# Uniformly rectifiable metric spaces satisfy the weak constant density
condition

Jared Krandel

###### Abstract.

The weak constant density condition is a quantitative regularity property
originally investigated by David and Semmes in their foundational work on
uniform rectifiability. Roughly speaking, a space satisfies this condition if
in most balls, the space supports a measure with nearly constant density in a
neighborhood of scales and locations. In this paper, we prove that uniformly
rectifiable metric spaces satisfy the weak constant density condition,
extending a theorem of David and Semmes. In order to do this, we first prove a
slightly strengthened version of a separate theorem of David and Semmes which
weakly controls the oscillation of the means of $L^{2}$ functions.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Oscillation of means of $L^{2}$ functions
  4. 4 Bi-Lipschitz images satisfy the weak constant density condition
  5. 5 Stability of the weak constant density condition under big pieces

##  1\. Introduction

The relation between the rectifiability properties and density properties of
sets and measures has long been a topic of interest in geometric measure
theory. For a metric space $X$, we say that $E\subseteq X$ with
$\mathscr{H}^{n}(E)<\infty$ is $n$-rectifiable if $E$ can be covered
$\mathscr{H}^{n}$ almost everywhere by a countable union of Lipschitz images
of subsets of $\mathbb{R}^{n}$. In Euclidean space, rectifiable sets give a
natural generalization of $C^{1}$ manifolds. If $E\subseteq X$ is
$n$-rectifiable, then for $\mathscr{H}^{n}$ almost every $x\in E$,

(1.1) $\lim_{r\rightarrow 0}\frac{\mathscr{H}^{n}(B(x,r)\cap E)}{(2r)^{n}}=1$

as holds for an $n$-dimensional submanifold of Euclidean space (See [Kir94]
Theorem 9 for a proof of (1.1)). In fact, Preiss showed that if
$E\subseteq\mathbb{R}^{d}$, then the above condition also implies
$n$-rectifiability and hence gives a characterization of rectifiability
equivalent to coverings by Lipschitz images [Pre87].

The weak constant density condition (WCD) is one of several conditions meant
to provide an analog of (1.1) in the world of uniform rectifiability pioneered
by David and Semmes in their foundational works [DS91] and [DS93].

######  Definition 1.1 (uniform $n$-rectifiability).

We say that a set $E\subseteq\mathbb{R}^{d}$ is uniformly $n$-rectifiable if
there exists a constant $C_{0}>0$ such that $E$ is Ahlfors
$(C_{0},n)$-regular, i.e., for all $x\in E$ and
$0<r<\mathop{\mathrm{missing}}{diam}(E)$,

(1.2) $C_{0}^{-1}r^{n}\leq\mathscr{H}^{n}(B(x,r)\cap E)\leq C_{0}r^{n},$

and $E$ has Big Pieces of Lipschitz images of $\mathbb{R}^{n}$ (BPLI), i.e.,
there exist constants $L,\theta>0$ such that for all $x\in E$ and
$0<r<\mathop{\mathrm{missing}}{diam}(E)$, there exists an $L$-Lipschitz map
$f:B(0,r)\subseteq\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$ such that

(1.3) $\mathscr{H}^{n}(B(x,r)\cap E\cap f(B(0,r)))\geq\theta r^{n}.$

One can think of this as a stronger form of $n$-rectifiability in which one
requires a uniform percentage of the measure of each ball to be covered by a
Lipschitz image. David and Semmes introduced the WCD as a way of quantifying
(1.1) by requiring that in almost every ball, there exists a measure supported
on the set with nearly constant density nearby.

######  Definition 1.2 (Weak constant density condition, Carleson sets and
measures).

Let $E\subseteq\mathbb{R}^{d}$ be Ahlfors $n$-regular, let
$C_{0},\epsilon_{0}>0$, and define

(1.7) $\displaystyle\mathscr{G}_{\text{cd}}(C_{0},\epsilon_{0})$
$\displaystyle=\Set{(x,r)\in
E\times\mathbb{R}^{+}}{\begin{array}[]{l}\exists\text{ Ahlfors
$(C_{0},n)$-regular $\mu$},\ \mathrm{spt}(\mu)=E,\\\ \forall y\in B(x,r),\
0<t\leq r,\\\ |\mu(E\cap B(y,t))-t^{n}|\leq\epsilon_{0}r^{n}\end{array}},$
(1.8) $\displaystyle\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$
$\displaystyle=E\times\mathbb{R}^{+}\setminus\mathscr{G}_{\text{cd}}(C_{0},\epsilon_{0}).$

We say that $E$ satisfies the weak constant density condition if there exists
$C_{0}>0$ such that for all $\epsilon_{0}>0$,
$\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$ is a Carleson set. That is,
there exists a constant $C_{1}>0$ such that for all $z\in E$ and
$0<r<\mathop{\mathrm{missing}}{diam}(E)$,

$\int_{B(z,r)}\int_{0}^{r}\chi_{\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})}(x,t)d\mathscr{H}^{n}(x)\frac{dt}{t}\leq
C_{1}r^{n}.$

If this holds, we say that
$\chi_{\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})}d\mathscr{H}^{n}(x)\frac{dt}{t}$
is a Carleson measure and say that
$\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$ is $C_{1}$-Carleson.

For related quantitative conditions involving densities, see [CGLT16], [AH22],
and [TT15]. The work of David, Semmes, and Tolsa combine to prove the
following Theorem:

######  Theorem 1.3.

Let $E\subseteq\mathbb{R}^{d}$ be Ahlfors $n$-regular. Then $E$ is uniformly
$n$-rectifiable if and only if $E$ satisfies the WCD.

David and Semmes proved the forward implication in Chapter 6 of [DS91] using a
characterization of uniform rectifiability (condition C2 of [DS91]) more
closely related to the boundedness of singular integral operators. We will say
more about this when we discuss our result.

They proved the reverse implication only in the case $n=1,2,$ and $d-1$. Their
proof uses the fact that if a measure is very close to having constant density
in a large neighborhood of scales and locations, then its support is well-
approximated by the support of an $n$-uniform measure, a measure $\mu$ for
which there exists $c>0$ such that $\mu(B(x,r))=cr^{n}$ for all
$x\in\mathrm{spt}(\mu)$ and $r>0$. Because uniform measures in Euclidean space
are completely classfied for $n=1,2$ (they are all multiples of Hausdorff
measure on a plane) and for $n=d-1$ (they are Hausdorff measure on products of
planes and light cones [KP87]), David and Semmes are able to show that a WCD
set is very close to flat on most balls which are good for the WCD. The
absence of a classification for uniform measures in intermediate dimensions
prevented a direct adaptation of their arguments. However, Tolsa completed the
proof of the reverse direction in Theorem 1.3 in [Tol15] by replacing elements
of David and Semmes’s argument specific to their examples of uniform measures
with general flatness properties of uniform measures derived by Preiss [Pre87]
in addition to new arguments using the Riesz transform.

In general, classifying uniform measures is a difficult open problem, but see
[Nim22] for an interesting family of examples. For further studies of uniform
measures in Euclidean spaces see [Pre87] (and [DL08] for a more gentle
presentation of Preiss), [KP02], [Nim17], and [Nim19]. For research into
uniform measures in the Heisenberg group see [CMT20] and [Mer22] and for a
related result in $\ell_{\infty}^{3}$, see [Lor03].

Given the fact that (1.1) is valid even in rectifiable metric spaces, it is
natural to ask whether there is some extension of the above theory to
uniformly rectifiable metric spaces, i.e., metric spaces which are Ahlfors
$n$-regular and have big pieces of Lipschitz images of subsets of
$\mathbb{R}^{n}$. In this paper, we extend a piece of this theory by proving
the following theorem.

######  Theorem A.

Uniformly $n$-rectifiable metric spaces satisfy the WCD.

Our proof is made possible by the recent work of Bate, Hyde, and Schul [BHS23]
which adapted a substantial portion of the theory of uniformly rectifiable
subsets of Euclidean spaces to metric spaces. While our argument uses this
theory, it does not follow David and Semmes’s original proof very closely.
Roughly speaking, David and Semmes proved the Euclidean version of Theorem A
by showing that for most balls centered on an $n-$uniformly rectifiable set
$E$, there exists a $d$-plane $P$ such that the pushforward of Hausdorff
measure for $E$ onto $P$ must be very close to symmetric. They do this by
showing that the non-symmetric balls contribute substantially to the value of
a Carleson measure defined using integrals over $E$ of a family of smooth odd
functions designed to detect asymmetry.

We prove Theorem A by first proving that bi-Lipschitz images satisfy the WCD.
Then, using the fact that uniformly rectifiable spaces have very big pieces of
bi-Lipschitz images in Banach spaces proven by Bate, Hyde, and Schul [BHS23],
we adapt the bi-Lipschitz image arguments to the general case. To handle the
bi-Lipschitz image case, we first prove Lemma 3.5, a form of quantitative
Lebesgue differentiation theorem for $L^{2}$ functions very similar to
theorems considered by David and Semmes (see [DS93] Lemma IV.2.2.14, Corollary
IV.2.2.19, etc.), although our proof proceeds by contradiction and uses a
compactness argument, a method which differs significantly from their proofs.
We apply this lemma to the Jacobian of the metric derivative of our bi-
Lipschitz function $f:\mathbb{R}^{n}\rightarrow\Sigma$ to control the
variation of its averages over neighborhoods of scales and locations and
receive control over the variation of the Hausdorff measure of $\Sigma$ using
the area formula. To the knowledge of the author, this gives a new proof of
the WCD even in the Euclidean case.

We note here that the naive converse of Theorem 1.3 is false: There exist
Ahlfors regular metric spaces which satisfy the WCD, yet are not uniformly
rectifiable. Indeed, the metric space
$(X,d)=(\mathbb{R},|\cdot|_{\text{Euc}}^{1/2})$ is in fact $2$-uniform:
$\mathscr{H}^{2}(B(x,r))=cr^{2}$ for all $x\in\mathbb{R}$ and $r\geq 0$, hence
$X$ satisfies the weak constant density condition, yet $X$ is purely
$2$-unrectifiable. Some different examples of this failure are given by Bate
[Bat23]. He proves that every $1$-uniform metric measure space is either
$\mathbb{R}$, a particular union of disjoint circles of radius $d$, or a
purely unrectifiable “limit” of the circle spaces. These last two spaces are
examples of $1$-uniform spaces which are not uniformly rectifiable.

Analyzing connectedness plays a special role in the proof because any
$1$-uniform connected component must be locally isometric to $\mathbb{R}$,
implying any connected $1$-uniform space is itself $\mathbb{R}$. From these
examples, it seems reasonable to think that some connectedness and topological
conditions are necessary hypotheses for any type of converse to hold. It also
follows from work of Schul [Sch07], [Sch09], and Fassler and Violo [FV23] (see
also [Hah05]) that any Ahlfors $1$-regular connected subset of a metric space
is uniformly $1$-rectifiable, although perhaps adding some form of weaker
hypothesis could provide an interesting converse to our result in the one-
dimensional case using Bate’s classification.

