Density of smooth maps for fractional Sobolev spaces <span class="mathjax-formula">$W^{s, p}$</span> into <span class="mathjax-formula">$\ell $</span> simply connected manifolds when <span class="mathjax-formula">$s \ge 1$</span>

CONFLUENTES MATHEMATICI

Pierre BOUSQUET, Augusto C. PONCE, and Jean VAN SCHAFTINGEN Density of smooth maps for fractional Sobolev spacesW^s,pinto`simply connected manifolds whens>1

Tome 5, n^o2 (2013), p. 3-22.

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5, 2 (2013) 3-22

DENSITY OF SMOOTH MAPS FOR FRACTIONAL SOBOLEV SPACES W^s,p INTO ` SIMPLY CONNECTED

MANIFOLDS WHEN s>1

PIERRE BOUSQUET, AUGUSTO C. PONCE, AND JEAN VAN SCHAFTINGEN

Abstract. Given a compact manifoldNⁿ⊂R^νand real numberss>1 and 16p <∞, we prove that the classC^∞(Q^m;Nⁿ) of smooth maps on the cube with values into Nⁿ is strongly dense in the fractional Sobolev space W^s,p(Q^m;Nⁿ) whenNⁿ isbspcsimply connected. Forspinteger, we prove weak sequential density ofC^∞(Q^m;Nⁿ) whenNⁿ is sp−1 simply connected. The proofs are based on the existence of a retraction ofR^ν onto Nⁿexcept for a small subset ofNⁿand on a pointwise estimate of fractional derivatives of composition of maps inW^s,p∩W^1,sp.

1. Introduction

In this paper we discuss results and open questions related to the density of smooth maps in Sobolev spaces with values into a manifold. For this purpose, let Nⁿ be a compact manifold of dimension nimbedded in the Euclidean space R^ν. For anys >0 and 16p <+∞, we define the class of Sobolev maps defined on the unitmdimensional cubeQ^mwith values intoNⁿ,

W^s,p(Q^m;Nⁿ) =

u∈W^s,p(Q^m;R^ν) :u∈Nⁿ a.e. .

Whens =k is an integer,W^s,p(Q^m;R^ν) is the standard Sobolev space equipped with the norm

kuk_Ws,p(Q^m)=kuk_Lp(Q^m)+

k

X

j=1

kD^juk_Lp(Q^m).

Whens is not an integer,s =k+σwith k ∈Nand 0 < σ <1. In this case, by u∈W^s,p(Q^m;R^ν) we mean thatu∈W^k,p(Q^m;R^ν) and

[D^ku]_Wσ,p(Q^m)= Z

Q^m

Z

Q^m

|D^ku(x)−D^ku(y)|^p

|x−y|^m+σp dxdy ^1/p

<+∞, and the associated norm is given by

kukW^s,p(Q^m)=kukL^p(Q^m)+

k

X

j=1

kD^jukL^p(Q^m)+ [D^ku]W^σ,p(Q^m).

The fractional Sobolev spacesW^s,p(Q^m;R^ν) arise in the trace theory of Sobolev spaces of integer order. For example, the trace is a continuous linear operator from W^1,p(Q^m;R^ν) onto W^1−1p,p(∂Q^m;R^ν) [12, Theorem 1.I].

We first address the question of strong density of smooth maps: given u ∈ W^s,p(Q^m;Nⁿ), does there exist a sequence inC^∞(Q^m;Nⁿ) which converges tou with respect to the strong topology induced by theW^s,p norm?

A naive approach consists in applying a standard regularization argument. This works well for maps inW^s,p(Q^m;R^ν) and shows thatC^∞(Q^m;R^ν) is strongly dense in that space. When R^ν is replaced by Nⁿ, the conclusion is less clear since the convolution of a mapu∈W^s,p(Q^m;Nⁿ) with a smooth kernelϕtyields a map with

Math. classification:58D15, 46E35, 46T20.

Keywords: Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces;

simply connectedness.

3

values in the convex hull ofNⁿ. In this case, one might try to projectϕ_t∗uinto the manifoldNⁿ. This is indeed possible for sp>m.

Ifsp > m, then by the Morrey-Sobolev imbedding,W^s,p is continuously imbed- ded intoC⁰. Thus, every mapu∈W^s,p(Q^m;Nⁿ) has a continuous representative andϕ_t∗uconverges uniformly ast tends to zero. In particular,

t→0lim sup

x∈Q^m

dist (ϕ_t∗u(x), Nⁿ) = 0. (1.1) Hence, one may project ϕt∗uback toNⁿ since the nearest point projection Π is well defined and smooth on a neighborhood ofNⁿ.

