Necessary condition for rectifiability involving Wasserstein distance $W_2$

TL;DR

This paper establishes a necessary condition for the rectifiability of Radon measures using Wasserstein distance-based flatness coefficients, complementing previous sufficiency results to characterize rectifiable measures.

Contribution

It introduces a new necessary condition for rectifiability involving Wasserstein distance, completing a characterization when combined with prior sufficiency results.

Findings

Necessary condition for rectifiability via $eta_2$ coefficients

Characterization of rectifiable measures using Wasserstein distance

Bridging flatness coefficients with geometric measure theory

Abstract

A Radon measure $\mu$ is $n$-rectifiable if $\mu\ll\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give a necessary condition for rectifiability in terms of the so-called $\alpha_2$ numbers -- coefficients quantifying flatness using Wasserstein distance $W_2$. In a recent article we showed that the same condition is also sufficient for rectifiability, and so we get a new characterization of rectifiable measures.

Equations

$$\mu ({\mathbb{R}}^d\setminus \bigcup _i{f}_i({\mathbb{R}}^n))=0,$$

$$W_p(\mu ,\nu )={\left(\underset{\pi }{\inf }{\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}|x-y{|}^{p}d\pi (x,y)\right)}^{1/p},$$

$$c^{-1}\mathrm{dist}(x,\partial B(0,3){)}^2\le \phi (x)\le c\mathrm{dist}(x,\partial B(0,3){)}^2,$$

$$|\nabla \phi (x)|\le c\mathrm{dist}(x,\partial B(0,3)),$$

$${\phi }_B(y)=\phi \left(\frac{y-x}{r}\right).$$

$${\alpha }_{\mu ,p,L}(B)=\frac{1}{r\mu (B{)}^{1/p}}{W}_p({\phi }_B\mu ,a_{B,L}{\phi }_B{{\mathcal{H}}^n|}_L),$$

$${\alpha }_{\mu ,p}(B)=\underset{L}{\inf }{\alpha }_{\mu ,p,L}(B),$$

$$\mu\big{(}B(x,r)\cap g(B^{n}(0,r))\big{)}\geq\theta r^{n}.$$

$${\int }_0^R{\int }_B{\alpha }_{\mu ,p}(x,r{)}^{2}d\mu (x)\frac{dr}{r}\le C{R}^n.$$

$${\int }_0^1{\alpha }_{\mu ,2}(x,r{)}^2\frac{dr}{r}<\infty .$$

$$\alpha_{\mu,p}(B)\leq\bigg{(}\frac{\mu(3B)}{\mu(B)}\bigg{)}^{1/p-1/q}\alpha_{\mu,q}(B).$$

$$\underset{r\to 0^{+}}{\mathrm{limsup}}\frac{\mu (B(x,2r))}{\mu (B(x,r))}<\infty \text{for μ-a.e. x∈Rd,}$$

$${\beta }_{\mu ,p}(x,r)=\underset{L}{\inf }{\left(\frac{1}{r^n}{\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,L)}{r}\right)}^{p}d\mu (y)\right)}^{1/p},$$

$${\Theta }^{n,\ast }(x,\mu )=\underset{r\to 0^{+}}{\mathrm{limsup}}\frac{\mu (B(x,r))}{r^n},{\Theta }_{\ast }^n(x,\mu )=\underset{r\to 0^{+}}{\mathrm{liminf}}\frac{\mu (B(x,r))}{r^n},$$

$${\int }_0^1{\beta }_{\mu ,2}(x,r{)}^2\frac{dr}{r}<\infty \text{for μ-a.e. x∈Rd.}$$

$$0<{\Theta }^{n,\ast }(x,\mu )<\infty \text{for μ-a.e. x∈Rd,}$$

$${\Theta }^{n,\ast }(x,\mu )>0\text{and}{\Theta }_{\ast }^n(x,\mu )<\infty \text{for μ-a.e. x∈Rd.}$$

$$\begin{gathered} b{\beta }_{\mu ,2}(x,r{)}^2= \\ \underset{L}{\inf }\frac{1}{r^n}\left({\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,L)}{r}\right)}^{2}d\mu (y)+{\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,\mathrm{supp}\mu )}{r}\right)}^{2}d{{\mathcal{H}}^n|}_L(y)\right). \end{gathered}$$

$${\int }_0^{1}b{\beta }_{\mu ,2}(x,r{)}^2\frac{dr}{r}<\infty .$$

$$F_B(\mu ,\nu )=\sup \left\{\left|\int \varphi d\mu -\int \varphi d\nu \right|:\varphi \in {\text{Lip}}_1(B)\right\},$$

$${\mathrm{Lip}}_1(B)=\{\varphi :\mathrm{Lip}(\varphi )\le 1,\mathrm{supp}\varphi \subset B\}.$$

$${\alpha }_{\mu }(B)=\underset{c,L}{\inf }\frac{1}{r\mu (B)}{F}_B(\mu ,c{{\mathcal{H}}^n|}_L),$$

$${\int }_0^1{\alpha }_{\mu }(x,r{)}^2\frac{dr}{r}<\infty \text{for μ-a.e. x∈Rd.}$$

$${\int }_{R^{\prime }}{\int }_0^1{\alpha }_{\mu ,2}(x,r{)}^2\frac{dr}{r}d\mu (x)<\infty .$$

$$\mu\big{(}\mathbb{R}^{d}\setminus\bigcup_{i}\Gamma_{i}\big{)}=0.$$

$$\sum _{\begin{array}{c}Q\in \mathsf{Tree}\end{array}}{\alpha }_{{\nu }_Q,2}({\tilde{B}}_Q{)}^2\ell (Q{)}^n<\infty ,$$

$$\sum _{\begin{array}{c}Q\in \mathsf{Tree}\end{array}}{\alpha }_{\mu ,2}({\tilde{B}}_Q{)}^2\ell (Q{)}^n<\infty .$$

$$\measuredangle (L_1,L_2)={\mathrm{dist}}_H(L_1^{\prime }\cap B(0,1),L_2^{\prime }\cap B(0,1)),$$

$$T_{\ast }\nu (A)=\nu (T^{-1}(A)).$$

$$a=\frac{{\mu }_1({\mathbb{R}}^d)}{{\mu }_2({\mathbb{R}}^d)}.$$

Full text

1. Introduction

Let $1\leq n\leq d$ be integers. We say that a Radon measure $\mu$ on $\mathbb{R}^{d}$ is $n$-rectifiable if there exist countably many Lipschitz maps $f_{i}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$ such that

[TABLE]

and moreover $\mu$ is absolutely continuous with respect to $n$-dimensional Hausdorff measure $\mathscr{H}^{n}$. A set $E\subset\mathbb{R}^{d}$ is $n$-rectifiable if the measure ${\left.\kern-1.2pt\mathscr{H}^{n}\right|_{E}}$ is $n$-rectifiable. We will often omit $n$ and just write “rectifiable”.

The study of rectifiable sets and measures lies at the very heart of geometric measure theory. We refer the reader to [Mat95, Chapters 15–18] for some classical characterizations of rectifiability involving densities, tangent measures, and projections. The aim of this paper is to prove a necessary condition for rectifiability involving the so-called $\alpha_{2}$ coefficients.

1.1. $\alpha_{p}$ numbers

Coefficients $\alpha_{p}$ were introduced by Tolsa in [Tol12]. In order to define them, we recall the definition of Wasserstein distance.

Let $1\leq p<\infty$, and let $\mu,\nu$ be two probability Borel measures on $\mathbb{R}^{d}$ satisfying $\int|x|^{p}\ d\mu<\infty,\ \int|x|^{p}\ d\nu<\infty$. The Wasserstein distance $W_{p}$ between $\mu$ and $\nu$ is defined as

[TABLE]

where the infimum is taken over all transport plans between $\mu$ and $\nu$, i.e. Borel probability measures $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ satisfying $\pi(A\times\mathbb{R}^{d})=\mu(A)$ and $\pi(\mathbb{R}^{d}\times A)=\nu(A)$ for all measurable $A\subset\mathbb{R}^{d}$. The same definition makes sense if instead of probability measures we consider $\mu,\ \nu,\ \pi$ of the same total mass.

Wasserstein distances are a way to measure the cost of transporting one measure to another, and they are of fundamental importance to the theory of optimal transport. For more information see for example [Vil03, Chapter 7] or [Vil08, Chapter 6].

The idea behind $\alpha_{p}$ numbers is to quantify how far a given measure is from being a flat measure, that is, from being of the form $c{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L}}$ for some constant $c>0$ and some $n$-plane $L$. In order to measure it locally (say, in a ball $B$), we introduce the following auxiliary function.

Let $\varphi\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\rightarrow[0,1]$ be a radial Lipschitz function satisfying $\varphi\equiv 1$ in $B(0,2)$, $\operatorname{supp}\varphi\subset B(0,3)$, and for all $x\in B(0,3)$

[TABLE]

for some constant $c>0$. For example, one could take $\varphi(x)=\phi(|x|)$ where $\phi\mathrel{\mathop{\ordinarycolon}}[0,\infty)\to[0,1]$ is such that $\phi(r)=1$ for $0\leq r\leq 2$, $\phi(r)=0$ for $r\geq 3$, and $\phi(r)=(3-r)^{2}$ for $2<r<3$. Given a ball $B=B(x,r)\subset\mathbb{R}^{d}$ we set

[TABLE]

$\varphi_{{B}}$ can be seen as a regularized characteristic function of $B$.

For $1\leq p<\infty$, a Radon measure $\mu$ on $\mathbb{R}^{d}$, a ball $B=B(x,r)\subset\mathbb{R}^{d}$ with $\mu(B)>0$, and an $n$-plane $L$ intersecting $B$, we define

[TABLE]

where $a_{{B,L}}=(\int\varphi_{B}\ d\mu)/(\int\varphi_{B}\ d{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L}})$. We will usually omit the subscripts and just write $a$. We define also

[TABLE]

where the infimum is taken over all $n$-planes $L$ intersecting $B$. For a ball $B=B(x,r)$ we will sometimes write $\alpha_{\mu,p}(x,r)$ instead of $\alpha_{\mu,p}(B)$, and we will do the same with all the other coefficients introduced below.

