# Ill-posedness of the Novikov equation in the critical Besov space
$B^{1}_{\infty,1}(\mathbb{R})$

Jinlu Li1, Yanghai Yu2,111E-mail<EMAIL_ADDRESS>yuyanghai214@sina.com(Corresponding author<EMAIL_ADDRESS>and Weipeng
Zhu3
1 School of Mathematics and Computer Sciences, Gannan Normal University,
Ganzhou 341000, China
2 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002,
China
3 School of Mathematics and Big Data, Foshan University, Foshan, Guangdong
528000, China

Abstract: It is shown that both the Camassa-Holm and Novikov equations are
ill-posed in $B_{p,r}^{1+1/p}(\mathbb{R})$ with
$(p,r)\in[1,\infty]\times(1,\infty]$ in [14] and well-posed in
$B_{p,1}^{1+1/p}(\mathbb{R})$ with $p\in[1,\infty)$ in [35]. Recently, the
ill-posedness for the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$
has been proved in [15]. In this paper, we shall solve the only left an
endpoint case $r=1$ for the Novikov equation. More precisely, we prove the
ill-posedness for the Novikov equation in $B^{1}_{\infty,1}(\mathbb{R})$ by
exhibiting the norm inflation phenomena.

Keywords: Novikov equation; Ill-posedness; Critical Besov space.

MSC (2010): 35Q53, 37K10.

##  1 Introduction

Vladimir Novikov [29] investigated the question of integrability for Camassa-
Holm type equations of the form

$(1-\partial_{x}^{2})u_{t}=P\left(u,u_{x},u_{xx},u_{xxx},\cdots\right),$

where $P$ is a polynomial of $u$ and its $x$-derivatives. Using as test for
integrability the existence of an infinite hierarchy of (quasi-) local higher
symmetries, he produced about 20 integrable equations with quadratic
nonlinearities that include the Camassa-Holm $(\mathrm{CH})$ equation

$\displaystyle(1-\partial_{x}^{2})u_{t}=uu_{xxx}+2u_{x}u_{xx}-3uu_{x}$ (1.1)

and the Degasperis-Procesi $(\mathrm{DP})$ equation

$\displaystyle(1-\partial_{x}^{2})u_{t}=uu_{xxx}+3u_{x}u_{xx}-4uu_{x}.$ (1.2)

Moreover, he produced about 10 integrable equations with cubic nonlinearities
that include the following new equation (we may call it the Novikov equation
(NE))

$\displaystyle(1-\partial_{x}^{2})u_{t}=u^{2}u_{xxx}+3uu_{x}u_{xx}-4u^{2}u_{x}.$
(1.3)

The Camassa-Holm equation was originally derived as a bi-Hamiltonian system by
Fokas and Fuchssteiner [12] in the context of the KdV model and gained
prominence after Camassa-Holm [2] independently re-derived it from the Euler
equations of hydrodynamics using asymptotic expansions. (1.1) is completely
integrable [2, 5] with a bi-Hamiltonian structure [4, 12] and infinitely many
conservation laws [2, 12]. Also, it admits exact peaked soliton solutions
(peakons) of the form $ce^{-|x-ct|}$ with $c>0$, which are orbitally stable
[8] and models wave breaking (i.e., the solution remains bounded, while its
slope becomes unbounded in finite time [3, 6, 7]). The Degasperis-Procesi
equation with a bi-Hamiltonian structure is integrable [10] and has traveling
wave solutions [21]. Although DP is similar to CH in several aspects, these
two equations are truly different. One of the novel features of DP different
from CH is that it has not only peakon solutions [10] and periodic peakon
solutions [36], but also shock peakons [22] and the periodic shock waves [11].

For the Novikov equation, Hone-Wang [20] derived the Lax pair which is given
by

$\displaystyle\left(\begin{array}[]{l}\psi_{1}\\\ \psi_{2}\\\
\psi_{3}\end{array}\right)_{x}=U(m,\lambda)\left(\begin{array}[]{l}\psi_{1}\\\
\psi_{2}\\\
\psi_{3}\end{array}\right),\quad\left(\begin{array}[]{l}\psi_{1}\\\
\psi_{2}\\\
\psi_{3}\end{array}\right)_{t}=V(m,u,\lambda)\left(\begin{array}[]{l}\psi_{1}\\\
\psi_{2}\\\ \psi_{3}\end{array}\right),$

where $m=u-u_{xx}$ and the matrices $U$ and $V$ are defined by

$\displaystyle U(m,\lambda)=\left(\begin{array}[]{ccc}0&\lambda m&1\\\
0&0&\lambda m\\\ 1&0&0\end{array}\right)\quad\text{and}\quad
V(m,u,\lambda)=\left(\begin{array}[]{lrr}\frac{1}{3\lambda^{2}}-uu_{x}&\frac{u_{x}}{\lambda}-\lambda
mu^{2}&u_{x}^{2}\\\
\frac{u}{\lambda}&-\frac{2}{3\lambda^{2}}&-\frac{u_{x}}{\lambda}-\lambda
mu^{2}\\\
-u^{2}&\frac{u}{\lambda}&\frac{1}{3\lambda^{2}}+uu_{x}\end{array}\right).$

NE possesses peakon traveling wave solutions [18, 19, 13], which on the real
line are given by the formula $u(x,t)=\pm\sqrt{c}e^{-|x-ct|}$ where $c>0$ is
the wave speed. In fact, NE possesses multi-peakon traveling wave solutions on
both the line and the circle. More precisely, on the line the $n$-peakon

$u(x,t)=\sum_{j=1}^{n}p_{j}(t)e^{-\left|x-q_{j}(t)\right|}$

is a solution to $\mathrm{NE}$ if and only if the positions
$\left(q_{1},\ldots,q_{n}\right)$ and the momenta
$\left(p_{1},\ldots,p_{n}\right)$ satisfy the following system of $2n$
differential equations:

$\left\\{\begin{aligned} \frac{\mathrm{d}q_{j}}{\mathrm{d}t}&=u^{2}(q_{j}),\\\
\frac{\mathrm{d}p_{j}}{\mathrm{d}t}&=-u(q_{j})u_{x}(q_{j})p_{j}.\end{aligned}\right.$

Furthermore, it has infinitely many conserved quantities. Like CH, the most
important quantity conserved by a solution $u$ to NE is its $H^{1}$-norm
$\|u\|_{H^{1}}^{2}=\int_{\mathbb{R}}(u^{2}+u_{x}^{2})\mathrm{d}x.$

The well-posedness of the Camassa-Holm type equations has been widely
investigated during the past 20 years. The local well-posedness for the Cauchy
problem of CH [23, 30, 9, 24] and NE [16, 17, 28, 31, 32, 33, 34] in Sobolev
and Besov spaces $B_{p,r}^{s}(\mathbb{R})$ with $s>\max\\{1+1/{p},3/{2}\\}$
and $(p,r)\in[1,\infty]\times[1,\infty)$ has been established. In our recent
papers [26, 27], we established the ill-posedness for CH in
$B^{s}_{p,\infty}(\mathbb{R})$ with $p\in[1,\infty]$ by proving the solution
map starting from $u_{0}$ is discontinuous at $t=0$ in the metric of
$B^{s}_{p,\infty}(\mathbb{R})$. Guo-Liu-Molinet-Yin [14] established the ill-
posedness for the Camassa-Holm type equations in $B_{p,r}^{1+1/p}(\mathbb{R})$
with $(p,r)\in[1,\infty]\times(1,\infty]$ by proving the norm inflation, which
implies that $B_{p,1}^{1+1/p}$ is the critical Besov space for both CH and NE.
Ye-Yin-Guo [35] obtained the local well-posedness for the Camassa-Holm type
equation in critical Besov spaces $B^{1+1/p}_{p,1}(\mathbb{R})$ with
$p\in[1,\infty)$. We should mention that the well-posedness for DP in
$B^{1}_{\infty,1}(\mathbb{R})$ has been established in our recent paper [25].
Very recently, Guo-Ye-Yin [15] obtained the ill-posedness for CH in
$B^{1}_{\infty,1}(\mathbb{R})$ by constructing a special initial data which
leads to the norm inflation. However, their initial data seems to be invalid
when proving the ill-posedness for NE in $B^{1}_{\infty,1}(\mathbb{R})$. To
the best of our knowledge, whether NE is well-posed or not in in
$B^{1}_{\infty,1}(\mathbb{R})$ is still an open problem. We shall present the
negative result in this paper.

Setting $\Lambda^{-2}=(1-\partial^{2}_{x})^{-1}$, then $\Lambda^{-2}f=G*f$
where $G(x)=\frac{1}{2}e^{-|x|}$ is the kernel of the operator $\Lambda^{-2}$.
We can transform the Novikov equation into the following transport type
equation

$\begin{cases}u_{t}+u^{2}u_{x}=\mathbf{P}_{1}(u)+\mathbf{P}_{2}(u),\\\
u(x,t=0)=u_{0}(x),\end{cases}$ (1.4)

where

$\mathbf{P}_{1}(u)=-\frac{1}{2}\Lambda^{-2}u_{x}^{3}\quad\text{and}\quad\mathbf{P}_{2}(u)=-\partial_{x}\Lambda^{-2}\left(\frac{3}{2}uu^{2}_{x}+u^{3}\right).$
(1.5)

We can now state our main result as follows.

######  Theorem 1.1.

For any $n\in\mathbb{Z}^{+}$ large enough, there exist $u_{0}$ and $T>0$ such
that the Novikov equation has a solution $u\in\mathcal{C}([0,T);H^{\infty})$
satisfying

$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}}\leq\frac{1}{\log\log
n}\quad\text{but}\quad\|u(t_{0})\|_{B^{1}_{\infty,1}}\geq{\log\log
n}\quad\text{with}\quad t_{0}\in\left(0,\frac{1}{\log n}\right].$

Since the norm inflation implies discontinuous of the data-to-solution at the
trivial function $u_{0}\equiv 0$, Theorem 1.1 demonstrates that

######  Corollary 1.1.

The Cauchy problem for the Novikov equation is ill-posed in
$B^{1}_{\infty,1}(\mathbb{R})$ in the sense of Hadamard.

