# Factorization for entropy production of the Eikonal equation and regularity

Andrew Lorent111Department of Mathematical Sciences, University of Cincinnati,
Cincinnati, OH 45221, USA. Email<EMAIL_ADDRESS>Guanying Peng222Department
of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA
01609, USA. Email<EMAIL_ADDRESS>
###### Abstract

The Eikonal equation arises naturally in the limit of the second order Aviles-
Giga functional whose $\Gamma$-convergence is a long standing challenging
problem. The theory of entropy solutions of the Eikonal equation plays a
central role in the variational analysis of this problem. Establishing fine
structures of entropy solutions of the Eikonal equation, e.g. concentration of
entropy measures on $\mathcal{H}^{1}$-rectifiable sets in $2$D, is arguably
the key missing part for a proof of the full $\Gamma$-convergence of the
Aviles-Giga functional. In the first part of this work, for
$p\in\left(1,\frac{4}{3}\right]$ we establish an $L^{p}$ version of the main
theorem of [GL20]. Specifically we show that if $m$ is a solution to the
Eikonal equation, then $m\in B^{\frac{1}{3}}_{3p,\infty,{\rm loc}}$ is
equivalent to all entropy productions of $m$ being in $L^{p}_{{\rm loc}}$.
Given the main result of [GL20], this result also shows that as a consequence
of a weak form of the Aviles-Giga conjecture (namely the conjecture that all
solutions to the Eikonal equation whose entropy productions are in
$L^{p}_{{\rm loc}}$ are rigid) - the rigidity/flexibility threshold of the
Eikonal equation is exactly the space $B^{\frac{1}{3}}_{3,\infty,{\rm loc}}$.
In the second part of this paper, under the assumption that all entropy
productions are in $L^{p}_{{\rm loc}}$, we establish a factorization formula
for entropy productions of solutions of the Eikonal equation in terms of the
two Jin-Kohn entropies. A consequence of this formula is control of all
entropy productions by the Jin-Kohn entropies in the $L^{p}$ setting - this is
a strong extension of the main result of [LP18].

##  1 Introduction

###  1.1 The Aviles-Giga functional and the Eikonal equation

The Aviles-Giga functional is a second order functional that (subject to
appropriate boundary conditions) models phenomena from thin film blistering to
smectic liquid crystals, and is also the most natural higher order
generalization of the Cahn-Hilliard functional. It is defined as

$AG_{\varepsilon}(u)=\int_{\Omega}\left(\varepsilon\left|\nabla^{2}u\right|^{2}+\frac{\left(1-\left|\nabla
u\right|^{2}\right)^{2}}{\varepsilon}\right)\;dx$

for $u\in W^{2,2}(\Omega)$ over a bounded domain
$\Omega\subset{\mathbb{R}}^{2}$, where $\nabla^{2}u$ is the Hessian matrix of
the scalar-valued function $u$ and $\varepsilon>0$ is a small parameter. The
Aviles-Giga conjecture for the $\Gamma$-limit of $AG_{\varepsilon}$ is one of
the central conjectures in the theory of $\Gamma$-convergence and has
attracted a great deal of attention, yet remains open; see for example [AG87,
AG96, ADLM99, DMKO01, DLO03, CDL07, Pol07, GL20]. What makes the Aviles-Giga
conjecture much more challenging than the $\Gamma$-convergence of the Cahn-
Hilliard functional is the cubic power scaling in the former, which makes the
$BV$ function theory inapplicable.

One of the foundational theorems established for the Aviles-Giga functional is
the compactness [ADLM99, DMKO01]. Specifically, given a sequence
$\\{u_{\varepsilon}\\}\subset W^{2,2}(\Omega)$ such that
$\sup_{\varepsilon}AG_{\varepsilon}(u_{\varepsilon})<\infty$, it has a
subsequence that converges strongly in $W^{1,3}$ to some limiting function $u$
(here we are stating the compactness result in [ADLM99]; the version proved in
[DMKO01] is slightly different). The limiting function $u$ must satisfy the
Eikonal equation given by

$\left|\nabla u\right|=1\quad\text{ a.e in }\Omega.$

In two dimensions, the above Eikonal equation can be equivalently formulated
in terms of vector fields $m:\Omega\to{\mathbb{R}}^{2}$ as

$|m|=1\text{ a.e.},\;\;\;\;\;\operatorname{div}m=0\text{ in
}\mathcal{D}^{\prime}(\Omega)$ (1)

by identifying $m=\nabla^{\perp}u$. This formulation of the Eikonal equation
in $2$D is enlightening in that one can view (1) as a scalar conservation law
in $1$D. This was first observed by the authors of [DMKO01], who introduced
the concept of entropies as a central tool for the analysis of the Aviles-Giga
functional (the implicit use of the concept of entropies in this setting
appeared in [JK00, ADLM99]). In [DMKO01], entropies for the Eikonal equation
(1) are defined as vector fields $\Phi\in
C^{\infty}_{c}({\mathbb{R}}^{2};{\mathbb{R}}^{2})$ such that
$\operatorname{div}\Phi(m)\equiv 0$ if $m$ is a smooth solution to (1). Such
entropies can be characterized explicitly; see (81). This is completely
analogous to entropies for hyperbolic conservation laws. For solutions
$m=\nabla^{\perp}u=\lim_{\varepsilon\rightarrow 0}\nabla
u_{\varepsilon}^{\perp}$ with
$\sup_{\varepsilon}AG_{\varepsilon}(u_{\varepsilon})<\infty$, it can be shown
that $\operatorname{div}\Phi(m)$ are finite Radon measures, called _entropy
measures_ , and (if $m$ has the additional property that $m\in BV$) they
detect the jumps in $m$. As such, the function space for the
$\Gamma$-convergence of the Aviles-Giga functional is a subset of the space
$\mathcal{A}(\Omega)$ consisting of _entropy solutions_ of the Eikonal
equation, i.e. weak solutions $m$ to (1) such that
$\operatorname{div}\Phi(m)\in\mathcal{M}(\Omega)$ for all entropies $\Phi$,
where $\mathcal{M}(\Omega)$ is the set of finite Radon measures on $\Omega$.
It is thus natural to understand the space $\mathcal{A}(\Omega)$ for the
purpose of the full proof of the $\Gamma$-convergence of the Aviles-Giga
functional.

The lack of understanding of fine structures of the space
$\mathcal{A}(\Omega)$ constitutes one of the major obstacles in the study of
the Aviles-Giga functional. Analogous issues arise in the context of a closely
related micromagnetics energy [RS01, ARS02, AKLR02, DKMO02, RS03] and in the
study of large deviation principles for some stochastic processes, where the
limiting equations are scalar conservation laws [BBMN10]. Roughly speaking it
is expected that vector fields $m\in\mathcal{A}(\Omega)$ exhibit properties
similar to those enjoyed by $BV$ functions and that entropy measures
$\operatorname{div}\Phi(m)$ are concentrated on a one-dimensional rectifiable
set on which $m$ has left and right traces. The most progress to date in this
direction is due to De Lellis and Otto [DLO03], who showed that the points of
positive one-dimensional density of entropy measures do form an
$\mathcal{H}^{1}$-rectifiable set $J$. However their result leaves open
concentration of entropy measures on this set $J$. Indeed, a major conjecture
raised in [DLO03] is the following

######  Conjecture 1 (De Lellis-Otto).

For any $m\in\mathcal{A}(\Omega)$, $\operatorname{div}\Phi(m)$ is supported on
an $\mathcal{H}^{1}$ $\sigma$-finite rectifiable set $J$ for all entropies
$\Phi$ and

$\operatorname{div}\Phi(m)=\left[\eta\cdot\left(\Phi\left(m^{+}\right)-(\Phi\left(m^{-}\right)\right)\right]\mathcal{H}^{1}_{\lfloor
J},$

where $\eta$ is the unit vector normal to $J$ and $m^{\pm}$ are the traces of
$m$ on the two sides of $J$.

It is expected that such concentration of entropy measures on the
$\mathcal{H}^{1}$-rectifiable set $J$, if resolved, will be a crucial step
towards the full proof of the Aviles-Giga conjecture. Note that very recently
Marconi has resolved the analogous versions of Conjecture 1 for Burgers
equation [Mar20b] and the micromagnetics functional [Mar20a] using a powerful
Lagrangian representation method.

###  1.2 On the threshold regularity for the Eikonal equation

As described in the introduction of [CS19], a recurring theme in modern non-
linear PDE is to understand the threshold between rigidity and flexibility
that often occurs at the limiting regularity required for weak solutions of
PDE to satisfy additional derived equations by virtue of some form of the
chain rule. In [CS19] the list of examples presented includes entropy
solutions of hyperbolic conservation laws and incompressible Euler equations.
The authors further note that it is an interesting future direction to
understand the rigidity/flexibility threshold on Besov or Sobolev scale for
non-linear PDE.

The notion of entropies for the Aviles-Giga functional and the Eikonal
equation is closely connected to (and indeed was inspired by) entropies for
hyperbolic conservation laws. For $|m|=1$, it is sufficient to define
entropies on $\mathbb{S}^{1}$. It is easy to check that the restrictions to
$\mathbb{S}^{1}$ of entropies defined in [DMKO01] satisfy

$e^{it}\cdot\frac{d}{dt}\Phi(e^{it})=0.$ (2)

Thus, following [GL20] we define the set of entropies to be

$ENT:=\left\\{\Phi\in
C^{2}\left(\mathbb{S}^{1};\mathbb{R}^{2}\right):\Phi\text{ satisfies
}(\ref{cpa1.7})\right\\}.$ (3)

The derived equations for the Eikonal equation are therefore the equations
$\operatorname{div}\Phi(m)\equiv 0$ for all $\Phi\in ENT$. In [DLI15]
(building on previous work [Ign12]) the authors showed that for $m$ satisfying
the Eikonal equation (1), the Sobolev regularity $m\in W^{\frac{1}{3},3}_{{\rm
loc}}$ is sufficient to allow something like the chain rule (or more
accurately a substitute for the chain rule) to hold and hence conclude that
all the derived equations hold. By the fundamental result of [JOP02], this is
strong enough information to conclude that $m$ is rigid in the sense that it
has only vortex singularities; see Conjecture 4 below for the precise
statement. On the other hand, in [GL20, Theorem 2.6] it is shown that for weak
solutions $m$ to the Eikonal equation (1), the Besov regularity $m\in
B^{\frac{1}{3}}_{3,\infty,{\rm loc}}(\Omega)$ is equivalent to
$\operatorname{div}\Phi(m)$ being locally finite measures for all $\Phi\in
ENT$, and thus there is no rigidity for $m$ under such regularity. These
results leave the question of the critical regularity threshold between
rigidity/flexibility for the Eikonal equation, and our first main result
provides an initial step in this direction.

Our first theorem is an $L^{p}$ version of the main theorem in [GL20]. To
state this result, we first make some definitions. Recall that we denote by
$\mathcal{M}(\Omega)$ the set of finite Radon measures on $\Omega$. For
$\mu\in\mathcal{M}(\Omega)$, let $\|\mu\|$ denote its total variation measure.
Next we recall [GL20, Definition 2.5] with some extensions for our setting:

######  Definition 2.

We say that a vector field $m$ solving (1) satisfies the _kinetic formulation_
if there exists a Radon measure $\sigma\in\mathcal{M}_{{\rm
loc}}(\Omega\times{\mathbb{R}}/2\pi\mathbb{Z})$ such that

$e^{is}\cdot\nabla_{x}\mathds{1}_{e^{is}\cdot
m(x)>0}=\partial_{s}\sigma\quad\text{ in
}\mathcal{D}^{\prime}\left(\Omega\times\mathbb{R}/2\pi\mathbb{Z}\right).$ (4)

We call the measure $\sigma$ the _kinetic measure_.

Further, we say that $m$ satisfies the _$L^{p}$ kinetic equation_ for some
$1\leq p<\infty$ if it satisfies (4) and there exists a family of measures
$\left\\{\sigma_{x}\right\\}\subset\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z})$ for
$\mathcal{L}^{2}$-a.e. $x\in\Omega$ such that

$\langle\sigma(x,s),f(s)\zeta(x)\rangle=\int_{\Omega}\left(\int_{\mathbb{R}/2\pi\mathbb{Z}}f(s)d\sigma_{x}(s)\right)\zeta(x)\,dx\qquad\forall
f\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right),\>\zeta\in C^{0}_{c}(\Omega)$
(5)

and

$\nu(x):=\|\sigma_{x}\|_{\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z})}\in
L^{p}_{{\rm loc}}(\Omega).$ (6)

When (5) and (6) hold, following the notation of [AFP00, Definition 2.27], we
write $\sigma=\mathcal{L}^{2}\otimes\sigma_{x}$. Our first main result is the
following

######  Theorem 3.

Let $\Omega\subset{\mathbb{R}}^{2}$ be an open set and
$m:\Omega\rightarrow{\mathbb{R}}^{2}$ satisfy (1). Then for any
$1<p\leq\frac{4}{3}$, the following are equivalent:

  1. (A)

$m\in B^{\frac{1}{3}}_{3p,\infty,{\rm loc}}(\Omega)$;

  2. (B)

$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$ for all $\Phi\in
ENT$;

  3. (C)

$m$ satisfies the $L^{p}$ kinetic equation in the sense of Definition 2.

Note that for $p>\frac{4}{3}$, we are unable to establish the implication
$(C)\Longrightarrow(A)$ in Theorem 3. However, for $p\geq\frac{4}{3}$, knowing
only two special entropy productions of $m$ (those of the Jin-Kohn entropies
given in (7) and (8) in the following subsection) are in $L^{p}_{{\rm loc}}$
is sufficient to establish the $B^{\frac{1}{3}}_{4,\infty,{\rm loc}}$
regularity for $m$ (see Proposition 33). Our interest in Theorem 3 is two-
fold. Firstly, concerning Conjecture 1, for $m\in\mathcal{A}(\Omega)$, it is
not even known if entropy measures are singular with respect to the Lebesgue
measure and a proof of this fact would represent the first progress on this
long standing problem. The following conjecture represents an even more
accessible goal in this program:

######  Conjecture 4.

Let $m:\Omega\rightarrow{\mathbb{R}}^{2}$ satisfy (1). Assume
$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$ for some $p>1$ and
all $\Phi\in ENT$, then $\operatorname{div}\Phi(m)\equiv 0$ for all $\Phi\in
ENT$, and hence $m$ is rigid, i.e. $m$ is locally Lipschitz outside a locally
finite set of points, and in any neighborhood containing only one singularity
$m$ forms a vortex.

Our Theorem 3 gives equivalent formulations for $\operatorname{div}\Phi(m)\in
L^{p}_{{\rm loc}}(\Omega)$ for all $\Phi\in ENT$, in particular in terms of
the Besov regularity $m\in B^{\frac{1}{3}}_{3p,\infty,{\rm loc}}(\Omega)$.
Under this Besov regularity, many powerful tools become available. In
particular, in Theorem 5, we establish explicit formulas for entropy
productions and the kinetic measure $\sigma$ in terms of two special
entropies. It is likely that one can extract very detailed information out of
the explicit formulas, which will lead to rigidity of $m$ as stated in
Conjecture 4.

Secondly, it is known that the Eikonal equation enjoys no rigidity for $m\in
B^{\frac{1}{3}}_{3,\infty,{\rm loc}}$. Thus a consequence of (a) Theorem 3 and
(b) a proof of Conjecture 4 would provide the threshold of
rigidity/flexibility for the Eikonal equation at the Besov scale.
Specifically, let $\mathcal{S}_{m}$ denote the points of approximate
discontinuity of $m$ (see [AFP00, Definition 3.63]), then

$(a)\,\&\,(b)\Longrightarrow m\in\begin{cases}B^{\frac{1}{3}}_{3,\infty,{\rm
loc}}(\Omega)&\mathcal{S}_{m}\text{ can be
}\mathcal{H}^{1}\text{-rectifiable};\\\ B^{\frac{1}{3}}_{q,\infty,{\rm
loc}}(\Omega),q>3&\mathcal{S}_{m}\text{ can only be isolated points and }m\\\
&\text{forms a vortex around each point in }\mathcal{S}_{m}.\end{cases}$

###  1.3 Control of all entropies by two special ones

Our next main results explore the connections between general entropies and
two special entropies. Building from earlier work of Aviles and Giga [AG87],
Jin and Kohn [JK00] introduced two fundamental entropies
$\Sigma_{1},\Sigma_{2}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ given by

$\Sigma_{1}(m)=\left(m_{2}\left(1-m_{1}^{2}-\frac{m_{2}^{2}}{3}\right),m_{1}\left(1-m_{2}^{2}-\frac{m_{1}^{2}}{3}\right)\right)$
(7)

and

$\Sigma_{2}(m)=\left(-m_{1}\left(1-\frac{2m_{1}^{2}}{3}\right),m_{2}\left(1-\frac{2m_{2}^{2}}{3}\right)\right).$
(8)

These two entropies restricted to $\mathbb{S}^{1}$ satisfy (2), and thus
belong to the set $ENT$ defined in (3). They play fundamental roles in the
$\Gamma$-limit conjecture for the Aviles-Giga functional. Indeed, the
conjecture in [ADLM99] is that the $\Gamma$-limit of the Aviles-Giga
functional is (up to a constant) the total mass of the entropy measure

$\mu=\left\|\begin{array}[]{c}\operatorname{div}\Sigma_{1}(\nabla^{\perp}u)\\\
\operatorname{div}\Sigma_{2}(\nabla^{\perp}u)\end{array}\right\|,$

which is indeed controlled by the energy, and coincides with the cubic jump
cost when $\nabla u\in BV$. A necessary condition for this $\Gamma$-limit
conjecture to hold is that the two special entropy measures
$\operatorname{div}\Sigma_{j}(m)$ control all entropy measures, and our
remaining results are in this spirit.

To state our next result, for $m:\Omega\to{\mathbb{R}}^{2}$, $x\in\Omega$ and
$0<\varepsilon<\mathrm{dist}\\{x,\partial\Omega\\}$, we define the function

$\mathcal{P}_{m}^{\varepsilon}(x):=\varepsilon^{-3}\int_{B_{\varepsilon}(0)}\left|D^{z}m(x)\right|^{3}dz,$
(9)

where we denote $D^{z}m(x)=m(x+z)-m(x)$. Let us take a moment to explain the
function $\mathcal{P}_{m}^{\varepsilon}$. This function arises naturally in
the control of entropy productions of $m$ satisfying (1). Indeed, letting
$m_{\varepsilon}$ be a regularization of $m$, direct computations show that
the main contribution of $\operatorname{div}\Phi(m_{\varepsilon})$ comes from
terms of the form $|\nabla
m_{\varepsilon}|\left(1-|m_{\varepsilon}|^{2}\right)$ for any $\Phi\in ENT$
(see (3)). A commutator argument further gives $|\nabla
m_{\varepsilon}|\left(1-|m_{\varepsilon}|^{2}\right)\lesssim\mathcal{P}_{m}^{\varepsilon}$.
On the other hand, if $m\in B^{\frac{1}{3}}_{3p,\infty}$ for some $p>1$, then
from Lemma 9 we have
${\|\mathcal{P}_{m}^{\varepsilon}\|}_{L^{p}}\lesssim|m|_{B^{\frac{1}{3}}_{3p,\infty}}^{3}$.
Thus, putting everything together, if $m$ is sufficiently regular (which is
the case if $m\in B^{\frac{1}{3}}_{3p,\infty}$ for some $p>1$), one can pass
to the limit as $\varepsilon\to 0$ to deduce

${\|\operatorname{div}\Phi(m)\|}_{L^{p}}\lesssim{\|\mathcal{P}_{m}\|}_{L^{p}}\lesssim|m|_{B^{\frac{1}{3}}_{3p,\infty}}^{3}$
(10)

for some $\mathcal{P}_{m}$ which is a weak limit of
$\mathcal{P}_{m}^{\varepsilon}$. As such, the function
$\mathcal{P}_{m}^{\varepsilon}$ plays a crucial role in controlling entropy
productions of $m$ by its Besov norm, and we will make extensive use of it.

In [GL20], Ghiraldin and Lamy constructed a special class of entropies
$\\{\Phi_{f}\\}\subset ENT$ parameterized by $f\in
C^{0}({\mathbb{R}}/2\pi\mathbb{Z})$ (see (23)–(25) in Section 4). They proved
the existence of the kinetic measure $\sigma$ (recalling (4)) for any $m$
satisfying (1) and $\operatorname{div}\Phi_{f}(m)\in\mathcal{M}_{{\rm
loc}}(\Omega)$ for all $f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z})$, i.e. the
family of entropies $\left\\{\Phi_{f}\right\\}$ is rich enough to generate the
kinetic equation (4). Our next result gives explicit formulas for
$\operatorname{div}\Phi_{f}(m)$ and the kinetic measure $\sigma$ in terms of
$\operatorname{div}\Sigma_{j}(m)$, $j=1,2$, under suitable assumptions.

