#  $C^{1,\nicefrac{{1}}{{3}}-}$ very weak solutions to the two dimensional
Monge–Ampère equation

Wentao Cao, Jonas Hirsch, and Dominik Inauen  Wentao Cao, Academy for
Multidisciplinary Studies, Capital Normal University, West 3rd Ring North Road
105, Beijing, 100048 P.R. China<EMAIL_ADDRESS>Jonas Hirsch,
Institut für Mathematik, Universität Leipzig, D-04109, Leipzig, Germany.
<EMAIL_ADDRESS>Dominik Inauen, Institut für
Mathematik, Universität Leipzig, D-04109, Leipzig, Germany.
<EMAIL_ADDRESS>
###### Abstract.

For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-
dimensional Monge–Ampère equation with regularity $C^{1,\theta}$ are dense in
the space of continuous functions. This result is shown by a convex
integration scheme involving a subtle decomposition of the defect at each
stage. The decomposition diagonalizes the defect and, in addition,
incorporates some of the leading-order error terms of the first perturbation,
effectively reducing the required amount of perturbations to one.

##  1\. Introduction

This work is concerned with very weak solutions of the Monge–Ampère equation
on a simply connected, bounded open set $\Omega\subset\mathbb{R}^{2}$

$\mathcal{D}\text{et}D^{2}v=f\text{ in }\Omega\,.$ (1.1)

Here, $\mathcal{D}\text{et}D^{2}v$ is the distribution

$\mathcal{D}\text{et}D^{2}v:=\partial^{2}_{12}\left(\partial_{1}v\partial_{2}v\right)-\frac{1}{2}\partial_{22}^{2}\left((\partial_{1}v)^{2}\right)-\frac{1}{2}\partial_{11}^{2}\left((\partial_{2}v)^{2}\right)=-\frac{1}{2}\mathrm{curl\,curl}(\nabla
v\otimes\nabla v)\,,$

introduced in [12] as the _very weak Hessian_. It is well defined for $v\in
W^{1,2}_{loc}(\Omega)$ and a quick approximation argument shows that it agrees
with $\det D^{2}v$ for $v\in W^{2,2}_{loc}(\Omega)$. As in [3, 14],
distributional solutions to (1.1) are called _very weak solutions_ of the
Monge-Ampère equation. The following is our main result.

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, simply connected and
smooth domain and $f\in L^{p}(\Omega)$ for $p\geq\frac{3}{2}$. For any
$\theta<\frac{1}{3}$ the set of $C^{1,\theta}(\bar{\Omega})$ solutions of
(1.1) is dense in $C^{0}(\bar{\Omega})$. More precisely, for any
$\underline{v}\in C^{0}(\bar{\Omega})$ and $\epsilon>0$ there exists
distributional solution $\overline{v}\in C^{1,\theta}(\bar{\Omega})$ of (1.1)
satisfying

$\|\underline{v}-\overline{v}\|_{0}<\epsilon\,.$

This result improves upon the main theorems in [14] and [3], where the above
theorem is shown for $\theta<\frac{1}{7}$ and $\theta<\frac{1}{5}$
respectively. It shows a type of flexibility of the space of solutions to
(1.1) exhibited more generally by differential relations satisfying the
$h$-principle, a term introduced in [10] and proven to be present in various
problems across fields such as geometry, topology, PDE and also fluid dynamics
(see for example [10], [9],[16],[1]). In most of these instances (and in
particular for the main theorems of [14, 3] and the present work), this
flexibility is shown by means of a convex integration scheme, a method
originally introduced by J. Nash in [15] for the construction of counter-
intuitive isometric embeddings of Riemannian manifolds into Euclidean space.

###  1.1. Connection to isometric embeddings and reduction to convex
integration

Recall that an isometric embedding of a Riemannian manifold $(\mathcal{M},g)$
into $\mathbb{R}^{m}$ is an injective $C^{1}$-map
$u:\mathcal{M}\to\mathbb{R}^{m}$ such that the induced metric $u^{\sharp}e$
agrees with $g$. In local coordinates this leads to the system

$\partial_{i}u\cdot\partial_{j}u=g_{ij}\,,1\leq
i,j\leq\mathrm{dim}\mathcal{M}$

sometimes simply written $\nabla u^{T}\nabla u=g$. The connection between
isometric embeddings of two-dimesional Riemannian manifolds into
$\mathbb{R}^{3}$ and the Monge–Ampère equation is intimate: finding a (local)
$C^{2}$-regular isometric embedding of a two-dimensional domain into
$\mathbb{R}^{3}$ is equivalent to solving the Darboux equation, a fully
nonlinear equation of Monge–Ampère type (see for example [11]). As observed in
[14], a connection remains on the level of weaker solutions, i.e., isometric
embeddings of regularity $C^{1,\theta}$ and very weak solutions of the Monge-
Ampère equation of regularity $C^{1,\theta}$ respectively. Indeed, recall that
the kernel of the operator curlcurl on a simply connected domain is given by
matrices of the form $\mathrm{sym}\nabla w:=\frac{1}{2}(\nabla w+\nabla
w^{T})$ for $w:\Omega\to\mathbb{R}^{2}$. Thus, solving (1.1) is equivalent to
solving the system

$\displaystyle\frac{1}{2}\nabla v\otimes\nabla v+\mathrm{sym}\nabla w=A$ (1.2)
$\displaystyle-\mathrm{curl\,curl}A=f$ (1.3)

in the unknowns $v:\Omega\to\mathbb{R}$, $w:\Omega\to\mathbb{R}^{2}$ and
$A:\Omega\to\mathbb{R}^{2\times 2}_{sym}$. The second equation is simply a
compatibility condition and is easily solved: one can set $A=u\,\mathrm{Id}$
for a function $u$ such that $-\Delta u=f$. On the other hand, for $A=0$,
equation (1.2) can be seen as the condition for the family
$\phi_{t}:\Omega\to\mathbb{R}^{3}$ given by $\phi_{t}(x)=(x+t^{2}w(x),tv(x))$
to ”preserve lengths upto second order” (i.e., the induced metric
$\nabla\phi_{t}^{T}\nabla\phi_{t}$ agrees with the identity metric upto terms
of order $t^{2}$), or in other words, to form a second order infinitesimal
isometryaaaFor a more elaborate explanation of this connection and further
motivations to study very weak solutions to the Monge-Ampère equation, see for
example the introductions in [14] and [3].. Motivated by this connection,
Lewicka–Pakzad observed in [14] that an adaptation of the convex integration
process of [6] for the construction of $C^{1,\theta}$ isometric embeddings can
be used to construct solutions of (1.2). Such a process generates a solution
as the limit of a sequence of ”subsolutions”, i.e., in the present case, maps
$(v_{q},w_{q})$ such that

$\mathcal{D}_{q}:=A-\frac{1}{2}\nabla v_{q}\otimes\nabla
v_{q}-\mathrm{sym}\nabla w_{q}$

is positive definite. The construction of $(v_{q+1},w_{q+1})$ given
$(v_{q},w_{q})$, termed ”stage”, aims at decreasing the defect
$\mathcal{D}_{q}$ and consists in first _decomposing_ the defect into a sum of
rank-one matrices (called ”primitive metrics” in [15] and subsequent works)
and then _perturbing_ $(v_{q},w_{q})$ by adding ”corrugations” which cancel
these primitive metrics upto an error of lower order. The corrugations
oscillate at a high frequency which increases with $q$ but have amplitudes
which decrease with $q$. If set-up correctly, the sequence
$\|v_{q+1}-v_{q}\|_{1}$ tends to zero while $\|v_{q+1}-v_{q}\|_{2}$ blows up
in a controlled way. This yields the convergence in some Hölder space
$C^{1,\theta}$ to a solution of the problem. The number of corrugations used
in a stage influences the size of $\theta$ (see e.g. [16] for detailed
explanation): the three corrugations used in [14], [6] yield a regularity
$\theta<\frac{1}{7}$. For the isometric embedding problem, a change of
coordinates was introduced in [8] to diagonalize the defect. This reduced the
required number of corrugations to two, allowing the construction of local
isometric embeddings with regularity $C^{1,\theta}$ for $\theta<\frac{1}{5}$
(a result which was extended to global isometric embeddings in [4]). For the
Monge–Ampère equation it was observed in [3] that instead of introducing a
change of coordinate one can use the term $\mathrm{sym}\nabla w$ to
diagonalize the defect directly by solving

$\mathcal{D}-\mathrm{sym}\nabla w=r\mathrm{Id}$ (1.4)

for $w$ and $r$, where $\mathcal{D}$ represents the defect and $r$ is the
desired coefficient function. This gives the reduction to two corrugations and
therefore regularity $\theta<\frac{1}{5}$.

###  1.2. Strategy of proof

As in [14], [3] we prove Theorem 1.1 by constructing solutions to (1.2) by the
convex integration scheme outlined above. However, to get to Hölder regularity
$\theta<\frac{1}{3}$ we need to decrease the rate with which the frequencies
of the oscillations grow along the sequence. We achieve this by a
decomposition which simultaneously diagonalizes the defect and in addition
incorporates (some of) the leading-order error terms of the first perturbation
(see (1.5)). Even though we still need to add two perturbations to cancel the
defect, their frequency can be chosen smaller, effectively reducing the number
of perturbations to one.

The idea of absorbing the error terms coming from the oscillations into the
decomposition first appears in [13] in the context of isometric embeddings
into Euclidean space with high codimension, where the author makes use of the
freedom of high codimension to absorb such errors with arbitrary precision.
This allows for the construction of isometric embeddings of class
$C^{1,\theta}$ for all $\theta<1$, given that the metric $g$ is regular enough
and the codimension high enough (this idea is also used for less regular
metrics and smaller codimension in [5, 7, 2]).

Since in the present context we do not have the freedom of codimension, the
best we can hope for is to absorb the error terms of the first oscillation. In
practice this involves solving an equation of the form

$\mathcal{D}-\mathrm{sym}\nabla w+E(r)=r\mathrm{Id}$ (1.5)

for $w$ and $r$, where $E(r)$ represents the error terms of the first
perturbation (compare to (1.4)). However, due to the structure of these error
terms, a direct analogue of the decomposition in [13] does not yield the
required improvementbbbThe same is true for codimension one isometric
embeddings.. Instead, the observation that the fast oscillation of the first
perturbation only occurs in one direction leads us to absorb only specific
parts of the error terms in the decomposition. Their algebraic structure is
chosen such that it is possible to trade one “fast” derivative for a “slow”
one. Equation (1.5) is then approached by a Picard scheme. This yields the
crucial decomposition Lemma 3.2, which is one of the main technical
contributions of this paper. The leading-order error terms of the first
perturbation which are not absorbed by the decomposition are then matched
exactly by the second perturbation.

###  1.3. Organization of the paper

After some preliminaries on Hölder spaces and mollification in Section 2 we
show in Section 3 how the decomposition Lemma 3.2 leads to a proof of the
crucial inductive building block, the ”stage” proposition. We then show in
Section 4 how the stage proposition can be applied to prove the main theorem.
Finally we prove the decomposition lemma in Section 5.

##  2\. Preliminaries

In this section we introduce some notations, function spaces and basic lemmas
which are needed for the proof of Theorem 1.1

###  2.1. Hölder norms and interpolation

Let $\Omega\subset\mathbb{R}^{2}$ be as above. In the following, the maps $f$
can be real valued, vector valued, or tensor valued. In every case, the target
is equipped with the Euclidean norm, denoted by $|f(x)|$. The Hölder norms are
then defined as follows:

$\|f\|_{0}=\sup_{\Omega}|f|,~{}~{}\|f\|_{m}=\sum_{j=0}^{m}\max_{|I|=j}\|\partial^{I}f\|_{0},,$

where $I$ denotes a multi-index, and, for $\alpha\in]0,1]$ and
$k\in\mathbb{N}$,

$[f]_{0,\alpha}=\sup_{x\neq
y}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}},~{}~{}[f]_{k,\alpha}=\max_{|I|=k}\sup_{x\neq
y}\frac{|\partial^{I}f(x)-\partial^{I}f(y)|}{|x-y|^{\alpha}}\,.$

Then the Hölder norms are given as

$\|f\|_{k,\alpha}=\|f\|_{k}+[f]_{k,\alpha}.$

We sometimes abbreviate $[f]_{k,\alpha},\|f\|_{k,\alpha}$ as $[f]_{k+\alpha}$
and $\|f\|_{k+\alpha}$ respectively. We recall the standard interpolation
inequality

$[f]_{p}\leq C\|f\|_{0}^{1-\frac{p}{d}}[f]_{d}^{\frac{p}{d}}$ (2.1)

for any $d>p\geq 0$ and the Leibniz rule

$\|fg\|_{p}\leq C(\|f\|_{p}\|g\|_{0}+\|f\|_{0}\|g\|_{p}),$ (2.2)

where the constants only depend on $p,d$ and $p$ respectively. We also collect
two classical estimates on the Hölder norms of compositions. A proof can be
found in [8].

