Describing Mathieu Lewin
email: mathieu.lewin@math.cnrs.fr

DESCRIBING LACK OF COMPACTNESS IN SOBOLEV SPACES

Keywords: oscillations, Example 1b: concentration (blow-up), Example 1a: vanishing

We review the concentration-compactness method and the profile decomposition in Sobolev spaces, which allow to describe the possible lack of compactness of sequences in these spaces.

In these notes we consider a bounded sequence u = {u n } in L 2 (R d ) and we describe its possible behavior (up to subsequences) in a rather detailed way. We typically think of a sequence u n which converges to u weakly in L 2 (R d ) but which does not converge for the strong topology. One says that u = {u n } exhibits a lack of compactness. Considering instead u nu, we can think of a sequence u n 0 weakly in L 2 (R d ) but such that, for instance, R d |u n | 2 = λ > 0 for all n.

Example 2a: translation. Example 2b:

In Figure 1, we display four typical examples of such a sequence u = {u n } (the examples can easily be adapted to fix any L p (R d ) norm). In Example 1a, the sequence {u n } spreads out everywhere in space but since its mass is conserved, locally the mass must go to zero. The sequence is said to "vanish". In Example 1b, the sequence concentrates at one point (to be more precise |u n | 2 converges to a delta measure located at zero). For n large enough we essentially detect all the mass in any small neighborhood of the origin. In Example 2a, the sequence {u n } keeps a constant shape for all n but it runs off to infinity. If we want to see anything, we have to follow the sequence by "running after it". Finally, in Example 2b, the function oscillates so fast that (by the Riemann-Lebesgue Lemma) it converges weakly to zero in L 2 (R d ).

Examples 1a and 1b on the one side, and 2a and 2b on the other side are in some sense dual to each other. We can go from 1a to 1b (resp. 2a to 2b) by applying a Fourier transform. For instance, a sequence which oscillates very fast has a lack of compactness due to translations in Fourier space, and conversely. Similarly, a sequence which concentrates in Fourier space vanishes in direct space and conversely.

Another important remark is that the examples are all associated to the action of a non-compact group over L 2 (R d ). The group is (R + , ×) for Examples 1a and 1b and its action consists in dilating a function in the manner (λ • u)(x) = λ d/2 u(λx). The non-compactness is obtained by taking either λ → 0 + or λ → +∞. Writing λ = e t we can also see this as an action of the additive group (R, +). In Examples 2a and 2b the group is (R d , +) and it acts either by translating the function or by multiplying it by a phase factor, the two actions being the same up to a Fourier transform. The non-compactness of the sequence {u n } then arises from the noncompactness of the group itself.

One difference between the examples from the first line and the ones from the second is the behavior with respect to L p norms with p = 2. In the vanishing case (Example 1a) we have ||u n || L p (R d ) → 0 for all 2 < p ≤ ∞ and ||u n || L p (R d ) → ∞ for all 1 ≤ p < 2. In Example 1b the situation is exactly reversed. On the other hand, in Examples 2a and 2b the L p norms are all conserved for all p. This says that we can expect to use other L p norms to detect a sequence with vanishes or concentrates, but not the lack of compactness due to translations and oscillations.

There is a crucial difference between the examples of the first column and that of the second one. Examples 1a and 2a cannot happen in a bounded domain Ω ⊂ R d , it is important that the problem is settled over the infinite space R d and the noncompactness arises from the non-compactness of R d itself. Examples 1b and 2b can however perfectly happen in a bounded set Ω. Examples 1a and 2a are called locally compact, whereas Examples 1b and 2b are not locally compact. The difference can also be detected by looking at the derivative. In the first column, the derivative of the sequence {u n } stays bounded uniformly in n: For instance the sequence is bounded in the Sobolev space H 1 (R d ). In the cases of the second column, the function oscillates or blows up so fast that the derivative has to explode: We have ||∇u n || L 2 (Ω) → ∞, even in a (well-chosen) bounded domain Ω.

Of course, it is clear that we can combine the four examples as we want. We can add functions behaving differently or even compose the corresponding group actions. Think of a sequence u n which concentrates about a point x n such that |x n | → ∞.

In the beginning of the 80s, there has been a high activity in trying to describe the possible behavior of sequences which undergo a lack of compactness. The main idea was to prove that, in an appropriate sense, the four above examples are universal. Stating differently, a non-compact sequence should, up to a subsequence, be a (possibly infinite) sum of sequences having one or several of the above behaviors, up to a small error. Knowing this fact is very useful either in a contradiction argument (when we want to prove that the sequence has to be compact), or when the studied lack of compactness has a special physical meaning and has to be described more accurately.

The first to address these issues on a specific example were Sacks and Uhlenbeck [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF] in 1981 who dealt with a concentration phenomenon for harmonic maps. [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] then faced similar difficulties for some elliptic Partial Differential Equations (PDE) with a critical Sobolev exponent in 1983. The same year, Lieb proved in [START_REF] Lieb | On the lowest eigenvalue of the Laplacian for the intersection of two domains[END_REF] a useful lemma dealing with lack of compactness due to translations in the locally compact case. The most general method for dealing with locally compact problems was published by Lions [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF] in 1984 under the name "concentration-compactness" (it was announced before in [START_REF]Symétrie et compacité dans les espaces de Sobolev[END_REF]). Later in 1984-85, Struwe [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF] and, independently, Brezis and Coron [START_REF] Brezis | Convergence of solutions of H-systems or how to blow bubbles[END_REF] have provided the first "bubble decompositions" (that is they have shown that the sequence under study can be written as a sum of functions which concentrate at some points in space). In 1985, Lions adapted his concentration-compactness method to the nonlocal case [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The limit case. II[END_REF] (see also [START_REF]Applications de la méthode de concentration-compacité à l'existence de fonctions extrémales[END_REF]). These fundamental works from the 80s have generated many interesting further developments. The tools invented at that time are now of widespread use in the analysis of PDEs.

Brezis and Nirenberg

In these notes, we present a simple "Hilbert" version of this theory. First we consider a bounded sequence u = {u n } in H 1 (R d ) and describe in a rather detailed way its behavior in L p (R d ) for a subcritical 2 ≤ p < p * (with p * being the critical Sobolev exponent to be recalled below). More precisely, we will show that {u n } can be written as a sum of functions retaining their shape and escaping to infinity at different speeds (like in Example 2a), plus a remainder which can vanish in the sense of Example 1a. This is enough for many of the physical models encountered in practice. Most of the arguments of this section can be generalized to a Sobolev space W k,p (R d ) with k ≥ 1 and 1 < p < ∞. It is not essential that the underlying space is a Hilbert space, but several arguments are easier in this setting. An important point is that the sequence is locally compact, by the Rellich-Kondrachov Theorem. Then, in Section 6, we quickly touch upon the critical case where both Examples 1a, 1b and 2a can happen.

Our approach in these notes is a combination of ideas of Lieb [START_REF] Lieb | On the lowest eigenvalue of the Laplacian for the intersection of two domains[END_REF], of the concentration-compactness method of Lions [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF] and of the bubble decomposition as used for instance in [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF][START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF]. For a more general version of the concentrationcompactness method (in L 1 (R d ) or for measures), we refer to the original papers of Lions, or to the book of Struwe [START_REF]Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems[END_REF]. For a more abstract presentation, see the book of Tintarev and Fieseler [START_REF] Tintarev | Concentration Compactness: Functional-Analytic Grounds And Applications[END_REF]. The french speaking reader can refer to the book of Kavian [START_REF] Kavian | Introduction à la théorie des points critiques et applications aux problèmes elliptiques[END_REF], or to that of Cancès, Le Bris and Maday [START_REF] Cancès | Méthodes mathématiques en chimie quantique. Une introduction[END_REF] for an intuitive presentation of the method.

Finding bubbles

We consider a sequence u = {u n } bounded in L 2 (R d ) (later our sequence will be bounded in H 1 (R d )). Our goal is to detect pieces of mass which retain their shape for n large and, possibly, escape to infinity in the spirit of Example 2a in Figure 1. We call them 'bubbles'. For this reason, we consider all the possible weak limits, up to translations, of subsequences of {u n } and define the largest possible mass that these weak limits can have.

Definition 1 (Highest local mass of a sequence). Let u = {u n } be a bounded sequence in L 2 (R d ). We define the following number

(1) m(u) = sup R d |u| 2 : ∃{x k } ⊂ R d , u n k (• + x k ) u weakly in L 2 (R d ) .
Here we are interested in understanding the lack of compactness due to space translations which corresponds to the group action of (R d , +) on L 2 (R d ). There is a similar definition of m for any other group action. The combination of translations and dilations will be considered later in Section 6.

Remark 2. We have m(u) = m(u(• + x)) for any sequence of translations x = {x n } ⊂ R d (here u(• + x) is the sequence {u n (• + x n )}). Also, if u = {u n k } is a subsequence of u = {u n }, then m(u ) ≤ m(u). Remark 3. If {u n } is not only bounded in L 2 (R d ) but also in the Sobolev space H 1 (R d
), then we can replace the weak convergence in L 2 (R d ) by weak convergence in H 1 (R d ) in the definition of m(u) without changing anything.

Exercise 4. There does not necessarily exist a u ∈ L 2 (R d ) realizing the above supremum (that is such that

u n k (• + x n k )
u for a subsequence with R d |u| 2 = m(u)). We provide a counter-example. Let

ψ n = n -d/2 1 -1/n u(x/n) for some fixed u ∈ L 2 (R d ).
Define a new sequence {u n } as follows:

u 1 = ψ 1 , u 2 = ψ 1 , u 3 = ψ 2 , u 4 = ψ 1 , u 5 = ψ 2 , u 6 = ψ 3 , u 7 = ψ 1 ,
... and so on. Verify that m(u) = R d |u| 2 but that it is not "attained" by any weakly convergent subsequence.

Exercise 5. Verify that m(u) = 0 if u n (x) = n -d/2 u(x/n) (Example 1a), u n (x) = n d/2 u(nx) (Example 1b), or u n (x) = u(x) e in v•x (Example 2b). Here u is a fixed function in L 2 (R d ).
The purpose of m(u) is to detect the largest piece of mass in the sequence u = {u n }, which possibly escapes to infinity (when

|x n k | → ∞). If m(u) > 0, then we can find • a subsequence {u n k }; • a sequence of translations {x (1)
k } ⊂ R d ; • a function u (1) ∈ L 2 (R d ) such that m(u) ≥ R d |u (1) | 2 ≥ m(u) -ε > 0 (in particular u (1) = 0), such that u n k ( • + x (1)
k ) u (1) weakly in L 2 (R d ). Once we have found the first 'bubble' u (1) , we can go on and try to find the next one by considering the sequence r

(2)

k := u n k -u (1) ( • -x (1)
k ) and the corresponding m(r (2) ). Arguing by induction we can find by induction all the bubbles contained in the original sequence u = {u n }.