Ifsp=m, then the Morrey-Sobolev imbedding fails but property (1.1) remains true [31] since W^s,p injects continuously into the space VMO of functions with vanishing mean oscillation. This fact has been observed by Brezis and Nirenberg [9].

We may summarize as follows:

Theorem 1.1. — C^∞(Q^m;Nⁿ)is strongly dense inW^s,p(Q^m;Nⁿ)forsp>m.

The case wheresp < mis more subtle and the answer depends on the topology ofNⁿ. Even whenNⁿis the unit sphereSⁿthe approximation problem is not fully understood. For instance, consider the mapu: B³ →S² defined in the unit ball B³⊂R³by

u(x) = x

|x|.

Then, u ∈ W^s,p(B³;S²) for every s > 0 and p > 1 such that sp < 3, but u cannot be strongly approximated in W^s,p by smooth maps with values into S² when 2 6sp <3. This example originally due to Schoen and Uhlenbeck [31] for s = 1 can be adapted to the case whereS² is replaced by any compact manifold Nⁿ and for any value ofs[10, Theorem 3; 23, Theorem 4.4]:

Theorem 1.2. — If sp < m and π_bspc(Nⁿ) 6= {0}, then C^∞(Q^m;Nⁿ) is not strongly dense inW^s,p(Q^m;Nⁿ).

It seems that the topological conditionπ_bspc(Nⁿ)6={0}is the only obstruction to the strong density of smooth maps inW^s,p(Q^m;Nⁿ). This is indeed true when sis an integer by a remarkable result of Bethuel [3, Theorem 1; 17] fors= 1 which has been recently generalized by the authors [6, Theorem 4] for anys∈N(see also [14]):

Theorem 1.3. — For any s ∈ N∗, if sp < m and π_bspc(Nⁿ) = {0}, then C^∞(Q^m;Nⁿ)is strongly dense inW^s,p(Q^m;Nⁿ).

Some cases of non-integer values have been investigated. For instance when s= 1−1/pin the setting of trace spaces [4, 24] and also whens>1 and Nⁿ =Sⁿ [5, 10]. Brezis and Mironescu [8] have announced in a personal communication a solution to the question of strong density for any 0< s <1.

All these cases give an affirmative answer to the following:

Open Problem 1.4. — Let s6∈N∗. Whensp < m and π_bspc(Nⁿ) ={0}, is it true thatC^∞(Q^m;Nⁿ) is strongly dense inW^s,p(Q^m;Nⁿ)?

In this paper, we investigate Open problem 1.4 for`simply connected manifolds Nⁿ:

π0(Nⁿ) =· · ·=π`(Nⁿ) ={0}. (1.2) We prove the following:

Theorem 1.5. — Let s >1. If sp < m and if Nⁿ is bspcsimply connected, thenC^∞(Q^m;Nⁿ)is strongly dense inW^s,p(Q^m;Nⁿ).

Even in the case wheresis an integer — which is covered in full generality by Theorem 1.3 — the proof is simpler and has its own interest. We have been inspired by Hajłasz [15] who has proved Theorem 1.5 fors= 1.

Our proof of Theorem 1.5 is based on two main ingredients. The geometric tool (Proposition 2.1) gives a smooth retraction of the ambient space R^ν onto Nⁿ except for a small subset of Nⁿ. The analytic tool (Proposition 2.6) gives a pointwise estimate of the fractional derivative of η◦u, where η is a smooth map anduis aW^s,p map.

The counterpart of Theorem 1.5 for 0< s <1 requires different tools and will be investigated in a subsequent paper.

The second problem we adress in this paper concerns theweak sequential density of C^∞(Q^m;Nⁿ) in W^s,p(Q^m;Nⁿ): given u ∈ W^s,p(Q^m;Nⁿ), does there exist a sequence in C^∞(Q^m;Nⁿ) which is bounded in W^s,p(Q^m;Nⁿ) and converges tou in measure?

The casesp>mhas an affirmative answer due to the strong density of smooth maps. When sp < m, we find the same topological obstruction as for the strong density problem whenspis not an integer [3, Theorem 3]:

Theorem 1.6. — Ifsp < mis such thatsp6∈Nand ifC^∞(Q^m;Nⁿ)is weakly sequentially dense inW^s,p(Q^m;Nⁿ), thenπ_bspc(Nⁿ) ={0}.

From Theorem 1.3, it follows that for everys∈N∗such thatsp6∈Nthe problems of weak and strong density of smooth maps inW^s,p(Q^m;Nⁿ) are equivalent. We expect the same is true fors6∈N; the missing ingredient would be an affirmative answer to Open Problem 1.4.

