Valeria Banica

Nicolas Burq
email: nicolas.burq@universite-paris-saclay.fr

MICROLOCAL ANALYSIS OF SINGULAR MEASURES

HAL is

Introduction

The purpose of this article is the investigation of the singular part of scalar and vectorial measures from a microlocal analysis perspective.

We start by recalling some classical facts about the class M loc (R d , R m ) of locally bounded Radon measures on R d with values in R m (see for instance [START_REF] Rudin | Real and complex analysis[END_REF] or [START_REF] Matilla | Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability[END_REF]). The Radon-Nikodym Theorem allows to write the polar decomposition dµ = dµ d|µ| d|µ|, where the non negative measure |µ| is the total variation of the measure µ and the function dµ d|µ| ∈ L 1 (R d (d|µ|), S m-1 ) is the Radon-Nikodym derivative of µ with respect to |µ|, called the polar function of µ. Moreover, the Radon-Nikodym Theorem gives the Lebesgue decomposition of |µ| with respect to the Lebesgue measure, L d , so we have

dµ = gdL d + dµ d|µ| d|µ| s ,
where g ∈ L 1 (R d , R m ) and the positive measure |µ| s satisfies |µ| s ⊥ L d . Finally, the Radon-Nikodym Theorem applied to each component of µ with respect to L d implies that the singular part of µ is dµ s = dµ d|µ| d|µ| s , and in particular dµ d|µ| = dµs d|µ|s , µ s a.e. and |µ s | = |µ| s .

In the following we shall consider pseudo-differential operators and we have gathered in Section 6 the definitions and basic results from the theory we need. We denote by S k cl (R d ) the class of symbols of order k and by Ψ k cl (R d ) the class of classical pseudo-differential operators of order k. For u a temperate distribution on R d with values in R m , that is u = (u 1 , ..., u m ) with

u i ∈ S (R d , R), if A = (A ij ) is a n × m matrix of pseudo-differential operators then the distribution Au ∈ D (R d , R n ) is defined by (Au) j = k A jk u k , 1 ≤ j ≤ n.
For A ∈ Ψ k cl (R d ) we denote by a k : R d × R d → R its homogeneous principal symbol and similarly for matrices of pseudo-differential operators.

We recall that the classical wave front set introduced by Hörmander gives a proper characterisation of the singular support of a distribution, i.e. the complementary of points where the distribution is C ∞ . Moreover, for vector-valued distributions a generalisation of this concept is also available through the refined notion of polarisation wave front set introduced by Denker in [START_REF] Denker | On the Propagation of polarisation Sets for Systems of Real Principal Type[END_REF] (see also [START_REF] Gérard | Propagation de la polarisation pour des problèmes aux limites convexes pour les bicaractéristiques[END_REF]).

1.1. Main result. In this article we introduce an L 1 -regularity wave front set for vector valued temperate distributions. Definition 1.1. For u ∈ S (R d , R m ) we define the set

W F L 1 (u) ⊆ (T * R d \ {0}) × R m = R d × R d * × R m
by the following: (x 0 , ξ 0 , ω 0 ) / ∈ W F L 1 (u) if and only if there exist

A ∈ M 1×m (Ψ 0 cl (R d )), N ∈ N * , Q i ∈ Ψ 0 cl (R d ) and f i ∈ L 1 (R d , R) for 1 ≤ i ≤ N , satisfying (1.1) Au =
N i=1 Q i f i ,
with A elliptic at (x 0 , ξ 0 , ω 0 ) in the sense that a 0 (x 0 , ξ 0 ) ω 0 = 0.

Among the class M loc (R d , R m ) we shall consider the temperate Radon measures

M t (R d , R m ) := ∪ κ∈R + M κ (R d , R m ), M κ (R d , R m ) := {µ ∈ M loc (R d , R m ); ∃C > 0, ∀R > 1, |µ|(B(0, R)) ≤ CR κ }.
Our main result is a connection between the polarisation of the singular part of a measure and its wave front set.

Theorem 1.2. For µ ∈ M t (R d , R m ) we have the inclusion:

(1.2) x, dµ d|µ| (x) |µ|s-a.e ∈ Π 13 (W F L 1 (µ) \ 0 pol )).
where Π 13 denotes the projection with respect to the first and third variables, i.e. for |µ| s almost all x, ∃ξ ∈ S d-1 , x, ξ, dµ d|µ| (x) ∈ W F L 1 (µ) \ 0 pol .

Remark 1.3.

• Definition 1.1 is equivalent to:

(1.3) W F L 1 (u) = {(x, ξ, ω) ∈ R d × R d * × R m ; ω ∈ ∩ ker a 0 (x, ξ)},
where the intersection is over all possible

A ∈ M 1×m (Ψ 0 cl (R d )), N ∈ N * , Q i ∈ Ψ 0 cl (R d ) and f i ∈ L 1 (R d , R) for 1 ≤ i ≤ N , such that Au = N i=1 Q i f i . • We can obtain W F L 1 by considering in Definition 1.1 the existence of n, N ∈ N * , A ∈ M n×m (Ψ 0 cl (R d )), Q j ∈ M 1×N (Ψ 0 cl (R d )) and f j ∈ M N ×1 (L 1 (R d , R
)) such that (Au) j = Q j f j holds for 1 ≤ j ≤ n, and a 0 (x, ξ)ω = 0 R n . • In the case of scalar distributions Definition 1.1 of W F L 1 recovers the classical shape of a subset of T * R d \ {0} and the definition of ellipticity a 0 (x, ξ) = 0 is the usual one. • The set W F L 1 (u) is closed, conical in ξ, linear in ω, and always contains the trivial fiber

0 pol := R d × R d * × {0 R m }.
1.2. Consequences. Starting from Theorem 1.2 we shall deduce a series of results. For simplicity, we chose to work with temperate distributions and temperate Radon measures rather than distributions and locally bounded Radon measures, which would have been possible by simply adding a cut-off in the definition of W F L 1 , i.e. replacing Au by Aχu in (1.1). In the same vein, it is easy to see that for any

χ ∈ C ∞ c (R d ), µ ∈ M t (R d , R m ), (1.4) W F L 1 (χµ) = W F L 1 (µ) ∩ {(x, ξ, w) ∈ R d × R d * × R m , χ(x) = 0}. Indeed, let χ ∈ C ∞ c (R d ) equal to 1 on the support of χ. Then Aµ = Qf ⇒ χA(χµ) = χQf + χ[A, χ]µ,
and χ[A, χ] is a pseudo-differential operator of order -1 which sends M t to L 1 loc in view of Proposition 6.8. Also, for conciseness, we shall state the results involving hypothesis as equality (1.1) with only one term of type Ψ 0 cl L 1 (R d , R) in the right-hand side, but of course our results hold for finite sums of such terms.

First we get a proper micro-local characterisation of the singular support of a measure.

Theorem 1.4. For µ ∈ M t (R d , R m ) we have:

Π 1 (W F L 1 (µ) \ 0 pol )) = ∅ ⇐⇒ µ ∈ L 1 loc (R d , R m ), and moreover Π 1 (W F L 1 (µ) \ 0 pol )) ⊆ supp |µ| s , |µ| s (supp |µ| s \ Π 1 (W F L 1 (µ) \ 0 pol ))) = 0.
Next, we obtain an elliptic full L 1 -regularity result.

Theorem 1.5. Let A ∈ M m×m (Ψ 0 cl (R d )) be elliptic at x ∈ R d in the sense a 0 (x, ξ)ω = 0 R m , ∀ξ ∈ S d-1 , ∀ω ∈ R m * ,
and let µ ∈ M t (R d , R m ). Then the following holds:

Aµ ∈ Ψ 0 cl L 1 (R d , R) m =⇒ ∃V ∈ V(x), µ | V ∈ L 1 (R d , R m ). Corollary 1.6. Assume u ∈ S (R d , R m ), A ∈ M m×m (Ψ k cl (R d )) is elliptic with k > 0 for all x ∈ R d , then for |α| ≤ k we have ∂ α x u ∈ M t (R d , R m ), Au ∈ Ψ 0 cl L 1 (R d , R) m =⇒ ∂ α x u ∈ L 1 loc (R d , R). Also, if A ∈ M m×m (Ψ 0 cl (R d )) is elliptic for all x ∈ R d and if u ∈ BV (R d , R m
), the set of L 1 -functions with gradient a matrix-valued finite Radon measure, then we recover full L 1 regularity:

A∇u ∈ Ψ 0 cl L 1 (R d , R) d =⇒ u ∈ W 1,1 loc (R d , R m
). Previously known L 1 elliptic regularity results assuming u, P u ∈ L 1 loc , P elliptic operator of order k involve loss of derivatives or requires the use of Besov spaces B k,1 ∞ , see [12, Theorem 2.6 and Remark 2.7]. At the exact L 1 level, the following counterexample from [START_REF] Giaquinta | An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs[END_REF]Example 7.5] shows that, in general some loss is anavoidable: on the disk of radius one, consider the function

u(x) = log log(e|x| -1 ) ∈ W 1,1 (B 1 ), which satisfies ∆u = - 1 |x| 2 log 2 (e|x| -1 ) ∈ L 1 (B 1 ), D 2 u / ∈ L 1 (B 1 ).
Note that this is (of course!) not in contradiction with Theorem 1.5 as D 2 u is not a Radon measure. Theorem 1.5 eliminates the loss under the additional assumption that µ is a Radon measure. To the best of our knowledge, even for scalar equations (m = 1), this result is new. We want to emphasize that allowing 0 order pseudo-differential operators is not anecdotal: in fluid mechanics, 0-th order operators are ubiquitous and their lack of L ∞ (or equivalently L 1 ) boundedness is responsible for many pathological behaviours (see e.g. the double exponential growth of the complexity of solutions to two dimensional Euler equations [START_REF] Kiselev | Small scale creation for solutions of the incompressible two-dimensional Euler equation[END_REF]). Let us now give an example with such a fluid dynamics flavour.

