# Existence and uniqueness of limits at infinity for homogeneous Sobolev
functions

Pekka Koskela  Department of Mathematics and Statistics
P.O. Box 35
FI-40014 University of Jyväskylä, Finland<EMAIL_ADDRESS>and  Khanh
Nguyen  Faculty of Mathematics, Mechanics and Informatics, VNU University of
Science, Vietnam National University, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

We establish the existence and uniqueness of limits at infinity along infinite
curves outside a zero modulus family for functions in a homogeneous Sobolev
space under the assumption that the underlying space is equipped with a
doubling measure which supports a Poincaré inequality. We also characterize
the settings where this conclusion is nontrivial. Secondly, we introduce
notions of weak polar coordinate systems and radial curves on metric measure
spaces. Then sufficient and necessary conditions for existence of radial
limits are given. As a consequence, we characterize the existence of radial
limits in certain concrete settings.

###### 2020 Mathematics Subject Classification:

46E36, 31B25, 31B15.
Key words and phases: limit at infinity, Sobolev function, metric measure
space.

##  1\. Introduction

Let $(X,d,\mu)$ be a metric measure space with metric $d$ and Borel regular
measure $\mu$. A locally rectifiable curve $\gamma:[0,\infty)\to X$ is an
infinite curve if $\gamma\setminus B\neq\emptyset$ for all balls $B$. Then
$\int_{\gamma}ds=\infty$ if $\gamma$ is an infinite curve. We write
$\Gamma^{\infty}$ for the collection of all infinite curves and denote by
$\dot{N}^{1,p}(X),1\leq p<\infty,$ the collection of all locally integrable
functions that have a $p$-integrable upper gradient on $(X,d,\mu)$. Here the
notion of upper gradients is given in Section 2.2.

The aim of this paper is to study the existence and uniqueness of the limit

(1.1) $\lim_{t\to\infty}u(\gamma(t))$

for $\gamma\in\Gamma^{\infty}$ and for $u\in\dot{N}^{1,p}(X)$. We say that
the existence and uniqueness of (1.1) hold for $p$-a.e
$\gamma\in\Gamma^{\infty}$ if, for every $u\in\dot{N}^{1,p}(X)$, there exists
$c\in\mathbb{R}$ such that

$\lim_{t\to\infty}u(\gamma(t))\text{ exists and
}\lim_{t\to\infty}u(\gamma(t))=c\text{\rm\ \ for $p$-a.e
$\gamma\in\Gamma^{\infty}$}.$

Here the notion of $p$-a.e curve is given in Section 2.2.

Towards uniqueness, we employ an annular chain property at a given based point
$O$, see Definition 2.3. This chain property holds, for example, if there
exists a constant $C\geq 1$ for which any pair of points in $B(O,r)\setminus
B(O,r/2)$ can be joined by a curve in $B(O,Cr)\setminus B(O,r/C),$ see Section
2.5 for more details. This holds especially when $X$ is annularly quasiconvex
as defined in [HKST15, Section 8.3].

Our first result deals with the limit (1.1) for $p$-a.e
$\gamma\in\Gamma^{\infty}$.

######  Theorem 1.1.

Let $1\leq p<\infty$. Suppose that $(X,d,\mu)$ is a doubling metric measure
space that supports a $p$-Poincaré inequality. Assume that $X$ has the annular
chain property. Then the existence and uniqueness of (1.1) hold for $p$-a.e
$\gamma\in\Gamma^{\infty}$.

In Theorem 1.1, the uniqueness does not hold without some additional
assumption besides doubling and Poincaré whose definitions are given in
Section 2.3. For example, on a space with at least two ends one easily
constructs a Lipschitz function in $\dot{N}^{1,p}(X)$ so that the tail value
of two of the ends is a different constant. A tree in [BBGS17, NgWa20] (or a
weighted real line in [BBK06, BBS20]) is doubling, supports a Poincaré
inequality and has more than one end.

The conclusion of Theorem 1.1 is nontrivial only when the $p$-modulus of
$\Gamma^{\infty}$ is strictly positive. Let us suppose that $\mu$ is a
$Q$-Ahlfors regular measure as in Section 2.3 where $1<Q\leq p<\infty$. Then
the $p$-modulus of $\Gamma^{\infty}$ vanishes, and hence there exists
$u\in\dot{N}^{1,p}(X)$ such that $\lim_{t\to\infty}u(\gamma(t))=\infty$ for
every $\gamma\in\Gamma^{\infty}$, see [HKST15, Page 135]. One can actually
characterize the settings where the $p$-modulus of the family
$\Gamma^{\infty}$ is strictly positive. This is the content of Theorem 1.2
below.

Let $O$ be a fixed point in $X$ and let $A_{2^{j}}(O)$ be the annuli
$B(O,2^{j+1})\setminus B(O,2^{j})$ for $j\in\mathbb{N}$. We define

$\mathcal{R}_{p}(O):=\sum_{j\in\mathbb{N}}(2^{j})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{j}}(O))\text{\rm\
\ if $p>1$ and \ \
}\mathcal{R}_{1}(O):=\sup_{j\in\mathbb{N}}2^{j}\mu^{-1}(A_{2^{j}}(O)).$

The finiteness of $\mathcal{R}_{p}(O)$ is then a volume growth condition. A
reader familiar with classification theory should recognize this as a
condition towards $p$-hyperbolicity [HoKo01, Hol99, Gri99].

######  Theorem 1.2.

Let $1\leq p<\infty$. Suppose that $(X,d,\mu)$ is a complete doubling metric
measure space that supports a $p$-Poincaré inequality. Assume that $X$ has the
annular chain property at O. Then the following statements are equivalent:

  1. I.

$\mathcal{R}_{p}(O)<\infty$.

  2. II.

$\text{\rm Mod}_{p}(\Gamma^{\infty})>0$.

  3. III.

For every $u\in\dot{N}^{1,p}(X)$, there exists an infinite curve
$\gamma\in\Gamma^{\infty}$ such that $\lim_{t\to\infty}u(\gamma(t))$ exists.

The space $X=[0,\infty)\subset\mathbb{R}$ equipped with the Lebesgue measure
and the usual distance is doubling and supports a $p$-Poincaré inequality for
all $p\geq 1.$ It also has the annular chain property and there is (modulo
reparametrizations and translations) only one injective infinite curve.
Moreover, $\mathcal{R}_{p}(0)<\infty$ precisely when $p=1,$ in which case
every function in $\dot{N}^{1,1}(X)$ has a (unique) limit along this curve.
Hence the existence of a single “good” curve is the best one can obtain in
this generality. However, for $\mathbb{R}^{n},$ $n\geq 2,$ equipped with the
Lebesgue measure and Euclidean distance, $\mathcal{R}_{p}(0)<\infty$ precisely
when $p<n$ and one then obtains a unique limit along the radial half-line in
the direction $\xi\in S^{n-1},$ for almost every $\xi$ with respect to the
surface measure on the boundary $S^{n-1}$ of the unit ball. This is a
consequence of the existence of spherical (or polar) coordinates. Towards
establishing limits along a recognizable family of infinite curves, let us
introduce an abstract version of polar coordinates.

Let $\mathbb{S}$ be a nonempty set (a set of indices), $O\in X$ and consider
collections $\Gamma^{O}(\mathbb{S})$ consisting of
$\gamma_{\xi}\in\Gamma^{\infty},$ $\xi\in\mathbb{S},$ with
$\gamma_{\xi}(0)=O.$ We say that $X$ has a weak polar coordinate system at the
coordinate point $O$ if there exist a set $\mathbb{S}$ of indices with a Radon
probability measure $\sigma$ on $\mathbb{S}$, a choice of
$\Gamma^{O}(\mathbb{S}),$ a function $h:X\to\mathbb{[}0,\infty)$, and a
constant $\mathcal{C}>0$ such that

(1.2) $\int_{\mathbb{S}}\int_{\gamma^{O}_{\xi}}|f|h\
dsd\sigma\leq\mathcal{C}\int_{X}|f|d\mu\text{\rm\ \ for every integrable
function $f$. }$

We then call $h$ a coordinate weight and each
$\gamma^{O}_{\xi}\in\Gamma^{O}(\mathbb{S})$ a radial curve in the direction
$\xi\in\mathbb{S}$ from $O$. Towards the existence of (1.1) along radial
curves, consider a polar coordinate (1.2) at $O$. We set

$R_{p}(h,O):=\int_{X\setminus B(O,1)}h^{\frac{p}{1-p}}d\mu\text{\ \ if $p>1$
and\ \ }R_{1}(h,O):=\|h^{-1}\|_{L^{\infty}_{\mu}(X\setminus B(O,1))}.$

Given $\xi\in F\subset\mathbb{S}$ with
$\liminf_{t\to\infty}d(O,\gamma_{\xi}^{O}(t))>1$, there is $t_{\xi}>0$ such
that $d(O,\gamma_{\xi}^{O}(t_{\xi}))=1$ and $d(O,\gamma_{\xi}^{O}(t))>1$ for
all $t>t_{\xi}$. Let $\hat{\gamma}_{\xi}^{O}:[0,\infty)\to X$ be the infinite
curve starting from $\gamma_{\xi}^{O}(t_{\xi})$ defined by
$\hat{\gamma}_{\xi}^{O}(t):=\gamma_{\xi}^{O}(t+t_{\xi})$ for $t\geq 0$. The
collection of all these infinite curves $\hat{\gamma}_{\xi}^{O}$ with respect
to $\xi\in F$ satisfying $\liminf_{t\to\infty}d(O,\gamma_{\xi}^{O}(t))>1$ is
denoted by $\hat{\Gamma}^{O}(F)$.

In order to state our next result, we introduce the following properties:

  1. 1.

For every $u\in\dot{N}^{1,p}(X)$, there is a radial curve $\gamma$ such that
$\liminf_{t\to\infty}u(\gamma(t))$ is finite.

  2. 2.

The limit (1.1) exists for $\sigma$-a.e $\xi\in\mathbb{S}$, i.e. the limit

$\lim_{t\to\infty}u(\gamma^{O}_{\xi}(t))\text{\rm\ \ exists for $\sigma$-a.e
$\xi\in\mathbb{S}$, for every $u\in\dot{N}^{1,p}(X)$}.$

  3. 3.

The existence and uniqueness of (1.1) are obtained for $\sigma$-a.e
$\xi\in\mathbb{S}$, i.e. for every $u\in\dot{N}^{1,p}(X),$ there exists
$c\in\mathbb{R}$ such that

$\lim_{t\to\infty}u(\gamma^{O}_{\xi}(t))\text{\rm\ exists\ and \
}\lim_{t\to\infty}u(\gamma^{O}_{\xi}(t))=c\text{\rm\ \ for $\sigma$-a.e
$\xi\in\mathbb{S}$}.$

  4. 4.

