# MIcrolocal analysis of singular measures

Valeria Banica  Sorbonne Université, CNRS, Université de Paris, Laboratoire
Jacques-Louis Lions (LJLL), F-75005 Paris, France, and Institut Universitaire
de France (IUF<EMAIL_ADDRESS>and  Nicolas Burq
Laboratoire de mathÃ©matiques d’Orsay, CNRS, Université Paris-Saclay, Bât.
307, 91405 Orsay Cedex, France, and Institut Universitaire de France (IUF)
<EMAIL_ADDRESS>
###### Abstract.

The purpose of this article is to investigate the structure of singular
measures from a microlocal perspective. Motivated by the result of De
Philippis-Rindler [10], and the notions of wave cones of Murat-Tartar [19, 20,
26, 27] and of polarisation set of Denker [9] we introduce a notion of
$L^{1}$-regularity wave front set for scalar and vector distributions. Our
main result is a proper microlocal characterisation of the support of the
singular part of tempered Radon measures and of their polar functions at these
points. The proof is based on De Philippis-Rindler’s approach reinforced by
microlocal analysis techniques and some extra geometric measure theory
arguments. We deduce a sharp $L^{1}$ elliptic regularity result which appears
to be new even for scalar measures and which enlightens the interest of the
techniques from geometric measure theory to the study of harmonic analysis
questions. For instance we prove that
$\Psi^{0}L^{1}\cap\mathcal{M}_{loc}\subseteq L^{1}_{loc},$ and in particular
we obtain $L^{1}$ elliptic regularity results as $\Delta u\in
L^{1}_{loc},D^{2}u\in\mathcal{M}_{loc}\Longrightarrow D^{2}u\in L^{1}_{loc}.$
We also deduce several consequences including extensions of the results in
[10] giving constraints on the polar function at singular points for measures
constrained by a PDE, and of Alberti’s rank one theorem. Finally, we also
illustrate the interest of this microlocal approach with a result of
propagation of singularities for constrained measures.
MSC classification codes: primary 42B37 (Harmonic analysis and PDEs),
secondary 28B05 (Vector-valued set functions, measures and integrals), 35A18
(Wave front sets in context of PDEs), 35Jxx (Elliptic equations and elliptic
systems).

##  1\. Introduction

###  1.1. The framework

We start by recalling some classical facts about the class of locally bounded
Radon measures on $\mathbb{R}^{d}$ with values in $\mathbb{R}^{m}$,
$\mathcal{M}_{loc}(\mathbb{R}^{d},\mathbb{R}^{m})$ (see for instance [24] or
[17]). The Radon-Nikodym Theorem allows to write the polar decomposition
$d\mu=\frac{d\mu}{d|\mu|}d|\mu|$, where the non negative measure $|\mu|$ is
the total variation of the measure $\mu$ and the function
$\frac{d\mu}{d|\mu|}\in L^{1}_{loc}(\mathbb{R}^{d}(d|\mu|),\mathbb{S}^{m-1})$
is the Radon-Nikodym derivative of $\mu$ with respect to $|\mu|$, called the
polar function of $\mu$. Moreover, the Radon-Nikodym Theorem gives the
Lebesgue decomposition of $|\mu|$ with respect to the Lebesgue measure,
$\mathcal{L}^{d}$, so we have

$d\mu=gd\mathcal{L}^{d}+\frac{d\mu}{d|\mu|}d|\mu|_{s},$

where $g\in L^{1}(\mathbb{R}^{d},\mathbb{R}^{m})$ and the positive measure
$|\mu|_{s}$ satisfies $|\mu|_{s}\perp\mathcal{L}^{d}$. Finally, the Radon-
Nikodym Theorem applied to each component of $\mu$ with respect to
$\mathcal{L}^{d}$ implies that the singular part of $\mu$ is
$d\mu_{s}=\frac{d\mu}{d|\mu|\,}d|\mu|_{s}$, and in particular
$\frac{d\mu}{d|\mu|\,}=\frac{d\mu_{s}}{d|\mu|_{s}\,}$, $\mu_{s}$ a.e. and
$|\mu_{s}|=|\mu|_{s}$.

In the following discussion and results we shall consider pseudodifferential
operators and we have gathered in Section 6 the definitions and basic results
from the theory we need. We denote by $S^{k}_{\text{cl}}(\mathbb{R}^{d})$ the
class of symbols of order $k$ and by $\Psi^{k}_{\text{cl}}(\mathbb{R}^{d})$
the class of classical pseudodifferential operators of order $k$. For $u$ a
tempered distribution on $\mathbb{R}^{d}$ with values in $\mathbb{R}^{m}$,
that is $u=(u_{1},...,u_{m})$ with
$u_{i}\in\mathcal{S}^{\prime}(\mathbb{R}^{d},\mathbb{R})$, if $A=(A_{ij})$ is
a $n\times m$ matrix of pseudodifferential operators then the distribution
$Au\in\mathcal{D}^{\prime}(\mathbb{R}^{d},\mathbb{R}^{n})$ is defined by

$(Au)_{j}=\sum_{k}A_{jk}u_{k},\quad 1\leq j\leq n.$

For $A\in\Psi^{k}_{\text{cl}}(\mathbb{R}^{d})$ we denote by
$a_{k}:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ (or simply by
$a$) its homogeneous principal symbol and similarly for matrices of
pseudodifferential operators.

We recall now the following result of De Philippis and Rindler that connects
the polarisation of the singular part of an $\mathcal{A}$-constrained measure
and the wave cone of non-elliptic directions of the operator $\mathcal{A}$
introduced by Murat and Tartar [19, 20, 26, 27]:

$\Lambda_{\mathcal{A}}:=\underset{|\xi|=1}{\bigcup}\ker(\sum_{|\alpha|=k}A_{\alpha}\xi^{\alpha}).$

######  Theorem ([10, Theorem 1.1]).

Let $\mathcal{A}=\sum_{|\alpha|\leq
k}A_{\alpha}\partial^{\alpha},k\in\mathbb{N}$, $A_{\alpha}\in M_{n\times m}$,
and $\mu\in\mathcal{M}_{loc}(\mathbb{R}^{d},\mathbb{R}^{m})$. Then

$\mathcal{A}\mu\overset{\mathcal{D}^{\prime}}{=}0_{\mathbb{R}^{n}}\quad\Longrightarrow\quad\frac{d\mu}{d|\mu|}(x)\overset{|\mu|_{s}-a.e}{\in}\Lambda_{\mathcal{A}}.$

This result is impressive by its statement, its proof and its deep
consequences in geometric measure theory proved in [10] by taking the
constraint operator $\mathcal{A}$ to be $div,curl$ and $curlcurl$. We shall do
now two series of remarks, one with respect to elliptic constraints, and one
on the link with the wave front set notions.

First we notice that [10, Theorem 1.1], even for variable coefficients
differential operators, is only relevant for systems ($m\geq 2$), and non
elliptic equations, i.e. the operator $\mathcal{A}$ is not elliptic. Indeed,
for the scalar case, and non elliptic equation, [10, Theorem 1.1] gives no
information. On the other hand, if the differential operator $\mathcal{A}$ is
elliptic and of order $0$, even space dependent, i.e it is the multiplication
by an invertible space dependent matrix, it is straightforward that

$\mathcal{A}(x)\mu=f\in L^{1}\Rightarrow\mu=\mathcal{A}(x)^{-1}f\in L^{1},$

so $\mu$ has no singular part. Finally, if $\mathcal{A}(x,D_{x})$ is elliptic
of order $k>0$, $\mathcal{A}\mu=f\in\mathcal{M}$, then the pseudodifferential
calculus shows, after replacing $\mathcal{A}$ by $\mathcal{A}^{*}\mathcal{A}$
if $m\neq n$, that there exists an operator $\mathcal{B}\in M_{m\times
m}(\Psi^{-k}_{cl}(\mathbb{R}^{d}))$, such that
$\mathcal{B}\mathcal{A}=\text{Id}+\mathcal{R},$ where $\mathcal{R}$ maps
$\mathcal{S}^{\prime}(\mathbb{R}^{n})$ to $C^{\infty}(\mathbb{R}^{n})$. As a
consequence, we get

$\mu=\mathcal{B}\mathcal{A}\mu-\mathcal{R}\mu=\mathcal{B}f-\mathcal{R}\mu,\qquad
f\in\mathcal{M},$

and since pseudodifferential operators of negative order send $\mathcal{M}$ to
$L^{1}$ (see Corollary 6.7) we also get that $\mu$ has no singular part.

We now notice that the general scalar case, when $\Lambda_{\mathcal{A}}$
corresponds to the characteristic set, can be seen also by using Hörmander’s
theorem:

$\mu\notin L^{1}\mbox{ at }x\Rightarrow\mu\notin\mathcal{C}^{\infty}\mbox{ at
}x\Rightarrow\exists\xi,(x,\xi)\in
WF_{\mathcal{C}^{\infty}}\mu\overset{\mathcal{A}\mu=0}{\Rightarrow}a_{k}(x,\xi)=0.$

We recall that the classical wave front set of a distribution
$u\in\mathcal{D}(\mathbb{R}^{d},\mathbb{R})$, which can be written as:

$WF(u):=\\{(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d*};\,\mathbb{R}=\cap_{Au\in\mathcal{C}^{\infty},A\in\Psi(\mathbb{R}^{d})}\ker
a(x,\xi)\\},$

introduced by Hörmander gives a proper characterisation of the singular
support of a distribution, i.e. the complementary of points where the
distribution is $\mathcal{C}^{\infty}$. This lead us to looking for a wave
front set notion appropriate for the present context. For vectorial
distributions $u\in\mathcal{D}(\mathbb{R}^{d},\mathbb{R}^{m})$, a refinement
of the classical $\mathcal{C}^{\infty}$-regularity wave front set that takes
into account the polarization was introduced by Denker [9] following
suggestions by of Hörmander:

$WF_{pol}(u):=\\{(x,\xi,\omega)\in\mathbb{R}^{d}\times\mathbb{R}^{d*}\times\mathbb{R}^{m};\,\omega\in\cap_{Au\in\mathcal{C}^{\infty},A\in
M_{1\times m}(\Psi(\mathbb{R}^{d}))}\ker a(x,\xi)\\}.$

In particular, Proposition 2.7 in [9] on the action of the principal symbol on
the fiber of the polarization wave front set encouraged us to look for a
purely microlocal setting (see our main result Theorem 1.2). Moreover, results
of propagations of $C^{\infty}$ or $L^{2}$-based singularities are available
for the polarisation wave fronts sets in [9] (see also Gérard [12]).

Summarizing the last remarks, the starting motivation for us to introduce a
$L^{1}$\- wave front notion was to make a connection between De Phillipis-
Rindler results and the polarisation wave fronts introduced in microlocal
analysis by Dencker [9]. The proof of [10, Theorem 1.1], involving powerful
measure theory notions, will allow us to prove our main result Theorem 1.2 and
to treat critical $L^{1}$-cases that were not at reach by the classical
Hörmander theory.

###  1.2. Definition of a $L^{1}$-wave front set for distributions and the
main result

In this article we introduce an $L^{1}$-regularity wave front set for vector
valued tempered distributions. For simplicity, we chose to work with tempered
distributions and tempered Radon measures rather than distributions and
locally bounded Radon measures, which would have been possible by localising
our definitions of wave fronts with a cut off (see Remark 1.3 1)

######  Definition 1.1.

For $u\in\mathcal{S}^{\prime}(\mathbb{R}^{d},\mathbb{R}^{m})$ we define the
set

$WF_{L^{1}}(u)\subseteq(T^{*}\mathbb{R}^{d}\setminus\\{0\\})\times(\mathbb{R}^{m}\setminus\\{0\\}))=\mathbb{R}^{d}\times\mathbb{R}^{d*}\times\mathbb{R}^{m*}$

by the following: $(x_{0},\xi_{0},\omega_{0})\notin WF_{L^{1}}(u)$ if and only
if there exist $A\in M_{1\times m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$,
$N\in\mathbb{N}^{*},Q_{i}\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})$ and
$f_{i}\in L^{1}(\mathbb{R}^{d},\mathbb{R})$ for $1\leq i\leq N$, satisfying

(1.1) $Au=\sum_{i=1}^{N}Q_{i}f_{i},$

with $A$ elliptic at $(x_{0},\xi_{0},\omega_{0})$ in the sense that

$a_{0}(x_{0},\xi_{0})\,\omega_{0}\neq 0.$

This definition is equivalent to:

(1.2)
$WF_{L^{1}}(u)=\\{(x,\xi,\omega)\in\mathbb{R}^{d}\times\mathbb{R}^{d*}\times\mathbb{R}^{m*};\,\omega\in\cap\ker
a_{0}(x,\xi)\\},$

where the intersection is over all possible $A\in M_{1\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$,
$N\in\mathbb{N}^{*},Q_{i}\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})$ and
$f_{i}\in L^{1}(\mathbb{R}^{d},\mathbb{R})$ for $1\leq i\leq N$, such that
$Au=\sum_{i=1}^{N}Q_{i}f_{i}.$

Among the class $\mathcal{M}_{loc}(\mathbb{R}^{d},\mathbb{R}^{m})$ we shall
consider the tempered Radon measures

$\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{m}):=\cup_{\eta\in\mathbb{R}^{+}}\mathcal{M}_{\eta}(\mathbb{R}^{d},\mathbb{R}^{m}),$
$\mathcal{M}_{\eta}(\mathbb{R}^{d},\mathbb{R}^{m}):=\\{\mu\in\mathcal{M}_{loc}(\mathbb{R}^{d},\mathbb{R}^{m});\exists
C>0,\,\forall R>1,\,|\mu|(B(0,R))\leq CR^{\eta}\\}.$

Our main result is a connection between the support and the polarisation of
the singular part of a measure and its wave front set.

######  Theorem 1.2.

For $\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{m})$ we have the
inclusion:

(1.3)
$\Bigl{(}x,\frac{d\mu}{d|\mu|}(x)\Bigr{)}\overset{|\mu|_{s}-a.e}{\in}\Pi_{13}(WF_{L^{1}}(\mu)),$

where $\Pi_{13}$ denotes the projection with respect to the first and third
variables, i.e. for $|\mu|_{s}$ almost all $x$,

$\exists\xi\in\mathbb{S}^{d-1},\>\Bigl{(}x,\xi,\frac{d\mu}{d|\mu|}(x)\Bigr{)}\,{\in}\,WF_{L^{1}}(\mu).$

######  Remark 1.3.

  1. (1)

Instead of considering temperate distributions, we could consider general
distributions simply by adding a cut-off in the definition of $WF_{L^{1}}$
below i.e. replacing $Au$ by $A\chi u$ in (1.1). In the same vein, it is easy
to see that for any $\chi\in C^{\infty}_{c}(\mathbb{R}^{d})$,
$\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{m})$,

(1.4)
$WF_{L^{1}}(\chi\mu)=WF_{L^{1}}(\mu)\cap\\{(x,\xi,w)\in\mathbb{R}^{d}\times\mathbb{R}^{d*}\times\mathbb{R}^{m*,}\,\chi(x)\neq
0\\}.$

Indeed, let $\widetilde{\chi}\in C^{\infty}_{c}(\mathbb{R}^{d})$ equal to $1$
on the support of $\chi$. Then

$A\mu=Qf\Rightarrow\widetilde{\chi}A(\chi\mu)=\chi
Qf+\widetilde{\chi}[A,\chi]\mu,$

and $\tilde{\chi}[A,\chi]$ is a pseudodifferential operator of order $-1$
which sends $\mathcal{M}_{t}$ to $L^{1}_{loc}$ in view of Proposition 6.8.

  2. (2)

We can obtain $WF_{L^{1}}$ by considering in Definition 1.1 the existence of
$n,N\in\mathbb{N}^{*},A\in M_{n\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$, $Q^{j}\in M_{1\times
N}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$ and $f^{j}\in M_{N\times
1}(L^{1}(\mathbb{R}^{d},\mathbb{R}))$ such that $(Au)_{j}=Q^{j}f^{j}$ holds
for $1\leq j\leq n$, and $a_{0}(x,\xi)\omega\neq 0_{\mathbb{R}^{n}}$. Indeed,
if $a_{0}(x,\xi)\omega_{0}\neq 0_{\mathbb{R}^{n}}$, then there exists one line
such that $(a_{0}(x,\xi)\omega_{0})_{j_{0}}\neq 0$, and choosing this line as
the $M_{1\times m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$ test operator allows
to conclude.

  3. (3)

We can also define $WF_{L^{1}}$ using operators of order $k\in\mathbb{R}$ with
the same order $k$ for both $A$ and $Q_{i}$ (by replacing $A$ by
$(1-\Delta)^{-k/2}A$).

  4. (4)

In the case of scalar distributions Definition 1.1 of $WF_{L^{1}}$ recovers
the classical shape of a subset of $T^{*}\mathbb{R}^{d}\setminus\\{0\\}$ and
the definition of ellipticity $a_{0}(x,\xi)\neq 0$ is the usual one.

  5. (5)

The set $WF_{L^{1}}(u)$ together with the trivial fibers
($\omega=0_{\mathbb{R}^{m}}$) form a closed set, conical in $\xi$ and linear
in $\omega$.