The conclusion of Theorem 1.6 need not be true when sp is an integer. For instance, by a result of Bethuel [2, Theorem 3],C^∞(Q³;S²) is weakly sequentially dense inW^1,2(Q³;S²), even though it is not strongly dense by Theorem 1.2.

As a byproduct of the tools we use to prove Theorem 1.5, we establish the following:

Theorem 1.7. — Lets>1. Ifsp < mis such thatsp∈Nand ifNⁿ issp−1 simply connected, thenC^∞(Q^m;Nⁿ)is weakly sequentially dense inW^s,p(Q^m;Nⁿ).

This result is due to Hajłasz [15, Corollary 1] whens= 1; Hajłasz’s argument still applies forp= 1 although it is not explicitly stated in his paper. More recently, Hang and Lin [18, Corollary 8.6] proved an analogue of Theorem 1.7 under a weaker topological assumption for s = 1. To our knowledge, the only result concerning weak sequential density of smooth maps for non-integer values ofsdeals with the cases= 1/2,p= 2 andN =S¹ and is due to Rivière [29, Theorem 1.2].

Combining Theorem 1.2 and Theorem 1.7 we deduce thatC^∞(Q^m;Sⁿ) is weakly sequentially dense but not strongly dense inW^s,p(Q^m;Sⁿ) forn < mandsp=n.

Whensp∈N, we do not know whetherC^∞(Q^m;Nⁿ) is weakly sequentially dense inW^s,p(Q^m;Nⁿ) with no extra assumption on the compact manifoldNⁿ. The only results which are known in this sense concerns = 1: forp= 1 [16, Theorem 1.3;

27, Theorem I] and forp= 2 [28, Theorem I].

2. Main tools

2.1. Geometric tool. Our first tool is the construction of a retraction ofR^ν onto Nⁿ except for a small subset of Nⁿ. This is the only place where the topological assumption (1.2) concerning the`simply connectedness of the manifoldNⁿ comes into place.

Proposition 2.1. — If Nⁿ is ` simply connected, then for every 0 < 6 1 there exist a smooth functionη:R^ν→Nⁿ and a compact setK⊂Nⁿ such that

(i) for everyx∈Nⁿ\K,η(x) =x,

(ii) Hⁿ(K)6C^`+1, for some constantC >0 depending onNⁿ andν, (iii) for everyj ∈N∗,

kD^jηkL^∞(R^ν)6 C^{0 j},

for some constantC⁰ >0depending onNⁿ,ν andj.

The setKis chosen as theneighborhood of ann−`−1 dimensional dual skeleton of Nⁿ. This proposition is the smooth counterpart of Hajłasz’s construction of a Lipschitz continuous mapη [15, Section 4].

The proof of Proposition 2.1 relies on the existence of a triangulation of the manifoldNⁿ. It is more convenient to use a variant of the triangulation based on the decomposition ofNⁿ in terms of cubes rather than simplices.

A cubicationT ofNⁿ is a finite collection of closed sets coveringNⁿof the form Φ(σ) withσ∈ Qsuch that

(a) Φ : S

σ∈Q

σ→Nⁿ is a biLipschitz map,

(b) Qis a finite collection of cubes of dimensionmin some Euclidean spaceR^µ, such that two elements ofQare either disjoint or intersect along a common face of dimension`for some`∈ {0, . . . , n}.

Given ` ∈ {0, . . . , n}, we denote by T<sup>`</sup> the union of all ` dimensional faces of elements ofT; we callT<sup>`</sup> the`dimensional skeleton ofT.

We recall the following lemma [15, Lemma 1]:

Lemma 2.2. — LetT be a cubication of Nⁿ and let T^{`</sup> be the ` dimensional skeleton ofT. IfNⁿis`simply connected, then there exists a Lipschitz continuous functionη:R<sup>ν</sup>→N<sup>n</sup> such that for every x∈T<sup>`},η(x) =x.

Proof. —LetCT^`⊂R×R^ν denote the cone

(λ, λx)∈R×R^ν :λ∈[0,1] andx∈T^` .

Since CT^{`</sup> is contractible, there exists a continuous map ξ:R<sup>ν</sup> →CT<sup>`} such that for everyx∈T^`,ξ(x) = (1, x).

We may choose ξ to be uniformly continuous. Indeed, ifp : R^ν → R^ν is any Lipschitz function such that pcoincides with the identity onT^{`</sup> andpis constant outside some ball containing T<sup>`}, then for every x∈ T^`, ξ◦p(x) = (1, x) and, in addition,ξ◦pis uniformly continuous. Replacingξbyξ◦pif necessary, we assume in the sequel thatξitself is uniformly continuous.

