# On the mean radius of quasiconformal mappings

Alastair N. Fletcher  and  Jacob Pratscher

###### Abstract.

We study the mean radius growth function for quasiconformal mappings. We give
a new sub-class of quasiconformal mappings in $\mathbb{R}^{n}$, for $n\geq 2$,
called bounded integrable parameterization mappings, or BIP maps for short.
These have the property that the restriction of the Zorich transform to each
slice has uniformly bounded derivative in $L^{n/(n-1)}$. For BIP maps, the
logarithmic transform of the mean radius function is bi-Lipschitz. We then
apply our result to BIP maps with simple infinitesimal spaces to show that the
asymptotic representation is indeed quasiconformal by showing that its Zorich
transform is a bi-Lipschitz map.

##  1\. Introduction

The growth of quasiconformal mappings is a topic that has been well studied.
If we set

$L_{f}(x_{0},r)=\max_{|x-x_{0}|=r}|f(x)-f(x_{0})|$

and

$\ell_{f}(x_{0},r)=\min_{|x-x_{0}|=r}|f(x)-f(x_{0})|,$

then it is well-known that for quasiconformal mappings we have

$\limsup_{r\to 0}\frac{L_{f}(x_{0},r)}{\ell_{f}(x_{0},r)}\leq H$

for some $H\geq 1$. On the other hand, for a fixed $x_{0}$, the (real)
functions $L_{f}$ and $\ell_{f}$ are bounded above and below via Hölder-type
inequalities. The local distortion result given by [9, Theorem III.4.7] shows
that if $f$ is $K$-quasiconformal then there exist positive constants $A,B$
and and $r_{0}>0$ so that

$Ar^{K^{1/(n-1)}}\leq\ell_{f}(x_{0},r)\leq L_{f}(x_{0},r)\leq
Br^{K^{1/(1-n)}}$

for $0<r<r_{0}$.

Another quantity that is comparable to $L_{f}$ and $\ell_{f}$ on small scales
is the mean radius function that, as far as the authors are aware, was
introduced into the study of quasiregular mappings by Gutlyanskii et al in
[5]. We recall this notion here. Let $n\geq 2$, suppose that
$U\subset\mathbb{R}^{n}$ is a domain and $f:U\to\mathbb{R}^{n}$ is a non-
constant quasiregular mapping. Let $x_{0}\in U$ and
$r_{0}=\operatorname{dist}(x_{0},\partial U)$. For $0<r<r_{0}$, the mean
radius of the image of the ball
$B(x_{0},r)=\\{x\in\mathbb{R}^{n}:|x-x_{0}|<r\\}$ under $f$ is defined by

$\rho_{f}(x_{0},r)=\left(\frac{\operatorname{vol}_{n}f(B(x_{0},r))}{\Omega_{n}}\right)^{1/n},$

where $\operatorname{vol}_{n}E$ denotes the $n$-dimensional Lebesgue measure
of $E$ and $\Omega_{n}$ denotes the volume of the unit ball
$\mathbb{B}^{n}=B(0,1)\subset\mathbb{R}^{n}$.

Since

$\Omega_{n}\ell_{f}(x_{0},r)\leq\operatorname{vol}_{n}f(B(x_{0},r))\leq\Omega_{n}L_{f}(x_{0},r),$

it is clear that

$\ell_{f}(x_{0},r)\leq\rho_{f}(x_{0},r)\leq L_{f}(x_{0},r).$

Henceforth, we will assume that $x_{0}=0$. This makes no difference to our
discussion.

The mean radius was used by Gutlyanskii et al [5] to construct generalized
derivatives of quasiregular mappings, a topic which has been studied more
recently in papers of the first named author and Wallis [4] and the authors
[3]. In this paper, we will study the regularity properties of $\rho_{f}$
itself. It is clear from the definition that $\rho_{f}$ is increasing and,
moreover, it was shown in the proof of [4, Lemma 3.1] that $\rho_{f}$ is
continuous as a consequence of the Lusin (N) condition. It follows from a
standard result in analysis that, as a continuous monotonic real function,
$\rho_{f}$ is differentiable almost everywhere.

It will be convenient for our results to use the logarithmic transform of
$\rho_{f}$. Suppose $\rho_{f}:(0,r_{0})\to(0,\infty)$. Then the logarithmic
transform of $\rho_{f}$ is defined by the function
$\widetilde{\rho_{f}}:(-\infty,\ln r_{0})\to\mathbb{R}$ given by

$\widetilde{\rho_{f}}(t)=\ln\rho_{f}(e^{t}).$

We will typically use the $r$-variable for real functions, and the $t$
variable for logarithmic transforms.

As a generalization of the logarithmic transform to quasiregular maps, we will
also use the Zorich transform introduced by the authors in [3]. This will be
described in further detail below, but for a quasiconformal map $f$ fixing
$0$, this is defined via a particular Zorich map $\mathcal{Z}$ and the
functional equation $\mathcal{Z}\circ\widetilde{f}=f\circ\mathcal{Z}$. The
domain of definition of $\widetilde{f}$ is a half-beam
$B_{M}=Q\times(-\infty,M)$, where $\overline{Q}$ is a cuboid in
$\mathbb{R}^{n-1}$ of the form $[0,1]^{n-2}\times[0,2]$.

To illustrate why the logarithmic transform is a natural object to study in
the growth of quasiconformal mappings, consider the following example.

######  Example 1.1.

Let $f:\mathbb{C}\to\mathbb{C}$ be a radial quasiconformal mapping of the form
$f(re^{i\theta})=h(r)e^{i\theta}$ in polar coordinates, where
$h:[0,\infty)\to[0,\infty)$ is a strictly increasing homeomorphism satisfying
$h(0)=0$. Then via a direct computation we see that its complex dilatation is

$\mu_{f}=\frac{f_{r}+\frac{i}{r}f_{\theta}}{f_{r}-\frac{i}{r}f_{\theta}}=\frac{\frac{rh^{\prime}(r)}{h(r)}-1}{\frac{rh^{\prime}(r)}{h(r)}+1},$

wherever $h^{\prime}(r)$ exists. Since $h$ is an increasing homeomorphism, it
is differentiable almost everywhere, and so $\mu_{f}$ has this expression for
almost every $r$ value. Letting $\widetilde{h}$ be the logarithmic transform
of $h$, and as

$\widetilde{h}^{\prime}(t)=\frac{e^{t}h^{\prime}(e^{t})}{h(e^{t})},$

it is then clear that for $f$ to be quasiconformal, it is necessary that there
exist constants $0<C_{1}<C_{2}<\infty$ so that

$C_{1}\leq\widetilde{h}^{\prime}(t)\leq C_{2}$

for all $t$ where $\widetilde{h}^{\prime}$ is defined.

Radial maps are the nicest class of mappings from our viewpoint, since
$\ell_{f},L_{f}$ and $\rho_{f}$ will all coincide. In general, we expect other
factors to play a role in the behaviour of $\rho_{f}$. To that end, we build
towards the key definition of our paper.

Let $n\geq 2$. For vectors $v_{1},\ldots,v_{n-1}$ in $\mathbb{R}^{n}$, form
the $n$ by $n$ matrix $A$ where the top row consists of the unit vectors
$e_{1},\ldots,e_{n}$ in $\mathbb{R}^{n}$, and the $i$th row of $A$ consists of
entries from the vector $v_{i-1}$, for $2\leq i\leq n$. Then denote by
$\Pi(v_{1},\ldots,v_{n-1})$ the vector in $\mathbb{R}^{n}$ given by $\det A$.