  6. (6)

For conciseness, we shall state the results involving hypothesis as equality
(1.1) with only one term of type
$\Psi^{0}_{\text{cl}}\,L^{1}(\mathbb{R}^{d},\mathbb{R})$ in the right-hand
side, but of course our results hold for finite sums of such terms.

  7. (7)

It is very likely that our results remain true even after relaxing the
smoothness assumptions on the pseudodifferential operators or even for
paradifferential operators [18]. However, for the sake of simplicity, we
decided not to pursue this issue in this article.

###  1.3. Consequences

Starting from Theorem 1.2 we shall deduce a series of results. First we get a
proper microlocal characterisation of the singular support of a scalar or
vector measure.

######  Theorem 1.4.

For $\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{m})$ we have:

$\Pi_{1}(WF_{L^{1}}(\mu)))=\varnothing\iff\mu\in
L^{1}_{loc}(\mathbb{R}^{d},\mathbb{R}^{m}),$

and moreover

$\Pi_{1}(WF_{L^{1}}(\mu)))\subseteq\mbox{supp}\,|\mu|_{s},\quad|\mu|_{s}(\mbox{supp}\,|\mu|_{s}\setminus\Pi_{1}(WF_{L^{1}}(\mu))))=0.$

Next, we obtain a series of full $L^{1}$ elliptic regularity result, that are
the central applications of Theorem 1.2. We start with the scalar case.

######  Theorem 1.5.

Consider $\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R})$. Let
$k\in\mathbb{R}$ and $A\in\Psi^{k}_{\text{cl}}(\mathbb{R}^{d})$ be elliptic at
$x_{0}\in\mathbb{R}^{d}$ in the sense

$a_{0}(x_{0},\xi)\neq 0,\quad\forall\xi\in\mathbb{S}^{d-1},$

and let $B\in\Psi^{k}_{\text{cl}}(\mathbb{R}^{d})$. Assume that

$B\mu\in\mathcal{M}_{t}(\mathbb{R}^{n}),A\mu\in\Psi^{0}_{\text{cl}}\,L^{1}(\mathbb{R}^{d}).$

Then in a neighborhood of $x_{0}$,

$B\mu\in L^{1}(\mathbb{R}^{n}).$

More generally, for vector valued measures, we have the following result.

######  Theorem 1.6.

Consider $\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{l\times m})$. Let
$k\in\mathbb{R}$ and $A\in M_{m\times
m}(\Psi^{k}_{\text{cl}}(\mathbb{R}^{d}))$be elliptic at
$x_{0}\in\mathbb{R}^{d}$ in the sense

$\ker(a_{0}(x_{0},\xi))=\\{0_{\mathbb{R}^{m}}\\},\quad\forall\xi\in\mathbb{S}^{d-1},$

and let $B,C\in M_{m\times m}(\Psi^{k}_{\text{cl}}(\mathbb{R}^{d}))$ such that
the principal symbols of $A,B$ and $C$,

$a_{k}(x,\xi),b_{k}(x,\xi),c_{k}(x,\xi)\in M_{m\times m}$

satisfy

(1.5) $a_{k}(x,\xi)b_{k}(x,\xi)=c_{k}(x,\xi)a_{k}(x,\xi).$

Assume that

$B\mu\in\mathcal{M}_{t}(\mathbb{R}^{n},\mathbb{R}^{m}),A\mu\in\Psi^{0}_{\text{cl}}\,L^{1}(\mathbb{R}^{d},\mathbb{R}^{m}).$

Then in a neighborhood of $x_{0}$,

$B\mu\in L^{1}(\mathbb{R}^{n},\mathbb{R}^{m}).$

######  Remark 1.7.

A by-product of Theorem 1.6 with $A=\text{Id}$ is that counter-examples for
the lack of continuity on $L^{1}$ of $0$-th order operators can occur only for
operators $Q\in\Psi^{0}_{\text{cl}}$ sending an $L^{1}$ function outside
$\mathcal{M}_{t}$. Indeed, if $Qf=\mu\in\mathcal{M}_{t}$ we deduce from
Theorem 1.6 with $A=B=C=\text{ Id}$ that $\mu\in L^{1}_{loc}$.

######  Remark 1.8.

If $\mathcal{B}$ is scalar i.e. $\mathcal{B}=B(x,D_{x})\text{Id},$ then
assumption (1.5) is automatically satisfied with $B=C$ which shows that
Theorem 1.6 implies Theorem 1.5. As a consequence, in the particular case
$k\in\mathbb{N}$, $B\mu=\partial_{x}^{\alpha_{0}}$, for some
$\alpha\in\mathbb{N}^{d},|\alpha|=k$, we get (under the same ellipticity
assumption on ${A}$),

$\partial_{x}^{\alpha_{0}}\mu\in\mathcal{M}_{t}(\mathbb{R}^{n},\mathbb{R}^{m}),A\mu\in\Psi^{0}_{\text{cl}}\,L^{1}(\mathbb{R}^{d},\mathbb{R}^{m})\Longrightarrow\partial_{x}^{\alpha_{0}}\mu\in
L^{1}(\mathbb{R}^{d},\mathbb{R}^{m}).$

For instance, consider $A\in M_{m\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$ elliptic for all
$x\in\mathbb{R}^{d}$ and $u\in BV(\mathbb{R}^{d},\mathbb{R}^{m})$, the set of
$L^{1}$-functions with gradient a matrix-valued finite Radon measure. We
recover the full $L^{1}$ regularity:

$A\nabla
u\in\Psi^{0}_{\text{cl}}\,L^{1}(\mathbb{R}^{d},\mathbb{R}^{d})\Longrightarrow
u\in W^{1,1}_{\text{loc}}(\mathbb{R}^{d},\mathbb{R}^{m}).$

Previously known $L^{1}$ elliptic regularity results assuming $u,Pu\in
L^{1}_{loc}$, $P$ elliptic operator of order $k$ involve loss of derivatives
or requires the use of Besov spaces $B^{k,1}_{\infty}$, see [14, Theorem 2.6
and Remark 2.7]. At the exact $L^{1}$ level, the following counterexample from
[13, Example 7.5] shows that, in general some loss is anavoidable: on the disk
of radius one, consider the function

$u(x)=\log\log(e{|x|^{-1}})\in W^{1,1}(B_{1}),$

which satisfies

$\Delta u=-\frac{1}{|x|^{2}\log^{2}(e{|x|^{-1}})}\in L^{1}(B_{1}),\quad
D^{2}u\notin L^{1}(B_{1}).$

Note that this is (of course!) not in contradiction with Theorem 1.5 as
$D^{2}u$ is not a Radon measure. Theorems 1.5 and 1.6 eliminate the loss under
the additional assumption that $\mu$ is a Radon measure. To the best of our
knowledge, even for scalar equations ($m=1$), this result is new. Let us now
give an example with a fluid dynamics flavour.

######  Example 1.9.

Let $u\in BV_{loc}(\mathbb{R}^{d},\mathbb{R}^{d})$. Assume that both
$\operatorname{div}u$ and $\operatorname{curl}u$ are in
$\Psi^{0}_{\text{cl}}(L^{1}_{loc})$. Then

$\nabla u\in L^{1}_{loc}(\mathbb{R}^{d},\mathbb{R}^{d\times d}).$

###### Proof.

Replacing $u$ by $\chi u,\chi\in C^{\infty}_{0}$, we can assume that $u$ is
compactly supported. Now we have the following equation for any derivative
$\partial_{i}u$, which is a Radon measure by the BV assumption on $u$,

(1.6)
$A(\partial_{i}u):=\frac{1}{(1+|D_{x}|^{2})^{\frac{1}{2}}}\left(\begin{array}[]{cccccc}\partial_{1}&\partial_{2}&\partial_{3}&\cdots&\cdots&\partial_{d}\\\
\partial_{2}&-\partial_{1}&0&\cdots&\cdots&0\\\
0&\partial_{3}&-\partial_{2}&0&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&\partial_{d}&-\partial_{d-1}\\\
-\partial_{d}&0&0&\cdots&\cdots&\partial_{1}\end{array}\right)\partial_{i}u\\\
=\frac{\partial_{i}}{(1+|D_{x}|^{2})^{\frac{1}{2}}}\begin{pmatrix}\operatorname{div}u\\\
\operatorname{curl}u\end{pmatrix},$

so $A(\partial_{i}u)\in\Psi_{cl}^{0}\,L^{1}(\mathbb{R}^{d},\mathbb{R})^{d+1},$
and a straightforward calculation shows that the constant coefficients
operator $A\in M_{d+1\times d}(\Psi_{cl}^{0}(\mathbb{R}^{d}))$ is elliptic:

$\forall\xi\neq 0,\,\ker a_{0}(\xi)=\\{0_{\mathbb{R}^{d+1}}\\}.$

We deduce, using the second point in Remark 1.3 that
$WF_{L^{1}}(\partial_{i}u)=\varnothing$ and then Theorem 1.2 yields
$\partial_{i}u\in L^{1}_{loc}(\mathbb{R}^{d},\mathbb{R}^{d}).$ We could also
compose by $A^{*}$ and apply Theorem 1.6 to the elliptic operators $A^{*}A$,
and the choice $B=C=\text{ Id}$. ∎

######  Remark 1.10.

In the particular case in which both $\operatorname{div}u$ and
$\operatorname{curl}u$ are in $L^{1}_{loc}(\mathbb{R}^{d})$ the result follows
by applying Alberti’s rank one theorem that ensures the existence of
$a,b\in\mathbb{R}^{d}$ such that $\nabla u-a\otimes b|\nabla u|_{s}\in L^{1}$.
Indeed this implies that $\operatorname{div}u-a.b|\nabla u|_{s}$ and
$\operatorname{curl}u-a\wedge b|\nabla u|_{s}$ are in
$L^{1}_{loc}(\mathbb{R}^{d})$, so $a.b=a\wedge b=0$ thus $\nabla u\in L^{1}$.

We now turn to an extension of De Philippis and Rindler’s result [10, Theorem
1.1]. It was stated in [10, Remark 1.4 ] that the result extends easily to
variable coefficients differential operators and ”similar statements can be
obtained if $\mu$ solves some pseudo-differential equations” by using

$\Lambda_{\mathcal{A}}(x)=\underset{|\xi|=1}{\bigcup}\ker(a(x,\xi)).$

While the proof in [10] is indeed not much perturbed by considering variable
coefficients, we shall handle here the case of general pseudodifferential
operators that induce more serious issues due to their pseudo-local character.

######  Theorem 1.11.

Let $\mu\in\mathcal{M}(\mathbb{R}^{d},\mathbb{R}^{m})$. Let $A\in M_{n\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$ such that

$A\mu\overset{\mathcal{D}^{\prime}}{=}0_{\mathbb{R}^{n}}.$

Then

$\frac{d\mu}{d|\mu|}(x)\overset{|\mu|_{s}-a.e}{\in}\Lambda_{A}(x).$

For $A$ of order $k\geq 0$ and for $Q$ or order less than or equal to $k$, the
theorem extends to the more general equation $\mathcal{A}\mu=Qf$ for $f\in
L^{1}$. In the same spirit, for $Q$ or order strictly less than $k$ the result
extends to the more general equation $\mathcal{A}\mu=Q\sigma$ with
$\sigma\in\mathcal{M}_{t}$ that can be treated viewing it as
$(A,-Q)(\mu,\sigma)=0$ and using the fact that the principal symbol of
$(A,-Q)$ is $(a_{k},0)$.

We give here the short proof of Theorem 1.11 as a corollary of the main
theorem. Theorem 1.2 gives us the existence for $|\mu|_{s}$-almost all $x$ of
at least one $\xi_{x}\in\mathbb{R}^{d*}$ such that

$(x,\xi_{x},\frac{d\mu}{d|\mu|}(x))\in WF_{L^{1}}(\mu).$

As $A\mu\in\Psi^{0}_{cl}L^{1}$, from definition (1.2) of $WF_{L^{1}}(\mu)$ we
get $\frac{d\mu}{d|\mu|}(x)\in\ker a_{0}(x,\xi_{x})$, thus the conclusion of
Theorem 1.11.

We note that Theorem 1.11 does not imply trivially Theorem 1.2. Indeed, this
would be the case if one could easily get

(1.7)
$\mu_{s}(\cap_{A\mu\in\Psi^{0}L^{1}}\,\Lambda_{A}(x))\setminus\cup_{|\xi|=1}\cap_{A\mu\in\Psi^{0}L^{1}}\ker(a_{0}(x,\xi))=0.$

Thus the difference between the two theorems is encoded in this non-trivial
information which derives from Propositions 3.1-3.3.

Theorem 1.11, that allows $0$-th order pseudodifferential operators, is
relevant both for the scalar or vector valued measures and for elliptic
equations, as illustrated by the previous elliptic regularity results. $0$-th
order pseudodifferential operators appear naturally in harmonic analysis and
PDEs, let us quote just some of the most famous ones: the Hilbert and Riesz
transforms ($H=\frac{\nabla}{|\nabla|},R_{i}=\frac{\partial_{i}}{|\nabla|}$),
Hodge operator, Leray projector and the pressure operator
($\mathbb{P}=I-\nabla\frac{\nabla}{\Delta},p(u)=\sum
R_{i}R_{j}u_{i}u_{j}$)111Notice that homogeneous $0$-th order operators are
strictly speaking not pseudodifferential operator due to the singularity at
the origin in the $\xi$ variable. However, our theorems still apply in this
context after composing by a frequency cut-off $(1-\chi(D_{x}))$, $\chi\in
C^{\infty}_{0}$ equal to $1$ near $0$, which eliminates this singularity,
while the new operator obtained by composition with $(1-\chi(D_{x}))$ has the
same principal symbol.. We want to emphasize that allowing $0$-th order
pseudodifferential operators is not anecdotal: in fluid mechanics, $0$-th
order operators are ubiquitous and their lack of $L^{1}$ (or equivalently
$L^{\infty}$) boundedness is responsible for many pathological behaviours (see
e.g. the double exponential growth of the complexity of solutions to two
dimensional Euler equations [16]).

Theorem 1.1 in [10] implied several major consequences. Our new version can be
used to relax assumptions: for instance Theorem 1.11 shows that Alberti’s rank
one theorem for gradient of BV functions ([1], see also [8]) also holds for
measures whose curl is a combination of first order derivatives of $L^{1}$
functions:

(1.8) $\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{l\times
d}),\;\operatorname{curl}\mu=(\partial_{p}\mu_{i,q}-\partial_{q}\mu_{i,p})_{i\leq
l,p\neq q\leq
d}\in\Psi^{1}_{\text{cl}}L^{1}(\mathbb{R}^{d},\mathbb{R})^{d^{2}l}\\\
\Longrightarrow\frac{d\mu}{d|\mu|}(x)\overset{|\mu|_{s}-a.e}{\in}\\{a\otimes\xi;\>a\in\mathbb{R}^{l},\xi\in\mathbb{R}^{d*}\\}.$

The result in [10] has been extended in [5]; 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 microlocal perspective also allows to extend some invariance
properties for constrainted measures. We prove some propagation type results
in Section 5 which show that this point of view is useful and supports the
interest of defining the $L^{1}$\- wave front set notion.

###  1.4. 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 $|\mu|_{s}$-measure on which the inclusion (1.3)
fails. We first use Proposition 3.1 to construct at any point $x\in E$ an
elliptic operator $B_{x}$ at $\bigl{(}x,\frac{d\mu}{d|\mu|}(x)\bigr{)}$ that
smoothens $\mu$ 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 [10] relying on a precise description of tangent measures
near $\mu_{s}$-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 $\mu_{s}$ measure. These properties are apriori valid only
$|\mu|_{s}$-a.e., while the functions themselves depend on $x$; this issue is
solved in Proposition 3.3.

  * •

We need to handle the non locality of pseudodifferential operators, which are
only pseudo-local, and this requires some uniform temperance properties of the
scaled measures 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 $WF_{L^{1}}$. 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 1.6. In Section 5 we
consider the propagation of singularities results. In Section 6 are gathered
the definitions and basic results on pseudodifferential 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 [21], which are important when dealing with pseudodifferential operators.

Aknowledgements:  We are grateful to Guido de Philippis for the lectures he
gave in Paris in 2017 that allowed us to learn about these topics. We would
like to thank Frédéric Bernicot for enlightenments about weak $L^{1}$
estimates for pseudodifferential operators, and Radu Ignat for pointing us the
Remark 1.10. We would also like to thank the referee for the improving
suggestions.

We 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.

##  2\. Examples

In this section we give some examples of measures for which we can describe
the $L^{1}$ wave front, $WF_{L^{1}}$. We illustrate the relevance of both the
cotangent variable $\xi$ and the polarisation variable $\omega$ by providing
examples for which the dependance with respect to these variables is non
trivial.

######  Proposition 2.1.

  1. (1)

Contrarily to the usual wave fronts defined in microlocal analysis for
distributions, in general, the natural implication

(2.1) $WF_{L^{1}}(u)=\varnothing\Rightarrow u\in L^{1}_{loc},$

is not true. There exists distributions
$\mu\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ which are not in $L^{1}_{loc}$
but such that

$WF_{L^{1}}(\mu)=\varnothing.$

  2. (2)

Let $\mu\in L^{1}_{loc}(\mathbb{R}^{d},\mathbb{R}^{m})$. Then
$WF_{L^{1}}(\mu)=\varnothing$ (and according to Theorem 1.4 the converse is
true if $\mu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R}^{m})$).