$\text{\rm Mod}_{p}(\hat{\Gamma}^{O}(\mathbb{S}))>0$.

  5. 5.

If $F\subseteq\mathbb{S}$ satisfies $\sigma(F)>0,$ then $\text{\rm
Mod}_{p}(\hat{\Gamma}^{O}(F))>0.$

We show that each of $\\{1.,2.,3.,4.,5.\\}$ is “between”
$\mathcal{R}_{p}(O)<\infty$ and $R_{p}(h,O)<\infty.$

######  Theorem 1.3.

Let $1\leq p<\infty$. Suppose that $(X,d,\mu)$ is a doubling metric measure
space that supports a $p$-Poincaré inequality. Assume that $X$ has the annular
chain property at O. Suppose that $X$ has a weak polar coordinate system
$(\mathbb{S},\sigma,\Gamma^{O},h)$ at $O$ as in (1.2). Then

  1. I.

The condition $R_{p}(h,O)<\infty$ is sufficient for all of
$\\{1.,2.,3.,4.,5.\\}$.

  2. II.

The condition $\mathcal{R}_{p}(O)<\infty$ is necessary for each of
$\\{1.,2.,3.,4.,5.\\}$.

Theorem 1.3 leads us to compare the finiteness of $R_{p}(h,O)$ and
$\mathcal{R}_{p}(O).$ First of all, it immediately follows from Theorem 1.3
that the finiteness of $R_{p}(h,O)$ guarantees the finiteness of
$\mathcal{R}_{p}(O).$ We do not know a simple direct proof for this. On the
other hand, the finiteness of $\mathcal{R}_{p}(O)$ does not in general yield
the finiteness of $R_{p}(h,O).$ To see this, simply consider $\mathbb{R}^{n},$
$n\geq 2,$ with the Euclidean distance and Lebesgue measure, usual spherical
coordinates (with normalized measure on $S^{n-1}$) but with the coordinate
weight $h=\chi_{\mathbb{R}^{n}\setminus B(0,1)}.$ On the other hand, for
example, if

$h(x)\gtrsim\sum_{j\in\mathbb{N}}\frac{\mu(A_{2^{j}}(O))}{(2^{j})^{p}}\chi_{A_{2^{j}}(O)}(x)$

for all $x\in X,$ then $R_{p}(h,O)\lesssim\mathcal{R}_{p}(O).$

Let us next consider the Muckenhoupt $\mathcal{A}_{p}$-weighted space
$(\mathbb{R}^{n},d_{E},w)$, where $n\geq 2$ and $O$ is the origin. Here
$d_{E}$ is the Euclidean distance, and $w$ is a Muckenhoupt
$\mathcal{A}_{p}$-weight with the associate measure $d\mu=wdx$, i.e. there is
a constant $C\geq 1$ such that for every ball $B\subset\mathbb{R}^{n}$,

(1.3) $\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}wdx\right)^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}w^{\frac{1}{1-p}}dx\right)^{\frac{p}{p-1}}\leq
C\text{\rm\ \ if $p>1$,}$

and

(1.4) $\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}wdx\right)\|w^{-1}\|_{L^{\infty}(B)}\leq
C\text{\rm\ \ if $p=1$}.$

Then $\mu$ is doubling and supports a $p$-Poincaré inequality, see for
instance [HKM06]. The annular chain property at $O$ is satisfied and the usual
(normalized) spherical coordinate system satisfies (1.2) at $O.$ We show that
$R_{p}(h,O)\approx\mathcal{R}_{p}(O)$ in Example 4.3. In particular,
$R_{p}(h,O)<\infty$ if and only if $1\leq p<n$ for the unweighted space
$\mathbb{R}^{n}$. It follows that Theorem 1.3 recovers some of the conclusions
in [EKN22, Usp61].

Next, we consider a $Q$-Ahlfors regular space $X$ that supports a $p$-Poincaré
inequality, where $1\leq p<\infty$ and $1\leq Q<\infty.$ Suppose that $X$ has
the annular chain property at $O$ and satisfies the strong form

$\int_{X\setminus
B(O,1)}|f|d\mu=\int_{\mathbb{S}}\int_{1}^{\infty}|f(\gamma_{\xi}(t))|t^{Q-1}\
dtd\sigma(\xi)\text{\rm\ \ (for every integrable function $f$)}$

of our weak polar coordinate system. Then each of $\\{1.,2.,3.,4.,5.\\}$ is
equivalent to $1\leq p<Q$ if $1<Q<\infty$, and equivalent to $p=1$ if $Q=1$
since

(1.5) $\mathcal{R}_{p}(O)<\infty\implies\begin{cases}1\leq p<Q&\text{\rm\ \ if
\ \ }1<Q<\infty,\\\ p=1&\text{\rm\ \ if \ \ }Q=1.\end{cases}\implies
R_{p}(h,O)<\infty$

where

$R_{p}(h,O)\approx\begin{cases}\int_{1}^{\infty}r^{\frac{(Q-1)}{1-p}}dr&\text{\rm
if \ \ }p>1,\\\ \|r^{1-Q}\|_{L^{\infty}([1,\infty))}&\text{\rm if \ \
}p=1,\end{cases}\text{\ \ and\ \
}\mathcal{R}_{p}(O)\approx\begin{cases}\sum_{j\in\mathbb{N}}(2^{j})^{\frac{p-Q}{p-1}}&\text{\rm
if \ \ }p>1,\\\ \sup_{j\in\mathbb{N}}(2^{j})^{1-Q}&\text{\rm if \ \
}p=1.\end{cases}$

In particular, by [BaTy02, Ty20] every polarizable Carnot group $G$
(especially every group of Heisenberg type) admits a polar coordinate system
(2.3) that satisfies (1.2). Since the Haar measure on $G$ is $Q$-Ahlfors
regular for the homogeneous dimension $Q>1$ of $G$ and supports a $p$-Poincaré
inequality for each $p$, each of $\\{1.,2.,3.,4.,5.\\}$ is equivalent to
$1\leq p<Q<\infty$.

We then obtain the following characterization.

######  Corollary 1.4.

Let $1\leq p<\infty$ and let $1<Q<\infty$. Then

  1. I.

Each of $\\{1.,2.,3.,4.,5.\\}$ is equivalent to $R_{p}(h,O)<\infty$ (or
$\mathcal{R}_{p}(O)<\infty$) on the Muckenhoupt $\mathcal{A}_{p}$-weighted
Euclidean spaces.

  2. II.

Each of $\\{1.,2.,3.,4.,5.\\}$ is equivalent to $1\leq p<Q<\infty$ on
polarizable Carnot groups.

The organization of this paper is as follows. In Section 2, we introduce polar
coordinates and recall the basic notions on metric measure spaces. In Sections
3-4, proofs of Theorem 1.1-1.2-1.3 are given.

Throughout this paper, we use the following conventions. We denote by $O$ the
base point in the annular chain property and also refer by $O$ to the
coordinate point in the polar coordinate (1.2). The notation $A\lesssim B\
(A\gtrsim B)$ means that there is a constant $C>0$ only depending on the data
such that $A\leq C\cdot B\ (A\geq C\cdot B)$, and $A\approx B$ means that
$A\lesssim B$ and $A\gtrsim B$. For each locally integrable function $f$ and
for every measurable subset $A\subset X$ of positive measure, we let
$f_{A}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{A}fd\mu=\frac{1}{\mu(A)}\int_{A}fd\mu$.

##  2\. Polar coordinates and preliminaries

In Section 2.1, we introduce polar coordinates. We recall the basic notions of
modulus, doubling, Poincaré inequalities, and Lebesgue points in Sections
2.2-2.3-2.4.

###  2.1. Polar coordinates

Let $(X,d)$ be a metric space. A curve is a nonconstant continuous mapping
from an interval $I\subseteq\mathbb{R}$ into $X$. The length of a curve
$\gamma$ is denoted by $l(\gamma)$. A curve $\gamma$ is said to be a
rectifiable curve if its length is finite. Similarly, $\gamma$ is a locally
rectifiable curve if its restriction to each compact subinterval of $I$ is
rectifiable. Each rectifiable curve $\gamma$ will be parameterized by arc
length and hence the line integral over $\gamma$ of a Borel function $f$ on
$X$ is

$\int_{\gamma}fds=\int_{0}^{l(\gamma)}f(\gamma(t))dt.$

If $\gamma$ is locally rectifiable, then we set

$\int_{\gamma}fds=\sup\int_{\gamma^{\prime}}fds$

where the supremum is taken over all rectifiable subcurves $\gamma^{\prime}$
of $\gamma$. Let $\gamma:[0,\infty)\to X$ be a locally rectifiable curve,
parameterized by arc length. Then

$\int_{\gamma}fds=\int_{0}^{\infty}f(\gamma(t))dt.$

A locally rectifiable curve $\gamma$ is an infinite curve if $\gamma\setminus
B\neq\emptyset$ for all balls $B$. Then $\int_{\gamma}ds=\infty.$ We denote by
$\Gamma^{\infty}$ the collection of all infinite curves.

Let $\mathbb{S}$ be a nonempty set (a set of indices). Given a point $O$, we
consider collections $\Gamma^{O}(\mathbb{S})$ of infinite curves with
parameter space $\mathbb{S}$ starting from $O$, namely

$\Gamma^{O}(\mathbb{S})=\\{\gamma^{O}_{\xi}\in\Gamma^{\infty}:\gamma_{\xi}^{O}(0)=O,\xi\in\mathbb{S}\\}.$

We say that $(X,d,\mu)$ has a weak polar coordinate system at the coordinate
point $O$ if there is a choice of a pair $(\mathbb{S},\Gamma^{O}(\mathbb{S}))$
with a Radon probability measure $\sigma$ on $\mathbb{S}$, a coordinate weight
$h:X\to\mathbb{[}0,\infty)$, and a constant $\mathcal{C}>0$ such that

(2.1) $\int_{\mathbb{S}}\int_{\gamma^{O}_{\xi}}|f|\ h\
dsd\sigma\leq\mathcal{C}\int_{X}|f|d\mu\text{\rm\ \ for every integrable
function $f$.}$

Each $\gamma_{\xi}^{O}\in\Gamma^{O}(\mathbb{S})$ is called a radial curve with
respect to $\xi\in\mathbb{S}$ (starting from $O$). Notice that we have not
assumed $h$ to be strictly positive. In applications it will be important to
require $h$ to be strictly positive at least almost everywhere in the union of
the images of $\gamma_{\xi}^{O}.$

Let us give examples of weak polar coordinates. First of all, in the
$n$-dimensional Euclidean space $\mathbb{R}^{n}$, where $n\geq 2$, one has the
usual spherical coordinate system at a given point $O$: there is a constant
$C(n)>0$ depending on $n$ such that

(2.2)
$\int_{\mathbb{R}^{n}}f(x)dx=C(n)\int_{S^{n-1}}\int_{0}^{\infty}f(O+r\cdot\xi)r^{n-1}drd\mathcal{H}^{n-1}\text{\rm\
\ for every integrable function $f$}.$

Here $\mathcal{H}^{n-1}$ is the (normalized) $(n-1)$-Hausdorff measure and
$S^{n-1}$ is the unit sphere centered at $O$. For these coordinates, (2.1)
holds as an identity.

