The Jacobian determinant revisited

The Jacobian determinant revisited Ha¨ım Brezis∗†and Hoai-Minh Nguyen‡ January 20, 2010

Contents 1 Introduction

1

2 Theorems 1 and 2, and related topics 2.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . 2.2 The dipole construction. Further discussion around 2.3 On a conjecture of S. M¨ uller . . . . . . . . . . . . . 2.4 Proof of Theorem 2 . . . . . . . . . . . . . . . . . .

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3 Theorems 3 and 4, and optimality results 3.1 Proof of Theorem 3 . . . . . . . . . . . . . 3.2 Optimality results . . . . . . . . . . . . . 3.2.1 Proof of Theorem 4 . . . . . . . . 3.2.2 Optimality of Corollary 1 . . . . .

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A Appendix: An interpolation inequality

1

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29

Introduction

This paper is devoted to the study of the Jacobian determinant of a map g from Ω, a smooth bounded open subset of RN , into RN (N ≥ 2). More generally, Ω could be a smooth bounded open subset of an N -dimensional manifold. Starting with the seminal work of C. B. Morrey [27], Y. Reshetnyak [32], and J. Ball [1], it has been known that one can define the distributional Jacobian determinant Det(∇g) under fairly weak N2

assumption on g; in particular, it is defined for all maps g ∈ W 1, N +1 (Ω) and also for all ∗

Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA, [email protected] † Department of Mathematics, Technion, Israel Institute of Technology, 32.000 Haifa, Israel ‡ Courant Institute, New York University, 251 Mercer St., New York, NY, 10012, [email protected]

1

maps g ∈ L∞ (Ω) ∩ W 1,N −1 (Ω) (see e.g., [1], [2], [15], and [17]). Moreover n o N −1 N ∞ hDet(∇g), ψi ≤ C min k∇gk , kgk k∇gk k∇ψkL∞ , ∀ ψ ∈ Cc1 (Ω). L N2 LN −1

(1.1)

L N +1

Estimate (1.1) follows from the divergence structure of the Jacobian determinant which is originally due to Morrey [27, Lemma 4.6.4]. Namely if g is smooth, we have det(∇g) =

N N X X ∂ ∂gi Ci,j = [gi Ci,j ] ∂xj ∂xj

since

X ∂Ci,j ∂xj

j

∀ i = 1, . . . , N,

(1.2)

j=1

j=1

= 0,

∀ i = 1, 2, . . . , N.

∂gi Here (∇g) is the matrix whose components are (∇g)i,j = ∂x , and C = (Ci,j ) is the matrix j of cofactors of matrix (∇g). Consequently, for smooth g, we have Z hDet(∇g), ψi = − gi Fi (g, ψ), (1.3) Ω

where Fi (g, ψ) = det(∇g1 , . . . , ∇gi−1 , ∇ψ, ∇gi+1 , . . . , ∇gN ), for all i = 1, . . . , N , and for all ψ ∈ Cc1 (Ω). Here we use the fact that N X ∂ψ Ci,j = det(∇g1 , . . . , ∇gi−1 , ∇ψ, ∇gi+1 , . . . , ∇gN ), ∂xj

∀ i = 1, . . . , N.

j=1

In particular, if g is smooth on Ω and ψ ∈ Cc1 (Ω), the quantity Z gi Fi (g, ψ) is independent of i.

(1.4)

Ω

Set pN =

N2 N +1 ∗

and note that if g ∈ W 1,pN(Ω), then g ∈ LpN (Ω) by Sobolev’s inequality, with p∗N = N 2 since p1∗ = p1N − N1 = N12 . Therefore p1∗ + Np−1 = 1, and hence by H¨older’s inequality, N N

N

gi det(∇g1 , . . . , ∇gi−1 , ∇ψ, ∇gi+1 , . . . , ∇gN ) ∈ L1 (Ω),

∀ i = 1, . . . , N.

By density, it is easy to see that (1.3) still holds for g ∈ W 1,pN (Ω). Consequently Det(∇g) is a well-defined distribution given by (1.3) (independently of i). It is clear from (1.3) that

2

a) Det(∇g (k) ) converges to Det(∇g) in the distributional sense if g (k) converges to g in W 1,pN (Ω). A more striking well-known property is the fact that b) Det(∇g (k) ) converges to Det(∇g) in the distributional sense if g (k) converges weakly to g in W 1,p (Ω) for some p > pN . The standard argument goes as follows. Since g (k) converges weakly to g in W 1,p (Ω) and ∗ p > pN , we deduce that g (k) converges to g in LpN (Ω) (by the compactness of the embed∗ (k) (k) (k) (k) ding W 1,p (Ω) ⊂ LpN (Ω)). Next we see that det(∇g1 , . . . , ∇gi−1 , ∇ψ, ∇gi+1 , . . . , ∇gN ) is bounded in Lp/(N −1) ; therefore it converges weakly to some limit in Lp/(N −1) (Ω). In fact this limit is precisely det(∇g1 , . . . , ∇gi−1 , ∇ψ, ∇gi+1 , . . . , ∇gN ) (as can be seen by induction using repeatedly formula (1.3)). A very simple alternative proof, which also gives a rate of convergence, will be presented later (see i) of Theorem 1 applied with p = pN and q = p∗N ). More generally Det(∇g) is well-defined as a distribution, via formula (1.3), if g ∈ ∩ Lq (Ω) with N p−1 + 1q = 1 and N − 1 ≤ p ≤ ∞ (note that this formula is independent of i because the validity of (1.4) extends by density to this setting). A particular case is p = pN and q = p∗N . Another interesting case is p = N − 1 and q = +∞. The same method as above gives that W 1,p (Ω)

c) Det(∇g (k) ) → Det(∇g) in the distributional sense if g (k) → g in Lq (Ω) and g (k) * g weakly in W 1,p (Ω) with N p−1 + 1q = 1 and p > N − 1 (i.e., q < +∞). 2

In the special case where p = pN = NN+1 and q = p∗N = N 2 we see that if g (k) → g in ∗ LpN (Ω) and g (k) * g weakly in W 1,pN (Ω) then Det(∇g (k) ) → Det(∇g) in D0 (Ω).