Lemma 6 (Extracting bubbles). Let u = {u n } be a bounded sequence in L 2 (R d ) (resp. H 1 (R d )). Then there exists a (possibly empty or finite) sequence of functions {u (1) , u (2) , ...} in L 2 (R d ) (resp. H 1 (R d )) such that the following holds true: For any fixed ε > 0, there exists

• an integer J, • a subsequence {u n k } of {u n }, • space translations {x (j) k } k≥1 ⊂ R d for j = 1, ..., J with |x (j) k -x (j ) k | → ∞ as k → ∞, for each j = j , such that we can write (2) u n k = J j=1 u (j) ( • -x (j) k ) + r (J+1) k where r (J+1) k (• + x (j) k ) k→∞ 0 weakly in L 2 (R d ) (resp. H 1 (R d
)) for all j = 1, ..., J and m(r (J+1) ) ≤ ε.

In particular we have

(3) lim k→∞ R d |u n k | 2 - R d |r (J+1) k | 2 = J j=1 R d |u (j) | 2 . If u = {u n } is bounded in H 1 (R d
), then we also have

(4) lim k→∞ R d |∇u n k | 2 - R d |∇r (J+1) k | 2 = J j=1 R d |∇u (j) | 2 and (5) lim k→∞ R d |u n k | p - R d |r (J+1) k | p = J j=1 R d |u (j) | p for every 2 ≤ p < ∞ for d = 1, 2, ≤ 2d d-2 for d ≥ 3.
Note that the lemma is essentially an abstract result. It can be stated similarly in any Hilbert space on which is acting a non-compact group. Only the limit ( 5) is specific to the space H 1 (R d ) for p > 2. See [START_REF] Tintarev | Concentration Compactness: Functional-Analytic Grounds And Applications[END_REF] for a more abstract presentation in the same spirit.

The functions u (j) are all the possible weak limits of u up to translations and extraction of subsequences. The lemma says that any bounded sequence u = {u n } in L 2 (R d ) can be written as a linear combination of these limits translated in space, up to an error term r (J+1) k . This error term is not necessarily small in norm, because the remainder r (J+1) can exhibit all sorts of other compactness issues. But we know that m(r (J+1) ) is small. When {u n } is bounded in H 1 (R d ) then m(r (J+1) ) ≤ ε will tell us a lot since Examples 1b and 2b cannot happen in a locally compact setting. We will explain this in the next section.

Let us emphasize the fact that the bubbles u (j) do not depend on ε. They can be constructed once and for all for any fixed sequence u = {u n }. On the contrary, the number J of bubbles, the space translations x (j) k and the subsequence n k needed to achieve a given accuracy in the decomposition of u n k , all depend on ε.

Proof of Lemma 6. We start with the case where u = {u n } is bounded in L 2 (R d ). If m(u) = 0, there is nothing to prove and we may therefore assume that m(u) > 0. In this case we deduce from the definition of m(u) that there exists a subsequence n k , a sequence of translations {x [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] k } ⊂ R d , and a function u (1)

∈ L 2 (R d ) such that m(u) 2 ≤ R d |u (1) | 2 ≤ m(u) and u n k (• + x (1)
k ) u (1) weakly in L 2 (R d ). Defining r

(2) k := u n k -u (1) ( • -x (1)
k ), we obviously get that r

k (• + x (1) k )
0. We also have

||u n k || 2 L 2 = u (1) 2 L 2 + r (2) k 2 L 2 + 2 r (2) k (• + x (1)
k ), u (1) where the last term tends to 0 by the weak convergence of r

k . So we conclude that

lim k→∞ ||u n k || 2 L 2 -r (2) k 2 L 2 = u (1) 2 L 2 .
If m(r (2) ) = 0, we can stop here. Otherwise we go on and extract the next bubble. We find a subsequence and space translations such that r

(2) 2) ).

k (• + x (2) ) u (2) = 0, with m(r (2) ) 2 ≤ R d |u (2) | 2 ≤ m(r (
We also extract a further subsequence from u n k and x

k . To simplify the exposition, we use the same notation for all these subsequences. So we can write

u n k = u (1) ( • -x (1) k ) + u (2) ( • -x (2) k ) + r (3) k where r (3) k (•+x (2) k ) 0. If |x (2) k -x (1)
k | does not diverge, then we can extract another subsequence such that x

(1) k -x (2) k → v. We would then obtain u n k (• + x (1) k ) u (1) + u (2) (• + v), since r (3) k (• + x (1) k ) = r (3) k (• + v + x (2) k ) 0. This contradicts the fact that u n k (• + x (1) k )
u (1) by construction, unless u (2) = 0 which is not the case for us here. So we conclude that |x

(2) k -x (1) k | → ∞. Now we can use that u n k (•+x (1) k )
u (1) and deduce that r

(3) k (•+x (1) k ) 0, since u (2) (•+x (1) k -x (2)
k ) 0. Similarly as before, it can be checked that

lim k→∞ ||u n k || 2 L 2 -r (3) k 2 L 2 = u (1) 2 L 2 + u (2) 2 L 2 .
We can apply the previous argument ad infinitum, constructing the sequences u (j) and r (j) in the same fashion, except when we reach a remainder r

(J+1) k which is such that m(r (J+1) ) = 0. Our construction satisfies lim k→∞ ||u n k || 2 L 2 -r (J+1) k 2 L 2 = J j=1 u (j) 2 L 2
for all J, and in particular

j u (j) 2 L 2 ≤ lim sup n→∞ ||u n || 2 L 2 ≤ C.
Therefore we see that u (j)

L 2 → 0 as j → ∞. Recalling that m(r (j) ) ≤ 2 R d |u (j) | 2
by construction, we deduce that m(r (j) ) → 0 as well. Hence, for J large enough, m(r (J) ) must be smaller than any ε fixed in advance, and this concludes the proof of Theorem 6 in the L 2 case. When the sequence is bounded in H 1 (R d ), any weak limit in L 2 (R d ) is automatically also a weak limit in H 1 (R d ), hence (3) and (4) follow easily. For [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], however, we have to use the strong local compactness (Rellich-Kondrachov theorem). That is, when u n u weakly in H 1 (R d ), we can find a subsequence such that u n → u strongly locally in L 2 and almost everywhere. Then (5) follows from the missing term in Fatou's lemma, also called the Brezis-Lieb lemma [START_REF] Brezis | A relation between pointwise convergence of functions and convergence of functionals[END_REF]18], which precisely states that [START_REF] Cancès | Méthodes mathématiques en chimie quantique. Une introduction[END_REF] lim

n→∞ R d |u n (x)| p dx - R d |u(x) -u n (x)| p dx = R d |u(x)| p dx when u n (x) → u(x) almost everywhere.
Exercise 7. Write the full details of the proof in the H 1 (R d ) case.

Remark 8. The proof of (3) relies on the Hilbert space structure of L 2 (R d ) and the theory is much less nice in L p (R d ) with 1 ≤ p < ∞ and p = 2. For instance, in this case it is not true in general that u n u implies [START_REF] Cancès | Méthodes mathématiques en chimie quantique. Une introduction[END_REF]. From the Brezis-Lieb lemma the limit (6) follows from almost everywhere convergence, but the latter cannot be obtained from weak convergence in L p (R d ) without more information on the sequence. In Sobolev spaces, almost everywhere convergence is obtained in our proof by using strong local compactness. For a theory in L p (R d ) with p = 2, the best is to use concentration functions as in [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF], see Section 3 below. Remark 9. As is obvious from the above proof of Lemma 6, we can also choose the sequence {u (j) } to have that

(7) max j≥0 R d |u (j) | 2 ≥ m(u) -η
for any η > 0. It suffices to choose u (1) satisfying this property. Note that (7) is a maximum, since R d |u (j) | 2 → 0 as j → ∞, by construction. As we have mentioned in Exercise 4, m(u) is not necessarily 'attained'. We now prove that it is always attained for a subsequence.

Corollary 10 (m({u n }) is attained for a subsequence). Let u = {u n } be a bounded sequence in L 2 (R d ) and η > 0. There exists a subsequence u = {u n k }, {x k } ⊂ R d , u ∈ L 2 (R d ) such that R d |u| 2 = m(u ) ≥ m(u) -η and u n k ( • + x k ) u.
Proof. We apply Lemma 6, choosing specifically the sequence {u (j) } such as to have

max j≥0 R d |u (j) | 2 ≥ m(u) -η by Remark 9. Then we choose J large enough such that m r (J+1) < max R d |u (j) | 2 , j = 1, ..., J .
It is then clear that m(u) = max R d |u (j) | 2 , j = 1, ..., J .

Vanishing

In the previous section we have defined the highest mass that weak limits can have up to space translations. We have shown in Lemma 6 that any bounded sequence in L 2 (R d ) can be written as a linear combination of these bumps escaping to infinity, plus a remainder r (J+1) k which is such that m(r (J+1) ) is small. If we continue this construction infinitely many times we will have found all the bubbles and essentially arrive at a sequence r (∞) such that m(r (∞) ) = 0. What does this information tell us? Under some additional assumption on the derivatives, we will show that the remainder can only vanish, in a proper sense. We use here the Sobolev space H 1 (R d ) and other L p norms, as we have explained after the examples from Figure 1.

Lemma 11 (A subcritical estimate involving m(u)). Let u = {u n } be a bounded sequence in H 1 (R d ).
There exists a universal constant C = C(d) such that the following holds:

(8) lim sup n→∞ R d |u n | 2+ 4 d ≤ C m(u) 2 d lim sup n→∞ ||u n || 2 H 1 (R d ) .
The previous result gives an estimate on the size of Proof of Lemma 11. We only write the proof for d ≥ 3 and let the (very similar) cases d = 1, 2 as an exercise. We consider a tiling of the whole space R d by means of cubes, say

||u n || L 2+4/d (R d ) in terms of m(u).
R d = ∪ z∈Z d C z with C z = d j=1 [z j , z j + 1
). We then calculate by Hölder's inequality (9)

R d |u n | q = z∈Z d Cz |u n | q ≤ z∈Z d ||u n || θq L 2 (Cz) ||u n || (1-θ)q L p * (Cz) ,
where 1/q = θ/2 + (1θ)/2 * . We choose q in such a way that (1θ)q = 2. A simple calculation shows that q = 2 + 4/d and θq = 4/d. Note that this choice satisfies 2 < q < 2 * = 2 + 4/(d -2) (recall we are in the case d ≥ 3). By the Sobolev embedding in the cube C z , there exists a constant C such that

||u n || 2 L 2 * (Cz) ≤ C ||u n || 2 L 2 (Cz) + ||∇u n || 2 L 2 (Cz) .
Note that the constant C only depends on the volume of the cube, hence it is independent of z ∈ Z d . Inserting in [START_REF] Frank | Ground states of semi-linear PDE[END_REF], we obtain

R d |u n | q ≤ C sup z∈Z d ||u n || L 2 (Cz) 4 d z∈Z d Cz |u n | 2 + Cz |∇u n | 2 = C sup z∈Z d ||u n || L 2 (Cz) 4 d ||u n || 2 H 1 (R d ) .
Passing to the limit n → ∞, we deduce that

(10) lim sup n→∞ R d |u n | 2+ 4 d ≤ C lim sup n→∞ sup z∈Z d Cz |u n | 2 2 d lim sup n→∞ ||u n || 2 H 1 (R d ) .
We now claim that The sequence

u n (• + z n ) is bounded in H 1 (R d ).
Up to extraction of a subsequence, we have

u n k (• + z n k ) u weakly in H 1 (R d
) and, by the Rellich-Kondrachov compactness theorem, strongly in L 2 (C 0 ). We deduce that

lim n→∞ Cz n |u n | 2 = lim n k →∞ C0 |u n k (• + z n k )| 2 = C0 |u| 2 ≤ R d |u| 2 ≤ m(u).
This concludes the proof of Lemma 11.