In [BaTy02, Ty20], Balogh and Tyson produced a polar coordinate system at the
origin on polarizable Carnot groups $G$: there is a family of horizontal
curves $\gamma_{\xi}:[0,\infty)\to G$ where $\xi$ ranges over a certain
compact unit sphere $S\subset G$, and a positive Radon measure $\sigma$ on $S$
so that

(2.3)
$\int_{G}f(g)dg=\int_{S}\int_{0}^{\infty}f(\gamma_{\xi}(t))t^{Q-1}dtd\sigma(\xi)$

is valid for every integrable function $f$. Here the integral on the left is
taken with respect to the Haar measure in the group and $Q$ denotes the
homogeneous dimension of $G$. The unit sphere $S$ is the level set $\\{g\in
G:N(g)=1\\}$ for a certain homogeneous norm $N$ in $G$.

In [KNZ22, Lemma 3.1], a polar coordinate at the root $O$ on a $K$-regular
tree is given. Let $X$ be a $K$-regular tree with its boundary $\partial X.$
We equip $X$ with a radially weighted distance $\lambda$ and a radially
weighted measure $\mu$. Then there is a uniform measure $\nu$ on $\partial X$
such that

$\int_{X}fd\mu\approx\int_{\partial
X}\int_{[O,\xi)}f(x)\frac{K^{|x|}\mu(x)}{\lambda(x)}ds(x)d\nu(\xi)$

for every integrable function $f$. Here $[O,\xi):=\gamma_{\xi}$ is the unique
geodesic ray from the root $O$ to $\xi\in\partial X$.

In the above examples, one actually has two-sided estimates for the terms in
(2.1). This is not necessarily the case for our weak polar coordinate systems.
For example, let $X:=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:-1\leq x_{2}\leq
1\\}\cup\\{(x_{1},x_{2})\in\mathbb{R}^{2}:|x_{2}|\leq x_{1}\\}$ and equip it
with the Lebesgue measure and the Euclidean distance. Let $S_{1}=[-1,1]$ and
set $\gamma_{\xi}^{1}(t)=\begin{cases}(-t,t\cdot\xi),&0\leq t\leq 1\\\
(-t,\xi),&t>1\end{cases},$ when $\xi\in[-1,1]$. Define
$h_{1}(x_{1},x_{2})=\begin{cases}1,&x_{1}<-1\\\ 0,&{\rm
otherwise.}\end{cases}$ Then

$\int_{S_{1}}\int_{0}^{\infty}|f(\gamma_{\xi}^{1}(t))|\
h_{1}(\gamma_{\xi}^{1}(t))\
dtd\xi\leq\int_{X}|f(x_{1},x_{2})|dx_{1}dx_{2}\text{\rm\ \ for every
integrable function $f$}.$

This is a weak polar coordinate system. A second weak polar coordinate system
on $X$ is obtained by taking $S_{2}=[-\pi/4,\pi/4]$,
$\gamma_{\xi}^{2}(t)=e^{i\cdot\xi}$ and
$h_{2}(x_{1},x_{2})=\|(x_{1},x_{2})\|\chi_{\\{(x_{1},x_{2})\in X:x_{2}\geq
0\\}}(x_{1},x_{2})$. Then

$\int_{S_{2}}\int_{0}^{\infty}|f(\gamma_{\xi}^{2}(t))|\
h_{2}(\gamma_{\xi}^{2}(t))\
dtd\xi\leq\int_{X}|f(x_{1},x_{2})|dx_{1}dx_{2}\text{\rm\ \ for every
integrable function $f$}.$

For a third one, we take $S_{3}=S_{1}\cup S_{2},h_{3}=h_{1}+h_{2}$ and
$\gamma_{\xi}^{3}=\gamma_{\xi}^{1}\cup\gamma_{\xi}^{2}$. Then

$\int_{S_{3}}\int_{0}^{\infty}|f(\gamma_{\xi}^{3}(t))|\
h_{3}(\gamma_{\xi}^{3}(t))\
dtd\xi\leq\int_{X}|f(x_{1},x_{2})|dx_{1}dx_{2}\text{\rm\ \ for every
integrable function $f$}.$

In [KM98, Section 3], there is a weak polar coordinate system on the Cantor
$\infty$-diamond. Let us describe it. Let $E_{i}$ be the usual Cantor set in
the unit interval $[i,i+1]$ obtained by first taking out the middle interval
of length $1/3$ and leaving two intervals of length $1/3$ and then continuing
inductively. The Cantor $\infty$-diamond, denoted by $X$, is obtained by
replacing each of the complementary intervals of $E_{i}$ by a square having
that interval as one of its diagonals. Thus we have a line of diamonds along
the unit interval, and they are joined up by $E_{i}$. We consider the map
$F:[0,\infty)\times[-1,1]\to X$ defined by

$F(x,y)=\left(x,\delta(x)\tan\left(\pi\frac{y}{4}\right)\right).$

Here $\delta(x)$ is the distance from $x$ to $E:=\bigcup_{i=0}^{\infty}E_{i}$.
Then the map $F$ is simply the vertical projection on $E\times[-1,1]$ and it
is one-to-one, locally bi-Lipschitz on $([0,\infty)\setminus E)\times[-1,1]$.
The Jacobian of $F$ at $(x,y)$, denoted by $J_{F}(x,y)$, is
$\frac{\pi\delta(x)}{4\cos^{2}(\pi y/4)}$. Let $\Gamma$ be the family of
curves $\gamma_{y}$, $-1<y<1$, defined by

$\gamma_{y}(x)=F(x,y){\rm\ \ for\ all\ }y\in[-1,1].$

Then $\gamma_{y}$ is $4$-Lipschitz for each $y\in[-1,1]$. We denote by
$\sigma$ the $1$-Lebesgue measure on [-1,1]. By the change of variables
formula, we have

$\displaystyle\int_{\Gamma}\int_{\gamma}f\cdot(J_{F}\circ
F^{-1})dsd\left(\frac{\sigma}{2}\right)(\gamma)=$
$\displaystyle\frac{1}{2}\int_{[-1,1]}\left(\int_{[0,\infty)}(f\circ\gamma_{y})(x)\cdot(J_{F}\circ
F^{-1}\circ\gamma_{y})(x)\cdot|\dot{\gamma}_{y}(x)|dx\right)dy$
$\displaystyle\leq$ $\displaystyle 2\int_{[-1,1]}\int_{[0,\infty)}(f\circ
F)(x,y)\cdot J_{F}(x,y)dxdy$ $\displaystyle=$ $\displaystyle
2\int_{X}fd\mathcal{L}^{2}$

for each positive integrable function $f$ on $X$ with respect to the
$2$-Lebesgue measure $\mathcal{L}^{2}$. Here the inequality is obtained since
$\gamma_{y}$ is $4$-Lipschitz for each $y\in[-1,1]$ and $(J_{F}\circ
F^{-1}\circ\gamma_{y})(x)=(J_{F}\circ F^{-1})(F(x,y))=J_{F}(x,y)$.

Finally, we give a definition of Semmes-type families which are related to our
weak polar coordinate system.

######  Definition 2.1 (A Semmes type family of infinite curves).

Let $(X,d,\mu)$ be a metric measure space with metric $d$ and measure $\mu$.
Given $O\in X$, a family $\Gamma$ of infinite curves starting from $O$ is
called Semmes-type if there exist a constant $C>0$ and a Radon probability
measure $\sigma$ on $\Gamma$ such that

(2.4) $\int_{\Gamma}\int_{\gamma}f\ ds\ d\sigma(\gamma)\leq
C\int_{X}f(x)\frac{d(x,O)}{\mu(B(O,d(x,O)))}d\mu(x)$

for every positive measurable function $f$ on $X$ for which the right-hand
side of (2.4) is finite.

For instance, a family of all radial curves is a Semmes type family on
$\mathbb{R}^{n}$ by the usual spherical coordinate system (2.2) or on a
polarizable Carnot group by the coordinate system (2.3). This definition is
naturally modified from the existence of families of rectifiable curves
joining pairs of points on metric measure spaces by Semmes in [Sem96].

###  2.2. Modulus

Let $\Gamma$ be a family of curves in a metric measure space $(X,d,\mu)$.
Given $1\leq p<\infty$, the $p$-modulus of $\Gamma$, denoted $\text{\rm
Mod}_{p}(\Gamma)$, is defined by

$\text{\rm Mod}_{p}(\Gamma):=\inf\int_{X}\rho^{p}d\mu$

where the infimum is taken over all Borel functions $\rho:X\to[0,\infty]$
satisfying $\int_{\gamma}\rho ds\geq 1$ for every locally rectifiable curve
$\gamma\in\Gamma$. A family of curves is called $p$-exceptional if it has
$p$-modulus zero. We say that a property holds for $p$-a.e curve if the
collection of curves for which the property fails is $p$-exceptional.

Let $u$ be a locally integrable function. A Borel function
$\rho:X\to[0,\infty]$ is said to be an upper gradient of $u$ if

(2.5) $|u(x)-u(y)|\leq\int_{\gamma}\rho ds$

for every rectifiable curve $\gamma$ connecting $x$ and $y$. Then we have that
(2.5) holds for all compact subcurves of $\gamma\in\Gamma^{\infty}$. We say
that $\rho$ is a $p$-weak upper gradient of $u$ if (2.5) holds for $p$-a.e
rectifiable curve. In what follows, we denote by $g_{u}$ the minimal upper
gradient of $u$, which is unique up to measure zero and which is minimal in
the sense that $g_{u}\leq\rho$ a.e. for every $p$-integrable $p$-weak upper
gradient $\rho$ of $u$. In [Haj03], the existence and uniqueness of such a
minimal upper gradient are given.