Returning to the Zorich transform, $\widetilde{f}$ maps each slice
$Q\times\\{t\\}$, for $-\infty<t<M$, onto a hypersurface in the beam $B$. The
map $\widetilde{f}$ provides a parameterization for each image hypersurface
through $Q$. Recalling that

$\overline{Q}=\\{(x_{1},\ldots,x_{n-1}):0\leq x_{1},\ldots,x_{n-2}\leq 1,0\leq
x_{n-1}\leq 2\\},$

for $-\infty<t<M$, write $\gamma:Q\to\widetilde{f}(Q\times\\{t\\})$ so that

$\gamma(x_{1},\ldots,x_{n-1})=\widetilde{f}(x_{1},\ldots,x_{n-1},t).$

If $\gamma(Q)$ has a well-defined $(n-1)$-dimensional volume in $B$ (noting
that this is not necessarily the case for an arbitrary quasiconformal Zorich
transform $\widetilde{f}$), then it is given by

$\operatorname{vol}_{n-1}\gamma(Q)=\int_{Q}\left|\left|\Pi\left(\frac{\partial\gamma}{\partial
x_{1}},\ldots,\frac{\partial\gamma}{\partial
x_{n-1}}\right)\right|\right|_{2}dV,$

where $dV$ denotes the volume element on $Q$, and $||.||_{2}$ denotes the
usual $2$-norm.

######  Definition 1.2.

Let $n\geq 2$, let $e^{M}>0$ and let $f:B(0,e^{M})\to\mathbb{R}^{n}$ be
quasiconformal with $f(0)=0$. We say that $f$ h is a bounded integrable
parameterization map, or is a BIP map, if there exists $P>0$ such that for
every $t<M$, the parameterization $\gamma:Q\to\widetilde{f}(Q\times\\{t\\})$
via $\widetilde{f}$ satisfies

$\int_{Q}\left|\left|\Pi\left(\frac{\partial\gamma}{\partial
x_{1}},\ldots,\frac{\partial\gamma}{\partial
x_{n-1}}\right)\right|\right|^{n/(n-1)}_{2}dV\leq P.$

Since $n/(n-1)>1$, it follows that the BIP condition implies the images of the
various slices $\widetilde{f}(Q\times\\{t\\})$ have finite $(n-1)$-dimensional
volume, but it also implies stronger regularity of the parameterizations.

For our main result, we show that BIP mappings have improved regularity for
$\widetilde{\rho_{f}}$ than just almost everywhere differentiability.

######  Theorem 1.3.

Let $n\geq 2$, $K\geq 1$, $e^{M}>0$ and let $f:B(0,e^{M})\to\mathbb{R}^{n}$ be
a BIP $K$-quasiconformal map with $f(0)=0$. Then there exists $L$ depending
only on $n$, $K$ and $P$ such that $\widetilde{\rho_{f}}$ is $L$-bi-Lipschitz
on $(-\infty,M)$.

It follows from Theorem 1.3 that

$\frac{1}{L}\leq\widetilde{\rho_{f}}^{\prime}(t)\leq L$

for every $t\in(-\infty,M)$ at which $\widetilde{\rho_{f}}$ is differentiable,
recalling Example 1.1. As will be evident from the proof of Theorem 1.3, the
implied lower bound in the bi-Lipschitz behaviour of $\widetilde{\rho_{f}}$
always holds. The requirement of BIP maps is needed for the upper bound.

Theorem 1.3 may be applied in neighbourhoods of a fixed point of a
quasiregular mapping, provided that the local index $i(x_{0},f)=1$, that is,
that $f$ is injective in a neighbourhood of $x_{0}$. If the fixed point of $f$
is geometrically attracting, then by definition there exist $c>1$ and
$r_{1}>0$ such that $|f(x)|<c|x|$ for $|x|<r_{1}$. It is worth pointing out
that although $\rho_{f}(r)<cr$, it does not follow that
$\widetilde{\rho_{f}}^{\prime}(t)<1$: consider the simple example $f(x)=x/2$
for which $\widetilde{\rho_{f}}(t)=t-\ln 2$.

The idea behind the proof of Theorem 1.3 is as follows. We first need to show
that the Zorich transform $\widetilde{f}$ is not just quasiconformal in
$B_{M}$, but in fact quasisymmetric. This allows us to compare quantities of
the form $(\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0}))/t$ to
$n$-dimensional volumes of images of slices $Q\times[t_{0},t]$ under
$\widetilde{f}$. Subdividing these slices into small cuboids and using the
quasisymmetry of $\widetilde{f}$ allows us to compare the volume of the image
of a cuboid to its diameter raised to the $n$’th power. Summing over all such
cuboids and a judicious use of an $\ell^{p}$ inequality yields the lower
bound. The upper bound arises by using the sum over all cuboids as an
approximation for the expression in Definition 1.2. The BIP condition is then
needed to provide the upper bound.

We record the fact that the BIP condition is necessary for Theorem 1.3 in the
following proposition.

######  Proposition 1.4.

Suppose that $e^{M}>0$ and let $f:B(0,e^{M})\to\mathbb{R}^{2}$ be
quasiconformal, fix $0$ and so that for some $0<e^{t_{0}}<e^{M}$, the image of
the circle centred at $0$ of radius $e^{t_{0}}$ under $f$ is a non-rectifiable
curve. Then $\widetilde{\rho_{f}}$ is not bi-Lipschitz at $t_{0}$.

It is well-known that the image of a circle under a quasiconformal map need
not be rectifiable, for example, it could be a snowflake curve in
$\mathbb{R}^{2}$ or a snowball in $\mathbb{R}^{3}$. We refer to [7] for
snowballs and references in the literature addressing snowflake curves.

As an application of Theorem 1.3, we turn to infinitesimal spaces. Recall from
[5] that a generalized derivative of a quasiregular map $f:U\to\mathbb{R}^{n}$
at $x_{0}\in U$ is defined to be any local uniform limit of

$\frac{f(x_{0}+r_{k}x)-f(x_{0})}{\rho_{f}(r_{k})}$

as $r_{k}\to 0$. Of course, not every such sequence need have a limit, but the
quasiregular version of Montel’s Theorem implies there will be a subsequence
along which there is local uniform convergence. See [5, p.103] for a
discussion of this point. The collection of generalized derivatives of $f$ at
$x_{0}$ is called the infinitesimal space $T(x_{0},f)$.

In the special case where $T(x_{0},f)$ consists of only one mapping $g$, then
$f$ is called simple at $x_{0}$. For example, if $f$ is differentiable at
$x_{0}$, then $f$ is simple at $x_{0}$. If $f$ is simple at $x_{0}$, then $f$
has an asymptotic representation analogous to a first degree Taylor polynomial
approximation of an analytic function. As usual, we suppose
$x_{0}=f(x_{0})=0$. Then [5, Proposition 4.7] states that as $x\to 0$, we have

$f(x)\sim D(x):=\rho_{f}(|x|)g(x/|x|),$

where $p(x)\sim q(x)$ as $x\to 0$ means

$|p(x)-q(x)|=o(|p(x)|+|q(x)|).$

The map $D$ is called the asymptotic representative of $f$ at (in this case)
$0$. Our second main result concerns the asymptotic representative.

######  Theorem 1.5.

Let $n\geq 2$, $U\subset\mathbb{R}^{n}$ be a domain and suppose that
$f:U\to\mathbb{R}^{n}$ is a quasiregular map, $0\in U$, $i(0,f)=1$ and $f$ is
simple at $0$ and $f$ is a BIP quasiconformal map on some neighbourhood of
$0$. Then there exists $M\in\mathbb{R}$ such that $\widetilde{D}$ is bi-
Lipschitz in $B_{M}$. Moreover, $D$ is quasiconformal in $B(x_{0},e^{M})$.