(1.5)

As a consequence of c), we have d) Det(∇g (k) ) converges to Det(∇g) in the distributional sense if p > N − 1, g (k) * g weakly in W 1,p (Ω) and supk kg (k) kLq < +∞ for some q > q0 where q0 is defined by N −1 1 p + q0 = 1. The case q = +∞ and p = N − 1 is more delicate. Indeed, if g (k) * g weakly in W 1,N −1 (Ω), Fi (g (k) , ψ) is bounded in L1 (Ω) and converges only in the sense of measures to Fi (g, ψ), not in σ(L1 , L∞ ) , and this creates a difficulty since g ∈ L∞ (Ω) (g need not be continuous). Nevertheless, we will prove (see Theorem 1) that e) Det(∇g (k) ) converges to Det(∇g) in the distributional sense if g ∈ W 1,N −1 (Ω) ∩ L∞ (Ω) and (g (k) ) ⊂ W 1,N −1 (Ω) ∩ L∞ (Ω) are such that supk kg (k) kW 1,N −1 < +∞ and limk→0 kg (k) − gkL∞ = 0. Our first result is the following 3

Theorem 1. Let N ≥ 2, N − 1 ≤ p ≤ +∞, and 1 ≤ q ≤ +∞ be such that N p−1 + 1q = 1. We have, for all f, g ∈ W 1,p (Ω, RN ) ∩ Lq (Ω, RN ), and for all ψ ∈ Cc1 (Ω, R), i) hDet(∇f ), ψi − hDet(∇g), ψi ≤ CN,Ω kf − gkLq (k∇f kLp + k∇gkLp )N −1 k∇ψkL∞ and ii) hDet(∇f ), ψi − hDet(∇g), ψi ≤ CN,Ω k∇f − ∇gkLp (k∇f kLp + k∇gkLp )N −2 (kf kLq + kgkLq )k∇ψkL∞ . Hereafter in this paper, CN,Ω denotes a positive constant depending only on N and Ω; it can change from one place to another. Surprisingly, estimate i) in Theorem 1 seems to have gone unnoticed until now, although it illuminates the fact that Det(∇g) is continuous under weak convergence e.g. in W 1,p (Ω), p > pN . We also point out that the estimates in Theorem 1 (and Theorems 2, 3 below) can be written in terms of the Wasserstein metric hDet(∇f ), ψi − hDet(∇g), ψi . k Det(∇f ) − Det(∇g)kW = sup ψ∈Cc1 (Ω) k∇ψkL∞ ≤1

In the limiting case p = N − 1 and q = +∞, if N ≥ 3, one can replace the assumption g ∈ W 1,N −1 (Ω) ∩ L∞ (Ω) by g ∈ W 1,N −1 (Ω) ∩ BM O(Ω). We need to give a “robust” meaning to the quantity Det(∇g) (since it is not true anymore that |g||∇g|N −1 ∈ L1 (Ω)). Our argument combines the technique used in the proof of Theorem 1 with the theory of R. Coifman, P. L. Lions, Y. Meyer, and S. Semmes [14, Theorem II.1]. We postpone the precise definition of Det(∇g) and state our basic estimate. Theorem 2. Let N ≥ 3. For all f, g ∈ W 1,N −1 (Ω, RN ) ∩ BM O(Ω, RN ), and for all ψ ∈ Cc1 (Ω, R), we have i) hDet(∇f ), ψi − hDet(∇g), ψi ≤ CN,Ω kf − gkBM O (k∇f kLN −1 + k∇gkLN −1 )N −1 k∇ψkL∞ and ii) hDet(∇f ), ψi − hDet(∇g), ψi ≤ CN,Ω k∇f − ∇gkLN −1 (k∇f kLN −1 + k∇gkLN −1 )N −2 (kf kBM O + kgkBM O )k∇ψkL∞ . Theorems 1 and 2 will be proved in Sections 2.1 and 2.4.

4

Remark 1. In view of Theorem 2 the reader may wonder whether it is possible to improve Theorem 1 and replace k∇ψkL∞ by k∇ψkBM O . The answer is negative. There is no constant C such that, for all g ∈ Cc1 (Ω, RN ), and for all ψ ∈ Cc1 (Ω, R), −1 (1.6) hDet(∇g), ψi ≤ CkgkLq k∇gkN Lp k∇ψkBM O , where

N −1 p

+

1 q

= 1 and 1 ≤ N − 1 ≤ p ≤ ∞. The proof is presented in Section 2.2.

In Section 2.2 we discuss the concept of “dipole” which turns out to be a very effective tool in the study of distributional Jacobians concentrated on “thin” sets. The dipole construction was originally introduced by H. Brezis, J. M. Coron, and E. Lieb [8]. In Section 2.3 we present an example, involving dipoles, which is related to a conjecture of S. M¨ uller [30]. Remark 2. From Theorem 1 we deduce that if g (k) → g in Lq (Ω) and (k∇g (k) kLp )k∈N is bounded or if g (k) → g in W 1,p (Ω) and (kg (k) kLq )k∈N is bounded, then Det(∇g (k) ) converges to Det(∇g) in the sense of distributions. When 1 ≤ N − 1 ≤ p ≤ pN and 1 N −1 (k) k p ) (k) k q ) (k) → 0 L k∈N and (kg L k∈N are bounded, g p + q = 1, it may happen that (k∇g a.e., and Det(∇g (k) ) converges in the sense of distributions to a limit T different from 0, e.g. a derivative of a Dirac mass (see Section 2.2). Such an example was already constructed by B. Dacorogna and F. Murat [15, Proof of Theorem 1] for the special case p = pN and q = N 2 . The construction in the general case N − 1 ≤ p < pN is more delicate and uses dipoles. The second part of our paper is devoted to the search of an “optimal ” space (containing all the above cases) where one can define the Jacobian determinant (note, for example, N2

that neither W 1, N +1 (Ω) nor W 1,N −1 (Ω)∩L∞ (Ω) is a subset of the other). For this purpose it is convenient to work in the fractional Sobolev spaces W s,p (Ω). We postpone again the precise definition of Det(∇g) and state our basic estimate. N −1 ,N N

Theorem 3. Let N ≥ 2. For all f and g ∈ W we have |hDet(∇f ), ψi − hDet(∇g), ψi| ≤ CN,Ω |f − g|

W

N −1 ,N N

(Ω, RN ), and for all ψ ∈ Cc1 (Ω, R),
N −1 |f |N −1 N −1 ,N + |g| N −1 ,N k∇ψkL∞ . W

N

W

N

(1.7) We recall that for 0 < s < 1 and p > 1, 1 p |g(x) − g(y)|p := dx dy ∀ g ∈ Lp (Ω), N +sp |x − y| Ω Ω  s,p W (Ω) := g ∈ Lp (Ω); |g|W s,p < ∞ , Z Z

|g|W s,p (Ω)

and kgkW s,p := kgkLp + |g|W s,p 1

1

As usual, the space W 2 ,2 (Ω) is denoted H 2 (Ω). 5

∀ g ∈ W s,p (Ω).

Remark 3. The proof of Theorem 3, presented in Section 3.1, relies heavily on an idea of 1 J. Bourgain, H. Brezis, and P. Mironescu [6] (see also [7]) concerning maps in H 2 (Ω, S1 ) where Ω is the boundary of a domain in R3 . This idea was subsequently exploited by T. Rivi`ere [35], and F. Hang and F. H. Lin [20]. N −1

Remark 4. Estimate (1.7) applied with f = 0 asserts that ∀ g ∈ W N ,N (Ω, RN ) hDet(∇g), ψi ≤ CN,Ω |g|N N −1 k∇ψkL∞ ∀ ψ ∈ Cc1 (Ω, R). ,N W

(1.8)

N

Using Hahn-Banach it is standard to deduce from (1.8) that the distribution Det(∇g) has the form N X ∂ µi in D0 (Ω), Det(∇g) = ∂xi i=1

where µi , i = 1, . . . , N , are bounded Radon measures on Ω and kµi kM(Ω) ≤ CN,Ω |g|N

W

N −1 ,N N

.