We now use the previous result to characterize when m(u) = 0 holds, which is equivalent to saying that u n (• + x n ) 0 for all {x n } ⊂ R d . Intuitively this corresponds to vanishing as displayed in Example 1a of Figure 1 since u n must then converge to zero strongly in all L p (R d ), for all the sub-critical exponents 2 < p < 2 * . Lemma 12 (Vanishing). Let u = {u n } be a bounded sequence in H 1 (R d ). The following assertions are equivalent:

(i) m(u) = 0; (ii) for all R > 0, we have lim n→∞ sup x∈R d B(x,R) |u n | 2 = 0; (iii) u n → 0 strongly in L p (R d ) for all 2 < p < 2 * , where 2 * = 2d/(d -2) if d ≥ 3, 2 * = ∞ if d = 1, 2.
The interpretation of (ii) is that there is essentially no mass remaining for n large enough in any ball of fixed radius R, independently of the location of the center of the ball.

Proof of Lemma 12. We start by proving that (i) ⇒ (ii). Indeed, from the estimate [START_REF] Kavian | Introduction à la théorie des points critiques et applications aux problèmes elliptiques[END_REF] To show that (iii)

⇒ (i), we consider {x n k } ⊂ R d and u such that u n k (•+x n k ) u. Since ||u n k (• + x n k )|| L p (R d ) = ||u n k || L p (R d ) → 0 when 2 < p < 2 * , we have u n k (•+x n k ) → 0 strongly in L p (R d
), hence (by uniqueness of the weak limit) u = 0. Therefore m(u) = 0. Finally, the proof that (ii) ⇒ (iii) is a consequence of [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF]. We have ||u n || L 2+4/d (R d ) → 0 and the rest follows by interpolation, using that {u n } is bounded in L p (R d ) for 2 ≤ p ≤ 2 * by the Sobolev embedding.

Exercise 13. Write the proof for d = 1, 2. Generalize also Lemmas 11 and 12 to the case of a bounded sequence {u n } in W k,p (R d ) for 1 < p < ∞ and k a positive integer, with an appropriate definition of m(u).

Isolating bubbles in space

In applications it is sometimes useful to "isolate" the bubbles, which means that we write u n k as a sum of functions u (j) k of compact supports, each of them converging strongly to the weak limit u (j) of Lemma 6, and where the distance between the supports diverges. This step is not always necessary in practice in L 2 (R d ) or H 1 (R d ) but it is sometimes convenient. In addition, this is the preferred technique to be used in other spaces such as L p (R d ) or for non-negative measures. For instance when all the functions u n are non-negative, it might be convenient to ensure as well that the remainder r (J+1) k is non-negative (in the proof of Lemma 6 the remainder r (J+1) k is defined as a difference). Using localization functions is more in the spirit of Lions' concentration-compactness principle and this is an occasion for us to learn how to use the concentration functions of Levy.

The localization procedure can be done for any chosen J in the profile decomposition (2) of Lemma 6. In order to illustrate the main idea, we start with the case J = 1, that is when we have one sequence u n which converges weakly to some u (1) := u. We will come back to the general case later in Corollary 20.

Lemma 14 (Extracting the locally convergent part). Let {u n } be a sequence in

H 1 (R d ) such that u n u weakly in H 1 (R d ) and let 0 ≤ R k ≤ R k such that R k → ∞. Then there exists a subsequence {u n k } such that (12) |x|≤R k |u n k (x)| 2 dx → R d |u(x)| 2 dx and (13) R k ≤|x|≤R k |u n k (x)| 2 + |∇u n k (x)| 2 dx → 0 as k → ∞.
In particular, we have that u n k 1 B(0,R k ) → u strongly in L p (R d ) for all 2 ≤ p < 2 * (where 2 * is like in Lemma 12).

The interpretation of Lemma 14 is that we can isolate by means of a growing ball B(0, R k ) the part which converges strongly to u in sub-critical spaces L p (R d ) with 2 ≤ p < 2 * . In the annulus {R k ≤ |x| ≤ R k }, we have essentially nothing (by the convergence to zero in H 1 (R d )). In practice, we choose R k such that R k -R k → ∞. Since the original sequence {u n k } has been decomposed into two parts (the convergent part in the ball B(0, R k ) and the rest in the domain R d \ B(0, R k )), Lions used the word dichotomy [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF] to describe this procedure.

The intuition behind the decomposition of Lemma 14 is that we have to choose R k and R k growing slowly enough that we essentially only retain the mass of the weak limit u and not more. However, our statement is for convenience written the other way: We can choose R k and R k essentially as we want, but then the sequence n k has to go to infinity so fast that u n k has almost already converged to its local limit u in the ball B(0, R k ).

Proof of Lemma 14. We follow Lions [START_REF] Lions | Principe de concentration-compacité en calcul des variations[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF] and introduce the so-called Levy concentration functions [START_REF] Levy | Théorie de l'Addition des Variables Aléatoires, Monographies des Probabilités[END_REF] Q

n (R) := B(0,R) |u n | 2 , and K n (R) := B(0,R) |∇u n | 2 .
Note that Q n and K n are continuous non-decreasing functions on [0, ∞), such that

∀n ≥ 1, ∀R > 0, Q n (R) + K n (R) ≤ R d |u n | 2 + |∇u n | 2 ≤ C since {u n } is bounded in H 1 (R d ).
By the Rellich-Kondrachov Theorem, we have

Q n (R) = B(0,R) |u n | 2 → B(0,R) |u| 2 := Q(R)
for all R ≥ 0. We now recall a very useful result dealing with sequences of monotone functions.

Lemma 15 (Sequences of monotone functions). Let I be a (possibly unbounded) interval of R, and {f n } be a sequence of non-increasing non-negative functions on I. We assume that there exists a fixed function g, locally bounded on I, such that 0 ≤ f n ≤ g on I. Then there exists a subsequence {f n k } and a non-increasing function f such that f n k (x) → f (x) for all x ∈ I.

In the applications, g is often a constant on the whole interval I. By Lemma 15, up to extraction of a subsequence (for the sake of clarity we do not change notation), we may assume that K n (R) → K(R) for all R ≥ 0, and for some non-decreasing function K. We denote := lim R→∞ K(R).

It can be shown that ≥ R d |∇u| 2 but there is not always equality, except if we know that {u n } is bounded in H s (R d ) for some s > 1.

Consider now the given sequences R k and R k which diverge to ∞. We have

Q n (R k ) → Q(R k ) and Q n (R k ) → Q(R k ) as n → ∞,
and a similar property for K n . Extracting a subsequence, we may assume for instance that

|Q n k (R k )-Q(R k )|+|Q n k (R k )-Q(R k )|+|K n k (R k )-K(R k )|+|K n k (R k )-K(R k )| ≤ 1 k .
We then deduce that

B(0,R k ) |u n k | 2 - R d |u| 2 = |Q n k (R k ) -Q(∞)| ≤ 1 k + |x|≥R k |u| 2 -→ k→∞ 0 and that R k ≤|x|≤R k |u n k | 2 = Q n k (R k ) -Q n k (R k ) ≤ 1 k + Q(R k ) -Q(R k ) -→ k→∞ 0, R k ≤|x|≤R k |∇u n k | 2 = K n k (R k ) -K n k (R k ) ≤ 1 k + K(R k ) -K(R k ) -→ k→∞ 0,
where we have used that

K(R k ) -K(R k ) → -= 0 when k → ∞. Finally, we have 1 B(0,R k ) u n k u weakly in L 2 (R d ).
Since the norm also converges, we get that the convergence must be strong. By the Sobolev embeddings, {u n } is bounded in L p (R d ) for all 2 ≤ p < 2 * , hence so is 1 B(0,R k ) u n k . By interpolation, 1 B(0,R k ) u n k converges towards u strongly in L p (R d ) for all 2 ≤ p < 2 * . This concludes the proof of Lemma 14.

Exercise 16. Prove that for any bounded sequence u = {u n } in H 1 (R d ), ( 14)

m(u) = lim R→∞ lim sup n→∞ sup x∈R d B(x,R) |u n | 2 .
Exercise 17. Generalize Lemma 14 to the case of a bounded sequence u = {u n } in W k,p (R d ) for 1 < p < ∞ and k a positive integer.

In the previous lemma, we have isolated in the ball B(0, R k ) the part of the sequence u n k which converges to u strongly, by retaining only the appropriate mass. Unfortunately, the function

u n k 1 B(0,R k ) is not in H 1 (R d
) and it is in practice convenient to replace the characteristic function 1 B(0,R k ) by a smooth cut-off. This is the purpose of the next corollary.