  3. (3)

Let $\mu=(\mu_{1},\mu_{2}),\mu_{1}\in
L^{1}_{loc}(\mathbb{R}^{d};\mathbb{R}^{m_{1}})$ and
$\mu_{2}\in\mathcal{M}_{t}(\mathbb{R}^{d};\mathbb{R}^{m_{2}})$. Then

$(x,\xi,\omega)\in WF_{L^{1}}(\mu)\Leftrightarrow(x,\xi,\omega_{2})\in
WF_{L^{1}}(\mu_{2})\text{ and }\omega=(0_{\mathbb{R}^{m_{1}}},\omega_{2}).$

  4. (4)

Let $x_{0}\in\mathbb{R}^{d},\omega_{0}\in\mathbb{R}^{m*}$ and
$\mu=\omega_{0}\delta_{x_{0}}$. Then

$WF_{L^{1}}(\mu)=\\{(x_{0},\xi,\omega);\xi\in\mathbb{R}^{d*},\omega\in\mathbb{R^{*}}\omega_{0}\\}$

  5. (5)

Let $\Sigma$ a smooth $d_{1}$ dimensional embedded submanifold of
$\mathbb{R}^{d}$, and $\mu$ the scalar surface measure on $\Sigma$. Then
$WF_{L^{1}}(\mu)$ is the set of points
$(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d*}$ such that $x\in\Sigma$ and
$\xi$ is (co)-normal to $\Sigma$:

$\forall X\in T_{x}\Sigma,\langle\xi,X\rangle=0.$

  6. (6)

Let $(e_{p})_{p=1}^{m}$ be the canonical basis of $\mathbb{R}^{m}$. We
consider a sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ of pairwise disjoint points
such that $x_{n}\overset{n\rightarrow+\infty}{\longrightarrow}x_{0}$ and

$\mu=\sum_{n\in\mathbb{N}}2^{-n}\delta_{x_{n}}\omega_{n},\qquad\omega_{n}=e_{p},\text{
if }n\overset{}{\equiv}p\,(\text{mod }m).$

Then

$WF(\mu)=\bigcup_{n\geq
1}\\{x_{n}\\}\times\mathbb{R}^{d*}\times\mathbb{R^{*}}\omega_{n}\,\bigcup\,\\{x_{0}\\}\times\mathbb{R}^{d*}\times\mathbb{R}^{m^{*}}.$

###### Proof.

To prove (1), recall that general pseudodifferential 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 $f\in L^{1}_{comp}$,
$Q=D^{2}(-\Delta+1)^{-1}\in\Psi^{0}_{\text{cl}}$ such that $Qf=u\notin L^{1}$.
On one hand this implies by definition that $WF_{L^{1}}(u)=0_{\text{pol}}$,
while $u\notin L^{1}_{loc}$ (because $u$ is a compactly supported distribution
but is not in $L^{1}$). However, from Theorem 1.4, if we assume in addition
$u\in\mathcal{M}_{t}$, then property (2.1) is true.

Example (2) is trivial, as in the definition of $WF_{L^{1}}$, we can choose
for any $x_{0}\in\mathbb{R}^{d}$ and $\omega_{0}\in\mathbb{R}^{m*}$, the
multiplication operator $A=\chi(x)\,^{t}\omega_{0}$ with $\chi\neq 0$ in a
neighborhood of $x_{0}$.

To prove Example (3), we consider first $(x,\xi,\omega)$ with
$\omega=(\omega_{1},\omega_{2})$ such that $\omega_{1}\neq
0_{\mathbb{R}^{m_{1}}}$. Assume for example that the first coordinate of
$\omega_{1}$ does not vanish. Let

$A=\begin{pmatrix}1,0,\dots,0\end{pmatrix}\in M_{1\times m}(\mathbb{R}),$

and $f$ be the first component of the vector valued function $\mu_{1}$. Then
we have

$A\mu=f\in L^{1}_{\text{loc}}(\mathbb{R}^{d},\mathbb{R}),$

and the constant operator $A$ is elliptic at $(x,\xi,\omega)$ because the
first coordinate of $\omega_{1}$ does not vanish. We deduce that
$(x,\xi,\omega)\notin WF_{L^{1}}(\mu).$ Let now
$(x,\xi,0_{\mathbb{R}^{m_{1}}},\omega_{2})\notin WF_{L^{1}}(\mu).$ By
definition, there exists a $A,Q_{j},f_{j}$ as in Definition 1.1 such that $A$
is elliptic at $(x,\xi,0,\omega_{2})$ and

$A\mu=\sum_{j}Q_{j}f_{j}.$

which implies by linearity

$A\begin{pmatrix}0\\\
\mu_{2}\end{pmatrix}=\sum_{j}Q_{j}f_{j}-A\begin{pmatrix}\mu_{1}\\\
0\end{pmatrix}$

or equivalently

$A_{2}\mu_{2}=\sum_{j}Q_{j}f_{j}+A_{1}\mu_{1},\qquad A=(A_{1},A_{2}).$

Since by assumption $\mu_{1}\in L^{1}$, to conclude that
$(x,\xi,\omega_{2})\notin WF_{L^{1}}(\mu_{2})$. It remains to check that if
$A$ is elliptic at $(x,\xi,0_{\mathbb{R}^{m_{1}}},\omega_{2})$ then $A_{2}$ is
elliptic at $(x,\xi,\omega_{2})$, which is clear from

$a(x,\xi)\begin{pmatrix}0\\\
\omega_{2}\end{pmatrix}=a_{2}(x,\xi)(\omega_{2}).$

Conversely, let now $(x,\xi,\omega_{2})\notin WF_{L^{1}}(\mu_{2}).$ By
definition, there exists a $A_{2},Q_{j},f_{j}$ as in Definition 1.1 such that
$A_{2}$ is elliptic at $(x,\xi,\omega_{2})$ and

$A_{2}\mu_{2}=\sum_{j}Q_{j}f_{j}.$

With $A=(0_{\mathbb{R}^{m_{1}}},A_{2})$ we have for any
$\omega_{1}\in\mathbb{R}^{m_{1}}$:

$A\begin{pmatrix}\mu_{1}\\\ \mu_{2}\end{pmatrix}=A_{2}\mu_{2},\quad
a(x,\xi)(\omega_{1},\omega_{2})=a_{2}(x,\xi)\omega_{2}\neq 0,$

and we obtain that $(x,\xi,\omega_{1},\omega_{2})\notin WF_{L^{1}}(\mu).$

For example (4), it is clear from Example (2) that if $(x,\xi,\omega)\in
WF_{L^{1}}(\omega_{0}\delta_{x_{0}})$ then $x=x_{0}$. Let us assume that
$\omega\notin\mathbb{R}\omega_{0}$. Choosing for $A$ the multiplication by a
constant $1\times m$ vector orthogonal to $\omega_{0}$ but not to $\omega$, we
get $A\omega_{0}\delta_{x_{0}}=0$ and $a(x_{0},\xi)\omega\neq 0$ so
$(x_{0},\xi,\omega)\notin
WF_{L^{1}}(\omega_{0}\delta_{x_{0}}),\forall\xi\in\mathbb{R}^{d*}$. As a
consequence we have

$WF_{L^{1}}(\omega_{0}\delta_{x_{0}})\subseteq\\{(x_{0},\xi,\omega);\xi\in\mathbb{R}^{d*},\omega\in\mathbb{R^{*}}\omega_{0}\\}.$

The equality follows from the fact that
$WF_{L^{1}}(\omega_{0}\delta_{x_{0}})\neq\varnothing$ and
$WF_{L^{1}}(\omega_{0}\delta_{x_{0}})$ is invariant under rotations in the
$\xi$ variables, as so does $\hat{\delta}_{x_{0}}$.

For example (5), we perform a change of variables and are reduced to the case
when

$\Sigma=\\{(x_{1},0_{\mathbb{R}^{d_{2}}});x_{1}\in\mathbb{R}^{d_{1}}\\},\quad\mu=g(x_{1})\delta_{x_{2}=0_{\mathbb{R}^{d_{2}}}},\,g\in
C^{\infty}(\mathbb{R}^{d_{1}},\mathbb{R}),$

where the function $g$ is positive. Let us apply a pseudodifferential operator
$\chi(D_{x})$ where $\chi$ is supported in a conic neighborhood of a point
$\xi=(\xi_{1},\xi_{2})$ with $\xi_{1}\neq 0_{\mathbb{R}^{d_{1}}}$, small
enough so that on the support of $\chi$ we have $|\xi|\leq C|\xi_{1}|$. We
have

$\chi(D_{x})\mu=\frac{1}{(2\pi)^{d}}\int
e^{i(x_{1}-y_{1})\cdot\xi_{1}+x_{2}\cdot\xi_{2}}\chi(\xi)g(y_{1})dy_{1}d\xi,$

and integrations by parts using
$\frac{\partial}{\partial{y_{1}}}e^{-iy_{1}\xi_{1}}=-i\xi_{1}e^{-iy_{1}\xi_{1}}$
gain arbitrary inverse powers of $|\xi_{1}|$ (hence of $|\xi|$) which shows
that $\chi(D_{x})\mu\in C^{\infty}$. We deduce the inclusion

$WF_{L^{1}}(\mu)\subset\\{(x_{1},0_{\mathbb{R}^{d_{2}}},0_{\mathbb{R}^{d_{1}}},\xi_{2});x_{1}\in\mathbb{R}^{d_{1}},\xi_{2}\in\mathbb{R}^{d_{2}}\\}.$

To get the equality, we first remark that it is enough to study the case above
when $g=1$, because we can multiply all pseudodifferential operators by the
smooth function $g(x_{1})^{-1}$ to the right. According to Theorem 1.4,
$\Pi_{1}WF_{L^{1}}$ must have a non trivial intersection with the set
$\\{(x_{1},0_{\mathbb{R}^{d_{2}}});x_{1}\in\mathbb{R}^{d_{1}}\\}$ because
$\mu$ is not $L^{1}$ near this point. We get the equality simply because in
that particular case $g_{1}=1$, $WF_{L^{1}}$ is clearly invariant by rotations
of the variable $\xi_{2}$ as so does $\delta_{0_{\mathbb{R}^{d_{2}}}}$.

Let us now turn to Example (6). Since the points $x_{n}$ are pairwise disjoint
and converging to $x_{0}$, all the points $x_{n},n\geq 1$ are isolated in the
sequence and localising $\mu$ near each point $x_{n}$ (i.e. considering
$\mu_{n}=\chi_{n}\mu_{n}$, with $\chi_{n}=1$ near $x_{n}$ and $\chi_{n}=0$ in
a neighborhood of $x_{k},k\neq n$), we get, according to Example (4) that

$WF(\mu)\cap\\{(x,\xi,\omega);x\neq x_{0}\\}=\bigcup_{n\geq
1}\\{x_{n}\\}\times\mathbb{R}^{d*}\times\mathbb{R^{*}}\omega_{n}.$

To conclude, it remains to prove that

$WF(\mu)\cap\\{(x,\xi,\omega);x=x_{0}\\}=\\{x_{0}\\}\times\mathbb{R}^{d*}\times\mathbb{R}^{m*}.$

By linearity with respect to the $\omega$ variable, it is actually enough to
prove that

$(x_{0},\xi,e_{p})\in WF_{L^{1}}(\mu),\forall\xi\in\mathbb{R}^{d*},1\leq p\leq
m.$

We argue by contradiction. Assume there exists $\xi_{0}$ and $p$ such that
$(x_{0},\xi_{0},e_{p})\notin WF_{L^{1}}(\mu).$ Then there exists
$A,Q_{j}\in\Psi^{0}_{\text{cl}}$ matrices of pseudodifferential operators ,
$f_{j}\in L^{1}$ such that

$A\mu=\sum_{j}Q_{j}f_{j},\quad a_{0}(x_{0},\xi_{0})e_{p}\neq 0.$

We deduce with $\chi_{n}$ as above,

$A\chi_{n}\mu=\chi_{n}Qf+[A,\chi_{n}]\mu=Q_{n}f+g,g\in L^{1}.$

But for $n$ sufficiently large, $x_{n}$ is arbitrarily close to $x_{0}$, we
get

$a(x_{n},\xi_{0})e_{p}\neq 0.$

We deduce that

$(x_{n},\xi_{0},e_{p})\notin
WF_{L^{1}}(\chi_{n}\mu)=WF_{L^{1}}(2^{-n}\omega_{n}\delta_{x_{n}})=\\{x_{n}\\}\times\mathbb{R}^{d*}\times\mathbb{R^{*}}\omega_{n},$

which is a contradiction if we choose $n=p\,(\text{mod }m)$ so that
$\omega_{n}=e_{p}$. ∎

In example (6), we have $\mu=\mu_{s}$ and

$|\mu|_{s}=\sum_{n\in\mathbb{N}}2^{-n}\delta_{x_{n}},\qquad\frac{d\mu}{d|\mu|}(x)=\sum_{n\in\mathbb{N}}1_{x=x_{n}}\omega_{n}.$

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.

##  3\. Proof of Theorem 1.2

Let us recall that all the results are local. Indeed, following the argument
in Remark 1.3, we have

$A\mu=Qf\Rightarrow\widetilde{\chi}A(\chi\mu)=\chi
Qf+\widetilde{\chi}[A,\chi]\mu,$

and $\widetilde{\chi}[A,\chi]\mu\in L^{1}$. Using (1.4) and its explanation,
we shall assume without loss of generality that $\mu$ is compactly supported
222If one wants to prove directly Theorem 1.11, without using Theorem 1.2 but
starting from §3.3, then the localisation of the measure induce only a
commutator term in the right-hand-side, that is of the same type as the
initial constraint in view of the explanations of (1.4).. We argue by
contradiction, and we suppose that the conclusion of Theorem 1.2 does not
hold:

$|\mu|_{s}(E)>0,$

where

$E:=\\{x;\;\bigl{(}x,\frac{d\mu}{d|\mu|}(x)\bigr{)}\in(\Pi_{13}WF_{L^{1}}(\mu))^{\mathsf{c}}\\}.$

We start by proving a general property of $WF_{L^{1}}$ that will be the
starting point of our approach.

###  3.1. An elliptic characterisation of $\Pi_{13}(WF_{L^{1}})$

In the following Proposition we give a characterisation of the elements in
$(\Pi_{13}(WF_{L^{1}}(u))^{c}$ for
$u\in\mathcal{S}^{\prime}(\mathbb{R}^{d},\mathbb{R}^{m})$.

######  Proposition 3.1.

For $u\in\mathcal{S}^{\prime}(\mathbb{R}^{d},\mathbb{R}^{m})$ the set
$(\Pi_{13}(WF_{L^{1}}(u))^{c}$ coincides with

(3.1) $\Bigl{\\{}(x,\omega)\in\mathbb{R}^{d}\times\mathbb{R}^{m*};\>\exists
B\in M_{1\times m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))\mbox{ elliptic at
}(x,\omega),\\\
N\in\mathbb{N}^{*},Q_{i}\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}),\,f_{i}\in
L^{1}(\mathbb{R}^{d},\mathbb{R}),\forall 1\leq i\leq
N,\;Bu=\sum_{i=1}^{N}Q_{i}f_{i}\Bigr{\\}}.$

Here ellipticity at $(x,\omega)$ means

(3.2) $b_{0}(x,\xi)\omega\neq 0,\quad\forall\xi\in\mathbb{S}^{d-1}.$

###### Proof of Proposition 3.1.

Let $(x,\omega)\in(\Pi_{13}(WF_{L^{1}}(u)))^{c}$. Then by definition for all
$\xi\in\mathbb{S}^{d-1}$ we have the existence of
$N_{\xi}\in\mathbb{N}^{*},B_{\xi}\in M_{1\times
m}(\Psi^{0}_{\text{cl}}),Q_{\xi,i}\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})$ and
$f_{\xi,i}\in L^{1}(\mathbb{R}^{d},\mathbb{R}),\forall 1\leq i\leq N_{\xi}$
such that $B_{\xi}u=\sum_{i=1}^{N_{\xi}}Q_{\xi,i}f_{\xi,i}$, with a principal
symbol $b_{0,\xi}$ satisfying

$b_{0,\xi}(x,\xi)\omega\neq 0.$

By continuity we can find a conical neighborhood
$V_{\xi}\in\mathcal{V}_{c}(\xi)$ such that

$b_{0,\xi}(x,\eta)\omega\neq 0,\quad\forall\eta\in V_{\xi}.$

and moreover $\delta_{\xi}\in\\{\pm 1\\}$ such that

(3.3) $\delta_{\xi}b_{0,\xi}(x,\eta)\omega>0,\quad\forall\eta\in V_{\xi}.$

Let $\chi_{\xi}$ be a non-negative smooth cut-off with support in $V_{\xi}$,
valued one on a conical neighborhood $\tilde{V}_{\xi}\in\mathcal{V}_{c}(\xi)$
with $\tilde{V}_{\xi}\subset V_{\xi}$. As
$\cup_{\xi\in\mathbb{S}^{d-1}}\tilde{V}_{\xi}$ is a covering of
$\mathbb{S}^{d-1}$ we can extract a finite covering

(3.4) $\exists
L\in\mathbb{N}^{*},\quad\mathbb{S}^{d-1}\subset\cup_{l=1}^{L}\tilde{V}_{\xi_{l}}.$

Then we define
$B:=\sum_{l=1}^{L}Op(\chi_{\xi_{l}}\delta_{\xi_{l}})B_{\xi_{l}}\in M_{1\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}))$ with principal symbol

$b_{0}(x,\xi):=\sum_{l=1}^{L}\chi_{\xi_{l}}\delta_{\xi_{l}}b_{0,\xi_{l}}(x,\xi),$

satisfying

$Bu=\sum_{l\in
L}\sum_{i=1}^{N_{\xi_{l}}}Op(\chi_{\xi_{l}}\delta_{\xi_{l}})Q_{\xi_{l},i}f_{\xi_{l},i}.$

In particular the right-hand-side is a finite sum of terms $Qf$ with
$Q\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})$ and $f\in
L^{1}(\mathbb{R}^{d},\mathbb{R})$. We consider now

$b_{0}(x,\xi)\omega=\sum_{l=1}^{L}\chi_{\xi_{l}}\delta_{\xi_{l}}b_{0,\xi_{l}}(x,\xi)\omega.$

On one hand from (3.3) we deduce that all terms in the sum are non negative
for all $\xi\in\mathbb{S}^{d-1}$. On the other hand from (3.4) there exists at
least a set $\tilde{V}_{\xi_{j}}$ such that $\xi\in\tilde{V}_{\xi_{j}}$, 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 $\mathcal{N}_{0}$ with $|\mu|_{s}(\mathcal{N}_{0})=0$ such
that for all $x\in\mathcal{N}_{0}^{c}$, there exists a sequence
$r_{j}\rightarrow 0$ such that the following three assertions hold.

