The case where N = S 1 is somehow special ; one may attempt to answer these questions by lifting the maps u ∈ X. Here is a strategy: given u ∈ W s,p (Ω; S 1 ), one may try to find some ϕ ∈ W s,p (Ω; R) such that u = e iϕ . Then, hopefully, the path t ∈ [0, 1] → e itϕ will connect continuously u 0 ≡ 1 to u.

Moreover, if ϕ j are smooth R-valued functions on Ω such that ϕ j → ϕ in W s,p , then, hopefully, the smooth maps e iϕ j converge to u in W s,p (Ω; S 1 ).

We are thus naturally led to the study of the mapping

W s,p (Ω) ∋ ψ → f (ψ)
for "reasonable" functions f (e.g., f (x) = e ix -1), where Ω is either a smooth bounded domain or Ω = R n and s ≥ 1.

In a forthcoming paper [12], we will apply Theorem 1 to settle the above mentioned questions about W s,p (Ω; S 1 ) when s ≥ 1.

Another motivation for analysing composition and products in fractional Sobolev spaces comes from the study of nonlinear evolution equations (e.g. Schrödinger equation) in H s spaces; see e.g. T. Kato [START_REF] Kato | On nonlinear Schrödinger equations. II. H s solutions and unconditional well-posedness[END_REF] and the references therein. In fact, the Appendix in [START_REF] Kato | On nonlinear Schrödinger equations. II. H s solutions and unconditional well-posedness[END_REF] contains a result which is a special case of the Runst-Sickel lemma about products: it coincides with Lemma 5 below when q = 2.

Remark 1. The reader may wonder why we impose the additional condition u ∈ W 1,sp . At least for the case we are interested in, i.e. f (x) = e ix -1, this condition is also necessary in order to conclude that f (ψ) ∈ W s,p (R n ).

Indeed, assume that ψ ∈ W s,p and (e iψ -1) ∈ W s,p . Then (e iψ -1) ∈ W s,p ∩ L ∞ =⇒ (e iψ -1) ∈ W 1,sp (by Gagliardo-Nirenberg, see Corollary 2 below). Therefore, ie iψ Dψ ∈ L sp , so that Dψ ∈ L sp . Thus ψ ∈ W 1,sp . Remark 2. There is a vast literature about composition, starting with the result of Moser [START_REF] Moser | A rapidly convergent iteration method and nonlinear differential equations[END_REF] asserting that f (ψ) ∈ W m,p when ψ ∈ W m,p ∩L ∞ , f ∈ R and m is an integer. (See the historical comments at the end of section V). Unfortunately, for the application we have in mind, the lifting ϕ of an arbitrary u ∈ W s,p (Ω; S 1 ) need not belong to L ∞ . However, if s ≥ 1 and if the lifting ϕ exists in W s,p (Ω; R), it must belong to W 1,sp , by the above remark.

Surprisingly, Theorem 1 is new, but it is closely related and implies two earlier results having a similar flavour; see Adams-Frazier [START_REF] Adams | Composition operators on potential spaces[END_REF] and Runst-Sickel [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], Theorem 1, p. 345, and Remark 1, p. 348. Remark 3. When s is an integer, the proof of Theorem 1 is very easy via the standard Gagliardo-Nirenberg inequality [START_REF] Nirenberg | On elliptic partial differential equations[END_REF] (e.g. W k,p ∩ L ∞ ⊂ W ℓ,q , with ℓ < k, ℓq = kp). When s > 1, s is not an integer, our proof is quite involved. The standard form of the Gagliardo-Nirenberg inequality (e.g. W s,p ∩ L ∞ ⊂ W σ,q , with σ < s, σq = sp) does not suffice. We rely on a "microscopic" improvement (due to T. Runst [START_REF] Runst | Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type[END_REF]) of the Gagliardo-Nirenberg inequality, in the Triebel-Lizorkin scale, namely W s,p ∩ L ∞ ⊂ F σ q,ν for every ν. We present in Section III a more general form of the Gagliardo-Nirenberg inequality due to Oru [START_REF] Oru | Rôle des oscillations dans quelques problèmes d'analyse non-linéaire[END_REF]; see also P. Gérard, Y. Meyer and F. Oru [START_REF] Gérard | Inégalités de Sobolev précisées[END_REF] for a special case. We combine this with an important estimate on products of functions in the Triebel-Lizorkin spaces, due to T. Runst and W. Sickel (see [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF] and Section IV).

It would be interesting to find a more elementary argument which avoids this excursion into the F s p,q scale. The paper is organized as follows. In Section II we recall the definition of the Triebel-Lizorkin spaces F s p,q , their connection with the classical function spaces and some results needed in the proof of Theorem 1. In Section III we recall the general form of the Gagliardo-Nirenberg inequality, due to Oru [START_REF] Oru | Rôle des oscillations dans quelques problèmes d'analyse non-linéaire[END_REF]. Section IV deals with the Runst-Sickel lemma. This beautiful result contains all the usual statements about products in fractional Sobolev spaces: e.g., it implies that if u, v ∈ W s,p ∩ L ∞ then uv ∈ W s,p ∩ L ∞ , and if s ≥ 1, then uDv ∈ W s-1,p . More consequences of the Runst-Sickel lemma are presented in Section VI. Theorem 1 is proved in Section V.

Plan

Section I. Introduction Section II. Triebel-Lizorkin spaces and maximal inequalities Section III. A microscopic improvement of the Gagliardo-Nirenberg inequality Section IV. The Runst-Sickel lemma Section V. Proof of Theorem 1 Section VI. More about products

II. Triebel-Lizorkin spaces and maximal inequalities

We start by recalling the Littlewood-Paley decomposition of temperate distributions. Let ψ 0 ∈ C ∞ 0 (R n ) be such that 0 ≤ ψ 0 ≤ 1, ψ 0 (ξ) = 1 for |ξ| ≤ 1, ψ 0 (ξ) = 0 for |ξ| ≥ 2. Set ψ j (ξ) = ψ 0 (2 -j ξ) -ψ 0 (2 -j+1 ξ), j ≥ 1, and ϕ j = F -1 (ψ j ), j ≥ 0. Thus [START_REF] Adams | Composition operators on potential spaces[END_REF] ϕ j (x) = 2 nj ϕ 0 (2 j x) -2 n(j-1) ϕ 0 (2 j-1 x), j ≥ 1, and

k≤j ϕ k (x) = 2 nj ϕ 0 (2 j x), j ≥ 0. For f ∈ S ′ , set f j = f ⋆ ϕ j . We have f = j≥0 f j in S ′ . Definition ([34], 2.3.1). For -∞ < s < ∞, 0 < p ≤ ∞, 0 < q ≤ ∞, set F s p,q = {f ∈ S ′ ; f F s p,q = 2 sj f j (x) ℓ q L p (R n ) < ∞}. (2)
For 0 < p < ∞ or p = q = ∞, these are the standard Triebel-Lizorkin spaces F s p,q [START_REF] Triebel | Theory of function spaces[END_REF]. We have added the ˜to avoid confusions in the exceptional cases where they do not coincide. When 0 < p < ∞, different choices of ψ 0 yield equivalent quasi-norms ( [START_REF] Triebel | Theory of function spaces[END_REF], 2.3.5). The usual function spaces are special cases of these Triebel-Lizorkin spaces ( [START_REF] Triebel | Theory of function spaces[END_REF]):

a) L p = F 0 p,2 , 1 < p < ∞; b) W m,p = F m p,2 , m = 1, 2, ..., 1 < p < ∞; c) W s,p = F s p,p , 0 < s < ∞, s non-integer, 1 ≤ p < ∞; d) L s,p = F s p,2 , s ∈ R, 1 < p < ∞; e) L ∞ ⊂ F 0 ∞,∞ , i.e., (3) sup
j,x |f j (x)| ≤ C f L ∞ .
In this list, when 1 ≤ p < ∞, 0 < s < ∞, s non-integer, the W s,p are the Sobolev-Slobodeckij spaces. An equivalent norm on these spaces may be obtained as follows: let

s = k + σ, k integer, 0 < σ < 1. Then (4) f p W s,p ∼ f p L p + D k f p L p + R n R n |D k f (x) -D k f (y)| p |x -y| n+σp dx dy ([ 34
], 2.6.1). These spaces also coincide with the Besov spaces B s p,p (recall that s is not an integer). We warn the reader that, for p = 2, the spaces W s,p do not coincide with the Bessel potential spaces L s,p .

We will often use the trivial fact that, for fixed s and p, the space F s p,q increases with q.

The following result is well-known:

Lemma 1 ([35]). Let 0 < s < ∞, 1 < p < ∞, 1 < q < ∞. For every j ≥ 0, let f j ∈ S ′ be such that supp F (f j ) ⊂ B 2 j+2 . Then (5) j f j F s p,q ≤ C 2 sj f j (x) ℓ q L p (R n ) .
In the H s -spaces (p = q = 2), this result is proved in [START_REF] Chemin | Fluides parfaits incompressibles[END_REF], p. 21. We postpone the proof of Lemma 1 after the discussion of some maximal inequalities. Recall that, for any f ∈ L 1 loc , the maximal function M f is defined by

M f (x) = sup r>0 1 |B r (x)| B r (x)
|f (y)|dy.

For t > 0, set, for ϕ : R n → R,

(6) ϕ t (x) = t -n ϕ(x/t), x ∈ R n .
We recall some classical inequalities Lemma 2. We have: a) ( [START_REF] Stein | Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF], p. 13) for 1 < p ≤ ∞ and any function f , [START_REF] Stein | Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF], p. 55) for 1 < p < ∞, 1 < q < ∞, and any sequence of function (f j ), ( 8) [START_REF] Stein | Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF], p. 57) for any fixed ϕ ∈ S and any function f ,

(7) M f L p ∼ f L p ; b) ([
M f j (x) ℓ q L p (R n ) ≤ C f j (x) ℓ q L p (R n ) ; c) ([
(9) |f ⋆ ϕ t (x)| ≤ C M f (x), ∀t > 0, ∀x ∈ R n .
By ( 1), ( 2) and ( 9) we obtain the following Corollary 1. For every f ∈ L 1 loc we have

(10) |f j (x)| ≤ C M f (x), j ≥ 0, x ∈ R n , (11)
| j≤k f j (x)| ≤ C M f (x), k ≥ 0, x ∈ R n .
We now return to the Proof of Lemma 1. With f = j f j , we have

f k =   j f j   k =   j≥k-3 f j   k = j≥k-3 (f j ) k . Therefore f F s p,q = 2 sk j≥k-3 (f j ) k (x) ℓ q L p (R n ) = k 2 sqk j≥k-3 (f j ) k (x) q 1/q L p (R n ) ≤ C k 2 sqk j≥k-3 (f j ) k (x) q (j -k + 4) 2q 1/q L p (R n )
, by the Hölder inequality with exponents q and q ′ = q q -1 applied to the inner sum. We obtain, using [START_REF] Brezis | Topology and Sobolev spaces[END_REF], that

f F s p,q ≤ C j k≤j+3 2 sqk (j -k + 4) 2q |M f j (x)| q 1/q L p (R n ) ≤ C j 2 sqj |M f j (x)| q 1/q L p (R n ) = C 2 sj M f j (x) ℓ q L p (R n ) (12)
.

The desired conclusion is a consequence of ( 8) and (12).

III. A "microscopic" improvement of the Gagliardo-Nirenberg inequality

The main result of this section is that, in the Gagliardo-Nirenberg type inequalities for the spaces F s p,q , there is a gain in the "microscopic" parameter q; this gain is also called sometimes "precised" or "improved" Sobolev inequalities. Let us explain what we mean. In the context of Besov spaces, a typical Gagliardo-Nirenberg inequality asserts that

B s p,r ∩ L ∞ ⊂ B s/2 2p,2r , for 0 < s < ∞, 0 < p < ∞, 0 < r ≤ ∞
(see, e.g. [START_REF] Runst | Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type[END_REF], Lemma 2, p. 331).

Here, the value 2r of the microscopic parameter is optimal in general. By contrast, in the scale of F -spaces we have, given 0

< s < ∞, 0 < p < ∞, 0 < r ≤ ∞, F s p,r ∩ L ∞ ⊂ F s/2 2p,q for every 0 < q ≤ ∞ ([31], Lemma 1, p. 329).
A more general version of this phenomenon, due to Oru [START_REF] Oru | Rôle des oscillations dans quelques problèmes d'analyse non-linéaire[END_REF], is the following. Let

-∞ < s 1 < s 2 < ∞, 0 < q 1 , q 2 ≤ ∞, 0 < p 1 , p 2 ≤ ∞, 0 < θ < 1, and define s = θs 1 + (1 -θ)s 2 1 p = θ p 1 + 1 -θ p 2 .
Lemma 3. Under the above hypotheses we have, for every 0 < q ≤ ∞,

(13) f F s p,q ≤ C f θ F s 1 p 1 ,q 1 f 1-θ F s 2 p 2 ,q 2
, where C depends on s i , p i , θ and q.

For the convenience of the reader, we reproduce the proof of Oru, since it is not yet published.