This paper is structured as follows. In Section 2, we list some notations and
known results and recall some Lemmas which will be used in the sequel. In
Section 3 we present the proof of Theorem 1.1 by dividing it into several
parts: (1) Construction of initial data; (2) Key Estimation for Discontinuity;
(3) The Equation Along the Flow; (4) Norm inflation.

##  2 Preliminaries

Notation $C$ stands for some positive constant independent of $n$, which may
vary from line to line. The symbol $A\approx B$ means that $C^{-1}B\leq A\leq
CB$. We shall call a ball $B(x_{0},r)=\\{x\in\mathbb{R}:|x-x_{0}|\leq R\\}$
with $R>0$ and an annulus
$\mathcal{C}(0,r_{1},r_{2})=\\{x\in\mathbb{R}:0<r_{1}\leq|x|\leq r_{2}\\}$
with $0<r_{1}<r_{2}$. Given a Banach space $X$, we denote its norm by
$\|\cdot\|_{X}$. We shall use the simplified notation
$\|f,\cdots,g\|_{X}=\|f\|_{X}+\cdots+\|g\|_{X}$ if there is no ambiguity. We
will also define the Lipschitz space $C^{0,1}$ using the norm
$\|f\|_{C^{0,1}}=\|f\|_{L^{\infty}}+\|\partial_{x}f\|_{L^{\infty}}$. For
$I\subset\mathbb{R}$, we denote by $\mathcal{C}(I;X)$ the set of continuous
functions on $I$ with values in $X$. Sometimes we will denote $L^{p}(0,T;X)$
by $L_{T}^{p}X$. Let us recall that for all $f\in\mathcal{S}^{\prime}$, the
Fourier transform $\widehat{f}$, is defined by

$(\mathcal{F}f)(\xi)=\widehat{f}(\xi)=\int_{\mathbb{R}}e^{-ix\xi}f(x)\mathrm{d}x\quad\text{for
any}\;\xi\in\mathbb{R}.$

The inverse Fourier transform of any $g$ is given by

$(\mathcal{F}^{-1}g)(x)=\check{g}(x)=\frac{1}{2\pi}\int_{\mathbb{R}}g(\xi)e^{ix\cdot\xi}\mathrm{d}\xi.$

Next, we will recall some facts about the Littlewood-Paley decomposition and
the nonhomogeneous Besov spaces (see [1] for more details). Let
$\mathcal{B}:=\\{\xi\in\mathbb{R}:|\xi|\leq 4/3\\}$ and
$\mathcal{C}:=\\{\xi\in\mathbb{R}:3/4\leq|\xi|\leq 8/3\\}.$ Choose a radial,
non-negative, smooth function $\chi:\mathbb{R}\mapsto[0,1]$ such that it is
supported in $\mathcal{B}$ and $\chi\equiv 1$ for $|\xi|\leq 3/4$. Setting
$\varphi(\xi):=\chi(\xi/2)-\chi(\xi)$, then we deduce that $\varphi$ is
supported in $\mathcal{C}$. Moreover,

$\displaystyle\chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1\quad\mbox{ for any
}\xi\in\mathbb{R}.$

We should emphasize that the fact $\varphi(\xi)\equiv 1$ for $4/3\leq|\xi|\leq
3/2$ will be used in the sequel.

For every $u\in\mathcal{S^{\prime}}(\mathbb{R})$, the inhomogeneous dyadic
blocks ${\Delta}_{j}$ are defined as follows

$\displaystyle\Delta_{j}u=$ $\displaystyle 0,$ if $j\leq-2$;
$\displaystyle\Delta_{j}u=$
$\displaystyle\chi(D)u=\mathcal{F}^{-1}(\chi\mathcal{F}u),$ if $j=-1$;
$\displaystyle\Delta_{j}u=$
$\displaystyle\varphi(2^{-j}D)u=\mathcal{F}^{-1}\left(\varphi(2^{-j}\cdot)\mathcal{F}u\right),$
if $j\geq 0$.

Let $s\in\mathbb{R}$ and $(p,r)\in[1,\infty]^{2}$. The nonhomogeneous Besov
space $B^{s}_{p,r}(\mathbb{R})$ is defined by

$\displaystyle
B^{s}_{p,r}(\mathbb{R}):=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R}):\;\|f\|_{B^{s}_{p,r}(\mathbb{R})}:=\left\|2^{js}\|\Delta_{j}u\|_{L^{p}}\right\|_{\ell^{r}(j\geq-1)}<\infty\right\\}.$

The following Bernstein’s inequalities will be used in the sequel.

######  Lemma 2.1 (Lemma 2.1 in [1]).

Let $\mathcal{B}$ be a Ball and $\mathcal{C}$ be an annulus. There exist
constants $C>0$ such that for all $k\in\mathbb{N}\cup\\{0\\}$, any
$\lambda\in\mathbb{R}^{+}$ and any function $f\in L^{p}$ with $1\leq p\leq
q\leq\infty$, we have

$\displaystyle{\rm{supp}}\widehat{f}\subset\lambda\mathcal{B}\;\Rightarrow\;\|\partial_{x}^{k}f\|_{L^{q}}\leq
C^{k+1}\lambda^{k+(\frac{1}{p}-\frac{1}{q})}\|f\|_{L^{p}},$
$\displaystyle{\rm{supp}}\widehat{f}\subset\lambda\mathcal{C}\;\Rightarrow\;C^{-k-1}\lambda^{k}\|f\|_{L^{p}}\leq\|\partial_{x}^{k}f\|_{L^{p}}\leq
C^{k+1}\lambda^{k}\|f\|_{L^{p}}.$

######  Lemma 2.2 (Lemma 2.100 in [1]).

Let $1\leq r\leq\infty$, $1\leq p\leq p_{1}\leq\infty$ and
$\frac{1}{p_{2}}=\frac{1}{p}-\frac{1}{p_{1}}$. There exists a constant $C$
depending continuously on $p,p_{1}$, such that

$\left\|\left(2^{j}\left\|[\Delta_{j},v\partial_{x}]f\right\|_{L^{p}}\right)_{j}\right\|_{\ell^{r}}\leq
C\left(\|\partial_{x}v\|_{L^{\infty}}\|f\|_{B_{p,r}^{1}}+\|\partial_{x}f\|_{L^{p_{2}}}\|\partial_{x}v\|_{B_{p_{1},r}^{0}}\right).$

##  3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1.

###  3.1 Construction of Initial Data

Define a smooth cut-off function $\chi$ with values in $[0,1]$ which satisfies

$\displaystyle\chi(\xi)=\begin{cases}1,\quad\mathrm{if}\
|\xi|\leq\frac{1}{4},\\\ 0,\quad\mathrm{if}\ |\xi|\geq\frac{1}{2}.\end{cases}$

From now on, we set $\gamma:=\frac{17}{24}$ just for the sake of simplicity.
Letting

$\displaystyle n\in
16\mathbb{N}=\left\\{16,32,48,\cdots\right\\}\quad\text{and}\quad\mathbb{N}(n)=\left\\{k\in
8\mathbb{N}:\frac{n}{4}\leq k\leq\frac{n}{2}\right\\}.$

We introduce the following new notation which will be used often throughout
this paper

$\displaystyle\|f\|_{B^{k}_{\infty,1}\left(\mathbb{N}(n)\right)}=\sum_{j\in\mathbb{N}(n)}2^{k}\|\Delta_{j}f\|_{L^{\infty}},\quad
k\in\\{0,1\\}.$

Now, we can define the initial data $u_{0}$ by

$\displaystyle u_{0}$
$\displaystyle=n^{-\frac{1}{3}}\left(u^{\rm{H}}_{0}+u^{\rm{L}}_{0}\right),$

where

$\displaystyle u^{\rm{H}}_{0}$ $\displaystyle:=2^{-n}\log
n\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma),$
(3.6) $\displaystyle u^{\rm{L}}_{0}$
$\displaystyle:=\sum_{\ell\in\mathbb{N}(n)}\check{\chi}(x+2^{\ell+1}\gamma).$
(3.7)

Some Observations

  1. 1.

Obviously,

$\displaystyle\mathrm{supp}\
\widehat{u^{\rm{L}}_{0}}\subset\left\\{\xi\in\mathbb{R}:\
|\xi|\leq\frac{1}{2}\right\\}.$ (3.8)

  2. 2.

It is not difficult to check that

$\displaystyle\mathrm{supp}\
\mathcal{F}\left(\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\right)\subset\left\\{\xi\in\mathbb{R}:\
2^{\ell}\gamma-\frac{1}{2}\leq|\xi|\leq 2^{\ell}\gamma+\frac{1}{2}\right\\},$
(3.9)

which gives in turn

$\displaystyle\mathrm{supp}\
\mathcal{F}\left(\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\right)$
$\displaystyle\qquad\subset\left\\{\xi\in\mathbb{R}:\
2^{n}\gamma-2^{\ell}\gamma-\frac{1}{2}\leq|\xi|\leq
2^{n}\gamma+2^{\ell}\gamma+\frac{1}{2}\right\\}.$ (3.10)

Thus

$\displaystyle\mathrm{supp}\
\widehat{u^{\rm{H}}_{0}}\subset\left\\{\xi\in\mathbb{R}:\
\frac{4}{3}2^{n-1}\leq|\xi|\leq\frac{3}{2}2^{n-1}\right\\}.$ (3.11)

  3. 3.

Since $\check{\chi}$ is a Schwartz function, we have

$\displaystyle|\check{\chi}(x)|+|\partial_{x}\check{\chi}(x)|\leq
C(1+|x|)^{-M},\qquad M\gg 1.$ (3.12)

  4. 4.