######  Lemma 2.1.

Let $k,n\in\mathbb{N}$, $\alpha\in[0,1]$, $U\subset\mathbb{R}^{n}$ and
$\Omega\subset\mathbb{R}^{2}$ as above. Assume $\Psi:U\to\mathbb{R}$ and
$u:\Omega\to U$ are two $C^{k,\alpha}$ functions. Then there is a constant $C$
(depending only on $k$, $\alpha$, $\Omega$ and $U$) such that

$\displaystyle\left[\Psi\circ u\right]_{k,\alpha}$ $\displaystyle\leq
C[u]_{k,\alpha}\left([\Psi]_{1}+\|u\|_{0}^{k-1}[\Psi]_{k}\right)$
$\displaystyle\quad\quad+C[\Psi]_{k,\alpha}\left(\|u\|_{0}^{k-1}[u]_{k}\right)^{\frac{k+\alpha}{k}}\,,$
(2.3) $\displaystyle\left[\Psi\circ u\right]_{k,\alpha}$ $\displaystyle\leq
C\left([u]_{k,\alpha}[\Psi]_{1}+[u]_{1}^{k+\alpha}[\Psi]_{k+\alpha}\right)\,.$
(2.4)

###  2.2. Mollification estimates

In the proof of the inductive proposition in the next section we will
regularize maps $f\in C^{k,\alpha}(\bar{\Omega},\mathbb{R}^{N})$ by
convolution with a standard mollifier, i.e., a radially symmetric
$\varphi_{\ell}\in C^{\infty}_{c}(B_{\ell}(0))$ with $\int\varphi_{\ell}=1$,
where $\ell>0$ denotes the length-scale. The resulting regularized maps will
then have a smaller domain of definition. To counteract this, we first extend
$f$ to a map $\bar{f}\in
C^{k,\alpha}\left(\mathbb{R}^{2},\mathbb{R}^{N}\right)$ such that

$\|\bar{f}\|_{C^{k,\alpha}(\mathbb{R}^{2})}\leq
C\|f\|_{C^{k,\alpha}(\bar{\Omega})}\,,$

where the constant $C>0$ depends only on $k,\alpha,N$ and $\Omega$. Such a
procedure is given by Whitney’s extension theorem, see [17]. We then mollify
the resulting extensions at some length-scale $\ell>0$ to obtain
$\tilde{f}=\bar{f}\ast\varphi_{\ell}\in
C^{\infty}(\bar{\Omega},\mathbb{R}^{m})$. We will not further specify this
extension procedure from now on. Such a regularization of Hölder functions
enjoys the following estimates (for a proof, see [6],[8]).

######  Lemma 2.2.

For any $p,d\geq 0,$ and $0<\alpha\leq 1,$ we have

$\displaystyle[f*\varphi_{\ell}]_{p+d}\leq C\ell^{-d}[f]_{p},$
$\displaystyle\|f-f*\varphi_{\ell}\|_{p}\leq C\ell^{2-p}[f]_{2},\text{ if
}0\leq p\leq 2,$
$\displaystyle\|(fg)*\varphi_{\ell}-(f*\varphi_{\ell})(g*\varphi_{\ell})\|_{p}\leq
C\ell^{2\alpha-p}\|f\|_{\alpha}\|g\|_{\alpha}$

with constant $C$ depending only on $d,p,\beta,\varphi.$

###  2.3. Notations

As usual, $C$ will denote constants whose values can change from line to line.
If we wish to emphasize the dependents of the constant on various parameters,
say $k,\alpha$, we write $C(k,\alpha)$. Symmetric $2\times 2$ matrices are
denoted by $\mathcal{S}$, the $2\times 2$ identity matrix by $\mathrm{Id}$.
For any matrix $M$, $M_{ij}$ denotes the $ij$-th entry of $M$. For
$\xi=(\xi_{1},\xi_{2}),\,\eta=(\eta_{1},\eta_{2})\in\mathbb{R}^{2}$,
$\xi\otimes\eta$ denotes the $2\times 2$ matrix with entries
$(\xi\otimes\eta)_{ij}=\xi_{i}\eta_{j}$ and we abbreviate
$\xi\odot\eta=\frac{1}{2}(\xi\otimes\eta+\eta\otimes\xi)=\mathrm{sym}{\xi\otimes\eta}$.

##  3\. Stage Proposition

As explained in the introduction we want to construct solutions to (1.2) by
iteratively constructing a sequence of subsolutions $(v_{q},w_{q})$ converging
to an actual solution. In this section we state and prove the main building
block of the iteration, the ”stage proposition”.

For a given $\theta<\frac{1}{3}$ as in Theorem 1.1 we choose
$b>1,\,c>\frac{3}{2}$ such that

$c\geq 2b+\frac{1}{2b}-1,\quad\frac{1}{2bc}<\theta\,.$ (3.1)

Define a sequence of amplitudes

$\delta_{q}=a^{-b^{q}},\quad q\in\mathbb{N}$ (3.2)

and a sequence of frequencies of the oscillations

$\lambda_{q}=a^{cb^{q+1}},\quad q\in\mathbb{N},$ (3.3)

where $a\geq a_{0}$ is a large constant to be chosen. We initially take
sufficiently large $a_{0}$ such that

$\delta_{1}<1<\lambda_{q},\quad\delta_{q+1}\leq\frac{1}{2}\delta_{q}\quad\lambda_{q+1}\geq
2\lambda_{q}\,.$ (3.4)

We then have the following

######  Proposition 3.1.

Let $\theta<\frac{1}{3}$ and $b>1,c>\frac{3}{2}$ as in (3.1), and let
$\gamma\geq\frac{2}{3}$. There exists
$\alpha_{0}=\alpha_{0}(b,c)\in\,]0,\gamma[$ such that for any
$\alpha\in\,]0,\alpha_{0}[$, $D\in C^{0,\gamma}(\bar{\Omega},\mathcal{S})$ and
$B\geq 1$ there exist constants $\tau_{0}(\alpha,b,c,\Omega)\in\,]0,1[$,
$C_{0}(\alpha,b,c,\Omega,B)\geq 1$ and
$a_{0}(\alpha,b,c,\Omega,\|D\|_{0,\gamma},C_{0}(\alpha,b,c,\Omega,B))$ with
the following property.

Let $v_{q}\in C^{\infty}(\bar{\Omega}),w_{q}\in
C^{\infty}(\bar{\Omega},\mathbb{R}^{2})$ satisfy

$\displaystyle\|D-\frac{1}{2}\nabla v_{q}\otimes\nabla
v_{q}-\mathrm{sym}\nabla
w_{q}-\delta_{q+1}\mathrm{Id}\|_{0,\alpha}\leq\tau_{0}\delta_{q+1},$ (3.5)
$\displaystyle\|v_{q}\|_{1}+\|w_{q}\|_{1}\leq
B-\delta_{q}^{\nicefrac{{1}}{{2}}}\,,$ (3.6)
$\displaystyle\|v_{q}\|_{2}+\|w_{q}\|_{2}\leq
C_{0}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\,,$ (3.7)

where $\lambda_{q},\delta_{q}$ are defined as in (3.2), (3.3) respectively. If

$a\geq a_{0}(\alpha,b,c,\Omega,\|D\|_{0,\gamma},C_{0}),$ (3.8)

then there exist $v_{q+1}\in C^{\infty}(\bar{\Omega}),w_{q+1}\in
C^{\infty}(\bar{\Omega},\mathbb{R}^{2})$ such that (3.5)-(3.7) hold with $q$
replaced by $q+1$ and additionally

$\displaystyle\|v_{q}-v_{q+1}\|_{0}$
$\displaystyle\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{-1}\,,$ (3.9)
$\displaystyle\|v_{q}-v_{q+1}\|_{1}+\|w_{q}-w_{q+1}\|_{1}$ $\displaystyle\leq
C_{0}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,.$ (3.10)

###### Proof.

We divide the proof into several steps.

_Step 1: Parameters._ For a fixed $\alpha\in\,]0,\frac{1}{2}[$ to be chosen
below in Lemma 3.1 we define a mollification parameter $\sigma$ by

$\sigma^{2-2\alpha}:=\frac{\delta_{q+1}}{\delta_{q}\lambda_{q}^{2}}$ (3.11)

and the frequency parameter of the first oscillation by

$\mu:=\frac{\delta_{q+2}\lambda_{q+1}^{1-2\alpha}}{\delta_{q+1}}\,.$ (3.12)

Obviously, we have

$\sigma<1<\mu<\lambda_{q+1}.$

Moreover,

######  Lemma 3.1.

Let $b>1,c>\frac{3}{2}$ satisfy (3.1) and let the parameters $\sigma$ and
$\mu$ be defined as in (3.11) and (3.12) respectively. There exists a constant
$\alpha_{0}(b,c)\ll 1$ such that for any $0<\alpha<\alpha_{0}$ there exists a
constant $N_{0}(b,c,\alpha)\in\mathbb{N}$ such that for any $N_{0}\leq
N\in\mathbb{N},$ it holds

$\displaystyle\sigma^{-1}\leq\mu^{1-2\alpha}\,,$ (3.13)
$\displaystyle\sigma^{\frac{2}{3}-\alpha}\leq\delta_{q+2}\lambda_{q+1}^{-\alpha}\,,$
(3.14)
$\displaystyle\delta_{q+1}\left(\mu^{\alpha-1}\sigma^{-1}\right)^{N}\leq\delta_{q+2}\lambda_{q+1}^{-\alpha}\,.$
(3.15)

###### Proof.

Inserting the definitions of $\sigma$ and $\mu$, the estimate (3.13) is
implied by

$\frac{\delta_{q}\lambda_{q}^{2}}{\delta_{q+1}}\leq\frac{\delta_{q+2}^{(2-2\alpha)(1-2\alpha)}\lambda_{q+1}^{(1-2\alpha)^{2}(2-2\alpha)}}{\delta_{q+1}^{(1-2\alpha)(2-2\alpha)}}.$

Taking the logarithm in $a$ and dividing by $b^{q}\log a,$ we see that the
latter inequality follows from

$-1+2cb+b\leq-b^{2}(1-2\alpha)(2-2\alpha)+cb^{2}(1-2\alpha)^{2}(2-2\alpha)+b(1-2\alpha)(2-2\alpha),$

which follows by

$b>\frac{1}{(1-2\alpha)^{2}(1-\alpha)},\quad
c>\frac{b^{2}(1-2\alpha)(2-2\alpha)-b((1-2\alpha)(2-2\alpha)-1)-1}{2b(b(1-2\alpha)^{2}(1-\alpha)-1)}\,.$
(3.16)

Note that in the limit $\alpha\downarrow 0$ the above bounds read

$b>1,\quad c>\frac{1}{2b}+1\,.$

Thus, with (3.1), we can take $\alpha_{1}(b,c)$ small enough to make (3.13)
hold for $0<\alpha<\alpha_{1}$.