Corollary 18 (Splitting of a weakly convergent sequence in H

1 (R d )). Let u = {u n } be a sequence in H 1 (R d ) such that u n u weakly in H 1 (R d ) and let 0 ≤ R k ≤ R k such that R k → ∞. Then we have for a subsequence {u n k } lim k→∞ u n k -u (1) k -ψ (2) k H 1 (R d ) = 0
where u (1) = {u

k } and ψ (2) = {ψ

(2) k } are sequences in H 1 (R d ) such that • u (1) k → u weakly in H 1 (R d ) and strongly in L p (R d ) for all 2 ≤ p < 2 * ; • supp u (1) k ⊂ B(0, R k ) and supp ψ (2) k ⊂ R d \ B(0, R k ); • m r (2) ≤ m {u n k } ≤ m(u).
Proof. We apply Lemma 14 with R k /2 and 4R k . We obtain a subsequence {u n k } such that [START_REF]On singularity formation for the L 2 -critical Boson star equation[END_REF]

|x|≤R k /2 |u n k | 2 → R d |u| 2 , R k /2≤|x|≤4R k |u n k | 2 + |∇u n k | 2 → 0. Let χ : R + → [0, 1] be a smooth function such that 0 ≤ χ ≤ 2, χ |[0,1] ≡ 1 and χ |[2,∞) ≡ 0. We denote χ k (x) := χ(2|x|/R k ) and ζ k (x) = 1 -χ(|x|/R k ) and introduce u (1) k := χ k u n k and ψ (2) k := ζ k u n k . By (15), we clearly have u n k -u (1) k - ψ (2) k → 0 in H 1 (R d ) since this function has its support in the annulus {R k /2 ≤ |x| ≤ 2R k }. Furthermore lim k→∞ R d |u (1) k | 2 = lim k→∞ |x|≤R k /2 |u (1) k | 2 = R d |u| 2 , hence u (1) k u weakly in H 1 (R d ) and strongly in L p (R d ) for all 2 ≤ p < 2 * . Note that by construction 1 B(0,4R k ) ψ (2) k → 0 strongly in L 2 (R d ). It remains to show that m ψ (2) ≤ m {u n k } . If m ψ (2) = 0 there is nothing to prove. Assume that ψ (2) kj ( • -x j )
ψ for some subsequence and some ψ = 0. If for a subsequence (denoted the same for simplicity), we have |x j | ≤ 3R kj , then

B(x j , R kj ) ⊂ B(0, 4R kj ), hence ψ (2) kj ( • -x j )1 B(0,R k j ) 0 = ψ, a contradiction.
Hence it must hold |x j | ≥ 3R kj for all j sufficiently large. Note then that for j large enough, ζ kj ≡ 1 on the ball B(x j , R kj ), hence

ψ (2) kj ( • -x j )1 B(0,R k j ) = u n k j ( • -x j )1 B(0,R k j ) ψ, thus u n k j ( • -x j ) ψ weakly in H 1 (R d ). This proves that R d |ψ| 2 ≤ m {u n k } and the proof is complete. Remark 19. Here ψ (2) k = ζ k u n k is different from r (2) k
used in Lemma 6, although they behave essentially the same in the limit k → ∞. The fact that ψ

(2) k is proportional to u n k is sometimes useful in practical applications. For instance when

u n k ≥ 0 then ψ (2)
k is non-negative as well. We now consider a sequence u = {u n } and its weak limits u (j) up to translations, obtained from Lemma 6. We want to localize all the bubbles in disjoint balls receeding from each other, in the same spirit as in the previous result. The following result is proved in a very similar fashion as Lemma 14 and Corollary 18, using two concentration functions per bubble.

Theorem 20 (Splitting in arbitrarily many localized bubbles). Let u = {u n } be a bounded sequence in H 1 (R d ) and {u (j) } ⊂ H 1 (R d ) be the sequence given by Lemma 6. For any ε > 0 and any fixed sequence 0 ≤ R k → ∞, there exist

• J ≥ 0, • a subsequence {u n k }, • sequences of functions u (1) = {u (1) k }, ..., u (J) = {u (J) k }, ψ (J+1) = {ψ (J+1) k } in H 1 (R d ), • space translations x (1) = {x (1) k }, ..., x (J) = {x (J) k } in R d , such that (16) lim k→∞ u n k - J j=1 u (j) k ( • -x (j) k ) -ψ (J+1) k H 1 (R d ) = 0 where • u (j) k → u (j) = 0 weakly in H 1 (R d ) and strongly in L p (R d ) for all 2 ≤ p < 2 * ; • supp(u (j) k ) ⊂ B(0, R k ) for all j = 1, ..., J and all k; • supp(ψ (J+1) k ) ⊂ R d \ ∪ J j=1 B(x j k , 2R k ) for all k; • |x (i) k -x (j) k | ≥ 5R k for all i = j and all k; • m ψ (J+1) ≤ ε.
We emphasize again that the error term ψ (J+1) k is small in the sense that m ψ (J+1) is small, that is {ψ Results taking the same form as Corollary 20 are ubiquitous in the literature and they are often called profile or bubble decompositions of the sequence u = {u n }, see for instance [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF][START_REF] Brezis | Convergence of solutions of H-systems or how to blow bubbles[END_REF][START_REF]Solutions of Hartree-Fock equations for Coulomb systems[END_REF][START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF].

Exercise 21. Write the proof of Corollary 20, using 2J concentration functions (two per bubble), that is for the functions

|u n k ( • + x (j) k )| 2 and |∇u n k ( • + x (j) k )| 2 for j = 1, ..., J.

The concentration-compactness principle

In this section, we explain how to use the tools of Sections 1-3 in practice, following Lions [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF][START_REF]Solutions of Hartree-Fock equations for Coulomb systems[END_REF]. The "concentration-compactness principle" is a general method to prove the compactness of some specific sequences u = {u n } (possibly up to translations and/or dilations), which are usually minimizing sequences for some variational problem, Palais-Smale sequences for some nonlinear equations, or even time-dependent solutions to some dispersive nonlinear PDEs [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF][START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case[END_REF]. Concentrationcompactness is a general strategy which has to be developed in each practical situation, not an abstract theorem that can be applied as a black box. The main idea is to prove the compactness of the sequence u = {u n } by showing that it must stay "concentrated", meaning that it stays in one piece and does not split into two or more non-trivial bubbles, nor vanish. The way to prove this fact is by studying what would happen in the case of non-compactness, usually by showing that the corresponding energy would become the sum of the energies of the bubbles and that it would then be too high.

To describe the strategy, we assume that we are given an energy functional E defined on H 1 (R d ), which is continuous, bounded from below and coercive on [START_REF] Lieb | On the lowest eigenvalue of the Laplacian for the intersection of two domains[END_REF] S ≤ (λ) := u ∈ H 1 (R 3 ) :

R 3 |u| 2 ≤ λ .
for all λ ≥ 0. The corresponding minimization principle reads ( 18)

I(λ) := inf u∈S(λ) E(u).
where, this time, ( 19)

S(λ) := u ∈ H 1 (R 3 ) : R 3 |u| 2 = λ .
The constraint on the L 2 norm is only for illustration, other norms can be fixed. By coercivity, all the minimizing sequences u = {u n } are bounded in H 1 (R d ). The goal is to prove the existence of a minimizer for (18) and to give a criterion for the precompactness of all the minimizing sequences. As we have explained, a given minimizing sequence can undergo lack of compactness: It can vanish in the sense that m(u) = 0, or it can split into several pieces as in Corollary 20. The main idea is to describe in a rather detailed way the behavior of minimizing sequences in the case of lack of compactness and to find what would be the total energy of the system when this happens. If we can show that the energy is above I(λ), we arrive at a contradiction, hence the minimizing sequence must be precompact.

To deal with vanishing, we introduce a functional E van which is the original energy E to which all the subcritical terms which go to zero when u n → 0 in L p (R d ) for 2 < p < 2 * (by Lemma 12) are removed. In the applications, E van usually only contains the gradient terms. One then defines the corresponding minimization principle I van (λ) := inf u∈S(λ) E van (u). Another equivalent way to define I van (λ) is as follows:

(20)

I van (λ) := inf u={un}⊂S(λ) m(u)=0 lim inf n→∞ E(u n ).
To deal with the phenomenon of splitting ("dichotomy"), we introduce an energy E ∞ which is the original energy E to which we remove all the compact terms that converge to zero when u n 0 (but without assuming a priori that u n → 0 strongly in some L p (R d ) space), thinking of u n = u( •n v). In practical examples, this usually means removing the potential energy terms when they involve an external potential V tending to zero at infinity, and to only keep the terms which are translation-invariant. The corresponding minimization principle reads I ∞ (λ) := inf u∈S(λ) E ∞ (u). Another way to define I ∞ (λ) is:

(21) I ∞ (λ) := inf {un}⊂S(λ) un 0 lim inf n→∞ E(u n ).
Note that I ∞ (λ) ≤ I van (λ). By taking appropriate test functions, it can usually be proven that ∀0 ≤ λ ≤ λ, I(λ) ≤ I(λλ ) + I ∞/van (λ ). If I(0) = I van (0) = I ∞ (0) = 0 (which we always assume), we deduce that I(λ) ≤ I ∞/van (λ).

For models which are translation-invariant, we always get E ∞ = E. In this case we can only hope to prove the precompactness of minimizing sequences up to translations (that is, we want to show that there exists {x k } ⊂ R d and a subsequence such that u n k ( •-x k ) is compact). We start by explaining the general strategy when E is translation-invariant.

The first step is to show that vanishing does not occur, by proving that ( 22)

I(λ) = I ∞ (λ) < I van (λ) for all λ > 0.
This is our first energetic inequality to be proven, usually by means of a well-chosen test function. This implies that m(u) > 0 for every minimizing sequence u = {u n } of I(λ). Then, by definition of m(u), there exists a subsequence (denoted the same for clarity) and translations x (1) = {x

n } ⊂ R d such that u n ( • -x (1)
= 0. Using Lemma 6 and its proof, we can write

u n ( • -x (1) n ) = u (1) + r (2) n . One then shows that E(u n ) = E(u n ( • -x (1) n )) = E(u (1) ) + E(r (2)
n ) + o(1) using the splitting properties in (3), ( 4), [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] or similar results for other kinds of terms in the energy. Let us denote λ (1)

:= R d |u (1) | 2 > 0 and remark that lim n→∞ R d |r (2) n | 2 = λ -λ (1)
by [START_REF] Brezis | Convergence of solutions of H-systems or how to blow bubbles[END_REF]. Using

E(r (2) n ) ≥ I R d |r (2) n | 2
and passing to the limit (we need here the continuity of λ → I(λ)), we obtain ( 23) I(λ) ≥ E(u (1) ) + I(λλ (1) ) ≥ I(λ (1) ) + I(λλ (1) ).

Since the converse inequality is always true, there must be equality:

I(λ) = E(u (1) ) + I(λ -λ (1) ) = I(λ (1) ) + I(λ -λ (1) ).
In particular, u (1) is a minimizer of the problem corresponding to its own mass I(λ (1) ) and lim n→∞ E(r

n ) = I(λ -λ (1) ), that is r (2) = {r (2)
n } is a minimizing sequence for I(λλ (1) ).

If we know already that (24) I(λ) < I(λ ) + I(λλ ) for all 0 < λ < λ, then we conclude (since λ (1) > 0) that λ = λ (1) . Therefore {u n } is compact in L 2 (R d ) and u (1) is the sought-after minimizer. The compactness in H 1 (R d ) usually follows from the convergence E(u n ) → E(u (1) ).

