Hajer Bahouri

Albert Cohen

Gabriel Koch

A general wavelet-based profile decomposition in the critical embedding of function spaces *

 to the case where X is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces X and Y that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of X and Y satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces.

Introduction

The critical embedding of homogeneous Sobolev spaces in dimension d states that for 0 ≤ t < s and 1 ≤ p < q < ∞ such that d/p -d/q = s -t, one has Ẇ s,p (R d ) ⊂ Ẇ t,q (R d ).

(1.1)

The lack of compactness in this embedding can be described in terms of an asymptotic decomposition following Gérard [START_REF] Gérard | Description du défaut de compacité de l'injection de Sobolev[END_REF] who considered the case p = 2 and t = 0, and Jaffard [START_REF] Jaffard | Analysis of the lack of compactness in the critical Sobolev embeddings[END_REF] who considered general values p > 1 with the Riesz potential spaces Ḣs,p in replacement of Sobolev spaces Ẇ s,p , again with t = 0. Their results can be formulated in the following terms : a sequence (u n ) n≥0 bounded in Ḣs,p (IR d ) can be decomposed up to a subsequence extraction according to

u n = L l=1 h s-d/p l,n φ l • -x l,n h l,n + r n,L (1.2)
where (φ l ) l>0 is a family of functions in Ḣs,p (IR d ) and where lim L→+∞ lim sup n→+∞ r n,L L q = 0.

This decomposition is "asymptotically orthogonal" in the sense that for k = l | log(h l,n /h k,n )| → +∞ or |x l,n -x k,n |/h l,n → +∞, as n → +∞.

This type of decomposition was also obtained earlier in [START_REF] Brezis | Convergence of solutions of H-Systems or how to blow bubbles[END_REF] for a bounded sequence in H 1 0 (D, R 3 ) of solutions of an elliptic problem, with D the open unit disk of R 2 and in [START_REF] Struwe | A global compactness result for boundary value problems involving limiting nonlinearities[END_REF] and [START_REF] Solimini | A note on compactness-type properties with respect to Lorentz norms of bounded subset of a Sobolev space[END_REF] for the critical injections of W 1,2 (Ω) in Lebesgue space and of W 1,p (Ω) in Lorentz spaces respectively, with Ω a bounded domain of R d . They were also studied in [START_REF] Schindler | An abstract version of the concentration compactness principle[END_REF] in an abstract Hilbert space framework and in [START_REF] Ameur | Description du défaut de compacité de l'injection de Sobolev sur le groupe de Heisenberg[END_REF] in the Heisenberg group context.

The above mentioned references treat different types of examples of critical embedding by different methods. One of the motivations of the present paper is to identify some fundamental mechanisms that lead to such results for a general critical embedding X ⊂ Y, in a unified way. Here X and Y are generic homogeneous function spaces which, similar to the above particular cases, have the same scaling properties in the sense that for any function f and

h > 0 f (h•) X = h r f X and f (h•) Y = h r f Y , (1.3)
for the same value of r. In a similar way to Jaffard, we use wavelet bases in order to construct the functions φ l , yet in a somehow different and hopefully simpler way. Our construction is based on two basic key properties of wavelet expansions in the spaces X and Y , which may then be easily checked on particular pairs of spaces of interest. In particular, any critical embedding involving Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder or BMO spaces is covered by our approach.

The study of the lack of compactness in the critical embedding of Sobolev spaces supplies us with a large amount of information about solutions of nonlinear partial differential equations, both in the elliptic frame or the evolution frame. One has, for example,

• the pioneering works of P. -L. Lions [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF] and [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case. II[END_REF] for the sake of geometric problems,

• the description of bounded energy sequences of solutions to the defocusing semi-linear quintic wave equation, up to remainder terms small in energy norm in [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF],

• the characterization of the defect of compactness for Strichartz estimates for the Shrödinger equation in [START_REF] Keraani | On the defect of compactness for the Strichartz estimates of the Shrödinger Equation[END_REF],

• the understanding of features of solutions of nonlinear wave equations with exponential growth in [START_REF] Bahouri | On the lack of compactness in the 2D critical Sobolev embedding[END_REF],

• the sharp estimate of the time life-span of the focusing critical semi-linear wave equation by means of the size of energy of the Cauchy data in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF],

• the study of the bilinear Strichartz estimates for the wave equation in [START_REF] Tao | An inverse theorem for the bilinear L 2 Strichartz estimate for the wave equation[END_REF].

For further applications, we refer to [START_REF] Gallagher | Profile decomposition for the wave equation outside convex obstacles[END_REF], [START_REF] Gallagher | Profile decomposition for solutions of the Navier-Stokes equations[END_REF], [START_REF] Ibrahim | Comparaison des ondes linéaires et non linéaires à coefficients variables[END_REF], [START_REF] Majdoub | Qualitative study of the critical wave equation with a subcritical perturbation[END_REF], [START_REF] Laurent | On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold[END_REF] and the references therein.

Our results which cover a broad spectrum of spaces could be at the origin of several prospectus of similar types of regularity results for Navier-Stokes systems (as in [START_REF] Kenig | An alternative approach to the Navier-Stokes equations in critical spaces[END_REF][START_REF] Gallagher | A profile decomposition approach to the L ∞ t (L 3 x ) Navier-Stokes regularity criterion[END_REF]), qualitative study of non linear evolution equations or estimates of the span life of focusing semi-linear dispersive evolution equations.

Wavelet expansions

Wavelet decompositions of a function have the form

f = λ∈∇ d λ ψ λ , (1.4)
where λ = (j, k) concatenates the scale index j = j(λ) and space index k = k(λ): for d = 1, we have with the L 2 normalization,

ψ j,k = ψ λ = 2 j/2 ψ(2 j • -k), j ∈ Z, k ∈ Z,
where ψ is the so-called "mother wavelet". In higher dimension d > 1, one needs several generating functions ψ e for e ∈ E a finite set, so that setting ψ λ := (ψ e λ ) T e∈E and d λ = (d e λ ) e∈E , we can again write (1.4) with d λ ψ λ a finite dimensional inner product and

ψ j,k = ψ λ = 2 dj/2 ψ(2 j • -k), j ∈ Z, k ∈ Z d .
The index set ∇ in (1.4) is thus always defined as

∇ := Z × Z d .
Note that λ may also be identified to a dyadic cube

λ ∼ 2 -j (k + [0, 1] d ).
We shall sometimes use the notation

|λ| := j(λ),
for the scale level of λ. In all the sequel, we systematically normalize our wavelets in X which is equivalent to normalizing them in Y in view of (1.3):

ψ j,k = ψ λ = 2 rj ψ(2 j • -k). (1.5)
It is known that, in addition to be Schauder bases, wavelet bases are unconditional bases for "most" classical function spaces, including in particular the family of Besov and

c λ ψ λ X ≤ D λ∈E d λ ψ λ X . (1.6)
We refer to [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF][START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Meyer | Ondelettes et opérateurs[END_REF] for more details on the construction of wavelet bases and on the characterization of classical function spaces by expansions in such bases.

Main results

Our profile decomposition relies on two key assumptions concerning wavelet decompositions and the spaces X and Y .

In addition we always work under the general assumption that our wavelet basis (ψ λ ) λ∈Λ is an unconditional basis for both spaces X and Y . We therefore assume that (1.6) holds with some constant D for both norms.

Our first assumption involves the nonlinear projector that we define for each M > 0 as follows: if f ∈ X has the expansion in the wavelet basis given by (1.4), then

Q M f := λ∈E M d λ ψ λ , (1.7)
where

E M = E M (f ) is the subset of ∇ of cardinality M that corresponds to the M largest values of |d λ |.
Such a set always exists due to the fact that (ψ λ ) λ∈Λ is a Schauder basis for X, since this implies that for any η > 0 only finitely many coefficients d λ are larger than η in modulus. This set may however not be unique when some |d λ | are equal, in which case we may choose an arbitrary realization of such a set. Recall that we have assumed the normalization (1.5) making ψ λ X or ψ λ Y independent of λ, therefore E M also corresponds to the

M largest d λ ψ λ X or d λ ψ λ Y . Assumption 1: The nonlinear projection satisfies lim M →+∞ max f X ≤1 f -Q M f Y = 0.
(1.8)

The fact that the convergence of Q M f towards f in Y holds uniformly on the unit ball of X is tied to the nonlinear nature of the operator Q M : if instead we took Q M to be the projection onto a fixed M -dimensional space, then (1.8) would be in contradiction with the fact that the critical embedding of X into Y is not compact. As will be recalled further, nonlinear approximation theory actually allows for a more precise quantification of the above property in most cases of interest, through an estimate of the form max

f X ≤1 f -Q M f Y ≤ CM -s , M > 0
for some s > 0 and C only depending on the choice of X and Y . However, Assumption 1 alone will be sufficient for our purpose.

Our second assumption only concerns the behavior of wavelet expansions with respect to the X norm. It reflects the fact that this norm is stable with respect to certain operations such as "shifting" the indices of wavelet coefficients, as well as perturbating the value of these coefficients. This is expressed as follows.

Assumption 2: Consider a sequence of functions (f n ) n>0 which are uniformly bounded in X and may be written as

f n = λ∈∇ c λ,n ψ λ , (1.9)
and such that for all λ, the sequence c λ,n converges towards a finite limit c λ as n → +∞. Then, the series λ∈∇ c λ ψ λ converges in X with

λ∈∇ c λ ψ λ X ≤ C lim inf n→+∞ f n X , (1.10)
where C is a constant only depending on the space X and on the choice of the wavelet basis.

As will be recalled further, for practical choices of X such as Besov or Triebel-Lizorkin spaces, the X norm of a function is equivalent to the norm of its wavelet coefficients in a certain sequence space. This allows us to establish (1.10) essentially by invoking Fatou's lemma.

We are now in position to state the main theorem of this paper. For any function φ, not necessarily a wavelet, and any scale-space index λ = (j, k) we use the notation

φ λ := 2 rj φ(2 j • -k), (1.11)
for the version of φ scaled and translated according to λ.

Theorem 1.1 Assume that X and Y are two function spaces with the same scaling (1.3) and continuous embedding X ⊂ Y , and assume that there exists a wavelet basis (ψ λ ) λ∈∇ which is unconditional for both X and Y , and such that Assumptions 1 and 2 hold. Let (u n ) n>0 be a bounded sequence in X. Then, up to subsequence extraction, there exists a family of functions (φ l ) l>0 in X and sequences of scale-space indices (λ l (n)) n>0 for each l > 0 such that

u n = L l=1 φ l λ l (n) + r n,L , (1.12)
where

lim L→+∞ lim sup n→+∞ r n,L Y = 0.
The decomposition (1.12) is asymptotically orthogonal in the sense that for any k = l,

|j(λ k (n))-j(λ l (n))| → +∞ or |k(λ k (n))-2 j(λ k (n))-j(λ l (n)) k(λ l (n))| → +∞, as n → +∞. (1.13)
Moreover, we have the following for the specific case where X is a Besov or Triebel-Lizorkin space:

Theorem 1.2 The decomposition in Theorem 1.1 is stable in the sense that, for some τ = τ (X) we have

( φ l X ) l>0 ℓ τ ≤ CK. (1.14)
where C is a constant that only depends on X and on the choice of the wavelet basis and where K := sup n≥0 u n X .

Remark 1.3 For certain sequences (u n ) n>0 , it is possible that for any L > 0 the decomposition (1.12) only involves a finite number of profiles φ l for l = 1, • • • , L 0 , which means that φ l = 0 for l > L 0 . Inspection of our proof shows that the theorem remains valid in such a case, in the sense that

u n = L 0 l=1 φ l λ l (n) + r n , (1.15)
where lim n→+∞ r n Y = 0.

In particular, the sequence (u n ) n>0 is compact in Y if and only if φ l = 0 for all l > 0.

Remark 1.4 Inspection of our proof also shows that in Assumption 1, we may use for Q M a more general nonlinear projector than the one obtained by taking the M largest values of |d λ |. Generally speaking, we may consider a nonlinear projector Q M that has the general form (1.7), where the sets

E M = E M (f ) of cardinality M depend on f and satisfy E M (f ) ⊂ E M +1 (f ).
Such a generalization appears to be useful when treating certain types of embedding, see §3.