The simplicity assumption on $f$ implies via [5, Proposition 4.7] that
$\rho_{f}$ is asymptotically $d$-homogeneous, that is, there exists $d>0$ such
that for any $s>0$,

$\rho_{f}(st)\sim s^{d}\rho_{f}(t)$

as $t\to 0$. It should be noted that this conclusion does not imply that
$\rho_{f}$ itself is $d$-homogeneous: one can check that if $d>0$ then the two
dimensional radial map given in polar coordinates by

$(r,\theta)\mapsto\left(\frac{r^{d}}{\log(1/r)},\theta\right)$

is quasiconformal, but $\rho_{f}$ is not $d$-homogeneous.

The assumption that $f$ is BIP in a neighbourhood of $0$ is necessary in
Theorem 1.5. We illustrate this by means of an example.

######  Example 1.6.

We work in dimension two. There exists a quasiconformal map $h$ from the
square $[-1,1]^{2}$ to itself which is the identity on the boundary and so
that $h$ maps $[-1,1]\times\\{0\\}$ onto a non-rectifiable curve. Next, for
$M<0$, let $B_{M}=[0,2]\times(-\infty,M)$. For $d\in\mathbb{Z}$ and
$d<\min\\{M,-2\\}$, consider the square $S_{d}$ centred at $z_{d}=(1,d)$ with
sides parallel to the coordinate axes and of side-length $|d|^{-1}$. We define
$\widetilde{f}$ to be the identity outside $S_{d}$ in $B_{M}$. Let $A_{d}$ be
the linear map

$A_{d}(z)=\frac{z}{2|d|}+z_{d}$

and then for $z\in S_{d}$, we define
$\widetilde{f}(z)=A_{d}(h(A_{d}^{-1}(z)))$.

There exists a quasiconformal map $f:B(0,e^{M})\to\mathbb{R}^{2}$ whose Zorich
transform is $\widetilde{f}$. By construction, for $r>0$, $f$ is not the
identity in $B(0,r)$ on the union of the sets $\mathcal{Z}(S_{d})$, where
$d<\log r$. For such $d$, the diameter of $\mathcal{Z}(S_{d})$ is bounded
above by

$e^{d+1/2|d|}-e^{d-1/2|d|}<2\sinh\left(\frac{1}{2\ln\frac{1}{r}}\right)=o(1)$

as $r\to 0$. We conclude that $f(x)\sim x$ as $x\to 0$ and so $f$ has a simple
infinitesimal space consisting of the identity. However,
$\widetilde{\rho_{f}}$ is not bi-Lipschitz on any neighbourhood of $d$ by the
argument in Proposition 1.4, from which it follows that
$D(x)=\rho_{f}(|x|)x/|x|$ is not quasiconformal.

As a final remark, in [1], it was implicitly assumed that the map $D$ is
quasiconformal. Since our results above show that this is not always the case,
Theorem 1.5 can be viewed as providing an extra assumption necessary for the
results from that paper to hold.

##  2\. Preliminaries

Throughout, $B(x_{0},r)$ denotes the ball centred at $x_{0}\in\mathbb{R}^{n}$
of radius $r>0$. Its boundary is the sphere $S(x_{0},r)$ or, if $x_{0}=0$, we
may simply write $S(r)$. If $x\in\mathbb{R}^{n}$, then we write
$[x]_{n}\in\mathbb{R}$ for the $n$th coordinate.

We refer to standard references such as [9] for the definition and basic
properties of quasiregular and quasiconformal mappings in $\mathbb{R}^{n}$. In
particular, if $f$ is quasiregular, then $K_{O}(f),K_{I}(f)$ and $K(f)$ refer
to the outer dilatation, inner dilatation and maximal dilatation,
respectively. Bounded length distortion maps, BLD for short, are a sub-class
of quasiregular maps for which the finite length curves are mapped to curves
of finite length, with uniform control on the length distortion. BLD maps are
locally bi-Lipschitz. In some sense, BIP maps provide a generalization of BLD
maps.

We will use the following result.

######  Theorem 2.1 (Theorem 1.1, [2]).

Let $U\subset\mathbb{R}^{n}$ be a domain for $n\geq 2$ and let
$f:U\to\mathbb{R}^{n}$ be a non-constant quasiregular map. If $x\in U$, there
exist $C>1$ and $r_{0}>0$ such that for all $0<T\leq 1$ and $r\in(0,r_{0})$

$T^{-\mu}\leq\frac{L_{f}(x,r)}{\ell_{f}(x,Tr)}\leq C^{2}T^{-\nu}$

and

$\frac{T^{-\mu}}{C^{2}}\leq\frac{\ell_{f}(x,r)}{L_{f}(x,Tr)}\leq T^{-\nu},$

where $\mu=(i(x,f)/K_{I}(f))^{1/(n-1)}$, $\nu=(K_{O}(f)i(x,f))^{1/(n-1)}$ and
$C$ depends only on $n$, $K_{O}(f)$ and the local index $i(x,f)$.

###  2.1. The Zorich transform

The class of Zorich mappings provides a well-known quasiregular generalization
of the exponential function in the plane. These maps are strongly automorphic
with respect to a discrete group $G$ of isometries. This means that if
$\mathcal{Z}$ is a Zorich map, then $\mathcal{Z}(g(x))=\mathcal{Z}(x)$ for all
$g\in G$ and all $x\in\mathbb{R}^{n}$ and, moreover, if
$\mathcal{Z}(x)=\mathcal{Z}(y)$ then $y=g(x)$ for some $g\in G$.

For each $n\geq 2$, we will fix one particular Zorich map, which we denote
simply by $\mathcal{Z}$, with the following properties.

  1. (i)

For each $t\in\mathbb{R}$, $\mathcal{Z}$ maps the hyperplane
$H_{t}=\\{x\in\mathbb{R}^{n}:[x]_{n}=t\\}$ onto the sphere $S(e^{t})$.

  2. (ii)

On each hyperplane $H_{t}$, $\mathcal{Z}$ is locally bi-Lipschitz with
isometric distortion scaling with $e^{t}$. In fact,

$\mathcal{Z}(x_{1},\ldots,x_{n-1},t)=e^{t}\mathcal{Z}(z_{1},\ldots,x_{n-1},0).$

  3. (iii)

The group $G$ is generated by a translation subgroup of rank $n-1$ and a
finite group of rotations about the origin such that each $g\in G$ preserves
the $n$’th coordinate.

  4. (iv)

A fundamental set $B$ for the action of $G$ is given by a beam
$Q\times\mathbb{R}$, where the closure of $Q$ is a cuboid in
$\mathbb{R}^{n-1}$ of the form $[0,1]^{n-2}\times[0,2]$ and the sides of this
beam are identified via the group $G$.

For an explicit formula for such a Zorich map, together with its associated
group $G$, we refer to, for example, [3, Section 3]. If $f$ is quasiconformal
in a neighbourhood $U$ of the origin in $\mathbb{R}^{n}$ and $f(0)=0$, then we
may consider the Zorich transform $\widetilde{f}:\mathcal{Z}^{-1}(U)\cap B\to
B$ of $f$. This is defined via the relation

$f\circ\mathcal{Z}=\mathcal{Z}\circ\widetilde{f}.$

We often just assume the domain of $\widetilde{f}$ is

$B_{M}:=\\{x\in B:[x]_{n}<M\\},$

in the same way that we may assume that the domain of $f$ is
$\\{x:|x|<e^{M}\\}$.

We will use the fact that $B_{M}$ can be made into a metric space via the
quotient of the Euclidean metric under the group $G$ and consider the
appropriate topological notions of balls, convergence and so on in this metric
space. In particular, $\mathcal{Z}^{-1}:B_{M}\to\mathbb{R}^{n}$ is locally bi-
Lipschitz with this metric in the domain and the Euclidean metric in the
range.

