Sufficient condition for rectifiability involving Wasserstein distance $W_2$

TL;DR

This paper provides two new sufficient conditions for the rectifiability of Radon measures, using square functions of flatness coefficients involving Wasserstein distance $W_2$, and establishes their necessity, offering new characterizations.

Contribution

The paper introduces two novel sufficient conditions for rectifiability based on flatness coefficients, one involving Wasserstein distance $W_2$, and proves their necessity, enriching the understanding of measure rectifiability.

Findings

Both conditions are necessary and sufficient for rectifiability.

The $eta_2$ and $ ext{α}$ coefficients characterize rectifiability.

The $ ext{α}_2$ coefficients involving Wasserstein distance $W_2$ provide a new characterization.

Abstract

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $\alpha$ and $\beta_2$ numbers. The second one involves $\alpha_2$ numbers -- coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too -- the first one was shown to be necessary by Tolsa, while the necessity of the $\alpha_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.

Equations

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

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

$${\beta }_{\mu ,p}(x,r)=\underset{L}{\inf }{\left(\frac{1}{\mu (B(x,3r))}{\int }_{B(x,r)}{\left(\frac{\mathrm{dist}(y,L)}{r}\right)}^{p}d\mu (y)\right)}^{1/p}.$$

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

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

$${\alpha }_{\mu }(B)=\underset{c,L}{\inf }\frac{1}{r(B)\mu (3B)}{F}_B(\mu ,c{{\mathcal{H}}^n|}_L),$$

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

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

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

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

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

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

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

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

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

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

$${\alpha }_{\mu ,p}(B)=\underset{L}{\inf }\frac{1}{r(B)\mu (3B{)}^{1/p}}{W}_p({\phi }_B\mu ,a_{B,L}{\phi }_B{{\mathcal{H}}^n|}_L),$$

$$a_{B,L}=\frac{\int {\phi }_{B}d\mu }{\int {\phi }_{B}d{{\mathcal{H}}^n|}_L}.$$

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

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

$${\Theta }_{\mu }(B)=\frac{\mu (B)}{r(B{)}^n}.$$

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

$$\mathrm{dist}(A,B)=\underset{a\in A}{\inf }\underset{b\in B}{\inf }|a-b|.$$

$${\beta }_{\mu ,2}(B)\le {\alpha }_{\mu ,2}(B),$$

$${\alpha }_{\mu }(B)\le {\alpha }_{\mu ,1}(B)\le {\alpha }_{\mu ,2}(B).$$

$${\alpha }_{\mu ,2}(B{)}^{2}r(B{)}^2\mu (3B)=\int |x-y{|}^{2}d\pi (x,y)\ge {\int }_B\mathrm{dist}(x,L{)}^{2}d\mu \ge {\beta }_{\mu ,2}(B{)}^2\mu (3B)r(B{)}^2.$$

$$W_1(\mu ,\nu )=\underset{\mathrm{Lip}(f)\le 1}{\sup }\left|\int fd\mu -\int fd\nu \right|,$$

$$\begin{gathered} {\alpha }_{\mu }(B)r(B)\mu (3B)\le F_B(\mu ,a_{B,L}{{\mathcal{H}}^n|}_L)=\underset{\begin{array}{c}\mathrm{Lip}(f)\le 1\\ \mathrm{supp}(f)\subset B\end{array}}{\sup }\left|\int fd\mu -\int f{a}_{B,L}d{{\mathcal{H}}^n|}_L\right| \\ =\underset{\begin{array}{c}\mathrm{Lip}(f)\le 1\\ \mathrm{supp}(f)\subset B\end{array}}{\sup }\left|\int f{\phi }_{B}d\mu -\int f{\phi }_B{a}_{B,L}d{{\mathcal{H}}^n|}_L\right|\le \underset{\mathrm{Lip}(f)\le 1}{\sup }\left|\int f{\phi }_{B}d\mu -\int f{\phi }_B{a}_{B,L}d{{\mathcal{H}}^n|}_L\right| \\ =W_1({\phi }_B\mu ,a_{B,L}{\phi }_B{{\mathcal{H}}^n|}_L)={\alpha }_{\mu ,1}(B)r(B)\mu (3B). \end{gathered}$$

$${\beta }_{\mu ,1}(B)\le {\beta }_{\mu ,2}(B),$$

$${\beta }_{\mu ,1}(B)\lesssim {\alpha }_{\mu }(2B).$$

$${\beta }_{\mu ,2}(B_1)$$

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 $\mathcal{H}^{n}$. A set $E\subset\mathbb{R}^{d}$ is $n$-rectifiable if the measure ${\left.\kern-1.2pt\mathcal{H}^{n}\right|_{E}}$ is $n$-rectifiable. We will often omit $n$ and just write “rectifiable”.

Rectifiable sets and measures have been studied for many decades now, and the first results in this area are due to Besicovitch. See [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 sufficient condition for rectifiability involving the so-called $\alpha_{2}$ coefficients. In fact, we will show a bit more: a sufficient condition for rectifiability involving $\alpha$ and $\beta_{2}$ numbers. We introduce the whole menagerie of flatness-quantifying coefficients below.

1.1. $\beta$ and $\alpha$ coefficients

$\beta$ numbers were introduced by Jones in [Jon90], where they were used to characterize subsets of rectifiable curves. They were further developed by David and Semmes in [DS91, DS93]. For $1\leq p<\infty$ and a Radon measure $\mu$ on $\mathbb{R}^{d}$ we define

[TABLE]

where the infimum runs over all $n$-planes $L$ intersecting $B(x,r)$. The letter $h$ in the superscript stands for homogeneous and refers to the normalizing factor $r^{n}$. In our setting it will be more convenient to normalize by $\mu(B(x,3r))$ instead, and so we define

[TABLE]

For a ball $B=B(x,r)$ we will sometimes write $\beta_{\mu,p}(B)$ instead of $\beta_{\mu,p}(x,r)$, and we will do the same with all the other coefficients introduced below.

Another way to quantify flatness of measures is offered by $\alpha$ numbers, introduced by Tolsa in [Tol09]. To define them, we need a distance on the space of measures. Given Radon measures $\mu$ and $\nu$, and an open ball $B$, we set

[TABLE]

where

[TABLE]

The coefficient $\alpha$ of a measure $\mu$ in $B$ is defined as

[TABLE]

where the infimum runs over all $c\geq 0$ and all $n$-planes $L$.