In our situation, we have a better information about the structure of Det(∇g), namely, there exist N functions hi ∈ L1 (Ω), i = 1, . . . , N , such that khi kL1 ≤ CN,Ω |g|N N −1 ,N for W

i = 1, . . . , N , and Det(∇g) =

N X ∂ hi ∂xi

N

in D0 (Ω);

i=1

see Corollary 2 in Section 3.1. Such a property is a direct consequence of the divergence structure of Det(∇g) when g ∈ W 1,p ∩Lq with p and q as in Theorem 1, but it is already non obvious in the framework of Theorem 2. However, one cannot find such functions hi belonging to the Hardy space H1 (Ω) (with an estimate of the H1 -norm); see Remark 1. Remark 5. W. Sickel and A. Youssfi [37], [38] have also defined a distributional Jacobian determinant for maps in a space which resembles ours. They proved that Det(∇g) is wellN −1

defined as a distribution (in the dual of Cc1 (Ω)) if g ∈ HNN (Ω) where Hps denotes the s (F s is the standard usual Bessel-potential space. Recall (see e.g. [39]) that Hps = Fp,2 p,q s (= B s ); in addition F s (Ω) ⊂ F s (Ω) with Lizorkin-Triebel space) and W s,p = Fp,p p,p p,p p,2 strict inclusion if p > 2 (see e.g. [39, Proposition 2 on page 47]). Therefore, if N ≥ 3, N −1

the space HNN (Ω) considered by W. Sickel and A. Youssfi is strictly smaller than the 1

N −1

1

1

space W N ,N (Ω) we use (when N = 2, H22 (Ω) = W 2 ,2 (Ω) = H 2 (Ω)). Moreover, our N −1 proof is much simpler: it relies only on the fact that W N ,N is the trace space of W 1,N (and on Lemma 3 below, which is just an integration by parts), while their proof is quite sophisticated and involves paraproducts. Remark 6. We recover with Theorem 3 all the definitions of Jacobian determinants mentioned above, except the case N = 2, p = 1, and q = ∞. Indeed, we have i) W 1,p (Ω) ⊂ W bedding if p >

N −1 ,N N

N2 N +1

(Ω) with continuous embedding if p ≥

N2 N +1

and compact em-

(see e.g., [39, Section 3.3.1]) (this implies a) and b)). 6

N −1

ii) W 1,p (Ω) ∩ Lq (Ω) ⊂ W N ,N (Ω) with continuous embedding if ∞) except in the case N = 2, p = 1, and q = +∞. Moreover, kgk

W

with α = 1 −

1 N

N −1 ,N N

N −1 p

+ 1q = 1 (1 ≤ q ≤

≤ CkgkαW 1,p kgk1−α Lq ,

(1.9)

(see e.g., [10, Corollary 3.2]). This implies c), and e) for N ≥ 3.

iii) The case where g ∈ W 1,N −1 (Ω) ∩ BM O(Ω) and N ≥ 3 can also be covered by N −1 Theorem 3 using the fact that W 1,N −1 (Ω) ∩ BM O(Ω) ⊂ W N ,N (Ω) with kgk

W

N −1 ,N N

1−α ≤ CkgkαW 1,N −1 kgkBM O,

(1.10)

for α = 1 − N1 . Inequality (1.10) is probably known to the experts but we could not find a reference in the literature; therefore we have presented a proof in the Appendix. Our next result asserts that Theorem 3 is optimal in the framework of the spaces W s,p . More precisely, the distributional Jacobian is well-defined in W s,p (Ω) if and only if N −1 W s,p (Ω) ⊂ W N ,N (Ω). N −1

Theorem 4. Let s ∈ (0, 1) and p ∈ (1, +∞) be such that W s,p (Ω) 6⊂ W N ,N (Ω). Then ¯ RN ) and a function ψ ∈ C 1 (Ω, R) such that there exists a sequence (g (k) ) ⊂ C 1 (Ω, c lim kg (k) kW s,p = 0

k→∞

and

Z lim

k→∞ Ω

det(∇g (k) )ψ = +∞.

(1.11)

(1.12)

In order to prove Theorem 4 we consider all possible cases: N −1

i) s + N1 > max{1, Np } and then W s,p (Ω) ⊂ W N ,N (Ω) (by the fractional Sobolev embedding, see e.g. [39, page 196]), so that the distributional Jacobian is welldefined using Theorem 3. ii) s + N1 < max{1, Np } and then the distributional Jacobian is meaningless because one can construct a sequence (g (k) ) satisfying (1.11) and (1.12) (see Lemma 5 in Section 3.2.1). When p ≥ 2, this was already done by W. Sickel and A. Youssfi [38, Theorem 4]; our argument is very elementary and valid for all p ∈ (1, +∞). iii) the borderline case s +

1 N

= max{1, Np } is more delicate. If p ≤ N and s = N −1

N p

−

1 N

one knows that W s,p (Ω) ⊂ W N ,N (Ω) (see e.g. [39, page 196]). If p > N and s = 1 − N1 the distributional Jacobian is again meaningless because one can exhibit a sequence (g (k) ) satisfying (1.11) and (1.12) (see Lemma 5 in Section 3.2.1). The construction of g (k) in this case is quite subtle and involves several ingredients: a suggestion of L. Tartar used in [2, Counterexample 7.3], a device communicated to us by P. Mironescu [26], and the theory of Besov spaces [36]. 7

¯ ⊂ W NN−1 ,N (Ω) with continuous embedding Using Theorem 3 and the fact that C 0,α (Ω) ¯ RN ) with α > N −1 when α > NN−1 , we are able to define Det(∇g) for maps g ∈ C 0,α (Ω, N and obtain: Corollary 1. Let N ≥ 2 and

N −1 N

< α < 1. We have

¯ RN ) then for some positive constant CN,Ω,α , i) If g ∈ C 0,α (Ω, |hDet(∇g), ψi| ≤ CN,Ω,α |g|N C 0,α k∇ψkL∞ ,

∀ ψ ∈ Cc1 (Ω).

¯ RN ) converges to some g in C 0,α (Ω, ¯ RN ), then ii) If (g (k) ) ⊂ C 0,α (Ω, lim hDet(∇g (k) ), ψi = hDet(∇g), ψi,

k→∞

∀ ψ ∈ Cc1 (Ω).

Corollary 1 is optimal in the framework of H¨older spaces (see Proposition 4 in Section 3.2.2). In an earlier paper [11], we studied “minimal” assumptions in order to define Det(∇g) (as a distribution) for maps g from SN into itself. The condition g ∈ V M O(SN , SN ) ∩ N −1 W N ,N (SN , SN ) played there an essential role. As mentioned above, if g : RN → RN , N −1 we only need the condition g ∈ W N ,N (RN , RN ) (and the stringent V M O assumption is totally unnecessary). We point out that prior to this work, other authors were also concerned with the definition of a distribution Jacobian for maps g : Ω → SN −1 (we really mean SN −1 , not SN ), where Ω is a domain in RN , or an N -dimensional manifold. Of course, in this case Det(∇g) is a distribution concentrated on the singular set of g. F. Hang and F. H. Lin [20] (see also R. Jerrard and H. Soner [25]) assumed that g ∈ N −1 W N ,N (Ω, SN −1 ). More generally, J. Bourgain, H. Brezis, and P. Mironescu could even define Det(∇g) for all maps g in W s,p (Ω, SN −1 ) for any s ∈ (0, 1) with sp = N − 1. Finally, we mention that the Jacobian determinant was extensively studied in the literature see e.g., [27], [1], [2], [33], [34], [28], [29], [15], [24], [9], [14], [30], [19], [22], [23], [16], [17], [21], and references therein.