(i) $\underset{r\rightarrow
0}{\lim}\frac{\mathcal{L}^{d}(B(x,r))}{|\mu|_{s}(B(x,r))}=0,\;\underset{r\rightarrow
0}{\lim}\frac{|\mu|_{a}(B(x,r))}{|\mu|_{s}(B(x,r))}=0,\;\underset{r\rightarrow
0}{\lim}\mathop{\ooalign{$\int$\cr$-$}}_{B(x,r)}\left|\frac{d\mu}{d|\mu|}(y)-\frac{d\mu}{d|\mu|}(x)\right|d|\mu|_{s}=0,$

There exists $C,c>0$ such that the family of measures 333by $T^{x,r_{j}}$ we
denote the dilations $\tilde{x}\in\mathbb{R}^{d}\mapsto
x+r_{j}\tilde{x}\in\mathbb{R}^{d},$ and by $T^{x,r_{j}}_{\sharp}\sigma$ we
denote the push-forward of the measure $\sigma$ given by
$T^{x,r_{j}}_{\sharp}\sigma(A)=\sigma(x+r_{j}A)$.

$\mu_{j}:=\frac{(T^{x,r_{j}})_{\sharp}\mu}{|\mu_{s}|(B(x,r_{j}))}$

satisfies the temperance uniform in $j$ bounds:

(ii) $|\mu_{j}|(B(x,R))=\frac{|\mu|(B(x,Rr_{j}))}{|\mu|_{s}(B(x,r_{j}))}\leq
C\max\\{1,R^{d+\frac{1}{2}}\\},$

We can pass to the limit

(iii)
$\mu_{j}\underset{j\rightarrow\infty}{\overset{*}{\rightharpoonup}}\tilde{\nu}\in
Tan(x,\mu),\quad\tilde{\nu}=\frac{d\mu}{d|\mu|}(x)\nu,\quad\nu\in\,Tan(x,|\mu|),$

and the restriction of $\nu$ to $B_{\frac{1}{2}}$ is not trivial.

Assertion (i) is based on classical characterisation of singular points of a
measure (see e.g. [17, Theorem 2.12 (3)], [11, 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. [22, Section 2.3]) and the lower
bound (7.2).

Next we are going to prove the following property on the set
$E\setminus\mathcal{N}_{0}$.

######  Proposition 3.3.

There exists a set $\tilde{E}\subseteq E\setminus\mathcal{N}_{0}$ with
$|\mu|_{s}(E\setminus\tilde{E})=0$ such that for all $x\in\tilde{E}$, there
exists $B\in M_{1\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})),N\in\mathbb{N}^{*},Q_{i}\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}),f_{i}\in
L^{1}(\mathbb{R}^{d},\mathbb{R}),\forall 1\leq i\leq N$, satisfying the
identity

$Bu=\sum_{i=1}^{N}Q_{i}f_{i},$

and the elliptic condition at $\bigl{(}x,\frac{d\mu}{d|\mu|}(x)\bigr{)}$:

$b_{0}(x,\xi)\frac{d\mu}{d|\mu|}(x)\neq 0,\quad\forall\xi\in\mathbb{S}^{d-1}.$

Moreover, for the sequence $(r_{j})$ constructed in Lemma 3.2 we have the
following properties:

(iv) $\begin{gathered}\forall i\in\\{1,..,N_{x}\\},\quad\underset{r\rightarrow
0}{\lim}\frac{|f_{i}\mathcal{L}^{d}(B(x,r))|}{|\mu|_{s}(B(x,r))}=0,\\\ \exists
C_{x,i}>0,\>\frac{|f_{i}\mathcal{L}^{d}(B(x,Rr_{j}))|}{|\mu|_{s}(B(x,r_{j}))}\leq
C_{x,i}\max\\{1,R^{d+\frac{1}{2}}\\}.\end{gathered}$

###### Proof.

As $|\mu|_{s}$ is a finite Radon measure we have
$|\mu|_{s}(E\setminus\mathcal{N}_{0})=\sup\\{\mu_{s}(K);K\text{
compact}\subset E\setminus\mathcal{N}_{0}\\}$. Thus there exists a compact set
$K\subseteq E\mathcal{N}_{0}$ such that
$|\mu|_{s}(K)>\frac{1}{2}|\mu|_{s}(E)>0$.

For any $x\in K$ we have $x\in E$, so
$\bigl{(}x,\frac{d\mu}{d|\mu|}(x)\bigr{)}\in(\Pi_{13}WF_{L^{1}}(\mu))^{\mathsf{c}}$.
Thus we can apply Proposition 3.1 to get the existence of $B_{x}\in M_{1\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})),$ $N\in\mathbb{N}^{*},Q_{x,i}\in
M_{m\times 1}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})),f_{x,i}\in
L^{1}(\mathbb{R}^{d},\mathbb{R}),\forall 1\leq i\leq N$, such that

$B_{x}\mu=\sum_{i=1}^{N_{x}}Q_{x,i}f_{x,i},$

with the elliptic property at $\bigl{(}x,\frac{d\mu}{d|\mu|}(x)\bigr{)}$:

$b_{0,x}(x,\xi)\frac{d\mu}{d|\mu|}(x)\neq
0,\quad\forall\xi\in\mathbb{S}^{d-1}.$

This implies the existence of $r_{x,1}>0$ and $c_{x}>0$ such that

$|b_{0,x}(y,\xi)\frac{d\mu}{d|\mu|}(x)|>c_{x}>0,\quad\forall y\in
B(x,r_{x,1}),\quad\forall\xi\in\mathbb{S}^{d-1},$

In particular there exists $\delta_{x}>0$ such that

(3.5) $b_{0,x}(y,\xi)\omega\neq 0,\quad\forall y\in
B(x,r_{x,1}),\quad\forall\xi\in\mathbb{S}^{d-1},\quad\forall\omega\in\mathbb{S}^{m-1},\,\left\|\omega-\frac{d\mu}{d|\mu|}(x)\right\|\leq\delta_{x}.$

From Lemma 3.2 for all $x\in\mathcal{N}_{0}^{c}$, thus for all $x\in K$,

(3.6) $\underset{r\rightarrow
0}{\lim}\mathop{\ooalign{$\int$\cr$-$}}_{B(x,r)}\left|\frac{d\mu}{d|\mu|}(y)-\frac{d\mu}{d|\mu|}(x)\right|d|\mu|_{s}=0,$

with the average integral well-defined, meaning $|\mu|_{s}(B(x,r))>0,\,\forall
r,\,\forall x\in K$. Thus for all $x\in K$ we get the existence of $r_{x,2}$
such that for all $r\leq r_{x,2}$

(3.7)
$\mathop{\ooalign{$\int$\cr$-$}}_{B(x,r)}\left|\frac{d\mu}{d|\mu|}(y)-\frac{d\mu}{d|\mu|}(x)\right|d|\mu|_{s}\leq\frac{\delta_{x}}{2}.$

For all $x\in K$ and $r\leq r_{x,2}$ we have $|\mu|_{s}(B(x,r))>0$ and the set
defined by

$F_{x,r}:=\\{y\in
B(x,r),\,\left\|\frac{d\mu}{d|\mu|}(y)-\frac{d\mu}{d|\mu|}(x)\right\|\leq\delta_{x}\\},$

satisfies, in view of (3.7),

(3.8) $F_{x,r}\subseteq
B(x,r),\quad|\mu|_{s}(F_{x,r})>\frac{|\mu|_{s}(B(x,r))}{2}.$

We denote by $\mathcal{N}_{1}$ the set of null $|\mu|_{s}-$measure of the
zero-density points w.r.t. $|\mu|_{s}$ of the set $K$, thus

(3.9) $\forall x\in K\setminus\mathcal{N}_{1},\quad\exists
r_{x,3},\quad\forall r\leq
r_{x,3}\quad\frac{3}{4}|\mu|_{s}(B(x,r))\leq|\mu|_{s}(K\cap B(x,r)).$

The closed balls $\overline{B}(x,\frac{r_{x}}{2}),x\in
K\setminus\mathcal{N}_{1}$ with $r_{x}=\min\\{r_{x,1},r_{x,2},r_{x,3}\\}$ form
a covering of the bounded set $K\setminus\mathcal{N}_{1}$, so by Besicovitch’s
covering theorem ([17, Theorem 2.7]) we can extract a countable covering:
there exists $\\{x_{n}\\}_{n\in\mathbb{N}}\subseteq K\setminus\mathcal{N}_{1}$
such that

$K\setminus\mathcal{N}_{1}\subseteq\cup_{n\in\mathbb{N}}\overline{B}(x_{n},\frac{r_{x_{n}}}{2})\subseteq\cup_{n\in\mathbb{N}}B(x_{n},r_{x_{n}}).$

For all $n\in\mathbb{N}$ and $1\leq i\leq N_{x_{n}}$ there exists a set
$\mathcal{N}_{n,i}$ of null $|\mu|_{s}$-measure such that

$\underset{r\rightarrow
0}{\lim}\frac{|f_{x_{n},i}\mathcal{L}^{d}(B(x,r))|}{|\mu|_{s}(B(x,r))}=0,\>\forall
x\in(\mathcal{N}_{n,i})^{c}.$

This is due to the fact that $|\mu|_{s}\perp f_{x_{n},i}\mathcal{L}^{d}$ and
Theorem 2.12 (3) in [17].

It follows from (3.5) that

(3.10) $\begin{gathered}\forall x\in K\setminus\mathcal{N}_{1},\quad\exists
n\in\mathbb{N},\quad x\in B(x_{n},r_{x_{n}}),\quad\mbox{ and if
}\left\|\frac{d\mu}{d|\mu|}(x)-\frac{d\mu}{d|\mu|}(x_{n})\right\|\leq\delta_{x_{n}},\\\
\mbox{ then }\quad b_{0,x_{n}}(x,\xi)\frac{d\mu}{d|\mu|}(x)\neq
0\quad\forall\xi\in\mathbb{S}^{d-1}.\end{gathered}$

Now we define the sets

$\tilde{K}:=K\setminus(\mathcal{N}_{1}\cup_{n\in\mathbb{N}}\cup_{i=1}^{N_{x_{n}}}\mathcal{N}_{x_{n},i}),$

and

$\tilde{E}:=(\cup_{n\in\mathbb{N}}F_{x_{n},r_{x_{n}}})\cap\tilde{K}\subseteq
E\setminus\mathcal{N}_{0}.$

In view of (3.8)-(3.9) we have

(3.11)
$|\mu|_{s}(F_{x_{n},r_{x_{n}}}\cap\tilde{K})=|\mu|_{s}(F_{x_{n},r_{x_{n}}}\cap
K)\\\ \geq|\mu|_{s}(F_{x_{n},r_{x_{n}}})-|\mu|_{s}(B(x_{n},r_{x_{n}})\cap
K^{c})\geq(\frac{1}{2}-\frac{1}{4})|\mu|_{s}(B(x_{n},r_{x_{n}})),$

and since $|\mu|_{s}(K)>0$ we get at least one $n\in\mathbb{N}$ such that
$|\mu|_{s}(B(x_{n},r_{x_{n}}))>0$ so

$|\mu|_{s}(\tilde{E})>0.$

Then (3.10) and the definitions of $F_{x,r}$ and $\tilde{E}$ ensure us that

$\forall x\in\tilde{E},\quad\exists n\in\mathbb{N},\quad
b_{0,x_{n}}(x,\xi)\frac{d\mu}{d|\mu|}(x)\neq
0,\quad\quad\forall\xi\in\mathbb{S}^{d-1}.$

Therefore for $x\in\tilde{E}$ we define $B:=B_{x_{n}}$, which have the
elliptic property in the statement of the Lemma, we define $N:=N_{x_{n}}$, and
we have the first part of the Lemma:

$Bu=\sum_{i=1}^{N}Q_{x_{n},i}f_{x_{n},i}.$

Moreover, $f_{x_{n},i}$ satisfies the first assertion of (iv) as by definition
$\tilde{E}\subset\mathcal{N}_{x_{n},i}^{c}$. Then it follows that

$\exists R_{x,x_{n},i},\;\forall
r\in(0,R_{x,x_{n},i}),\>\frac{|f_{x_{n},i}\mathcal{L}^{d}(B(x,r))|}{|\mu|_{s}(B(x,r))}\leq
1.$

For $Rr_{j}\leq R_{x,x_{n},i}$ we deduce by combining also with Lemma 3.2 (ii)
that

(3.12)
$\frac{|f_{x_{n},i}\mathcal{L}^{d}(B(x,Rr_{j}))|}{|\mu|_{s}(B(x,r_{j}))}=\frac{|f_{x_{n},i}\mathcal{L}^{d}(B(x,Rr_{j}))|}{|\mu|_{s}(B(x,Rr_{j}))}\frac{|\mu|_{s}(B(x,Rr_{j}))}{|\mu|_{s}(B(x,r_{j}))}\\\
\leq\frac{|\mu|_{s}(B(x,Rr_{j}))}{|\mu|_{s}(B(x,r_{j}))}\leq
C\max\\{1,R^{d+\frac{1}{2}}\\}.$

On the other hand, by using Lemma 3.2 (i) we have for $Rr_{j}\geq
R_{x,x_{n},i}$

$\frac{|f_{x_{n},i}\mathcal{L}^{d}(B(x,Rr_{j}))|}{|\mu|_{s}(B(x,r_{j}))}\leq
C\frac{\|f_{x_{n},i}\|_{L^{1}}}{r_{j}^{-d}}\leq\frac{C^{\prime}}{R_{x,x_{n},i}^{d}}R^{d}.$

Therefore $f_{x_{n},i}$ satisfies also the last assertion of (iv).

Summarizing, we have shown that we can find $\tilde{E}\subset
E\setminus\mathcal{N}_{0}$ of positive $|\mu|_{s}-$measure, on which all the
conclusion of the Lemma are satisfied, excepted the fact that
$|\mu_{s}|(E\setminus\tilde{E})=0$. The argument above is valid also starting
from any subset of $E\setminus\mathcal{N}_{0}$ of positive
$|\mu|_{s}-$measure. Thus the conclusion of the Lemma are satisfied
$|\mu|_{s}-$a.e. on $E\setminus\mathcal{N}_{0}$. Indeed, let us we denote by
$E_{1}$ the subset of $E\setminus\mathcal{N}_{0}$ of the points for which the
conclusion of the Lemma are not satisfied. If $|\mu_{s}|(E_{1})>0$ then the
construction above would give the existence of a subset $\tilde{E}_{1}\subset
E_{1}$ of strictly positive $|\mu_{s}|-$measure, thus non-empty, on which the
conclusions of the Lemma are both true and false on $\widetilde{E}_{1}$.
Therefore $|\mu_{s}|(E_{1})=0$ 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 [10], using the elliptic type operator family from
Proposition 3.3 instead of an operator constraining the measure to vanish.