######  Theorem 5.

Let $\Omega\subset{\mathbb{R}}^{2}$ be an open set and
$m:\Omega\rightarrow{\mathbb{R}}^{2}$ satisfy (1), and let
$\mathcal{P}_{m}^{\varepsilon}$ be defined by (9). Assume for some sequence
$\varepsilon_{k}\rightarrow 0$ and $1\leq p<\infty$, there exists
$\mathcal{P}_{m}\in L^{p}_{{\rm loc}}(\Omega)$ such that

$\mathcal{P}_{m}^{\varepsilon_{k}}\rightharpoonup\mathcal{P}_{m}\quad\text{ in
}L^{p}_{{\rm loc}}(\Omega).$ (11)

Then $\operatorname{div}\Phi_{f}(m)\in L^{p}_{{\rm loc}}(\Omega)$ for all
$f\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$, where $\Phi_{f}\in ENT$ is
defined in (23). Further, it explicitly holds

$\displaystyle\operatorname{div}\Phi_{f}(m)=\frac{1}{2}\left(f\left(\theta+\frac{\pi}{2}\right)+f\left(\theta-\frac{\pi}{2}\right)-2\langle
f,1\rangle\right)e^{i2\theta}\cdot\operatorname{div}\Sigma(m)\quad\text{ a.e.
in }\Omega,$ (12)

where $\theta:\Omega\to[0,2\pi)$ satisfies $m(x)=e^{i\theta(x)}$ for a.e.
$x\in\Omega$, $\Sigma(m)=\left(\Sigma_{1}(m),\Sigma_{2}(m)\right)$ and
$\langle f,g\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}fg$. Equivalently, (12) can
be formulated as the disintegration of the kinetic measure
$\sigma=\mathcal{L}^{2}\otimes\sigma_{x}$, where

$\sigma_{x}=\frac{1}{2}\left(\delta_{\theta(x)+\frac{\pi}{2}}+\delta_{\theta(x)-\frac{\pi}{2}}-\frac{1}{\pi}\mathcal{L}^{1}\right)e^{i2\theta(x)}\cdot\operatorname{div}\Sigma(m)(x)\quad\text{
for a.e. }x\in\Omega.$ (13)

######  Remark 6.

By Lemmas 9, 10 and Theorem 3, the assumption (11) is equivalent to
$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$ for all $\Phi\in ENT$
and equivalent to $m\in B^{\frac{1}{3}}_{3p,\infty,{\rm loc}}(\Omega)$ for
$1<p\leq\frac{4}{3}$. When $p>\frac{4}{3}$, Lemmas 9 and 10 imply that

$m\in B^{\frac{1}{3}}_{3p,\infty,{\rm
loc}}(\Omega)\Longrightarrow\eqref{cpeq1.23}\Longrightarrow\operatorname{div}\Phi(m)\in
L^{p}_{{\rm loc}}(\Omega)\text{ for all }\Phi\in ENT,$

however we are unable to establish equivalence. Nevertheless, the conclusions
of Theorem 5 still hold under the assumption $\operatorname{div}\Phi(m)\in
L^{p}_{{\rm loc}}(\Omega)$ for all $\Phi\in ENT$ when $p>\frac{4}{3}$; see
Theorem 7 below. When $p=1$, we only have the implication
$\eqref{cpeq1.23}\Longrightarrow\operatorname{div}\Phi(m)\in L^{1}_{{\rm
loc}}(\Omega)$ for all $\Phi\in ENT$ by Lemma 10. Again we are unable to
establish equivalence in this case, nor do we have a proof of Theorem 5 with
(11) replaced by the assumption $\operatorname{div}\Phi(m)\in L^{1}_{{\rm
loc}}(\Omega)$ for all $\Phi\in ENT$.

Finally note that after writing up our factorization result, we became aware
of an upcoming similar but stronger result being established by Elio Marconi
[Mar21] by use of the theory of “Lagrangian representation” [BM17, BBM17,
Mar19, Mar20b, Mar20a].

As a consequence of the explicit formula (12), the entropy production
$\operatorname{div}\Phi_{f}(m)$ is controlled by $\operatorname{div}\Sigma(m)$
in a very precise manner for the class of entropies $\\{\Phi_{f}\\}$ under the
assumption (11). This formula could potentially have wider applications in the
study of the more “regular” part of entropy measures for general
$m\in\mathcal{A}(\Omega)$, i.e. the part concentrated outside the
$\mathcal{H}^{1}$-rectifiable set. Note that by similar methods to the proof
of Theorem 3, it is possible to show that
$\left|e^{i2\theta}\cdot\operatorname{div}\Sigma\left(m\right)\right|\geq\pi^{-5}\mathcal{P}_{m}$
a.e. in $\Omega$. As a somewhat direct application of the above Theorems 3 and
5, we have

######  Theorem 7.

Let $\Omega\subset{\mathbb{R}}^{2}$ be an open set and
$m:\Omega\rightarrow{\mathbb{R}}^{2}$ satisfy (1). Assume either

$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)\quad\text{ for some
}p>1\text{ and all }\Phi\in ENT$ (14)

or

$\operatorname{div}\Sigma_{j}(m)\in L^{p}_{{\rm loc}}(\Omega)\quad\text{ for
some }p\geq\frac{4}{3}\text{ and }j=1,2.$ (15)

Then $\operatorname{div}\Phi_{f}(m)\in L^{p}_{{\rm loc}}(\Omega)$ for all
$f\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$, and further (12) and (13)
hold true.

The statements of Theorems 5 and 7 are for the class of entropies
$\\{\Phi_{f}\\}$ essentially because this is a wide enough class to generate
the kinetic measure and the factorization has a neater form. In Proposition 29
we also establish the corresponding factorization formula for sufficiently
regular entropies $\Phi\in ENT$. Note that the conclusions of Theorem 7 under
the assumption (15) is a result to control _all_ entropies by the two special
entropies $\Sigma_{j}$ in the strong sense that we only need
$\operatorname{div}\Sigma_{j}(m)\in L^{p}_{{\rm loc}}(\Omega)$ to conclude
that $\operatorname{div}\Phi_{f}(m)\in L^{p}_{{\rm loc}}(\Omega)$ for all
$\Phi_{f}$ with the additional explicit formula (12). This is a strong
extension of the main result in [LP18], which can be formulated as an
immediate corollary:

######  Corollary 8 ([LP18]).

Let $\Omega\subset{\mathbb{R}}^{2}$ be an open set and
$m:\Omega\rightarrow{\mathbb{R}}^{2}$ satisfy (1). Assume
$\operatorname{div}\Sigma_{j}(m)=0$ in $\mathcal{D}^{\prime}(\Omega)$ for
$j=1,2$, then $\operatorname{div}\Phi_{f}(m)=0$ in
$\mathcal{D}^{\prime}(\Omega)$ for all $f\in
C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ and the kinetic measure $\sigma$
vanishes.

When $\sigma$ vanishes, the right-hand side of the kinetic equation (4) also
vanishes. It then follows from the main result of [JOP02] that $m$ is rigid in
the sense of Conjecture 4, and this recovers [LP18, Theorem 3]. Corollary 8
and Theorem 7 (in particular the formula (12)) give strong evidence towards
the $\Gamma$-limit for the Aviles-Giga functional conjectured in [ADLM99].

Acknowledgments. We would like to extend our deep thanks to Xavier Lamy for
innumerable very helpful conversations and specifically for the proof of
Proposition 13 and Lemmas 14 and 36, and for most of the ideas of the proof of
Proposition 33 and Lemmas 17 and 31. The proof of Proposition 13 helps
strengthen an earlier version of our Theorem 3. Also note that Proposition 30
(including Lemmas 34 and 35) was originally in the first posted version of
[LLP20] and as such is joint work of the three of us. A. L. gratefully
acknowledges the support of the Simons foundation, collaboration grant
#426900.

##  2 Proof sketch

###  2.1 Extension of entropies to $\overline{B}_{1}$ and commutator argument

A fundamental idea that we will use throughout is to try and understand
$\operatorname{div}\Phi(m)$ by understanding $\lim_{\varepsilon\rightarrow
0}\operatorname{div}\Phi\left(m_{\varepsilon}\right)$ for a regularization
$m_{\varepsilon}$ of $m$. By definition, entropies are designed to provide
additional derived equations via the chain rule and as such applying them to
$m_{\varepsilon}$ is a natural idea. However for $\Phi\in ENT$, it is
necessary to extend the definition to $\overline{B}_{1}$ in order to act them
on $m_{\varepsilon}$. One natural way to do this is to extend $\Phi$ through a
_radial extension_ (see (18)). Another way is through the original definition
of entropies in [DMKO01]. Here entropies are defined in ${\mathbb{R}}^{2}$ and
obtained from smooth functions $\varphi$ via the formula

$\Phi^{{{\color[rgb]{0,0,0}\varphi}}}(z)=\varphi(z)z+((iz)\cdot\nabla\varphi(z))iz\qquad\forall
z\in{\mathbb{R}}^{2}.$

The underlying principle is, through different extensions, we extract
different information from the limit of
$\operatorname{div}\Phi\left(m_{\varepsilon}\right)$, and when pieced
together, the information has strong consequences for the structure of
$\operatorname{div}\Phi\left(m\right)$.

In order to pass to the limit in
$\operatorname{div}\Phi\left(m_{\varepsilon}\right)$, an important trick is
the commutator argument of [CET94] that was first introduced to our setting in
[DLI15], and used later in [LP18, GL20, LLP20] with extensions. As explained
above (10), this commutator argument allows to control
$\operatorname{div}\Phi\left(m_{\varepsilon}\right)$ and pass to the limit as
$\varepsilon\to 0$ to establish an estimate like (10) under sufficient
regularity of $m$. This is a crucial idea in many of our arguments.

###  2.2 Ideas in the proof of Theorem 3

The proof of Theorem 3 goes in the loop $(A)\Longrightarrow(B)$,
$(B)\Longrightarrow(C)$ and $(C)\Longrightarrow(A)$ and uses many of the ideas
in [GL20]. The first two steps use rather straightforward ideas, and the step
$(C)\Longrightarrow(A)$ is the most involved requiring significant refinements
of the estimates used in [GL20]. For this last step, the starting point is to
create a function $\Delta_{\alpha}$ (as opposed to the $\Delta$ used in
[GL20]) that provides the right coercivity estimate towards the goal of
$B^{\frac{1}{3}}_{3p,\infty}$ regularity; see Lemma 17. On the other hand, as
observed in [GL20], (a regularized) $\Delta_{\alpha}$ enjoys a nice identity
due to the kinetic equation (4) (see (44)). Then using more delicate arguments
based on many ideas from [GL20] (what the authors called the interaction
estimate due to Varadhan (see [Tar08, Chapter 22]) as used in [GP13, GJO15]),
we establish in Lemmas 19 and 21 appropriate upper bound estimate for
$\Delta_{\alpha}$, which, coupled with the coercivity established in Lemma 17,
gives the desired $B^{\frac{1}{3}}_{3p,\infty}$ regularity.

###  2.3 Ideas in the proof of Theorem 5

Our starting point is to use computations from [LLP20] to establish in Lemma
26 the formula

$\operatorname{div}\Phi(m)=A^{\Phi}_{1}(m)e^{i2\theta}\cdot\operatorname{div}\Sigma(m)+A^{\Phi}_{2}(m)ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)\quad\text{
a.e. in }\Omega$ (16)

for general entropies $\Phi$. To this end, we use ideas presented in
Subsection 2.1, namely, using regularization of $m$ and appropriate extension
of $\Phi$ to $\overline{B}_{1}$ to compute
$\operatorname{div}\Phi(m_{\varepsilon})$, and passing to the limit through a
commutator argument. Such ideas are not hard to implement in the case $p>1$.
In the case $p=1$, it is a bit more involved and the technical aspect is
handled in Lemma 25.

The rest of the game involves showing
$ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)=0$ using a similar
contradiction argument as in [LLP20]. Specifically, assuming
$ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)\neq 0$ at a generic point $x$,
then estimates established for the coefficients $A^{\Phi}_{j}$ in Lemma 27
plugged into (16) would force $A^{\Phi}_{2}$ to satisfy a bound of the form
${\left|A_{2}^{\Phi}(m(x))\right|}\lesssim{\|\Phi_{\lfloor\mathbb{S}^{1}}\|}_{C^{2}(\mathbb{S}^{1})}$
for all entropies $\Phi$. This estimate cannot hold due to the fact that the
Hilbert transform (or conjugate function operator) on $\mathbb{S}^{1}$ is not
bounded from $C^{0}(\mathbb{S}^{1})$ to $L^{\infty}(\mathbb{S}^{1})$. This
contradiction shows $ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)=0$, and the
formula (16) reduces to more general version of (12) (see Proposition 29 for
the details). Finally, (12) is obtained by recognizing the class of entropies
$\\{\Phi_{f}\\}$ as harmonic entropies (first introduced in [LP18]) $\Phi\in
ENT$, and this is the content of Subsection 5.2.

##  3 Preliminaries

In this section we put together some notations and results that will be used
repeatedly in later sections. Throughout this paper, we use the notation
$A\lesssim B$ to indicate $A\leq cB$ for some constant $c$ independent of the
underlying domain or functions. Let $\rho\in C^{\infty}_{c}({\mathbb{R}}^{2})$
be the standard convolution kernel with $\int_{{\mathbb{R}}^{2}}\rho=1$, and
let
$\rho_{\varepsilon}(z)=\varepsilon^{-2}\rho\left(\frac{z}{\varepsilon}\right)$.
Given a function $f$ we denote $f_{\varepsilon}:=f*\rho_{\varepsilon}$.

Given a bounded Lipschitz domain $\Omega\subset{\mathbb{R}}^{n}$ and
$f:\Omega\rightarrow{\mathbb{R}}$, $z\in{\mathbb{R}}^{n}$, we define

$D^{z}f(x):=\begin{cases}f(x+z)-f(x)&\text{ if }x,x+z\in\Omega;\\\ 0&\text{
otherwise}.\end{cases}$

For any $s\in(0,1)$, $p,q\in[1,\infty]$, we set

${\left|f\right|}_{B^{s}_{p,q}(\Omega)}=\left\|t^{-s}\sup_{{\left|h\right|}\leq
t}{\|D^{h}f\|}_{L^{p}(\Omega)}\right\|_{L^{q}(\frac{dt}{t})},$

and the Besov space $B^{s}_{p,q}(\Omega)$ is the space of functions
$f\colon\Omega\to{\mathbb{R}}$ such that

${\|f\|}_{L^{p}(\Omega)}+{\left|f\right|}_{B^{s}_{p,q}(\Omega)}<\infty.$

######  Lemma 9.

Given $m\in B^{\frac{1}{3}}_{3p,\infty,{\rm loc}}(\Omega)$ for some
$1<p<\infty$, let $\mathcal{P}_{m}^{\varepsilon}$ be given in (9). Then for
any $U\subset\subset\Omega$ and $\varepsilon>0$ sufficiently small, we have

${\|\mathcal{P}_{m}^{\varepsilon}\|}_{L^{p}(U)}\leq|m|_{B^{\frac{1}{3}}_{3p,\infty}(U)}^{3}.$

In particular $\\{\mathcal{P}_{m}^{\varepsilon}\\}$ forms a bounded sequence
in $L^{p}_{{\rm loc}}(\Omega)$.

Lemma 9 is a trivial generalization of [LLP20, Lemma 8]. Its proof follows
exactly the same lines by replacing $4/3$ with $p$ in the proof of the latter,
and is thus skipped. The following lemma is a slight extension of [LLP20,
Proposition 10].

######  Lemma 10.

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1). Assume (11) holds true for some
sequence $\varepsilon_{k}\rightarrow 0$ and $1\leq p<\infty$. Then for any
$\Phi\in ENT$ (given in (3)), we have $\operatorname{div}\Phi(m)\in
L^{p}_{{\rm loc}}(\Omega)$ and

$\left|\operatorname{div}\Phi(m)(x)\right|\lesssim\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\mathcal{P}_{m}(x)\qquad\text{
for a.e. }x\in\Omega.$ (17)

###### Proof.

Since the proof is very similar to the proof of [LLP20, Proposition 10], we
only sketch it somewhat briefly. Given $\Phi\in ENT$, we introduce the
extension $\widetilde{\Phi}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ of $\Phi$
given by

$\widetilde{\Phi}(z):=\eta\left(\left|z\right|\right)\Phi\left(\frac{z}{\left|z\right|}\right),$
(18)

where $\eta:[0,\infty)\to{\mathbb{R}}$ is a smooth cut-off function with
$\eta=0$ in $[0,\frac{1}{2}]\cup[2,\infty)$ and $\eta(1)=1$. For a
regularization $m_{\varepsilon}$ of $m$, as in the proof of [LLP20,
Proposition 10] (originally from Steps 2 and 3 of the proof of [DLI15,
Proposition 3]) we have

$\displaystyle\operatorname{div}\widetilde{\Phi}\left(m_{\varepsilon}\right)$
$\displaystyle=\Psi\left(m_{\varepsilon}\right)\cdot\nabla\left(1-\left|m_{\varepsilon}\right|^{2}\right)$
$\displaystyle=\nabla\cdot\left(\Psi(m_{\varepsilon})(1-|m_{\varepsilon}|^{2})\right)-\nabla\cdot\Psi(m_{\varepsilon})\left(1-|m_{\varepsilon}|^{2}\right),$
(19)

where $\Psi(z)=\frac{-D\widetilde{\Phi}(z)z+\gamma(z)z}{2\left|z\right|^{2}}$
and $\gamma(z)=\frac{z^{\perp}\cdot D\widetilde{\Phi}(z)z^{\perp}}{|z|^{2}}$.

Given any open set $U\subset\subset\Omega$, for any $\zeta\in
C_{c}^{\infty}(U)$, an integration by parts using (3) and the estimate
$\|D\Psi\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}$
(see equation (35) in [LLP20]) yield (with $\varepsilon$ replaced by
$\varepsilon_{k}$)

$\displaystyle\left|\int_{U}\widetilde{\Phi}(m_{\varepsilon_{k}})\cdot\nabla\zeta
dx\right|$
$\displaystyle\lesssim\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\int_{U}\left|\nabla
m_{\varepsilon_{k}}\right|\left|1-\left|m_{\varepsilon_{k}}\right|^{2}\right|\left|\zeta\right|dx$
$\displaystyle\quad\,+\left|\int_{U}\left(1-\left|m_{\varepsilon_{k}}\right|^{2}\right)\Psi\left(m_{\varepsilon_{k}}\right)\cdot\nabla\zeta\,dx\right|.$
(20)

It is not hard to see that the left-hand side of (3) converges to
$\left|\int_{U}\Phi(m)\cdot\nabla\zeta dx\right|$, and, upon extraction of a
subsequence (not relabeled), the second term on the right-hand side tends to
zero as $k\to\infty$ by the Dominated Convergence Theorem. And by [LLP20,
Lemma 9] we have that $\left|\nabla
m_{\varepsilon}\right|\left|1-\left|m_{\varepsilon}\right|^{2}\right|\lesssim\mathcal{P}_{m}^{\varepsilon}$.
Passing to the limit as $k\to\infty$ in (3) and using (11), we obtain

$\left|\int_{U}\Phi\left(m\right)\cdot\nabla\zeta\,dx\right|\lesssim\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\int_{U}\mathcal{P}_{m}\left|\zeta\right|dx\leq\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\|\mathcal{P}_{m}\|_{L^{p}(U)}\|\zeta\|_{L^{p^{\prime}}(U)}$
(21)

for all $\zeta\in C^{\infty}_{c}(U)$.

When $1<p<\infty$, it follows from (21) and standard arguments that
$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$ and the pointwise
estimate (17) holds for all Lebesgue points of $\operatorname{div}\Phi(m)$ and
$\mathcal{P}_{m}$; see the end of the proof of [LLP20, Proposition 6] for the
details. When $p=1$, one can extend the estimate (21) to all $\zeta\in
C^{0}_{c}(U)$. It follows that $\operatorname{div}\Phi(m)\in\mathcal{M}_{{\rm
loc}}\left(\Omega\right)$ and, using [AFP00, Proposition 1.47], we infer

$\|\operatorname{div}\Phi(m)\|_{\mathcal{M}(U)}=\sup_{\zeta\in
C^{0}_{c}(U),\|\zeta\|_{C^{0}}\leq
1}\left|\langle\operatorname{div}\Phi(m),\zeta\rangle\right|\lesssim\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\|\mathcal{P}_{m}\|_{L^{1}(U)}$
(22)

for all open $U\subset\subset\Omega$. It is then standard to deduce from (22)
that $\|\operatorname{div}\Phi(m)\|$ is absolutely continuous with respect to
the Lebesgue measure with $L^{1}_{{\rm loc}}$ density, and the same holds for
$\operatorname{div}\Phi(m)$. Finally, for all Lebesgue points $x$ of
$|\operatorname{div}\Phi(m)|$ and $\mathcal{P}_{m}$, we deduce from (22) that

$\displaystyle|\operatorname{div}\Phi(m)(x)|$
$\displaystyle=\lim_{r\rightarrow
0}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B_{r}(x)}|\operatorname{div}\Phi(m)|\,dz$
$\displaystyle\lesssim\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\lim_{r\rightarrow
0}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B_{r}(x)}\mathcal{P}_{m}\,dz=\|\Phi\|_{C^{2}\left(\mathbb{S}^{1}\right)}\mathcal{P}_{m}(x),$

and this establishes (17) for $p=1$. ∎

######  Remark 11.

Note that Lemma 10 in particular gives

$\operatorname{div}\Sigma_{j}(m)\in L^{p}_{{\rm loc}}(\Omega)\quad\text{ for
}j=1,2,$

where $\Sigma_{j}$ are given in (7), (8), since $\Sigma_{j}$ agrees (up to a
factor) with $\Phi_{f}$ on $\mathbb{S}^{1}$ for $f=\cos(2t)$ or $f=\sin(2t)$;
see [GL20, Remark 3.2].

##  4 Proof of Theorem 3

In this section we prove Theorem 3 by establishing the loop of implications
$(A)\Longrightarrow(B)$, $(B)\Longrightarrow(C)$ and $(C)\Longrightarrow(A)$
in the following three subsections. This structure of proof is parallel to
that of the main theorem in [GL20]. The most involved step is the last step
$(C)\Longrightarrow(A)$, which requires significant refinements of the
estimates in the counterpart of the proof of [GL20, Theorem 2.6]. Note that
the proofs of $(A)\Longrightarrow(B)$ and $(B)\Longrightarrow(C)$ work for all
$1<p<\infty$, and the assumption $p\leq\frac{4}{3}$ in Theorem 3 is only
needed in the proof of $(C)\Longrightarrow(A)$.

###  4.1 Proof of $(A)\Longrightarrow(B)$ in Theorem 3

######  Proposition 12.

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1). Assume $m\in
B^{\frac{1}{3}}_{3p,\infty,{\rm loc}}(\Omega)$ for some $1<p<\infty$, then
$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$ for all $\Phi\in
ENT$.