Before proving Lemma 3, we state some interesting consequences:

Corollary 2. We have a) for 0 ≤ s 1 < s 2 < ∞, 1 < p 1 < ∞, 1 < p 2 < ∞, s = θs 1 + (1 -θ)s 2 , 1 p = θ p 1 + 1 -θ p 2 , ( 14
) f W s,p ≤ C f θ W s 1 ,p 1 f 1-θ W s 2 ,p 2 ; b) ([31], Lemma 1, p. 329) for 0 < s < ∞, 1 < p < ∞, 0 < q ≤ ∞, (15)
f F θs p/θ,q ≤ C f θ W s,p f 1-θ L ∞ .
In particular, we have

c) for 0 < s < ∞, 1 < p < ∞, 0 < θ < 1, (16) f W θs,p/θ ≤ C f θ W s,p f 1-θ L ∞ .
Remark 4. Inequality ( 14) is a special case of (13), with q = 2 when s is an integer, q = p otherwise, and similarly for q 1 and q 2 . Inequality ( 15) is a consequence of (13) for s 1 = 0 θ replaced by 1 -θ, p 1 = q 1 = ∞, s 2 = s, q 2 = 2 if s is an integer, q 2 = p otherwise. Here one uses in addition the fact that 16) is a special case of (15). Remark 5. There is something intriguing about inequality [START_REF] Escobedo | Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions[END_REF]. It is trivial when s < 1 (with C = 1) if one takes the usual Gagliardo norm [START_REF] Bethuel | Density of smooth functions between two manifolds in Sobolev spaces[END_REF]. It is also straightforward when both s and θs are integers. We do not know any elementary (i.e., without the Littlewood-Paley machinery) proof when s = 1. It would be of interest to establish [START_REF] Escobedo | Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions[END_REF] with control of the constant C, in particular when s ր 1. In view of the results in [START_REF] Bourgain | Another look at Sobolev spaces, in Optimal Control and Partial Equations[END_REF], one may expect an inequality of the form

f F 0 ∞,∞ ≤ C f L ∞ (inequality (3) above). Finally, (
f W s/2,2p ≤ C(p)(1 -s) 1/2p f 1/2 W s,p f 1/2 L ∞ as s ր 1,
if we take the Gagliardo norms (4). Remark 6. Inequality (15) may be viewed as an improvement of ( 16), since for 0 < q < min{2, p/θ} we have F θs p/θ,q ⊂ W θs,p/θ , F θs p/θ,q =W θs,p/θ . This improvement seems microscopic, however in our situation it is magnified and it plays a central role. A similar (microscopic) improvement of the Sobolev embeddings in the framework of Lorentz spaces which is magnified by the Trudinger inequality is presented in [START_REF] Brezis | A note on limiting cases of Sobolev embeddings and convolution inequalities[END_REF], [START_REF] Brezis | Laser beams and limiting cases of Sobolev inequality[END_REF]. Remark 7. We call the attention of the reader to the fact that some inequalities à la Gagliardo-Nirenberg are wrong, e.g., W 1,1 ∩ L ∞ is not contained in W θ,1/θ for 0 < θ < 1; see [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Remark D.1.

We now turn to the proof of Lemma 3. It relies on the following inequality:

Lemma 4. Let -∞ < s 1 < s 2 < ∞, 0 < q < ∞, 0 < θ < 1, and set s = θs 1 + (1 -θ)s 2 .
Then for every sequence (a j ) we have

(17) 2 sj a j ℓ q ≤ C 2 s 1 j a j θ ℓ ∞ 2 s 2 j a j 1-θ ℓ ∞ .
Remark 8. A special case of ( 17) is implicit in the proof of Theorem 1, p. 328, in [START_REF] Runst | Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type[END_REF]. For similar inequalities, see also [START_REF] Triebel | Theory of function spaces[END_REF], Theorem 2.7.1 or [START_REF] Jawerth | Some observations on Besov and Lizorkin-Triebel spaces[END_REF].

Proof of Lemma 4. Let C 1 = sup 2 s 1 j |a j |, C 2 = sup 2 s 2 j |a j |, so that C 1 ≤ C 2 .
We may assume C 1 > 0. Since s 1 < s 2 , there is some

j 0 > 0 such that min C 1 2 s 1 j , C 2 s s 2 j =      C 1 2 s 1 j , j ≤ j 0 C 2 2 s 2 j , j > j 0 . Since C 1 2 s 1 j 0 ≤ C 2 2 s 2 j 0 and C 2 2 s 1 (j 0 +1) ≤ C 1 2 s 1 (j 0 +1) we find that (18) C 2 ∼ C 1 2 (s 2 -s 1 )j 0 . Therefore (19) 2 s 1 j a j θ ℓ ∞ 2 s 2 j a j 1-θ ℓ ∞ ∼ C 1 2 (s 2 -s 1 )j 0 (1-θ) .
On the other hand, we have

a j ≤ min C 1 2 s 1 j , C 2 s s 2 j , so that (20) a j ≤ C 1 2 s 1 j for 0 ≤ j ≤ j 0 , a j ≤ C 2 2 s 2 j for j > j 0 .
It then follows that

2 sj a j ℓ q ≤   j≤j 0 C q 1 2 (s-s 1 )jq + j>j 0 C q 2 2 (s-s 2 )jq   1/q ≤ C   j≤j 0 C q 1 2 (s-s 1 )jq + j>j 0 C q 1 2 -θ(s 2 -s 1 )jq+(s 2 -s 1 )j 0 q   1/q , so that 2 sj a j ℓ q ≤ C C 1 2 (s 2 -s 1 )j 0 (1-θ)   j≤j 0 2 -(1-θ)(s 2 -s 1 )(j 0 -j)q + j>j 0 2 -θ(s 2 -s 1 )(j-j 0 )q   1/q .
Finally, we find that (21) 2 sj a j ℓ q ≤ C C 1 2 (s 2 -s 1 )j 0 (1-θ) , and ( 17) follows from ( 19) and ( 21).

Proof of Lemma 3. Since a j ℓ ∞ ≤ a j ℓ q , 0 < q ≤ ∞, we find that the r.h.s. of ( 13) is

≥ C f θ F s 1 p 1 ,∞ f 1-θ F s 2 p 2 ,∞
.

On the other hand,

f F s p,∞ ≤ f F s p,q
, 0 < q < ∞. It therefore suffices to prove (13) in the special case 0 < q < ∞, q 1 = q 2 = ∞.

In this case, we have

f F s p,q = 2 s j f j (x) ℓ q L p (R n ) (22)
≤ (by ( 17))

≤ C 2 s 1 j f j (x) θ ℓ ∞ 2 s 2 j f j (x) 1-θ ℓ ∞ L p (R n )
.

Using the Hölder inequality, ( 22) yields

f F s p,q ≤ C 2 s 1 j f j (x) ℓ ∞ θ L p 1 (R n ) 2 s 2 j f j (x) ℓ ∞ 1-θ L p 2 (R n ) = C f θ F s 1 p 1 ,∞ f 1-θ F s 2 p 2 ,∞
.

The proof of Lemma 3 is complete.

Remark 9. While talking about microscoping improvements in the F -scale, we call the attention of the reader to the following "improved" Sobolev embedding: ([19] or [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], p. 31).

W s,p ֒→ F σ r,q for every 0 < q ≤ ∞ if 0 ≤ σ < s and 1 r = 1 p - s -σ n > 0 (see

IV. The Runst-Sickel lemma

For the convenience of the reader, we split the statement into two parts; the first one contains the fundamental estimate, the other one deals with the continuity of the product.

Let 0 < s < ∞, 1 < q < ∞, 1 < p 1 ≤ ∞, 1 < p 2 ≤ ∞, 1 < r 1 ≤ ∞, 1 < r 2 ≤ ∞ be such that (23) 0 < 1 p = 1 p 1 + 1 r 2 = 1 p 2 + 1 r 1 < 1.
Lemma 5 ([32], p. 345). We have, for f ∈ F s p 1 ,q ∩ L r 1 and g ∈ F s p 2 ,q ∩ L r 2 , (24)

f g F s p,q ≤ C M f (x) 2 sj g j (x) ℓ q L p (R n ) + M g(x) 2 sj f j (x) ℓ q L p (R n ) and (25) f g F s p,q ≤ C f F s p 1 ,q g L r 2 + g F s p 2,q f L r 1 .
Proof. We start by noting that (25) follows from [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF]. Indeed, using the Hölder inequality we find

M f (x) 2 sj g j (x) ℓ q L p (R n ) + M g(x) 2 sj f j (x) ℓ q L p (R n ) ≤ 2 sj g j (x) ℓ q L p 2 (R n ) M f (x) L r 1 (R n ) + 2 sj f j (x) ℓ q L p 1 (R n ) M g(x) L r 2 (R n ) ≤ C( f F s p 1 ,q g L r 2 + g F s p 2 ,q f L r 1 )
, by [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]. We turn to the proof of [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF]. It relies on Lemma 1 which is valid since 1 < p < ∞ and 1 < q < ∞. We have

f g = k G k + j F j ,
where

G k = j≤k f j g k , F j = k<j g k f j . Since supp F (F j ) ⊂ B 2 j+2 and supp F (G k ) ⊂ B 2 k+2 , Lemma 1 yields (26) f g F s p,q ≤ C(A + B), with A = 2 sk G k (x) ℓ q L p (R n ) , B = 2 sk F j (x) ℓ q L p (R n ) .
We estimate, e.g., A:

A = 2 sk   j≤k f j (x)   g k (x) ℓ q L p (R n ) (27)
≤ by ( 11)

C M f j (x) 2 sk g k (x) ℓ q L p (R n ) .
We obtain [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF] by combining ( 26), ( 27) and the similar estimate for B.

We state the continuity part of this result in the three possible situations:

Corollary 3. We have that: a) for 1 < q < ∞, 0 < s < ∞, 1 < p 1 < ∞, 1 < p 2 < ∞, 1 < r 1 < ∞, 1 < r 2 < ∞, 0 < 1 p = 1 p 1 + 1 r 2 = 1 p 2 + 1 r 1 < 1, the map F s p 1 ,q ∩ L r 1 × F s p 2 ,q ∩ L r 2 ∋ (f, g) → f g ∈ F s p,q is continuous; b) for 1 < q < ∞, 0 < s < ∞, 1 < p < ∞, if f ℓ → f in F s p,q , f ℓ L ∞ ≤ C g ℓ → g in F s p,q , g ℓ L ∞ ≤ C then f ℓ g ℓ → f g in F s p,q ; c) for 1 < q < ∞, 0 < s < ∞, 1 < p 1 < ∞, 1 < r < ∞, 1 < p < ∞ such that 1 p = 1 p 1 + 1 r , if f ℓ → f in F s p 1 ,q , f ℓ L ∞ ≤ C g ℓ → g in F s p,q ∩ L r , then f ℓ g ℓ → f g in F s
p,q . Proof. a) follows directly from [START_REF] Mizohata | Lectures on the Cauchy problem[END_REF]. Some care is needed when one of the r ′ j s is ∞. We treat, e.g., case c). It clearly suffices to prove the following two assertions:

(i) if f ℓ → 0 in F s p 1 ,q and f ℓ L ∞ ≤ C, then f ℓ g → 0 for each g ∈ F s p,q ∩ L r . (ii) if g ℓ → 0 in F s p,q ∩ L r , f ℓ F s p 1 ,q ≤ C, f ℓ L ∞ ≤ C, then f ℓ g ℓ → 0.
Assertion (ii) is clear from [START_REF] Mizohata | Lectures on the Cauchy problem[END_REF]. We prove (i) using [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF]. We have ( 28)

f ℓ g F s p,q ≤ C f ℓ F s p 1 ,q g L r + M f ℓ (x) 2 sj g j (x) ℓ q L p (R n ) ≤ o(1) + C M f ℓ (x) 2 sj g j (x) ℓ q L p (R n )
.

Finally, by [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], [START_REF] Stein | Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF], the Runst-Sickel Lemma 5 and Corollary 3c), we obtain that

f ′ (u)Du ∈ F s-1 p,p = W s-1,p and that if u ℓ → u in W s,p ∩ W 1,sp , then f ′ (u ℓ )Du ℓ → f ′ (u)Du in W s-1,p .
Remark 10. The same proof yields the following variant of Theorem 1.

Theorem 1". Assume 1 < s < ∞, s non integer, 1 < p < ∞, 1 < q < ∞. Then, for every f ∈ R, the map

F s p,q ∩ W 1,sp ∋ u → f (u) ∈ F s p,q
is well-defined and continuous.

Remark 11. There is a natural strategy for proving Theorem 1: assume, e.g., that 1 < s < 2 and try to prove that f ′ (u)Du ∈ W s-1,p . Set s = 1 + σ. On the one hand, we have Du ∈ W σ,p ∩ L (1+σ)p . On the other hand, since u ∈ W 1,(1+σ)p , we find that f ′ (u) ∈ W 1,(1+σ)p ∩ L ∞ . By the "standard" Gagliardo-Nirenberg inequality, we obtain f ′ (u) ∈ W σ, 1+σ σ p ∩ L ∞ . The conclusion of Theorem 1 would follow if we can prove that

(34) U ∈ W σ,p ∩ L (1+σ)p V ∈ W σ, 1+σ σ p ∩ L ∞ =⇒ U V ∈ W σ,p .
Using the Gagliardo norm (4), we have to estimate (35)

R n R n |U (x + h)V (x + h) -U (x)V (x)| p |h| n+σp dx dh ≤ C R n R n |V (x)| p |U (x + h) -U (x)| p |h| n+σp dx dh + R n R n |U (x)| p |V (x + h) -V (x)| p |h| n+σp dx dh ≤ C V p L ∞ U p W σ,p + R n R n |U (x)| p |V (x + h) -V (x)| p |h| n+σp dx dh .
It is natural to estimate the last integral in (34) using the Hölder inequality with exponents 1 + σ and 1 + σ σ . We find

U V p W σ,p ≤ C V p L ∞ U p W σ,p + V p W σ, 1+σ σ p R n R n |U (x)| (1+σ)p |h| n dx dh 1 1+σ
.