##  2\. Preliminaries

Whenever we write $A\lesssim B$, we mean that there exists some constant $C$
independent of $A$ and $B$ such that $A\leq CB$. If we write
$A\lesssim_{a,b,c}B$ for some constants $a,b,c$, then we mean that the
implicit constant $C$ mentioned above is allowed to depend on $a,b,c$. We will
sometimes write $A\asymp_{a,b,c}B$ to mean that both $A\lesssim_{a,b,c}B$ and
$B\lesssim_{a,b,c}A$ hold. We use the notation $f:E\twoheadrightarrow F$ to
mean $f$ is a surjective map from $E$ to $F$.

Let $(X,d)$ be a metric space. For any subset $F\subseteq X$, integer $n\geq
0$, and constant $0<\delta\leq\infty$, we define

$\mathscr{H}^{n}_{\delta}(F)=\inf\Set{\sum\mathop{\mathrm{missing}}{diam}(E_{i})^{d}:F\subseteq\bigcup
E_{i},\ \mathop{\mathrm{missing}}{diam}(E_{i})<\delta}$

where $\mathop{\mathrm{missing}}{diam}(E)=\sup_{x,y\in E}d(x,y)$. The
Hausdorff $n$-measure of $F$ is defined as

$\mathscr{H}^{n}(F)=\lim_{\delta\rightarrow 0}\mathscr{H}^{m}_{\delta}(F).$

Occasionally, we will specify a subset $\Sigma\subseteq X$ and write
$\mathscr{H}_{\Sigma}^{n}=\mathscr{H}^{n}|_{\Sigma}$. For any
$\mathscr{H}^{n}$ measurable $A\subseteq X$ and measurable $f:A\subseteq
X\rightarrow\mathbb{R}$, we define

$\fint_{A}f=\frac{1}{\mathscr{H}^{n}(A)}\int_{A}f(x)d\mathscr{H}^{n}(x).$

We let $\mathcal{D}(\mathbb{R}^{n})$ denote the family of dyadic cubes in
$\mathbb{R}^{n}$. For $Q\in\mathcal{D}(\mathbb{R}^{n})$, we let $\ell(Q)$
denote the side length of $Q$. If $R\in\mathcal{D}(\mathbb{R}^{n})$ and
$k\in\mathbb{Z}$, we define

$\displaystyle\mathcal{D}(R)$
$\displaystyle=\Set{Q\in\mathcal{D}(\mathbb{R}^{n})}{Q\subseteq R},$
$\displaystyle\mathcal{D}_{k}(R)$
$\displaystyle=\Set{Q\in\mathcal{D}(R)}{\ell(Q)=2^{-k}\ell(R)}.$

We will also need a version of “cubes” associated to a metric space. David
[Dav88] introduced this idea first, and it was later generalized by [Chr90]
and [HM12]. The following formulation draws most from the latter two.

######  Theorem 2.1 (Christ-David cubes).

Let $X$ be a doubling metric space. Let $X_{k}$ be a nested sequence of
maximal $\rho^{k}$-nets for $X$ where $\rho<1/1000$ and let $c_{0}=1/500$. For
each $k\in\mathbb{Z}$ there is a collection $\mathscr{D}_{k}$ of “cubes,”
which are Borel subsets of $X$ such that the following hold.

  1. (i)

$X=\bigcup_{Q\in\mathscr{D}_{k}}Q$.

  2. (ii)

If $Q,Q^{\prime}\in\mathscr{D}=\bigcup\mathscr{D}_{k}$ and $Q\cap
Q^{\prime}\neq\emptyset$, then $Q\subseteq Q^{\prime}$ or $Q^{\prime}\subseteq
Q$.

  3. (iii)

For $Q\in\mathscr{D}$, let $k(Q)$ be the unique integer so that
$Q\in\mathscr{D}_{k}$ and set $\ell(Q)=5\rho^{k(Q)}$. Then there is $x_{Q}\in
X_{k}$ so that

$B(x_{Q},c_{0}\ell(Q))\subseteq Q\subseteq B(x_{Q},\ell(Q))$

and

$X_{k}=\\{x_{Q}:Q\in\mathscr{D}_{k}\\}.$

  4. (iv)

If $X$ is Ahlfors $n$-regular, then there exists $C\geq 1$ such that

$\mathscr{H}^{n}(\Set{x\in Q}{d(x,X\setminus
Q)\leq\eta\rho^{k}})\lesssim\eta^{1/C}\ell(Q)^{n}$

for all $Q\in\mathscr{D}$ and $\eta>0$.

In addition, we define

$B_{Q}=B(x_{Q},\ell(Q)).$

In analogy to the dyadic cube notation, for any $R\in\mathscr{D}$ and
$k\in\mathbb{Z}$ we also write

$\displaystyle\mathscr{D}(R)$ $\displaystyle=\Set{Q\in\mathscr{D}}{Q\subseteq
R},$ $\displaystyle\mathscr{D}_{k}(R)$
$\displaystyle=\Set{Q\in\mathscr{D}(R)}{\ell(Q)=\rho^{-k}\ell(R)}.$

We will actually prove a form of the WCD adapted to Christ-David cubes. The
following two lemmas will allow us to show that the cube WCD in Definition 4.1
implies the WCD from Definition 1.2. Recall from Theorem 2.1 that
$c_{0}=\frac{1}{500}$.

######  Lemma 2.2.

Let $X$ be a doubling metric space with doubling constant $C_{d}$. There
exists $N(C_{d})\in\mathbb{N}$ such that the following holds: There exist $N$
Christ-David systems of cubes $\\{\mathscr{D}_{i}\\}_{i=1}^{N}$ for $X$ such
that for any $x\in X,\ 0<t<\mathop{\mathrm{missing}}{diam}(X)$, there exists
$i\in\\{1,\ldots,N\\}$ and $Q\in\mathscr{D}_{i}$ with
$\ell(Q)\leq\frac{5}{\rho c_{0}}t$ such that $x\in\frac{c_{0}}{4}B_{Q}$ and
$t<\frac{c_{0}}{4}\ell(Q)$.

###### Proof.

Fix $\rho<\frac{1}{1000}$. For each $k$, let $\tilde{X}_{k}$ be a maximal
$c_{0}\rho^{k}$-net for $X$. We now iteratively construct maximal
$\rho^{k}$-nets $X_{k}^{1},X_{k}^{2},\ldots,X_{k}^{N},\ldots$ in the following
way. Let $X_{k}^{1}$ be a completion of a maximal $\rho^{k}$-separated subset
of $\tilde{X}_{k}$ to a maximal $\rho^{k}$-net for $X$. Given $X_{k}^{i}$ for
any $i>0$, construct $X_{k}^{i+1}$ by completing a maximal
$\rho^{k}$-separated subset of
$Y_{k}^{i}\vcentcolon=\tilde{X}_{k}\setminus(X_{k}^{1}\cup
X_{k}^{2}\cup\ldots\cup X_{k}^{i})$ to a maximal $\rho^{k}$-net for $X$. We
claim that this process terminates in $N(C_{d})$ steps, giving for each
$k\in\mathbb{Z}$ a collection of maximal $\rho^{k}$-nets
$X_{k}^{1},\ldots,X_{k}^{N}$. Indeed, let $B$ be a ball of radius $2\rho^{k}$.
By doubling, there exists $N(C_{d})<\infty$ such that
$\\#(B\cap\tilde{X}_{k})\leq N(C_{d})$. Suppose that $\frac{1}{2}B\cap
Y_{k}^{j}\not=\varnothing$ for some $j>0$. Then, because $X_{k}^{j+1}$ is
maximal, there exists some $x\in B\cap Y_{k}^{j}$ such that $x\in
X_{k}^{j+1}$. Therefore, $\\#(B\cap Y_{k}^{j+1})<\\#(B\cap Y_{k}^{j})$
whenever $\frac{1}{2}B\cap Y_{k}^{j}\not=\varnothing$. This means
$\frac{1}{2}B\cap Y_{k}^{N+1}=\varnothing$ for any such $B$, implying
$Y_{k}^{N+1}=\varnothing$ and $\tilde{X}_{k}\subseteq\cup_{i=1}^{N}X_{k}^{i}$
as desired.

We now show that the lemma follows from this. Recall that Theorem 2.1 takes as
input a collection $\\{X_{k}\\}_{k\in\mathbb{Z}}$ of maximal $\rho^{k}$-nets
for $X$ and outputs a system of cubes $\mathscr{D}$ such that every
$x^{\alpha}_{k}\in X_{k}$ is the “center” of a cube
$Q_{k}^{\alpha}\in\mathscr{D}$ with
$B_{X}(x_{k}^{\alpha},c_{0}5\rho^{k})=c_{0}B_{Q_{k}^{\alpha}}\subseteq
Q_{k}^{\alpha}$. We apply Theorem 2.1 to the collection
$\\{X_{k}^{i}\\}_{k\in\mathbb{Z}}$ for every $1\leq i\leq N$ and receive a
Christ-David system $\mathscr{D}_{i}$ such that each point
$\tilde{x}_{k}\in\tilde{X}_{k}$ is the center of some $Q\in\mathscr{D}_{i}$
for some $i$. So, let $x\in X$, $0<t<\mathop{\mathrm{missing}}{diam}(X)$, and
let $k\in\mathbb{Z}$ such that $c_{0}\rho^{k-1}\leq t<c_{0}\rho^{k}$. Because
$\tilde{X}_{k}$ is a maximal $c_{0}\rho^{k}$-net for $X$, there exists
$\tilde{x}_{k}\in\tilde{X}_{k}$ such that $d(x,\tilde{x}_{k})<c_{0}\rho^{k}$.
Because $\tilde{X}_{k}\subseteq\cup_{i=1}^{N}X_{k}^{i}$, there then exists
$1\leq i\leq N$ and $Q\in\mathscr{D}_{i}$ such that $\tilde{x}_{k}=x_{Q}$ so
that $x\in
B(x_{Q},c_{0}\rho^{k})=\frac{1}{5}B(x_{Q},c_{0}\ell(Q))=\frac{c_{0}}{5}B_{Q}$.
Similarly, $\frac{\rho c_{0}}{5}\ell(Q)\leq t<\frac{c_{0}}{5}\ell(Q)$. ∎

######  Lemma 2.3.

Let $X$ be Ahlfors $(C_{0},n)$-regular. If $X$ satisfies the cube WCD of
Definition 4.1, then $X$ satisfies the WCD.