Coefficients $\alpha_{p}$ were first defined in [Tol12] with the aim of characterizing uniformly rectifiable measures. The notion of uniform rectifiability, which can be seen as a more quantitative counterpart of rectifiability, was introduced by David and Semmes in [DS91, DS93]. We say that a measure $\mu$ is uniformly $n$-rectifiable if:

• (i)

it is $n$-AD-regular, i.e. there exists a constant $C$ such that for all $x\in\operatorname{supp}\mu$ and $0<r<\operatorname{diam}(\operatorname{supp}\mu)$ we have $C^{-1}r^{n}\leq\mu(B(x,r))\leq Cr^{n}$,

• (ii)

it has big pieces of Lipschitz images, i.e. there exist constants $\theta,L>0$ such that for any $x\in\operatorname{supp}\mu$ and $0<r<\operatorname{diam}(\operatorname{supp}\mu)$ we may find an $L$-Lipschitz mapping $g$ from the $n$-dimensional ball $B^{n}(0,r)\subset\mathbb{R}^{n}$ into $\mathbb{R}^{d}$ satisfying

[TABLE]

A trivial example of a uniformly rectifiable measure is the surface measure on a Lipschitz graph.

In [Tol12] Tolsa showed the following characterization of uniformly rectifiable measures:

Theorem 1.1** ([Tol12, Theorem 1.2]).**

Let $1\leq p\leq 2$. Suppose $\mu$ is an $n$-AD-regular measure on $\mathbb{R}^{d}$. Then, $\mu$ is uniformly rectifiable if and only if there exists $C>0$ such that for any ball $B=B(z,R)$ centered at $\operatorname{supp}\mu$ we have

[TABLE]

In this paper we prove a necessary condition for rectifiability of measures which is of similar spirit.

Theorem 1.2**.**

Let $\mu$ be an $n$-rectifiable measure on $\mathbb{R}^{d}$. Then for $\mu$-a.e. $x\in\mathbb{R}^{d}$

[TABLE]

In [Dąb19, Theorem 1.3] we show that (1.4) is also a sufficient condition for rectifiability (we use a slightly different version of $\alpha_{2}$, but it does not matter, see Remark 1.5). Putting the two results together, we get the following characterization.

Corollary 1.3**.**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then $\mu$ is $n$-rectifiable if and only if for $\mu$-a.e. $x\in\mathbb{R}^{d}$ we have

[TABLE]

Remark 1.4**.**

The characterization above is sharp in the following sense. Suppose $1\leq p\leq q<\infty$. Then it follows easily by Hölder’s inequality, definition of $\alpha_{p}$ numbers, and the fact that $\operatorname{supp}\varphi_{B}\subset 3B$, that

[TABLE]

Hence, for doubling measures, $\alpha_{p}$ numbers are increasing in $p$. It is well known that rectifiable measures are pointwise doubling, i.e.

[TABLE]

and so the finiteness of $\alpha_{2}$ square function (1.4) implies finiteness of $\alpha_{p}$ square function for any $1\leq p\leq 2.$ However, in general one cannot expect finiteness of $\alpha_{p}$ square function for $p>2$, see Remark 1.7. In other words, Theorem 1.2 cannot be improved.

Remark 1.5**.**

For technical reasons, in [Dąb19] we define $\alpha_{p}$ numbers normalizing by $\mu(3B)$ (i.e. in (1.3) we replace $\mu(B)$ with $\mu(3B)$). Of course, the $3B$-normalized coefficients are smaller than the $B$-normalized variant used here. Hence, if (1.4) is finite for $B$-normalized $\alpha_{2}$ numbers, then it is finite for $3B$-normalized $\alpha_{2}$ numbers, and so [Dąb19, Theorem 1.3] may be applied to get Corollary 1.3.

It is worthwhile to compare this result with other recent characterizations of rectifiability which all involve some sort of scale-invariant quantities measuring flatness.

1.2. $\beta_{p}$ numbers

The first flatness-quantifying coefficients to be defined were Jones’ $\beta$ numbers, originating in [Jon90, DS91, DS93]. For $1\leq p<\infty$ and a Radon measure $\mu$ on $\mathbb{R}^{d}$ set

[TABLE]

where the infimum runs over all $n$-planes $L$ intersecting $B(x,r)$. Let us also define upper and lower $n$-dimensional densities of a Radon measure $\mu$ at $x\in\mathbb{R}^{d}$ as

[TABLE]

respectively. If both quantities are equal, we set $\Theta^{n}(x,\mu)=\Theta^{n,*}(x,\mu)=\Theta_{*}^{n}(x,\mu)$ and we call it $n$-dimensional density.

In [Tol15] it was shown that for a rectifiable measure $\mu$ we have

[TABLE]

On the other hand, Azzam and Tolsa proved in [AT15] that if a Radon measure $\mu$ satisfies (1.7) and

[TABLE]

then $\mu$ is $n$-rectifiable. More recently, Edelen, Naber and Valtorta [ENV16] managed to weaken the assumption (1.8) to

[TABLE]

An alternative proof showing that (1.7) and (1.9) are sufficient for rectifiability is given in [Tol19].

Theorem 1.6** ([Tol15, AT15, ENV16]).**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then, $\mu$ is $n$-rectifiable if and only if (1.7) and (1.9) hold for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

Contrary to Corollary 1.3, some sort of assumptions on densities of measure seem to be unavoidable because $\beta_{2}$ numbers are “weaker” than $\alpha_{2}$ numbers (see Lemma 3.1). What we mean by that is the following: coefficients $\beta_{p}$ measure how close is $\operatorname{supp}\mu$ to being contained in an $n$-plane, and so they do not see holes or high concentrations of measure. Any measure with support contained in an $n$-plane will have all $\beta$ numbers equal to 0 – even Dirac mass! Moreover, due to the normalizing factor $r^{n}$ in (1.6), $\beta$ numbers do not charge higher dimensional measures properly (note that the $(n+1)$-dimensional Lebesgue measure satisfies (1.7)). Coefficients $\alpha_{p}$, on the other hand, penalize such phenomena.

The choice of $p=2$ in the above considerations is not arbitrary. Condition (1.7) with $\beta_{\mu,2}(x,r)$ replaced by $\beta_{\mu,p}(x,r)$ is necessary for rectifiability only for $1\leq p\leq 2$. On the other hand, (1.7) together with (1.8) imply rectifiability only for $p\geq 2$. See [Tol19] for relevant counterexamples. Still, if instead of (1.8) we assume that $\Theta_{*}^{n}(\mu,x)>0$ and $\Theta^{n,*}(\mu,x)<\infty$ for $\mu$-a.e. $x\in\mathbb{R}^{d}$, then the finiteness of $\beta_{p}$ square function for certain $p<2$ becomes sufficient for rectifiability, see [Paj97, BS16].

Remark 1.7**.**

The example from [Tol19] shows that one cannot expect finiteness of the $\alpha_{p}$ square function when $p>2$. Indeed, it is easy to see that $\alpha_{p}$ numbers bound from above $\beta_{p}$ numbers (see Lemma 3.1, the same proof works with arbitrary $1\leq p<\infty$). Tolsa gave an example of a rectifiable measure such that for all $p>2$ the square function involving $\beta_{p}$ in infinite almost everywhere. Hence, the $\alpha_{p}$ square function of that measure is also infinite almost everywhere.

Let us mention that modified versions of $\beta$ numbers are also used to study a competing notion of rectifiability for measures, the so-called Federer rectifiability. We say that a measure is $n$-rectifiable in the sense of Federer if it satisfies (1.1), and no absolute continuity with respect to $\mathscr{H}^{n}$ is required. Dropping the absolute continuity assumption makes such measures very difficult to characterize. A surprising example of a doubling, Federer $1$-rectifiable measure supported on the whole plane was found by Garnett, Killip and Schul [GKS10]. Nevertheless, for $n=1$ significant progress has been achieved in [Ler03, BS15, BS16, AM16, BS17, MO18]. See also a recent survey of Badger [Bad19].

Theorem 1.2 yields an easy corollary involving bilateral $\beta$ numbers. Set

[TABLE]

As shown in Lemma 3.1, if a ball $B(x,r)$ satisfies $\mu(B(x,r))\approx r^{n}$ (see Subsection 2.1 for the precise meaning of $\approx$ symbol), then coefficients $\alpha_{\mu,2}(x,r)$ bound from above $b\beta_{\mu,2}(x,r)$. Since for $n$-rectifiable measure $\mu$ we have $0<\Theta^{n}(\mu,x)<\infty$ $\mu$-almost everywhere, we immediately get the following.

Corollary 1.8**.**

Let $\mu$ be an $n$-rectifiable measure on $\mathbb{R}^{d}$. Then for $\mu$-a.e. $x\in\mathbb{R}^{d}$ we have

[TABLE]

1.3. $\alpha$ numbers

Another kind of coefficients quantifying flatness that has attracted a lot of interest are $\alpha$ numbers, first introduced in [Tol09]. Their definition is very similar to that of $\alpha_{p}$ coefficients, and in fact they can be seen as a variant of $\alpha_{1}$ numbers, see [Tol12, Section 5].

Like before, we define a distance on the space of Radon measures. Given Radon measures $\mu,\nu$, and an open ball $B$ we set

[TABLE]

where

[TABLE]

The coefficient $\alpha$ of a measure $\mu$ in a ball $B=B(x,r)$ is defined as

[TABLE]

where the infimum is taken over all $n$-planes $L$ and all $c\geq 0$ (we do not demand a priori that $\mu(B)=c{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L}}(B)$).