###  2.2. Quasisymmetric maps

A standard reference for quasisymmetric maps is Heinonen’s book [6]. A map
$f:X\to Y$ between metric spaces is called quasisymmetric if there exists a
homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that for all triples of
points $x,a,b\in X$, we have

$\frac{d_{Y}(f(x),f(a))}{d_{Y}(f(x),f(b))}\leq\eta\left(\frac{d_{X}(x,a)}{d_{X}(x,b)}\right).$

If we wish to specify the function $\eta$, then $f$ is said to be
$\eta$-quasisymmetric. A map $f:X\to Y$ is called weakly quasisymmetric if
there exists a constant $H\geq 1$ such that $d_{X}(x,a)\leq d_{X}(x,b)$
implies

$d_{Y}(f(x),f(a))\leq Hd_{Y}(f(x),f(b))$

for all triples of points $x,a,b\in X$. While in general weakly quasisymmetric
maps need not be quasisymmetric, they are in connected doubling spaces, see
[6, Theorem 10.19].

Quasisymmetry can be viewed as a global version of quasiconformality. In
particular, a quasisymmetric map $f:U\to V$ between domains in
$\mathbb{R}^{n}$ is quasiconformal. Conversely, the egg-yolk principle states
that a $K$-quasiconformal map $f:U\to V$ is $\eta$-quasisymmetric on each ball
$B(x_{0},\operatorname{dist}(x_{0},\partial U)/2)$, with $\eta$ depending only
on $n$ and $K$, see [6, Theorem 11.14].

###  2.3. $\ell^{p}$ estimates

It is well-known that if $x=(x_{1},\ldots,x_{N})\in\mathbb{R}^{N}$ and
$x_{i}\geq 0$ for $i=1,\ldots,N$, then for $1\leq p<q\leq\infty$ we have

$||x||_{q}\leq||x||_{p}\leq N^{1/p-1/q}||x||_{q},$

where $||x||_{p}$ denotes the usual $p$-norm

$||x||_{p}=\left(\sum_{i=1}^{N}x_{i}^{p}\right)^{1/p}.$

In particular, for $n\geq 2$, if $p=n-1$ and $q=n$, then

(2.1) $\sum_{i=1}^{N}x_{i}^{n}\geq
N^{1/(1-n)}\left(\sum_{i=1}^{N}x_{i}^{n-1}\right)^{n/(n-1)}.$

##  3\. Quasisymmetry

We will need the following result that shows the Zorich transform of a
quasiconformal map is quasisymmetric. This is a non-trivial result, as
$\mathcal{Z}^{-1}$ itself is not quasisymmetric on any punctured neighbourhood
of the origin.

######  Proposition 3.1.

Let $n\geq 2$, $K>1$ and $r_{0}>0$. If $f:B(0,r_{0})\to\mathbb{R}^{n}$ is a
$K$-quasiconformal map with $f(0)=0$, then there exists $M<\ln r_{0}$ such
that $\widetilde{f}$ is quasisymmetric in $B_{M}$.

###### Proof.

First, for any $M$, as $B_{M}$ is a connected doubling space, [6, Theorem
10.19] implies that it is sufficient to show that $\widetilde{f}$ is weakly
quasisymetric. Suppose for a contradiction that $\widetilde{f}$ is not weakly
quasisymmetric. Then there exists a sequence of triples of points
$x_{k},a_{k},b_{k}$ in $B_{M}$ such that

(3.1) $|x_{k}-a_{k}|\leq|x_{k}-b_{k}|,\quad\text{ but
}\quad|\widetilde{f}(x_{k})-\widetilde{f}(a_{k})|>k|\widetilde{f}(x_{k})-\widetilde{f}(b_{k})|.$

We will deal with large scales and small scales separately.

First, if $|\widetilde{f}(x_{k})-\widetilde{f}(b_{k})|$ does not converge to
$0$ then, passing to a subsequence if necessary, we have
$|\widetilde{f}(x_{k})-\widetilde{f}(a_{k})|\to\infty$. This implies that
$|[\widetilde{f}(x_{k})]_{n}-[\widetilde{f}(a_{k})]_{n}|\to\infty$. Now set
$t_{k}=|x_{k}-b_{k}|$, so $t_{k}\to\infty$ by (3.1). Without loss of
generality, suppose $[a_{k}]_{n}\leq[x_{k}]_{n}\leq[b_{k}]_{n}$. Then we have

(3.2)
$[b_{k}]_{n}\leq[x_{k}]_{n}+t_{k},\quad[a_{k}]_{n}\geq[x_{k}]_{n}-t_{k}.$

Next, set $s_{k}=[b_{k}]_{n}-[x_{k}]_{n}$. As $t_{k}^{2}\leq
s_{k}^{2}+(\operatorname{diam}Q)^{2}$ by Pythagoras and
$(\operatorname{diam}Q)^{2}=n+2$, we have

(3.3)
$\sqrt{1-\frac{(\operatorname{diam}Q)^{2}}{t_{k}^{2}}}\leq\frac{s_{k}}{t_{k}}\leq
1$

and in particular, $s_{k}/t_{k}\to 1$ as $k\to\infty$.

Next, set $y_{k}=\mathcal{Z}(x_{k})$, $p_{k}=\mathcal{Z}(a_{k})$ and
$q_{k}=\mathcal{Z}(b_{k})$. Using Theorem 2.1 and (3.2), we have

$\displaystyle|\widetilde{f}(x_{k})-\widetilde{f}(a_{k})|$
$\displaystyle=\left|\log\frac{f(y_{k})}{f(p_{k})}\right|$
$\displaystyle\leq\log\frac{L_{f}(0,e^{[x_{k}]_{n}})}{\ell_{f}(0,e^{[a_{k}]_{n}})}$
$\displaystyle\leq\log\frac{L_{f}(0,e^{[x_{k}]_{n}})}{\ell_{f}(0,e^{-t_{k}}e^{[x_{k}]_{n}})}$
$\displaystyle\leq\log\left(C^{2}(e^{-t_{k}})^{-\nu}\right),$

recalling that $\nu=K_{O}(f)^{1/(n-1)}$ and $C$ only depends on $n$ and
$K_{O}(f)$ since $f$ is quasiconformal. On the other hand, by Theorem 2.1 and
the definition of $s_{k}$ we have

$\displaystyle|\widetilde{f}(b_{k})-\widetilde{f}(x_{k})|$
$\displaystyle=\left|\log\frac{f(q_{k})}{f(y_{k})}\right|$
$\displaystyle\geq\log\frac{\ell_{f}(0,e^{[b_{k}]_{n}})}{L_{f}(0,e^{[x_{k}]_{n}})}$
$\displaystyle=\log\frac{\ell_{f}(0,e^{[b_{k}]_{n}})}{L_{f}(0,e^{-s_{k}}e^{[b_{k}]_{n}})}$
$\displaystyle\geq\log\left(\frac{(e^{-s_{k}})^{-\mu}}{C^{2}}\right),$

where $\mu=K_{I}(f)^{1/(1-n)}$. Hence by (3.3) we have

$\displaystyle\frac{|\widetilde{f}(b_{k})-\widetilde{f}(x_{k})|}{|\widetilde{f}(x_{k})-\widetilde{f}(a_{k})|}$
$\displaystyle\geq\frac{-2\log C+\mu s_{k}}{2\log C+\nu
t_{k}}\to\frac{\nu}{\mu}=(K_{O}(f)K_{I}(f))^{1/(1-n)}$

as $k\to\infty$. This provides a contradiction to (3.1).