By $\check{\chi}(0)=\frac{1}{2\pi}\int_{\mathbb{R}}\chi(x)\mathrm{d}x$, we
have

$\displaystyle\left\|\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{\ell+1}\gamma)\right\|_{L^{\infty}}\geq\check{\chi}^{3}(0)\geq\frac{1}{128\pi}.$
(3.13)

######  Lemma 3.1.

There exists a positive constant $C$ independent of $n$ such that

$\displaystyle
2^{n}\|u^{\rm{H}}_{0}\|_{L^{\infty}}+\|\partial_{x}u^{\rm{H}}_{0}\|_{L^{\infty}}\leq
C\log n,$ $\displaystyle\|u^{\rm{L}}_{0}\|_{C^{0,1}}\leq C,$
$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}}\leq Cn^{-\frac{1}{3}}\log n.$

###### Proof.

Due to (3.11)-(3.12), by Bernstein’s inequality, we have

$\displaystyle 2^{n}\|u^{\rm{H}}_{0}\|_{L^{\infty}}$
$\displaystyle+\|\partial_{x}u^{\rm{H}}_{0}\|_{L^{\infty}}+\log
n\|u^{\rm{L}}_{0}\|_{C^{0,1}}$ $\displaystyle\leq C\log
n\left\|\sum_{\ell\in\mathbb{N}(n)}\frac{1}{(1+|x+2^{\ell+1}\gamma|)^{M}}\right\|_{L^{\infty}}$
$\displaystyle\leq C\log n$

and

$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}}$ $\displaystyle\leq
Cn^{-\frac{1}{3}}\left(\|u^{\rm{H}}_{0}\|_{B^{1}_{\infty,1}}+\|u^{\rm{L}}_{0}\|_{B^{1}_{\infty,1}}\right)$
$\displaystyle\leq
Cn^{-\frac{1}{3}}\left(2^{n}\|u^{\rm{H}}_{0}\|_{L^{\infty}}+\|u^{\rm{L}}_{0}\|_{L^{\infty}}\right)$
$\displaystyle\leq Cn^{-\frac{1}{3}}\log n.$

This completes the proof of Lemma 3.1. ∎

###  3.2 Key Estimation for Discontinuity

The following Lemma is crucial for the proof of Theorem 1.1.

######  Lemma 3.2.

There exists a positive constant $c$ independent of $n$ such that

$\displaystyle\left\|u_{0}(\partial_{x}u_{0})^{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\geq
c(\log n)^{2},\qquad n\gg 1.$

###### Proof.

Obviously,

$\displaystyle n\cdot u_{0}(\partial_{x}u_{0})^{2}$
$\displaystyle=\underbrace{u^{\rm{L}}_{0}(\partial_{x}u^{\rm{H}}_{0})^{2}}_{=:\;\mathbf{I}_{1}}+\underbrace{u^{\rm{H}}_{0}(\partial_{x}u^{\rm{H}}_{0}+\partial_{x}u^{\rm{L}}_{0})^{2}}_{=:\;\mathbf{I}_{2}}+\underbrace{u^{\rm{L}}_{0}\left((\partial_{x}u^{\rm{L}}_{0})^{2}+2\partial_{x}u^{\rm{H}}_{0}\partial_{x}u^{\rm{L}}_{0}\right)}_{=:\;\mathbf{I}_{3}}.$

Next, we need to estimate the above three terms.

Estimation of $\mathbf{I}_{2}$. Using Lemma 3.1 yields

$\displaystyle\left\|\mathbf{I}_{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\leq
Cn\|\mathbf{I}_{2}\|_{L^{\infty}}\leq
n\|u^{\rm{H}}_{0}\|_{L^{\infty}}\|\partial_{x}u^{\rm{H}}_{0},\partial_{x}u^{\rm{L}}_{0}\|^{2}_{L^{\infty}}\leq
Cn2^{-n}(\log n)^{2}.$

Estimation of $\mathbf{I}_{3}$. Notice that the support conditions (3.11) and
(3.8), one has

$\displaystyle\Delta_{j}\mathbf{I}_{3}=0\quad\text{for}\;j\in\mathbb{N}(n)\quad\Rightarrow\quad\left\|\mathbf{I}_{3}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}=0.$

Estimation of $\mathbf{I}_{1}$. Now we focus on the estimation of
$\mathbf{I}_{1}$. Obviously,

$\displaystyle\partial_{x}u^{\rm{H}}_{0}$ $\displaystyle=-\gamma\log
n\sum_{\ell\in\mathbb{N}(n)}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad+2^{-n}\log
n\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}.$

We decompose $\mathbf{I}_{1}$ into three terms

$\displaystyle\mathbf{I}_{1}=(\log
n)^{2}\left(\mathbf{I}_{11}+\mathbf{I}_{12}-\mathbf{I}_{13}\right),$

where

$\displaystyle\mathbf{I}_{11}$
$\displaystyle=\gamma^{2}u^{\rm{L}}_{0}\left(\sum_{\ell\in\mathbb{N}(n)}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\right)^{2},$
$\displaystyle\mathbf{I}_{12}$
$\displaystyle=2^{-2n}u^{\rm{L}}_{0}\left(\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}\right)^{2},$
$\displaystyle\mathbf{I}_{13}$ $\displaystyle=2\gamma
2^{-n}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad\times\sum_{\ell\in\mathbb{N}(n)}\cos\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}.$

Easy computations give that

$\displaystyle\|\mathbf{I}_{12}\|_{L^{\infty}}$ $\displaystyle\leq
C2^{-2n}\|u^{\rm{L}}_{0}\|_{L^{\infty}}\left\|\sum_{\ell\in\mathbb{N}(n)}\partial_{x}\Big{(}\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)\Big{)}\right\|^{2}_{L^{\infty}}$
$\displaystyle\leq
C2^{-2n}\left\|\sum_{\ell\in\mathbb{N}(n)}\frac{2^{\ell}}{(1+|x+2^{\ell+1}\gamma|)^{M}}\right\|^{2}_{L^{\infty}}$
$\displaystyle\leq C2^{-n}.$

Similarly, we have $\|\mathbf{I}_{13}\|_{L^{\infty}}\leq C2^{-\frac{n}{2}}.$
Thus

$\displaystyle\left\|\mathbf{I}_{12},\mathbf{I}_{13}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\leq
Cn2^{-\frac{n}{2}}.$

By the simple equality
$\sin^{2}(a)\cos^{2}(b)=\frac{1}{4}(1-\cos(2a))(1+\cos(2b))$, we break
$\mathbf{I}_{11}$ down into some easy-to-handle terms

$\displaystyle\mathbf{I}_{11}=\gamma^{2}\sum_{i=1}^{5}\mathbf{I}_{11i},\quad\text{where}$
$\displaystyle\mathbf{I}_{111}$
$\displaystyle=\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{112}$
$\displaystyle=\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{113}$
$\displaystyle=-\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{n+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\cdot\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{114}$
$\displaystyle=-\frac{1}{4}u^{\rm{L}}_{0}\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{n+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\cdot\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\cdot\check{\chi}^{2}(x+2^{\ell+1}\gamma),$
$\displaystyle\mathbf{I}_{115}$
$\displaystyle=u^{\rm{L}}_{0}\sum_{\ell,j\in\mathbb{N}(n)\atop\ell\neq
j}\Big{(}\sin\left(2^{n}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\cos\left(2^{\ell}\gamma(x+2^{\ell+1}\gamma)\right)\cdot\check{\chi}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad\times\sin\left(2^{n}\gamma(x+2^{j+1}\gamma)\right)\cdot\cos\left(2^{j}\gamma(x+2^{j+1}\gamma)\right)\cdot\check{\chi}(x+2^{j+1}\gamma)\Big{)}.$

Notice that the support conditions (3.9) and (2), one has

$\displaystyle\Delta_{j}\mathbf{I}_{112}=\Delta_{j}\mathbf{I}_{113}=\Delta_{j}\mathbf{I}_{114}=0\quad\text{for}\;j\in\mathbb{N}(n),$

which implies directly that

$\displaystyle\left\|\mathbf{I}_{112},\mathbf{I}_{113},\mathbf{I}_{114}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}=0.$

Using Lemma 3.1, we have

$\displaystyle\|\mathbf{I}_{115}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\leq Cn\|\mathbf{I}_{115}\|_{L^{\infty}}\leq
Cn\|u^{\rm{L}}_{0}\|_{L^{\infty}}\sum_{\ell,j\in\mathbb{N}(n)\atop\ell\neq
j}\left\|\check{\chi}(x+2^{\ell+1}\gamma)\cdot\check{\chi}(x+2^{j+1}\gamma)\right\|_{L^{\infty}}$
$\displaystyle\leq
Cn\sum_{j>\ell\in\mathbb{N}(n)}\left\|(1+|x+2^{j+1}\gamma|^{2})^{-M}(1+|x+2^{\ell+1}\gamma|^{2})^{-M}\right\|_{L^{\infty}}$
$\displaystyle\leq
Cn\sum_{j>\ell\in\mathbb{N}(n)}\left\|(1+|x|^{2})^{-M}(1+|x-(2^{j+1}-2^{\ell+1})\gamma|^{2}))^{-M}\right\|_{L^{\infty}}$
$\displaystyle\leq
Cn\sum_{j>\ell\in\mathbb{N}(n)}\left(\gamma(2^{j}-2^{\ell})\right)^{-2M}$
$\displaystyle\leq Cn2^{-\frac{nM}{2}},$

where we have separated $\mathbb{R}$ into two different regions
$\\{x:|x|\leq\gamma(2^{j}-2^{\ell})\\}$ and
$\\{x:|x|>\gamma(2^{j}-2^{\ell})\\}$.

Finally, we can break $\mathbf{I}_{111}$ down into two parts, where the first
part contributes the main part.