Corollary 1.7. Let u ∈ BV loc (R d , R d ).
Assume that both div u and curl u are in

L 1 loc (R d ) (or Ψ 1 (L 1 )). Then ∇u ∈ L 1 loc (R d , R d×d ).
Proof. Replacing u by χu, χ ∈ C ∞ 0 , we can assume that u is compactly supported. Now we have the following equation for any derivative ∂ i u, which is a Radon measure by the BV assumption on u,

(1.5) A(∂ i u) := 1 (1 + |D x | 2 ) 1 2         ∂ 1 ∂ 2 ∂ 3 • • • • • • ∂ d ∂ 2 -∂ 1 0 • • • • • • 0 0 ∂ 3 -∂ 2 0 • • • 0 • • • • • • • • • • • • • • • • • • 0 0 0 • • • ∂ d -∂ d-1 -∂ d 0 0 • • • • • • ∂ 1         ∂ i u = ∂ i (1 + |D x | 2 ) 1 2 div u curl u , so A(∂ i u) ∈ Ψ 0 cl L 1 (R d , R) d+1
, and a straightforward calculation shows that the constant coefficients operator A ∈ M d+1×d (Ψ 0 cl (R d )) is elliptic:

∀ξ = 0, ker a 0 (ξ) = {0 R d+1 }.
We deduce that W F L 1 (∂ i u) = 0 pol and then Theorem 1.2 yields

∂ i u ∈ L 1 loc (R d , R d ).
Remark 1.8. A by-product of Theorem 1.5 with A = Id is that counter-examples for the lack of continuity on L 1 of zero-order operators can occur only for operators

Q ∈ Ψ 0 cl sending an L 1 function outside M t . Indeed, if Qf = µ ∈ M t we deduce from Theorem 1.5 with A = Id that µ ∈ L 1 loc .
Contrarily to the usual wave fronts defined in microlocal analysis for distributions, in general, the natural implication

(1.6) W F L 1 (u) = 0 pol ⇒ u ∈ L 1 loc ,
is not true. Indeed, recall that general pseudo-differential operators of order 0 are not bounded on L 1 . Taking for example the function u = D 2 w and f from (6.6), we have

f ∈ L 1 comp , Q = D 2 (-∆ + 1) -1 ∈ Ψ 0 cl such that Qf = u / ∈ L 1 .
On one hand this implies by definition that W F L 1 (u) = 0 pol , while u / ∈ L 1 loc (because u is a compactly supported distribution but is not in L 1 ). However, from Theorem 1.4, if we assume in addition u ∈ M t , then property (1.6) is true.

We end with the following extension of De Philippis and Rindler's result [9, Theorem 1.1] from the case of constant coefficient differential operators A = |α|≤k A α ∂ α , A α ∈ R n×m of order at least 1 to the case of pseudo-differential operators of nonnegative order, with moreover an extra-degree of freedom on the constraint. This was the starting motivation for us to introduce the L 1 -wave front notion.

Theorem 1.9. Let µ ∈ M(R d , R m ). Let A ∈ M n×m (Ψ 0 cl (R d )) such that Aµ ∈ Ψ 0 cl L 1 (R d , R) n . Then dµ d|µ| (x) |µ|s-a.e ∈ |ξ|=1
ker(a 0 (x, ξ)).

As a consequence, for A of order k > 0 and for Q or order at most k, we can extend the theorem to the more general equation Aµ = Qf . Also, for Q or order strictly less than k we can extend to the more general equation Aµ = Qσ with σ ∈ M t that can be treated viewing it as (A, -Q)(µ, σ) = 0 and using the fact that the principal symbol of (A,

-Q) is (a k , 0).
The version of Theorem 1.9 in [START_REF] Philippis | On the structure of A-free measures and applications[END_REF] implies several major consequences, and our new version allows to relax the assumptions therein. In particular, Theorem 1.9 shows that Alberti's rank one theorem for gradient of BV functions also holds for measures whose curl is a combination of derivatives of L 1 functions

(1.7) µ ∈ M t (R d , R l×d ), curl µ = (∂ p µ i,q -∂ q µ i,p ) i≤l,p =q≤d ∈ Ψ 1 cl L 1 (R d , R) d 2 l =⇒ dµ d|µ| (x) |µ|s-a.e ∈ {a ⊗ ξ; a ∈ R l , ξ ∈ R d * }.
The result in [START_REF] Philippis | On the structure of A-free measures and applications[END_REF] has been extended in [START_REF] Arroyo-Rabasa | Dimensional estimates and rectifiability for measures satisfying linear PDE constraints[END_REF]; it would be interesting to see whether these results of dimensional estimates and rectifiability can be viewed and analyzed from the microlocal analysis point of view.

Finally, the micro-local perspective also allows to extend some invariance properties for constrainted measures. We prove some propagation type results in Section 5. However our results in this direction are only preliminary.

1.3. Sketch of the proof of Theorem 1.2. The proof of Theorem 1.2 goes as follows.

Arguing by contradiction we consider the set E of positive |µ| s -measure on which the inclusion (1.2) fails. We first use Proposition 3.1 to construct at any point x ∈ E an elliptic operator B x at x, dµ d|µ| (x) that smoothens µ in the L 1 -sense of (3.1). Then we use the equation in (3.1) as a starting point to implement an elliptic regularity strategy à la De Philippis-Rindler [START_REF] Philippis | On the structure of A-free measures and applications[END_REF] relying on a precise description of tangent measures near µ s -generic points. The main difficulties in this approach are the following:

• We need some regularity properties of the L 1 functions f i in (3.1) with respect to the µ s measure. These properties are apriori valid only |µ| s -a.e., while the functions themselves depend on x; this issue is solved in Lemma 3.3. • We need to handle the non locality of pseudo-differential operators, which are only pseudo-local, and this requires some temperance properties of the scaled functions in the definition of the tangent measures. • Finally we need to handle the limiting case of 0-th order operators with no gain of regularity.

The article is structured as follows. In Section 2 we give some natural examples illustrating the relevance of W F L 1 . Section 3 contains the proof of the main result, namely Theorem 1.2. In Section 4 we give the short proofs that Theorem 1.2 implies Theorems 1.4, 1.5 and 1.9. In Section 5 we consider the propagation of singularities results. In Section 6 are gathered the definitions and basic results on pseudo-differential operators that we use throughout the article. The last section contains general results of temperance properties for measures, obtained from the construction by Preiss in [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF], which are important when dealing with pseudo-differential operators.

Aknowledgements: Both authors are grateful to the Institut Universitaire de France for the ideal research conditions offered by their memberships. The first author was also partially supported by the French ANR project SingFlows ANR-18-CE40-0027, while the second author was also partially supported by ANR project ISDEEC ANR-16-CE40-0013. We would like to thank Frédéric Bernicot for enlightenments about weak L 1 estimates for pseudo-differential operators.

Examples

In this section we give some examples of measures for which we can describe the L 1 wave front, W F L 1 . We illustrate the relevance of both the cotangent variable ξ and the polarisation variable ω by providing examples for which the dependance with respect to these variables is non trivial. Proposition 2.1.

(

) Let µ ∈ L 1 loc (R d , R m ). Then W F L 1 (µ) = 0 pol (and according to Theorem 1.4 the converse is true if µ ∈ M t (R d , R m )). (2) Let µ = (µ 1 , µ 2 ), µ 1 ∈ L 1 loc (R d ; R m 1 ) and µ 2 ∈ M t (R d ; R m 2 ). Then (x, ξ, ω) ∈ W F L 1 (µ) ⇔ (x, ξ, ω 2 ) ∈ W F L 1 (µ 2 ) and ω = (0 R m 1 , ω 2 ). (3) Let x 0 ∈ R d , ω 0 ∈ R m * and µ = ω 0 δ x 0 . Then W F L 1 (µ) \ 0 pol = {(x 0 , ξ, ω); ξ ∈ R d * , ω ∈ R * ω 0 } (4) Let Σ a smooth d 1 dimensional embedded submanifold of R d , 1
and µ the scalar surface measure on Σ.

Then W F L 1 (µ) is the set of points (x, ξ) ∈ R d × R d * such that x ∈ Σ and ξ is (co)-normal to Σ: ∀X ∈ T x Σ, ξ, X = 0.
(5) Let (e p ) m p=1 be the canonical basis of R m . We consider a sequence {x n } n∈N of pairwise disjoint points such that x n n→+∞ -→ x 0 and

µ = n∈N 2 -n δ xn ω n , ω n = e p , if n ≡ p (mod m). Then W F (µ) \ 0 pol = n≥1 {x n } × R d * × R * ω n {x 0 } × R d * × R m * .
Proof. Example (1) is trivial, as in the definition of W F L 1 , we can choose for any x 0 ∈ R d and ω 0 ∈ R m * , the multiplication operator A = χ(x) t ω 0 with χ = 0 in a neighborhood of x 0 .

To prove Example (2), we consider first (x, ξ, ω) with ω = (ω 1 , ω 2 ) such that ω 1 = 0 R m 1 . Assume for example that the first coordinate of ω 1 does not vanish. Let A = 1, 0, . . . , 0 ∈ M 1×m (R), and f be the first component of the vector valued function µ 1 . Then we have

Aµ = f ∈ L 1 loc (R d , R
), and the constant operator A is elliptic at (x, ξ, ω) because the first coordinate of ω 1 does not vanish. We deduce that (x, ξ, ω) /

∈ W F L 1 (µ). Let now (x, ξ, 0 R m 1 , ω 2 ) / ∈ W F L 1 (µ)
. By definition, there exists a A, Q j , f j as in Definition 1.1 such that A is elliptic at (x, ξ, 0, ω 2 ) and Aµ = j Q j f j .

which implies by linearity

A 0 µ 2 = j Q j f j -A µ 1 0 or equivalently A 2 µ 2 = j Q j f j + A 1 µ 1 , A = (A 1 , A 2 ).
Since by assumption

µ 1 ∈ L 1 , to conclude that (x, ξ, ω 2 ) / ∈ W F L 1 (µ 2 ). It remains to check that if A is elliptic at (x, ξ, 0 R m 1 , ω 2 ) then A 2 is elliptic at (x, ξ, ω 2 ), which is clear from a(x, ξ) 0 ω 2 = a 2 (x, ξ)(ω 2 ). Conversely, let now (x, ξ, ω 2 ) / ∈ W F L 1 (µ 2 )
. By definition, there exists a A 2 , Q j , f j as in Definition 1.1 such that A 2 is elliptic at (x, ξ, ω 2 ) and

A 2 µ 2 = j Q j f j . With A = (0 R m 1 , A 2 ) we have for any ω 1 ∈ R m 1 : A µ 1 µ 2 = A 2 µ 2 , a(x, ξ)(ω 1 , ω 2 ) = a 2 (x, ξ)ω 2 = 0,
and we obtain that (x, ξ, ω

1 , ω 2 ) / ∈ W F L 1 (µ). For example (3), it is clear from Example (1) that if (x, ξ, ω) ∈ W F L 1 (ω 0 δ x 0 ) \ 0 pol then x = x 0 .
Let us assume that ω / ∈ Rω 0 . Choosing for A the multiplication by a constant 1 × m vector orthogonal to ω 0 but not to ω, we get Aω 0 δ x 0 = 0 and a(x 0 , ξ)ω = 0 so

(x 0 , ξ, ω) / ∈ W F L 1 (ω 0 δ x 0 ), ∀ξ ∈ R d * .
As a consequence we have

W F L 1 (ω 0 δ x 0 ) \ 0 pol ⊆ {(x 0 , ξ, ω); ξ ∈ R d * , ω ∈ R * ω 0 }.
The equality follows from the fact that W F L 1 (ω 0 δ x 0 ) = 0 pol and W F L 1 (ω 0 δ x 0 ) is invariant under rotations in the ξ variables, as so does δx 0 . For example (4), we perform a change of variables and are reduced to the case when