The notion of upper gradients is due to Heinonen and Koskela [HeKo98], we
refer interested readers to [Björn, Haj03, HeKo98, Sha00] for a more detailed
discussion on upper gradients.

###  2.3. Doubling and Poincaré inequalities

Let $(X,d)$ be a metric space. A Borel regular measure $\mu$ is called
doubling if every ball in $X$ has finite positive measure and if there exists
a constant $C\geq 1$ such that for all balls $B(x,r)$ with radius $r>0$ and
center at $x\in X$,

$\mu(B(x,2r))\leq C\mu(B(x,r)).$

Let $1\leq Q<\infty$. A Borel regular measure $\mu$ is said to be $Q$-Ahlfors
regular if there exists a constant $C\geq 1$ such that for all balls $B(x,r)$
with radius $r>0$ and center at $x\in X$,

$\frac{r^{Q}}{C}\leq\mu(B(x,r))\leq Cr^{Q}.$

Hence if $\mu$ is $Q$-Ahlfors regular for some $1\leq Q<\infty$, then $\mu$ is
a doubling measure.

Let $1\leq p<\infty$. We say that a measure $\mu$ supports a $p$-Poincaré
inequality if every ball in $X$ has finite positive measure and if there exist
constants $C>0$ and $\lambda\geq 1$ such that

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,r)}|u-u_{B(x,r)}|d\mu\leq
Cr\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\lambda\cdot
B(x,r)}\rho^{p}d\mu\right)^{\frac{1}{p}}$

for all balls $B(x,r)$ with radius $r>0$ and center at $x\in X$, and for all
pairs $(u,\rho)$ satisfying (2.5) such that $u$ is integrable on balls. Here
$\lambda\cdot B(x,r):=B(x,\lambda\cdot r)$ and $\lambda$ is called the scaling
constant of $p$-Poincaré inequality or the scaling factor of $p$-Poincaré
inequality. Since $u\in\dot{N}^{1,p}(X)$ is integrable on balls, the
$p$-Poincaré inequality makes sense for any pair $(u,\rho_{u})$ where
$\rho_{u}$ is an upper gradient of $u$.

###  2.4. Lebesgue points

A point $x\in X$ is called a Lebesgue’s point of $u$ if $\lim_{r\to
0}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,r)}|u(y)-u(x)|d\mu(y)=0$ where $B(x,r)$ is the
ball with radius $r$ and center at $x$. Let $N_{u}$ be a set of all points
$x\in X$ such that $x$ is not a Lebesgue point of $u$.

######  Theorem 2.2 (Lebesgue differentiation theorem, see for instance
[HKST15, Page 77]).

We have $\mu(N_{u})=0$ for every locally integrable function $u$ on $X$.

###  2.5. Chain conditions

In this paper, we employ the following annular chain property.

######  Definition 2.3.

Let $\lambda\geq 1$. We say that $X$ satisfies an annular $\lambda$-chain
condition at $O$ if the following holds. There are constants $c_{1}\geq
1,c_{2}\geq 1,\delta>0$ and a finite number $M<\infty$ so that given $r>0$ and
points $x,y\in B(O,r)\setminus B(O,r/2)$, one can find balls
$B_{1},B_{2},\ldots,B_{k}$ with the following properties:

  1. 1.

$k\leq M$.

  2. 2.

$B_{1}=B(x,r/(\lambda c_{1}))$, $B_{k}=B(y,r/(\lambda c_{1}))$ and the radius
of each $B_{i}$ is $r/(\lambda c_{1})$ for $1\leq i\leq k$.

  3. 3.

$B_{i}\subset B(O,c_{2}r)\setminus B(O,r/c_{2})$ for $1\leq i\leq k$.

  4. 4.

For each $1\leq i\leq k-1$, there is a ball $D_{i}\subset B_{i}\cap B_{i+1}$
with radius $\delta r$.

If $X$ satisfies an annular $\lambda$-chain condition at $O$ for every
$\lambda\geq 1$, we say that $X$ has the annular chain property.

If $X$ satisfies an annular $\lambda$-chain condition at $O$ for some
$\lambda\geq 1$ then it also satisfies an annular $\lambda^{\prime}$-chain
condition at $O$ for all $\lambda^{\prime}$ with
$1\leq\lambda^{\prime}\leq\lambda$ by taking
$c_{1}(\lambda^{\prime}):=c_{1}(\lambda)\frac{\lambda}{\lambda^{\prime}}\geq
1$. Here $c_{1}(\lambda)$ is the constant $c_{1}$ with respect to $\lambda$ as
in Definition 2.3. It follows that $X$ has the annular chain property if and
only if there is a sequence $\lambda_{k}$ with
$\lim_{k\to+\infty}\lambda_{k}=+\infty$ such that $X$ satisfies an annular
$\lambda_{k}$-chain condition at $O$ for each $k\in\mathbb{N}$.

######  Lemma 2.4.

Let $\mu$ be doubling on $(X,d)$. Suppose that there is a constant $c_{0}\geq
1$ so that for every $r>0$, each pair $x,y$ of points in $B(O,r)\setminus
B(O,r/2)$ can be joined by a curve in $B(O,c_{0}r)\setminus B(O,r/c_{0})$.
Then $X$ has the annular chain property.

###### Proof.

We follow ideas in the proof of [HaKo00, Theorem 7.2]. Let $\lambda\geq 1$.
Let $\gamma_{x,y}$ be a curve in $B(O,c_{0}r)\setminus B(O,r/c_{0})$ joining
$x,y\in B(O,r)\setminus B(O,r/2)$. We consider the collection of all balls
$B(w,r/(100\lambda))$ with $w\in B(O,c_{0}r)\setminus B(O,r/c_{0})$. As $\mu$
is doubling, by the $5B$-covering lemma, we find a cover of $\gamma_{x,y}$
consisting of $k$ of these balls, say $D_{1},D_{2},\ldots,D_{k},$ with $k$
depending only on $c_{0},\lambda$ and the doubling constant, so that the
following properties hold:

  1. (1)

$\gamma_{x,y}\subset\bigcup_{i=1}^{k}5D_{i}$.

  2. (2)

$\\{D_{i}\\}_{i=1}^{k}$ are pairwise disjoint.

  3. (3)

$5D_{i}\bigcap 5D_{i+1}\neq\emptyset$ for $1\leq i\leq k-1$.

Let $B_{i}:=20D_{i}$. Then the four properties as in Definition 2.3 can be
checked to hold for a subcollection of the balls $B_{i}$ when
$\delta=1/(100\lambda),c_{1}=1/5,c_{2}=10c_{0}$. ∎

######  Corollary 2.5.

If $X$ is annularly quasiconvex as defined in [HKST15, Section 8.3], then $X$
has the annular chain property.

It especially follows from Corollary 2.5 together with [HKST15, Theorem 9.4.1]
that every complete metric measure space that is $Q$-Ahlfors regular and
supports a $p$-Poincaré inequality for some $1\leq p<Q$ has the annular chain
property.

##  3\. Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Throughout this section, we always
assume that $(X,d,\mu)$ is doubling and supports a $p$-Poincaré inequality
with scaling factor $\lambda\geq 1$, and that $u\in\dot{N}^{1,p}(X)$ with
$\rho_{u}$ a $p$-integrable upper gradient of $u$. We assume that $X$
satisfies an annular $\lambda$-condition at $O$. Let $B(x,r)$ be the ball with
radius $r>0$ and center at $x\in X$. We set $\tau B(x,r):=B(x,\tau r)$,
$A_{\lambda 2^{j+1}}:=B(O,\lambda 2^{j+2})\setminus B(O,\lambda 2^{j+1})$,
$c_{2}\cdot A_{\lambda 2^{j+1}}:=B(O,c_{2}\lambda 2^{j+3})\setminus
B(O,2^{j}/(c_{2}\lambda))$ for $\tau>0$, $j\in\mathbb{N}$ where $c_{2}$ is as
in Definition 2.3. Let us begin with useful lemmas, established using ideas
from [HeKo98, HaKo95, HaKo00].

######  Lemma 3.1.

Let $1\leq p<\infty$ and $j\in\mathbb{N}$. Suppose that $E,F$ are two subsets
of $A_{\lambda 2^{j+1}}$ such that $|u(x)-u(y)|\geq 1$ for all $x\in E,y\in
F$, and that $|u(x)-u_{B(x,2^{j}/c_{1})}|\leq{1}/{5}$,
$|u(y)-u_{B(y,2^{j}/c_{1})}|\leq{1}/{5}$ for some $x\in E$, $y\in F$, where
$c_{1}\geq 1$ is as in Definition 2.3. Then

(3.1) $\mu(c_{2}\cdot A_{\lambda 2^{j+1}})\lesssim(2^{j})^{p}\int_{c_{2}\cdot
A_{\lambda 2^{j+1}}}\rho_{u}^{p}d\mu.$

###### Proof.

We have that
$1\leq|u(x)-u(y)|\leq{1}/{5}+|u_{B(x,2^{j}/c_{1})}-u_{B(y,2^{j}/c_{1})}|+{1}/{5}.$
Hence $1\lesssim|u_{B(x,2^{j}/c_{1})}-u_{B(y,2^{j}/c_{1})}|.$ Let
$\\{B_{i}\\}_{i=1}^{k}$ statisfy the four properties in Definition 2.3. Since
$\mu$ is doubling and supports a $p$-Poincaré inequality, it follows that

$\displaystyle 1$
$\displaystyle\lesssim|u_{B(x,2^{j}/c_{1})}-u_{B(y,2^{j}/c_{1})}|\lesssim\sum_{i=1}^{k}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i}}|u-u_{B_{i}}|d\mu\lesssim\sum_{i=1}^{k}2^{j}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\lambda
B_{i}}\rho_{u}^{p}d\mu\right)^{\frac{1}{p}}.$

Hence there is an index $i$ such that $1\lesssim
2^{j}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\lambda
B_{i}}\rho_{u}^{p}d\mu\right)^{\frac{1}{p}}.$ By the doubling property, we
obtain that

$\mu(c_{2}\cdot A_{\lambda 2^{j+1}})\lesssim\mu(\lambda
B_{i})\lesssim(2^{j})^{p}\int_{\lambda
B_{i}}\rho_{u}^{p}d\mu\leq(2^{j})^{p}\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu$

which is (3.1). ∎

######  Lemma 3.2.

Let $1\leq p<\infty$ and $j\in\mathbb{N}$. Suppose that $E$ is a subset of
$A_{\lambda 2^{j+1}}$ such that $|u(x)-u_{B(x,2^{j}/c_{1})}|>{1}/{5}$ holds
for every $x\in E$, where $c_{1}\geq 1$ is as in Definition 2.3. Then