2

Theorems 1 and 2, and related topics

2.1

Proof of Theorem 1

It suffices to prove the results for f and g smooth. Set g = (g1 , . . . , gN ) and f = (f1 , . . . , fN ). Write N X det(∇f ) − det(∇g) = Xi , i=1

where

  X1 = det(∇(f1 − g1 ), ∇f2 , . . . , ∇fN ), 

XN = det(∇g1 , ∇g2 , . . . , ∇gN −1 , ∇(fN − gN )), 8

and for i = 2, . . . , N − 1, Xi = det(∇g1 , . . . , ∇gi−1 , ∇(fi − gi ), ∇fi+1 , . . . , ∇fN ). Applying (1.3) and H¨ older’s inequality yields Z −i i−1 ∞ Xi ψ ≤ kfi − gi kLq k∇f kN Lp k∇gkLp k∇ψkL . Ω

This implies i). To prove ii), it suffices to note that, by (1.3), Z  N −2   p p p q q X ψ ≤ k∇f − ∇g k k∇f k + k∇gk kf k + kgk i i i L L L L L k∇ψkL∞ . Ω



2.2

The dipole construction. Further discussion around Theorem 1

The concept of dipole plays an important role in this section and we recall its construction. Fix a smooth map ω : RN −1 → SN −1 such that ω(y) = N = (0, . . . , 0, 1) for |y| ≥ 1 and ω covers SN −1 exactly once (ω has a degree 1 if RN −1 is identified with SN −1 via a stereographic projection). Consider the cone Q0 in RN defined by Q0 = {x = (x0 , z) ∈ RN −1 × R;

L|x0 | < 1 with 0 < z < L}, ρz

with height L and spherical base of radius ρ ≤ L. The maps f0 : RN −1 × (−∞, L) → RN is defined by   0   ω Lx − N if (x0 , z) ∈ Q0 , 0 ρz f0 (x , z) = (2.1)   0 otherwise. Next we perform a symmetry Σ about the hyperplane {(x0 , L); x0 ∈ RN −1 } and we obtain a map f : RN → RN with support in the region Q = Q0 ∪ Σ(Q0 ). When needed we will write fL,ρ instead of f . The map f is smooth except at the points P = (0, . . . , 0) and N = (0, . . . , 0, 2L). Moreover (see the computation below), f ∈ W 1,p (RN ) for all 1 ≤ p < N and f ∈ L∞ (RN ); f does not belong to W 1,N (RN ). Such an f is a good example of a map which enters in framework of Theorem 1. Multiplying f by an appropriate factor (and keeping the same notation f ) we obtain, via a standard computation, Det(∇f ) = δP − δN .

(2.2) 1

More generally if ν is an integer we may glue ν copies of ω in a ball of radius R ≈ ν N −1 . After rescaling we obtain a map ω ν : RN −1 → SN −1 such that  supp ω ν ⊂ B1 , the unit ball of RN −1 ,      1 (2.3) k∇ω ν kL∞ . ν N −1 ,      ν ω covers SN −1 exactly ν times. 9

Hereafter in this section, a . b means a ≤ Cb for some C > 0 depending only on p, q N , and Ω, a & b means b . a and a ≈ b means a . b and b . a. The corresponding f ν (defined via (2.1)) satisfies Det(∇f ν ) = ν(δP − δN ). Note that vol Q ≈ ρN −1 L, and thus ∀ q ∈ [1, ∞), kf kqLq ≈ ρN −1 L

(2.4)

kf kL∞ = 2.

(2.5)

while On the other hand, we have in Q0 ,  Lx0   L L|x0 |  |∇f0 (x0 , z)| . ∇ω + ρz ρz ρz 2  Lx0   L 1   Lx0  2L + ≤ ∇ω ≤ ∇ω , ρz ρz z ρz ρz since ρ ≤ L. Therefore we have, provided p < N , Z |∇f (x0 , z)|p dx0 dz . ρN −1−p L.

(2.6)

Q

For later reference note that, by (2.3), Z p |∇f ν |p . ρN −1−p Lν N −1 Q

and in particular Z

|∇f ν |N −1 . Lν.

(2.7)

Q

We may also glue a sequence of disjoint dipoles ([Pi , Ni ])i∈N placed on the xN -axis, with ρi = Li = 12 |Pi − Ni |. For any p < N we obtain a map f ∈ W 1,p (RN ) ∩ L∞ (RN ) satisfying ∞ X Det(∇f ) = (δPi − δNi ) in D0 (RN )

(2.8)

i=1

P P provided i |Pi − Ni | < +∞ and i |Pi − Ni |N −p < +∞. Note that the RHS in (2.8) is a distribution and is not a measure (more precisely Det(∇f ) belongs to the dual of Cc1 ). We may now state a result mentioned in Remark 2.

10

Proposition 1. Assume p and q satisfy 1 ≤ N − 1 ≤ p ≤ pN

(2.9)

N −1 1 + = 1. p q

(2.10)

and

Then there exists a sequence g (k) in Cc∞ (RN ) such that  supp g (k) ⊂ B(0, rk ) with rk → 0 as k → ∞,          k∇g (k) kLp ≤ C,   kg (k) kLq ≤ C,        det(∇g (k) ) →

∂ ∂xN δ0

in D0 (RN ) as k → ∞.

Proof. We distinguish two cases: Case 1: N − 1 < p ≤ pN and N 2 ≤ q < ∞. Case 2: p = N − 1 and q = ∞. Case 1: N − 1 < p ≤ pN < N and N 2 ≤ q < ∞. Set gL,ρ =

1 1

(2L) N

fL,ρ ,

so that, by (2.2), we have Det(∇gL,ρ ) =

1 (δP − δN ). 2L

From (2.4) and (2.6) we obtain p

k∇gL,ρ kpLp . ρN −1−p L1− N and

q

kgL,ρ kqLq . ρN −1 L1− N . It follows from (2.9) and (2.10) that γ=

1 − p/N q/N − 1 = ≥ 1. p − (N − 1) N −1

Finally we choose ρ = Lγ and then

  k∇gL,ρ kLp . 1, kgL,ρ kLq . 1,     ρ ≤ L provided L ≤ 1,      supp g ⊂ B(0, L). L,ρ 11

(2.11)

Moreover, by (2.11), ∂ δ0 as L → 0. ∂xN The desired result is obtained after changing xN into −xN and smoothing gL,Lγ by convolution with a sequence of mollifiers. Det(∇gL,Lγ ) → −

Case 2: p = N − 1 and q = ∞. Here we set ν gL,ν = fL,L ,

so that Det(∇gL,ν ) = ν(δP − δN ). From (2.5) and (2.7), we have −1 k∇gL,ν kN . Lν LN −1

and k∇gL,ν kL∞ ≤ 2. Finally we choose L =

1 2ν

and we see that Det(∇g 1 ,ν ) → − 2ν

∂ δ0 as ν → ∞. ∂xN

We conclude as above.



