where C is independent of ε and of g.

For variants of Theorem 5, see Remark 8 in Section 5.

The plan of the paper is the following: 1. Proof of Theorems 1 and 2 when sp < 1 2. Proof of Theorem 1 when sp ≥ 1 and of Theorem 2 when sp ≥ n 3. Proof of Theorem 3 when sp ≥ 2 4. Examples of obstruction in Theorems 2 and 3 5. Control of lifting in the H s -norm for s → < 1 2 and application to Ginzburg-Landau Appendix A. A characterization of W s,p (Ω; R) when sp < 1

Theorem 2. Assume n ≥ 2, 0 <s<1 and 1 <p<∞. The answer to the lifting question in W s,p is:

a) positive if sp < 1, b) negative if 1 ≤ sp<n, c) positive if sp ≥ n.
Theorem 3. Assume n ≥ 2, 1 ≤ s<∞ and 1 <p<∞. The answer to the lifting question in W s,p is:

a) negative if sp < 2, b) positive if sp ≥ 2.
In these statements "positive" means that every u ∈ W s,p (Ω; S 1 ) may be written as u = e iϕ for some ϕ ∈ W s,p (Ω; R) and "negative" means that for some u's in W s,p (Ω; S 1 ) there is no ϕ ∈ W s,p (Ω; R) such that u = e iϕ .

As a simple consequence of the theorems when p = 2, i.e., for H s = W s,2 ,w eh a v e Corollary 1. When n =1the answer to the lifting problem in H s is always positive.

When n ≥ 2 the answer to the lifting problem in H s is:

a) positive if 0 <s<1/2, b) negative if 1/2 ≤ s<1, c) positive if s ≥ 1.
The proof of Theorems 1 and 2 when sp < 1 turns out to be quite involved (even for the H s case, s<1/2, and even when n = 1). It relies on a characterization, due to G. Bourdaud [1] (see also the earlier paper of R. Devore and V. A. Popov [1]), of the W s,p space when sp < 1; for the convenience of the reader, and also because we make use of sharp estimates, we have presented a proof in a separate section, Appendix A.

In view of the Corollary for n ≥ 2, a function u ∈ H 1/2 (Ω; S 1 ) need not have a lifting ϕ ∈ H 1/2 (Ω; R); however, it has a lifting ϕ in H s , ∀s<1/2. We prove (see Appendix E) Theorem 4. Assume Q is a cube in R n ,n ≥ 1. For every u ∈ H s (Q; S 1 ) with 0 <s<1/2 one may find a ϕ in H s such that u = e iϕ and satisfying the (optimal) estimate ϕ H s ≤ C(1 -2s) -1/2 u H s with C independent of u and independent of s (for s near 1/2). Such an estimate is useful in deriving bounds for the Ginzburg-Landau functional when the boundary condition belongs to H 1/2 . For example, let Q b eac u b eo fR n ,n ≥ 1, and let Ω = Q × (0, 1). For any function g ∈ H 1/2 (Q; C), set

E ε (u)= 1 2 Ω |∇u| 2 + 1 4ε 2 Ω (|u| 2 -1) 2 ,
where ∇ denotes the full gradient (in (x, t)).

Theorem 5. For every g ∈ H 1/2 (Q; S 1 ) we have, for ε>0,

E ε = Min u∈H 1 g (Ω)
1. Proof of Theorems 1 and 2 when sp < 1.

Here, the assumption that Ω is simply connected is not needed since we may always extend the given function by a constant outside Ω; the resulting function still belongs to W s,p since sp < 1 (this is a well-known fact, see e.g. Lions-Magenes [1], Section 1.11 when p = 2 and the references therein; it is also a consequence of the characterization of W s,p in Appendix A). Thus, we may assume that Ω = (0, 1) n and we use the same notation as in Appendix A.

Let u ∈ W s,p (Ω; S 1 ). For each j =0, 1,..., consider the function U j ∈E j defined by

U j (x)= E j (u)(x) |E j (u)(x)| if E j (u)(x) =0 1i f E j (u)(x)=0 .
Clearly U j → u a.e. on Ω (since E j (u) → u a.e. and |u| = 1 a.e.) For each j =0, 1,... we construct a real-valued function ϕ j ∈E j such that (1.1) e iϕ j = U j on Ω, (1.2) |ϕ j -ϕ j-1 |≤C|U j -U j-1 | on Ω.

Note that (1.2) can be achieved by induction on j, for example with C = π/2. On the other hand, observe that for every ξ, η, ζ ∈ C with |ζ| = 1, we have

(1.3) ξ |ξ| - η |η| ≤ 4(|ζ -ξ| + |ζ -η|)
with the convention that 0 0 = 1 (consider separately the case where |ξ|, |η|≥1/2 and the case where either |ξ| < 1/2o r|η| < 1/2).

Applying (1.3) to ξ = E j (u)(x), η = E j-1 (u)(x) and ζ = u(x) we obtain a.e. on Ω (1.4)

|U j -U j-1 |≤4(|u -E j (u)| + |u -E j-1 (u)|).
Combining this with (1.2) yields

(1.5) |ϕ j -ϕ j-1 |≤C(|u -E j (u)| + |u -E j-1 (u)|)
and thus

(1.6)

j≥1 2 spj ϕ j -ϕ j-1 p L p ≤ C j≥0 2 spj u -E j (u) p L p .
Applying Theorem A.1 and Corollary A.1 in Appendix A, we conclude that ϕ j → ϕ in L p with ϕ ∈ W s,p , e iϕ = u, and

(1.7) ϕ W s,p ≤ C u W s,p .

We may always assume (by adding to ϕ an integer multiple of 2π) that

| Ω ϕ|≤2π.
Thus, we have constructed a function ϕ ∈ W s,p such that e iϕ = u and

(1.8) ϕ L p + ϕ W s,p ≤ C(1 + u W s,p ).

Remark 1. One should observe the linear dependence while in the continuous case there is no bound whatsoever for ϕ L ∞ in terms of u L ∞ ; see also Remark 3 where we show that there is no bound for ϕ in H 1/2 in terms for u H 1/2 in one dimension despite the fact that every u ∈ H 1/2 has a (unique) lifting in H 1/2 .

Remark 2. The function ϕ constructed above also belongs to every L q ,q < ∞. This may be easily seen by observing that u ∈ W s,p ∩ L ∞ ⊂ W σ,q for every σ<swith σq = sp (by the Gagliardo-Nirenberg inequality, see Appendix D). Therefore ϕ belongs to every such W σ,q . Choosing σ close to zero we obtain a q which is arbitrarily large.

2. Proof of Theorem 1 when sp ≥ 1 and of Theorem 2 when sp ≥ n.

When sp > 1 in Theorem 1 or sp > n in Theorem 2, u is continuous by the Sobolev imbedding theorem and, locally, we may consider ϕ = -i log u which is well-defined and singlevalued. To conclude, we rely on a lemma about composition: Lemma 1. Assume n ≥ 1, 0 <s<∞ and 1 <p<∞. Let v ∈ W s,p (Ω) ∩ L ∞ (Ω) and let Φ ∈ C ∞ . Then Φ • v ∈ W s,p (Ω).

The proof is very simple when 0 <s<1 (using the definition of W s,p and the fact that Φ is Lipschitz on the range of v). This lemma is also well-known when s is an integer, with the help of the Gagliardo-Nirenberg inequality. When s>1 is not an integer the argument is more delicate; we refer to Escobedo [1] and Lemma C.1 in Appendix C.

We now turn to the proof of Theorem 1 when s =1 /p; the proof of Theorem 2 when s = n/p is identical and we omit it. Set I =Ω=(0, 1). By standard trace theory there is some ũ ∈ W s+1/p,p (I 2 ; R 2 ) such that ũ(x, 0) = u(x).

Since u takes its values into S 1 one may expect that, near I ×{0},ũ takes its values "close" to S 1 . This is not true for a general extension ũ. However, special extensions have that property. For example ũ(x, y)= 1 2y

x+y x-y u(t)dt
(u is extended by symmetry to the interval (-2, +2)) has the property that ũ ∈ W s+1/p,p , and moreover, |ũ(x, y)|→1 uniformly in x as y → 0. This is a consequence of the fact that W s,p ⊂ VMO in the limiting case of the Sobolev imbedding (see e.g. Boutet de Monvel-Berthier, Georgescu and Purice [1], [2], Brezis and Nirenberg [1]). Similarly, any harmonic extension ũ of u in I 2 has also the same property (see Brezis and Nirenberg [2], Appendix 3). If we consider v =ũ/|ũ| in a neighborhood ω of I ×{0} in I 2 we have an extension v of u such that v ∈ W s+1/p,p (ω; S 1 ).

Here, we have used again Lemma 1.

Let us now explain how to complete the proof of the theorem when p =2 , i.e., u ∈ H 1/2 (I; S 1 ). From the above discussion we have some extension v of u, with v ∈ H 1 (ω; S 1 ).

Applying the theorem of Bethuel and Zheng we may write v = e iψ for some ψ ∈ H 1 (ω; R) and then ϕ = ψ | I has the required properties.

We now turn to the general case. Here, we shall use the following lemma about products in fractional Sobolev spaces. Its proof is presented in Appendix D when Ω = R n (see Lemma D.2). The case of a smooth domain Ω follows by extending the functions to R n . Lemma 2. Assume s ≥ 1 and 1 <p<∞. Let

f, g ∈ W s,p (Ω; R) ∩ L ∞ (Ω; R)
where Ω is a smooth bounded domain in R n . Then fDg ∈ W s-1,p (Ω).

Proof of Theorem 1 completed. We recall that there is a neighborhood Q of I ×{0} in I 2 and an extension v of u such that v ∈ W s+(1/p),p (Q; S 1 ).

Applying once more the same construction we find some

w ∈ W s+(2/p),p (U; S 1 )
where U is a neighborhood of Q ×{ 0} in Q × I. (This construction is possible since (s +1/p)p = 2, so that we are again in a limiting case for the Sobolev imbedding and thus v ∈ VMO. Iterating this construction we find some

ζ ∈ W s+(k/p),p (G; S 1 ) where G is a domain in R k+1 . Consider the first integer k ≥ 1 such that s +(k/p) ≥ 1.
This choice of k implies that

s + j p < 1,j =0, 1,... ,k-1,
so that, at each step, standard trace theory applies (recall that a function in W s,p has an extension in W s+1/p,p provided s is not an integer).

¿From the Gagliardo-Nirenberg inequality (see Lemma D.1) we have

ζ ∈ W 1,k+1 (G; S 1 ).
Applying the theorem of Bethuel and Zheng, we may write

(2.1) ζ = e iψ
for some ψ ∈ W 1,k+1 (G; R). Differentiating (2.1) we find Dψ = -i ζDζ.

By Lemma 2 we have Dψ ∈ W s+(k/p)-1,p (G) and hence ψ ∈ W s+(k/p),p (G).

Taking back traces we obtain ϕ = ψ | I ∈ W s,p (I) and u = e iϕ .

Remark 3. In one dimension we have established that every u ∈ H

ϕ n (x)=      0 for -1 <x<0 2πnx for 0 <x<1/n 2π for 1/n<x<1.
Clearly ϕ n H 1/2 → +∞ (since ϕ n → ϕ = 1 (0,1) in L 2 and ϕ does not belong to H 1/2 ).

In fact, a more precise computation left to the reader shows that ϕ n H 1/2 ≥ c(log n) 1/2 with c>0. On the other hand the reader will easily check (for example by scaling) that e iϕ n -1 H 1/2 remains bounded. The same conclusion holds when H 1/2 is replaced by W 1/p,p with any p,1<p<∞.

Remark 4. As we have just pointed out there is no control of ϕ in H 1/2 in terms of e iϕ in H 1/2 . There is, however, (in dimension one), an estimate for (ϕ -ϕ) in the space

H 1/2 + W 1,1
, equipped with its usual norm, in terms of e iϕ H 1/2 . Here is the argument, working for simplicity with periodic functions. We may also assume (by density) that ϕ is smooth. Observe that the dual of

H 1/2 + W 1,1 is H -1/2 ∩ W -1,∞ . Given any T ∈ H -1/2 ∩ W -1,∞ , write T = ψ ′ + c for some ψ ∈ H 1/2 ∩ L ∞ and some constant c. Then T,ϕ-ϕ = ψ ′ ,ϕ-ϕ = -ψ,ϕ ′ .
But if we set u = e iϕ , then ϕ ′ = -iūu ′ and thus

| T,ϕ-ϕ | = | ψ,iūu ′ | = | u ′ ,iψū | ≤ u H 1/2 ψu H 1/2 .
Recall that H 1/2 ∩ L ∞ is an algebra (see e.g. Appendix D) and that

ψu H 1/2 ≤ C( ψ H 1/2 + ψ L ∞ )( u H 1/2 + u L ∞ ) ≤ C T H -1/2 ∩W -1,∞ ( u H 1/2 +1).
We conclude that

ϕ-ϕ H 1/2 +W 1,1 ≤ C u H 1/2 ( u H 1/2 +1).
The same estimate holds in higher dimensions if u belongs to the closure of C ∞ ( Ω; S 1 ) in H 1/2 (Ω; S 1 ); however, the argument is much more delicate and will be presented in our forthcoming paper, Bourgain, Brezis and Mironescu [1].

3. Proof of Theorem 3 when sp ≥ 2.

The case s = 1 in Theorem 3 coincides with the theorem of Bethuel and Zheng. For the sake of completeness we present a proof which is simpler than the original one (see also Carbou [1] for a similar idea).