(3.2) $\mu(E)\lesssim(2^{j})^{p}\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu.$

###### Proof.

Let $x\in E$. Set $B_{i}=B(x,2^{j-i}/c_{1})$, $i\in\mathbb{N}$. By Theorem
2.2, we may assume that every $x\in E$ is a Lebesgue point of $u$. Since $\mu$
is doubling and supports a $p$-Poincaré inequality, we have that

(3.3) $\displaystyle\sum_{i=0}^{\infty}2^{-i}\approx\frac{1}{5}<$
$\displaystyle|u(x)-u_{B(x,2^{j}/c_{1})}|\leq\sum_{i=0}^{\infty}|u_{B_{i}}-u_{B_{i+1}}|\lesssim\sum_{i=0}^{\infty}2^{j-i}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\lambda
B_{i}}\rho_{u}^{p}d\mu\right)^{\frac{1}{p}}.$

Thus there is an index $i_{x}$ such that $2^{-i_{x}}\lesssim
2^{j-i_{x}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\lambda
B_{i_{x}}}\rho_{u}^{p}d\mu\right)^{\frac{1}{p}}$ and so

(3.4) $\mu(\lambda B_{i_{x}})\lesssim(2^{j})^{p}\int_{\lambda
B_{i_{x}}}\rho_{u}^{p}d\mu.$

Hence $E$ has a cover $\\{\lambda B_{i_{x}}:\eqref{eq3.8}\text{\rm\
holds}\\}_{x\in E}$. Using the $5B$-covering lemma, there is a pairwise
disjoint collection $\\{\lambda B_{i_{x_{k}}}\\}_{k=1}^{\infty}$ such that
$E\subset\bigcup_{k=1}^{\infty}5\lambda B_{i_{x_{k}}}$. By (3.4), since $\mu$
is doubling, we obtain that

$\mu(E)\leq\sum_{k=1}^{\infty}\mu(5\lambda
B_{i_{x_{k}}})\lesssim\sum_{k=1}^{\infty}\mu({\lambda
B_{i_{{x_{k}}}}})\lesssim\sum_{k=1}^{\infty}(2^{j})^{p}\int_{\lambda
B_{i_{x_{k}}}}\rho_{u}^{p}d\mu\leq(2^{j})^{p}\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu$

which is (3.2). Here the last inequality of above is given since $\\{\lambda
B_{i_{x_{k}}}\\}_{k=1}^{\infty}$ are pairwise disjoint in $c_{2}\cdot
A_{\lambda 2^{j+1}}$. The claim follows. ∎

###### Proof of Theorem 1.1.

Let
$A:=\left\\{\gamma\in\Gamma^{\infty}:\int_{\gamma}\rho_{u}ds=\infty\right\\}.$
We then obtain from [HKST15, Lemma 5.2.8] that $\text{\rm Mod}_{p}(A)=0$ since
$u\in\dot{N}^{1,p}(X)$, and hence $\lim_{t\to\infty}u(\gamma(t))\text{\rm\ \
exists for all $\gamma\in\Gamma^{\infty}\setminus A$}.$ It remains to show the
uniqueness of $c$ in (1.1) for $p$-a.e $\gamma\in\Gamma^{\infty}$. We argue by
contradiction. By adding a suitable constant to $u$ and finally by multiplying
$u$ by another suitable constant, we may assume that there exist two
subfamilies of $\Gamma^{\infty}$, denoted $\Gamma_{E},\Gamma_{F}$, and
$\delta>0$ such that

(3.5) $\text{\rm Mod}_{p}(\Gamma_{E})\geq\delta>0,\text{\rm
Mod}_{p}(\Gamma_{F})\geq\delta>0,\lim_{t\to\infty}u(\gamma(t))\geq
2,\lim_{t\to\infty}u(\gamma^{\prime}(t))\leq 0\text{\rm\ \ for all
$\gamma\in\Gamma_{E},\gamma^{\prime}\in\Gamma_{F}$.}$

We assume that each curve $\gamma$ in these two curve families is
parameterized by arc length. Set $\Gamma^{j}_{E}=\\{\gamma\in\Gamma_{E}:\
u(\gamma(t))\geq\frac{3}{2}\text{\rm\ for all\ \ }t\geq\lambda 2^{j}\\}.$ Then
$\Gamma_{E}=\bigcup_{j}\Gamma_{E}^{j}$ and hence, by the subadditivity of the
modulus,

$\text{\rm Mod}_{p}(\Gamma_{E})\leq\sum_{j}\text{\rm
Mod}_{p}(\Gamma_{E}^{j}).$

Hence (3.5) gives the existence of $j_{E}$ and $\delta_{E}>0$ so that
$\text{\rm Mod}_{p}(\Gamma_{E}^{j_{E}})\geq\delta_{E}.$ By arguing analogously
for $F,$ we find $j_{F}$ so that $\text{\rm
Mod}_{p}(\Gamma_{F}^{j_{F}})\geq\delta_{F}>0$ and
$u(\gamma(t))\leq\frac{1}{2}$ when $\gamma\in\Gamma_{F}^{j_{F}}$ and
$t\geq\lambda 2^{j_{F}}.$ Let $j\geq\max\\{j_{E},j_{F}\\}.$ We define sets
$E_{j}$, $F_{j}$ by setting

$E_{j}:=\left\\{A_{\lambda
2^{j+1}}\bigcap\left(\bigcup_{\gamma\in\Gamma_{E}^{j_{E}}}\gamma\right)\right\\}$

and

$F_{j}:=\left\\{A_{\lambda
2^{j+1}}\bigcap\left(\bigcup_{\gamma\in\Gamma_{F}^{j_{F}}}\gamma\right)\right\\}.$

Then $u(x)\geq 3/2$ for all $x\in E_{j}$ and $u(x)\leq 1/2$ for all $x\in
F_{j}$. Moreover, $(\lambda 2^{j+1})^{p}\text{\rm
Mod}_{p}(\Gamma_{E}^{j_{E}})\leq\mu(E_{j})$ and $(\lambda
2^{j+1})^{p}\text{\rm Mod}_{p}(\Gamma_{F}^{j_{F}})\leq\mu(F_{j})$ since every
curve in these families has a subcurve of length no less than $\lambda
2^{j+1}$ in $A_{\lambda 2^{j+1}}$ and hence $\chi_{E_{j}}/(\lambda 2^{j+1})$
and $\chi_{F_{j}}/(\lambda 2^{j+1})$ are admissible functions for computing
$\text{\rm Mod}_{p}(\Gamma_{E}^{j_{E}})$ and $\text{\rm
Mod}_{p}(\Gamma_{F}^{j_{F}}),$ respectively.

Notice that
$1\leq|u(x)-u(y)|\leq|u(x)-u_{B(x,2^{j}/c_{1})}|+|u_{B(x,2^{j}/c_{1})}-u_{B(y,2^{j}/c_{1})}|+|u_{B(y,2^{j}/c_{1})}-u(y)|$
for all $x\in E_{j},y\in F_{j}$ where $c_{1}$ is as in Definition 2.3. We will
consider three cases corresponding to Lemma 3.1-3.2. Applying Lemma 3.1 for
the pair $(E_{j},F_{j})$, the estimate (3.1) with $1\leq|u(x)-u(y)|$ for all
$x\in E_{j},y\in F_{j}$ gives

$\mu(c_{2}\cdot A_{\lambda 2^{j+1}})\lesssim(2^{j})^{p}\int_{c_{2}\cdot
A_{\lambda 2^{j+1}}}\rho_{u}^{p}d\mu$

if $|u(x)-u_{B(x,2^{j}/c_{1})}|\leq\frac{1}{5}$ and
$|u(y)-u_{B(y,2^{j}/c_{1})}|\leq\frac{1}{5}$ hold for some $x\in E_{j},y\in
F_{j}$. Applying Lemma 3.2 for each $E_{j},F_{j}$, the estimate (3.2) gives

$\min\\{\mu(E_{j}),\mu(F_{j})\\}\lesssim(2^{j})^{p}\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu$

if either $|u(x)-u_{B(x,2^{j}/c_{1})}|>\frac{1}{5}$ holds for every $x\in
E_{j}$ or $|u(y)-u_{B(y,2^{j}/c_{1})}|>\frac{1}{5}$ holds for every $y\in
F_{j}$. Since both $E_{j}$ and $F_{j}$ are subsets of $c_{2}\cdot A_{\lambda
2^{j+1}}$, the above estimates imply that

$\min\\{\mu(E_{j}),\mu(F_{j})\\}\lesssim(2^{j})^{p}\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu.$

From our upper estimates on $\text{\rm Mod}_{p}(\Gamma_{E}^{j_{E}})$ and on
$\text{\rm Mod}_{p}(\Gamma_{F}^{j_{F}})$ from above we conclude that

$\min\\{\text{\rm Mod}_{p}(\Gamma_{E}^{j_{E}}),\text{\rm
Mod}_{p}(\Gamma_{F}^{j_{F}})\\}\lesssim\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu.$

By inserting the strictly positive lower bounds for these two modulus, we
conclude that

$0<\min\\{\delta_{E},\delta_{F}\\}\lesssim\int_{c_{2}\cdot A_{\lambda
2^{j+1}}}\rho_{u}^{p}d\mu.$

Since $u\in\dot{N}^{1,p}(X),$ a contradiction follows by letting $j$ tend to
infinity. ∎

##  4\. Proofs of Theorems 1.2-1.3

We begin with a sequence of auxiliary lemmas whose proofs rely on arguments
similar to those in [EKN22, KNZ22, Ngu22]. Fix $1\leq p<\infty$.

######  Lemma 4.1.

Let $(X,d,\mu)$ be a doubling metric measure space that supports a
$p$-Poincaré inequality. Assume that $X$ has the annular chain property at O.
Suppose that $X$ has a weak polar coordinate system at $O$ as in (1.2). If
$R_{p}(h,O)<\infty$, then the existence and uniqueness of (1.1) hold for
$\sigma$-a.e $\xi\in\mathbb{S}$.