Since Nⁿ is ` simply connected, the identity map in N<sup>n</sup> is homotopic to a continuous map in N<sup>n</sup> which is constant on T<sup>`</sup> [33, Section 6]. More precisely, there exist a continuous mapH : [0,1]×Nⁿ→Nⁿ anda∈Nⁿ such that

(a) for everyx∈T^`, H(0, x) =a, (b) for everyx∈Nⁿ,H(1, x) =x.

SinceH is constant on{0} ×T^{`</sup>,H induces a continuous quotient mapH :CT<sup>`}→ Nⁿ defined for every (λ, λx) ∈ CT^{`</sup> by H(λ, λx) = H(λ, x). Then, H ◦ξ is a uniformly continuous map with values into N<sup>n</sup> which coincides with the identity map onT<sup>`}.

Using a standard approximation argument, we may construct a Lipschitz map having the same properties. We present the argument for the sake of completeness.

Givenι >0, let θ:R^ν →[0,1] be a Lipschitz continuous function supported in a neighborhood ofT^` such that

(a⁰) for everyx∈T^`,θ(x) = 1,

(b⁰) for everyx∈suppθ,|x−H◦ξ(x)|6ι.

SinceH◦ξis uniformly continuous, there exists a Lipschitz approximationh:R^ν → R^ν such that for every x∈R^ν,

|h(x)−H◦ξ(x)|6ι.

Then, for everyx∈R^ν,

H◦ξ(x)− θ(x)x+ (1−θ(x))h(x) 6ι.

SinceH◦ξ(x)∈Nⁿ, it follows that

θ(x)x+ (1−θ(x))h(x)∈Nⁿ+B^ν_ι,

where B^ν_ι is the closed ball inR^ν of radiusι centered at 0. Choosingι such that the nearest point projection Π :Nⁿ+B^ν_ι →Nⁿ is well-defined and smooth, then we have the conclusion by takingη:R^ν→Nⁿ defined forx∈R^ν by

η(x) = Π θ(x)x+ (1−θ(x))h(x) .

The proof is complete.

We shall also use dual skeletons associated to a cubication T given by a map Φ : S

σ∈Q

σ→Nⁿ.We first define dual skeletons for a cube inRⁿ. Letj∈ {0, . . . , n}.

When the center of the cube is 0 and the faces are parallel to the coordinate axes, the dual skeleton of dimension j is the set of points in the cube which have at least n−j components equal to zero. By using an isometry, we can define the dual skeleton of a cube of dimensionn in R^µ in general position. Then, the dual skeleton of dimension j of a family Q of cubes as above is simply the union of the dual skeletons of dimension j of each cube. Finally, the dual skeleton L^j of dimensionj of the cubicationT ofNⁿis the image by Φ of thej dimensional dual skeleton ofQ.

The following lemma implies the homotopy equivalence between the skeletonT^{`</sup> of the manifold N<sup>n</sup> and the complement of the dual skeleton L<sup>n−`−1} in Nⁿ. We are particularly interested in the pointwise estimates of the homotopyf:

Lemma 2.3. — Let ` ∈ {0, . . . , n−1}, let T be a cubication of N<sup>n</sup> and let L<sup>n−`−1</sup>be then−`−1dimensional dual skeleton ofT. Then, there exists a locally Lipschitz continuous function

f : [0,1]×(Nⁿ\L^n−`−1)→Nⁿ such that

(i) for everyt∈[0,1]and for everyx∈T^{`</sup>,f(t, x) =x, (ii) for everyx∈N<sup>n</sup>\L<sup>n−`−1},f(0, x) =xandf(1, x)∈T^{`</sup>, (iii) for everyt∈[0,1]and for everyx∈N<sup>n</sup>\L<sup>n−`−1},

|∂tf(t, x)|6C, and

|∂xf(t, x)|6 C⁰ dist (x, L^n−`−1),

for some constants C, C⁰>0depending onn,`, Nⁿ andT.

Proof. —We first establish the result when the manifold Nⁿ is replaced by the cube [−1,1]ⁿ andL<sup>n−`−1</sup> is the dual skeleton of dimension n−`−1 of [−1,1]ⁿ. Following [33], we consider for everyx∈[−1,1]ⁿ,

|x|`= min

S⊂{1,...,n}

|S|=`+1

maxi∈S |xi|.

In particular, for every x ∈ [−1,1]ⁿ, x ∈ L<sup>n−`−1</sup> if and only if |x|<sub>`</sub> = 0. The functionx∈[−1,1]ⁿ7→ |x|` is Lipschitz continuous of constant 1.