###### Proof.

From Lemma 9 we know $\\{\mathcal{P}^{\varepsilon}_{m}\\}$ forms a bounded
sequence in $L^{p}_{{\rm loc}}(\Omega)$. As $1<p<\infty$, upon extraction of a
subsequence, we have
$\mathcal{P}^{\varepsilon_{k}}_{m}\rightharpoonup\mathcal{P}_{m}$ in
$L^{p}_{{\rm loc}}(\Omega)$ for some $\mathcal{P}_{m}$. This allows us to
apply Lemma 10 to deduce for all $\Phi\in ENT$ that
$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$. ∎

###  4.2 Proof of $(B)\Longrightarrow(C)$ in Theorem 3

######  Proposition 13 (Lamy).

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1). Assume
$\operatorname{div}\Phi(m)\in L^{p}_{{\rm loc}}(\Omega)$ for some $1<p<\infty$
and all $\Phi\in ENT$, then $m$ satisfies the $L^{p}$ kinetic equation in the
sense of Definition 2.

The proof of the above proposition relies crucially on the following Lemma 14,
which is an $L^{p}$ version of [GL20, Lemma 3.4]. Note that the proof of
[GL20, Lemma 3.4] is a bit unclear in that the estimates for tensor products
established there are not sufficient to apply the Riesz representation
theorem. Nevertheless as shown to us by Lamy, the generalized Riesz
representation theorem stated in [DU77] overcomes these obstacles. In the
following we include this complete proof adapted to our $L^{p}$ setting. In
Appendix B, we also include the complete proof of [GL20, Lemma 3.4] since this
result is needed in our Theorem 5 and the complete proof has not appeared
elsewhere. We warmly thank Lamy for these proofs presented.

To begin with, we recall a class of entropies $\\{\Phi_{f}\\}\subset ENT$
constructed in [GL20]. Given $f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z})$,
identifying $\mathbb{S}^{1}\cong\mathbb{R}/2\pi\mathbb{Z}$ and
$\mathbb{R}^{2}\cong\mathbb{C}$, we define

$\Phi_{f}(e^{it}):=-i\varphi_{f}(t-\pi/2)+i\varphi_{f}(t+\pi/2),$ (23)

where

$\varphi_{f}(t):=\int_{0}^{t}\psi_{f}(s)ie^{is}ds$ (24)

and

$\psi_{f}(t):=\int_{0}^{t}\left[f(s)-\langle f,1\rangle-2\langle
f,\cos\rangle\cos(s)-2\langle f,\sin\rangle\sin(s)\right]ds,$ (25)

and $\langle\cdot,\cdot\rangle$ denotes the inner product on
$L^{2}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ defined by $\langle
f,g\rangle:=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)g(t)\,dt$.

######  Lemma 14 (Lamy).

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1) and $1<p<\infty$. Assume
$\operatorname{div}\Phi_{f}(m)\in L^{p}(\Omega)$ for all $f\in
C^{0}(\mathbb{S}^{1})$, where $\Phi_{f}$ is defined in (23). Then there exists
$\sigma\in L^{p}_{w}(\Omega;\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z}))$, where
the subscript $w$ stands for weak measurability, such that

$\langle\operatorname{div}\Phi_{f}(m),\zeta\rangle=\langle\sigma(x,s),f(s)\zeta(x)\rangle\qquad\forall
f\in C^{0}({\mathbb{R}}/2\pi\mathbb{Z}),\>\zeta\in L^{p^{\prime}}(\Omega),$
(26)

where $\langle\cdot,\cdot\rangle$ denotes duality in the appropriate sense.
Consequently, $\sigma$ satisfies (5) and (6).

###### Proof.

Given Banach spaces $X$ and $Y$, let $\mathcal{L}(X;Y)$ denote the space of
bounded linear operators from $X$ to $Y$. Given $\zeta\in C^{1}_{c}(\Omega)$,
define the linear operator
$T_{\zeta}:C^{0}(\mathbb{R}/2\pi\mathbb{Z})\to{\mathbb{R}}$ to be

$T_{\zeta}(f):=\langle\operatorname{div}\Phi_{f}(m),\zeta\rangle.$

The estimate

$|T_{\zeta}(f)|\leq\lVert\Phi_{f}\rVert_{L^{\infty}}\lVert\nabla\zeta\rVert_{L^{\infty}}\leq
C\lVert\nabla\zeta\rVert_{L^{\infty}}\lVert f\rVert_{L^{\infty}}$

shows that
$T_{\zeta}\in\mathcal{L}(C^{0}(\mathbb{R}/2\pi\mathbb{Z});{\mathbb{R}})$ for
all $\zeta\in C^{1}_{c}(\Omega)$. For the convenience of the proof of Lemma 36
in Appendix B, let $\mathcal{Z}=L^{p^{\prime}}(\Omega)$, so that
$\mathcal{Z}^{*}=L^{p}(\Omega)$, and define

$\displaystyle\mathbb{W}_{\mathcal{Z}}:=\left\\{T_{\zeta}:\zeta\in
C^{1}_{c}(\Omega),{\|\zeta\|}_{\mathcal{Z}}\leq 1\right\\}.$

By hypothesis, for each fixed $f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z})$ we have

$\sup_{\zeta\in C_{c}^{1}(\Omega),\|\zeta\|_{\mathcal{Z}}\leq
1}\langle\operatorname{div}\Phi_{f}(m),\zeta\rangle<\infty.$

So by Banach-Steinhaus’ uniform boundedness principle applied to
$\mathbb{W}_{\mathcal{Z}}$ we have

$\displaystyle\langle\operatorname{div}\Phi_{f}(m),\zeta\rangle\leq
C{\|f\|}_{C^{0}(\mathbb{R}/2\pi\mathbb{Z})}{\|\zeta\|}_{\mathcal{Z}}\qquad\forall
f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z}),\>\zeta\in C^{1}_{c}(\Omega).$ (27)

By extension, the above estimate (27) also holds for all
$\zeta\in\mathcal{Z}$, and therefore the map

$\displaystyle T\colon
C^{0}(\mathbb{R}/2\pi\mathbb{Z})\to\mathcal{Z}^{*},\quad
f\mapsto\operatorname{div}\Phi_{f}(m)$ (28)

is a bounded linear operator.

Since $1<p<\infty$, the space $L^{p}(\Omega)$ is reflexive. Thanks to [DU77,
Theorem VI.2.1], there exists an $\mathcal{Z}^{*}$-valued Borel measure $G$ on
$\mathbb{R}/2\pi\mathbb{Z}$ such that

$Tf=\int_{\mathbb{R}/2\pi\mathbb{Z}}f\,dG.$ (29)

We use this representation to define $\langle\sigma,\psi\rangle$ for a
convenient class of functions $\psi(x,s)$. First consider $\psi$ a finite
linear combination of the form

$\displaystyle\psi(x,s)=\sum_{j}{\mathds{1}}_{E_{j}}(s)\zeta_{j}(x),\qquad
E_{j}\subset\mathbb{R}/2\pi\mathbb{Z}\text{ Borelian,
}\zeta_{j}\in\mathcal{Z},$

and set

$\displaystyle\langle\sigma,\psi\rangle:=\sum_{j}\langle
G(E_{j}),\zeta_{j}\rangle.$ (30)

Assuming without loss of generality that the $E_{j}$’s are disjoint and non-
negligible, for such $\psi$ we have

$\displaystyle\langle\sigma,\psi\rangle$
$\displaystyle\leq\sum_{j}{\|G\|}(E_{j}){\|\zeta_{j}\|}_{\mathcal{Z}}$
$\displaystyle\leq{\|G\|}(\mathbb{R}/2\pi\mathbb{Z})\max_{j}{\|\zeta_{j}\|}_{\mathcal{Z}}={\|T\|}_{\mathcal{L}(C^{0}(\mathbb{R}/2\pi\mathbb{Z});\mathcal{Z}^{*})}{\|\psi\|}_{L^{\infty}(\mathbb{R}/2\pi\mathbb{Z};\mathcal{Z})}.$
(31)

As a consequence the linear form $\sigma$ admits a unique linear continuous
extension (still denoted $\sigma$) to the space of all limits of $\psi$’s of
the above form in the $L^{\infty}(\mathbb{R}/2\pi\mathbb{Z};\mathcal{Z})$
norm. Hence $\sigma$ can in particular be considered as a continuous linear
form on $C^{0}(\mathbb{R}/2\pi\mathbb{Z};\mathcal{Z})$. Moreover, since for
$f={\mathds{1}}_{E}$ and $\zeta\in\mathcal{Z}$, the definition (30) gives

$\displaystyle\langle\sigma(x,s),f(s)\zeta(x)\rangle=\langle\int
f\,dG,\zeta\rangle,$

this formula must also be valid for $f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z})$
and we deduce that

$\displaystyle\langle\sigma(x,s),f(s)\zeta(x)\rangle$
$\displaystyle=\langle\int
f\,dG,\zeta\rangle\overset{(\ref{eqokl2})}{=}\langle
Tf,\zeta\rangle\overset{(\ref{fineqa1})}{=}\langle\operatorname{div}\Phi_{f}(m),\zeta\rangle\qquad\forall
f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z}),\>\zeta\in\mathcal{Z},$

and this establishes (26).

Since $C^{0}(\mathbb{R}/2\pi\mathbb{Z};L^{p^{\prime}}(\Omega))\supset
L^{p^{\prime}}(\Omega;C^{0}(\mathbb{R}/2\pi\mathbb{Z}))$, we can identify
$\sigma$ with an element of

$\displaystyle
L^{p^{\prime}}(\Omega;C^{0}(\mathbb{R}/2\pi\mathbb{Z}))^{*}\approx
L^{p}_{w}(\Omega;\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z}))$

(for this duality result see e.g. [Edw65, Theorem 8.20.3] – recall the
subscript $w$ stands for weak measurability). Thus for almost all
$x\in\Omega$,
$\sigma_{x}(\cdot):=\sigma(\cdot,x)\in\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z})$,
and

$\displaystyle\left(\int_{\Omega}{\|\sigma(\cdot,x)\|}_{\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z})}^{p}\,dx\right)^{\frac{1}{p}}\leq{\|\sigma\|}_{*}<\infty.$

This completes the proof of (5) and (6). ∎

###### Proof of Proposition 13.

For any open set $U\subset\subset\Omega$, the fact that $m$ satisfies the
kinetic equation (4) and $\sigma$ from Lemma 14 forms the kinetic measure for
$m$ in $U$ follows in a reasonably straightforward way; see Subsection 3.1 in
[GL20]. The additional structure of $\sigma$ in (5) and (6) is also given in
Lemma 14. ∎

######  Remark 15.

Under the assumptions of Theorem 3, define the “grand measure” $\mu_{m}$ for
$m$ to be the least upper bound of all entropy measures (see [AFP00,
Definition 1.68]), i.e.

$\mu_{m}:=\bigvee_{\Phi\in ENT,\|\Phi\|_{C^{2}}\leq
1}\|\operatorname{div}\Phi(m)\|.$ (32)

Then calling

$(B^{\prime})\,\,\text{The grand measure }\mu_{m}\in L^{p}_{{\rm
loc}}(\Omega),$

a little more work using the proof of Proposition 12 and the estimate (17)
shows that $(A)\Longrightarrow(B^{\prime})$. And since $(B^{\prime})$ is
stronger than $(B)$ we have $(B^{\prime})\Longrightarrow(C)$ from Proposition
13. Therefore combining these with Proposition 16 in the next subsection, we
can replace $(B)$ by $(B^{\prime})$ in Theorem 3, and in particular $(B)$ and
$(B^{\prime})$ are equivalent. This is analogous to part of the main theorem
of [GL20].

###  4.3 Proof of $(C)\Longrightarrow(A)$ in Theorem 3

######  Proposition 16.

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1). Assume $m$ satisfies the
$L^{p}$ kinetic equation in the sense of Definition 2 for some
$1<p\leq\frac{4}{3}$, then $m\in B^{\frac{1}{3}}_{3p,\infty,{\rm
loc}}(\Omega)$ .

We start by introducing the analog of $\Delta$ used in Subsection 3.2 in
[GL20], which provides the right coercivity estimate in our case; see (37)
below. To this end, we first define, for $\alpha>0$, the function
$\varphi_{\alpha}(t):{\mathbb{R}}\to{\mathbb{R}}$ which satisfies

$\displaystyle\bullet\;\varphi_{\alpha}\text{ is odd and
}\pi\text{-periodic};$
$\displaystyle\bullet\;\varphi_{\alpha}(t)=t^{\alpha}\text{ for
}t\in[0,\pi/4];$ $\displaystyle\bullet\;\varphi_{\alpha}>0\text{ in
}(0,\pi/2)\text{ and is smooth in }(0,\pi).$ (33)

In the following we will take $\alpha=3p-3$ so that $0<\alpha\leq 1$ for
$1<p\leq\frac{4}{3}$. In the sequel the functions will be defined in terms of
$m$. Since $m$ is fixed, we will not explicitly show this dependence. We
largely follow the notation of Section 3 in [GL20]. Define $\theta(x)$ to be

$\theta:\Omega\to[0,2\pi),\quad m(x)=e^{i\theta(x)}\text{ for a.e.
}x\in\Omega.$ (34)

Next, for $(x,t)\in\Omega\times{\mathbb{R}}/2\pi\mathbb{Z}$ we define

$\chi\left(x,t\right):=\mathds{1}_{e^{it}\cdot m(x)>0}.$ (35)

Finally, for $h\in{\mathbb{R}}$ and $e\in\mathbb{S}^{1}$, we define

$\displaystyle\Delta_{\alpha}\left(x,h,e\right):=\int_{0}^{2\pi}\int_{0}^{2\pi}\varphi_{\alpha}(s-t)\sin(s-t)D^{he}\chi(x,t)D^{he}\chi(x,s)\,dt\,ds,$
(36)

where $D^{he}\chi(x,\cdot)=\chi(x+he,\cdot)-\chi(x,\cdot)$. Our next lemma
plays the role of [GL20, Lemma 3.8].

######  Lemma 17.

Let $m:\Omega\to\mathbb{S}^{1}$ and $\alpha>0$. For all $x\in\Omega$,
$e\in\mathbb{S}^{1}$ and $|h|<\mathrm{dist}\\{x,\partial\Omega\\}$, we have

$\Delta_{\alpha}(x,h,e)\geq c(\alpha)\,|D^{he}m(x)|^{3+\alpha}.$ (37)

###### Proof.

Since the proof follows almost the same lines of the proof of [GL20, Lemma
3.8], we only sketch the main ingredients focusing on the differences. Given
$x\in\Omega$, $e\in\mathbb{S}^{1}$ and
$|h|<\mathrm{dist}\\{x,\partial\Omega\\}$, exactly the same arguments as those
at the beginning of the proof of [GL20, Lemma 3.8] show that it is sufficient
to assume $m(x+he)=e^{-i\beta}$ and $m(x)=e^{i\beta}$ for some
$\beta\in[0,\pi/2]$. Thus (37) becomes

$\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)\geq
c(\alpha)\left(2\sin(\beta)\right)^{3+\alpha}$

for

$\displaystyle\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right):=$
$\displaystyle\int_{0}^{2\pi}\int_{0}^{2\pi}\varphi_{\alpha}(s-t)\sin(s-t)$
$\displaystyle\cdot\left(\mathds{1}_{e^{it}\cdot
e^{-i\beta}>0}-\mathds{1}_{e^{it}\cdot
e^{i\beta}>0}\right)\left(\mathds{1}_{e^{is}\cdot
e^{-i\beta}>0}-\mathds{1}_{e^{is}\cdot e^{i\beta}>0}\right)\,dt\,ds.$

It is in turn sufficient to show

$\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)\geq\tilde{c}(\alpha)\beta^{3+\alpha}.$
(38)

Following carefully the calculations in [GL20, Lemma 3.8], we have

$\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)=8\int_{0}^{2\beta}\varphi_{\alpha}(\omega)(2\beta-\omega)\sin(\omega)\,d\omega\quad\text{
for }\beta\in[0,\pi/4]$ (39)

and

$\displaystyle\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)$
$\displaystyle=8\int_{0}^{\pi-2\beta}\varphi_{\alpha}(\omega)(2\beta-\omega)\sin(\omega)\,d\omega$
$\displaystyle\quad+8\int_{\pi-2\beta}^{\frac{\pi}{2}}\varphi_{\alpha}(\omega)(\pi-2\omega)\sin(\omega)\,d\omega\quad\text{
for }\beta\in[\pi/4,\pi/2].$

Recall that $\varphi_{\alpha}(\omega)=\omega^{\alpha}$ for
$\omega\in[0,\pi/4]$. Thus for $\beta\in\left[0,\pi/8\right]$, we have (after
a change of variable $\omega=2\beta v$)

$\displaystyle\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)$
$\displaystyle\overset{(\ref{eqgn1})}{=}8(2\beta)^{2+\alpha}\int_{0}^{1}v^{\alpha}(1-v)\sin(2\beta
v)\,dv$
$\displaystyle\gtrsim(2\beta)^{3+\alpha}\int_{0}^{1}v^{1+\alpha}(1-v)\,dv\gtrsim\beta^{3+\alpha}.$
(40)

For $\beta\in[\pi/8,\pi/4]$, recalling (4.3), it is clear that

$\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)\geq
8\int_{0}^{\frac{\pi}{4}}\varphi_{\alpha}(\omega)\left(\frac{\pi}{4}-\omega\right)\sin(\omega)\,d\omega=c_{1}>0.$

For $\beta\in[\pi/4,\pi/2]$, it is clear that
$2\beta-\omega\geq\frac{\pi}{2}-\omega$ and
$\pi-2\omega\geq\frac{\pi}{2}-\omega$ for all $\omega\in[0,\pi/2]$. Thus we
have

$\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)\geq
8\int_{0}^{\frac{\pi}{2}}\varphi_{\alpha}(\omega)\left(\frac{\pi}{2}-\omega\right)\sin(\omega)\,d\omega=c_{2}>0.$

It follows that
$\Xi_{\alpha}\left(e^{-i\beta},e^{i\beta}\right)\geq\min\\{c_{1},c_{2}\\}\gtrsim\beta^{3+\alpha}$
for $\beta\in[\pi/8,\pi/2]$. This together with (4.3) establishes (38) and
hence (37). ∎

Next we need a technical lemma.

######  Lemma 18.

Let $m:\Omega\to\mathbb{S}^{1}$ and $q\in
L^{\infty}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$. Recalling $\chi(x,t)$
defined in (35), for any $x_{0},x_{1}\in\Omega$ we have

$\left|\int_{0}^{2\pi}\left(\chi(x_{0},s)-\chi(x_{1},s)\right)q(s)\,ds\right|\lesssim\|q\|_{L^{\infty}\left(\mathbb{R}/2\pi\mathbb{Z}\right)}\left|m(x_{0})-m(x_{1})\right|.$
(41)

In addition, denoting by $\chi_{\varepsilon}(x,t)$ the convolution of
$\chi(x,t)$ in the $x$ variable, i.e.
$\chi_{\varepsilon}=\chi\ast\rho_{\varepsilon}$ for a smooth approximation of
the identity $\rho_{\varepsilon}(x)$, we have for $x\in\Omega$ and
$\varepsilon<\mathrm{dist}\\{x,\partial\Omega\\}$

$\left|\int_{0}^{2\pi}\left(\chi_{\varepsilon}(x,s)-\chi(x,s)\right)q(s)\,ds\right|\lesssim\|q\|_{L^{\infty}\left(\mathbb{R}/2\pi\mathbb{Z}\right)}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B_{\varepsilon}(x)}|m(z)-m(x)|\,dz.$
(42)

###### Proof.

Note that for any $a\in\mathbb{R}$ we have $\min\left\\{\left|a-2\pi
k\right|:k\in\mathbb{Z}\right\\}\leq\pi$. So given $x_{0},x_{1}\in\Omega$
there exists $k_{0}=k_{0}(x_{0},x_{1})\in\mathbb{Z}$ such that
$\left|\theta(x_{1})-\theta(x_{0})-2\pi k_{0}\right|\leq\pi$, where $\theta$
satisfies (34). Define $\widetilde{\theta}(x_{0})=\theta(x_{0})+2\pi k_{0}$,
so that $|\widetilde{\theta}(x_{0})-\theta(x_{1})|\leq\pi$. It is easily
checked that $2-2\cos(t)\gtrsim t^{2}$ for $|t|\leq\pi$. It follows that

$\left|\widetilde{\theta}(x_{0})-\theta(x_{1})\right|^{2}\lesssim
2-2\cos\left(\widetilde{\theta}(x_{0})-\theta(x_{1})\right)=\left|m(x_{0})-m(x_{1})\right|^{2}.$
(43)

As the function $q$ is $2\pi$-periodic, we have

$\displaystyle\left|\int_{0}^{2\pi}\left(\chi(x_{0},s)-\chi(x_{1},s)\right)q(s)ds\right|$
$\displaystyle=\left|\int_{\widetilde{\theta}(x_{0})-\frac{\pi}{2}}^{\widetilde{\theta}(x_{0})+\frac{\pi}{2}}q(s)\,ds-\int_{\theta(x_{1})-\frac{\pi}{2}}^{\theta(x_{1})+\frac{\pi}{2}}q(s)\,ds\right|$
$\displaystyle\lesssim\|q\|_{L^{\infty}\left(\mathbb{R}/2\pi\mathbb{Z}\right)}\left|\widetilde{\theta}(x_{0})-\theta(x_{1})\right|$
$\displaystyle\overset{(\ref{eqaux11})}{\lesssim}\|q\|_{L^{\infty}\left(\mathbb{R}/2\pi\mathbb{Z}\right)}\left|m(x_{0})-m(x_{1})\right|,$

which establishes (41).