In view of Proposition 1 one may wonder whether there exists some g, satisfying the assumptions of Theorem 1, such that Det(∇g) =

∂ δa ∂xN

with a ∈ Ω.

(2.12)

The answer is negative. Here is the reason. Without loss of generality, we Q may assume −1 that 0 ∈ [−1, 1]N ⊂ Ω and a is the origin. Consider ψε (x1 , . . . , xN ) = ψ1,ε (xN ) N i=1 ψ2 (xi ) where ψ1,ε (xN ) = εψ1 (xN /ε), ψ1 , ψ2 are smooth, 0 ≤ ψ1 ≤ 1, ψ10 (0) = −1, supp ψ1 ⊂ (−1, 1), 0 ≤ ψ2 ≤ 1, supp ψ2 ⊂ (−1, 1), and ψ2 = 1 in (−1/2, 1/2). Then, by (2.12), hDet(∇g), ψε i = 1 ∀ 0 < ε < 1. Using (1.3), we write N X ∂ Det(∇g) = hi in D0 (Ω) with hi ∈ L1 (Ω) for i = 1, . . . , N ∂xi

(2.13)

i=1

and we deduce that Z N −1 Z X |hi | + hDet(∇g), ψi . ε i=1

|hN | → 0 as ε → 0.

|xN | 0 depending only on N and Ω, a & b means b . a and a ≈ b means a . b and b . a. Proof of Lemma 4. Let f˜ and g˜ be extensions of f and g to RN such that kf˜i k

W

N −1 ,N N (RN )

. kfi k

W

N −1 ,N N (Ω)

k˜ gi k

,

W

N −1 ,N N (RN )

. kgi k

W

N −1 ,N N (Ω)

,

and kf˜i − g˜i k

W

. kfi − gi k

N −1 ,N N (RN )

W

N −1 ,N N (Ω)

,

for i = 1, . . . , N with f = (f1 , . . . , fN ) and g = (g1 , . . . , gN ). Let u and v be the extensions by average of g˜ and f˜ to Ω × [0, 1) i.e., Z Z u(x, r) = g˜(y) dy and v(x, r) = f˜(y) dy, B(x,r)

B(x,r)

where B(x, r) denotes the ball B(x, r) = {y ∈ RN ; |y − x| < r}. We have, by standard trace theory, k∇ui kLN (Ω×(0,1)) . kgi k

W

N −1 ,N N (Ω)

k∇vi kLN (Ω×(0,1)) . kfi k

,

W

N −1 ,N N (Ω)

and k∇ui − ∇vi kLN (Ω×(0,1)) . kgi − fi k

W

N −1 ,N N (Ω)

.

Let ϕ ∈ Cc1 (Ω × [0, 1)) be such that k∇ϕkL∞ (Ω×[0,1)) . k∇ψkL∞ (Ω) . Since |Di (u)| .

N Y

|∇uj |,

j=1

and
|Di (u) − Di (v)| . |∇u − ∇v| |∇u|N −1 + |∇v|N −1 , 19

,

it follows from Lemma 3 and H¨ older’s inequality, that N Z Y kgj k det(∇g)ψ ≤ CN,Ω Ω

j=1

W

N −1 ,N N

k∇ψkL∞ ,

and Z Z det(∇g)ψ det(∇f )ψ − Ω

Ω

≤ CN,Ω kf − gk

W

Finally, we have Z Z Z   det(∇f )ψ = det ∇(f − f ) ψ Ω

Ω


N −1 kf kN −1 N −1 ,N + kgk N −1 ,N k∇ψkL∞ .

N −1 ,N N

W

W

N

Z

Z

Ω

Ω



Z

det ∇(g −

det(∇g)ψ =

and

N

Ω

 g) ψ,

Ω

and the conclusion follows.



Based on Lemma 3, we can give a “robust” definition of Det(∇g) when g ∈ W

N −1 ,N N

(Ω).

N −1

Definition 2. Let N ≥ 2Rand g ∈ W N ,N (Ω, RN ). For any ψ ∈ Cc1 (Ω, R) we define ¯ RN ) such that hDet ∇g, ψi as the limit of Ω det(∇g (k) )ψ for any sequence (g (k) ) ⊂ C 1 (Ω, g (k) → g in W

N −1 ,N N

(Ω).

This object is well-defined according to Lemma 4 and the fact that for any g ∈ ¯ such that g (k) → g in W NN−1 ,N (Ω). W (Ω), there exists (g (k) ) ⊂ C 1 (Ω) N −1 ,N N

It is clear that Theorem 3 is a consequence of Lemma 3 and Definition 2. Our next result provides a fundamental representation of the distribution Det(∇g) (which might also serve as an alternative definition for Det(∇g)). Proposition 3. Let N ≥ 2, g ∈ W

N −1 ,N N

hDet(∇g), ψi =

(Ω, RN ), and ψ ∈ Cc1 (Ω, R). Then N +1 Z X i=1

Di (u)∂i ψ,

(3.4)

Ω×(0,1)

for any extensions u ∈ W 1,N (Ω × (0, 1), RN ) and ϕ ∈ Cc1 (Ω × [0, 1), R) of g and ψ, where Di (u) ∈ L1 (Ω × (0, 1)), 1 ≤ i ≤ N + 1, is defined in Lemma 3. ¯ Proof. Let u(k) be a sequence in C 1 (Ω×[0, 1], RN ) such that u(k) → u in W 1,N (Ω×(0, 1)). By trace theory we know that g (k) = u(k) |Ω×{0} → g in W

N −1 ,N N

(Ω, RN ) as k → ∞.

From Definition 2 we deduce that Z det(∇g (k) )ψ → hDet(∇g), ψi as k → ∞. Ω

20

On the other hand, we have by Lemma 3 Z det(∇g

(k)

)ψ =

Ω

N +1 Z X i=1

Di (u(k) )∂i ϕ.

Ω×(0,1)

Passing to the limit as k → ∞ we obtain the desired conclusion.



From Proposition 3 we can deduce some information about the structure of the distribution Det(∇g) (compare with (1.3) in the case g ∈ W 1,p (Ω, RN ) ∩ Lq (Ω, RN ) with N −1 1 p + q = 1). N −1

Corollary 2. Let N ≥ 2 and g ∈ W N such that khkL1 ≤ CΩ,N kgkN N −1 ,N and W

,N

(Ω, RN ). Then there exist h = (hi ) ∈ [L1 (Ω)]N

N

Det(∇g) =

N X ∂ hi in D0 (Ω). ∂xi i=1

N −1 ,N N

Moreover, for g (1) and g (2) ∈ W kh(1) − h(2) kL1 ≤ CΩ,N kg (1) − g (2) k

W

Det(∇g (m) ) =

(Ω), there exist h(1) and h(2) ∈ [L1 (Ω)]N such that (1) k (2) k N −1 and N −1 ,N (kg N −1 ,N + kg N −1 ,N ) W

N

W

N

N X ∂ (m) h in D0 (Ω) ∂xi i

N

for m = 1, 2.

i=1

Proof. We only prove the first statement of Corollary 2. The second statement follows by the same method. Let u be the extension of g as in Lemma 4. Then kukW 1,N (Ω×(0,1)) ≤ CΩ,N kgk

W

N −1 ,N N (Ω)

.