$\displaystyle\mathbf{I}_{111}$
$\displaystyle=\sum_{\ell\in\mathbb{N}(n)}\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{\ell+1}\gamma)$
$\displaystyle\quad+\sum_{\ell,j\in\mathbb{N}(n)\atop\ell\neq
j}\cos\big{(}2^{\ell+1}\gamma(x+2^{\ell+1}\gamma)\big{)}\check{\chi}^{2}(x+2^{\ell+1}\gamma)\check{\chi}(x+2^{j+1}\gamma)$
$\displaystyle:=\mathbf{I}_{1111}+\mathbf{I}_{1112}.$

Due to (3.9) and the support condition of $\varphi$ and for all
$k\in\mathbb{Z}$

$\displaystyle\varphi(2^{-k}\xi)\equiv
1\quad\text{for}\quad\xi\in\mathcal{C}_{k}\equiv\left\\{\xi\in\mathbb{R}^{d}:\
\frac{4}{3}2^{k}\leq|\xi|\leq\frac{3}{2}2^{k}\right\\},$

we have

${\dot{\Delta}_{j}\mathbf{I}_{1111}=\mathcal{F}^{-1}\left(\varphi(2^{-j}\cdot)\mathcal{F}\mathbf{I}_{1111}\right)=}\begin{cases}\cos\big{(}2^{j+1}\gamma(x+2^{j+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{j+1}\gamma),&\text{if}\;\ell=j,\\\
0,&\text{if}\;\ell\neq j,\end{cases}$

which combining (3.13) implies that

$\displaystyle\|\mathbf{I}_{1111}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle=\sum_{j\in\mathbb{N}(n)}\left\|\cos\big{(}2^{j+1}\gamma(x+2^{j+1}\gamma)\big{)}\check{\chi}^{3}(x+2^{j+1}\gamma)\right\|_{L^{\infty}}\geq
cn.$

Following the same procedure as $\mathbf{I}_{115}$, we get

$\displaystyle\|\mathbf{I}_{1112}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}\leq
Cn2^{-\frac{nM}{2}},$

Gathering all the above estimates, we obtain that for large enough $n$

$\displaystyle
n\|u_{0}(\partial_{x}u_{0})^{2}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\geq\|\mathbf{I}_{1}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}-\|\mathbf{I}_{2}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\geq C(\log
n)^{2}\left(\|\mathbf{I}_{1111}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}-\|\mathbf{I}_{12},\mathbf{I}_{13},\mathbf{I}_{115},\mathbf{I}_{1112}\|_{B^{0}_{\infty,1}(\mathbb{N}(n))}-n2^{-n}\right)$
$\displaystyle\geq cn(\log n)^{2}.$

This completes the proof of Lemma 3.2. ∎

######  Remark 3.1.

Setting $u_{0}=n^{-\frac{1}{2}}u^{\mathrm{H}}_{0}$ where $u^{\mathrm{H}}_{0}$
is given by (3.6) and following the same argument as the proof of Lemma 3.2,
we can establish

$\displaystyle\left\|u_{0}\right\|_{B^{1}_{\infty,1}(\mathbb{R})}\leq
C_{1}n^{-\frac{1}{2}}\log
n\quad\text{and}\quad\left\|(\partial_{x}u_{0})^{2}\right\|_{B^{0}_{\infty,1}\left(\mathbb{N}(n)\right)}\geq
c_{2}(\log n)^{2},\qquad n\gg 1.$

###  3.3 The Equation Along the Flow

Given a Lipschitz velocity field $u$, we may solve the following ODE to find
the flow induced by $u^{2}$:

$\displaystyle\quad\begin{cases}\frac{\mathrm{d}}{\mathrm{d}t}\phi(t,x)=u^{2}(t,\phi(t,x)),\\\
\phi(0,x)=x,\end{cases}$ (3.14)

which is equivalent to the integral form

$\displaystyle\phi(t,x)=x+\int^{t}_{0}u^{2}(\tau,\phi(\tau,x))\mathrm{d}\tau.$
(3.15)

Considering

$\displaystyle\quad\begin{cases}\partial_{t}v+u^{2}\partial_{x}v=P,\\\
v(0,x)=v_{0}(x),\end{cases}$ (3.16)

then, we get from (3.16) that

$\displaystyle\partial_{t}(\Delta_{j}v)+u^{2}\partial_{x}\Delta_{j}v$
$\displaystyle=R_{j}+\Delta_{j}P,$

with
$R_{j}=[u^{2},\Delta_{j}]\partial_{x}v=u^{2}\Delta_{j}\partial_{x}v-\Delta_{j}(u^{2}\partial_{x}v)$.

Due to (3.14), then

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left((\Delta_{j}v)\circ\phi\right)$
$\displaystyle=R_{j}\circ\phi+\Delta_{j}P\circ\phi,$

which means that

$\displaystyle\Delta_{j}v\circ\phi=\Delta_{j}v_{0}+\int^{t}_{0}R_{j}\circ\phi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}P\circ\phi\mathrm{d}\tau.$
(3.17)

###  3.4 Norm Inflation

For $n\gg 1$, we have for $t\in[0,1]$

$\displaystyle\|u\|_{C^{0,1}}\leq C\|u_{0}\|_{C^{0,1}}\leq
Cn^{-\frac{1}{3}}\log n.$

To prove Theorem 1.1, it suffices to show that there exists
$t_{0}\in(0,\frac{1}{\log n}]$ such that

$\displaystyle\|u(t_{0},\cdot)\|_{B^{1}_{\infty,1}}\geq\log\log n.$ (3.18)

We prove (3.18) by contraction. If (3.18) were not true, then

$\displaystyle\sup_{t\in(0,\frac{1}{\log
n}]}\|u(t,\cdot)\|_{B^{1}_{\infty,1}}<\log\log n.$ (3.19)

We divide the proof into two steps.

Step 1: Lower bounds for $(\Delta_{j}u)\circ\phi$

Now we consider the equation along the Lagrangian flow-map associated to
$u^{2}$. Utilizing (3.17) to (1.4) yields

$\displaystyle(\Delta_{j}u)\circ\phi$
$\displaystyle=\Delta_{j}u_{0}+\int^{t}_{0}R^{1}_{j}\circ\phi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}F\circ\phi\mathrm{d}\tau$
$\displaystyle\quad+\int^{t}_{0}\big{(}\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\big{)}\mathrm{d}\tau+t\Delta_{j}E_{0},$

where

$\displaystyle R^{1}_{j}=[u^{2},\Delta_{j}]\partial_{x}u,\qquad
F=-\Lambda^{-2}\left(\frac{1}{2}(\partial_{x}u)^{3}+\partial_{x}(u^{3})\right),$
$\displaystyle
E=-\frac{3}{2}\partial_{x}\Lambda^{-2}\left(u(\partial_{x}u)^{2}\right)\quad\text{with}\quad
E_{0}=-\frac{3}{2}\partial_{x}\Lambda^{-2}\left(u_{0}(\partial_{x}u_{0})^{2}\right).$

Due to Lemma 3.2, we deduce

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E_{0}\|_{L^{\infty}}$
$\displaystyle\approx\sum_{j\in\mathbb{N}(n)}\|\Delta_{j}\partial_{x}E_{0}\|_{L^{\infty}}$
$\displaystyle\geq
C\sum_{j\in\mathbb{N}(n)}\left\|\Delta_{j}[u_{0}(\partial_{x}u_{0})^{2}]\right\|_{L^{\infty}}$
$\displaystyle\geq c(\log n)^{2}.$ (3.20)

Notice that (3.15), then we have for $t\in(0,\frac{1}{\log n}]$,

$\displaystyle\frac{1}{2}\leq|\partial_{x}\phi|\leq 2,$

thus

$\displaystyle\|f(t,\phi(t,x))\|_{L^{\infty}}=\|f(t,\cdot)\|_{L^{\infty}}.$

Then, using the commutator estimate from Lemma 2.2, we have

$\displaystyle\sum_{j\geq-1}2^{j}\|R^{1}_{j}\circ\phi\|_{L^{\infty}}$
$\displaystyle\leq C\sum_{j\geq-1}2^{j}\|R^{1}_{j}\|_{L^{\infty}}$
$\displaystyle\leq
C\|\partial_{x}(u^{2})\|_{B^{0}_{\infty,1}}\|u\|_{B^{1}_{\infty,1}}$
$\displaystyle\leq C\|u\|_{C^{0,1}}\|u\|^{2}_{B^{1}_{\infty,1}}$
$\displaystyle\leq Cn^{-\frac{1}{3}}(\log n)^{3}.$ (3.21)

Also, we have

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}F\circ\phi\|_{L^{\infty}}$
$\displaystyle\leq C\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}F\|_{L^{\infty}}$
$\displaystyle\leq C\|(\partial_{x}u)^{3}+\partial_{x}(u^{3})\|_{L^{\infty}}$
$\displaystyle\leq C\|u\|^{3}_{C^{0,1}}\leq C\|u_{0}\|^{3}_{C^{0,1}}\leq
Cn^{-1}(\log n)^{3}.$ (3.22)

Combining (3.4)-(3.4) and using Lemmas 3.1-3.2 yields

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|(\Delta_{j}u)\circ\phi\|_{L^{\infty}}$
$\displaystyle\geq
t\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E_{0}\|_{L^{\infty}}-\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}$
$\displaystyle\quad-Cn^{-\frac{1}{3}}(\log
n)^{3}-C\|u_{0}\|_{B^{1}_{\infty,1}}$ $\displaystyle\geq
ct\log^{2}n-\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}-Cn^{-\frac{1}{3}}(\log
n)^{3}.$

Step 2: Upper bounds for $\Delta_{j}E\circ\phi-\Delta_{j}E_{0}$

By easy computations,

$\partial_{x}\Lambda^{-2}[u^{2}u_{x}^{3}+2u\partial_{x}(u^{2}u_{x})u_{x}]=\partial_{x}\Lambda^{-2}[2u^{2}u_{x}^{3}+\partial_{x}(u^{3}u_{x}^{2})]=2\partial_{x}\Lambda^{-2}(u^{2}u_{x}^{3})+\Lambda^{-2}(u^{3}u_{x}^{2})-u^{3}u_{x}^{2},$

then we find that

$\displaystyle\partial_{t}E+u^{2}\partial_{x}E$
$\displaystyle=-\frac{3}{2}\left(\partial_{x}\Lambda^{-2}\partial_{t}(uu_{x}^{2})+u^{2}\partial_{x}^{2}\Lambda^{-2}(uu_{x}^{2})\right)$
$\displaystyle=\mathrm{J}+\frac{3}{2}\left(\partial_{x}\Lambda^{-2}[u^{2}u_{x}^{3}+2u\partial_{x}(u^{2}u_{x})u_{x}]+u^{3}u_{x}^{2}-u^{2}\Lambda^{-2}(uu_{x}^{2})\right)$
$\displaystyle=\mathrm{J}+\mathrm{K},$ (3.23)

where

$\displaystyle\mathrm{J}=-\frac{3}{2}\partial_{x}\Lambda^{-2}\left(\mathbf{P}_{1}(u)u^{2}_{x}+\mathbf{P}_{2}(u)u^{2}_{x}+2\partial_{x}\mathbf{P}_{1}(u)uu_{x}+2\partial_{x}\mathbf{P}_{2}(u)uu_{x}\right),$
$\displaystyle\mathrm{K}=\frac{3}{2}\left(2\partial_{x}\Lambda^{-2}(u^{2}u_{x}^{3})+\Lambda^{-2}(u^{3}u_{x}^{2})-u^{2}\Lambda^{-2}(uu_{x}^{2})\right).$