Letφ`: [−1,1]<sup>n</sup>\L<sup>n−`−1</sup>→T<sup>`</sup>be defined for every x∈[−1,1]<sup>n</sup> by φ<sub>`</sub>(x) = (y₁, . . . , y_n),

where

yi=

(sgnxi if|xi|>|x|`, x<sub>i</sub>/|x|` if|xi|<|x|`.

The homotopyf : [0,1]×([−1,1]ⁿ\L<sup>n−`−1</sup>)→[−1,1]<sup>n</sup> defined by f(t, x) = (1−t)x+tφ`(x)

has the required properties.

In order to prove the existence of the homotopyf for a general compact manifold Nⁿ, we perform the above construction in every cube of a given cubication Φ :

S

σ∈Q

σ→Nⁿ. If two cubesσ1andσ2 inQhave a non empty intersection, then the corresponding mapsφ_{`,1</sub> andφ<sub>`,2} coincide on the common faceσ₁∩σ₂. Hence, we can glue together the locally Lipschitz continuous maps obtained for each cube so as to obtain a global map f₀ which is defined on the entire collection of cubes in Q. The conclusion follows by taking

f(t, x) = Φ f₀(t,Φ⁻¹(x))

.

We now prove a counterpart of Proposition 2.1 for a Lipschitz continuous map η:

Lemma 2.4. — Let`∈ {0, . . . , n−1}, let T be a cubication ofN<sup>n</sup>andL<sup>n−`−1</sup> be then−`−1dimensional dual skeleton ofT and letι >0be such that the nearest point projectionΠ ontoN<sup>n</sup> is smooth on N<sup>n</sup>+B<sup>ν</sup><sub>2ι</sub>. IfN<sup>n</sup> is `simply connected, then for every0< 61there exists a Lipschitz continuous mapη:R^ν →Nⁿsuch that

(i) η= Πon(Nⁿ+B_ι^ν)\Π⁻¹(L^n−`−1+B^ν), (ii) for everyx∈R^ν,

|Dη(x)|6 C⁰⁰ .

for some constantC⁰⁰>0depending onNⁿ,T andν.

Proof. —Letf be the map given by Lemma 2.3. The extension f¯: {0} ×L^n−`−1

∪ [0,1]×(Nⁿ\L^n−`−1)

→Nⁿ defined by

f¯(t, x) =

(x ift= 0, f(t, x) if 0< t61, is continuous.

Let Π be the nearest point projection onto Nⁿ and denote by Π :R^ν → R^ν a smooth extension of Π. The image of Π need not be contained in the manifoldNⁿ.

Letθ:R^ν →[0,2] be a Lipschitz continuous function such that (a) for everyx∈Nⁿ+B_ι^ν, θ(x) = 2,

(b) for everyx∈R^ν\(Nⁿ+B_2ι^ν),θ(x) = 0.

Given 0< 61, letd:Nⁿ+B_2ι^ν →Rbe defined by d(x) = 1

dist (Π(x), L^n−`−1).

Letλ: [0,+∞)→[0,1] be a Lipschitz continuous function such that (a⁰) for everyt6¹2 and for everyt>2,λ(t) = 0,

(b⁰) λ(1) = 1.

Denote byη:R^ν →Nⁿ the function given by Lemma 2.2. Let η:R^ν →Nⁿ be the map defined by

η(x) =







f λ(θ(x)d(x)),Π(x)

ifx∈Nⁿ+B_2ι^ν and θ(x)d(x)>1, η◦f λ(θ(x)d(x)),Π(x)

ifx∈Nⁿ+B_2ι^ν and θ(x)d(x)61, η Π(x)

ifx6∈Nⁿ+B_2ι^ν.

We first check thatη is continuous. For this purpose we only need to consider the borderline cases:

(1) x∈Nⁿ+B^ν_2ι andθ(x)d(x) = 1,

(2) x∈∂(Nⁿ+B_2ι^ν).

In the first case, since λ(1) = 1, f(1,·)∈ T^{`</sup> andη is the identity map on T<sup>`}, we have

f λ(θ(x)d(x)),Π(x)

=f(1,Π(x))

=η f(1,Π(x))

=η◦f λ(θ(x)d(x)),Π(x) . In the second case,θ(x) = 0. Sinceλ(0) = 0 andf(0,·) is the identity map onNⁿ,

f λ(θ(x)d(x)),Π(x)

= Π(x) = Π(x), whence

η◦f λ(θ(x)d(x)),Π(x)

=η Π(x) . We now check that property (i) holds. Indeed, if

x∈(Nⁿ+B^ν_ι)\Π⁻¹(L^n−`−1+B^ν), thenθ(x) = 2 andd(x)>1. Thus,

λ(θ(x)d(x)) = 0.