(3.5)

Let ψ ∈ Cc1 (Ω, R) and ζ ∈ C 1 ([0, 1], R) be such that ζ = 1 on [0, 1/4] and ζ = 0 on [1/2, 1]. Set ϕ(x, xN +1 ) = ψ(x)ζ(xN +1 ). By Proposition 3, we have hDet(∇g), ψi =

N Z X

∂ψ Di (u)ζ + ∂x i Ω×(0,1)

i=1

Set ˜ i (x) = h

Z

Z DN +1 (u)ψ Ω×(0,1)

∂ζ . ∂xN +1

1

Di (u)ζ(xN +1 ) dxN +1

∀ 1 ≤ i ≤ N,

0

and ˜ N +1 (x) = h

Z

1

DN +1 (u)ζ 0 (xN +1 ) dxN +1 .

0

˜ i belongs to L1 (Ω) for i = 1, . . . , N + 1. Moreover we have By Fubini we know that h hDet(∇g), ψi =

N Z X i=1

˜ i ∂ψ + h ∂xi Ω

21

Z Ω

˜ N +1 ψ, h

i.e., Det(∇g) = −

N X ∂ ˜ ˜ N +1 . hi + h ∂xi i=1

˜ N +1 as a divergence of an L1 vector-field. The conclusion follows from (3.5) by writing h  Remark 10. In view of Corollary 2, the conclusion of Proposition 2 remains valid for N −1 every g ∈ W N ,N (Ω), in particular one cannot have Det(∇g) = ∂x∂N δa for some a ∈ Ω if g∈W

N −1 ,N N

3.2

Optimality results

(Ω).

In this section, a . b means a ≤ Cb for some C > 0 independent of k, a & b means b . a and a ≈ b means a . b and b . a. 3.2.1

Proof of Theorem 4

Theorem 4 is consequence of the following lemma as explained in the Introduction. Lemma 5. Let N ≥ 2, s ∈ (0, 1), and p ∈ (1, ∞) be such that i) either s+

n No 1 < max 1, , N p

ii) or s=1−

1 and p > N N

¯ RN ) and a function ψ ∈ C 1 (Ω, R) such that Then there exist a sequence (g (k) ) ⊂ C 1 (Ω, c limk→∞ kg (k) kW s,p = 0 and Z lim det(∇g (k) )ψ = +∞. k→∞ Ω

Remark 11. Let s = NN−1 and p ∈ (1, +∞). We deduce from Lemma 5 and Theorem 3 that W s,p (Ω) ⊂ W s,N (Ω) if and only if p = N . Indeed, suppose that W s,p (Ω) ⊂ N −1 W N ,N (Ω). Consider the case p < N . Applying i) of Lemma 5 and the closed graph theorem we have kg (k) kW s,N ≤ Ckg (k) kW s,p → 0 as k → ∞, and

Z lim

k→∞ Ω

det(∇g (k) )ψ = +∞.

On the other hand, we deduce from Theorem 3 that Z ∞ det(∇g (k) )ψ ≤ Ckg (k) kN W s,N k∇ψkL → 0 as k → ∞, Ω

22

which contradicts the previous assertion. When p > N , we apply ii) of Lemma 5 and obtain again a contradiction. P. Mironescu [26] has established the same property in the general case: let s ∈ (0, 1) and p, q ∈ (1, +∞), then W s,p (Ω) ⊂ W s,q (Ω) if and only if p = q. There is a sharp difference with the situation in the Bessel potential spaces Hps . For such spaces it is known (see e.g. [39, Theorem on page 196]) that Hps (Ω) ⊂ Hqs (Ω) when N −1

p ≥ q. As a consequence Det(∇g) is well-defined when g ∈ Hp N (Ω) and p > N , since N −1

N −1

Hp N (Ω) ⊂ HNN (Ω) ⊂ W N −1 W N ,p (Ω) when p > N .

N −1 ,N N

(Ω); by contrast Det(∇g) is meaningless on the space

Proof of Lemma 5. Without loss of generality, one may assume that (−4, 4)N ⊂ Ω. We distinguish three cases: Case 1: p ≤ N and s + 1/N < N/p. Case 2: p > N and s + 1/N < 1. Case 3: p > N and s = 1 − 1/N . Case 1: We use the same notation as in the proof of Remark 1, then we set 1

hε (x) = ε− N g(x/ε). We know (see (2.19)) that Z det(∇hε )ψ = Ω

N X i=1

αi

∂ψ (0) + O(ε) with α1 = 1. ∂xi

On the other hand, |hε |pW s,p =

1

Z Z

p

εN

Ω

Ω

|g(x/ε) − g(y/ε)|p dx dy ≈ εN −sp−p/N → 0 as ε → 0. |x − y|N +sp

Indeed, recall that s + 1/N < N/p. To obtain the desired conclusion take g (k) (x) = ε−γ hε (x) with 0 < γ sufficiently small and ε = 1/k, and then choose any function ψ ∈ ∂ψ ∂ψ Cc∞ (Ω) such that ∂x (0) > 0 and ∂x (0) = 0 for i = 2, . . . , N . 1 i Case 2 can be deduced from Case 3. However the proof of Case 2 is simple, while the proof of Case 3 is tricky. Since the proof of Case 3 borrows some ideas from the proof of Case 2, we have included both proofs for the convenience of the reader. (k)

(k)

Case 2: Let 0 < s < α < NN−1 . For k  1, define g (k) = (g1 , . . . , gN ) : Ω → RN as follows (k) gi (x) = k −α sin(kxi ) ∀ 1 ≤ i ≤ N − 1, and (k)

gN (x) = k −α xN

N −1 Y i=1

23

cos(kxi ).

We have det(∇g

(k)

)=k

(N −1)(1−α)−α

N −1 Y

cos2 (kxi ) ≥ 0.

(3.6)

i=1

Since k∇g (k) kL∞ . k 1−α and kg (k) kL∞ . k −α , it follows by interpolation that |g (k) |C 0,α (Ω) ¯ . 1.

(3.7)

Let ψ ∈ Cc1 (Ω, R) be such that supp ψ ⊂ (1/5, 4/5)N , ψ ≥ 0, ψ = 1 on (1/4, 3/4)N . We have Z

det(∇g (k) )ψ dx ≥

Z

k (N −1)(1−α)−α

N −1 Y

(1/4,3/4)N

Ω

cos2 (kxi ) dx & k (N −1)(1−α)−α . (3.8)

i=1

It follows from (3.8) that Z lim

k→∞ Ω

det(∇g (k) ) ψ dx = +∞.

Since α > s, we have kg (k) kW s,p . kg (k) kC 0,α (Ω) ¯ and the conclusion follows. Case 3: Fix k  1. Let

2

n` = k N 8` for 1 ≤ ` ≤ k.

(3.9)

It is clear that n`+1 ≥ 2n`

∀ ` = 1, . . . , k − 1,

(3.10)

{n` ; ` = 1, . . . , k} ∩ {z ∈ R; 2m−1 ≤ |z| < 2m } has at most one element, ∀ m ∈ N, (3.11) and

N2

min |ni − nj | ≥ k N −1 .