We prove the following sufficient condition for rectifiability in terms of $\alpha$ and $\beta_{2}$ square functions.

Theorem 1.1**.**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Suppose that

[TABLE]

and

[TABLE]

Then $\mu$ is $n$-rectifiable.

Since Tolsa has shown in [Tol15] that (1.3) and (1.4) are also necessary conditions for rectifiability, we immediately get the following characterization.

Corollary 1.2**.**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then, $\mu$ is $n$-rectifiable if and only if (1.3) and (1.4) hold for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

A number of similar characterizations has been shown in recent years. First, recall that upper and lower $n$-dimensional densities of a Radon measure $\mu$ at $x\in\mathbb{R}^{d}$ are

[TABLE]

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

In [Tol15] it was shown that a rectifiable measure $\mu$ satisfies

[TABLE]

On the other hand, Azzam and Tolsa proved in [AT15] that if a Radon measure $\mu$ satisfies ($\ref*{eq:main theorem assumption on betas}^{h}$) and $0<\Theta^{n,*}(x,\mu)<\infty$ for $\mu$-a.e. $x\in\mathbb{R}^{d}$, then $\mu$ is $n$-rectifiable. More recently, Edelen, Naber and Valtorta [ENV16] managed to weaken the assumption on densities to

[TABLE]

An alternative proof showing that ($\ref*{eq:main theorem assumption on betas}^{h}$) and (1.5) are sufficient for rectifiability was later given in [Tol19].

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

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then, $\mu$ is $n$-rectifiable if and only if ($\ref*{eq:main theorem assumption on betas}^{h}$) and (1.5) hold for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

There is a reason why the theorems above are stated for $\beta_{2}$ numbers (as opposed to $\beta_{p}$ numbers with $p\neq 2$). The conditions (1.4) and ($\ref*{eq:main theorem assumption on betas}^{h}$) with $\beta_{2}$ numbers replaced by $\beta_{p}$ are necessary for rectifiability only for $1\leq p\leq 2$. On the other hand, conditions ($\ref*{eq:main theorem assumption on betas}^{h}$) and $0<\Theta^{n,*}(\mu,x)<\infty$ $\mu$-almost everywhere are sufficient for rectifiability only if $p\geq 2$. Relevant counterexamples were given in [Tol19]. However, if we assume more on densities (namely that $\Theta_{*}^{n}(\mu,x)>0$ and $\Theta^{n,*}(\mu,x)<\infty$ $\mu$-almost everywhere), then the finiteness of $\beta_{p}$ square function for certain $p<2$ becomes sufficient for rectifiability, see [Paj97, BS16].

It is also worth mentioning that appropriate versions of $\beta$ numbers give rise to various necessary and/or sufficient conditions for the so called Federer rectifiability of measures. 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 $\mathcal{H}^{n}$ is required. This notion is more difficult to characterize than the one we work with, as illustrated by the surprising example of Garnett, Killip and Schul [GKS10]. In the case $n=1$ significant progress has been achieved in [Ler03, BS15, BS16, AM16, BS17, MO18]. An excellent overview of the problem is given in the survey [Bad19].

Concerning $\alpha$ numbers, as we already mentioned, Tolsa showed in [Tol15] that (1.3) is necessary for rectifiability. Is it also sufficient? Azzam, David, and Toro proved in [ADT16] that if $\mu$ is doubling, then some condition related to (1.3) is sufficient for rectifiability. In [Orp18] Orponen showed that for $n=d=1$ a variant of (1.3) is sufficient for rectifiability (which in this case is equivalent to absolute continuity with respect to $\mathcal{H}^{1}$). Finally, Azzam, Tolsa and Toro [ATT18] proved that a measure $\mu$ satisfying (1.3) which is also pointwise doubling, i.e. such that

[TABLE]

is rectifiable.

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

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Then, $\mu$ is $n$-rectifiable if and only if (1.3) and (1.6) hold for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

Also in [ATT18], the authors construct a purely $1$-unrectifiable measure on $\mathbb{R}^{2}$ satisfying (1.3). This shows that, for general $n$ and $d$, (1.3) on its own is not a sufficient condition for rectifiability.

Finally, let us mention that in [TT15, Tol17] rectifiable sets and measures are characterized using yet another kind of square functions. They involve 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], valid for arbitrary $n$, require $0<\Theta^{n}_{*}(\mu,x)\leq\Theta^{n,*}(\mu,x)<\infty$ for $\mu$-a.e $x\in\mathbb{R}^{d}$. On the other hand, in [Tol17] it was shown that for $n=1$ analogous results hold under weaker assumption $0<\Theta^{1,*}(x,\mu)<\infty$ for $\mu$-a.e. $x\in\mathbb{R}^{d}$.

1.2. $\alpha_{p}$ coefficients

Coefficients $\alpha_{p}$ were introduced by Tolsa in [Tol12]. They can be thought of as a generalization of $\alpha$ numbers – in fact, under relatively mild assumptions, one has $\alpha_{\mu}(B)\approx\alpha_{\mu,1}(B)$, see Lemma 3.1 and [Tol12, Lemma 5.1]. As in the case of $\alpha$ coefficients, in order to define $\alpha_{p}$ numbers we need a metric on the space of measures.

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 $\mu,\ \nu,$ and $\pi$ of the same total mass. For more information on Wasserstein distance see for example [Vil03, Chapter 7] or [Vil08, Chapter 6].

Similarly as $\alpha$ numbers, $\alpha_{p}$ numbers quantify how far is a given measure from being a flat measure, that is, from being of the form $c{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}}$ 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 $c>0$. Given a ball $B=B(x,r)\subset\mathbb{R}^{d}$ we set

[TABLE]

$\varphi_{B}$ should be thought as a regularized characteristic function of $B$. For $1\leq p<\infty$, a Radon measure $\mu$ on $\mathbb{R}^{d}$, and a ball $B\subset\mathbb{R}^{d},$ we define

[TABLE]

where the infimum is taken over all $n$-planes $L$ intersecting $B$, and

[TABLE]

Coefficients $\alpha_{p}$ were introduced in [Tol12] with the aim of characterizing uniformly rectifiable measures. Uniform rectifiability, introduced by David and Semmes in [DS91, DS93], is a stronger, more quantitative version of rectifiability. Can $\alpha_{p}$ numbers be used to characterize rectifiability also in the non-uniform case? Driven by this question, our main motivation for proving Theorem 1.1 was to get the following sufficient condition for rectifiability.