The "binding inequality" ( 24) is our second inequality to be proven in order to get compactness up to translations. In practice [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF] may be hard to prove without more information on I(λ ) and I(λλ ). One possible strategy is to go on with our extraction of bubbles and apply the whole argument to r (2) = {r [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] n }, which is a minimizing sequence for I(λλ (1) ), assuming of course that λ (1) < λ. The sequence r (2) cannot vanish by [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF], hence there is a u (2) = 0 such that, up to a subsequence, r

n ( • -x (2) n ) u (2)
= 0. As before u (2) must be a minimizer of the problem corresponding to its own mass, λ (2) = R d |u (2) | 2 > 0 and we can write r

(2) n = u (2) + r (3)
n where {r

(3)
n } is a minimizing sequence for I(λλ (1)λ (2) ). We obtain I(λ) = I(λ (1) ) + I(λ (2) ) + I(λλ (1)λ (2) )

where I(λ (1) ) and I(λ (2) ) admit u (1) and u (2) as minimizers. In principle one could go on and extract as many bubbles as possible (or even apply Lemma 6 directly).

It is however usually enough to stop as soon as there is a contradiction, which is often the case when there are two bubbles. The argument is as follows. We have

I(λ) ≤ I(λ (1) + λ (2) ) + I(λ -λ (1) -λ (2)
) ≤ I(λ (1) ) + I(λ (2) ) + I(λλ (1)λ (2) )

and since the term on the left equals the one on the right, there must be equality everywhere. In particular ) ) + I(λ (2) ).

I(λ (1) + λ (2) ) = I(λ ( 1
In order to get a contradiction we have to prove the strict inequality. We have gained here that both I(λ (1) ) and I(λ (2) ) are known to have minimizers. Therefore we see that it suffices to prove [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF] for any λ > 0 under the additional assumption that both I(λ ) and I(λλ ) admit minimizers. In practice this last step requires to place the two minimizers far away at a distance R and to evaluate rather precisely the energy expansion as R → ∞ in order to show that the two bubbles attract each other.

The previous method was devoted to the case of a translation-invariant system, E = E ∞ . When E = E ∞ (for instance when there is an external potential V ), there is an additional step at the beginning of the method. One start by showing that

I(λ) < I ∞ (λ)
for all λ > 0 which implies that a given minimizing sequence u = {u n } cannot have a vanishing weak limit up to subsequences: u n u (1) = 0. Here there is no space translation at the first step. We then go on as before and write u n = u (1) + r [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] n and show that the energy splits as

E(u n ) = E(u (1) ) + E ∞ (r (2) n ) + o (1)
. Note that r

(2) n 0 hence the local terms disappear and we get E ∞ . Arguing as before we find I(λ) = I(λ (1) )+I ∞ (λ-λ (1) ). The rest of the proof is similar to what we have said before and the binding inequality that must be proven now reads: I(λ (1) + λ (2) ) < I(λ (1) ) + I ∞ (λ (2) ) where I(λ (1) ) and I ∞ (λ (2) ) can both be assumed to have minimizers.

For many concrete examples of the strategy described in this section, see [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF].

Application: existence of Gagliardo-Nirenberg optimizers

In this section we explain how to use the previous strategy in order to show the existence of optimizers for the Gagliardo-Nirenberg inequality. 1 . We recall that we 1 The original proof of existence is due to Weinstein [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF] who used that one can restrict the problem to radial functions by rearrangement inequalities [18] and then a compactness lemma of Strauss [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF] for radial functions in H 1 (R d ). Here we give a direct argument inspired of [START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The locally compact case, Part II[END_REF] have [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. II[END_REF] ||u||

2 L p (R d ) ≤ C d,p R d |u(x)| 2 dx θ R d |∇u(x)| 2 dx 1-θ where 2 ≤ p < 2 * = ∞ for d = 1, 2, 2d d-2 for d ≥ 3, θ = 2d -(d -2)p 2p .
In our convention C d,p is the best constant in [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. II[END_REF], which means that (26)

C d,p = sup u∈H 1 (R d ) ||u|| 2 L p (R d ) ||u|| 2θ L 2 (R d ) ||∇u|| 2(1-θ) L 2 (R d )
The norms are all raised to the power two to simplify some later expressions. For p = 2 we simply have θ = 1 and C d,p = 1 and the inequality is an equality for every function. Our goal in this section is to investigate the existence and uniqueness of optimizers for 2 < p < 2 * . The result stays valid for p = 2 * in dimensions d ≥ 3 but the existence in this critical case requires more advanced arguments discussed in the next section.

Theorem 22 (Optimizers for the Gagliardo-Nirenberg inequality). Let 2 < p < 2 * in dimension d ≥ 1. Then the supremum in (26) is attained. The optimizers are unique up to translations, dilations and multiplication by a constant. That is, they all take the form

(27) x → aQ(bx -z)
for some a, b = 0 and z ∈ R d , where Q is the unique positive radial solution to the nonlinear Schrödinger equation

(28) -∆Q -Q p-1 + Q = 0 in L 2 (R d ).
Here we only discuss the existence part of the theorem and give some short comments and references about the rest of the statement at the end of the section. It is possible to study directly the optimization problem [START_REF]Solutions of Hartree-Fock equations for Coulomb systems[END_REF] in the form of a quotient (see for instance [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Frank | Ground states of semi-linear PDE[END_REF]) but we prefer to use extensive quantities, that is, to only involve integrals raised to the power one. Those behave better when a function splits into two bubbles. We introduce the following auxiliary minimization problem: [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF] I(λ) := inf

u∈H 1 (R d ) R d |u(x)| p dx=λ R d |∇u(x)| 2 dx + R d |u(x)| 2 dx .
The parameter λ is redundant and its role will appear more clearly later. Writing u = λ 1/p v we immediately conclude that ( 30)

I(λ) = λ 2 p I(1).
In addition I(λ) has minimizers if and only I(1) has some and they are related by the previous simple multiplication.

The following result says that the minimization problem I(λ) in ( 29) is just the original Gagliardo-Nirenberg optimization problem [START_REF]Solutions of Hartree-Fock equations for Coulomb systems[END_REF], written in a different manner.

Lemma 23. We have

(31) I(1) = 1 θ θ (1 -θ) 1-θ C d,p
.

where C d,p is the Gagliardo-Nirenberg constant in [START_REF]Solutions of Hartree-Fock equations for Coulomb systems[END_REF]. In addition, minimizers for I(1) coincide exactly with the optimizers for (26), appropriately rescaled.

Proof. From Hölder's inequality for numbers we have

a + b ≥ a θ b 1-θ θ θ (1 -θ) 1-θ
and therefore we obtain ( 32)

I(1) ≥ u 2θ L 2 (R d ) ∇u 2(1-θ) L 2 (R d ) θ θ (1 -θ) 1-θ ≥ 1 θ θ (1 -θ) 1-θ C d,p .
This proves in particular that I(1) > 0. Let us fix any u ∈ H 1 (R d ) such that

R d |u(x)| p dx = 1 and define u α (x) = α d/p u(αx) which is still normalized in L p (R d ).
By definition of I(1), we obtain

I(1) ≤ α 2θ R d |∇u(x)| 2 dx + α -2(1-θ) R d |u(x)| 2 dx.
If we minimize over α, we find

I(1) ≤ ||u|| I(1) ≤ 1 θ θ (1 -θ) 1-θ C d,p
.

Recalling [START_REF] Tao | CBMS Regional Conference Series in Mathematics[END_REF], we obtain the stated equality [START_REF]Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems[END_REF]. Our proof also shows how to relate the optimizers of the two problems via the above rescaling.

We now discuss the existence of minimizers for I(λ), using the techniques developed in the previous sections.

Lemma 24 (Existence). The minimization problem I(λ) admits minimizers for all λ > 0. All the minimizing sequences are precompact in H 1 (R d ), up to translations.

Proof. It suffices to consider I(1) by [START_REF]Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems[END_REF]. Consider a minimizing sequences u = {u n } for I(1). Then u n 2

H 1 (R d ) → I(1) and therefore u n is bounded in H 1 (R d ).
Define m(u) as in the previous sections. By definition of I(1) we have ||u n || L p (R d ) = 1 which can therefore not tend to zero. We conclude from Lemma 12 that m(u) > 0, that is, the sequence cannot vanish. Therefore there exists a subsequence n k and space translations

x k ∈ R d such that u n k (• + x k ) u = 0 weakly in H 1 (R d ).
Since our problem is translation-invariant, we can assume without loss of generality that x k ≡ 0. For simplicity of notation we also assume that u n u weakly, that is, we replace the original sequence by the subsequence. From the local strong compactness we may as well assume that u n (x) → u(x) almost everywhere and strongly in L 2 loc (R d ). From [START_REF] Brézis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] in Lemma 6 we have

(33) 1 = R d |u n (x)| p dx = R d |u(x)| p dx :=λ + R d |u n (x) -u(x)| p dx + o(1)
and therefore, using (3) and ( 4),

||u n || 2 H 1 (R d ) = ||u|| 2 H 1 (R d ) + ||u n -u|| 2 H 1 (R d ) + o(1) ≥ ||u|| 2 H 1 (R d ) + I R d |u n -u| p + o(1) ≥ ||u|| 2 H 1 (R d ) + I(1 -λ) + o( 1
). In the third line we have used that R d |u n -u| p → 1λ and the continuity of I which follows from the explicit formula [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF]. Passing to the limit, we obtain ( 34)

I(1) ≥ ||u|| 2 H 1 (R d ) + I(1 -λ) ≥ I(λ) + I(1 -λ) = I(1) λ 2 p + (1 -λ) 2 p
.

The right side is always greater than I(1) > 0 for λ ∈ (0, 1), which cannot be. On the other hand we already know that λ > 0 since u = 0. We conclude that λ = 1 and the series of inequalities in [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF] then imply that

I(1) = ||u|| 2 H 1 (R d ) , that is, u is an optimizer. Also, we have ||u n || 2 H 1 (R d ) → ||u|| 2 H 1 (R d ) , hence the convergence is strong in H 1 (R d ).
Once we know that I(1) has a minimizer u, usual perturbations arguments show that satisfies the nonlinear equation ( 35)

-∆u + u = I(1)u p-1 .

Here I(1) plays a role of a Lagrange multiplier due to the constraint R d |u| p = 1. In addition, we remark that |u| is a minimizer if u is one, hence we may always assume that u ≥ 0. Letting u = αQ with α = I(1) 1/(2-p) we see that Q solves [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF]. The uniqueness (up to translations) of non-negative solutions to the equation ( 28) is proved in [START_REF] Coffman | Uniqueness of the ground state solution for ∆u-u+u 3 = 0 and a variational characterization of other solutions[END_REF][START_REF] Kwong | Uniqueness of positive solutions of ∆u -u + u p = 0 in R n[END_REF][START_REF] Mcleod | Uniqueness of positive radial solutions of ∆u+f (u) = 0 in R n . II[END_REF][START_REF] Tao | CBMS Regional Conference Series in Mathematics[END_REF][START_REF] Frank | Ground states of semi-linear PDE[END_REF].

The results from all the previous sections easily generalize to H s (R d ), at the expense of using localization methods for fractional Laplacians as described for instance in [14, App. A] and [15, App. B].