Unfortunately, the last integral diverges, but we are "close" to convergence. In fact, we suspect that (34) is wrong.

It is here that the microscopic improvement of the Gagliardo-Nirenberg inequality Lemma 3, combined with the Runst-Sickel Lemma 5, magically saves the proof. We make use, in an essential way, of the additional information that V = f ′ (u) ∈ F σ 1+σ σ p,p .

We conclude this section with a brief survey of earlier results dealing with composition. a) if 0 < s ≤ 1, 1 < p < ∞, f (0) = 0, f Lipschitz, then u ∈ W s,p =⇒ f (u) ∈ W s,p (trivial for s < 1; see [START_REF] Marcus | Nemytskij operators on Sobolev spaces[END_REF] and [START_REF] Marcus | Complete characterization of functions which act via superposition on Sobolev spaces[END_REF]

for s = 1); b) if s = n/p, 1 < p < ∞, f ∈ R, where m = s, if s is an integer [s] + 1, otherwise , then u ∈ W s,p =⇒ f (u) ∈ W s,p .
This result is explicitely stated in [START_REF] Brezis | Composition in fractional Sobolev spaces[END_REF]; G. Bourdaud has pointed out that it may also be derived from a result of T. Runst and W. Sickel, see p. 345 in [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], combined with a result in [START_REF] Jawerth | Some observations on Besov and Lizorkin-Triebel spaces[END_REF] which asserts that, when s = n/p, W s,p ֒→ F θs p/θ,q for 0 < θ < 1 and every 0 < q < ∞ (see Remark 9 above); [START_REF] Mizohata | Lectures on the Cauchy problem[END_REF] for p = 2 and [START_REF] Peetre | Interpolation of Lipschitz operators and metric spaces[END_REF] for the general case; d) if 1 < s < n/p, we have to impose additional restrictions on u. Indeed, if 1 + 1/p < s < n/p, the only C 2 f 's that act on W s,p are of the form f (t) = ct; see [START_REF] Dahlberg | A note on Sobolev spaces, in Harmonic Analysis in Euclidean Spaces[END_REF] for s integer and [START_REF] Runst | Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type[END_REF], Theorem 3.2, p. 319, for a general s. For 1 < s < n/p, it follows from Remark 1 in the Introduction that R does not act on W s,p , since [START_REF] Peetre | Interpolation of Lipschitz operators and metric spaces[END_REF], [START_REF] Escobedo | Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions[END_REF]; e) an improvement is that, for f as above and 0 < σ < 1 we have u ∈ W s,p ∩ W σ,sp/σ =⇒ f (u) ∈ W s,p ; see [START_REF] Brezis | Composition in fractional Sobolev spaces[END_REF]. This result implies the previous one, since W s,p ∩ L ∞ ֒→ W σ,sp/σ (by Corollary 2); f) a finer result asserts that, for f as above, we have u ∈ W s,p ∩ F 1 sp,q (with q ≤ 2 sufficiently small depending on s and p) =⇒ f (u) ∈ W s,p ; see [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], Theorem 1, p. 345. This hypothesis on u is weaker than the previous one, since W s,p ∩ W σ,sp/σ ֒→ F 1 sp,q for all q > 0, by Lemma 3. This result is contained in Theorem 1, since F 1 sp,q ֒→ W 1,sp = F 1 sp,2

c) if s > n/p, 1 < p < ∞, f (0) = 0 and f ∈ C m , then u ∈ W s,p =⇒ f (u) ∈ W s,p ; see
W s,p ⊂ W 1,sp . A standard additional condition on u is u ∈ L ∞ : if f (0) = 0 and f ∈ C m , then u ∈ W s,p ∩ L ∞ =⇒ f (u) ∈ W s,p ; see
as soon as q ≤ 2 (recall that F s p,q increases with q). However, when p ≤ 2 or 1 < s < 2, Runst and Sickel point out in Remark 1, p. 348 that the above smallness condition on q is precisely q ≤ 2. This means that Runst and Sickel had established Theorem 1 when p ≤ 2 or 1 < s < 2; g) in the framework of Bessel potential spaces L s,p = {f = G s ⋆ g ; g ∈ L p , Ĝs (ξ) = (1 + |ξ| 2 ) -s/2 } = F s p,2 , there are various similar results about composition, starting with [START_REF] Meyer | Régularité des solutions des équations aux dérivées partielles non linéaires[END_REF], [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF] when s > n/p, [START_REF] Rauch | Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension[END_REF], [START_REF] Alinhac | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF] and [START_REF] Chemin | Fluides parfaits incompressibles[END_REF] for H s ∩ L ∞ when s ≥ 1. The ultimate result for s ≥ 1 was obtained by Adams-Frazier in [START_REF] Adams | Composition operators on potential spaces[END_REF]: if 1 ≤ s < ∞, 1 < p < ∞, f ∈ R, then u ∈ L s,p ∩ L 1,sp =⇒ f (u) ∈ L s,p . This is a special case (q = 2) of Theorem 1" since L 1,sp = W 1,sp . h) Other questions concerning composition in Sobolev spaces have been investigated e.g. in [START_REF] Bourdaud | Le calcul fonctionnel dans les espaces de Sobolev[END_REF], [START_REF] Bourdaud | Fonctions qui opèrent sur les espaces de Sobolev[END_REF], [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF].

VI. More about products

In this last Section, we state some natural results about products which may be derived from the Runst-Sickel lemma.

Let 1 < p < ∞, 0 < s < ∞, 1 < r < ∞, 0 < θ < 1, 1 < t < ∞ be such that 1 r + θ t = 1 p .

Lemma 6. For f ∈ W s,t ∩ L ∞ , g ∈ W θs,p ∩ L r , we have f g ∈ W θs,p and (36)

f g W θs,p ≤ C f L ∞ g W θs,p + g L r f θ W s,t f 1-θ L ∞ .
In the special case s > 1, θ = s -1 s , we have r = sp and we obtain the following Corollary 4. If 1 < s < ∞, 1 < p < ∞ and f ∈ W s,p ∩ L ∞ , g ∈ W s-1,p ∩ L sp , then f g ∈ W s-1,p and (37)

f g W s-1,p ≤ C f L ∞ g W s-1,p + g L sp f 1-1/s W s,p f 1/s L ∞ .
In particular, if f, g ∈ W s,p ∩ L ∞ , then Dg ∈ W s-1,p ∩ L sp , so that Corollary 4 contains as a special case the following result Corollary 5 ([7], Lemma 2). If 1 < s < ∞, 1 < p < ∞ and f, g ∈ W s,p ∩ L ∞ , then f Dg ∈ W s-1,p . Remark 12. Clearly, Corollary 5 implies the well-known assertion that W s,p ∩ L ∞ is an algebra.

Proof of Lemma 6. Let q = 2 if θs is an integer, q = p otherwise. By [START_REF] Dahlberg | A note on Sobolev spaces, in Harmonic Analysis in Euclidean Spaces[END_REF], we find that f ∈ F θs t/θ,q and (38)

f F θs t/θ,q ≤ C f θ W s,t f 1-θ L ∞ .
From the Runst-Sickel lemma, we deduce that f g ∈ F θs p,q and f g W θs,p = f g F θs p ,q ≤ C f L ∞ g F θs p,q

+ g L r f F θs t/θ,q ≤ C f L ∞ g W θs,p + g L r f θ W s,t f 1-θ L ∞ .

Acknowledgements. The authors thank G. Bourdaud and F.Planchon for interesting discussions. The first author (H.B.) is partially supported by a European Grant ERB FMRX CT98 0201. He is also a member of the Institut Universitaire de France. Part of this work was done when the second author (P.M.) was visiting Rutgers University; he thanks the Mathematics Department for its invitation and hospitality.

Set

F ℓ (x) = M f ℓ (x) 2 sj g j (x) ℓ q .

Then clearly [START_REF] Peetre | Interpolation of Lipschitz operators and metric spaces[END_REF] |F ℓ (x)| ≤ C 2 sj g j (x) ℓ q ∈ L p .

On the other hand, F s p 1 ,q ֒→ L p 1 (see, e.g., [START_REF] Triebel | Theory of function spaces[END_REF], 2.3.2, or [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], Proposition 2.2.1, p. 29). It follows from the maximal inequality [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] that M f ℓ → 0 in L p 1 and, up to a subsequence, that M f ℓ → 0 a.e. Then (i) follows from ( 28) and ( 29) by dominated convergence.

V. Proof of Theorem 1

The conclusion is well-known when s is an integer (this uses the standard Gagliardo-Nirenberg inequalities).

Assume s non integer. Clearly, the map

is well-defined and continuous, since f (0) = 0, f is Lipschitz and W s,p ֒→ L p . Thus it suffices to prove that the map

is well-defined and continuous. With m = [s] + 1 ≥ 2, we obtain, using [START_REF] Chemin | Fluides parfaits incompressibles[END_REF], that the inclusion

is continuous. Applying Theorem 1 to the integer s = m -1 ≥ 1, we find that

On the other hand, we clearly have that

Using [START_REF] Runst | Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type[END_REF] and the Gagliardo-Nirenberg type inequality (15) (with q = p, s = m -1,

), we obtain [START_REF] Stein | Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF] if u ℓ → u in W s,p ∩ W 1,sp , then f ′ (u ℓ ) → f ′ (u) in F s-1 sp s-1 ,p and f ′ (u ℓ ) L ∞ ≤ C.


ON SOME QUESTIONS OF TOPOLOGY FOR S 1 -VALUED FRACTIONAL SOBOLEV SPACES HAIM BREZIS (1), (2) AND PETRU MIRONESCU (3)

I. Introduction

The purpose of this paper is to describe the homotopy classes (i.e., path-connected components) of the space W s,p (Ω; S 1 ). Here, 0 < s < ∞, 1 < p < ∞, Ω is a smooth, bounded, connected open set in R N and W s,p (Ω; S 1 ) = { u ∈ W s,p (Ω; S 1 ); |u| = 1 a.e.}.

Our main results are

Theorem 1. If sp < 2, then W s,p (Ω; S 1 ) is path-connected.

Theorem 2. If sp 2, then W s,p (Ω; S 1 ) and C 0 ( Ω; S 1 ) have the same homotopy classes in the sense of [START_REF] Brezis | Topology and Sobolev spaces[END_REF]. More precisely: a) each u ∈ W s,p (Ω; S 1 ) is W s,p -homotopic to some v ∈ C ∞ ( Ω; S 1 ); b) two maps u, v ∈ C ∞ ( Ω; S 1 ) are C 0 -homotopic if and only if they are W s,p -homotopic.

Here a simple consequence of the above results Corollary 1. If 0 < s < ∞, 1 < p < ∞ and Ω is simply connected, then W s,p (Ω; S 1 ) is path-connected. Indeed, when sp < 2 this is the content of Theorem 1. When sp 2, we use a) of Theorem 2 to connect u 1 , u 2 ∈ W s,p (Ω; S 1 ) to v 1 , v 2 ∈ C ∞ ( Ω; S 1 ); since Ω is simply connected, we may write v j = e iϕ j for ϕ j ∈ C ∞ ( Ω; R) and then we connect

v 1 to v 2 via e i [(1-t)ϕ 1 +tϕ 2 ] .
When M is a compact connected manifold, the study of the topology of W 1,p (Ω; M ) was initiated in Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF] (see also White [26] for some related questions). In particular, these authors proved Theorems 1 and 2 in the special case s = 1. The analysis of homotopy

Typeset by A M S-T E X 1 classes for an arbitrary manifold M and s = 1 was subsequently tackled by Hang -Lin [START_REF] Hang | Topology of Sobolev mappings II[END_REF]. The passage to W s,p introduces two additional difficulties: a) when s is not an integer, the W s,p norm is not "local"; b) when s 2 (or more generally s > 1 + 1 p ), gluing two maps in W s,p does not yield a map in W s,p .

In our proofs, we exploit in an essential way the fact that the target manifold is S 1 . (The case of a general target is widely open.) In particular, we use the existence of a lifting of W s,p unimodular maps when s 1 and sp 2 (see Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]). Another important tool is the following Composition Theorem (Brezis -Mironescu [START_REF] Brezis | composition and products in fractional Sobolev spaces[END_REF]). If f ∈ C ∞ (R; R) has bounded derivatives and s 1, then ϕ -→ f • ϕ is continuous from W s,p ∩ W 1,sp into W s,p . Remark 1. A very elegant and straightforward proof of this Composition Theorem has been given by V.Maz'ya and T.Shaposhnikova [START_REF] Maz | An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces[END_REF].

A related question is the description, when sp 2, of the homotopy classes of W s,p (Ω; S 1 ) in terms of lifting. Here is a partial result Theorem 3. We have a) if s 1, N 3, and 2 sp < N , then [u] s,p = {ue iϕ ; ϕ ∈ W s,p (Ω; R) ∩ W 1,sp (Ω; R)}; b) if sp N , then [u] s,p = {ue iϕ ; ϕ ∈ W s,p (Ω; R)}.