We thus apply Proposition 3.3 and as $|\mu|_{s}(\tilde{E})>0$ we can choose
now $x_{0}\in\tilde{E}$ so that properties in Lemma 3.2 and Proposition-3.3
are valid with $x=x_{0}$. We denote

$\omega_{0}:=\frac{d\mu}{d|\mu|}(x_{0}).$

From Lemma 3.2 (iii) we have the existence of a sequence $r_{j}$ converging to
zero such that

$\mu_{j}\underset{j\rightarrow\infty}{\overset{*}{\rightharpoonup}}\frac{d\mu}{d|\mu|}(x)\,\nu,$

with $\nu\in Tan(x,|\mu|)$ a positive measure with $\nu(B_{\frac{1}{2}})>0$.
Then we also have (cf Theorem 2.12 (3) in [17] applied with $\mu_{s}\perp
g_{R}\mathcal{L}^{d}$, where $g_{R}(x):=R\,g(Rx)$ and $R>0$)

$\nu_{j}:=\frac{(T^{x_{0},r_{j}})_{\sharp}|\mu|_{s}}{|\mu|_{s}(B(x_{0},r_{j}))}\underset{j\rightarrow\infty}{\overset{*}{\rightharpoonup}}\nu,$

and as $\nu_{j}\perp\mathcal{L}^{d}$, there exists sets $E_{j}\subseteq
B_{\frac{1}{2}}$ where $\nu_{j}\llcorner B_{\frac{1}{2}}$ concentrates and
$\mathcal{L}^{d}$ vanishes:

$\nu_{j}(E_{j})=\nu_{j}(B_{\frac{1}{2}}),\quad\mathcal{L}^{d}(E_{j})=0.$

By using the full information at $(x_{0},\omega_{0})$ given by Lemma 3.2 we
shall get Proposition 3.4 below. Then the two assertions of Proposition 3.4
give us the following contradiction:

$0<\nu(B_{\frac{1}{2}})=\underset{j\rightarrow\infty}{\lim}\nu_{j}(E_{j})\leq\underset{j\rightarrow\infty}{\lim}|\nu_{j}-\nu|(E_{j})+\nu(E_{j})\leq\underset{j\rightarrow\infty}{\lim}|\nu_{j}-\nu|(B_{\frac{1}{2}})=0.$

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

$\nu\llcorner
B_{\frac{1}{2}}\Lt\mathcal{L}^{d},\quad\underset{j\rightarrow\infty}{\lim}|\nu_{j}-\nu|(B_{\frac{1}{2}})=0.$

###### Proof.

Let $\chi$ be a smooth cut-off function equal to $1$ on $B_{\frac{1}{2}}$ and
$0$ outside $B_{\frac{3}{4}}$, satisfying $\int\chi(x)dx=1$. The conclusion of
the Proposition follows if we show that $\chi\nu\in L^{1}(\mathbb{R}^{d})$ and
on a subsequence

(3.13) $\chi\nu_{j}\overset{L^{1}(\mathbb{R}^{d})}{\longrightarrow}\chi\nu.$

As $\nu_{j}\underset{j\rightarrow\infty}{\overset{*}{\rightharpoonup}}\nu$, it
is thus enough to prove that $\\{\chi\nu_{j}\\}$ is precompact in
$L^{1}_{loc}(\mathbb{R}^{d})$.

To get the $L^{1}_{loc}(\mathbb{R}^{d})$-precompactness of $\\{\chi\nu_{j}\\}$
we shall consider the operator $B$ from Proposition 3.3, and use its
ellipticity property through an appropriate inversion argument.

Step 1: The inversion formula.

As $x_{0}\in\tilde{E}$, the properties in Proposition 3.3 are satisfied for
$x_{0}$ and in particular we obtain the existence of $B\in M_{1\times
m}(\Psi^{0}_{\text{cl}}(\mathbb{R}^{d})),Q\in\Psi^{0}_{\text{cl}}(\mathbb{R}^{d}),f\in
L^{1}(\mathbb{R}^{d},\mathbb{R})$ satisfying the identity

$B\mu=Qf,$

the elliptic condition at $(x_{0},\omega_{0})$:

(3.14) $b_{0}(x_{0},\xi)\omega_{0}\neq 0,\quad\forall\xi\in\mathbb{S}^{d-1},$

and moreover $f$ satisfies the properties (iv) in Proposition 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 $\xi$ 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
$\frac{(T^{x_{0},r_{j}})_{\sharp}\cdot}{|\mu|_{s}(B_{r_{j}}(x_{0}))}$ and get

$\tilde{B}^{j}\mu_{j}=\tilde{Q}^{j}f_{j},$

where

(3.15)
$\begin{gathered}f_{j}\mathcal{L}^{d}:=\frac{(T^{x_{0},r_{j}})_{\sharp}f\mathcal{L}^{d}}{|\mu|_{s}(B(x_{0},r_{j}))},\quad
f_{j}(x)=\frac{(r_{j})^{d}}{|\mu|_{s}(B(x_{0},r_{j}))}f(x_{0}+r_{j}x),\\\
\tilde{b}^{j}(y,\xi):=b(x_{0}+r_{j}y,\frac{\xi}{r_{j}}),\quad\tilde{q}^{j}(y,\xi):=q(x_{0}+r_{j}y,\frac{\xi}{r_{j}}).\end{gathered}$

We first microlocalize in frequency to suppress the low frequencies by
applying to the left $(1-\chi(D_{x}))$, and get

(3.16)
$B^{j}\mu_{j}:=(1-\chi(D_{x}))\tilde{B}^{j}\mu_{j}=(1-\chi(D_{x}))\tilde{Q}^{j}f_{j}=:Q^{j}(f_{j})$

with symbols (remark that the following exact formula follows from the
independence of $\tilde{b},\tilde{q}$ with respect to the $x$ variable and
(6.2)),

$b^{j}(y,\xi)=(1-\chi(\xi))b(x_{0}+r_{j}y,\frac{\xi}{r_{j}}),\quad
q^{j}(y,\xi):=(1-\chi(\xi))q(x_{0}+r_{j}y,\frac{\xi}{r_{j}}).$

Notice that the symbols $b^{j}$ and $q^{j}$ are in
$S^{0}_{\text{cl}}(\mathbb{R}^{d})$ with semi-norms in
$S^{0}_{\text{cl}}(\mathbb{R}^{d})$ uniformly bounded in $j$. Indeed, from
(3.16) we get

(3.17)
$|\partial_{y}^{\alpha}\partial_{\xi}^{\beta}b^{j}(y,\xi)|=\Bigl{|}\sum_{\delta+\gamma=\beta}\frac{\beta!}{\delta!\gamma!}r_{j}^{|\alpha|-|\gamma|}\partial_{\xi}^{\delta}(1-\chi(\xi))\partial_{y}^{\alpha}\partial_{\xi}^{\gamma}b(x_{0}+r_{j}y,\frac{\xi}{r_{j}})\Bigr{|}\\\
\leq
Cr_{j}^{-|\beta|}(1+\frac{|\xi|}{r_{j}})^{-|\beta|}1_{\frac{1}{2}\leq|\xi|}+C\sum_{\genfrac{}{}{0.0pt}{1}{\delta\neq
0}{\delta+\gamma=\beta}}r_{j}^{-|\gamma|}(1+\frac{|\xi|}{r_{j}})^{-|\gamma|}1_{\frac{1}{2}\leq|\xi|\leq
2}\leq C(1+|\xi|)^{-|\beta|}$

Now we localize in space

$B^{j}\chi\mu_{j}=Q^{j}\chi f_{j}+[\chi,Q^{j}]f_{j}-[\chi,B^{j}]\mu_{j}.$

Next we shall apply a smoothing operator $\zeta_{\epsilon}(D_{x})$ where
$\zeta_{\epsilon}(\xi)=\widehat{\chi}(\epsilon\xi)$. Since

$\zeta_{\epsilon}(D_{x})\chi\nu_{j}=\widehat{\chi}(\epsilon
D_{x})(\chi\nu_{j})=\chi_{\epsilon}\star(\chi\nu_{j})\overset{\epsilon\rightarrow
0}{\rightharpoonup}\chi\nu,\qquad\chi_{\epsilon}(x):=\frac{1}{\epsilon^{d}}\chi(\frac{x}{\epsilon}),$

we can choose a sequence $\epsilon_{j}<\frac{1}{8}$ converging to $0$ fast
enough so that

$u_{j}:=\zeta_{\epsilon_{j}}(D_{x})\chi\nu_{j}=\chi_{\epsilon_{j}}\star(\chi\nu_{j})\underset{j\rightarrow\infty}{\overset{*}{\rightharpoonup}}\chi\nu.$

Thus to get $\\{\chi\nu_{j}\\}$ precompact in $L^{1}_{loc}(\mathbb{R}^{d})$ it
is enough to get $\\{u_{j}\\}$ precompact in $L^{1}_{loc}(\mathbb{R}^{d})$. We
have by applying $\zeta_{\epsilon_{j}}(D_{x})$:

(3.18) $B^{j}\omega_{0}u_{j}=B^{j}V_{j}+{Q}^{j}g_{j}$
$-[\zeta_{\epsilon_{j}}(D_{x}),B^{j}]\chi\mu_{j}+[\zeta_{\epsilon_{j}}(D_{x}),Q^{j}]\chi
f_{j}+\zeta_{\epsilon_{j}}(D_{x})([\chi,Q^{j}]f_{j}-[\chi,B^{j}]\mu_{j}),$

where

$V_{j}:=\zeta_{\epsilon_{j}}(D_{x})(\omega_{0}\chi\nu_{j}-\chi\mu_{j}),\qquad
g_{j}=\zeta_{\epsilon_{j}}(D_{x})\chi f_{j}$

Next we consider the family of pseudodifferential operators

$B^{j}_{\omega_{0}}:=B^{j}\omega_{0},$

of symbols

$b^{j}_{\omega_{0}}(y,\xi):=b^{j}(y,\xi)\omega_{0}=(1-\chi(\xi))b(x_{0}+r_{j}y,\frac{\xi}{r_{j}})\omega_{0},$

which are bounded in $S_{cl}^{0}(\mathbb{R}^{d})$ uniformly with respect to
$j$. We split:

$b^{j}_{\omega_{0}}(y,\xi)=(1-\chi(\xi))b_{0}(x_{0}+r_{j}y,{\xi})\omega_{0}+(1-\chi(\xi))(b-b_{0})(x_{0}+r_{j}y,\frac{\xi}{r_{j}})\omega_{0}.$

From (3.14) we get that the principal symbols $b_{0,l}$ satisfy

$|b_{0}(y,\xi)\omega_{0}|\geq c>0,$

for all $\xi\in\mathbb{S}^{d-1}$ and $y$ close to $x_{0}$, and following the
same lines as in (3.17), the symbol

$(1-\chi(\xi))(b-b_{0})(x_{0}+r_{j}y,\frac{\xi}{r_{j}})\omega_{0}$

is uniformly in $S^{-1}$ and bounded by $Cr_{j}(1+|\xi|)^{-1}$. So for $j$
large enough, $B^{j}_{\omega_{0}}$ is elliptic on $B_{1}$ uniformly with
respect to $j$, in the sense:

$\exists\,J,c>0,\quad|b^{j}_{\omega_{0}}(y,\xi)\omega_{0}|\geq c,\quad\forall
j\geq J,\forall|\xi|\geq 1,\forall y\in B_{1}.$

and we can approximately invert it locally as follows. Let us consider
$\tilde{\chi}\in C^{\infty}_{0}(B_{1})$ equal to $1$ on $B_{\frac{7}{8}}$ thus
in particular near the support of $\chi$. For $j\geq J$ we define

$p^{j}(x,\xi)=\frac{\chi(x)(1-\chi({\xi}))}{b^{j}_{\omega_{0}}(x,\xi)},$

which is in $S^{0}_{\text{cl}}$ uniformly with respect to $j$, so that, by
symbolic calculus of pseudodifferential operators, and by adding the
additional cutt-off $\tilde{\chi}$ for later use,

$P^{j}\tilde{\chi}B^{j}_{\omega_{0}}\tilde{\chi}=\chi(x)(1-\chi({D_{x}}))+\widetilde{R}^{j}=\chi(x)+R^{j},$

where $\tilde{\chi}R^{j}\tilde{\chi}=R^{j}$ and the family of operators
$R^{j}$ is a family of pseudodifferential operators uniformly bounded in
$\Psi_{cl}^{-1}$, i.e.

(3.19)
$\forall\alpha,\beta,\sup_{j}\sup_{x,\xi}(1+|\xi|)^{1+|\beta|}|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}(r^{j}(x,\xi))|<+\infty.$

As $u_{j}=\chi_{\epsilon_{j}}\star(\chi\nu_{j})$ is supported in an
$\epsilon_{j}<\frac{1}{8}$ neighborhood of $B_{\frac{3}{4}}$ we have
$u_{j}=\tilde{\chi}u_{j}$. Then we can apply $\tilde{\chi}P^{j}\tilde{\chi}$
to (3.18) to get:

(3.20)
$u_{j}=-R^{j}(u_{j})+\tilde{\chi}P^{j}\tilde{\chi}(B^{j}V_{j}+Q^{j}g_{j})$
$-\tilde{\chi}P^{j}\tilde{\chi}[\zeta_{\epsilon_{j}}(D_{x}),B^{j}]\chi\mu_{j}+\tilde{\chi}P^{j}\tilde{\chi}[\zeta_{\epsilon_{j}}(D_{x}),Q^{j}]\chi
f_{j}+\tilde{\chi}P^{j}\tilde{\chi}\zeta_{\epsilon_{j}}(D_{x})([\chi,Q^{j}]f_{j}-[\chi,B^{j}]\mu_{j}).$

Step 2: The compactness arguments.

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

We start with the term $R^{j}(u_{j})=\tilde{\chi}R^{j}\tilde{\chi}u_{j}$ in
(3.20). From (3.19) and Corollary 6.7 applied with $\delta=-1$ the operators
$R^{j}$ are bounded from $\mathcal{M}_{0}$ to $W^{1-\eta,1}(\mathbb{R}^{d})$
for $0<\eta<1$, uniformly in $j$. Also, by the weak convergence of $u_{j}$ we
get that $\tilde{\chi}u_{j}$ is uniformly bounded in $\mathcal{M}_{0}$. Then
Proposition 6.4 with $s=1-\eta$ implies that $\\{R^{j}(u_{j})\\}$ is pre-
compact in $L^{1}_{loc}(\mathbb{R}^{d})$.

The same argument allows to handle the contribution of
$\tilde{\chi}P^{j}\tilde{\chi}[\zeta_{\epsilon_{j}}(D_{x}),Q^{j}]\chi f_{j}$,
by using this time that $\chi f_{j}$ is uniformly bounded in $\mathcal{M}_{0}$
due to the first assertion in Proposition 3.3 (iv), and also the fact that the
symbol $\zeta(\epsilon_{j}\xi)$ of the operator $\zeta_{\epsilon_{j}}(D_{x})$
is in $S^{0}_{\text{cl}}(\mathbb{R}^{d})$ uniformly with respect to $j$. The
sequence
$\\{\tilde{\chi}P^{j}\tilde{\chi}[\zeta_{\epsilon_{j}}(D_{x}),B^{j}]\chi\mu_{j}\\}$
is also precompact in $L^{1}$ by the same argument.

Let us now study the sequence
$\\{\tilde{\chi}P^{j}\tilde{\chi}\zeta_{\epsilon_{j}}(D_{x})[\chi,B^{j}]\mu_{j}\\}.$
Here the main difference with respect to the previous analysis is that
$\mu_{j}$ is not necessarily bounded in $\mathcal{M}_{0}$. However, according
to Lemma 3.2 (ii), $\mu_{j}$ is bounded in
$\mathcal{M}_{d+1/2}(\mathbb{R}^{d})$ only, i.e. its mass on balls of radius
$R>1$ can grow at most like $R^{d+\frac{1}{2}}$. From Proposition 6.8, we get
that the family of operators

$\tilde{\chi}P^{j}\tilde{\chi}\zeta_{\epsilon_{j}}(D_{x})[\chi,B^{j}]\langle
x\rangle^{d+\frac{1}{2}}$

is uniformly in $j$ bounded from $\mathcal{M}_{0}$ to $W^{1-\epsilon,1}$ for
an $\epsilon\in(0,1)$. Then the boundedness of $\langle
x\rangle^{-(d+\frac{1}{2})}\mu_{j}$ in $\mathcal{M}_{0}$ from Lemma 3.2 (ii)
and Proposition 6.4 give that
$\\{\tilde{\chi}P^{j}\tilde{\chi}\zeta_{\epsilon_{j}}(D_{x})[\chi,B^{j}]\mu_{j}\\}$
is precompact in $L^{1}_{loc}(\mathbb{R}^{d})$.

We get similarly the relative compactness of the sequence
$\\{\tilde{\chi}P^{j}\tilde{\chi}\zeta_{\epsilon_{j}}(D_{x})[\chi,Q^{j}]f_{j}\\}$
by using the boundedness of $\langle x\rangle^{-(d+\frac{1}{2})}f_{j}$ in
$\mathcal{M}_{0}$ which follows from the second assertion of Proposition 3.3
(iv).