Proof of the Bethuel-Zheng theorem. The idea is to assume that ϕ is known and to derive some consequences. Writing u = u 1 + iu 2 with u 1 = cos ϕ and u 2 = sin ϕ we have The strategy is now to find ϕ by solving (3.1) with the help of a generalized form of Poincaré's lemma,

Lemma 3. Let 1 ≤ p<∞ and let f ∈ L p (Ω; R n ).
The following properties are equivalent:

a) there is some ϕ ∈ W 1,p (Ω; R) such that f = Dϕ, b) one has (3.2) ∂f i ∂x j = ∂f j ∂x i ∀ i, j, 1 ≤ i, j ≤ n
in the sense of distributions, i.e.,

f i ∂ψ ∂x j = f j ∂ψ ∂x i ∀ ψ ∈ C ∞ 0 (Ω).
We emphasize that the assumption that Ω is simply connected is needed in this lemma.

Proof of Lemma 3. The implication a) ⇒ b) is obvious. To prove the converse, let f be the extension of f by 0 outside Ω and let fε = ρ ε ⋆ f where (ρ ε ) is a sequence of mollifiers. The fε 's satisfy (3.2) on every compact subset of Ω (for ε sufficiently small). In particular, on every smooth simply connected domain ω ⊂ Ω with compact closure in Ω, there is a function ψ ε such that Dψ ε = fε in ω.

(Here we have used the standard Poincaré lemma). Passing to the limit we obtain some ψ ∈ W 1,p (ω) such that Dψ = f in ω. Finally, we write Ω as an increasing union of ω n as above and obtain a corresponding sequence ψ n . In the limit we find some ϕ ∈ L 1 loc (Ω) with Dϕ = f in Ω. Using the regularity of Ω and a standard property of Sobolev spaces (see e.g. Maz'ja [1], Corollary in Section 1.1.11) we conclude that ϕ ∈ W 1,p (Ω).

Proof of the Bethuel-Zheng theorem completed. We will first verify condition b) of the lemma for

(3.3) f = u 1 Du 2 -u 2 Du 1
i.e.,

f i = u 1 ∂u 2 ∂x i -u 2 ∂u 1 ∂x i .
Formally, property (3.2) is clear. Indeed, if u 1 and u 2 are smooth, then

∂f i ∂x j - ∂f j ∂x i =2 ∂u 1 ∂x j ∂u 2 ∂x i - ∂u 1 ∂x i ∂u 2 ∂x j .
On the other hand, if we differentiate the relation

|u| 2 = u 2 1 + u 2 2 =1
we find

(3.4) u 1 ∂u 1 ∂x i + u 2 ∂u 2 ∂x i =0 ∀ i =1, 2,... ,n.
Thus, in R 2 , the vector ( ∂u 1 ∂x i , ∂u 2 ∂x i ) is orthogonal to (u 1 ,u 2 ). It follows that the vectors ( ∂u 1 ∂x i , ∂u 2 ∂x i ) and ( ∂u 1 ∂x j , ∂u 2 ∂x j ) are colinear and therefore

(3.5) det ∂u 1 ∂x i ∂u 2 ∂x i ∂u 1 ∂x j ∂u 2 ∂x j = ∂u 1 ∂x i ∂u 2 ∂x j - ∂u 1 ∂x j ∂u 2 ∂x i =0.
Hence (3.2) holds. To make this argument rigorous we rely on the density of smooth functions in the Sobolev space W 1,p (Ω; R) (see e.g. Adams [1], Chap. III or Brezis [1], Chap. IX): there are sequences (u 1n ) and (u 2n )i nC ∞ (Ω; R) such that u 1n → u 1 and

u 2n → u 2 in W 1,p (Ω; R) and u 1n L ∞ ≤ 1, u 2n L ∞ ≤ 1.
[Warning: We do not claim that u n =( u 1n ,u 2n ) takes its values in S 1 . The density of C ∞ ( Ω; N )inW 1,p (Ω; N ), where N is a compact manifold without boundary, e.g. N = S 1 , is a delicate matter which has been extensively studied by Bethuel [1]. As a matter of fact, the Bethuel-Zheng theorem can be used to prove the density of C ∞ ( Ω; S 1 )i nW ∂x j . Multiplying (3.6) by ψ ∈ C ∞ 0 (Ω), integrating by parts and passing to the limit (using the fact that p ≥ 2) we obtain

- Ω (f i ∂ψ ∂x j -f j ∂ψ ∂x i )=2 Ω ( ∂u 1 ∂x j ∂u 2 ∂x i - ∂u 1 ∂x i ∂u 2 ∂x j )ψ.
On the other hand (3.4) and (3.5) hold a.e. (even for any u ∈ W 1,p (Ω; S 1 ), 1 ≤ p<∞) It follows that f satisfies b) of Lemma 3, and therefore there is some ϕ ∈ W 1,p (Ω; R) such that f = Dϕ.

We will now prove that this ϕ is essentially the one in the conclusion of the Bethuel-Zheng theorem.

Recall that if g, h ∈ W 1,p (Ω) ∩ L ∞ (Ω) with 1 ≤ p<∞, then gh ∈ W 1,p and

∂ ∂x i (gh)=g ∂h ∂x i + h ∂g ∂x i . Set v = ue -iϕ , so that v ∈ W 1,p and Dv = e -iϕ (Du -iDϕ)=ue -iϕ (ūDu -iDϕ) = ue -iϕ (ūDu -if )=ue -iϕ (u 1 Du 1 + u 2 Du 2 ) = 0 by (3.4).
We deduce that v is a constant and since |v| = 1 we may write v = e iC for some constant C ∈ R. Hence u = e i(ϕ+C) and the function ϕ + C has the desired properties.

We now turn to the proof of Theorem 3 when sp ≥ 2. In fact, we have a more precise statement: Lemma 4. Assume n ≥ 1,s ≥ 1, 1 <p<∞ and sp ≥ 2. Then any u ∈ W s,p (Ω; S 1 ) may be lifted as u = e iϕ with ϕ ∈ W s,p (Ω; R) ∩ W 1,sp (Ω; R).

Proof. Observe that W s,p ∩ L ∞ ⊂ W 1,sp
by the Gagliardo-Nirenberg inequality (see Lemma D.1). Since sp ≥ 2 we may apply the Bethuel-Zheng theorem and write u = e iϕ for some ϕ ∈ W 1,sp (Ω; R). Using Lemma 2 we find that Dϕ = -iūDu ∈ W s-1,p , so that ϕ ∈ W s,p .

Examples of obstruction in Theorems 2 and 3.

We start with an example of obstruction in Theorem 2, i.e., when 0 <s<1 and 1 ≤ sp < n.

Lemma 5. Assume n ≥ 2. Given any s and any p with 0 <s<1, 1 <p<∞, and 1 ≤ sp < n, there is some u ∈ W s,p (Ω; S 1 ) which cannot be lifted, i.e., for this u no ϕ ∈ W s,p (Ω; R) exists such that u = e iϕ .

Proof. Without loss of generality we may assume that Ω is the unit ball. Let

ψ(x)= 1 |x| α with n -sp p ≤ α< n -sp sp and let u = e iψ .
We claim that

(4.1) u ∈ W s,p (Ω; S 1 ).
Indeed it is clear that ψ ∈ W 1,q ∀ q with 1 <q< n α +1 , and thus ψ ∈ W σ,q ∀ σ with 0 <σ<1 , ∀ q with 1 <q< n α +1 .

Since u ∈ L ∞ , we also know, by the Gagliardo-Nirenberg inequality (see Lemma D.1 in Appendix D), that

u ∈ W t,r ∀ t ∈ (0, 1) ∀ r ∈ (1, ∞) with tr < n α +1
.

In particular, we may choose t = s and r = p since sp < n/(α + 1), i.e., (4.1) holds.

Next we claim that there is no ϕ ∈ W s,p (Ω; R) such that u = e iϕ . Assume, by contradiction, that such ϕ exists. Set

η = 1 2π (ϕ -ψ),
so that η takes its values in Z and

η ∈ W s,p loc (Ω\{0}; Z)
(because ψ is smooth on Ω\{0}). Since sp ≥ 1a n dΩ \{0} is connected we conclude, using Lemma B.1 in Appendix B, that η is a constant. Thus ψ ∈ W s,p (Ω; R). Note that, by scaling,

A(r)= B r B r |ψ(x) -ψ(y)| p |x -y| n+sp dxdy satisfies A(1) = r β A(r) with β =( α + s)p -n ≥ 0 (by assumption on α). If A(1) < ∞, then A(1) = 0 (by letting r → 0). But this is impossible. Thus A(1) = ∞, i.e., ψ/ ∈ W s,p . A contradiction.
A topological obstruction. There is an alternative example of obstruction to lifting, which is of topological nature.

Consider first the case n = 2. Set

(4.2) u(x)= x |x| on the unit ball Ω of R 2 .

Since |Du(x)|≤C/|x|

we see that u ∈ W 1,q (Ω; S 1 ) for every q<2 and therefore u ∈ W s,p (Ω; S 1 ) for every s ∈ (0, 1) and every p ∈ (1, ∞) with sp < 2 (by the Gagliardo-Nirenberg inequality; see Lemma D.1), If, in addition, we assume sp ≥ 1 then there is no ϕ ∈ W s,p (Ω; R) such that u = e iϕ . Indeed set

Ω ′ =Ω\([0, 1] ×{0})
and θ ∈ (0, 2π) with e iθ = u.

Clearly θ ∈ C ∞ (Ω ′ ) and θ has a jump of 2π along the segment [0, 1] ×{0}. Assume, by contradiction, that u has a lifting ϕ ∈ W s,p (Ω; R). Arguing as above we would conclude that θ ∈ W s,p (Ω; R) but this is impossible since θ has a jump of 2π along the segment (0, 1) ×{0} and such a function cannot belong to W s,p with sp ≥ 1.

When n ≥ 3, the same construction as above with

u(x)= (x 1 ,x 2 ) (x 2 1 + x 2 2 ) 1/2 x =(x 1 ,x 2 ,... ,x n )
provides an example of a function u ∈ W s,p (Ω; S 1 ) for every s ∈ (0, 1) and every p ∈ (1, ∞) with sp < 2 and which has no lifting in W s,p when sp ≥ 1. However, this example does not reach the optimal condition sp<nwhen n ≥ 3.

Remark 5. The topological obstruction provides an example of loss of regularity in lifting.

To explain the phenomenon consider the simple case where p = 2. Recall (see Corollary 1) that if u ∈ H s (Ω; S 1 ) with 1/2 <s<1, then, in general, u has no lifting in H s . From the positive part in Corollary 1 one knows that u has a lifting in H (1/2-ε) . Roughly speaking, we lose (s -1/2) derivative in the lifting.

Open Problem: When n ≥ 3 the precise loss of regularity in lifting is not fully understood. For simplicity consider the case n = 3 and p = 4. First a summary of the known results:

a) If s<1/4, any u ∈ W s,4 has a lifting in W s,4 . b) If s ≥ 3/4, any u ∈ W s,4 has a lifting in W s,4 . c) If 1/4 ≤ s<3/4 some u's in W s,4 have no lifting in W s,4 . d)
The topological example provides an example of a function u ∈ W s,4 ∀ s<1/2, and this u has no lifting even in W 1/4,4 .

It would be interesting to find an example of a function u ∈ W s,4 ∀s<3/4 which has no lifting even in W 1/4,4 .

Finally, case b) in Theorem 3 relies on Lemma 6. Assume n ≥ 2. Given any s and any p with s ≥ 1 and 1 <p<∞ with sp < 2, there is some u ∈ W s,p (Ω; S 1 ) which cannot be lifted by a function ϕ ∈ W s,p (Ω; R).

Proof. Use the topological example u above. It is easy to see that u ∈ W s,p ∀s ∈ (0, ∞), ∀p ∈ (1, ∞) with sp < 2. This u has no lifting even in W 1/p,p .

Control of lifting in the H

s -norm for s → < 1
2 and application to Ginzburg-Landau.

We return to the particular issue of lifting a function u ∈ H s (Ω; S 1 ) when s<1/2 and s → 1/2. Recall (see Corollary 1) that, for every s<1/2, u admits a lifting ϕ ∈ H s (Ω; R), i.e., (5.1)

u = e iϕ
We also know (see (1.7)) that we may find a ϕ ∈ H s such that

ϕ H s ≤ C s u H s .
Our aim is to find an optimal control for the constant C s as s → 1/2. Such a control will then be used in the study of the Ginzburg-Landau energy E ε as ε → 0.

If we follow the proof in Section 1 we obtain a ϕ as a limit of sequence ϕ j such that (5.2)

j≥1 4 sj ϕ j -ϕ j-1 2
L 2 ≤ C j≥0 4 sj u -E j (u) 2 L 2
where here, and in what follows, C without a subscript s denotes a constant which remains bounded as s → 1/2. Following the proof of Corollary 1 we obtain

(5.3) j≥1 4 sj E j (ϕ) -E j-1 (ϕ) 2 L 2 ≤ C j≥1 4 sj ϕ j -ϕ j-1 2 L 2 .
We also recall (see Step 3 in Appendix A) that

(5.4)

j≥0 4 sj u -E j (u) 2 L 2 ≤ C u 2 H s .
Combining (5.2), (5.3) and (5.4) yields

(5.5)

j≥1 4 sj E j (ϕ) -E j-1 (ϕ) 2 L 2 ≤ C u 2 H s .
Finally we know (see Corollary A.2 in Appendix A) that

(5.6) ϕ H s ≤ C s   j≥1 4 sj E j (ϕ) -E j-1 (ϕ) 2 L 2   1/2
and the optimal constant C s for the inequality (5.6) is of the order of (1 -2s) -1 . Hence we deduce that the ϕ constructed by this technique satisfies

(5.7) ϕ H s ≤ C(1 -2s) -1 u H s .
In fact, there is a more refined construction of lifting which yields a better estimate. For simplicity we work in a cube Q of R d , d ≥ 1; for more general domains see Remark E.2 in Appendix E.