Next, using $\int_{{\mathbb{R}}^{2}}\rho_{\varepsilon}=1$ and
$\rho_{\varepsilon}\lesssim 1/\varepsilon^{2}$, we find

$\displaystyle\left|\int_{0}^{2\pi}\left(\chi_{\varepsilon}(x,s)-\chi(x,s)\right)q(s)ds\right|$
$\displaystyle\quad\quad=\left|\int_{0}^{2\pi}\left(\int_{B_{\varepsilon}(x)}\rho_{\varepsilon}(x-z)\left(\chi(z,s)-\chi(x,s)\right)dz\right)q(s)\,ds\right|$
$\displaystyle\quad\quad=\left|\int_{B_{\varepsilon}(x)}\rho_{\varepsilon}(x-z)\int_{0}^{2\pi}\left(\chi(z,s)-\chi(x,s)\right)q(s)\,ds\,dz\right|$
$\displaystyle\quad\quad\overset{\eqref{eqaux1}}{\lesssim}\|q\|_{L^{\infty}(\mathbb{R}/2\pi\mathbb{Z})}\cdot\frac{1}{\varepsilon^{2}}\int_{B_{\varepsilon}(x)}|m(z)-m(x)|\,dz,$

which establishes (42). ∎

###### Proof of Proposition 16.

Following [GL20] we introduce the regularization of $\Delta_{\alpha}$ defined
in (36) in the $x$ variable. Specifically, for $x\in\Omega,e\in\mathbb{S}^{1}$
and $|h|<\mathrm{dist}\\{x,\partial\Omega\\}$ define

$\displaystyle\Delta^{\varepsilon}_{\alpha}\left(x,h,e\right)$
$\displaystyle:=\int_{0}^{2\pi}\int_{0}^{2\pi}\varphi_{\alpha}(s-t)\sin(s-t)D^{he}\chi_{\varepsilon}(x,t)D^{he}\chi_{\varepsilon}(x,s)\,dt\,ds,$

for $\varepsilon>0$ sufficiently small, where recall that
$\chi_{\varepsilon}=\chi\ast\rho_{\varepsilon}$ for a smooth approximation of
the identity $\rho_{\varepsilon}(x)$. Assume without loss of generality that
$e=e_{1}$, and the general case can be dealt with by rotation. In the sequel
we omit the dependence of all the quantities on $e$. For all $x\in\Omega$ and
$|h|+\varepsilon<\mathrm{dist}\\{x,\partial\Omega\\}$, the computations in
[GL20, Lemma 3.9] using the kinetic equation (4) give

$\partial_{h}\Delta_{\alpha}^{\varepsilon}(x,h)=I_{\alpha}^{\varepsilon}(x,h)+\nabla\cdot
A_{\alpha}^{\varepsilon}(x,h),$ (44)

where, denoting $\sigma_{\varepsilon}=\sigma\ast\rho_{\varepsilon}$ and
$\chi_{\varepsilon}^{h}(x,s)=\chi_{\varepsilon}(x+he_{1},s)$ with the same
meaning extended to all other applicable functions, we have

$\displaystyle I_{\alpha}^{\varepsilon}(x,h)$
$\displaystyle=-2\int_{0}^{2\pi}\sigma_{\varepsilon}^{h}(x,t)\int_{0}^{2\pi}\varphi^{\prime}_{\alpha}(s-t)\chi_{\varepsilon}(x,s)\sin(s)\,ds\,dt$
$\displaystyle\quad+2\int_{0}^{2\pi}\sigma_{\varepsilon}(x,t)\int_{0}^{2\pi}\varphi^{\prime}_{\alpha}(s-t)\chi_{\varepsilon}^{h}(x,s)\sin(s)\,ds\,dt,$
(45)

and

$\displaystyle A_{\alpha,1}^{\varepsilon}(x,h)$
$\displaystyle=2\iint_{\left[0,2\pi\right]\times\left[0,2\pi\right]}\varphi_{\alpha}(s-t)\sin(s)\cos(t)\,\chi_{\varepsilon}^{h}(x,s)D^{h}\chi_{\varepsilon}(x,t)\,ds\,dt,$
$\displaystyle A_{\alpha,2}^{\varepsilon}(x,h)$
$\displaystyle=2\iint_{\left[0,2\pi\right]\times\left[0,2\pi\right]}\varphi_{\alpha}(s-t)\sin(s)\sin(t)\,\chi_{\varepsilon}(x,s)\chi^{h}_{\varepsilon}(x,t)\,ds\,dt.$

In the following, we consider an arbitrary fixed domain
$U\subset\subset\Omega$ so that there exist intermediate domains
$U\subset\subset\Omega^{\prime}\subset\subset\Omega^{\prime\prime}\subset\subset\Omega$.

Step 1. We will show that for any fixed
$|h|<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}$ and
a.e. $x\in\Omega^{\prime}$, we have (recalling $\theta(x)$ given in (34))

$\lim_{\varepsilon\rightarrow
0}I^{\varepsilon}_{\alpha}(x,h)=I_{\alpha}(x,h),$ (46)
$\displaystyle\lim_{\varepsilon\rightarrow 0}A_{\alpha,i}^{\varepsilon}(x,h)$
$\displaystyle=A_{\alpha,i}(x,h)\quad\text{ for }i=1,2,$ (47)

and

$\lim_{\varepsilon\rightarrow
0}\Delta^{\varepsilon}_{\alpha}(x,h)=\Delta_{\alpha}(x,h),$ (48)

where

$\displaystyle I_{\alpha}(x,h)=$
$\displaystyle-2\int_{0}^{2\pi}\left(\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\varphi^{\prime}_{\alpha}(s-t)\sin(s)\,ds\right)d\sigma_{x}^{h}(t)$
$\displaystyle+2\int_{0}^{2\pi}\left(\int_{\theta^{h}(x)-\frac{\pi}{2}}^{\theta^{h}(x)+\frac{\pi}{2}}\varphi^{\prime}_{\alpha}(s-t)\sin(s)\,ds\right)d\sigma_{x}(t),$
(49) $\displaystyle A_{\alpha,1}(x,h)$
$\displaystyle=2\iint_{\left[0,2\pi\right]\times\left[0,2\pi\right]}\varphi_{\alpha}(s-t)\sin(s)\cos(t)\,\chi^{h}(x,s)D^{h}\chi(x,t)\,ds\,dt,$
$\displaystyle A_{\alpha,2}(x,h)$
$\displaystyle=2\iint_{\left[0,2\pi\right]\times\left[0,2\pi\right]}\varphi_{\alpha}(s-t)\sin(s)\sin(t)\,\chi(x,s)\chi^{h}(x,t)\,ds\,dt,$
(50)

and $\Delta_{\alpha}(x,h)=\Delta_{\alpha}(x,h,e_{1})$ is given in (36).

Proof of Step 1. For any $\psi\in
C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$, define
$\mathcal{F}_{\psi}(x):=\int_{0}^{2\pi}\psi(t)\,d\sigma_{x}(t)$ for a.e.
$x\in\Omega$. It follows that

$\left|\mathcal{F}_{\psi}(x)\right|\leq\|\psi\|_{C^{0}({\mathbb{R}}/2\pi\mathbb{Z})}\nu(x)\quad\text{
for a.e. }x\in\Omega\text{ and all }\psi\in
C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right),$ (51)

where recall from (6) that
$\nu(x)=\|\sigma_{x}\|_{\mathcal{M}({\mathbb{R}}/2\pi\mathbb{Z})}\in
L^{p}_{{\rm loc}}(\Omega)$. Thus $\mathcal{F}_{\psi}\in L^{p}_{{\rm
loc}}(\Omega)$ for all $\psi\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$.
By definition of $\sigma_{\varepsilon}$ and (5) we have

$\displaystyle\int_{0}^{2\pi}\psi(t)\sigma_{\varepsilon}(x,t)\,dt=\int_{0}^{2\pi}\psi(t)\int_{B_{\varepsilon}(x)}\rho_{\varepsilon}(x-z)\,d\sigma(z,t)$
$\displaystyle\quad\quad\quad\quad\quad\quad=\int_{B_{\varepsilon}(x)}\left(\int_{0}^{2\pi}\psi(t)\,d\sigma_{z}(t)\right)\rho_{\varepsilon}(x-z)\,dz=\int_{B_{\varepsilon}(x)}\mathcal{F}_{\psi}(z)\rho_{\varepsilon}(x-z)\,dz.$
(52)

Combining (4.3) with (51), we obtain that, for all Lebesgue points $x$ of
$\nu$ with $\nu(x)<\infty$, there exists $\varepsilon_{0}(x)>0$ such that

$\left|\int_{0}^{2\pi}\psi(t)\sigma_{\varepsilon}(x,t)\,dt\right|\leq\|\psi\|_{C^{0}({\mathbb{R}}/2\pi\mathbb{Z})}\int_{B_{\varepsilon}(x)}\nu(z)\rho_{\varepsilon}(x-z)\,dz\leq
C\|\psi\|_{C^{0}({\mathbb{R}}/2\pi\mathbb{Z})}\nu(x)$

for all $\psi\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ and all
$\varepsilon\in\left(0,\varepsilon_{0}(x)\right)$. It follows that

$\|\sigma_{\varepsilon}(x,\cdot)\|_{L^{1}\left(\mathbb{R}/2\pi\mathbb{Z}\right)}\leq
C\nu(x)<\infty\quad\text{ for all
}\varepsilon\in\left(0,\varepsilon_{0}(x)\right).$ (53)

Let $\\{\psi_{j}\\}_{j}$ be a countable dense subset of
$C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$, and $\Lambda\subset\Omega$ be
the intersection of the Lebesgue points of $\mathcal{F}_{\psi_{j}}$ for all
$j$ and the Lebesgue points of $\nu$ with $\nu(x)<\infty$. We know
$|\Omega\setminus\Lambda|=0$. It follows from (4.3) that

$\lim_{\varepsilon\to
0}\int_{0}^{2\pi}\psi_{j}(t)\sigma_{\varepsilon}(x,t)\,dt=\mathcal{F}_{\psi_{j}}(x)=\int_{0}^{2\pi}\psi_{j}(t)\,d\sigma_{x}(t)\quad\text{
for all }j\text{ and all }x\in\Omega^{\prime}\cap\Lambda.$ (54)

By density of $\\{\psi_{j}\\}_{j}$ in
$C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ and (53), one can extend (54) to
all $\psi\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ to deduce that

$\sigma_{\varepsilon}(x,\cdot)\mathcal{L}^{1}\rightharpoonup\sigma_{x}(\cdot)\quad\text{
in }\mathcal{M}\left(\mathbb{R}/2\pi\mathbb{Z}\right)\text{ as
}\varepsilon\rightarrow 0\text{ for all }x\in\Omega^{\prime}\cap\Lambda.$ (55)

We will use (55) to first find the limit of the second term of
$I^{\varepsilon}_{\alpha}$ (recalling (4.3)) as $\varepsilon\rightarrow 0$. By
(42) of Lemma 18, for all $t\in\mathbb{R}/2\pi\mathbb{Z}$ we have

$\displaystyle\left|\int_{0}^{2\pi}\varphi_{\alpha}^{\prime}(s-t)\sin(s)\chi^{h}(x,s)ds-\int_{0}^{2\pi}\varphi_{\alpha}^{\prime}(s-t)\sin(s)\chi_{\varepsilon}^{h}(x,s)ds\right|$
$\displaystyle\quad\lesssim\|\varphi_{\alpha}^{\prime}\|_{C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B_{\varepsilon}(x)}|m^{h}(z)-m^{h}(x)|\,dz,$

and hence for all Lebesgue points $x$ of $m^{h}$ we have

$\int_{0}^{2\pi}\varphi_{\alpha}^{\prime}(s-t)\sin(s)\chi_{\varepsilon}^{h}(x,s)ds\rightarrow\int_{0}^{2\pi}\varphi_{\alpha}^{\prime}(s-t)\sin(s)\chi^{h}(x,s)ds\quad\text{
in }C_{t}^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right).$

This together with (53) and (55) shows

$\displaystyle\lim_{\varepsilon\rightarrow
0}\iint_{\left[0,2\pi\right]\times\left[0,2\pi\right]}\varphi_{\alpha}^{\prime}(s-t)\sin(s)\chi_{\varepsilon}^{h}(x,s)\sigma_{\varepsilon}(x,t)ds\,dt$
$\displaystyle\quad\quad\quad\quad=\iint_{\left[0,2\pi\right]\times\left[0,2\pi\right]}\varphi_{\alpha}^{\prime}(s-t)\sin(s)\chi^{h}(x,s)ds\,d\sigma_{x}(t)\quad\text{
for a.e. }x\in\Omega^{\prime}.$

In exactly the same way we can deal with the first term in
$I^{\varepsilon}_{\alpha}$ and hence establish (46). Finally, (47) and (48)
follow from (42) and straightforward estimates. This completes the proof of
Step 1.

Proof of Proposition 16 completed. Let

$\alpha=3p-3.$ (56)

We take a smooth nonnegative cut off function $\gamma\in
C^{\infty}_{c}(\Omega^{\prime})$ with $\|\gamma\|_{L^{\infty}(\Omega)}\leq 1$
and $\gamma\equiv 1$ on $\overline{U}$. Using (44), the Dominated Convergence
Theorem and Step 1, we have for all
$|h|<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}$ that

$\displaystyle\int_{\Omega}\gamma(x)\Delta_{\alpha}(x,h)\,dx$
$\displaystyle\quad\overset{(\ref{eqokl50.3})}{=}\lim_{\varepsilon\rightarrow
0}\int_{\Omega}\gamma(x)\Delta^{\varepsilon}_{\alpha}(x,h)\,dx$
$\displaystyle\quad\overset{(\ref{partial_h})}{=}\lim_{\varepsilon\rightarrow
0}\left(\int_{0}^{h}\int_{\Omega}\gamma(x)I^{\varepsilon}_{\alpha}(x,\tilde{h})dx\,d\tilde{h}-\int_{0}^{h}\int_{\Omega}\nabla\gamma(x)\cdot
A_{\alpha}^{\varepsilon}(x,\tilde{h})dx\,d\tilde{h}\right)$
$\displaystyle\quad\overset{(\ref{eqokl62}),(\ref{cpeq298.2})}{=}\int_{0}^{h}\int_{\Omega}\gamma(x)I_{\alpha}(x,\tilde{h})dx\,d\tilde{h}-\int_{0}^{h}\int_{\Omega}\nabla\gamma(x)\cdot
A_{\alpha}(x,\tilde{h})dx\,d\tilde{h}.$ (57)

In the following Lemmas 19 and 21, we will show

$\displaystyle\left|\int_{\Omega}I_{\alpha}(x,h)\,\gamma(x)\,dx\right|\leq
C\left(\left\lVert\gamma\left(\left|D^{h}m\right|^{\alpha}+\left|D^{-h}m\right|^{\alpha}\right)\right\rVert_{L^{p^{\prime}}(\Omega)}+\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}|h|\right)\|\nu\|_{L^{p}(\Omega^{\prime\prime})}$
(58)

and

$\left|\int_{\Omega}A_{\alpha}(x,h)\cdot\nabla\gamma(x)\,dx\right|\leq
C\int_{\Omega}\left|D^{h}m\right|\left|\nabla\gamma\right|\,dx.$ (59)

We combine the lower bound established in (37) of Lemma 17 with the upper
bounds (58) and (59) and apply them to (4.3) to deduce that

$\displaystyle\int_{\Omega}\gamma(x)\left|D^{h}m(x)\right|^{3+\alpha}\,dx$
$\displaystyle\leq
C\left|h\right|\sup_{|\tilde{h}|\in\left[0,|h|\right]}\left\lVert\left(\left|D^{\tilde{h}}m\right|^{\alpha}+\left|D^{-\tilde{h}}m\right|^{\alpha}\right)\gamma\right\rVert_{L^{p^{\prime}}(\Omega)}\left\lVert\nu\right\rVert_{L^{p}(\Omega^{\prime\prime})}$
$\displaystyle+C\left|h\right|^{2}\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}\left\lVert\nu\right\rVert_{L^{p}(\Omega^{\prime\prime})}$
$\displaystyle+C\left|h\right|\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}\sup_{|\tilde{h}|\in\left[0,|h|\right]}\int_{\Omega^{\prime}}\left|D^{\tilde{h}}m\right|\,dx.$
(60)

Recalling (56), we have $\alpha p^{\prime}=3+\alpha=3p$, and since
$0\leq\gamma\leq 1$, we have

$\displaystyle\left\lVert\gamma\left|D^{h}m\right|^{\alpha}\right\rVert_{L^{p^{\prime}}(\Omega)}=\left(\int_{\Omega}\gamma^{p^{\prime}}\left|D^{h}m\right|^{3p}\,dx\right)^{\frac{1}{p^{\prime}}}\leq\left(\int_{\Omega}\gamma\left|D^{h}m\right|^{3p}\,dx\right)^{\frac{1}{p^{\prime}}}.$

Thus it follows from Young’s inequality that

$\displaystyle\left|h\right|\sup_{\left\\{\tilde{h}:\left|\tilde{h}\right|\leq\left|h\right|\right\\}}\left\lVert\gamma\left|D^{\tilde{h}}m\right|^{\alpha}\right\rVert_{L^{p^{\prime}}(\Omega)}$
$\displaystyle\quad\leq\left|h\right|\sup_{\left\\{\tilde{h}:\left|\tilde{h}\right|\leq\left|h\right|\right\\}}\left(\int_{\Omega}\gamma\left|D^{\tilde{h}}m\right|^{3p}\,dx\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\quad\leq
C(\delta)|h|^{p}+\delta\sup_{\left\\{\tilde{h}:\left|\tilde{h}\right|\leq\left|h\right|\right\\}}\left(\int_{\Omega}\gamma\left|D^{\tilde{h}}m\right|^{3p}\,dx\right),\quad\forall\delta>0.$
(61)

Next by [GL20, Theorem 2.6], we know $m\in B^{\frac{1}{3}}_{3,\infty,{\rm
loc}}(\Omega)$. Thus, for all $|h|$ sufficiently small we have

$\int_{\Omega^{\prime}}\left|D^{h}m\right|\,dx\leq C\left\lVert
D^{h}m\right\rVert_{L^{3}(\Omega^{\prime})}\leq C|h|^{\frac{1}{3}}.$ (62)

Putting (4.3) and (62) into (4.3) and using $3+\alpha=3p$, we readily deduce

$\displaystyle\int_{\Omega}\gamma\left|D^{h}m\right|^{3p}\,dx$
$\displaystyle\leq
C\left(C(\delta)|h|^{p}+\delta\sup_{\left\\{\tilde{h}:\left|\tilde{h}\right|\leq\left|h\right|\right\\}}\left(\int_{\Omega}\gamma\left|D^{\tilde{h}}m\right|^{3p}\,dx\right)\right)\left\lVert\nu\right\rVert_{L^{p}(\Omega^{\prime\prime})}$
$\displaystyle\quad+C\left|h\right|^{2}\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}\left\lVert\nu\right\rVert_{L^{p}(\Omega^{\prime\prime})}+C|h|^{\frac{4}{3}}\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}.$

Taking $\delta>0$ sufficiently small and recalling
$p\in\left(1,\frac{4}{3}\right]$, we obtain, for all
$0<\xi<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}$,

$\displaystyle\sup_{\left\\{h:\left|h\right|\leq\xi\right\\}}\int_{\Omega}\gamma\left|D^{h}m\right|^{3p}\,dx$
$\displaystyle\leq\frac{1}{2}\sup_{\left\\{h:\left|h\right|\leq\xi\right\\}}\sup_{\left\\{\tilde{h}:\left|\tilde{h}\right|\leq\left|h\right|\right\\}}\int_{\Omega}\gamma\left|D^{\tilde{h}}m\right|^{3p}dx+C\sup_{\left\\{h:\left|h\right|\leq\xi\right\\}}\left|h\right|^{p}$
$\displaystyle=\frac{1}{2}\sup_{\left\\{h:\left|h\right|\leq\xi\right\\}}\int_{\Omega}\gamma\left|D^{h}m\right|^{3p}dx+C\xi^{p}.$

Recalling $\gamma\equiv 1$ on $\overline{U}$, we deduce from the above that

$\sup_{\left\\{h:\left|h\right|\leq\xi\right\\}}\int_{U}\gamma\left|D^{h}m\right|^{3p}\,dx\leq
C\xi^{p}\quad\text{ for all
}0<\xi<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}.$

For larger values of $\xi$, the boundedness of $m$ implies that
$\xi^{-\frac{1}{3}}\sup_{|h|\leq\xi}\lVert D^{h}m\rVert_{L^{3p}(U)}$ is
bounded. It follows that $m\in B^{\frac{1}{3}}_{3p,\infty}(U)$ for all
$U\subset\subset\Omega$. ∎

######  Lemma 19.

Let $I_{\alpha}(x,h)$ be given in (4.3). For domains
$U\subset\subset\Omega^{\prime}\subset\subset\Omega^{\prime\prime}\subset\subset\Omega$
and any $\gamma\in C^{\infty}_{c}(\Omega^{\prime})$, the estimate (58) holds
for all
$|h|<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}$.

To simplify notation, define

$\mathcal{W}_{\alpha}(x,s):=\int_{0}^{2\pi}\varphi_{\alpha}(s-t)d\sigma_{x}(t).$
(63)

It is clear that

$\|\mathcal{W}_{\alpha}(x,\cdot)\|_{L^{\infty}(\mathbb{R}/2\pi\mathbb{Z})}\leq\|\varphi_{\alpha}\|_{C^{0}}\,\nu(x)\quad\text{
for a.e. }x\in\Omega,$ (64)

where recall that
$\nu(x)=\|\sigma_{x}\|_{\mathcal{M}(\mathbb{R}/2\pi\mathbb{Z})}\in L^{p}_{{\rm
loc}}(\Omega)$. We first do some computations.