Tolsa showed in [Tol15] that given a rectifiable measure $\mu$ we have

[TABLE]

One might ask if (1.10) is also a sufficient condition for rectifiability. Partial answers to that question were given in [ADT16] and [Orp18]. Very recently Azzam, Tolsa and Toro [ATT18] proved that a measure $\mu$ satisfying (1.10) which is also pointwise doubling, i.e. such that (1.5) holds, is rectifiable. Since rectifiable measures satisfy (1.5), the following characterization holds.

Theorem 1.9** ([Tol15, ATT18]).**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then $\mu$ is $n$-rectifiable if and only if (1.10) and (1.5) hold for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

In the same paper authors construct a purely unrectifiable measure satisfying (1.10), and so the pointwise doubling assumption (1.5) cannot be omitted. Let us remark that in the characterization from Corollary 1.3 we do not need to assume any doubling property.

We mention briefly yet another kind of square functions used to describe rectifiability. [TT15] and [Tol17] are devoted to the so-called $\Delta$ numbers, defined as $\Delta_{\mu}(x,r)=|\frac{\mu(B(x,r))}{r^{n}}-\frac{\mu(B(x,2r))}{(2r)^{n}}|$. The results from [TT15] characterize rectifiable measures satisfying $0<\Theta^{n}_{*}(\mu,x)\leq\Theta^{n,*}(\mu,x)<\infty$ for $\mu$-a.e $x\in\mathbb{R}^{d}$. In [Tol17] it was shown that for $n=1$ analogous results hold under the weaker assumption $0<\Theta^{1,*}(x,\mu)<\infty$ for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

1.4. Localizing Theorem 1.2 and Organization of the Paper

Theorem 1.2 follows easily from the following lemma.

Lemma 1.10**.**

Let $\mu$ be an $n$-rectifiable measure on $\mathbb{R}^{d}$, and let $\Gamma\subset\mathbb{R}^{d}$ be an $n$-dimensional $1$-Lipschitz graph. Suppose $R\in\mathcal{D}_{\Gamma}$ with $\ell(R)=1$ (see (2.2) for the defintion of $\mathcal{D}_{\Gamma}$). Then, for any $\varepsilon>0$, there exists a set $R^{\prime}\subset R$ such that $\mu(R^{\prime})\geq(1-\varepsilon)\mu(R)$ and

[TABLE]

Proof of Theorem 1.2 using Lemma 1.10.

Let $\mu$ be $n$-rectifiable. It is well known that if one replaces Lipschitz images in (1.1) by Lipschitz graphs, or $C^{1}$ manifolds, the definition of rectifiability remains unchanged (see e.g. [Mat95, Theorem 15.21]). Each $C^{1}$ manifold is contained in a countable union of (possibly rotated) Lipschitz graphs $\Gamma$ with $\operatorname{\text{Lip}}(\Gamma)\leq 1$. Hence, there exists a countable family of $n$-dimensional $1$-Lipschitz graphs $\Gamma_{i}$ such that

[TABLE]

Each $\Gamma_{i}$ is a countable union of dyadic $\Gamma_{i}$-cubes $R_{i}^{j}\in\mathcal{D}_{\Gamma_{i}}$ satisfying $\ell(R_{i}^{j})=1$. Clearly, $\mu(\mathbb{R}^{d}\setminus\bigcup_{i,j}R_{i}^{j})=0$.

Now, denote the set of $x$ where (1.4) does not hold by $\mathcal{B}$, and suppose that $\mu(\mathcal{B})>0$. Then, there exists $R_{i}^{j}$ such that $\mu(\mathcal{B}\cap R_{i}^{j})>0$. Let $\varepsilon>0$ be such that $\mu(\mathcal{B}\cap R_{i}^{j})>2\varepsilon\mu(R_{i}^{j})$. Applying Lemma 1.10 to $R_{i}^{j}$ and $\varepsilon$ as above we reach a contradiction. Thus, $\mu(\mathcal{B})=0$. ∎

The rest of the article is dedicated to proving Lemma 1.10. Let us give a brief outline of the proof.

We introduce the necessary tools in Section 2. In Section 3 we show various estimates of $\alpha_{2}$ coefficients, usually relying heavily on the results from [Tol12]. In Section 4 we define a family of measures $\{\nu_{Q}\}_{Q\in\mathcal{D}_{\Gamma}}$, where $\nu_{Q}\ll{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{\Gamma}}$, and each $\nu_{Q}$ approximates $\mu$ in some ball around $Q$. Roughly speaking, $\nu_{Q}$ is defined by projecting the measure of Whitney cubes onto the graph $\Gamma$ – but only those Whitney cubes whose sidelength is not much bigger than $\ell(Q)$. Then, we construct a tree of good cubes satisfying

[TABLE]

where $\widetilde{B}_{Q}$ are balls with the same center as the corresponding cube $Q$. The stopping region of the tree of good cubes is small. In Section 5 we use the estimate above to show that actually

[TABLE]

Using the inequality above, we prove (1.11) with $R^{\prime}=R\setminus\bigcup_{Q\in\mathsf{Stop}(\mathsf{Tree})}Q$. This finishes the proof of Lemma 1.10.

Acknowledgements

The author would like to thank Xavier Tolsa for all his help and guidance. He acknowledges the support of the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), and also partial support from 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MINECO, Spain).

2. Preliminaries

2.1. Notation

Throughout the paper we will write $A\lesssim B$ whenever $A\leq CB$ for some constant $C$, the so-called “implicit constant”. All such implicit constants may depend on dimensions $n,d$, and we will not track this dependence. If the implicit constant depends also on some other parameter $t$, we will write $A\lesssim_{t}B$. The notation $A\approx B$ means $A\lesssim B\lesssim A$, and $A\approx_{t}B$ means $A\lesssim_{t}B\lesssim_{t}A$. Moreover, if symbols $\lesssim$ or $\approx$ appear in the assumptions of a lemma, then the implicit constant of the proven estimate will depend on the implicit constants from the assumptions (see Lemma 3.1 for example).

We denote by $B(z,r)\subset\mathbb{R}^{d}$ an open ball with center at $z\in\mathbb{R}^{d}$ and radius $r>0$. Given a ball $B$, its center and radius are denoted by $z(B)$ and $r(B)$, respectively. If $\lambda>0$, then $\lambda B$ is defined as a ball centered at $z(B)$ of radius $\lambda r(B)$.

Given two $n$-planes $L_{1},L_{2}$, let $L_{1}^{\prime}$ and $L_{2}^{\prime}$ be the respective parallel $n$-planes passing through [math]. Then,

[TABLE]

where $\operatorname{dist}_{H}$ stands for Hausdorff distance between two sets. $\measuredangle(L_{1},L_{2})$ can be seen as a “sine of the angle between $L_{1}$ and $L_{2}$,” and we always have $\measuredangle(L_{1},L_{2})\in[0,1]$.

Given an $n$-plane $L$, we will denote the orthogonal projection onto $L$ by $\Pi_{L}$.

For a Borel measure $\nu$ on $\mathbb{R}^{d}$ and a Borel map $T\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\mathbb{R}^{d}$, we denote by $T_{*}\nu$ the pushforward of $\nu$, that is, a measure on $\mathbb{R}^{d}$ such that for all Borel $A\subset\mathbb{R}^{d}$

[TABLE]

In expressions of the form $W_{p}(\mu_{1},a\mu_{2})$, the letter $a$ will always mean the unique constant for which the total mass of $a\mu_{2}$ is equal to that of $\mu_{1}$. In other words,

[TABLE]

It may happen that $a$ appears in the same line several times, and every time refers to a different quantity. We hope that this will not cause too much confusion.

Let us once and for all fix measure $\mu$, an $n$-dimensional $1$-Lipschitz graph $\Gamma$, and a small constant $0<\varepsilon\ll 1$ for which we are proving Lemma 1.10. We fix also a coordinate system such that $\Gamma=\{(x,A(x))\mathrel{\mathop{\ordinarycolon}}x\in\mathbb{R}^{n}\}\subset\mathbb{R}^{d},$ where $A\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\rightarrow\mathbb{R}^{d-n}$ is a $1$-Lipschitz map.

We will denote by $L_{0}$ the subspace of $\mathbb{R}^{d}$ formed by the points whose last $d-n$ coordinates are zeros, so that $\Gamma$ is a graph over $L_{0}$. We will write $\Pi_{0}$ and $\Pi_{\Gamma}$ to denote projections onto $L_{0}$ and $\Gamma,$ respectively, orthogonal to $L_{0}$. For the sake of convenience, instead of dealing with the usual surface measure on $\Gamma$ we will work with

[TABLE]

which is comparable to ${\left.\kern-1.2pt\mathscr{H}^{n}\right|_{\Gamma}}$.

Given a ball $B\subset\mathbb{R}^{d}$ centered at $\Gamma$ denote by $L_{B}$ an $n$-plane minimizing $\alpha_{\sigma,2}(B)$ (note that for an open ball $B$, it could happen that $L_{B}\cap B=\varnothing$). Concerning the existence of minimizers, it follows easily from the fact that $W_{2}$ metrizes weak convergence of measures (see e.g. [Vil08, Theorem 6.9]), from good compactness properties of weak convergence, and from the fact that the minimizing sequence is of the special form $\varphi_{B}a_{B,L_{k}}{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L_{k}}}$. There may be more than one minimizing plane; if that happens, we simply choose one of them.

For any Radon measure $\nu$ such that $\nu(B)>0$ we set

[TABLE]

Clearly, $\widehat{\alpha}_{\nu,2}(B)\geq\alpha_{\nu,2}(B)$. We will show that

[TABLE]

which implies (1.11).