Layout

The effective construction of the decomposition is addressed in Section §2, together with the proof of Theorem 1.1.

In Section §3, we discuss examples of X and Y with critical embedding for which Assumptions 1 and 2 can be proved. This includes all previously treated cases, and many others such as the embedding of Sobolev, Besov and Triebel-Lizorkin spaces into Lebesgue, Lorentz, BMO and Hölder spaces, or into other Sobolev, Besov and Triebel-Lizorkin spaces.

Finally, in §4, we prove the stability Theorem 1.2 for both setting of Besov and Triebel-Lizorkin spaces.

2 Construction of the decomposition and proof of Theorem 1.1

In this section, we place ourselves under the assumptions of Theorem 1.1. Let (u n ) n>0 be a bounded sequence in the space X and define

K := sup n>0 u n X < +∞.
The decomposition construction and the proof of Theorem 1.1 proceed in several steps.

Step 1: rearrangements. We first introduce the wavelet decompositions

u n = λ∈∇ d λ,n ψ λ .
(2.1)

For each n > 0, we consider the non-increasing rearrangement (d m,n ) m>0 of (d λ,n ) λ∈∇ according to their moduli. We may therefore write

u n = m>0 d m,n ψ λ(m,n) . (2.2)
Using the nonlinear projector Q M defined by (1.7), we further split this expansion into

u n = M m=1 d m,n ψ λ(m,n) + R M u n , (2.3)
with

R M u n = u n -Q M u n .
Combining Assumption 1 with the boundedness of (u n ) n>0 in X, we infer that lim

M →+∞ sup n>0 R M u n Y = 0. (2.4)
Our next observation is that if (ψ λ ) λ∈∇ is an unconditional basis of X then the coefficients d m,n are uniformly bounded: indeed, (1.6) implies that the rank one projectors

P µ : f = λ∈Λ d λ ψ λ → P µ f := d µ ψ µ ,
satisfy the uniform bound

P µ X→X ≤ D, µ ∈ ∇.
Since we have assumed that our wavelets are normalized in X, for example according to ψ µ X = 1 for all µ ∈ ∇, we thus have sup

λ,n |d λ,n | = sup m,n |d m,n | ≤ DK.
Up to a diagonal subsequence extraction procedure in n, we may therefore assume that for all m > 0, the sequence (d m,n ) n>0 converges towards a finite limit that depends on m,

d m = lim n→+∞ d m,n .
Note that (|d m |) m>0 is a non-increasing sequence since all sequences (|d m,n |) m>0 are non-increasing. We may thus write

u n = M m=1 d m ψ λ(m,n) + t n,M ,
where

t n,M := M m=1 (d m,n -d m )ψ λ(m,n) + R M u n .
Step 2: construction of approximate profiles. We construct the profiles φ l as limit of sequences φ l,i obtained by the following algorithm. At the first iteration i = 1, we set

φ 1,1 = d 1 ψ, λ 1 (n) := λ(1, n), ϕ 1 (n) := n. (2.5)
Assume that after iteration i -1, we have constructed

L -1 functions (φ 1,i , • • • , φ L-1,i ) and scale- space index sequences (λ 1 (n), • • • , λ L-1 (n)) with L ≤ i, as well as an increasing sequence of positive integers ϕ i-1 (n) such that i-1 m=1 d m ψ λ(m,ϕ i-1 (n)) = L-1 l=1 φ l,i λ l (ϕ i-1 (n)) .
At iteration i we shall use the i-th component d i ψ λ(i,ϕ i-1 (n)) to either modify one of these functions or build a new one according to the following dichotomy. (i) First case: assume that we can extract ϕ i (n) from ϕ i-1 (n) such that for l = 1, • • • , L -1 at least one of the following holds:

lim n→+∞ |j(λ l (ϕ i (n))) -j(λ(i, ϕ i (n)))| = +∞, (2.6)
or lim

n→+∞ |k(λ(i, ϕ i (n))) -2 j(λ(i,ϕ i (n)))-j(λ l (ϕ i (n))) k(λ l (ϕ i (n)))| = +∞. (2.7)
In such a case, we create a new profile and scale-space index sequence by defining

φ L,i := d i ψ, λ L (n) := λ(i, n),
and we set φ l,i = φ l,i-

1 for l = 1, • • • , L -1.
(ii) Second case: assume that for some subsequence ϕ i (n) of ϕ i-1 (n) and some l ∈ {1, • • • , L-1} both (2.6) and (2.7) do not hold. Then it is easy checked that j(λ l (ϕ

i (n))) -j(λ(i, ϕ i (n))) and k(λ(i, ϕ i (n))) -2 j(λ(i,ϕ i (n)))-j(λ l (ϕ i (n))) k(λ l (ϕ i (n))
) only take a finite number of values as n varies.

Therefore, up to an additional subsequence extraction, we may assume that there exists numbers a and b such that for all n > 0,

j(λ(i, ϕ i (n))) -j(λ l (ϕ i (n))) = a, (2.8) and k(λ(i, ϕ i (n))) -2 j(λ(i,ϕ i (n)))-j(λ l (ϕ i (n))) k(λ l (ϕ i (n))) = b.
(2.9)

We then update the function φ l,i-1 according to

φ l,i = φ l,i-1 + d i 2 ar ψ(2 a • -b).
(2.10)

and φ l ′ ,i = φ l ′ ,i-1 for l ′ ∈ {1, • • • , L -1} and l ′ = l.
From this construction, and after extracting a diagonal subsequence which eventually coincides with a subsequence of ϕ i (n) for each i, we see that for each value of M there exists

L = L(M ) ≤ M such that M m=1 d m ψ λ(m,n) = L l=1 φ l,M λ l (n) .
More precisely, for each l = 1, • • • , L, we have

φ l,M λ l (n) = m∈E(l,M ) d m ψ λ(m,n) ,
where the sets E(l, M

) for i = 1, • • • , L constitute a disjoint partition of {1, • • • , M }. Note that E(l, M ) ⊂ E(l, M + 1) with #(E(l, M + 1)) ≤ #(E(l, M )) + 1.
Similarly, the number of profiles L(M ) grows at most by 1 as we move from M to M + 1. As explained in Remark 1.2, it is possible that L(M ) terminates at some maximal value L 0 . Finally, note that for any m, m ′ ∈ E l,M we have that

j(λ(m, n)) -j(λ(m ′ , n)) = a(m, m ′ ), (2.11)
and

k(λ(m, n)) -2 j(λ(m,n))-j(λ(m ′ ,n)) k(λ(m ′ , n)) = b(m, m ′ ), (2.12)
where a(m, m ′ ) and b(m, m ′ ) do not depend on n.

Step 3: construction of the exact profiles. We now want to define the functions φ l as the limits in X of φ l,M as M → +∞. For this purpose, we shall make use of Assumption 2, combined with the scaling property (1.3) of the X norm and the fact that (ψ λ ) λ∈∇ is an unconditional basis.

For some fixed l and M such that l ≤ L(M ), let us define the functions

g l,M := m∈E(l,M ) d m ψ λ(m) , f l,M,n := m∈E(l,M ) d m,n ψ λ(m) ,
with λ(m) := λ(m, 1). From the scaling property (1.3) and the properties (2.11) and (2.12), we find that

f l,M,n X = m∈E(l,M ) d m,n ψ λ(m,n) X .
Since m∈E(l,M ) d m,n ψ λ(m,n) is a part of the expansion of u n , we thus find that

f l,M,n X ≤ DK,
where D is the constant in (1.6) and K := sup n>0 u n X . Invoking Assumption 2, we therefore find that g l,M converges in X towards a limit g l as M → +∞. We finally notice that, by construction, the g l,M are rescaled versions of the φ l,M : there exists A > 0 and B ∈ IR d such that

φ l,M = 2 Ar g l,M (2 A • -B).
By (1.3), we therefore conclude that φ l,M converges in X towards a limit

φ l := 2 Ar g l (2 A • -B) as M → +∞.
Step 4: conclusion of the proof. For any given L > 0, we may write

u n = L l=1 φ l λ l (n) + r n,L ,
where, for any value of M such that L ≤ L(M ), the remainder r n,L may be decomposed into

L l=1 (φ l,M λ l (n) -φ l λ l (n) ) + L l=1 m∈E(l,M ) (d m,n -d m )ψ λ(m,n) + L(M ) l=L+1 m∈E(l,M ) d m,n ψ λ(m,n) + R M u n . (2.13)
Note that each of these terms depend on the chosen value of M but their sum r n,L is actually independent of M . We rewrite this decomposition as

r n,L = r 1 (n, L, M ) + r 2 (n, L, M ),
where r 1 and r 2 stand for the first and last two terms in (2.13), respectively. By construction, all values of m which appear in the third term of (2.13) are between L + 1 and M . Therefore the last two terms in (2.13) may be viewed as a partial sum of

R L u n = m>L d m,n ψ λ(m,n) .
Since we have assumed that (ψ λ ) λ∈∇ is an unconditional basis for Y , we may therefore write

r 2 (n, L, M ) Y ≤ D R L u n Y .
According to Assumption 1, which is expressed by (2.4), the right hand side converges to 0 as L → +∞ uniformly in n and therefore

lim L→+∞ sup n,M r 2 (n, L, M ) Y = 0.
We now consider the first two terms in(2.13). For the first term, we have

L l=1 (φ l,M λ l (n) -φ l λ l (n) ) X ≤ L l=1 φ l,M λ l (n) -φ l λ l (n) X = L l=1 φ l,M -φ l X .
Therefore, for any fixed L, this term goes to 0 in X as M → +∞. For the second term, we first notice that for any fixed L and M such that L ≤ L(M ), all values of m which appear in this term are less or equal to M . Since we have assumed that (ψ λ ) λ∈∇ is an unconditional basis for X, it follows that

L l=1 m∈E(l,M ) (d m,n -d m )ψ λ(m,n) X ≤ D M m=1 (d m,n -d m )ψ λ(m,n) X ≤ CD M m=1 |d m,n -d m |,
where C = ψ X = ψ λ X for all λ ∈ ∇. Therefore for any any fixed L and M such that L ≤ L(M ), this term goes to 0 in X as n → +∞. Combining these observations, we find that for any fixed L and any ε > 0, there exists M and n 0 such that for all n ≥ n 0 ,

r 1 (n, L, M ) X ≤ ε.
By continuous embedding, the same holds for r 1 (n, L, M ) Y . Since M was arbitrary in the decomposition (2.13) of r n,L , we obtain that lim

L→+∞ lim sup n→+∞ r n,L Y = 0,
which concludes the proof of the theorem.

Examples

Our main result applies to a large range of critical embedding. Specifically, we consider (i) For the space X: spaces of Besov type Ḃs p,a or Triebel-Lizorkin type Ḟ s p,a with 1 ≤ p < ∞ and 1 ≤ a ≤ ∞.

(ii) For the space Y : spaces of Besov type Ḃt q,b , Triebel-Lizorkin type Ḟ t q,b , Lebesgue type L q , Lorentz type L q,b , and the space BM O, with 1 ≤ q ≤ ∞ and 1 ≤ b ≤ ∞.

Note that Lebesgue spaces may be thought of as a particular case of Triebel-Lizorkin spaces since L q = Ḟ 0 q,2 , yet we treat them separetely since several results that we invoke further have been proved in an isolated manner for the specific case of Lebesgue spaces.