We then have

η(x) =f(0,Π(x)) = Π(x).

It remains to establish property (ii). Indeed, if x 6∈ Nⁿ +B_2ι^ν, then η(x) = η Π(x)

and the conclusion follows sinceηand Π(x) are both Lipschitz continuous, with Lipschitz constants independent of . If x ∈ Nⁿ +B^ν_2ι and θ(x)d(x) < ¹₂, then η(x) =η◦Π(x) and the estimate follows similarly. Finally, ifx∈Nⁿ+B_2ι^ν andθ(x)d(x)> ¹2, then

dist (Π(x), L^n−`−1)>

4. By the chain rule and the estimates given by Lemma 2.3,

|Dη(x)|6C1

1

+ 1

dist (Π(x), L^n−`−1)

.

Combining both estimates, we get the conclusion. The proof of the lemma is

complete.

We now have all tools to prove Proposition 2.1.

Proof of Proposition 2.1. — Letϕ:R^ν →Rbe a smooth map supported in the unit ballB^ν₁. For everyt >0, letϕt:R^ν →R be the function defined forx∈R^ν byϕt(x) = _t¹νϕ(^x_t). Letι >0 as in the previous lemma.

Given 0< 61, letζ:R^ν→[0,1] be a smooth function such that (a) for everyx∈Nⁿ\(L^n−`−1+B₂^ν),ζ(x) = 1,

(b) for everyx6∈(Nⁿ+B_ι^ν)\Π⁻¹(L^n−`−1+B^ν),ζ(x) = 0, (c) for everyj∈N∗,

kD^jζkL^∞(R^ν)6 C₁ ^j , whereC1>0 depends onj.

Letη:R^ν→Nⁿbe the Lipschitz continuous map given by the previous lemma and lett >0 to be chosen below.

By property (b) and by Lemma 2.4 (i), the function ζη+ (1−ζ)ϕt∗η

is smooth inR^ν and for every j∈N∗ there existsC₂>0 such that D^j ζη+ (1−ζ)ϕ_t∗η

_L∞(

R^ν)6C₂ 1 + 1

t^j + 1 ^j

. (2.1)

Moreover, by property (a) and by Lemma 2.4 (i), for everyx∈Nⁿ\(L^n−`−1+B₂^ν), ζη+ (1−ζ)ϕt∗η

(x) =η(x) = Π(x) =x.

By Lemma 2.4 (ii) we have for everyt >0,

kϕ_t∗η−ηk_L∞(R^ν)6tkDηk_L∞(R^ν)6tC3

. Taking

t= ι C₃,

it follows from the previous estimate that the image ofζη+(1−ζ)ϕt∗ηis contained inNⁿ+B_ι^ν. Hence, the functionη:R^ν→Nⁿ,

η= Π◦ ζη+ (1−ζ)ϕt∗η

, is well-defined and smooth. Property (i) holds with

K=Nⁿ∩(L^n−`−1+B₂^ν).

Property (ii) also holds sinceKis a neighborhood ofL^n−`−1inNⁿwhose radius is of the order of. By estimate (2.1), property (iii) is also satisfied. This completes

the proof of the proposition.

2.2. Analytic tool. In this section we establish pointwise estimates of derivatives and fractional derivatives of the map η◦u, where η is a smooth function and u belongs toW^s,p∩L^∞. In the case wheresis an integer, this estimate follows from the classical chain rule for higher order derivatives:

Proposition 2.5. — Letk∈N∗. Ifu∈W^k,p(Q^m;R^ν)∩W^1,kp(Q^m;R^ν), then for everyj∈ {1, . . . , k}there exists a measurable functionGj∈L^p(Q^m)such that for every smooth mapη:R^ν→R^ν,

|D^j(η◦u)|6[η]_Cj(R^ν)Gj.

Moreover,kG_jk_Lp(Q^m)is bounded from above by a constant depending onk,p,m, kukW^k,p(Q^m)andkukW^1,kp(Q^m).

We use the following notation:

[η]_Cj(R^ν)=

j

X

i=1

kDⁱηk_L^∞_(Rν).

Proof. —We first observe that η◦u∈W^k,p(Q^m;R^ν). By the chain rule,

|D^j(η◦u)(x)|6C1 j

X

i=1

|Dⁱη(u(x))| X

16t16···6ti6j, t₁+···+ti=j

|D^t1u(x)| · · · |D^tiu(x)|

6C1[η]C^j(R^ν) j

X

i=1

X

16t16···6ti6j, t₁+···+ti=j

|D^t1u(x)| · · · |D^tiu(x)|.