Theorem 4. For every u ∈ H s (Q; S 1 ) with 0 <s<1/2 one may construct a ϕ ∈ H s (Q; R) satisfying (5.1) and the (optimal) estimate

(5.8) ϕ H s ≤ C(1 -2s) -1/2 u H s ,
where C is independent of u and independent of s as s → 1/2.

The reason why the previous construction does not yield the correct asymptotic as s → 1/2 is due to "edge-singularities" at the nodes of our dyadic partitions P j .T o overcome this, we rely on an argument of translations which is explained in Appendix E where we present the proof of Theorem 4. That type of argument has been exploited earlier in slightly different contexts (for instance in comparing the usual and dyadic BMO-norms, see Garnett and Jones [1]).

The next result is an application to the Ginzburg-Landau functional. Let Q be a cube of R d , d ≥ 1, and let Ω = Q × (0, 1). For any function g ∈ H 1/2 (Q; C) set

H 1 g (Ω) = u(x, t):Ω→ C; Ω |∇u| 2 dxdt < ∞ and u(x, 0) = g(x)o nQ , E ε (u)= 1 2 Ω |∇u| 2 + 1 4ε 2 Ω (|u| 2 -1) 2 ,
where ∇ denotes the full gradient (in (x, t)).

Theorem 5. For every g ∈ H 1/2 (Q; S 1 ) we have, for ε>0,

(5.9)

E ε = Min u∈H 1 g (Ω) E ε (u) ≤ C log(1/ε) g 2 H 1/2
where C is independent of ε and of g.

Proof. Let s = s(ε) < 1/2 to be specified. It follows from Theorem 4 (applied to g) that g = e iϕ for some ϕ ∈ H s (Q; R) satisfying (5.10)

ϕ H s ≤ C(1 -2s) -1/2 g H 1/2 .
Denote ϕ δ a δ-smoothing of ϕ (with δ to be chosen later). Thus, we have

(5.11) ϕ -ϕ δ L 2 (Q) ≤ Cδ s ϕ H s (Q) ≤ Cδ s (1 -2s) -1/2 g H 1/2 (Q)
also, by (5.10),

(5.12)

ϕ δ H 1/2 (Q) ≤ Cδ s-1/2 ϕ H s (Q) ≤ C(1 -2s) -1/2 δ s-1/2 g H 1/2 (Q) . Taking (5.13) 1 -2s ∼ (log 1/δ) -1
we conclude that (5.14)

ϕ δ H 1/2 (Q) ≤ C(log 1/δ) 1/2 g H 1/2 (Q) .
Let φδ denote some harmonic extension of ϕ δ to Ω with

(5.15) φδ H 1 (Ω) ≤ C(log 1/δ) 1/2 g H 1/2 (Q)
and set

(5.16) G δ = e i φδ so that

(5.17)

G δ H 1 (Ω) ≤ C(log 1/δ) 1/2 g H 1/2 (Q) .
Let P denote some harmonic extension of (g -e iϕ δ ) to Ω satisfying the following three estimates ¿From (5.17) and (5.18) we have

P H 1 (Ω) ≤ C g -e iϕ δ H 1/2 (Q) ≤ C( g H 1/2 (Q) + ϕ δ H 1/2 (Q) ) ≤ C(log 1/δ) 1/2 g H 1/2 (Q) by (5.14), (5.18) (5.19) P L ∞ (Ω) ≤ C g -e iϕ δ L ∞ (Q) ≤ C, and
P L 2 (Ω) ≤ C g -e iϕ δ L 2 (Q) ≤ C ϕ -ϕ δ L 2 (Q) ≤ Cδ 1/2 (log 1/δ) 1/2 g H 1/2 (Q)
(5.22) u 2 H 1 (Ω) ≤ C log(1/δ) g 2 H 1/2 (Q) .
On the other hand, using (5.19) we find

|u| 2 -1 ≤ C |u|-1 |u| +1 ≤ C |u|-1 and since |u|-1 = |u|-|G δ | ≤|u -G δ | = |P |
we are led to

(5.23)

Ω (|u| 2 -1) 2 ≤ C Ω |P | 2 ≤ Cδ(log 1/δ) g H 1/2 (Q) by (5.20).
Combining (5.22) and (5.23) we obtain

E ε (u) ≤ C(1 + δ/ε 2 ) log(1/δ) g 2 H 1/2 (Q) .
Choosing δ = ε 2 yields the desired estimate (5.9).

Remark 6. In dimension d =1,E ε remains bounded as ε → 0 since we may write g = e iϕ with some ϕ ∈ H 1/2 and then take u = e i φ where φ is some harmonic extension of ϕ.

However, the bound for E ε depends on g, not just on g H 1/2 (see also Remark 3).

Remark 7. In dimension d ≥ 2, estimate (5.9) is optimal. This may be seen, for example in dimension d = 2, by choosing for g the topological example described in Section 4,

g(x)= x |x| on Q.
We claim that E ε ≥ α log(1/ε) for some constant α>0. Indeed we may write for any u ∈ H 1 g (Ω),

E ε (u) ≥ α 1 1/2 dr r 1 2 |∇ σ u| 2 + 1 4ε (|u| 2 -1) 2 dσ
where Σ r = {(x, t) ∈ Ω;|x| 2 + t 2 = r 2 } and ∇ σ denote the tangential gradient on Σ r .W e then invoke the lower bound

1 2 r |∇ σ u| 2 + 1 4ε 2 r (|u| 2 -1) 2 ≥ c(log 1/ε)
which is known for a 2-dimensional flat disk (see Bethuel, Brezis and Hélein [1], Theorem V.3) and can be transported to Σ r by a smooth diffeomorphism.

The fact that (5.9) is optimal when d ≥ 2 shows in turn that (5.8) is also optimal for d ≥ 2. Indeed an estimate of the form ϕ H s ≤ o((1 -2s) -1/2 ) in place of (5.8), would yield E ε ≤ o(log 1/ε), which is impossible. When d = 1, estimate (5.8) is still optimal, but this requires a separate argument (see Remark E.1 in Appendix E).

Remark 8. Theorem 4 is still valid for a general smooth domain Q in R d (without any topological assumption); see Remark E.2 in Appendix E. As a result, Theorem 5 is also true in that situation. In Theorem 5 we may also take for Ω any smooth bounded domain in R d+1 ,d ≥ 1 and Q = ∂Ω; this is a consequence of the fact that Theorem 4 is still valid when Q is a smooth d-dimensional manifold (see Remark E.2 in Appendix E). In that case a more elementary (and simple) proof of (5.9) was obtained recently by T. Rivière [START_REF]Dense subsets of H 1/2 (S 2 ; S 1 ), Global analysis and Geometry[END_REF]. Estimate (5.9) plays a fundamental role in the asymptotic analysis (as ε → 0) of Ginzburg-Landau minimizers (see Rivière [1], [2], Lin and Rivière [1], Sandier [1] and also the forthcoming paper Bourgain, Brezis and Mironescu [1]).

APPENDIX A

A characterization of W s,p (Ω; R) when sp < 1

Let Ω = (0, 1) n .F o rj =0 , 1,... we denote by P j the dyadic partition of Ω into 2 jn cubes of side 2 -j and by E j the space of functions from Ω into R (or C) which are constant on each cube of P j . Given a function f ∈ L p (Ω) we consider the function f j = E j (f ) ∈E j defined as follows: every x ∈ Ω belongs to one of the cubes, say Q j (x), of the partition P j and we set

f j (x)=E j (f )(x)= Q j (x)
f.

Clearly we have

(A.1) E j (f ) L p ≤ f L p ∀ j, (A.
2) E j (f ) → f in L p and a.e. as j →∞ .

Theorem A.1. Assume sp < 1. Then

f p W s,p ∼ j≥1 2 spj E j (f ) -E j-1 (f ) p L p ∼ j≥0 2 spj f -E j (f ) p L p .
Remark A.1. Theorem A.1 is due to G. Bourdaud [1] (see his Théorème 5 with m =0 and also the earlier paper of R. Devore and V. A. Popov [1]). It gives a positive answer to a conjecture of H. Triebel [1] (Conjecture 1) for the Haar system {h (-1,0) j } in the spaces B s p,p = W s,p . The parameter ℓ = -1+1-0 = 0 and (for s>0), the condition s<ℓ+(1/p) is indeed sp < 1. For the convenience of the reader, and also because we are interested in the behaviour of the sharp constants in the norm equivalence as sp → 1, we present below a proof of Theorem A.1.

We have also made use of the Corollary A.1. Assume sp < 1 and let (ϕ j ) j=0,1,... be a sequence of functions on Ω such that

(A.3) ϕ j ∈E j ∀j =0, 1 ... and
(A.4) j≥1 2 spj ϕ j -ϕ j-1 p L p < ∞.
Then ϕ j → ϕ in L p and ϕ ∈ W s,p with

(A.5) ϕ p W s,p ≤ C j≥1 2 spj ϕ j -ϕ j-1 p L p .
Remark A.2. Here f W s,p denotes the standard semi-norm,

f p W s,p = Ω Ω |f (x) -f (y)| p |x -y| n+sp dxdy.
To work with a norm it suffices to add | f |.

Proof of Corollary A.1. From (A.4) we see that ϕ j is a Cauchy sequence in L p and thus ϕ j → ϕ in L p . In order to prove that ϕ ∈ W s,p it suffices, in view of Theorem A.1, to check that

(A.6) j≥1 2 spj E j (ϕ) -E j-1 (ϕ) p L p < ∞.
Note that

(A.7) E j (ϕ) -E j-1 (ϕ)=E j (ϕ -ϕ j ) -E j-1 (ϕ -ϕ j-1 )+ϕ j -ϕ j-1
and thus

(A.8) E j (ϕ) -E j-1 (ϕ) L p ≤ ϕ -ϕ j L p + ϕ -ϕ j-1 L p + ϕ j -ϕ j-1 L p
On the other hand, if we write

ϕ j -ϕ =(ϕ j -ϕ j+1 )+(ϕ j+1 -ϕ j+2 )+••• , we see that ϕ j -ϕ L p ≤ k≥j ϕ k -ϕ k+1 L p
so that, by (A.8), we have

(A.9) E j (ϕ) -E j-1 (ϕ) L p ≤ 3 k≥j ϕ k -ϕ k-1 L p .
Thus, by Hölder,

E j (ϕ) -E j-1 (ϕ) L p ≤ 3 k≥j (k -j +1) ϕ k -ϕ k-1 L p 1 (k -j +1) ≤ 3   k≥j (k -j +1) p ϕ k -ϕ k-1 p L p   1/p   k≥j 1 (k -j +1) p ′   1/p ′ and therefore (A.10) E j (ϕ) -E j-1 (ϕ) p L p ≤ C k≥j (k -j +1) p ϕ k -ϕ k-1 p L p . Consequently j≥1 2 spj E j (ϕ) -E j-1 (ϕ) p L p ≤ C j≥1 k≥j 2 spj (k -j +1) p ϕ k -ϕ k-1 p L p = C k≥1 2 spk ϕ k -ϕ k-1 p L p a k (A.11)
where

a k = 1≤j≤k 2 sp(j-k) (k -j +1) p =2 sp 1≤ℓ≤k ℓ p 2 spℓ ≤ a ∞ =2 sp ∞ ℓ=1 ℓ p 2 spℓ .
We deduce from (A.11) and Theorem A.1 that ϕ ∈ W s,p and

ϕ p W s,p ≤ C j≥1 2 spj ϕ j -ϕ j-1 p L p . Proof of Theorem A.1. Set X = f p W s,p Y = j≥1 2 spj E j (f ) -E j-1 (f ) p L p Z = j≥0 2 spj f -E j (f ) p L p .
We will prove that Y ∼ Z and Z ≤ CX without assuming sp < 1. That condition enters only to prove that X ≤ CY .

Step 1: Y ≤ Z Proof. We have, since

E j-1 (f ) ∈E j-1 ⊂E j , E j (E j-1 (f )) = E j-1 (f )
and thus

|E j (f ) -E j-1 (f )| = |E j (f ) -E j (E j-1 (f ))|. Therefore E j (f ) -E j-1 (f ) L p ≤ f -E j-1 (f ) L p
and the estimate Y ≤ Z follows.

Step 2: Z ≤ CY . Here the condition sp < 1 is not used; it suffices to have s>0.

Proof. Set ϕ j = E j (f ); as in the proof of Corollary A.1 we obtain

f -ϕ j L p ≤ k≥j+1 ϕ k -ϕ k-1 L p
and, by Hölder,

f -ϕ j L p ≤   k≥j+1 (k -j) p ϕ k -ϕ k-1 p L p   1/p   k≥j+1 1 (k -j) p ′   1/p ′ . Thus f -ϕ j p L p ≤ C k≥j+1 (k -j) p ϕ k -ϕ k-1 p L p and consequently j≥0 2 spj f -ϕ j p L p ≤ C j≥0 k≥j+1 2 spj (k -j) p ϕ k -ϕ k-1 p L p = C k≥1 2 spk a k ϕ k -ϕ k-1 p L p
where

a k = 0≤j≤k-1 2 sp(j-k) (k -j) p ≤ a ∞ = ∞ ℓ=1 ℓ p 2 spℓ < ∞. Hence Z ≤ Ca ∞ Y.
Step 3: Z ≤ CX. Here, again, the condition sp < 1 is not used.