Exercise 25. Let u be an optimizer for I(1) and define u(t) := (u+tχ) u+tχ -1

L p (R d ) with χ ∈ H 1 (R d ).
Expand the H 1 (R d ) norm of u(t) to first order in t and deduce that u must solve (35) in H -1 (R d ).

Exercise 26 (The NLS functional). Let us consider the minimization problem

J(λ) := inf u∈H 1 (R d ) R d |u| 2 =λ R d |∇u(x)| 2 dx - R d |u(x)| p dx . Show that J(λ) > -∞ for 2 ≤ p < 2+4/d but that J(λ) = -∞ for 2+4/d < p < 2 * .
In the first case, compute J(λ) in terms of the Gagliardo-Nirenberg constant C d,p . Prove finally that J(λ) admits a minimizer and that all minimizing sequences are precompact, by following a similar strategy as for I(λ).

The Sobolev inequality and its optimizers

The arguments used to prove the existence of optimizers for the Gagliardo-Nirenberg inequality do not apply to the critical case p = 2 * = 2d/(d -2) in dimensions d ≥ 3. The Sobolev inequality has an additional scaling invariance due to dilations and, since it does not involve the L 2 norm at all, it is not possible to work in H 1 (R d ). Our goal in this section is to provide a version of Lemma 11 in the critical case and this will force us to take dilations into account, in addition to translations. But before we start with a simple proof of the Sobolev inequality taken from [START_REF] Chemin | Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques[END_REF], which will guide us in our reasoning. Later we use the convention that S d,s is the best constant in this inequality. For s = 1 we have S d,s = C d,2 * , the constant introduced in (26).

Proof. For shortness we denote

K := R d |k| 2s | u(k)| 2 dk 1/2 := ||u|| Ḣs (R d ) .
We start by writing 6.2. Lack of compactness in Ḣs (R d ). In order to adapt the arguments used before in the subcritical case, we now have to deal with the lack of compactness due to dilations, in addition to translations. This leads us to introducing the following concept. For a bounded sequence u = {u n } in L p * (R d ) with p * = 2d d-2s , we introduce the highest (critical) mass that weak limits can have, up to translations, dilations, and extraction of a subsequence:

R d |u(x)| 2d d-2s dx = d -2s d + 2s R d ∞ 0 λ 2d d-2s -1 1(|u(x)| ≥ λ) dλ dx = d -2s d + 2s ∞ 0 λ 2d d-2s -1 |{|u(x)| ≥ λ}| dλ. ( 37
(39) m (u) = R d |u| p * : α -d/p * k u n k • + x k α k u .
The following is the corresponding adaptation of Lemma 11 to the critical case. The argument is then the same as before.

The following is the equivalent of Lemma 12 in the sub-critical case.

Corollary 29 (Vanishing in the critical case). Let u = {u n } be a bounded sequence in Ḣs (R d ), with d > 2s. Then m (u) = 0 if and only if u n → 0 strongly in L p * (R d ).

A similar result has appeared first in [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF].

6.3. Existence of optimizers for the Sobolev inequality. Using the above result, we can deduce the existence of an optimizer for the Sobolev inequality (36) in a few lines, using the exact same argument as in Section 5 in the case of subcritical p.

Corollary 30 (Existence of a minimizer for the Sobolev inequality). The best constant S d,s in (36) is attained and all the minimizing sequences are precompact up to translations and dilations.

Proof. We introduce the minimization problem similar to the sub-critical case (29): This is because a sequence bounded in Ḣs (R d ) is also bounded in H s loc (R d ) hence we may assume, after extracting a subsequence, that it converges almost everywhere, so as to apply the Brezis-Lieb convergence [START_REF] Cancès | Méthodes mathématiques en chimie quantique. Une introduction[END_REF]. We also have which proves again that λ = 1 since 2/p * < 1. Hence u is an optimizer and u n → u strongly in Ḣs (R d ).

I(λ) := inf R d |u| p * =λ
Arguments of this type have been used for instance in [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]. It is possible to extend several of the results of the previous sections to the critical case (for instance the profile decomposition, due to Gérard [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF]), with of course the addition of the possible lack of compactness due to dilations.

Figure 1 .

 1 Figure 1. Four typical behaviors of a sequence {u n} ⊂ L 2 (R d ) with u n 0 while R d |u n | 2 = λ for all n. In the examples, u is a fixed (smooth) function in L 2 (R d ), such that R d |u| 2 = λ.

  Note that the exponent 2 + 4/d is always smaller than the critical Sobolev exponent p * , which equals 2 + 4/(d -2) in dimensions d ≥ 3 and +∞ in dimensions d = 1, 2. The proof relies on the Sobolev inequality. Lemma 11 is due to Lions [23, Lemma I.1].

  Cz |u n | 2 ≤ m(u), which will end the proof of Lemma 11. Indeed, consider a sequence z = {z n } ⊂ R d such that lim n→∞ Cz n |u n | 2 = lim sup n→∞ sup z∈Z d Cz |u n | 2 .

  of the proof of Lemma 11, we deduce that lim sup n→∞ sup z∈Z d Cz |u n | 2 = 0 where C z = d j=1 [z j , z j + 1) are cubes covering the whole space. Since a ball of radius R can be covered by a finite union of such cubes, the result follows.

}}

  does not contain a local mass larger than ε. In general the mass is not necessarily small since the sequence can still undergo vanishing. However, by Lemma 11, the subcritical norms of ψ

6. 1 . 2 L

 12 The Sobolev inequality.Theorem 27 (Sobolev inequality). For every s > 2d in dimension d ≥ 1, there exists a constant S d,s < ∞ such that(36) ||u|| 2d d-2s (R d ) ≤ S d,s R d |k| 2s | u(k)| 2 dk,for every u ∈ S (R d ) such that |{|u| ≥ λ}| < ∞ for all λ > 0.

) 2 ||u|| 2 = 2 dλ 2 R d |w| 2 ≤ 1 λ 2 |k|≥a | u| 2 .|

 22222 Now we will give an estimate on |{|u(x)| ≥ λ}| for any fixed λ. We write u = v + w where v(k) = u(k)χ(k/a), with a a parameter depending on λ to be determined later and 0 ≤ χ ≤ 1 a radial localizing function satisfying χ |B1 ≡ 1 and χ |R d \B2 ≡ 0.We use that|{|u(x)| ≥ λ}| ≤ |{|v(x)| ≥ λ/2}| + |{|w(x)| ≥ λ/2}|and choose a to ensure ||v|| L ∞ ≤ λ/2 which will give |{|v(x)| ≥ λ/2}| = 0. Indeed, we have||v|| L ∞ ≤ (2π) Ḣs (R d ) = C K a λ/(2CK) ⇐⇒ a = C (λ/K) -2s. For the term involving w, we write|{|w(x)| ≥ λ/2}| ≤1Coming back to (37), this gives u| 2 dk dλ = CK 4s d-2s |k| 2s | u| 2 dk = CK 2d d-2s .

Lemma 28 ( 1 ad/p * n u n y + x n a n 2 m

 2812 A critical estimate involving m (u)). Let d > 2s > 0. We have for every bounded sequenceu = {u n } ⊂ Ḣs (R d ) d ) .Proof. We follow the argument used in the proof of Theorem 27 and only give a different estimate of v n L ∞ with v n = u n χ(•/a), in terms of m (u) instead of K. Indeed, we havev n L ∞ = (2π) -d/2 sup x∈R d R d a d χ(ay)u n (xy) dy = (2π) -d/2 supIn the last line we used that χ is radial. For every n, let a n and x n realizing the above supremum. Since the function y → is bounded in L p * (R d ), we may extract a subsequence and assume that 1a d/p * n u n • + x n a n u.By definition of m (u) we have ||u|| L p * (R d ) ≤ m (u) 1/p * . On the other hand, lim (u) 1/p * .

||u|| 2 Ḣs (R d ) = λ 2 p

 22 * I(1).A minimizing sequence u = {u n } must satisfy m (u) > 0 by Lemma 28. Hence we can find a translation and a dilation such that, after extracting a subsequence, 1a d/p * n u n • + x n a n u = 0. Writing u n = a d/p * n u (a n (• + x n )) + r n we have R d |u n | p * = R d |u| p * :=λ>0 + R d |r n | p * + o(1).

||u n || 2 Ḣs 2 p

 22 (R d ) = ||u|| 2 Ḣs (R d ) + ||r n || 2 Ḣs (R d ) + o(1)since Ḣs (R d ) is a Hilbert space. Therefore,I(1) = lim n→∞ ||u n || 2 Ḣs (R d ) ≥ ||u|| 2 Ḣs (R d ) + I(1λ) ≥ λ * + (1λ)


Keywords: Noncircular, improper, widely linear, Single Antenna Interference Cancellation (SAIC), Binary signals, AM, ASK, BPSK, MSK, GMSK, demodulation, equalization, detection, beamforming, direction finding, MIMO, radar, waveform diversity

With new generation of Active Digital Radar

Antenna, there is a renewal of waveform generation and processing approaches, and new strategies can be explored to optimize waveform design and waveform analysis and to benefit of all potential waveform diversity. Among these strategies, building and exploitation of the Noncircularity of waveforms is a promising issue. Up to the middle of the nineties, most of the signals encountered in practice are assumed to be second order (SO) circular (or proper), with a zero second correlation function. However, in numerous operational contexts such as in radio communications, the observed signals are either SO noncircular (or improper) or jointly SO noncircular with a particular signal to estimate, to detect or to demodulate, with some information contained in the second correlation function of the signals. Exploitation of this information in the processing of SO noncircular signals may generate dramatic gain in performance with respect to conventional processing and opens new perspective in signal processing. The purpose of this paper is to present a short overview of the interest of taking into account the potential SO noncircularity of the signals in signal processing and to describe the potential interest of SO noncircular waveforms for radar applications.

I. INTRODUCTION

Up to the middle of the nineties, most of the signals encountered in practice are assumed to be SO circular [START_REF] Picinbono | On circularity[END_REF], which means that their second correlation function is zero. Despite of the fact that many signals used in practical situations are indeed SO circular, some of them, mainly borrowed from the radio communications context, are SO noncircular [START_REF] Picinbono | On circularity[END_REF] and contain information in their second correlation function. It is in particular the case for rectilinear signals such as Amplitude Modulated (AM), Amplitude Shift Keying (ASK) or Binary Phase Shift Keying (BPSK) signals [START_REF] Proakis | Digital communications[END_REF]. It is also the case for signals which become quasirectilinear (i.e. a complex filtered version of a rectilinear signal) after a simple derotation operation such as Offset Quadrature Amplitude Modulations (OQAM), Minimum Shift Keying (MSK) or Gaussian MSK (GMSK) modulations [START_REF] Proakis | Digital communications[END_REF]. Moreover, in the context of MIMO wireless systems [START_REF]Space-Time Wireless systems : From Array processing to MIMO communications[END_REF] and whatever the chosen constellation, some Spatio-Temporal Block Coding (STBC) schemes, such as the Alamouti's scheme [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF], generate observations which become jointly SO noncircular with the transmitted symbols. For observations which are either SO noncircular or jointly SO noncircular with a given input signal, exploitation of the information contained in the second correlation function of the signals may generate dramatic gain in performance with respect to conventional processing. This may even open new perspectives in signal processing.