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

$\tilde{\chi}P^{j}\tilde{\chi}(B^{j}V_{j}+Q^{j}g_{j}).$

For this we first note that we have the following property:

(3.21) $\lim_{j\rightarrow+\infty}\|V_{j}\|_{L^{1}}+\|g_{j}\|_{L^{1}}=0.$

The $L^{1}-$convergence of $V_{j}$’s follows from Lemma 3.2 (i)-(ii) exactly
as in [10, (2.8)]. For sake of completeness let us recall the short argument.
For $\epsilon_{j}<\frac{1}{8}$, the function $V_{j}$ is supported in
$B_{3/4}+B_{1/8}\subseteq B_{7/8}$, and

$\int_{B_{7/8}}|V_{j}|(x)dx\leq\int_{B_{1}}|\omega_{0}\chi\nu_{j}-\chi\mu_{j}|\leq\frac{\bigl{|}\omega_{0}T^{x_{0},r_{j}}_{\sharp}|\mu|_{s}-T^{x_{0},r_{j}}_{\sharp}\mu\bigr{|}(B_{1})}{|\mu|_{s}(B(x_{0},r_{j})}$
$\leq\frac{\bigl{|}\omega_{0}|\mu|_{s}-\mu_{s}\bigr{|}B(x_{0},r_{j})}{|\mu|_{s}(B(x_{0},r_{j}))}+\frac{|\mu_{a}|B(x_{0},r_{j})}{|\mu|_{s}(B(x_{0},r_{j})}\leq\mathop{\ooalign{$\int$\cr$-$}}_{B(x_{0},r_{j})}\Bigl{|}\frac{d\mu}{d|\mu|}(x_{0})-\frac{d\mu}{d|\mu|}(x)\Bigr{|}+\frac{|\mu_{a}|B(x_{0},r_{j})}{|\mu|_{s}(B(x_{0},r_{j})}.$

Then we deduce from Lemma 3.2 (i) the $L^{1}-$convergence of $V_{j}$ to zero.
On the other hand, from Proposition 3.3 (iv), we get that $\chi f_{j}$, and
hence also $g_{j}=\chi_{j}\star\chi f_{j}$, converges to $0$ in $L^{1}$.

Now we can use $L^{1}-L^{1,\infty}$ estimates for the $0-$order operators
$\tilde{\chi}P^{j}\tilde{\chi}B^{j}_{l}$ and
$\tilde{\chi}P^{j}\tilde{\chi}Q^{j}$ that are Calderon-Zygmund operators, the
fact that the bound depends only on a finite number of semi-norms [7], that
these semi-norms are uniformly bounded in $j$, and the convergence (3.21) to
get a convergence in measure:

(3.22) $\sup_{\lambda\geq
0}\lambda\mathcal{L}^{d}(\\{|\tilde{\chi}P^{j}\tilde{\chi}(B^{j}V_{j}+Q^{j}g_{j})|>\lambda\\})\\\
\leq\sup_{\lambda\geq
0}\lambda\mathcal{L}^{d}(\\{\tilde{\chi}P^{j}\tilde{\chi}V_{j}|>\frac{\lambda}{2}\\})+\sup_{\lambda\geq
0}\lambda\mathcal{L}^{d}(\\{|\tilde{\chi}P^{j}\tilde{\chi}Q^{j}g_{j}|>\frac{\lambda}{2}\\})\\\
\leq
C\|V_{j}\|_{L^{1}}+C\|g_{j}\|_{L^{1}}\overset{j\rightarrow\infty}{\longrightarrow}0.$

To conclude, we are going to use following result from [10].

######  Lemma 3.5 ([10, Lemma 2.2]).

Consider $\\{h_{j}\\}$ a sequence of $L^{1}-$functions supported in $B_{1}$
satisfying:

  1. a)

The sequence $h_{j}$ converges weakly to $0$,
$h_{j}\overset{*}{\rightharpoonup}0$ in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$

  2. b)

The negative part of $h_{j}$ tends to $0$ in measure

$\forall\lambda>0,\lim_{j\rightarrow+\infty}\mathcal{L}^{d}(\\{h_{j}^{-}>\lambda\\})=0$

  3. c)

The sequence of negative parts $h_{j}^{-}$ is equi-integrable,

$\lim_{\mathcal{L}^{d}(E)\rightarrow
0}\sup_{j\in\mathbb{N}}\int_{E}h_{j}^{-}dx=0$

Then

$h_{j}\overset{L^{1}_{loc}(B_{1})}{\longrightarrow}0.$

###### Proof.

Let us recall for completeness the short proof from [10]. Let $\varphi\in
C^{\infty}_{0}(B_{1})$ with $0\leq\varphi\leq 1$. Since $h_{j}$ is supported
in $B_{1}$, it is enough to prove

$\lim_{j\rightarrow+\infty}\int_{B_{1}}\varphi|h_{j}|dx=0.$

We write

$\int_{B_{1}}\varphi|h_{j}|dx=\int_{B_{1}}\varphi h_{j}dx+2\int_{B_{1}}\varphi
h_{j}^{-}dx\leq\int_{B_{1}}\varphi h_{j}dx+2\int_{B_{1}}h_{j}^{-}dx.$

As $h_{j}\overset{*}{\rightharpoonup}0$ it is enough to show that the last
integral converges to $0$. Then from the equi-integrability, for any
$\varepsilon>0$ there exists $\delta>0$ such that

$\mathcal{L}^{d}(E)\leq\delta\Rightarrow\sup_{j\in\mathbb{N}}\int_{E}h_{j}^{-}dx\leq\varepsilon.$

From the convergence in measure we have

$\exists J;\forall j\geq
J,\mathcal{L}^{d}(\\{h_{j}^{-}>\varepsilon\\}\leq\delta.$

We deduce that for $j\geq J$

$\int_{B_{1}}h^{-}_{j}dx\leq\int_{\\{h_{j}^{-}>\varepsilon\\}\cap
B_{1}}h^{-}_{j}+\int_{\\{h_{j}^{-}\leq\varepsilon\\}\cap
B_{1}}h^{-}_{j}\leq\varepsilon(1+\mathcal{L}^{d}(B_{1})).$

∎

We now check that

$h_{j}:=\tilde{\chi}P^{j}\tilde{\chi}(B^{j}V_{j}+Q^{j}g_{j}),$

satisfies the assumptions of Lemma 3.5:

  1. 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 $0$,

  2. b)

from (3.22), $h_{j}$ and hence also $h_{j}^{-}$ tends to $0$ in measure,

  3. c)

as $u_{j}\geq 0$, the negative part of $h_{j}$ is bounded by
$(\chi\nu_{j}-h_{j})^{+}$ which we already proved it is relatively compact in
$L^{1}_{loc}(\mathbb{R}^{d})$. 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 $L^{1}_{loc}(\mathbb{R}^{d})$, and finishes the proof of Proposition 3.4
and hence of Theorem 1.2. ∎

##  4\. Proof of Theorems 1.4, 1.5, and 1.6

###### Proof of Theorem 1.4.

The first assertion of Theorems 1.4 is equivalent to

$\mu\in\mathcal{M}_{t},\Pi_{13}(WF_{L^{1}}(\mu)))=\varnothing\iff\mu\in
L^{1}_{loc}.$

If $\mu\in L^{1}_{loc}$ then all points
$(x,\omega)\in\mathbb{R}^{d}\times\mathbb{R}^{m}$ are in the complementary of
the projection of the wave front set, $\Pi_{13}(WF_{L^{1}}(\mu)))$ since (3.1)
is satisfied with $B=\chi\in C^{\infty}_{c}(\mathbb{R}^{d})$ equal to $1$ near
$x$. Conversely, if $\Pi_{13}(WF_{L^{1}}(\mu)))=\varnothing$ then Theorem 1.2
ensures that $\mu\in L^{1}$.

Concerning the second assertion, let us first consider
$x\notin\mbox{supp}\,|\mu|_{s}$. Then there exists $\chi\in
C^{\infty}_{c}(\mathbb{R}^{d})$ equal to $1$ near $x$ and such that
$\chi\mu\in L^{1}$. The first assertion ensures that
$WF_{L^{1}}(\chi\mu)=\varnothing$, so using (1.4) we get
$x\notin\Pi_{1}(WF_{L^{1}}(\mu)))$. So we have the inclusion
$\Pi_{1}(WF_{L^{1}}(\mu)))\subseteq\mbox{supp}\,|\mu|_{s}.$ On the other hand
Theorem 1.2 gives us the existence of a set $N$ of null $|\mu|_{s}$-measure
such that for $x\in N^{c}$ we have
$(x,\frac{d\mu}{d|\mu|}(x))\in\Pi_{13}(WF_{L^{1}}(\mu)))$ so in particular
$x\in\Pi_{1}(WF_{L^{1}}(\mu)))$. Therefore we get the last assertion.

∎

###### Proof of Theorems 1.5 and 1.6.

Theorem 1.5 is clearly a consequence of Theorem 1.6. By the same argument as
in the proof of the last point in Remark 1.3, we can assume that $\mu$ is
supported in a neighborhood of $x_{0}$ where the elliptic assumption holds.
Now, by standard elliptic regularity we can invert $A$ on the support of $\mu$
and get that

$\mu\in W^{k-\epsilon,1},\,\forall\epsilon>0,$

the $\epsilon$-loss coming from the defect of $L^{1}$-boundedness of $0$-th
order operators. We now come back to

$A(B\mu)=(AB-
CA)\mu+CA\mu\in\Psi^{k}(W^{1-2\epsilon,1})+\Psi^{k}(L^{1})\subset\Psi^{k}(L^{1}),$

(by choosing $2\epsilon<1$) because according to (1.5), $AB-CA\in\Psi^{2k-1}$.
We deduce from the third point in Remark 1.3 that
$WF_{L^{1}}(B\mu)=\varnothing$. From Theorem 1.4 we get Theorem 1.6.

∎

##  5\. On the singular part of elementary constrained measures and more

In [10] was considered the question of elementary constrained measures, i.e.
measures of the form $\mu_{0}=\omega_{0}\nu$ with
$\omega_{0}\in\mathbb{R}^{m*},\nu\in\mathcal{M}_{t}(\mathbb{R}^{d},\mathbb{R})$,
constrained to vanish under the action of a first order linear constant
coefficient operator $\sum_{|\alpha|=1}A_{\alpha}\partial_{x}^{\alpha}$,
$A_{\alpha}\in\mathbb{R}^{n\times m}$, satisfying

$C:=\\{\xi\in\mathbb{R}^{d},\sum_{|\alpha|=1}A_{\alpha}\omega_{0}\,\xi^{\alpha}=0\\}\neq\\{0_{\mathbb{R}^{d}}\\}.$

It was noticed in [10] that in this case, passing in Fourier the equation
yields $\mbox{supp }\hat{\nu}\subseteq C$ and $\nu$ is invariant in the
directions orthogonal to $C$. Indeed, for $\tilde{x}\in C^{\perp}$, using that
$\widehat{\nu}$ is supported in $C$,

$\nu(x+\tilde{x})=\int
e^{i(x+\tilde{x})\xi}d\hat{\nu}(\xi)=\int_{C}e^{i(x+\tilde{x})\xi}d\hat{\nu}(\xi)=\nu(x).$

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 ($n=m=1$,
$\partial_{t}^{2}-\Delta$) for which the characteristic manifold is

$C=\\{(\tau,\eta)\in\mathbb{R}^{d+1};\tau^{2}=|\eta|^{2}\\},$

whose orthogonal set $C^{\perp}$ is reduced to $\\{0_{\mathbb{R}^{d+1}}\\}$,
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 $WF_{L^{1}}$, 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 [6]). The
results in this section can be seen as a very first step toward of a general
theory of microlocal defect measures at the $L^{1}$ level, which is still
quite far away!

We first notice that the special choice $\mu_{0}=\omega_{0}\nu$ reduces the
study to a system of $n$ scalar equations on the measure $\nu$,

$A_{j}\nu=\sum_{|\alpha|=1}A^{j}_{\alpha}\omega_{0}\partial_{x}^{\alpha}\nu=0,$

where $A^{j}_{\alpha}$ is the $j$-th line of the matrix $A_{\alpha}$, and the
invariance along $C^{\perp}$ is just the invariance of $\nu$ by each of the
$n$ transport equations $A_{j}$. When studying propagation of singularities
for systems, the natural extension of scalar equations (see [9, Sections 3 &
4] and [6] 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 $\mathbb{R}^{d}$

$A:=\sum_{i=1}^{d}b_{i}(x)\,\partial_{x_{i}},\quad^{t}A:=A+\operatorname{div}(b)=\sum_{i=1}^{d}\partial_{x_{i}}\circ
b_{i},$

a function $H\in C(\mathbb{R}^{d},M_{m\times m}(\mathbb{R}))$ and
$\mu_{0}\in\mathcal{M}_{\text{loc}}(\mathbb{R}^{d},\mathbb{R}^{m})$ solution
of the system444Abusing notations we still denote by $A$ the vector field
$A\text{Id}_{\mathbb{R}^{m}}$ and ${}^{t}A=\sum_{i}\partial_{x_{i}}\circ
b_{i}\,\text{Id}_{\mathbb{R}^{m}}$ its transpose.

(5.1)
$^{t}A\mu_{0}+H\mu_{0}=\operatorname{div}(b\mu_{0})+H\mu=\Bigl{(}\sum_{i=1}^{d}\partial_{x_{j}}(b_{i}\,\mu_{0,k})+\sum_{p=1}^{m}h_{k,p}\,\mu_{0,p}\Bigr{)}_{k=1,\dots,m}=0_{\mathbb{R}^{m}}.$

Then the set

$Z=\\{\bigl{(}x,\frac{d\mu_{0}}{d|\mu_{0}|}(x)\bigr{)},x\in\mathbb{R}^{d}\\},\qquad(\text{
resp.
}Z_{s}=\\{\bigl{(}x,\frac{d\mu_{0,s}}{d|\mu_{0,s}|}(x)\bigr{)},x\in\mathbb{R}^{d}\\})$

is $|\mu_{0}|$-a.e. (resp. $|\mu_{0,s}|$-a.e) invariant by the flow

$(x_{0},\omega_{0})\mapsto\bigl{(}x(s),\frac{\omega(s)}{\|\omega(s)\|}\bigr{)},$

where $x(s)=\phi(s,x_{0})$ and
$\omega(s)=\widetilde{\phi}(s,x_{0},\omega_{0})$ are defined by

$\dot{x}(s)=b(x(s)),\;x(0)=x_{0},\qquad\dot{\omega}(s)=-H(x(s))\omega(s),\;\omega(0)=\omega_{0}.$

###### Proof.

The proof is easy: we just solve the equation! More precisely, we solve the
associated equation

(5.2) $\partial_{s}\mu+\operatorname{div}(b\mu)+H\mu=0,\mu_{|_{s=0}}=\mu_{0}$

Since by the duality method the solution to this equation is unique, if
$\mu_{0}$ solves (5.1), then the unique solution to (5.2) is given by
$\mu(s)\equiv\mu_{0}$. Let us now assume that $\mu$ solves (5.2) and define
$\nu$ by

$\nu=\phi(-s,\cdot)_{\sharp}\mu\Leftrightarrow\mu=\phi(s,\cdot)_{\sharp}\nu.$

Here we abuse slightly notations and denote by $\phi(s,\cdot)_{\sharp}$ the
push forward of the measure $\mu$ by the map

$(s,x)\mapsto(s,\phi(s,x)),$

defined by

$\langle\phi(s,\cdot)_{\sharp}\mu,\psi\rangle=\langle\mu,\psi(s,\phi(s,x))\rangle,$

for any $\psi\in C^{\infty}_{0}(\mathbb{R}^{d+1},\mathbb{R}^{m}).$ By using
the definition of $\phi$ we get

$\langle\partial_{s}\mu+\operatorname{div}(b\mu)+H\mu,\psi\rangle=-\langle\mu,(\partial_{1}+b.\nabla_{x})\psi\rangle+\langle
H\mu,\psi\rangle$
$=-\langle\nu,(\partial_{1}\psi+b.\nabla_{x}\psi)_{|_{(s,\phi(s,x))}}\rangle+\langle
H\mu,\psi\rangle=-\langle\nu,\partial_{s}(\psi(s,\phi(s,x)))\rangle+\langle
H\mu,\psi\rangle$
$=\langle\partial_{s}\nu,\psi_{|_{(s,\phi(s,x))}}\rangle+\langle
H(\cdot)\,\phi(s,\cdot)_{\sharp}\nu,\psi\rangle=\langle\phi(s,\cdot)_{\sharp}(\partial_{s}\nu+H(\phi(s,x))\nu),\psi\rangle.$

We deduce that $\mu$ solves (5.2) if and only if
$\nu=\phi(-s,\cdot)_{\sharp}\mu(s,\cdot)$ solves

$\partial_{s}\nu+H(\phi)\nu=0,\quad\nu_{|_{s=0}}=\mu_{0}.$

To solve this equation we apply the variation of parameter method and compute,
with $C(s,x)$ to be defined,

$\partial_{s}(C(s,x)\,\nu)(s,x)=\left(\partial_{s}C(s,x)-C(s,x)H(\phi(s,x)\right)\nu(s,x).$

Let us now define $C$ as the solution to the differential equation

$\partial_{s}C(s,x)=C(s,x)H(\phi(s,x)),\quad C_{|_{s=0}}=\text{Id}.$

Remark that $C$ is invertible as $C^{-1}$ is the solution to

$\partial_{s}C^{-1}(s,x)=-H(\phi(s,x))C^{-1}(s,x),\quad{C^{-1}}_{|_{s=0}}=\text{
Id}.$

We get then

$C(s,x)\nu(t,\cdot)=C(0,x)\nu(0,\cdot)=\mu_{0}.$

Summarising, we proved

$\mu_{0}=\mu(s,\cdot)=\phi(s,\cdot)_{\sharp}\nu(s,\cdot)=\phi(s,\cdot)_{\sharp}\left(C^{-1}(s,x)\mu_{0}\right)\Leftrightarrow\phi(-s,\cdot)_{\sharp}\mu_{0}=C^{-1}(s,x)\mu_{0}.$