###### Proof.

We first note that if an $a_{Q}>0$ as in (4.3) exists, then $(2C_{0})^{-1}\leq
a_{Q}\leq 2C_{0}$ by Ahlfors regularity. Therefore, whenever
$|\mathscr{H}^{n}(X\cap B(y,r))-a_{Q}r^{n}|\leq\epsilon_{0}\ell(Q)$, we have

$|(a_{Q})^{-1}\mathscr{H}^{n}(X\cap
B(y,r))-r^{n}|=(a_{Q})^{-1}|\mathscr{H}^{n}(X\cap
B(y,r))-a_{Q}r^{n}|\leq(2C_{0})^{-1}\epsilon_{0}\ell(Q).$

This means that one can replace $\mathscr{H}^{n}$ with a multiple of
$\mathscr{H}^{n}$ and $a_{Q}r^{n}$ with $r^{n}$ in the definition of the cube
WCD at the cost of increasing $\epsilon_{0}$. Therefore, it suffices to show
that the complement of

$\mathscr{G}_{0}(C_{0},\epsilon_{0})=\Set{(x,t)\in
X\times\mathbb{R}^{+}}{\begin{array}[]{l}\exists a_{(x,t)}>0,\text{ such that
}\forall y\in B(x,t),\ 0<r\leq t,\\\ |\mathscr{H}^{n}(X\cap
B(y,r))-a_{(x,t)}r^{n}|\leq\epsilon_{0}t^{n}\end{array}}$

is a Carleson set. In order to show this, we apply Lemma 2.2 to $X$ and
receive a finite number of Christ-David systems
$\\{\mathscr{D}_{i}\\}_{i=1}^{N}$ with $N$ depending only on $n$ and $C_{0}$
such that for any $x\in X,\ 0<t<\mathop{\mathrm{missing}}{diam}(X)$, there
exists $i\in\\{1,\ldots,N\\}$ and $Q\in\mathscr{D}_{i}$ with $\ell(Q)\lesssim
t$ such that $x\in\frac{c_{0}}{4}B_{Q}$ and $t<\frac{c_{0}}{4}\ell(Q)$. It
follows that if if $Q\in\mathscr{G}_{\text{cd}}(C_{0},\epsilon_{0})$, then
$(x,t)\in\mathscr{G}_{0}(C_{0},C(n)\epsilon_{0})$ for any
$x\in\frac{c_{0}}{4}B_{Q}$ and $\frac{\rho^{2}c_{0}}{4}\ell(Q)\leq
t<\frac{c_{0}}{4}\ell(Q)$ by choosing $a_{(x,t)}=a_{Q}$. Therefore, if $X$
satisfies the cube WCD then $\mathscr{D}_{i}$ has a Carleson packing condition
for its bad set, implying a Carleson condition for the bad balls of the WCD
with a larger choice of $\epsilon_{0}$ and with larger Carleson constant. ∎

##  3\. Oscillation of means of $L^{2}$ functions

In this section, we review necessary facts about wavelets and prove Lemma 3.5,
one of our main tools for the proof of Theorem A.

######  Definition 3.1.

We follow the presentation of [Tol12]. Given
$h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and $Q\in\mathcal{D}(\mathbb{R}^{n})$,
define

$\Delta_{Q}h(x)=\begin{cases}\fint_{P}h(z)dz-\fint_{Q}h(z)dz&\text{ if $x\in
P$, where $P$ is a child of $Q$},\\\ 0&\text{ otherwise}\end{cases}$

If $h\in L^{2}(\mathbb{R}^{n})$, then

$h=\sum_{Q\in\mathcal{D}(\mathbb{R}^{n})}\Delta_{Q}h\quad\text{ and }\quad
h\chi_{Q}=\fint_{Q}h+\sum_{R\subseteq Q}\Delta_{R}h$

where the sums converge in $L^{2}$ and
$\langle\Delta_{Q}h,\Delta_{Q^{\prime}}h\rangle_{L^{2}}=0$ when
$Q\not=Q^{\prime}$ so that
$\|h\|_{2}=\sum_{Q\in\mathcal{D}}\|\Delta_{Q}h\|_{2}^{2}$. One can view
$\Delta_{Q}h$ as a projection of $h$ onto the subspace of $L^{2}$ formed by
the Haar wavelets $h_{Q}^{\epsilon},\
\epsilon\in\\{0,1\\}^{n}\setminus\\{(0,0,\ldots,0)\\}$ associated to $Q$. For
any $k\in\mathbb{N}$, define

(3.1)
$\Delta_{k}^{h}(Q)^{2}=\sum_{j=0}^{k}\sum_{R\in\mathcal{D}_{j}(Q)}\|\Delta_{R}h\|_{2}^{2}.$

######  Remark 3.2 (Properties of $\Delta_{k}^{h}$).

Notice that if $h\in L^{\infty}(\mathbb{R}^{n})$, then $\Delta^{h}_{k}$ has a
form of geometric lemma since

$\sum_{Q\subseteq Q_{0}}\Delta^{h}_{k}(Q)^{2}=\sum_{Q\subseteq
Q_{0}}\sum_{j=0}^{k}\sum_{R\in\mathcal{D}_{j}(Q)}\|\Delta_{R}h\|_{2}^{2}\lesssim_{k,n}\sum_{R\subseteq
Q_{0}}\|\Delta_{R}h\|_{2}^{2}\lesssim_{\|h\|_{\infty}}\ell(Q_{0})^{n}.$

This gives the Carleson condition

(3.2) $\sum_{\begin{subarray}{c}Q\subseteq Q_{0}\\\
\Delta_{k}^{h}(Q)>\delta\ell(Q)^{n/2}\end{subarray}}\ell(Q)^{n}\lesssim_{\delta}\sum_{Q\subseteq
Q_{0}}\Delta_{k}^{h}(Q)^{2}\lesssim_{k,n,\|h\|_{\infty}}\ell(Q_{0})^{n}.$

$\Delta^{h}_{k}$ also scales appropriately in the following manner: Let
$Q,\tilde{Q}\in\mathcal{D}(\mathbb{R}^{n})$ and let
$T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be the affine map sending
$\tilde{Q}$ onto $Q$ by

(3.3) $T(x)=x_{Q}+\left(\frac{x-x_{\tilde{Q}}}{\ell(\tilde{Q})}\right)\ell(Q)$

where $x_{\tilde{Q}}$ is the center of $\tilde{Q}$. Let $\tilde{h}\in
L^{2}(\tilde{Q})$ and set
$h=\left(\frac{\ell(\tilde{Q})}{\ell(Q)}\right)^{n/2}\tilde{h}\circ T^{-1}$, a
scaled and translated version of $\tilde{h}$ satisfying
$\|h\|_{2}=\|\tilde{h}\|_{2}$ because, by change of variables,

$\displaystyle\int
h^{2}=\frac{\ell(\tilde{Q})^{n}}{\ell(Q)^{n}}\int(\tilde{h}\circ
T^{-1})^{2}=\frac{\ell(\tilde{Q})^{n}}{\ell(Q)^{n}}\int\tilde{h}^{2}\frac{\ell(Q)^{n}}{\ell(\tilde{Q})^{n}}=\int\tilde{h}^{2}.$

Similarly, notice that if $V\subseteq Q$ and $\tilde{V}\subseteq\tilde{Q}$
with $T(\tilde{V})=V$, then

$\displaystyle\|\Delta_{V}h\|_{2}^{2}$
$\displaystyle=\int_{V}(\Delta_{V}h(x))^{2}dx=\int_{T(\tilde{V})}(\Delta_{\tilde{V}}\tilde{h}(T^{-1}(x)))^{2}dx$
$\displaystyle=\left(\frac{\ell(Q)}{\ell(\tilde{Q})}\right)^{n}\int_{\tilde{V}}(\Delta_{\tilde{V}}\tilde{h})^{2}dx=\left(\frac{\ell(Q)}{\ell(\tilde{Q})}\right)^{n}\|\Delta_{\tilde{V}}\tilde{h}\|_{2}^{2}$

which gives
$\Delta_{k}^{h}(Q)^{2}=\left(\frac{\ell(Q)}{\ell(\tilde{Q})}\right)^{n}\Delta_{k}^{\tilde{h}}(\tilde{Q})^{2}$.

######  Definition 3.3 (normed balls).

Given $L>0$, we define the set of norms on $\mathbb{R}^{n}$ which are $L$-bi-
Lipschitz to the Euclidean norm by

$\mathcal{N}_{L}=\\{\|\cdot\|\ :\ L^{-1}\|x\|\leq|x|\leq L\|x\|\\}.$

Given a dyadic cube $Q\in\mathcal{D}(\mathbb{R}^{n})$ and $L>0$, we define a
collection of $L$-bi-Lipschitz normed balls inside $Q$ by

$\mathcal{B}_{L}(Q)=\\{B_{\|\cdot\|}(x,r)\subseteq Q\ :\
\|\cdot\|\in\mathcal{N}_{L},\ r\geq L^{-1}\ell(Q)\\}.$

The following lemma gives a form of compactness result for
$\mathcal{B}_{L}(Q)$.

######  Lemma 3.4.

Fix $Q\in\mathcal{D}(\mathbb{R}^{n})$ and $L>0$. Let
$B_{j}=B_{\|\cdot\|_{j}}\in\mathcal{B}_{L}(Q)$ for $j\in\mathbb{N}$. There
exists a subsequence of $B_{j}=B_{\|\cdot\|_{j}}(x_{j},r_{j})$ and a normed
ball $B=B_{\|\cdot\|}(x,r)\in\mathcal{B}_{L}(Q)$ for which the following
holds: For every $\eta>0$, there exists $j_{0}\in\mathbb{N}$ such that for all
$j\geq j_{0}$,

(3.4) $B_{\|\cdot\|}(x,(1-5\eta)r)\subseteq B_{j}\subseteq
B_{\|\cdot\|}(x,(1+5\eta)r).$

###### Proof.

Because $\|\cdot\|_{j}\in\mathcal{N}_{L}$ for all $j$, the functions
$f_{j}:B(0,1)\rightarrow\mathbb{R}$ given by $f_{j}(x)=\|x\|_{j}$ are an
equicontinuous, uniformly bounded family of continuous functions. Therefore,
they subconverge uniformly to some limit function
$f:B(0,1)\rightarrow\mathbb{R}$. It is straightforward to show this function
gives a norm $\|\cdot\|$ when extended homogeneously to $\mathbb{R}^{n}$. By
passing to further subsequences, we can assume that $x_{j}\rightarrow x\in Q$
and $r_{j}\rightarrow r$ with $L^{-1}\ell(Q)\leq r$. We set
$B=B_{\|\cdot\|}(x,r)$ and fix $\eta>0$ as in the statement of the lemma. By
the convergences assumed, we can take $j_{0}$ large enough such that for
$j\geq j_{0}$, we have $\|x_{j}-x\|\leq\eta\min\\{r,r_{j}\\}$,
$|r_{j}-r|\leq\eta\min\\{r,r_{j}\\}$, and

$\displaystyle(1-\eta)\|x\|\leq\|x\|_{j}\leq(1+\eta)\|x\|,\text{ for all
$x\in\mathbb{R}^{n}$},$

We now aim to prove (3.4). Let $y\in B_{j}$. Then

$\displaystyle\|y-x\|$
$\displaystyle\leq\|y-x_{j}\|+\|x_{j}-x\|\leq(1+2\eta)\|y-x_{j}\|_{j}+\eta r$
$\displaystyle\leq(1+2\eta)(1+\eta)r+\eta r\leq(1+5\eta)r.$

so that $y\in B_{\|\cdot\|}(x,(1+5\eta)r)$. On the other hand, if $z\in
B_{\|\cdot\|}(x,(1-5\eta)r)$, then

$\displaystyle\|z-x_{j}\|_{j}$
$\displaystyle\leq\|z-x\|_{j}+\|x-x_{j}\|_{j}\leq(1+2\eta)\|z-x\|+(1+2\eta)\|x-x_{j}\|$
$\displaystyle\leq(1+2\eta)(1-5\eta)r_{j}+(1+2\eta)\eta r_{j}\leq r_{j}$

showing $z\in B_{j}$. ∎

######  Lemma 3.5.

For all $\epsilon,M,L>0,\ n\in\mathbb{N}$, there exist
$k(\epsilon,M,L,n)\in\mathbb{N}$ and $\delta(\epsilon,M,L,n)>0$ such that the
following holds: Let $h\in L^{2}(\mathbb{R}^{n})$ with $h\geq 0$ and
$\|h\|_{2}\leq M$. Let $Q\in\mathcal{D}(\mathbb{R}^{n})$. If
$\Delta^{h}_{k}(Q)\leq\delta\ell(Q)^{n/2}$, then for any normed ball
$B\in\mathcal{B}_{L}(Q)$, we have