Inequality (3.14) is equivalent to

$\frac{\delta_{q+1}^{\frac{2}{3}-\alpha}}{\delta_{q}^{\frac{2}{3}-\alpha}\lambda_{q}^{\frac{4}{3}-2\alpha}}\leq\delta_{q+2}^{2-2\alpha}\lambda_{q+1}^{-\alpha(2-2\alpha)}\,.$

Taking the logarithm in $a$ and dividing by $b^{q}\log a$ yields

$-(b-1)(\frac{2}{3}-\alpha)-2cb(\frac{2}{3}-\alpha)\leq-2b^{2}(1-\alpha)-2cb^{2}\alpha(1-\alpha),$

which is ensured by

$\alpha+b\alpha(1-\alpha)<\frac{2}{3},\quad
c>\frac{2b^{2}(1-\alpha)-(b-1)(\frac{2}{3}-\alpha)}{2b(\frac{2}{3}-\alpha-b\alpha(1-\alpha))}\,.$
(3.17)

Letting $\alpha\downarrow 0$, we get the above bounds as

$0<\frac{2}{3},\quad c>\frac{3}{2}b+\frac{1}{2b}-\frac{1}{2}\,.$

Thus, noting (3.1), we can take $\alpha_{2}(b,c)$ small enough to make (3.14)
hold for $0<\alpha<\alpha_{2}$.

It remains to show (3.15). By the definition of $\mu$ and $\sigma$, it is
equivalent to get

$\left(\frac{\delta_{q+2}^{\alpha-1}\lambda_{q+1}^{(1-2\alpha)(\alpha-1)}}{\delta_{q+1}^{\alpha-1}}\frac{\delta_{q}^{\frac{1}{2-2\alpha}}\lambda_{q}^{\frac{1}{1-\alpha}}}{\delta_{q+1}^{\frac{1}{2-2\alpha}}}\right)^{N}\leq\frac{\delta_{q+2}\lambda_{q+1}^{-\alpha}}{\delta_{q+1}}$

We also take the logarithm in $a$, divide $b^{q}\log a,$ and then get

$\displaystyle
N[b^{2}(1-\alpha)-cb^{2}(1-2\alpha)(1-\alpha)-\frac{1}{2-2\alpha}+\frac{cb}{1-\alpha}-b(1-\alpha)+\frac{b}{2-2\alpha}]$
$\displaystyle\leq-b^{2}-\alpha cb^{2}+b,$

which will be implied by

$b>\frac{1}{(1-\alpha)^{2}(1-2\alpha)},\quad
c>\frac{(b-1)(1+2b(1-\alpha)^{2})}{2b(b(1-\alpha)^{2}(1-2\alpha)-1)},$ (3.18)

and

$N\geq N_{0}:=1+\frac{2(1-\alpha)(b^{2}-b+\alpha
cb^{2})}{2cb[b(1-\alpha)^{2}(1-2\alpha)-1]-(b-1)(1+2b(1-\alpha)^{2})}\,.$
(3.19)

Note that letting $\alpha\downarrow 0,$ the bounds in (3.18) tend to

$b>1,\quad c>1+\frac{1}{2b}.$

Hence there exists $\alpha_{3}(b,c)$ small enough to make (3.18) hold for
$0<\alpha<\alpha_{3}$. Hence for any $0<\alpha<\alpha_{3},$ $N_{0}$ makes
sense and then for any $N\geq N_{0}$, (3.15) hold.

Finally, taking $\alpha_{0}=\min\\{\alpha_{1},\,\alpha_{2},\,\alpha_{3}\\}$
and $N_{0}$ as in (3.19) completes the proof. ∎

_Step 2: Mollification._ Let $\\{\varphi_{\sigma}\\}_{\sigma>0}$ be a standard
radially symmetric mollification kernel, and set
$v=v_{q}\ast\varphi_{\sigma}$, $w=w_{q}\ast\varphi_{\sigma}$ for $\sigma$
defined in (3.11) with $\alpha<\alpha_{0}$. From Lemma 2.2 we find for
$l=0,1,2$

$\displaystyle\|v_{q}-v\|_{l}+\|w_{q}-w\|_{l}$ $\displaystyle\leq
C\sigma^{2-l}\left(\|v_{q}\|_{2}+\|w_{q}\|_{2}\right)\leq
CC_{0}\sigma^{2-l}\sigma^{-1+\alpha}\delta_{q+1}^{\nicefrac{{1}}{{2}}}$ (3.20)
$\displaystyle\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{1-l+\nicefrac{{\alpha}}{{2}}}\,,$
(3.21)

and

$\|v\|_{2}+\|w\|_{2}\leq\|v_{q}\|_{2}+\|w_{q}\|_{2}\leq
C_{0}\delta_{q}^{\nicefrac{{1}}{{2}}}\lambda_{q}\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{-1+\nicefrac{{\alpha}}{{2}}}$
(3.22)

if $a\geq a_{0}(b,c,\alpha,C_{0})$ is large enough. More generally, for any
$l\geq 2$ and $\beta\in[0,1[$ it holds

$\|v\|_{l,\beta}+\|w\|_{l,\beta}\leq
C(l,\beta)\sigma^{2-(l+\beta)}\left(\|v_{q}\|_{2}+\|w_{q}\|_{2}\right)\leq
C(l,\beta)\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{-l+1-\beta+\nicefrac{{\alpha}}{{2}}}\,.$
(3.23)

From (3.6) we moreover get

$\|\nabla v\|_{0}\leq\|\nabla v_{q}\|_{0}\leq B.$ (3.24)

_Step 3: Decomposition._ Now define

$H=\frac{D\ast\varphi_{\sigma}-\frac{1}{2}\nabla v\otimes\nabla
v-\mathrm{sym}\nabla
w}{\delta_{q+1}}-\frac{\delta_{q+2}}{\delta_{q+1}}\mathrm{Id}\,.$ (3.25)

This is the rescaled defect we want to decompose. As explained in the
introduction however, we want incorporate some of the error terms resulting
from the first perturbation (see (3.45)–(3.46)) in the decomposition. The
error terms we want to absorb have the form

$\displaystyle E_{1}(x,a,M)$ $\displaystyle:=\frac{\gamma_{1}(\mu
x_{1})}{\mu}\sqrt{a}(M-M_{22}e_{2}\otimes e_{2})$ (3.26) $\displaystyle
E_{2}(x,a)$ $\displaystyle:=-\frac{\gamma_{3}(\mu x_{1})}{\mu}\nabla a\odot
e_{1}$ (3.27) $\displaystyle E_{3}(x,a)$
$\displaystyle:=-\frac{\gamma_{1}^{2}(\mu
x_{1})}{2\mu^{2}}\left(\nabla\sqrt{a}\otimes\nabla\sqrt{a}-\left(\partial_{2}\sqrt{a}\right)^{2}e_{2}\otimes
e_{2}\right)\,.$ (3.28)

where $\mu\geq 1$ is defined in (3.12), $M\in
C^{\infty}(\bar{\Omega},\mathcal{S})$, $a\in C^{\infty}(\Omega,]0,+\infty[)$
will be the coefficient of the decomposition, and

$\gamma_{1}(t)=\frac{1}{\pi}\sin(2\pi
t),\quad\gamma_{2}(t)=-\frac{1}{4\pi}\sin(4\pi
t),\quad\gamma_{3}=\frac{1}{2}\gamma_{1}\dot{\gamma_{1}}+\gamma_{2}\,.$ (3.29)

The functions $\gamma_{1},\gamma_{2}$ are responsible for the oscillatory
nature of the perturbations we will add (see (3.45),(3.46)) and are chosen as
in [14], [3]. They satisfy the crucial identity
$\frac{1}{2}\dot{\gamma_{1}}^{2}+\dot{\gamma_{2}}=1$ which is used below. In
the following we think of $\mu$ and $M$ as given and $a$ as variable. We thus
abbreviate

$E(a):=E_{1}(x,a,M)+E_{2}(x,a)+E_{3}(x,a)$ (3.30)

Then we have the following decomposition lemma.

######  Lemma 3.2.

For every $N\in\mathbb{N}$ and $\alpha\in\,]0,1[$ there exist constants
$\tau_{*}(N,\alpha,\Omega)\in\,]0,1[$ and $C(N,\alpha,\Omega)>0$ such that the
following holds for any $\sigma\in\,]0,1[$. If $\mu\geq 1$, and $H,M\in
C^{\infty}(\bar{\Omega},\mathcal{S})$ are such that

$\displaystyle\|H-\mathrm{Id}\|_{\alpha}+\mu^{\alpha-1}\sigma^{-1}\leq\tau_{*}\,,$
(3.31) $\displaystyle\|H\|_{l,\alpha}\leq C(l,\alpha)\sigma^{-l}\text{ for any
}l\in\mathbb{N},$ (3.32) $\displaystyle\|M\|_{l,\beta}\leq
C(l,\beta)\sigma^{-l-1-\beta}\text{ for any }l\in\mathbb{N},\beta\in[0,1[$
(3.33)

for some constants $C(l,\alpha),C(l,\beta)\geq 1$ then there exist maps
$\\{a^{(k)}\\}_{k=0}^{N}\subset C^{\infty}(\bar{\Omega},]0,+\infty[)$,
$\\{w^{(k)}\\}_{k=0}^{N}\subset C^{\infty}(\bar{\Omega},\mathbb{R}^{2})$ such
that for $k=1,\ldots,N$

$a^{(k)}\mathrm{Id}=H-\mathrm{sym}\nabla w^{(k)}+E(a^{(k-1)})\,.$ (3.34)

with estimates

$\displaystyle\|a^{(k)}-1\|_{0,\alpha}+\|w^{(k)}\|_{1,\alpha}$
$\displaystyle\leq\frac{1}{2}$ (3.35)
$\displaystyle\|a^{(k)}-1\|_{l,\alpha}+\|w^{(k)}\|_{l+1,\alpha}$
$\displaystyle\leq C(l,\alpha,\Omega)\sigma^{-1}\mu^{l-1}\quad\text{ for
}1\leq l\leq N+3-k$ (3.36)
$\displaystyle\|E(a^{(k)})-E(a^{(k-1)})\|_{0,\alpha}$ $\displaystyle\leq
C(N,\alpha,\Omega)\left(\mu^{\alpha-1}\sigma^{-1}\right)^{k+1}\,.$ (3.37)

######  Remark 3.1.

Due to the appearance of $\nabla a^{(k)}$ in $E(a^{(k)})$ one cannot use a
fixed point theorem to get the limit of $a^{(k)}$. Therefore, we only use
finitely many times iteration (3.34).

###### Proof.

The proof is in Section 5. ∎

To apply the decomposition lemma, we fix $\alpha<\alpha_{0}$ and $N\geq N_{0}$
such that Lemma 3.1 holds. With such $\alpha$, after taking $a$ large, we get
the following.

######  Lemma 3.3.

If $\tau_{0}<\frac{\tau_{*}}{6}$ and $a\geq a_{0}(\alpha,b,c,\Omega)$ are
chosen large enough, then it holds

$\displaystyle\|H-\mathrm{Id}\|_{0,\alpha}\leq\frac{\tau_{*}}{2}$ (3.38)
$\displaystyle\|H-\mathrm{Id}\|_{l,\alpha}\leq C(l,\alpha)\sigma^{-l},\text{
for }1\leq l\in\mathbb{N}.$ (3.39)

###### Proof.

Thanks to (3.5) and (3.7), we can estimate with Lemma 2.2

$\displaystyle\|H-\mathrm{Id}\|_{0,\alpha}$
$\displaystyle\leq\frac{1}{\delta_{q+1}}\|(D-\frac{1}{2}\nabla
v_{q}\otimes\nabla v_{q}-\mathrm{sym}\nabla
w_{q}-\delta_{q+1}\mathrm{Id})\ast\varphi_{\sigma}\|_{0,\alpha}+C\frac{\delta_{q+2}}{\delta_{q+1}}$
$\displaystyle\quad\quad+\frac{1}{2\delta_{q+1}}\|\nabla v\otimes\nabla
v-\left(\nabla v_{q}\otimes\nabla
v_{q}\right)\ast\varphi_{\sigma}\|_{0,\alpha}$ $\displaystyle\leq
2\tau_{0}+\frac{C}{\delta_{q+1}}\sigma^{2-\alpha}\|v_{q}\|_{2}^{2}\leq
3\tau_{0}\,,$

if $a\geq a_{0}(\tau_{0},C_{0})$ is chosen large enough. Moreover, using (2.2)
and Lemma 2.2 we infer

$\displaystyle\|H-\mathrm{Id}\|_{l,\alpha}$
$\displaystyle\leq\frac{C(l,\alpha)}{\delta_{q+1}}\left(\tau_{0}\delta_{q+1}\sigma^{-l}+\|v_{q}\|_{2}^{2}\sigma^{2-(l+\alpha)}\right)\leq
C(l,\alpha)\sigma^{-l}\,.\qed$

Moreover, from Lemma 3.1, it is easy to get

$\mu^{\alpha-1}\sigma^{-1}\leq\mu^{-\alpha}<\frac{\tau_{*}}{2}$ (3.40)

provided that $a$ is sufficiently large depending on $\tau_{*}$, i.e. on
$b,c,\alpha,\Omega$. Set

$M:=\delta_{q+1}^{-\nicefrac{{1}}{{2}}}D^{2}v\,,$ (3.41)

for which we have (thanks to (3.23))

$\|M\|_{l,\beta}\leq C(l,\beta)\sigma^{-l-1-\beta+{\alpha}}\leq
C(l,\beta)\sigma^{-l-1-\beta}$ (3.42)

for $l\in\mathbb{N}$, $\beta\in[0,1[$. In view of (3.38)–(3.39), (3.40) and
(3.42), we can therefore apply Lemma 3.2 to get maps
$\\{a^{(k)}\\}_{k=0}^{N},\\{w^{(k)}\\}_{k=0}^{N}$ satisfying the conclusions
of the lemma. In particular,

$a^{(N)}\mathrm{Id}=H-\mathrm{sym}\nabla
w^{(N)}+E(a^{(N-1)})=a^{(N)}(e_{1}\otimes e_{1}+e_{2}\otimes e_{2})\,$ (3.43)

with $M$ of $E_{1}$ defined in (3.41).