This implies that the polarisation $\frac{d\mu_{0}}{d|\mu_{0}|}$ of $\mu_{0}$
at $\phi(s,x)$ is colinear to $C^{-1}(s,x)$ applied to the polarisation of
$\mu_{0}$ at $x$ (colinear rather than equal because the Jacobian determinant
of the change of variables $x\mapsto\phi(s,x)$ 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 $\mathbb{R}^{2}$ and
$\mu=(\mu_{1},\mu_{2})\in\mathcal{M}_{loc}(\mathbb{R}^{2},\mathbb{R}^{m_{1}+m_{2}})$
solution to the coupled system

(5.3) $\begin{pmatrix}{}^{t}A_{1}&0\\\
0&{}^{t}A_{2}\end{pmatrix}\begin{pmatrix}\mu_{1}\\\
\mu_{2}\end{pmatrix}+\begin{pmatrix}H_{11}&H_{12}\\\
H_{21}&H_{22}\end{pmatrix}\begin{pmatrix}\mu_{1}\\\ \mu_{2}\end{pmatrix}=0$

where we now assume for simplicity that the matrices $H_{jk}$ is $C^{\infty}$.
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 $x\in\mathbb{R}^{2}$, and that $\mu$ solves (5.3). Then for
$j\in\\{1,2\\}$ the sets

$Z_{j}=\\{\bigl{(}x,\frac{d\mu_{j}}{d|\mu_{j}|}(x)\bigr{)},x\in\mathbb{R}^{d}\\}$

are $|\mu_{j,s}|$-a.e. invariant by the flows

$(x_{0},\omega_{0})\mapsto(x_{j}(s),\frac{\omega_{j}(s)}{\|\omega_{j}(s)\|}),$

where

$\dot{x}_{j}(s)=b_{j}(s),\;x_{j}(0)=x_{0},\quad\dot{\omega}_{j}(s)=-H_{jj}(x_{j}(s))\,\omega_{j}(s),\;\omega_{j}(0)=\omega_{0}.$

In other words, as far as their singular parts are concerned, the propagation
formulas for $\mu_{j,s}$ 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 [3] 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) $^{t}A_{j}\,\mu_{j,s}+H_{jj}\,\mu_{j,s}=0.$

We work near a point $x_{0}\in\mathbb{R}^{2}$ and can replace $\mu_{j}$ by
$\chi(x)\mu_{j}$, $\chi\in C^{\infty}_{0}(\mathbb{R}^{2})$ equal to $1$ near
$x_{0}$ and after a linear change of variables, we can assume that

$A_{1}(x_{0})=\,^{t}A_{1}(x_{0})=\frac{\partial}{\partial x_{1}},\quad
A_{2}(x_{0})=\,^{t}A_{2}(x_{0})=\frac{\partial}{\partial x_{2}}.$

Consider now a smooth cut off $\zeta_{1}(\xi)$ vanishing near $0$, homogeneous
of degree $0$ outside $\\{\|\xi\|\geq 1\\}$ and equal to $1$ in a small
conical neighborhood of $(0,1)\cup(0,-1)$. The first step is the following
Lemma.

######  Lemma 5.4.

Near $x_{0}$ we have

$(1-\zeta_{1})(D_{x})\mu_{1}\in W^{1-\epsilon,1}(\mathbb{R}^{2}).$

Notice also that this implies $\mu_{1,s}=(\zeta_{1}(D_{x})\mu_{1})_{s}$.

###### Proof.

Let $\zeta_{0}\in C^{\infty}_{0}(\mathbb{R}^{2})$ equal to $1$ near $0$. Since
$\zeta_{0}(D_{x})\in\Psi^{-N}$, we have from Corollary 6.7:

$(1-\zeta_{1})\zeta_{0}(D_{x})\mu_{1}\in W^{1-\epsilon,1}(\mathbb{R}^{2}).$

It remains to study

$(1-\zeta_{1})(1-\zeta_{0})(D_{x})\mu_{1}.$

Near $x_{0}$ the principal symbol $a_{1}(x,\xi)$ of the operator ${}^{t}A_{1}$
is close to its value at $(x_{0},\xi)$ which is $i\xi_{1}$, hence it is
invertible in a neighborhood of the support of
$(1-\zeta_{1})(1-\zeta_{0})(\xi)$. As a consequence, we can define

$p(x,\xi)=\frac{\chi(x)(1-\zeta_{1})(1-\zeta_{0})(\xi)}{a_{1}(x,\xi)},$

with $\chi\in C^{\infty}_{0}$ smooth, equal to $1$ near $x_{0}$. Applying
$p(x,D_{x})$ to the equation satisfied by $\mu_{1}$,

(5.5) $^{t}A_{1}\,\mu_{1}=-H_{11}\,\mu_{1}-H_{12}\,\mu_{2},$

we get by symbolic calculus

$\chi(x)(1-\zeta_{1})(1-\zeta_{0})(D_{x})\mu_{1}=R_{1}\mu_{1}+R_{2}\mu_{2},$

where $R_{j}$ are matrices of pseudodifferential operators of order $-1$,
which implies by Corollary 6.7 that

$\chi(x)(1-\zeta_{1})(1-\zeta_{0})(D_{x})\mu_{1}\in
W^{1-\epsilon,1}(\mathbb{R}^{2}).$

∎

Let now $\widetilde{\zeta_{1}}$ be a cut-off equal to $1$ in a neighborhood of
the support of $\zeta_{1}$. Applying $\zeta_{1}(D_{x})$ to equation (5.5) we
get

(5.6) $^{t}A_{1}\,\zeta_{1}(D_{x})\mu_{1}+H_{11}\,\zeta_{1}(D_{x})\mu_{1}$
$=-H_{12}\,\zeta_{1}(D_{x})\mu_{2}+[^{t}A_{1},\zeta_{1}(D_{x})]\mu_{1}+[H_{11},\zeta_{1}(D_{x})]\mu_{1}+[H_{12},\zeta_{1}(D_{x})]\mu_{2},$

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 used555This is where we
use the smoothness of the $A_{j,k}$; this smoothness could be relaxed to
Hölder continuity or even log continuity
$\|A(x)-A(y)\|\leq\frac{C}{|\log^{\alpha}(\|x-y\|)|},\alpha>1,\|x-y\|\leq\frac{1}{2}.$
. Let us now study the first term. The function $\zeta_{1}$ is supported in a
small conical neighborhood of $(0,1)\cup(0,-1)$. We can now choose a smooth
function $\zeta_{2}$ vanishing near $0$, homogeneous of degree $0$ outside
$\\{\|\xi\|\geq 1\\}$ and equal to $1$ in a small conical neighborhood of
$(1,0)\cup(-1,0)$, and which vanishes on the support of $\zeta_{1}$. This is
where we use crucially that we are working in $\mathbb{R}^{2}$: in higher
dimensions, the cut-off $\zeta_{1}$ would be required to vanish near the
characteristic manifold of $X_{1}$, which at $x_{0}$ is

$C_{1}=\\{\xi;\xi_{1}=0\\}$

while the cut-off $\zeta_{2}$ is required to vanish on the characteristic
manifold of $X_{2}$ which is at $x_{0}$,

$C_{2}=\\{\xi;\xi_{2}=0\\}.$

In dimension $2$, these two manifolds intersect at $0_{\mathbb{R}^{2}}$. In
higher dimension they intersect along the plane

$C_{1,2}=\\{\xi;\xi_{1}=\xi_{2}=0\\},$

and consequently such a choice for $\zeta_{2}$ (equal to $1$ near $C_{1}$ but
vanishing near $C_{2}$, apart from a neighborhood of $0$) is possible only in
dimension $2$.

$\zeta_{2}=1$$\zeta_{2}=1$$\zeta_{1}=1$$\zeta_{1}=1$$\xi_{1}$$\xi_{2}$ Figure
1. The cut-off functions

Applying Lemma 5.4 exchanging the roles of $\mu_{1}$ and $\mu_{2}$ (and the
roles of the variables $x_{1}$ and $x_{2}$), we get by Proposition 6.6 that

$(1-\zeta_{2})(D_{x})\mu_{2}\in
W^{1-\epsilon,1}(\mathbb{R}^{2})\Rightarrow\zeta_{1}(D_{x})\mu_{2}=\zeta_{1}(D_{x})(1-\zeta_{2})(D_{x})\mu_{2}\in
W^{1-2\epsilon,1}(\mathbb{R}^{2}),$

and consequently near $x_{0}$,

$H_{12}\,\zeta_{1}(D_{x})\mu_{2}\in L^{1}.$

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

$[^{t}A_{1},\zeta_{1}(D_{x})]=-(\nabla_{x}a_{1}.\nabla_{\xi}\zeta_{1})(x,D_{x})+R,\quad
R\in\Psi^{-1}.$

Now $R\mu_{1}\in W^{1-\epsilon,1}(\mathbb{R}^{2})$, and since $\zeta_{1}=1$ on
a neighborhood of the support of $\widetilde{\zeta_{1}}$, we get that
$\nabla_{\xi}\zeta_{1}$ is supported where $(1-\widetilde{\zeta_{1}})=1$ and
consequently we have

$(\nabla_{x}a_{1}.\nabla_{\xi}\zeta_{1})(x,D_{x})=(\nabla_{x}a_{1}.\nabla_{\xi}\zeta_{1}(1-\widetilde{\zeta_{1}}))(x,D_{x}),$

therefore Lemma 5.4 with $\tilde{\zeta}_{1}$ and Proposition 6.6 imply

$(\nabla_{x}a_{1}.\nabla_{\xi}\zeta_{1})(x,D_{x})\mu_{1}=(\nabla_{x}a_{1}.\nabla_{\xi}\zeta_{1})(x,D_{x})(1-\widetilde{\zeta_{1}})(D_{x})\mu_{1}\in
W^{1-2\epsilon,1}(\mathbb{R}^{2}).$

Summarizing, we have obtained from (5.6) that in a neighborhood of $x_{0}$

${}^{t}A_{1}\,\zeta_{1}(D_{x})\mu_{1}+H_{11}\,\zeta_{1}(D_{x})\mu_{1}\in
L^{1}(\mathbb{R}^{d}).$

We can now revisit the proof of Lemma 5.1, with $\mu_{0}$ replaced by
$\zeta_{1}(D_{x})\mu_{1}$ (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

$\phi_{1}(-s,\cdot)^{\sharp}\bigl{(}\zeta_{1}(D_{x})\mu_{1}\bigr{)}-C_{1}^{-1}(s,x)\bigl{(}\zeta_{1}(D_{x})\mu_{1}\bigr{)}\in
L^{1}_{\text{loc}},$

Passing to the singular parts (from Lemma 5.4
$\mu_{1,s}=\zeta_{1}(D_{x})\mu_{1})_{s}$), we get near $x_{0}$ (and for small
$s$)

$\phi_{1}(-s,\cdot)^{\sharp}\mu_{1,s}=C_{1}^{-1}(s,x)\mu_{1,s},$

which implies Theorem 5.2 (and also (5.4)) for $\mu_{1}$ and small $s$. The
general case is obtained by iterating in $s$. The proof for $\mu_{2}$ is
similar.

∎

##  6\. Pseudodifferential operators

In this section we have gathered basic facts about pseudodifferential
operators. We refer to [15, Chapter XVIII] for a general presentation or [25,
Section VI.6] for a presentation closer to our needs of the pseudodifferential
calculus (see also [2]). Let us recall the definitions of the symbol classes.
Here we adopt the following convention:

$\forall\alpha\in\mathbb{N}^{d},|\alpha|=\sum_{j=1}^{d}\alpha_{j},\quad\alpha!=\prod_{j=1}^{d}\alpha_{j}!,\quad\partial_{x}^{\alpha}=\frac{\partial^{\alpha_{1}}}{\partial
x_{1}^{\alpha_{1}}}\circ\dots\circ\frac{\partial^{\alpha_{d}}}{\partial
x_{d}^{\alpha^{d}}}.$

We now define the class of symbols of order $k$ by

(6.1) $S^{k}(\mathbb{R}^{d})=\\{a\in C^{\infty}(\mathbb{R}^{3d});\\\
\forall\alpha,\beta,\gamma\in\mathbb{N}^{d},\sup_{x,y,\xi\in\mathbb{R}^{d}}|\partial_{x}^{\alpha}\partial_{y}^{\beta}\partial_{\xi}^{\gamma}a(x,y,\xi)|(1+|\xi|)^{-k+|\gamma|}=:\|a\|_{k,\alpha,\beta,\gamma}<+\infty\\}.$

The constants $\|a\|_{\alpha,\beta,\gamma}$ 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 $S^{k}_{\text{cl}}$
of symbols admiting homogeneous principal symbol i.e. there exists in addition
$\chi\in C^{\infty}_{c}(\mathbb{R}^{d})$ and $\tilde{a}_{k}\in
C^{\infty}(\mathbb{R}^{d}\times\mathbb{S}^{d})$ such that:

$a-(1-\chi)(\xi)|\xi|^{k}\tilde{a}_{k}(x,\frac{\xi}{|\xi|})\in S^{k-1}.$

The function $a_{k}(x,\xi):=|\xi|^{k}\tilde{a}_{k}(x,\frac{\xi}{|\xi|})$ is
the principal symbol of $a$.

To any symbol $a\in S^{k}_{\text{cl}}(\mathbb{R}^{d})$ we can associate an
operator on the temperate distributions set
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ by the formula

$a(x,y,D_{x})u(x)=\text{Op}(a)u(x)=\frac{1}{(2\pi)^{d}}\int
e^{i(x-y)\cdot\xi}a(x,y,\xi)u(y)dyd\xi,$

where this integral is defined as an oscillatory integral. We shall denote
$Op(a)\in\Psi^{k}_{\text{cl}}(\mathbb{R}^{d})$. Remark that if $a$ do not
depend on the variable $x$, we have

(6.2) $\text{Op}(a)u(x)=\mathcal{F}^{-1}\left(\int
e^{-iy\cdot\xi}a(y,\xi)u(y)dyd\xi\right)(x).$

An operator $Op(a)$ is said to be elliptic at a point $(x_{0},\xi_{0})$ if the
principal symbol $a_{k}(x,y,\xi)$ is non zero at the point
$(x_{0},x_{0},\frac{\xi_{0}}{|\xi_{0}|})$.

######  Definition 6.1.

For any $a\in S^{k}(\mathbb{R}^{d})$ and any sequence $a_{i}\in
S^{k_{i}}(\mathbb{R}^{d})$ with $k_{0}=k>k_{1}>\dots$, we write

$a\sim\sum_{i}a_{i}\Leftrightarrow\forall M,(a-\sum_{i=0}^{N}a_{i})\in
S^{k_{N+1}}.$

The basic properties of symbolic calculus of pseudodifferential operators are
summarized in the following

######  Proposition 6.2.

We have the following symbolic calculus properties

  * •

For any $a\in S^{k}_{\text{cl}}(\mathbb{R}^{d})$, there exists
$\widetilde{a}\in S^{k}_{\text{cl}}(\mathbb{R}^{d})$ not depending on the
variable $y$. (resp. $\widetilde{\widetilde{a}}\in
S^{k}_{\text{cl}}(\mathbb{R}^{d})$ not depending on the variable $x$), such
that

$Op(a)=Op(\widetilde{a}),(\text{resp. }Op(a)=Op(\widetilde{\widetilde{a}}),$

with

$\widetilde{a}(x,\xi)\sim\sum_{N}\sum_{|\alpha|\leq
N}\frac{i^{|\alpha|}}{\alpha!}(\partial_{y}^{\alpha}\partial_{\xi}^{\alpha}a(x,y,\xi))\mid_{y=x},$
$\widetilde{\widetilde{a}}(y,\xi)\sim\sum_{N}\sum_{|\alpha|\leq
N}\frac{(-i)^{|\alpha|}}{\alpha!}(\partial_{x}^{\alpha}\partial_{\xi}^{\alpha}a(x,y,\xi))\mid_{x=y}.$

In particular,

$\tilde{a}_{k}(x,\xi)=a(x,x,\xi),\quad\widetilde{\widetilde{a}}_{k}(y,\xi)=a(y,y,\xi),$

and $Op(a)$ is elliptic at a point $(x_{0},\xi_{0})$ if and only if
$Op(\tilde{a})$ is elliptic at $(x_{0},\xi_{0})$ in the sense that
$\tilde{a}_{k}(x_{0},\xi_{0})\neq 0$ (and similarly for the right
quantization).