Theorem 3 is due to Rubinstein -Sternberg [START_REF] Rubinstein | Homotopy classification of minimizers of the Ginzurg-Landau energy and the existence of permanent currents[END_REF] in the special case where s = 1, p = 2 and Ω is the solid torus in R 3 . When 0 < s < 1, N 3 and 2 sp < N , there is no such simple description of [u] s,p . For instance, using the "non-lifting" results in Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], it is easy to see that [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF] s,p ⊃ = {e iϕ ; ϕ ∈ W s,p (Ω; R)}.

Here is an example: if

N = 3, Ω = B 1 , 0 < s < 1, 1 < p < ∞, 2 sp < 3, then a) u(x) = e 1/|x| α ∈ [1] s,p ;
b) there is no ϕ ∈ W s,p (B 1 ; R) such that u = e iϕ for α satisfying 3-sp p α < 3-sp sp . However, we conjecture the following result Conjecture 1. Assume that 0 < s < 1, 1 < p < ∞, N 3 and 2 sp < N . Then [u] s,p = u{e iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p . We will prove below (see Corollary 2) that "half" of Conjecture 1 holds, namely

[u] s,p ⊃ u{e iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p .
In a different but related direction, we establish some partial results concerning the density of C ∞ ( Ω; S 1 ) into W s,p (Ω; S 1 ).

Theorem 4. We have, for 0 < s < ∞, 1 < p < ∞:

a) if sp < 1, then C ∞ ( Ω; S 1 ) is dense in W s,p (Ω; S 1 ); b) if 1 sp < 2, N 2, then C ∞ ( Ω; S 1 ) is not dense in W s,p (Ω; S 1 ); c) if sp N , then C ∞ ( Ω; S 1 ) is dense in W s,p (Ω; S 1 ); d) if s 1 and sp 2, then C ∞ ( Ω; S 1 ) is dense in W s,p (Ω; S 1 ).
There is only one missing case for which we make the following

Conjecture 2. If 0 < s < 1, 1 < p < ∞, N 3, 2 sp < N , then C ∞ ( Ω; S 1 ) is dense in W s,p (Ω; S 1 ).
This problem is open even when Ω is a ball in R 3 . We will prove below the equivalence of Conjectures 1 and 2.

Parts of Theorem 4 were already known. Part a) is due to Escobedo [START_REF] Escobedo | Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions[END_REF]; so is part b), but in this case the idea goes back to Schoen -Uhlenbeck [START_REF] Schoen | Boundary regularity and the Dirichlet problem for harmonic maps[END_REF] (see also Bourgain -Brezis -Mironescu [START_REF] Bourgain | On the structure of the space H 1/2 with values into the circle[END_REF]). For s = 1, part c) is due to Schoen -Uhlenbeck [START_REF] Schoen | Boundary regularity and the Dirichlet problem for harmonic maps[END_REF]; their argument can be adapted to the general case (see, e.g., Brezis -Nirenberg [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF] or Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF]). The only new result is part d). The proof relies heavily on the Composition Theorem and Theorems 2 and 3. We do not know any direct proof of d). We also mention that for s = 1 and Ω = B 1 , Theorem 4 was established by Bethuel -Zheng [START_REF] Bethuel | Density of smooth functions between two manifolds in Sobolev spaces[END_REF]. For a general compact connected manifold M and for s = 1, the question of density of C ∞ ( Ω; M ) into W 1,p (Ω; M ) was settled by Bethuel [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF] and Hang -Lin [START_REF] Hang | Topology of Sobolev mappings II[END_REF]. Remark 2. In Theorems 2 and 4, one may replace Ω by a manifold with or without boundary. The statements are unchanged. However, the argument in the proof of Theorem 1 does not quite go through to the case of a manifold without boundary. Nevertheless, we make the following Conjecture 3. Let Ω be a manifold without boundary with dim Ω 2. Then W s,p (Ω; M ) is path-connected for every 0 < s < ∞, 1 < p < ∞ with sp < 2, and for every compact connected manifold M .

Note that the condition dim Ω 2 is necessary, since W s,p (S 1 ; S 1 ) is not path-connected when sp 1.

Finally, we investigate the local path-connectedness of W s,p (Ω; S 1 ). Our main result is

Theorem 5. Let 0 < s < ∞, 1 < p < ∞. Then W s,p (Ω; S 1 ) is locally path-connected.
Consequently, the homotophy classes coincide with the connected components and they are open and closed.

The heart of the matter in the proof is the following

Claim. Let 0 < s < ∞, 1 < p < ∞.
Then there is some δ > 0 such that, if ||u-1|| W s,p < δ, then u may be connected to 1 in W s,p .

As a consequence of Theorem 5, we have

Corollary 2. Let 0 < s < 1, 1 < p < ∞. Then [u] s,p ⊃ {ue iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p = u {e iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p .
Equality in Corollary 2 follows from the well-known fact that W s,p ∩ L ∞ is an algebra. The inclusion is a consequence of the fact that, clearly, we have Proof. By Corollary 2, we have

[u] s,p ⊃ u{e iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p .
We prove that the reverse inclusion follows from Conjecture 1. By Proposition 1 a) below, we may take u = 1. Let v ∈ [1] s,p . By Theorem 5, there is some

ε > 0 such that ||v -w|| W s,p < ε ⇒ w ∈ [1] s,p . Let (w n ) ⊂ C ∞ ( Ω; S 1 ) be such that w n → v in W s,p
and ||w n -v|| W s,p < ε. By Theorem 2 b), we obtain that w n and 1 are homotopic in C 0 ( Ω; S 1 ). Thus w n = e iϕ n for some globally defined smooth ϕ n . Hence

v ∈ {e iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p .
Conversely, assume that Conjecture 2 holds. Let u ∈ W s,p (Ω; S 1 ). By Theorem 2 a), there is some w ∈ C ∞ ( Ω; S 1 ) such that w ∈ [u] s,p . By Proposition 1 b), we have u w ∈

[1] s,p . Thus u w ∈ {e iϕ ; ϕ ∈ W s,p (Ω; R)} W s,p , so that clearly u w ∈ {e iϕ ; ϕ ∈ C ∞ ( Ω; R)} W s,p . Finally, u ∈ {we iϕ ; ϕ ∈ C ∞ ( Ω; R)} W s,p
, i.e. u may be approximated by smooth maps.

In the same vein, we raise the following Open Problem 1. Let Ω be a manifold with or without boundary. Is W s,p (Ω; M ) locally path-connected for every s, p and every compact manifold M ?

The case s = 1 can be settled using the methods of Hang -Lin [START_REF] Hang | Topology of Sobolev mappings II[END_REF]. We will return to this question in a subsequent work; see Brezis -Mironescu [11].

The reader who is looking for more open problems may also consider the following Open Problem 2. Let Ω ⊂ R 2 be a smooth bounded domain. Assume 0 < s < ∞, 1 < p < ∞ and 1 sp < 2 (this is the range where C ∞ ( Ω; S 1 ) is not dense in W s,p (Ω; S 1 )). Set R 0 = {u ∈ W s,p (Ω; S 1 ); u is smooth except a finite number of points}.

(Here, the number and location of singular points is left free). Is R 0 dense in W s,p (Ω; S 1 )?

Comment. R 0 is known to be dense in W s,p (Ω; S 1 ) in many cases, e.g.:

a) s = 1 and 1 p < 2; see Bethuel-Zheng [START_REF] Bethuel | Density of smooth functions between two manifolds in Sobolev spaces[END_REF] b) s = 1 -1/p and 2 < p < 3; see Bethuel [START_REF] Bethuel | Approximation in trace spaces defined between manifolds[END_REF] c) s = 1/2 and p = 2; see Rivière [START_REF] Rivière | Dense subsets of H 1/2 (S 2 ; S 1 )[END_REF].

The paper is organized as follows Lemma 1. Let 0 < σ < 1, 1 < p < ∞, σp < 1. Then any v ∈ W σ,p (∂ω; S 1 ) has an extension w ∈ W σ+1/p,p (ω; S 1 ).

The proof is given in Appendix A; see Lemma A.1. It relies heavily on the lifting results in Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF].

Returning to the proof of Case 2, with w given by Lemma 1, set

ũ =      u, in Ω w, in ω a, in R n \ (Ω ∪ ω)
.

Clearly, ũ ∈ W s,p loc (R N ; S 1 ) and ũ is constant outside some compact set. As in the proof of Theorem 6, we may use ũ to connect u to a, since once more we have sp < N .

Case 3: sp = 1, N 2
The idea is the same as in the previous case; however, there is an additional difficulty, since in the limiting case s = 1/p the trace theory is delicate -in particular, tr W 1/p,p = L p (unless p = 1). Instead of trace, we work with a notion of "good restriction" developed in Appendix B; when s = 1/2, p = 2, the space of functions in H 1/2 having 0 as good restriction on the boundary coincides with the space H 1/2 00 of Lions -Magenes [START_REF] Lions | Problèmes aux limites non homogènes[END_REF] (see Theorem 11.7, p. 72).

Our aim is to prove that any u ∈ W 1/p,p (Ω; S 1 ) can be connected to a constant a ∈ S 1 .

Step 1: we connect u ∈ W 1/p,p (Ω; S 1 ) to some u 1 ∈ W 1/p,p (Ω; S 1 ) having a good restriction on ∂Ω Let ε > 0 be such that the projection Π onto ∂Ω be well-defined and smooth in the set {x ∈ R N ; dist (x, ∂Ω) < 2ε)}. For 0 < δ < ε, set Σ δ = {x ∈ Ω; dist (x, ∂Ω) = δ}. By Fubini, for a.e. 0 < δ < ε , we have [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF] u| P δ ∈ W 1/p,p (Σ δ ) and

Σ δ Ω |u(x) -u(y)| p |x -y| N +1 dy ds x < ∞.
By Lemma B.5, this implies that u has a good restriction on Σ δ , and that Rest u| P δ = u| P δ a.e. on Σ δ . Let any 0 < δ < ε satisfying (1). For 0 < λ < δ, let Ψ λ be the smooth inverse of Π| P λ : Σ λ → ∂Ω. Let also Ω λ = {x ∈ Ω; dist (x, ∂Ω) > λ}. Consider a continuous family of diffeomorphisms Φ t : Ω → Ω tδ , 0 t 1, such that Φ 0 = id and Φ

t | ∂Ω = Ψ tδ . Then t → u • Φ t is a homotopy in W 1/p,p . Moreover, if u t = u • Φ t , then u 0 = u and u 1 | ∂Ω = u| P δ • Ψ δ | ∂Ω . By (1)
, u 1 has a good restriction on ∂Ω.

Step 2: we extend

u 1 to R N Let ω = {x ∈ R N \ Ω; dist (x; ∂Ω) < ε}. As in Case 2, we fix some a ∈ S 1 and set v = u 1 , on ∂Ω a, on Λ .
Clearly, v ∈ W 1/p,p (∂ω), so that v ∈ W σ,p (∂ω) for 0 < σ < 1/p. We fix any 0 < σ < 1/p. By Lemma 1, there is some w ∈ W σ+1/p,p (ω; S 1 ) such that w| ∂ω = v. We define

ũ1 =      u 1 , in Ω w, in ω a, in R N \(Ω ∪ ω)
.

We claim that ũ1 ∈ W

1/p,p loc (R N ; S 1 ). Obviously, ũ ∈ W 1/p,p loc (R N \Ω). It remains to check that ũ1 ∈ W 1/p,p (Ω ∪ ω). This is a consequence of Lemma 2. Let 0 < s < 1, 1 < p < ∞, sp 1 and ρ > s. Let u 1 ∈ W s,p (Ω) and w ∈ W ρ,p (ω). Assume that u 1 has a good restriction Rest u 1 | ∂Ω on ∂Ω and that tr w| ∂Ω = Rest u 1 | ∂Ω . Then the map u 1 , in Ω w, in ω belongs to W s,p (Ω ∪ ω).
Clearly, in the proof of Lemma 2 it suffices to consider the case of a flat boundary. When Ω = (-1, 1) N -1 × (0, 1) and ω = (-1, 1) N -1 × (-1, 0), the proof of Lemma 2 is presented in Appendix B; see Lemma B.4.

Returning to Case 3 and applying Lemma 2 with s = 1/p, ρ = σ + 1/p, we obtain that ũ1 ∈ W 1/p,p loc (R N ). As in the two previous cases, this means that u 1 is W 1/p,p -homotopic to a constant.

Case 4: 1 sp < 2, N = 1
In this case, Ω is an interval. Recall the following result proved in Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] (Theorem 1): if Ω is an interval and sp 1, then for each u ∈ W s,p (Ω; S 1 ) there is some ϕ ∈ W s,p (Ω; R) such that u = e iϕ . Recall also that, when sp N , then C ∞ (R; R) functions f with bounded derivatives operate on W s,p ; that is, the map ϕ → f •ϕ is continuous from W s,p into itself (see, e.g., Peetre [START_REF] Peetre | Interpolation of Lipschitz operators and metric spaces[END_REF] for sp > N , Runst -Sickel [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF], Corollary 2 and Remark 5 in Section 5.3.7 or Brezis -Mironescu [START_REF] Brezis | Composition in fractional Sobolev spaces[END_REF] when sp = N ; this is also a consequence of the Composition Theorem). By combining these two results, we find that the homotopy t → e i(1-t)ϕ connects u = e iϕ to 1.

The proof of Theorem 1 is complete.

III. Proof of Theorems 2 and 3

We start with some useful remarks. For u ∈ W s,p (Ω; S 1 ), let [u] s,p denote its homotopy class in W s,p .