######  Lemma 20.

Under the assumptions of Lemma 19, we have

$\int_{\Omega}I_{\alpha}(x,h)\gamma(x)dx=2\Xi_{1}(h)-4\Xi_{2}(h),$ (65)

where

$\displaystyle\Xi_{1}(h)$
$\displaystyle=\int_{\Omega}\left(\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}^{h}\left(x,s\right)\cos(s)\,ds-\int_{\theta^{h}(x)-\frac{\pi}{2}}^{\theta^{h}(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}\left(x,s\right)\cos(s)\,ds\right)\gamma(x)\,dx$
(66)

and

$\displaystyle\Xi_{2}(h)$
$\displaystyle=\int_{\Omega}\left(\sin\left(\theta(x)+\frac{\pi}{2}\right)\mathcal{W}_{\alpha}^{h}\left(x,\theta(x)+\frac{\pi}{2}\right)\right.$
$\displaystyle\quad\quad\quad\quad-\left.\sin\left(\theta^{h}(x)+\frac{\pi}{2}\right)\mathcal{W}_{\alpha}\left(x,\theta^{h}(x)+\frac{\pi}{2}\right)\right)\gamma(x)\,dx.$
(67)

###### Proof.

Using integration by parts and $\pi$-periodicity of $\varphi_{\alpha}$ gives

$\displaystyle\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\varphi^{\prime}_{\alpha}(s-t)\sin(s)\,ds$
$\displaystyle=2\varphi_{\alpha}\left(\theta(x)+\frac{\pi}{2}-t\right)\sin\left(\theta(x)+\frac{\pi}{2}\right)-\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\varphi_{\alpha}(s-t)\cos(s)\,ds.$
(68)

Thus

$\displaystyle\int_{0}^{2\pi}\left(\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\varphi^{\prime}_{\alpha}(s-t)\sin(s)\,ds\right)d\sigma^{h}_{x}(t)$
$\displaystyle\overset{(\ref{cpeq14.3}),(\ref{cpeq14.2})}{=}2\sin\left(\theta(x)+\frac{\pi}{2}\right)\mathcal{W}_{\alpha}^{h}\left(x,\theta(x)+\frac{\pi}{2}\right)-\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}^{h}(x,s)\cos(s)\,ds.$
(69)

And in the same way we have

$\displaystyle\int_{0}^{2\pi}\left(\int_{\theta^{h}(x)-\frac{\pi}{2}}^{\theta^{h}(x)+\frac{\pi}{2}}\varphi^{\prime}_{\alpha}(s-t)\sin(s)\,ds\right)d\sigma_{x}(t)$
$\displaystyle=2\sin\left(\theta^{h}(x)+\frac{\pi}{2}\right)\mathcal{W}_{\alpha}\left(x,\theta^{h}(x)+\frac{\pi}{2}\right)-\int_{\theta^{h}(x)-\frac{\pi}{2}}^{\theta^{h}(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}(x,s)\cos(s)\,ds.$
(70)

Putting (4.3), (4.3), (4.3), (66), (20) together establishes (65). ∎

###### Proof of Lemma 19.

Assume $1<p\leq\frac{4}{3}$, and thus $0<\alpha=3p-3\leq 1$. We first show

$\displaystyle\left|\Xi_{1}(h)\right|$ $\displaystyle\leq
C\left(\left\lVert\gamma\left(\left|D^{h}m\right|^{\alpha}+\left|D^{-h}m\right|^{\alpha}\right)\right\rVert_{L^{p^{\prime}}(\Omega)}+\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}|h|\right)\|\nu\|_{L^{p}(\Omega^{\prime\prime})}.$
(71)

A change of variable gives

$\displaystyle\int_{\Omega}\left(\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}^{h}\left(x,s\right)\cos(s)\,ds\right)\gamma(x)\,dx$
$\displaystyle\quad\quad=\int_{\Omega}\left(\int_{\theta^{-h}(x)-\frac{\pi}{2}}^{\theta^{-h}(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}\left(x,s\right)\cos(s)\,ds\right)\gamma(x-he_{1})\,dx,$
(72)

and thus

$\displaystyle\left|\int_{\Omega}\left(\int_{\theta(x)-\frac{\pi}{2}}^{\theta(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}^{h}\left(x,s\right)\cos(s)\,ds\right)\gamma(x)\,dx\right.$
$\displaystyle\quad\quad\quad-\left.\int_{\Omega}\left(\int_{\theta^{-h}(x)-\frac{\pi}{2}}^{\theta^{-h}(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}\left(x,s\right)\cos(s)\,ds\right)\gamma(x)\,dx\right|$
$\displaystyle\quad\overset{(\ref{cpeq221.4})}{=}\left|\int_{\Omega}\left(\int_{\theta^{-h}(x)-\frac{\pi}{2}}^{\theta^{-h}(x)+\frac{\pi}{2}}\mathcal{W}_{\alpha}\left(x,s\right)\cos(s)\,ds\right)\left(\gamma(x-he_{1})-\gamma(x)\right)\,dx\right|$
$\displaystyle\quad\overset{(\ref{cpeq14.4})}{\lesssim}\|\nabla\gamma\|_{L^{\infty}(\Omega)}\|\nu\|_{L^{p}\left(\Omega^{\prime\prime}\right)}|h|.$
(73)

Therefore, recalling the definition of $\chi$ in (35), we have

$\displaystyle\left|\Xi_{1}(h)-\int_{\Omega}\left(\int_{0}^{2\pi}\left(\chi^{-h}(x,s)-\chi^{h}(x,s)\right)\mathcal{W_{\alpha}}(x,s)\cos(s)ds\right)\gamma(x)\,dx\right|$
$\displaystyle\quad\overset{(\ref{cpeq172}),(\ref{cpeq221.45})}{\lesssim}\|\nabla\gamma\|_{L^{\infty}(\Omega)}\|\nu\|_{L^{p}\left(\Omega^{\prime\prime}\right)}|h|.$
(74)

As $0<\alpha\leq 1$, it follows from (41) and (64) that

$\displaystyle\left|\int_{0}^{2\pi}\left(\chi^{-h}(x,s)-\chi^{h}(x,s)\right)\mathcal{W_{\alpha}}(x,s)\cos(s)ds\right|$
$\displaystyle\quad\quad\lesssim\nu(x)\left|m(x+he_{1})-m(x-he_{1})\right|$
$\displaystyle\quad\quad\leq\nu(x)\left(\left|D^{h}m(x)\right|^{\alpha}+\left|D^{-h}m(x)\right|^{\alpha}\right)\quad\text{
for }a.e.\ x\in\Omega^{\prime}.$ (75)

Putting (4.3) and (4.3) together gives (71).

Next we show

$\left|\Xi_{2}(h)\right|\leq
C\left(\left\lVert\gamma\left(\left|D^{h}m\right|^{\alpha}+\left|D^{-h}m\right|^{\alpha}\right)\right\rVert_{L^{p^{\prime}}(\Omega)}+\left\lVert\nabla\gamma\right\rVert_{L^{\infty}(\Omega)}|h|\right)\|\nu\|_{L^{p}(\Omega^{\prime\prime})}.$
(76)

In the same way as (4.3), from (20) we have

$\left|\Xi_{2}(h)-\widetilde{\Xi}_{2}(h)\right|\lesssim\|\nabla\gamma\|_{L^{\infty}(\Omega)}\|\nu\|_{L^{p}\left(\Omega^{\prime\prime}\right)}|h|,$
(77)

where

$\displaystyle\widetilde{\Xi}_{2}(h)$
$\displaystyle:=\int_{\Omega}\left(\sin\left(\theta^{-h}(x)+\frac{\pi}{2}\right)\mathcal{W}_{\alpha}\left(x,\theta^{-h}(x)+\frac{\pi}{2}\right)\right.$
$\displaystyle\quad\quad\quad\quad-\left.\sin\left(\theta^{h}(x)+\frac{\pi}{2}\right)\mathcal{W}_{\alpha}\left(x,\theta^{h}(x)+\frac{\pi}{2}\right)\right)\gamma(x)\,dx.$

It is easy to check that for $0<\alpha\leq 1$ we have

$\left|\varphi_{\alpha}(t_{1})-\varphi_{\alpha}(t_{2})\right|\leq
C(\alpha)\left|t_{1}-t_{2}\right|^{\alpha}\quad\text{ for all
}t_{1},t_{2}\in{\mathbb{R}}.$

Thus

$\displaystyle\left|\mathcal{W}_{\alpha}\left(x,s_{1}\right)-\mathcal{W}_{\alpha}\left(x,s_{2}\right)\right|$
$\displaystyle\overset{(\ref{cpeq14.3})}{\leq}C(\alpha)\left|s_{1}-s_{2}\right|^{\alpha}\nu(x)\quad\text{
for all }s_{1},s_{2}\in{\mathbb{R}}\text{ and a.e. }x\in\Omega.$ (78)

It follows that for $0<\alpha\leq 1$ we have

$\displaystyle\left|\sin\left(s_{1}\right)\mathcal{W}_{\alpha}(x,s_{1})-\sin\left(s_{2}\right)\mathcal{W}_{\alpha}(x,s_{2})\right|$
$\displaystyle\quad\quad\leq\left|\mathcal{W}_{\alpha}(x,s_{1})-\mathcal{W}_{\alpha}(x,s_{2})\right|+\left|\mathcal{W}_{\alpha}(x,s_{1})\right|\left|\sin(s_{1})-\sin(s_{2})\right|$
$\displaystyle\quad\quad\overset{(\ref{cpeq14.6}),(\ref{cpeq14.4})}{\leq}C\left|s_{1}-s_{2}\right|^{\alpha}\nu(x)\quad\text{
for all bounded }s_{1},s_{2}\text{ and a.e. }x\in\Omega.$

By $2\pi$-periodicity of $\sin(\cdot)\mathcal{W}_{\alpha}(x,\cdot)$, we may
assume without loss of generality that
$|\theta^{h}(x)-\theta^{-h}(x)|\leq\pi$. It follows from the above estimate
and (43) that

$\displaystyle\left|\widetilde{\Xi}_{2}(h)\right|$ $\displaystyle\leq
C\int_{\Omega}\nu(x)\left|m(x-he_{1})-m(x+he_{1})\right|^{\alpha}\gamma(x)\,dx$
$\displaystyle\leq
C\int_{\Omega}\nu(x)\left(\left|D^{-h}m(x)\right|^{\alpha}+\left|D^{h}m(x)\right|^{\alpha}\right)\gamma(x)\,dx.$
(79)

Putting (4.3) together with (77) establishes (76). Finally, putting (71) and
(76) together with (65) establishes (58). ∎

######  Lemma 21.

Let $A_{\alpha}(x,h)$ be given in (4.3). For domains
$U\subset\subset\Omega^{\prime}\subset\subset\Omega^{\prime\prime}\subset\subset\Omega$
and any $\gamma\in C^{\infty}_{c}(\Omega^{\prime})$, the estimate (59) holds
for all
$|h|<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}$.

###### Proof.

First we need to change $A_{\alpha,2}$ into a more convenient form to get
estimates. Since $\varphi_{\alpha}$ is odd, we have

$\displaystyle\iint\varphi_{\alpha}(s-t)\sin(s)\sin(t)\,\chi(x,s)\chi(x,t)\,dt\,ds$
$\displaystyle\quad\quad\overset{\tilde{s}=t,\tilde{t}=s}{=}\iint\varphi_{\alpha}\left(\tilde{t}-\tilde{s}\right)\sin\left(\tilde{s}\right)\sin\left(\tilde{t}\right)\,\chi(x,\tilde{s})\chi(x,\tilde{t})\,d\tilde{t}\,d\tilde{s}$
$\displaystyle\quad\quad=-\iint\varphi_{\alpha}\left(\tilde{s}-\tilde{t}\right)\sin\left(\tilde{s}\right)\sin\left(\tilde{t}\right)\,\chi(x,\tilde{s})\chi(x,\tilde{t})\,d\tilde{t}\,d\tilde{s},$

and thus
$\iint\varphi_{\alpha}(s-t)\sin(s)\sin(t)\,\chi(x,s)\chi(x,t)\,dt\,ds=0$. So
we have

$\displaystyle
A_{\alpha,2}(x,h)\overset{(\ref{eq:A_alpha})}{=}2\iint\varphi_{\alpha}(s-t)\sin(s)\sin(t)\,\chi(x,s)D^{h}\chi(x,t)\,dt\,ds.$

By $\pi$-periodicity of $\varphi_{\alpha}$, it is easy to check that
$f^{x,h}(t):=\int_{\theta^{h}(x)-\frac{\pi}{2}}^{\theta^{h}(x)+\frac{\pi}{2}}\varphi_{\alpha}(s-t)\sin(s)\cos(t)\,ds$
is $2\pi$-periodic with zero average on $[0,2\pi]$. Further it is clear that
$\|f^{x,h}\|_{L^{\infty}({\mathbb{R}}/2\pi\mathbb{Z})}\lesssim\|\varphi_{\alpha}\|_{C^{0}}$.
Using Lemma 18 we deduce that

$\displaystyle\left|A_{\alpha,1}(x,h)\right|\overset{(\ref{eq:A_alpha})}{=}2\left|\int_{0}^{2\pi}f^{x,h}(t)D^{h}\chi(x,t)\,dt\right|\overset{(\ref{eqaux1})}{\leq}C\left|D^{h}m(x)\right|\quad\text{
for a.e. }x\in\Omega^{\prime}.$

And in the same way we have

$\left|A_{\alpha,2}(x,h)\right|\leq C\left|D^{h}m(x)\right|\quad\text{ for
a.e. }x\in\Omega^{\prime}.$

This establishes (59). ∎

##  5 Proof of Theorem 5

In this section we give the proof of Theorem 5. In Subsection 5.1 we first
compute the entropy production $\operatorname{div}\Phi(m)$ for a wide class of
$\Phi\in ENT$ in terms of the two special entropy productions
$\operatorname{div}\Sigma_{j}(m)$; see (109) of Proposition 29 below. This
constitutes the major step of the proof. Then in Subsection 5.2, using the
specific structure of the entropies $\Phi_{f}$, we realize that the general
formula (109) turns into the more specific one (12) for this class of
entropies, which in turn gives the explicit characterization of the kinetic
measure $\sigma$ as in (13).

###  5.1 Factorization for general entropies

In this subsection, we establish in Proposition 29 a more general version of
the formula (12) for sufficiently regular entropies $\Phi\in ENT$. Let
$E:L^{2}(\mathbb{S}^{1})\rightarrow L^{2}(B_{1})$ be the continuous linear
operator uniquely determined by its action on Fourier modes

$E\psi_{k}(re^{i\theta})=r^{\left|k\right|}e^{ik\theta}\quad\text{ for
}\psi_{k}\left(e^{i\theta}\right)=e^{ik\theta},k\in\mathbb{Z}.$ (80)

######  Remark 22.

The operator $E$ is a continuous operator from $C^{4}(\mathbb{S}^{1})$ to
$C^{3}(\overline{B}_{1})$. This is a well known fact and is also shown at the
beginning of the proof of [LLP20, Lemma 16]. We include an argument for the
convenience of the reader. From (80) we have that
$\|E\psi_{k}\|_{C^{3}(\overline{B}_{1})}\lesssim 1+|k|^{3}$. Given
$\psi=\sum_{k}c_{k}(\psi)\psi_{k}\in C^{4}(\mathbb{S}^{1})$, by Parseval’s
identity, we have
$\sum_{k}\left(k^{4}|c_{k}(\psi)|\right)^{2}=\sum_{k}\left|c_{k}\left(\psi^{(4)}\right)\right|^{2}=\frac{1}{2\pi}\lVert\psi^{(4)}\rVert_{L^{2}(\mathbb{S}^{1})}^{2}\leq\lVert\psi^{(4)}\rVert_{C^{0}(\mathbb{S}^{1})}^{2}$.
It follows that

$\sum_{|k|\geq 1}\left|c_{k}(\psi)\right|\left\lVert
E\psi_{k}\right\rVert_{C^{3}(\overline{B}_{1})}\lesssim\sum_{|k|\geq
1}k^{4}\left|c_{k}(\psi)\right|\cdot|k|^{-1}\leq
C\left\lVert\psi^{(4)}\right\rVert_{C^{0}(\mathbb{S}^{1})},$

from which we deduce $\left\lVert
E\psi\right\rVert_{C^{3}(\overline{B}_{1})}\leq
C\lVert\psi\rVert_{C^{4}(\mathbb{S}^{1})}$.

As the linear span of $\\{\psi_{k}\\}_{k\in\mathbb{Z}}$ is dense in
$C^{4}(\mathbb{S}^{1})$ and $\Delta(E\psi_{k})=0$ for all $k\in\mathbb{Z}$, it
follows from Remark 22 that

$\Delta(E\psi)=0\quad\text{ in }B_{1}\text{ for all }\psi\in
C^{4}(\mathbb{S}^{1}).$

In the following we refer to $E$ as the _harmonic extension operator_.

######  Proposition 23.

Let $m$ satisfy (1). Assume (11) holds true for some sequence
$\varepsilon_{k}\rightarrow 0$ and $1\leq p<\infty$. Given any $\psi\in
C^{4}(\mathbb{S}^{1})$ and $\varphi=E\psi$ its harmonic extension to
$\overline{B}_{1}$, the harmonic entropy given by

$\Phi^{E\psi}(z)=\Phi^{\varphi}(z)=\varphi(z)z+\left(iz\cdot\nabla\varphi(z)\right)iz\quad\text{
for all }z\in\overline{B}_{1}$ (81)

satisfies $\operatorname{div}\Phi^{E\psi}(m)\in
L^{{{\color[rgb]{0,0,1}p}}}_{{\rm loc}}(\Omega)$. Further it explicitly holds

$\operatorname{div}\Phi^{E\psi}(m)=\mathcal{A}_{1}\psi(m)e^{i2\theta}\cdot\operatorname{div}\Sigma(m)+\mathcal{A}_{2}\psi(m)ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)\quad\text{
a.e. in }\Omega,$ (82)

where $\theta:\Omega\to[0,2\pi)$ satisfies $m(x)=e^{i\theta(x)}$ a.e. in
$\Omega$, and $\mathcal{A}_{1},\mathcal{A}_{2}:C^{4}(\mathbb{S}^{1})\to
C^{0}(\mathbb{S}^{1})$ are the Fourier multiplier operators given by

$\mathcal{A}_{1}\psi_{k}=\frac{ik}{2}(k^{2}-1)\psi_{k}$

and

$\mathcal{A}_{2}\psi_{k}=-\frac{\left|k\right|}{2}(k^{2}-1)\psi_{k}$

for $\psi_{k}(e^{i\theta})=e^{ik\theta}$ for all $k\in\mathbb{Z}$.

######  Remark 24.

The Fourier multiplier operators $\mathcal{A}_{1},\mathcal{A}_{2}$ are defined
for complex-valued functions on $\mathbb{S}^{1}$, but in (82) we only use
their restrictions to real-valued functions. In the sequel we often implicitly
make the identification ${\mathbb{R}}^{2}\cong\mathbb{C}$.

Proposition 23 follows directly from the following Lemmas 26 and 27. First we
need a technical lemma.

######  Lemma 25.

Let $m$ satisfy (1) and denote by $m_{\varepsilon}=m\ast\rho_{\varepsilon}$.
Assume (11) holds true for some sequence $\varepsilon_{k}\rightarrow 0$ and
$1\leq p<\infty$. Then for any $\mathcal{Q}\in
C^{0}\left(\overline{B}_{1}\right)$, upon extraction of a subsequence, we have

$\lim_{k\to\infty}\int_{\Omega}\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\zeta\,dx=\int_{\Omega}\mathcal{Q}\left(m\right)\operatorname{div}\Sigma_{j}\left(m\right)\zeta\,dx\quad\text{
for all }\zeta\in C^{\infty}_{c}(\Omega),j=1,2,$ (83)

where $\Sigma_{1}$ and $\Sigma_{2}$ are given in (7) and (8), respectively.