Turning now to the small scales case, assume that
$|\widetilde{f}(x_{k})-\widetilde{f}(b_{k})|\to 0$ and hence
$t_{k}=|x_{k}-b_{k}|\to 0$. As $f$ is $K$-quasiconformal, $\widetilde{f}$ is
$K^{\prime}$-quasiconformal, where $K^{\prime}\leq K[K(\mathcal{Z})]^{2}$. We
may define a sequence of $K^{\prime}$-quasiconformal maps
$g_{k}:\overline{\mathbb{B}^{n}}\to\overline{\mathbb{B}^{n}}$ via

$g_{k}(x)=\frac{\widetilde{f}(x_{k}+|x_{k}-b_{k}|x)-\widetilde{f}(x_{k})}{L_{\widetilde{f}}(x_{k},|x_{k}-b_{k}|)}.$

By construction, $\sup_{|x|=1}|g_{k}(x)|=1$ for all $k$. Since the family
$\\{g_{k}\\}$ is normal by the quasiregular version of Montel’s Theorem (see
[8, Theorem 4]), it follows that there exists $\delta>0$ such that

(3.4) $\inf_{|x|=1}|g_{k}(x)|>\delta$

for all $k$.

However, as $|x_{k}-a_{k}|\leq|x_{k}-b_{k}|$ and
$|\widetilde{f}(x_{k})-\widetilde{f}(a_{k})|>k|\widetilde{f}(x_{k})-\widetilde{f}(b_{k})|$,
it follows that, with $y_{k}=(b_{k}-x_{k})/|b_{k}-x_{k}|$, we have

$\displaystyle|g_{k}(y_{k})|$
$\displaystyle=\frac{\widetilde{f}(b_{k})-\widetilde{f}(x_{k})}{L_{\widetilde{f}}(x_{k},|x_{k}-b_{k}|)}$
$\displaystyle\leq\frac{|\widetilde{f}(a_{k})-\widetilde{f}(x_{k})|}{kL_{\widetilde{f}}(x_{k},|x_{k}-b_{k}|)}$
$\displaystyle\leq\frac{L_{\widetilde{f}}(|x_{k}-a_{k}|)}{kL_{\widetilde{f}}(|x_{k}-b_{k}|)}\to
0$

as $k\to\infty$. This is a contradiction to (3.4).

We conclude that $\widetilde{f}$ is weakly quasisymmetric, and hence
quasisymmetric, on $B_{M}$. ∎

We will assume henceforth that $\widetilde{f}$ is defined in $B_{M}$, and
identify $B_{M}$ with $Q\times(-\infty,M)$. Given $t_{0}<M-1$ and $t<1$,
consider the slice

$S_{t}=Q\times[t_{0},t_{0}+t]\subset B_{M}.$

We subdivide $S_{t}$ into boxes in the following way. Subdivide $Q$ into
$(n-1)$-dimensional cubes $T_{i}$ of side length $(\lfloor 1/t\rfloor)^{-1}$
with edges parallel to the unit vectors $e_{1},\ldots,e_{n-1}$. Finally, we
set

(3.5) $P_{i}=T_{i}\times[t_{0},t_{0}+t]$

for $i=1,\ldots,N$.

We collect important geometric information about the quasisymmetric images of
$P_{i}$ in the following lemma. Recall that we denote by
$\operatorname{vol}_{p}$ the $p$-dimensional volume.

######  Lemma 3.2.

Let $\widetilde{f}:B_{M}\to B$ be $\eta$-quasisymmetric and let $t\in(0,1/2)$.
Then the number of boxes in the subdivision of $S_{t}$ is $N$, where
$N/t^{1-n}\to 2$ as $t\to 0$. Moreover, there exists a constant $C_{1}>1$
depending only on $n$ and $\eta$ so that

$\frac{1}{C_{1}}\leq\frac{\operatorname{vol}_{n}\widetilde{f}(P_{i})}{(\operatorname{diam}\widetilde{f}(P_{i}))^{n}}\leq
C_{1},$

for each box $P_{i}$ constructed as above.

###### Proof.

The process of subdividing $Q$ into $(n-1)$-dimensional cubes of side length
$(\lfloor 1/t\rfloor)^{-1}$ yields $N=O(t^{1-n})$ of them as

(3.6) $t\leq(\lfloor 1/t\rfloor)^{-1}\leq\frac{t}{1-t}\leq 2t,$

$\operatorname{vol}_{n-1}Q=2$ by construction and
$\operatorname{vol}_{n-1}Q=N(\lfloor 1/t\rfloor)^{1-n}$. Note that as $t\to
0$, $(\lfloor 1/t\rfloor)^{-1}/t\to 1$ and so $N/t^{(1-n)}\to 2$.

Next, let $x_{0}$ be the centroid of $P_{i}$. By (3.6) we have

(3.7) $\frac{t}{2}\leq\operatorname{dist}(x_{0},\partial P_{i}).$

Let $y,z$ be any two points on $\partial P_{i}$. Then since $\widetilde{f}$ is
$\eta$-quasisymmetric, we have

$\frac{|\widetilde{f}(y)-\widetilde{f}(z)|}{|\widetilde{f}(y)-\widetilde{f}(x_{0})|}\leq\eta\left(\frac{|y-z|}{|y-x_{0}|}\right).$

Choosing $y,z$ so that $|\widetilde{f}(y)-\widetilde{f}(z)|$ realizes the
diameter of $\widetilde{f}(P_{i})$, we obtain via (3.7) that

(3.8)
$\operatorname{diam}\widetilde{f}(P_{i})\leq\operatorname{dist}(\widetilde{f}(x_{0}),\partial\widetilde{f}(P_{i}))\cdot\eta\left(\frac{t\sqrt{n}}{t/2}\right)=\operatorname{dist}(\widetilde{f}(x_{0}),\partial\widetilde{f}(P_{i}))\cdot\eta(2\sqrt{n}).$

Since

$B(\widetilde{f}(x_{0}),\operatorname{dist}(\widetilde{f}(x_{0}),\partial\widetilde{f}(P_{i})))\subset\widetilde{f}(P_{i})\subset
B(\widetilde{f}(x_{0}),\operatorname{diam}\widetilde{f}(P_{i})),$

it follows that

$\Omega_{n}\left(\operatorname{dist}(\widetilde{f}(x_{0}),\partial\widetilde{f}(P_{i}))\right)^{n}\leq|\widetilde{f}(P_{i})|\leq\Omega_{n}\left(\operatorname{diam}\widetilde{f}(P_{i})\right)^{n}.$

Combining this with (3.8), we obtain

$\frac{\Omega_{n}}{\eta(2\sqrt{n})}\leq\frac{|\widetilde{f}(P_{i})|}{(\operatorname{diam}\widetilde{f}(P_{i}))^{n}}\leq\Omega_{n},$

as required. ∎

######  Definition 3.3.

Given a box $P_{i}$ constructed as above, denote by
$\nu(\widetilde{f}(P_{i}))$ the minimum distance between images under
$\widetilde{f}$ of opposite pairs of faces of $P_{i}$.