Proposition 1. Let 0 < s < ∞, 1 < p < ∞. For u, v ∈ W s,p (Ω; S 1 ), we have a) u[v] s,p = [uv] s,p ; b) [u] s,p = [v] s,p ⇔ [uv] s,p = [1] s,p ; c) [u] s,p [v] s,p = [uv] s,p .
The proof relies on two well-known facts:

W s,p ∩ L ∞ is an algebra; moreover, if u n → u, v n → v in W s,p and ||u n || L ∞ C, ||v n || L ∞ C, then u n v n → uv in W s,p . Here is, for example, the proof of c) (using a)). Let first u 1 ∈ [u] s,p , v 1 ∈ [v] s,p . If U, V are homotopies connecting u 1 to u and v 1 to v, then U V connects u 1 v 1 to uv; thus [u] s,p [v] s,p ⊂ [uv] s,p . Conversely, if w ∈ [uv] s,p , then w ∈ u[v] s,p (by a)), so that wū ∈ [v] s,p . Therefore, w = u(wū) ∈ [u] s,p [v] s,p .
We next recall the degree theory for W s,p maps; see Brezis -Li -Mironescu -Nirenberg [START_REF] Brezis | Degree and Sobolev spaces[END_REF] for the general case, White [START_REF] Triebel | Theory of function spaces[END_REF] when s = 1 or Rubinstein -Sternberg [START_REF] Rivière | Dense subsets of H 1/2 (S 2 ; S 1 )[END_REF] for the space H 1 (Ω; S 1 ) and Ω the solid torus in R 3 . Let 0 < s < ∞, 1 < p < ∞ be such that sp 2. Let u ∈ W s,p (S 1 × Λ; S 1 ), where Λ is some open connected set in R k . Clearly, for a.e. λ ∈ Λ, u (•, λ) ∈ W s,p (S 1 ; S 1 ). For any such λ, u (•, λ) is continuous, so that it has a winding number (degree) deg u (•, λ) . The main result in [START_REF] Brezis | Degree and Sobolev spaces[END_REF] asserts that, if sp 2, then this degree is constant a.e. and stable under W s,p convergence.

In the particular case where s 1, there is a formula

deg (u(•, λ)) = 1 2π S 1 u (x, λ) ∧ ∂u ∂τ (x, λ) ds x , where u ∧ v = u 1 v 2 -u 2 v 1 .
It then follows that, if s 1 and sp 2, we have

deg (u| S 1 ×Λ ) = Λ S 1 u(x, λ) ∧ ∂u ∂τ (x, λ) ds x dλ.
Clearly, the above result extends to domains which are diffeomorphic to S 1 × Λ. In the sequel, we are interested in the following particular case: let Γ be a simple closed smooth curve in Ω and, for small ε > 0, let Γ ε be the ε-tubular neighborhood of Γ. We fix an orientation on Γ.

Let Φ : S 1 × B ε → Γ ε be a diffeomorphism such that Φ| S 1 ×{0} : S 1 × {0} → Γ be an orientation preserving diffeomorphism; here B ε is the ball of radius ε in R N -1 . Then we may define deg (u|

Γ ε ) = deg (u • Φ| S 1 ×B ε ); this integer is stable under W s,p convergence.
We now prove b) of Theorem 2, which we restate as

Proposition 2. Let 0 < s < ∞, 1 < p < ∞, sp 2. Let u, v ∈ C ∞ ( Ω; S 1 ). Then [u] s,p = [v] s,p if and only if u and v are C 0 -homotopic.
Proof. Using Proposition 1, we may assume v = 1. Suppose first that u ∈ C ∞ ( Ω; S 1 ) and 1 are C 0 -homotopic. Then u and 1 are W s,p -homotopic. Indeed, when s = 1, this is proved in Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF], Proposition A.1; however, their proof works without modification for any s. We sketch an alternative proof: since u and 1 are C 0 -homotopic, there is some

ϕ ∈ C ∞ ( Ω; R) such that u = e iϕ . Then t → e i (1-t) ϕ connects u to 1 in W s,p .
Conversely, assume that the smooth map u is W s,p -homotopic to 1. By continuity of the degree, we then have deg (u|

Γ ε ) = 0 for each Γ. Since u is smooth, we obtain 0 = deg (u| Γ ε ) = deg (u| Γ ) = 1 2π Γ u ∧ ∂u ∂τ ds.
Thus the closed form X = u ∧ Du has the property that Γ X • τ ds = 0 for any simple closed smooth curve Γ. By the general form of the Poincaré lemma, there is some ϕ ∈ C ∞ ( Ω; R) such that X = Dϕ. One may easily check that u = e i(ϕ+C) for some constant C. Then t → e i(1-t) (ϕ+C) connects u to 1 in C 0 ( Ω; S 1 ).

We now turn to the proof of the remaining assertions in Theorems 2 and 3.

Case 1: sp N, N 2

Step 1:

each u ∈ W s,p (Ω; S 1 ) can be connected to a smooth map v ∈ C ∞ ( Ω; S 1 )
This is proved in Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF], Proposition A.2, for s = 1 and p N ; their arguments apply to any s and any p such that sp N . The main idea originates in the paper Schoen -Uhlenbeck [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF]; see also Brezis -Nirenberg [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF], [START_REF] Brezis | Degree Theory and BMO, Part II: Compact manifolds with boundaries[END_REF].

Step 2: we have

[u] s,p = {ue iϕ ; ϕ ∈ W s,p (Ω; R)} Let ϕ ∈ W s,p (Ω; R). Then t -→ ue i(1-t)ϕ connects ue iϕ to u in W s,p . (Recall that, if f ∈ C ∞ (R; R)
has bounded derivatives and sp N , then the map ϕ → f • ϕ is continuous from W s,p into itself.) This proves "⊃". To prove the reverse inclusion, by Proposition 1, it suffices to show that [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF] s,p ⊂ {e iϕ ; ϕ ∈ W s,p (Ω; R) }.

Let v ∈ [1] s,p . For each x ∈ Ω, let B x ⊂ Ω be a ball containing x. We recall the following lifting result from Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] (Theorem 2): if U is simply connected in R N and sp N , then for each w ∈ W s,p (U ; S 1 ) there is some ψ ∈ W s,p (U ; R) such that w = e iψ . Thus, for each x ∈ Ω there is some ϕ x ∈ W s,p (B x ; R) such that v| B x = e iϕ x . Note that , in B x ∩ B y , we have ϕ x -ϕ y ∈ W s,p (B x ∩ B y ; 2πZ). Therefore, ϕ x -ϕ y ∈ V M O (B x ∩ B y ; 2πZ), since sp N . It then follows that ϕ x -ϕ y is constant a.e. on B x ∩ B y ; see Brezis -Nirenberg [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF], Section I.5.

By a standard continuation argument, we may thus define a (multi-valued) argument ϕ for v in the following way: fix some x 0 ∈ Ω. For any x ∈ Ω, let γ be a simple smooth path from x 0 to x. Then, for ε > 0 sufficiently small, there is a unique function

ϕ γ ∈ W s,p (γ ε ; R) such that v| γ ε = e iϕ γ and ϕ γ | B ε (x 0 ) = ϕ x 0 | B ε (x 0 ) ; here, γ ε is the ε-tubular neighborhood of γ. We then set ϕ| B ε (x) = ϕ γ | B ε (x) .
We actually claim that ϕ is single-valued. This follows from

Lemma 3. Assume that 0 < s < ∞, 1 < p < ∞, sp N, N 2. If w ∈ W s,p (S 1 × B 1 ; S 1 ) is such that deg (w| S 1 ×B 1 ) = 0, then there is some ψ ∈ W s,p (S 1 × B 1 ) such that w = e iψ .
Here, B 1 is the unit ball in R N -1 . The proof of Lemma 3 is presented in Appendix C; see Lemma C.1.

Returning to the claim that ϕ is single-valued, we have that deg (v| Γ ε ) = 0 for each Γ, since v ∈ [1] s,p . By Lemma 3, a standard argument implies that ϕ is single-valued.

The proof of Theorems 2 and 3 when sp N is complete.

Case 2: s 1, 1 < p < ∞, N 3, 2 sp < N Step 1: we have [u] s,p = {ue iϕ ; ϕ ∈ W s,p (Ω; R) ∩ W 1,sp (Ω; R)}
For "⊃", we use the Composition Theorem mentioned in the Introduction, which implies that t → ue i(1-t)ϕ connects ue iϕ to u in W s,p .

For "⊂" it suffices to prove that [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF]

s,p ⊂ {e iϕ ; ϕ ∈ W s,p (Ω; R) ∩ W 1,sp (Ω; R)}. We proceed as in Case 1, Step 2. Let v ∈ [1] s,p .
The corresponding lifting result we use is the following (see Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Lemma 4): if s 1, sp 2 and U is simply connected in R N , then for each w ∈ W s,p (U ; S 1 ) there is some ψ ∈ W s,p (U ; R) ∩ W 1,sp (U ; R) such that w = e iψ . As in Case 1, for each x there is some ϕ

x ∈ W s,p (B x ; R) ∩W 1,sp (B x ; R) such that v| B x = e iϕ x . Since ϕ x -ϕ y ∈ W 1,1 (B x ∩B y ; 2πZ),
we find that ϕ x -ϕ y is constant ae. on B x ∩ B y (see [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Theorem B.1.). These two ingredients allow the construction of a multi-valued phase ϕ ∈ W s,p ∩ W 1,sp for v. To prove that ϕ is actually single-valued, we rely on

Lemma 4. Assume that s 1, 1 < p < ∞, N 3, 2 sp < N . If w ∈ W s,p (S 1 × B 1 ; S 1 ) is such that deg (w| S 1 × B 1 ) = 0, then there is some ψ ∈ W s,p (S 1 × B 1 ; R) ∩ W 1,sp (S 1 × B 1 ; R) such that v = e iψ .
The proof of Lemma 4 is given in Appendix C; see Lemma C.2.

The proof of Step 1 is complete.

Step 2: assume s 1, 1 < p < ∞, sp 2; then, for each u ∈ W s,p (Ω; S 1 ), there is some [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Lemmas D.1 and D.2). Let ϕ ∈ W s,p (Ω; R) ∩ W 1,sp (Ω; R) be any solution of ∆ϕ = div X in Ω. By the Composition Theorem, we then have e -iϕ ∈ W s,p (Ω; S 1 ), and thus v = ue -iϕ ∈ W s,p (Ω; S 1 ). We claim that v ∈ C ∞ (Ω; S 1 ). Indeed, let B be any ball in Ω. Since s 1 and sp 2, there is some ψ ∈ W s,p (B; R) ∩ W 1,sp (B; R) such that u| B = e iψ . It then follows that X| B = Dψ. Thus ∆ϕ = ∆ψ in B, i.e., ψ -ϕ is harmonic in B. Since in B we have v = ue -iϕ = e i(ψ-ϕ) , we obtain that v ∈ C ∞ (B), so that the claim follows.

v ∈ W s,p (Ω; S 1 ) ∩ C ∞ (Ω; S 1 ) such that v ∈ [u] s,p Consider the form X = u ∧ Du. Then X ∈ W s-1,p (Ω) ∩ L sp (Ω) (see Bourgain -Brezis -Mironescu
Using Step 1 and the equality v = ue -iϕ , we obtain that v ∈ [u] s,p .

Step 3: for each u ∈ W s,p (Ω; S 1 ), there is some

w ∈ C ∞ ( Ω; S 1 ) such that w ∈ [u] s,p
In view of Step 2, it suffices to consider the case where u ∈ W s,p (Ω; S 1 ) ∩ C ∞ (Ω; S 1 ) . We use the same homotopy as in Step 1, Case 3, in the proof of Theorem 1: t → u • Φ t , where Φ t is a continuous family of diffeomorphisms Φ t : Ω → Ω tδ such that Φ 0 = id.

Clearly, v = u • Φ 1 ∈ C ∞ ( Ω; S 1 ).
The conclusions of Theorems 2 and 3 when s 1, 1 < p < ∞, N 3, 2 sp < N follow from Proposition 2 and Steps 1 and 3.

We now complete the proof of Theorem 2 with

Case 3: 0 < s < 1, 1 < p < ∞, N 3, 2 sp < N
In this case, all we have to prove is that, for each u ∈ W s,p (Ω; S 1 ) , there is some v ∈ C ∞ ( Ω; S 1 ) such that v ∈ [u] s,p . The ideas we use in the proof are essentially due to Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF] (see §1.3, "Filling" a hole).

We may assume that u is defined in a neighborhood O of Ω; this is done by extending u by reflections across the boundary of Ω-the extended map is still in W s,p since 0 < s < 1. We next define a good covering of Ω: let ε > 0 be small enough; for x ∈ R N , we set

C x N = {x + εl + (0, ε) N ; l ∈ Z N and x + εl + (0, ε) N ⊂ O}.
Define also C x j , j = 1, ..., N -1, by backward induction : C x j is the union of faces of cubes in C x j+1 . By Fubini, for a.e. x ∈ R N , we have u| C x j ∈ W s,p , j = 1, ..., N -1, in the following sense: since 1/p < s < 1, we have tr u| C x N -1 ∈ W s-1/p,p for all x. However, for a.e. x, we have the better property tr u| C x N -1 = u| C x N -1 ∈ W s,p . For any such x, we have tr u|

C x N -1 C x N -2
∈ W s-1/p,p , but once more for a.e. such x we have the better property

tr u| C x N -1 C x N -2 = u| C x N -2 ∈ W s,p
, and so on. (See Appendix E for a detailed discussion). We fix any x having the above property and we drop from now on the superscript x.