2.2. $\Gamma$-cubes

We denote by $\mathcal{D}_{\mathbb{R}^{n}},\mathcal{D}_{\mathbb{R}^{d}}$ the dyadic lattices on $L_{0}$ and $\mathbb{R}^{d}$, respectively. We assume the cubes to be half open-closed, i.e. of the form

[TABLE]

where $i=n$ for $\mathcal{D}_{\mathbb{R}^{n}}$, $i=d$ for $\mathcal{D}_{\mathbb{R}^{d}}$, and $k_{1},\ \dots,\ k_{i},\ j,$ are arbitrary integers. The sidelength of $Q$ as above will be denoted by $\ell(Q)=2^{-j}$.

The dyadic lattice on $\Gamma$ is defined as

[TABLE]

The elements of $\mathcal{D}_{\Gamma}$ will be called $\Gamma$-cubes, or just cubes. For every $Q\in\mathcal{D}_{\Gamma}$ and the corresponding $Q_{0}\in\mathcal{D}_{\mathbb{R}^{n}}$ we define the sidelength of $Q$ as $\ell(Q)=\ell(Q_{0})$, and the center of $Q$ as $z_{Q}=\Pi_{\Gamma}(z_{Q_{0}}),$ where $z_{Q_{0}}$ is the center of $Q_{0}$. We set

[TABLE]

where $\Lambda=\Lambda(n)>1$ is a constant fixed during the proof. We define also

[TABLE]

Recall that $L_{B_{Q}}$ is the $n$-plane minimizing $\alpha_{\sigma,2}(B_{Q})$, and that $\varphi_{B_{Q}}$ was defined in (1.2). The “$V$” in $V(Q)$ stands for “vertical”, since $V(Q)$ is a sort of vertical cube. Note also that $Q\subset B_{Q}\subset\widetilde{B}_{Q}$ and $r(B_{Q})\approx\ell(Q)$.

Given $P\in\mathcal{D}_{\Gamma}$, we will write $\mathcal{D}_{\Gamma}(P)$ to denote the family of $Q\in\mathcal{D}_{\Gamma}$ such that $Q\subset P$.

Remark 2.1**.**

Let us fix $R\in\mathcal{D}_{\Gamma}$ with $\ell(R)=1$ for which we are proving Lemma 1.10. Note that for $x\in R$ and $0<r<1$ computing $\alpha_{\mu,2}(x,r)$ involves only ${\left.\kern-1.2pt\mu\right|_{B}}$, where $B$ is some ball containing $R$. Thus, when proving (2.1), we may and will assume that $\mu$ is a finite, compactly supported measure.

For every $e\in\{0,1\}^{n}$ consider the translated dyadic grid on $L_{0}$

[TABLE]

and the corresponding translated dyadic grid on $\Gamma$

[TABLE]

Let us also define the translated dyadic lattice on $\mathbb{R}^{d}$

[TABLE]

The union of all translated dyadic grids on $\Gamma$ will be called an extended grid on $\Gamma$:

[TABLE]

For each $Q\in\widetilde{\mathcal{D}}_{\Gamma}$ we define $B_{Q},\ \varphi_{Q}$ etc. in the same way as for $Q\in\mathcal{D}_{\Gamma}$.

The main reason for introducing the extended grid is to use a variant of the well-known one-third trick, which was already used in this context by Okikiolu [Oki92].

Lemma 2.2**.**

There exists $k_{0}=k_{0}(n,\Lambda)>0$ such that for every $Q\in\mathcal{D}_{\Gamma}$ with $\ell(Q)\leq 2^{-k_{0}}$ there exists $P_{Q}\in\widetilde{\mathcal{D}}_{\Gamma}$ satisfying $\ell(P_{Q})=2^{k_{0}}\ell(Q)$ and $3\widetilde{B}_{Q}\subset V(P_{Q})$.

Proof.

First, we remark that for every $j\geq 0$ and for every $x\in L_{0}$ there exists $e\in\{0,1\}^{n}$ and $P\in\mathcal{D}^{e}_{\mathbb{R}^{n}}$ with $\ell(P)=2^{-j}$ and $x\in\frac{2}{3}P$. For a nice proof of this fact see [Ler03, Section 3].

Now, consider the point $\Pi_{0}(z_{Q})$. If we take $P\in\mathcal{D}^{e}_{\mathbb{R}^{n}}$ with $\ell(P)=2^{k_{0}}\ell(Q)$ such that $\Pi_{0}(z_{Q})\in\frac{2}{3}P$, we see that the $n$-dimensional ball $B^{n}(\Pi_{0}(z_{Q}),9\Lambda\operatorname{diam}(Q))$ is contained in $P$ as soon as $\frac{2^{k_{0}}}{3}\ell(Q)\geq 9\Lambda\operatorname{diam}(Q).$

It follows that for $P_{Q}\in\mathcal{D}^{e}_{\Gamma}$ such that $\Pi_{0}(P_{Q})=P$ we have $3\widetilde{B}_{Q}\subset V(P_{Q})$. ∎

It may happen that the cube $P_{Q}\in\widetilde{\mathcal{D}}_{\Gamma}$ from the lemma above is not unique, so let us just fix one for each $Q\in\mathcal{D}_{\Gamma}$. The direction $e\in\{0,1\}^{n}$ such that $P_{Q}\in\mathcal{D}^{e}_{\Gamma}$ will be denoted by $e(Q)$, and the integer $k$ such that $\ell(P_{Q})=2^{k_{0}}\ell(Q)=2^{-k}$ will be denoted by $k(Q)$.

We will use later on the fact that

[TABLE]

2.3. Whitney cubes

A very useful tool for approximating the measure $\mu$ close to $\Gamma$ are Whitney cubes. For each $e\in\{0,1\}^{n}$ we consider the decomposition of $\mathbb{R}^{d}\setminus\Gamma$ into a family $\mathcal{W}^{e}$ of Whitney dyadic cubes from $\mathcal{D}^{e}_{\mathbb{R}^{d}}$. That is, the elements of $\mathcal{W}^{e}\subset\mathcal{D}^{e}_{\mathbb{R}^{d}}$ are pairwise disjoint, their union equals $\mathbb{R}^{d}\setminus\Gamma$, and there exist dimensional constants $K>20,D_{0}\geq 1$ such that for every $Q\in\mathcal{W}^{e}$

1. a)

$10Q\subset\mathbb{R}^{d}\setminus\Gamma$, 2. b)

$KQ\cap\Gamma\neq\varnothing$, 3. c)

there are at most $D_{0}$ cubes $Q^{\prime}\in\mathcal{W}^{e}$ such that $10Q\cap 10Q^{\prime}\neq\varnothing$. Furthermore, for such cubes $Q^{\prime}$ we have $\ell(Q^{\prime})\approx\ell(Q)$.

For the proof see [Ste70, Chapter VI, §1] or [Gra08, Appendix J]. Moreover, it is not difficult to construct Whitney cubes in such a way that if $y\in\Gamma,\ Q\in\mathcal{W}^{e}$ and $B(y,r)\cap Q\neq\varnothing$, then

[TABLE]

see [Tol15, Section 2.3] for details. We set

[TABLE]

and also, for every $Q\in\mathcal{D}_{\Gamma}$ satisfying $\ell(Q)\leq 2^{-k_{0}}$,

[TABLE]

Remark 2.3**.**

It follows immediately from the definition of $k(Q)$ that if $P\in\mathcal{W}_{Q}$, then

[TABLE]

2.4. Constants and Parameters

For reader’s convenience, we collect here all the constants that appear in the proof. We indicate what depends on what, and when each constant gets fixed.

Recall that measure $\mu$ and Lipschitz graph $\Gamma$ were fixed at the very beginning, in Subsection 2.1, and also that $\operatorname{\text{Lip}}(\Gamma)\leq 1$. Moreover, in Remark 2.1 we fixed $R\in\mathcal{D}_{\Gamma}$ with $\ell(R)=1$, and without loss of generality we assumed that $\mu$ is finite and compactly supported.

• •

$0<\varepsilon\ll 1$ is a constant from the assumptions of Lemma 1.10 and it was fixed in Subsection 2.1,

• •

$\Lambda$ is an absolute constant from the definition of $\widetilde{B}_{Q}=\Lambda B_{Q}$, it is fixed in (5.2) (actually, one can take $\Lambda=9\sqrt{2}$),

• •

$k_{0}=k_{0}(n,\Lambda)$ is an integer from Lemma 2.2,

• •

$\varepsilon_{0}=\varepsilon_{0}(n)$ is the constant from Lemma 3.2,

• •

$K$ and $D_{0}$ are dimensional constants from the definition of Whitney cubes,

• •

$\lambda=\lambda(k_{0},K,n,d)>3$ is fixed in Lemma 5.1, more precisely in equation (5.1) (one can choose e.g. $\lambda=C(n,d)\,K\,2^{k_{0}}$),

• •

$M=M(\varepsilon,\lambda,\Lambda,n,d,\mu)\gg 1$ is chosen in Lemma 4.2.

3. Estimates of $\alpha_{2}$ Coefficients

We begin by showing the relationship between $b\beta_{2}$ and $\alpha_{2}$ coefficients.

Lemma 3.1**.**

Suppose that $\nu$ is a Radon measure, $B$ is a ball satisfying $\nu(B)\approx r(B)^{n}$, and $L$ is a plane minimizing $\alpha_{\nu,2}(B)$. Then

[TABLE]

Proof.