(3.12)

i6=j

(k)

(k)

Define g (k) = (g1 , . . . , gN ) : Ω → RN as follows (k)

gi (x) =

k X `=1

1 N −1 N

and (k)

gN (x) = xN

sin(n` xi ) ∀ 1 ≤ i ≤ N − 1,

(3.13)

(` + 1)1/N

n`

k X `=1

1 N −1 N

n`

(` +

N −1 Y

cos(n` xi ).

(3.14)

1)1/N i=1

Fix ψ1 ∈ Cc1 (R) suchQthat supp ψ1 ⊂ (0, 1), ψ1 ≥ 0, ψ1 = 1 in (1/4, 3/4). Define ψ : Ω → R by ψ(x1 , . . . , xN ) = N i=1 ψ1 (xi ). We claim that Z

det(∇g (k) )ψ & ln k

Ω

24

(3.15)

and kg (k) kp W

N −1 ,p N

≈ 1.

(3.16)

Assuming that (3.15) and (3.16) hold, we deduce that h(k) = (ln k)−1/(2N ) g (k) satisfies all the requirements. Hence it remains to prove (3.15) and (3.16). Step 1: Proof of (3.15). From (3.13) and (3.14), it is clear that 1 N `i





det(∇g (k) ) = 

k X



(` + 1)1/N ` =1 i

N −1 Y i=1

n

 cos(n`i xi ) ×

i

 ×

k X

`N =1

N −1 Y

1 n

N −1 N `N

(`N +

 cos(n`N xj ) .

1)1/N j=1

This implies det(∇g (k) ) =

k X `=1

N −1 Y 1 cos2 (n` xi ) (` + 1) i=1

+

N −1 h Y

1

X (`1 ,...,`N )6=(`,...,`)

for

`=1,...,k

n

N −1 N `N

(`N + 1)1/N

i=1

1

n`Ni (`i + 1)1/N

i cos(n`i xi ) cos(n`N xi ) . (3.17)

On the other hand, 1 Z N −1 h N i Y n`i 1 ψ(x) cos(n`i xi ) cos(n`N xi ) dx N −1 1/N (`i + 1) n N (`N + 1)1/N Ω i=1 `N

. and

1

1

N −1 Y

n`Ni

n`NN (`N + 1)1/N

i=1

(`i + 1)1/N

N −1

Z

0

1

Z

0

1

ψ1 (xi ) cos(n`i xi ) cos(n`N xi ) dxi (3.18)

ψ1 (xi ) cos(n`i xi ) cos(n`N xi ) dxi . min{1/|n`i − n`N |, 1}.

25

(3.19)

Since `j ≥ 1 for 1 ≤ j ≤ N , it follows from (3.18) and (3.19) that 1 Z N −1 N Y n`i 1 ψ(x) cos(n`i xi ) cos(n`N xi ) dx N −1 1/N (`i + 1) n N (`N + 1)1/N Ω i=1 `N

.

1

N −1 h Y

n`Ni

i=1

n`NN

1

i min{1/|n`i − n`N |, 1} . (3.20)

If (`1 , . . . , `N ) = 6 (`, . . . , `) for all ` = 1, . . . , k, then there exists 1 ≤ i ≤ N − 1 such that `i 6= `N . Hence from (3.20) and (3.10), we have, if (`1 , . . . , `N ) 6= (`, . . . , `) for all ` = 1, . . . , k, 1 Z N −1 Y n`Ni 1 x ) dx ψ(x) cos(n x ) cos(n i i ` ` i N N −1 1/N (` + 1) i n N (`N + 1)1/N Ω i=1 `N

1

.

n`Ni n

1 1 . (3.21) N −1 , |n`i − n`N | |n`i − n`N | N

1 N `N

for some 1 ≤ i ≤ N − 1. Since Z ψ Ω

N −1 Y

cos2 (n` xi ) & 1 ∀ ` = 1, . . . , k,

i=1

we deduce from (3.17) and (3.21) that Z

det(∇g (k) )ψ &

Ω

k X `=1

1 1 − Ck N max N −1 . i,j |n − n | N (` + 1) i j

Claim (3.15) now follows from (3.12). Step 2: Proof of (3.16). Let TN = [−π, π]N be the N -dimensional torus. We will prove that kg (k) k

W

≈ 1.

N −1 ,p N (TN )

(3.22)

This will imply (3.16) since kg (k) k

W

N −1 ,p N (Ω)

≈ kg (k) k

W

26

N −1 ,p N (TN )

.

For this purpose we define R0 = {0 = (0, . . . , 0) ∈ ZN }, Rj = {r = (r1 , . . . , rN ) ∈ ZN ; 2j−1 ≤ maxm=1,...,N |rm | < 2j },

∀ j ∈ N+ .

We recall (see e.g. [36, Theorems on pages 167 and 168]) that for 0 < s < 1 and 1 < p < +∞,

X

p

p

X

p

X X X



2spj ar eir·x p N , ar eir·x s,p N ≤ C2 C1 2spj ar eir·x p N ≤ j∈N

r∈Rj

L (T )

W

r∈ZN

(T )

r∈Rj

j∈N

L (T )

(3.23) for some positive constants C1 and C2 depending only on N , s, and p. We claim that (k) kgi kp N −1 ,p W N (TN )

≈

k X `=1

1 k sin(n` xi )kpLp (TN ) p/N (1 + `)

∀ 1 ≤ i ≤ N − 1.

(3.24)

P 1 in` xi ir·x for some [e − e−in` xi ], we P have g (k) = Indeed, since sin(n` xi ) = 2i re r∈ZN a P ir·x (ar ). Moreover from (3.11), for each j ∈ N, either r∈Rj ar e = 0 or r∈Rj ar eir·x = 1 sin(n` xi ) for some ` with n` ≈ 2j . Therefore, (3.24) follows from (3.23). N −1 n`

N

(`+1)1/N

Using the fact that p > N we deduce from (3.24) that (k)

kgi kp

W

N −1 ,p N (TN )

≈1

∀ 1 ≤ i ≤ N − 1.

(3.25)

Similarly, we have (k) kgN /xN kp N −1 ,p W N (TN )

≈

k X `=1

−1

NY

p 1

cos(n` xi ) p N ≈ 1.

p/N L (T ) (1 + `) `=1

Assertion (3.22) now follows from (3.25) and (3.26).

(3.26) 

Remark 12. Statement i) of Lemma 5 follows from Cases 1 and 2, statement ii) follows from Case 3. 3.2.2

Optimality of Corollary 1

Corollary 1 is optimal in the following sense: ¯ RN ) and a function Proposition 4. Let N ≥ 2. There exist a sequence (g (k) ) ⊂ C 1 (Ω, (k) 1 (k) ψ ∈ Cc (Ω, R) such that limk→∞ kg k 0, N −1 ¯ = 0, supk kg k N −1 ,N < +∞, and C

Z lim

k→∞ Ω

N

(Ω)

det(∇g (k) )ψ > 0

27

W

N

Proof. We use the same notation as in the proof of Lemma 5, Case 3. As in the proof of the Case 3 of Lemma 5, we have kg (k) kN W

N −1 ,N N

≈

k X `=1

1 ≈ ln k. (1 + `)

(3.27)

We claim that kg (k) k

C 0,

N −1 ¯ N (Ω)

. 1.