Step 1: we connect u to some smoother map u 1 Let k = [sp], so that 2 k N -1. Since u| C k ∈ W s,p and sp k, there is a neighborhood ω of C k in C k+1 and an extension ũ ∈ W s+1/p,p (ω; S 1 ) of u| C k . This extension is first obtained in each cube C ⊂ C k+1 starting from u| ∂C (see Brezis -Nirenberg [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF], Appendix 3, for the existence of such an extension). We next glue together all these extensions to obtain ũ; ũ belongs to W s+1/p,p since 1/p < s + 1/p < 1 + 1/p. Moreover, the explicit construction in [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF] yields some ũ ∈ C ∞ (ω\C k ). We next extend ũ to C k+1 in the following way: for each C ⊂ C k+1 , let Σ C be a convex smooth hypersurface in C ∩ ω. Since Σ C is k-dimensional and k 2, ũ| Σ C may be extended smoothly in the interior of Σ C as an S 1 -valued map (here, we use the fact that π k (S 1 ) = 0). Let ũC be such an extension. Then the map v = ũ, outside the Σ C 's ũC , inside Σ C belongs to W s+1/p,p (C k+1 ). To summarize, we have found some v ∈ W s+1/p,p (C k+1 ; S 1 ) such that v|

C k = u| C k .
Pick any s < s 1 < min {s + 1/p, 1} and let p 1 be such that s 1 p 1 = sp + 1 (note that 1 < p 1 < ∞). By Gagliardo -Nirenberg (see, e.g., Runst [START_REF] Runst | Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type[END_REF], Lemma 1, p.329 or Brezis -Mironescu [START_REF] Brezis | composition and products in fractional Sobolev spaces[END_REF], Corollary 3), we have

W s+1/p,p ∩ L ∞ ⊂ W s 1 ,p 1 . Thus v ∈ W s 1 ,p 1 (C k+1 ).
We complete the construction of the smoother map u 1 in the following way:

if k = N -1, then v is defined in C N and we set u 1 = v; if k < N -1, we extend v to C N with the help of Lemma 5. Let 0 < s 1 < ∞, 1 < p 1 < ∞, 1 < s 1 p 1 < N, [s 1 p 1 ] j < N . Then any v ∈ W s 1 ,p 1 (C j ; S 1 ) has an extension u 1 ∈ W s 1 ,p 1 (C N ; S 1 ) such that u 1 | C l ∈ W s 1 ,p 1 for l = j, ..., N -1.
When s 1 = 1, Lemma 5 is due to Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF], Section 1.3, "Filling" a hole; for the general case, see Lemma D.3 in Appendix D.

We summarize what we have done so far: if k = [sp], then there are some s 1 , p 1 such that s < s 1 < 1, 1 < p 1 < ∞, s 1 p 1 = sp + 1 and a map u 1 ∈ W s 1 ,p 1 (C N ; S 1 ) such that u 1 | C j ∈ W s 1 ,p 1 , j = k, ..., N -1 and u 1 | C k = u| C k . By Gagliardo -Nirenberg and the Sobolev embeddings, we have in particular u 1 | C j ∈ W s,p , j = k, ..., N -1. Finally, u and u 1 are W s,p -homotopic by

Lemma 6. Let 0 < s < 1, 1 < p < ∞, 1 < sp < N, [sp] j < N . If u| C l ∈ W s,p , u 1 | C l ∈ W s,
p , l = j, ..., N , and u| C j = u 1 | C j , then u and u 1 are W s,p -homotopic.

The case s = 1 is due to Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF]; the proof of Lemma 6 in the general case is presented in the Appendix D-see Lemma D.4.

Step 2: induction on [sp]

If k = [sp] = N -1, we have connected in the previous step u to u 1 ∈ W s 1 ,p 1 (C N ; S 1 ), where s < s 1 < 1, 1 < p 1 < ∞ and s 1 p 1 = sp + 1 N . Using Case 1 (i.e., sp N ) from this section, u 1 may be connected in W s 1 ,p 1 (and thus in W s,p , by Gagliardo -Nirenberg and the Sobolev embeddings) to some v ∈ C ∞ ( Ω; S 1 ). This case is complete.

If k = [sp] = N -2, then [s 1 p 1 ] = N -1.
By the previous case, u 1 can be connected in W s 1 ,p 1 (and thus in W s,p ) to some v ∈ C ∞ ( Ω; S 1 ). Clearly, the general case follows by induction.

The proof of Theorems 2 and 3 is complete.

We end this section with two simple consequences of the above proofs; these results supplement the description of the homotopy classes.

Corollary 4. Let 0 < s < ∞, 1 < p < ∞, sp 2, N 2. For u, v ∈ W s,p (Ω; S 1 ) , we have [u] s,p = [v] s,p ⇔ deg (u| Γ ε ) = deg (v| Γ ε ) for every Γ. Corollary 5. Let 0 < s 1 , s 2 < ∞, 1 < p 1 , p 2 < ∞, s 1 p 1 2, s 2 p 2 2, N 2. For u, v ∈ W s 1 ,p 1 (Ω; S 1 ) ∩ W s 2 ,p 2 (Ω; S 1 ) , we have [u] s 1 ,p 1 = [v] s 1 ,p 1 ⇔ [u] s 2 ,p 2 = [v] s 2 ,p 2 .
Clearly, Corollary 5 follows from Corollary 4. As for Corollary 4, let

u 1 , v 1 ∈ C ∞ ( Ω; S 1 ) be such that [u 1 ] s,p = [u] s,p and [v 1 ] s,p = [v] s,p . Then, by Theorem 2 b), (2) [u] s,p = [v] s,p ⇔ [u 1 ] s,p = [v 1 ] s,p ⇔ [u 1 ] C 0 = [v 1 ] C 0 ⇔ deg (u 1 | Γ ) = deg (v 1 | Γ ), ∀Γ.
Moreover, we have

(3) deg (u 1 | Γ ) = deg (v 1 | Γ ) ⇔ deg (u 1 | Γ ε ) = deg (v 1 | Γ ε ) ⇔ deg (u| Γ ε ) = deg (v| Γ ε ), ∀Γ,
by standard properties of the degree.

We obtain Corollary 4 by combining ( 2) and (3).

IV. Proof of Theorem 4

According to the discussion in the Introduction, we only have to prove part d). Let s 1, 1 < p < ∞, N 3, 2 sp < N . Let u ∈ W s,p (Ω; S 1 ) . By Theorem 2 a), there is some v ∈ C ∞ ( Ω; S 1 ) such that v ∈ [u] s,p . By Theorem 3 b), there is some 1,sp . By the Composition Theorem, the sequence of smooth maps (ve -iϕ n ) converges to u in W s,p (Ω; S 1 ) .

ϕ ∈ W s,p (Ω; R) ∩ W 1,sp (Ω; R) such that v = ue iϕ . Let (ϕ n ) ⊂ C ∞ ( Ω; R) be such that ϕ n → ϕ in W s,p ∩ W
The proof of Theorem 4 is complete.

V. Proof of Theorem 5

We start this section with a discussion on the stability of the degree: recall that if sp 2, then deg (u| Γ ε ) is well-defined and stable under W s,p convergence. However, while the condition sp 2 is optimal for the existence of the degree (see Brezis -Li -Mironescu -Nirenberg [START_REF] Brezis | Degree and Sobolev spaces[END_REF], Remark 1), the stability of the degree of W s,p maps holds under (the weaker assumption of) W s 1 ,p 1 convergence, where s 1 p 1 1. This property and Corollary 4 suggest the following generalization of Theorem 5

Theorem 7. Let 0 < s < ∞, 1 < p < ∞, 0 < s 1 < s, 1 < p 1 < ∞, 1 s 1 p 1 sp.
Then for each u ∈ W s,p (Ω; S 1 ) there is some δ > 0 such that {v ∈ W s,p (Ω; S 1 ) ; ||v -u|| W s 1 ,p 1 < δ} ⊂ [u] s,p . Note that W s,p (Ω; S 1 ) ⊂ W s 1 ,p 1 (Ω; S 1 ) , by Gagliardo -Nirenberg and the Sobolev embeddings, so that Theorem 5 follows from Theorem 7 when sp 2 (when sp < 2, there is nothing to prove, by Theorem 1).

Proof of Theorem 7

Step 1: reduction to special values of s, s 1 , p, p 1

We claim that it suffices to prove Theorem 7 when

(4) 0 < s 1 < s < 1 -(N -1)/p, 1 < p < ∞, 1 < p 1 < ∞, sp = 2, s 1 p 1 = 1, N 2.
Indeed, assume Theorem 7 proved for all the values of s, s 1 , p, p 1 satisfying (4). Let 0 < s 0 < ∞, 1 < p 0 < ∞, N 2 be such that s 0 p 0 2 (when N = 1 or s 0 p 0 < 2, there is nothing to prove). Let u ∈ W s 0 ,p 0 and let s, s 1 , p, p 1 satisfy (4) and the additional condition s < s 0 . By Gagliardo -Nirenberg and the Sobolev embeddings, there is some δ 0 > 0 such that [START_REF] Bourgain | On the structure of the space H 1/2 with values into the circle[END_REF] M ={v ∈ W s 0 ,p 0 (Ω; S 1 ) ; ||v -u|| W s 0 ,p 0 < δ 0 } ⊂ {v ∈ W s,p (Ω; S 1 ) ; ||v -u|| W s 1 ,p 1 < δ}.

By the special case of Theorem 7, we have v ∈ M ⇒ v ∈ [u] s,p . By Corollary 5, we obtain M ⊂ [u] s 0 ,p 0 , i.e., [u] s 0 ,p 0 is open.

In conclusion, it suffices to prove Theorem 7 under assumption (4). Moreover, by Proposition 1 we may assume u = 1.

Step 2: construction of a good covering We fix a small neighborhood O of Ω. By reflections across the boundary of Ω, we may associate to each u ∈ W s,p (Ω; S 1 ) an extension ũ ∈ W s,p (O; S 1 ) satisfying [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] ||ũ -

ṽ|| W s,p (O) C 1 ||u -v|| W s,p (Ω)
and ( 7)

||ũ -ṽ|| W s 1 ,p 1 (O) C 1 ||u -v|| W s 1 ,p 1 (Ω) .
In this section, C 1 , C 2 , ... denote constants independent of u, v, ....

We fix some small ε > 0. By Lemma E.2 in Appendix E, for each v ∈ W s,p (Ω; S 1 ) there is some x ∈ R N (depending possibly on v) such that the covering C x N has the properties [START_REF] Brezis | Degree and Sobolev spaces[END_REF] v| C x j ∈ W s,p , j = 1, ..., N -1 and

(9) ||v| C x 1 -1|| W s 1 ,p 1 (C x 1 ) C 2 ||v -1|| W s 1 ,p 1 (O) C 2 C 1 ||v -1|| W s 1 ,p 1 (Ω)
(the last inequality follows from ( 7)).

While x may depend on v, the covering C x N has two features independent of v: [START_REF] Brezis | composition and products in fractional Sobolev spaces[END_REF] the number of squares in C x 2 has a uniform upper bound K;

if C 1 , C 2 are two squares in C x 2 ,
there is a path of squares in C x 2 each one having an edge in common with its neighbours, connecting (11)

C 1 to C 2 .
Step 3: choice of δ We rely on Lemma 7. Let C = (0, ε) 2 and 0 < s 1 < 1, 1 < p 1 < ∞, s 1 p 1 = 1. Then for each δ 1 > 0 there is some δ 2 > 0 such that every map v ∈ W s 1 ,p 1 (∂C; S 1 ) satisfying [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF] ||v -1|| W s 1 ,p 1 (∂C) < δ 2 has a lifting ϕ ∈ W s 1 ,p 1 (∂C; R) such that

(13) ||ϕ|| W s 1 ,p 1 (∂C) < δ 1 .
Clearly, in Lemma 7, C may be replaced by the unit disc. For the unit disc, the proof of Lemma 7 is given in Appendix C; see Lemma C.3.

In particular, if [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF] holds, then we have [START_REF] Escobedo | Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions[END_REF] ||ϕ|| L 1 (∂C) < C 3 δ 1 for some C 3 independent of the δ ′ s.

We now take δ 1 such that (15)

δ 1 < πε/C 3 .
With δ 2 provided by Lemma 7, we choose

(16) δ = min {δ 2 /C 0 , δ 2 /C 1 C 2 }.
Step 4: construction of a global lifting for v|

C x 1 Let v ∈ W s,p (Ω; S 1 ) satisfy ||v -1|| W s 1 ,p 1 < δ. Since δ δ 2 /C 1 C 2 , (9)
implies that the conclusion of Lemma 7 holds for v| ∂C and every square C in C x 2 . Thus, for every C ∈ C x 2 , v| ∂C has a lifting ϕ C satisfying ( 14) and ϕ C ∈ W s 1 ,p 1 (∂C).

We claim that ϕ C ∈ W s,p (∂C). The statement being local, it suffices to prove that ϕ C ∈ W s,p (L), where L is the union of three edges in ∂C. Since L is Lipschitz homeomorphic with an interval, by Theorem 1 in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] there is some ψ ∈ W s,p (L) such that v = e iψ in L (here we use 0 < s < 1 and sp = 2 1). In L, we have ψ -ϕ C ∈ (W s,p + W s 1 ,p 1 ) (L; 2πZ); thus ψ -ϕ C is constant a.e. in L (see [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Remark B.3), so that the claim follows.