Utilizing (3.17) to (3.4) yields

$\displaystyle\Delta_{j}E\circ\phi-\Delta_{j}E_{0}=\int^{t}_{0}[u^{2},\Delta_{j}]\partial_{x}E\circ\phi\mathrm{d}\tau+\int^{t}_{0}\Delta_{j}(\mathrm{J}+\mathrm{K})\circ\phi\mathrm{d}\tau.$

Using the commutator estimate from Lemma 2.2, one has

$\displaystyle 2^{j}\|[u^{2},\Delta_{j}]\partial_{x}E\|_{L^{\infty}}$
$\displaystyle\leq
C(\|\partial_{x}(u^{2})\|_{L^{\infty}}\|E\|_{B^{1}_{\infty,\infty}}+\|\partial_{x}E\|_{L^{\infty}}\|u\|_{B^{1}_{\infty,\infty}})$
$\displaystyle\leq C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5}.$

Due to the facts

$\displaystyle\|\Lambda^{-2}f\|_{L^{\infty}}\leq\|f\|_{L^{\infty}}\quad\text{and}\quad\|\partial_{x}\Lambda^{-2}f\|_{L^{\infty}}\leq\|f\|_{L^{\infty}}\quad\Rightarrow\quad\|\partial^{2}_{x}\Lambda^{-2}f\|_{L^{\infty}}\leq
2\|f\|_{L^{\infty}},$

then we have

$\displaystyle
2^{j}\|\Delta_{j}\mathrm{J}\|_{L^{\infty}}\approx\|\partial_{x}\mathrm{J}\|_{L^{\infty}}\leq
C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5}.$

Similarly,

$\displaystyle 2^{j}\|\Delta_{j}\mathrm{K}\|_{L^{\infty}}\leq
C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5}.$

Then, we deduce that

$\displaystyle 2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}\leq
C\|u\|^{5}_{C^{0,1}}\leq Cn^{-\frac{5}{3}}(\log n)^{5},$

which leads to

$\displaystyle\sum_{j\in\mathbb{N}(n)}2^{j}\|\Delta_{j}E\circ\phi-\Delta_{j}E_{0}\|_{L^{\infty}}\leq
Cn^{-\frac{2}{3}}(\log n)^{5}.$

Combining Step 1 and Step 2, then for $t=\frac{1}{\log n}$, we obtain for
$n\gg 1$

$\displaystyle\|u(t)\|_{B^{1}_{\infty,1}}$
$\displaystyle\geq\|u(t)\|_{B^{1}_{\infty,1}(\mathbb{N}(n))}$
$\displaystyle\geq
C\sum_{j\in\mathbb{N}(n)}2^{j}\|(\Delta_{j}u)\circ\phi\|_{L^{\infty}}$
$\displaystyle\geq ct(\log n)^{2}-Cn^{-\frac{2}{3}}(\log
n)^{5}-Cn^{-\frac{1}{3}}(\log n)^{3}$ $\displaystyle\geq\log\log n,$

which contradicts the hypothesis (3.19). Thus, Theorem 1.1 is proved.
$\square$

##  4 Discussion

By the clever Lagrangian coordinate transformation used in [15] and
constructing a new initial data, we prove that the Novikov equation is ill-
posed in critical Besov spaces $B^{1}_{\infty,1}(\mathbb{R})$. Thus our
results (Theorem 1.1 and Corollary 1.1) indicate that the local well-posedness
and ill-posedness for the Novikov equation in all critical Besov spaces
$B^{1+1/p}_{p,r}(\mathbb{R})$ have been solved completely. Since the Novikov
equation has cubic nonlinear term, we expect that norm inflation is stemmed
from the worst term $u(\partial_{x}u)^{2}$, which is different from the
quadratic term $(\partial_{x}u)^{2}$ for the Camassa-Holm equation. Our new
idea is to construct a initial data which includes two parts, one of whose
Fourier transform is supported at high frequencies and the other is supported
at low frequencies. Then the cubic nonlinear term $u(\partial_{x}u)^{2}$ will
generate the low-high-high frequency interaction, which contributes a large
quantity lead to the norm inflation. Lastly, we should mention that, by
dropping the low frequency term, the initial data
$u_{0}=2^{-\frac{n}{2}}u_{0}^{\mathrm{H}}$ can be as an example which leads to
the norm inflation for the Camassa-Holm equation (see Remark 3.1) .

## Acknowledgments

J. Li is supported by the National Natural Science Foundation of China
(11801090 and 12161004) and Jiangxi Provincial Natural Science Foundation
(20212BAB211004). Y. Yu is supported by the National Natural Science
Foundation of China (12101011) and Natural Science Foundation of Anhui
Province (1908085QA05). W. Zhu is supported by the National Natural Science
Foundation of China (12201118) and Guangdong Basic and Applied Basic Research
Foundation (2021A1515111018).

## Data Availability

No data was used for the research described in the article.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg, 2011.
  * [2] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
  * [3] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier, 50 (2000), 321-362.
  * [4] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.
  * [5] A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.
  * [6] A. Constantin, J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
  * [7] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
  * [8] A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
  * [9] R. Danchin, A few remarks on the Camassa-Holm equation, Differ. Integral Equ., 14 (2001), 953-988.
  * [10] A. Degasperis, D. Holm, A. Hone, A new integral equation with peakon solutions. Theoret. Math. Phys. 133 (2002), 1463-1474.
  * [11] J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J. 56 (2007), 87-177.
  * [12] A. Fokas, B. Fuchssteiner, Symplectic structures, their bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/1982) 47-66.
  * [13] K. Grayshan, A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232.
  * [14] Z. Guo, X. Liu, L. Molinet, Z. Yin, Ill-posedness of the Camassa-Holm and related equations in the critical space, J. Differ. Equ., 266 (2019), 1698-1707.
  * [15] Y. Guo, W. Ye, Z. Yin, Ill-posedness for the Cauchy problem of the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$, J. Differ. Equ., 327 (2022), 127-144.
  * [16] A. A. Himonas, C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449–479.
  * [17] A. Himonas, C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equ., 19 (2014) 161-200.
  * [18] A. Himonas, D. Matzavinos, An ab-family ofequations with peakon traveling waves, Proc. Amer. Math. Soc., 144 (2016), 3797–3811.
  * [19] A. Hone, H. Lundmark, J. Szmigielski, Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.
  * [20] A. Hone, J. Wang, Integrable peakon equations with cubic nonlinearity, J Phys A., 41 (2008) 372002: 1-11.
  * [21] J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl. 306 (2005), 72-82.
  * [22] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.
  * [23] Y. Li, P.J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162:1 (2000) 27-63.
  * [24] J. Li, Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differ. Equ., 261 (2016), 6125-6143.
  * [25] J. Li, Y. Yu, W. Zhu, Well-posedness and continuity properties of the Degasperis-Procesi equation in critical Besov space. Monatsh. Math., (2022) doi.org/10.1007/s00605-022-01691-4
  * [26] J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differ. Equ., 306 (2022), 403-417.
  * [27] J. Li, Y. Yu, W. Zhu, Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces, J. Evol. Equ., 22:29 (2022) 11pp.
  * [28] L. Ni, Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differ. Equ., 250 (2011), 3002-3021.
  * [29] V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A., 42 (2009), 342002.
  * [30] G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. 46 (3) (2001), 309-327.
  * [31] X. Wu, Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Classe Sci. Ser. V 11 (2012), 707-727.
  * [32] X. Wu, Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137.
  * [33] W. Yan, Y. Li, Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differ. Equ., 253 (2012), 298-318.
  * [34] W. Yan, Y. Li, Y. Zhang, The Cauchy problem for the Novikov equation, NoDEA-Nonlinear Differ. Equ. Appl., 20 (2013), 1157–1169.
  * [35] W. Ye, Z. Yin, Y. Guo, A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces $B^{1+1/p}_{p,1}$, $1\leq p<\infty$, arXiv, 2021.
  * [36] Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal. 212 (2004), 182-194.


# Bi-Lipschitz invariance of
planar $BV$\- and $W^{1,1}$-extension domains

Miguel García-Bravo  ,  Tapio Rajala  and  Zheng Zhu  University of Jyvaskyla
Department of Mathematics and Statistics
P.O. Box 35 (MaD)
FI-40014 University of Jyvaskyla
Finland<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove that a bi-Lipschitz image of a planar $BV$-extension domain is also a
$BV$-extension domain, and that a bi-Lipschitz image of a planar
$W^{1,1}$-extension domain is again a $W^{1,1}$-extension domain.

###### Key words and phrases:

Sobolev extension, BV-extension

###### 2000 Mathematics Subject Classification:

Primary 46E35.

The first two authors acknowledge the support from the Academy of Finland,
grant no. 314789. The third author was supported by the Academy of Finland via
the Centre of Excellence in Analysis and Dynamics Research, grant no. 323960.