Σ = {(x 1 , 0 R d 2 ); x 1 ∈ R d 1 }, µ = g(x 1 )δ x 2 =0 R d 2 , g ∈ C ∞ (R d 1 , R),
where the function g is positive. Let us apply a pseudo-differential operator χ(D x ) where χ is supported in a conic neighborhood of a point ξ = (ξ 1 , ξ 2 ) with ξ 1 = 0 R d 1 , small enough so that on the support of χ we have |ξ| ≤ C|ξ 1 |. We have

χ(D x )µ = 1 (2π) d e i(x 1 -y 1 )•ξ 1 +x 2 •ξ 2 χ(ξ)g(y 1 )dy 1 dξ,
and integrations by parts using ∂ ∂y 1 e -iy 1 ξ 1 = -iξ 1 e -iy 1 ξ 1 gain arbitrary inverse powers of

|ξ 1 | (hence of |ξ|) which shows that χ(D x )µ ∈ C ∞ . We deduce the inclusion W F L 1 (µ) ⊂ {(x 1 , 0 R d 2 , 0 R d 1 , ξ 2 ); x 1 ∈ R d 1 , ξ 2 ∈ R d 2 }.
To get the equality, we first remark that it is enough to study the case above when g = 1, because we can multiply all pseudo-differential operators by the smooth function g(x 1 ) -1 to the right. According to Theorem 1.4, Π 1 W F L 1 must have a non trivial intersection with the set

{(x 1 , 0 R d 2 ); x 1 ∈ R d 1 } because µ is not L 1 near
this point. We get the equality simply because in that particular case g 1 = 1, W F L 1 is clearly invariant by rotations of the variable ξ 2 as so does δ 0 R d 2 . Let us now turn to Example [START_REF] Burq | Mesures de défaut de compacité, application au système de Lamé[END_REF]. Since the points x n are pairwise disjoint and converging to x 0 , all the points x n , n ≥ 1 are isolated in the sequence and localising µ near each point x n (i.e. considering µ n = χ n µ n , with χ n = 1 near x n and χ n = 0 in a neighborhood of

x k , k = n), we get, according to Example (3) that W F (µ) \ 0 pol ∩ {(x, ξ, ω); x = x 0 } = n≥1 {x n } × R d * × R * ω n .
To conclude, it remains to prove that

W F (µ) \ 0 pol ∩ {(x, ξ, ω); x = x 0 } = {x 0 } × R d * × R m * .
By linearity with respect to the ω variable, it is actually enough to prove that

(x 0 , ξ, e p ) ∈ W F L 1 (µ), ∀ξ ∈ R d * , 1 ≤ p ≤ m.
We argue by contradiction. Assume there exists ξ 0 and p such that (x 0 , ξ 0 , e p ) /

∈ W F L 1 (µ). Then there exists A, Q j ∈ Ψ 0 cl matrices of pseudo-differential operators , f j ∈ L 1 such that Aµ = j Q j f j , a 0 (x 0 , ξ 0 )e p = 0.
We deduce with χ n as above,

Aχ n µ = χ n Qf + [A, χ n ]µ = Q n f + g, g ∈ L 1 .
But for n sufficiently large, x n is arbitrarily close to x 0 , we get a(x n , ξ 0 )e p = 0.

We deduce that

(x n , ξ 0 , e p ) / ∈ W F L 1 (χ n µ) = W F L 1 (2 -n ω n δ xn ) = {x n } × R d * × R * ω n ,
which is a contradiction if we choose n = p (mod m) so that ω n = e p .

In example (5), we have µ = µ s and

|µ| s = n∈N 2 -n δ xn , dµ d|µ| (x) = n∈N 1 x=xn ω n .
As a consequence, this example shows that in Theorem 1.2, the inclusion is, in general not an equality. This is an indication that the wave front we define in the present work, though sufficient for the applications of our present work, might need to be refined for further applications.

Proof of Theorem 1.2

All the results being local, using (1.4), we shall assume without loss of generality that µ is compactly supported. We argue by contradiction, and we suppose that the conclusion of Theorem 1.2 does not hold:

|µ| s (E) > 0,
where

E := {x; x, dµ d|µ| (x) ∈ (Π 13 W F L 1 (µ) \ 0 pol ) c }.
We start by proving a general property of W F L 1 that will be the starting point of our approach.

3.1. An elliptic characterisation of W F L 1 . In the following Proposition we give a characterisation of the elements in (Π 13 (W

F L 1 (u) \ 0 pol )) c for u ∈ S (R d , R m ). Proposition 3.1. For u ∈ S (R d , R m ) the set (Π 13 (W F L 1 (u) \ 0 pol )) c coincides with (3.1) (x, ω) ∈ R d × R m ; ∃B ∈ M 1×m (Ψ 0 cl (R d )) elliptic at (x, ω), N ∈ N * , Q i ∈ Ψ 0 cl (R d ), f i ∈ L 1 (R d , R), ∀1 ≤ i ≤ N, Bu = N i=1 Q i f i .
Here ellipticity at (x, ω) means

(3.2) b 0 (x, ξ)ω = 0, ∀ξ ∈ S d-1 . Proof of Proposition 3.1. Let (x, ω) ∈ (Π 13 (W F L 1 (u) \ 0 pol )) c .
Then by definition for all ξ ∈ S d-1 we have the existence of

N ξ ∈ N * , B ξ ∈ M 1×m (Ψ 0 cl ), Q ξ,i ∈ Ψ 0 cl (R d ) and f ξ,i ∈ L 1 (R d , R), ∀1 ≤ i ≤ N ξ such that B ξ u = N ξ i=1 Q ξ,i f ξ,i , with a principal symbol b 0,ξ satisfying b 0,ξ (x, ξ)ω = 0.
By continuity we can find a conical neighborhood

V ξ ∈ V c (ξ) such that b 0,ξ (x, η)ω = 0, ∀η ∈ V ξ .
and moreover δ ξ ∈ {±1} such that

(3.3) δ ξ b 0,ξ (x, η)ω > 0, ∀η ∈ V ξ .
Let χ ξ be a non-negative smooth cut-off with support in V ξ , valued one on a conical neighborhood Ṽξ ∈ V c (ξ) with Ṽξ ⊂ V ξ . As ∪ ξ∈S d-1 Ṽξ is a covering of S d-1 we can extract a finite covering

(3.4) ∃L ∈ N * , S d-1 ⊂ ∪ L l=1 Ṽξ l .
Then we define

B := L l=1 Op(χ ξ l δ ξ l )B ξ l ∈ M 1×m (Ψ 0 cl (R d )) with principal symbol b 0 (x, ξ) := L l=1 χ ξ l δ ξ l b 0,ξ l (x, ξ), satisfying Bu = l∈L N ξ l i=1 Op(χ ξ l δ ξ l )Q ξ l ,i f ξ l ,i .
In particular the right-hand-side is a finite sum of terms Qf with

Q ∈ Ψ 0 cl (R d ) and f ∈ L 1 (R d , R). We consider now b 0 (x, ξ)ω = L l=1 χ ξ l δ ξ l b 0,ξ l (x, ξ)ω.
On one hand from (3.3) we deduce that all terms in the sum are non negative for all ξ ∈ S d-1 . On the other hand from (3.4) there exists at least a set Ṽξ j such that ξ ∈ Ṽξ j , thus using again (3.3) we get (3.2).

3.2.

Preliminaries. We start with a lemma gathering several general measure properties. Lemma 3.2. There exists a set N 0 with |µ| s (N 0 ) = 0 such that for all x ∈ N c 0 , there exists a sequence r j → 0 such that the following three assertions hold.

(i) lim r→0 L d (B(x, r)) |µ| s (B(x, r)) = 0, lim j→0 |µ| a (B(x, r)) |µ| s (B(x, r)) = 0, lim r→0 - B(x,r) dµ d|µ| (y) - dµ d|µ| (x) d|µ| s = 0,
There exists C, c > 0 such that the family of measures 1

µ j := (T x,r j k ) µ s |µ| s (B(x, r j k ))
satisfies the temperate uniform in j bounds:

(ii) |µ j |(B(x, R)) = |µ| s (B(x, Rr j )) |µ| s (B(x, r j )) ≤ C max{1, R d+ 1 2 }, |µ j |(B 1
3

) ≥ c > 0
We can pass to the limit

(iii) µ j * j→∞ ν ∈ T an(x, µ) = dµ d|µ| (x) T an(x, |µ|),
and since

|µ j |(B 1
3

) ≥ c > 0, the restriction of ν to B 1 2
is not trivial.

1 by T x,r j we denote the dilations x ∈ R d → x + rj x ∈ R d . and by T x,r j σ we denote the push-forward of the measure σ given by T x,r j σ(A) = σ(x + rjA).

Assertion (i) is based on classical characterisation of singular points of a measure (see e.g. [15, Theorem 2.12 (3)], [7, Corollary 1.7.1]). Assertion (ii) is shown in Proposition 7.1. Assertion (iii) is given by the classical properties of tangent measures (see e.g. [17, Section 2.3]) and the lower bound (7.2).

Next we are going to prove the following property on the set E \ N 0 .

Lemma 3.3. There exists a set Ẽ ⊆ E \ N 0 with |µ| s (E \ Ẽ) = 0 such that for all x ∈ Ẽ, there exists

B ∈ M 1×m (Ψ 0 cl (R d )), N ∈ N * , Q i ∈ Ψ 0 cl (R d ), f i ∈ L 1 (R d , R), ∀1 ≤ i ≤ N , satisfying the identity Bu = N i=1 Q i f i ,
and the elliptic condition at x, dµ d|µ| (x) :

b 0 (x, ξ) dµ d|µ| (x) = 0, ∀ξ ∈ S d-1 .
Moreover, for the sequence (r j ) constructed in Lemma 3.2 we have the following properties:

(iv) ∀i ∈ {1, .., N x }, lim r→0 |f i L d (B(x, r))| |µ| s (B(x, r)) = 0, ∃C x,i > 0, |f i L d (B(x, Rr j ))| |µ| s (B(x, r j )) ≤ C x,i max{1, R d+ 1 2 }. Proof. As |µ| s is a finite Radon measure we have |µ| s (E \ N 0 ) = sup{µ s (K); K compact ⊂ E \ N 0 }. Thus there exists a compact set K ⊆ E such that |µ| s (K) > 1 2 |µ| s (E) > 0. For any x ∈ K we have x ∈ E, so x, dµ d|µ| (x) ∈ (Π 13 W F L 1 (µ) \ 0 pol ) c .
Thus we can apply Proposition 3.1 to get the existence of

B x ∈ M 1×m (Ψ 0 cl (R d )), N ∈ N * , Q x,i ∈ M m×1 (Ψ 0 cl (R d )), f x,i ∈ L 1 (R d , R), ∀1 ≤ i ≤ N , such that B x µ = Nx i=1 Q x,i f x,i ,
with the elliptic property at x, dµ d|µ| (x) :

b 0,x (x, ξ) dµ d|µ| (x) = 0, ∀ξ ∈ S d-1 .
This implies the existence of r x,1 > 0 and c x > 0 such that

|b 0,x (y, ξ) dµ d|µ| (x)| > c x > 0, ∀y ∈ B(x, r x,1 ), ∀ξ ∈ S d-1 ,
In particular there exists δ x > 0 such that