Theorem 1.3**.**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$. Suppose that

[TABLE]

Then $\mu$ is $n$-rectifiable.

Theorem 1.3 follows immediately from Theorem 1.1 because, as shown in Lemma 3.1, $\alpha_{2}$ numbers bound from above both $\alpha$ and $\beta_{2}$ numbers. In [Dąb19] we show that (1.7) is also a necessary condition for rectifiability, and so we get the following characterization.

Corollary 1.4**.**

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]

We would like to stress that, compared to Theorem A and Theorem B, the characterization above does not make any additional assumptions on densities or on doubling properties of the measure.

The organization of the paper, as well as the general strategy of the proof, are outlined in Section 2. For now, let us just say that Lemma 5.1, our main lemma, can be seen as a technical, more quantitative version of Theorem 1.1. If one prefers working with homogeneous coefficients $\beta_{\mu,2}^{h}$ and $\alpha_{\mu}^{h}$ (where $\alpha_{\mu}^{h}(x,r)=\frac{\mu(B(x,3r))}{r^{n}}\alpha_{\mu}(x,r)$), then a possible “homogenized” modification of Lemma 5.1 is discussed in Remark 5.4. However, it is clear that “homogenized” (i.e. with $\alpha$ and $\beta_{2}$ numbers replaced by their homogeneous counterparts) versions of Theorem 1.1 and Theorem 1.3 are not true (unless we assume more about densities) – think of Lebesgue measure on $\mathbb{R}^{d}$.

Acknowledgements

The author would like to express his deep gratitude to Xavier Tolsa for all his help and guidance. He acknowledges the support from 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 the Catalan Agency for Management of University and Research Grants (2017-SGR-0395), and from the Spanish Ministry of Science, Innovation and Universities (MTM-2016-77635-P).

2. Sketch of the proof

The proof of Theorem 1.1 is organized as follows. In Section 3 we provide basic estimates of $\alpha$ and $\beta$ coefficients, while in Section 4 we recall the definition and some properties of the David-Mattila lattice, which will be used further on. In Section 5 we formulate the main lemma. Given an appropriate David-Mattila cube $R_{0}$, the main lemma provides us with a Lipschitz graph $\Gamma$ such that we have $\mu\ll{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{\Gamma}}$ on a large chunk of $\Gamma\cap R_{0}$, and $\mu(\Gamma\cap R_{0})\geq\frac{1}{2}\mu(R_{0})$. In the same section we show how to use the main lemma to prove Theorem 1.1. Everything that follows is dedicated to proving the main lemma.

In Section 6 we perform the usual stopping time argument. We define the family of stopping cubes $\mathsf{Stop}$, comprising high density cubes $\mathsf{HD}$, low density cubes $\mathsf{LD}$, big angle cubes $\mathsf{BA}$ (cubes whose best approximating planes form a big angle with $L_{0}$, the best approximating plane of $R_{0}$), big square function cubes $\mathsf{BS}$ (cubes with a big portion of points for which the square functions are larger than a certain threshold), and far cubes $\mathsf{F}$ (cubes with a big portion of $R_{\mathsf{Far}}$, points that are far from certain best approximating planes). Cubes not contained in any of the stopping cubes form the $\mathsf{Tree}$. Next, we show various good properties of cubes from the $\mathsf{Tree}$, as well as estimate the measure of cubes from $\mathsf{BS}$ and $\mathsf{F}$ (it is easy).

Section 7 is devoted to constructing the Lipschitz graph $\Gamma$. One possible way to do it would be to use the tools from [DT12] – this was done for example in [AT15, ATT18]. In this paper we decided to use another well-known method, dating back at least to [DS91] and [Lég99]. We follow the way it was applied in [CMT18] and [Tol14]. It consists of showing that $R_{0}\setminus\bigcup_{Q\in\mathsf{Stop}}Q$ forms a graph of a Lipschitz map $F$ defined on a subset of $L_{0}$, and then carefully extending $F$ to the whole $L_{0}$. The remaining part of the paper is dedicated to showing that the measure of stopping cubes is small.

In Section 8 we first show that cubes from $\mathsf{Tree}$ lie close to $\Gamma$ (the graph of $F$), and then use this property to estimate the measure of low density cubes. Roughly speaking, we may cover (almost all) $\mathsf{LD}$ cubes with a family of (almost) disjoint balls satisfying $B\cap\Gamma\approx r(B)^{n}$, and such that the densities $\Theta_{\mu}(B)$ are low. Small measure of $\mathsf{LD}$ easily follows. It is crucial that we have finiteness of the $\beta_{2}$ square function (1.4), as it lets us estimate the size of $R_{\mathsf{Far}}$ (see Lemma 6.6). This approach to bounding the measure of low density cubes comes from [AT15].

In Section 9 we define a measure $\nu$ supported on $\Gamma$. We show that $\nu$ is very close to $\mu$ in the sense of distance $F_{B}(\mu,\nu)$, so that the $\alpha_{\nu}$ numbers are close to $\alpha_{\mu}$. The measure $\nu$ is then used in Section 10 to estimate the size of high density set. The general idea is to consider $f$ – the density of $\nu$ with respect to ${\left.\kern-1.2pt\mathcal{H}^{n}\right|_{\Gamma}}$, and then to bound the $L^{2}$ norm of $|f-c_{0}|$, where $c_{0}$ is a certain constant. We do it using the smallness of $\alpha_{\mu}$ square function (1.3), the fact that $\nu$ approximates $\mu$ well, and an appropriate type of Paley-Littlewood result (see (10.8)). Estimating $\lVert f-c_{0}\rVert_{L^{2}}$ requires a lot of work, but once we have it, it is not very difficult to bound the measure of $\mathsf{HD}$ cubes. Roughly speaking, high density cubes correspond to big values of $f$, and those we can control since $\lVert f-c_{0}\rVert_{L^{2}}$ is small. This method of estimating $\mathsf{HD}$ is due to [ATT18], where a similar approach from [Tol17] was refined and simplified.

Finally, in Section 11 we bound the size of big angle cubes $\mathsf{BA}$. First, we show that this amounts to estimating $\lVert\nabla F\rVert_{L^{2}}$ (recall that $F$ is the Lipschitz map whose graph is $\Gamma$). Using Dorronsoro’s theorem, this reduces to estimating the $\beta_{\sigma,1}$ square function, where $\sigma$ is the surface measure on $\Gamma$. This could be done using the smallness of either $\beta_{\mu,2}$ or $\alpha_{\mu}$ square functions. For us it was easier to deal with $\alpha_{\mu}$, due to all the estimates from Section 9.