The purpose of this paper is to present a brief overview of the interest of taking into account the potential SO noncircularity of the signals in signal processing and to describe the potential interest of noncircular waveforms in radar applications.

II. SECOND ORDER NONCIRCULAR OBSERVATIONS

The SO statistics of a (N x 1) complex random vector process x(t) are completely defined by the two correlation

matrices R x (t, τ) = Δ E[x(t) x(t -τ) † ] and C x (t, τ) = Δ E[x(t) x(t - τ) T ],
for all the values of (t, τ), where † and T are the transposition conjugation and the transposition operation respectively. The vector x(t) is said to be SO noncircular [START_REF] Picinbono | On circularity[END_REF] if the pseudo-correlation matrix [START_REF] Nesser | Proper Complex random processes with applications to information theory[END_REF], C x (t, τ), is not zero for at least one couple (t, τ). Such situations may for instance occur in the presence of rectilinear signals such as BPSK signals used in secondary radar or quasi-rectilinear signals such as GMSK signals used in the GSM network. Extension of the SO noncircularity concept to higher order may be found in [1] [6].

III. SO NONCIRCULARITY EXPLOITATION THROUGH OTIMAL WIDELY LINEAR FILTERING IMPLEMENTATION

A. Widely linear filtering

It is shown in [START_REF] Chevalier | Optimal array processing for non stationary signals[END_REF] that for stationary observations, whose complex envelope is necessarily SO circular [START_REF] Picinbono | On circularity[END_REF], SO optimal complex filters are linear and Time Invariant (TI), i.e. of the form y(t) = Δ w † x(t), where w is a TI complex filter. However, for a nonstationary observation, whose complex envelope may be SO noncircular, SO optimal complex filters are Time Variant (TV) and, under some conditions of SO noncircularity presented in [START_REF] Chevalier | Optimal array processing for non stationary signals[END_REF] [8], Widely Linear (WL) [START_REF] Picinbono | Widely linear estimation with complex data[END_REF], i.e. of the form [START_REF] Brown | Conjugate linear filtering[END_REF] and the first works about their SO optimality in noncircular contexts [START_REF] Chevalier | Optimal array processing for non stationary signals[END_REF] [START_REF] Picinbono | Widely linear estimation with complex data[END_REF], there has been an increasing interest for such filters in signal processing. The main applications of WL filtering currently concern Co-Channel Interference (CCI) cancellation of noncircular interferences, equalization of frequency selective channels for noncircular transmissions and optimization of MIMO systems using either SO noncircular constellations or some particular "noncircular" STBC schemes.

y(t) = Δ w 1 (t) † x(t) + w 2 (t) † x(t)

B. Cochannel Interference Rejection

1) Cellular F/TDMA radio communications systems SO optimal WL filters have been used for CCI rejection in F/TDMA radio communications cellular networks using either rectilinear modulations such as BPSK signals [START_REF] Chevalier | New Insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by non circular interferences -Application to SAIC[END_REF] or quasirectilinear modulations such as MSK [START_REF] Chevalier | New Insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by non circular interferences -Application to SAIC[END_REF] or GMSK [START_REF] Chevalier | New Insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by non circular interferences -Application to SAIC[END_REF][START_REF] Trigui | Performance bounds for cochannel interference cancellation within the current GSM standard[END_REF] signals. In such contexts, it has been shown in [START_REF] Chevalier | New Insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by non circular interferences -Application to SAIC[END_REF] that SO optimal WL filters allow the processing of up to P = 2N -1 rectilinear or quasi-rectilinear interferences from an array of N sensors. This proves in particular the capability of these filters to do Single Antenna Interference Cancellation (SAIC) (P = 1 for N = 1). For this reason, the 3G Americas [START_REF] Austin | SAIC and synchronised networks for increased GSM capacity[END_REF] presents the SAIC technology as a great improvement for GSM mobile station receivers, already standardized and allowing significant network's capacity gains for the GSM system. Figure 1 illustrates the behavior of both the conventional receiver and the optimal receiver, implementing the SAIC concept, when a BPSK useful signal is corrupted by a BPSK interference. Contrary to the conventional receiver, which compensates the phase of the useful signal before taking the real part of the observation, the SAIC receiver puts the interference on the y-axis before taking the real part of the observation, which eliminates the latter. 2) CDMA radiocommunications systems SO optimal WL filters have been proposed to mitigate Multiple Acces Interference (MAI) in CDMA radio communications networks using either BPSK modulation [START_REF] Gelli | Blind widely linear multiuser detection[END_REF][START_REF] Buzzi | A new family of MMSE multiuser receivers for interference suppression in DS/CDMA systems employing BPSK modulation[END_REF][START_REF] Tulino | Asymptotic analysis of improved linear receivers for BPSK-CDMA subject to fading[END_REF][START_REF] Buzzi | Performance analysis for the improved linear multiuser detectors in BPSK-modulated DS-CDMA systems[END_REF] or OQPSK modulation [START_REF] Mirbagheri | An enhanced widely linear CDMA receiver with OQPSK modulation[END_REF], through the implementation of WL Multi-user Detection (MUD) schemes. For Direct Sequence Spread Spectrum (DSSS) BPSK signals , it is shown in [START_REF] Tulino | Asymptotic analysis of improved linear receivers for BPSK-CDMA subject to fading[END_REF][START_REF] Buzzi | Performance analysis for the improved linear multiuser detectors in BPSK-modulated DS-CDMA systems[END_REF] that the use of optimal WL filters for MUD allows to double the capacity of the system in terms of number of users for a given spreading factor. For OQPSK signals, which concern IS'95 and CDMA 2000 standard, the use of optimal WL filters strongly improves the performance of the receivers.

3) Detection systems

Very recently, optimal WL filters have been applied to the problem of detection of either a random signal [START_REF] Schreier | Detection and Estimation of improper complex random signals[END_REF] or a known deterministic signal with unknown parameters [START_REF] Chevalier | GLRT-Based array receivers to detect a known signal corrupted by noncircular interferences[END_REF] corrupted by SO noncircular interferences. Several new GLRT-based detectors associated to several choices of unknown desired signal parameters have been developed in [START_REF] Chevalier | GLRT-Based array receivers to detect a known signal corrupted by noncircular interferences[END_REF] for binary coded signals and arbitrary SO noncircular interferences. One of this new detector is also well-suited for synchronization purposes in radio communications [START_REF] Chevalier | Second-order optimal array receivers for synchronization of BPSK, MSK and GMSK signals corrupted by noncircular interferences[END_REF]. All the proposed detectors allow to perform SAIC in the presence of a rectilinear interference as it is illustrated by figure 2. The latter shows the variations of the Non Detection Probability of the desired BPSK signal at the output of several conventional (Ci) and optimal detectors (Nci) as a function of the input SNR in the presence of a BPSK interference such that INR = SNR + 20 dB, for a false alarm rate equal to 0.001, for a signal containing 64 symbols and for N = 1 sensor. Phase difference between the sources is equal to 45°. Note the non detection of the signal by the conventional receivers and the detection of the latter by the optimal receivers.

4) MVDR Beamforming for passive listening

For passive listening applications, two WL MVDR beamformers have been proposed recently [START_REF] Chevalier | Widely linear MVDR beamformer for the reception of an unknown signal corrupted by noncircular interferences[END_REF] to improve the performance of the well-known Capon's beamformer [START_REF] Capon | High resolution frequency wave number spectrum analysis[END_REF][START_REF] Capon | Multidimensional maximum likelihood processing of a large aperture seismic array[END_REF] in the presence of SO noncircular interferences. The first one is TI whereas the second one, TV, allows to process SO noncircular interferences having a nonzero frequency offset or carrier residue. The proposed TI MVDR beamformer allow the processing of up to 2(N -1) rectilinear interferences from an array of N sensors.

C. Egalisation of Linear Frequency Selective channels

1) Single carrier links

In [START_REF] Gerstacker | Receivers with widely linear processing for frequency-selective channels[END_REF] several MMSE equalization schemes based on WL processing are proposed for rectilinear and quasi-rectilinear single carrier signals propagating through frequency selective channels, generating SO noncircular Inter-Symbol Interference (ISI) at the reception. It is shown in particular that WL equalizers outperforms Linear ones in such contexts.

2) DMT or OFDM links

Optimal Narrow-Band Interference (NBI) resistant WL receivers/equalizers have also been proposed in [START_REF] Darsena | Widely linear equalization and blind channel identification for interference-contaminated multicarrier systems[END_REF] and [START_REF] Darsena | Universal Linear Precoding for NBIproof Widely Linear Equalization in MC systems[END_REF] for Discrete Multitone (DMT) and Orthogonal Frequency Division Multiplex (OFDM) systems using SO noncircular constellations. Reference [START_REF] Darsena | Universal Linear Precoding for NBIproof Widely Linear Equalization in MC systems[END_REF] considers both Cyclic-Prefix (CP) and zero-padded based multicarrier (MC) systems whereas [START_REF] Darsena | Widely linear equalization and blind channel identification for interference-contaminated multicarrier systems[END_REF] limits to CP systems.

3) UWB links

In [START_REF] Parihar | Equalization for DS-UWB Systems -Part I : BPSK modulation[END_REF] and [START_REF] Parihar | Equalization for DS-UWB Systems -Part II : 4BOK modulation[END_REF], WL equalizers are proposed to mitigate ISI in the context of direct-sequence based Ultra-Wideband (DS-UWB) systems using either BPSK [START_REF] Parihar | Equalization for DS-UWB Systems -Part I : BPSK modulation[END_REF] or 4BOK [START_REF] Parihar | Equalization for DS-UWB Systems -Part II : 4BOK modulation[END_REF] modulations. The use of WL receivers allows very powerful high rate and short range transmission from the standardized UWB wave-forms under investigation for wireless personal area networks.