(3.5) b 0,x (y, ξ)ω = 0, ∀y ∈ B(x, r x,1 ), ∀ξ ∈ S d-1 , ∀ω ∈ S m-1 , ω - dµ d|µ| (x) ≤ δ x .
From Lemma 3.2 for all x ∈ N c 0 , thus for all x ∈ K,

(3.6) lim r→0 - B(x,r) dµ d|µ| (y) - dµ d|µ| (x) d|µ| s = 0,
with the average integral well-defined, meaning |µ| s (B(x, r)) > 0, ∀r, ∀x ∈ K. Thus for all x ∈ K we get the existence of r x,2 such that for all r ≤ r x,2

(3.7) - B(x,r) dµ d|µ| (y) - dµ d|µ| (x) d|µ| s ≤ δ x 2 .
For all x ∈ K and r ≤ r x,2 we have |µ| s (B(x, r)) > 0 and the set defined by

F x,r := {y ∈ B(x, r), dµ d|µ| (y) - dµ d|µ| (x) ≤ δ x },
satisfies, in view of (3.7),

(3.8) F x,r ⊆ B(x, r), |µ| s (F x,r ) > |µ| s (B(x, r)) 2 .
We denote by N 1 the set of null |µ| s -measure of the zero-density points w.r.t. |µ| s of the set K, thus

(3.9) ∀x ∈ K \ N 1 , ∃r x,3 , ∀r ≤ r x,3 3 4 |µ| s (B(x, r)) ≤ |µ| s (K ∩ B(x, r)).
The closed balls B(x, rx 2 ), x ∈ K \ N 1 with r x = min{r x,1 , r x,2 , r x,3 } form a covering of the bounded set K \ N 1 , so by Besicovitch's covering theorem ([15, Theorem 2.7]) we can extract a countable covering: there exists

{x n } n∈N ⊆ K \ N 1 such that K \ N 1 ⊆ ∪ n∈N B(x n , r xn 2 ) ⊆ ∪ n∈N B(x n , r xn ).
For all n ∈ N and 1 ≤ i ≤ N xn there exists a set N n,i of null |µ| s -measure such that

lim r→0 |f xn,i L d (B(x, r))| |µ| s (B(x, r)) = 0, ∀x ∈ (N n,i ) c .
This is due to the fact that |µ| s ⊥ f xn,i L d and Theorem 2.12 (3) in [START_REF] Matilla | Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability[END_REF]. It follows from (3.5) that (3.10)

∀x ∈ K \ N 1 , ∃n ∈ N, x ∈ B(x n , r xn ), and if dµ d|µ| (x) - dµ d|µ| (x n ) ≤ δ xn , then b 0,xn (x, ξ) dµ d|µ| (x) = 0 ∀ξ ∈ S d-1 .

Now we define the sets

K := K \ (N 1 ∪ n∈N ∪ Nx n i=1 N xn,i ), and Ẽ := (∪ n∈N F xn,rx n ) ∩ K ⊆ E \ N 0 .
In view of (3.8)-(3.9) we have

(3.11) |µ| s (F xn,rx n ∩ K) = |µ| s (F xn,rx n ∩ K) ≥ |µ| s (F xn,rx n ) -|µ| s (B(x n , r n ) ∩ K c ) ≥ ( 1 2 - 1 4
)|µ| s (B(x n , r x )), and since |µ| s (K) > 0 we get at least one n ∈ N such that |µ| s (B(x n , r x )) > 0 so

|µ| s ( Ẽ) > 0.
Then (3.10) and the definitions of F x,r and Ẽ ensure us that

∀x ∈ Ẽ, ∃n ∈ N, b 0,xn (x, ξ) dµ d|µ| (x) = 0, ∀ξ ∈ S d-1 .
Therefore for x ∈ Ẽ we define B := B xn , which have the elliptic property in the statement of the Lemma, we define N := N xn , and we have the first part of the Lemma:

Bu = N i=1 Q xn,i f xn,i .
Moreover, f xn,i satisfies the first assertion of (iv) as by definition Ẽ ⊂ N c xn,i . Then it follows that

∃R x,xn,i , ∀r ∈ (0, R x,xn,i ), |f xn,i L d (B(x, r))| |µ| s (B(x, r)) ≤ 1.
For Rr j ≤ R x,xn,i we deduce by combining also with Lemma 3.2 (ii) that

(3.12) |f xn,i L d (B(x, Rr j ))| |µ| s (B(x, r j )) = |f xn,i L d (B(x, Rr j ))| |µ| s (B(x, Rr j ))
|µ| s (B(x, Rr j )) |µ| s (B(x, r j ))

≤ |µ| s (B(x, Rr j )) |µ| s (B(x, r j )) ≤ C max{1, R d+ 1 2 }.
On the other hand, by using Lemma 3.2 (i) we have for

Rr j ≥ R x,xn,i |f xn,i L d (B(x, Rr j ))| |µ| s (B(x, r j )) ≤ C f xn,i L 1 r -d j ≤ C R d x,xn,i R d .
Therefore f xn,i satisfies also the last assertion of (iv).

Summarizing, we have shown that we can find Ẽ ⊂ E \ N 0 of positive |µ| s -measure, on which all the conclusion of the Lemma are satisfied, excepted the fact that |µ s |(E \ Ẽ) = 0. The argument above is valid also starting from any subset of E \ N 0 of positive |µ| s -measure. Thus the conclusion of the Lemma are satisfied |µ| s -a.e. on E \N 0 . Indeed, let us we denote by E 1 the subset of E \ N 0 of the points for which the conclusion of the Lemma are not satisfied. If |µ s |(E 1 ) > 0 then the construction above would give the existence of a subset Ẽ1 ⊂ E 1 of strictly positive |µ s |-measure, thus non-empty, on which the conclusions of the Lemma are both true and false on E 1 . Therefore |µ s |(E 1 ) = 0 and the full conclusion of the Lemma follows.

3.3. The contradiction argument. We go back to the contradiction argument. From now on, our proof is inspired from the approach in [START_REF] Philippis | On the structure of A-free measures and applications[END_REF], using the elliptic type operator family from Lemma 3.3 instead of an operator constraining the measure to vanish.

We thus apply Lemma 3.3 and as |µ| s ( Ẽ) > 0 we can choose now x 0 ∈ Ẽ so that properties in Lemmas 3.2-3.3 are valid with x = x 0 . We denote

ω 0 := dµ d|µ| (x 0 ).
From Lemma 3.2 (iii) we have that

µ j * j→∞ dµ d|µ| (x) ν,
with ν ∈ T an(x, |µ|) a positive measure with ν j (B 1 2

) > 0. Then we also have (cf Theorem 2.12 (3) in [START_REF] Matilla | Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability[END_REF] applied with µ s ⊥ g R L d , where g R (x) := R g(Rx) and R > 0)

ν j := (T x 0 ,r j ) |µ| s |µ| s (B(x 0 , r j )) * j→∞ ν. In particular 0 < lim j→∞ ν j (B 1
3

) ≤ ν(B 1 2
), and as ν j ⊥ L d , there exists sets

E j ⊆ B 1 2
where ν j B 1 2 concentrates and L d vanishes:

ν j (E j ) = ν j (B 1 2
), L d (E j ) = 0.

By using the full information at (x 0 , ω 0 ) given by Lemma 3.

χν j L 1 (R d ) -→ χν.
As ν j * j→∞ ν, it is thus enough to prove that {χν j } is precompact in L 1 loc (R d ). To get the L 1 loc (R d )-precompactness of {χν j } we shall consider the operator B from Lemma 3.3, and use its ellipticity property through an appropriate inversion argument.

Step 1: The inversion formula.

As x 0 ∈ Ẽ, the properties in Lemma 3.3 are satisfied for x 0 and in particular we obtain the existence of

B ∈ M 1×m (Ψ 0 cl (R d )), Q ∈ Ψ 0 cl (R d ), f ∈ L 1 (R d , R) satisfying the identity Bµ = Qf,
the elliptic condition at (x 0 , ω 0 ):

(3.14) b 0 (x 0 , ξ)ω 0 = 0, ∀ξ ∈ S d-1 ,
and moreover f satisfies the properties (iv) in Lemma 3.3. For simplicity we have considered only one term in the right-hand side instead of a finite sum, as finitely many such terms can be treated similarly. According to Proposition 6.2, we can pass from left quantization to right quantization, i.e. we shall work with symbols that depend on y and ξ and are independent of the variable x, and the ellipticity property (3.14) remains valid for them at y = x 0 . For simplicity we call these symbols again b l and q. We apply the zoom

(T x 0 ,r j ) • |µ|s(Br j (x 0 ))
and get Bj µ j = Qj f j , where

f j L d := (T x 0 ,r j ) f L d |µ| s (B(x 0 , r j )) , f j (x) = (r j ) d |µ| s (B(x 0 , r j )) f (x 0 + r j x), bj (y, ξ) := b(x 0 + r j y, ξ r j ), qj (y, ξ) := q(x 0 + r j y, ξ r j ). (3.15)
We first microlocalize in frequency to suppress the low frequencies by applying to the left (1 -χ(D x )), and get

(3.16) B j µ j := (1 -χ(D x )) Bj µ j = (1 -χ(D x )) Qj f j =: Q j (f j )
with symbols (remark that the following exact formula follows from the independence of b, q with respect to the x variable and (6.2)), b j (y, ξ) = (1 -χ(ξ))b(x 0 + r j y, ξ r j ), q j (y, ξ) := (1 -χ(ξ))q(x 0 + r j y, ξ r j ).

Notice that the symbols b j and q j are in S 0 cl (R d ) with semi-norms in S 0 cl (R d ) uniformly bounded in j. Indeed, from (3.16) we get

(3.17) |∂ α y ∂ β ξ b j (y, ξ)| = δ+γ=β β! δ!γ! r |α|-|γ| j ∂ δ ξ (1 -χ(ξ))∂ α y ∂ γ ξ b(x 0 + r j y, ξ r j ) ≤ Cr -|β| j (1 + |ξ| r j ) -|β| 1 1 2 ≤|ξ| + C δ =0 δ+γ=β r -|γ| j (1 + |ξ| r j ) -|γ| 1 1 2 ≤|ξ|≤2 ≤ C(1 + |ξ|) -|β|
Now we localize in space

B j χµ j = Q j χf j + [χ, Q j ]f j -[χ, B j ]µ j .
Next we shall apply a smoothing operator ζ (D x ) where ζ (ξ) = χ( ξ). Since

ζ (D x )χν j = χ( D x )(χν j ) = χ (χν j ) →0 χν, χ (x) := 1 d χ( x ),
we can choose a sequence j < 1 8 converging to 0 fast enough so that u j := ζ j (D x )χν j = χ j (χν j ) * j→∞ χν.