###### Proof.

Let $u\in\dot{N}^{1,p}(X)$. By Theorem 1.1, there exists $c\in\mathbb{R}$ such
that

(4.1) $\lim_{t\to\infty}u(\gamma(t))\text{\rm\ \ exists and \ \
}\lim_{t\to\infty}u(\gamma(t))=c\text{\rm\ \ for $p$-a.e
$\gamma\in\Gamma^{+\infty}$.}$

Let $F$ be the collection of all $\xi\in\mathbb{S}$ such that
$\lim_{t\to\infty}u(\gamma_{\xi}^{O})$ does not exist or exists but is not
equal to $c$. It suffices to prove that

(4.2) $\sigma(F)=0.$

We first set $F_{\rm bad}:=\\{\xi\in
F:\liminf_{t\to\infty}d(O,\gamma_{\xi}^{O})\leq 1\\}$. Let
$\gamma_{\xi}^{O}\in F_{\rm bad}$. By the definition of infinite curves, we
have $\gamma_{\xi}^{O}\setminus B(O,r)\neq\emptyset$ for all $r>0$ and hence
there is a sequence $\\{t_{n}\\}_{n\in\mathbb{N}}$ such that
$d(O,\gamma_{\xi}^{O}(t_{n}))\geq n$. From
$\liminf_{t\to\infty}d(O,\gamma_{\xi}^{O})\leq 1$, it follows that there is
also (after passing to a subsequence) a sequence
$\\{s_{n}\\}_{n\in\mathbb{N}}$ with $\lim_{n\to\infty}s_{n}=\infty$ such that
$d(O,\gamma_{\xi}^{O}(s_{n}))\leq 1$ and $s_{n}\in(t_{n},t_{n+1})$ for all
$n\in\mathbb{N}$. Roughly speaking, $\gamma_{\xi}^{O}$ oscillates infinitely
often between the unit ball and far away parts of our space. Therefore, it is
clear that

$\int_{\gamma_{\xi}^{O}}\chi_{B(O,2)\setminus B(O,1)}ds=\infty$

for each $\gamma_{\xi}^{O}\in F_{\rm bad}$. Also, $h(x)>0$ almost everywhere
in the complement of $B(O,1)$ because $R_{p}(h,O)<\infty.$ We then obtain that
for all $m\in\mathbb{N}$,

$\displaystyle m\ \sigma(F_{\rm bad})\leq$ $\displaystyle\int_{F_{\rm
bad}}\int_{\gamma_{\xi}^{O}}\chi_{B(O,2)\setminus B(O,1)}dsd\sigma(\xi)$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{S}}\int_{\gamma_{\xi}^{O}}\chi_{B(O,2)\setminus
B(O,1)}dsd\sigma(\xi)$ $\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{X}\frac{\chi_{B(O,2)\setminus B(O,1)}}{h}d\mu$
$\displaystyle\leq$ $\displaystyle\mathcal{C}[\mu(B(O,2)\setminus
B(O,1))]^{1/p}\left(\int_{B(O,2)\setminus
B(O,1)}h^{\frac{p}{1-p}}d\mu\right)^{\frac{p-1}{p}}$ (4.3) $\displaystyle\leq$
$\displaystyle\mathcal{C}[\mu(B(O,2)\setminus
B(O,1))]^{1/p}R_{p}^{\frac{p-1}{p}}(h,O)$

if $p>1$, and similarly

(4.4) $m\ \sigma(F_{\rm bad})\leq\mathcal{C}\mu(B(O,2)\setminus B(O,1))\
R_{1}(h,O)$

by (1.2) and the Hölder inequality, and where $\mathcal{C}>0$ is the constant
of our weak polar coordinate system (1.2). Here the weak polar coordinate
system (1.2) can be applied since $\frac{\chi_{B(O,2)\setminus B(O,1)}}{h}$ is
integrable. Letting $m\to\infty$, the above estimates give

(4.5) $\sigma(F_{\rm bad})=0$

for $1\leq p<\infty$ since the right-hand side of (4.3)-(4.4) is bounded.

Let $\gamma_{\xi}^{O}\in F\setminus F_{\rm bad}$. Then
$\liminf_{t\to\infty}d(O,\gamma_{\xi}^{O}(t))>1$, and so there is $t_{\xi}>0$
such that $d(O,\gamma_{\xi}^{O}(t_{\xi}))=1$ and $\gamma^{O}_{\xi}(t)\cap
B(O,1)=\emptyset$ for all $t>t_{\xi}$. Let us define $\hat{\gamma}_{\xi}^{O}$,
a subcurve of $\gamma_{\xi}^{O},$ by setting
$\hat{\gamma}_{\xi}^{O}(t)=\gamma_{\xi}^{O}(t+t_{\xi})$ for all $t\geq 0$. Let
$\hat{\Gamma}^{O}(F)$ be the collection of all these infinite subcurves
$\hat{\gamma}_{\xi}^{O}$ of $\gamma_{\xi}^{O}$ with respect to $\xi\in F$
satisfying $\liminf_{t\to\infty}d(O,\gamma_{\xi}^{O}(t))>1$. Then, for all
$\hat{\gamma}_{\xi}^{O}\in\hat{\Gamma}^{O}(F)$,

(4.6) $\hat{\gamma}_{\xi}^{O}\cap B(O,1)=\emptyset$

and

$\lim_{t\to\infty}u(\gamma_{\xi}^{O}(t))\text{\rm\ \ does not exist or is not
equal to $c$}$

since $\xi\in F$. By (4.1), we also have

(4.7) $\text{\rm Mod}_{p}(\hat{\Gamma}^{O}(F))=0.$

Let $g$ be admissible for computing $\text{\rm Mod}_{p}(\hat{\Gamma}^{O}(F))$.
Then we may assume that $g=0$ on $B(O,1)$ since $\hat{\gamma}_{\xi}^{O}\cap
B(O,1)=\emptyset$ by (4.6). Suppose first that $p>1.$ By the same arguments as
for $\eqref{eq4.3-1205}$-$\eqref{eq4.4-1205}$, we obtain that

$\sigma(F\setminus F_{\rm
bad})\leq\mathcal{C}\left(\int_{X}g^{p}d\mu\right)^{1/p}R_{p}^{\frac{p-1}{p}}(h,O).$

Since $g$ is arbitrary, it follows that

$\sigma(F\setminus F_{\rm bad})\leq\mathcal{C}\left(\text{\rm
Mod}_{p}(\hat{\Gamma}^{O}(F))\right)^{1/p}R_{p}^{\frac{p-1}{p}}(h,O).$

Combining with (4.5), we conclude that

(4.8) $\sigma(F)\leq\mathcal{C}\left(\text{\rm
Mod}_{p}(\hat{\Gamma}^{O}(F))\right)^{1/p}R_{p}^{\frac{p-1}{p}}(h,O).$

As $R_{p}(h,O)<\infty$ and $\text{\rm Mod}_{p}(\hat{\Gamma}^{O}(F))=0$ by
(4.7), we conclude that $\eqref{eq4.2-1205}$ holds. The case $p=1$ follows via
an analogous argument. ∎

######  Lemma 4.2.

If $\mathcal{R}_{p}(O)=\infty$, then there is $u\in\dot{N}^{1,p}(X)$ such that
$\liminf_{t\to\infty}u(\gamma(t))=\infty$ for every
$\gamma\in\Gamma^{\infty}$.

###### Proof.

By $\mathcal{R}_{p}(O)=\infty$, there exists a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ (or $\\{i_{k}\\}_{k=1}^{\infty}$) with

(4.9) $n_{1}<n_{2}<\ldots$ (or $i_{1}<i_{2}<\ldots$) such that
$\lim_{k\to\infty}n_{k}=\infty$ (or $\lim_{k\to\infty}i_{k}=\infty$),

(4.10)
$\sum_{i=n_{k}}^{n_{k+1}}(2^{i})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})>2^{k}\text{\rm\
\ \, if $p>1$ $\big{(}$or \ \ \ }2^{i_{k}}\mu^{-1}(A_{2^{i_{k}}})>2^{k}\text{\
\ \ \rm if $p=1\big{)}$}.$

Here $A_{2^{i}}:=B(O,2^{i+1})\setminus B(O,2^{i})$. Let

$g_{p}(x)=\sum_{k=1}^{\infty}\left(\sum_{i=n_{k}}^{{n_{k+1}}}\frac{(2^{i})^{\frac{1}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}{\sum_{i=n_{k}}^{n_{k+1}}(2^{i})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}\chi_{A_{2^{i}}}(x)\right)\text{\rm\
if $p>1$\ }\big{(}{\rm or\ \
}g_{1}(x)=\sum_{k=1}^{\infty}2^{-i_{k}}\chi_{A_{2^{i_{k}}}}(x)\big{)}.$

We define $u(x):=\inf\int_{\gamma_{O,x}}g_{p}ds$ for $x\in X$ where the
infimum is taken over all rectifiable curves $\gamma_{O,x}$ connecting $O$ and
$x$. Then $g_{p}$ is an upper gradient of $u$, see for instance [HKST15, Page
188-189]. We have the fact (see for instance [HKST15, Proposition 5.1.11])
that $\int_{\gamma_{O,x}\cap A_{2^{i}}}ds\geq\text{\rm diam}(\gamma_{O,x}\cap
A_{2^{i}})\gtrsim 2^{i}$ for every $\gamma_{O,x}$ with $d(O,x)\geq 2^{i+1}$.
Here $\text{\rm diam}(\gamma_{O,x}\cap A_{2^{i}})$ is the diameter of
$\gamma_{O,x}\cap A_{2^{i}}$. We let $\gamma\in\Gamma^{\infty}$ and let $N>1$.
For all $x\in\gamma$ with $d(O,x)=N$, we have that

$\displaystyle u(x)=$
$\displaystyle\inf_{\gamma_{O,x}}\int_{\gamma_{O,x}}g_{p}ds=\inf_{\gamma_{O,x}}\sum_{2^{n_{k+1}}\leq
N}\sum_{i=n_{k}}^{n_{k+1}}\frac{(2^{i})^{\frac{1}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}{\sum_{i=n_{k}}^{n_{k+1}}(2^{i})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}\int_{\gamma_{O,x}\cap
A_{2^{i}}}ds$ $\displaystyle\gtrsim$ $\displaystyle\sum_{2^{n_{k+1}}\leq
N}\sum_{i=n_{k}}^{n_{k+1}}\frac{(2^{i})^{\frac{1}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}{\sum_{i=n_{k}}^{n_{k+1}}(2^{i})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}2^{i}=\sum_{2^{n_{k+1}}\leq
N}1\to\infty\text{\rm\ \ as $N\to\infty$\ \ \ if $p>1$,}$