###### Proof.

From equations (25) and (26) in [LLP20] we have

$\operatorname{div}\Sigma_{1}\left(m_{\varepsilon}\right)=\left(\partial_{1}m_{\varepsilon
2}+\partial_{2}m_{\varepsilon
1}\right)\left(1-\left|m_{\varepsilon}\right|^{2}\right),\quad\operatorname{div}\Sigma_{2}\left(m_{\varepsilon}\right)=\left(\partial_{2}m_{\varepsilon
2}-\partial_{1}m_{\varepsilon
1}\right)\left(1-\left|m_{\varepsilon}\right|^{2}\right).$

It follows from [LLP20, Lemma 9] that
$\left|\operatorname{div}\Sigma_{j}\left(m_{\varepsilon}\right)(x)\right|\lesssim\mathcal{P}^{\varepsilon}_{m}(x)$
for all $x\in\Omega$, $0<\varepsilon<\mathrm{dist}\\{x,\partial\Omega\\}$ and
$j=1,2$. Given $U\subset\subset\Omega$, by assumption (11), we know in
particular that $\\{\mathcal{P}^{\varepsilon_{k}}_{m}\\}\subset L^{1}(U)$ is a
bounded sequence that converges weakly to $\mathcal{P}_{m}\in L^{1}(U)$. By
the Dunford-Pettis Theorem (see [AFP00, Theorem 1.38]), the sequence
$\\{\mathcal{P}^{\varepsilon_{k}}_{m}\\}$ is equiintegrable in $U$. By [AFP00,
Proposition 1.27], it is clear that
$\\{\operatorname{div}\Sigma_{j}(m_{\varepsilon_{k}})\\}\subset L^{1}(U)$ is
also bounded and equiintegrable in $U$, and thus by the Dunford-Pettis Theorem
again, upon extraction of a subsequence which is not relabeled, it converges
weakly to some $W_{j}\in L^{1}(U)$. We claim that
$W_{j}=\operatorname{div}\Sigma_{j}(m)$. Indeed, by Lemma 10 and Remark 11, we
know $\operatorname{div}\Sigma_{j}(m)\in L^{p}_{{\rm loc}}(\Omega)$. Further,
we have

$\displaystyle-\int_{U}\Sigma_{j}\left(m\right)\cdot\nabla\zeta\,dx$
$\displaystyle=\lim_{k\rightarrow\infty}-\int_{\Omega}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\cdot\nabla\zeta\,dx$
$\displaystyle=\lim_{k\to\infty}\int_{\Omega}\operatorname{div}\Sigma_{j}(m_{\varepsilon_{k}})\zeta\,dx=\int_{\Omega}W_{j}\,\zeta\,dx\quad\text{
for all }\zeta\in C^{\infty}_{c}(U),$

which shows $W_{j}=\operatorname{div}\Sigma_{j}(m)$, $j=1,2$. Therefore we
deduce that

$\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\rightharpoonup\operatorname{div}\Sigma_{j}\left(m\right)\quad\text{
in }L^{1}_{{\rm loc}}(\Omega)\text{ for }j=1,2.$ (84)

Next we establish (83). As $\mathcal{Q}$ is continuous and, upon extraction of
another subsequence (not relabeled), $m_{\varepsilon_{k}}\to m$ a.e., it
follows that $\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\to\mathcal{Q}(m)$
a.e. in $\Omega$. Given $U\subset\subset\Omega$, by Egorov’s theorem there
exists a sequence of subsets $U_{l}\subset U$ such that $\left|U\setminus
U_{l}\right|\to 0$ as $l\to\infty$ and

$\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\to\mathcal{Q}(m)\quad\text{ in
}L^{\infty}(U_{l})\text{ as }k\rightarrow\infty\text{ for all }l.$ (85)

So for fixed $l$, it follows from (84) and (85) that

$\lim_{k\rightarrow\infty}\int_{U_{l}}\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\zeta\,dx=\int_{U_{l}}\mathcal{Q}\left(m\right)\operatorname{div}\Sigma_{j}\left(m\right)\zeta\,dx\quad\text{
for all }\zeta\in C^{\infty}_{c}(U).$ (86)

Now we have

$\displaystyle\left|\int_{U}\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\zeta\,dx-\int_{U}\mathcal{Q}\left(m\right)\operatorname{div}\Sigma_{j}\left(m\right)\zeta\,dx\right|$
$\displaystyle\quad\leq\left|\int_{U_{l}}\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\zeta\,dx-\int_{U_{l}}\mathcal{Q}\left(m\right)\operatorname{div}\Sigma_{j}\left(m\right)\zeta\,dx\right|$
$\displaystyle\quad\quad+\left|\int_{U\setminus
U_{l}}\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\zeta\,dx\right|+\left|\int_{U\setminus
U_{l}}\mathcal{Q}\left(m\right)\operatorname{div}\Sigma_{j}\left(m\right)\zeta\,dx\right|$
$\displaystyle\quad\leq\left|\int_{U_{l}}\mathcal{Q}\left(m_{\varepsilon_{k}}\right)\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\zeta\,dx-\int_{U_{l}}\mathcal{Q}\left(m\right)\operatorname{div}\Sigma_{j}\left(m\right)\zeta\,dx\right|$
$\displaystyle\quad\quad+C\int_{U\setminus
U_{l}}\left|\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\right|\,dx+C\int_{U\setminus
U_{l}}\left|\operatorname{div}\Sigma_{j}\left(m\right)\right|\,dx.$ (87)

By equiintegrability of
$\\{\operatorname{div}\Sigma_{j}(m_{\varepsilon_{k}})\\}$ and
$\left|U\setminus U_{l}\right|\to 0$, we know

$\lim_{l\to\infty}\sup_{k}\int_{U\setminus
U_{l}}\left|\operatorname{div}\Sigma_{j}\left(m_{\varepsilon_{k}}\right)\right|\,dx=0.$
(88)

Therefore, by first taking the limit $l\to\infty$ and then taking the limit
$k\to\infty$ in (5.1), we readily deduce (83) from (88) and (86). ∎

######  Lemma 26.

Let $m$ satisfy (1). Assume (11) holds true for some sequence
$\varepsilon_{k}\rightarrow 0$ and $1\leq p<\infty$. Given any $\varphi\in
C^{3}(\overline{B}_{1})$ satisfying $\Delta\varphi=0$, the harmonic entropy
$\Phi^{\varphi}$ defined in (81) satisfies
$\operatorname{div}\Phi^{\varphi}(m)\in L^{p}_{{\rm loc}}(\Omega)$. Further it
explicitly holds

$\operatorname{div}\Phi^{\varphi}(m)=A^{\varphi}_{1}(m)e^{i2\theta}\cdot\operatorname{div}\Sigma(m)+A^{\varphi}_{2}(m)ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)\quad\text{
a.e. in }\Omega,$ (89)

where $\theta:\Omega\to[0,2\pi)$ satisfies $m(x)=e^{i\theta(x)}$ a.e. in
$\Omega$, and $A_{1}^{\varphi},A_{2}^{\varphi}\in C^{0}(\overline{B}_{1})$ are
given by

$\displaystyle A^{\varphi}_{1}(z)$
$\displaystyle:=\left(z_{1}^{2}-z_{2}^{2}\right)\left(\frac{3}{2}\partial_{12}\varphi(z)-\frac{z_{1}}{2}\partial_{222}\varphi(z)+\frac{z_{2}}{2}\partial_{122}\varphi(z)\right)$
$\displaystyle\quad\quad+2z_{1}z_{2}\left(-\frac{3}{2}\partial_{11}\varphi(z)+\frac{z_{1}}{2}\partial_{122}\varphi(z)-\frac{z_{2}}{2}\partial_{112}\varphi(z)\right)$
(90)

and

$\displaystyle A^{\varphi}_{2}(z)$
$\displaystyle:=-2z_{1}z_{2}\left(\frac{3}{2}\partial_{12}\varphi(z)-\frac{z_{1}}{2}\partial_{222}\varphi(z)+\frac{z_{2}}{2}\partial_{122}\varphi(z)\right)$
$\displaystyle\quad\quad+\left(z_{1}^{2}-z_{2}^{2}\right)\left(-\frac{3}{2}\partial_{11}\varphi(z)+\frac{z_{1}}{2}\partial_{122}\varphi(z)-\frac{z_{2}}{2}\partial_{112}\varphi(z)\right).$
(91)

###### Proof.

Let $m_{\varepsilon}=m\ast\rho_{\varepsilon}$. By [LLP20, Lemma 18] we have
that

$\operatorname{div}\Phi^{\varphi}(m_{\varepsilon})=\operatorname{div}\left(\left(|m_{\varepsilon}|^{2}-1\right)B(m_{\varepsilon})\right)+\partial_{2}B_{1}(m_{\varepsilon})\operatorname{div}\Sigma_{1}(m_{\varepsilon})-\partial_{1}B_{1}(m_{\varepsilon})\operatorname{div}\Sigma_{2}(m_{\varepsilon}),$
(92)

where $B=(B_{1},B_{2})$ and

$\displaystyle B_{1}(z)$
$\displaystyle=\partial_{1}\varphi(z)-\frac{1}{2}z_{1}\partial_{22}\varphi(z)+\frac{1}{2}z_{2}\partial_{12}\varphi(z),$
$\displaystyle B_{2}(z)$
$\displaystyle=\partial_{2}\varphi(z)-\frac{1}{2}z_{2}\partial_{11}\varphi(z)+\frac{1}{2}z_{1}\partial_{12}\varphi(z).$

So, using $\Delta\varphi=0$, we have

$\displaystyle\partial_{1}B_{1}(z)$
$\displaystyle=\partial_{11}\varphi(z)-\frac{1}{2}\partial_{22}\varphi(z)-\frac{1}{2}z_{1}\partial_{122}\varphi(z)+\frac{1}{2}z_{2}\partial_{112}\varphi(z)$
$\displaystyle=\frac{3}{2}\partial_{11}\varphi(z)-\frac{1}{2}z_{1}\partial_{122}\varphi(z)+\frac{1}{2}z_{2}\partial_{112}\varphi(z),$
$\displaystyle\partial_{2}B_{1}(z)$
$\displaystyle=\partial_{12}\varphi(z)-\frac{1}{2}z_{1}\partial_{222}\varphi(z)+\frac{1}{2}\partial_{12}\varphi(z)+\frac{1}{2}z_{2}\partial_{122}\varphi(z)$
$\displaystyle=\frac{3}{2}\partial_{12}\varphi(z)-\frac{1}{2}z_{1}\partial_{222}\varphi(z)+\frac{1}{2}z_{2}\partial_{122}\varphi(z).$

Thus we can rewrite (92) as

$\displaystyle\operatorname{div}\Phi^{\varphi}(m_{\varepsilon})$
$\displaystyle=Q^{\varphi}_{1}(m_{\varepsilon})\operatorname{div}\Sigma_{1}(m_{\varepsilon})+Q^{\varphi}_{2}(m_{\varepsilon})\operatorname{div}\Sigma_{2}(m_{\varepsilon})-\operatorname{div}\left(\left(1-{\left|m_{\varepsilon}\right|}^{2}\right)B(m_{\varepsilon})\right),$
(93)

where

$Q^{\varphi}_{1}(z)=\frac{3}{2}\partial_{12}\varphi(z)-\frac{1}{2}z_{1}\partial_{222}\varphi(z)+\frac{1}{2}z_{2}\partial_{122}\varphi(z)$
(94)

and

$Q^{\varphi}_{2}(z)=-\frac{3}{2}\partial_{11}\varphi(z)+\frac{1}{2}z_{1}\partial_{122}\varphi(z)-\frac{1}{2}z_{2}\partial_{112}\varphi(z).$
(95)

Hence for any test function $\zeta\in C_{c}^{1}(\Omega)$ it holds

$\displaystyle-\int_{\Omega}\Phi^{\varphi}(m)\cdot\nabla\zeta\,dx=\lim_{\varepsilon\to
0}\int_{\Omega}\operatorname{div}\Phi^{\varphi}(m_{\varepsilon})\zeta\,dx$
$\displaystyle\quad\quad\quad\quad\quad\overset{(\ref{xxeqa12})}{=}\lim_{\varepsilon\to
0}\int_{\Omega}(1-{\left|m_{\varepsilon}\right|}^{2})B(m_{\varepsilon})\cdot\nabla\zeta\,dx+\lim_{\varepsilon\to
0}\sum_{j=1,2}\int_{\Omega}\operatorname{div}\Sigma_{j}(m_{\varepsilon})Q^{\varphi}_{j}(m_{\varepsilon})\zeta\,dx.$
(96)

The first term in the right-hand side is zero by the Dominated Convergence
Theorem. From Lemma 25 we also have

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{\Omega}\operatorname{div}\Sigma_{j}(m_{\varepsilon})Q^{\varphi}_{j}(m_{\varepsilon})\zeta\,dx\overset{(\ref{eqfv30})}{=}\int_{\Omega}\operatorname{div}\Sigma_{j}(m)Q^{\varphi}_{j}(m)\zeta\,dx\quad\text{
for }j=1,2.$

Plugging this into (5.1) we obtain

$-\int_{\Omega}\Phi^{\varphi}(m)\cdot\nabla\zeta\,dx=\int_{\Omega}\operatorname{div}\Sigma_{1}(m)Q^{\varphi}_{1}(m)\zeta\,dx+\int_{\Omega}\operatorname{div}\Sigma_{2}(m)Q^{\varphi}_{2}(m)\zeta\,dx\quad\text{
for all }\zeta\in C^{1}_{c}(\Omega).$

By Lemma 10 and Remark 11,
$\operatorname{div}\Phi^{\varphi}(m),\operatorname{div}\Sigma_{j}(m)\in
L^{p}_{{\rm loc}}(\Omega)$. Thus we infer that

$\operatorname{div}\Phi^{\varphi}(m)=Q^{\varphi}_{1}(m)\operatorname{div}\Sigma_{1}(m)+Q^{\varphi}_{2}(m)\operatorname{div}\Sigma_{2}(m)\quad\text{
a.e. in }\Omega.$ (97)

Letting $Q^{\varphi}(m):=\left(\begin{array}[]{c}Q_{1}^{\varphi}(m)\\\
Q_{2}^{\varphi}(m)\end{array}\right)$ and
$\mathcal{R}(\theta):=\left(\begin{array}[]{cc}\cos(\theta)&\sin(\theta)\\\
-\sin(\theta)&\cos(\theta)\end{array}\right)$, it follows that

$\displaystyle\operatorname{div}\Phi^{\varphi}(m)$
$\displaystyle\overset{(\ref{eqggbba1})}{=}Q^{\varphi}(m)\cdot\operatorname{div}\Sigma(m)$
$\displaystyle=\left(\mathcal{R}(2\theta)Q^{\varphi}(m)\right)\cdot\left(\mathcal{R}(2\theta)\operatorname{div}\Sigma(m)\right)$
$\displaystyle=\left(\begin{array}[]{c}e^{i2\theta}\cdot Q^{\varphi}(m)\\\
ie^{i2\theta}\cdot
Q^{\varphi}(m)\end{array}\right)\cdot\left(\begin{array}[]{c}e^{i2\theta}\cdot\operatorname{div}\Sigma(m)\\\
ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)\end{array}\right)$ (102)
$\displaystyle=\left(e^{i2\theta}\cdot
Q^{\varphi}(m)\right)\left(e^{i2\theta}\cdot\operatorname{div}\Sigma(m)\right)+\left(ie^{i2\theta}\cdot
Q^{\varphi}(m)\right)\left(ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)\right).$
(103)

Note that

$e^{i2\theta}\cdot
Q^{\varphi}(m)=\left(\begin{array}[]{c}m_{1}^{2}-m_{2}^{2}\\\
2m_{1}m_{2}\end{array}\right)\cdot Q^{\varphi}(m),\quad ie^{i2\theta}\cdot
Q^{\varphi}(m)=\left(\begin{array}[]{c}-2m_{1}m_{2}\\\
m_{1}^{2}-m_{2}^{2}\end{array}\right)\cdot Q^{\varphi}(m).$ (104)

Now putting (94), (95), (5.1) and (104) together gives (89)–(26). ∎

######  Lemma 27.

Define the operators $\mathcal{A}_{j}:C^{4}(\mathbb{S}^{1})\to
C^{0}(\mathbb{S}^{1}),j=1,2$ by

$\mathcal{A}_{j}\psi:=A^{E\psi}_{j\,\lfloor\mathbb{S}^{1}}\quad\text{ for all
}\psi\in C^{4}(\mathbb{S}^{1}),$ (105)

where $A_{j}$ are defined in (26)-(26). Further let
$\mathcal{B}_{1},\mathcal{B}_{2}:C^{4}(\mathbb{S}^{1})\to
C^{0}(\mathbb{S}^{1})$ be the Fourier multiplier operators characterized by

$\mathcal{B}_{1}\psi_{k}=\frac{ik}{2}(k^{2}-1)\psi_{k}\quad\text{ for all
}k\in\mathbb{Z}$ (106)

and

$\mathcal{B}_{2}\psi_{k}=-\frac{\left|k\right|}{2}(k^{2}-1)\psi_{k}\quad\text{
for all }k\in\mathbb{Z},$

where $\psi_{k}(e^{i\theta})=e^{ik\theta}$. Then we have

$\mathcal{A}_{j}\psi=\mathcal{B}_{j}\psi\quad\text{ for }j=1,2\text{ and for
all }\psi\in C^{4}(\mathbb{S}^{1}).$ (107)

######  Remark 28.

As $\varphi\mapsto A^{\varphi}_{j}$ is linear, the definition of
$\mathcal{A}_{j}$ in (105) can be naturally extended to complex-valued $\psi$,
and in particular $\psi_{k}$. We will make use of this implicitly in the
sequel.

###### Proof of Lemma 27.

Using almost exactly the same arguments as in Remark 22 one can show that
$\mathcal{A}_{j},\mathcal{B}_{j}$ are continuous operators from
$C^{4}(\mathbb{S}^{1})$ to $C^{0}(\mathbb{S}^{1})$. Thus, to establish (107),
it suffices to show $\mathcal{A}_{j}\psi_{k}=\mathcal{B}_{j}\psi_{k}$ for all
$k\in\mathbb{Z}$ and $j=1,2$ as the linear span of
$\\{\psi_{k}\\}_{k\in\mathbb{Z}}$ is dense in $C^{4}(\mathbb{S}^{1})$.

Let $\varphi_{k}=E\psi_{k}$. From (80) we have $\varphi_{k}(z)=z^{k}$ for
$k\geq 0$, and thus

$\partial_{1}\varphi_{k}=kz^{k-1}\quad\text{ and
}\quad\partial_{2}\varphi_{k}=ikz^{k-1}.$ (108)

So for $k\geq 3$ we have

$\displaystyle\frac{3}{2}\partial_{12}\varphi_{k}-\frac{z_{1}}{2}\partial_{222}\varphi_{k}+\frac{z_{2}}{2}\partial_{122}\varphi_{k}$
$\displaystyle\quad=\frac{3}{2}ik(k-1)z^{k-2}+\frac{z_{1}}{2}ik(k-1)(k-2)z^{k-3}-\frac{z_{2}}{2}k(k-1)(k-2)z^{k-3}$
$\displaystyle\quad=\frac{3}{2}ik(k-1)z^{k-2}+\frac{ik(k-1)(k-2)}{2}(z_{1}+iz_{2})z^{k-3}$
$\displaystyle\quad=\frac{ik(k-1)(k+1)}{2}z^{k-2}$

and

$\displaystyle-\frac{3}{2}\partial_{11}\varphi_{k}+\frac{z_{1}}{2}\partial_{122}\varphi_{k}-\frac{z_{2}}{2}\partial_{112}\varphi_{k}$
$\displaystyle\quad=-\frac{3}{2}k(k-1)z^{k-2}-\frac{z_{1}}{2}k(k-1)(k-2)z^{k-3}-\frac{z_{2}}{2}ik(k-1)(k-2)z^{k-3}$
$\displaystyle\quad=-\frac{3}{2}k(k-1)z^{k-2}-\frac{k(k-1)(k-2)}{2}(z_{1}+iz_{2})z^{k-3}$
$\displaystyle\quad=-\frac{k(k-1)(k+1)}{2}z^{k-2}.$

Hence, it follows from (26), (26) and the above computations that

$\displaystyle A_{1}^{\varphi_{k}}$
$\displaystyle=\left(z_{1}^{2}-z_{2}^{2}\right)\frac{ik(k-1)(k+1)}{2}z^{k-2}-2z_{1}z_{2}\frac{k(k-1)(k+1)}{2}z^{k-2}$
$\displaystyle=\frac{ik(k^{2}-1)}{2}\left(z_{1}^{2}-z_{2}^{2}+2iz_{1}z_{2}\right)z^{k-2}=\frac{ik(k^{2}-1)}{2}z^{k},$

and

$\displaystyle A_{2}^{\varphi_{k}}$
$\displaystyle=-2z_{1}z_{2}\frac{ik(k-1)(k+1)}{2}z^{k-2}-\left(z_{1}^{2}-z_{2}^{2}\right)\frac{k(k-1)(k+1)}{2}z^{k-2}$
$\displaystyle=-\frac{k(k^{2}-1)}{2}\left(z_{1}^{2}-z_{2}^{2}+2iz_{1}z_{2}\right)z^{k-2}=-\frac{k(k^{2}-1)}{2}z^{k}.$

Thus we have

$\mathcal{A}_{1}\psi_{k}=\frac{ik(k^{2}-1)}{2}\psi_{k}=\mathcal{B}_{1}\psi_{k},\quad\mathcal{A}_{2}\psi_{k}=-\frac{k(k^{2}-1)}{2}\psi_{k}=\mathcal{B}_{2}\psi_{k}\quad\text{
for all }k\geq 3.$

For $0\leq k\leq 2$, $\mathcal{A}_{j}\psi_{k}=\mathcal{B}_{j}\psi_{k}$ can be
checked directly. For $k<0$, note that $E\psi_{k}=\overline{E\psi_{-k}}$ and
thus
$A_{j}^{E\psi_{k}}=A_{j}^{\overline{E\psi_{-k}}}=\overline{A_{j}^{E\psi_{-k}}}$.
Using this and the formulas established for $k>0$, one can check directly that
$\mathcal{A}_{j}\psi_{k}=\mathcal{B}_{j}\psi_{k}$ for $k<0$. Thus we have
established $\mathcal{A}_{j}\psi_{k}=\mathcal{B}_{j}\psi_{k}$ for all
$k\in\mathbb{Z}$ and $j=1,2$. ∎

Now combining Lemmas 26 and 27 completes the proof of Proposition 23. Next we
show that the term $ie^{i2\theta}\cdot\operatorname{div}\Sigma(m)$ must indeed
vanish a.e. due to the specific structure of its coefficient
$\mathcal{A}_{2}\psi$, and thus the factorization formula (82) has only the
first term on the right-hand side. This constitutes the most important step
towards the proof of Theorem 5.

######  Proposition 29.

Let $m$ satisfy (1). Assume (11) holds true for some sequence
$\varepsilon_{k}\rightarrow 0$ and $1\leq p<\infty$. Given any $\psi\in
C^{4}(\mathbb{S}^{1})$, the harmonic entropy $\Phi^{E\psi}$ given by (81)
satisfies $\operatorname{div}\Phi^{E\psi}(m)\in L^{p}_{{\rm loc}}(\Omega)$ and
the formula

$\operatorname{div}\Phi^{E\psi}(m)=\mathcal{A}_{1}\psi(m)e^{i2\theta}\cdot\operatorname{div}\Sigma(m)\quad\text{
a.e. in }\Omega,$ (109)

where $\theta:\Omega\to[0,2\pi)$ satisfies $m(x)=e^{i\theta(x)}$ a.e. in
$\Omega$. Further, it explicitly holds

$\mathcal{A}_{1}\psi=-\frac{1}{2}\left(\psi^{(3)}+\psi^{\prime}\right)\quad\text{
for all }\psi\in C^{4}(\mathbb{S}).$ (110)

###### Proof.

Let $\widetilde{\mathcal{B}}_{1}:C^{3}(\mathbb{S}^{1})\to
C^{0}(\mathbb{S}^{1})$ be the continuous linear operator defined by
$\widetilde{\mathcal{B}}_{1}\psi:=-\frac{1}{2}\left(\psi^{(3)}+\psi^{\prime}\right)$
for all $\psi\in C^{3}(\mathbb{S}^{1})$. From (107) and (106), it is clear
that
$\mathcal{A}_{1}\psi_{k}=\mathcal{B}_{1}\psi_{k}=\widetilde{\mathcal{B}}_{1}\psi_{k}$
for all $k\in\mathbb{Z}$. As $\mathcal{A}_{1}$ and
$\widetilde{\mathcal{B}}_{1}$ are both continuous from $C^{4}(\mathbb{S}^{1})$
to $C^{0}(\mathbb{S}^{1})$ and the linear span of
$\\{\psi_{k}\\}_{k\in\mathbb{Z}}$ is dense in $C^{4}(\mathbb{S}^{1})$, it
follows that (110) holds for all $\psi\in C^{4}(\mathbb{S})$.