(3.5) $\left|\fint_{B}h-\fint_{Q}h\right|\leq\epsilon\left|\fint_{Q}h\right|.$

###### Proof.

Suppose the conclusion of the lemma is false. Then there exist
$\epsilon,M,L,n$ and a sequence of maps $\tilde{h}_{j}\in
L^{2}(\mathbb{R}^{n})$ with $\tilde{h}_{j}\geq 0$ and
$\|\tilde{h}_{j}\|_{2}\leq M$, cubes
$\tilde{Q}_{j}\in\mathcal{D}(\mathbb{R}^{n})$, and normed balls
$\tilde{B}_{j}\in\mathcal{B}_{L}(\tilde{Q}_{j})$ so that (3.5) does not hold
for $\tilde{h}_{j},\tilde{Q}_{j},\tilde{B}_{j}$, yet
$\Delta_{j}^{\tilde{h}_{j}}(\tilde{Q}_{j})\leq\frac{1}{j}\ell(\tilde{Q}_{j})^{n/2}$.
For any $j\in\mathbb{N}$, let $T_{j}$ be the affine transformation sending
$\tilde{Q}_{j}$ onto $Q=[0,1]^{n}$ as in (3.3) and define a scaled and
translated copy of $\tilde{h}_{j}$ called $h_{j}:Q\rightarrow\mathbb{R}$ as in
Remark 3.2 by

$h_{j}=\chi_{Q}\left(\frac{\ell(\tilde{Q}_{j})}{\ell(Q)}\right)^{n/2}\tilde{h}_{j}\circ
T_{j}^{-1}$

Let $c_{j}=\int_{Q}h_{j}$ and observe that
$\Delta_{j}^{h_{j}}(Q)=\left(\frac{\ell(Q)}{\ell(\tilde{Q}_{j})}\right)^{n/2}\Delta_{j}^{\tilde{h}_{j}}(Q_{j})\leq\frac{1}{j}\ell(Q)^{n/2}=\frac{1}{j}$
by Remark 3.2. In addition, we define $B_{j}=T_{j}(\tilde{B}_{j})$ to be the
appropriately translated and scaled copy of $\tilde{B}_{j}$. By the weak
compactness of bounded closed balls in $L^{2}$, there exists some $h\in
L^{2}(Q)$ such that $h_{j}\rightharpoonup h$ in $L^{2}$ for some subsequence
of $h_{j}$. By further refining subsequences and using Lemma 3.4, we can
further assume the subsequence is chosen so that a limiting normed ball
$B=B_{\|\cdot\|}(x,r)\in\mathcal{B}_{L}(Q)$ as in the lemma’s conclusion
exists. Let $c=\int_{Q}h$.

We will first show that $h=c$ by showing that $\Delta_{V}h=0$ for all
$V\subseteq Q$. By weak convergence we have

(3.6) $c_{j}=\int_{Q}h_{j}\rightarrow\int_{Q}h=c.$

Write $h_{j}=c_{j}+\sum_{R\subseteq Q}\Delta_{R}h_{j}$ and
$h=c+\sum_{R\subseteq Q}\Delta_{R}h$. Fix $V\subseteq Q$ and observe

$\displaystyle\int_{Q}h_{j}\Delta_{V}h$
$\displaystyle=\int_{Q}\left(c_{j}+\sum_{R\subseteq
Q}\Delta_{R}h_{j}\right)\Delta_{V}h=c_{j}\int_{Q}\Delta_{V}h+\sum_{R\subseteq
Q}\langle\Delta_{R}h_{j},\Delta_{V}h\rangle$
$\displaystyle=\langle\Delta_{V}h_{j},\Delta_{V}h\rangle.$

where the final equality follows since
$\langle\Delta_{R}f_{1},\Delta_{V}f_{2}\rangle=0$ whenever $f_{1},f_{2}\in
L^{2}$ and $R\not=V$. Similarly, we have

$\displaystyle\int_{Q}h\Delta_{V}h=\langle
h,\Delta_{V}h\rangle=\langle\Delta_{V}h,\Delta_{V}h\rangle=\|\Delta_{V}h\|_{2}^{2}$

Using weak convergence again, we get

$\displaystyle\langle\Delta_{V}h_{j},\Delta_{V}h\rangle=\int_{Q}h_{j}\Delta_{V}h\rightarrow\int_{Q}h\Delta_{V}h=\|\Delta_{V}h\|_{2}^{2}.$

Using Cauchy-Schwarz, we can now conclude that
$\|\Delta_{V}h\|_{2}\leq\lim_{j}\|\Delta_{V}h_{j}\|_{2}$. We claim that
$\|\Delta_{V}h\|_{2}=0$. Indeed, if $j$ is sufficiently large, then both
$V\in\mathcal{D}_{j}(Q)$ and $\|\Delta_{V}h\|_{2}\leq
2\|\Delta_{V}h_{j}\|_{2}$. This means

$\|\Delta_{V}h\|_{2}^{2}\leq 4\|\Delta_{V}h_{j}\|^{2}\leq
4\sum_{k=0}^{j}\sum_{R\in\mathcal{D}_{k}(Q)}\|\Delta_{R}h_{j}\|_{2}^{2}=4\Delta_{j}^{h_{j}}(Q)\leq\frac{4}{j}.$

for all large $j$. This shows that $\Delta_{V}h=0$ for all $V\subseteq Q$,
hence $h=c$ as desired.

We will now show how this leads to a contradiction. Let $\eta>0$ and choose
$j$ large enough so that

$B_{1}\vcentcolon=B_{\|\cdot\|}(x,(1-5\eta)r)\subseteq B_{j}\subseteq
B_{\|\cdot\|}(x,(1+5\eta)r)=\vcentcolon B_{2}.$

Using the fact that $h_{j}\geq 0$ (this is the first use of this hypothesis),

(3.7) $\int_{B_{1}}h_{j}\leq\int_{B_{j}}h_{j}\leq\int_{B_{2}}h_{j}$

so that

$\frac{\mathscr{L}^{n}(B_{1})}{\mathscr{L}^{n}(B_{j})}\fint_{B_{1}}h_{j}-\fint_{Q}h_{j}\leq\fint_{B_{j}}h_{j}-\fint_{Q}h_{j}\leq\frac{\mathscr{L}^{n}(B_{2})}{\mathscr{L}^{n}(B_{j})}\fint_{B_{2}}h_{j}-\fint_{Q}h_{j}.$

Because
$\mathscr{L}^{n}(B_{1})\leq\mathscr{L}^{n}(B_{j})\leq\mathscr{L}^{n}(B_{2})$
and
$\frac{\mathscr{L}^{n}(B_{2})}{\mathscr{L}^{n}(B_{1})}\leq(1+c^{\prime}(n)\eta)$,
we can assume without loss of generality that

$\limsup_{j}\left|\fint_{B_{j}}h_{j}-\fint_{Q}h_{j}\right|\leq\limsup_{j}\left|\frac{\mathscr{L}^{n}(B_{2})}{\mathscr{L}^{n}(B_{1})}\fint_{B_{2}}h_{j}-\fint_{Q}h_{j}\right|\leq\left|(1+c^{\prime}\eta)c-c\right|\lesssim_{n}\eta
c.$

Since this holds for all $\eta>0$, we get
$\limsup_{j}\left|\fint_{B_{j}}h_{j}-\fint_{Q}h_{j}\right|=0$. On the other
hand, by hypothesis

$\left|\fint_{B_{j}}h_{j}-\fint_{Q}h_{j}\right|>\epsilon\left|\fint_{Q}h_{j}\right|$

for all $j$ so that
$\limsup_{j}\left|\fint_{B_{j}}h_{j}-\fint_{Q}h_{j}\right|>\epsilon c\geq 0$,
giving a contradiction. ∎

######  Remark 3.6.

Suppose that we only want to conclude (3.5) with normed balls
$B\in\mathcal{B}_{L}(Q)$ replaced by $Q^{\prime}\in\mathcal{D}_{j}(Q)$ for
$j\leq k\in\mathbb{N}$. The following stronger condition holds even without
the positivity assumption for $h$: Let
$\alpha:\mathcal{D}(\mathbb{R}^{n})\rightarrow\mathcal{D}(\mathbb{R}^{n})$
where $\alpha(Q)\in\mathcal{D}_{j}(Q)$. There exists $C_{0}(k,n)>0$ such that

(3.8) $\sum_{Q\subseteq
R}\left|\fint_{\alpha(Q)}h-\fint_{Q}h\right|^{2}\ell(Q)^{n}\leq
C_{0}\|h\|_{2}^{2}.$

The proof is straightforward: because $\alpha(Q)\in\mathcal{D}_{j}(Q)$, there
is a chain of at most $k+1$ cubes $\alpha(Q)=Q_{j}\subseteq
Q_{j-1}\subseteq\ldots\subseteq Q_{0}=Q$ such that $Q_{j+1}$ is a child of
$Q_{j}$. Therefore, we can use the triangle inequality to write

$\displaystyle\left|\fint_{\alpha(Q)}h-\fint_{Q}h\right|^{2}\ell(Q)^{n}\lesssim_{k,n}\sum_{i=1}^{j}\left|\fint_{Q_{i}}h-\fint_{Q_{i-1}}h\right|^{2}\ell(Q_{i-1})^{n}\leq\sum_{i=1}^{j}\|\Delta_{Q_{i}}h\|_{2}^{2}.$

Because each cube $Q^{\prime}\subseteq R$ can appear in at most
$N(n,k)<\infty$ chains of the above type, this gives

$\left|\fint_{\alpha(Q)}h-\fint_{Q}h\right|^{2}\ell(Q)^{n}\lesssim_{n,k}\sum_{Q\subseteq
R}\sum_{i=1}^{j(Q)}\|\Delta_{Q_{i}}h\|_{2}^{2}\lesssim_{n,k}\sum_{Q\subseteq
R}\|\Delta_{R}h\|_{2}^{2}=\|h\|_{2}^{2}.$

The reader should also see [DS93] Lemma IV.2.2.14 for a version of this
statement where $\alpha(Q)$ is only required to be $N$-close to $Q$ rather
than contained in $Q$.

######  Corollary 3.7 (cf. [DS93] Corollary IV.2.2.19).

Let $L,\epsilon,M>0$ and let $h\in L^{\infty}(\mathbb{R}^{n})$ with
$\|h\|_{\infty}\leq M$. Let

$\mathscr{G}_{h}(L,\epsilon)=\Set{Q\in\mathcal{D}(\mathbb{R}^{n})}{\left|\fint_{B}h-\fint_{Q}h\right|\leq\epsilon\text{
for all $B\in\mathcal{B}_{L}(Q)$}}.$

Then
$\mathscr{B}_{h}(L,\epsilon)=\mathcal{D}(\mathbb{R}^{n})\setminus\mathscr{G}_{h}(L,\epsilon)$
is $C(M,L,n,\epsilon)$-Carleson.