(3.28)

Assuming (3.28) holds, we deduce from (3.15), (3.27), and (3.28) that h(k) = (ln k)−1/N g (k) satisfies all the requirements. Hence it remains to prove (3.28). Let Ψ ∈ Cc∞ (RN ) be such that Ψ is radial, √ √ supp Ψ ⊂ {x ∈ RN ; 1/2 ≤ |x| ≤ 2} and Ψ > 0 in {x ∈ RN ; 1/ 2 ≤ |x| ≤ 2}. P −1 P ∞ −m x) Define Φj (x) = Ψ(2−j x) if j = 1, 2, . . . and Φ0 (x) = 1− ∞ m=−∞ Ψ(2 j=1 Φj (x). We recall (see e.g. [36, Theorem on 168]) that for 0 < s < 1,

X

C1 sup 2sj Φj (r)ar eir·x j∈N

L∞ (TN )

r∈ZN

X

≤ ar eir·x

C s (TN )

r∈ZN

X

≤ C2 sup 2sj Φj (r)ar eir·x j∈N

L∞ (TN )

r∈ZN

, (3.29)

for some positive constants C1 and C2 depending only on N and s. We claim that (k)

kgi k

C

N −1 N (TN )

1 k sin(n` xi )kL∞ (TN ) . 1, (1 + `)1/N `=1,...,k

. sup

∀ 1 ≤ i ≤ N − 1,

(3.30)

P (k) 1 in` xi ir·x for some (a ). i ], we have g Indeed, since sin(n` xi ) = 2i [e − e−in` xP = r∈ZN aP re r i ir·x Moreover from (3.9), for each j ∈ N, either r∈ZN Φj (r)ar e = 0 or r∈ZN Φj (r)ar eir·x = Φj ((n` ,0,...,0)) N −1 N (`+1)1/N

sin(n` xi ) for some ` with n` ≈ 2j (because Φj is radial). Since |Φj | ≤ 1, (3.30)

n`

follows from (3.29). Similarly, (k) kgN /xN k N −1 N C N (T )

−1

NY

1

. sup cos(n` xi ) L∞ (TN ) . 1. 1/N `=1,...,k (1 + `) `=1

Hence by the same method used in the proof of the Case 3 of Lemma 5, we have kg (k) k

C 0,

N −1 ¯ N (Ω)

. 1. 

28

A

Appendix: An interpolation inequality

Lemma A1. Let N ≥ 1, θ ∈ (0, 1), s > 0, and p > 1. Suppose that g ∈ W s,p (RN ) ∩ BM O(RN ), then g ∈ W θs,p/θ (RN ) and 1−θ kgkW θs,p/θ (RN ) ≤ c(N, p, θ, s)kgkθW s,p (RN ) kgkBM . O(RN )

(A1)

Here we use the following BM O-norm: Z Z Z kgkBM O(RN ) := sup |g(x) − g(y)| dx dy + sup |g| dx, |Q|1 Q

Z |Q| denotes the volume of Q and Q

1 := |Q|

Z . Q

The proof of Lemma A1 uses two basic ingredients: i) An estimate due to Oru [31]. The proof of this estimate is not readily available; we refer the reader to the proof reproduced in [10]. Related results also appeared in [13] and [12]. ii) A characterization of BM O functions in terms of their Littlewood-Paley decomposition (see e.g., [39]). Proof of Lemma A1. In this proof we will use the standard notion of Littlewood-Paley theory (see e.g. [39]). Let Ψ ∈ Cc∞ (RN ) be such that √ √ supp Ψ ⊂ {x ∈ RN ; 1/2 ≤ |x| ≤ 2} and Ψ > 0 in {x ∈ RN ; 1/ 2 ≤ |x| ≤ 2}. Define Φj (x) = Ψ(2−j x) For u ∈

S 0 (RN )

P

∞ −k k=−∞ Ψ(2 x)

−1

if j = 1, 2, . . . and Φ0 (x) = 1−

(the space of tempered distributions), set uj = F −1 Φj Fu,

where F denotes the Fourier transform. We have (see e.g., [39, page 51]) ( kgkW s,p ≈

kgkF˜ s

if s 6∈ N+ ,

kgkF˜ s

otherwise,

p,p p,2

and kgkBM O ≈ kgkF˜ 0 . ∞,2

Here  Z kgkF˜ s

p,q

:= 

RN

∞ X

sqj

2

j=0

29

q

|gj (x)|

p q

1

p

dx ,

P∞

j=1 Φj (x).

where gj := F −1 Φj Fg, for −∞ < s < +∞, 0 < p ≤ +∞, and 0 < q ≤ +∞ with the usual notation for p = +∞ or q = +∞. On the other hand, [10, Lemma 3.1] asserts that kf kF˜ps0,q . kf kθF˜ s1 kf k1−θ , s F˜ 2 0

p1 ,q1

0

(A2)

p2 ,q2

for −∞ < s1 , s2 < +∞, 0 < q0 , q1 , q2 ≤ +∞, 0 < p1 , p2 ≤ +∞, 0 < θ < 1, and (s0 , p0 ) such that s0 = θs1 + (1 − θ)s2 , θ 1−θ 1 = + . p0 p1 p2 Applying (A2) with (s1 , p1 , q1 ) = (s, p, p) if s 6∈ N+ and (s1 , p1 , q1 ) = (s, p, 2) otherwise, (s2 , p2 , q2 ) = (0, +∞, +∞), and (s0 , p0 , q0 ) = (θs, p/θ, p/θ) if θs 6∈ N+ and (s0 , p0 , q0 ) = (θs, p/θ, 2) otherwise, we have kgkW θs,p/θ ≤ c(N, p, θ, s)kgkθW s,p kgk1−θ F˜ 0

.

+∞,+∞

Since kgkF˜ 0

+∞,+∞

= sup sup |gj (x)| ≤ sup x

j

x

∞ X

|gj (x)|2

1 2

= kgkF˜ 0

∞,2

≈ kgkBM O ,

j=0

the conclusion follows.



Lemma A2. Let N ≥ 1, θ ∈ (0, 1), s > 0, p > 1, and Ω be a smooth bounded open subset of RN . Suppose that g ∈ W s,p (Ω) ∩ BM O(Ω) then g ∈ W θs,p/θ (Ω) and 1−θ kgkW θs,p/θ ≤ c(N, p, θ, s, Ω)kgkθW s,p kgkBM O.

Proof. Let G be an extension of g to RN such that kGkW s,p (RN ) . kgkW s,p (Ω) and kGkBM O(RN ) . kgkBM O(Ω) . By Lemma A1, G ∈ W θs,p/θ (RN ) and kGkW θs,p/θ (RN ) ≤ c(N, p, θ, s, Ω)kGkθW s,p (RN ) kGk1−θ . BM O(RN ) The conclusion follows.



Acknowledgments. We are very grateful to P. Mironescu for sharing with us a device [26], used in the proof of Lemma 5, which led us to the full statement of Theorem 4. We thank P. Bousquet for calling our attention to the work of W. Sickel and A. Youssfi. We also thank A. Cohen and P. Bousquet for interesting discussions. Part of this work was done when the second author visited the Institute for Advanced Study and Rutgers University; he thanks these Mathematics Departments for the hospitality. The first author is partially supported by NSF Grant DMS-0802958.

30

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