Proof. Recall that Q j (x) is the cube in the partition P j containing the point x. Write

f (x) -E j (f )(x)=f (x)- Q j (x) f (y)dy = Q j (x) (f (x) -f (y))dy =2 nj Q j (x) (f (x) -f (y))dy
and thus, by Hölder,

|f (x) -E j (f )(x)| p ≤ 2 nj Q j (x) |f (x) -f (y)| p dy. Therefore (A.12) f -E j (f ) p L p ≤ 2 nj Ω dx Q j (x) |f (x) -f (y)| p dy, so that Z = j≥0 2 spj f -E j (f ) p L p ≤ j≥0 2 (n+sp)j Ω dx Q j (x) |f (x) -f (y)| p dy = Ω Ω |f (x) -f (y)| p |x -y| n+sp a(x, y)dxdy,
where a(x, y)=|x -y| n+sp j≥0 2 (n+sp)j 1 Q j (x) (y) and 1 denotes the characteristic function. Clearly a(x, y) ≤ (4n) (n+sp)/2 ∀x, y ∈ Ω and the conclusion follows.

Step 4: X ≤ CY when sp < 1.

Proof. For h ∈ R n set (δ h f )(x)=f (x + h) -f (x),x∈ Ω h =Ω∩ (Ω -h).
A quantity equivalent to X is (A.13)

X ′ = |h|<1 dh |h| n+sp Ω h |(δ h f )(x)| p dx.

We will use the following two lemmas

Lemma A.1. We have, with some constant C (depending only on p, α and β), for all h ∈ R n and all j ≥ 1

δ h f p L p (Ω h ) ≤ C   j k=1 2 α(j-k)p δ h (f k -f k-1 ) p L p (Ω h ) + ∞ k=j+1 2 β(k-j)p f k -f k-1 p L p (Ω)   ,
where α>0 and β>0 will be chosen later.

Proof. As above, write

f = f 0 + k≥1 (f k -f k-1 )
and thus

δ h f = k≥1 δ h (f k -f k-1 ), so that δ h f L p (Ω h ) ≤ j k=1 δ h (f k -f k-1 ) L p (Ω h ) +2 ∞ k=j+1 f k -f k-1 L p (Ω) ,
and the conclusion follows from Hölder's inequality.

Lemma A.2. We have, for all h ∈ R n and all ψ ∈E k ,k ≥ 1,

(A.14) δ h ψ p L p (Ω h ) ≤ C|h|2 k ψ p L p (Ω)
where C depends only on p and n.

Proof. Write ψ = Q∈P k a Q 1 Q and thus δ h ψ = Q a Q (δ h 1 Q ).
Therefore, by Hölder

|δ h ψ| p ≤   Q |a Q | p |δ h 1 Q |     Q |δ h 1 Q |   p-1 . But Q |δ h 1 Q |≤2 and thus (A.15) Ω h |δ h ψ| p ≤ C Q |a Q | p Ω h |δ h 1 Q |.
On the other hand (A. 16)

Ω h |δ h 1 Q |≤|Q\(Q -h)| + |(Q -h)\Q|≤C |h| 2 (n-1)k and (A.17) ψ p L p (Ω) = 1 2 nk Q |a Q | p .
Combining (A.15), (A.16) and (A.17) yields (A.14).

Proof of

Step 4 completed. In view of (A.13) we have

X ≤ C ∞ j=1 1 2 j <|h|< 1 2 j-1 dh |h| n+sp Ω h |(δ h f )(x)| p dx.
Combining this with Lemma A.1 we find

X ≤ C(I 1 + I 2 )
where (A.18)

I 1 = ∞ j=1 1 2 j <|h|< 1 2 j-1 2 (n+sp)j j k=1 2 α(j-k)p δ h (f k -f k-1 ) p L p (Ω h ) dh and (A.19) I 2 = ∞ j=1 1 2 j <|h|< 1 2 j-1 2 (n+sp)j ∞ k=j+1 2 β(k-j)p f k -f k-1 p L p (Ω) dh.
The estimate for I 2 is very simple since

I 2 ≤ C ∞ j=1 ∞ k=j+1 2 spj 2 β(k-j)p f k -f k-1 p L p = C ∞ k=2 2 spk b k f k -f k-1 p L p where b k = k-1 j=1 2 sp(j-k) 2 β(k-j)p ≤ b = ∞ ℓ=1 2 (β-s)ℓp < ∞
provided we choose 0 <β<s. Therefore I 2 ≤ CY .

To estimate I 1 we apply Lemma A.2 with ψ =( f k -f k-1 ). Inserting (A.14) in (A.18) we obtain

I 1 ≤ C ∞ j=1 2 spj j k=1 2 (k-j) 2 α(j-k)p f k -f k-1 p L p = Cc ∞ k=1 2 spk f k -f k-1 p L p with c = ∞ ℓ=0 2 (sp-1+αp)ℓ < ∞,
provided we choose 0 <α<(1 -sp)/p (this is the only place where we use the assumption sp < 1). Thus we have proved that I 1 ≤ CY and the proof of Step 4 is complete.

Returning to Theorem A.1 it is a natural question to ask how the norm-equivalence deteriorates when sp → 1. It was already observed that the inequality

j≥1 2 spj ∆ j f p L p ≤ C f p W s,p ,
where ∆ j f = E j (f ) -E j-1 (f ), is independent of the assumption sp < 1. Concerning the other direction, one has the following more precise result when sp is close to 1.

Proposition A.1. Assume sp < 1. Then (A.20) f W s,p ≤ C s(1 -sp)   j≥1 2 spj ∆ j f p L p   1/p
where C is an absolute constant.

Proof. Following the proof of Step 4 with α =(1-sp)/2p and β = s/2 and using the fact that

∞ ℓ=1 2 -aℓ ≤ ∞ 0 dx 2 ax = C/a, we obtain X ≤ 1+ C αp ′ + C βp ′ p-1 (I 1 + I 2 )
and then

I 2 ≤ C(1 + 1 sp )Y I 1 ≤ C 1 -sp Y.
Combining these inequalities yields (A.20).

In particular, with p = 2, we find Corollary A.2. For 1/4 <s<1/2 we have

f H s ≤ C(1 -2s) -1   j≥1 4 sj ∆ j f 2 L 2   1/2
where C is an absolute constant.

The dependence in (1 -2s) -1 for s → 1/2 in Corollary A.2 is optimal as can be seen from the following example.

Lemma A.3. Let 0 <s< 1 2 . Let Ω=( -1, 1) equipped with standard dyadic partition {P j } and

f = (log 1 x )χ [0<x<1]
.

Then (i) f H s (1 -2s) -3/2 (ii)( j≥1 4 js ∆ j f 2 L 2 ) 1/2 ∼ (1 -2s) -1/2 .
Proof.

(i)

f 2 H s = |f (x + h) -f (x)| 2 |h| 1+2s dxdh ≥ x<0<x+h h -(1+2s) (log 1 x + h ) 2 dxdh ≥ j 4 js -2 -j -2 -j+1 (log 1 x ) 2 dx ∼ j j 2 2 -j(1-2s) ∼ (1 -2s) -3 .
(ii) We need to evaluate the increments ∆ j f . Let I ∈P j-1 ,

I =[a, a +2 -(j-1) ] ⊂ [0, 1].
Thus the value of

|∆ j f | on I is (A.21) 2 j a+2 -j a f - a+2 -j+1 a+2 -j f =2 j F (a +2 -j+1 )+F (a) -2F (a +2 -j )
where

F (x)=x log 1 x + x.
For a =0,

(A.22) (A.21) = 2 j |F (2 -j+1 ) -2F (2 -j )| =2 j |2 -j+1 (j -1) -2 -j+1 j =2.
For a = r2 -(j-1) ,r ≥ 1

(A.23) (A.21) 2 j 4 -j F ′′ L ∞ (I) =2 -j 1 x L ∞ (I) ∼ 1 r .
It follows in particular from (A.22), (A.23) that

∆ j f 2 2 ≤ C2 -j r≥1 r -2 = C2 -j 4 js ∆ j f 2 2 ≤ C 2 -j(1-2s) ∼ (1 -2s) -1 .
APPENDIX B Functions in W s,p (Ω; Z) are constant when sp≥1.

A continuous function on a connected space with values into Z must be constant. Functions in the Sobolev space W s,p with sp ≥ 1 have the same property although they need not be continuous. More precisely we have

Theorem B.1. Assume Ω is a connected open set in R n ,n ≥ 1. Let 0 <s<∞ and 1 <p<∞ be such that (B.1) sp ≥ 1,
including s =1and p =1. Then any function f ∈ W s,p (Ω; Z) must be constant.

Remark B.1. Hardt, Kinderlehrer and Lin [1] have stated the same conclusion when s =1 /2 and p = 2 with a sketch of proof. Bethuel and Demengel [1] have also obtained the same result when sp > 1 and the proof we present follows their argument with an additional ingredient to cover the case sp =1.

Proof. It is convenient to split the proof into two steps:

Step 1: the case n = 1.

If sp > 1, the conclusion is obvious since f is continuous by the Sobolev imbedding theorem. If sp = 1, a borderline for the Sobolev imbedding, f need not be continuous, but f is VMO (see e.g. Brezis and Nirenberg [1], Section I.2). Therefore, the essential range of f is connected (see Brezis and Nirenberg [1], Section I.5) and thus f is constant. For the convenience of the reader we reproduce the argument. Set

f ε (x)= B ε (x) f (y)dy and note that dist(f ε (x), Z) ≤ B ε (x) |f (y) -f ε (x)|dy → 0
uniformly in x as ε → 0 (since f ∈ VMO). On the other hand f ε (Ω) is connected and consequently there is some integer

k ε ∈ Z such that f ε -k ε L ∞ → 0a s ε → 0. It follows that k ε → k as ε → 0 with k ∈ Z and f = k a.e. on Ω.
Step 2: the case n ≥ 2.

It suffices to prove that f is locally constant a.e. and thus we may assume, without loss of generality, that Ω = (0, 1) n . For a.e. x ′ =( x 1 ,... ,x i-1 ,x i+1 ,...x n ) in (0, 1) n-1 the function

(B.2) t → ψ(t)=f (x 1 ,...x i-1 ,t,x i+1 ,...x n )
belongs to W s,p (0, 1). This is a consequence of the fact that an equivalent norm for W s,p (R n )(0 <s<1) is given by

| f | p = f p L p + n i=1 1 0 R n |f (x + te i ) -f (x)| p t 1+sp dxdt
where (e i ) denotes the canonical basis of R n (see e.g. Adams [1], p.208-214). Applying

Step 1 we know that for a.e. x ′ ∈ (0, 1) n-1 the function ψ is constant. To complete

Step 2 we rely on the following simple measure theoretical lemma (see e.g. Lemma 2 in Brezis, Li, Mironescu and Nirenberg [1])

Lemma B.1. Let Ω=(0, 1) n and let f be a measurable function on Ω such that for each 1 ≤ i ≤ n and for a.e. x ′ =( x 1 ,... ,x i-1 ,x i+1 ,...x n ) in (0, 1) n-1 the function ψ defined in (B.2) is constant a.e. on (0, 1). Then f is constant a.e. on Ω.

Remark B.2. Assumption (B.1) cannot be weakened. Indeed, the characteristic function of any smooth domain ω compactly contained in Ω belongs to W s,p for any s, p with sp < 1.

Remark B.3. The conclusion of Theorem B.1 is still valid if f :Ω→ Z is a sum of functions in different Sobolev space, i.e., f = k i=1 f i with f i ∈ W s i ,p i (Ω; R) and s i p i ≥ 1 for all i. The proof is identical to the one we have presented above. In particular the conclusion holds if f ∈ H 1/2 + W 1,1 ; this fact will be used in our forthcoming paper Bourgain, Brezis and Mironescu [1].

APPENDIX C Composition in fractional Sobolev spaces

We investigate here the question whether Φ • v belongs to W s,p (Ω) when v belongs to W s,p (Ω) and Φ is smooth. For simplicity we consider only the case Ω = R n . Of course, here, we also assume that Φ(0) = 0. The case of a domain can be treated by extending the functions to R n .

Lemma C.1. Let 0 <s<∞ and 1 <p<∞. Assume

(C.1) v ∈ W s,p (Ω) ∩ L ∞ (Ω). Then (C.2) Φ • v ∈ W s,p (Ω).
Proof. When s is an integer the conclusion is easy via the Gagliardo-Nirenberg inequality.

For example, when s =2

D 2 (Φ • v)=Φ ′ (v)D 2 v +Φ ′′ (v)(Dv) 2 ∈ L p since W 2,p ∩L ∞ ⊂ W 1,2p
by the Gagliardo-Nirenberg inequality. A similar argument holds for higher order derivatives.

We now turn to the case where s is fractional. The conclusion is obvious when 0 <s<1. Suppose now that 1 <s<2. One has to show that

D(Φ • v)=Φ ′ (v)Dv ∈ W s-1,p .
This would require a lemma about products which eludes us.

Instead of this strategy one relies on a characterization of W s,p via finite differences.

Set (δ h u)(x)=u(x + h) -u(x), so that (δ 2 h u)(x)=u(x +2h) -2u(x + h)+u(x). Then (C.3) u ∈ W s,p ⇔ |δ 2 h u(x)| p |h| n+sp dhdx < ∞,
(see Triebel [2], p.110).