The critical embedding for such spaces imposes that t < s together with the scaling

r = d p -s = d q -t, (3.1)
where t = 0 if Y is of Lebesgue or Lorentz type, and t = 0 and q = ∞ if Y = BM O. It also imposes some relations between the fine tuning indices a and b. For example for s > 0 and p, q such that d p -s = d q , the space Ḃs p,a embeds continuously into L q,b if b ≥ a. Note that for non integer t > 0 the Hölder space Ċt coincides with the Besov space Ḃt ∞,∞ , and that for all integer m ≥ 0, the Sobolev space Ẇ m,p coincides with the Triebel-Lizorkin space Ḟ m p,2 when 1 < p < ∞. In particular L p = Ḟ 0 p,2 for 1 < p < ∞. For p = 1, it is known that Ḟ 0 1,2 coincides with the Hardy space H 1 which is a closed subspace of L 1 . We refer to [START_REF] Adams | Sobolev spaces. Pure and Applied Mathematics[END_REF] and [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF] for an introduction to all such spaces. It is known that properly constructed wavelet bases are unconditional for all such spaces, see in particular [START_REF] Meyer | Ondelettes et opérateurs[END_REF]. In addition, Besov and Triebel-Lizorkin spaces, as well as BM O, may be characterized by simple properties on wavelet coefficients. More precisely, for f = λ∈∇ d λ ψ λ and wavelets normalized according to (1.5) with r given by (3.1), we have the following norm equivalences (see [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF][START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Meyer | Ondelettes et opérateurs[END_REF]):

(i) For Besov spaces, f Ḃt q,b ∼ j∈Z ( |λ|=j |d λ | q ) b/q 1/b , (3.2)
with the standard modification when q = ∞ or b = ∞.

(ii) For Triebel-Lizorkin spaces,

f Ḟ t q,b ∼ λ∈∇ |d λ χ λ | b 1/b L q , ( 3.3)
where

χ λ = 2 dj/q χ (2 j • -k) with χ = χ [0,1] d for λ ∼ (j, k). When b = ∞, λ∈∇ |d λ χ λ | b 1/b should be replaced by sup j∈Z |λ|=j |d λ χ λ |. (iii) For BM O, f BM O ∼ max λ∈∇ 2 d|λ| µ⊂λ |d µ | 2 2 -d|µ| 1/2 , (3.4)
where by definition µ ⊂ λ means that 2

-j(µ) ([0, 1] d + k(µ)) ⊂ 2 -j(λ) ([0, 1] d + k(λ)).
Note that due to the discretization of the scale-space index d λ , the above equivalent norms do not exactly satisfy the scaling relation (1.3). These norm equivalences readily imply that (ψ λ ) λ∈Λ is an unconditional basis for such spaces. Note that there exists no simple wavelet characterization of Lorentz spaces L q,b when b = q. However the unconditionality of (ψ λ ) λ∈Λ in such spaces follows by interpolation of Lebesgue spaces for any 1 < b, q < ∞.

We now need to discuss the validity of Assumptions 1 and 2, for such choices of spaces. We first discuss Assumption 2 which is only concerned with the space X. Since we assumed here that X is of Besov type Ḃs p,a or Triebel-Lizorkin type Ḟ s p,a , we may use equivalent norms given by (3.2) and (3.3). Therefore, the X norm of

f n = λ∈∇ c λ,n ψ λ is either equivalent to j∈Z ( |λ|=j |c λ,n | p ) a/p 1/a , or λ∈∇ |c λ,n χ λ | a 1/a L p .
In both cases, we may invoke Fatou's lemma to conclude that for the limit sequence (c λ ), we have

j∈Z ( |λ|=j |c λ | p ) a/p 1/a ≤ lim inf n→+∞ j∈Z ( |λ|=j |c λ,n | p ) a/p 1/a ,
and

λ∈∇ |c λ χ λ | a 1/a L p ≤ lim inf n→+∞ λ∈∇ |c λ,n χ λ | a 1/a L p .
Therefore, Assumption 2 holds for all Besov and Triebel-Lizorkin spaces.

We next discuss Assumption 1, for some specific examples of pairs X and Y which satisfy the critical embedding property. The study of the nonlinear projector Q M is an important chapter of approximation theory. The process of approximating a function

f = λ∈∇ d λ ψ λ , by a function of the form λ∈E M c λ ψ λ ,
with #(E M ) ≤ M is sometimes called best M -term approximation, and has been studied extensively. The most natural choice is to take for E M the indices corresponding to the largest coefficients |d λ | and to set c λ = d λ , which corresponds to our definition of Q M . However as already mentioned in Remark 1.4, other more relevant choices could be used if necessary for proving the validity of Assumption 1 for certain pairs (X, Y ) and a specific instance will be mentioned below.

The study of the convergence of Q M f towards f is particularly elementary in the case where X = Ḃs p,p and Y = Ḃt q,q , with 1 p -1 q = s-t d . Indeed, according to (3.2), we have for such spaces

f Ḃs p,p ∼ (d λ ) λ∈∇ ℓ p and f Ḃt q,q ∼ (d λ ) λ∈∇ ℓ q ,
and therefore for any f ∈ X, using the decreasing rearrangement (d m ) m>0 of the |d λ |, we obtain

f -Q M f Ḃt q,q ∼ λ / ∈E M |d λ | q 1/q = m>M |d m | q 1/q ≤ |d M | 1-p/q m>M |d m | p 1/q ≤ M -1 M m=1 |d m | p 1/p-1/q m>M |d m | p 1/q ≤ M -(1/p-1/q) m>0 |d m | p 1/p ≤ M -s-t d (d λ ) λ∈∇ ℓ p ∼ M -s-t d f Ḃs p,p .
We have thus proved that sup

f Ḃs p,p ≤1 f -Q M f Ḃt q,q ≤ CM -σ , σ := s -t d > 0, (3.5)
which shows that Assumption 1 holds in such a case. For other choices of X and Y , the study of best M -term approximation is more involved and we just describe the available results without proof.

The case of the embedding of the Besov space X = Ḃs p,p into the Lebesgue space Y = L q , with q < ∞ and 1 p -1 q = s d > 0 has first been treated in [START_REF] Devore | Compression of wavelet decompositions[END_REF] -see also [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF] and [START_REF] Devore | Nonlinear approximation[END_REF] -where it was proved that sup

f Ḃs p,p ≤1 f -Q M f L q ≤ CM -σ , σ := s d > 0. (3.6)
Therefore Assumption 1 also holds in such a case. Note that when q ≤ 2, one has continuous embedding of Ḃ0 q,q in L q and therefore (3.6) may be viewed as a consequence of (3.5), however this is no more the case when 2 ≤ q < ∞, yet (3.6) still holds.

A finer result, that may be obtained by interpolation techniques, states that, with the same relations between p and q, the Besov space Ḃs p,q -which is strictly larger than Ḃs p,p is continuously embedded in L q , and one may therefore ask if Assumption 1 is still valid in such a case. A positive answer was given in [START_REF] Kyriasis | Nonlinear approximation and interpolation spaces[END_REF] for the more general embedding of X = Ḃs p,q into Y = Ḟ t q,b with 1 p -1 q = s-t d , where b ∈]0, ∞] is arbitrary: we have the convergence estimate sup

f Ḃs p,q ≤1 f -Q M f Ḟ t q,b ≤ CM -σ , σ := s -t d > 0, (3.7)
and therefore Assumption 1 is again valid. Note that L q = Ḟ 0 q,2 is a particular case. For Besov spaces, the critical embedding of X = Ḃs p,a into Y = Ḃt q,b with 1 p -1 q = s-t d is known to hold whenever a ≤ b (it is an immediate consequence of the norm equivalence (3.2)). The study of best M -term approximation in this context was done in [START_REF] Kyriasis | Nonlinear approximation and interpolation spaces[END_REF], where the following result was proved: there exists a nonlinear projector Q M of the form (1.7), such that when 1 a -1 b ≥ s-t d , one has sup

f Ḃs p,a ≤1 f -Q M f Ḃt q,b ≤ CM -σ , σ := s -t d > 0. ( 3.8)
The set E M (f ) used in the definition of Q M is however not generally based on picking the M largest |d λ |, which is not a problem for our purposes as already mentioned in Remark 1.4. Therefore, Assumption 1 is valid for such pairs. In this last example, the restriction 1 a -1 b ≥ s-t d is stronger than a ≤ b which is sufficient for the critical embedding. However, we may still obtain the validity of Assumption 1 when a < b by a general trick which we shall re-use further: introduce an auxiliary space Z with continuous embedding

X ⊂ Z ⊂ Y, (3.9)
such that Assumption 1 either holds for the embedding between X and Z, or between Z and Y , which immediately implies the validity of Assumption 1 between X and Y . In the present case we take Z = Ḃs p,a with t < s < s,

1 p - 1 p = s - s d .
The continuous embeddings (3.9) clearly hold. In addition, when s sufficiently close to t, we have that 1 a -1 b ≥ s-t d , so that Assumption 1 is valid for the pair (Z, Y ) according to (3.8), and thus also for (X, Y ). Remark 3.1 It is not difficult to check that Assumption 1 does not hold for the embedding of X = Ḃs p,a into Y = Ḃt q,a , and we also conjecture that the profile decomposition does not generally exist for such an embedding. As an example, consider a = ∞, and a sequence (u n ) n>0 obtained by piling up one wavelet at each scale j = 0, • • • , n at position k = 0:

u n = n j=0 2 rj ψ(2 j •).
All wavelets in u n contribute equally to the X and Y norm (which is equivalent to the supremum of the coefficients, equal to 1) and the extraction of profiles with asymptotically orthogonal scale-space localization seems impossible.

The above trick based on the intermediate space Z may be used to prove Assumption 1 for other types of critical embeddings:

• Embedding of Besov spaces into BM O:

Ḃs p,p ⊂ BM O, s = d p > 0,
which includes as a particular case the well known embedding Ḣd/2 ⊂ BM O, and may be easily proved from the wavelet characterization (3.2) and (3.4). Choosing Z = Ḃs p,p for any 0 < s < s and p such that s = d p , we clearly have the continuous embeddings (3.9). In addition, Assumption 1 is valid for the pair (X, Z) according to (3.5), and thus also for (X, Y ).

• Embedding of Besov spaces into Lorentz spaces:

Ḃs p,a ⊂ L q,b , 1 p - 1 q = s d > 0,
which is valid for any a ≤ b. If a < b, we may introduce for any 0 < s < s

Z = Ḃs p,b , 1 p - 1 q = s d > 0,
so that we have the continuous embeddings (3.9). In addition, we have already proved that Assumption 1 holds for the pair (X, Z). It therefore holds for the pair (X, Y ). One may easily check that Assumption 1 does not hold for the embedding of X = Ḃs p,a into Y = L q,a , and conjecture that the profile decomposition does not generally exist for such an embedding, by an argument analogous to the one in Remark 3.1.

• Embedding of Triebel-Lizorkin spaces into Triebel-Lizorkin or Besov spaces: for any a, b > 0, consider X = Ḟ s p,a and

Y = Ḟ t q,b with 1 p -1 q = s-t d . It is known -see [29] -that X is continuously embedded into Z = Ḃs p,p , s < s and 1 p - 1 p = s - s d .
If we assume t < s < s, we have the continuous embeddings (3.9). Moreover, we have already proved that Assumption 1 holds for the pair (Z, Y ). It therefore holds for the pair (X, Y ). The same type of reasoning allows to prove Assumption 1 for the embedding of X = Ḟ s p,a into Y = Ḃt q,b with b > p.

Remark 3.2 It is easily seen that if X and Y are a pair of spaces such that both Assumptions 1 and 2 hold for a certain wavelet basis (ψ λ ), then the corresponding vector fields spaces (X) d and (Y ) d also satisfy the same assumptions for the vector valued wavelet basis

ψ λ,i := ψ λ e i , λ ∈ ∇, i = 1, • • • , d,
where e i = (0, • • • , 0, 1, 0, • • • , 0) is the canonical basis vector, and the wavelet coefficients d λ are defined accordingly as vectors.

Stability of the decomposition

We finally want to show that the decomposition is stable in the sense that the sum of the φ l X raised to an appropriate power remains bounded. In our discussion, we distinguish between the cases where X is a Besov or Triebel-Lizorkin space. We first address the Besov case.

Theorem 4.1 Assume that X = Ḃs p,a with 1 ≤ p < ∞ and 1 ≤ a ≤ ∞. We then have

( φ l X ) l>0 ℓ τ ≤ CK, τ := max{p, a}. (4.1)
where C is a constant that only depends on X and on the choice of the wavelet basis and where K := sup n≥0 u n X .