###### Proof.

Let $\tilde{h}=h+\|h\|_{\infty}+1$, let $\epsilon,M,L>0$, and choose
$k,\delta>0$ such that the conclusion of Lemma 3.5 holds. Let

$\mathscr{B}_{\tilde{h}}^{\prime}=\Set{Q\in\mathcal{D}(\mathbb{R}^{n})}{\exists
B\in\mathcal{B}_{L}(Q),\
\left|\fint_{B}\tilde{h}-\fint_{Q}\tilde{h}\right|>\epsilon\left|\fint_{Q}\tilde{h}\right|}$

and fix $R\in\mathcal{D}$. By Lemma 3.5,
$Q\in\mathscr{B}_{\tilde{h}}^{\prime}$ implies
$\Delta_{k}^{\tilde{h}}(Q)>\delta\ell(Q)^{n/2}$ so that by (3.2).

$\sum_{\begin{subarray}{c}Q\subseteq R\\\
Q\in\mathscr{B}_{\tilde{h}}\end{subarray}}\ell(Q)^{n}\leq\sum_{\begin{subarray}{c}Q\subseteq
R\\\
\Delta_{k}^{\tilde{h}}(Q)>\delta\ell(Q)^{n/2}\end{subarray}}\ell(Q)^{n}\lesssim_{\delta,k,n,M}\ell(R)^{n}.$

The result follows since
$\fint_{B}\tilde{h}-\fint_{Q}\tilde{h}=\fint_{B}h-\fint_{Q}h$ and
$|\fint_{Q}\tilde{h}|\leq 2M+1$ so that
$\mathscr{B}_{h}(L,(2M+1)\epsilon)\subseteq\mathscr{B}_{\tilde{h}}^{\prime}$.
∎

##  4\. Bi-Lipschitz images satisfy the weak constant density condition

In this section, we use the tools from Section 3 to prove that metric spaces
which are bi-Lipschitz images of Euclidean spaces satisfy the WCD. In this
section and the next, we will use the following version of the WCD adapted to
Christ-David cubes using only $\mathscr{H}^{n}$.

######  Definition 4.1 (Cube weak constant density condition).

Let $(X,d)$ be an Ahlfors $n$-regular metric space, $\mathscr{D}$ be a system
of Christ-David cubes for $X$, and let $C_{0},\epsilon_{0}>0$. Define

(4.3) $\displaystyle\mathscr{G}_{\text{cd}}(C_{0},\epsilon_{0})$
$\displaystyle=\Set{Q\in\mathscr{D}}{\begin{array}[]{l}\exists a_{Q}>0,\text{
such that }\forall y\in\frac{c_{0}}{2}B_{Q},\
0<r\leq\frac{c_{0}}{2}\ell(Q),\\\
|\mathscr{H}^{n}(B(y,r))-a_{Q}r^{n}|\leq\epsilon_{0}\ell(Q)^{n}\end{array}},$
(4.4) $\displaystyle\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$
$\displaystyle=\mathscr{D}\setminus\mathscr{G}_{\text{cd}}(C_{0},\epsilon_{0}).$

We say that $X$ satisfies the cube WCD if there exists $C_{0}>0$ such that for
all choices of system $\mathscr{D}$ and $\epsilon_{0}>0$,
$\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$ is Carleson.

See Lemma 2.3 for a proof that this version of the WCD implies the version
given in Definition 1.2. We will also need to review some of the theory of
rectifiability in metric spaces.

######  Definition 4.2 (metric derivatives, jacobians).

Let $f:\mathbb{R}^{n}\rightarrow\Sigma$ be $L$-Lipschitz. We say a seminorm on
$\mathbb{R}^{n}$ $|Df|(x)$ is a metric derivative of $f$ at $x$ if

$\lim_{y,z\rightarrow x}\frac{d(f(y),f(z))-|Df|(x)(y-z)|}{|y-x|+|z-x|}=0.$

Given a seminorm $s$ on $\mathbb{R}^{n}$, define $\mathscr{J}(s)$, the
jacobian of $s$, by

$\mathscr{J}(s)=\alpha(n)n\left(\int_{{\mathbb{S}}^{n-1}}(s(x))^{-n}d\mathscr{H}^{n-1}(x)\right)^{-1}.$

Kircheim used these ideas to prove the following metric analogs of
Rademacher’s theorem and the area formula for Lipschitz maps from
$\mathbb{R}^{n}$ into metric spaces.

######  Theorem 4.3 (cf. [Kir94] Theorem 2, Corollary 8).

Let $f:\mathbb{R}^{n}\rightarrow\Sigma$ be $L$-Lipschitz and let
$\mathscr{J}_{f}(x)=\mathscr{J}(|Df|(x))$. A metric derivative for $f$ exists
at $\mathscr{L}^{n}$ almost every $x\in\mathbb{R}^{n}$. In addition, for any
Lebesgue integrable function $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$,

$\int_{\mathbb{R}^{n}}g(x)\mathscr{J}_{f}(x)d\mathscr{L}^{n}(x)=\int_{\Sigma}\left(\sum_{x\in
f^{-1}(y)}g(x)\right)d\mathscr{H}^{n}(y).$

In their work on Lipschitz analogs of Sard’s Theorem, Azzam and Schul
developed the following quantitative measure of how far a function $f$ is from
being given by a seminorm.

######  Definition 4.4.

Let $f:\mathbb{R}^{n}\rightarrow X$ and $Q\in\mathcal{D}(\mathbb{R}^{n})$.
Define

$\mathrm{md}_{f}(Q)=\frac{1}{\ell(Q)}\inf_{\|\cdot\|}\sup_{x,y\in
Q}\bigg{|}d(f(x),f(y))-\|x-y\|\bigg{|}$

They used this to prove the following metric quantitative differentiation
theorem.

######  Theorem 4.5 ([AS14] Theorem 1.1).

Let $f:\mathbb{R}^{n}\rightarrow X$ be an $L$-Lipschitz function. Let
$\delta>0$. Then for each $R\in\mathcal{D}(\mathbb{R}^{n})$,

$\sum_{\begin{subarray}{c}Q\in\mathcal{D}(R)\\\ \mathrm{md}_{f}(3Q)>\delta
L\end{subarray}}\ell(Q)^{n}\leq C(\delta,n)\ell(R)^{n}.$

Finally, we will need to extend the standard system of dyadic cubes.

######  Definition 4.6 (one-third trick lattices).

The following family of dyadic systems were introduced by Okikiolu [Oki92].
For any $e\in\\{0,1\\}^{n}$ and cube $Q_{0}\in\mathcal{D}(\mathbb{R}^{n})$,
define the shifted dyadic lattice

$\displaystyle\mathcal{D}_{j}^{e}(Q_{0})$
$\displaystyle=\Set{Q+\frac{\ell(Q)}{3}e}{Q\in\mathcal{D}_{j}(Q_{0})},$
$\displaystyle\mathcal{D}^{e}(Q_{0})$ $\displaystyle=\bigcup_{j\geq
0}\mathcal{D}_{j}^{e}(\mathbb{R}^{n})$

and set

$\tilde{\mathcal{D}}(Q_{0})=\bigcup_{e\in\\{0,1\\}^{n}}\mathcal{D}^{e}(Q_{0}).$

$\tilde{\mathcal{D}}(Q_{0})$ has the following property: For any $x\in Q_{0}$
and $j\geq 0$, there exists $Q\in\tilde{\mathcal{D}}(Q_{0})$ such that
$x\in\frac{2}{3}Q$ (See [Ler03] Proposition 3.2).

We now begin setting up the proof of the WCD for bi-Lipschitz images. We use
the following good family of dyadic cubes from our collection of dyadic trees
$\tilde{\mathcal{D}}$ to do analysis in the domain of our bi-Lipschitz maps.

######  Definition 4.7 ($L$-good $I_{Q}$).

Let $f:[0,1]^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz. Fix
$Q\in\mathscr{D}(\Sigma)$. We call a dyadic cube $I_{Q}\in\tilde{\mathcal{D}}$
$L$-good for $Q$ if the following hold:

  1. (i)

$\ell(I_{Q})\asymp_{L}\ell(Q)$,

  2. (ii)

$3B_{Q}\subseteq f(I_{Q})$,

where the implicit constant in (i)(i) is independent of $Q$ and $I_{Q}$.

Using the special property of the one-third trick lattices and the definition
of bi-Lipschitz maps, the following lemma is standard.

######  Lemma 4.8.

Let $f:[0,1]^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz. For each
$Q\in\mathscr{D}(\Sigma)$ with $\ell(Q)\lesssim_{L}1$, there exists an
$L$-good $I_{Q}\in\tilde{\mathcal{D}}$.

For $k\in\mathbb{N},\ \delta>0$, define

(4.7) $\displaystyle\mathscr{G}_{\Sigma}(k,\delta)$
$\displaystyle=\Set{Q\in\mathscr{D}}{\begin{array}[]{l}\exists\ L\text{-good
}I_{Q}\in\tilde{\mathcal{D}}\text{ with }\\\
\Delta_{k}^{\mathscr{J}_{f}}(I_{Q})\leq\delta\ell(I_{Q})^{n/2},\
\mathrm{md}_{f}(I_{Q})\leq\delta\end{array}},$ (4.8)
$\displaystyle\mathscr{B}_{\Sigma}(k,\delta)$
$\displaystyle=\mathscr{D}\setminus\mathscr{G}_{\Sigma}(k,\delta).$

######  Lemma 4.9.

Let $f:[0,1]^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz and let
$\epsilon>0$. There exist $k(\epsilon,L,n),\ \delta(\epsilon,L,n)>0$ such that
the following holds: For any $Q\in\mathscr{G}_{\Sigma}(k,\delta)$ there exists
a constant $c_{Q}\asymp_{L,n}1$ such that for any normed ball
$B\in\mathcal{B}_{L}(I_{Q})$,

$|\mathscr{H}^{n}(f(B))-c_{Q}\mathscr{L}^{n}(B)|\leq\epsilon
c_{Q}\mathscr{L}^{n}(B)$

###### Proof.

Let $I_{Q}\in\tilde{\mathcal{D}}$ for $Q$ be as in (4.7), let $\epsilon>0$,
and assume $k,\delta$ are small enough to satisfy the hypotheses of Lemma 3.5
with respect to $0\leq\mathscr{J}_{f}\in L^{\infty}$ and $\epsilon$. Then
$\Delta_{k}^{\mathscr{J}_{f}}(I_{Q})\leq\delta$ implies that

$\left|\fint_{B}\mathscr{J}_{f}-\fint_{Q}\mathscr{J}_{f}\right|\leq\epsilon\left|\fint_{Q}\mathscr{J}_{f}\right|.$

for any normed ball $B\in\mathcal{B}_{L}(I_{Q})$. By setting
$c_{Q}=\fint_{Q}\mathscr{J}_{f}$ and using the area formula, we get the
desired inequality. ∎

######  Lemma 4.10.

Let $f:[0,1]^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz, let $C_{0}$ be
a regularity constant for $\Sigma$, and let $\epsilon_{0}>0$. There exist
$k(\epsilon_{0},L,n),\ \delta(\epsilon_{0},L,n)>0$ such that
$\mathscr{G}_{\Sigma}(k,\delta)\subseteq\mathscr{G}_{\text{cd}}(2C_{0},\epsilon_{0})$.
In fact, for any $Q\in\mathscr{G}_{\Sigma}(k,\delta)$, there exists a constant
$(2C_{0})^{-1}\leq a_{Q}\leq 2C_{0}$ such that for any $y\in B_{Q},\
0<r\leq\ell(Q)$, we have

(4.9) $|\mathscr{H}^{n}(B(y,r))-a_{Q}r^{n}|\leq\epsilon_{0}\ell(Q)^{n}.$

That is, the condition on cubes in
$\mathscr{G}_{\text{cd}}(2C_{0},\epsilon_{0})$ is attained with a multiple of
$\mathscr{H}^{n}$.