Now comparing (109) with (82), it suffices to show
$ie^{i2\theta}\cdot\operatorname{div}\Sigma\left(m\right)=0$ a.e. in $\Omega$.
To this end, we argue in a very similar way to the proof of [LLP20, Theorem
1]. Let $\mathcal{X}\subset C^{4}(\mathbb{S}^{1})$ denote a countable dense
subset. Let $\mathcal{G}\subset\Omega$ be the set of all points $x\in\Omega$
at which $\mathcal{P}_{m}(x)<\infty$ and
$|\operatorname{div}\Sigma_{j}(m)(x)|<\infty$ for $j=1,2$, and both:

  * •

the explicit expression of $\operatorname{div}\Phi^{E\psi}(m)$ given by (82),

  * •

its control in terms of ${\|\Phi^{E\psi}\|}_{C^{2}(\mathbb{S}^{1})}$ given by
(17),

hold for all $\psi\in\mathcal{X}$. By Proposition 23 and Lemma 10, the set
$\mathcal{G}$ has full measure in $\Omega$. From (81) we have
$\Phi^{E\psi}(e^{i\theta})=\psi(e^{i\theta})e^{i\theta}+\psi^{\prime}(e^{i\theta})ie^{i\theta}$,
and thus
$\|\Phi^{E\psi}\|_{C^{2}(\mathbb{S}^{1})}\lesssim\|\psi\|_{C^{3}(\mathbb{S}^{1})}$.
Therefore the estimate (17) becomes

$\left|\operatorname{div}\Phi^{E\psi}(m)(x)\right|\lesssim\|\psi\|_{C^{3}\left(\mathbb{S}^{1}\right)}\mathcal{P}_{m}(x)\quad\text{
for all }x\in\mathcal{G},\psi\in\mathcal{X}.$ (111)

And we know from (110) that
$\|\mathcal{A}_{1}\psi\|_{C^{0}\left(\mathbb{S}^{1}\right)}\lesssim\|\psi\|_{C^{3}\left(\mathbb{S}^{1}\right)}$
for all $\psi\in C^{4}(\mathbb{S}^{1})$. Suppose for some
$x_{0}\in\mathcal{G}$ we have
$ie^{i2\theta(x_{0})}\cdot\operatorname{div}\Sigma\left(m\right)(x_{0})\not=0$,
then it follows from (82) and (111) that

$\displaystyle\left|\mathcal{A}_{2}\psi(m(x_{0}))\right|$
$\displaystyle\lesssim\left|ie^{i2\theta(x_{0})}\cdot\operatorname{div}\Sigma\left(m\right)(x_{0})\right|^{-1}\left(\mathcal{P}_{m}(x_{0})+\left|\operatorname{div}\Sigma(m)(x_{0})\right|\right)\|\psi\|_{C^{3}\left(\mathbb{S}^{1}\right)}$
$\displaystyle=C(m,x_{0})\|\psi\|_{C^{3}\left(\mathbb{S}^{1}\right)}\quad\text{
for all }\psi\in\mathcal{X}.$ (112)

As $\mathcal{X}\subset C^{4}(\mathbb{S}^{1})$ is dense in
$C^{4}(\mathbb{S}^{1})$ and both sides of (5.1) depend on $\psi$ continuously
in the $C^{4}$ topology, we deduce that

$\left|\mathcal{A}_{2}\psi(m(x_{0}))\right|\lesssim
C(m,x_{0})\|\psi\|_{C^{3}\left(\mathbb{S}^{1}\right)}\quad\text{ for all
}\psi\in C^{4}(\mathbb{S}^{1}).$

The remainder of the argument follows very closely the proof of [LLP20,
Theorem 1] and thus we sketch it somewhat briefly. We identify functions on
$\mathbb{S}^{1}$ with $2\pi$-periodic functions on $\mathbb{R}$, and
$m(x_{0})\in\mathbb{S}^{1}$ with its argument
$\theta_{0}\in\mathbb{R}/2\pi\mathbb{Z}$. So we have
$\left|\mathcal{A}_{2}\psi(\theta_{0})\right|\lesssim
C(m,x_{0})\|\psi\|_{C^{3}\left(\mathbb{S}^{1}\right)}$ for all $\psi\in
C^{4}(\mathbb{S}^{1})$. This estimate turns into the stronger estimate

$\lVert\mathcal{A}_{2}\psi\rVert_{L^{\infty}(\mathbb{S}^{1})}\lesssim
C(m,x_{0})\|\psi\|_{C^{3}(\mathbb{S}^{1})}\quad\text{ for all }\psi\in
C^{4}(\mathbb{S}^{1})$ (113)

because the multiplier operator $\mathcal{A}_{2}$ commutes with translations
of the variable. Now decomposing
$\mathcal{A}_{2}=\mathcal{A}_{2}^{1}+\mathcal{A}_{2}^{2}$, where
$\mathcal{A}_{2}^{1}\psi_{k}=-\frac{|k|^{3}}{2}\psi_{k}$ and
$\mathcal{A}_{2}^{2}\psi_{k}=\frac{|k|}{2}\psi_{k}$ for all $k\in\mathbb{Z}$,
a simple argument shows that $\mathcal{A}_{2}^{2}$ can be extended to a
continuous linear operator from $C^{3}(\mathbb{S}^{1})$ to
$C^{0}(\mathbb{S}^{1})$. Thus the estimate (113) gives

$\|\mathcal{A}_{2}^{1}\psi\|_{L^{\infty}(\mathbb{S}^{1})}\lesssim\widetilde{C}\|\psi\|_{C^{3}(\mathbb{S}^{1})}\quad\text{
for all }\psi\in C^{4}(\mathbb{S}^{1}).$ (114)

This estimate is of exactly the same type as the estimate (51) in [LLP20],
which is shown to lead to a contradiction because of well known properties of
the Hilbert transform; see the arguments below equation (51) in [LLP20] for
the details. The contradiction from the estimate (114) shows that
$ie^{i2\theta(x)}\cdot\operatorname{div}\Sigma\left(m\right)(x)=0$ for all
$x\in\mathcal{G}$, and thus establishes (109). ∎

###  5.2 Harmonic extensions for $\Phi_{f}$

In this subsection we construct specific extensions to $\overline{B}_{1}$ of
the entropies $\Phi_{f}$ defined by (23) on $\mathbb{S}^{1}$. For $f$
sufficiently regular, these extensions are harmonic entropies and enjoy the
nice formula (109) established in Proposition 29. This fact coupled with the
special structure of the entropies $\Phi_{f}$ leads to the explicit structure
of the kinetic measure $\sigma$ in (13).

Given $f\in L^{2}({\mathbb{R}}/2\pi\mathbb{Z})$, let $a_{k}(f)$ and $b_{k}(f)$
denote the standard Fourier coefficients of $f$, i.e.

$a_{k}(f)=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\cos(kt)\,dt,\qquad
b_{k}(f)=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\sin(kt)\,dt\qquad\text{for }k\geq
1.$

We define $\xi_{f}\colon\overline{B}_{1}\to{\mathbb{R}}$ by

$\xi_{f}=\sum_{k\geq
1}\frac{(-1)^{k}}{k(1-2k)(1+2k)}\left(-a_{2k}(f)\varphi^{1}_{2k}+b_{2k}(f)\varphi^{2}_{2k}\right),$
(115)

where $\varphi_{k}=\varphi_{k}^{2}+i\varphi_{k}^{1}$ and
$\varphi_{k}(z)=z^{k}$ for $z\in\mathbb{C}$ and $k\geq 0$. So explicitly in
polar coordinates we have

$\varphi^{1}_{k}=r^{k}\sin(k\theta),\quad\varphi^{2}_{k}=r^{k}\cos(k\theta).$

Note that this choice of indices for $\varphi_{k}^{1}$ and $\varphi_{k}^{2}$
is apparently natural from some of the following computations; see e.g. (A)
and (A).

######  Proposition 30.

For all $f\in L^{2}({\mathbb{R}}/2\pi\mathbb{Z})$, the function $\xi_{f}$
given by (115) belongs to $C^{2}(\overline{B}_{1})$ and solves
$\Delta\xi_{f}=0$ in $B_{1}$. The entropy given by

$\Phi^{\xi_{f}}(z)=\xi_{f}(z)z+((iz)\cdot\nabla\xi_{f}(z))iz\qquad\forall
z\in\overline{B}_{1}$ (116)

extends the entropy $\Phi_{f}$ defined by (23) on $\mathbb{S}^{1}=\partial
B_{1}$, up to a linear term:

$\Phi^{\xi_{f}}(z)=\Phi_{f}(z)+\left(2\sum_{k\geq
2}\frac{b_{k}(f)}{k}\right)z\qquad\forall z\in\partial B_{1}.$ (117)

If moreover $f\in H^{\ell}({\mathbb{R}}/2\pi\mathbb{Z})$ for some
$\ell\in\mathbb{N}$, then the series (115) defining $\xi_{f}$ converges in
$C^{\ell+2}(\overline{B}_{1})$.

The proof of Proposition 30 follows from direct calculations showing its
validity for the Fourier modes $f(t)=\cos(kt),\sin(kt)$, and from standard
estimates on Fourier coefficients ensuring that the claimed convergence and
regularity hold.

###### Proof of Proposition 30.

First notice that since $z^{k}=\varphi^{2}_{k}(z)+i\varphi^{1}_{k}(z)$ for
$z\in\mathbb{C}$, we have

$\|\nabla^{\ell}\varphi^{j}_{k}\|_{C^{0}\left(\overline{B}_{1}\right)}\lesssim
k^{\ell}\qquad\forall k\geq 1,$ (118)

and since by Parseval’s identity the Fourier coefficients $a_{k}(f)$,
$b_{k}(f)$ belong to $\ell^{2}$, we have

$\left\|\frac{-a_{2k}(f)\varphi^{1}_{2k}+b_{2k}(f)\varphi^{2}_{2k}}{k(1-2k)(1+2k)}\right\|_{C^{2}(\overline{B}_{1})}\overset{\eqref{xxeqa1}}{\lesssim}\frac{1}{k}({\left|a_{2k}(f)\right|}+{\left|b_{2k}(f)\right|})\in\ell^{1}.$

Hence the series (115) converges in $C^{2}(\overline{B}_{1})$, and using
Hölder’s inequality for the following first inequality we obtain

${\|\xi_{f}\|}_{C^{2}(\overline{B}_{1})}\lesssim{\|(a_{k}(f))\|}_{\ell^{2}}+{\|(b_{k}(f))\|}_{\ell^{2}}\lesssim{\|f\|}_{L^{2}({\mathbb{R}}/2\pi\mathbb{Z})}.$
(119)

As all terms of the series (115) are harmonic functions, it follows that
$\Delta\xi_{f}=0$ in $B_{1}$.

If $f\in H^{\ell}({\mathbb{R}}/2\pi\mathbb{Z})$ then the sequences
$(k^{\ell}a_{k}(f))$, $(k^{\ell}b_{k}(f))$ belong to $\ell^{2}$ since they
are, up to constants, the Fourier coefficients of $f^{(\ell)}$. Thus

$\left\|\frac{-a_{2k}(f)\varphi^{1}_{2k}+b_{2k}(f)\varphi^{2}_{2k}}{k(1-2k)(1+2k)}\right\|_{C^{\ell+2}(\overline{B}_{1})}\overset{\eqref{xxeqa1}}{\lesssim}\frac{1}{k}(k^{\ell}{\left|a_{2k}(f)\right|}+k^{\ell}{\left|b_{2k}(f)\right|})\lesssim\lVert
f^{(\ell)}\rVert_{L^{2}(\mathbb{R}/2\pi\mathbb{Z})},$ (120)

and the series (115) converges in $C^{\ell+2}(\overline{B}_{1})$.

It remains to prove that the harmonic entropy $\Phi^{\xi_{f}}$ indeed extends
$\Phi_{f}$, i.e. we have (117). Since we have just shown in (119) that the
linear map $f\mapsto\xi_{f}$ is continuous
$L^{2}({\mathbb{R}}/2\pi\mathbb{Z})\to C^{2}(\overline{B}_{1})$, we have in
particular (recall (116)) that for any $z\in\mathbb{S}^{1}$ the linear map
$f\mapsto\Phi^{\xi_{f}}(z)$ is continuous in
$L^{2}({\mathbb{R}}/2\pi\mathbb{Z})$. By the construction of $\Phi_{f}$
(recalling (23), (24), (25)) the two terms in the right-hand side of (117)
depend also linearly and continuously on $f\in
L^{2}({\mathbb{R}}/2\pi\mathbb{Z})$. Therefore it is sufficient to establish
(117) for $f(t)=\cos(kt),\sin(kt)$ for all $k\geq 0$. For $k\geq 2$ this
follows from lengthy but direct computations, to be found in Appendix A. And
for $k\in\\{0,1\\}$ it can be checked directly that both sides of (117)
vanish. ∎

######  Lemma 31.

Let $f\in C^{2}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ and $\xi_{f}$ be the
function defined by (115). Then

$\mathcal{A}_{1}\xi_{f\lfloor\mathbb{S}^{1}}(e^{i\theta})=\frac{1}{2}f\left(\theta+\frac{\pi}{2}\right)+\frac{1}{2}f\left(\theta-\frac{\pi}{2}\right)-\langle
f,1\rangle,$ (121)

where recall that $\langle\cdot,\cdot\rangle$ is the inner product on
$L^{2}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$, i.e. $\langle
f,g\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}f\,g\,dt$.

###### Proof.

Given $f\in C^{2}({\mathbb{R}}/2\pi\mathbb{Z})$, it follows from the
continuity of $\mathcal{A}_{1}$ from $C^{4}(\mathbb{S}^{1})$ to
$C^{0}(\mathbb{S}^{1})$ (recalling Lemma 27) and the estimate (120) that

$\lVert\mathcal{A}_{1}\xi_{f\lfloor\mathbb{S}^{1}}\rVert_{C^{0}(\mathbb{S}^{1})}\lesssim\lVert\xi_{f\lfloor\mathbb{S}^{1}}\rVert_{C^{4}(\mathbb{S}^{1})}\overset{(\ref{iteq2})}{\lesssim}\lVert
f\rVert_{C^{2}(\mathbb{R}/2\pi\mathbb{Z})}.$

It is clear that the right-hand side of (121) also depends continuously on $f$
in the $C^{2}$ topology, and both sides of (121) are linear in $f$. Thus it
suffices to show (121) for $f=f^{j}_{k}$, $j=1,2$, $k\geq 0$, where

$f^{1}_{k}(t)=\cos(kt),\quad f^{2}_{k}(t)=\sin(kt),$ (122)

as the linear span of $\\{f^{j}_{k}\\}$ is dense in
$C^{2}({\mathbb{R}}/2\pi\mathbb{Z})$.

Note that since $\psi_{2k}=\varphi_{2k}^{2}+i\varphi_{2k}^{1}$ and

$\mathcal{A}_{1}\psi_{2k}\overset{(\ref{eqggbba6}),(\ref{AequalB})}{=}\frac{i2k(4k^{2}-1)}{2}\psi_{2k}=k(4k^{2}-1)\left(-\varphi_{2k}^{1}+i\varphi_{2k}^{2}\right),$

we have

$\mathcal{A}_{1}\varphi_{2k}^{2}=-k(4k^{2}-1)\varphi_{2k}^{1}\quad\text{ and
}\quad\mathcal{A}_{1}\varphi_{2k}^{1}=k(4k^{2}-1)\varphi_{2k}^{2}.$ (123)

For $f=f_{2k+1}^{j}(t)$, $j=1,2$, $k\geq 0$ and $f\equiv 1$, it is clear that
$\xi_{f}\overset{(\ref{eq:xif})}{=}0$, and thus both sides of (121) vanish.
From (115), for $k\geq 1$, we have
$\xi_{f_{2k}^{1}}=\frac{(-1)^{k+1}}{k(1-4k^{2})}\varphi^{1}_{2k}$ and
$\xi_{f_{2k}^{2}}=\frac{(-1)^{k}}{k(1-4k^{2})}\varphi^{2}_{2k}$. Thus it
follows from (123) that

$\displaystyle\mathcal{A}_{1}\xi_{f_{2k}^{1}\lfloor\mathbb{S}^{1}}(e^{i\theta})$
$\displaystyle=\frac{(-1)^{k+1}}{k(1-4k^{2})}\cdot
k(4k^{2}-1)\varphi^{2}_{2k}(e^{i\theta})=(-1)^{k}\cos(2k\theta)$
$\displaystyle=\frac{1}{2}f_{2k}^{1}\left(\theta+\frac{\pi}{2}\right)+\frac{1}{2}f_{2k}^{1}\left(\theta-\frac{\pi}{2}\right),$

and

$\displaystyle\mathcal{A}_{1}\xi_{f_{2k}^{2}\lfloor\mathbb{S}^{1}}(e^{i\theta})$
$\displaystyle=\frac{(-1)^{k+1}}{k(1-4k^{2})}\cdot
k(4k^{2}-1)\varphi^{1}_{2k}(e^{i\theta})=(-1)^{k}\sin(2k\theta)$
$\displaystyle=\frac{1}{2}f_{2k}^{2}\left(\theta+\frac{\pi}{2}\right)+\frac{1}{2}f_{2k}^{2}\left(\theta-\frac{\pi}{2}\right).$

Thus we have established (121) for all Fourier modes and this concludes the
proof of the lemma. ∎

###  5.3 Proof of Theorem 5 completed

###### Proof of Theorem 5.

First note that by Lemma 10 and Remark 11 we have
$\operatorname{div}\Phi_{f}(m),\operatorname{div}\Sigma_{j}(m)\in L^{p}_{{\rm
loc}}(\Omega)$. For any $f\in C^{2}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$, as
harmonic extension is unique, we deduce from Propositions 29, 30 and Lemma 31
that

$\displaystyle\operatorname{div}\Phi_{f}(m)(x)$
$\displaystyle\overset{(\ref{eq:harmext})}{=}\operatorname{div}\Phi^{\xi_{f}}(m)(x)$
$\displaystyle\overset{(\ref{eqmma5}),(\ref{eqoolla1})}{=}\left(\frac{1}{2}f\left(\theta(x)+\frac{\pi}{2}\right)+\frac{1}{2}f\left(\theta(x)-\frac{\pi}{2}\right)-\langle
f,1\rangle\right)e^{2i\theta(x)}\cdot\operatorname{div}\Sigma(m)(x)$

for a.e. $x\in\Omega$. Integrating by parts gives

$\displaystyle-\int_{\Omega}\Phi_{f}(m)(x)\cdot\nabla\zeta(x)\,dx$
$\displaystyle\quad\quad=\int_{\Omega}\left(\frac{1}{2}f\left(\theta(x)+\frac{\pi}{2}\right)+\frac{1}{2}f\left(\theta(x)-\frac{\pi}{2}\right)-\langle
f,1\rangle\right)e^{2i\theta(x)}\cdot\operatorname{div}\Sigma(m)(x)\,\zeta(x)\,dx$
(124)

for all $f\in C^{2}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ and all $\zeta\in
C^{\infty}_{c}(\Omega)$. For any fixed $\zeta\in C^{\infty}_{c}(\Omega)$, the
left- and right-hand sides of (5.3) depend continuously and linearly on $f$ in
the $C^{0}$ topology (recalling again the construction of $\Phi_{f}$ from
(23), (24), (25)). By density of $C^{2}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$
in $C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ it follows that (5.3) holds
true for all $f\in C^{0}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$ and all
$\zeta\in C_{c}^{\infty}(\Omega)$. As
$\operatorname{div}\Phi_{f}(m),\operatorname{div}\Sigma(m)\in L^{p}_{{\rm
loc}}(\Omega)$, we readily deduce (12) from (5.3).

Finally, to establish (13), note that the existence of $\sigma$ and its
disintegration $\sigma=\mathcal{L}^{2}\otimes\sigma_{x}$ follow from Lemma 14
in the case $1<p<\infty$. When $p=1$, by Lemma 10, the estimate (17) holds.
Given $A\subset\subset\Omega$, for any pairwise disjoint decomposition
$A=\bigcup_{\alpha}A_{\alpha}$ and any choice of $\\{\Phi_{\alpha}\\}\subset
ENT$ with $\|\Phi_{\alpha}\|_{C^{2}(\mathbb{S}^{1})}\leq 1$, it follows from
(17) that
$\sum_{\alpha}\int_{A_{\alpha}}\left|\operatorname{div}\Phi_{\alpha}(m)\right|dx\lesssim\int_{A}\mathcal{P}_{m}\,dx$.
Thus it follows from (32) that $\mu_{m}(A)\leq\int_{A}\mathcal{P}_{m}\,dx$ for
all $A\subset\subset\Omega$. This implies that $\mu_{m}$ is absolutely
continuous with respect to the Lebesgue measure and its density is bounded
above by a constant multiple of $\mathcal{P}_{m}\in L^{1}_{{\rm
loc}}(\Omega)$, and thus $\mu_{m}\in L^{1}_{{\rm loc}}(\Omega)$. The existence
of $\sigma$ and its disintegration $\sigma=\mathcal{L}^{2}\otimes\sigma_{x}$
then follow from Lemmas 36 and 32. Now putting (135) and (5) together gives
the equivalence of (12) and (13). ∎

######  Lemma 32.

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1). Assume $\mu_{m}\in L^{p}_{{\rm
loc}}(\Omega)$ for some $1\leq p<\infty$, where $\mu_{m}$ is given in (32).
Then there exists a family of measures
$\left\\{\sigma_{x}\right\\}\subset\mathcal{M}\left(\mathbb{R}/2\pi\mathbb{Z}\right)$
for $\mathcal{L}^{2}$-a.e. $x\in\Omega$ satisfying (5) and (6).