Proof: Fix an arbitrary L > 0 and let M be such that L ≤ L(M ) as in Step 4 of §2. For l = 1, • • • , L, we recall the approximate profiles

φ l,M λ l (n) = m∈E(l,M ) d m ψ λ(m,n) ,
and we also define φ l,M,n :=

m∈E(l,M ) d m,n ψ λ(m,n) ,
which are disjoint part of the wavelet expansion of u n . More precisely, we have

u n = L l=1 φ l,M,n + m>M d m,n ψ λ(m,n) .
We next claim that if E 1 , • • • , E L are disjoints finite sets in ∇, then for any coefficient sequence (d λ ), one has

L l=1 λ∈E l d λ ψ λ τ X 1/τ ≤ C L l=1 λ∈E l d λ ψ λ X , (4.2)
where C is a constant that only depends on X and on the choice of the wavelet basis, and with the standard modification of the sum to the power 1/τ by a supremum in the left hand side when τ = ∞. Before proving this claim, we first show that it leads to the conclusion of the proof. Indeed, for l = 1, • • • , L, the functions φ l,M,n are linear combinations of wavelets with indices in disjoint finite sets E 1 , • • • , E L (that vary with n), and therefore according to (4.2), when τ < ∞,

L l=1 φ l,M,n τ X 1/τ ≤ C L l=1 φ l,M,n X .
Using the unconditionality inequality (1.6), we thus find that for all n > 0

L l=1 φ l,M,n τ X 1/τ ≤ CK,
up to a multiplication of the constant C by D. Since φ l,M λ l (n) -φ l,M,n X → 0 as n → ∞, it follows that for any ε > 0 we have

L l=1 φ l,M λ l (n) τ X 1/τ ≤ CK + ε,
for n sufficiently large. By the scaling invariance (1.3) we thus find that

L l=1 φ l,M τ X 1/τ ≤ CK.
Letting M go to +∞, we obtain the same inequality for the exact profiles

L l=1 φ l τ X 1/τ ≤ CK,
and we thus conclude that (4.1) holds, by letting L → +∞. The case τ = ∞ is treated in an exact similar way, replacing the sum to the power 1/τ by a supremum.

It remains to prove (4.2). We actually claim that this property holds with constant C = 1 if we take for • X the equivalent norm given by (3.2). This is obvious when p = a = τ since this equivalent norm is then simply the ℓ τ norm of the wavelet coefficients. When p = a, we distinguish between the cases p < a and p > a. We denote by

E j,l := {λ ∈ E l ; |λ| = j}, so that E l = ∪ j∈Z E l,j .
First consider the case τ = a > p. We then have, when a < ∞,

L l=1 λ∈E l d λ ψ λ a X = j∈Z Z L l=1 λ∈E l,j |d λ | p a/p ≥ j∈Z Z L l=1 λ∈E l,j |d λ | p a/p = L l=1 j∈Z Z λ∈E l,j |d λ | p a/p = L l=1 λ∈E l d λ ψ λ a X
, where for the inequality we have simply used the fact that a/p > 1. Therefore (4.2) holds. When a = ∞, we obtain the same result by writing Theorem 4.2 Assume that X = Ḟ s p,a with 1 ≤ p < ∞ and 1 ≤ a ≤ ∞. We then have

( φ l X ) l>0 ℓ p ≤ CK. (4.3)
where C is a constant that only depends on X and on the choice of the wavelet basis and where K := sup n≥0 u n X .

Proof: We only give the proof in the case a < ∞, the case a = ∞ being treated by the same type of arguments up to notational changes. Fix an arbitrary L > 0 and let M be such that L ≤ L(M ) as in Step 4 of §2. By the unconditionality (1.6) of the wavelet basis with respect to X, we first observe that

M m=1 d m,n ψ λ(m,n) X ≤ DK.
It follows that for any ε > 0, we have

M m=1 d m ψ λ(m,n) X ≤ DK + ε.
for n sufficiently large. Recall that the sum inside the norm may be rewritten in terms of the approximate profiles:

M m=1 d m ψ λ(m,n) = L l=1 m∈E(l,M ) d m ψ λ(m,n) = L l=1 φ l,M λ l (n) .
We associate to the approximate profile φ l,M a piecewise constant function χ l,M defined by

χ l,M λ l (n) := m∈E(l,M ) |d m χ λ(m,n) | a ,
where χ λ = 2 dj/p χ (2 j • -k) with χ = χ [0,1] d for λ ∼ (j, k). Thus according to the wavelet charac- terization (3.3) of Triebel-Lizorkin spaces, we have c

IR d | χ l,M λ l (n) (x)| p/a dx ≤ φ l,M λ l (n) p X ≤ C IR d | χ l,M λ l (n) (x)| p/a dx, as well as c IR d | L l=1 χ l,M λ l (n) (x)| p/a dx ≤ M m=1 d m,n ψ λ(m,n) p X ≤ C IR d | L l=1 χ l,M λ l (n) (x)| p/a dx,
where 0 < c ≤ C only depend on the choice of the wavelet basis. In the case where a ≤ p, we obviously have

L l=1 IR d | χ l,M λ l (n) (x)| p/a dx ≤ IR d | L l=1 χ l,M λ l (n) (x)| p/a dx.
It therefore follows that

L l=1 φ l,M p X = L l=1 φ l,M λ l (n) p X ≤ C c M m=1 d m,n ψ λ(m,n) p X ≤ C c (DK + ε) p .
Since this holds for any ε > 0, and L > 0 and M such that L ≤ L(M ), we therefore obtain (4.3) by a limiting argument, up to renaming (C/c) 1/p D into C.

In order to reach the same conclusion in the case p < a, we need to exploit the "asymptotic orthogonality" of the scales λ l (n) as expressed by (1.13) in the statement of Theorem (1.1). For this purpose, let us define

Ω n := Supp L l=1 χ l,M λ l (n) = ∪ L l=1 Supp( χ l,M λ l (n) ).
For any x ∈ Ω n , we denote by l * the number in {1, • • • , L} such that

χ l * ,M λ l * (n) (x) = max l=1,•••,L χ l,M λ l (n) (x).
Note that l * depends both of x and n. We claim that a consequence of (1.13) is that the function χ l * ,M λ l * (n) tends to dominate all other χ l,M λ l (n) at the point x as n → +∞ in the following uniform sense: We may then conclude the proof as in the case a ≤ p. It remains to prove (4.4). Our first observation is that the asymptotic orthogonality of the scales λ l (n) expressed by (1.13), shows that for a given x the profile scales |λ l (n)| for those l ∈ {1, • • • , L} such that x ∈ Supp( χ l,M λ l (n) ) tend to get far apart as n grows. Indeed, these λ l (n) do not get far apart in space since the supports of χ l,M λ l (n) all contain the same point x. We introduce l * the number that maximizes |λ l (n)| among all those l ∈ {1, • • • , L} such that x ∈ Supp( χ l,M λ l (n) ). Similar to l * , the number l * depends both of x and n. From the previous observation, we know that for any arbitrarily large B > 0, there exists n 0 such that for all n ≥ n 0 , we have

|λ l * (n)| ≥ |λ l (n)| + B, (4.5)
for all l ∈ {1, • • • , L} such that x ∈ Supp( χ l,M λ l (n) ) and l = l * . Moreover we may choose this n 0 independent of the selected point x for the same B > 0.

We claim that as n grows χ l * ,M λ l * (n) tends to dominate all other χ l,M λ l (n) at the point x as n → +∞, in the sense that Combining this observation with (4.5), we easily obtain (4.6). 2

Gabriel Koch Mathematical Institute 24-29 St Giles' Oxford, OX1 3LB, England e-mail: koch@maths.ox.ac.uk

2

 2 l d λ ψ λ X = sup j∈Z Z L l=1 λ∈E l,j |d λ | p 1/p ≥ sup j∈Z Z sup l≤L λ∈E l,j |d λ | p 1/p = sup l≤L sup j∈Z Z λ∈E l,j |d λ | p 1/p = sup l≤L λ∈E l d λ ψ λ X .We next consider the case τ = p > a. To treat this case where p < ∞ by hypothesis, we introduce the notation b j,l := λ∈E l,j |d λ | p a/p , and we remark that (4.2) is then equivalent to j,l | p/a a/p , which trivially holds by applying the triangle inequality in ℓ p/a .Our last result addresses the Triebel-Lizorkin case.

4 )

 4 λ l (n) (x) -χ l * ,M λ l * (n)Before proving this claim, let us show how it leads us to the conclusion of the theorem. We observe that (4.4) also means that | χ l * ,M λ l * (n) | p/a tends to dominate all other | χ l,M λ l (n) | p/a at the point x as n → +∞. Therefore, for any ε > 0, we have for n large enoughL l=1 | χ l,M λ l (n) (x)| p/a ≤ (1 + ε)| χ l * ,M λ l * (n) (x)| p/a ≤ (1 + ε)| L l=1 χ l,M λ l (n) (x)| p/a ,for all x ∈ Ω n , and thusL l=1 IR d | χ l,M λ l (n) (x)| p/a dx ≤ (1 + ε) IR d | L l=1χ l,M λ l (n) (x)| p/a dx.

6 )

 6 (n) (x) L l=1 χ l,M λ l (n) (x) -χ l * ,Mλ l *(n) This clearly implies (4.4) (and shows that l * = l * for n large enough).In order to prove (4.6), we observe that if x ∈ Supp( χ l,M λ l (n) ) for some l ∈ {1, • • • , L}, we maythen frame χ l,M λ l (n) (x) according to c2 adj l (n) p ≤ χ l,M λ l (n) (x) ≤ C2 adJ l (n) p where j l (n) := min m∈E(l,M ) |λ(m, n)| and J l (n) := max m∈E(l,M ) |λ(m, n)|,and wherec := |d M | a and C := M m=1 |d m | a .The constants c and C of course depend on L and M which are fixed at that stage. Note that from the construction of the profile there exists A > 0 (that also depends on L and M ) such that for alll ∈ {1, • • • , L} |λ l (n)| -A ≤ j l (n) ≤ J l (n) ≤ |λ l (n)| + A,and therefore, up to a modification in the constants c and C we may write c2 ad|λ l (n)| p ≤ χ l,M λ l (n) (x) ≤ C2 ad|λ l (n)| p .

  Triebel-Lizorkin spaces: for such spaces X there exists a constant D such that for any finite subset E ⊂ ∇ and coefficients vectors (c λ ) λ∈E and (d λ ) λ∈E such that |c λ | ≤ |d λ | for all λ, one has

	λ∈E

The third author was supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).


Habib Ammari

Hajer Bahouri

David Dos

Santos Ferreira

Isabelle Gallagher
email: isabelle.gallagher@math.jussieu.fr

STABILITY ESTIMATES FOR AN INVERSE SCATTERING PROBLEM AT HIGH FREQUENCIES

Keywords: . 2000 Mathematics Subject Classification. 35R30, 35B30. Key words and phrases. Inverse scattering problem, Lipschitz stability, resolution

We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that, in the case of a radial potential supported sufficiently near the boundary, infinite resolution can be achieved from measurements of the near-field operator in the monotone case.