D. MIMO systems optimization

In [START_REF] Mattera | Widely Linear MMSE equaliser for MIMO linear time-dispersive channel[END_REF], [START_REF] Mattera | Widely Linear Decision-Feedback Equalizer for Time-Dispersive linear MIMO channels[END_REF] and [START_REF] Gerstacker | Equalization concepts for Alamouti's space-time block code[END_REF], WL equalizers are proposed to mitigate ISI in the context of MIMO systems using either BPSK constellation [START_REF] Mattera | Widely Linear MMSE equaliser for MIMO linear time-dispersive channel[END_REF], [START_REF] Mattera | Widely Linear Decision-Feedback Equalizer for Time-Dispersive linear MIMO channels[END_REF] or STBC schemes, such as the Alamouti's scheme, which generate observation jointly SO noncircular with the transmitted symbols [START_REF] Gerstacker | Equalization concepts for Alamouti's space-time block code[END_REF]. Strong gain in performance with respect to conventional linear equalizers are proved. In [START_REF] Buzzi | Widely linear reception strategies for layered space-time wireless communications[END_REF] and [START_REF] Witzke | Linear and Widely linear filtering applied to iterative detection of generalized MIMO signals[END_REF], WL receivers are proposed to mitigate interferences generated by some spatial multiplexing schemes involving either SO noncircular constellations [START_REF] Buzzi | Widely linear reception strategies for layered space-time wireless communications[END_REF] or transmitted symbols and their complex conjugates [START_REF] Witzke | Linear and Widely linear filtering applied to iterative detection of generalized MIMO signals[END_REF]. Finally reference [START_REF] Sterle | Widely linear MMSE transceivers for MIMO channels[END_REF] proposes optimized WL precoder schemes from rectilinear constellations and Channel state information knowledge at the transmitter.

IV. SO NONCIRCULARITY EXPLOITATION FOR PARAMETERS

ESTIMATION

The SO noncircularity of the signals may also be exploited for some parameter estimation problems without the implementation of WL filters. Such a property has been exploited for direction finding of rectilinear [START_REF] Gounon | Localisation angulaire de signaux non circulaires[END_REF] [START_REF] Charge | A non-circular source direction finding method using polynomial rooting[END_REF] or SO noncircular signals [START_REF] Abeida | MUSIC-like Estimation of Direction of Arrival for Noncircular Sources[END_REF], where up to 2N -1 or 2(N -1) rectilinear signals may be processed from an array of N sensors. In [START_REF] Grellier | Analytical blind channel identification[END_REF] it is shown that for SO noncircular signals, blind channel identification is possible from SO statistics only. Finally several algorithms for blind frequency offset estimation of SO noncircular single carrier [START_REF] Ciblat | Performance analysis of blind carrier frequency offset estimators for non circular transmissions through frequency-selective channels[END_REF] [40] and OFDM [START_REF] Fusco | ML-based symbol timing and frequency offset estimation for OFDM systems with noncircular transmissions[END_REF] links through frequency selective channels have been proposed recently.

V. NONCIRCULAR WAVEFORM EXPLOITATION FOR RADAR SYSTEMS

Radar waveforms have been classically optimized relatively to ambiguity function for target detection (good range resolution; sufficient energy on target, non sensitivity to target Doppler offset, robustness to clutter and jamming,…). For the time being, Frequency Modulation waveforms are widely used and preferred to Phase Coding ones for Radar Applications. But noncircularity properties of phase coded radar waveforms have not yet been exploited, and this breakthrough coming from radiocommunication could modify balance between frequency & phase coding choice.

From this matter of fact, we have to explore classical phase coded waveform : BPSK, Pseudo-Noise coded (PN) sequences, Multi-Phase codes, … .

Waveforms based on Binary phase-coded sequences, like BPSK, are used in Radar but their noncircularity properties are not exploited. In radio communications, this Binary phasecoded sequences are exploited to transmit information in a multi-user environment or for discretion purposes (e.g. for spread-spectrum communication system using binary PSK modulation, M-sequences are applied as sub-baud codes for bandwidth expansion), but for radar applications, these codes are used in an other framework for "pulse compression" to increase range resolution. Furthermore in radar, target motion gives rise to a Doppler frequency offset on the received signals. It is thus necessary not only to analyse the coded waveforms auto-correlation functions, but also to study the ambiguity function ) , ( f t X in time-frequency (t, f) plan, introduced by P. M. Woodward in 1953, to evaluate the performance under various Doppler frequency returns :

∫ +∞ ∞ - - + = τ τ τ π d e t s s f t X ft i 2 * ) ( ) ( ) , ( (1)
Radar Engineering will select s(t) that will give an ambiguity function as close as possible to the ideal "thumbtack" response, with most of the weight around the origin in the (t,f) plane :

) 0 , 0 ( ) , ( X f t X ≤ (2) with 1 ) 0 , 0 ( ) , ( 2 2 = = ∫ ∫ +∞ ∞ - +∞ ∞ - X dtdf f t X (3)
The ambiguity should still be small in some strip of the (t,f) plane containing the delay axis and because of the sidelobe on the Doppler axis, the strip must be narrower than l/(MT) in the Doppler direction.

Considering binary phase-coded sequences, like BPSK (with 0 and π phase coding of the carrier waveform), Barker sequences have been introduced [START_REF] Barker | Group synchronisation of binary digital systems[END_REF], as their aperiodic autocorrelations are all 1/M or 0, for k = 1, 2, ...,M -1. However, while the ambiguity function is small on the delay axis, for all known Barker sequences there are substantial ridges away from the zero Doppler axis. Poor performance of Barker sequences is mainly due because they are all short sequences (there are no Barker sequences of length greater than 13, some elements of this proof in [START_REF] Fredman | Polynomes reciproques: conjecture d'Erdos en norme L 4 , taille des autocorrelalions et inexistence des codes de Barker[END_REF]), but exploitation of their non-circularity in anti-jamming or DOA could improve their advantages and other performances that could be attractive.

Pseudo-Noise coded (PN) sequences have been also studied as alternative of Barker sequences. The fact, that we have longer sequence could be also an advantage for noncircularity exploitation. They are generated by maximum length shift-register sequences, and have the property that the cyclic autocorrelation sidelobe levels are very small, equal to -1/M for 1 ≤ k ≤ M -1. If we consider their aperiodic autocorrelations, their sidelobes are at most

M M / ) log(
(see [START_REF] Mceliece | Correlation properties of sets of sequences derived from irreducible cyclic codes[END_REF]). The low level of cyclic autocorrelation Ambiguity function sidelobes (-1/M) only occurs for the zero Doppler case, while the remainder of the plot is similar to the aperiodic performance. The idea is that subsequences of long Msequences might be sufficiently random to have good autocorrelation properties, as studied in [START_REF] Lindholm | An analysis of the pseudo-randomness properties of subsequences of long m-sequences[END_REF] where some properties of such subsequences (e.g. moments, mean weights) are calculated.

In practice, the ambiguity performance of binary phase coded sequences is lower than linear Frequency Modulation (FM) chirp signals. This is why Shift-register sequences have been deeply analyzed algebraically and investigated experimentally by simulation. For example, Bartee and Wood [START_REF] Bartee | Coding for tracking Radar ranging[END_REF] have studied interlacing several short M-sequences having relatively prime lengths to obtain long codes having good short-term correlation properties to promote rapid acquisition in a radar ranging application. The general problem area involves the pseudo-randomness properties and/or weight distributions of segments from M-sequences. Most of the well-known good pseudo-random properties of M-sequences pertain to averages over an entire period of the M-sequence. The detailed weight distribution of segments from M-sequences is not well known. Bartee and Wood [START_REF] Bartee | Coding for tracking Radar ranging[END_REF] have considered an exhaustive comparison of all M-sequences of lengths up to and including M = 2 14 -1 = 16 383 for a wide range of subsequence lengths to select "optimum" Msequences. For a given subsequence length, M, and fixedlength M-sequences the "optimum" M-sequence, in the sense of Bartee and Wood [START_REF] Bartee | Coding for tracking Radar ranging[END_REF], is essentially that one whose minimum weight M-tuple has greatest weight. In one of their examples, a particular-system performance parameter varies over more than five orders of magnitude from worst-case Msequence to best-case M-sequence. Unfortunately, Bartee and Wood found no systematic relationship between their set of optimum M-sequences and the corresponding generating functions.

Alternatively, Lindholm has considered the moments of the distribution of weights of M-tuples from any given Msequence, and develop a systematic relationship between the moments of the distribution and the characteristic polynomial via the notion of coset functions. In addition, the coset function approach yields a systematic relationship between the distributions of the entire family of M-sequences of any given length. The first few moments of a distribution are sufficient to describe the main geometrical features of the distribution.

As in communication, noncircularity could be exploited on Multi-Phase Codes. In Radar, the design of Multi-Phase codes has focused on small aperiodic autocorrelation, rather than maximum sidelobes minimization of the whole ambiguity function. For instance, Golomb [START_REF] Golomb | Generalised Barker sequences[END_REF] has developed a 6-phase analogues of Barker codes. But, as with Barker codes, the maximum Ambiguity function sidelobes of these codes, away from the delay axis, are high.

Performance of Phase coded sequences should be now considered in the new framework of noncircularity, as it is done in radio communications. As we have seen in the first part of the paper, we could accept degraded performance of ambiguity function if we exploit noncircularity to improve Radar detection, anti-jamming (twice more number of interferences sources that could be rejected, hostile interference or interferences from other co-localized sensors that share same frequency plan) and DOA (better resolution than MUSIC with circularity assumption) performances by new processing chains exploiting statistics contained in the complex symmetric unconjugated spatial covariance matrix, and not only the classical covariance matrix. A potential improvement could be obtained if the two covariance matrices are used simultaneously. Phase coded waveforms can be used also for data transmission (plots & tracks) in netted radar system (in a dense network of co-localized radars, Phase coded waveforms could be used for communication modes between radars).

VI. CONCLUSION

The area of waveform design enjoys a new attention due to advances in "software radar" concept (Digital Array, use of software in Front-End,…). Classically, radar waveforms have been designed to optimize ambiguity function for target detection properties (good range resolution; sufficient energy on target, non sensitivity to target Doppler offset, robustness to clutter and jamming,…). In practice, Frequency Modulation have been preferred to Phase Coding for Radar Applications. But for the time being, noncircularity properties of phase coded radar waveforms have not been exploited as it has been done in radio communications. We propose to study new phase coded waveforms and exploit statistics contained in the complex symmetric unconjugated spatial covariance matrix (pseudocovariance matrix), to improve radar performances (detection, anti-jamming, DOA,…) and to allow communication, with the same phase coded waveform, between co-localized radar sensors (information on targets plots/tracks are exchanged between radar sensors in a collaborative way).

In that respect, it highlights the importance of designing waveforms with diverse objectives in terms of global radar performances and not with only classical ambiguity function constraints.

Exploitation of radar waveforms noncircularity property could be also extended to be applied for multi-static, MIMO and Ultra-Wideband Radars applications.

Concept of Waveform Diversity is a new challenging project that will promote new solutions and paradigms for advanced agile waveform resources management as explained in [START_REF] Barbaresco | New Agile Waveforms Based on Mathematics & Resources Management of Waveform Diversity[END_REF].

As Woodward wrote in his seminal book, I could end with the following statement : "The reader may feel some disappointment, not unshared by the writer, that the basic question of what to transmit remains substantially unanswered."

Figure 1 -

 1 Figure 1 -SAIC concept for BPSK signal and interference

Figure 2 -

 2 Figure 2 -Non detection probability as a function of SNR