Thus, having estimated the measure of the stopping region, the proof of the main lemma is finished.

Notation

Throughout the paper we will write $A\lesssim B$ whenever $A\leq CB$ for some constant $C$. All such implicit constants may depend on dimensions $n,d$, and on constants $A_{0},C_{0}$, which will be fixed in Section 4. If the implicit constant depends also on some other parameter $t$, we will write $A\lesssim_{t}B$. The notation $A\approx B$ means $A\lesssim B\lesssim B$, and $A\approx_{t}B$ means $A\lesssim_{t}B\lesssim_{t}B$.

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 $\lambda r(B)$.

For a ball $B$ and measure $\mu$, we define the $n$-dimensional density of $\mu$ at $B$ as

[TABLE]

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 an angle between $L_{1}$ and $L_{2}$.

Given an affine subspace $L\subset\mathbb{R}^{d}$, we will denote the orthogonal projection onto $L$ by $\Pi_{L}$. The orthogonal projection onto $L^{\perp}$ will be denoted by $\Pi^{\perp}_{L}$.

Given a set $A\subset\mathbb{R}^{d}$, we denote by $\mathds{1}_{A}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\to\{0,1\}$ a characteristic function of $A$.

Finally, for sets $A,B\subset\mathbb{R}^{d}$ we define

[TABLE]

3. Estimates of $\alpha$ and $\beta$ coefficients

In this section we provide some basic estimates of $\alpha$ and $\beta$ coefficients. We begin by showing the relationship between $\alpha,\beta_{2}$, and $\alpha_{2}$ coefficients.

Lemma 3.1**.**

Suppose that $\mu$ is a Radon measure, and $B$ is a ball intersecting $\operatorname{supp}\mu$. Then

[TABLE]

and

[TABLE]

Proof.

To see $\beta_{\mu,2}(B)\leq\alpha_{\mu,2}(B)$, let $L$ be a minimizing plane for $\alpha_{\mu,2}(B)$ and $\pi$ be a minimizing transport plan between $\varphi_{{B}}\mu$ and $a_{B,L}\varphi_{{B}}{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}}$, where $a_{B,L}=({\int\varphi_{B}\ d\mu})/({\int\varphi_{B}\ d{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}}})$ is as in the definition of $\alpha_{\mu,2}(B)$. Then, by the definition of a transport plan, and the fact that $\varphi_{B}\equiv 1$ on $B$,

[TABLE]

For the estimate involving $\alpha$ numbers we will use the so-called Kantorovich duality for $W_{1}$ Wasserstein distance. It states that

[TABLE]

see [Vil08, Remark 6.5] for more information.

Let $L$ be a minimizing plane for $\alpha_{\mu,1}(B)$, and let $a_{B,L}$ be as in the definition of $\alpha_{\mu,1}(B)$. Since $\varphi_{B}\equiv 1$ in $B$, it follows from the definition of $\alpha_{\mu}$ that

[TABLE]

The estimate $\alpha_{\mu,1}(B)\leq\alpha_{\mu,2}(B)$ follows immediately by the Cauchy-Schwarz inequality and the fact that $\int\varphi_{{B}}\ d\mu\leq\mu(3B)$. ∎

Lemma 3.2**.**

Suppose that $\mu$ is a Radon measure on $\mathbb{R}^{d}$, and that $B\subset\mathbb{R}^{d}$ is a ball satisfying $\mu(3B)\approx\mu(6B)$. Then

[TABLE]

and

[TABLE]

Moreover, given balls $B_{1}\subset B_{2}$ such that $r(B_{1})\approx r(B_{2})$ and $\mu(3B_{1})\approx\mu(3B_{2})$ we have

[TABLE]

Proof.

The first estimate is a direct consequence of the Cauchy-Schwarz inequality.

To prove the second estimate, let $L_{B}$ be the minimizing plane for $\beta_{\mu,1}(B)$. The estimate follows if we consider the 1-Lipschitz function $\phi=\psi\operatorname{dist}(x,L_{B}),$ where $\psi$ is $r(B)^{-1}$-Lipschitz, $\psi\equiv 1$ on $B$, and $\operatorname{supp}(\psi)\subset 2B$.

The last two inequalities follow immediately from the definitions of $\beta_{\mu,2}$ and $\alpha_{\mu}$. ∎

Lemma 3.3**.**

Suppose that $\mu$ is a Radon measure, $B$ is a ball with $\mu(B)>0$, $L$ an $n$-plane intersecting $0.9B$, and assume that $c$ minimizes $F_{B}(\mu,\,c{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}})$. Then

[TABLE]

Furthermore, there exists $\varepsilon>0$ such that if $\mu(0.9B)\approx\mu(3B)$, and $F_{B}(\mu,\,c{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}})\leq\varepsilon\mu(3B)r(B),$ then

[TABLE]

Proof.

Let $r=r(B)$ and consider $\Phi(x)=(r-|x-z(B)|)_{+}\in\operatorname{\text{Lip}}_{1}(B)$. It is not difficult to see that on a significant portion (say, a half) of the $n$-dimensional ball $L\cap B$ we have $\Phi(x)\approx r$, and so

[TABLE]

If we had $c\gg\mu(B)r^{-n}$, then

[TABLE]

But in that case the constant $\tilde{c}=0$ would be better than $c$, since we always have $F_{B}(\mu,0)\leq\mu(B)r,$ and thus we reach a contradiction with optimality of $c$.

Now, assume further that $F_{B}(\mu,\,c{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}})\leq\varepsilon\mu(3B)r,$ and $\mu(0.9B)\approx\mu(3B),$ so that $\int\Phi(x)\ d\mu(x)\approx\mu(3B)r$. If we had $c\ll\mu(3B)r^{-n}$, then

[TABLE]

Thus, we reach a contradiction with $F_{B}(\mu,\,c{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L}})\leq\varepsilon\mu(3B)r.$ ∎

Lemma 3.4**.**

Suppose that $\mu$ is a Radon measure on $\mathbb{R}^{d}$, and that $B_{1},B_{2}\subset\mathbb{R}^{d}$ are concentric balls satisfying $B_{1}\subset 0.9B_{2},\ \mu(B_{1})\approx\mu(3B_{2})\approx r(B_{1})^{n}\approx r(B_{2})^{n}$. Let $L_{\beta}$ be the $n$-plane minimizing $\beta_{\mu,2}(B_{2})$, and $L_{\alpha},\ c>0,$ be the $n$-plane and constant minimizing $\alpha_{\mu}(B_{2})$. Suppose further that $L_{\alpha},L_{\beta}$ intersect $0.9B_{1}$. Then

[TABLE]

Proof.