##  1\. Introduction

Let ${\Omega}\subset{\mathbb{R}}^{n}$ be a domain. For $1\leq p\leq\infty$, we
define the Sobolev space $W^{1,p}({\Omega})$ by setting

$W^{1,p}:=\\{u\in L^{p}({\Omega})\,:\,\nabla u\in
L^{p}({\Omega};{{{\mathbb{R}}}^{n}})\\},$

where $\nabla u$ means the weak (distributional) derivative of $u$. The
Sobolev space $W^{1,p}({\Omega})$ is equipped with the norm

$\|u\|_{W^{1,p}({\Omega})}:=\|u\|_{L^{p}({\Omega})}+\||\nabla
u|\|_{L^{p}({\Omega})}.$

We say that ${\Omega}$ is a $W^{1,p}$-extension domain if there exists a
bounded extension operator $T\colon W^{1,p}({\Omega})\to
W^{1,p}({{{\mathbb{R}}}^{n}})$, meaning that for every $u\in
W^{1,p}({\Omega})$ we have $T(u)\big{|}_{\Omega}\equiv u$ and

$\|T(u)\|_{W^{1,p}({{{\mathbb{R}}}^{n}})}\leq C\|u\|_{W^{1,p}({\Omega})},$

where the constant $C$ is independent of $u$. The minimal possible constant
$C$ above is denoted by $\|T\|$. Based on results in [7, 8], for
$1<p\leq\infty$, whenever ${\Omega}$ is a $W^{1,p}$-extension domain, we can
construct a bounded linear extension operator $T\colon W^{1,p}({\Omega})\to
W^{1,p}({{{\mathbb{R}}}^{n}})$. By [12], for planar bounded simply connected
$W^{1,1}$-extension domains ${\Omega}$, we can also construct a bounded linear
extension operator $T\colon W^{1,1}({\Omega})\to W^{1,1}({\mathbb{R}}^{2})$.
The classical results due to Calderón and Stein [3, 15] tell us that Lipschitz
domains are $W^{1,p}$-extension domains, for every $1\leq p\leq\infty$. Later,
Jones [9] defined a class of so-called $(\epsilon,\delta)$-domains which are a
generalization of Lipschitz domains. He also proved that these domains are
$W^{1,p}$-extension domains for every $1\leq p\leq\infty$. Moreover, in the
works [11, 12, 14], a geometric characterization of planar bounded simply
connected $W^{1,p}$-extension domains was established.

For arbitrary $u\in W^{1,p}({{{\mathbb{R}}}^{n}})$ with $1\leq p\leq\infty$,
the inequality

$|u(x)-u(y)|\leq|x-y|\left(CM[|\nabla u|](x)+CM[|\nabla u|](y)\right)$

holds on every Lebesgue point of $u$, where the constant $C$ is independent of
$u$ and $M[|\nabla u|]$ denotes the Hardy-Littlewood maximal function of
$|\nabla u|$. Motivated by this estimate, Hajłasz defined the so-called
Hajłasz-Sobolev space $M^{1,p}({\Omega})$ which consists of all functions
$u\in L^{p}({\Omega})$ such that there exists a function $0\leq g\in
L^{p}({\Omega})$ satisfying the inequality

$|u(x)-u(y)|\leq|x-y|(g(x)+g(y))$ (1.1)

for every $x,y\in{\Omega}\setminus F$ where the exceptional set $F$ satisfies
$|F|=0$. We use $\mathcal{D}_{p}(u)$ to denote the class of all nonnegative
functions $g\in L^{p}({\Omega})$ which satisfy the inequality (1.1). The
Hajłasz-Sobolev space $M^{1,p}({\Omega})$ is then equipped with the norm

$\|u\|_{M^{1,p}({\Omega})}:=\|u\|_{L^{p}({\Omega})}+\inf_{g\in\mathcal{D}_{p}(u)}\|g\|_{L^{p}({\Omega})}.$

For $1\leq p\leq\infty$, one always has $M^{1,p}({\Omega})\subset
W^{1,p}({\Omega})$ and the inclusion is strict for $p=1$, see [13]. By [6],
for $1<p\leq\infty$, the equality $M^{1,p}({\Omega})=W^{1,p}({\Omega})$ holds
for a bounded $W^{1,p}$-extension domain ${\Omega}$. Similarly to
$W^{1,p}$-extension domains, a domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is
said to be an $M^{1,p}$-extension domain, if there exists a bounded extension
operator $T\colon M^{1,p}({\Omega})\to M^{1,p}({{{\mathbb{R}}}^{n}})$ (the
existence of such an operator implies the existence of a linear one for the
cases $1\leq p<\infty$ by [8]). Observe that since $M^{1,\infty}({\Omega})$
consists of Lipschitz functions, by Kirszbraun theorem, every domain is a
$M^{1,\infty}$-extension domain. Furthermore, in [8], it is proved that for
$1\leq p<\infty$ we have that ${\Omega}$ is an $M^{1,p}$-extension domain if
and only if ${\Omega}$ is Ahlfors $n$-regular. We say that a domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is Ahlfors $n$-regular, if for every
$x\in{\Omega}$ and $0<r<1$, we have

$|B(x,r)\cap{\Omega}|\geq c|B(x,r)|$

with a constant $0<c<1$ independent of $x$ and $r$. Combining these two
results, one can prove that for $1<p<\infty$, a domain
${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is a $W^{1,p}$-extension domain if and
only if $W^{1,p}({\Omega})=M^{1,p}({\Omega})$ and ${\Omega}$ is Ahlfors
$n$-regular (see [7, Theorem 5]).

In the case that ${\Omega}$ and ${\Omega}^{\prime}$ are bi-Lipschitz
equivalent, one can easily check that
$M^{1,p}({\Omega})=M^{1,p}({\Omega}^{\prime})$ isomorphically, and also the
fact that ${\Omega}$ is Ahlfors $n$-regular if and only if ${\Omega}^{\prime}$
is Ahlfors $n$-regular. Moreover, a domain is a $W^{1,\infty}$-extension
domain if and only if it is locally quasiconvex and local quasiconvexity is
bi-Lipschitz invariant. We say that a set $E\subset{\mathbb{R}}^{n}$ is
quasiconvex, if there exists a constant $c\geq 1$ such that for every $x,y\in
E$, there exists a rectifiable curve $\gamma\subset E$ connecting $x$ and $y$
with the length controlled from above by $c|x-y|$. In such case $E$ is called
$c$-quasiconvex. These last observations lead to the following theorem,
presented by Hajłasz, Koskela and Tuominen in [7].

######  Theorem 1.1.

If ${\Omega}$ and ${\Omega}^{\prime}$ are bi-Lipschitz equivalent, for
$1<p\leq\infty$, ${\Omega}$ is a $W^{1,p}$-extension domain if and only if
${\Omega}^{\prime}$ is a $W^{1,p}$-extension domain.

Since $M^{1,1}({\Omega})$ is strictly included in $W^{1,1}({\Omega})$ for
arbitrary domains ${\Omega}$, the above method does not work for the case
$p=1$. In the same paper, Hajłasz, Koskela and Tuominen raised the question.

######  Question 1.2.

Is Theorem 1.1 true for $p=1$?

A partial affirmative answer to this question was provided in [10, Corollary
1.3]. There, it was shown that Theorem 1.1 holds for bounded simply connected
planar domains also in the case $p=1$, and also for $BV$-functions. In [10] it
was also conjectured that the hypothesis of simply connectivity was
superfluous. In this paper we will show that they were right. We will extend
these previous results and answer Question 1.2 positively for general bounded
planar domains. We will do this by first resolving the question for
$BV$-functions via decomposition of sets of finite perimeter into Jordan
domains, and then employing a recent result from [4] to pass to
$W^{1,1}$-functions. Both, the proof in [10] and our proof, rely on the
results of Väisälä [16] and on the quasiconvexity of the connected open
components of the complement of planar $BV$-extension domains. The difference
is that in [10] the bi-Lipschitz function was extended to the complement by
using the fact that the bi-Lipschitz map can be extended to a small
neighbourhood. Here we use a decomposition of sets of finite perimeter and the
bi-Lipschitz invariance of the quasiconvexity of the holes, see Section 2 for
the definitions and results needed for this approach.

Recall that the space of functions of bounded variation $BV({\Omega})$ is
defined by setting

$BV({\Omega}):=\\{u\in L^{1}({\Omega})\ :\|Du\|({\Omega})<\infty\\}$

where

$\|Du\|({\Omega})=\sup\left\\{\int_{\Omega}u{\rm div}(v)dx\,:\,v\in
C_{0}^{\infty}({\Omega};{{{\mathbb{R}}}^{n}}),|v|\leq 1\right\\}$

means the total variation of $u$ on ${\Omega}$. The function space
$BV({\Omega})$ is equipped with the norm

$\|u\|_{BV({\Omega})}:=\|u\|_{L^{1}({\Omega})}+\|Du\|({\Omega}).$

Note that $\|Du\|$ is a Radon measure on ${\Omega}$ that is defined for every
set $F\subset{\Omega}$ as

$\|Du\|(F):=\inf\\{\|Du\|(U)\,:\,F\subset U,U\ {\rm is\ open}\\}.$

A domain ${\Omega}\subset{{{\mathbb{R}}}^{n}}$ is said to be a $BV$-extension
domain, if there exists a bounded extension operator $T\colon BV({\Omega})\to
BV({{{\mathbb{R}}}^{n}})$ with $T(u)\big{|}_{\Omega}\equiv u$ and an absolute
constant $C>0$ so that

$\|T(u)\|_{BV({{{\mathbb{R}}}^{n}})}\leq C\|u\|_{BV({\Omega})}$

for every $u\in BV({\Omega})$. By a result in [10], a $W^{1,1}$-extension
domain is also a $BV$-extension domain. A typical example showing that the
converse is not true is the slit disk in the plane.

In Section 2 we will prove the following result.

######  Theorem 1.3.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded $BV$-extension domain and
$f\colon\Omega\to\Omega^{\prime}$ a bi-Lipschitz map. Then $\Omega^{\prime}$
is also a $BV$-extension domain.

Let us show how Theorem 1.3 implies the same result for $W^{1,1}$-extension
domains. We use the recent characterization of $W^{1,1}$-extension domains
among bounded $BV$-extension domains that was proven by the first and second
named authors in [4].

######  Theorem 1.4.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded $BV$-extension domain. Then
$\Omega$ is a $W^{1,1}$-extension domain if and only if the set

$\partial\Omega\setminus\bigcup_{i\in I}\overline{\Omega_{i}}$

is purely $1$-unrectifiable, where $\\{\Omega_{i}\\}_{i\in I}$ are the
connected components of $\mathbb{R}^{2}\setminus\overline{\Omega}$.

Recall that a set $H\subset\mathbb{R}^{2}$ is called purely $1$-unrectifiable
if for every Lipschitz map $f\colon\mathbb{R}\to\mathbb{R}^{2}$ we have
$\mathcal{H}^{1}(H\cap f(\mathbb{R}))=0$.