Let $\pi$ be a minimizing transport plan between $\varphi_{{B}}\nu$ and $a_{B,L}\varphi_{{B}}{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L}}$ (where $a_{B,L}$ is as in the definition of $\alpha_{\nu,2}(B)$; note that $a_{B,L}\gtrsim 1$ since $\nu(B)\approx r(B)^{n}$). Then, by the definition of a transport plan, and the fact that $\varphi_{B}\equiv 1$ on $B$,

[TABLE]

∎

Recall that $\Gamma$ is an $n$-dimensional $1$-Lipschitz graph that was fixed in Subsection 2.1, $\sigma=(\Pi_{\Gamma})_{*}{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L_{0}}}$, and that $L_{Q}$ is the plane minimizing $\alpha_{\sigma,2}(B_{Q})$. The next lemma states that $\Gamma$-cubes $Q$ whose best approximating planes $L_{Q}$ form big angle with $L_{0}$ have large $\alpha_{2}$ numbers. In consequence, there are very few cubes of this kind (in fact, they form a Carleson family).

Lemma 3.2**.**

There exists $\varepsilon_{0}=\varepsilon_{0}(n)>0$ such that for every $Q\in\widetilde{\mathcal{D}}_{\Gamma}$ with $\measuredangle(L_{Q},L_{0})>1-\varepsilon_{0}$ we have

[TABLE]

Proof.

Suppose $Q\in\widetilde{\mathcal{D}}_{\Gamma}$. Take $x_{k}\in 0.5B_{Q}\cap\Gamma,\ k=1,\dots,n,$ such that $|x_{k}-z_{Q}|=0.5r(B_{Q}),$ and the vectors $\{\Pi_{0}(x_{k}-z_{Q})\}_{k}$ form an orthogonal basis of $L_{0}$. Set $B_{0}=B(z_{Q},\eta r(B_{Q})),\ B_{k}=B(x_{k},\eta r(B_{Q}))$, where $\eta=\eta(n)<0.01$ is a small dimensional constant that will be chosen later. Clearly, for all $k=0,\dots,n$ we have $B_{k}\subset B_{Q}$.

If $L_{Q}$ does not intersect one of the balls, say $B_{k}$, then by Lemma 3.1

[TABLE]

Now suppose that $L_{Q}$ intersects all $B_{k}$. Then, since $B_{k}$ are all centered at $\Gamma$, $\Gamma$ is $1$-Lipschitz, and $x_{k}$ were chosen appropriately, it is easy to see that for $\eta=\eta(n)$ and $\varepsilon_{0}=\varepsilon_{0}(n)$ small enough we have $\measuredangle(L_{Q},L_{0})\leq 1-\varepsilon_{0}$. ∎

The following two lemmas will let us compare $\alpha_{2}$ coefficients at similar scales, so that we can pass from the integral form of $\alpha_{2}$ square function (1.4) to its dyadic variant.

Lemma 3.3** ([Tol12, Lemma 5.3]).**

Let $\nu$ be a finite measure supported inside the ball $B^{\prime}\subset\mathbb{R}^{d}$. Let $B\subset\mathbb{R}^{d}$ be another ball such that $3B\subset B^{\prime}$, with $r(B)\approx r(B^{\prime})$ and $\nu(B)\approx\nu(B^{\prime})\approx r(B)^{n}$. Let $L$ be an $n$-plane which intersects $B$ and let $f\mathrel{\mathop{\ordinarycolon}}L\rightarrow[0,1]$ be a function such that $f\equiv 1$ on $3B$, $f\equiv 0$ on $L\setminus B^{\prime}$. Then

[TABLE]

Recall that $\widehat{\alpha}_{\mu,2}(B)=\alpha_{\mu,2,L_{B}}(B)$.

Lemma 3.4**.**

*Let $\nu$ be a Radon measure on $\mathbb{R}^{d}$, $B_{1},B_{2}\subset\mathbb{R}^{d}$ be balls centered at $\Gamma$ with $3B_{1}\subset B_{2},\ r(B_{1})\approx r(B_{2}),\ \nu(B_{1})\approx\nu(3B_{2})\approx r(B_{2})^{n}$. Then we have *

[TABLE]

Proof.

We begin by noting that since $\nu(3B_{1})\lesssim\nu(B_{1})$, we have $\widehat{\alpha}_{\nu,2}(B_{1})\lesssim 1$. As a result, it suffices to prove the lemma under the assumption ${\alpha}_{\sigma,2}(B_{2})\leq\delta$ for some small constant $\delta>0$ which will be fixed later on.

For brevity of notation set $\varphi_{i}=\varphi_{B_{i}},\ L_{i}=L_{B_{i}}$ for $i=1,2.$ We want to apply Lemma 3.3 with $B=B_{1},\,B^{\prime}=3B_{2},\,\nu=\varphi_{2}\nu,\,L=L_{2},\,f={\left.\kern-1.2pt\varphi_{2}\right|_{L}}.$ What needs to be checked is that $B_{1}\cap L_{2}\neq\varnothing$. If this intersection were empty, we would have by Lemma 3.1

[TABLE]

Thus, if $B_{1}\cap L_{2}=\varnothing$, then ${\alpha}_{\sigma,2}(B_{2})\gtrsim 1$ and we arrive at a contradiction with ${\alpha}_{\sigma,2}(B_{2})\leq\delta$ for $\delta$ small enough.

So the assumptions of Lemma 3.3 are met and we get

[TABLE]

Similarly, taking $\nu=\varphi_{2}\sigma$ and $B=B_{1},\,B^{\prime}=3B_{2},\,L=L_{2},\,f={\left.\kern-1.2pt\varphi_{2}\right|_{L}}$ it follows that

[TABLE]

Using the triangle inequality, the scaling of $W_{2}$, the fact that $L_{1}$ minimizes $\alpha_{\sigma,2}(B_{1})$, and the inequalities above, we arrive at

[TABLE]

Dividing both sides by $r(B_{1})^{1+n/2}$ yields

[TABLE]

∎

For technical reasons we define a modified version of $\alpha_{2}$ coefficients. For any $Q\in\widetilde{\mathcal{D}}_{\Gamma}$ set

[TABLE]

where $\varepsilon_{0}$ is as in Lemma 3.2, and

[TABLE]

Recall that $\sigma=(\Pi_{\Gamma})_{*}{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L_{0}}}\approx{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{\Gamma}}$.

Lemma 3.5**.**

Let $\nu\ll\sigma$, $B\subset\mathbb{R}^{d}$ be a ball, $Q\in\widetilde{\mathcal{D}}_{\Gamma}$. Suppose they satisfy $3B\subset V(Q)\cap B_{Q},\ r(B)\approx\ell(Q),\ \nu(B)\approx\nu(Q)\approx\ell(Q)^{n}$. Then

[TABLE]

Proof.

Since $\nu(B)>0$ and $\operatorname{supp}\nu\subset\Gamma$, we certainly have $\sigma(3B)\approx r(B)^{n}$. Moreover, our assumptions imply that $\nu(3B)\approx\nu(B)$, and so $\widehat{\alpha}_{\nu,2}(B)\lesssim 1$. Thus, we may argue in the same way as in the beginning of the proof of Lemma 3.4 to conclude that, without loss of generality, $L_{Q}\cap B\neq\varnothing$. Similarly, we may assume that $\measuredangle(L_{Q},L_{0})\leq 1-\varepsilon_{0}$, because otherwise it would follow from Lemma 3.2 that $\alpha_{\sigma,2}(B_{Q})$ is big.

Now, since $\measuredangle(L_{Q},L_{0})\leq 1-\varepsilon_{0}$, we get that $V(Q)\cap L_{Q}\subset\kappa B_{Q}$ for some constant $\kappa$ depending on $\varepsilon_{0}$; we may assume $\kappa>10$.

We use Lemma 3.3 twice, first with $B=B,\ B^{\prime}=\kappa B_{Q},\ \nu=\psi_{Q}\nu,\ L=L_{Q},\ f={\left.\kern-1.2pt\psi_{Q}\right|_{L}}$, and then with $B=B,\ B^{\prime}=\kappa B_{Q},\ \nu=\varphi_{Q}\sigma,\ L=L_{Q},\ f={\left.\kern-1.2pt\varphi_{Q}\right|_{L}}$, to obtain

[TABLE]

By the triangle inequality, the scaling of $W_{2}$, the fact that $L_{B}$ minimizes $\alpha_{\sigma,2}(B)$, and the estimates above we get

[TABLE]

Dividing both sides by $r(B)^{1+n/2}$ yields the desired result. ∎

We will need an estimate which is a slight modification of [Tol12, Lemma 6.2]. In order to formulate it, let us introduce the usual martingale difference operator. Recall that if $P\in\mathcal{D}_{\Gamma}^{e}$ for some $e\in\{0,1\}^{n}$, then $P^{\prime}\in\mathcal{D}_{\Gamma}^{e}$ is a child of $P$ if $P^{\prime}\subset P$ and $\ell(P^{\prime})=\frac{1}{2}\ell(P)$. Children of $P\in\mathcal{D}_{\mathbb{R}^{n}}^{e}$ are defined analogously.

Given $g\in L^{1}_{loc}(\sigma)$ and $P\in\mathcal{D}_{\Gamma}^{e}$ we set

[TABLE]

Given $h\in L^{1}_{loc}({\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L_{0}}})$ and $P\in\mathcal{D}^{e}_{\mathbb{R}^{n}}$ we define analogously $\Delta_{P}h(x)$:

[TABLE]

Recall that for $g\in L^{2}(\sigma)$ we have

[TABLE]

in the sense of $L^{2}(\sigma)$, and

[TABLE]

for details see e.g. [Dav91, Part I] or [Gra08, Section 5.4.2].

Let us introduce also some additional vocabulary. We will say that a family of cubes $\mathsf{Tree}\subset\mathcal{D}_{\Gamma}^{e}$ is a tree with root $R_{0}$ if it satisfies:

1. (T1)

$R_{0}\in\mathsf{Tree}$, and for every $Q\in\mathsf{Tree}$ we have $Q\subset R_{0}$, 2. (T2)

for every $Q\in\mathsf{Tree}$ such that $Q\neq R_{0}$, the parent of $Q$ also belongs to $\mathsf{Tree}$.