######  Lemma 3.4.

Let $\widetilde{f}:B_{M}\to B$ be $\eta$-quasisymmetric and let $t\in(0,1/2)$.
There exists a constant $C_{2}>1$ depending only on $n$ and $\eta$ such that
if $P_{i}$ is a box constructed as in (3.5) with parameter $t$, then

$1\leq\frac{\operatorname{diam}(\widetilde{f}(P_{i}))}{\nu(\widetilde{f}(P_{i}))}\leq
C_{2}.$

###### Proof.

The lower bound here is trivial. Given $P_{i}$, find $x,y\in P_{i}$ which
realize the minimum in the definition of $\nu(\widetilde{f}(P_{i}))$.
Necessarily, $x$ and $y$ must lie in opposite faces of the boundary of
$P_{i}$. Applying the quasisymmetry of $\widetilde{f}$ twice, for any $p,q\in
P_{i}$ we have

$\frac{|\widetilde{f}(p)-\widetilde{f}(q)|}{|\widetilde{f}(x)-\widetilde{f}(y)|}=\frac{|\widetilde{f}(p)-\widetilde{f}(q)|}{|\widetilde{f}(x)-\widetilde{f}(q)|}\cdot\frac{|\widetilde{f}(x)-\widetilde{f}(q)|}{|\widetilde{f}(x)-\widetilde{f}(y)|}\leq\eta\left(\frac{|p-q|}{|x-q|}\right)\eta\left(\frac{|x-q|}{|x-y|}\right).$

We choose $p,q$ so that their images realize
$\operatorname{diam}\widetilde{f}(P_{i})$ and, without loss of generality, we
may assume that $|x-q|\geq t/2$ (if not, switch the roles of $x$ and $y$).
From (3.6), we have $|x-q|\leq 2t\sqrt{n}$, $|p-q|\leq 2t\sqrt{n}$ and
$|x-y|\geq t$, from which it follows that

$\frac{|\widetilde{f}(p)-\widetilde{f}(q)|}{|\widetilde{f}(x)-\widetilde{f}(y)|}\leq\eta(4\sqrt{n})\eta(2\sqrt{n}),$

completing the proof. ∎

######  Lemma 3.5.

Let $\widetilde{f}:B_{M}\to B$ be $\eta$-quasisymmetric and let $t\in(0,1/2)$.
There exists a constant $C_{3}$ depending only on $n$ and $\eta$ such that if
$P_{i}$ is a box constructed as in (3.5) with parameter $t$, and $Q_{i}$
denotes the base of the box, that is, $Q_{i}=P_{i}\cap(Q\times\\{t_{0}\\})$,
then

$\frac{\nu(\widetilde{f}(P_{i}))^{n-1}}{\operatorname{vol}_{n-1}\widetilde{f}(Q_{i})}\leq
C_{3}.$

###### Proof.

Given $Q_{i}$, let $x_{0}$ be its centroid. As $Q_{i}$ is an
$(n-1)$-dimensional cube of side length $(\lfloor 1/t\rfloor)^{-1}$, it
follows from (3.6) that

(3.9) $\frac{t}{2}\leq\operatorname{dist}(x_{0},\partial Q_{i})\leq
t\sqrt{n-1}.$

Here, we denote by $\partial Q_{i}$ the $(n-2)$-dimensional faces on the edge
of $Q_{i}$. Let $y,z\in\partial Q_{i}$ be such that $\widetilde{f}(y)$ and
$\widetilde{f}(z)$ realize the minimum distance between images under
$\widetilde{f}$ of opposite pairs of faces of $\partial Q_{i}$. Then
$|\widetilde{f}(y)-\widetilde{f}(z)|\geq\nu(\widetilde{f}(P_{i}))$. By the
quasisymmetry of $\widetilde{f}$ and (3.9), it follows that

$\frac{\nu(\widetilde{f}(P_{i}))}{|\widetilde{f}(x_{0})-\widetilde{f}(y)|}\leq\frac{|\widetilde{f}(y)-\widetilde{f}(z)|}{|\widetilde{f}(x_{0})-\widetilde{f}(y)|}\leq\eta\left(\frac{|y-z|}{|x_{0}-y|}\right)\leq\eta\left(\frac{2t\sqrt{n-1}}{t/2}\right)=\eta(4\sqrt{n-1}).$

We conclude that

$\nu(\widetilde{f}(P_{i}))\leq\eta(4\sqrt{n-1})\operatorname{dist}(\widetilde{f}(x_{0}),\partial
Q_{i}).$

Thus for $s=\nu(\widetilde{f}(P_{i}))/\eta(4\sqrt{n-1})$, the ball
$B(\widetilde{f}(x_{0}),s)$ does not meet $\widetilde{f}(\partial Q_{i})$. It
follows that

$\operatorname{vol}_{n-1}\widetilde{f}(Q_{i})\geq\frac{\Omega_{n-1}\nu(\widetilde{f}(P_{i}))^{n-1}}{\eta(4\sqrt{n-1})^{n-1}},$

from which the lemma follows. ∎

##  4\. Volume comparison

######  Lemma 4.1.

Let $\widetilde{f}:B_{M}\to B$ be $\eta$-quasisymmetric and let $t\in(0,1/2)$.
Let $S_{t}=Q\times[t_{0},t_{0}+t]$ and denote by $V_{t}$ the volume
$\operatorname{vol}_{n}S_{t}$. Then there exist $C_{4}>1$, depending only on
$n$ and $\eta$, and $t_{1}>0$ so that for $t\in(0,t_{1})$ we have

$\frac{1}{C_{4}}\leq\frac{\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})}{V_{t}}\leq
C_{4}.$

###### Proof.

By the definition of $\widetilde{\rho_{f}}$, we have

(4.1)
$\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})=\log\rho_{f}(e^{t_{0}+t})-\log\rho_{f}(e^{t_{0}})=\log\left(\frac{\rho_{f}(e^{t_{0}}e^{t})}{\rho_{f}(e^{t_{0}})}\right).$

Intepreting $\rho_{f}(s)$ as the normalized volume
$\operatorname{vol}_{n}f(B(0,s))/\Omega_{n}$, (4.1) yields

$\displaystyle\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})$
$\displaystyle=\log\left(\frac{\operatorname{vol}_{n}f(B(0,e^{t_{0}}e^{t}))}{\operatorname{vol}_{n}f(B(0,e^{t_{0}}))}\right)$
(4.2)
$\displaystyle=O\left(\frac{\operatorname{vol}_{n}f(B(0,e^{t_{0}}e^{t}))-\operatorname{vol}_{n}f(B(0,e^{t_{0}}))}{\operatorname{vol}_{n}f(B(0,e^{t_{0}}))}\right)$

as $t\to 0$. Denoting by $A_{t}$ the annulus $\\{x:e^{t_{0}}\leq|x|\leq
e^{t_{0}}e^{t}\\}$, we see that the numerator on the right hand side of (4) is
$\operatorname{vol}_{n}f(A_{t})$. Moreover, $V_{t}$ is related to $A_{t}$ via
$V_{t}=\operatorname{vol}_{n}\mathcal{Z}^{-1}(f(A_{t}))$.

Now, $\mathcal{Z}^{-1}$ is locally bi-Lipschitz away from the origin, but we
want an estimate for $\mathcal{Z}^{-1}$ that will work for all small $t_{0}$.
To that end, we observe that on a punctured neighbourhood of the origin, we
have

$\mathcal{Z}^{-1}(x)=\mathcal{Z}^{-1}\left(\frac{x}{\ell_{f}(e^{t_{0}})}\right)-\left[\ln\frac{1}{\ell_{f}(e^{t_{0}})}\right]e_{n}.$

Since $f$ is quasiconformal, it has finite linear distortion at $0$, that is,
there exist $r_{0}>0$ and a constant $H\geq 1$ such that

$\frac{L_{f}(r)}{\ell_{f}(r)}\leq H$

for $0<r<r_{0}$. Hence if $e^{t_{0}}<r_{0}$, we may conclude that the image of
$f(A_{t})$ under the map $x/\ell_{f}(e^{t_{0}})$ is contained in the ring
$R_{H}=\\{x:1\leq|x|\leq H\\}$. Since this set is compact in
$\mathbb{R}^{n}\setminus\\{0\\}$, $\mathcal{Z}^{-1}$ is $\alpha$-bi-Lipschitz
on $R_{H}$, for some $\alpha\geq 1$. Finally, the map
$x-\left[\ln\frac{1}{\ell_{f}(e^{t_{0}})}\right]e_{n}$ is a translation, and
hence just an isometry. Putting this together, we obtain

(4.3)
$\frac{\operatorname{vol}_{n}f(A_{t})}{\alpha^{n}\ell_{f}(e^{t_{0}})^{n}}\leq
V_{t}\leq\frac{\alpha^{n}\operatorname{vol}_{n}f(A_{t})}{\ell_{f}(e^{t_{0}})^{n}}.$