_Step 4: First perturbation._ Set

$r=\delta_{q+1}a^{(N)}\,,\quad w_{c}=\delta_{q+1}w^{(N)}$ (3.44)

and define the first perturbation

$\displaystyle\tilde{v}=v+\frac{\sqrt{r}}{\mu}\gamma_{1}(\mu x_{1})$ (3.45)
$\displaystyle\tilde{w}=w+w_{c}-\frac{\sqrt{r}}{\mu}\gamma_{1}(\mu
x_{1})\nabla v+\frac{r}{\mu}\gamma_{2}(\mu x_{1})e_{1}\,.$ (3.46)

A quick computation gives

$\nabla\tilde{v}=\nabla
v+\sqrt{r}\dot{\gamma_{1}}e_{1}+\frac{\gamma_{1}}{\mu}\nabla\sqrt{r}$ (3.47)

and

$\displaystyle\nabla\tilde{w}$ $\displaystyle=\nabla w+\nabla
w_{c}-\sqrt{r}\dot{\gamma_{1}}\nabla v\otimes
e_{1}+r\dot{\gamma_{2}}e_{1}\otimes e_{1}$
$\displaystyle\quad\quad-\frac{\gamma_{1}}{\mu}\nabla
v\otimes\nabla\sqrt{r}-\frac{\sqrt{r}}{\mu}\gamma_{1}D^{2}v+\frac{\gamma_{2}}{\mu}e_{1}\otimes\nabla
r\,.$ (3.48)

Consequently

$\displaystyle\frac{1}{2}\nabla\tilde{v}\otimes\nabla\tilde{v}$
$\displaystyle=\frac{1}{2}\nabla v\otimes\nabla
v+\frac{1}{2}r\dot{\gamma_{1}}^{2}e_{1}\otimes
e_{1}+\sqrt{r}\dot{\gamma_{1}}\nabla v\odot e_{1}$
$\displaystyle\quad\quad+\frac{\gamma_{1}}{\mu}\nabla\sqrt{r}\odot\nabla
v+\frac{\gamma_{1}\dot{\gamma_{1}}}{\mu}\sqrt{r}e_{1}\odot\nabla\sqrt{r}$
$\displaystyle\quad\quad+\frac{\gamma_{1}^{2}}{2\mu^{2}}\nabla\sqrt{r}\otimes\nabla\sqrt{r}\,,$

and therefore

$\displaystyle\frac{1}{2}\nabla\tilde{v}\otimes\nabla\tilde{v}+\mathrm{sym}\nabla\tilde{w}$
$\displaystyle=\frac{1}{2}\nabla v\otimes\nabla v+\mathrm{sym}\nabla
w+\mathrm{sym}\nabla w_{c}+re_{1}\otimes e_{1}$
$\displaystyle\quad+\frac{\gamma_{3}}{\mu}\nabla r\odot
e_{1}-\frac{\sqrt{r}}{\mu}\gamma_{1}D^{2}v+\frac{\gamma_{1}^{2}}{2\mu^{2}}\nabla\sqrt{r}\otimes\nabla\sqrt{r}$

recalling the definition of $\gamma_{3}$ in (3.29) and
$\frac{1}{2}\dot{\gamma_{1}}^{2}+\dot{\gamma_{2}}\equiv 1$. But from (3.34) we
have

$\mathrm{sym}\nabla w_{c}=\delta_{q+1}H-r\mathrm{Id}+\delta_{q+1}E(a^{(N-1)})$

so that

$\displaystyle\frac{1}{2}\nabla\tilde{v}\otimes\nabla\tilde{v}+\mathrm{sym}\nabla\tilde{w}$
$\displaystyle=\frac{1}{2}\nabla v\otimes\nabla v+\mathrm{sym}\nabla
w+\delta_{q+1}H$
$\displaystyle\quad\quad+\delta_{q+1}\left(E(a^{(N-1)})-E(a^{(N)}))\right)$
$\displaystyle\quad\quad+\left(\frac{\sqrt{r}}{\mu}\gamma_{1}\partial^{2}_{22}v-\frac{\gamma_{1}^{2}}{2\mu^{2}}\left(\partial_{2}\sqrt{r}\right)^{2}-r\right)e_{2}\otimes
e_{2}$
$\displaystyle=D\ast\varphi_{\sigma}-\delta_{q+2}\mathrm{Id}+\delta_{q+1}\left(E(a^{(N-1)})-E(a^{(N)})\right)$
$\displaystyle\quad\quad+\left(\frac{\sqrt{r}}{\mu}\gamma_{1}\partial^{2}_{22}v-\frac{\gamma_{1}^{2}}{2\mu^{2}}\left(\partial_{2}\sqrt{r}\right)^{2}-r\right)e_{2}\otimes
e_{2}$ (3.49)

_Step 5: Second perturbation._ For the second perturbation we set

$s=r-\frac{\sqrt{r}}{\mu}\gamma_{1}\partial^{2}_{22}v+\frac{\gamma_{1}^{2}}{2\mu^{2}}\left(\partial_{2}\sqrt{r}\right)^{2}$
(3.50)

and let

$\displaystyle
v_{q+1}=\tilde{v}+\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}\left(\lambda_{q+1}x_{2}\right)$
(3.51) $\displaystyle
w_{q+1}=\tilde{w}-\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}\left(\lambda_{q+1}x_{2}\right)\nabla\tilde{v}+\frac{s}{\lambda_{q+1}}\gamma_{2}\left(\lambda_{q+1}x_{2}\right)e_{2}\,.$
(3.52)

We will show below in (3.58) that these choices are well-defined. As above we
find

$\displaystyle\frac{1}{2}\nabla v_{q+1}\otimes\nabla
v_{q+1}+\mathrm{sym}\nabla w_{q+1}=$
$\displaystyle\frac{1}{2}\nabla\tilde{v}\otimes\nabla\tilde{v}+\mathrm{sym}\nabla\tilde{w}+se_{2}\otimes
e_{2}+\frac{\gamma_{3}}{\lambda_{q+1}}\nabla s\odot e_{2}$
$\displaystyle\quad-\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}D^{2}\tilde{v}+\frac{\gamma_{1}^{2}}{2\lambda_{q+1}^{2}}\nabla\sqrt{s}\otimes\nabla\sqrt{s}$
$\displaystyle=$ $\displaystyle
D\ast\varphi_{\sigma}-\delta_{q+2}\mathrm{Id}+\delta_{q+1}\left(E(a^{(N-1)})-E(a^{(N)})\right)$
$\displaystyle\quad+\frac{\gamma_{3}}{\lambda_{q+1}}\nabla s\odot
e_{2}-\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}D^{2}\tilde{v}$
$\displaystyle\quad+\frac{\gamma_{1}^{2}}{2\lambda_{q+1}^{2}}\nabla\sqrt{s}\otimes\nabla\sqrt{s}\,.$
(3.53)

In particular,

$\displaystyle D-\frac{1}{2}\nabla v_{q+1}\otimes\nabla
v_{q+1}-\mathrm{sym}\nabla w_{q+1}-\delta_{q+2}\mathrm{Id}$ (3.54)
$\displaystyle=$ $\displaystyle
D-D\ast\varphi_{\sigma}-\delta_{q+1}\left(E(a^{(N-1)})-E(a^{(N)}\right)$
$\displaystyle\quad-\frac{\gamma_{3}}{\lambda_{q+1}}\nabla s\odot
e_{2}-\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}D^{2}\tilde{v}$
$\displaystyle\quad+\frac{\gamma_{1}^{2}}{2\lambda_{q+1}^{2}}\nabla\sqrt{s}\otimes\nabla\sqrt{s}\,,$
(3.55)

It now remains to show that the $\alpha$-Hölder norm of the right hand side is
less than $\sigma_{0}\delta_{q+2}$, that (3.7) holds for $q+1$, and that
(3.9)–(3.10) hold provided $a$ and $C_{0}$ are chosen large enough and
$\alpha$ small enough. In the following estimates, the constants $C$ may
depend on $\alpha,b,c,N,\Omega$ and $B$, _but not on_ $C_{0}$ or $a$.

_Step 6: Estimates on $r,\tilde{v},\tilde{w}$._ Using (3.35) and (3.36) we
find

$\|r-\delta_{q+1}\|_{0,\alpha}\leq\frac{1}{2}\delta_{q+1}\,,\quad\|r-\delta_{q+1}\|_{l,\alpha}\leq
C\delta_{q+1}\mu^{l-1}\sigma^{-1}\,\text{ for }l=1,2,3\,$

and with the help of Lemma 2.1

$\|\sqrt{r}-\delta_{q+1}^{\nicefrac{{1}}{{2}}}\|_{0,\alpha}\leq\frac{1}{2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,,\quad\|\sqrt{r}-\delta_{q+1}^{\nicefrac{{1}}{{2}}}\|_{l,\alpha}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l-1}\sigma^{-1}\,\text{ for
}l=1,2,3\,.$

From there we get

$\|\tilde{v}-v\|_{0}\leq\frac{3}{2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{-1}\,,$

and

$\|\tilde{v}-v\|_{l}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\left(\mu^{l-2}\sigma^{-1}+\mu^{l-1}\right)\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l-1}\,\text{ for }l=1,2,$ (3.56)

since $\mu^{\alpha-1}\sigma^{-1}<1$ by (3.40). For $\tilde{w}$ we estimate for
$l=1,2$

$\displaystyle\|\frac{\sqrt{r}}{\mu}\gamma_{1}(\mu x_{1})\nabla v\|_{l}$
$\displaystyle\leq
C\left(\mu^{-1}\|\sqrt{r}\|_{l,\alpha}+\|\sqrt{r}\|_{0,\alpha}\left(\mu^{l-1}+\mu^{-1}\|v\|_{l+1}\right)\right)$
$\displaystyle\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\left(\mu^{l-2}\sigma^{-1}+\mu^{l-1}+\mu^{-1}\sigma^{-l+{\alpha}}\right)\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l-1}\,,$

again due to $\mu^{\alpha-1}\sigma^{-1}<1$. With (3.47),(3.23), we have

$\|\frac{r}{\mu}\gamma_{2}(\mu x_{1})e_{1}\|_{l}\leq
C\mu^{-1}\delta_{q+1}\left(\mu^{l-1}\sigma^{-1}+\mu^{l}\right)\leq
C\delta_{q+1}\mu^{l-1}$

for $l=1,2$, then in combination with (3.36),

$\|\tilde{w}-w\|_{l}\leq
C\delta_{q+1}\sigma^{-1}\mu^{l-2}+C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l-1}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l-1}\,$ (3.57)

for $l=1,2$.