By iterating (T2), we can actually see that if $Q\in\mathsf{Tree}$, then all the intermediate cubes $Q\subset P\subset R_{0}$ also belong to $\mathsf{Tree}$.

The stopping region of $\mathsf{Tree}$, denoted by $\mathsf{Stop}(\mathsf{Tree})$, is the family of all the cubes $P\in\mathcal{D}^{e}_{\Gamma}(R_{0})$ satisfying:

1. (S)

$P\not\in\mathsf{Tree}$, but the parent of $P$ belongs to $\mathsf{Tree}$.

It is easy to see that the cubes from $\mathsf{Stop}(\mathsf{Tree})$ are pairwise disjoint, and that they are maximal descendants of $R_{0}$ not belonging to $\mathsf{Tree}$. Moreover, for every $x\in R_{0}$ we have either $x\in P$ for some $P\in\mathsf{Stop}(\mathsf{Tree})$, or $x\in Q_{k}$ for a sequence of cubes $\{Q_{k}\}_{k}\subset\mathsf{Tree}$ satisfying $\ell(Q_{k})\xrightarrow{k\to\infty}0.$

The following lemma is a modified version of [Tol12, Lemma 6.2].

Lemma 3.6**.**

Let $\nu$ be a Radon measure on $\Gamma$ of the form $\nu=g\sigma$, with $g\in L^{1}(\sigma),\ 0\leq g\leq C$ for some $C>1$. Consider a cube $Q\in\widetilde{\mathcal{D}}_{\Gamma}$ and a tree $\mathcal{\mathsf{Tree}}$ with root $Q$. Suppose that for all $P\in\mathcal{\mathsf{Tree}}$ we have $C^{-1}\ell(P)^{n}\leq\nu(P)\leq C\ell(P)^{n}$. Then, we have

[TABLE]

and

[TABLE]

In the proof we will use [Tol12, Remark 3.14]. It can be thought of as a flat counterpart of Lemma 3.6 – it is valid for more general measures $\nu$ (even more general then what we state below), but at the price of assuming $\Gamma=L_{0}\simeq\mathbb{R}^{n}$.

Lemma 3.7** (simplified [Tol12, Remark 3.14]).**

Suppose $Q\in\mathcal{D}_{\mathbb{R}^{n}}$ is a dyadic cube in $\mathbb{R}^{n}$ and $\mathcal{\mathsf{Tree}}$ is a tree with root $Q$. Consider a measure $\nu=g{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{Q}}$ such that $\nu(P)\approx\ell(P)^{n}$ for $P\in\mathsf{Tree}$. Then,

[TABLE]

Remark 3.8**.**

The definition of a tree of dyadic cubes in [Tol12, p. 492] is slightly more restrictive than the one we adopted. Apart from conditions (T1) and (T2), they also satisfy

• (T3)

if $Q\in\mathsf{Tree}$, then either all the children of $Q$ belong to $\mathsf{Tree}$, or none of them.

Equivalently, if $Q\in\mathsf{Tree}$, and $Q$ is not the root, then all the brothers of $Q$ also belong to $\mathsf{Tree}$. To underline the difference between the two notions, sometimes the terms coherent and semicoherent family of cubes are used. The former refers to trees satisfying (T1–T3), the latter to those satisfying (T1–T2).

Nevertheless, [Tol12, Remark 3.14] cited above is true for both coherent and semicoherent families of cubes. That is, property (T3) is never used in the proof of either [Tol12, Remark 3.14] or the preceding “key lemma” [Tol12, Lemma 3.13].

We are finally ready to prove Lemma 3.6.

Proof of Lemma 3.6.

Let $L=L_{Q}$. If $\measuredangle(L,L_{0})>1-\varepsilon_{0}$, then by Lemma 3.2 and the definition of $\widetilde{\alpha}_{\nu,2}(Q)$

[TABLE]

and we are done. Now assume that $\measuredangle(L,L_{0})\leq 1-\varepsilon_{0}$.

Let $\tilde{\Pi}_{L}$ be the projection from $\mathbb{R}^{d}$ onto $L$, orthogonal to $L_{0}$. We also consider the flat measure $\sigma_{L}=(\tilde{\Pi}_{L})_{*}\sigma=(\tilde{\Pi}_{L})_{*}{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L_{0}}}=c_{L}{\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L}}$ (recall that $\Pi_{\Gamma}$ is a projection orthogonal to $L_{0}$, so that $\tilde{\Pi}_{L}\circ\Pi_{\Gamma}=\tilde{\Pi}_{L}$). Define $g_{0}\mathrel{\mathop{\ordinarycolon}}L_{0}\rightarrow\mathbb{R}$ as $g_{0}=g\circ\Pi_{\Gamma}$.

By triangle inequality

[TABLE]

The first term from the right hand side is estimated by $\alpha_{\sigma,2}(B_{Q})$:

[TABLE]

We estimate the second term from the right hand side of (3.7) using the fact that ${\left.\kern-1.2pt\Pi_{0}\right|_{L\cap V(Q)}}\mathrel{\mathop{\ordinarycolon}}L\cap V(Q)\rightarrow L_{0}\cap V(Q)$ is bilipschitz, with a constant depending on $\varepsilon_{0}$ (because $\measuredangle(L,L_{0})\leq 1-\varepsilon_{0}$):

[TABLE]

By Lemma 3.7 we have

[TABLE]

where $\mathcal{\mathsf{Tree}}_{\mathbb{R}^{n}}\subset\mathcal{D}_{\mathbb{R}^{n}}$ is the tree formed by cubes $P^{\prime}=\Pi_{0}(P),\ P\in\mathcal{\mathsf{Tree}}$, and $L^{2}(L_{0})=L^{2}({\left.\kern-1.2pt\mathscr{H}^{n}\right|_{L_{0}}})$.

Using (3.7) and the estimates above we get

[TABLE]

We conclude the proof of (3.5) by noting that for each $P\in\mathcal{\mathsf{Tree}}$

[TABLE]

The estimate (3.6) follows trivially from the fact that if $e\in\{0,1\}^{n}$ is such that $Q\in\mathcal{D}_{\Gamma}^{e}$, then

[TABLE]

∎

We would like to use Lemma 3.6 also on measures with unbounded density. An approximation argument allows us to get rid of the boundedness assumption, at least if we assume additionally that $\nu(B_{P})\leq C\ell(P)^{n}$ for $P\in\mathsf{Tree}$.

Lemma 3.9**.**

Let $\nu=g\sigma$ with $g\in L^{1}(\sigma),\ g\geq 0$. Consider a cube $Q\in\widetilde{\mathcal{D}}_{\Gamma}$ and a tree $\mathcal{\mathsf{Tree}}$ with root $Q$. Suppose there exists $C>1$ such that for all $P\in\mathcal{\mathsf{Tree}}$ we have $C^{-1}\ell(P)^{n}\leq\nu(P)\leq\nu(B_{P})\leq C\ell(P)^{n}$. Then, we have

[TABLE]

and

[TABLE]

We divide the proof into smaller pieces. Let $\mathsf{Stop}=\mathsf{Stop}(\mathsf{Tree})$. First, we define the set of good points as

[TABLE]

Note that the points from $x\in G$ are not contained in any stopping cube, and so there are arbitrarily small cubes $P\in\mathcal{\mathsf{Tree}}$ containing $x$. We introduce the following approximating measure:

[TABLE]

It is clear that for $Q\in\mathsf{Tree}\cup\mathsf{Stop}$ we have $\tilde{\nu}(Q)=\nu(Q)$. Moreover, for $Q\in\mathsf{Tree}$

[TABLE]

On the other hand, each $S\in\mathsf{Stop}$ is a child of some $Q\in\mathsf{Tree}$, so that

[TABLE]

Lemma 3.10**.**

We have

[TABLE]

Proof.

It is trivial that for $x\in S\in\mathsf{Stop}$ the density is constant and

[TABLE]

On the other hand, by the definition of $\tilde{\nu}$, for $\sigma$-a.e. $x\in G$ we have $\frac{d\tilde{\nu}}{d\sigma}(x)=\frac{d\nu}{d\sigma}(x)=g(x)$. Moreover, for $\sigma$-a.e. $x\in G$ we have a sequence of cubes $Q_{j}\in\mathcal{\mathsf{Tree}}$ such that $\ell(Q_{j})=2^{-j}$ and $x\in Q_{j}$. Note that there exists some integer $j_{0}>0$ (depending on dimension) such that

[TABLE]

It follows that

[TABLE]

Thus,

[TABLE]

∎

Let $\tilde{g}\in L^{1}(\sigma)\cap L^{\infty}(\sigma)$ be such that $\tilde{\nu}=\tilde{g}\sigma$. Applying Lemma 3.6 to $\tilde{\nu}$ yields

[TABLE]

and

[TABLE]

Observe that for $P\in\mathsf{Tree}$ we have

[TABLE]

Indeed, for $x\notin P$ both quantities are equal to zero. For $x\in P^{\prime}\subset P$, where $P^{\prime}$ is a child of $P$, we have $P^{\prime}\in\mathsf{Tree}\cup\mathsf{Stop}$, and so

[TABLE]

Hence, (3.9) follows immediately from (3.13).

Since for $S\in\mathsf{Stop}$ we have $\tilde{\nu}(S)=\nu(S)$, we can use (3.14) to transform (3.12) into

[TABLE]

In order to reach (3.8) and finish the proof of Lemma 3.9, we only need to show how to pass from the estimate on $\widetilde{\alpha}_{\tilde{\nu},2}(Q)$ (3.15) to one on $\widetilde{\alpha}_{{\nu},2}(Q)$.

Proof of Lemma 3.9.