Thus to get {χν

j } precompact in L 1 loc (R d ) it is enough to get {u j } precompact in L 1 loc (R d ).
We have by applying ζ j (D x ):

(3.18) B j ω 0 u j = B j V j + Q j g j -[ζ j (D x ), B j ]χµ j + [ζ j (D x ), Q j ]χf j + ζ j (D x )([χ, Q j ]f j -[χ, B j ]µ j ), where V j := ζ j (D x )(ω 0 χν j -χµ j ), g j = ζ j (D x
)χf j Next we consider the family of pseudo-differential operators

B j ω 0 := B j ω 0 , of symbols b j ω 0 (y, ξ) := b j (y, ξ)ω 0 = (1 -χ(ξ))b(x 0 + r j y, ξ r j
)ω 0 , which are bounded in S 0 cl (R d ) uniformly with respect to j. We split:

b j ω 0 (y, ξ) = (1 -χ(ξ))b 0 (x 0 + r j y, ξ)ω 0 + (1 -χ(ξ))(b -b 0 )(x 0 + r j y, ξ r j )ω 0 .
From (3.14) we get that the principal symbols b 0,l satisfy

|b 0 (y, ξ)ω 0 | ≥ c > 0,
for all ξ ∈ S d-1 and y close to x 0 , and following the same lines as in (3.17), the symbol

(1 -χ(ξ))(b -b 0 )(x 0 + r j y, ξ r j )ω 0
is uniformly in S -1 and bounded by Cr j (1 + |ξ|) -1 . So for j large enough, B j ω 0 is elliptic on B 1 uniformly with respect to j, in the sense: thus in particular near the support of χ. For j ≥ J we define

∃ J, c > 0, |b j ω 0 (y, ξ)ω 0 | ≥ c, ∀j ≥ J,
p j (x, ξ) = χ(x)(1 -χ(ξ)) b j ω 0 (x, ξ) ,
which is in S 0 cl uniformly with respect to j, so that, by symbolic calculus of pseudodifferential operators, and by adding the additional cutt-off χ for later use,

P j χB j ω 0 χ = χ(x)(1 -χ(D x ) + R j = χ(x) + R j ,
where χR j χ = R j and the family of operators R j is a family of pseudo-differential operators uniformly bounded in Ψ -1 cl , i.e. (3.19) ∀α, β, sup

j sup x,ξ (1 + |ξ|) 1+|β| |∂ α x ∂ β ξ (r j (x, ξ))| < +∞.
As u j = χ j (χν j ) is supported in an j < 1 8 neighborhood of B 3 4

we have u j = χu j .

Then we can apply χP j χ to (3.18) to get:

(3.20)

u j = -R j (u j ) + χP j χ(B j V j + Q j g j ) -χP j χ[ζ j (D x ), B j ]χµ j + χP j χ[ζ j (D x ), Q j ]χf j + χP j χζ j (D x )([χ, Q j ]f j -[χ, B j ]µ j ).
Step 2: The compactness arguments.

To get Proposition 3.4 we want to prove that {u j } is precompact in L 1 loc (R d ), and we have to study the contributions of each terms.

We start with the term R j (u j ) = χR j χu j in (3.20). From (3.19) and Corollary 6.7 applied with δ = -1 the operators R j are bounded from M 0 to W 1-,1 (R d ) for 0 < κ < 1, uniformly in j. Also, by the weak convergence of u j we get that χu j is uniformly bounded in M 0 . Then Proposition 6.4 with s = 1 -κ implies that {R j (u j )} is pre-compact in

L 1 loc (R d ).
The same argument allows to handle the contribution of χP j χ[ζ j (D x ), Q j ]χf j , by using this time that χf j is uniformly bounded in M 0 due to the first assertion in Lemma 3.3 (iv), and also the fact that the symbol ζ( j ξ) of the operator ζ j (D x ) is in S 0 cl (R d ) uniformly with respect to j. The sequence { χP j χ[ζ j (D x ), B j ]χµ j } is also precompact in L 1 by the same argument.

Let us now study the sequence { χP j χζ j (D x )[χ, B j ]µ j }. Here the main difference with respect to the previous analysis is that µ j is not necessarily bounded in M 0 . However, according to Lemma 3.2 (ii), µ j is bounded in M d+1 (R d ) only, i.e. its mass on balls of radius R > 1 can grow at most like R d+ 1 2 . From Proposition 6.8, we get that the family of operators χP j χζ j (D x )[χ, B j ] x d+ 1 2 is uniformly in j bounded from M 0 to W 1-,1 for an ∈ (0, 1). Then the boundedness of x -(d+ 1 2 ) µ j in M 0 from Proposition 3.2 (ii) and Proposition 6.4 give that { χP j χζ j (D x )[χ, B j ]µ j } is precompact in L 1 loc (R d ). We get similarly the relative compactness of the sequence { χP j χζ j (D x )[χ, Q j ]f j } by using the boundedness of x -(d+ 1 2 ) f j in M 0 which follows from the second assertion of Lemma 3.3 (iv).

It remains to study the second term in (3.20):

χP j χ(B j V j + Q j g j ).
For this we first note that we have the following property:

(3.21) lim j→+∞ V j L 1 + g j L 1 = 0.
The L 1 -convergence of V j 's follows from Lemma 3.2 (i)-(ii) exactly as in [9, (2.8)]. For sake of completeness let us recall the short argument. For j < 1 8 , the function V j is supported in B 3/4 + B 1/8 ⊆ B 7/8 , and

B 7/8 |V j |(x)dx ≤ B 1 |ω 0 χν j -χµ j | ≤ ω 0 T x 0 ,r j |µ| s -T x 0 ,r j µ (B 1 )
|µ| s (B(x 0 , r j )

≤ ω 0 |µ| s -µ s B(x 0 , r j ) |µ| s (B(x 0 , r j )) + |µ a |B(x 0 , r j ) |µ| s (B(x 0 , r j ) ≤ - B(x 0 ,r j ) dµ d|µ| (x 0 ) - dµ d|µ| (x) + |µ a |B(x 0 , r j ) |µ| s (B(x 0 , r j ) .
Then we deduce from Lemma 3.2 (i)-(ii) the L 1 -convergence of V j to zero. On the other hand, from Lemma 3.3 (iv), we get that χf j , and hence also g j = χ j χf j , converges to 0 in L 1 . Now we can use L 1 -L 1,∞ estimates for the 0-order operators χP j χB j l and χP j χQ j that are Calderon-Zygmund operators, the fact that the bound depends only on a finite number of semi-norms [START_REF] Coifman | Au-delà des opérateurs pseudo-différentiels[END_REF], that these semi-norms are uniformly bounded in j, and the convergence (3.21) to get a convergence in measure:

(3.22) sup λ≥0 λL d ({| χP j χ(B j V j + Q j g j )| > λ}) ≤ sup λ≥0 λL d ({ χP j χV j | > λ 2 }) + sup λ≥0 λL d ({| χP j χQ j g j | > λ 2 }) ≤ C V j L 1 + C g j L 1 j→∞ -→ 0.
To conclude, we are going to use following result from [START_REF] Philippis | On the structure of A-free measures and applications[END_REF]. a) The sequence h j converges weakly to 0,

h j * 0 in D (R d )
b) The negative part of h j tends to 0 in measure

∀λ > 0, lim j→+∞ L d ({h - j > λ}) = 0 c) The sequence of negative parts h - j is equi-integrable, lim L d (E)→0 sup j∈N E h - j dx = 0 Then h j L 1 loc (B 1 )
-→ 0.

Proof. Let us recall for completeness the short proof from [START_REF] Philippis | On the structure of A-free measures and applications[END_REF].

Let ϕ ∈ C ∞ 0 (B 1 ) with 0 ≤ ϕ ≤ 1. Since h j is supported in B 1 , it is enough to prove lim j→+∞ B 1 ϕ|h j |dx = 0.
We write

B 1 ϕ|h j |dx = B 1 ϕh j dx + 2 B 1 ϕh - j dx ≤ B 1 ϕh j dx + 2 B 1 h - j dx.
As h j * 0 it is enough to show that the last integral converges to 0. Then from the equi-integrability, for any ε > 0 there exists δ > 0 such that

L d (E) ≤ δ ⇒ sup j∈N E h - j dx ≤ ε.
From the convergence in measure we have

∃J; ∀j ≥ J, L d ({h - j > ε} ≤ δ. We deduce that for j ≥ J B 1 h - j dx ≤ {h - j >ε}∩B 1 h - j + {h - j ≤ε}∩B 1 h - j ≤ ε(1 + L d (B 1 )).
We now check that h j := χP j χ(B j V j + Q j g j ), satisfies the assumptions of Lemma 3.5: a) from (3.21) we have that V j and g j are converging to 0 in L 1 , thus h j converges weakly to 0, b) from (3.22), h j and hence also h - j tends to 0 in measure, c) as u j ≥ 0, the negative part of h j is bounded by (χν j -h j ) + which we already proved it is relatively compact in L 1 loc (R d ). As a consequence, the negative part of h j is equi-integrable. Thus by applying Lemma 3.5 we deduce that the sequence {h j } converges to 0 in L 1 loc (B 1 ). This concludes the fact that u j converges in L 1 loc (R d ), and finishes the proof of Proposition 3.4 and hence of Theorem 1.2. 4. Proof of Theorems 1.4, 1.5 and 1.9

Proof of Theorem 1.4. The first assertion of Theorems 1.4 is equivalent to

Π 13 (W F L 1 (µ) \ 0 pol )) = ∅ ⇐⇒ µ ∈ L 1 loc . If µ ∈ L 1 loc then all points (x, ω) ∈ R d × R m are in the complementary of the projection of the wave front set, Π 13 (W F L 1 (µ) \ 0 pol )) since (3.1) is satisfied with B = χ ∈ C ∞ c (R d ) equal to 1 near x. Conversely, if Π 13 (W F L 1 (µ) \ 0 pol )) = ∅ then Theorem 1.2 ensures that µ ∈ L 1 .
Concerning the second assertion, let us first consider x / ∈ supp |µ| s . Then there exists χ ∈ C ∞ c (R d ) equal to 1 near x and such that χµ ∈ L 1 . The first assertion ensures that W F L 1 (χµ) = 0 pol , so using (1.4) we get x / ∈ Π 1 (W F L 1 (µ) \ 0 pol )). So we have the inclusion Π 1 (W F L 1 (µ)\0 pol )) ⊆ supp |µ| s . On the other hand Theorem 1.2 gives us the existence of a set N of null |µ| s -measure such that for x ∈ N c we have (x, dµ d|µ| (x)) ∈ Π 13 (W F L 1 (µ) \ 0 pol )) so in particular x ∈ Π 1 (W F L 1 (µ) \ 0 pol )). Therefore we get the last assertion.

Proof of Theorem 1.5. Let Ṽ a neighborhood of x on which the elliptic property remains valid, and χ ∈ C ∞ c (R d ) a smooth localisation supported in Ṽ , equal to 1 on V ∈ V(x).