_Step 7: Estimates on $s$, $v_{q+1}$, $w_{q+1}$._ By (3.23) and
$\sigma^{-\alpha}\leq\mu^{\alpha}$ we can estimate

$\displaystyle\|\frac{s}{\delta_{q+1}}-1\|_{0,\alpha}$
$\displaystyle\leq\|a^{(N)}-1\|_{0,\alpha}+\delta_{q+1}^{-\nicefrac{{1}}{{2}}}\left(\mu^{\alpha-1}\|v\|_{2}+\mu^{-1}\|v\|_{2,\alpha}\right)+\mu^{\alpha-2}\|\sqrt{a^{(N)}}\|_{1,\alpha}^{2}$
$\displaystyle\leq\frac{1}{2}+C\left(\mu^{\alpha-1}\sigma^{{\alpha}-1}+\mu^{-1}\sigma^{-1-{\alpha}}\right)+C\mu^{\alpha-2}\sigma^{-2}$
$\displaystyle\leq\frac{1}{2}+C\mu^{-\alpha}\leq\frac{3}{4}$

if $a$ is large enough. It follows that

$\|s-\delta_{q+1}\|_{0,\alpha}\leq\frac{3}{4}\delta_{q+1}\text{ and
}\|\sqrt{s}-\delta_{q+1}^{\nicefrac{{1}}{{2}}}\|_{0,\alpha}\leq\frac{3}{4}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,,$
(3.58)

and hence $v_{q+1}$ and $w_{q+1}$ are well-defined. For the derivatives we
find for $l=1,2$

$\displaystyle\|\frac{s}{\delta_{q+1}}-1\|_{l,\alpha}$
$\displaystyle\leq\|a^{(N)}-1\|_{l,\alpha}+C\delta_{q+1}^{-\nicefrac{{1}}{{2}}}\mu^{-1}\left(\left(\|\sqrt{a^{(N)}}\|_{l,\alpha}+\mu^{l+\alpha}\right)\|v\|_{2}+\|v\|_{l+2,\alpha}\right)$
$\displaystyle\quad+C\left(\mu^{l+\alpha-2}\|\sqrt{a^{(N)}}\|_{1,\alpha}^{2}+\mu^{-2}\|\sqrt{a^{(N)}}\|_{l+1,\alpha}\|\sqrt{a^{(N)}}\|_{1,\alpha}\right)$
$\displaystyle\leq
C\mu^{l-1}\sigma^{-1}+C\delta_{q+1}^{-\nicefrac{{1}}{{2}}}\mu^{-1}\left(\mu^{l-1}\sigma^{-1}+\mu^{l+\alpha}\right)\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{{\alpha}-1}$
$\displaystyle\quad+C\delta_{q+1}^{-\nicefrac{{1}}{{2}}}\mu^{-1}\sigma^{-(l+\alpha)}\cdot\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{{\alpha}-1}+C\left(\mu^{l+\alpha-2}\sigma^{-2}+\mu^{l-2}\sigma^{-2}\right)$
$\displaystyle\leq C\mu^{l-1+\alpha}\sigma^{-1}\leq
C\mu^{l-\alpha}\leq\mu^{l}\,,$

if $a$ is large enough by (3.13). Thus

$\|s-\delta_{q+1}\|_{l,\alpha}\leq\delta_{q+1}\mu^{l}\,,\quad\|\sqrt{s}-\delta_{q+1}^{\nicefrac{{1}}{{2}}}\|_{l,\alpha}\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l}\,$
(3.59)

for $l=1,2$. This yields the estimates

$\|v_{q+1}-\tilde{v}\|_{0}\leq\frac{3}{2}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{-1}\,,$
(3.60)

and

$\|v_{q+1}-\tilde{v}\|_{l}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{-1}\left(\lambda_{q+1}^{l}+\mu^{l}\right)\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{l-1}\,$ (3.61)

for $l=1,2$, where we have used $\lambda_{q+1}>\mu.$ Concerning the estimate
for $w_{q+1}$ we have for $l=1,2$

$\displaystyle\|w_{q+1}-\tilde{w}\|_{l}$ $\displaystyle\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{-1}\left(\lambda_{q+1}^{l}+\mu^{l}+\delta_{q+1}^{\nicefrac{{1}}{{2}}}\mu^{l}+\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{-l+\nicefrac{{\alpha}}{{2}}}\right)$
$\displaystyle\quad+C\delta_{q+1}\lambda_{q+1}^{-1}\left(\lambda_{q+1}^{l}+\mu^{l}\right)$
$\displaystyle\leq C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{l-1}\,.$
(3.62)

Combining with (3.22) and (3.56), (3.57) this yields

$\|v_{q+1}\|_{2}+\|w_{q+1}\|_{2}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\left(\sigma^{-1+\nicefrac{{\alpha}}{{2}}}+\mu+\lambda_{q+1}\right)\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}\,.$

Moreover,

$\displaystyle\|v_{q}-v_{q+1}\|_{0}$
$\displaystyle\leq\|v_{q}-v\|_{0}+\|v-\tilde{v}\|_{0}+\|\tilde{v}-v_{q+1}\|_{0}$
$\displaystyle\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\left(\sigma+\frac{3}{2}\mu^{-1}+\frac{3}{2}\lambda_{q+1}^{-1}\right)\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{-1}$

for $a$ large enough in view of

$\sigma<\sigma^{1-\alpha}\leq\left(\frac{\delta_{q+1}}{\delta_{q}}\right)^{\nicefrac{{1}}{{2}}}\lambda_{q}^{-1}\,.$

Lastly, we find

$\|v_{q+1}-v_{q}\|_{1}+\|w_{q+1}-w_{q}\|_{1}\leq\delta_{q+1}^{\nicefrac{{1}}{{2}}}\sigma^{\nicefrac{{\alpha}}{{2}}}+C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\,.$

This gives (3.7), (3.9) and (3.10) provided $C_{0}$ is large enough (depending
on $\alpha,b,c,\Omega,B$), and also (3.6)

$\|v_{q+1}\|_{1}+\|w_{q+1}\|_{1}\leq\|v_{q}\|_{1}+\|w_{q}\|_{1}+\|v_{q+1}-v_{q}\|_{1}+\|w_{q+1}-w_{q}\|_{1}\leq
B-\delta_{q}^{\nicefrac{{1}}{{2}}}+C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\leq
B-\delta_{q+1}^{\nicefrac{{1}}{{2}}}$

by taking $a$ large enough.

_Step 8: Estimates on $\mathcal{E}_{1},\mathcal{E}_{2}$ and
$\mathcal{E}_{3}$._ We are left to show (3.5) for $q+1$. By (3.55) we have

$D-\frac{1}{2}\nabla v_{q+1}\otimes\nabla v_{q+1}-\mathrm{sym}\nabla
w_{q+1}-\delta_{q+2}\mathrm{Id}=\mathcal{E}_{1}+\mathcal{E}_{2}+\mathcal{E}_{3}$

for

$\displaystyle\mathcal{E}_{1}$ $\displaystyle=D-D\ast\varphi_{\sigma}$
$\displaystyle\mathcal{E}_{2}$
$\displaystyle=-\delta_{q+1}\left(E(a^{(N-1)})-E(a^{(N)})\right)$
$\displaystyle\mathcal{E}_{3}$
$\displaystyle=-\frac{\gamma_{3}}{\lambda_{q+1}}\nabla s\odot
e_{2}-\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}D^{2}\tilde{v}+\frac{\gamma_{1}^{2}}{2\lambda_{q+1}^{2}}\nabla\sqrt{s}\otimes\nabla\sqrt{s}\,,$

and we estimate each term separately. First, by Lemma 2.2, $\sigma<1$ and
$\gamma>\frac{2}{3},$

$\|\mathcal{E}_{1}\|_{0,\alpha}\leq
C\sigma^{\gamma-\alpha}\|D\|_{0,\gamma}\leq
C\sigma^{\frac{2}{3}-\alpha}\|D\|_{0,\gamma}\,,$ (3.63)

which combining with (3.14) will leads to

$\|\mathcal{E}_{1}\|_{0,\alpha}\leq
C\delta_{q+2}\lambda_{q+1}^{-\alpha}\|D\|_{0,\gamma}\leq\frac{\tau_{0}}{3}\delta_{q+2},$

if $a$ is large enough additionally depending on $\|D\|_{0,\gamma}$. Next, we
find from (3.37)

$\|\mathcal{E}_{2}\|_{0,\alpha}\leq
C\delta_{q+1}\left(\mu^{\alpha-1}\sigma^{-1}\right)^{N+1}\leq\frac{\tau_{0}}{3}\delta_{q+2}\,,$
(3.64)

by (3.15) and taking $a$ large enough. Lastly, using (3.59) and (3.56), we can
estimate

$\|\frac{\gamma_{3}}{\lambda_{q+1}}\nabla s\odot e_{2}\|_{0,\alpha}\leq
C\delta_{q+1}\lambda_{q+1}^{\alpha-1}\mu^{\alpha}\sigma^{-1}\,,$
$\|\frac{\sqrt{s}}{\lambda_{q+1}}\gamma_{1}D^{2}\tilde{v}\|_{0,\alpha}\leq
C\delta_{q+1}\left(\lambda_{q+1}^{\alpha-1}\mu+\lambda_{q+1}^{-1}\mu^{1+\alpha}\right)\leq
C\delta_{q+1}\lambda^{\alpha-1}\mu\,,$

and

$\|\frac{\gamma_{1}^{2}}{2\lambda_{q+1}^{2}}\nabla\sqrt{s}\otimes\nabla\sqrt{s}\|_{0,\alpha}\leq
C\delta_{q+1}\lambda_{q+1}^{\alpha-2}\mu^{2\alpha}\sigma^{-2}\,.$

Then using $\mu^{\alpha-1}\sigma^{-1}<1$ and $\lambda_{q+1}\geq\mu$, we get

$\|\mathcal{E}_{3}\|_{0,\alpha}\leq
C\delta_{q+1}\lambda^{\alpha-1}_{q+1}\mu\leq\frac{\tau_{0}}{3}\delta_{q+2}\,,$

by (3.12) and taking $a$ sufficiently large. ∎

##  4\. Proof of Theorem 1.1

Let $\underline{v}\in C^{0}(\bar{\Omega})$, $f\in L^{p}(\Omega)$, $\epsilon>0$
and $\theta<\frac{1}{3}$ be given. By extending and mollifying we can assume
$\underline{v}\in C^{\infty}(\bar{\Omega})$. Then Theorem 1.1 will be proven
by seeking $C^{1,\theta}$ solutions to (1.1) $C^{0}$ neighbourhood of
$\underline{v},$ which is realised by solving the system (1.2) and (1.3) in
the

We first solve (1.3). Let $u\in W^{2,p}(\Omega)$ be the potential theoretic
solution of $-\Delta u=f$ on $\Omega$. By Morrey’s embedding it holds $u\in
C^{0,\gamma}(\bar{\Omega})$ for $\gamma=2-\frac{2}{p}\geq\frac{2}{3}$ and

$\|u\|_{0,\gamma}\leq C\|f\|_{L^{p}}.$

We then fix $b>1,c>\frac{3}{2}$ such that $\frac{1}{2bc}<\theta$, some
$\alpha<\alpha_{0}(b,c,\gamma)<\gamma$ and define

$A=(u+\kappa)\mathrm{Id},\text{ with
}\kappa=\frac{\|u\|_{0,\gamma}+\|\underline{v}\|^{2}_{1,\gamma}+100}{\tau_{0}}>1.$

where $\tau_{0}=\tau_{0}(\alpha,b,c,\Omega)<1$ is from Proposition 3.1, then

$\|A\|_{0,\gamma}\leq\|u\|_{0,\gamma}+\kappa,\quad-\mathrm{curl}~{}\mathrm{curl}A=f.$

Next we shall apply Proposition 3.1 to solve (1.2). To this end fix

$\displaystyle B=$ $\displaystyle\|\underline{v}\|_{1}+2$ $\displaystyle
a\geq$ $\displaystyle
a_{0}(\alpha,b,c,\Omega,\|A\|_{0,\gamma},C_{0}(\alpha,b,c,\Omega,B))$ (4.1)