(or that
$u(x)=\inf_{\gamma_{O,x}}\int_{\gamma_{O,x}}g_{1}ds=\inf_{\gamma_{O,x}}\sum_{2^{i_{k}}\leq
N}2^{-i_{k}}\int_{\gamma_{O,x}\cap A_{2^{i_{k}}}}ds\gtrsim\sum_{2^{i_{k}}\leq
N}2^{-i_{k}}2^{i_{k}}=\sum_{2^{i_{k}}\leq N}1\to\infty$ as $N\to\infty$).
Hence $\liminf_{t\to\infty}u(\gamma(t))=\infty$ for every
$\gamma\in\Gamma^{\infty}$. It remains to show that $g_{p}$ is $p$-integrable.
Using (4.9)-(4.10), we have that

$\displaystyle\int_{X}g_{p}^{p}d\mu=\sum_{k=1}^{\infty}\sum_{i=n_{k}}^{{n_{k+1}}}\int_{A_{2^{i}}}\left(\frac{(2^{i})^{\frac{1}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}{\sum_{i=n_{k}}^{n_{k+1}}(2^{i})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})}\right)^{p}d\mu=\sum_{k=1}^{\infty}\frac{1}{\left(\sum_{i=n_{k}}^{n_{k+1}}(2^{i})^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^{i}})\right)^{p-1}}\leq\sum_{k=1}^{\infty}\frac{1}{2^{k(p-1)}}$

if $p>1$ (or that
$\int_{X}g_{1}d\mu=\sum_{k=1}^{\infty}\int_{A_{2^{i_{k}}}}2^{-i_{k}}d\mu=\sum_{k=1}2^{-i_{k}}\mu(A_{2^{i_{k}}})\leq\sum_{k=1}^{\infty}\frac{1}{2^{k}}).$
The claim follows. ∎

######  Example 4.3.

If $h(x)=|x-O|^{n-1}w(x)\chi_{\mathbb{R}^{n}\setminus B(O,1)}(x)$, where $w$
is a classical Muckenhoupt $\mathcal{A}_{p}$-weight on $\mathbb{R}^{n}$ and
where $n\geq 2$, then $R_{p}(h,O)\approx\mathcal{R}_{p}(O)$.

###### Proof.

Let $A_{2^{i}}=B(O,2^{i+1})\setminus B(O,2^{i})$. Notice that

$R_{p}(h,O)=\sum_{i=0}^{\infty}\int_{A_{2^{i}}}|x-O|^{\frac{p(n-1)}{1-p}}w^{\frac{1}{1-p}}(x)dx\text{
if $p>1$ $\big{(}$ or\
}R_{1}(h,O)=\sup_{i\in\mathbb{N}}\left\||x-O|^{1-n}w^{-1}(x)\right\|_{L^{\infty}(A_{2^{i}})}\big{)}.$

Since $|x-O|\approx 2^{i}$ for any $x\in A_{2^{i}}$ and
$|A_{2^{i}}|\approx(2^{i})^{n}$, it follows that

$R_{p}(h,O)\approx\sum_{i=0}^{\infty}(2^{i})^{\frac{p(n-1)}{1-p}+n}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{A_{2^{i}}}w^{\frac{1}{1-p}}(x)dx\text{\ \ \rm if
$p>1$ $\big{(}$ or \
}R_{1}(h,O)\approx\sup_{i\in\mathbb{N}}\left(2^{i})^{1-n}\|w^{-1}(x)\right\|_{L^{\infty}(A_{2^{i}})}\text{\
\ \rm if $p=1$}\big{)}.$

Because $w$ is a Muckenhoupt $\mathcal{A}_{p}$-weight, we have from
(1.3)-(1.4) that

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{A_{2^{i}}}w^{\frac{1}{1-p}}(x)dx\right)\approx\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{A_{2^{i}}}w(x)dx\right)^{\frac{1}{1-p}}\approx(2^{i})^{\frac{-n}{1-p}}\mu^{\frac{1}{1-p}}(A_{2^{i}})\text{
if }p>1,$

$\big{(}$or that
$\left\|w^{-1}(x)\right\|_{L^{\infty}(A_{2^{i}})}\approx\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{A_{2^{i}}}w(x)dx\right)^{-1}\approx(2^{i})^{n}\mu^{-1}(A_{2^{i}})\big{)}.$
Inserting these into the above formula of $R_{p}(h,O)$, we obtain the claim. ∎

######  Lemma 4.4.

Let $(X,d,\mu)$ be a doubling metric measure space that supports a
$p$-Poincaré inequality. Assume that $X$ is complete. If
$\mathcal{R}_{p}(O)<\infty$ then $\text{\rm Mod}_{p}(\Gamma^{\infty})>0$.

###### Proof.

Since $X$ is complete and doubling, and supports a $p$-Poincaré inequality,
there exists a geodesic metric $\hat{d}$ that is biLipschitz equivalent to
$d$, see for instance in [HKST15, Corollary 8.3.16]. Let
$\mathcal{R}_{p}(\hat{d},O)$ be the version of $\mathcal{R}_{p}(O)$ in
$(X,\hat{d},\mu)$ and $B_{\hat{d}}(x,r)$ be the ball with radius $r$ and
center at $x$ on $(X,\hat{d})$. It follows that
$\mathcal{R}_{p}(\hat{d},O)<\infty$ since $\mathcal{R}_{p}(O)<\infty$. Since
$\hat{d}$ is geodesic we are allowed to employ Theorem 2.10 in [HoKo01]. This
result gives us a constant $C>0$ such that for all $i\in\mathbb{N}$

$C\leq{\rm
Cap}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))\left(\mathcal{R}_{p}(\hat{d},O)\right)^{p-1}$

when $p>1$ $\big{(}$or otherwise

$C\leq{\rm
Cap}_{1}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))\mathcal{R}_{1}(\hat{d},O)\big{)}.$

Here ${\rm Cap}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))$ is the quantity

${\rm
Cap}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i})):=\inf\int_{X}g_{u}^{p}d\mu$

where the infimum is taken over all functions $u:X\to\mathbb{R}$ with the
minimal upper gradient $g_{u}$ such that $u|_{B_{\hat{d}}(O,1)}\equiv 1$ and
$u|_{X\setminus B_{\hat{d}}(O,2^{i})}\equiv 0$. It follows from
$\mathcal{R}_{p}(\hat{d},O)<\infty$ that ${\rm
Cap}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))\geq\delta>0$ for all
$i\in\mathbb{N}$, and for some $\delta$ only depending on $C$ and
$\mathcal{R}_{p}(\hat{d},O)$. By [HeKo98, Page 12], we obtain that

${\rm Cap}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))\leq\text{\rm
Mod}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))\text{\rm\ \ for all
$i\in\mathbb{N}$.}$

Here $\text{\rm Mod}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))$ is the
$p$-modulus of the family of all retifiable curves connecting
$B_{\hat{d}}(O,1)$ and $X\setminus B_{\hat{d}}(O,2^{i})$. Then $\text{\rm
Mod}_{p}(B_{\hat{d}}(O,1),B_{\hat{d}}(O,2^{i}))\geq\delta>0\text{\rm\ \ for
all $i\in\mathbb{N}$}.$ It follows that $X$ is $p$-hyperbolic in the sense of
[Sha21, Definition 2.4]. Hence [Sha21, Theorem 4.2] yields that $\text{\rm
Mod}_{p}(\Gamma^{\infty})>0$. The claim follows. ∎

###### Proof of Theorem 1.2.

${\rm I.}\Rightarrow{\rm II.}$ is given by Lemma 4.4.

${\rm II.}\Rightarrow{\rm III.}$ is given by Theorem 1.1.

${\rm III.}\Rightarrow{\rm I.}$ is given by Lemma 4.2. ∎

###### Proof of Theorem 1.3.

$R_{p}(h,O)<\infty\Rightarrow 3.$ is given by Lemma 4.1.

$3.\Rightarrow 2.\Rightarrow 1.$ is trivial.

$1.\Rightarrow\mathcal{R}_{p}(O)<\infty$ is given by Lemma 4.2.

$R_{p}(h,O)<\infty\Rightarrow 4.$ is given by the estimate (4.8).

$R_{p}(h,O)<\infty\Rightarrow 5.$ is given by the estimate (4.8).

$4.\Rightarrow\mathcal{R}_{p}(O)<\infty$ is given by Lemma 4.2 together with
Theorem 1.1.

$5.\Rightarrow\mathcal{R}_{p}(O)<\infty$ is given by Lemma 4.2 together with
Theorem 1.1. Indeed, if $\mathcal{R}_{p}(O)=\infty$ then $\text{\rm
Mod}_{p}(\hat{\Gamma}^{O}(F))=0$ for all subsets $F\subseteq\mathbb{S}$ with
$\sigma(F)>0$. ∎

## Acknowledgements

We thank the anonymous referee for a careful reading and thoughtful comments
on the paper. Both authors have been supported by the Academy of Finland Grant
number 323960.