Introduction

The first aim of this paper is to establish Lipschitz stability results for the inverse scattering problem of determining the low-frequency component (lower than the operating frequency) of the compactly supported potential from scattering or near-field measurements. It is known that, in general, the problem is exponentially unstable [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF][START_REF] Alessandrini | Examples of instability in inverse boundary-value problems[END_REF][START_REF] Alessandrini | Open issues of stability for the inverse conductivity problem[END_REF][START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF][START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF]. However, taking advantage of a priori information may improve stability and give accurate reconstruction algorithms [START_REF] Ammari | Direct reconstruction methods in ultrasound imaging of small anomalies[END_REF][START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF][START_REF] Ammari | The generalized polarization tensors for resolved imaging. Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements[END_REF]. The Lipschitz stability results proved in this paper together with the recent analysis of the local convergence of the nonlinear Landweber iteration in [START_REF] De Hoop | Local analysis of inverse problems: Hölder stability and iterative reconstruction[END_REF] show that the low-frequency component of the potential can be determined from the data in a linearly stable way. Moreover, they precisely quantify the resolution limit, which is defined as the characteristic size of the smallest oscillations in the potential that can be stably recovered from the data. Since Rayleigh's work, it has been admitted that the resolution limit in inverse scattering is of order π over the operating frequency [START_REF] Born | Principles of Optics[END_REF]. This is nothing else than a direct application of the uncertainty principle in inverse scattering [START_REF] Bertero | Resolution and super-resolution in inverse diffraction[END_REF][START_REF] Chen | Inverse scattering via Heisenberg's uncertaintly principle[END_REF][START_REF] Donoho | Uncertaintly principles and signal recovery[END_REF][START_REF] Slepian | Some comments on Fourier analysis, uncertainty and modeling[END_REF]. It is well-known that if the support of the potential is a point support, then the reconstructed location of the point potential from the scattering data has finite size of order of the Rayleigh resolution limit [START_REF] Ammari | An Introduction to Mathematics of Emerging Biomedical Imaging[END_REF][START_REF] Bertero | Resolution and super-resolution in inverse diffraction[END_REF]. Having this in mind, the results of this paper prove that the Fourier transform of the potential can be reconstructed in a linearly stable way for all frequencies (dual variable to the space one) smaller than the operating frequency, and therefore, justify the notion of resolution limit. More intriguingly, again in view of [START_REF] De Hoop | Local analysis of inverse problems: Hölder stability and iterative reconstruction[END_REF], they prove that the stability of the reconstruction of the potential increases at high operating frequencies.

The second aim of the paper is to show that infinite resolution can be achieved from near-field measurements. Here, the near-field operator approximates Sommerfeld's radiation condition and is equivalent to the measurements of the Cauchy data at a finite distance. Moreover, if the potential is supported near the boundary, then infinite resolution can be achieved in the monotone case. In fact, a Lipschitz stability result holds for both the low and high frequency components of the potential. It should be noted that the scattering amplitude can be recovered from the near-field operator. However, approximating the near-field operator from the scattering amplitude is a severely ill-posed problem [START_REF] Karp | A convergent 'farfield' expansion for two-dimensional radiation functions[END_REF][START_REF] Novikov | A multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0[END_REF][START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF][START_REF] Taylor | Estimates for approximate solutions to acoustic inverse scattering problems[END_REF]] and therefore cannot be of any practical and realistic use. It was shown in [START_REF] Karp | A convergent 'farfield' expansion for two-dimensional radiation functions[END_REF] that in order to compute the near-field operator from the scattering amplitude one needs to differentiate the scattering amplitude an infinite number of times.

The results of this paper extend to medium scattering the recent results in [START_REF] Ammari | Multistatic imaging of extended targets[END_REF][START_REF] Ammari | Resolution and stability analysis in full aperture, linearized conductivity and wave imaging[END_REF][START_REF] Zhao | Analysis of the response matrix for an extended target[END_REF], where a stability and resolution analysis was performed for linearized conductivity and wave imaging problems. They can be also used to justify the hopping (or continuation in the frequency) reconstruction algorithms proposed in [START_REF] Bao | Regularity and stability for the scattering map of a linearized inverse medium problem[END_REF][START_REF] Chen | Inverse scattering via Heisenberg's uncertaintly principle[END_REF][START_REF] Chen | On the inverse scattering problem for the Helmholtz equation in one dimension[END_REF][START_REF] Coifman | An improved operator expansion algorithm for direct and inverse scattering computations[END_REF].

In connection with our results in this paper, we also refer to the works by Isakov [START_REF] Isakov | Increased stability in the Cauchy problem for some elliptic equations[END_REF] and Isakov and Kindermann [START_REF] Isakov | Subspaces of stability in the Cauchy problem for the Helmholtz equation[END_REF], Bao, Lin, and Triki [START_REF] Bao | A multi-frequency inverse source problem[END_REF][START_REF] Bao | Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data[END_REF], Nagayasu, Uhlmann and Wang [START_REF] Nagayasu | Depth dependent stability estimates in electrical impedance tomography[END_REF][START_REF] Nagayasu | Increasing stability in an inverse problem for the acoustic equation[END_REF], as well as Derveaux, Papanicolaou, and Tsogka [START_REF] Derveaux | Resolution and denoising in near-field imaging[END_REF]. In [START_REF] Isakov | Increased stability in the Cauchy problem for some elliptic equations[END_REF][START_REF] Isakov | Subspaces of stability in the Cauchy problem for the Helmholtz equation[END_REF], an evidence of increasing stability in wave imaging when frequency is growing was given. In [START_REF] Bao | A multi-frequency inverse source problem[END_REF], stability estimates for the inverse source problem were established and the conversion of the logarithmic type stability to a Lispchitz one first proved. Numerical results to illustrate the stability of the source reconstruction problem were presented in [START_REF] Bao | Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data[END_REF]. In [START_REF] Bao | On the stability of an inverse problem for the wave equation[END_REF], Lipschitz stability estimates for the time-dependent wave equation were obtained. In [START_REF] Nagayasu | Depth dependent stability estimates in electrical impedance tomography[END_REF], a stability estimate for a linearized conductivity problem was derived and its dependence on the depth of the inclusion highlighted. In [START_REF] Nagayasu | Increasing stability in an inverse problem for the acoustic equation[END_REF], it is shown that the ill-posedness of the inverse acoustic problem decreases when the frequency increases and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz type for large frequencies. In [START_REF] Derveaux | Resolution and denoising in near-field imaging[END_REF], the enhancement of resolution in the near-field was studied and numerically illustrated. Our results in this paper confirm these important observations in a quite general situation and precisely quantify them.

Our paper is organized as follows. Section 1 is devoted to the stability of the reconstruction of the potential from the scattering amplitude (called also far-field pattern) in the high frequency regime. Theorem 2 proves that the low-frequency component of the potential can be determined in a stable way from the scattering amplitude. The threshold frequency determines the resolution limit. Section 2 extends the results of Section 1 to the nearfield measurements. Theorem 3 shows that the same results as those in Section 1 hold for reconstructing the potential from measurements of the near-field operator. In Section 3 we show that we gain infinite resolution for potentials supported near the boundary. If the potential is supported near the boundary, then infinite resolution can be achieved in the monotone case. Theorem 4 provides a Lipschitz stability result for both the low and high frequency components of the potential. Finally, in Appendix A, we provide useful results on Bessel's functions.

Finally, we mention that the letter C will be used to denote a universal constant which may vary from line to line. We also use A B to denote an estimate of the form A ≤ CB for some constant C. We also use the classical notation x = 1 + |x| 2 .

1. Far field pattern 1.1. Definitions and notations. Let Ω be a bounded domain in the Euclidean space R d of dimension d ≥ 2, let q ∈ C ∞ 0 (R d ) be a real-valued potential supported in Ω. We use the classical notation D = -i∂ for derivatives and consider the Helmholtz equation with potential

D 2 u -λ 2 u + qu = 0 (1.1) at frequency λ ∈ R * + := R + \ {0}.
Plane waves e iλx•ω propagating along the direction ω in S d-1 are solutions of the free Helmholtz equation

D 2 u -λ 2 u = 0. (1.2)
Here S d-1 denotes the unit sphere in R d . More generally, plane waves generate the set of solutions in the space of tempered distributions, S ′ (R d ), of the free Helmholtz equation: all solutions of (1.2) with polynomial growth are superpositions of elementary plane waves e iλx•ω when ω varies on the sphere S d-1 . It will be useful later on to adopt Melrose's notation in [START_REF] Melrose | Geometric scattering theory[END_REF] to designate these solutions: for g ∈ C ∞ (S d-1 ), we shall write

Φ 0 (λ)g(x) = S d-1 e iλx•ω g(ω) dω.
Obviously Φ 0 (λ)g is a solution of (1.2) which belongs to S ′ (R d ).

To guarantee the uniqueness of solutions to the Helmholtz equation (1.1), one can impose conditions on the behavior of solutions at infinity. More precisely, we are interested in solutions which can be decomposed

u = e iλx•ω + u scat = u in + u scat (1.3)
as the sum of an incoming planar wave and a scattered wave satisfying Sommerfeld's radiation condition

∂ ∂|x| -iλ u scat = o 1 |x| d-1 2
as |x| → +∞, (1.4) uniformly with respect to the direction θ = x |x| at fixed frequency λ ∈ R * + . The following result from [START_REF] Melrose | Geometric scattering theory[END_REF]Lemma 2.4] holds.

Proposition 1.1. There exists a unique solution to the Helmholtz equation (1.1) with potential q ∈ C ∞ 0 (Ω) of the form (1.5) ϕ q (x, ω, λ) = e iλx•ω + ϕ scat q (x, ω, λ), where the scattered wave ϕ scat q satisfies Sommerfeld's radiation condition (1.4) and which is given by

(1.6) ϕ scat q = -R q (λ)(e iλx•ω q).
Here R q denotes the meromorphic continuation of the perturbed resolvent. Furthermore ϕ q depends smoothly on (x, ω, λ) ∈ R d × S d-1 × R * + and is bounded.

We choose to denote Φ q (λ)g = S d-1 ϕ q (x, ω, λ)g(ω) dω the operator with kernel ϕ q (x, ω, λ) given by (1.5).

Theorem 1. The scattered wave in the solution (1.3) to the Helmholtz equation (1.1) given by Proposition 1.1 assumes the form

ϕ scat q (x) = e iλ|x| |x| d-1 2 a q x |x| , ω, λ + O 1 |x| d+1 2 as |x| → +∞, (1.7)
where a q is a smooth function on

S d-1 × S d-1 × R * + and a q x |x| , ω, λ = - 1 2iλ λ 2πi d-1 2
q(y)ϕ q (y, ω, λ)e -iλ x |x| •y dy.

Proof. We consider the Green function G λ corresponding to the free Helmholtz equation

(D 2 y -λ 2 )G λ (x, y) = δ(x -y), (1.8)
subject to Sommerfeld's radiation condition (1.4), with δ being the Dirac delta function. Let R > 0 be large enough so that the ball of radius R contains the support of q. By definition of the Green function, for all |x| ≤ R we have

u(x) = |y|≤R (D 2 y -λ 2 )G λ (x, y)u(y) dy,
and if u is a solution to the Helmholtz equation (1.1) we deduce by Green's formula that for any

|x| ≤ R u(x) = - |y|≤R G λ (x, y)q(y)u(y) dy - |y|=R ∂G λ ∂r (x, y) u(y) -G λ (x, y) ∂u ∂r (y) dσ(y).
Along the same lines, it is possible to derive a similar identity for the plane wave

u in (x) = e iλω•x u in (x) = - |y|=R ∂G λ ∂r (x, y) u in (y) -G λ (x, y) ∂u in (y) ∂r (y) dσ(y),
taking into account the fact that (D 2λ 2 )u in = 0. Subtracting the two identities gives the following representation formula for the scattered wave

u scat = u -u in u scat (x) = - |y|≤R G λ (x, y) q(y) u(y) dy - |y|=R ∂G λ ∂r (x, y) u scat (y) -G λ (x, y) ∂u scat ∂r (y) dσ(y).
The Green function of the free Helmholtz equation is explicitly given by

G λ (x, y) = 1 4i λ 2π d-2 2 |x -y| -d-2 2 H (1) d/2-1 λ|x -y| , (1.9)
where H

G λ (x, y) = 1 2iλ λ 2πi d-1 2 e iλ|x-y| |x -y| -d-1 2 1 + O 1 λ|x -y| . (1.10) Since we have |x -y| = |y| - x • y |y| + O |x| 2 |y|
we find for |y| large enough and fixed λ:

e iλ|x-y| = e iλ |y| e -iλ y |y| •x 1 + O |x| 2 |y| .
We therefore obtain for fixed x, λ and large R = |y|

G λ (x, y) = 1 2iλ λ 2πi d-1 2 |y| -d-1 2 e iλ |y| e -iλ y |y| •x 1 + O 1 R . (1.11)
Analogously, we have

∂G λ ∂r (x, y) = iλc d (λ)|y| -d-1 2 e iλ|y| e -iλ y |y| •x 1 + O 1 R , with c d (λ) = 1 2iλ (λ/2πi) d-1
2 . This leads to

u scat (x) = - |y|≤R G λ (x, y)q(y)u(y) dy + c d (λ) R d-1 2 |y|=R ∂u scat ∂r -iλu scat e iλR-iλ y |y| •x 1 + O 1 R dσ(y).
Sommerfeld's radiation condition implies that the second right-hand side term tends to zero when R tends to infinity, so we get

u scat (x) = -G λ (x, y)q(y)u(y) dy .
Using once again the asymptotic formula (1.11) together with the fact that the Green function is symmetric we get

u scat (x) = - 1 2iλ λ 2πi d-1 2 e iλ|x| |x| d-1 2 q(y)u(y)e -iλ x |x| •y 1 + O |y| 2 |x| dy.
To summarize, we have that

u scat (x) = e iλ|x| |x| d-1 2 a q x |x| , ω, λ + O 1 |x| d+1 2 , with a q x |x| , ω, λ = - 1 2iλ λ 2πi d-1 2 q(y)u(y)e -iλ x |x| •y dy.
This proves that the scattered part of any solution of Helmholtz' equation subject to Sommerfeld's radiation condition takes the form announced in Theorem 1.