Since sp > 1 and v| C x 1 ∈ W s,p , ϕ C ∈ W s,p , we may redefine v| C x 1 and ϕ C on null sets in order to have continuous functions. We claim that the function ϕ(y) = ϕ C (y), if y ∈ C is well-defined on C x 1 (and thus continuous and W s,p ). By (11), it suffices to prove that, if b) the map

w 1 = u, in Ω + ṽ, in Ω - belongs to W s,p (Ω); c) the map w 2 = u -ṽ, in Ω + 0,
in Ω - belongs to W s,p (Ω).

Proof. Recall that, if U is a smooth or cube-like domain, then an equivalent (semi-) norm on W s,p (U ) is given by

(B.2) f -→    N j=1 ∞ 0 {x∈U ; x+te j ∈U } f (x + te j ) -f (x)| p t sp+1 dxdt    1/p
(see, e.g., Triebel [START_REF] Triebel | Theory of function spaces[END_REF]).

Clearly, both b) and c) imply that v ∈ W s,p (Q). Conversely, for v ∈ W s,p (Q) we have to prove the equivalence of (B.1), b) and c). We consider the norm given by (B.2). Taking into account the fact that w 1 , w 2 belong to W s,p in Ω + and Ω -, we see that

(B.3) w 1 ∈ W s,p (Ω) ⇔ J = Ω + 0 -1 |u(x) -ṽ(x)| p (x N -t) sp+1 dtdx < ∞ and (B.4) w 2 ∈ W s,p (Ω) ⇔ J < ∞.
The lemma follows from the obvious inequality 1 -2 -sp sp I J 1 sp I.

We now assume in addition that sp 1 and derive the following Corollary B.1. Let 0 < s < 1, 1 < p < ∞ be such that sp 1. Then, for every u ∈ W s,p (Ω + ) we have a) for each 0 t 0 < 1, there is at most one function v defined on Q such that the maps

w t 0 1 = u, in Q × (t 0 , 1) ṽ, in Q × (-1, t 0 )
and

w t 0 2 = u -ṽ, in Q × (t 0 , 1) 0, in Q × (-1, t 0 ) belong to W s,p (Ω);
b) for a.e. 0 t 0 < 1, the function v = u (•, t 0 ) has the property that w t 0 1 , w t 0 2 ∈ W s,p (Ω).

(As usual, the uniqueness of v is understood a.e.)

The above corollary suggests the following Definition: let 0 < s < 1, 1 < p < ∞, sp 1, 0 t 0 < 1. Let u ∈ W s,p (Ω + ) and let v be a function defined on Q. Then v is the downward good restriction of u to {x N = t 0 } if w t 0 1 , w t 0 2 ∈ W s,p (Ω); we then write v = Rest u| - x N =t 0 . Similarly, for 0 < t 0 < 1 we may define an upward good restriction Rest u| +

x N =t 0 = v as the unique function v defined on Q satisfying the two equivalent conditions a)

W t 0 1 = ṽ, in Q × (t 0 , 1) u, in Q × (0, t 0 ) ∈ W s,p (Ω + ) and b) W t 0 2 = 0, in Q × (t 0 , 1) u -ṽ, in Q × (0, t 0 ) ∈ W s,p (Ω + ).
If v is both an upward and a downward good restriction, we call it a good restriction and we write v = Rest u| x N =t 0 .

Corollary B.2. Let 0 < s < 1, 1 < p < ∞, sp 1. Let u ∈ W s,p (Ω + ).
Then, for a.e. 0 < t 0 < 1, we have Rest u| x N =t 0 = u (•, t 0 ).

Remark B.1. If sp > 1, then functions u ∈ W s,p (Ω + ) have traces for all 0 t 0 1. However, these traces need not be good restrictions. Here is an example: For N = 2, one may prove that the map x → (x -1/2e 1 )/|x -1/2e 1 | belongs to W s,p (Ω) if 0 < s < 1, 1 < p < ∞, sp < 2. However, if sp > 1, its trace

tr u| x 2 =0 = 1, if x 1 > 1/2 -1, if x 1 < 1/2
does not belong to W s,p (0, 1), so that it is not a good restriction.

Remark B.2. In the limiting case s = 1/p, functions in W s,p do not have traces. However, they do have good restrictions a.e.

Here is yet another simple consequence of Lemma B.1

Corollary B.3. Let 0 < s < 1, 1 < p < ∞, sp 1. Let u ± ∈ W s,p (Ω ± ) be such that Rest u + | - x N =0 = Rest u -| + x N =0 .
Then the map

w = u + , in Ω + u -,
in Ω - belongs to W s,p .

The following results explain the connections between good restrictions and traces.

Lemma B.2. Let 0 < s < 1, 1 < p < ∞, sp > 1. Let u ∈ W s,p (Ω + ). Assume that there exists v = Rest u| - x N =0 . Then v = tr u| x N =0 .
Proof. Let w = u -ṽ, in Ω + 0, in Ω - . By Lemma B.1, we have w ∈ W s,p (Ω). By trace theory and continuity of the trace, we have 0 = tr w| x N =0 , so that tr u| x N =0 = v.

Lemma B.3. Let 0 < s < 1, 1 < p < ∞, sp 1. Let u ∈ W s+1/p,p (Ω + ). Then, considered as a W s,p function, u has a good downward restriction to {x N = 0} which coincides with tr u| x N =0 .

Proof. Let v = tr u| x N =0 . Then v ∈ W s,p (Q), by the trace theory. By Lemma B.1, it remains to prove that (B.5)

Ω + |u(x) -ṽ(x)| p x sp N dx < ∞.
Assume first that s + 1/p = 1. Then (B.5) follows from the well-known Hardy inequality

(B.6) Q 1 0 |u(x ′ , t) -u(x ′ , 0)| p t p dtdx C Du p L p , ∀u ∈ W 1,p (Ω + ).
Consider now the case where s + 1/p = 1. Let σ = s + 1/p. We are going to prove that (B.7)

Ω + |u(x) -ṽ(x)| p x sp N dx C u p W σ,p
for some convenient equivalent (semi-) norm on W σ,p . It is useful to consider the norm (B.8)

f →    N j=1 ∞ 0 {x∈U ; x+te j ∈U, x+2te j ∈U } |f (x + 2te j ) -2f (x + te j ) + f (x)| p t σp+1 dxdt    1/p
(see, e.g., Triebel [START_REF] Schoen | Boundary regularity and the Dirichlet problem for harmonic maps[END_REF]).

For any

x ′ ∈ Q such that u x ′ = u(x ′ , •) ∈ W σ,p (0, 1), the map f x ′ (t) = u(x ′ , t), if t > 0 v(x ′ ), if t < 0
belongs to W σ,p (-1, 1), by standard trace theory. Moreover, for any such x ′ we have (B.9)

f x ′ p W σ,p (-1,1) C u x ′ p W σ,p (0,1) , i.e. ∞ 0 {h∈(-1,1); h+t∈(-1,1), h+2t∈(-1,1)} |f x ′ (h + 2t) -2f x ′ (h + t) + f x ′ (h)| p t σp+1 dhdt C ∞ 0 {h∈(0,1); h+t∈(0,1), h+2t∈(0,1)} |u x ′ (h + 2t) -2u x ′ (h + t) + u x ′ (h)| p t σp+1 dhdt.
In particular, (B.10)

I = 1/2 0 -t -2t |f x ′ (h + 2t) -2f x ′ (h + t) + f x ′ (h)| p t σp+1 dhdt C u x ′ p W σ,p . Since (B.11) I C 1/3 0 |u(x ′ , t) -v(x ′ )| p t σp dt = C 1/3 0 |u(x ′ , t) -v(x ′ )| p t sp+1 dt,
we find that (B.12)

1/3 0 |u(x ′ , t) -v(x ′ )| p t sp+1 dt C u x ′ p W σ,p .
On the other hand, we clearly have (B.13)

1 1/3 |u(x ′ , t) -v(x ′ )| p t sp+1 dt C u x ′ p L p + C|v(x ′ )| p .
By combining (B.12), (B.13) and integrating with respect to x ′ , we obtain (B.7). The proof of Lemma B.3 is complete.

A simple consequence of Lemma B.3 is the following

Lemma B.4. Let 0 < s < 1, 1 < p < ∞, sp 1 and ρ > s. Let u 1 ∈ W s,p (Ω + ) and u 2 ∈ W ρ,p (Ω -). Assume that u 1 has a good downward restriction v = Rest u 1 | - x N =0 and that v = tr u 2 | x N =0 . Then the map w = u 1 , in Ω + u 2 ,
in Ω - variable t. Indeed, for a.e. x ∈ B 1 , we have u ∈ W s,p (S 1 ×{x}; S 1 ) and deg (u| S 1 ×{x} ) = 0. In particular, for any such x the map u| S 1 ×{x} has a continuous lifting η x . On the other hand, for a.e. x ∈ B 1 we have ψ x = ψ(•, x) ∈ W s,p loc (R×{x}; R). Thus, with λ x (t) = η x (e it ), we find that for a.e. x ∈ B 1 the function ψ x -λ x is continuous and 2πZ -valued; therefore it is a constant. Since λ x is 2π-periodic, so is ψ x for a.e. x ∈ B 1 . We obtain that ψ is 2π-periodic in the variable t. Thus the map ϕ : S 1 × B 1 → R, ϕ(e it , x) = ψ(t, x) is well-defined and belongs to W s,p (S 1 × B 1 ; R). Moreover, we clearly have u = e iϕ .

In the same vein, we have

Lemma C.2. Let s 1, 1 < p < ∞, N 3, 2 sp < N . Let u ∈ W s,p (S 1 × B 1 ; S 1 ) be such that deg (u| S 1 ×B 1 ) = 0. Then there is some ϕ ∈ W s,p (S 1 ×B 1 ; R)∩W 1,sp (S 1 ×B 1 ; R) such that u = e iϕ .
The proof is similar to that of Lemma C.1; one has to use Lemma 4 in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] instead of Theorem 2 in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF].

Lemma C.3. Let 1 < p < ∞ and δ 1 > 0. Then there is some δ 2 > 0 such that every v ∈ W 1/p,p (S 1 ; S 1 ) satisfying v -1 W 1/p,p (S 1 ) < δ 2 has a global lifting ϕ ∈ W 1/p,p (S 1 ; R) such that ϕ W 1/p,p (S 1 ) < δ 1 .
Proof. Recall that if I is an interval, then every w ∈ W 1/p,p (I; S 1 ) has a lifting ψ ∈ W 1/p,p (I; R) (see Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Theorem 1). Moreover, this lifting may be chosen to be (locally) continuous with respect to w, i.e. for every w 0 ∈ W 1/p,p (I; S 1 ) there is some δ 0 > 0 such that in the set {w; w -w 0 W 1/p,p (I;S 1 ) < δ 0 } there is a lifting w → ψ continuous for the W 1/p,p norm. (This assertion can be established using the same argument as in Step 7 of the proof of Theorem 4 in Brezis -Nirenberg [START_REF] Brezis | Degree Theory and BMO, Part I: Compact manifolds without boundaries[END_REF]; it can also be derived from the explicit construction of ψ in the proof of Theorem 1 in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]; see also Boutet de Monvel-Berthier -Georgescu -Purice [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] when p = 2).

Let I = [-2π, 2π]. To each v ∈ W 1/p,p (S 1 ; S 1 ) we associate the map w ∈ W 1/p,p (I; S 1 ), w(t) = v(e it ). By the above considerations, for every δ 3 > 0 there is some δ 4 > 0 such that, if v -1 W 1/p,p (S 1 ) < δ 4 , then w has a lifting ψ such that ψ W 1/p,p (I) < δ 3 . We claim that ψ is 2π-periodic if δ 3 is small enough. Indeed, the function ξ(t) = ψ(t -2π) -ψ(t) belongs to W 1/p,p ([0, 2π]; 2πZ), so that ξ is constant a.e. (see [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], Theorem B.1). Since ξ L 1 ψ L 1 < Cδ 3 , we have ξ = 0 (i.e. ψ is 2π-periodic) if Cδ 3 < 2π.

Thus, for δ 3 small enough, the map ϕ(e it ) = ψ(t) is well-defined, belongs to W 1/p,p and satisfies ϕ W 1/p,p (S 1 ) < δ 1 and u = e iϕ .

Appendix D. Filling a hole -the fractional case

We adapt to fractional Sobolev spaces the technique of Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF], Section 1.3.

The first two results are preparations for the proofs of Lemmas 5,6 and 8 (see Lemmas D.3, D.4 and D.5 below).

Lemma D.1. Let 0 < s < 1, 1 < p < ∞, 1 < sp < N . Let C = (-1, 1) N and u ∈ W s,p (∂C). Then ũ ∈ W s,p (C); here, ũ(x) = u(x/|x|) and | | is the L ∞ norm in R N . Moreover, the map u → ũ is continuous from W s,p (∂C) into W s,p (C).

Proof. Clearly, we have ũ L p (C) C 0 u L p (∂C) . Thus it suffices to prove, for the Gagliardo semi-norms in W s,p , the inequality Corollary E.1. Let 0 < s < 1, 1 < p < ∞, sp > 1. Let u ∈ W s,p . Then, for a.e.

x ∈ C N , tr u| C N -1 (x) ∈ W s,p . Moreover, for a.e. x ∈ C N , tr u| C N -1 (x) has a trace on C N -2 (x) which belongs to W s,p , and so on.