Set $r=r(B_{1})$. It follows easily by (3.5) that $c\lesssim\mu(B_{2})r(B_{2})^{-n}\approx 1$, and so $F_{B_{1}}(\mu,c{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L_{\beta}}})\lesssim r\mu(B_{1})$. Thus, without loss of generality, we may assume that $\beta_{\mu,2}(B_{2})+\alpha_{\mu}(B_{2})<\varepsilon$ for some small $\varepsilon>0$.

By the triangle inequality, we have

[TABLE]

and so our aim is to estimate the second term from the right hand side.

Let $x_{\alpha}\in L_{\alpha}\cap\bar{B_{1}}$ and $x_{\beta}\in L_{\beta}$ be such that

[TABLE]

Without loss of generality we may assume that $x_{\alpha}=0$, so that $L_{\alpha}$ is a linear subspace. Denote $L^{\prime}_{\beta}=L_{\beta}-x_{\beta}$. It follows by basic linear algebra that for $x\in L_{\alpha}\cap B_{1}$

[TABLE]

Note that by the above and the triangle inequality

[TABLE]

On the other hand, by our choice of $x_{\beta}$, $|x_{\beta}|\leq\operatorname{dist}(x,L_{\beta})$ for all $x\in L_{\alpha}\cap B_{1}$. Together with the triangle inequality and the identity (3.8) this gives

[TABLE]

We put the two estimates above together to get

[TABLE]

Now, observe that, by the definition of $\measuredangle(L_{\alpha},L_{\beta})$, for every $x\in L_{\alpha}$ we have $|\Pi^{\perp}_{L^{\prime}_{\beta}}(x)|\leq|x|\measuredangle(L_{\alpha},L_{\beta})$. Moreover, there exists a subspace $\ell\subset L_{\alpha}$ on which the equality is achieved, i.e. for all $x\in\ell$ we have $|\Pi^{\perp}_{L^{\prime}_{\beta}}(x)|=|x|\measuredangle(L_{\alpha},L_{\beta})$. Consider a cone around $\ell$:

[TABLE]

Since $0\in B_{1}\cap K\cap L_{\alpha}$, it is easy to see that $\mathcal{H}^{n}(B_{1}\cap K\cap L_{\alpha})\gtrsim r^{n}$, which in turn implies that for some small constant $0<\delta\ll 1$ (depending on the implicit constant in the previous inequality and dimension) we have

[TABLE]

Moreover, for $x\in B_{1}\cap K\cap L_{\alpha}\setminus B(0,\delta r)$ we have

[TABLE]

Hence, using the above, (3.10), and (3.9) yields

[TABLE]

Now, consider $\phi\in\operatorname{\text{Lip}}_{1}(B_{2})$ such that $\phi(x)\approx\operatorname{dist}(x,L_{\beta})$ in $B_{1}$, and $\phi(x)\lesssim\operatorname{dist}(x,L_{\beta})$ in $B_{2}$. Then,

[TABLE]

(3.11) and the calculation above give $\measuredangle(L_{\alpha},L_{\beta})\lesssim\beta_{\mu,2}(B_{2})+\alpha_{\mu}(B_{2})<\varepsilon$. Let $\Pi\mathrel{\mathop{\ordinarycolon}}L_{\alpha}\to\mathbb{R}^{d}$ be the orthogonal projection onto $L^{\prime}_{\beta}$, and $i\mathrel{\mathop{\ordinarycolon}}L_{\alpha}\to{\mathbb{R}^{d}}$ an embedding. We have

[TABLE]

Thus, $\Pi$ is a linear isomorphism onto $L^{\prime}_{\beta}$, with a bound on Jacobian

[TABLE]

It follows that for any $f\in\operatorname{\text{Lip}}_{1}(B_{1})$ we have

[TABLE]

Taking supremum over all $f\in\operatorname{\text{Lip}}_{1}(B_{1})$, dividing by $r^{n+1}$, using (3.11), (3.12), the fact that $\mu(B_{1})\approx r^{n}$, and that $c\lesssim 1$, yields the desired inequality:

[TABLE]

∎

4. The David-Mattila cubes

In the proof of Theorem 1.1 we will use the lattice of “dyadic cubes” constructed by David and Mattila [DM00].

Lemma 4.1** ([DM00, Theorem 3.2, Lemma 5.28]).**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$, $E=\operatorname{supp}\mu.$ For any constants $C_{0}>1$, $A_{0}>5000C_{0}$ there exists a sequence of partitions of $E$ into Borel subsets $Q,\ Q\in\mathcal{D}_{k}$, with the following properties:

• (a)

For each integer $k\geq 0$, $E$ is the disjoint union of the “cubes” $Q$, $Q\in\mathcal{D}_{k}$, and if $k<l$, $Q\in\mathcal{D}_{l}$, and $R\in\mathcal{D}_{k}$, then either $Q\cap R=\varnothing$ or else $R\subset Q$.

• (b)

The general position of the cubes $Q$ can be described as follows. For each $k\geq 0$ and each cube $Q\in\mathcal{D}_{k}$, there is a ball $B(Q)=B(z_{Q},r(Q)),$ such that

[TABLE]

*and the balls $5B(Q),Q\in\mathcal{D}_{k},$ are disjoint. *

Remark 4.2**.**

The cubes of David and Mattila have many other useful properties, most notably the so-called small boundaries. We will not need them, however.

For any $Q\in\mathcal{D}\mathrel{\mathop{\ordinarycolon}}=\bigcup_{k\geq 0}\mathcal{D}_{k}$ we denote by $\mathcal{D}(Q)$ the family of $P\in\mathcal{D}$ such that $P\subset Q$. Given $Q\in\mathcal{D}_{k}$ we set $J(Q)=k$ and $\ell(Q)=56C_{0}A_{0}^{-k}$. Note that $r(Q)\approx\ell(Q).$ We define also $B_{Q}=28B(Q)=B(z_{Q},28\,r(Q)),$ so that

[TABLE]

Denote by $\mathcal{D}^{db}$ the family of doubling cubes, i.e. $Q\in\mathcal{D}$ satisfying

[TABLE]

If the constants $C_{0},\,A_{0}$ in Lemma 4.1 are chosen of the form $A_{0}=C(C_{0})^{K},\ K\geq 3n+5$, and $C_{0}=C_{0}(K,n)$ is large enough, then the following lemma holds.