Then Definition 1.1 yields Ṽ × R d * × R m * ⊆ W F L 1 (µ) c , which in view of (1.4) implies W F L 1 (χµ) = 0 pol . Then Theorem 1.4 implies χµ ∈ L 1 loc , thus µ | V ∈ L 1 .
Proof of Theorem 1.9. We consider first the case Q 1 = ... = Q N = Id. Theorem 1.2 gives us the existence for |µ| s -almost all x of at least one

ξ x ∈ R d * such that (x, ξ x , dµ d|µ| (x)) ∈ W F L 1 (µ).
As Aµ ∈ L 1 , from definition (1.3) of W F L 1 (µ) we get a 0 (x, ξ x ) dµ d|µ| (x) = 0, thus the conclusion of Theorem 1.9.

The case

Q i = Id for 1 ≤ i ≤ p and Q i = Id for p < i ≤ N follows by considering the equation (A, -Q 1 , ..., -Q p )(µ, f 1 , ..., f p ) = N i=p+1 f i ∈ L 1 .
Thus we can use the conclusion from the first case with the extended measure σ = (µ, f 1 dL d , ..., f p dL d ) as by uniqueness of Radon-Nikodym decomposition we have σ = (g, f 1 , ..., f p )dL d + ( dµ d|µ| , 0 R p )d|µ| s , d|σ| s = d|µ| s and dσ d|σ| = ( dµ d|µ| , 0 R p ).

On the singular part of elementary constrained measures and more

In [START_REF] Philippis | On the structure of A-free measures and applications[END_REF] was considered the question of elementary constrained measures, i.e. measures of the form µ 0 = ω 0 ν with ω 0 ∈ R m * , ν ∈ M t (R d , R), constrained to vanish under the action of a first order linear constant coefficient operator

|α|=1 A α ∂ α x , A α ∈ R n×m , satisfying C := {ξ ∈ R d , |α|=1 A α ω 0 ξ α = 0} = {0 R d }.
It was noticed in [START_REF] Philippis | On the structure of A-free measures and applications[END_REF] that in this case, passing in Fourier the equation yields supp ν ⊆ C and ν is invariant in the directions orthogonal to C. Indeed, for x ∈ C ⊥ , using that ν is supported in C,

ν(x + x) = e i(x+x)ξ dν(ξ) = C e i(x+x)ξ dν(ξ) = ν(x).
However, this property gives useful informations only for first order operators as in this case, the set C is a vector space. For higher order operators the example of a simple scalar wave equation

(n = m = 1, ∂ 2 t -∆) for which the characteristic manifold is C = {(τ, η) ∈ R d+1 ; τ 2 = |η| 2 }, whose orthogonal set C ⊥ is reduced to {0 R d+1 },
shows that this invariance property may provide no information. In this section, we give a few elements toward the understanding of more general cases, providing information about the structure of polarisation of the singular part of a constrained measure in non-elementary constrained measure cases. We shall not use our wave front W F L 1 , but rely rather on propagation of singularities ideas introduced previously for the study of systems of PDE's, and in particular systems of wave equations (see [START_REF] Burq | Mesures de défaut de compacité, application au système de Lamé[END_REF]). We first notice that the special choice µ 0 = ω 0 ν reduces the study to a system of n scalar equations on the measure ν,

A j ν = |α|=1 A j α ω 0 ∂ α x ν = 0,
where A j α is the j-th line of the matrix A α , and the invariance along C ⊥ is just the invariance of ν by each of the n transport equations A j . When studying propagation of singularities for systems, the natural extension of scalar equations (see [START_REF] Denker | On the Propagation of polarisation Sets for Systems of Real Principal Type[END_REF]Sections 3 & 4] and [START_REF] Burq | Mesures de défaut de compacité, application au système de Lamé[END_REF] for boundary value problems) is to study systems with diagonal (or at least diagonalisable) scalar principal parts, and we start with an elementary result in this simpler case. Lemma 5.1. Consider a smooth vector field on R d

A := d i=1 b i (x) ∂ x i , t A := A + div(b) = d i=1 ∂ x i • b i , a function H ∈ C(R d , M m×m (R)) and µ 0 ∈ M loc (R d , R m ) solution of the system 2 (5.1) t Aµ 0 + Hµ 0 = div(bµ 0 ) + Hµ = d i=1 ∂ x j (b i µ 0,k ) + m p=1 h k,p µ 0,p k=1,...,m = 0 R m .
Then the set

Z = { x, dµ 0 d|µ 0 | (x) , x ∈ R d } is |µ 0 |-a.e.
invariant by the flow

(x 0 , ω 0 ) → x(s), ω (s) ω(s)
, where x(s) = φ(s, x 0 ) and ω(s) = φ(s, x 0 , ω 0 ) are defined by

ẋ(s) = b(x(s)), x(0) = x 0 , ω(s) = -H(x(s))ω(s), ω(0) = ω 0 .
Proof. The proof is easy: we just solve the equation! More precisely, we solve the associated equation

(5.2) ∂ s µ + div(bµ) + Hµ = 0, µ | s=0 = µ 0 2
Abusing notations we still denote by A the vector field AId R m and t A = i ∂x i • bi Id R m its transpose.

Since by the duality method the solution to this equation is unique, if µ 0 solves (5.1), then the unique solution to (5.2) is given by µ(s) ≡ µ 0 . Let us now assume that µ solves (5.2) and define ν by

ν = φ(-s, •) µ ⇔ µ = φ(s, •) ν.
Here we abuse slightly notations and denote by φ(s, •) the push forward of the measure µ by the map (s, x) → (s, φ(s, x)), defined by φ(s, •) µ, ψ = µ, ψ(s, φ(s, x)) ,

for any ψ ∈ C ∞ 0 (R d+1 , R m )
. By using the definition of φ we get

∂ s µ + div(bµ) + Hµ, ψ = -µ, (∂ 1 + b.∇ x )ψ + Hµ, ψ = -ν, (∂ 1 ψ + b.∇ x ψ) | (s,φ(s,x)) + Hµ, ψ = -ν, ∂ s (ψ(s, φ(s, x))) + Hµ, ψ = ∂ s ν, ψ | (s,φ(s,x)) + H(•) φ(s, •) ν, ψ = φ(s, •) (∂ s ν + H(φ(s, x))ν), ψ .
We deduce that µ solves (5.2) if and only if ν = φ(-s, •) µ(s, •) solves

∂ s ν + H(φ)ν = 0, ν | s=0 = µ 0 .
To solve this equation we apply the variation of parameter method and compute, with C(s, x) to be defined,

∂ s (C(s, x) ν)(s, x) = (∂ s C(s, x) -C( s, x)H(φ(s, x)) ν(s, x).
Let us now define C as the solution to the differential equation

∂ s C(s, x) = C(s, x)H(φ(s, x)), C | s=0 = Id.
Remark that C is invertible as C -1 is the solution to

∂ s C -1 (s, x) = -H(φ(s, x))C -1 (s, x), C -1 | s=0 = Id. We get then C(s, x)ν(t, •) = C(0, x)ν(0, •) = µ 0 . Summarising we proved µ 0 = µ(s, •) = φ(s, •) ν(s, •) = φ(s, •) C -1 (s, x)µ 0 ⇔ φ(-s, •) µ 0 = C -1 (s, x)µ 0 .
This implies that the polarisation dµ 0 d|µ 0 | of µ 0 at φ(s, x) is colinear to C -1 (s, x) applied to the polarisation of µ 0 at x (colinear rather than equal because the Jacobian determinant of the change of variables x → φ(s, x) is not necessarilly equal to 1 and C -1 is not necessarily an isometry). Now, we turn to a more complicated setting involving a first order non diagonal equation. We consider two smooth vector fields A 1 , A 2 given as previously by functions b 1 and b 2 on

R 2 and µ = (µ 1 , µ 2 ) ∈ M loc (R 2 , R m 1 +m 2 ) solution to the coupled system (5.3) t A 1 0 0 t A 2 µ 1 µ 2 + H 11 H 12 H 21 H 22 µ 1 µ 2 = 0
where we now assume for simplicity that the matrices H jk is C ∞ . We can now state our result which is reminiscent of propagation of singularities type results.

Theorem 5.2. Assume that the vector fields A 1 and A 2 are linearly independent at each point x ∈ R 2 . Then for j ∈ {1, 2} the sets

Z j = { x, dµ j d|µ j | (x) , x ∈ R d }
are |µ j,s |-a.e. invariant by the flows

(x 0 , ω 0 ) → (x j (s), ω j (s) ω j (s) ), where ẋj (s) = b j (s), x j (0) = x 0 , ωj (s) = -H jj (x j (s)) ω j (s), ω j (0) = ω 0 .
In other words, as far as their singular parts are concerned, the propagation formulas for µ j,s are obtained by forgetting the coupling terms H 12 and H 21 in the equation.

Remark 5.3. For conciseness, we chose to work with smooth vector fields A j . It is however very likely that this kind of results holds for Lipschitz vector fields. It would be interesting to apply the methods developed in [START_REF] Ambrosio | Continuity equations and ODE flows with non-smooth velocity[END_REF] to deal with lower regularity.

Proof. We reduce the proof to Lemma 5.1 by showing that locally we have the uncoupling property:

(5.4) t A j µ j,s + H jj µ j,s = 0.

We work near a point x 0 ∈ R 2 and can replace µ j by χ(x)µ j , χ ∈ C ∞ 0 (R 2 ) equal to 1 near x 0 and after a linear change of variables, we can assume that

A 1 (x 0 ) = t A 1 (x 0 ) = ∂ ∂x 1 , A 2 (x 0 ) = t A 2 (x 0 ) = ∂ ∂x 2 .
Consider now a smooth cut off ζ 1 (ξ) vanishing near 0, homogeneous of degree 0 outside { ξ ≥ 1} and equal to 1 in a small conical neighborhood of (0, 1) ∪ (0, -1). The first step is the following Lemma.