Recall that if $\measuredangle(L_{Q},L_{0})>1-\varepsilon_{0}$, then $\widetilde{\alpha}_{\nu,2}(Q)=1$, but at the same time $\alpha_{\sigma,2}(B_{Q})\gtrsim 1$ by Lemma 3.2, so this case is trivial. Suppose $\measuredangle(L_{Q},L_{0})\leq 1-\varepsilon_{0}$. We define a transport plan between $\psi_{Q}\tilde{\nu}$ and $\psi_{Q}\nu$:

[TABLE]

and we estimate

[TABLE]

From the triangle inequality, the bound above, and (3.15), we get that

[TABLE]

∎

4. Approximating Measures

We will construct a family of measures on $\Gamma$ that will approximate $\mu$. For every Whitney cube $P\in\mathcal{W}^{e}$ we define $g_{P}\mathrel{\mathop{\ordinarycolon}}\Gamma\rightarrow\mathbb{R}$ as

[TABLE]

Note that $\int g_{P}\ d\sigma=\mu(P)$.

Given $e\in\{0,1\}^{n},\ k\in\mathbb{Z}$, we define the following measures supported on $\Gamma$:

[TABLE]

Moreover, for every $Q\in\mathcal{D}_{\Gamma}$ with $\ell(Q)\leq 2^{-k_{0}}$ we set

[TABLE]

Note that, since we assume $\mu$ is finite and compactly supported (see Remark 2.1), all the measures $\nu^{e},\ \nu^{e}_{k},$ are also finite and compactly supported.

We defined $\nu_{Q}$ in such a way that, for “good” $Q\in\mathcal{D}_{\Gamma}$, the measures ${\left.\kern-1.2pt\mu\right|_{B_{Q}}}$ and ${\left.\kern-1.2pt\nu_{Q}\right|_{B_{Q}}}$ are close in the $W_{2}$ distance. This will be shown in Section 5. The rest of this section is dedicated to the construction of a tree of “good cubes”.

Recall that $R\in\mathcal{D}_{\Gamma}$ is a $\Gamma$-cube fixed in Remark 2.1, and $0<\varepsilon\ll 1$ is a small constant fixed in Subsection 2.1.

Lemma 4.1**.**

Let $\lambda>3$. Then, there exist a big constant $M=M(\varepsilon,\lambda,\Lambda,n,d,\mu)\gg 1$ and a tree of good cubes $\mathcal{\mathsf{Tree}}=\mathsf{Tree}(\lambda,\varepsilon,M)\subset\mathcal{D}_{\Gamma}(R)$ with root $R$, such that for every $Q\in\mathcal{\mathsf{Tree}}$ we have

[TABLE]

the stopping region $\mathsf{Stop}=\mathsf{Stop}(\mathsf{Tree})$ is small:

[TABLE]

and $\widehat{\alpha}_{\nu_{Q},2}(\widetilde{B}_{Q})^{2}$ satisfy the packing condition:

[TABLE]

We split the proof into several small lemmas. First, we define auxiliary families of good cubes in $\mathcal{D}^{e}_{\Gamma}$ using a standard stopping time argument.

For each $e\in\{0,1\}^{n}$ there exists a finite collection of cubes $\{R_{i}^{e}\}\subset\mathcal{D}^{e}_{\Gamma}$ such that $\ell(R_{i}^{e})=1,\ R_{i}^{e}\cap R\neq\varnothing.$ Set $R^{e}=\bigcup_{i}R_{i}^{e}$. Let $M\gg 1$ be constant to be fixed later on, and set

[TABLE]

$\mathsf{HD}$ and $\mathsf{LD}$ stand for “high density” and “low density”. Let $\mathsf{Stop}^{e}\subset\mathcal{D}^{e}_{\Gamma}$ be the family of maximal with respect to inclusion cubes from $\mathsf{HD}_{\nu,0}^{e}\cup\mathsf{HD}_{\mu,0}^{e}\cup\mathsf{LD}_{0}^{e}$, and set $\mathsf{HD}_{\nu}^{e}=\mathsf{HD}_{\nu,0}^{e}\cap\mathsf{Stop}^{e},\ \mathsf{HD}_{\mu}^{e}=\mathsf{HD}_{\mu,0}^{e}\cap\mathsf{Stop}^{e},\ \mathsf{LD}^{e}=\mathsf{LD}_{0}^{e}\cap\mathsf{Stop}^{e}$. Note that cubes from $\mathsf{Stop}^{e}$ are pairwise disjoint. We define $\mathcal{\mathsf{Tree}}^{e}$ as the family of those cubes from $\bigcup_{i}\mathcal{D}^{e}_{\Gamma}(R^{e}_{i})$ which are not contained in any cube from $\mathsf{Stop}^{e}$. Actually, this might not be a tree, but it is a finite collection of trees with roots $R_{i}^{e}$.

Lemma 4.2**.**

For $M=M(\varepsilon,\lambda,\Lambda,n,d,\mu)$ big enough, we have for all $e\in\{0,1\}^{n}$

[TABLE]

Proof.

Let $e\in\{0,1\}^{n}$. It is easy to see that the measure of $\mathsf{LD}^{e}$ is small: for every $Q\in\mathsf{LD}^{e}$ we have $\mu(Q)\leq M^{-1}\sigma(Q)$, so

[TABLE]

To estimate the measure of $\mathsf{HD}_{\mu}^{e}$, define for some big $N\gg 1$

[TABLE]

Since $\mu$ is $n$-rectifiable, the density $\Theta^{n}(x,\mu)$ exists, and is positive and finite $\mu$-a.e. Moreover, recall that $\mu(\mathbb{R}^{d})$ is finite. This implies that for $N=N(\mu,\varepsilon,n)$ big enough

[TABLE]

We will show that, if $M$ is chosen big enough, then for all $Q\in\mathsf{HD}^{e}_{\mu}$ we have $Q\subset H_{N}$. Indeed, let $x\in Q\in\mathsf{HD}^{e}_{\mu}$. Then $B(x,2\lambda r(\widetilde{B}_{Q}))\supset\lambda\widetilde{B}_{Q}$, and so

[TABLE]

for $M$ big enough with respect to $N,\lambda,\Lambda,n$. Moreover, note that for $Q\in\mathsf{HD}^{e}_{\mu}$ we have

[TABLE]

and so taking $M$ big enough (depending on $\mu(\mathbb{R}^{d}),\lambda,\Lambda,n$) we can ensure that all $Q\in\mathsf{HD}^{e}_{\mu}$ satisfy $2\lambda r(\widetilde{B}_{Q})<1$. Thus, $x\in H_{N}$, and we conclude that

[TABLE]

Since $\nu^{e}$ is a finite $n$-rectifiable measure, we can argue in the same way as above to get

[TABLE]

Smallness of $\mu(\bigcup_{Q\in\mathsf{HD}_{\nu}^{e}}Q)$ follows from the fact that ${\left.\kern-1.2pt\mu\right|_{\Gamma}}\leq\nu^{e}$. Putting this together with (4.3) and (4.4) we get

[TABLE]

We take $M$ so big that the above holds for all $e\in\{0,1\}^{n}$, and the proof is finished. ∎

For each $e\in\{0,1\}^{n},\ k=0,1,2,\dots,$ let $g^{e}_{k}$ be the density of $\nu^{e}_{k}$ with respect to $\sigma$. Note that, due to the definition of $\mathsf{Tree}^{e}$, for any $Q\in\mathsf{Tree}^{e}$ we have

[TABLE]

Hence, given a cube $Q\in\mathcal{\mathsf{Tree}}^{e}$ with $\ell(Q)=2^{-k}$, we can estimate $\widetilde{\alpha}_{\nu^{e}_{k},2}(Q)^{2}$ using Lemma 3.9 (applied to $\nu^{e}_{k}$ and $\mathsf{Tree}=\{P\in\mathcal{\mathsf{Tree}}^{e}\ \mathrel{\mathop{\ordinarycolon}}\ P\subset Q\}$) to get

[TABLE]

The following lemma states that the right hand side of this estimate can be made independent of $k$.

Lemma 4.3**.**

For all $Q\in\mathsf{Tree}^{e}$ with $\ell(Q)=2^{-k},\ k\geq 0,$ we have

[TABLE]

Moreover,

[TABLE]

Proof.

We claim that for $P\in\mathcal{\mathsf{Tree}}^{e}$ with $\ell(P)\leq 2^{-k}$ (in particular, for $P\in\mathcal{\mathsf{Tree}}^{e}$ such that $P\subset Q$) we have

[TABLE]

Indeed, for $x\not\in P$ both sides of (4.8) are zero. For $x\in P^{\prime}\subset P$, where $P^{\prime}\in\mathcal{\mathsf{Tree}}^{e}\cup\mathsf{Stop}^{e}$ is a child of $P$, we have

[TABLE]

The Whitney cubes $S$ in the sums above above satisfy $\ell(S)>2^{-k}\geq\ell(P)$, and moreover we have $\Pi_{\Gamma}(S)\in\mathcal{D}^{e}_{\Gamma}$. Hence, we either have $P\cap\Pi_{\Gamma}(S)=P$ or $P\cap\Pi_{\Gamma}(S)=\varnothing$. The same is true for $P^{\prime}$. Moreover, we have $P\cap\Pi_{\Gamma}(S)\neq\varnothing$ if and only if $P^{\prime}\cap\Pi_{\Gamma}(S)\neq\varnothing$. It follows that the right hand side above is equal to

[TABLE]

Thus $\Delta^{\sigma}_{P}g^{e}_{k}=\Delta^{\sigma}_{P}g^{e}_{0}.$ Using this equality, and also the fact that $\nu^{e}_{k}\leq\nu^{e}$, we transform (4.5) into

[TABLE]

Concerning (4.7), it is an immediate consequence of (3.9) when we apply Lemma 3.9 to $\nu^{e}_{0}$ and the trees $\{Q\in\mathsf{Tree}^{e}\ \mathrel{\mathop{\ordinarycolon}}\ Q\subset R^{e}_{i}\}$ (recall that the union of such trees gives the entire $\mathsf{Tree}^{e}$). ∎

We finally define $\mathcal{\mathsf{Tree}}$ as the collection of cubes $Q\in\mathcal{D}_{\Gamma}$ such that for every $e\in\{0,1\}^{n}$ there exists $P\in\mathcal{\mathsf{Tree}}^{e}$ satsfying $\ell(P)=\ell(Q)$ and $P\cap Q\not=\varnothing$. It is easy to check that $\mathcal{\mathsf{Tree}}$ is indeed a tree, and that the stopping cubes $\mathsf{Stop}=\mathsf{Stop}(\mathsf{Tree})$ satisfy $\bigcup_{Q\in\mathsf{Stop}}Q\subset\bigcup_{e}\bigcup_{Q\in\mathsf{Stop}^{e}}Q$. Thus,

[TABLE]

Moreover, $\mathcal{\mathsf{Tree}}\subset\mathcal{\mathsf{Tree}}_{(0,\dots,0)}$, so for all $Q\in\mathcal{\mathsf{Tree}}$

[TABLE]

The only thing that remains to be shown is the packing condition (4.1).