## References

  * [Ty20] J. Tyson, _Polar coordinates in Carnot groups II_ , arXiv:2208.04913.
  * [BaTy02] Z. Balogh, J. Tyson, _Polar coordinates in Carnot groups_. Math. Z. 241 (2002), no. 4, 697–730.
  * [Björn] A. Björn, J. Björn, _Nonlinear potential theory on metric spaces_. EMS Tracts in Mathematics 17. European Mathematical Society (EMS), Zürich, 2011, xii+403.
  * [BBGS17] A. Björn, J. Björn, J. T. Gill, N. Shanmugalingam, _Geometric analysis on Cantor sets and trees._ J. Reine Angew. Math. 725 (2017), 63–114.
  * [BBS20] A. Björn, J. Björn, N. Shanmugalingam, _Locally p-admissible measures on $\mathbb{R}$._ J. Funct. Anal. 278 (2020), no. 4, 108344, 17 pp.
  * [BBK06] J. Björn, S. Buckley, S. Keith, _Admissible measures in one dimension._ Proc. Amer. Math. Soc. 134 (2006) 703–705.
  * [EKN22] S. Eriksson-Bique, P. Koskela, K. Nguyen, _On limits at infinity of weighted Sobolev functions._ J. Funct. Anal. 283 (2022), no. 10, 40 Pages.
  * [Gri99] A. Grigor’yan, _Analytic and geometric background of recurrence and non- explosion of the Brownian motion on Riemannian manifolds._ Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135-249.
  * [Haj03] P. Hajłasz, _Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces_. (Paris, 2002), 173-218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.
  * [HaKo95] P. Hajłasz, P. Koskela, _Sobolev meets Poincaré._ C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 10, 1211–1215.
  * [HaKo00] P. Hajłasz, P. Koskela, _Sobolev met Poincaré._ Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101 pp.
  * [HeKo98] J. Heinonen, P. Koskela, _Quasiconformal maps in metric spaces with controlled geometry._ Acta Math. 181 (1998), no. 1, 1–61.
  * [HKM06] J. Heinonen, T. Kilpeläinen, O. Martio, _Nonlinear potential theory of degenerate elliptic equations._ Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, NY, 2006. xii+404 pp.
  * [HKST15] J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson, _Sobolev spaces on metric measure spaces. An approach based on upper gradients_. New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015. xii+434 pp.
  * [Hol99] I. Holopainen _Volume growth, Green’s functions, and parabolicity of ends._ Duke Math. J. 97 (1999), no. 2, 319-346.
  * [HoKo01] I. Holopainen, P. Koskela, _Volume growth and parabolicity._ Proc. Amer. Math. Soc. 129 (2001), no. 11, 3425–3435.
  * [KM98] P. Koskela, P. MacManus, _Quasiconformal mappings and Sobolev spaces._ Studia Math. 131 (1998), no. 1, 1–17.
  * [KNZ22] P. Koskela, K. Nguyen, Z. Wang, _Trace and density results on regular trees_. Potential Anal. 57 (2022), no. 1, 101–128.
  * [Ngu22] K. Nguyen, _Classification criteria for regular trees_. Ann. Fenn. Math. 47(2022), 3–21.
  * [NgWa20] K. Nguyen, Z. Wang, _Admissibility versus Ap-conditions on regular trees._ Anal. Geom. Metr. Spaces 8 (2020), no. 1, 92–105.
  * [Sem96] S. Semmes, _Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities._ Selecta Math. (N.S.) 2 (1996), no. 2, 155–295.
  * [Sha00] N. Shanmugalingam, _Newtonian spaces: an extension of Sobolev spaces to metric measure spaces._ Rev. Mat. Iberoamericana 16 (2000), no. 2, 243-279.
  * [Sha21] N. Shanmugalingam, _$p$ -hyperbolicity of ends and families of paths in metric spaces._ Fractal geometry and stochastics VI, 191–205, Progr. Probab., 76, Birkhäuser/Springer, Cham, 2021.
  * [Usp61] S. V. Uspenskǐǐ, _Imbedding theorems for classes with weights_. Trudy Math. Inst. Stklov. 60 (1961), 282-303.


We test the effectiveness of using pre-trained embeddings as teachers on two
different tasks, datasets, and models with four different pre-trained
embeddings as follows.

##  0.1 Tasks, Datasets, Models, and Metrics

2*OpenMIC | None | VGGish | OpenL3 | PaSST | PANNs
---|---|---|---|---|---
| mAP | F1 | mAP | F1 | mAP | F1 | mAP | F1 | mAP | F1
CP ResNet* [koutini2021receptive] | .819 | .809 | - | - | - | - | - | - | - | -
SS CP ResNet* [koutini2021receptive] | .831 | .822 | - | - | - | - | - | - | - | -
TeacherLR | - | - | .803 | .799 | .803 | .798 | .858 | .837 | .853 | .834
KD (w/ mask) ** | - | - | .829 | .820 | .823 | .813 | .851 | .834 | .848 | .823
EAsTCos-Diff | - | - | .838 | .824 | .838 | .820 | .837 | .822 | .836 | .814
EAsTFinal | - | - | .842 | .828 | .835 | .822 | .847 | .830 | .849 | .828
EAsTAll | - | - | .836 | .823 | .835 | .822 | .845 | .827 | .845 | .827
EAsTKD | - | - | .836 | .825 | .836 | .821 | .852 | .834 | .857 | .831
|  |  |  |  |  |  |  |  |  |
2*MagnaTagATune | None | VGGish | OpenL3 | PaSST | PANNs
| mAP | AUC | mAP | AUC | mAP | AUC | mAP | AUC | mAP | AUC
FCN† [choi2016automatic] | .429 | .900 | - | - | - | - | - | - | - | -
Mobile FCN | .437 | .905 | - | - | - | - | - | - | - | -
TeacherLR | - | - | .433 | .903 | .403 | .890 | .473 | .917 | .460 | .911
KD | - | - | .447 | .911 | .439 | .907 | .454 | .912 | .448 | .909
EAsTCos-Diff | - | - | .446 | .906 | .438 | .907 | .453 | .912 | .453 | .911
EAsTFinal | - | - | .454 | .912 | .447 | .910 | .459 | .912 | .449 | .909
EAsTAll | - | - | .455 | .911 | .452 | .911 | .458 | .913 | .457 | .911
EAsTKD | - | - | .441 | .908 | .437 | .904 | .461 | .915 | .459 | .912
Table 1: Results on OpenMIC (above) and MagnaTagATune (below) dataset for
different models regularized with different pre-trained embeddings. Best
performances are in bold, and best results excluding the teachers are
underlined. *Reported results [koutini2021receptive], SS means being trained
with shake-shake regularization [gastaldi2017shake]. **When using KD, the
missing labels in OpenMIC were masked to avoid potentially adding more
training data. †Results from the open-source re-implementation [won2020eval].

###  0.1.1 Musical Instrument Classification with OpenMIC

Musical instrument classification is a multi-label classification problem. We
use the OpenMIC dataset [humphrey2018openmic], which provides weakly labeled
audio snippets of length [10]s. Following prior work [gururani2019attention,
chen2023music], we use the suggested test set and randomly split 15% of the
training data as the validation set, resulting in 12,692 training
observations, 2,223 validation observations, and 5085 test observations. To
ensure a consistent sample rate, the audio is resampled to [32]kHz
[koutini2021receptive, chen2023music]. As the dataset is not completely
labeled, i.e., parts of the labels are missing and not labeled as positive or
negative, the missing labels are masked out when computing the loss function
as suggested in previous work [chen2023music, koutini2021receptive,
koutini2022efficient].

We use receptive field regularized ResNet (CP-ResNet) [koutini2021receptive]
for this task, as it reaches state-of-the-art performance when trained only on
the OpenMIC dataset (i.e., neither trained with transfer learning nor trained
with any knowledge distillation). CP-ResNet has a ResNet-like structure
[he2016deep] with an added hyper-parameter $\rho$ to control the maximum
receptive field of the ResNet. We set $\rho=7$ to match the setting which
provides the best results in the original work [koutini2021receptive].

The results are reported with the metrics mean Average Precision (mAP) and
F1-score. The F1-score is calculated in a macro fashion, which means that for
each instrument, the F1-score is computed for both the positive labels and the
negative labels and then averaged, and the final F1-score is the mean of the
F1-scores of all instruments. The detection threshold for the prediction is
set to 0.4 following previous work [koutini2021receptive].

###  0.1.2 Music Auto-Tagging with MagnaTagATune

Similar to musical instrument classification, music auto-tagging is also a
multi-label classification problem. We use the MagnaTagATune dataset
[law2009evaluation] for this task, which comes with audio clips of
approximately [29.1]s. Following previous work, we use only the top 50 labels
and exclude all the songs without any positive label from the dataset
[won2019toward, kim2018sample]. For comparability, the data split is adopted
from previous work, with audio files in the directories ’0’ to ’b’ being the
training set, ’c’ being the validation set, and ’e’ and ’f’ being the test set
[won2020data, won2020eval], resulting in 15,247 training clips, 1,529
validation clips, and 4,332 test clips.

We apply a modified fully convolutional neural network (FCN)
[choi2016automatic] to this task. It is the simplest model among the benchmark
models for the MagnaTagATune dataset [won2020eval] and consists of several
convolution and max-pooling layers. To further reduce the complexity of the
model, we apply the MobileNet-like modification [howard2017mobilenets] to the
network by breaking the $3\times 3$ convolutions into depth-wise separable
convolutions and $1\times 1$ convolutions.

The results are evaluated with mAP and ROC-AUC.

##  0.2 Pre-trained Embeddings

###  0.2.1 VGGish

VGGish [hershey2017cnn] is a widely used embedding in MIR, with a VGG network
[simonyan2014very] being trained on a large number of Youtube videos. The
open-source PyTorch implementation is used to extract VGGish
features111https://github.com/harritaylor/torchvggish. Last accessed on April
4, 2023. which by default extracts 128 principle components and then quantizes
them to [8]bit. The time resolution is [960]ms.

###  0.2.2 OpenL3

The OpenL3 embedding [cramer2019look, arandjelovic2017look] is trained on a
music subset of AudioSet [gemmeke2017audio] in a self-supervised paradigm. The
audio embeddings are extracted using the open-source Python package
OpenL3222https://github.com/marl/openl3/tree/main. Last accessed on April 4,
2023 with the dimensionality being 512. To keep consistent with VGGish, the
time resolution is set to [960]ms.

###  0.2.3 PaSST

PaSST [koutini2022efficient] is a 7-layer transformer trained on AudioSet for
acoustic event detection. It applies the structure of a vision transformer
[dosovitskiy2021an, touvron2021training] and proposes the technique of
Patchout to make the training efficient. We use the open-source
code333https://github.com/kkoutini/PaSST/tree/main. Last accessed on April 4,
2023. released by the authors to extract the 768-dimensional embeddings. The
time resolution is also set to [960]ms.

###  0.2.4 PANNs

PANNs [kong2020panns] include several convolutional neural networks and are
also trained on AudioSet for acoustic event detection. We use the default
CNN14 model from the official
repository444https://github.com/qiuqiangkong/audioset_tagging_cnn. Last
accessed on April 4, 2023. . The embedding dimensionality is 2048. Different
from other embeddings, PANNs provide only one global embedding for each clip
of audio. Pilot experiments have shown that extracting the embeddings for
short segments and concatenating them does not improve performance.

##  0.3 Systems Overview

The following systems are evaluated for comparison: 1em1em1em1em1em1em

Baseline: CP ResNet (on OpenMIC) and Mobile FCN (on MagnaTagATune) trained
without any extra regularization loss.

TeacherLR: logistic regression on the pre-trained embeddings (averaged along
the time axis), which can be seen as one way to do transfer learning by
freezing the whole model except for the classification head.

KD: classical knowledge distillation where the soft targets are generated by
the logistic regression.

EAsTCos-Diff (for Embeddings-As-Teachers): feature space regularization as
proposed by Hung and Lerch that uses cosine distance difference and
regularizes only the final feature map [hung2022feature_classification].

EAsTFinal and EAsTAll: proposed systems based on distance correlation as the
distance measure, either regularizing only at the final stage or at all
stages, respectively.

EAsTKD: a combination of classical knowledge distillation and our method of
using embeddings to regularize the feature space. The feature space
regularization is done only at the final stage. We perform a search of
$\lambda$ for each of the EasT systems and choose the best-performing value on
the validation set.555For all the hyperparameters, please refer to the config
files in our GitHub.