Lemma 5.4. Near x 0 we have

(1 -ζ 1 )(D x )µ 1 ∈ W 1-,1 (R 2 ).
Notice also that this implies

µ 1,s = (ζ 1 (D x )µ 1 ) s . Proof. Let ζ 0 ∈ C ∞ 0 (R 2 ) equal to 1 near 0. Since ζ 0 (D x ) ∈ Ψ -N
, we have from Corollary 6.7:

(1

-ζ 1 )ζ 0 (D x )µ 1 ∈ W 1-,1 (R 2 ).
It remains to study

(1 -ζ 1 )(1 -ζ 0 )(D x )µ 1 .
Near x 0 the principal symbol a 1 (x, ξ) of the operator t A 1 is close to its value at (x 0 , ξ) which is iξ 1 , hence it is invertible in a neighborhood of the support of (1

-ζ 1 )(1 -ζ 0 )(ξ).
As a consequence, we can define

p(x, ξ) = χ(x)(1 -ζ 1 )(1 -ζ 0 )(ξ) a 1 (x, ξ) ,
with χ ∈ C ∞ 0 smooth, equal to 1 near x 0 . Applying p(x, D x ) to the equation satisfied by µ 1 , (5.5)

t A 1 µ 1 = -H 11 µ 1 -H 12 µ 2 ,
we get by symbolic calculus

χ(x)(1 -ζ 1 )(1 -ζ 0 )(D x )µ 1 = R 1 µ 1 + R 2 µ 2 ,
where R j are matrices of pseudo-differential operators of order -1, which implies by Corollary 6.7 that

χ(x)(1 -ζ 1 )(1 -ζ 0 )(D x )µ 1 ∈ W 1-,1 (R 2 ).
Let now ζ 1 be a cut-off equal to 1 in a neighborhood of the support of ζ 1 . Applying ζ 1 (D x ) to equation (5.5) we get

(5.6) t A 1 ζ 1 (D x )µ 1 + H 11 ζ 1 (D x )µ 1 = -H 12 ζ 1 (D x )µ 2 + [ t A 1 , ζ 1 (D x )]µ 1 + [H 11 , ζ 1 (D x )]µ 1 + [H 12 , ζ 1 (D x )]µ 2 ,
Now the key point is that the r.h.s. is an L 1 function in a neighborhood of x 0 . Indeed, it is clear for the two last term as they are operators of order -1 applied to measures so Corollary 6.7 can be used 3 . Let us now study the first term. The function ζ 1 is supported in a small conical neighborhood of (0, 1) ∪ (0, -1). We can now choose a smooth function ζ 2 vanishing near 0, homogeneous of degree 0 outside { ξ ≥ 1} and equal to 1 in a small conical neighborhood of (1, 0) ∪ (-1, 0), and which vanishes on the support of ζ 1 . This is where we use crucially that we are working in R 2 : in higher dimensions, the cut-off ζ 1 would be required to vanish near the characteristic manifold of X 1 , which at x 0 is

C 1 = {ξ; ξ 1 = 0}
while the cut-off ζ 2 is required to vanish on the characteristic manifold of X 2 which is at x 0 ,

C 2 = {ξ; ξ 2 = 0}.
In dimension 2, these two manifolds intersect at 0 R 2 . In higher dimension they intersect along the plane

C 1,2 = {ξ; ξ 1 = ξ 2 = 0},
3 This is where we use the smoothness of the A j,k ; this smoothness could be relaxed to Holder continuity or even log continuity

A(x) -A(y) ≤ C -log α (|x -y|) , α > 1, x -y ≤ 1 2 .
and consequently such a choice for ζ 2 (equal to 1 near C 1 but vanishing near C 2 , apart from a neighborhood of 0) is possible only in dimension 2. Applying Lemma 5.4 exchanging the

ζ 2 = 1 ζ 2 = 1 ζ 1 = 1 ζ 1 = 1 ξ 1 ξ 2 Figure 1.
The cut-off functions roles of µ 1 and µ 2 (and the roles of the variables x 1 and x 2 ), we get by Proposition 6.6 that

(1 -ζ 2 )(D x )µ 2 ∈ W 1-,1 (R 2 ) ⇒ ζ 1 (D x )µ 2 = ζ 1 (D x )(1 -ζ 2 )(D x )µ 2 ∈ W 1-2 ,1 (R 2 ),
and consequently near x 0 ,

H 12 ζ 1 (D x )µ 2 ∈ L 1 .
Finally, to study the second term in the r.h.s. of (5.6), we apply the symbolic calculus formula (6.4) which shows that

[ t A 1 , ζ 1 (D x )] = -(∇ x a 1 .∇ ξ ζ 1 )(x, D x ) + R, R ∈ Ψ -1 . Now Rµ 1 ∈ W 1-,1 (R 2
), and since ζ 1 = 1 on a neighborhood of the support of ζ 1 , we get that ∇ ξ ζ 1 is supported where (1 -ζ 1 ) = 1 and consequently we have

(∇ x a 1 .∇ ξ ζ 1 )(x, D x ) = (∇ x a 1 .∇ ξ ζ 1 (1 -ζ 1 ))(x, D x ),
therefore Lemma 5.4 with ζ1 and Proposition 6.6 imply

(∇ x a 1 .∇ ξ ζ 1 )(x, D x )µ 1 = (∇ x a 1 .∇ ξ ζ 1 )(x, D x )(1 -ζ 1 )(D x )µ 1 ∈ W 1-2 ,1 (R 2 ).
Summarizing, we have obtained from (5.6) that in a neighborhood of x 0

t A 1 ζ 1 (D x )µ 1 + H 11 ζ 1 (D x )µ 1 ∈ L 1 (R d ).
We can now revisit the proof of Lemma 5.1, with µ 0 replaced by ζ 1 (D x )µ 1 (wich, according to Lemma 5.4 is also a measure), with the only modification that we now have an L 1 r.h.s. We get

φ 1 (-s, •) ζ 1 (D x )µ 1 -C -1 1 (s, x) ζ 1 (D x )µ 1 ∈ L 1 loc
, Passing to the singular parts (from Lemma 5.4 µ 1,s = ζ 1 (D x )µ 1 ) s ), we get near x 0 (and for small s) φ 1 (-s, •) µ 1,s = C -1 1 (s, x)µ 1,s , which implies Theorem 5.2 (and also (5.4)) for µ 1 and small s. The general case is obtained by iterating in s. The proof for µ 2 is similar.

Pseudo-differential operators

In this section we have gathered basic facts about pseudo-differential operators. We refer to [13, Chapter XVIII] for a general presentation or [START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF]Section VI.6] for a presentation closer to our needs of the pseudo-differential calculus (see also [START_REF] Alinhac | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF]). Let us recall the definitions of the symbol classes. Here we adopt the following convention:

∀α ∈ N d , |α| = d j=1 α j , α! = d j=1 α j !, ∂ α x = ∂ α 1 ∂x α 1 1 • • • • • ∂ α d ∂x α d d .
We now define the class of symbols of order k by

(6.1) S k (R d ) = {a ∈ C ∞ (R 3d ); ∀α, β, γ ∈ N d , sup x,y,ξ∈R d |∂ α x ∂ γ y ∂ β ξ a(x, y, ξ)|(1 + |ξ|) -k+|β| = a k,α,β,γ < +∞}.
The constants a α,β,γ are called semi-norms of the symbol a. Most of the time, the symbols we shall consider will not depend on the y variable, but it is convenient to allow this dependence. For simplicity we shall sometimes only consider the subclass S k cl of symbols admiting homogeneous principal symbol i.e. there exists in addition χ

∈ C ∞ c (R d ) and ãk ∈ C ∞ (R d × S d ) such that: a -(1 -χ)(ξ)|ξ| k ãk (x, ξ |ξ| ) ∈ S m-1 .
The function a k (x, ξ) := |ξ| k ãk (x, ξ |ξ| ) is the principal symbol of a. To any symbol a ∈ S k cl (R d ) we can associate an operator on S (R d ) by the formula

a(x, y, D x )u(x) = Op(a)u(x) = 1 (2π) d e i(x-y)•ξ a(x, y, ξ)u(y)dydξ,
where this integral is defined as an oscillatory integral. We shall denote Op(a) ∈ Ψ k cl (R d ). Remark that if a do not depend on the variable x, we have (6.2) Op(a)u(x) = F -1 e -iy•ξ a(y, ξ)u(y)dydξ (x).

An operator Op(a) is said to be elliptic at a point (x 0 , ξ 0 ) if the principal symbol a k (x, y, ξ) is non zero at the point (x 0 , x 0 , ξ 0 |ξ 0 | ). Definition 6.1. For any a ∈ S k (R d ) and any sequence

a i ∈ S k i (R d ) with k 0 = k > k 1 > . . . , we write a ∼ i a i ⇔ ∀M, (a - N i=0 a i ) ∈ S k N +1 .
The basic properties of symbolic calculus of pseudo-differential operators are summarized in the following Proposition 6.2. We have the following symbolic calculus properties

• For any a ∈ S k cl (R d ),
(x, ξ) ∼ N |α|≤N i |α| α! (∂ α y ∂ α ξ a(x, y, ξ)) | y=x , a(y, ξ) ∼ N |α|≤N (-i) |α| α! (∂ α x ∂ α ξ a(x, y, ξ)) | x=y .
In particular, ãk (x, ξ) = a(x, x, ξ), a k (y, ξ) = a(y, y, ξ),

and Op(a) is elliptic at a point (x 0 , ξ 0 ) if and only if Op(ã) is elliptic at (x 0 , ξ 0 ) in the sense that ãk (x 0 , ξ 0 ) = 0 (and similarly for the right quantization). • The formal L 2 adjoint of a pseudo-differential operator, Op(a), is the pseudodifferential operator Op(a * ), with a * (x, y, ξ) = a(y, x, ξ).

• For any a ∈ S k cl (R d ), b ∈ S k(R d ), there exists c ∈ S k+ k(R d ) such that Op(a) • Op(b) = Op(c).
Furthermore, if the symbols a, b and c depend only on the x variable (by the previous results we can reduce the analysis to this case), we have

(6.3) c = a b ∼ N |α|=N i |α| α! ∂ α ξ (a(x, ξ))∂ α x (b(x, ξ)).
Remark 6.3. From the explicit formula a b(x, ξ) = 1 (2π) d e i(x-y)•(η-ξ) a(x, η)b(y, ξ)dydη, we can actually get a quantitative version of (6.3). Namely, for any N 0 , each semi-norm

(6.4) a b - N ≤N 0 |α|=N i |α| α! ∂ α ξ (a(x, ξ))∂ α x (b(x, ξ)) k-N 0 -1,α,β
is bounded by a product of a finite number of semi-norms of a and b.

Recal that the Sobolev space W s,p (R d ) is defined by

W s,p (R d ) = {u ∈ S (R d ); (1 -∆) s/2 u ∈ L p (R d )},
and that we have the following property.

Proposition 6.4. For any 1 ≤ p ≤ +∞ and any s > 0

, χ ∈ C ∞ 0 (R d ), the application u ∈ W s,p (R d ) → χu ∈ L p (R d ) is compact.
Pseudo-differential operators are bounded on L p , and more generally on W s,p , for 1 < p < +∞. More precisely, we have the following result. In general, pseudo-differential operators are not continuous on L 1 and L ∞ , the basic example being the Hilbert transform (smoothed out near ξ = 0), associated to the symbol

a(x, y, ξ) = χ(ξ), χ ∈ C ∞ (R), χ | (-∞,-1) = 0, χ | (1,+∞) = 1.
We also have the following counter example (see the introduction). Let χ ∈ C ∞ 0 (R 2 ) equal to 1 near 0, and

(6.5) v(x) := log log(e|x| -1 ) ∈ W 1,1 (B 1 ), w := χv ∈ W 1,1 comp (R 2 ) ⇒ (-∆ + 1)w = χ(x) |x| 2 log 2 (e|x| -1 ) + [χ, ∆]v + w = f ∈ L 1 comp (B 1 ), D 2 w / ∈ L 1 loc .
We thus get (6.6)

D 2 (-∆ + 1) -1 f / ∈ L 1 loc , f ∈ L 1 comp .
However, by the dual estimate of the Lemma in [19, Section VI.5.3.1], we have the following weaker result (we shall use only the case p = 1). Proposition 6.6. Let δ ∈ R, > 0 and A = Op(a) ∈ Ψ δ . Then for all 1 ≤ p ≤ +∞, s ∈ R the operator A is continuous from W s,p (R d ) to W s-δ-,p (R d ). Furthermore, its norm is bounded by a finite number of semi-norms

∃N (d, ); Op(a) L(W s,p (R d ),W s-δ-,p (R d ) ≤ C sup |α|+|γ|+|β|≤N (d)
a -,α,β,γ .