The key observation is that δ 2 h (Φ • v) can be estimated in terms of δ 2 h v and δ h v. This is the purpose of our next computation.

Set

X = v(x +2h) Y = v(x + h) Z = v(x). Then (C.4) Φ(X) -Φ(Y )=Φ ′ (Y )(X -Y )+O(|X -Y | 2 ) and (C.5) Φ(Z) -Φ(Y )=Φ ′ (Y )(Z -Y )+O(|Z -Y | 2 ). Since δ 2 h (Φ • v)(x) = (Φ(X) -Φ(Y )) + (Φ(Z) -Φ(Y )), one finds (C.6) |δ 2 h (Φ • v)(x)|≤C(|δ 2 h v(x)| + |δ h v(x + h)| 2 + |δ h v(x)| 2 ). Consequently (C.7) |δ 2 h (Φ • v)(x)| p |h| n+sp ≤ C |δ 2 h v(x)| p |h| n+sp + C |δ h v(x)| 2p |h| n+sp .
The first term on the righthand side of (C.7) is finite since v ∈ W s,p and for the second term we observe that

|δ h v(x)| 2p |h| n+sp = v 2p W s 2 ,2p ≤ C v p L ∞ v p W s,p
by the Gagliardo-Nirenberg inequality (see Lemma D.1). Hence we have proved that Φ • v ∈ W s,p . The same argument extends to a general s>2, s non integer (see e.g. Escobedo [1]).

APPENDIX D

Gagliardo-Nirenberg inequalities and products in fractional Sobolev spaces

We establish here some Gagliardo-Nirenberg type inequalities used in the paper. We also present a proof of Lemma 2 concerning products in fractional Sobolev spaces. These results are presumably known to the experts. For simplicity we work on R n ; the case of a domain can be treated by extending the functions to R n .

Lemma D.1. Let 0 <s<∞, 1 <p<∞. Assume u ∈ W s,p (R n ) ∩ L ∞ (R n ). Then (D.1) u ∈ W r,q , ∀ r ∈ (0,s) with q = sp r , and
(D.2) | u| W r,q ≤ C u 1-(r/s) L ∞ | u| r/s W s,p ,
provided that either (i) both r, s are non integers or (ii) r is an integer.

Here, we use the following semi-norm on W s,p (see e.g. Triebel [2]):

| u| W s,p = D s u L p , if s is an integer ( |δ M h u(x)| p |h| n+sp dxdh) 1/p
if s is not an integer (as usual, M>sis any integer).

Proof of Lemma D.1. It is convenient to observe that, for every s ∈ (0, ∞) and every p ∈ (1, ∞),

(D.3) | u| p W s,p (R n ) ∼ S n-1 dσ y•σ=0 | u(tσ + y)| p W s,p (R) dy.
(When s is not an integer, (D.3) is clear. When s is an integer, (D.3) follows from the fact that the function

A → S n-1 |A(σ, σ, ..., σ)| p dσ 1 p
is a norm on the space of s-linear symmetric forms on R n .) Using (D.3) one sees that the proof of (D.2) reduces to the one-dimensional case.

Also, note that the desired inequality (D.2) is clear when both s and r are not integers. Indeed, in this case, we have, for M>s(and hence M>r )

| u| q W r,q = |δ M h u(x)| q |h| n+rq dxdh ≤ δ M h u q-p L ∞ |δ M h u(x)| p |h| n+rq dxdh ≤ C u q-p L ∞ | u| p W s,p .
Therefore, it suffices to establish (D.2) for n = 1 and s ≥ 1. We follow the proof of Nirenberg [1]. By the Sobolev imbedding theorem, we have (since sp > 1),

W s,p ([0, 1]) ⊂ W r,q ([0, 1]). Hence (D.4) | u| W r,q ([0,1]) ≤ C( u L p ([0,1]) + | u| W s,p ([0,1]) ),u ∈ W s,p ([0, 1]).
It then follows that

(D.5) | u| W r,q ([0,1]) ≤ C( u L ∞ ([0,1]) + | u| W s,p ([0,1]) ),u ∈ W s,p ([0, 1]).
By scaling, we find

| u| q W r,q ([0,ℓ]) ≤ C(ℓ 1-sp u q L ∞ ([0,ℓ]) + ℓ ( s r -1)(sp-1) | u| q W s,p ([0,ℓ]) ), (D.6) = C(A(ℓ)+B(ℓ)),u ∈ W s,p ([0,ℓ]).
It clearly suffices to prove (D.2) in [0, ∞) and we may assume that u W s,p = 1. Fix some ε>0. We construct inductively a sequence of disjoint intervals

I 1 ,I 2 ,... such that [0, +∞)=I 1 ∪ I 2 ∪••• as follows:
We compare A(ε) and B(ε). If B(ε) ≥ A(ε), then we take I 1 =[0,ε) and next construct I 2 . Otherwise, note that lim ℓ→∞ A(ℓ) = 0, lim ℓ→∞ B(ℓ)=∞ (unless u ≡ 0, which is not the case). Hence there is some ε<ℓ<∞ such that A(ℓ)=B(ℓ). It then follows that

| u| q W r,q ([0,ℓ]) ≤ C u q-p L ∞ ([0,ℓ]) | u| p W s,p ([0,ℓ]) .
In this case we take I 1 =[ 0 ,ℓ). We next start the above procedure from the endpoint of I 1 . Since at each step we have |I j |≥ε, we clearly cover in this way [0, ∞) with a sequence of intervals. Denote the first type of intervals by I j and the second type by K j . Using the assumption that r is an integer we have

| u| q W r,q ([0,∞)) = I j | u| q W r,q (I j ) + K j ••• ≤ Cε ( s r -1)(sp-1) I j | u| q W s,p (I j ) + C u q-p L ∞ (R) K j | u| p W s,p (K j ) .
Note that, since q>p , we have

I j | u| p W s,p (I j ) ≤ 1 ⇒ I j | u| q W s,p (I j ) ≤ 1. Hence (D.7) | u| q W r,q ([0,∞]) ≤ Cε ( s r -1(sp-1) + C u q-p L ∞ (R) | u| p W s,p (R) .
We conclude by letting ε → 0 in (D.7) (the constants C are of ε).

Remark D.1. The conclusion of Lemma D.1 fails when s = 1 and p = 1. For example

W 1,1 (R) ∩ L ∞ (R) is not contained in W 1/2,2 (R)-because this would imply the inequality u W 1/2,2 ≤ C u W 1,1
which is clearly wrong (use for example the sequence in Remark 3).

Remark D.2. In the general case (no restrictions on r and s), the conclusions of Lemma D.1 are still true (the remaining case, i.e., s integer and r non integer, is treated in T.

Runst [1], Lemma 5.2.1).

We next prove a regularity result for products in Sobolev spaces.

Lemma D.2. Let n ≥ 1, 1 <s<∞, 1 <p<∞. Let u, v ∈ W s,p (R n ) ∩ L ∞ (R n ). Then uDv ∈ W s-1,p (R n ).
Proof of Lemma D.2. If s is an integer, the conclusion follows easily from the Gagliardo-Nirenberg inequality. We henceforth assume that s is not an integer. We use a Littlewood-Paley decomposition technique (see e.g. Bony [1], Alinhac and Gérard [1] or Chemin [1]). Let ψ 0 ∈ C ∞ 0 (R n ) be such that ψ 0 (ξ)=1 i f|ξ|≤1 and ψ o (ξ)=0 i f|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.

For f ∈S ′ , let f j = f * ϕ j , so that f = j≥0 f j in S ′ .

We have uDv = (r j + s j ), where

r j = u j k≤j-1
Dv k and s j = Dv j k≤j u k .

Since clearly

k≤j ϕ k L 1 ≤ C, k≤j Dϕ k L 1 ≤ C2 j , ∀j ≥ 0, we obtain (D.8) k≤j v k L q ≤ C v L q , ∀q, (D.9) k≤j Dv k L q ≤ C2 j v L q , ∀q,
and the same inequalities hold for u. Therefore, (D.10)

r j p L p ≤ C u j p L p k≤j-1 Dv k p L ∞ ≤ C2 jp u j p L p v p L ∞ .
On the other hand, v j = k≤j+2 (v j ) k , since, for k ≥ j +3,

F ((v j ) k )=F (v)ψ j ψ k =0.
Therefore,

Dv j L q = k≤j+2 D(v j ) k L q ≤ C2 j v j L q , ∀q,
by (D.9) applied to v j . Consequently, (D.11)

s j p L p ≤ C u p L ∞ Dv j p L p ≤ C2 jp v j p L p u p L ∞ .
We now recall two basic facts about W σ,p , σ>0, σ non integer, 1 <p<∞. Let f ∈ W σ,p and let f j = f * ϕ j as above. Then (D.12)

f p W σ,p ∼ j≥0 2 σjp f j p L p
(see e.g. Triebel [2], p. 46).

Conversely, let g j be a sequence in L p such that suppF (g j ) ⊂ B 2 j . Then (D.13)

j≥0 g j p W σ,p ≤ C j≥0 2 σjp g j p L p
(see e.g. Chemin [1], p. 27). Using (D.10), (D.11) and (D.12) (with σ = s), we find

(D.14) j≥0 2 (s-1)jp r j + s j p L p ≤ C u p L ∞ v p W s,p + v p L ∞ u p W s,p .
Since suppF (r j + s j ) ⊂ B 2 j+3 , (D.13) (applied with σ = s -1 and g j = r j + s j ) combined with (D.14) yields that uDv ∈ W s-1,p and that

(D.15) uDv W s-1,p ≤ C( u L ∞ v W s,p + v L ∞ u W s,p ).
Remark D.3. As a consequence of Lemma D.2, we derive the well-known fact that W s,p ∩ L ∞ is an algebra.

APPENDIX E Behaviour of the H s -norm of lifting for s

→ < 1
2 . Proof of Theorem 4 We return to the particular issue of lifting of an unimodular function F in H s , s< 1 2 . As we have pointed out in Section 5 the construction described in Appendix A of a lifting (E.1) F = e iϕ ,ϕ∈ H s does not lead to the optimal estimate of ϕ H s when s → 1 2 . Our aim is to prove Theorem E.1. Let Q be a cube of R d ,d ≥ 1. For every F ∈ H s (Q; S 1 ) with 0 <s<1/2 one may construct a ϕ ∈ H s (Q; R) satisfying (E.1) and the (optimal) estimate

(E.2) ϕ H s ≤ C(1 -2s) -1/2 F H s
where C is a constant independent of F and independent of s as s → 1/2.

Proof. Given an unimodular H s -function F on a cube, say Q =[ 0 , 1 2 ] d ⊂ R d ,w em a y extend F to a 1-periodic unimodular function in H s loc (R d ) by the usual procedure of reflections and periodic continuation. Hence, we may assume F ∈ H s (T d ; S 1 ), where T d = d-dim torus. This setting is particularly convenient to perform our translation averaging. On Ω=T d , we fix again a system {P j } j=0,1,2,... of refining dyadic partitions (thus the atoms of P j are d-intervals of size ∼ 2 -j ) and denote E j the corresponding expectation operators. Denote also τ θ the shift operators on T d . We perform the following construction. Given F ∈ H s (Ω; S 1 ), denote F θ = F • τ θ and ϕ[θ] the lifting of F θ gotten from the construction described in Section 1 (with fixed P j 's). Thus

(E.3) F θ = e iϕ[θ] and F = e i(ϕ[θ]•τ -θ )
and ϕ[θ] • τ -θ = ϕ is a lifting for F . Thus Theorem 4 will follow immediately from the next statement.