Lemma 4.3** ([DM00, Lemma 5.31]).**

Let $R\in\mathcal{D}$ and $Q\subset R$ be cubes such that all the intermediate cubes $S,\ Q\subsetneq S\subsetneq R,$ are non-doubling, i.e. $S\in\mathcal{D}\setminus\mathcal{D}^{db}$. Then,

[TABLE]

Remark 4.4**.**

The constant $2d$ in (4.2) can be replaced by any positive constant if $C_{0},\,A_{0}$ are chosen suitably. See [DM00, (5.30)] for details.

It will be convenient for us to work with cubes satisfying a doubling condition stronger than (4.1). To introduce them we need a version of [Tol14, Lemma 2.8]. For reader’s convenience, we provide the proof below.

Lemma 4.5**.**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$ and $\alpha>1$ be some constant. Then, for $\mu$-a.e. $x\in\mathbb{R}^{d}$ there exists a sequence $r_{j}\to 0$ such that for every $j$ we have

[TABLE]

Proof.

Consider the set $Z\subset\operatorname{supp}\mu$ of points such that for $x\in Z$ there does not exist a sequence of radii $r_{j}\to 0$ satisfying (4.3). We want to show that $\mu(Z)=0$. Let

[TABLE]

Clearly $Z=\bigcup_{j}Z_{j}$, and so it suffices to prove $\mu(Z_{j})=0$ for all $j\geq 0$.

Let $B_{0}$ be an arbitrary ball of radius $2^{-j}$ centered at $Z_{j}$, and choose some integer $k\geq d$. For each $x\in Z_{j}\cap B_{0}$ we set $B_{x}=B(x,\alpha^{-k}2^{-j})$. Observe that, by the definition of $Z_{j}$, for $h=0,\dots,k-1$ we have

[TABLE]

Thus,

[TABLE]

Now, we use Besicovitch covering theorem to choose points $\{x_{m}\}\subset Z_{j}\cap B_{0}$ such that $\bigcup_{m}B_{x_{m}}$ covers $Z_{j}\cap B_{0}$, and moreover $\sum_{m}\mathds{1}_{B_{x_{m}}}\leq C_{d}$. The bounded intersection property implies that $N\mathrel{\mathop{\ordinarycolon}}=\#\{x_{m}\}<\infty$, and more precisely

[TABLE]

where $\omega_{d}$ stands for the volume of a $d$-dimensional ball. Hence,

[TABLE]

Consequently, we may use (4.4) and the fact that $\bigcup_{m}B_{x_{m}}\supset Z_{j}\cap B_{0}$ to obtain

[TABLE]

Since $k$ can be chosen as large as we wish, this gives $\mu(Z_{j}\cap B_{0})=0$. But $B_{0}$ was an arbitrary ball, and so $\mu(Z_{j})=0$. ∎

We may use the lemma above to show the following.

Lemma 4.6**.**

There exists a constant $C=C(d,C_{0},A_{0})$ such that for $\mu$-a.e. $x\in\mathbb{R}^{d}$ there exists a sequence of cubes $Q_{j}\in\mathcal{D}^{db}$ satisfying $x\in Q_{j}$, $\ell(Q_{j})\to 0$, and

[TABLE]

Proof.

Let $\alpha=2\,C_{0}^{2}\,A_{0}^{k+1}$, where $k$ is a constant that will be fixed later on. Consider a sequence of balls $B(x,r_{j})$ given by Lemma 4.5. Fix some $j$. Let $Q$ be the smallest cube satisfying $x\in Q$ and $B(x,r_{j})\subset 100B(Q).$ We have

[TABLE]

It is easy to check that, with the choice of $\alpha$ we made at the beginning, we have

[TABLE]

where $R$ is the $k$-th ancestor of $Q$, i.e. $Q\subset R$ and $J(Q)-J(R)=k$.

Now, if all the intermediate cubes $S,\ Q\subsetneq S\subsetneq R,$ were non-doubling, then by (4.2) and Lemma 4.5 we would have

[TABLE]

For $k=k(d,C_{0},A_{0})$ big enough the constant on the right hand side is smaller than $1$, and so we reach a contradiction. It follows that one of the intermediate cubes $S$ is doubling. Thus,

[TABLE]

Setting $Q_{j}=S$ finishes the proof. ∎

We will call the cubes satisfying (4.5) strongly doubling, and the family of all such cubes will be denoted by $\mathcal{D}^{sdb}$. We fix constants $C_{0}$ and $A_{0}$ so that all of the above holds, and from now on we will treat them as absolute constants. We will not mention dependence on them in our estimates.

Finally, for each $Q\in\mathcal{D}$ we define $L_{Q}$ to be the $n$-plane minimizing $\beta_{\mu,2}(3B_{Q})$, and $c_{Q}\geq 0$ to be the constant minimizing $\alpha_{\mu}(3B_{Q})$.

5. Main Lemma

Given $\varepsilon>0$ and $r>0$ let us define the set of “good points”:

[TABLE]

Using this notation we may formulate our main lemma.

Lemma 5.1**.**

Let $\mu$ be a finite Radon measure on $\mathbb{R}^{d}$. There exists a small dimensional constant $\varepsilon_{0}>0$ such that the following holds: suppose that $R_{0}\in\mathcal{D}^{sdb}$ satisfies

[TABLE]

Then, there exists a set $R_{G}\subset R_{0}$, and a Lipschitz map $F\mathrel{\mathop{\ordinarycolon}}L_{R_{0}}\rightarrow L_{R_{0}}^{\perp}$ (recall that $L_{R_{0}}$ denotes the $n$-dimensional plane minimizing $\beta_{\mu,2}(3B_{R_{0}})$), such that for

[TABLE]

we have $R_{G}\subset\Gamma$,

[TABLE]

and ${\left.\kern-1.2pt\mu\right|_{R_{G}}}$ is absolutely continuous with respect to $\mathcal{H}^{n}$.

Several remarks are in order.