Proof of Lemma E.2. In order to avoid long computations, we treat only the case j = 1, N = 2. The general case does not bring any additional difficulty. Let C ∈ C 1 (x); denote its lower (resp. upper, left, right) edge by C l (resp. C u , C L , C R ). By (E.1), we have u| C l ∈ W s,p for a.e. x ∈ C 2 and, for x in a fat set, C∈C 1 (x) u p (here, we take the usual Gagliardo norm in W s,p (R 2 )). We have The proof of Lemma E.2 is complete.

I = C 2 m∈Z 2

  [u] s,p ⊃ {ue iϕ ; ϕ ∈ W s,p (Ω; R)} and of the closedness of the homotopy classes. Another consequence of Theorem 5 is Corollary 3. Conjecture 1 ⇔ Conjecture 2.

(C 1 τττ

 1 ( u p W s,p (∂C) + u p L p (∂C) ).We have(D.2) C C |ũ(x) -ũ(y)| p |x -y| N +sp dxdy = |u(x) -u(y)| p |τ x -σy| N +sp τ N -1 σ N -1 ds x ds y dτ dσ. N -1 σ N -1 |τ x -σy| N +sp dτ dσ C 2 /|x -y| N +sp . N -1 (λτ ) N -1 |τ x -λτ y| N +sp dλdτ = N -sp-1 λ N -1 |x -λy| N +sp dλdτ I 1 + I 2 , |x -y| N +sp dλdτ C 4 /|x -y| N +sp .

W

  s,p (C l ) const. |u| p 1 . Similar statements hold for the other edges.It remains to control the cross -integrals in the Gagliardo norm, e.g. to prove(E.3) I = C 2 C∈C 1 (x) C l C L |u(y) -u(z)| p |y -z| 2+sp dydz const. u p W s,p

+ τ e 1 )

 1 + m 1 e 1 + m 2 e 2 + τ e 1 ) -u(x + m 1 e 1 + m 2 e 2 + σe 2 )| p |τ e 1 -σe 2 | 2+sp dσdτ dx -u(y + σe 2 )| p |τ e 1 -σe 2 | 2+sp dσdτ dy ) -u(z -τ e 1 + σe 2 )| p |τ e 1 -σe 2 | 2+sp dσdτ dz R 2 R 2 |u(z + h) -u(z)| p |h| 2+sp dhdz = u p W s,p .

Acknowledgements. The first author (H.B.) warmly thanks Yanyan Li for useful discussions. He is partially supported by a European Grant ERB FMRX CT980201, and is also a member of the Institut Universitaire de France. This work was initiated when the second author (P.M.) was visiting Rutgers University; he thanks the Mathematics Department for its invitation and hospitality. It was completed while both authors were visiting the Isaac Newton Institute in Cambridge, which they also wish to thank.

When sp < 1, we have the following more general result Theorem 6. If s > 0, 1 < p < ∞, sp < 1 and M is a compact manifold, then W s,p (Ω; M ) is path-connected.

Proof. Fix some a ∈ M . For u ∈ W s,p (Ω; M ), let ũ = u, in Ω a, in R N \Ω .

Since sp < 1, we have ũ ∈ W s,p loc (R N ; M ). Let U (t, x) = ũ (x/(1-t)), 0 t < 1, x ∈ Ω and U (1, x) ≡ a. Then clearly U ∈ C ([0, 1]; W s,p (Ω; M )) and U connects u to the constant a (here we use only sp < N ).

Case 2: 1 < sp < 2, N 2

In this case one could adapt the tools developed in Brezis -Li [START_REF] Brezis | Topology and Sobolev spaces[END_REF], but we prefer a more direct approach.

Let ε > 0 be such that the projection onto ∂Ω be well-defined and smooth in the region {x ∈ R N ; dist (x, ∂Ω) < 2ε}. Let ω = {x ∈ R N \ Ω; dist (x, ∂Ω) < ε}. We have ∂ω = ∂Ω ∪ Λ, where Λ = {x ∈ R N \Ω; dist (x, ∂Ω) = ε}.

Since 1 < sp < 2, we have 1/p < s < 1+1/p; thus, for u ∈ W s,p we have tr u ∈ W s-1/p,p . Let u ∈ W s,p (Ω; S 1 ). Fix some a ∈ S 1 and define v ∈ W s-1/p,p (∂ω; S 1 ) by v = tr u, on ∂ω a, on Λ .

We use the following extension result. (The first result of this kind is due to Hardt -Kinderlehrer -Lin [START_REF] Hardt | Stable defects of minimizers of constrained variational principles[END_REF]; it corresponds to our lemma when σ = 1 -1/p, p < 2.)

by [START_REF] Escobedo | Some remarks on the density of regular mappings in Sobolev classes of S M -valued functions[END_REF]. It follows that

which implies l = 0 by ( 15) and ( 16).

In conclusion, v| C x 1 has a global lifting ϕ ∈ W s,p (C x 1 ; R).

Step 5: construction of a good extension w of v| C x 1 Let ϕ 2 ∈ W s+1/p,p (C x 2 ; R) be an extension of ϕ, ϕ 3 ∈ W s+2/p,p (C x 3 ; R) an extension of ϕ 2 , and so on; let ϕ N ∈ W s+(N -1)/p,p (C x N ; R) be the final extension. Note that these extensions exist since s < 1 + (N -1)/p, so that trace theory applies. We set w = e iϕ N ∈ W s+(N -1)/p,p (C x N ; S 1 ). Since (s + (N -1)/p) • p = N + 1 > N , we obtain by Theorem 3 that w ∈ [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF] s+(N -1)/p,p . By Corollary 5, we also have w ∈ [START_REF] Bethuel | The approximation problem for Sobolev maps between two manifolds[END_REF] s,p .

We complete the proof of Theorem 7 by proving

We rely on the following variant of Lemma 6 Lemma 8. Let 0 < s < 1, 1 < p < ∞, 1 < sp < N, [sp] j < N . Let v, w ∈ W s,p (C N ; S 1 ) be such that v| C l ∈ W s,p , w| C l ∈ W s,p , l = j, ..., N -1. Assume that v| C j and w| C j are W s,p -homotopic. Then v and w are W s,p -homotopic.

The proof of Lemma 8 is given Appendix D; see Lemma D.5.

When N 3, we are going to apply Lemma 8 with j = 2. In order to prove that v| C 2 and w| C 2 are W s,p -homotopic, it suffices to find, for each C ∈ C 2 , a homotopy U C from v| C to w| C preserving the boundary condition on ∂C; we next glue together these homotopies (this works since 0 < s < 1). We construct U C using the lifting: since sp = 2 = dim C and C is simply connected, by Theorem 2 in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] there is some ψ ∈ W s,p (C; R) such that v = e iψ in C. By taking traces, we find that v| ∂C = e itr ψ = e iϕ C ; thus tr ψ -ϕ C ∈ (W s-1/p,p + W s,p )(∂C; 2πZ). Therefore, tr ψ -ϕ C is constant a.e., by Remark B.3 in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]. We may assume that tr

When N = 2, the above argument proves directly (i.e., without the help of Lemma 8) that w ∈ [v] s,p .

The proof of Theorem 7 is complete.

Appendix A. An extension lemma

In this appendix, we investigate, in a special case, the question whether a map in W σ,p (∂ω; S 1 ) admits an extension in W σ+1/p,p (ω; S 1 ).

Let ω be a smooth bounded domain in R N . Then every v ∈ W σ,p (∂ω; S 1 ) has an extension w ∈ W σ+1/p,p (ω; S 1 ).

Proof. We distinguish two cases: σ 1 -1/p and σ > 1 -1/p.

Case σ 1-1/p: since σp < 1, v may be lifted in W σ,p (see Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF]), i.e. there is some ψ ∈ W σ,p (∂ω; R) such that v = e iψ . Let ϕ ∈ W σ+1/p,p (ω; R) be an extension of ψ. Then w = e iϕ ∈ W σ+1/p,p (ω; S 1 ) (since σ + 1/p 1 and x → e ix is Lipchitz). Clearly, w has all the required properties.

Case σ > 1 -1/p: the argument is similar, but somewhat more involved. The proof in [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] actually yields a lifting which is better than W σ,p ; more specifically, this lifting ψ belongs to W tσ,p/t for 0 < t 1, see Remark 2, p.41, in the above reference. On the other hand, since σ > 1 -1/p, we have t = p/(σp + 1) < 1. For this choice of t, we obtain that v has a lifting ψ ∈ W σ,p ∩ W 1-1/(σp+1),σp+1 . This ψ has an extension ϕ ∈ W σ+1/p,p ∩ W 1,σp+1 . By the Composition Theorem stated in the Introduction, the map w = e iϕ belongs to W σ+1/p,p (ω; S 1 ). Clearly, we have tr w = v.

Remark A.1. The special case p < 2 and σ = 1 -1/p was originally treated by Hardt -Kinderlehrer -Lin [START_REF] Hardt | Stable defects of minimizers of constrained variational principles[END_REF] via a totally different method. Their argument extends to the case p < 2 and σp < 1, but does not seem to apply when p 2.

Appendix B. Good restrictions

In this appendix, we describe a natural substitute for the trace theory when s = 1/p; it is known that the standard trace theory is not defined in this limiting case.

For simplicity, we consider mainly the case of a flat boundary. However, we state Lemma B.5 (used in the proof of Theorem 1) for a general domain. We start by introducing some

Then for u ∈ W s,p (Ω + ) and for any function v defined on Q, the following assertions are equivalent:

Proof. Let u 3 ∈ W s+1/p,p (Ω -) be an extension of v. Then w = w 1 + w 2 , where

By Lemma B.3 and the assumption

. By Corollary B.3, we find that w 1 ∈ W s,p (Ω). It remains to prove that w 2 ∈ W s,p (Ω). Let σ = min {ρ, s + 1/p, 1}. Then w 2 ∈ W σ,p (Ω), by standard trace theory. Thus w 2 ∈ W s,p (Ω). u| Σ δ ∈ W 1/p,p (Σ δ ) and

b) for any such δ, u has a good restriction to Σ δ which coincides (a.e. on Σ δ ) with u| Σ δ .

Appendix C. Global lifting

In this appendix, we investigate the existence of a global lifting in some domains with non-trival topology.

Here, B 1 is the unit ball in R N -1 .

, where "loc" refers only to the variable t. By Theorem 2 in Bourgain -Brezis -Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF], there is some ψ ∈ W s,p loc (R × B 1 ; R) such that v = e iψ . We claim that ψ is 2π-periodic in the On the other hand, we have (D.6)

We obtain (D. The proof of Lemma D.1 is complete.

Proof. Let u = v| ∂C . It clearly suffices to prove the lemma in the special case w = ũ. In this case, let, for 0 t < 1,

and g = f -ũ. Then f, ũ ∈ W s,p loc (R N ), so that g ∈ W s,p loc (R N ). Since g = 0 outside C, we actually have g ∈ W s,p (R N ). Thus

We introduce a useful notation: let u ∈ W s 1 ,p 1 (C k ), where 0 < s 1 < 1, 1 < p 1 < ∞, 1 < s 1 p 1 < N . We extend, for each C ∈ C k+1 , u| ∂C to C as in Lemma D.1. Let ũ be the map obtained by gluing these extensions. We next extend ũ to C k+2 in the same manner, and so on, until we obtain a map defined in

Proof. We take u 1 = H j (v). We may use repeatedly Lemma D.1, since for l = j + 1, ..., N we have 1 < s 1 p 1 < l.

Proof. We argue by backward induction on j. If j = N -1, then for each C ∈ C N Lemma D.2 provides a W s,p -homotopy of u| C and u 1 | C preserving the boundary condition. By gluing together these homotopies we find that u and u 1 are W s,p -homotopic (here we use 1/p < s < 1). Suppose now that the conclusion of the lemma holds for j + 1; we prove it for j, assuming that j [sp]. By assumption, u and H j+1 (u| C j+1 ) are W s,p -homotopic, and so are u 1 and H j+1 (u 1 | C j+1 ). It suffices therefore to prove that v = H j+1 (u| C j+1 ) and

The proof of Lemma D.4 is complete.

) be such that v| C l ∈ W s,p , w| C l ∈ W s,p , l = j, ..., N -1. Assume that v| C j and w| C j are W s,p -homotopic. Then v and w are W s,p -homotopic.

Proof. By Lemma D.4, v and H j (v| C j ) (respectively w and H j (w| C j )) are W s,phomotopic. If U connects v| C j to w| C j in W s,p , then as in the proof of Lemma D.4, we obtain that t → H j (U (t)) connects H j (v| C j ) to H j (w| C j ) in W s,p . Thus v and w are W s,p -homotopic.

Appendix E. Slicing with norm control

In this section, we prove the existence of good coverings for W s,p maps. The arguments are rather standard.

Without loss of generality, we may consider maps defined in R N . Throughout this section, we assume ε = 1, i.e. we consider a covering with cubes of size 1. We start by introducing some useful notations: for x ∈ C N = (0, 1) N and for j = 1, ..., N -1, let

For a fixed set Λ ⊂ {1, .., N } such that |Λ| = j, let also (see, e.g., Triebel [START_REF] Schoen | Boundary regularity and the Dirichlet problem for harmonic maps[END_REF]). An obvious computation yields, for the usual Gagliardo (semi-) norm on C Λ j (x), Lemma E.1. Let 0 < s < 1, 1 < p < ∞ and u ∈ W s,p . Then C |u| p j , ∀x ∈ A.

Remark E.1. Here, u| C j (x) are restrictions, not traces. However, when sp > 1 we may replace restrictions by traces, by a standard argument. We obtain