Note that $a_{0}(\alpha,b,c,\Omega,\|D\|_{0,\gamma},C_{0})$ in Proposition 3.1
is monotonically increasing in $\|D\|_{0,\gamma}$. Hence for any choice of $a$
according to (4.1) it follows also

$a\geq
a_{0}(\alpha,b,c,\Omega,\|\delta_{1}\kappa^{-1}A\|_{0,\gamma},C_{0}(\alpha,b,c,\Omega,B)),$

where $\delta_{1}=a^{-b}<1$. Therefore, setting

$\bar{A}={\delta_{1}}{\kappa}^{-1}A,\quad
v_{0}=\delta_{1}^{\nicefrac{{1}}{{2}}}{\kappa}^{-\nicefrac{{1}}{{2}}}\underline{v},\quad
w_{0}=0\,,$

gives $\bar{A}\in C^{0,\gamma}(\bar{\Omega},\mathcal{S}),v_{0}\in
C^{\infty}(\bar{\Omega}),w_{0}\in C^{\infty}(\bar{\Omega},\mathbb{R}^{2})$ and

$\displaystyle\|\bar{A}-\frac{1}{2}\nabla v_{0}\otimes\nabla
v_{0}-\mathrm{sym}\nabla
w_{0}-\delta_{1}\mathrm{Id}\|_{0,\alpha}\leq\tau_{0}\delta_{1}$
$\displaystyle\|v_{0}\|_{1}+\|w_{0}\|_{1}\leq
B-\delta_{0}^{\nicefrac{{1}}{{2}}},$
$\displaystyle\|v_{0}\|_{2}+\|w_{0}\|_{2}\leq\delta_{0}^{\nicefrac{{1}}{{2}}}\lambda_{0}.$

We can therefore iteratively apply Proposition 3.1 with $D=\bar{A}$ to find
sequences $\\{v_{q}\\}_{q\in\mathbb{N}}\subset C^{\infty}(\bar{\Omega})$,
$\\{w_{q}\\}_{q\in\mathbb{N}}\subset C^{\infty}(\bar{\Omega},\mathbb{R}^{2})$
satisfying (3.5)–(3.10) for every $q$. Observe that by interpolation (2.1)

$\|v_{q+1}-v_{q}\|_{1,\theta}\leq
C\|v_{q+1}-v_{q}\|_{1}^{1-\theta}\|v_{q+1}-v_{q}\|_{2}^{\theta}\leq
C\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q+1}^{\theta}=Ca^{-b^{q+1}\left(\nicefrac{{1}}{{2}}-\theta
cb\right)}\,.$

Since $\theta<\frac{1}{2bc}$ by assumption this shows that $\\{v_{q}\\}$ is
Cauchy in $C^{1,\theta}(\bar{\Omega})$ and therefore converges to some
$v_{\infty}$. Moreover, from (3.10) we see that $w_{q}\to w_{\infty}$ in
$C^{1}$. Passing to the limit in (3.5) shows that

$\bar{A}=D=\frac{1}{2}\nabla v_{\infty}\otimes\nabla
v_{\infty}+\mathrm{sym}\nabla w_{\infty}\,.$

We now claim that if $a$ is chosen large enough (depending on $\kappa$ and
$\epsilon$) it holds

$\|v_{0}-v_{\infty}\|_{0}<\delta_{1}^{\nicefrac{{1}}{{2}}}{\kappa}^{-\nicefrac{{1}}{{2}}}\epsilon\,.$
(4.2)

Indeed, from (3.9), we get

$\|v_{0}-v_{\infty}\|_{0}\leq\sum_{q=0}^{\infty}\delta_{q+1}^{\nicefrac{{1}}{{2}}}\lambda_{q}^{-1}\,.$

Since $\delta_{q+1}\leq\delta_{1}$ is monotonically decreasing, (4.2) is
implied by

$\sum_{q=0}^{\infty}\lambda_{q}^{-1}<\frac{\epsilon}{\sqrt{\kappa}}\,,$

which is easily achieved by choosing $a$ large enough. Setting

$\bar{v}=\delta_{1}^{-\nicefrac{{1}}{{2}}}{\kappa}^{\nicefrac{{1}}{{2}}}v_{\infty}$

we have

$\|\bar{v}-\underline{v}\|_{0}=\delta_{1}^{-\nicefrac{{1}}{{2}}}{\kappa}^{\nicefrac{{1}}{{2}}}\|v_{0}-v_{\infty}\|_{0}\leq\epsilon,$

and

$\mathcal{D}\text{et}D^{2}\bar{v}=-\delta_{1}^{-1}\kappa~{}\mathrm{curl}~{}\mathrm{curl}\bar{A}=-\mathrm{curl}~{}\mathrm{curl}A=f,$

then finish the proof.

##  5\. Proof of Lemma 3.2

The proof proceeds by inductive construction of $a^{(k)},w^{(k)}$ satisfying
(3.34), (3.35) and (3.36), and then we show that (3.37) holds. In order to do
so we need slightly subtler estimates (see (5.9), (5.10)).

First of all, $a^{(0)}\in C^{\infty}(\bar{\Omega}),w^{(0)}\in
C^{\infty}\left(\bar{\Omega},\mathbb{R}^{2}\right)$ satisfying

$H-\mathrm{sym}\nabla w^{(0)}=a^{(0)}\mathrm{Id}$ (5.1)

can be easily found by Cauchy transform in complex analysis, which has also
been established in [3] in annother way. Indeed, fix a smooth bounded open set
$\tilde{\Omega}$ such that $\Omega\Subset\tilde{\Omega}$. The Cauchy transform

$\mathcal{C}[h](z):=-\frac{1}{\pi}\int_{\tilde{\Omega}}\frac{h(\omega)}{\omega-z}\,d\omega$

satisfies

$\partial_{\bar{z}}\mathcal{C}[h]=h\text{ (where
$\partial_{\bar{z}}=\frac{1}{2}(\partial_{x_{1}}+i\partial_{x_{2}}$))}$

and

$\|\mathcal{C}[h]\|_{l+1,\alpha}\leq C(l,\alpha)\|h\|_{l,\alpha}$ (5.2)

for any $l\in\mathbb{N}$, $f\in C^{l,\alpha}(\tilde{\Omega})$. Moreover, on
$C^{l,\alpha}_{c}(\tilde{\Omega})$, it commutes with any differential operator
of order $l$ (see for example Appendix A in [8]). Extend $H$ given in (3.25)
to $H\in C^{\infty}_{c}(\tilde{\Omega},\mathbb{R}^{2\times 2})$, let
$\hat{H}=H_{11}-H_{22}+i(H_{12}+H_{21})$ and set

$\displaystyle w^{(0)}$ $\displaystyle=\frac{1}{2}\mathcal{C}[\hat{H}]$
$\displaystyle a^{(0)}$ $\displaystyle=H_{22}-\partial_{2}w_{2}^{(0)}$

A quick computation then shows that (5.1) is satisfied. Moreover, using (5.2)
for $h=\widehat{H-\mathrm{Id}}$ we get

$\|a^{(0)}-1\|_{0,\alpha}+\|w^{(0)}\|_{1,\alpha}\leq
C\|H-\mathrm{Id}\|_{0,\alpha}\leq C\tau_{*}<\frac{1}{4}$ (5.3)

for $\tau_{*}$ small enough, and

$\|\partial^{I}(a^{(0)}-1)\|_{0,\alpha}+\|\partial^{I}w^{(0)}\|_{1,\alpha}\leq
C\|\partial^{I}(h-\mathrm{Id})\|_{0,\alpha}\leq C(N,\alpha)\sigma^{-|I|}$
(5.4)

for all multi-indices $I$ with $|I|\leq N+3$.

Now fix a cutoff function $\chi\in C^{\infty}_{c}(\tilde{\Omega})$ with
$\chi\equiv 1$ on $\Omega$ and define inductively $a^{(k)},w^{(k)},H^{(k)}$ by

$\displaystyle H^{(k)}$ $\displaystyle=H+\chi E(a^{(k-1)})$ (5.5)
$\displaystyle w^{(k)}$
$\displaystyle=\frac{1}{2}\mathcal{C}[\widehat{H^{(k)}}]$ (5.6) $\displaystyle
a^{(k)}$ $\displaystyle=H_{22}-\partial_{2}w_{2}^{(k)}\,$ (5.7)

which then yields (3.34) due to $E(a^{(k-1)})_{22}=0$. We now

Claim: the following estimates hold:

$\displaystyle\|a^{(k)}-1\|_{0,\alpha}+\|w^{(k)}\|_{1,\alpha}$
$\displaystyle\leq\frac{1}{2},$ (5.8)
$\displaystyle\|\partial_{2}^{l}\left(a^{(k)}-1\right)\|_{0,\alpha}+\|\partial_{2}^{l}w^{(k)}\|_{1,\alpha}$
$\displaystyle\leq C(k+1,\alpha)\sigma^{-l}\quad\text{ for }0\leq l\leq
N+3-k,$ (5.9)

and

$\begin{split}&\|\partial^{j+1}_{1}\partial_{2}^{l}\left(a^{(k)}-1\right)\|_{0,\alpha}+\|\partial_{1}^{j}\partial_{2}^{l+1}w^{(k)}\|_{1,\alpha}\\\
\leq&C(k+1,\alpha)\sigma^{-(l+1)}\mu^{j}\quad\text{ for }1\leq l+j+1\leq
N+3-k\,.\end{split}$ (5.10)

Observe that in (5.10) we manage to trade a $\partial_{1}$ derivative on the
coefficient $a^{(k)}$ (which is large due to the fast oscillation appearing in
$E$) for a “slower” derivative in $x_{2}$-direction. This is thanks to the
expression (3.26) for $a^{(k)}$, since now for example

$\partial_{1}a^{(k)}=\partial_{1}H_{22}-\partial_{1}\partial_{2}w^{(k)}_{2}=\partial_{1}H_{22}-\frac{1}{2}\partial_{1}\mathcal{C}[\partial_{2}\widehat{H^{(k)}}]_{2}\,,$

so

$\|\partial_{1}a^{(k)}\|_{0,\alpha}\leq
C\left(\|H\|_{1,\alpha}+\|\mathcal{C}[\partial_{2}\widehat{H^{(k)}}]\|_{1,\alpha}\right)\leq
C\left(\|H\|_{1,\alpha}+\|\partial_{2}H^{(k)}\|_{0,\alpha}\right)\,.$

It is for this reason that we choose to subtract the $e_{2}\otimes e_{2}$
components in the definition of the error terms in (3.26) and (3.28), since
this leads to expression (5.7).