Definition 1.2. We define the scattering amplitude associated with the po-

tential q ∈ C ∞ 0 (R d ) by the smooth function a q : S d-1 × S d-1 × R * + → C given by a q (θ, ω, λ) = - 1 2iλ λ 2πi d-1 2
q(y)ϕ q (y, ω, λ)e -iλθ•y dy. (1.12)

We denote

A q (λ)g(θ) = S d-1 a q (θ, ω, λ)g(ω) dω, θ ∈ S d-1 ,
the corresponding operator with kernel a q (θ, ω, λ).

It is easy to get an asymptotic expansion of

Φ q (λ)g(x) = S d-1 e iλx•ω + ϕ scat q (x, ω, λ) g(ω) dω
as |x| → +∞ using the stationary phase and Theorem 1:

(1.13) Φ q (λ)g = 2π λ|x| d-1 2
e -iλ|x| e i(d-1) π 4 g( -θ)

+ e iλ|x| e -i(d-1) π 4 g(θ) + λ 2π d-1 2 A q (λ)g(θ) + O 1 |x| with θ = x/|x| ∈ S d-1
. The operator which maps the coefficient of e -iλ|x| into the coefficient of e iλ|x| is given by

g(-θ) → i -d+1 g(θ) + λi 2π d-1 2 A q (λ)g(θ) .
This is, after renormalization and composition with the antipodal map, the scattering matrix [START_REF] Melrose | Geometric scattering theory[END_REF].

Definition 1.3. The scattering matrix is the operator given by

S q (λ) = Id + λi 2π d-1 2 A q (λ).
Integration by parts allows to relate the values of the potential inside the domain with the scattering matrix S q (λ).

Lemma 1.4. We have the following identities

(q 1 -q 2 )u 1 u 2 dx = -2iλ 2π λ d-1 S d-1 g 1 g 2 -S q 1 (λ)g 1 S q 2 (λ)g 2 dω, (q 1 -q 2 )u 1 u 2 dx = -2iλ 2π λi d-1 S d-1 ǧ2 S q 1 (λ)g 1 -ǧ1 S q 2 (λ)g 2 dω,
for any pair of solutions 1

u 1 = Φ q 1 (λ)g 1 , u 2 = Φ q 2 (λ)g 2 ,
to the Helmholtz equations (1.1) related to the potentials q 1 , q 2 .

Proof. Let R be large enough so that the ball of radius R contains the support of both potentials q 1 , q 2 . By Green's formula, we have

|x|≤R (q 1 -q 2 )u 1 u 2 dx = |x|=R ∂ r u 1 u 2 -u 1 ∂ r u 2 dσ(x)
and using the asymptotic formula (1.13) on the functions u 1 = Φ q 1 (λ)g 1 and u 2 = Φ q 2 (λ)g 2 , we deduce

∂ r u 1 u 2 -u 1 ∂ r u 2 = -2iλ 2π λR d-1 g 1 (-θ) g 2 (-θ) -S q 1 (λ)g 1 (θ) S q 2 (λ)g 2 (θ) + O 1 R d , which implies |x|≤R (q 1 -q 2 )u 1 u 2 dx = -2iλ 2π λ d-1 S d-1 g 1 g 2 -S q 1 g 1 S q 2 g 2 dθ + O 1 R .
Letting R tend to infinity provides the first identity.

The proof of the second identity is similar, since

∂ r u 1 u 2 -u 1 ∂ r u 2 = -2iλ 2π λR d-1 g 1 (-θ) S q 2 g 2 (θ) -S q 1 (λ)g 1 (θ) g 2 (-θ) + O 1 R d .
This completes the proof of the lemma.

Choosing q 1 = q 2 = q in Lemma 1.4, we obtain the following properties of the scattering matrix:

t S q (λ) = P • S q (λ) • P, S q (λ) * S q (λ) = Id recalling that P g = ǧ is the antipodal map. Here, t denotes the transpose and * the transpose conjugate. Besides, using the first relation together with Lemma 1.4, we finally get

(q 1 -q 2 )u 1 u 2 dx = 2iλ 2π λi d-1 2 S d-1 ǧ2 A q 1 (λ) -A q 2 (λ) g 1 dω, (1.14)
for any pair of solutions

u 1 = Φ q 1 (λ)g 1 , u 2 = Φ q 2 (λ)g 2 ,
to the Helmholtz equations (1.1) with Sommerfeld's radiation condition (1.4) related to the potentials q 1 , q 2 . 1.2. Stability estimates at high frequencies. The first stability result we shall prove in this paper states that low frequencies (i.e. smaller than 2λ) may be recovered in a stable way from the scattering amplitude.

Theorem 2. For all ε, M, R > 0 and all α > d there exist C ε , λ 0 > 0 such that the following stability estimate holds true. Let q 1 , q 2 ∈ C ∞ 0 (R d ) be two potentials supported in the ball centered at 0 and of radius R such that q 1 L ∞ , q 2 L ∞ ≤ M . Then for all λ ≥ λ 0 ,

(1.15) |ξ|≤(2-ε)λ ξ -α ( q 1 -q 2 )(ξ) 2 dξ ≤ C ε λ 3 a q 1 -a q 2 2 L 2 + C ε λ -2 q 1 -q 2 2 L ∞ .
The proof of Theorem 2 relies on estimates on solutions to the Helmoltz equations stated below, as well as on the following lemma, stating that one can relate the Fourier transform of the potential and the scattering amplitude.

Lemma 1.5. For all M, R > 0 there exist constants C, λ 0 > 0 such that for all potentials q ∈ C ∞ 0 (R d ) supported in the ball B(0, R) and satisfying q L ∞ ≤ M, we have the following approximation:

a q (θ, ω, λ) + 1 2iλ λ 2πi d-1 2 q λ(θ -ω) ≤ C λ d-5 2 q 2 L 2 (1.16)
for all λ ≥ λ 0 .

Lemma 1.5 is based on the following classical estimate on the free resolvent R 0 (λ). We refer to [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF] for an exposition by Burq of an elementary proof due to Zworski. This type of a priori estimates have a long history and go back to the work of Agmon [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF] on weighted estimates on the resolvent and the limiting absorption principle.

Proposition 1.6. Let χ ∈ C ∞ 0 (R d )
, there exists a constant C > 0 such that for all λ > 1 we have

χR 0 (λ)χ L(L 2 (R d )) ≤ Cλ -1 .
Proof of Lemma 1.5. We start with the expression defining the scattering amplitude (see Definition 1.2):

a q (θ, ω, λ) = - 1 2iλ λ 2πi d-1 2
q(y)ϕ q (y, ω, λ)e -iλθ•y dy so by (1.5) we deduce that

a q (θ, ω, λ) + 1 2iλ λ 2πi d-1 2 q λ(θ -ω) = - 1 2iλ λ 2πi d-1 2 q(x)ϕ scat q (x, ω, λ) e -iλx•θ dx.
The resolvent identity reads (see for instance Formula (2.3) in [START_REF] Melrose | Geometric scattering theory[END_REF])

(1.17) R q (λ) = R 0 (λ) -R 0 (λ) q R q (λ) which implies that if χ ∈ C ∞ 0 (R d
) is a cutoff function which equals 1 on the ball B(0, R), then χR q (λ)χ = χR 0 (λ)χ -χR 0 (λ)χ q χR q (λ)χ .

If we apply this identity to -qe iλx•ω , we get in view of (1.6)

χϕ scat q = -χR 0 (λ)χ qe iλx•ω ) + χR 0 (λ)χ qχϕ scat q
and using the estimate of Proposition 1.6, we obtain

χϕ scat q L 2 ≤ Cλ -1 q L 2 + Cλ -1 q L ∞ χϕ scat q L 2 .
Taking λ ≥ 2CM we deduce

χϕ scat q L 2 ≤ 2Cλ -1 q L 2 .
Using Cauchy-Schwarz's inequality, we get

q ϕ scat q e -iλx•ω dx ≤ 2Cλ -1 q 2 L 2
and this completes the proof of the lemma.

Note that the main ingredient in the proof of Lemma 1.5 consists in combining the estimate for the non-perturbed resolvent (Proposition 1.6) with the resolvent identity (1.17), along with the fact that q is compactly supported.

This lemma is the basis of the reconstruction of the potential from the scattering amplitude at high frequencies, since one can choose θ, ω ∈ S d-1 such that λ(θω) = ξ for any fixed frequency ξ with |ξ| ≤ 2λ and let λ tend to infinity. We use a similar approach to prove a stability estimate at high frequencies.

Proof of Theorem 2. Again, we start with the expression defining the scattering amplitude and then express the potential in terms of the scattering amplitude: for j ∈ {1, 2} we write

q j λ(θ -ω) = -2i(2πi) d-1 2 λ -d-3
2 a q j (θ, ω, λ)q j (x)ϕ scat q j (x, ω, λ) e -iλx•θ dx.

Taking the difference of the two expressions yields

(1.18) ( q 1 -q 2 ) λ(θ -ω) ≤ 2(2π) d-1 2 λ -d-3 2 (a q 1 -a q 2 )(θ, ω, λ) + q 1 -q 2 L 2 ϕ scat q 1 L 2 (|x|≤R) + q 2 L 2 ϕ scat q 1 -ϕ scat q 2 L 2 (|x|≤R)
where we recall that the supports of q 1 , q 2 are contained in the ball of center 0 and radius R > 0. As in the proof of Lemma 1.5 we have

ϕ scat q 1 L 2 (|x|≤R) ≤ Cλ -1 q 1 L 2 . (1.19)
In light of (1.6), the difference of the two scattered waves satisfies

ϕ scat q 1 -ϕ scat q 2 = -R q 1 (λ) -R q 2 (λ) (q 1 e iλx•ω -R q 2 (λ) (q 1 -q 2 )e iλx•ω
and by the resolvent identity (1.17), the first term on the right-hand side reads

R q 2 (λ)(q 1 -q 2 )R q 1 (λ)(q 1 e iλx•ω = -R q 2 (λ)(q 1 -q 2 ) ϕ scat q 1 . Let χ ∈ C ∞ 0 (R d
) be a cutoff function which equals 1 on the ball B(0, R), then χ(ϕ scat q 1ϕ scat q 2 ) = -χR q 2 (λ)χ (q 1q 2 )ϕ scat q 1

-χR q 2 (λ)χ (q 1q 2 )e iλx•ω .

Therefore along the same lines as in the proof of Lemma 1.5, the resolvent identity (1.17) and Proposition 1.6 give rise to

(1.20) ϕ scat q 1 -ϕ scat q 2 L 2 (|x|≤R) λ -1 q 1 -q 2 L ∞ ϕ scat q 1 L 2 (|x|≤R) + λ -1 q 1 -q 2 L 2 .
Indeed invoking the resolvent identity (1.17), we infer that χR q 2 (λ)χ = χR 0 (λ)χ -χR 0 (λ)χ q 2 χR q 2 (λ)χ .