The above result is a fairly standard application of disintegration/slicing
for the measure $\sigma$ constructed in Lemma 36 (or Lemma 14 when $p>1$ and
in this case the disintegration of $\sigma$ is already included in Lemma 14).
By showing that the push-forward of $\sigma$ into $\Omega$ via the projection
onto $\Omega$ is absolutely continuous under the assumption $\mu_{m}\in
L^{p}_{{\rm loc}}$, it could also be deduced from [AFP00, Theorem 2.28].
However since establishing this is almost as involved as a direct proof we
choose the latter and include the proof in Appendix C.

##  6 Proof of Theorem 7

The proof of Theorem 7 under the assumption (14) is a direct application of
Theorem 3. Under the assumption (15), on the other hand, the proof makes use
of the following

######  Proposition 33.

Let $m$ satisfy (1) and (15). Then $m\in B^{\frac{1}{3}}_{4,\infty,{\rm
loc}}(\Omega)$.

###### Proof.

The proof relies on the div-curl inequality

${\left|\int E\wedge B\right|}\lesssim
pp^{\prime}\left({\|E\|}_{L^{p^{\prime}}}{\|\operatorname{div}B\|}_{W^{-1,p}}+{\|B\|}_{L^{p^{\prime}}}{\|\operatorname{div}E\|}_{W^{-1,p}}\right),$
(125)

valid for all $p\in(1,\infty)$ and compactly supported bounded vector fields
$E,B\colon{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$; see [GL20, Lemma 4.2]. Given
$U\subset\subset\Omega^{\prime}\subset\subset\Omega^{\prime\prime}\subset\subset\Omega$
and $h\in{\mathbb{R}}^{2}$ with
$|h|<\mathrm{dist}\\{\Omega^{\prime},\partial\Omega^{\prime\prime}\\}$, we
apply the estimate (125) to

$E=\chi D^{h}\Sigma_{1}(m),\quad B=\chi D^{h}\Sigma_{2}(m),$

where $\chi\in C^{\infty}_{c}(\Omega^{\prime})$ is a cut-off function with
$0\leq\chi\leq 1$ and $\chi\equiv 1$ on $\overline{U}$. First by [LP18, Lemma
7] and noting that $m_{1}=\Sigma_{1}\left(m\right)\cdot
e_{2}-\Sigma_{2}\left(m\right)\cdot e_{1}$ and
$m_{2}=\Sigma_{1}\left(m\right)\cdot e_{1}+\Sigma_{2}\left(m\right)\cdot
e_{2}$, we have

$E\wedge
B\gtrsim\chi^{2}{\left|D^{h}\Sigma\right|}^{4}\gtrsim\chi^{2}{\left|D^{h}m\right|}^{4},$

where recall that $\Sigma=\left(\Sigma_{1},\Sigma_{2}\right)$.

Next we wish to estimate the right-hand side of (125). Note that the two terms
are symmetrical so it is enough to estimate
${\|E\|}_{L^{p^{\prime}}}{\|\operatorname{div}B\|}_{W^{-1,p}}$. We start with
${\|\operatorname{div}B\|}_{W^{-1,p}}$. Given any test function $\zeta\in
C^{\infty}_{c}(\Omega^{\prime})$ it holds

$-\int_{\Omega}B\cdot\nabla\zeta\,dx=\int_{\Omega}\nabla\chi\cdot
D^{h}\Sigma_{2}(m)\,\zeta\,dx+\int_{\Omega}\chi\operatorname{div}(D^{h}\Sigma_{2}(m))\,\zeta\,dx.$
(126)

Now estimating each term we have

$\displaystyle\int_{\Omega}\chi\operatorname{div}(D^{h}\Sigma_{2}(m))\,\zeta\,dx$
$\displaystyle=\int_{\Omega}\operatorname{div}\Sigma_{2}(m)\,D^{-h}(\chi\zeta)\,dx$
$\displaystyle\leq{\|\operatorname{div}\Sigma_{2}(m)\|}_{L^{p}(\Omega^{\prime\prime})}{\|D^{-h}(\chi\zeta)\|}_{L^{p^{\prime}}(\Omega)}$
$\displaystyle\lesssim{\|\operatorname{div}\Sigma_{2}(m)\|}_{L^{p}(\Omega^{\prime\prime})}{\|\chi\|}_{C^{1}}{\|\zeta\|}_{W^{1,p^{\prime}}(\Omega^{\prime})}{\left|h\right|},$

which shows that

${\|\chi\operatorname{div}\left(D^{h}\Sigma_{2}(m)\right)\|}_{W^{-1,p}(\Omega^{\prime})}\lesssim{\|\chi\|}_{C^{1}}{\|\operatorname{div}\Sigma_{2}(m)\|}_{L^{p}(\Omega^{\prime\prime})}{\left|h\right|}.$
(127)

The other term of the right-hand side of (126) can be estimated similarly:

$\displaystyle\int_{\Omega}\nabla\chi\cdot D^{h}\Sigma_{2}(m)\,\zeta\,dx$
$\displaystyle=\int_{\Omega}\Sigma_{2}\left(m\right)\cdot
D^{-h}(\zeta\nabla\chi)\,dx$
$\displaystyle\lesssim{\|\Sigma_{2}(m)\|}_{L^{\infty}(\Omega)}{\|\nabla(\zeta\nabla\chi)\|}_{L^{p^{\prime}}(\Omega)}{\left|h\right|}$
$\displaystyle\lesssim{\|\chi\|}_{C^{2}}{\|\zeta\|}_{W^{1,p^{\prime}}(\Omega^{\prime})}{\left|h\right|},$

which gives

${\|\nabla\chi\cdot
D^{h}\Sigma_{2}(m)\|}_{W^{-1,p}(\Omega^{\prime})}\lesssim{\|\chi\|}_{C^{2}}{\left|h\right|}.$
(128)

Thus

$\|\operatorname{div}B\|_{W^{-1,p}(\Omega^{\prime})}\overset{(\ref{eqand20}),(\ref{eqand10}),(\ref{eqand11})}{\lesssim}\left(1+{\|\operatorname{div}\Sigma_{2}(m)\|}_{L^{p}(\Omega^{\prime\prime})}\right){\|\chi\|}_{C^{2}}{\left|h\right|}.$

Gathering the above, and using the notation $\Sigma=(\Sigma_{1},\Sigma_{2})$,
we obtain from (125) that

$\displaystyle\int_{\Omega}\chi^{2}{\left|D^{h}m\right|}^{4}\,dx$
$\displaystyle\lesssim pp^{\prime}{\|\chi\|}_{C^{2}}{\|\chi
D^{h}\Sigma(m)\|}_{L^{p^{\prime}}(\Omega)}\left(1+{\|\operatorname{div}\Sigma(m)\|}_{L^{p}(\Omega^{\prime\prime})}\right){\left|h\right|}.$
(129)

Let us take $p=\frac{4}{3}$ and hence $p^{\prime}=4$. Since $\chi\leq 1$ we
have $\left|\chi\right|^{4}\leq\left|\chi\right|^{2}$. Further since $\Sigma$
is smooth we have that
$\left|D^{h}\Sigma\left(m\right)\right|\lesssim\left|D^{h}m\right|$. Thus

$\left|\chi(x)D^{h}\Sigma\left(m(x)\right)\right|^{4}\lesssim\left(\chi(x)\right)^{2}\left|D^{h}m(x)\right|^{4}\quad\text{
for a.e. }x$

and it follows that

$\|\chi
D^{h}\Sigma\left(m\right)\|_{L^{4}(\Omega)}\lesssim\left(\int_{\Omega}\chi^{2}\left|D^{h}m\right|^{4}dx\right)^{\frac{1}{4}}.$

Applying this to (129), dividing through and using $\chi\equiv 1$ on
$\overline{U}$ gives

$\displaystyle\left(\int_{U}{\left|D^{h}m\right|}^{4}\,dx\right)^{\frac{3}{4}}$
$\displaystyle\lesssim{\|\chi\|}_{C^{2}}\left(1+{\|\operatorname{div}\Sigma(m)\|}_{L^{\frac{4}{3}}(\Omega^{\prime\prime})}\right){\left|h\right|}.$

This shows that $m\in B^{\frac{1}{3}}_{4,\infty}(U)$ for all
$U\subset\subset\Omega$ and thus completes the proof. ∎

###### Proof of Theorem 7.

If (14) holds, then we have $m\in B^{\frac{1}{3}}_{3r,\infty,{\rm
loc}}(\Omega)$ for $r=\min\\{p,\frac{4}{3}\\}$ from Theorem 3. On the other
hand, if (15) holds, then $m\in B^{\frac{1}{3}}_{4,\infty,{\rm loc}}(\Omega)$
by Proposition 33. Thus, under the assumptions (14) or (15), we have $m\in
B^{\frac{1}{3}}_{3r,\infty,{\rm loc}}(\Omega)$ for
$r=\min\\{p,\frac{4}{3}\\}$. From Lemma 9, we know (11) holds in $L^{r}_{{\rm
loc}}(\Omega)$, and thus it follows from Theorem 5 that (12) and (13) hold.
Further, under the assumption (15), we have $\operatorname{div}\Phi_{f}(m)\in
L^{p}_{{\rm loc}}(\Omega)$ because of the formula (12). This completes the
proof. ∎

##  Appendix A Computations needed in the proof of Proposition 30

In this appendix we check that (117) holds for $f=f^{j}_{k}$ given in (122),
$j=1,2$, $k\geq 2$.

######  Lemma 34.

Let $k\geq 2$. For $f=f^{j}_{k}$, the entropies $\Phi_{f}$ defined in (23)
satisfy

$\displaystyle\Phi_{f^{1}_{k}}(e^{it})$
$\displaystyle=\frac{i\cos(k\frac{\pi}{2})}{k}\left[\frac{e^{i(k+1)t}}{k+1}+\frac{e^{-i(k-1)t}}{k-1}\right],$
$\displaystyle\Phi_{f^{2}_{k}}(e^{it})$
$\displaystyle=-\frac{2}{k}e^{it}+\frac{\cos(k\frac{\pi}{2})}{k}\left[\frac{e^{i(k+1)t}}{k+1}-\frac{e^{-i(k-1)t}}{k-1}\right].$

###### Proof of Lemma 34.

For $f=f^{1}_{k}$ we have

$\displaystyle\psi_{f}(t)$
$\displaystyle=\int_{0}^{t}\cos(ks)\,ds=\frac{1}{k}\sin(kt),$
$\displaystyle\varphi_{f}(t)$
$\displaystyle=\frac{1}{k}\int_{0}^{t}\sin(ks)ie^{is}\,ds=\frac{1}{2k}\int_{0}^{t}(e^{iks}-e^{-iks})e^{is}\,ds$
$\displaystyle=\frac{1}{2k}\int_{0}^{t}(e^{i(k+1)s}-e^{-i(k-1)s})\,ds=\frac{1}{2k}\left[\frac{e^{i(k+1)t}-1}{i(k+1)}+\frac{e^{-i(k-1)t}-1}{i(k-1)}\right],$
$\displaystyle\Phi_{f}(e^{it})$
$\displaystyle=-i\varphi_{f}(t-\frac{\pi}{2})+i\varphi_{f}(t+\frac{\pi}{2})$
$\displaystyle=\frac{1}{2k}\left[\frac{e^{i(k+1)t}}{k+1}(-e^{-i(k+1)\frac{\pi}{2}}+e^{i(k+1)\frac{\pi}{2}})+\frac{e^{-i(k-1)t}}{k-1}(-e^{i(k-1)\frac{\pi}{2}}+e^{-i(k-1)\frac{\pi}{2}})\right]$
$\displaystyle=\frac{i\cos(k\frac{\pi}{2})}{k}\left[\frac{e^{i(k+1)t}}{k+1}+\frac{e^{-i(k-1)t}}{k-1}\right].$

For $f=f^{2}_{k}$ we have

$\displaystyle\psi_{f}(t)$
$\displaystyle=\int_{0}^{t}\sin(ks)\,ds=\frac{1}{k}(1-\cos(kt)),$
$\displaystyle\varphi_{f}(t)$
$\displaystyle=\frac{1}{k}\int_{0}^{t}(1-\cos(ks))ie^{is}\,ds=\frac{i}{2k}\int_{0}^{t}(2e^{is}-e^{i(k+1)s}-e^{-i(k-1)s})\,ds$
$\displaystyle=\frac{1}{2k}\left[2(e^{it}-1)-\frac{e^{i(k+1)t}-1}{k+1}+\frac{e^{-i(k-1)t}-1}{k-1}\right],$
$\displaystyle\Phi_{f}(e^{it})$
$\displaystyle=-i\varphi_{f}(t-\frac{\pi}{2})+i\varphi_{f}(t+\frac{\pi}{2})$
$\displaystyle=\frac{i}{2k}\Bigg{[}2e^{it}(-e^{-i\frac{\pi}{2}}+e^{i\frac{\pi}{2}})$
$\displaystyle\quad-\frac{e^{i(k+1)t}}{k+1}(-e^{-i(k+1)\frac{\pi}{2}}+e^{i(k+1)\frac{\pi}{2}})+\frac{e^{-i(k-1)t}}{k-1}(-e^{i(k-1)\frac{\pi}{2}}+e^{-i(k-1)\frac{\pi}{2}})\Bigg{]}$
$\displaystyle=-\frac{2}{k}e^{it}+\frac{\cos(k\frac{\pi}{2})}{k}\left[\frac{e^{i(k+1)t}}{k+1}-\frac{e^{-i(k-1)t}}{k-1}\right].$

∎

######  Lemma 35.

Let $k\geq 2$. For $f=f^{j}_{k}$, the harmonic entropies $\Phi^{\xi_{f}}$
defined in (116) satisfy

$\displaystyle\Phi^{\xi_{f^{1}_{k}}}(e^{it})$
$\displaystyle=\frac{i\cos(k\frac{\pi}{2})}{k}\left[\frac{e^{i(k+1)t}}{k+1}+\frac{e^{-i(k-1)t}}{k-1}\right]=\Phi_{f^{1}_{k}}(e^{it}),$
$\displaystyle\Phi^{\xi_{f^{2}_{k}}}(e^{it})$
$\displaystyle=\frac{\cos(k\frac{\pi}{2})}{k}\left[\frac{e^{i(k+1)t}}{k+1}-\frac{e^{-i(k-1)t}}{k-1}\right]=\Phi_{f^{2}_{k}}(e^{it})+\frac{2}{k}e^{it}.$

###### Proof of Lemma 35.

Recall that $\Phi^{\xi}$ is given by

$\Phi^{\xi}(z)=\xi(z)z+((iz)\cdot\nabla\xi(z))iz.$ (130)

For $f=f^{j}_{k}$ we have

$\displaystyle\xi_{f^{j}_{k}}$ $\displaystyle=0\qquad\text{if }k\text{ is
odd},$ $\displaystyle\xi_{f^{1}_{2k}}$
$\displaystyle=\frac{(-1)^{k+1}}{k(1-2k)(1+2k)}\varphi^{1}_{2k},$
$\displaystyle\xi_{f^{2}_{2k}}$
$\displaystyle=\frac{(-1)^{k}}{k(1-2k)(1+2k)}\varphi^{2}_{2k},$ (131)

where $\varphi^{j}_{k}$ are the harmonic polynomials given in polar
coordinates by $\varphi^{1}_{k}=r^{k}\sin(k\theta)$,
$\varphi^{2}_{k}=r^{k}\cos(k\theta)$. Hence

$\displaystyle\Phi^{\xi_{f^{j}_{k}}}$
$\displaystyle\overset{\eqref{eqad11},\eqref{eqad12}}{=}0\qquad\text{if
}k\text{ is odd},$ $\displaystyle\Phi^{\xi_{f^{1}_{2k}}}$
$\displaystyle\overset{\eqref{eqad11},\eqref{eqad12}}{=}\frac{(-1)^{k+1}}{k(1-2k)(1+2k)}\Phi^{\varphi^{1}_{2k}},$
$\displaystyle\Phi^{\xi_{f^{2}_{2k}}}$
$\displaystyle\overset{\eqref{eqad11},\eqref{eqad12}}{=}\frac{(-1)^{k}}{k(1-2k)(1+2k)}\Phi^{\varphi^{2}_{2k}},$
(132)

and it remains to compute $\Phi^{\varphi^{j}_{2k}}(e^{it})$.

For $\xi=\varphi^{1}_{k}$ we find

$\displaystyle\Phi^{\varphi^{1}_{k}}(e^{it})$
$\displaystyle=\varphi^{1}_{k}e^{it}+(-\sin
t\,\partial_{1}\varphi^{1}_{k}+\cos t\,\partial_{2}\varphi^{1}_{k})ie^{it}$
$\displaystyle\overset{\eqref{appeqd3}}{=}\sin(kt)e^{it}+(-k\sin
t\sin((k-1)t)+k\cos t\cos((k-1)t))ie^{it}$
$\displaystyle=\sin(kt)e^{it}+k\cos(kt)ie^{it}$
$\displaystyle=\frac{i}{2}e^{it}\left[-e^{ikt}+e^{-ikt}+ke^{ikt}+ke^{-ikt}\right]$
$\displaystyle=\frac{i}{2}\left[(k-1)e^{i(k+1)t}+(k+1)e^{-i(k-1)t}\right],$
(133)

and for $\xi=\varphi^{2}_{k}$,

$\displaystyle\Phi^{\varphi^{2}_{k}}(e^{it})$
$\displaystyle=\varphi^{2}_{k}e^{it}+(-\sin
t\,\partial_{1}\varphi^{2}_{k}+\cos t\,\partial_{2}\varphi^{2}_{k})ie^{it}$
$\displaystyle\overset{\eqref{appeqd3}}{=}\cos(kt)e^{it}+(-k\sin
t\cos((k-1)t)-k\cos t\sin((k-1)t))ie^{it}$
$\displaystyle=\cos(kt)e^{it}-k\sin(kt)ie^{it}$
$\displaystyle=\frac{1}{2}e^{it}\left[e^{ikt}+e^{-ikt}-ke^{ikt}+ke^{-ikt}\right]$
$\displaystyle=\frac{1}{2}\left[-(k-1)e^{i(k+1)t}+(k+1)e^{-i(k-1)t}\right].$
(134)

Gathering the above we obtain

$\displaystyle\Phi^{\xi_{f^{1}_{2k}}}$
$\displaystyle\overset{\eqref{eqad13},\eqref{eqad14}}{=}\frac{i(-1)^{k}}{2k}\left[\frac{e^{i(2k+1)t}}{2k+1}+\frac{e^{-i(2k-1)t}}{2k-1}\right],$
$\displaystyle\Phi^{\xi_{f^{2}_{2k}}}$
$\displaystyle\overset{\eqref{eqad13},\eqref{eqad15}}{=}\frac{(-1)^{k}}{2k}\left[\frac{e^{i(2k+1)t}}{2k+1}-\frac{e^{-i(2k-1)t}}{2k-1}\right],$

which, gathering all the cases ($k$ even or odd), and recalling the
expressions found in Lemma 34 for $\Phi_{f^{j}_{k}}$, proves Lemma 35. ∎

##  Appendix B Proof of [GL20, Lemma 3.4]

######  Lemma 36 (Ghiraldin-Lamy).

Let $m:\Omega\to{\mathbb{R}}^{2}$ satisfy (1). Assume
$\operatorname{div}\Phi_{f}(m)\in\mathcal{M}(\Omega)$ for all $f\in
C^{0}(\mathbb{R}/2\pi\mathbb{Z})$. Then there exists
$\sigma\in\mathcal{M}(\Omega\times\mathbb{R}/2\pi\mathbb{Z})$ such that

$\langle\operatorname{div}\Phi_{f}(m),\zeta\rangle=\langle\sigma(x,s),f(s)\zeta(x)\rangle\qquad\forall
f\in C^{0}(\mathbb{R}/2\pi\mathbb{Z}),\>\zeta\in C^{0}_{c}(\Omega),$ (135)

where $\langle\cdot,\cdot\rangle$ denotes the duality between measures and
functions.

###### Proof.

Since the proof follows almost exactly the same lines as that of Lemma 14, we
only sketch it briefly focusing on the differences. Letting
$\mathcal{Z}=C^{0}_{c}(\Omega)$ and thus
$\mathcal{Z}^{*}=\mathcal{M}(\Omega)$, exactly as in the proof of Lemma 14,
the Banach-Steinhaus’ uniform boundedness principle implies the map

$\displaystyle T\colon
C^{0}(\mathbb{R}/2\pi\mathbb{Z})\to\mathcal{Z}^{*},\quad
f\mapsto\operatorname{div}\Phi_{f}(m)$ (136)

is a bounded linear operator.

Again by [DU77, Theorem VI.2.1], there exists an
$\mathcal{M}(\Omega)^{**}$-valued Borel measure $G$ on
$\mathbb{R}/2\pi\mathbb{Z}$ such that

$\langle\varphi^{*},Tf\rangle=\int_{\mathbb{R}/2\pi\mathbb{Z}}f\,d\langle
G,\varphi^{*}\rangle\qquad\forall\varphi^{*}\in\mathcal{Z}^{**}.$ (137)

As in the proof of Lemma 14, for $\psi$ a finite linear combination of the
form

$\displaystyle\psi(x,s)=\sum_{j}{\mathds{1}}_{E_{j}}(s)\zeta_{j}(x),\qquad