Lemma E.1. We have

T d ϕ[θ] H s dθ ≤ C(1 -2s) -1/2 F H s .
Proof. We show in fact that

(E.4) ϕ[θ] 2 H s dθ ≤ C(1 -2s) -1 F 2 H s .
The lefthand side of (E.4) equals Writing

|ϕ[θ] -τ h ϕ[θ]| 2 (x) |h| 2s+d dxdhdθ ∼ j≥0 2 (2s+d)j |h|∼2 -j ϕ[θ] -τ h ϕ[θ]
(E.6) ϕ = E j ϕ + j ′ >j ∆ j ′ ϕ (∆ j ′ = E j ′ -E j ′ -1 ) estimate (E.7) ϕ -τ h ϕ 2 2 E j ϕ -τ h E j ϕ 2 2 + j ′ >j (j ′ -j) 2 ∆ j ′ ϕ 2 2 .
Recall inequality (1.5) in Section 1

(E.8) |ϕ j -ϕ j-1 |≤C(|F θ -E j (F θ )| + |F θ -E j-1 (F θ )|).
Hence, since ϕ j = E j (ϕ j ), we have

∆ j ϕ 2 ≤ E j (ϕ -ϕ j ) 2 + E j-1 (ϕ -ϕ j-1 ) 2 + ϕ j -ϕ j-1 2 (E.9) ≤ C j ′ ≥j ϕ j ′ -ϕ j ′ -1 2 ≤ C j ′ ≥j-1 F θ -E j ′ (F θ ) 2 ≤ C j ′ ≥j-1 (j ′ -j +2) ∆ j ′ F θ 2 (E.10) and estimate in (E.7) (E.11) ∆ j ′ ϕ 2 2 ≤ C j ′′ ≥j ′ -1 (j ′′ -j ′ +2) 4 ∆ j ′′ F θ 2 2 .
Thus the contribution of the second term in (E.7) is bounded by

j≥0 2 (2s+d)j |h|∼2 -j j ′ >j (j ′ -j) 2 ∆ j ′ ϕ 2 2 dhdθ ≤ C dθ j≥0 2 2sj j ′′ +2≥j ′ >j (j ′ -j) 2 (j ′′ -j ′ +2) 4 ∆ j ′′ F θ 2 2
≤ C dθ

j ′′ >0 2 2sj ′′ ∆ j ′′ F θ 2 2 . (E.12)
Recalling the proof of Theorem A1 (in particular the inequality Y ≤ CX independent of the assumption 2s<1) we have

(E.13) (E.12) ≤ C dθ F θ 2 H s ≤ C F 2 H s .
Thus the θ-integration is irrelevant here. The main point is the contribution of the first term E j ϕ -τ h E j ϕ 2 2 in (E.5), thus

(E.14) j≥0 2 (2s+d)j |h|∼2 -j |E j ϕ -τ h E j ϕ| 2 dθdhdx. Estimate (E.15) |E j ϕ -τ h E j ϕ|≤ j ′ ≤j |∆ j ′ ϕ -τ h ∆ j ′ ϕ|. Write (E.16) ∆ j ′ ϕ = I∈P j ′ a I χ I .
Then, for |h| < 2 -j , one easily verifies that (E.17)

|∆ j ′ ϕ -τ h ∆ j ′ ϕ|≤ I∈P j ′ |a I ||χ I -τ h χ I |≤C(|∆ j ′ ϕ| * P 2 -j ′ )χ j ′ ,2 -j
The role of the θ-translation is that we introduced an extra variable to estimate (E.24). Write F as a Fourier series in

T d F = n∈Z d F (n)e inx . Then ∆ j (F θ )= F (n)e inθ ∆ j (e in. ) (E.25) |(|∆ j F θ | * P ε )(x)| 2 ≤ F (n)e inθ ∆ j (e in.
|∆ j F θ | * P ε 2 L 2 θ ≤ | F (n)| 2 ∆ j (e in. ) 2 ∞ | F (n)| 2 (1 ∧|n|2 -j ) 2 .
To estimate (E.24), perform first the θ-integration using Cauchy-Schwarz and (E.27). This gives, recalling (E.19) (E.28)

j≥0 2 2sj j ′ α ≤j,j ′ α ≤j ′′ α ,j ′ 1 ≤j ′ 2 2 j ′ 1 -j (j ′′ 1 -j ′ 1 + 1)(j ′′ 2 -j ′ 2 +1) n | F (n)| 2 (1 ∧|n|2 -j ′′ 1 ) 2 1/2 n | F (n)| 2 (1 ∧|n|2 -j ′′ 2 ) 2 1/2 .
To evaluate (E.28), denote

ℓ α = j ′′ α -j ′ α ≥ 0( α =1, 2) (E.29) m = j ′ 2 -j ′ 1 ≥ 0 (E.30) so that (E.31) (E.28) = m,ℓ 1 .ℓ 2 ≥0 (ℓ 1 + 1)(ℓ 2 +1) j ′ 1 2 j ′ 1 j≥j ′ 1 2 (2s-1)j . n | F (n)| 2 (1 ∧|n|2 -j ′ 1 -ℓ 1 ) 2 1/2 n | F (n)| 2 (1 ∧|n|2 -j ′ 1 -m-ℓ 2 ) 2 1/2 .
Applying Cauchy-Schwarz for the j

′ 1 -summation (E.32) (E.31) ≤ C m,ℓ 1 ,ℓ 2 (ℓ 1 + 1)(ℓ 2 + 1)(1 -2s) -1 n,j ′ 1 | F (n)| 2 2 2sj ′ 1 (1 ∧|n|2 -j ′ 1 -ℓ 1 ) 2 1/2 n,j ′ 1 | F (n) 2 2 2sj ′ 1 (1 ∧|n|2 -j ′ 1 -m-ℓ 2 ) 2 1/2 . Writing (E.33) j 2 2sj (1 ∧|n|2 -j-ℓ ) 2 ∼ 2 -2sℓ (1 + |n|) 2s it follows that (E.34) (E.32) ≤ C 1 -2s m,ℓ 1 ,ℓ 2 (ℓ 1 + 1)(ℓ 2 + 1)2 -s(ℓ 1 +ℓ 2 +m) n | F (n)| 2 (1 + |n|) 2s ≤ C(1 -2s) -1 F 2 H s .
Since (E.5) is bounded by the sum of (E.13) and (E.34), this proves Lemma E.1.

Remark E.1. The optimality of the bound (E.2) when d = 2 was proved in Remark 7. The case d ≥ 3 is similar by choosing

g(x)= (x 1 ,x 2 ) (x 2 1 + x 2 2 ) 1/2 x =(x 1 ,x 2 ,... ,x d )
and proceeding as in the 2-dimensional case. The optimality of (E.2) when d = 1 is more delicate and will be established in the forthcoming paper Bourgain, Brezis and Mironescu [1].

Remark E.

Q α ; R). For θ ∈ R d with |θ| <δ , δ sufficiently small, F θ = F • τ θ is well defined on α Q α has a lifting ϕ[θ]
. The proof of Lemma E.1 described above can be adapted and yields

|θ <δ ϕ[θ] H s dθ ≤ C(1 -2s) -1/2 F H s .
Theorem E.1 is also valid if the cube Q is replaced by a smooth d-dimensional manifold M, d ≥ 1, say without boundary. The dyadic partition of Q is replaced by some dyadic "triangulation" of M. The shift operators τ θ are replaced by a finite family {S i (t)}, 1 ≤ i ≤ N of 1-parameter group of transformations on M such that, at each x ∈ M, the generators V i (x)= d dt S i (t)x | t=0 span the tangent space T x (M). Such a family can be easily constructed as integral curves for the differential equations ẋ(t)=V i (x(t)) and the vectorfields V i (x) are obtained via local coordinates and a partition of unity. The shift operators τ θ are replaced by the shifts along the S i , i.e., σ θ =Π i S i (t i ), where θ =( t 1 ,t 2 ,... ,t N ), and then

F θ = F • σ θ . Adapting the proof of Lemma E.1 we find θ∈R N ,|θ|<1 ϕ[θ] dθ ≤ C(1 -2s) -1/2 F H s .

APPENDIX F Martingale representation and lifting in H s,p

The question of representation and lifting can be raised in other function spaces. For instance, in the H s,p space.

Recall the definition of the H s,p -norm (0 <s<1)

(F.1) f H s,p = |f (x + h) -f (x)| 2 |h| 2s+d dh p/2 dx 1/p .
This space is a bit more delicate to deal with then W s,p . The natural martingale counterpart of (F.1) is given by

(F.2) 2 2js |∆ j f | 2 1/2 p
where ∆ j f = E j (f ) -E j-1 (f ) and E j is the conditional expectation operator with respect to P j (as before). This situation is a bit different from W s,p . We show the following Proposition F.1. (i) We have

(F.3) 4 js |∆ j f | 2 1/2 p ≤ C|f H s,p
(ii) If sp < 1 and p ≥ 2, then the converse inequality holds

(F.4) f H s,p ≤ C 4 js |∆ j f | 2 1/2 p (iii) Inequality (F.4) fails for s> 1 2 .
Proposition F.1 leaves some cases unanswered and they will possibly be addressed elsewhere. Again, Proposition F.1 is relevant to the question of Triebel [1] concerning the representation of Besov and Sobolev spaces in the Haar-system. It implies that for the spaces H s,p = F s p,2 , the conjecture is valid if ps < 1,p ≥ 2 but fails for s> 1 2 .

In the proof of Proposition F.1, we will make use of some standard martingale inequalities (which the reader may find in Garsia [1] for instance).

Proposition F.2. We have

(F.5) E j (g j ) p ≤ C p |g j | p for 1 ≤ p<∞ and (F.6) |E j (g j )| 2 1/2 p ≤ C p |g j | 2 1/2 p for 1 <p<∞.
In both statements, the sequence {g j } consists of arbitrary functions.

Remark F.1. In (F.5), (F.6), the expectation operators E j may get replaced by convolution operator P 2 -j for instance, where P ε stands for the usual Poisson kernel (cf. Stein [l]).

Proof of Proposition F.1.

(i) By (F.6) (F.7) 4 js |∆ j f | 2 1/2 p ≤ C 4 js |f -E j-1 (f )| 2 1/2 p . Again |(f -E j-1 (f ))(x)|≤2 jd |h|<2 -j |f (x) -f (x + h)|dh |f -E j-1 (f )| 2 ≤ 2 jd |h|<2 -j |f -τ h f | 2 dh. (F.8)
where τ h is the translation operator. Substituting (F.8) in (F.7) implies (F.9) Use the general inequality (see Remark F.1) (F.17) j P 2 -j g j q ≤ C q j |g j | q for 1 ≤ q<∞.

(F.7) ≤ dh |f -τ h f | 2 |h|<2 -j 4 js 2 jd 1/2 p ∼ |f -τ h f | 2 |h| -(d+2s) dh 1/2 p = f H s,p . (ii) Write (F.10) |f -τ h f | 2 |h| -(d+2s) dh ∼ j 2 j(d+2s) |h|∼2 -j |f -τ h f | 2 dh. Fix j. Estimate |f -τ h f |≤|f j -τ h f j | + |f -f j | + τ h |f -f j | |f -τ h f | 2 j ′ <j (j -j ′ ) 2 |∆ j ′ f -τ h (∆ j ′ f )| 2 + |f -f j | 2 + τ h |f -f j | 2 (F.
Thus, since p ≥ 2, letting q = p/2 in (F.17 Denote T ℓ p the norm of (F.20). We estimate T ℓ p , 2 ≤ p, by interpolation between 2 and some large q.

Fixing 2 <q<∞, we may bound

T ℓ ḡ L q L 2 h ℓ 2 ≤ E j |g j |.2 (j+ℓ)d/2 χ [|h|<2 -(j+ℓ)] L q L 2 h ℓ 2 + τ h (E j |g j |).2 (j+ℓ)d/2 χ [|h|<2 -(j+ℓ) ] L q L 2 h ℓ 2
=(F.22) + (F.23).

Thus, invoking (F.6) (F.24) (F.22) ∼ (E j |g j |) 2 1/2 q ≤ C q ḡ L q ℓ 2 .

Also, since q>2 and using inequalities (F.17 ≤ C ḡ L q ℓ 2 . (F.25) Thus T ℓ ḡ L q L 2 h ℓ 2 ≤ C q ḡ L q ℓ 2 , i.e.

(F.26) T ℓ q ≤ C q for 2 ≤ q<∞.

Next, for p = 2, a direct calculation gives

T ℓ ḡ L 2 x L 2 h ℓ 2 = j 2 (j+ℓ)d |h|<2 -(j+ℓ)
|(E j g j )(x) -(E j g j )(x + h)| 2 dxdh Since sp < 1, we may take ε sufficiently small to ensure boundedness of the factor in (F.34), leading again to the bound ( 4 j ′ s |∆ j f | 2 1/2 p .

This establishes inequality (F.4).

(iii)T a k ed = 1 and define (F.35) f j =2 -js Consequently, letting δ → 0, we see that inequality (F.4) cannot hold for s> 1 2 . This completes the proof of Proposition F.1.

There is the following application of Proposition F.1 to the lifting problem of unimodular functions.

Corollary F.1. Let s>0,sp < 1,p ≥ 2 and F ∈ H s,p (Ω; S 1 ), where Ω is a cube in R d .

Then (F.44)

F = e iϕ for some ϕ ∈ H s,p (Ω).

Remark F.2. The other cases not covered by the corollary have not been investigated.

Proof. The function ϕ is constructed as in the W s,p -case (see Section 1). ¿From Proposition F.1, (i), (ii) and similar calculations as in the W

H 1 g

 1 (Ω = {u(x, t):Ω→ C ; Ω |∇u| 2 dxdt < ∞ and u(x, 0) = g(x)o nQ},

Du 1 =

 1 -(sin ϕ)Dϕ = -u 2 Dϕ and Du 2 = (cos ϕ)Dϕ = u 1 Dϕ. Hence (3.1) Dϕ = u 1 Du 2 -u 2 Du 1 .

  by (5.11). (5.20) Define (5.21) u = G δ + P so that by construction u |t=0 = g on Q.

  2 2 dhdθ. (E.5) Denote ϕ[θ]b yϕ for simplicity. Fix j.

4

  [START_REF]Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type[END_REF] and substituting (F.11) in (F.10), we get the following contributions(F.10) ≤ C j ′ <j 2 j(d+2s) (j -j ′ ) 2 |h|∼2 -j |∆ j ′ f -τ h (∆ j ′ f )| 2 dh (F.12) + j 4 js |f -f j | 2 (F.13) + j 4 js [P 2 -j * (|f -f j | 2 )(j ′ -j) 2 |∆ j ′ f | 2 js [P 2 -j * (|f -f j | 2 )]

2 p 2 p.

 22 ), it follows (F.16) ≤ C j 4 js |f -f j | Contribution of (F.12)Denoting ℓ = j -j ′ ≥ 0, write (F.19) (F.12) 1/2 p ≤ ℓ≥0 ℓ2 ℓs j ′ 4 j ′ s 2 (j ′ +ℓ)d |h|≤2 -(j ′ +ℓ) |∆ j ′ f -τ h (∆ j ′ f )| 2 dh 1/To bound (F.19), fix ℓ and consider the map(F.20) T ℓ : L p ℓ 2 T ℓ ḡ = T ℓ ({g j })={(E j g j -τ h E j g j )2 (j+ℓ)d/2 χ [|h|<2 -(j+ℓ) ] }Thus the components of T ℓ ḡ are functions of x and h.