  * •

The formal $L^{2}$ adjoint of a pseudodifferential operator, $Op(a)$, is the
pseudodifferential operator $Op(a^{*})$, with

$a^{*}(x,y,\xi)=\overline{a(y,x,\xi)}.$

  * •

For any $a\in S^{k}_{\text{cl}}(\mathbb{R}^{d}),b\in
S^{\tilde{k}}(\mathbb{R}^{d})$, there exists $c\in
S^{k+\tilde{k}}(\mathbb{R}^{d})$ such that

$Op(a)\circ Op(b)=Op(c).$

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) $c=a\sharp
b\sim\sum_{N}\sum_{|\alpha|=N}\frac{i^{|\alpha|}}{\alpha!}\partial_{\xi}^{\alpha}(a(x,\xi))\partial_{x}^{\alpha}(b(x,\xi)).$

######  Remark 6.3.

From the explicit formula

$a\sharp b(x,\xi)=\frac{1}{(2\pi)^{d}}\int
e^{i(x-y)\cdot(\eta-\xi)}a(x,\eta)b(y,\xi)dyd\eta,$

we can actually get a quantitative version of (6.3). Namely, for any $N_{0}$,
each $S^{k+\tilde{k}-N_{0}-1}(\mathbb{R}^{d})$ semi-norm of

(6.4) $a\sharp b-\sum_{N\leq
N_{0}}\sum_{|\alpha|=N}\frac{i^{|\alpha|}}{\alpha!}\partial_{\xi}^{\alpha}(a(x,\xi))\partial_{x}^{\alpha}(b(x,\xi))$

is bounded by a product of a finite number of semi-norms of $a$ and $b$.

Recal that the Sobolev space $W^{s,p}(\mathbb{R}^{d})$ is defined by

$W^{s,p}(\mathbb{R}^{d})=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{d});(1-\Delta)^{s/2}u\in
L^{p}(\mathbb{R}^{d})\\},$

and that we have the following property.

######  Proposition 6.4.

For any $1\leq p\leq+\infty$ and any $s>0$, $\chi\in
C^{\infty}_{0}(\mathbb{R}^{d})$, the application

$u\in W^{s,p}(\mathbb{R}^{d})\mapsto\chi u\in L^{p}(\mathbb{R}^{d})$

is compact.

Pseudodifferential operators are bounded on $L^{p}$, and more generally on
$W^{s,p}$, for $1<p<+\infty$. More precisely, we have the following result.

######  Proposition 6.5 (see e.g. [25, Section VI.5.2] ).

Let $A=Op(a)\in\Psi^{0}_{\text{cl}}$. Then, for all $1<p<+\infty$,
$s\in\mathbb{R}$, the operator $A$ is continuous on $W^{s,p}(\mathbb{R}^{d})$.
Furthermore, its norm is bounded by a finite number of semi-norms

$\exists N(d);\|Op(a)\|_{\mathcal{L}(W^{s,p}(\mathbb{R}^{d}))}\leq
C\sup_{|\alpha|+|\beta|+|\gamma|\leq N(d)}\|a\|_{0,\alpha,\beta,\gamma}.$

In general, pseudodifferential operators are not continuous on $L^{1}$ and
$L^{\infty}$, the basic example being the Hilbert transform (smoothed out near
$\xi=0$), associated to the symbol

$a(x,y,\xi)=\chi(\xi),\chi\in
C^{\infty}(\mathbb{R}),\chi\mid_{(-\infty,-1)}=0,\chi\mid_{(1,+\infty)}=1.$

We also have the following counter example (see the introduction). Let
$\chi\in C^{\infty}_{0}(\mathbb{R}^{2})$ equal to $1$ near $0$, and

(6.5) $u(x):=\log\log(e{|x|^{-1}})\in W^{1,1}(B_{1}),w:=\chi u\in
W^{1,1}_{comp}(\mathbb{R}^{2})\\\
\Rightarrow(-\Delta+1)w=\frac{\chi(x)}{|x|^{2}\log^{2}(e{|x|^{-1}})}+[\chi,\Delta]u+w=f\in
L^{1}_{comp}(B_{1}),\quad D^{2}w\notin L^{1}_{loc}.$

We thus get

(6.6) $\frac{D^{2}}{(-\Delta+1)^{-1}}f\notin L^{1}_{loc},\qquad f\in
L^{1}_{comp}.$

However, by the dual estimate of the Lemma in [25, Section VI.5.3.1], we have
the following weaker result (we shall use only the case $p=1$).

######  Proposition 6.6.

Let $\delta\in\mathbb{R},\epsilon>0$ and $A=Op(a)\in\Psi^{\delta}$. Then for
all $1\leq p\leq+\infty,s\in\mathbb{R}$ the operator $A$ is continuous from
$W^{s,p}(\mathbb{R}^{d})$ to $W^{s-\delta-\epsilon,p}(\mathbb{R}^{d})$.
Furthermore, its norm is bounded by a finite number of semi-norms

$\exists
N(d,\epsilon),\,\|Op(a)\|_{\mathcal{L}(W^{s,p}(\mathbb{R}^{d}),W^{s-\delta-\epsilon,p}(\mathbb{R}^{d})}\leq
C\sup_{|\alpha|+|\beta|+|\gamma|\leq N(d)}\|a\|_{\delta,\alpha,\beta,\gamma}.$

######  Corollary 6.7.

Let $\delta\in\mathbb{R},\eta>0$ and $A=Op(a)\in\Psi^{\delta}$. Then, for all
$s\in\mathbb{R}$, the operator $A$ is continuous from
$\mathcal{M}_{0}(\mathbb{R}^{d})$ to $W^{-\delta-\eta,1}(\mathbb{R}^{d})$.
Furthermore, its norm is bounded by a finite number of semi-norms

$\exists
N(d,\eta),\,\|Op(a)\|_{\mathcal{L}(\mathcal{M}_{0}(\mathbb{R}^{d}),W^{-\delta-\eta,1}(\mathbb{R}^{d})}\leq
C\sup_{|\alpha|+|\beta|+|\gamma|\leq N(d)}\|a\|_{\delta,\alpha,\beta,\gamma}.$

Indeed, from the continuous inclusion for all $\eta>0$,

$W^{\eta/2,\infty}(\mathbb{R}^{d})\subset C^{0}(\mathbb{R}^{d}),$

we deduce by duality the continuous inclusion

$\mathcal{M}_{0}\subset W^{-\eta/2,1}(\mathbb{R}^{d}),$

and Corollary 6.7 follows from Proposition 6.6 with $s=-\eta/2$ and
$\epsilon=\eta/2$.

We end this section with a result involving weights and bounded measures.

######  Proposition 6.8.

Let $\delta\in\mathbb{R}$ $\epsilon>0$, $A\in\Psi^{\delta}$ and $\chi\in
C^{\infty}_{0}(\mathbb{R}^{d})$. Then, for any $k\in\mathbb{R}^{+}$, the
operator

$\chi A(1+|x|^{k})$

is continuous from $\mathcal{M}_{0}(\mathbb{R}^{d})$ to
$W^{-\delta-\epsilon,1}(\mathbb{R}^{d})$. Furthermore, its norm is bounded by
a finite number of semi-norms of $a$:

$\exists N(d,\epsilon),\,\|\chi
A(1+|x|^{k})\|_{\mathcal{L}(\mathcal{M}_{0},W^{-\delta,1}(\mathbb{R}^{d}))}\leq
C(1+\sup_{|\alpha|+|\beta|+|\gamma|\leq
N(d)}\|a\|_{\delta,\alpha,\beta,\gamma}).$

###### 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 unity

$1=\sum_{p\geq 0}\phi_{p}(x),\,\phi_{0}\in
C^{\infty}_{0}(\mathbb{R}^{d}),\,\forall p\geq
1,\phi_{p}(x)=\phi(2^{-p}x),\,\phi\in
C^{\infty}_{0}(\\{\frac{1}{2}<\|x\|<2\\}),$

and write

$\chi A(1+|x|^{k})=\sum_{p\geq 0}\chi A(1+|x|^{k})\phi_{p}(x).$

According to Corollary 6.7 each term is bounded from
$\mathcal{M}_{0}(\mathbb{R}^{d})$ to $W^{-\delta-\epsilon,1}(\mathbb{R}^{d})$,
and we just have to check that the series of the norms is summable. Consider
$K_{p}(x,y)$ the kernel of the operator $\chi A(1+|x|^{k})\phi_{p}(x)$:

(6.7) $K_{p}(x,y)=\frac{1}{(2\pi)^{d}}\int
e^{i(x-y)\cdot\xi}a(x,y,\xi)d\xi\,\chi(x)(1+|y|^{k})\phi_{p}(y).$

Remark that on the support of this kernel, $\|x\|\leq C,\|y\|\sim 2^{p-1},$
and consequently, for $k$ large enough, $\|x-y\|\sim 2^{p}$. Integrating by
parts $N$ times in (6.7) using the identity

$L(e^{i(x-y)\cdot\xi})=-e^{i(x-y)\cdot\xi},\qquad
L=\frac{i(x-y)\cdot\nabla_{\xi}}{\|x-y\|^{2}},$

we get

(6.8) $K_{p}(x,y)=\frac{1}{(2\pi)^{d}}\int
e^{i(x-y)\cdot\xi}L^{N}(a(x,y,\xi))d\xi\,\chi(x)(1+|y|^{k})\phi_{p}(y),$

and consequently

$\chi
A(1+|x|^{k})\phi_{p}(x)=\text{Op}(a_{N,p}),\,a_{N,p}=L^{N}\bigl{(}a(x,y,\xi))\chi(x)\phi_{p}(y)(1+|y|^{k})\in
S^{\delta-N}(\mathbb{R}^{d}),$

with semi-norms in $S^{\delta-N}$ bounded by

$\|a_{N,p}\|_{\delta-N,\alpha,\beta,\gamma}\leq
C_{N,\alpha,\beta,\gamma}2^{p(k-N)}\sum_{|\alpha^{\prime}|\leq|\alpha|,|\beta^{\prime}|\leq|\beta|,|\gamma^{\prime}|\leq|\gamma|+N}\|a\|_{\delta,\alpha^{\prime},\beta^{\prime},\gamma^{\prime}}.$

We deduce from Corollary 6.7 that $\chi A(1+|x|^{k})\phi_{p}$ is bounded from
$\mathcal{M}_{0}(\mathbb{R}^{d})$ to $W^{N-\delta-\eta,1}(\mathbb{R}^{d})$ by
$C_{N,\eta}2^{p(k-N)}$ and we conclude by choosing $N>k$.

∎

##  7\. Temperance

To deal with pseudodifferential 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 [21, Theorem 2.5]). A
slightly weaker upper-bound (7.1) can be found in [23, Proposition 10.5] with
$\eta=2$. As we also need the lower bound we include its short proof here.

######  Proposition 7.1.

Let $\nu$ be a compactly supported bounded non negative Radon measure on
$\mathbb{R}^{d}$, and let $\\{r_{j}\\}_{j\in\mathbb{N}}$ a sequence convergent
to zero. Then for $\nu$-a.e. points $x_{0}$ and every $\eta>0$, there is a
subsequence of $\\{r_{j_{k}}\\}_{k\in\mathbb{N}}$ and $C,c>0$ such that

$\nu_{k}:=\frac{(T^{x_{0},r_{j_{k}}})_{\sharp}\nu}{\nu(B(x_{0},r_{j_{k}}))},\qquad
T^{x_{0},r_{j_{k}}}:y\mapsto x_{0}+r_{j_{k}}x,$

satisfies the uniform bounds for $k\in\mathbb{N}$ and $R>1$:

(7.1)
$\nu_{k}(B(x_{0},R))=\frac{\nu(B(x_{0},Rr_{j_{k}}))}{\nu(B(x_{0},r_{j_{k}}))}\leq
CR^{d+\eta},$ (7.2)
$\nu_{k}(B(x_{0},R^{-1}))=\frac{\nu(B(x_{0},R^{-1}r_{j_{k}}))}{\nu(B(x_{0},r_{j_{k}}))}\geq
cR^{-(d+\eta)}.$

######  Remark 7.2.

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 $\nu=fd\mathcal{L}^{d}$, at all Lebesgue points of $f$, i.e. for
$\mathcal{L}^{d}$-almost every $x_{0}$ we have

$\lim_{r\rightarrow 0}\frac{1}{C_{d}r^{d}}\int_{B(x_{0},r)}f(x)dx=f(x_{0}).$

Since $0<|f(x_{0})|<+\infty$ for $fd\mathcal{L}^{d}$-almost every $x_{0}$ we
get easily that in (7.1) -(7.2) we can replace $R^{\pm(d+\eta)}$ by $R^{\pm
d}$ (without the logarithmic loss). We do not know if it is the case for
general $\nu$, or even for doubling measures $\nu$.

###### Proof.

We basically follow the construction of tangent measures in [21] with minor
differences, but since the temperance 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 $R$

(7.3) $\nu(A)=\frac{1}{\mathcal{L}^{d}(B(0,R))}\int\nu(A\cap B(x,R))dx.$

We consider for $r>0,\delta>0,k\in\mathbb{N}$ the set,

$E_{r,k,\delta}:=\\{x\in\mathbb{R}^{d},\nu(B(x,2^{k}r))\geq\beta_{k,\delta}\,\nu(B(x,r))\\},$

where

$\beta_{k,\delta}:=\frac{2^{d}(2^{k}+1)^{d})\,\nu(K)}{\delta},$

$K$ is a compact set where $\nu$ is supported, and $\delta>0$ to be chosen
later.

Applying (7.3) to $A=E_{r,k,\delta}$, $R=\frac{r}{2}$, we get

$\nu(E_{r,k,\delta})=\frac{1}{\omega_{d}(\frac{r}{2})^{d}}\int\nu(E_{r,k,\delta}\cap
B(x,\frac{r}{2}))dx,$

where $\omega_{d}=\mathcal{L}^{d}(B(0,1))$. If $E_{r,k,\delta}\cap
B(x,\frac{r}{2})\neq\varnothing$ then we can use a point $z$ in this
intersection to get

(7.4) $\nu(E_{r,k,\delta}\cap
B(x,\frac{r}{2}))\leq\nu(B(x,\frac{r}{2}))\leq\nu(B(z,r))\\\
\leq\frac{1}{\beta_{r,k,\delta}}\nu(B(z,2^{k}r))\leq\frac{1}{\beta_{r,k,\delta}}\nu(B(x,(2^{k}+1)r)).$

Therefore for any $r>0,\delta>0,k\in\mathbb{N}$, and using again (7.3) with
$A=K$ we obtain

(7.5)
$\nu(E_{r,k,\delta})\leq\delta\frac{1}{\omega_{d}((2^{k}+1)r)^{d}\nu(K)}\int\nu(B(x,(2^{k}+1)r))dx=\delta,$

and on the complementary set ${}^{c}E_{r,k,\delta}$ we have

(7.6)
$\frac{\nu(B(x,2^{k}r))}{\nu(B(x,r))}\leq\beta_{k,\delta}=\frac{2^{d}(2^{k}+1)^{d}\,\nu(K)}{\delta},$

while on the complementary set ${}^{c}E_{r2^{-k},k,\delta}$ we have similarly

(7.7)
$\frac{\nu(B(x,r))}{\nu(B(x,2^{-k}r))}\leq\beta_{k,\delta}=\frac{2^{d}(2^{k}+1)^{d}\,\nu(K)}{\delta}.$

Let $r_{j}$ be a sequence convergent to zero and $\epsilon>0$. Now we consider

$A_{\\{r_{j}\\},\epsilon}:=\cup_{i=1}^{\infty}\cap_{j=i}^{\infty}A_{r_{j},\epsilon},$

where
$A_{r_{j},\epsilon}:=\cup_{k=1}^{\infty}\bigl{(}E_{r_{j},k,a_{k}\epsilon}\cup
E_{r_{j}2^{-k},k,a_{k}\epsilon}\bigr{)}$ and $a_{k}=k^{-1-\gamma},\gamma>0$,
summable so that in view of (7.5) we have $\nu(A_{r_{j},\epsilon})\leq
C\epsilon$. In particular $\nu(A_{\\{r_{j}\\},\epsilon})\leq C\epsilon$. For
any point $x\notin A_{\\{r_{j}\\},\epsilon}$ we get for all $i\in\mathbb{N}$
the existence of $j\geq i$ such that $x\notin A_{r_{j},\epsilon}$. This yields
a subsequence $r_{j_{n}}$ such that $x\notin A_{r_{j_{n}},\epsilon}$ for all
$n\in\mathbb{N}$, thus $x\notin E_{r_{j_{n}},k,a_{k}\epsilon}$ and $x\notin
E_{r_{j_{n}}2^{-k},k,a_{k}\epsilon}$ for all $n,k\in\mathbb{N}$. Then,
renaming this sequence $r_{j}$ for simplicity, from (7.6)-(7.7) we get

$\frac{\nu(B(x,2^{k}r_{j}))}{\nu(B(x,r_{j}))}\leq\frac{2^{kd}}{k^{1+\gamma}\epsilon},\quad\frac{\nu(B(x,2^{-k}r_{j}))}{\nu(B(x,r_{j}))}\geq\frac{k^{1+\gamma}\epsilon}{2^{kd}},\quad\forall
j,k\in\mathbb{N}.$

As a consequence for all $x\notin A_{\\{r_{j}\\},\epsilon}$ there is a
subsequence along which a non-zero tangent measure at $x$ is obtained. By
defining

$A_{\\{r_{j}\\}}:=\cap_{n=1}^{\infty}A_{\\{r_{j}\\},\frac{1}{n}},$

we get a set of zero measure such that for all $x\notin A_{\\{r_{j}\\}}$, thus
in particular $x\notin A_{\\{r_{j}\\},\frac{1}{n_{x}}}$ for some
$n_{x}\in\mathbb{N}$, 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 $R\geq 1$. ∎

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