###### Proof.

First, we note that if a constant $a_{Q}$ such as in (4.9) exists, it must
satisfy $(2C_{0})^{-1}\leq a_{Q}\leq 2C_{0}$ for small enough $\epsilon_{0}$
because $\Sigma$ is $(C_{0},n)$-regular. Let $I_{Q}$ be as in (4.7) and let
$\epsilon>0$. By Lemma 4.9, we can choose $k$ large enough and $\delta>0$
small enough so there exists $c_{Q}\asymp_{L,n}1$ such that for any
$B\in\mathcal{B}_{2L}(I_{Q})$

(4.10) $|\mathscr{H}^{n}(f(B))-c_{Q}\mathscr{L}^{n}(B))|\leq\epsilon
c_{Q}\mathscr{L}^{n}(B).$

In addition, the fact that $\mathrm{md}_{f}(I_{Q})\leq\delta$ implies that
there exists a norm $\|\cdot\|_{Q}$ such that

(4.11) $\sup_{x,y\in
I_{Q}}\big{|}d(f(x),f(y))-\|x-y\|_{Q}\big{|}\leq\delta\ell(I_{Q}).$

Let $c_{\|\cdot\|_{Q}}\asymp_{L,n}1$ be such that
$\mathscr{L}^{n}(B_{\|\cdot\|_{Q}}(0,r))=c_{\|\cdot\|_{Q}}r^{n}$. We set

$a_{Q}=c_{Q}c_{\|\cdot\|_{Q}}$

and begin the proof of (4.9).

Let $y_{0}=f^{-1}(y)$. We claim that there exists a constant $c_{1}(n,L)>0$
such that

(4.12) $B_{1}\vcentcolon=B_{\|\cdot\|_{Q}}(y_{0},(1-c_{1}\delta)r)\subseteq
f^{-1}(B(y,r))\subseteq B_{\|\cdot\|_{Q}}(y_{0},(1+c_{1}\delta)r)=\vcentcolon
B_{2}.$

For the first inclusion, let $x_{0}\in B_{1}$. By (4.11),

$d(f(x_{0}),f(y_{0}))\leq\|x_{0}-y_{0}\|_{Q}+3\delta\ell(I_{Q})\leq(1-c_{1}\delta)r+C(L,n)\delta
r<r$

where the final inequality holds if $c_{1}$ is large enough. Similarly, let
$z_{0}\in f^{-1}(B(y,r))\subseteq I_{Q}$. Then

$\|z_{0}-y_{0}\|_{Q}\leq d(f(z_{0}),f(y_{0}))+\delta\ell(3I_{Q})\leq
r+C(L,n)\delta r\leq(1+c_{1}\delta)r$

with the same restriction on $\delta$ as above This finishes the proof of
(4.12). Because $3B_{Q}\subseteq f(I_{Q})$, we immediately have that
$B_{1},B_{2}\subseteq I_{Q}$ for small enough $\delta$ so that
$B_{1},B_{2}\in\mathcal{B}_{2L}(I_{Q})$. Using (4.10), this implies the
existence of a constant $c_{2}(n,L)$ so that

$\mathscr{H}^{n}(B(y,r))\leq\mathscr{H}^{n}(f(B_{2}))\leq
c_{Q}(1+\epsilon)c_{\|\cdot\|_{Q}}(1+c_{1}\delta)^{n}r^{n}\leq
a_{Q}r^{n}+c_{2}(\epsilon+\delta)\ell(Q)^{n}.$

A similar computation using $\mathscr{H}^{n}(f(B_{1}))$ gives a similar lower
bound for $\mathscr{H}^{n}(B(y,r))$. This shows that

$\left|\mathscr{H}^{n}_{\Sigma}(B(y,r))-a_{Q}r^{n}\right|\leq
c_{2}(\epsilon+\delta)\ell(Q)^{n}$

By choosing $\epsilon$ small enough, then $k$ large enough and $\delta$ small
enough, we get the conclusion of the lemma. ∎

######  Lemma 4.11.

Let $f:[0,1]^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz. For any
$k,\delta>0$, $\mathscr{B}_{\Sigma}(k,\delta)$ is $C(k,\delta,n,L)$-Carleson.

###### Proof.

Let $R\in\mathscr{D}$. By Lemma 4.8, Remark 3.2, and Theorem 4.5 we have

$\displaystyle\sum_{\begin{subarray}{c}Q\in\mathscr{B}(k,\delta)\\\ Q\subseteq
R\end{subarray}}\ell(Q)^{n}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}Q\subseteq R\\\
\Delta_{k}^{\mathscr{J}_{f}}(I_{Q})>\delta\ell(I_{Q})^{n/2}\end{subarray}}\ell(Q)^{n}+\sum_{\begin{subarray}{c}Q\subseteq
R\\\
\mathrm{md}_{f}(I_{Q})>\delta\end{subarray}}\ell(Q)^{n}+\sum_{\begin{subarray}{c}Q\subseteq
R\\\ \ell(Q)>C(L)\end{subarray}}\ell(Q)^{n}$
$\displaystyle\lesssim_{L,n}\sum_{\begin{subarray}{c}I_{Q}\subseteq I_{R}\\\
\Delta_{k}^{\mathscr{J}_{f}}(I_{Q})>\delta\ell(I_{Q})^{n/2}\end{subarray}}\ell(I_{Q})^{n}+\sum_{\begin{subarray}{c}I_{Q}\subseteq
I_{R}\\\
\mathrm{md}_{f}(I_{Q})>\delta\end{subarray}}\ell(I_{Q})^{n}+C(L,n)\ell(R)^{n}$
$\displaystyle\lesssim_{k,\delta,L,n}\ell(I_{R})^{n}+\ell(R)^{n}\lesssim_{L,n}\ell(R)^{n}.\qed$

######  Theorem 4.12.

The WCD holds for any bi-Lipschitz image of $[0,1]^{n}$.

###### Proof.

Let $f$ be $L$-bi-Lipschitz $f:[0,1]^{n}\twoheadrightarrow\Sigma$. Choose
$C_{0}(L,n)$ such that $\Sigma$ is Ahlfors $(C_{0},n)$-regular and let
$\epsilon_{0}>0$. Choose $k$ large enough and $\delta$ small enough with
respect to $\epsilon_{0},L,n$ so that
$\mathscr{G}_{\Sigma}(k,\delta)\subseteq\mathscr{G}_{\text{cd}}(2C_{0},\epsilon_{0})$.
That is, the conclusion of Lemma 4.10 holds. Then
$\mathscr{B}_{\text{cd}}(\epsilon_{0},2C_{0})\subseteq\mathscr{B}_{\Sigma}(k,\delta)$.
Lemma 4.11 implies that $\mathscr{B}_{\Sigma}(k,\delta)$ is
$C(\epsilon_{0},L,n)$-Carleson, implying
$\mathscr{B}_{\text{cd}}(2C_{0},\epsilon_{0})$ is also
$C(\epsilon_{0},L,n)$-Carleson which says exactly that $\Sigma$ satisfies the
WCD. ∎

##  5\. Stability of the weak constant density condition under big pieces

The goal of this section is to prove Theorem A: uniformly $n$-rectifiable
metric spaces satisfy the WCD. We will prove this via a stability argument.
That is, we will use the fact that uniformly rectifiable metric spaces have
big pieces of bi-Lipschitz images (in fact, very big pieces of bi-Lipschitz
images with worsening constants) to transfer our bi-Lipschitz image estimates
to the uniformly rectifiable case. The primary tool for this argument is the
following abstract analog of the John-Nirenberg-Stromberg theorem.

######  Lemma 5.1 ([BHS23] Lemma 4.2.8, [DS93] Lemma IV.1.12).

Let $X$ be an Ahlfors $n$-regular metric space and $\mathscr{D}$ a system of
Christ-David cubes for $X$. Let $\alpha:\mathscr{D}\rightarrow[0,\infty)$ be
given and suppose there are $N,\eta>0$ such that

(5.1) $\mathscr{H}^{n}\left(\Set{x\in R}{\sum_{\begin{subarray}{c}Q\subseteq
R\\\ x\in Q\end{subarray}}\alpha(Q)\leq N}\right)\geq\eta\ell(R)^{n}$

for all $R\in\mathscr{D}$. Then,

$\sum_{Q\subseteq R}\alpha(Q)\ell(Q)^{n}\lesssim_{N,\eta}\ell(R)^{n}$

for all $R\in\mathscr{D}$.

For our application, we will take
$\alpha(Q)=\chi_{\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})}(Q)$ where
$\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$ is as in (4.4). If we can show
that (5.1) holds for this choice of $\alpha$, then we will conclude

(5.2) $\sum_{Q\subseteq
R}\chi_{\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})}(Q)\ell(Q)^{n}=\sum_{\begin{subarray}{c}Q\subseteq
R\\\
Q\in\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})\end{subarray}}\ell(Q)^{n}\lesssim_{N,\eta}\ell(R)^{n}$

which is exactly the desired Carleson packing condition for
$\mathscr{B}_{\text{cd}}(C_{0},\epsilon_{0})$. We will need the following
result from Bate, Hyde, and Schul’s paper which states that uniformly
rectifiable metric spaces have very big pieces of bi-Lipschitz images.

######  Theorem 5.2 (cf. [BHS23] Theorem B, Proposition 9.0.2).

Let $\epsilon>0$ and let $X$ be uniformly $n$-rectifiable. There is an $L\geq
1$ depending only on $\epsilon,n$, the Ahlfors regularity constant for $X$,
and the BPLI constants for $X$ such that for each $x\in X$ and $r>0$ there
exists $F\subseteq B(x,r)$, satisfying $\mathscr{H}^{n}_{X}(B(x,r)\setminus
F)\leq\epsilon r^{d}$ and an $L$-bi-Lipschitz map
$g:F\rightarrow\mathbb{R}^{n}$.

######  Remark 5.3.

If we embed $X$ isometrically into $\ell_{\infty}$, then we can take the map
$g^{-1}:g(F)\rightarrow F\subseteq\ell_{\infty}$ above and extend it to an
$L^{\prime}(L,n)$-bi-Lipschitz map $f:\mathbb{R}^{n}\rightarrow\ell_{\infty}$
satisfying the same conclusions with respect to the isometric embedding of
$X$. (See [BHS23] Lemma 4.3.2 for a proof.)