Remark 5.2**.**

Assumption (5.2) is implied by a somewhat more natural condition

[TABLE]

Remark 5.3**.**

The constant $\frac{1}{2}$ in (5.3) can be replaced by any $\delta\in(0,1)$, as long as we allow $\varepsilon_{0}$ to depend on $\delta$. Naturally, $\varepsilon_{0}(\delta)\to 0$ as $\delta\to 1$.

Remark 5.4**.**

Recall that we defined homogeneous $\beta$ numbers $\beta_{\mu,2}^{h}$ in (1.2). One could similarly define $\alpha_{\mu}^{h}(x,r)=\frac{\mu(B(x,3r))}{r^{n}}\alpha_{\mu}(x,r).$ Careful inspection of the proof of Lemma 5.1 (see Remark 5.6) shows the following. If instead of (5.1) we define for $Q\in\mathcal{D}$

[TABLE]

and we replace the assumption (5.2) by $\mu\left(R_{0}\setminus G^{\varepsilon_{0}}_{R_{0}}\right)\leq\varepsilon_{0}\,\mu(3B_{R_{0}}),$ then the conclusion of Lemma 5.1 still holds. In other words, if the homogeneous square functions in some initial cube $R_{0}$ are small relative to density of $\mu$ in the initial cube, then $\mu$ is rectifiable on a large chunk of $R_{0}$.

Let us show how Lemma 5.1 may be used to prove Theorem 1.1.

Proof of Theorem 1.1 using Lemma 5.1.

To show that $\mu$ is $n$-rectifiable it suffices to show that for any $E\subset\mathbb{R}^{d}$ satisfying $\mu(E)>0$ there exists $F\subset E$ with $\mu(F)>0$ and such that ${\left.\kern-1.2pt\mu\right|_{F}}$ is rectifiable. Let us fix $E\subset\mathbb{R}^{d}$ with $\mu(E)>0$.

Let $\varepsilon_{0}>0$ be so small that Lemma 5.1 holds. Note that by the assumption on the finiteness of $\alpha$ and $\beta$ square functions (1.3), (1.4), we have

[TABLE]

In particular, $\mu$-almost all of $E$ is contained in $\bigcup_{r>0}G^{\varepsilon_{0}}_{r}$. By Lebesgue differentiation theorem, for $\mu$-almost every $x\in E\cap G^{\varepsilon_{0}}_{r}$

[TABLE]

Taking into account that for $s<r$ we have $G^{\varepsilon_{0}}_{s}\supset G^{\varepsilon_{0}}_{r}$, it follows that for $\mu$-almost every $x\in E$

[TABLE]

Choose some $x\in E$ such that the above and Lemma 4.6 hold. Let $r_{0}>0$ be so small that $\mu(B(x,r)\cap E\cap G^{\varepsilon_{0}}_{r})>(1-\varepsilon_{0})\mu(B(x,r))$ for all $r<r_{0}$.

Using Lemma 4.6 we may choose $R_{0}\in\mathcal{D}^{sdb}$ such that $x\in R_{0}$ and $\tilde{r}\mathrel{\mathop{\ordinarycolon}}=2r(B_{R_{0}})<r_{0}$. We have $R_{0}\subset B(x,\tilde{r})\subset 3B_{R_{0}}$, and so

[TABLE]

Hence, $R_{0}$ satisfies the assumptions of Lemma 5.1. We obtain a Lipschitz graph $\Gamma$ and a set $R_{G}\subset R_{0}\cap\Gamma$ such that $\mu(R_{G})\geq 0.5\mu(R_{0}),$ and ${\left.\kern-1.2pt\mu\right|_{R_{G}}}$ is absolutely continuous with respect to $\mathcal{H}^{n}$. On the other hand, arguing as above, and using the fact that $R_{0}$ is doubling, we see that $\mu(R_{0}\setminus E)\leq\varepsilon_{0}\mu(3B_{R_{0}})\leq C_{0}\varepsilon_{0}\mu(R_{0})<0.5\mu(R_{0})$, assuming $\varepsilon_{0}<0.5\,C_{0}^{-1}.$

It follows that $\mu(R_{G}\cap E)\geq\mu(R_{G})-\mu(R_{0}\setminus E)>0,$ and ${\left.\kern-1.2pt\mu\right|_{R_{G}\cap E}}$ is $n$-rectifiable. Setting $F=R_{G}\cap E$ concludes the proof.

∎

The rest of the paper is dedicated to proving Lemma 5.1. We fix $R_{0}\in\mathcal{D}^{sdb}$ satisfying (5.2). The constant $\varepsilon_{0}$ will be chosen later on. To simplify notation, we set $G=G^{\varepsilon_{0}}_{r(R_{0})},\ B_{0}=B_{R_{0}},\ r_{0}=r(B_{0}),\ z_{0}=z_{R_{0}},\ c_{0}=c_{R_{0}}$ (where $c_{R_{0}}$ is a constant minimizing $\alpha_{\mu}(3B_{0})$), $L_{0}=L_{R_{0}},$ (where $L_{R_{0}}$ is an $n$-plane minimizing $\beta_{\mu,2}(3B_{0})$), and $\Pi_{0}=\Pi_{L_{0}}$.

Remark 5.5**.**

Without loss of generality we may (and will) assume that

[TABLE]

so that (using the strong doubling property of $R_{0}$ (4.5))

[TABLE]

Indeed, if we consider the normalized measure $\nu=\mu/\Theta_{\mu}(3B_{0})$, then: $\Theta_{\nu}(3B_{0})=1$; for any ball $B$ with $\mu(B)>0$ we have $\alpha_{\mu}(B)=\alpha_{\nu}(B),\ \beta_{\mu,2}(B)=\beta_{\nu,2}(B)$; and if the assumptions of Lemma 5.1 were satisfied for $\mu$, then they are also satisfied for $\nu$. Sets $\Gamma$ and $R_{G}$ constructed for $\nu$ will also have all the desired properties when applied to $\mu$.

Remark 5.6**.**

The reduction to case $\Theta_{\mu}(3B_{0})=1$ performed above is one of the main reasons why we decided to work with non-homogeneous (i.e. normalized by $\mu(3B)$) $\alpha$ and $\beta$ coefficients. If we assumed a priori that $\Theta_{\mu}(3B_{0})=1$, then we could replace $\alpha_{\mu}$ and $\beta_{\mu,2}$ numbers in (5.1) by $\alpha^{h}_{\mu}$ and $\beta^{h}_{\mu,2}$, and then carry on with the proof without making any changes. Roughly speaking, throughout most of the proof we work with cubes $Q$ satisfying $\mu(3B_{Q})\approx\ell(Q)^{n}\Theta_{\mu}(3B_{0}),$ so that $\alpha_{\mu}^{h}(3B_{Q})\approx\alpha_{\mu}(3B_{Q})\Theta_{\mu}(3B_{0})$ and $\beta_{\mu,2}^{h}(3B_{Q})\approx\beta_{\mu,2}(3B_{Q})\Theta_{\mu}(3B_{0})^{1/2}$ – see Remark 6.2.

