low occupational status might thus be more likely to overwhelm their coping resources owing to the cumulative effects of these two sources of stress. In addition, gender differences in coping strategies, including the above-mentioned proneness to rumination, might depend upon occupational status, so that these differences might be less marked in individuals of high status [START_REF] Narayanan | Stress in the workplace: a comparison of gender and occupations[END_REF][START_REF] Torkelson | The role of gender and job level in coping with occupational stress[END_REF]. In accordance with this hypothesis, a large cohort study recently found neuroticism (i.e. a personality trait associated with poor emotion regulation) to predict cardiovascular mortality in women of low socioeconomic status, but not in men regardless of their socioeconomic status [START_REF] Hagger-Johnson | Neuroticism and cardiovascular disease mortality: socioeconomic status modifies the risk in women (UK Health and Lifestyle Survey)[END_REF].

Strengths and limitations

To our knowledge, this is the largest study to address this question and the first one to show that gender moderates the association between perceived stress and high BP. Strengths of the study are its prospective design, a large sample size allowing subsample analyses and the wide set of covariates considered, including a measure of depressive mood. For women, the potential role of menopause was taken into account. Some limitations should also be acknowledged. First, BP was not measured at several successive consultations. However, high BP has been associated with cardiovascular mortality in the IPC cohort study [START_REF] Thomas | Cardiovascular risk as defined in the 2003 European blood pressure classification: the assessment of an additional predictive value of pulse pressure on mortality[END_REF], as well as in other cohorts [START_REF] Kelly | Hypertension subtype and risk of cardiovascular disease in Chinese adults[END_REF][START_REF] Dorjgochoo | Relation of blood pressure components and categories and all-cause, stroke and coronary heart disease mortality in urban Chinese women: a population-based prospective study[END_REF][START_REF] Takashima | Long-term risk of BP values above normal for cardiovascular mortality: a 24-year observation of Japanese aged 30 to 92 years[END_REF]. Furthermore, taking the use of antihypertensive drugs as an alternative endpoint, baseline perceived stress remained positively associated with future high BP among women of medium occupational status. Second, some potential confounders were not measured, such as diet, salt consumption, ethnicity, social support, and personality variables. Likewise, mental health was not fully explored and the QD2A does not allow making the diagnosis of clinical depression. As regards considered confounders, their measures might have been too crude or changes might have occurred during the follow-up in relation to perceived stress. However, these changes are unlikely to account for the prospective association of baseline perceived stress with high BP at follow-up, as this association remained significant even when adjusting for confounders measured at follow-up. Third, a large sample size ensures statistical power but not clinical significance because even small differences may reach statistical significance. Finally, the IPC cohort is not be representative of the general French population. Study recruitment was limited to the Paris area and two thirds of the participants were men, which potentially limits the generalizability of our results. Compared with Paris area inhabitants, individuals in the IPC cohort were less likely to live alone, and more likely to have a professional activity and a higher occupational status (http://www.recensement.insee.fr/home.action). In addition, the present participants had asked for two preventive medical examinations, and thus may presumably display increased interest in their own health. For instance, they were more likely to have a normal BMI and to be non-smokers than the individuals who had only one health check-up at the IPC center. However, the incidence of high BP in the present sample was consistent with the incidence of hypertension in France [START_REF] Meneton | High incidence of hypertension in middle-aged French adults in the late 1990s[END_REF]. Furthermore, these potential selection biases are unlikely to account for the relationships we found within the present sample, especially those characterized by the interactions of stress with gender and occupational status.

Perspectives

Although previous studies failed to establish stress as a risk factor for hypertension, the present results suggest that overlooking the moderating role of gender and occupational status might have blurred the relationship between stress and hypertension. From a clinical perspective, they suggest that perceived stress could be considered as a risk factor for hypertension in women of lower occupational status.

Further studies are warranted to elucidate the mechanisms of this association, as such knowledge may eventually inform prevention strategies.

[ 1 .

 1 03-1.38]; p=0.016). In sensitivity analyses, this association remained significant after further adjustment for menopausal status (OR [CI]: 1.20 [1.04-1.39]; p=0.01) or after adjustment for covariates measured at the follow-up visit instead of covariates measured at baseline (OR [CI]: 1.15 [1.01-1.32]; p=0.04). Similar results were also obtained if retired participants were not included (OR [CI]: 1.19 [1.02-1.40]; p=0.03).

Figure 1 .

 1 Figure 1. Association between baseline perceived stress and high BP (see text) at

Figure

  Figure S1. Flow chart of the study population selection

Table 1 . Characteristics of study participants.

 1

		Men	Women	p
		(n=13,652)	(n=6,114)
	CONTINUOUS VARIABLES *	Mean (SD)	Mean (SD)
	Age (years)	46.09 (8.87)	48.35 (10.01)	<0.001
	Perceived Stress (PSS-4)	3.38 (2.77)	4.18 (2.94)	<0.001
	Depressive mood (QD2A)	1.13 (2.07)	1.94 (2.64)	<0.001
	Perceived health status (10-point scale)	7.82 (1.54)	7.49 (1.69)	<0.001
	Systolic BP (mmHg)	122.82 (9.18)	119.00 (10.61) <0.001
	Diastolic BP (mmHg)	74.71 (7.13)	72.03 (7.67)	<0.001
	Heart rate (beats per minute)	60.70 (9.14)	64.31 (9.31)	<0.001
	Fasting glycemia (mmol/L)	5.42 (0.68)	5.15 (0.64)	<0.001
	Interval between visits (years)	5.89 (2.12)	5.66 (2.21)	<0.001
	DISCRETE VARIABLES *	N (%)	N (%)
	High BP at follow-up (see text)			<0.001
	No	10,937 (80.1)	5,055 (82.7)
	Yes	2,715 (19.9)	1,059 (17.3)
	Occupational status
	High	6,837 (50.1)	1,473 (24.1)	<0.001
	Medium	3,317 (24.3)	2,707 (44.3)
	Low	1,616 (11.8)	294 (4.8)
	Unemployed	1,803 (13.2)	923 (15.1)
	Unpaid occupation	79 (0.6)	717 (11.7)
	Living status			<0.001
	Living alone	2,762 (20.2)	1,810 (29.6)

* Unless otherwise specified, figures indicate baseline values

Table 2 . Associations between each variable at baseline and high BP (see text) at follow-up in fully adjusted models.