Let

Gj =C1 j

X

i=1

X

16t₁6···6ti6j, t1+···+ti=j

|D^t1u| · · · |D^tiu|

Since the map u in the statement belongs toW^k,p(Q^m;R^ν)∩W^1,kp(Q^m;R^ν), it follows from the Gagliardo-Nirenberg interpolation inequality [13, 25] that

Dⁱu∈L^jpi(Q^m).

By Hölder’s inequality, we deduce thatGj ∈L^p(Q^m).

We now establish a counterpart of the previous proposition for the fractional derivative introduced by Maz⁰ya and Shaposhnikova [22]. More precisely, given 0< σ <1, 16p <+∞, a domain Ω⊂R^mand a measurable functionu: Ω→R^ν, define forx∈Ω,

D^σ,pu(x) = Z

Ω

|u(x)−u(y)|^p

|x−y|^m+σp dy ^1/p

.

We extend this definition for anys >0 such thats /∈Nas follows:

D^s,pu=D^σ,p(D^ku),

where k=bscis the integral part of sand σ=s− bscis the fractional part of s.

Using this notation, we have

[D^ku]W^σ,p(Ω)=kD^s,pukL^p(Ω).

Proposition 2.6. — Let s > 1 be such thats 6∈ N. If u ∈ W^s,p(Q^m;R^ν)∩ W^1,sp(Q^m;R^ν), then there exists a measurable functionH∈L^p(Q^m)such that for every smooth mapη :R^ν→R^ν,

|D^s,p(η◦u)|6[η]^σ_Ck+1(R^ν)[η]^1−σ_Ck(

R^ν)H.

Moreover,kHkL^p(Q^m)is bounded from above by some constant depending ons,p, m,kukW^s,p(Q^m) andkukW^1,sp(Q^m).

This proposition implies a theorem of Brezis and Mironescu [8, Theorem 1.1]

concerning the boundedness of the composition operator from W^s,p∩W^1,sp into W^s,p. A more elementary proof of the same result has been provided by Maz⁰ya and Shaposhnikova [22]; our proof of Proposition 2.6 is based on their strategy.

We begin with the following pointwise estimate of Maz⁰ya and Shaposhnikova [22, Lemma]:

Lemma 2.7. — Letq>1. Ifv∈W_loc^1,q(R^m;R^ν), then forx∈R^m, D^σ,qv(x)^q

6C M|Dv|^q(x)^σ

M|v|^q(x)1−σ

, for some constantC >0 depending onmandq.

The maximal function associated to a nonnegative function f ∈ L¹_loc(R^m) is defined forx∈R^mby

Mf(x) = sup

ρ>0

1

|B_ρ^m| Z

B_ρ^m(x)

f.

For completeness, we prove Lemma 2.7 using a property of the maximal function due to Hedberg [20, Lemma]:

Lemma 2.8. — Letf ∈L¹_loc(R^m)be a nonnegative function and letδ >0. For everyx∈R^mandρ >0,

Z

B^m_ρ(x)

f(y)

|y−x|^m−δ dy6Cρ^δMf(x), Z

R^m\B_ρ^m(x)

f(y)

|y−x|^m+δ dy6 C

ρ^δMf(x), for some constantC >0 depending onmandδ.

Proof. —We briefly sketch the proof of Hedberg for the first estimate. The proof of the second one is similar. One has

Z

B^m_ρ(x)

f(y)

|y−x|^m−δ dy=

∞

X

i=0

Z

B^m

ρ2−i(x)\B^m

ρ2−i−1(x)

f(y)

|x−y|^m−δ dy

6C1ρ^δ

∞

X

i=0

2^−δiMf(x)6C2ρ^δMf(x).

Proof of Lemma 2.7. — Letρ >0. By Hardy’s inequality [21, Section 1.3], Z

B_ρ^m(x)

|v(x)−v(y)|^q

|x−y|^m+σq dy6C1

Z

B^m_ρ(x)

|Dv(y)|^q

|x−y|^{m−(1−σ)q} dy.

Thus, by Hedberg’s lemma, Z

B^m_ρ(x)

|v(x)−v(y)|^q

|x−y|^m+σq dy6C2ρ^(1−σ)qM|Dv|^q(x).

Since

|v(x)−v(y)|^q 6C3 |v(x)|^q+|v(y)|^q , by an explicit integral computation and by Hedberg’s lemma,

Z

R^m\B^m_ρ(x)

|v(x)−v(y)|^q

|x−y|^m+σq dy6 C4

ρ^σq |v(x)|^q+M|v|^q(x) 6 C5

ρ^σqM|v|^q(x).