We now begin setting up the proof of Theorem A. Fix a uniformly
$n$-rectifiable metric space $X$ with regularity constant $C_{0}$ and a system
of Christ-David cubes $\mathscr{D}$ for $X$. Let $\epsilon_{0}>0$ and
$R\in\mathscr{D}(X)$. By applying Theorem 5.2 to the ball $3B_{R}$, we get an
$L$-bi-Lipschitz map
$f:\mathbb{R}^{n}\twoheadrightarrow\Sigma\subseteq\ell_{\infty}$ such that

(5.3)
$\mathscr{H}^{n}_{X}(3B_{R}\setminus\Sigma)\leq\frac{\epsilon_{0}}{4}\ell(R)^{n}.$

We will only need to use $f$ near where it parameterizes $3B_{R}$, so it
suffices to consider $f|_{I_{R}}$ where $I_{R}$ is $L$-good for $R$ (See
Definition 4.7). We can assume without loss of generality that
$I_{R}=[0,1]^{n}$ so that the results of the previous section for bi-Lipschitz
images of $[0,1]^{n}$ apply to $f$.

Because $\Sigma$ has such large intersection with $3B_{R}$, we can use the
following lemma to find a substantial subset $\tilde{R}\subseteq R$ such that
for every $x\in\tilde{R}$, every cube $Q\subseteq R$ with $x\in Q$ has very
large intersection with $\Sigma$.

######  Lemma 5.4.

Let $X$ be a doubling metric space with a system of Christ-David cubes
$\mathscr{D}$. Let $\epsilon>0$, $F\subseteq X\text{ measurable}$, and let
$R\in\mathscr{D}$ be such that $\mathscr{H}^{n}(R\setminus
F)\leq\epsilon\mathscr{H}^{n}(R)$. Define

(5.4) $\tilde{R}=\Set{x\in R}{\begin{array}[]{l}\text{For all
$Q\in\mathscr{D}$ such that $x\in Q\subseteq R$, }\\\ \mathscr{H}^{n}(Q\cap
F)\geq(1-2\epsilon)\mathscr{H}^{n}(Q)\end{array}}.$

We have $\mathscr{H}^{n}(\tilde{R})\geq\epsilon\mathscr{H}^{n}(R)$.

###### Proof.

This proof is essentially contained in the proof of Lemma IV.2.2.38 in [DS93],
but we need to be more precise than them about the constant $\epsilon$. If
$x\in R\setminus\tilde{R}$, then $x$ is contained in some cube $Q$ such that
$\mathscr{H}^{n}(Q\cap F)\leq(1-2\epsilon)\mathscr{H}^{n}(Q)$. Let
$\\{Q_{i}\\}_{i}$ be a maximal disjoint family of such cubes so that
$R\setminus\tilde{R}=\bigcup_{i}Q_{i}$. Then

(5.5) $\displaystyle\mathscr{H}^{n}((R\setminus\tilde{R})\cap F)$
$\displaystyle=\sum_{i}\mathscr{H}^{n}(Q_{i}\cap
F)\leq(1-2\epsilon)\sum_{i}\mathscr{H}^{n}(Q_{i})$
$\displaystyle\leq(1-2\epsilon)\mathscr{H}^{n}(R\setminus\tilde{R})\leq(1-2\epsilon)\mathscr{H}^{n}(R).$

On the other hand,

(5.6) $\displaystyle\mathscr{H}^{n}((R\setminus\tilde{R})\cap F)$
$\displaystyle=\mathscr{H}^{n}((R\cap
F)\setminus\tilde{R})\geq\mathscr{H}^{n}(R\cap F)-\mathscr{H}^{n}(\tilde{R})$
$\displaystyle\geq(1-\epsilon)\mathscr{H}^{n}(R)-\mathscr{H}^{n}(\tilde{R}).$

Combining (5.5) and (5.6) and rearranging gives

$\mathscr{H}^{n}(\tilde{R})\geq(1-\epsilon)\mathscr{H}^{n}(R)-(1-2\epsilon)\mathscr{H}^{n}(R)=\epsilon\mathscr{H}^{n}(R).\qed$

While this lemma allows us to control the measure of the part of $X$ outside
of $\Sigma$, we will also use separate control of the maximal distance of
points in $Q\in\mathscr{D}(R)$ from $\Sigma$ as measured by the following
quantity.

######  Definition 5.5.

Let $(Z,d)$ be a metric space and suppose $X,Y\subseteq Z$. For $x\in X$ and
$0<r<\mathop{\mathrm{missing}}{diam}(X)$, define

$I_{X,Y}(x,r)=\frac{1}{r}\sup_{\begin{subarray}{c}y\in X\cap B(x,r)\\\
\mathop{\mathrm{missing}}{dist}(y,Y)\leq
r\end{subarray}}\mathop{\mathrm{missing}}{dist}(y,Y)$

The following lemma gives Carleson control over cubes where $I_{X,Y}$ is
large.

######  Lemma 5.6 ([BHS23] Lemma 4.2.6).

Let $(Z,d)$ be a metric space with $X,Y\subseteq Z$ Ahlfors
$(C_{0},n)$-regular subsets and $\mathscr{D}$ a system of Christ-David cubes
for $X$. For any $\delta>0$, the set
$\Set{Q\in\mathscr{D}}{I_{X,Y}(3B_{Q})>\delta}$ is
$C(C_{0},n,\delta)$-Carleson.

We can now define the good family of descendants of $R$ we want to consider.
Let $E=f^{-1}(X)$. For any $k\in\mathbb{N}$ and $\delta>0$, consider the
following three conditions applicable to
$Q\in\mathscr{D}(R)\subseteq\mathscr{D}(X)$:

  1. (i)

$\mathscr{H}^{n}(Q\setminus\Sigma)\leq\frac{\epsilon_{0}}{2}\mathscr{H}^{n}(Q)$,

  2. (ii)

$I_{X,\Sigma}(3B_{Q})\leq\delta$,

  3. (iii)

$\exists L\text{-good }I_{Q}\in\tilde{\mathcal{D}}(\mathbb{R}^{n})$ for which
the following hold:

    1. (a)

$\Delta_{k}^{\mathscr{J}_{f}\cdot\chi_{E}}(I_{Q})\leq\delta\ell(I_{Q})^{n/2}$,

    2. (b)

$\mathrm{md}_{f}(I_{Q})\leq\delta$.

Define

$\displaystyle\mathscr{G}_{R,f}(k,\delta)$
$\displaystyle=\Set{Q\in\mathscr{D}(R)}{\ Q\text{ satisfies \ref{i:big-int},
\ref{i:small-dist}, and \ref{i:good-IQ}}},$
$\displaystyle\mathscr{B}_{R,f}(k,\delta)$
$\displaystyle=\mathscr{D}\setminus\mathscr{G}_{R,f}(k,\delta).$

We first show that $\mathscr{G}_{R,f}(k,\delta)$ cubes are good for the WCD
for $X$. The reader should compare the following lemma with Lemma 4.10.

######  Lemma 5.7.

Let $X$ be uniformly $n$-rectifiable with regularity constant $C_{0}$, let
$\epsilon_{0}>0$, and let $R\in\mathscr{D}(X)$. Let
$f:\mathbb{R}^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz and satisfy
(5.3). There exist $k,\delta>0$ dependent on $C_{0},\epsilon_{0},n,L$ such
that
$\mathscr{G}_{R,f}(k,\delta)\subseteq\mathscr{G}_{\text{cd}}(2C_{0},\epsilon_{0})$.

###### Proof.

Let $Q\in\mathscr{G}_{R,f}(k,\delta)$, $y\in\frac{c_{0}}{2}B_{Q}$,
$0<r\leq\frac{c_{0}}{2}\ell(Q)$ and let $I_{Q}$ be the cube guaranteed from
condition (iii). Notice that $B(y,r)\subseteq c_{0}B_{Q}$. By condition (ii),
there exists $y_{0}\in\Sigma$ satisfying $|y-y_{0}|\leq 3\delta\ell(Q)$. Let
$r_{\pm}=r\pm 3\delta\ell(Q)$ so that

$B_{\Sigma}(y_{0},r_{-})\cap X\subseteq B_{X}(y,r)\cap\Sigma\subseteq
B_{\Sigma}(y_{0},r_{+})\cap X.$

Since $\mathrm{md}_{f}(I_{Q})\leq\delta$, the proof of Lemma 4.10,
specifically of (3.4), shows that there exists a norm $\|\cdot\|_{Q}$ and
$c_{1}(n,L)>0$ such that the balls

$\displaystyle B_{1}$
$\displaystyle=B_{\|\cdot\|_{Q}}(f^{-1}(y_{0}),(1-c_{1}\delta)r_{-}),$
$\displaystyle B_{2}$
$\displaystyle=B_{\|\cdot\|_{Q}}(f^{-1}(y_{0}),(1+c_{1}\delta)r_{+})$

satisfy $B_{1},B_{2}\in\mathcal{B}_{2L}(I_{Q})$ and

(5.7) $B_{1}\cap E\subseteq f^{-1}(B_{X}(y,r)\cap\Sigma)\subseteq B_{2}\cap
E.$

Let $\epsilon>0$. By taking $\delta$ small enough and $k$ large enough so that
the hypotheses of Lemma 3.5 are satisfied for $\mathscr{J}_{f}\chi_{E}$,
$\Delta_{k}^{\mathscr{J}_{f}\chi_{E}}(I_{Q})\leq\delta\ell(I_{Q})^{n/2}$ gives

$\left|\fint_{B}\mathscr{J}_{f}\chi_{E}-\fint_{Q}\mathscr{J}_{f}\chi_{E}\right|\leq\epsilon\left|\fint_{Q}\mathscr{J}_{f}\chi_{E}\right|$

for any normed ball $B\in\mathcal{B}_{2L}(I_{Q})$. Set
$c_{Q}=\left|\fint_{Q}\mathscr{J}_{f}\chi_{E}\right|$. After rearranging and
applying the area formula, this becomes

(5.8) $\left|\mathscr{H}^{n}(f(B\cap
E))-c_{Q}\mathscr{L}^{n}(B)\right|\leq\epsilon c_{Q}\mathscr{L}^{n}(B).$

Let $\mathscr{L}^{n}(B_{\|\cdot\|_{Q}}(0,r))=c_{\|\cdot\|_{Q}}r^{n}$ and set
$a_{Q}=c_{Q}c_{\|\cdot\|_{Q}}$. Combining (5.7) and (5.8) gives

$\displaystyle\mathscr{H}^{n}(B_{X}(y,r)\cap\Sigma)$
$\displaystyle\leq\mathscr{H}^{n}(f(B_{2}\cap E))\leq
c_{Q}(1+\epsilon)\mathscr{L}^{n}(B_{2})=a_{Q}(1+\epsilon)(1-c_{1}\delta)^{n}r_{+}^{n}$
$\displaystyle=a_{Q}(1+\epsilon)(1-c_{1}\delta)^{n}(r+3\delta\ell(Q))^{n}$
$\displaystyle\leq a_{Q}r^{n}+C(n,L)(\epsilon+\delta)\ell(Q)^{n}.$

A similar argument using $B_{1}$ gives a similar lower bound so that

$|\mathscr{H}^{n}(B_{X}(y,r)\cap\Sigma)-a_{Q}r^{n}|\leq
C(n,L)(\epsilon+\delta)\ell(Q)^{n}.$

Finally, using (i), we have

$\displaystyle|\mathscr{H}^{n}(B_{X}(y,r))-a_{Q}r^{n}|$
$\displaystyle\leq\mathscr{H}^{n}(B_{X}(y,r)\setminus\Sigma)+|\mathscr{H}^{n}(B_{X}(y,r))\cap\Sigma-
a_{Q}r^{n}|$ $\displaystyle\leq\mathscr{H}^{n}(c_{0}B_{Q}\cap
X\setminus\Sigma)+C(n,L)(\epsilon+\delta)\ell(Q)^{n}$
$\displaystyle\leq\frac{\epsilon_{0}}{2}\ell(Q)^{n}+C(n,L)(\epsilon+\delta)\ell(Q)^{n}\leq\epsilon_{0}\ell(Q)^{n}$

where the final inequality follows by first fixing $\epsilon$ sufficiently
small in terms of $\epsilon_{0},L,n$ then $\delta$ small and $k$ large in
terms of $\epsilon,\epsilon_{0},n,L$. ∎

We now show that $\mathscr{B}_{R,f}(k,\delta)$ is not too big. The reader
should compare this lemma with Lemma 4.11.