Thus applying this identity to (q 1q 2 )ϕ scat q 1 , we obtain

χR q 2 (λ)χ (q 1 -q 2 )ϕ scat q 1 L 2 ≤ Cλ -1 q 1 -q 2 L ∞ ϕ scat q 1 L 2 (|x|≤R) + Cλ -1 q 2 L ∞ χR q 2 (λ)χ (q 1 -q 2 )ϕ scat q 1 L 2 , which implies for λ > 2CM χR q 2 (λ)χ (q 1 -q 2 )ϕ scat q 1 L 2 ≤ 2Cλ -1 q 1 -q 2 L ∞ ϕ scat q 1 L 2 (|x|≤R) .
We have also

χR q 2 (λ)χ (q 1 -q 2 )e iλx•ω L 2 λ -1 q 1 -q 2 L 2 ,
which achieves the proof of Estimate (1.20).

Taking into account those bounds, we obtain in view of (1.18) the estimate

(1.21) ( q 1 -q 2 ) λ(θ -ω) λ -d-3 2 (a q 1 -a q 2 )(θ, ω, λ) + λ -1 q 1 -q 2 L ∞ . We denote r(ξ) = 1 -|ξ| 2 when |ξ| ≤ 1. For ξ ∈ R d with |ξ| ≤ 2λ, choose η ∈ ξ ⊥ with norm |η| = r(ξ/2λ). The vectors θ = η + ξ 2λ , ω = η - ξ 2λ ,
have length one, taking the square of (1.21) and integrating the resulting estimate with respect to η yields

( q 1 -q 2 )(ξ) 2 λ -2 q 1 -q 2 2 L ∞ + λ -d+3 r(ξ/2λ) -d+2 |η|=r(ξ/2λ) η⊥ξ a q 1 -a q 2 η + ξ 2λ , η - ξ 2λ , λ 2 dη.
When |ξ| ≤ (2ε)λ we have r(ξ/2λ) ≥ ε(4ε)/2 therefore multiplying the previous inequality by ξ -α ≤ 1 and integrating with respect to ξ we get

|ξ|≤(2-ε)λ ξ -α ( q 1 -q 2 )(ξ) 2 dξ ≤ C ε λ -2 q 1 -q 2 2 L ∞ + C ε λ 3 |ξ|≤1 |η|=r(ξ) η⊥ξ (a q 1 -a q 2 ) η + ξ, η -ξ, λ 2 dη dξ.
We consider the following 2d -2 dimensional submanifold of S 2d-1 Σ = (ξ, η) ∈ S 2d-1 : ξ, η = 0 and the following diffeomorphism

ϕ : Σ → S d-1 × S d-1 (ξ, η) → (ξ + η, η -ξ), with inverse ϕ -1 : S d-1 × S d-1 → Σ (θ, ω) → θ -ω 2 , θ + ω 2
for which we have

Σ F (η + ξ, η -ξ) dη ∧ dξ = S d-1 ×S d-1 F (θ, ω) ϕ -1 * (dη ∧ dξ) = 2 -d S d-1 ×S d-1 F (θ, ω) dω ∧ dθ.
Then we finally get

|ξ|≤(2-ε)λ ξ -α ( q 1 -q 2 )(ξ) 2 dξ ≤ C ε λ 3 S d-1 ×S d-1 a q 1 θ, ω, λ -a q 2 θ, ω, λ 2 dθ dω + C ε λ -2 q 1 -q 2 2 L ∞
and this completes the proof of our estimate.

In particular, we recover the uniqueness of the potential from the scattering amplitude at high frequencies.

Corollary 1.7. If for all (θ, ω, λ) belonging to S d-1 × S d-1 × R * + , we have a q 1 (θ, ω, λ) = a q 2 (θ, ω, λ) then q 1 = q 2 .

Near field pattern

(D 2 -λ 2 + q)u = 0 in Ω, (∂ ν -iλ)u| ∂Ω = f ∈ L 2 (∂Ω). (2.1)
This problem has a unique solution u ∈ H 1 (Ω) for all f ∈ L 2 (∂Ω). Writing a variational formulation of (2.1) and using a unique continuation argument shows the uniqueness of a solution to (2.1). The existence follows from Fredholm's alternative [START_REF] Melenk | On generalized finite element methods[END_REF].

Remark 2.1. Other classical boundary conditions are either Dirichlet or Neumann boundary conditions. However, one has to make the additional assumption that λ 2 is not a Dirichlet (or Neumann) eigenvalue of D 2 + q, for the Dirichlet problem corresponding to (2.1) to have a unique solution for all f ∈ H 1 2 (∂Ω). Unfortunately, this does not make sense if one wants to take the high frequency limit λ → ∞. To bypass this difficulty, we study the boundary problem (2.1). Indeed, this condition is natural. It approximates Sommerfeld's radiation condition at high frequencies [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Keller | Exact nonreflecting boundary conditions[END_REF].

Remark 2.2. It follows from [START_REF] Melenk | On generalized finite element methods[END_REF] that the unique solution u ∈ H 1 (Ω) of (2.1) satisfies

Du L 2 (Ω) + λ u L 2 (Ω) ≤ C qu L 2 (Ω) + f L 2 (∂Ω)
for some constant C independent of λ. Therefore, as λ → ∞, we have

Du L 2 (Ω) + λ 2 u L 2 (Ω) ≤ C f L 2 (∂Ω) .
Definition 2.3. The near field pattern is the map

Nq(λ) : L 2 (∂Ω) → L 2 (∂Ω) f → u| ∂Ω .
The typical inverse problem on the near field pattern is whether it uniquely determines the potential q. This was solved (for smooth potentials) by Sylvester and Uhlmann [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF] in dimension d ≥ 3 in the case of the Dirichletto-Neumann map. Reconstruction methods were proposed by Nachmann in [START_REF] Nachman | Reconstruction from boundary measurements[END_REF] and stability issues were studied by Alessandrini [START_REF] Alessandrini | Open issues of stability for the inverse conductivity problem[END_REF]. It was shown by Mandache in [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation[END_REF] that the logarithmic stability result of Alessandrini in [START_REF] Alessandrini | Open issues of stability for the inverse conductivity problem[END_REF] is optimal.

Stability estimates at high frequencies.

The following result is the counterpart of Theorem 2 in the near-field context. Theorem 3. For all M, R > 0 and all α > d there exist C, λ 0 > 0 such that the following stability estimate holds true. Let q 1 , q 2 ∈ C ∞ 0 (R d ) be two potentials supported in the ball centered at 0 and of radius R such that q 1 L ∞ , q 2 L ∞ ≤ M . Then for all λ ≥ λ 0 ,

(2.2) |ξ|≤2λ ξ -α q 1 -q 2 (ξ) 2 dξ ≤ C Nq 1 -Nq 2 2 + Cλ -2 q 1 -q 2 2 L ∞ .
Proof. We start with two solutions u 1 , u 2 of the equations (D 2λ 2 + q j )u j = 0 and computing

Ω (q 1 -q 2 )u 1 u 2 dx = Ω ∆u 1 u 2 dx - Ω u 1 ∆u 2 dx = ∂Ω (∂ ν -iλ)u 1 u 2 dσ - ∂Ω u 1 (∂ ν -iλ)u 2 dσ yields the formula Ω (q 1 -q 2 )u 1 u 2 dx = ∂Ω u 2 Nq 1 (λ)u 1 -u 1 Nq 2 (λ)u 2 dσ.
Choosing q 1 = q 2 shows that the application Nq j is symmetric, and using this additional information, we get the formula

Ω (q 1 -q 2 )u 1 u 2 dx = ∂Ω u 2 Nq 1 (λ) -Nq 2 (λ) u 1 dσ.
Let us write q = q 1q 2 and choose

u 1 = ϕ q 1 = e iλx•θ 1 + ϕ scat q 1
and u 2 = ϕ q 2 = e iλx•θ 2 + ϕ scat q 2 , with θ 1 , θ 2 ∈ S d-1 . Recall from the proof of Lemma 1.5 that the scattered waves satisfy the estimate

ϕ scat q 1 L 2 (B(0,R)) + ϕ scat q 2 L 2 (B(0,R)) ≤ Cλ -1 . It follows that Ω e iλx•(θ 1 +θ 2 ) q(x) dx = ∂Ω Nq 1 (λ) -Nq 2 (λ) u 1 u 2 dσ - Ω q(x)(ϕ scat q 1 + ϕ scat q 2 + ϕ scat q 1 ϕ scat q 2 ) dx
and therefore

q λ(θ 1 + θ 2 ) ≤ Nq 1 (λ) -Nq 2 (λ) u 1 L 2 (∂Ω) u 2 L 2 (∂Ω) + C λ q L ∞ ≤ C Nq 1 (λ) -Nq 2 (λ) + C λ q L ∞ for all (θ 1 , θ 2 ) ∈ S d-1 × S d-1 .
We notice that

S d-1 × S d-1 → B(0, 2) (θ 1 , θ 2 ) → θ 1 + θ 2
is a submersion when θ 1 , θ 2 are not colinear. This implies that

q(ξ) ≤ C Nq 1 (λ) -Nq 2 (λ) + C λ q L ∞ , ξ ∈ B(0, 2λ). (2.3)
Multiplying this estimate by ξ -α /2, taking the square and integrating on the ball B(0, 2λ) completes the proof of the theorem.

3. The case of a potential located near the boundary 3.1. Definitions and notations. In this section we show, in the model case of the unit disk, that if the potential is supported close to the boundary of the disk, then a larger range of frequencies may be recovered by the near field Nq(λ) than in the general case treated in the previous two sections. More precisely, introducing radial coordinates (x 1 , x 2 ) = (r cos θ, r sin θ), with (r, θ) ∈ R + × [0, 2π[, we consider the following model problem in two space dimensions:

(3.1) (D 2 -λ 2 + q λ )u n = 0 in B = x ∈ R 2 , |x| ≤ 1 , ∂ r -iλ u n|∂B = e inθ .
We suppose that q λ is a smooth, radial function, with support included in D κ λ for some fixed constant κ > 0, where (3.2)

D κ λ = r ∈ [0, 1], 1 -κλ -1 < r < 1 .
In the following for simplicity we shall drop the index λ in the notation of the potential.

The following result shows that in the monotone case, one can improve on the frequency band recovered in the general case (see Theorems 2 and 3). We recall that Nq(λ) denotes the operator norm of Nq(λ) in L(L 2 ).

Theorem 4. Let q 1 and q 2 be two smooth radial potentials supported on D κ λ as defined in (3.2) and such that q 1 ≥ q 2 . There are positive constants λ 0 and C such that the following holds. Let λ → K(λ) be any function such that λ ≤ K(λ). Then the following stability estimate is valid for all λ ≥ λ 0 :

|ξ|≤K(λ) ( q 1 -q 2 )(ξ) 2 dξ ≤ CK 2 (λ) λ 4 Nq 1 (λ) -Nq 2 (λ) 2 + C 2 q 1 ,q 2 λ 4 q 1 -q 2 2 L ∞ , where C q 1 ,q 2 = max q 1 L ∞ , q 2 L ∞ , q 1 L ∞ q 2 L ∞ .
Remark 3.1. One has trivially that |ξ|≤K(λ)

( q 1 -q 2 )(ξ) 2 dξ ≤ CK 2 (λ) q 1 -q 2 2 L ∞ ≤ CK 2 (λ) λ 2 q 1 -q 2 2
L ∞ , so the estimate provided in Theorem 4 is of a different nature.

Remark 3.2. The proof of Theorem 4 is presented in Section 3.2, as an immediate consequence of Lemma 3.3 proved in Section 3.3. That lemma relates the Laplace transform of a function to the near field operator. It holds in much more generality than Theorem 4, without the additional assumption that q 1q 2 ≥ 0. However we are unable to relate the Fourier transform of a function to its Laplace transform in general (this fact is well-known to be difficult and in general very unstable); for nonnegative functions however the relation is very easy and enables us to conclude.

3.2.

Proof of Theorem 4. We start by stating a lemma, proved in Section 3.3, which relates the Laplace transform of a function to the near field operator. It is stated in the framework of general, non radial functions. We define the operator T : g → T g by

T g(r) = g(1 -r)
as well as the Laplace transform L: L(g)(t) = R g(s) e -s t ds.