By (4) and (4.3) we have

$\displaystyle\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})$
$\displaystyle=O\left(\frac{\operatorname{vol}_{n}f(A_{t})}{\operatorname{vol}_{n}f(B(0,e^{t_{0}}))}\right)$
$\displaystyle=O\left(\frac{\ell_{f}(e^{t_{0}})^{n}V_{t}}{\operatorname{vol}_{n}f(B(0,e^{t_{0}}))}\right)$
$\displaystyle=O(V_{t}),$

where we use the fact that $\ell_{f}(e^{t_{0}})^{n}$ is comparable to
$\operatorname{vol}_{n}f(B(0,e^{t_{0}}))$ for all small enough $e^{t_{0}}$.
This completes the proof. ∎

##  5\. Proof of Theorem 1.3

Assuming $\widetilde{\rho_{f}}$ is locally $L$-bi-Lipschitz at every
$t\in(-\infty,M)$, for some $L$ depending on $n,K$ and $P$, it follows that
$\widetilde{\rho_{f}}$ is globally $L$-bi-Lipschitz on $(-\infty,M)$. Hence it
is enough to show that $\widetilde{\rho_{f}}$ is locally $L$-bi-Lipschitz.
Since the quasisymmetry function $\eta$ of $\widetilde{f}$ depends only on the
maximal dilatation $K$ of $f$ (and the maximal dilatation of the fixed map
$\mathcal{Z}$), at any place below where a constant depends on $\eta$, it
depends on $K$.

In the following subsections, we will show that bounds exist for

$\frac{\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})}{t}$

as $t\to 0$ for $t>0$. The arguments below can be easily modified to take into
account the case where $t<0$, and so we focus solely on the case when $t>0$.

###  5.1. The lower bound

Suppose $\widetilde{f}$ satisfies the hypotheses of Theorem 1.3, $\epsilon>0$
is small and for some $t_{0}<M-1$, some $0<t_{2}<1/2$ and all $0<t<t_{2}$, we
have

$\frac{\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})}{t}<\epsilon.$

Our goal here is to show that if $\epsilon$ is too small, we obtain a
contradiction.

Set $S_{t}=Q\times[t_{0},t_{0}+t]$ and let
$V_{t}=\operatorname{vol}_{n}\widetilde{f}(S_{t})$. By Lemma 4.1 it follows
that for $0<t<\min\\{t_{1},t_{2}\\}$ we have

(5.1) $V_{t}\leq C_{4}\epsilon t.$

Next, cover $Q\times\\{t_{0}\\}$ by $(n-1)$-dimensional cubes of side length
$(\lfloor 1/t\rfloor)^{-1}$ and form the $n$-dimensional boxes $P_{i}$, for
$i=1,\ldots,N$. Since the boxes $P_{i}$ cover $S_{t}$ with overlaps only on
their boundaries, we have

$V_{t}=\sum_{i=1}^{N}\operatorname{vol}_{n}\widetilde{f}(P_{i}).$

By Lemma 3.2, this yields

$V_{t}\geq
C_{1}^{-1}\sum_{i=1}^{N}\left(\operatorname{diam}\widetilde{f}(P_{i})\right)^{n}.$

Applying (2.1) we obtain

(5.2) $V_{t}\geq
C_{1}^{-1}N^{1/(1-n)}\left(\sum_{i=1}^{N}\left(\operatorname{diam}\widetilde{f}(P_{i})\right)^{n-1}\right)^{n/(n-1)}.$

Next, let $\mathcal{P}:B\to Q$ denote the orthogonal projection from the beam
$B$ onto $Q$. It is clear that

$\operatorname{diam}\mathcal{P}(\widetilde{f}(P_{i}))\leq\operatorname{diam}\widetilde{f}(P_{i})$

and hence that

(5.3)
$\operatorname{vol}_{n-1}\mathcal{P}(\widetilde{f}(P_{i}))\leq\frac{\Omega_{n-1}(\operatorname{diam}\widetilde{f}(P_{i}))^{n-1}}{2^{n-1}}.$

Now, we must have

$Q\subset\bigcup_{i=1}^{N}\mathcal{P}(\widetilde{f}(P_{i})),$

for otherwise $\partial f(B(0,e^{t_{0}}))$ would not separate $0$ and
$\infty$. Hence

(5.4)
$\sum_{i=1}^{N}\operatorname{vol}_{n-1}\mathcal{P}(\widetilde{f}(P_{i}))\geq\operatorname{vol}_{n-1}Q=2.$

Combining (5.2), (5.3) and (5.4), we obtain

(5.5) $V_{t}\geq
C_{1}^{-1}N^{1/(1-n)}\left(2^{n}\Omega_{n-1}^{-1}\right)^{n/(n-1)}\geq
C_{5}N^{1/(1-n)},$

where $C_{5}$ depends only on $n$ and $\eta$. Since $N/t^{1-n}\to 2$ as $t\to
0$, by (5.1) and (5.5) we have

$C_{4}\epsilon t\geq 2^{1/(1-n)}C_{5}t,$

which yields a contradiction if $\epsilon$ is small enough.

###  5.2. The upper bound

Suppose $\widetilde{f}$ satisfies the hypotheses of Theorem 1.3, $\epsilon>0$
is small and for some $t_{0}<M-1$, some $0<t_{2}<1/2$ and all $0<t<t_{2}$, we
have

$\frac{\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})}{t}>\frac{1}{\epsilon}.$

Our goal here is to show that if $\epsilon$ is too small, we obtain a
contradiction.

As with the lower bound, set $S_{t}=Q\times[t_{0},t_{0}+t]$ and let
$V_{t}=\operatorname{vol}_{n}\widetilde{f}(S_{t})$. By Lemma 4.1 it follows
that for $0<t<\min\\{t_{1},t_{2}\\}$ we have

(5.6) $V_{t}\geq\frac{t}{C_{4}\epsilon}.$

Next, cover $Q\times\\{t_{0}\\}$ by $(n-1)$-dimensional cubes of side length
$(\lfloor 1/t\rfloor)^{-1}$ and form the $n$-dimensional boxes $P_{i}$, for
$i=1,\ldots,N$ with base $Q_{i}$. Since the boxes $P_{i}$ cover $S_{t}$ with
overlaps only on their boundaries, we have

$V_{t}=\sum_{i=1}^{N}\operatorname{vol}_{n}\widetilde{f}(P_{i}).$

By Lemma 3.2, this yields

$V_{t}\leq
C_{1}\sum_{i=1}^{N}\left(\operatorname{diam}\widetilde{f}(P_{i})\right)^{n}.$

Then by Lemma 3.4, we obtain

$V_{t}\leq
C_{1}C_{2}^{n}\sum_{i=1}^{N}\left(\nu(\widetilde{f}(P_{i}))\right)^{n}.$

Now applying Lemma 3.5, we see that

(5.7) $V_{t}\leq
C_{1}C_{2}^{n}C_{3}^{n/(n-1)}\sum_{i=1}^{N}\left(\operatorname{vol}_{n-1}\widetilde{f}(Q_{i})\right)^{n/(n-1)}.$

Next, we use the fact that $f$ is a BIP map. Setting

$\gamma_{t}(x_{1},\ldots,x_{n-1})=\widetilde{f}(x_{1},\ldots,x_{n-1},t),$

then for all $t<M$,

$\int_{Q}\left|\left|\Pi\left(\frac{\partial\gamma_{t}}{\partial
x_{1}},\ldots,\frac{\partial\gamma_{t}}{\partial
x_{n-1}}\right)\right|\right|^{n/(n-1)}_{2}dV\leq P.$

As a shorthand, for $y_{i}\in Q$ and $t<M$, we write $\Pi(y_{i})$ for the
vector

$\Pi\left(\frac{\partial\gamma_{t}}{\partial
x_{1}}(y_{i}),\ldots,\frac{\partial\gamma_{t}}{\partial
x_{n-1}}(y_{i})\right).$