We conclude that

D^σ,qv(x)^q

6C₂ρ^(1−σ)qM|Dv|^q(x) + C5

ρ^σqM|v|^q(x).

Minimizing the right-hand side with respect toρ, we deduce the pointwise estimate.

The following lemma is implicitly proved in [22, Section 2]:

Lemma 2.9. — Let 0 < σ < 1, 1 6 p < +∞ and i ∈ N∗. If for every α ∈ {1, . . . , i},v_α∈L^qα(R^m)andDv_α∈L^rα(R^m), where 1< r_α< q_αand

1−σ q_α + σ

r_α +

i

X

β=1 β6=α

1 q_β = 1

p,

then Qⁱ

α=1

vα

W^σ,p(R^m)6C

i

X

α=1

kvαk^1−σ_Lqα(

R^m)kDvαk^σ_Lrα(R^m) i

Y

β=1 β6=α

kvβk_Lqβ(R^m)

,

for some constantC >0 depending onm, σ,r1, . . . ,ri,q1, . . . ,qi. Proof. —We first consider the case of dimensionm= 1. Note that

i

Q

α=1

vα(x)−

i

Q

α=1

vα(y) 6

i

X

α=1

v1(x)· · ·v_α−1(x) vα(x)−vα(y)

vα+1(y)· · ·vi(y) . (2.2) Thus, the left-hand side is bounded from above by a sum of functions of the form

fα(x)|vα(x)−vα(y)|f_α(y).

By the Fundamental theorem of Calculus, for everyx, y∈R,

|vα(x)−vα(y)|62|x−y|M|v_α⁰|(x).

Thus, for everyρ >0, Z

B¹_ρ(x)

|vα(x)−vα(y)|^p

|x−y|^1+σp (f_α(y))^pdy6C1 M|v⁰_α|(x)^p Z

B_ρ¹(x)

(f_α(y))^p

|x−y|^{1−(1−σ)p}dy.

By Hedberg’s lemma, we get Z

B_ρ¹(x)

|vα(x)−vα(y)|^p

|x−y|^1+σp (f_α(y))^pdy6C2ρ^(1−σ)p M|v⁰_α|(x)p

M(f_α)^p(x).

Next, we write

|vα(x)−vα(y)|^p(f_α(y))^p6C3 |vα(x)|^p(f_α(y))^p+|vα(y)|^p(f_α(y))^p . By Hedberg’s lemma, we also have

Z

R\B¹_ρ(x)

|vα(x)−vα(y)|^p

|x−y|^1+σp (f_α(y))^pdy6 C4

ρ^σp |vα(x)|^pM(f_α)^p(x) +M|vαf_α|^p(x) .

We conclude that Z

R

|vα(x)−vα(y)|^p

|x−y|^1+σp (f_α(y))^pdy

6C₂ρ^(1−σ)p M|v⁰_α|(x)^p

M(f_α)^p(x) + C₄

ρ^σp |vα(x)|^pM(f_α)^p(x) +M|vαf_α|^p(x) . Minimizing the right hand side with respect toρ, we then get

Z

R

|vα(x)−vα(y)|^p

|x−y|^1+σp (f_α(y))^pdy 6C5 M|v_α⁰|(x)σp

(M(f_α)^p(x))^σ |vα(x)|^pM(f_α)^p(x) +M|vαf_α|^p(x)1−σ

. Thus,

Z

R

Z

R

(fα(x))^p|vα(x)−vα(y)|^p

|x−y|^1+σp (f_α(y))^pdxdy 6C₅

Z

R

(fα)^p M|v_α⁰|σp

M(f_α)^pσ

|vα|^pM(f_α)^p+M|vαf_α|^p1−σ

. (2.3)

Let _q¹

α

=

α−1

P

β=1 1

q_β and _q¹

α =

i

P

β=α+1 1

q_β, so that by assumption, 1

qα

+ σ rα

+1−σ qα

+ 1 q_α =1

p. By Hölder’s inequality,

Z

R

(fα)^p M|v⁰_α|^σp

|vα|^(1−σ)pM(f_α)^p 6kf_αk^p

L^qα(R)kM|v⁰_α|k^σp_Lrα(

R)kvαk^(1−σ)p_Lqα(

R)k(M(f_α)^p)^1/pk^p

L^qα(R). We estimate the right hand side as follows. By Hölder’s inequality,

kfαk_L^q_α(

R)6

α−1

Y

β=1

kv_βk_Lqβ(R).