For any function θ → f (θ) we call c k (f ) its Fourier transform, for k ∈ Z:

c k (f )(r) = 2π 0 f (r, θ) e -i k θ dθ.
Lemma 3.3. Let q 1 and q 2 be two functions supported on D κ λ . There are three positive constants λ 0 , K and C such that if t ≥ 2Kλ > 2Kλ 0 , then

L c k T (r(q 1 -q 2 ) (t) ≤ C λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q 1 -q 2 L ∞ where C q 1 ,q 2 = max q 1 L ∞ , q 2 L ∞ , q 1 L ∞ q 2 L ∞ .
Proof of Theorem 4. Let q = q 1 -q 2 . Since q is radial, we have q(x) = Q(|x|) and q(ξ) = Q(|ξ|) with for all ρ > 0,

Q(ρ) = 2π 0 ∞ 0 e -irρ cos θ Q(r) rdrdθ.
In particular recalling that Q is nonnegative, we find that for all ξ ∈ R 2 ,

| q(ξ)| ≤ 2π ∞ 0 Q(r) rdr ≤ 2π ∞ 0 e ζ 0 (1-r) e -ζ 0 (1-r) Q(r) rdr
for any ζ 0 ∈ R. Then we can apply Lemma 3.3 to the particular case of a radial function, so choosing k = 0 and ζ 0 = 3Kλ 0 we infer that

| q(ξ)| ≤ Ce ζ 0 κ/λ L T (rQ) (ζ 0 ) (3.3) ≤ Ce 3Kκ λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q 1 -q 2 L ∞ .
The end of the proof of Theorem 4 follows easily by taking the L 2 norm in ξ.

3.3.

Proof of Lemma 3.3. The method of proof follows the ideas developed in the proofs of Theorems 2 and 3, adapting the estimates to our special situation where the potentials are located near the boundary. The heart of the matter consists to approximate the solutions to Helmholtz equation (3.1) by separable solutions in radial coordinates involving Bessel functions, which allows in light of Debye's formula to relate the Laplace transform of the potential to its near field operator.

More precisely, we shall look for solutions to (3.1) under the following form, for n ∈ Z:

u n (r, θ) = J |n| (λr) λ(J ′ |n| (λ) -iJ |n| (λ)) e inθ + v n = z n (r, λ) e inθ + v n ,
where J n is a Bessel function of the first kind (see Appendix A), solution to

∂ 2 r + 1 r ∂ r - n 2 r 2 + 1 J n (r) = 0.
Suppose that for ℓ ∈ {1, 2}, u ℓ n solves (3.1) with potential q ℓ . Then writing

u ℓ n (r, θ) = z n (r, λ) e inθ + v ℓ n (r, θ), we find that (D 2 -λ 2 + q ℓ )v ℓ n = -q ℓ z n e inθ in B, (∂ r -iλ)v ℓ n|∂B = 0.
Due to Property (A.6) we have if 1λ -1 ≤ r ≤ 1 and |n| ≥ Kλ for K large, that λ z n (r, λ) is bounded, for λ ≥ λ 0 , by a constant depending only on K and λ 0 . For now on we shall denote by C(λ 0 , K) such a constant, which may change from line to line.

Therefore, arguing as in the proof of Lemma 1.5 and taking advantage of the fact that on the support of q ℓ , r varies in an interval of size λ -1 , we find as soon as λ is large enough compared to q ℓ L ∞

(3.4) v ℓ n L 2 ≤ C(λ 0 , K) λ q ℓ (r, θ) z n (r, λ) L 2 ≤ C(λ 0 , K) λ 5 2 q ℓ L ∞ .
Now going back to the computations of Section 2 we consider two positive integers n and m such that n, m ≥ Kλ, and we write

B (q 1 -q 2 )u 1 n u 2 -m dx = ∂B (N q 1 (λ) -Nq 2 (λ)
e inθ e -imθ dθ.

Therefore decomposing

u 1 n (r, θ) = z n (r, λ) e inθ + v 1 n (r, θ) and u 2 -m (r, θ) = z m (r, λ) e -imθ + v 2 -m (r, θ), we get 2π 0 Nq 1 (λ) -Nq 2 (λ) e inθ e -imθ dθ = B (q 1 -q 2 )e i(n-m)θ z n (r, λ) z m (r, λ) rdr dθ + B (q 1 -q 2 )e inθ z n (r, λ) v 2 -m rdr dθ (3.5) + B (q 1 -q 2 )e -imθ z m (r, λ) v 1 n rdr dθ + B (q 1 -q 2 )v 1 n v 2 -m rdr dθ.
As λ z n (r, λ) is bounded by a constant C(λ 0 , K), we have (recalling that q 1 and q 2 are compactly supported in an interval in r of size λ -1 )

B (q 1 -q 2 )e inθ z n (r, λ) v 2 -m r dr dθ ≤ C(λ 0 , K) λ 3 2 q 1 -q 2 L ∞ v 2 -m L 2
and similarly

B (q 1 -q 2 )e -imθ z m (r, λ) v 1 n r dr dθ ≤ C(λ 0 , K) λ 3 2 q 1 -q 2 L ∞ v 1 n L 2 . Finally B (q 1 -q 2 )v 1 n v 2 -m rdr dθ ≤ q 1 -q 2 L ∞ v 1 n L 2 v 2 -m L 2 .
Thus by (3.4), we get from (3.5), for n, m ≥ Kλ > Kλ 0 ,

B (q 1q 2 )e i(n-m)θ z n (r, λ) z m (r, λ) rdr dθ

≤ 2π 0 Nq 1 (λ) -Nq 2 (λ) e inθ e -imθ dθ + C q 1 ,q 2 λ 4 q L ∞ ,
where q = q 1q 2 and

C q 1 ,q 2 = C(λ 0 , K) max q 1 L ∞ , q 2 L ∞ , q 1 L ∞ q 2 L ∞ .
This gives rise to

(3.7) B (q 1 -q 2 )e i(n-m)θ z n (r, λ) z m (r, λ) rdr dθ ≤ Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 4 q L ∞ , which leads in light of (A.6) to B (q 1 -q 2 )e i(n-m)θ e -(n+m)(1-r) rdr dθ ≤ C(λ 0 , K) λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q L ∞
, for all n, m ≥ Kλ. In conclusion we have for any k ∈ Z, any ℓ ∈ N and any j ≥ 2Kλ > 2Kλ 0 , (3.8) B (q 1q 2 )e -i kθ e -j(1-r) rdr dθ ≤ C(λ 0 , K) λ 2 Nq 1 (λ) -Nq 2 (λ) + C q 1 ,q 2 λ 2 q L ∞ . The conclusion follows from (3.3). Lemma 3.3 is proved.

Concluding remarks

In this paper we have shown that the low-frequency component of the potential can be determined in a stable way from the scattering measurements and justified the resolution limit. We have also proved that in the near-field we have in the monotone case infinite resolution in reconstructing the potential near the boundary. We think that the result holds in the general case. However, its proof seems to be out of reach. In fact, even though a sampling (or interpolation) formula for the Laplace transform does exist [START_REF] Boumenir | Sampling eigenvalues in Hardy spaces[END_REF][START_REF] Rybkin | A new interpolation formula for the Titchmarsh-Weyl mfunction[END_REF], making norm-estimates similar to those in Theorem 4 is very challenging. Our results can be extended in many directions. It would be very interesting to study the limited-view case and show, as in [START_REF] Ammari | Limited view resolving power of conductivity imaging from boundary measurements[END_REF], that we recover infinite resolution from near-field measurements on the overlap of the source and receiver apertures. Another challenging problem is to understand how probe interaction can improve local resolution by converting where z is any point of the complex plane cut along the segment ] -∞, 0]. The motivation for introducing the Hankel functions is that the linear combination of J n (z) and Y n (z) have very simple asymptotic expansions for large |z|: it is thus well-known that H (1) n (z) = 2 πz (2πn tanh α) (2πn tanh α)

1 2 1 + O 1 n , J ′ n (n sech α) = (tanh α)
1 2 e -n(α-tanh α) (4πn) Without loss of generality we may assume that n ∈ N, then defining cosh α 1 = n λr and cosh α 2 = n λ , it is easy to see that under the above assumptions (λ ≪ n and r ∼ 1), we have necessarily cosh α i ≫ 1 for i ∈ {1, 2}, hence α i ≫ 1. This implies cosh α i ∼ e α i , sinh α i ∼ e α i , and tanh α i ∼ 1, which gives rise to Re z n (λ, r) = J n (n sech α 1 )J ′ n (n sech α 2 ) λ(J ′2 n (n sech α 2 ) + J 2 n (n sech α 2 )) and Im z n (λ, r) = J n (n sech α 1 )J n (n sech α 2 ) λ(J ′2 n (n sech α 2 ) + J 2 n (n sech α 2 )) • Finally, taking advantage of (A.5) we get Re z n (λ, r) ∼ e -n(α 1 -α 2 ) λ ∼ e -n log( 1

r ) λ •
The computation is identical for Im z n (λ, r), so using the fact that r is near to 1, we obtain the desired conclusion.

1

 1 is the Hankel function of first kind and order d/2 -1 (cf. Appendix A). The asymptotic behavior of Hankel functions (A.3) implies that for |y| large enough

2. 1 .

 1 Definitions and notations. Instead of considering the Helmholtz equation on the whole Euclidean space (with Sommerfeld's radiation condition) we focus on the Cauchy problem with Robin boundary condition on a bounded open set Ω ⊂ R d with smooth boundary

  |z| → ∞. Furthermore, we have the following Debye formulas whose proof can be found for instance in[START_REF] Olver | Asymptotics and special functions[END_REF] Chapter 9.4] and [1, Chapter 9]:J n (n sech α) ∼ e -n(α-tanh α)

5 )

 5 J n (n sech α) = e -n(α-tanh α)

1 n

 1 as n → ∞, where sech z denotes the hyperbolic secant of z defined by sech z = 1 cosh z with cosh z the hyperbolic cosine.Debye's formula gives rise to the following asymptotic behavior for the function introduced in Section 3:z n (λ, r) = J |n| (λr) λ(J ′ |n| (λ) -iJ |n| (λ))defined for n ∈ Z. Let us prove that for |n| ≫ λ and r ∼ 1,

We use the notation ǧ(ω) = g(-ω) = P g(ω) for the antipodal map.

Acknowledgements. We are very grateful to J. Sjöstrand for taking the time to discuss some properties of the Laplace transform with us.

This work was supported by the ERC Advanced Grant Project MULTIMOD-267184.

evanescent modes of the potential to propagating ones [START_REF] Carney | Near-field tomography, in Inside out: inverse problems and applications[END_REF]. These problems will be the subject of forthcoming works.

Appendix A. Bessel functions

Bessel's equation arises when finding separable solutions to the Helmholtz equation in spherical coordinates, and writes as follows:

It is well known (see for instance [START_REF] Lebedev | Special functions and their applications[END_REF][START_REF] Olver | Asymptotics and special functions[END_REF] and the references therein) that one of the solutions of Bessel's equation is the entire function J n (z) known as the Bessel function of the first kind of order n, and defined for arbitrary z ∈ C by the convergent series

in the case where n ∈ N and by

To find a general solution of Bessel's equation (A.1), we need a second solution of (A.1) which is linearly independent of J n (z). For such a solution, we usually choose Y n (z) the Bessel function of the second kind which is entire in the complex plane cut along the segment ] -∞, 0] and defined for n ∈ N by

where ψ(m + 1) = -γ + 1 + 1 2 + ... + 1 m , γ being the Euler constant. We also define Y -n (z) = (-1) n Y n (z).

Since J n and Y n are linearly independent, the general expression for solutions of (A.1) is a linear combination of Bessel functions of the first and second kinds, i.e, u(z) = A J n (z) + B Y n (z), where A and B are constants.

Another basis of solutions to the differential equation (A.1) is given by the Bessel functions of the third kind or Hankel functions, denoted by H n . These functions are defined by the formulas (A.2)

H (1) n (z) = J n (z) + iY n (z) and H (2) n (z) = J n (z) -iY n (z),