Suppose that $\Omega\subset\mathbb{R}^{2}$ is a bounded $W^{1,1}$-extension
domain. By Theorem 1.4 the set

$H=\partial\Omega\setminus\bigcup_{i\in I}\overline{\Omega_{i}}$

is purely $1$-unrectifiable and so is the image $H^{\prime}=f(H)$ under a bi-
Lipschitz map $f\colon\Omega\to\Omega^{\prime}$ that is extended to the
closures $\overline{\Omega}$ and $\overline{\Omega}^{\prime}$ as a bi-
Lipschitz map. Hence, recalling that as a $W^{1,1}$-extension domain $\Omega$
is also a $BV$-extension domain, Theorem 1.3 implies that
$\Omega^{\prime}=f(\Omega)$ is also a $BV$-extension domain. Now, from Theorem
1.4 we conclude that $\Omega^{\prime}$ is a $W^{1,1}$-extension domain. We
have then established the following.

######  Corollary 1.5.

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded $W^{1,1}$-extension domain and
$f\colon\Omega\to\Omega^{\prime}$ a bi-Lipschitz map. Then $\Omega^{\prime}$
is a $W^{1,1}$-extension domain.

Let us remark that one could also prove Corollary 1.5 with a similar proof as
we provide for Theorem 1.3 in Section 2; via the invariance of quasiconvexity
of the components of the complement, a characterization of $W^{1,1}$-extension
domains as the domains with the strong extension property for sets of finite
perimeter and by slightly pushing the boundary of the extension of a Jordan
domain away from the boundary of $\partial\Omega$, see [4] for more details on
these tools. This alternative approach indicates that if we were able to prove
the bi-Lipschitz invariance of $BV$-extension domains in higher dimensions,
and were able to push the boundaries of sets of finite perimeter away from
$\partial\Omega$ in a controlled manner, the bi-Lipschitz invariance of
$W^{1,1}$-extension domains would follow. However, at the moment we are not
able to complete such proof. An alternative approach for trying to solve the
higher dimensional case could be to use the characterization of
$W^{1,1}$-functions from [5], similar to $M^{1,1}$.

##  2\. Proof of Theorem 1.3

We will prove Theorem 1.3 by using the bi-Lipschitz invariance of the
quasiconvexity of the connected components of
$\mathbb{R}^{2}\setminus\overline{\Omega}$, which are referred to as the
_holes_ of $\overline{\Omega}$. We recall the following result of Väisälä [16,
Corollary 4.11].

######  Theorem 2.1.

Let $G\subset\mathbb{R}^{2}$ be a bounded continuum such that each hole of $G$
is $c$-quasiconvex and let $f\colon G\to\mathbb{R}^{2}$ be $L$-bilipschitz.
Then each hole of $G^{\prime}=f(G)$ is $c^{\prime}$-quasiconvex with
$c^{\prime}(c,L)$.

In order to use the Theorem 2.1 we need to observe that the holes of a bounded
planar $BV$-extension domain are quasiconvex. This was established for simply
connected domains in [10, Theorem 1.1], and the proof works with minor
modifications in the more general case considered here, see [4, Lemma 5.2].

######  Lemma 2.2.

Suppose that $\Omega\subset\mathbb{R}^{2}$ is a bounded $BV$-extension domain.
Then there exists a constant $c>0$ so that each hole of $\overline{\Omega}$ is
$c$-quasiconvex.

The invariance of quasiconvexity of the holes is easier to use for the
boundaries of sets of finite perimeter, rather than for $BV$-functions. The
passage from $BV$-functions to sets of finite perimeter is provided by Lemma
2.3, which is a combination of the works [2, 10]. Before stating it we need to
recall some definitions.

A Lebesgue measurable subset $E\subset\mathbb{R}^{n}$ has finite perimeter in
$\Omega$ if $\chi_{E}\in BV(\Omega)$, where $\chi_{E}$ denotes the
characteristic function. We set $P(E,\Omega)=\|D\chi_{F}\|(\Omega)$ and call
it the perimeter of $E$ in $\Omega$. We will say that $\Omega$ has the
extension property for sets of finite perimeter if there exists $C>0$ so that
for every set $E\subset\Omega$ of finite perimeter in $\Omega$ one may find
$\widetilde{E}\subset\mathbb{R}^{n}$ of finite perimeter in $\mathbb{R}^{n}$
such that $\widetilde{E}\cap\Omega=E$, modulo a measure zero set, and
$P(\widetilde{E},\mathbb{R}^{n})\leq CP(E,\Omega)$.

######  Lemma 2.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain. Then the following are
equivalent:

  1. (1)

$\Omega$ is a $BV$-extension domain.

  2. (2)

$\Omega$ has the extension property for sets of finite perimeter.

One more tool that we use is the decomposition of planar sets of finite
perimeter into Jordan domains.

We say that $\Gamma\subset\mathbb{R}^{2}$ is a Jordan curve if
$\Gamma=\gamma([a,b])$ for some $a,b\in\mathbb{R}$, $a<b$, and some continuous
map $\gamma$, injective on $[a,b)$ and such that $\gamma(a)=\gamma(b)$. The
Jordan curve theorem assures that $\Gamma$ splits
$\mathbb{R}^{2}\setminus\Gamma$ into exactly two connected components, a
bounded one and an unbounded one that we denote by $\text{int}(\Gamma)$ and
$\text{ext}(\Gamma)$ respectively. A set $U$ whose boundary $\partial U$ is a
Jordan curve is called a Jordan domain.

For technical reasons we also add to the class of Jordan curves the formal
“Jordan” curves $J_{0}$ and $J_{\infty}$, whose interiors are $\mathbb{R}^{2}$
and the empty set respectively and for which we set
$\mathcal{H}^{1}(J_{0})=\mathcal{H}^{1}(J_{\infty})=0$.

For a measurable set $E\subset{{{\mathbb{R}}}^{n}}$, we denote by
$\partial^{M}E$ its essential boundary, which consists of points such that
both $E$ and ${{{\mathbb{R}}}^{n}}\setminus E$ have positive upper density on
them, that is

$\partial^{M}E=\left\\{x\in\mathbb{R}^{n}\,:\,\limsup_{r\searrow
0}\frac{|E\cap B(x,r)|}{|B(x,r)|}>0\;\text{and}\;\limsup_{r\searrow
0}\frac{|(\mathbb{R}^{n}\setminus E)\cap B(x,r)|}{|B(x,r)|}>0\right\\}.$

A set $E\subset\mathbb{R}^{n}$ of finite perimeter is called decomposable, if
there exist sets $A,B\subset\mathbb{R}^{n}$ of positive Lebesgue measure such
that $E=A\cup B$, $A\cap B=\emptyset$, and
$P(E,\mathbb{R}^{n})=P(A,\mathbb{R}^{n})+P(B,\mathbb{R}^{n})$. A set is called
indecomposable, if it is not decomposable.

The following was proven in [1, Corollary 1].

######  Theorem 2.4.

Let $E\subset\mathbb{R}^{2}$ have finite perimeter. Then, there exists a
unique decomposition of $\partial^{M}E$ into rectifiable Jordan curves
$\\{C_{i}^{+},C_{k}^{-}\,:\,i,k\in\mathbb{N}\\}$, modulo
$\mathcal{H}^{1}$-measure zero sets, such that

  1. (1)

Given $\text{int}(C_{i}^{+})$, $\text{int}(C_{k}^{+})$, $i\neq k$, they are
either disjoint or one is contained in the other; given
$\text{int}(C_{i}^{-})$, $\text{int}(C_{k}^{-})$, $i\neq k$, they are either
disjoint or one is contained in the other. Each $\text{int}(C_{i}^{-})$ is
contained in one of the $\text{int}(C_{k}^{+})$.

  2. (2)

$P(E,\mathbb{R}^{2})=\sum_{i}\mathcal{H}^{1}(C_{i}^{+})+\sum_{k}\mathcal{H}^{1}(C_{k}^{-})$.

  3. (3)

If $\text{int}(C_{i}^{+})\subset\text{int}(C_{j}^{+})$, $i\neq j$, then there
is some rectifiable Jordan curve $C_{k}^{-}$ such that
$\text{int}(C_{i}^{+})\subset\text{int}(C_{k}^{-})\subset\text{int}(C_{j}^{+})$.
Similarly, if $\text{int}(C_{i}^{-})\subset\text{int}(C_{j}^{-})$, $i\neq j$,
then there is some rectifiable Jordan curve $C_{k}^{+}$ such that
$\text{int}(C_{i}^{-})\subset\text{int}(C_{k}^{+})\subset\text{int}(C_{j}^{-})$.

  4. (4)

Setting $L_{j}=\\{i\,:\,\text{int}(C_{i}^{-})\subset\text{int}(C_{j}^{+})\\}$
the sets $Y_{j}=\text{int}(C_{j}^{+})\setminus\bigcup_{i\in
L_{j}}\text{int}(C_{i}^{-})$ are pairwise disjoint , indecomposable and
$E=\bigcup_{j}Y_{j}$.

Since sets of finite perimeter are defined via the total variation seminorm of
$BV$-functions, they are understood modulo $2$-dimensional measure zero sets.
In particular, the last equality in (4) of Theorem 2.4 is modulo measure zero
sets. To make precise the change of representatives, we use below the notation
$A\Delta B:=(A\setminus B)\cup(B\setminus A)$ for the symmetric difference
between subsets $A,B\subset\mathbb{R}^{2}$.

With the auxiliary tools now recalled, we are ready to prove the main result
of this paper.

###### Proof of Theorem 1.3.

Suppose $f\colon\Omega\to\Omega^{\prime}$ is $L$-bi-Lipschitz. First notice
that $f$ extends to $\overline{\Omega}\to\overline{\Omega^{\prime}}$ as a bi-
Lipschitz map. Let $\Omega_{i}^{\prime}$ be a connected component of
$\mathbb{R}^{2}\setminus\overline{\Omega^{\prime}}$ and $\Omega_{i}$ the
connected component of $\mathbb{R}^{2}\setminus\overline{\Omega}$ for which
$f(\partial\Omega_{i})=\partial\Omega_{i}^{\prime}$. Then by Lemma 2.2 there
exists a constant $c>0$ so that each hole ${\Omega_{i}}$ is $c$-quasiconvex.
Therefore, by Theorem 2.1 also each ${\Omega_{i}^{\prime}}$ is
$c^{\prime}$-quasiconvex, where the quasiconvexity constant $c^{\prime}$ does
not depend on $i$. Obviously, each $\overline{\Omega_{i}^{\prime}}$ is also
$c^{\prime}$-quasiconvex.