 2 OR is given per 5-point increment for the PSS-4 and per unit for the other continuous variables. † P<0.001; ‡ P<0.01; § P<0.05.

	INDEPENDENT VARIABLES	Men	Women
	CONTINUOUS VARIABLES *	OR [95%CI]	OR [95%CI]
	Age (years)	1.05 † [1.04-1.05] 1.06 † [1.05-1.07]
	Perceived Stress (PSS-4)	0.97 [0.88-1.07]	1.20 § [1.03-1.38]
	Depressive mood (QD2A)	1.02 [0.99-1.04]	0.97 [0.94-1.01]
	Perceived health status (10-point scale)	1.02 [0.99-1.05]	0.97 [0.92-1.01]
	Heart rate (beats per minute)	1.03 † [1.03-1.04] 1.03 † [1.02-1.04]
	Fasting glycemia (mmol/L)	1.01 † [1.00-1.01] 1.00 [1.00-1.01]
	Interval between visits (years)	1.09 † [1.06-1.12] 1.09 † [1.05-1.13]
	DISCRETE VARIABLES	OR [95%CI]	OR [95%CI]
	Occupational status
	High	Reference	Reference
	Medium	1.06 [0.95-1.18]	1.14 [0.95-1.37]
	Low	1.32 † [1.15-1.53] 1.41 § [1.00-1.99]
	Unemployed	1.01 [0.85-1.19]	0.91 [0.69-1.20]
	Unpaid occupation	1.51 [0.87-2.63]	1.19 [0.93-1.53]
	Living alone (vs. not living alone)	1.10 [0.98-1.24]	0.96 [0.81-1.13]
	BMI
	<18.5	0.65 [0.36-1.17]	0.75 [0.48-1.18]
	18.5-24.9	Reference	Reference
	25-29.9	1.57 † [1.44-1.73] 1.92 † [1.62-2.26]
	≥30	2.59 † [2.17-3.09] 2.59 † [1.98-3.37]
	Smoking status

*

Table 3 . Association between baseline perceived stress and high BP (see text) at follow-up across occupational categories in women. Odds ratios (OR) are given per 5-point increment of the 4-item Perceived Stress Scale score (ie, the difference between the 25th and the 75th percentile).

 3

	OCCUPATIONAL STATUS	OR [95%CI]
	High	0.89 (0.63-1.25)
	Medium	1.35 ‡ (1.09-1.67)
	Low	2.00 § (1.06-3.78)
	Unemployed	1.27 (0.83-1.94)
	Unpaid occupation	1.03 (0.69-1.52)
	CI, confidence interval.
	‡ P<0.01; § P<0.05.

Acknowledgments

We thank the Caisse Nationale d'Assurance Maladie des Travailleurs Salariés (CNAM-TS, France) and the Caisse Primaire d'Assurance Maladie de Paris (CPAM-P, France) for helping make this study possible.

Emmanuel Wiernik was supported by a grant from GESTES / Région Île-de-France

Conflicts of Interest

Czernichow has received speaker or consulting fees from Servier, Novo, HRA Pharma, AstraZeneca and Sanofi. Tabassome Simon has received unrestricted research grants from AstraZeneca, Daiichi-Sankyo, Eli-Lilly, Glaxo-Smith-Kline, MSD, Novartis, Pfizer, Sanofi-Aventis and Servier; speaker and consulting fees from AstraZeneca, Bayer-Schering, Eli-Lilly and Sanofi-Aventis. Nicolas Danchin has received research grants from AstraZeneca, Daiichi-Sankyo, Eli-Lilly, Glaxo-Smith-Kline, MSD, Novartis, Pfizer, Sanofi-Aventis, Servier and The Medicines Company; advisory panels or lecture fees from AstraZeneca, Bayer, Boehringer-Ingelheim, Daiichi-Sankyo, Eli-Lilly, Glaxo-Smith-Kline, Novo-Nordisk, Roche, Sanofi-Aventis and Servier. Frédéric Limosin had received fees from Bristol-Myers-Squibb, Euthérapie, Lundbeck and Otsuka Pharmaceuticals France. Silla M. Consoli had received fees from Pfizer, MSD, Servier, Gilead, Novo Nordisk, AstraZeneca, Pierre Fabre and Janssen. Cédric Lemogne has received grants from Pasteur Mutualité,

Text S1. English version of the 4-item Perceived Stress Scale (PSS-4)

Instructions

The questions in this scale ask you about your feelings and thoughts during the last month. In each case, please indicate with a check how often you felt or thought a certain way.

1. In the last month, how often have you felt that you were unable to control the important things in your life? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often 2. In the last month, how often have you felt confident about your ability to handle your personal problems? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often 3. In the last month, how often have you felt that things were going your way? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often 4. In the last month, how often have you felt difficulties were piling up so high that you could not overcome them? ___0=never ___1=almost never ___2=sometimes ___3=fairly often ___4=very often Scoring PSS-4 scores are obtained by reverse coding the positive items, e.g., 0=4, 1=3, 2=2, etc. and then summing across all 4 items. Items 2 and 3 are the positively stated items.