Now we show the claim by induction. Note that the claim holds for $k=0$, since
$\sigma^{-1}\leq\mu$ (due to $\mu^{\alpha-1}\sigma^{-1}<\tau_{*}<1$). Assume
that the claim is true for $k-1$ case. Then $E(a^{(k-1)})$ (and therefore
$H^{(k)}$, $w^{(k)}$ and $a^{(k)}$) is well-defined thanks to (5.8). Let us
show that (5.8)-(5.10) hold for $k$. Firstly, we find

$\displaystyle\|a^{(k)}-1\|_{0,\alpha}+\|w^{(k)}\|_{1,\alpha}$
$\displaystyle\leq$ $\displaystyle
C\left(\|H-\mathrm{Id}\|_{0,\alpha}+\|\chi\|_{0,\alpha}\|E(a^{(k-1)})\|_{0,\alpha}\right)$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}+C\|E(a^{(k-1)})\|_{0,\alpha}$

because of (5.3). So we shall estimate $\|E(a^{(k-1)})\|_{0,\alpha}$. By the
induction assumption (5.8)-(5.10) with $k$ replaced by $k-1$, we immediately
have

$\|\partial^{j_{1}}_{1}\partial^{l_{1}}_{2}a^{(k-1)}\ldots\partial^{j_{m}}_{1}\partial_{2}^{l_{m}}a^{(k-1)}\|_{0,\alpha}\leq
C\mu^{j_{1}-1}\sigma^{-(l_{1}+1)}\mu^{j_{2}}\sigma^{-l_{2}}\cdots\mu^{j_{m}}\sigma^{-l_{m}}\leq
C\mu^{j}\sigma^{-(l+1)}\,$

for any $j_{1}>j_{2}\geq\ldots\geq j_{m}$ and $l_{1}\geq\ldots\geq l_{m}$ with
$\sum_{i}j_{i}=j+1$, $\sum_{i}l_{i}=l$, using $k,m\leq N+3$. Since
$\|(a^{(k-1)})^{-m/2}\|_{0,\alpha}\leq C$ for $m\leq N+3$, we therefore find
by Faà di Bruno’s formula

$\displaystyle\|\partial_{2}^{l}\sqrt{a^{(k-1)}}\|_{0,\alpha}$
$\displaystyle\leq C(N,\alpha)\sigma^{-l}\quad\quad\text{ for
}l=0,\ldots,N+3-(k-1)$ (5.11)
$\displaystyle\|\partial_{1}^{j+1}\partial_{2}^{l}\sqrt{a^{(k-1)}}\|_{0,\alpha}$
$\displaystyle\leq C(N,\alpha)\mu^{j}\sigma^{-(l+1)}\quad\text{ for }0\leq
l+j+1\leq N+3-(k-1).$ (5.12)

Thus we can estimate

$\begin{split}&\|E_{1}(a^{(k-1)})\|_{0,\alpha}\leq
C\left(\mu^{\alpha-1}\|M\|_{0}+\mu^{-1}\|M\|_{0,\alpha}\right)\\\
\leq&C\left(\mu^{\alpha-1}\sigma^{-1}+\mu^{-1}\sigma^{-1-\alpha}\right)\leq
C\mu^{\alpha-1}\sigma^{-1}\,,\end{split}$ (5.13)

due to (3.33), (5.11) and $\sigma^{-1}\leq\mu$, and similarly

$\|E_{2}(a^{(k-1)})\|_{0,\alpha}\leq C\mu^{\alpha-1}\sigma^{-1}$ (5.14)

by the induction assumption. Also

$\|E_{3}(a^{(k-1)})\|_{0,\alpha}\leq C\mu^{\alpha-2}\sigma^{-2}\leq
C\mu^{\alpha-1}\sigma^{-1}\,,$ (5.15)

by (5.11) and (5.12), so that (5.8) is true for sufficiently small
$\tau_{*}(N,\alpha)$. As for the estimate (5.9), we have

$\displaystyle\|\partial_{2}^{l}\left(a^{(k)}-1\right)\|_{0,\alpha}+\|\partial_{2}^{l}w^{(k)}\|_{1,\alpha}$
$\displaystyle\leq
C\left(\|\partial_{2}^{l}(H-\mathrm{Id})\|_{0,\alpha}+\|\partial^{l}_{2}E(a^{(k-1)})\|_{0,\alpha}\right)$
$\displaystyle\leq
C\left(\sigma^{-l}+\sum_{i}\|\partial_{2}^{l}E_{i}(a^{(k-1)})\|_{0,\alpha}\right)\,.$

By the induction assumptions and (3.32), (3.33), (5.11), (5.12), we get

$\displaystyle\|\partial_{2}^{l}E_{1}(a^{(k-1)})\|_{0,\alpha}$
$\displaystyle\leq
C(k,\alpha)\left(\mu^{\alpha-1}\|\partial^{l}_{2}(\sqrt{a^{(k-1)}}M)\|_{0}+\mu^{-1}\|\partial^{l}_{2}(\sqrt{a^{(k-1)}}M)\|_{0,\alpha}\right)$
$\displaystyle\leq
C(k,\alpha)\left(\mu^{\alpha-1}\sigma^{-l-1}+\mu^{-1}\sigma^{-l-1-\alpha}\right)\leq
C(k,\alpha)\mu^{\alpha-1}\sigma^{-1-l}\,,$ (5.16)

and

$\|\partial_{2}^{l}E_{2}(a^{(k-1)})\|_{0,\alpha}\leq
C(k,\alpha)\mu^{\alpha-1}\sigma^{-(l+1)}\leq
C(k,\alpha)\mu^{\alpha-1}\sigma^{-1}\sigma^{-l}\,,$ (5.17)

and a similar estimate for $E_{3}(a^{(k-1)})$. Putting these estimates
together gives (5.9). The estimate (5.10) follows analogously, using the
product rule for the $\partial_{1}$ derivatives. Consequently, the claim is
proved.

It now remains to show (3.37). We also proceed by inductively showing that for
any $k=1,\ldots,N$ it holds

$\displaystyle\|a^{(k)}-a^{(k-1)}\|_{0,\alpha}+\|w^{(k)}-w^{(k-1)}\|_{1,\alpha}$
$\displaystyle\leq C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\,,$ (5.18)
$\begin{split}&\|\partial_{2}^{l}\left(a^{(k)}-a^{(k-1)}\right)\|_{0,\alpha}+\|\partial_{2}^{l}\left(w^{(k)}-w^{(k-1)}\right)\|_{1,\alpha}\\\
\leq&C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-l}\quad\text{ for
}0\leq l\leq N+3-k\end{split}$ (5.19)

and

$\begin{split}&\|\partial^{j+1}_{1}\partial_{2}^{l}\left(a^{(k)}-a^{(k-1)}\right)\|_{0,\alpha}+\|\partial_{1}^{j}\partial_{2}^{l+1}\left(w^{(k)}-w^{(k-1)}\right)\|_{1,\alpha}\\\
\leq&C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-(l+1)}\mu^{j},\quad\text{
for }1\leq l+j+1\leq N+3-k\,.\end{split}$ (5.20)

Indeed, observe that

$a^{(1)}-a^{(0)}=\partial_{2}\left(w^{(1)}-w^{(0)}\right)=\frac{1}{2}\partial_{2}\mathcal{C}[\widehat{E(a^{(0)})}]\,$

so that the estimates for $k=1$ follow from the corresponding ones above (see
(5.13)–(5.15), resp. (5.16)–(5.17)). Assume therefore that (5.18)–(5.20) hold
for $k$. From

$\displaystyle a^{(k+1)}-a^{(k)}$
$\displaystyle=\partial_{2}\left(w^{(k+1)}-w^{(k)}\right)=\frac{1}{2}\partial_{2}\mathcal{C}\left[\chi\left(\widehat{E(a^{(k)})}-\widehat{E(a^{(k-1)})}\right)\right]$
$\displaystyle=\frac{1}{2}\mathcal{C}\left[\partial_{2}\left(\chi\left(\widehat{E(a^{(k)})}-\widehat{E(a^{(k-1)})}\right)\right)\right],\,$
(5.21)

it is apparent that in order to show (5.18)–(5.20) we need to estimate

$\|\partial_{2}^{l}\left(E(a^{(k)})-E(a^{(k-1)})\right)\|_{0,\alpha}\text{ and
}\|\partial_{1}^{j}\partial_{2}^{l+1}\left(E(a^{(k)})-E(a^{(k-1)})\right)\|_{0,\alpha}$

in the given ranges of indices. To do this, recall that by definition (3.30)
we have

$E(a^{(k)})-E(a^{(k-1)})=\sum_{i=1}^{3}\left(E_{i}(a^{(k)})-E_{i}(a^{(k-1)})\right)=\sum_{i=1}^{3}\frac{\bar{\gamma}_{i}(\mu
x_{1})}{\mu}A_{i},$

where

$\displaystyle\bar{\gamma}_{1}=\gamma_{1},\quad A_{1}=$
$\displaystyle(\sqrt{a^{(k)}}-\sqrt{a^{(k-1)}})(M-M_{22}e_{2}\otimes e_{2}),$
$\displaystyle\bar{\gamma}_{2}=-\gamma_{3},\quad A_{2}=$
$\displaystyle\nabla({a^{(k)}}-{a^{(k-1)}})\odot e_{1},$
$\displaystyle\bar{\gamma}_{3}=-\frac{\gamma_{1}^{2}}{2\mu},\quad A_{3}=$
$\displaystyle\nabla\sqrt{a^{(k)}}\otimes\nabla\sqrt{a^{(k)}}-\nabla\sqrt{a^{(k-1)}}\otimes\nabla\sqrt{a^{(k-1)}}$
$\displaystyle+\left((\partial_{2}\sqrt{a^{(k-1)}})^{2}-(\sqrt{a^{(k)}})^{2}\right)e_{2}\otimes
e_{2}.$

By the Leibniz rule we get for each term and indices $0\leq m+n\leq N$

$\|\partial_{1}^{m}\partial_{2}^{n}\left(\frac{\bar{\gamma}_{i}(\mu
x_{1})}{\mu}A_{i}\right)\|_{0,\alpha}\leq
C(N,\alpha)\sum_{j=0}^{m}\mu^{m-j-1}\left(\mu^{\alpha}\|\partial_{1}^{j}\partial_{2}^{n}A_{i}\|_{0}+\|\partial_{1}^{j}\partial_{2}^{n}A_{i}\|_{0,\alpha}\right).$
(5.22)

Now for the second term, by induction assumption we immediately find

$\displaystyle\|\partial_{1}^{m}\partial_{2}^{n}\left(\frac{\bar{\gamma}_{2}(\mu
x_{1})}{\mu}A_{2}\right)\|_{0,\alpha}\leq$ $\displaystyle
C(N,\alpha)\sum_{l=1}^{m}\mu^{m-j-1}\mu^{\alpha}\sigma^{-(n+1)}\mu^{j}(\mu^{\alpha-1}\sigma^{-1})^{k}$
$\displaystyle\leq$ $\displaystyle
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k+1}\mu^{m}\sigma^{-n}\,.$

To estimate the other terms, we write

$\sqrt{a^{(k)}}-\sqrt{a^{(k-1)}}=\frac{a^{(k)}-a^{(k-1)}}{\sqrt{a^{(k)}}+\sqrt{a^{(k-1)}}}.$

Thanks to (5.8), (5.11) and (5.12), it holds

$\displaystyle\|\partial_{2}^{l}\left(\sqrt{a^{(k)}}+\sqrt{a^{(k-1)}}\right)^{-1}\|_{0,\alpha}$
$\displaystyle\leq C(N,\alpha)\sigma^{-l}$
$\displaystyle\|\partial_{1}^{j+1}\partial_{2}^{l}\left(\sqrt{a^{(k)}}+\sqrt{a^{(k-1)}}\right)^{-1}\|_{0,\alpha}$
$\displaystyle\leq C(N,\alpha)\sigma^{-(l+1)}\mu^{j}\,.$

Thus by induction assumption we find

$\begin{split}\|\partial_{2}^{l}\left(\sqrt{a^{(k)}}-\sqrt{a^{(k-1)}}\right)\|_{0,\alpha}&\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-l}\\\
\|\partial_{1}^{j+1}\partial_{2}^{l}\left(\sqrt{a^{(k)}}-\sqrt{a^{(k-1)}}\right)\|_{0,\alpha}&\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-(l+1)}\mu^{j}\,.\end{split}$
(5.23)

Note that

$\displaystyle A_{3}=$
$\displaystyle\nabla\sqrt{a^{(k)}}\otimes\left(\nabla\sqrt{a^{(k)}}-\nabla\sqrt{a^{(k-1)}}\right)+\left(\nabla\sqrt{a^{(k)}}-\nabla\sqrt{a^{(k-1)}}\right)\otimes\nabla\sqrt{a^{(k-1)}}$
$\displaystyle\quad-\left(\partial_{2}\sqrt{a^{(k)}}-\partial_{2}\sqrt{a^{(k-1)}}\right)\left(\partial_{2}\sqrt{a^{(k)}}+\partial_{2}\sqrt{a^{(k-1)}}\right)e_{2}\otimes
e_{2}\,.$

Using (5.23) and product rule, we can derive

$\|\partial^{j}_{1}\partial_{2}^{n}A_{3}\|_{0,\alpha}\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-(n+1)-1}\mu^{j}\,.$

Thus with (5.22) we find

$\displaystyle\|\partial_{1}^{m}\partial_{2}^{n}\left(\frac{\bar{\gamma}_{3}(\mu
x_{1})}{\mu}A_{3}\right)\|_{0,\alpha}$ $\displaystyle\leq
C(N,\alpha)\sum_{j=0}^{m}\mu^{m-j-2+\alpha}(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-(n+1)-1}\mu^{j}$
$\displaystyle\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k+1}\mu^{m-1}\sigma^{-(n+1)}$
$\displaystyle\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k+1}\mu^{m}\sigma^{-n}\,.$

As for the first term, by assumption (3.33) and (5.23) we can estimate for
$\alpha\in[0,\beta[$

$\|\partial_{1}^{j}\partial_{2}^{n}A_{1}\|_{0,\alpha}\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\sigma^{-n-1-\alpha}\mu^{j}\,,$

which yields

$\displaystyle\|\partial_{1}^{m}\partial_{2}^{n}\left(\frac{\bar{\gamma}_{1}(\mu
x_{1})}{\mu}A_{1}\right)\|_{0,\alpha}$ $\displaystyle\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\mu^{m-1}\sigma^{-n-1}\left(\mu^{\alpha}+\sigma^{-\alpha}\right)$
$\displaystyle\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k}\mu^{m-1+\alpha}\sigma^{-n-1}$
$\displaystyle=C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k+1}\mu^{m}\sigma^{-n}\,.$

Combining the above estimates hence gives

$\|\partial_{1}^{m}\partial_{2}^{n}\left(E(a^{(k)})-E(a^{(k-1)})\right)\|_{0,\alpha}\leq
C(N,\alpha)(\mu^{\alpha-1}\sigma^{-1})^{k+1}\mu^{m}\sigma^{-n}\,,$ (5.24)

which together with the expression (5.21) shows that (5.18)–(5.20) hold for
every $k=0,\ldots,N$. Consequently (3.37) follows from (5.24). The proof is
then complete.