Corollary 6.7. Let δ ∈ R, κ > 0 and A = Op(a) ∈ Ψ δ . Then, for all s ∈ R, the operator A is continuous from M 0 (R d ) to W -δ-κ,1 (R d ). Furthermore, its norm is bounded by a finite number of semi-norms

∃N (d, ); Op(a) L(M 0 (R d ),W -δ-κ,1 (R d ) ≤ C sup |α|+|γ|+|β|≤N (d) a -κ,α,β,γ .
Indeed, from the continuous inclusion for all > 0,

W κ/2,∞ (R d ) ⊂ C 0 (R d ),
we deduce by duality the continuous inclusion

M 0 ⊂ W -κ/2,1 (R d ),
and Corollary 6.7 follows from Proposition 6.6 with s = -κ/2 and = κ/2. We end this section with a result involving weights and bounded measures.

Proposition 6.8. Let δ ∈ R > 0, A ∈ Ψ δ and χ ∈ C ∞ 0 (R d ).
Then, for any k ∈ R + , the operator χA(1 + |x| k ) is continuous from M 0 (R d ) to W -δ-,1 (R d ). Furthermore, its norm is bounded by a finite number of semi-norms of a: According to Corollary 6.7 each term is bounded from M 0 (R d ) to W -δ-,1 (R d ), and we just have to check that the series of the norms is summable. Consider K(x, y) the kernel of the operator χA(1 + |x| k )φ k (x):

(6.8) K(x, y) = 1 (2π) d e i(x-y)•ξ χ(x)a(x, y, ξ)φ p (y)(1 + |y| k )dξ.

Remark that on the support of this kernel, x ≤ C, y ≥ 2 p-1 , and consequently, for k large enough, x -y ∼ 2 p . Integrating by parts N times in (6.9) using the identity L(e i(x-y)•ξ = -e i(x-y)•ξ , L = i(x -y) • ∇ ξ x -y 2 , we get (6.9) K(x, y) = 1 (2π) d e i(x-y)•ξ L N (a(x, y, ξ))(χ(x)φ p (y) (

Temperance

To deal with pseudo-differential operators in the passage to the limit when defining tangent measures, we need, in the definition of tangent measures, a temperance property which is actually satisfied on a set of full measure. This property is implicit in the construction by Preiss (see [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF]Theorem 2.5]). Proposition 7.1. Let ν be a compactly supported bounded non negative Radon measure on R d , and let {r j } j∈N a sequence convergent to zero. Then for ν-a.e. points x 0 and every κ > 0, there is a subsequence of {r j k } k∈N and C, c > 0 such that ν k := (T x 0 ,r j k ) ν ν(B(x, r j k )) , T x 0 ,r j k : x → x 0 + r j k x, satisfies the uniform bounds for k ∈ N and R > 1: Since 0 < |f (x 0 )| < +∞ for f dL d -almost every x 0 we get easily that in (7.1) -(7.2) we can replace R ±(d+κ) by R ±d (without the logarithmic loss). We do not know if it is the case for general ν, or even for doubling measures ν.

Proof. We basically follow the construction of tangent measures in [START_REF] Preiss | Geometry of measures in R n : distribution, rectifiability, and densities[END_REF] with minor differences, but since the temperate bound (7.1) does not seem to appear anywhere explicitly in the literature, we give below the complete proof.

We can use Fubini and get for any set A and radius R Therefore for any r > 0, δ > 0, k ∈ N, and using again (7.3) with A = K we obtain Let r j be a sequence convergent to zero and > 0. Now we consider

A {r j }, := ∪ ∞ i=1 ∩ ∞ j=i A r j , ,
where A r j , := ∪ ∞ k=1 E r j ,k,a k ∪ E r j 2 -k ,k,a k and a k = k -1-γ , γ > 0, summable so that in view of (7.5) we have ν(A r j , ) ≤ C . In particular ν(A {r j }, ) ≤ C . For any point x / ∈ A {r j }, we get for all i ∈ N the existence of j ≥ i such that x / ∈ A r j , . This yields a subsequence r jn such that x / ∈ A r jn , for all n ∈ N, thus x / ∈ E r jn ,k,a k and x / ∈ E r jn 2 -k ,k,a k for all n, k ∈ N. Then, renaming this sequence r j for simplicity, from (7.6)-(7.7) we get ν(B(x, 2 k r j )) ν(B(x, r j )) ≤ 2 kd k 1+γ , ν(B(x, 2 -k r j ))

ν(B(x, r j )) ≥ k 1+γ 2 kd , ∀j, k ∈ N.

As a consequence for all x / ∈ A {r j }, there is a subsequence along which a non-zero tangent measure at x is obtained. By defining

A {r j } := ∩ ∞ n=1 A {r j }, 1 n ,
we get a set of zero measure such that for all x / ∈ A {r j } , thus in particular x / ∈ A {r j }, 1 nx for some n x ∈ N, there is a subsequence along which a non-zero tangent measure at x is obtained and the bounds (7.1)-(7.2) are satisfied for R = 2 k , and hence for all R ≥ 1.

2 ) = 0 . 1 2Î 2 ) 2 and 0 outside B 3 4 ,

 201224 2 we shall get Proposition 3.4 below. Then the two assertions of Proposition 3.4 give us the following contradiction: 0 < ν(B 1 2 ) = lim j→∞ ν j (E j ) ≤ lim j→∞ |ν j -ν|(E j ) + ν(E j ) = lim j→∞ |ν j -ν|(B 1 Therefore to end the proof of Theorem 1.2 it remains to show Proposition 3.4. Proposition 3.4. At least on a subsequence we have ν B L d , lim j→∞ |ν j -ν|(B 1 = 0.Proof. Let χ be a smooth cut-off function equal to 1 on B 1 satisfying χ(x)dx = 1. The conclusion of the Proposition follows if we show that χν ∈ L 1 (R d ) and on a subsequence(3.13)

Lemma 3 . 5 ([ 9 ,

 359 Lemma 2.2]). Consider {h j } a sequence of L 1 -functions supported in B 1 satisfying:

  there exists a ∈ S k cl (R d ) not depending on the variable y. (resp. a ∈ S k cl (R d ) not depending on the variable x), such that Op(a) = Op( a), (resp. Op(a) = Op( a), with a

Proposition 6 . 5 (

 65 see e.g.[START_REF] Stein | Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals[END_REF] Section VI.5.2] ). Let A = Op(a) ∈ Ψ 0 cl . Then, for all 1 < p < +∞, s ∈ R, the operator A is continuous on W s,p (R d ). Furthermore, its norm is bounded by a finite number of semi-norms∃N (d); Op(a) L(W s,p (R d )) ≤ C sup |α|+|γ|+|β|≤N (d)a 0,α,β,γ .

( 6 . 7 ) 1 +

 671 ∃N (d, );χ 2 A(1 + |x| k ) L(M 0 ;W -δ,1 (R d )) ≤ C(1 + sup |α|+|γ|+|β|≤N (d)a -,α,β,γ ).Proof. With respect to Corollary 6.7, the only new point is the presence of the weight and of the cut-off. We use a dyadic partition of unity1 = p≥0 φ p (x), φ 0 ∈ C ∞ 0 (R d ), ∀p ≥ 1, φ p (x) = φ(2 -p x), φ ∈ C ∞ 0 |x| k ) = p≥0 χA(1 + |x| k )φ p (x).

1 +

 1 |y| k )dξ, and consequentlyχA(1 + |x| k )φ k (x) = op(a N ), a N = L N a(x, y, ξ))(χ(x)φ p (y)(1 + |y| k )) ∈ S -N (R d ), with semi-norms in S -N bounded by a N -N,α,β,γ ≤ C N,α,β,γ 2 -kN |α |≤|α,|β |≤|β,|γ |≤|γ|+N | a 0,α ,β ,γ . We deduce that [χ, B]φ k = χBφ k = χ 1 [χ, B]φ k is bounded from M(R d ) to W 1,1 (R d ) by C N 2 k(p-N )and we conclude by choosing N > p.

(7. 1 )Remark 7 . 2 . 1 C

 1721 ν k (B(x, R)) = ν(B(x, Rr j k )) ν(B(x, r j k )) ≤ CR d+κ , (7.2) ν k (B(x, R -1 )) = ν(B(x, R -1 r j k )) ν(B(x, r j k )) ≥ cR -(d+κ) . The proof gives actually a more precise bound (with a logarithmic loss) R d log a (1 + R), a > 1. On the other hand, for absolutely continuous measures ν = f dL d , at all Lebesgue points of f , i.e. for L d -almost every x 0 we have lim r→0 d r d B(x 0 ,r) f (x)dx = f (x 0 ).

1 L

 1 d (B(0, R)) ν(A ∩ B(x, R))dx.We consider for r > 0, δ > 0, k ∈ N the set,E r,k,δ := {x ∈ R d , ν(B(x, 2 k r)) ≥ β k,δ ν(B(x, r))},whereβ k,δ := 2 d (2 k + 1) d ) ν(K) δ ,K is a compact set where ν is supported, and δ > 0 to be chosen later. Applying (7.3) to A = E r,k,δ , R = r 2 , we get ν(E r,k,δ ) = 1 ω d ( r 2 ) d ν(E r,k,δ ∩ B(x, r2))dx,whereω d = L d (B(0, 1)). If E r,k,δ ∩B(x,r2 ) = ∅ then we can use a point z in this intersection to get(7.4) ν(E r,k,δ ∩ B(x, r2)) ≤ ν(B(x, r2)) ≤ ν(B(z, r))≤ 1 β r,k,δν(B(z, 2 k r)) ≤ 1 β r,k,δ ν(B(x, (2 k + 1)r)).

(7. 5 )

 5 ν(E r,k,δ ) ≤ δ 1 ω d ((2 k + 1)r) d ν(K) ν(B(x, (2 k + 1)r))dx = δ,and on the complementary set c E r,k,δ we have(7.6) ν(B(x, 2 k r)) ν(B(x, r)) ≤ β k,δ = 2 d (2 k + 1) d ν(K) δ ,while on the complementary set c E r2 -k ,k,δ we have similarly(7.7) ν(B(x, r)) ν(B(x, 2 -k r)) ≤ β k,δ = 2 d (2 k + 1) d ν(K) δ .

  ∀|ξ| ≥ 1, ∀y ∈ B 1 . and we can approximately invert it locally as follows. Let us consider χ ∈ C ∞

	0 (B 1 ) equal
	to 1 on B 7
	8