  ), (F.6)(F.23) ≤ C j (E j |g j |) 2 * P 2 -(j+ℓ) |g j |) 2 * P 2 -(j+ℓ)

≤ 2 =

 2 C2 -ℓ/2 ḡ L 2 ℓ 2 . (F.29)The estimate (F.28) simply results from the fact that for I ∈P j and |h| < 2 -(j+ℓ) (F.30)χ I (x) -χ I (x + h) L 2 x ≤ C2 (-d-1)j/2-j+ℓ C2 -ℓ/2

2 j r=1 (- 1 )ε 2 1/ 2 p= R 1 / 2 . 1 2. 1 2

 2r=11221211 r χ I r where P j = {I 1 ,... ,I 2 j }.Fix a large integer R and let {j r } r=1,... ,R be a lacunary sequence. r f j r where ε r = ±1 are independent signs. Thus ∆ j r f = ε r f j r and trivially (F.37)4 js |∆ j f |Next, take δ>0 a small number and write(F.38) |f -τ h f | 2 |h| -(1+2s) dh ≥ R r=1 (δ2 -j r ) -(1+2s) |h|<δ2 -j r |f -τ h f | 2 dh.Averaging over the ± signs ε r in (F.36) permits us clearly to ensure that(F.39) (F.38) ≥ r (δ2 -j r ) -(1+2s) |h|<δ2 -j r |f j r -τ h f j r | 2 dh.Recalling (F.35), one sees that(F.39) ≥ c r (δ2 -j r ) -(1+2s) (δ2 -j r )4 -j r s I∈P j r χ [dist (x,∂I)< δ2 -j r ]Fixing δ>0 and letting R>R (δ) be sufficiently large, the reader will easily convince himself that (F.43) (F.42) ≥ cδ -s (δR) 1/2 = cδ -s .(F.37).

  1/2 (Ω; S 1 ) admits a lifting ϕ ∈ H 1/2 (Ω; S 1 ). Moreover, this lifting is unique modulo an additive constant (see Appendix B) and the map u → ϕ is continuous from H 1/2 into H 1/2 (this can be established using the same argument as in Step 7 of the proof of Theorem 4 in Brezis-Nirenberg[1]).

Surprisingly there is no bound whatsoever for ϕ H 1/2 in terms of u H 1/2 . Here is an example of a sequence (ϕ n ) such that ϕ n H 1/2 → +∞ while e iϕ n H 1/2 ≤ C.O n Ω=(-1, +1) consider the sequence of functions ϕ n defined by

  2. Theorem E.1 is still valid if the cube Q is replaced by a smooth domain Ω in R d ,d ≥ 2 (without any topological assumption on Ω). The proof can be modified as follows. Consider a neighborhood Ωo f Ω and a function still denoted F , F ∈ H s ( Ω; S 1 ) which extends the original F (this can be done by the standard procedure of local reflexion across the boundary). Next, construct a finite sequence of disjoint cubes (Q α ), having the same size, and such that Ω ⊂

α Q α ⋐ Ω. The construction described in Section 1 is still valid on α Q α and we obtain a lifting ϕ ∈ H s ( α

  2 -dj / 2 . =2 j ′ s ∆ j ′ f so that, by (F.32) (F.19) ≤ ℓ≥0 ℓ2 ℓs T ℓ {g j ′ } L p

	¿From (F.29), (F.31) Interpolating 2 <p<q, it results from (F.26), (F.31) that T ℓ 2 ≤ C2 -ℓ/2 . (F.32) T ℓ p <C ε 2 -ℓ( 1 p -ε) for all ε>0. Returning to (F.19), we define thus (F.33) h ℓ 2 g j ′ L 2 ≤ C ε ℓ≥0 ℓ2 ℓs 2 -ℓ( 1 p -ε) {g j ′ } L p ℓ 2 . (F.34)

  s,p -estimate, we obtain (with the notations from Section 1)ϕ H s,p ≤ C 4 js |∆ j ϕ| 2 ′ >j 4 js (j ′ -j) 2 |ϕ j ′ -ϕ j ′ -1 | 2 ′ >j 4 js (j ′ -j) 2 |F -E j ′ -1 F | 2 ′′ ≥j ′ >j 4 js (j ′ -j) 2 (j ′′ -j ′ +1) 2 |∆ j ′′ F | 2 ′′ s |∆ j ′′ F | 2

			1/2
			p
				1/2			1/2
	(F.45)	≤ C	4 js E j (ϕ -ϕ j )| 2	+	4 js |ϕ j -ϕ j-1 | 2	p
				p
		≤ C by (F.6)	4 js |ϕ -ϕ j | 2	1/2	+	4 js |ϕ j -ϕ j-1 | 2	1/2
				p			p
					1/2
	(F.46)	≤ C
					p
	(F.47)	by(1.5) ≤ C				1/2
						p
						1/2
	(F.48)	≤ C
						p
		≤ C	4 j 1/2
		j ′′		p
	(F.49)	≤ C F H s,p .

j j j

Acknowledgment: The second 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. We thank Y. Meyer and G. Bourdaud for providing us useful references.

where χ j ′ ,2 -j denotes the characteristic function of the set (E.18) {x; dist (x, ∂I) ≤ 2 -j for some I ∈P j ′ } and P ε denotes the usual Poisson-kernel for instance. Thus

Substituting (E.17) in (E.15) implies (since

Next,

and again from inequality (E.8)

We


Num Analyse

Érique

Theorem 1. Assume f ∈ W 1,p (Ω), 1 ≤ p < ∞ and let ρ ∈ L 1 (R n ), ρ ≥ 0. Then Ω Ω |f (x) -f (y)| p |x -y| p ρ(x -y)dxdy ≤ C f p W 1,p ρ L 1
where C depends only on p and Ω.

Next, we take a sequence (ρ n ) of radial mollifiers, i.e.

ρ n (x) = ρ n (|x|)

ρ n ≥ 0, ρ n (x)dx = 1 lim n→∞ δ ∞
ρ n (r)r n-1 dr = 0 for every δ > 0.

Theorem 2. Assume f ∈ L p (Ω), 1 < p < ∞. Then

lim n→∞ Ω Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dxdy = K p,N f p W 1,p ,
with the convention that f W 1,p = ∞ if f / ∈ W 1,p . Here K p,N depends only on p and N .

When p = 1 we have the following variants Theorem 3. Assume f ∈ W 1,1 . Then

lim n→∞ Ω Ω |f (x) -f (y)| |x -y| ρ n (x -y)dxdy = K 1,N f W 1,1
where K 1,N depends only on N .

Theorem 3'. Assume f ∈ L 1 (Ω). Then f ∈ BV (Ω) if and only if

lim inf n→∞ Ω Ω |f (x) -f (y)| |x -y| ρ n (x -y)dxdy < ∞,
and then

C 1 f BV ≤ lim inf n→∞ Ω Ω |f (x) -f (y)| |x -y| ρ n (x -y)dxdy ≤ lim sup n→∞ Ω Ω |f (x) -f (y)| |x -y| ρ n (x -y)dxdy ≤ C 2 f BV . (1)
Here C 1 and C 2 depend only on Ω, and

f BV = Ω |∇f | = Sup    Ω f divϕ ϕ ∈ C ∞ 0 (Ω; R N ), |ϕ(x)| ≤ 1 on Ω    .
Remark 1. In dimension N = 1 we can prove that for every f ∈ BV

lim n→∞ Ω Ω |f (x) -f (y)| |x -y| ρ n (x -y)dxdy = K 1,1 Ω |∇f |,
where K 1,1 is the same constant as in Theorem 3. We do not know whether the same conclusion holds when N ≥ 2 (even for a special sequence of mollifiers).

Here are some simple consequences of the above results (and their proofs), where K denotes various constants depending only on p and N .

Corollary 1. Assume f ∈ W 1,p (Ω) with 1 ≤ p < ∞. Then Ω |f (x) -f (y)| p |x -y| p ρ n (x -y) dy -→ K p,N |∇f (x)| p in L 1 (Ω). Corollary 2. Assume f ∈ L p , 1 < p < ∞. Then lim ε→0 ε f p W 1-ε,p = K f p W 1,p . Corollary 3. Assume f ∈ L p , 1 < p < ∞. Then lim ε→0 ε N |x-y|<ε |f (x) -f (y)| p |x -y| p dxdy = K f p W 1,p . Corollary 4. Assume f ∈ L p , 1 < p < ∞. Then lim ε→0 1 |logε| |x-y|>ε |f (x) -f (y)| p |x -y| N+p dxdy = K f p W 1,p .
Remark 2. P. Mironescu and I. Shafrir [] have studied related limits, e.g., when N = 1 and f ∈ BV (0, 1)

lim ε→0 1 |logε| |x-y|>ε |f (x) -f (y)| 2 |x -y| 2 dxdy.
The case where f ∈ BV (Ω) is not fully satisfactory; we have only partial results, for example

Corollary 5. Assume f ∈ L 1 . Then C 1 f BV ≤ lim inf ε→0 ε Ω Ω |f (x) -f (y)| |x -y| N+1-ε dxdy ≤ lim sup ε→0 ε Ω Ω |f (x) -f (y)| |x -y| N+1-ε dxdy ≤ C 2 f BV .
Remark 3. In particular when f = χ A is the characteristic function of a measurable set A having finite perimeter, then

χ A BV ≤ C lim inf ε→0 ε Ω\A A dxdy |x -y| N+1-ε
and in view of the isoperimetric inequality

|A||Ω \ A| (N-1)/2N ≤ C lim inf ε→0 ε Ω\A A dxdy |x -y| N+1-ε If A is a measurable subset of Ω ⊂ R N , N ≥ 1, such that Ω\A A dxdy |x -y| N+1 < ∞,
then either |A| = 0 or |Ω \ A| = 0. This fact was already established in Bourgain, Brezis and Mironescu [START_REF] Adams | Sobolev spaces[END_REF] (Appendix B) with a different proof (see also Bourgain, Brezis and Mironescu [START_REF] Bourgain | Lifting in Sobolev spaces[END_REF] and Brezis []).

Proofs

Proof of Theorem 1. By standard extension we may always assume that f ∈ W 1,p (R N ) and then there is some constant C = C(p, N ) such that (2)

R N |f (x + h) -f (x)| p dx 1/p ≤ |h| f W 1,p (R N ) ≤ C|h| f W 1,p (Ω) ,
for all f ∈ W 1.p and h ∈ R N (see, e.g., Brezis [], Proposition IX.3). By (2), we obtain

Ω Ω |f (x) -f (y)| p |x -y| p ρ(x -y)dxdy ≤ R N 1 |h| p R N |f (x + h) -f (x)|dxdh p ≤ C p f p W 1,p R n ρ(h)dh = C p f p W 1,p ρ 1 L .
Proof of Theorem 2. For f ∈ L p , let

F n (x, y) = |f (x) -f (y)| |x -y| ρ 1/p n (x -y).
Assuming first that f ∈ W 1,p , we have to prove that

(3) lim n→∞ F n p L p = K f p W 1,p ,
for some K = K p,N . By Theorem 1, we have, for any n and f, g

∈ W 1,p , (4)
F n -G n L p ≤ C f -g W 1,p ,
for some constant C independent of n, f and g. Therefore it suffices to establish (3) for f in some dense subset of W 1,p , e.g., for f ∈ C 2 ( Ω).

Fix some f ∈ C 2 ( Ω). Then

|f (x) -f (y)| |x -y| = |(∇f )(x) • x -y |x -y| + 0(|x -y|).
For each fixed x ∈ Ω,

Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dy = |x-y|<dist(x,∂Ω) |f (x) -f (y)| p |x -y| p ρ n (x -y)dy + |x-y|≥dist(x,∂Ω) |f (x) -(y)| p |x -y| p ρ n (x -y)dy. (5)
Clearly, the last integral in (5) tends to 0 as n → ∞. On the other hand, with R = dist(x, ∂Ω), we have

|y-x|<R |f (x) -f (y)| p |x -y| p ρ n (x -y)dy = R 0 ρ n (r) |y-x|=r (∇f )(x). x -y |x -y| p + (|x -y|) p ds y dr = R 0 ρ n (r) |ω|=r (|(∇f )(x) • ω |ω| | p + 0(r p ))ds w dr = R 0 |S N-1 |K|∇f (x)| p r N-1 ρ n (r)dr + 0( R 0 r n+p-1 ρ n (r)dr),
where

K = K p,n = ω∈S n-1 |ω N | p ds ω /|S n-1 | = ω∈S N -1 |ω N | p ds ω Therefore, (6)
Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dy -→ K|∇f (x)| p , ∀x ∈ Ω. If L is such that |f (x) -f (y)| ≤ L|x -y|, ∀x, y ∈ Ω, then (7)
Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dy ≤ L p , ∀x ∈ Ω.
Hence (3) for f ∈ C 2 ( Ω) follows by dominated convergence from ( 6) and (7).