We may suppose that $t$ is small enough that the partition of
$Q\times\\{t_{0}\\}$ given by $Q_{1},\ldots,Q_{N}$ yields

(5.8) $\operatorname{vol}_{n-1}\widetilde{f}(Q_{i})\leq
2||\Pi(y_{i})||_{2}\operatorname{vol}_{n-1}(Q_{i})\leq
2^{n-1}||\Pi(y_{i})||_{2}t^{n-1},$

where $y_{i}\in Q_{i}$ is chosen so that all the partial derivatives of
$\gamma_{t_{0}}$ exist at $y_{i}$, and the last inequality follows from (3.6).
We may also suppose that $t$ is chosen small enough that

(5.9) $\sum_{i=1}^{N}||\Pi(y_{i})||_{2}^{n/(n-1)}t^{n-1}\leq
2\int_{Q}\left|\left|\Pi\left(\frac{\partial\gamma_{t}}{\partial
x_{1}},\ldots,\frac{\partial\gamma_{t}}{\partial
x_{n-1}}\right)\right|\right|^{n/(n-1)}_{2}dV.$

It now follows from (5.7), (5.8) and (5.9) that

$\displaystyle V_{t}$ $\displaystyle\leq
C_{1}C_{2}^{n}C_{3}^{n/(n-1)}\sum_{i=1}^{N}\left(\operatorname{vol}_{n-1}\widetilde{f}(Q_{i})\right)^{n/(n-1)}$
$\displaystyle\leq
2^{n}C_{1}C_{2}^{n}C_{3}^{n/(n-1)}\sum_{i=1}^{N}\left(||\Pi(y_{i})||_{2}^{n/(n-1)}t^{n-1}\right)t$
$\displaystyle\leq
2^{n+1}C_{1}C_{2}^{n}C_{3}^{n/(n-1)}\left(\int_{Q}\left|\left|\Pi\left(\frac{\partial\gamma_{t}}{\partial
x_{1}},\ldots,\frac{\partial\gamma_{t}}{\partial
x_{n-1}}\right)\right|\right|^{n/(n-1)}_{2}dV\right)t$ $\displaystyle\leq
C_{6}Pt,$

where $C_{6}$ depends only on $n$ and $K$. Combining this with (5.6), we have

$\frac{t}{C_{4}\epsilon}\leq C_{6}Pt,$

which yields a contradiction for $\epsilon$ small enough.

We emphasize that in both the upper and lower bounds the contradiction is
obtained by finding $\epsilon$ to be too small relative to constants that
depend only on $n$, $K$ and $P$. Hence $\widetilde{\rho_{f}}$ is locally
$L$-bi-Lipschitz for some $L$ depending only on $n$, $K$ and $P$, which
completes the proof of Theorem 1.3.

##  6\. Non-rectifiable images

###### Proof of Proposition 1.4.

Here we show in dimension two that if the image of a cross-section of the beam
is a non-rectifiable curve, then $\widetilde{\rho_{f}}$ is not bi-Lipschitz.
Recall that $\widetilde{f}$ is defined in $[0,2]\times(-\infty,M)$, so suppose
that $\gamma(x):=\widetilde{f}(x,t_{0})$ parameterizes a non-rectifiable
curve. In particular, its Hausdorff $1$-measure is infinite.

Using the notation established above, we have by Lemma 3.2 that

$V_{t}=\sum_{i=1}^{N}\operatorname{vol}_{n}\widetilde{f}(P_{i})\geq\frac{1}{C_{1}}\sum_{i=1}^{N}\left(\operatorname{diam}\widetilde{f}(P_{i})\right)^{2}.$

Since $\gamma$ has infinite Hausdorff $1$-measure, it follows that given
$\epsilon>0$ we can find $t_{1}>0$ so that if $0<t<t_{1}$, we have

$\sum_{i=1}^{N}\operatorname{diam}\widetilde{f}(P_{i})>\frac{1}{\epsilon}.$

By (2.1) and (3.6) we then have

$V_{t}\geq\frac{1}{C_{1}N}\left(\sum_{i=1}^{N}\operatorname{diam}\widetilde{f}(P_{i})\right)^{2}>\frac{t}{C_{1}\epsilon}$

for $0<t<t_{1}$. An application of Lemma 4.1 then shows that

$\frac{\widetilde{\rho_{f}}(t_{0}+t)-\widetilde{\rho_{f}}(t_{0})}{t}>\frac{1}{C_{1}C_{4}\epsilon}$

for $0<t<t_{1}$. Choosing $\epsilon$ small enough shows that
$\widetilde{\rho_{f}}$ is not bi-Lipschitz.

∎

##  7\. Simple infinitesimal spaces

###### Proof of Theorem 1.5.

Recall the asymptotic representative

$D(x)=\rho_{f}(x)g(x/|x|),$

where $g$ is the unique element in $T(x_{0},f)$. By [5, Proposition 4.18],

(7.1) $g(x)=|x|^{d}g(x/|x|)$

for some $d>0$, and where $g:S^{n-1}\to\mathbb{R}^{n}$ is BLD. In our case,
$g$ is bijective onto its image, as $i(x_{0},f)=1$, and so $g|_{S^{n-1}}$ is
bi-Lipschitz. Since $g$ is quasiconformal, fixes $0$ and preserves the measure
of the unit ball, there exists $C=C(K)\geq 1$ so that $g(S^{n-1})$ is
contained in the ring $R=\\{x\in\mathbb{R}^{n}:1/C\leq|x|\leq C\\}$. As
$\mathcal{Z}$ is bi-Lipschitz on $Q\times\\{0\\}$ and $\mathcal{Z}^{-1}$ is
bi-Lipschitz on $R$, and in particular on $g(S^{n-1})$, it follows that

$\widetilde{g}:Q\times\\{0\\}\to B$

is bi-Lipschitz. Again letting $\mathcal{P}:B\to Q$ be the orthogonal
projection, we see from (7.1) that $\widetilde{g}:B\to B$ is given by

$\widetilde{g}(x)=\widetilde{g}(\mathcal{P}(x))+(0,\ldots,0,d[x]_{n}).$

Since the restriction of $\widetilde{g}$ to each slice $Q\times\\{t\\}$ is bi-
Lipschitz, and since $\widetilde{g}$ is quasiconformal, it follows from the
bounded linear distortion that $\widetilde{g}$ is itself bi-Lipschitz.

Now, the Zorich transform of $D$ is given by

$\widetilde{D}(x)=\widetilde{g}(\mathcal{P}(x))+(0,\ldots,0,\widetilde{\rho_{f}}([x]_{n})).$

By Theorem 1.3, $\widetilde{\rho_{f}}$ is bi-Lipschitz on $(-\infty,M)$. For
every $t<M$, it follows that there exists $t^{\prime}\in\mathbb{R}$ such that

$\widetilde{D}|_{Q\times\\{t\\}}=\widetilde{g}|_{Q\times\\{t^{\prime}\\}}.$

Writing $h(t)=\widetilde{\rho_{f}}(t)/d$, the map
$\widetilde{g}^{-1}\circ\widetilde{D}$ is nothing other than

$(x_{1},\ldots,x_{n-1},x_{n})\mapsto(x_{1},\ldots,x_{n-1},h(x_{n})).$

Since $h$ is bi-Lipschitz on $(-\infty,M)$, it follows that
$\widetilde{g}^{-1}\circ\widetilde{D}$ is bi-Lipschitz on $B_{M}$. Since
$\widetilde{g}$ is bi-Lipschitz, we conclude that $\widetilde{D}$ is also bi-
Lipschitz and thus that $D$ itself is quasiconformal. ∎

## References

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