Lemma 4.4**.**

We have

[TABLE]

Proof.

Recall that in Lemma 2.2 we defined a constant $k_{0}>0$ such that for any $Q\in\mathcal{D}_{\Gamma},\ \ell(Q)\leq 2^{-k_{0}}$, there exists a cube $P_{Q}\in\widetilde{\mathcal{D}}_{\Gamma}$ satisfying $3\widetilde{B}_{Q}\subset V(P_{Q}),\ \ell(P_{Q})=2^{k_{0}}\ell(Q)$. Since there are only finitely many $Q\in\mathsf{Tree}$ with $\ell(Q)>2^{-k_{0}}$, we may ignore them in the estimates that follow.

Suppose $Q\in\mathcal{\mathsf{Tree}}$ and $\ell(Q)\leq 2^{-k_{0}}$, let $P_{Q}$ be as above. Recall that $\nu_{Q}=\nu^{e(Q)}_{k(Q)}$, where $e=e(Q),\ k=k(Q)$ are such that $P_{Q}\in\mathcal{D}^{e}_{\Gamma}$ and $\ell(P_{Q})=2^{-k}$.

We defined $\mathcal{\mathsf{Tree}}$ in such a way that necessarily $P_{Q}\in\mathcal{\mathsf{Tree}}^{e}$. It follows from Lemma 3.5 applied with $\nu=\nu_{Q},\ B=\widetilde{B}_{Q},\ Q=P_{Q},$ that

[TABLE]

We use (4.6) and the inequality above to obtain

[TABLE]

Taking into account that each $P_{Q}\in\mathcal{\mathsf{Tree}}^{e}$ may correspond to only a bounded number of $Q\in\mathcal{\mathsf{Tree}}$, and that $\ell(Q)\approx_{k_{0}}\ell(P_{Q})$, we get

[TABLE]

The first sum from the right hand side is finite because $\sigma$ is uniformly rectifiable, see Theorem 1.1. We estimate the second sum by changing the order of summation:

[TABLE]

The third sum is treated similarly:

[TABLE]

Thus,

[TABLE]

∎

5. From Approximating Measures to $\mu$

To prove Lemma 1.10 we need to pass from the estimates on $\widehat{\alpha}_{\nu_{Q},2}(\widetilde{B}_{Q})$ shown in Lemma 4.1 to estimates on $\widehat{\alpha}_{\mu,2}(B_{Q})$.

Recall that $K>20$ is the constant such that for all Whitney cubes $Q\in\mathcal{W}^{e}$ we have $KQ\cap\Gamma\neq\varnothing$, and $k_{0}=k_{0}(n,\Lambda)$ is an integer from Lemma 2.2.

Lemma 5.1**.**

There exists $\lambda=\lambda(k_{0},K,n,d)>3$ such that if $M=M(\varepsilon,\lambda,\Lambda,n,d,\mu)$ and $\mathsf{Tree}=\mathsf{Tree}(\lambda,M,\varepsilon)$ are as in Lemma 4.1, then for all $Q\in\mathsf{Tree}$ with $\ell(Q)\leq 2^{-k_{0}}$

[TABLE]

Proof.

Let $Q\in\mathcal{\mathsf{Tree}}$ with $\ell(Q)\leq 2^{-k_{0}}$. We will define an auxiliary measure $\mu_{Q}$. Set

[TABLE]

It is easy to check that

[TABLE]

for $\lambda=\lambda(k_{0},K,n,d)$ big enough (e.g. $\lambda=C(n,d)K2^{k_{0}}$ works). It is crucial that all cubes in $I_{Q}$ have sidelength bounded by $2^{k_{0}}\ell(Q)$, otherwise no such $\lambda$ would exist.

Recall that the functions $g_{P}(x)=\frac{\mu(P)}{\ell(P)^{n}}\mathds{1}_{\Pi_{\Gamma}(P)}(x),\ P\in\mathcal{W}_{Q},$ were used to define $\nu_{Q}$ at the beginning of Section 4. Let

[TABLE]

Note that for $P\in\mathcal{W}_{Q}\setminus I_{Q}$ we have $a_{P}=0$. The measure $\mu_{Q}$ is defined as

[TABLE]

First, let us show that if $\Lambda$ (the constant from the definition of $\widetilde{B}_{Q}=\Lambda B_{Q}$) is big enough, then ${\left.\kern-1.2pt\mu\right|_{3B_{Q}}}={\left.\kern-1.2pt\mu_{Q}\right|_{3B_{Q}}}$. We need to check the following: if $P\in\mathcal{W}^{e(Q)}$ is such that $P\cap 3B_{Q}\neq\varnothing$, then $P\in I_{Q}$ and $a_{P}=1$.

Note that for all such $P$ we have

[TABLE]

and so $P\in\mathcal{W}_{Q}$. Furthermore, the fact that $P\cap 3B_{Q}\neq\varnothing$ and (2.4) imply that $P\subset 9B_{Q}$. Since $\Pi_{\Gamma}$ is $\sqrt{2}$-Lipschitz continuous, and $B_{Q}$ is centered at $\Gamma$, we get that for $\Lambda$ big enough (e.g. $\Lambda=9\sqrt{2}$)

[TABLE]

We conclude that $P\in I_{Q}$ and $a_{P}=1$, and so,

[TABLE]

Set $L=L_{\widetilde{B}_{Q}}$. We will apply Lemma 3.3 with $\nu=\mu_{Q},\ B_{1}=B_{Q},\ B_{2}=\lambda\widetilde{B}_{Q},\ L=L,$ and $f=\varphi_{\widetilde{B}_{Q}}$. Notice that $\operatorname{supp}\mu_{Q}\subset\lambda\widetilde{B}_{Q}$ by (5.1). Moreover, using the same trick as in the beginning of the proof of Lemma 3.4, we may assume that $L\cap B_{Q}\neq\varnothing$. Since $\mu_{Q}(B_{Q})\approx_{M}\mu_{Q}(\lambda\widetilde{B}_{Q})\approx_{M}\ell(Q)^{n}$ by Lemma 4.1, and $r(\lambda\widetilde{B}_{Q})=\lambda\Lambda r(B_{Q})$, the assumptions of Lemma 3.3 are met, and we get that

[TABLE]

Applying the triangle inequality yields

[TABLE]

To estimate $W_{2}(\mu_{Q},\varphi_{\widetilde{B}_{Q}}\nu_{Q})$ we define the following transport plan:

[TABLE]

Then,

[TABLE]

Putting together (5.3), (5.4), (5.5), and the estimate above, we get

[TABLE]

Finally, we use the triangle inequality, the estimate $\mu(3B_{Q})\approx_{M}\sigma(B_{Q})\approx r(B_{Q})^{n}$, and the fact that $L_{Q}$ minimizes $\alpha_{\sigma,2}(B_{Q})$, to get

[TABLE]

and so the proof is complete. ∎

We are ready to finish the proof of Lemma 1.10.

Proof of Lemma 1.10.

Recall that $R$ is a $\Gamma$-cube with $\ell(R)=1$, and $\varepsilon>0$ is an arbitrary small constant, and that they were both fixed in Subsection 2.1. Let $\lambda,\ M,\ \mathsf{Tree},$ and $\mathsf{Stop}$ be as in Lemma 5.1 and Lemma 4.1. Set

[TABLE]

By Lemma 4.1, we have $\mu(R^{\prime})\geq(1-\varepsilon)\mu(R)$. Our aim is to show that

[TABLE]

For any $x\in R^{\prime}$ we have arbitrarily small cubes from $\mathsf{Tree}$ containing $x$. Hence, for any $k\geq k_{0}+3,\ r\in(2^{-k},2^{-k+1}],$ we have $3B(x,r)\subset B_{Q}$ for the cube $Q\in\mathcal{\mathsf{Tree}}$ containing $x$ and satisfying $\ell(Q)=2^{-k+3}$. Thus, by Lemma 3.4,

[TABLE]

Integrating both sides with respect to $r$ yields

[TABLE]

The inequality above holds for all $x\in Q\cap R^{\prime}$, so

[TABLE]

Summing over all $Q\in\mathcal{\mathsf{Tree}}$ with $\ell(Q)=2^{-k+3}$, and then over all $k\geq k_{0}+3$, we get

[TABLE]

On the other hand, for any $r>0$ we have

[TABLE]

so

[TABLE]

Thus, in order to prove Lemma 1.10, it suffices to show that the sums on the right hand side of (5.6) are finite.

The finiteness of

[TABLE]

follows by Theorem 1.1. To estimate the other sum we apply Lemma 5.1:

[TABLE]

The first sum is finite by Lemma 4.1, the second by Theorem 1.1. Concerning the last sum, we may estimate it in the following way:

[TABLE]

Thus,

[TABLE]

∎