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# Ordering the processes with indefinite causal order

Stanislav Filatov and Marcis Auzinsh
University of Latvia, Department of Physics
Raina boulevard 19, LV-1586, Riga, Latvia
E-mail<EMAIL_ADDRESS>
###### Abstract

We show a method of describing processes with indefinite causal order (ICO) by
a definite causal order. We do so by relabeling the processes that take place
in the circuit in accordance with the basis of measurement of control qubit.
Causal nonseparability is aleviated at a cost of nonlocality of the acting
processes. This result highlights the key role of superposition in creating
the paradox of ICO. We also draw attention to the issue of growing
incompatibility of language in its current form (especially the logical
structures it embodies) with the quantum logic.

##  1 Introduction

Indefinite Causal Order (ICO) implemented through Quantum Switch has seen a
lot of attention recently [1, 2, 3]. Apart from practical advances [4, 5, 6]
the approach implicitly claims a fundamental point as well: ICO setups, being
inherently unorderable, hint at the inherent unorderability and acausality of
quantum mechanics and therefore the nature. We would like to refine this point
and show that it is not valid universally, but only within a particularly
chosen reference frame.

First of all we consider the case of a particle being in spatial superposition
and show in what sense it occupies two places simultaneously and in what sense
it is in a particular place. Then we expand the analogy to the case of ICO
processes and show how to order them. Finally, remarks about linguistic
character of the difficulties of locating and ordering the superpositions in
quantum mechanics are presented along with conclusions. We use ordinary-
language descriptions along with technical throughout the text in order to
highlight the ways in which the paradox of acausality arises.

##  2 Analogy with spatial superposition

Let us look at a particle entering an interferometer which has two arms.
Depending on the conditions (state of the particle, setup of an
interferometer) particle may take the arm A, arm B or a superposition of arms
(we use bold to depict the names of the arms)111We will not be using bold in
some descriptions when the emphasis of naming is not that important. When
conditions are right for the particle to take the arm A, we say “Particle is
in arm A and not in arm B”. In language of quantum mechanics we may say “The
state of the system is $\ket{A}$ and not the one orthogonal to it, $\ket{B}$”.
When the conditions are right for the particle to take a superpositional
route, say $\ket{(A+B)/\sqrt{2}}$, we may be tempted to say that particle
takes two arms simultaneously and that there is no particular “arm” where
particle is. Nevertheless, in the language of quantum mechanics there is
precise expression for the state of the system “The state of the system is
$\ket{(A+B)/\sqrt{2}}$ and not the one orthogonal to it,
$\ket{(A-B)/\sqrt{2}}$”. Translating this to our language we may say “Particle
is in arm $\mathbf{(A+B)/\sqrt{2}}$ and not in arm
$\mathbf{(A-B)/\sqrt{2}}$.”222This sentence should be read as “Particle is in
arm “A plus B divided by square root 2” and not in arm “A minus B divided by
square root 2.”” Text in bold denotes the name of the arm.

We would like to stress that just like in the first example, in the second
example there is particular “place” or “arm” in which the particle definitely
is and particular “place” or “arm” in which the particle definitely is not.
That existence may be confirmed experimentally in both cases. The difference
between the examples is that our everyday macroscopic intuition allows to call
localized arm A an “arm”, but does not allow to call delocalized arm
$\mathbf{(A+B)/\sqrt{2}}$ an “arm”. In the next section we will show how this
analogy may be applied to ICO processes.

##  3 Ordering the ICO

We assume familiarity of reader with general properties of ICO and Quantum
Switch, for a detailed description of ICO processes one is referred to the
aforementioned literature, here we will concentrate on the key moments.
Instead of two arms A and B, there are two processes A and B through which a
particle might pass. Moreover, the processes are connected between each other
so that A may happen first, then B, then final measurement of the attributes
of the particle; or B first, then A, then final measurement. Important,
although A and B are arbitrary processes (that might even include measurement
and collapse in a sense), they are connected coherently, i.e. A and B don’t
involve irreversible processes (see the descriptions of experimental setups in
the cited literature). Finally, the order “A then B” vs “B then A” is related
to the state of control qubit (usually degree of freedom of the same particle
that goes through the circuit). When the state of control qubit is in
superposition between the states corresponding to particular order, indefinite
causal order of processes A and B appears because the particle now
simultaneously passes through “A then B” and “B then A” processes.

For clarity of our argument we restrict the “processes” we are referring to in
this work to unitary operations. Although extension of our argument onto
Completely Positive Trace-preserving maps using Stinespring dilation [7] seems
possible, it would require even more non-intuitive relabelling of the
processes and would clout the essence of our reasoning. We leave this
extension for the future.

In order to untangle the situation one should note that, although the
processes A and B might be very complex, essentially there are just two
possibilities of what might happen “A then B” or “B then A”. Moreover, two
possibilities are absolutely determined by what process happens first. Namely,
“A then B” could be described as “A happens first”. Because the system
consists of just two processes, and the particle passes through both, this
statement is enough to deduce the whole situation “A then B”.

Before we start employing BraKet-notation for processes we should describe the
Hilbert-space in which they reside. We follow the authors of original ICO
scheme [2] and researchers of superpositions of quantum evolutions [8] in
applying the Choi-Jamiolkowski isomorphism [9, 10] to obtain that Hilbert
space. The isomorphism represents the linear operator mapping states in input
Hilbert-space $\mathcal{H}_{in}$ onto states in output Hilbert-space
$\mathcal{H}_{out}$ as a vector in Hilbert-space
$\mathcal{H}_{out}\otimes\mathcal{H}_{out}$. Then $\ket{A}$ (“process named
A”) will reside in H-space (“pr” stands for “process”)

$\displaystyle\mathcal{H}_{pr}^{A}=\mathcal{H}_{in}^{A}\otimes\mathcal{H}_{out}^{A}$
(1)

$\ket{B}$ (“process named B”) will reside in H-space

$\displaystyle\mathcal{H}_{pr}^{B}=\mathcal{H}_{in}^{B}\otimes\mathcal{H}_{out}^{B}$
(2)

And therefore both processes $\ket{A}$ and $\ket{B}$ will reside in H-space

$\displaystyle\mathcal{H}_{pr}^{AB}=\mathcal{H}_{pr}^{A}\otimes\mathcal{H}_{pr}^{B}$
(3)

We could, if we wanted to, shrink the H-space to the one spanned by $\ket{A}$
and $\ket{B}$ to obtain 2-dimensional H-space reflecting two possible answers
to the question “Which process hapenned first?”

We should also mention orthogonality of $\ket{A}$ and $\ket{B}$ that is
assumed in this analysis. To order the processes those processes must be
different (in language of quantum mechanics the vectors representing them must
be orthogonal). That is why there is no sense of order to the same process
repeated twice and hence no need of ordering. This line of reasoning doesn’t
rule out the case of partial overlap. We must admit that we don’t have a
rigorous proof why such cases may or may not always be reduced to a
combination of aforementioned extreme cases of zero and full overlap. But we
have a feeling that it can be done in the following way.

Let us say $\ket{V}$ and $\ket{W}$ are mutually non-orthogonal representations
of some processes V and W. Let’s say V precedes W. Such a description will not
be an accurate ordering of the processes because there is a part in each of V
and W that doesn’t have a sense of orderedness to it. This part is projection
Proj($\ket{V};\ket{W}$). Subtracting the projection from one of the vectors
and scaling them appropriately we obtain two orthogonal vectors that have
notion of orderedness applicable to them. The apparent order of the processes
in our circuit is only applicable to these distilled and mutually orthogonal
vectors.

Now let us try to describe the ordinary-language situation “first A then B”
situation in quantum language “$\ket{A}$ happens first, not its orthogonal
$\ket{B}$, then $\ket{B}$ happens, not its orthogonal $\ket{A}$”. Situation
“first B then A” is described analogously. In order to describe in a similar
way the superpositional case we need to look at the control qubit, because its
state fully determines the path of the particle through the processes A and B.
Let us say that state $\ket{a}$ of control qubit corresponds to “first A then
B”, other way around for $\ket{b}$. In other words, the state of control qubit
uniquely determines the first process through which the particle undergoes.
Then the state of control qubit $\ket{(a+b)/\sqrt{2}}$ also uniquely
determines the first process through which the particle undergoes. Let us say
it is $\ket{(A+B)/\sqrt{2}}$. The first process (as we have discussed)
uniquely determines the second process - which is orthogonal to it (in this
case $\ket{(A-B)/\sqrt{2}}$). In quantum language we would say
“$\ket{(A+B)/\sqrt{2}}$ happens first, not its orthogonal
$\ket{(A-B)/\sqrt{2}}$, then $\ket{(A-B)/\sqrt{2}}$ happens, not its
orthogonal $\ket{(A+B)/\sqrt{2}}$”. That translates to ordinary language
“first $\mathbf{(A+B)/\sqrt{2}}$ then $\mathbf{(A-B)/\sqrt{2}}$”. This is
essentially the recipe for ordering ICO processes: following the whole
particle through the circuit and noticing that superpositional states of
control qubits uniquely determine paths in Hilbert-space. Extensions to the
circuits with more than two processes should be possible as well.

Interpretation of the meaning of the new name of the proces being
$\mathbf{(A+B)/\sqrt{2}}$ beyond realization that it is a superposition is
even more challenging than in the case of superposed slits. Possible
interpretation could be that at every instance of time inside the process
$\ket{(A+B)/\sqrt{2}}_{process}$ the particle “sees” the space in the same way
it “sees” it when it passes the slits in the analogous superposition
$\ket{(A+B)/\sqrt{2}}_{slit}$. This is the cost of being able to order the ICO
processes: the processes become delocalized and it is difficult to understand
what each of them does, but this does not undermine the fact that ICO
processes may be ordered. In fact, such a tradeoff has Bohr-like complementary
feeling to it.

##  4 Conclusions

We have presented a way to order ICO processes. Therefore we believe that ICO
processes cannot be considered an argument in favour of fundamental
unorderability or acausality of quantum mechanics and of nature and this
question has to wait for frame-of-reference independent arguments. In fact,
very rarely, when speaking about ICO, particularity of frame of reference is
discussed at all (see [8] for an example of explicit statement of the frame of
reference of ICO processes and [11] for the introduction of Quantum
Equivalence Principle in the context of ICO). It seems that vagueness of our
everyday words when we interpret quantum superposition extends to vagueness
when we interpret more intricate consequences of quantum superposition,
creating paradoxical situations. The more we try to interpret quantum
mechanics, the more is evident the incompatibility of logical structures at
the root of our current language with the logic of quantum mechanics. Perhaps
quantum mechanics has to enter the area of interest of linguists in order to
overcome this difficulty.

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