In order to complete the proof of Theorem 2, to suffices to prove that, if f ∈ L p and lim inf

n→∞ Ω Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dxdy < ∞, then f ∈ W 1,p .
Recall that, for some C 1 > 0,

f W 1,p ≤ C 1 sup Ω f ∂ i ϕ ; ϕ ∈ C ∞ 0 (Ω), ϕ L p1 ≤ 1, i = 1, ..., N .
Fix some ϕ ∈ C ∞ 0 (Ω) and some i ∈ {1, ..., n} and consider the functions

x H n -→ (y-x)•e i ≥0 f (x) -f (y) |x -y| ρ n (x -y)ϕ(y)dy.
On the one hand, we have

Ω |H n (x)|dx ≤ Ω Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dxdy 1/p Ω Ω ρ n (x -y)|ϕ(x -y)| p ′ dxdy 1/p = ϕ L p ′ Ω Ω f (x) -f (y)| p |x -y| p ρ n (x-)dxdy 1/p . (8)
As above,

H n (x) n→∞ -→ f (x) (∇ϕ)(x) • ωds ω = {ω ∈ S N-1 , •ω i ≥ 0}f (x)(∇ϕ)(x)• ωds ω = C 2 f (x)(∇ϕ) • e i = C 2 f (x)∂ i ϕ(x), {w ∈ § N-1 ; w i ≥ 0}
for some C 2 > 0 depending only on N .

Therefore, by combining (8) and(9) we find

| Ω f (x)∂ i ϕ(x)dx| ≤ C 3 lim inf n→∞ Ω Ω |f (x) -f (y)| p |x -y| p ρ n (x -y)dxdy 1/p ϕ L p ,
for any ϕ ∈ C ∞ 0 (Ω). The proof of Theorem 2 is complete.

Proof of Corollary 1. The conclusion is clear when f ∈ C 2 ( Ω). For a general ∈ W 1,p , the statement follows by density using [START_REF] Brezis | Analyse fontionnelle[END_REF].

The proof of Theorem 3 is the same as the first part of the proof of Theorem 2, since smooth functions are dense in W 1,1 .

Proof of Theorem 3. The last inequality in (1) is proved as in Theorem 1. The first inequality in ( 1) is proved as in the second part of the proof of Theorem 2 (using duality).

Finally, we return to Remark 1 and prove that , for f ∈ L 1 ((0, 1)), ( 10) lim

n→∞ 1 0 1 0 |f (x) -f (y)| |x -y| ρ n (x -y)dxdy = f BV Clearly, if f ∈ BV ((0, 1)), then |f (x + h) -f (x)|dx ≤ |h| f BV ,
and therefore, as in the proof of Theorem 1, (11) lim sup

n→∞ 1 0 1 0 |f (x) -f (y)| |x -y| ρ n (x -y)dxdy ≤ f BV . Assume now that f ∈ L 1 is such that lim inf n→∞ 1 0 1 0 |f (x) -f (y)| |x -y| ρ n (x -y)dxdy < ∞
Then, for any fixed ϕ ∈ C ∞ 0 ((0, 1)) with |ϕ| ≤ 1, we have

1 0 1 0 |f (x) -f (y)| |x -y| ρ n (x -y)dxdy ≤ 1 0 1 0 f (x) -f (y) y -x ρ n (x -y)ϕ(y)dy = 1 0 1 0 f (x)ρ n (x -y) ϕ(y) -ϕ(x) y -x dydx. (12)
As in the proof of Theorem 3, we find that

1 0 f (x)ϕ ′ (x)dx ≤ lim inf n→∞ 1 0 1 0 |f (x) -f (y)| |x -y| ρ n (x -y)dxdy, ∀ϕ ∈ C ∞ 0 (Ω) with|ϕ| ≤ 1. ( 13
)
Equality (10) follows from (11) and (13).

The case of a sequence (f n )

In the previous sections, f was a fixed function. Throughout this section, we assume that (f n ) is a sequence of L p functions satisying the uniform estimate ( 14)

Ω Ω |f n (x) -f n (y)| p |x -y| p ρ n (x -y)dxdy ≤ C 0 ,
where Ω is a smooth bounded domain in R N , 1 ≤ p < ∞, and (ρ n ) is a sequence of radial mollifiers. Without loss of generality, we may also assume the normalization condition (15) Ω f n (x)dx = 0, ∀n.

Theorem 4. Assume ( 14), ( 15) and

(16) For each n, the function t ∈ (0, ∞) → ρ n (t)is non-increasing.
Then the sequence (f n ) is relatively compact in L p and (up to a subsequence) we may assume that

f n → f in L p . Moreover, a) if 1 < p < ∞, then f ∈ W 1,p and f p W 1,p ≤ C(p, Ω)C 0 ; b) if p = 1, then f ∈ BV and f BV ≤ C(Ω)C 0 .
Remark 4. In view of Theorems 2 and 3, the additional assumption (16) may seem artificial. Actually, it is possible to slightly weaken (16); for example we may assume

(17) ρ n (t) ≥ C 1 ρ n (s), ∀ n, ∀t ≤ s,
Form some C 1 independent of n, t, s.

However, the conclusions of Theorem 4 fail for bf general ρ ′ n s. We shall give below a counterexample where the sequence (f n ) need not be relatively compact in L p (Counterexample 2).

Here are two examples of interest

Corollary 6. For 1 ≤ p < ∞, let (f ε ) be a family of L p functions such that |x-y|<ε |f ε (x) -f ε (y)| p |x -y| p dxdy ≤ C 0 ε N .
Then, up to a subsequence,

(f ε) converges in L p to some f ∈ W 1,p ( for1 < p < ∞) or f ∈ BV ( forp = 1). Corollary 7. For1 < p < ∞, let f ε ∈ W 1-ε,p . Assume that ε f ε p W 1-ε,p ≤ C 0 .
Then, up to a subsequence, (f ε) converges in L p (and, in fact, in W 1-δ,p , for all δ > 0) to some f ∈ W 1,p .

Proof of Theorem 4

The heart of the proof consists of showing that (f n ) is relatively compact in L p . The rest is done as in the second part of the proof of Theorem 2.

Without loss of generality, we may assume that Ω = R N and that supp f n ⊂ B, a ball in R N of diameter 1. This can be achieved by extendeing each function f n by reflection across the boundary in a neighborhood of ∂Ω. Using the monotonicity assumption (16), we see that assumption (14) still holds.

In order to prove compactness in L p , we rely on the Riesz-Fréchet-Kolmogorov theorem(see, e.g., Brezis [], Théorème IV.25) or rather its proof: let, for δ > 0, Φ δ be the mollifier

Φ δ = 1 |B δ (0)| χ B δ (0) . Then (f n ) is relatively compact in L p (Ω) if and only if (18) f n L p ≤ C and (19) lim δ→0 (lim sup n→∞ f n -f n * Φ δ L p ) = 0.
For each n and t > 0, let

F n (t) = ω∈S N -1 R N |f n (x + tω) -f n (x)| p dxdσ = 1 t N-1 |h|=t R N |f n (x + h) -f n (x)| p dxdσ.
Unsing the triangle inequality, we obtain (20)

F n (2t) ≤ 2 p F n (t).
In terms of F n , assumption (14) can be expressed as (21)

1 0 t N-1 F n (t) t p ρ n (t)dt ≤ C 0 . We claim that (22) |f n (x)| p dx ≤ C 1 0 t N-1 F n (t)dt and (23) |f n (x) -(f n * Φ δ )(x)| p dx ≤ Cδ -N δ 0 t N-1 F n (t)dt,
for some C independent of n and δ.

We prove for example (23):

|f n (x) -(f n * Φ δ )(x)| p dx = |f n (x) - 1 |S N-1 |δ n |y-x|<δ f n (y)dy|dx p = 1 (|S N-1 |δ N ) p | |y-x|<δ (f n (x) -f n (y))dy| p dx ≤ 1 |S N-1 | δ -N |y-x|<δ |f n (x) -f n (y)| p dxdy = 1 |S N-1 | δ -N |h|<δ ( |f n (x + h) -f n (x)| p dx)dh = Cδ -N δ 0 t N-1 F n (t)dt.
The proof of ( 22) is similar, since

f n (x) = f n (x)- B f r (y)dy.
We are going to establish below the key inequality

(24) δ -N δ 0 t N-1 F n (t) t p dt ≤ C( δ 0 t N-1 F n (t) t p ρ n (t)dt)/( |x|<δ ρ n (x)dx).
Assume (24) has been moved, then we proceed as follows since.

lim

N→∞ |x|<δ ρ n (x)dx = 1,
by combining ( 14) with (24) we find

(25) δ -N δ 0 ddt N-1 F n (t) t p dt ≤ C for n ≥ n δ .
In particular, we have

(26) δ -N δ 0 t N-1 F n (t)dt ≤ Cδ p for n ≥ n δ
Inequalities (18), (19)-and thus the conclusion of Theorem 4-follow from ( 22), ( 23) and (26).

It remains to establish inquality (24). Note that it is a particular case (g(t) = F n (t) t p , h(t) = ρ n (t)) of the following varian of an inequality due to Chebyshev:

Lemma. Let g, h : (0, δ) → R + . Assume that g(t) ≤ g(t/2), t ∈ (0, δ), and that h is non-increasing.

Then, for some C = C(N ) > 0,

δ 0 t N-1 g(t)h(t)dt ≥ Cδ -N δ 0 t N-1 g(t)dt δ 0 t N-1 h(t)dt.
Proof of the lemma: It suffices to consider the case δ = 1; the general case follows by scaling. We have

1 0 t N-1 g(t)h(t)dt = j≥0 1/2 j 1/2 j+1 t N-1 g(t)h(t)dt = j≥0 1 2 Nj 1 1/2 S N-1 g( s 2 j )h( s 2 j )ds = 1 1/2 j≥0 1 2 Nj g( s 2 j )h( s 2 j )ds, ( 27
)
and a similar equality holds for 1 0 t N-1 g(t)dt. We recall the classical Chebyshev inequality: if G, H : X → R, u a positive measure on X and (G(x)-G(y))(H(x)-H(y)) ≥ 0, ∀x, y ∈ X,then In particular, if α j ≥ 0 and the sequences (a j ), (b j ) have the same nonotonicity, then (28) α j a j b j ≥ 1 α j α j a j α j b j .

Since for each s ∈ (1/2, 1), the sequence (g( s 2 j )) and (h( s 2 j ) are non-decreasing, (28) with

α j = 1 2 N j yields (29) j≥0 1 2 Nj g( s 2 j )h( s 2 j ) ≥ C j≥0 1 2 Nj g( s 2 j ) j≥0 1 2 N j h( s 2 j ).
Now clearly, for each s ∈ (1/2, 1) and each j ≥ 1,

1 2 Nj h( s 2 j ) ≥ 1 2 Nj h( s 2 j ) ≥ C 1/2 j-1 t N-1 h(t)dt,
for some C independent of j, so that 1/2 j (30)

j≥0 1 2 Nj h( s 2 j ) ≥ C 1 0 t N-1 h(t)dt.
It follows from (2g) and (30) that (31)

j≥0 1 2 Nj g( s 2 j )h( s 2 j ) ≥ C 1 0 t N-1 h(t)dt j≥0 1 2 Nj g( s 2 j ).
Inserting (31) into (27), we find

1 0 t N-1 g(t)j(t)dt ≥ C 1 0 t N-1 j(t)dt 1 1/2 j≥0 1 2 Nj g( s 2 j )ds = C 1 0 t N-1 h(t)dt 1 0 g(t)dt.
the proof of Theorem4 is complete.

Returning to Corollary7, we still have to prove that, for any δ > 0, we have ε > 0

f ε W 1-s,p ≤ C
Considering the same functions f ε(t), as above (relative to the parameter ε instead of n we have to prove that (32)

1 0 F ε(t) t (1-δ)p+1 dt ≤ C, for smallε > 0, under the assumption (33) ε 1 0 F ε(t) t (1-ε)p+1 dt ≤ C.
the proof is similar to that of the Lemma, so we just sketch it. We start by rewritting (32) and (33) as (34)

1 0 1 t 1-sp F ε(t) t p dt ≤ C and (35) 1 0 1 t 1-sp F ε(t) t p ε t (s-ε)p dt ≤ C.
We continue as in the proof of the lemma, with g(t) = F ε(t) t p and h(t) = ε t (s-ε)p , and take 0 < ε < δ.

We finally find (36)

1 0 1 t 1-sp F ε(t) t p ε t ( s -ε)p dt ≥ C 1 0 1 t 1-sp F ε(t) t p dt
for some C depending possibly on S, but not on ε. We take Ω = (0, 1).

Fix some function f ∈ L p loc (R), non-constant, periodical of period 1, such that Finally, the functions f n = ng n satisfy the desired inequality (37) and f -N -L p ∼ n.

Counterexample 2: the sequence (g n ) constructed above is bounded in l p , is not relatively compact in L p , and yet it satisfies

Remark 5 .

 5 If we renorm the W s,p (Ω) spaces by yields |f | p W σ,p ≤ C|f | p W s,p .0 < σ < s ≤ 1 for some constant C independent of s and σ. Counterexample 1: a sequence (f n ) unbounded in L p and a sequence of radial mollifiers(ρ n ) such that (37) Ω Ω |f n (x) -f n (y)| p |x -y| p ρ n (x -y)dxdy ≤ C.

1 0fp = 1 0 1 0 1 2 n such that 1 0 1 n -δ n , 1 n 1 n -δ n ,- 1 n

 111111111 (x)dx = 0( e.g. , f (x) = sin (2πx)).Define g n (x) = f (nx),so that g n p L |f (x)| p dx = C.Clearly,|g n (x = 1 n ) -g n (x)| p dx = 0.Since that translations are continous in l p , we may find some 0 < δ n <|g n (x + h) -g n (x)| p dx ≤ 1 n2p for |h ± 1 n | < δ n . Let ρ -N = 1 4δ n (χ +δ n ) + χ (-=δ n ) ). Then clearly Ω Ω |g n (x) -g n (y)| |x -y| p ρ n (x -y)dxdy ≤ C n p .

  Ω Ω |g n (x) -g n (y)| p |x -y| p ρ n (x -y)dxdy ≤ C.