Suppose that $\Omega$ is a bounded $BV$-extension domain. By Lemma 2.3, we
only need to prove that having the extension property for sets of finite
perimeter is invariant under the bi-Lipschitz map $f$. Let
$E^{\prime}\subset\Omega^{\prime}$ be a set of finite perimeter. Then
$E=f^{-1}(E^{\prime})$ is also a set of finite perimeter with

$P(E,\Omega)\leq LP(E^{\prime},\Omega^{\prime}).$ (2.1)

Let $\widetilde{E}$ be the perimeter extension of $E$ to the whole
$\mathbb{R}^{2}$ with

$P(\widetilde{E},{\mathbb{R}}^{2})\leq CP(E,{\Omega}).$ (2.2)

By Theorem 2.4, there exists a class of Jordan curves
$\\{\widetilde{C}_{i}^{+},\widetilde{C}_{k}^{-}\\}_{i,k\in\mathbb{N}}$ with

$\mathcal{H}^{2}(F)=0$

for the symmetric difference

$F:=\widetilde{E}\Delta\left(\bigcup_{i}\text{int}(\widetilde{C}_{i}^{+})\setminus\bigcup_{k}\text{int}(\widetilde{C}_{k}^{-})\right),$

and

$P(\widetilde{E},\mathbb{R}^{2})=\sum_{i}\mathcal{H}^{1}(\widetilde{C}_{i}^{+})+\sum_{k}\mathcal{H}^{1}(\widetilde{C}^{-}_{k})$
(2.3)

Take
$J\in\\{\text{int}(\widetilde{C}_{i}^{+})\\}_{i}\cup\\{\text{int}(\widetilde{C}_{k}^{-})\\}_{k}.$
We will extend each $J^{\prime}=f(J\cap\Omega)\subset\Omega^{\prime}$ to the
whole $\mathbb{R}^{2}$ in order to define the final extension set
$\widetilde{E}^{\prime}$ of $E^{\prime}$. Consider a homeomorphism
$\gamma\colon\mathbb{S}^{1}\to\partial J$, given by the fact that $J$ is a
Jordan domain. The set $\gamma\setminus\overline{\Omega}$ consists of (at
most) countably many open arcs $\gamma_{i}$ with endpoints
$x_{i},y_{i}\in\partial\Omega_{j(i)}$ for some $j(i)$. Observe that since
$|x_{i}-y_{i}|\leq\mathcal{H}^{1}(\gamma_{i})$, we always have

$\mathcal{H}^{1}(\gamma\cap\overline{\Omega})+\sum_{i}|x_{i}-y_{i}|\leq\mathcal{H}^{1}(\gamma\cap\overline{\Omega})+\mathcal{H}^{1}(\gamma\setminus\overline{\Omega})\leq\mathcal{H}^{1}(\gamma).$
(2.4)

Let us write $I_{i}=\gamma^{-1}(\gamma_{i})$ for each $i$ and use the
$c^{\prime}$-quasiconvexity of $\overline{\Omega_{j(i)}^{\prime}}$ to find a
curve $\gamma_{i}^{\prime}\subset\overline{\Omega_{j(i)}^{\prime}}$ joining
$f(x_{i})$ to $f(y_{i})$ with

$\ell(\gamma_{i}^{\prime})\leq c^{\prime}|f(x_{i})-f(y_{i})|.$ (2.5)

For convenience, we use the parametrization $\gamma_{i}^{\prime}\colon
I_{i}\to\mathbb{R}^{2}$ so that
$\gamma_{i}^{\prime-1}(x_{i})=\gamma^{-1}(x_{i})$ and
$\gamma_{i}^{\prime-1}(y_{i})=\gamma^{-1}(y_{i})$. The combination of
$f(\gamma\cap\overline{\Omega})$ with the curves $\gamma_{i}^{\prime}$ results
in a continuous curve $\gamma^{\prime}\colon\mathbb{S}^{1}\to\mathbb{R}^{2}$
defined as

$\gamma^{\prime}(t)=\begin{cases}f(\gamma(t)),&\text{if
}\gamma(t)\in\overline{\Omega},\\\ \gamma_{i}^{\prime}(t),&\text{if }t\in
I_{i}.\end{cases}$

Now, we define the extension domain $\widetilde{J}^{\prime}$ of
$J^{\prime}=f(\Omega\cap J)$ as the union of all the connected components of
$\mathbb{R}^{2}\setminus\gamma^{\prime}$ that intersect $J^{\prime}$. Then,
combining (2.5) with (2.4) and the fact that $f$ is $L$-bi-Lipschitz, gives

$\begin{split}P(\widetilde{J}^{\prime},\mathbb{R}^{2})&\leq\mathcal{H}^{1}(\gamma^{\prime})\\\
&=\mathcal{H}^{1}(f(\gamma\cap\overline{\Omega}))+\sum_{i}\mathcal{H}^{1}(\gamma_{i}^{\prime})\\\
&\leq\mathcal{H}^{1}(f(\gamma\cap\overline{\Omega}))+\sum_{i}c^{\prime}|f(x_{i})-f(y_{i})|\\\
&\leq
L\mathcal{H}^{1}(\gamma\cap\overline{\Omega})+c^{\prime}L\sum_{i}\mathcal{|}x_{i}-y_{i}|\\\
&\leq c^{\prime}L\mathcal{H}^{1}(\gamma)\\\
&=c^{\prime}LP(J,\mathbb{R}^{2}).\end{split}$ (2.6)

We finally set our extension of $E^{\prime}$ to be

$\widetilde{E}^{\prime}=\bigcup_{J\in\\{\text{int}(\widetilde{C}_{i}^{+})\\}_{i}}\widetilde{J}^{\prime}\setminus\bigcup_{J\in\\{\text{int}(\widetilde{C}_{k}^{-})\\}_{k}}\widetilde{J}^{\prime}.$

Since the decomposition of $\widetilde{E}$ was left unchanged inside $\Omega$,
we have that

$(\widetilde{E}^{\prime}\cap\Omega^{\prime})\Delta E^{\prime}=f(F\cap\Omega)$

has zero $2$-dimensional measure, since $F$ is measure-zero and $f$ is bi-
Lipschitz. Hence, $\widetilde{E^{\prime}}$ is indeed an extension of
$E^{\prime}$

Now, summing the estimate (2.6) over all the Jordan curves and using (2.3),
(2.2), and (2.1), we get

$\displaystyle P(\widetilde{E}^{\prime},\mathbb{R}^{2})$
$\displaystyle\leq\sum_{J\in\\{\text{int}(\widetilde{C}_{i}^{+}),\text{int}(\widetilde{C}_{k}^{-})\\}_{i,k}}P(\widetilde{J^{\prime}},\mathbb{R}^{2})$
$\displaystyle\leq
c^{\prime}L\sum_{J\in\\{\text{int}(\widetilde{C}_{i}^{+}),\text{int}(\widetilde{C}_{k}^{-})\\}_{i,k}}P(J,\mathbb{R}^{2})$
$\displaystyle=c^{\prime}LP(\widetilde{E},\mathbb{R}^{2})$ $\displaystyle\leq
Cc^{\prime}LP(E,\Omega)$ $\displaystyle\leq
Cc^{\prime}L^{2}P(E^{\prime},\Omega^{\prime}).$

This shows that $\widetilde{E}^{\prime}$ is a perimeter extension of $E$ as
required. Thus, by Lemma 2.3 we conclude that the domain $\Omega^{\prime}$ is
a $BV$-extension domain. ∎

## References

  * [1] L. Ambrosio, V. Caselles, S. Masnou and J.-M. Morel, _Connected components of sets of finite perimeter and applications to image processing._ J. Eur. Math. Soc. 3, 39–92 (2001).
  * [2] Y. D. Burago and V. G. Maz’ya, _Certain questions of potential theory and function theory for regions with irregular boundaries_ (Russian), Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 3 (1967) 152pp.
  * [3] A. P. Calderón, _Lebesgue spaces of differentiable functions and distributions_ , in Proc. Symp. Pure Math., Vol $IV$, 1961, 33-49.
  * [4] M. García-Bravo and T. Rajala, _Strong $BV$-extension and $W^{1,1}$-extension domains_, preprint 2021.
  * [5] P. Hajłasz, _A new characterization of the Sobolev space_ , Stud. Math. 159 (2003), no. 2, 263–275.
  * [6] P. Hajłasz, _Sobolev spaces on an arbitrary metric space_ , Potential Anal. 5 (1996), no. 4, 403-415.
  * [7] P. Hajłasz, P. Koskela and H. Tuominen, _Sobolev embeddings, extensions and measure density condition_ , J. Funct. Anal. 254 (2008), no. 5, 1217–1234.
  * [8] P. Hajłasz, P. Koskela and H. Tuominen, _Measure density and extendability of Sobolev functions_ , Rev. Mat. Iberoam. 24 (2008), no. 2, 645–669.
  * [9] P. W. Jones, _Quasiconformal mappings and extendability of Sobolev functions_ , Acta. Math. 47 (1981), 71-88.
  * [10] P. Koskela, M. Jr. Miranda and N. Shanmugalingam, _Geometric properties of planar $BV$-extension domains._ Around the research of Vladimir Maz’ya. $I$, 255–272, Int.Math.Ser.(N.Y.), 11, Springer, New York, 2010.
  * [11] P. Koskela, T. Rajala and Y. Zhang, _A geometric characterization of planar Sobolev extension domains._ arXiv:1502.04139v6.
  * [12] P. Koskela, T. Rajala and Y. Zhang, _Planar $W^{1,1}$-extension domains_, preprint.
  * [13] P. Koskela and E. Saksman _Pointwise characterizations of Hardy-Sobolev functions_ , Math. Res. Lett. 15 (2008), 727–744.
  * [14] P. Shvartsman, _On Sobolev extension domains in ${{{\mathbb{R}}}^{n}}$_, J. Funct. Anal. 258 (2010), no. 7, 2205-2245.
  * [15] E. M. Stein, _Singular integrals and differentiability properties of functions_ , Princeton University Press, Princeton, New Jersey, 1970.
  * [16] J. Väisälä, _Holes and maps of Euclidean spaces_ , Conform. Geom. Dyn. 12 (2008) 58–66.