######  Lemma 5.8.

Let $X$ be uniformly $n$-rectifiable with regularity constant $C_{0}$, let
$\epsilon_{0},\delta,k>0$, and let $R\in\mathscr{D}(X)$. Let
$f:\mathbb{R}^{n}\twoheadrightarrow\Sigma$ be $L$-bi-Lipschitz and satisfy
(5.3). There exist constants $N,\eta>0$ dependent on
$k,\delta,\epsilon_{0},n,L$ such that

(5.9) $\mathscr{H}^{n}\left(\Set{x\in R}{\sum_{\begin{subarray}{c}Q\subseteq
R\\\ x\in Q\end{subarray}}\chi_{\mathscr{B}_{R,f}(k,\delta)}\leq
N}\right)\geq\eta\ell(R)^{n}.$

###### Proof.

Define

$\displaystyle\mathscr{B}_{1}$ $\displaystyle=\Set{Q\subseteq
R}{I_{X,\Sigma}(3B_{Q})>\delta},$ $\displaystyle\mathscr{B}_{2}$
$\displaystyle=\Set{Q\subseteq R}{\text{there is no $I_{Q}$ satisfying
\ref{i:good-IQ}}}.$

Lemma 5.6 shows that $\mathscr{B}_{1}$ is $C(\delta,C_{0},n)$-Carleson and
Lemma 4.11 shows that $\mathscr{B}_{2}$ is $C(k,\delta,n,L)$-Carleson. Let
$\tilde{R}$ be as in (5.4) and write $\tilde{R}=\bigcup_{i}Q_{i}$ for a
maximal disjoint set of cubes. By Chebyshev’s inequality,

$\displaystyle\mathscr{H}^{n}\left(\Set{x\in\tilde{R}}{\sum_{\begin{subarray}{c}Q\subseteq
R\\\ x\in Q\end{subarray}}\chi_{\mathscr{B}_{R,f}(k,\delta)}(Q)\geq N}\right)$
$\displaystyle\leq\frac{1}{N}\int_{\tilde{R}}\sum_{\begin{subarray}{c}Q\subseteq
R\\\ x\in Q\end{subarray}}\chi_{\mathscr{B}_{R,f}(k,\delta)}(Q)$
$\displaystyle\lesssim\frac{1}{N}\int_{\tilde{R}}\sum_{\begin{subarray}{c}Q\subseteq
Q_{i}\subseteq\tilde{R}\\\ x\in
Q\end{subarray}}\chi_{\mathscr{B}_{1}}(Q)+\chi_{\mathscr{B}_{2}}(Q)$
$\displaystyle\lesssim\frac{1}{N}\sum_{i}\left[\sum_{\begin{subarray}{c}Q\subseteq
Q_{i}\\\
Q\in\mathscr{B}_{a}\end{subarray}}\ell(Q)^{n}+\sum_{\begin{subarray}{c}Q\subseteq
Q_{i}\\\ Q\in\mathscr{B}_{b}\end{subarray}}\ell(Q)^{n}\right]$
$\displaystyle\lesssim_{k,\delta,C_{0},\epsilon_{0},n,L}\frac{1}{N}\sum_{i}\ell(Q_{i})^{n}\leq\frac{1}{N}\ell(R)^{n}.$

The result follows by taking $N$ sufficiently large and using Lemma 5.4. ∎

We finally observe that these pieces combine to prove Theorem A:

###### Proof of Theorem A.

Choose $R\in\mathscr{D}(X)$ and apply Theorem 5.2 to get an $L$-bi-Lipschitz
map $f:\mathbb{R}^{n}\twoheadrightarrow\Sigma\subseteq\ell_{\infty}$
satisfying (5.3). Fix $k$ large enough and $\delta$ small enough in terms of
$C_{0},\epsilon_{0},n,L$ so that
$\mathscr{G}_{R,f}(k,\delta)\subseteq\mathscr{G}_{\text{cd}}(2C_{0},\epsilon_{0})$.
That is, the conclusion of Lemma 5.7 holds. Then we have
$\mathscr{B}_{\text{cd}}(2C_{0},\epsilon_{0})\cap\mathscr{D}(R)\subseteq\mathscr{B}_{R,f}(k,\delta)$
so that
$\chi_{\mathscr{B}_{\text{cd}}(2C_{0},\epsilon_{0})}(Q)\leq\chi_{\mathscr{B}_{R,f}(k,\delta)}(Q)$
for all $Q\subseteq R$. Lemma 5.8 gives the existence of $N,\eta$ independent
of $R$ so that

$\mathscr{H}^{n}\left(\Set{x\in R}{\sum_{\begin{subarray}{c}Q\subseteq R\\\
x\in Q\end{subarray}}\chi_{\mathscr{B}_{\text{cd}}(2C_{0},\epsilon_{0})}\leq
N}\right)\geq\eta\ell(R)^{n}.$

By Lemma 5.1, this implies $\mathscr{B}_{\text{cd}}(2C_{0},\epsilon_{0})$ is
Carleson, implying $X$ satisfies the WCD. ∎

## References

  * [AH22] Jonas Azzam and Matthew Hyde.  The weak lower density condition and uniform rectifiability.  Ann. Fenn. Math., 47(2):791–819, 2022.
  * [AS14] Jonas Azzam and Raanan Schul.  A quantitative metric differentiation theorem.  Proc. Amer. Math. Soc., 142(4):1351–1357, 2014.
  * [Bat23] David Bate.  On 1-regular and 1-uniform metric measure spaces.  Youtube, 2023.  https://www.youtube.com/watch?v=9zXjfKeWGv4&t=2096s.
  * [BHS23] David Bate, Matthew Hyde, and Raanan Schul.  Uniformly rectifiable metric spaces: Lipschitz images, bi-lateral weak geometric lemma and corona decompositions.  arXiv preprint, arXiv:2306.12933, 2023.
  * [CGLT16] Vasileios Chousionis, John Garnett, Triet Le, and Xavier Tolsa.  Square functions and uniform rectifiability.  Trans. Amer. Math. Soc., 368(9):6063–6102, 2016.
  * [Chr90] M. Christ.  A T(b) theorem with remarks on analytic capacity and the Cauchy integral.  Colloq. Math., 60/61(2):601–628, 1990.
  * [CMT20] Vasilis Chousionis, Valentino Magnani, and Jeremy T. Tyson.  On uniform measures in the Heisenberg group.  Adv. Math., 363:106980, 42, 2020.
  * [Dav88] Guy David.  Morceaux de graphes lipschitziens et intégrales singulières sur une surface.  Rev. Mat. Iberoamericana, 4(1):73–114, 1988.
  * [DL08] Camillo De Lellis.  Rectifiable sets, densities and tangent measures.  Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.
  * [DS91] Guy David and Stephen Semmes.  Singular integrals and rectifiable sets in ${\bf R}^{n}$: Beyond Lipschitz graphs.  Astérisque, (193):152, 1991.
  * [DS93] Guy David and Stephen Semmes.  Analysis of and on uniformly rectifiable sets, volume 38 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1993.
  * [FV23] Katrin Fassler and Ivan Yuri Violo.  On various carleson-type geometric lemmas and uniform rectifiability in metric spaces.  arXiv preprint, arXiv:2310.10519, 2023.
  * [Hah05] Immo Hahlomaa.  Menger curvature and Lipschitz parametrizations in metric spaces.  Fund. Math., 185(2):143–169, 2005.
  * [HM12] T. Hytönen and H. Martikainen.  Non-homogeneous ${T}b$ theorem and random dyadic cubes on metric measure spaces.  J. Geom. Anal., 22(4):1071–1107, 2012.
  * [Kir94] Bernd Kirchheim.  Rectifiable metric spaces: local structure and regularity of the Hausdorff measure.  Proc. Amer. Math. Soc., 121(1):113–123, 1994.
  * [KP87] Oldřich Kowalski and David Preiss.  Besicovitch-type properties of measures and submanifolds.  J. Reine Angew. Math., 379:115–151, 1987.
  * [KP02] Bernd Kirchheim and David Preiss.  Uniformly distributed measures in Euclidean spaces.  Math. Scand., 90(1):152–160, 2002.
  * [Ler03] Gilad Lerman.  Quantifying curvelike structures of measures by using $l_{2}$ Jones quantities.  Comm. Pure Appl. Math., 56(9):1294–1365, 2003.
  * [Lor03] Andrew Lorent.  Rectifiability of measures with locally uniform cube density.  Proc. London Math. Soc. (3), 86(1):153–249, 2003.
  * [Mer22] Andrea Merlo.  Geometry of 1-codimensional measures in Heisenberg groups.  Invent. Math., 227(1):27–148, 2022.
  * [Nim17] A. Dali Nimer.  A sharp bound on the Hausdorff dimension of the singular set of a uniform measure.  Calc. Var. Partial Differential Equations, 56(4):Paper No. 111, 31, 2017.
  * [Nim19] A. Dali Nimer.  Uniformly distributed measures have big pieces of Lipschitz graphs locally.  Ann. Acad. Sci. Fenn. Math., 44(1):389–405, 2019.
  * [Nim22] A. Dali Nimer.  Conical 3-uniform measures: a family of new examples and characterizations.  J. Differential Geom., 121(1):57–99, 2022.
  * [Oki92] Kate Okikiolu.  Characterization of subsets of rectifiable curves in $\mathbf{R}^{n}$.  Journal of the London Mathematical Society, s2-46(2):336–348, 1992.
  * [Pre87] David Preiss.  Geometry of measures in ${\bf R}^{n}$: distribution, rectifiability, and densities.  Ann. of Math. (2), 125(3):537–643, 1987.
  * [Sch07] Raanan Schul.  Ahlfors-regular curves in metric spaces.  Ann. Acad. Sci. Fenn. Math., 32(2):437–460, 2007.
  * [Sch09] Raanan Schul.  Bi-Lipschitz decomposition of Lipschitz functions into a metric space.  Rev. Mat. Iberoam., 25(2):521–531, 2009.
  * [Tol12] Xavier Tolsa.  Mass transport and uniform rectifiability.  Geom. Funct. Anal., 22(2):478–527, 2012.
  * [Tol15] Xavier Tolsa.  Uniform measures and uniform rectifiability.  J. Lond. Math. Soc. (2), 92(1):1–18, 2015.
  * [TT15] Xavier Tolsa and Tatiana Toro.  Rectifiability via a square function and Preiss’ theorem.  Int. Math. Res. Not. IMRN, (13):4638–4662, 2015.

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651

E-mail address<EMAIL_ADDRESS>
Jared Krandel was partially supported by the National Science Foundation under
Grants No. DMS-1763973.