Now, the claim we made in Remark 5.4 follows because the modified assumption (involving $G^{\varepsilon}_{Q}$) allows us to make the reduction $\Theta_{\mu}(3B_{0})=1$.

6. Stopping cubes

This section is dedicated to performing the stopping time argument. We will show basic properties of the resulting tree of cubes, and estimate the size of two families of stopping cubes.

The stopping conditions involve parameters $A\gg 1,\ \tau\ll 1,\ \theta\ll 1$, which depend on dimension and which will be fixed later on. The constant $\varepsilon_{0}$ is fixed at the very end of the proof, and depends on $A,\ \tau,\ \theta$.

We define the following subfamilies of $\mathcal{D}(R_{0})$:

• •

$\mathsf{HD}_{0}$ (“high density”), which contains cubes $Q\in\mathcal{D}(R_{0})$ satisfying

[TABLE]

• •

$\mathsf{LD}_{0}$ (“low density”), which contains cubes $Q\in\mathcal{D}(R_{0})$ satisfying

[TABLE]

• •

$\mathsf{BS}_{0}$ (“big square functions”), which contains cubes $Q\in\mathcal{D}(R_{0})\setminus(\mathsf{LD}_{0}\cup\mathsf{HD}_{0})$ satisfying

[TABLE]

Let $\mathsf{Stop}_{0}$ be the family of maximal (and thus disjoint) cubes from $\mathsf{HD}_{0}\cup\mathsf{LD}_{0}\cup\mathsf{BS}_{0}$, and let $\mathsf{Tree}_{0}\subset\mathcal{D}(R_{0})$ be the family of cubes that are not contained in any $Q\in\mathsf{Stop}_{0}$. In particular, $\mathsf{Stop}_{0}\not\subset\mathsf{Tree}_{0}$.

Recall that $L_{Q}$ is an $n$-plane minimizing $\beta_{\mu,2}(3B_{Q})$. We define

[TABLE]

We introduce two more families of stopping cubes:

• •

$\mathsf{BA}_{0}$ (“big angles”), which contains cubes $Q\in\mathcal{D}(R_{0})\setminus\mathsf{Stop}_{0}$ satisfying

[TABLE]

• •

$\mathsf{F}_{0}$ (“far”), which consists of $Q\in\mathcal{D}(R_{0})\setminus(\mathsf{Stop}_{0}\cup\mathsf{BA}_{0})$ satisfying

[TABLE]

Let $\mathsf{Stop}\subset\mathcal{D}(R_{0})$ be the family of maximal (and thus disjoint) cubes from $\mathsf{Stop}_{0}\cup\mathsf{BA}_{0}\cup\mathsf{F}_{0}$. Set $\mathsf{HD}=\mathsf{HD}_{0}\cap\mathsf{Stop},\ \mathsf{LD}=\mathsf{LD}_{0}\cap\mathsf{Stop},\ \mathsf{BS}=\mathsf{BS}_{0}\cap\mathsf{Stop},\ \mathsf{BA}=\mathsf{BA}_{0}\cap\mathsf{Stop},\ \mathsf{F}=\mathsf{F}_{0}\cap\mathsf{Stop}.$ We define $\mathsf{Tree}\subset\mathsf{Tree}_{0}$ as the family of cubes that are not contained in any $Q\in\mathsf{Stop}$. Note that $\mathsf{Stop}\not\subset\mathsf{Tree}$. For $P\in\mathcal{D}$ we set $\mathsf{Tree}_{0}(P)=\mathsf{Tree}_{0}\cap\mathcal{D}(P),\ \mathsf{Tree}(P)=\mathsf{Tree}\cap\mathcal{D}(P)$.

6.1. Properties of cubes in $\mathsf{Tree}$

Lemma 6.1**.**

The following estimates hold:

[TABLE]

Proof.

All estimates except for (6.5) follow immediately from the stopping time conditions. (6.5) holds for $R_{0}$ because $R_{0}\in\mathcal{D}^{sdb}$. To see it for $Q\in\mathsf{Tree}_{0}\cup\mathsf{Stop}_{0},\ Q\neq R_{0}$, note that the parent of $Q$, denoted by $R$, satisfies $R\in\mathsf{Tree}_{0}$, and so $\mu(100B_{Q})\leq\mu(3B_{R})\leq A\ell(R)^{n}\approx A\ell(Q)^{n}$. ∎

Remark 6.2**.**

Note that, by (6.4) and (6.5), for $Q\in\mathsf{Tree}_{0}\cup\mathsf{Stop}_{0}\setminus\mathsf{LD}_{0}$ we have $\beta_{\mu,2}(3B_{Q})\approx_{A,\tau}\beta^{h}_{\mu,2}(3B_{Q})$ and $\alpha_{\mu}(3B_{Q})\approx_{A,\tau}\alpha^{h}_{\mu}(3B_{Q})$.

Lemma 6.3**.**

Let $R\in\mathsf{Tree}_{0}$. Then

[TABLE]

Moreover, for any $x\in 3B_{0}$

[TABLE]

Proof.

Let $Q\in\mathsf{Tree}_{0}(R)$. By the definition of $G$, for any $z\in 4B_{Q}\cap G$ we have

[TABLE]

It is easy to see that for $300r(Q)\leq r\leq 400r(Q)$ we have $3B_{Q}\subset B(z,r)\subset 25B_{Q}$, and that $\mu(9B_{Q})\approx_{A,\tau}\mu(B(z,3r))\approx_{A,\tau}\mu(100B_{Q})$. Using (3.4) with $B_{1}=3B_{Q}$ and $B_{2}=B(z,r)$ yields

[TABLE]

Integrating with respect to $r$ gives us for every $z\in 4B_{Q}\cap G$

[TABLE]

To see (6.11), let $x\in 3B_{0}$ and choose some $P\in\mathsf{Tree}_{0}$ satisfying $x\in 3B_{P}$. By (6.6) we may pick $z\in P\cap G$. It is clear that for all cubes $Q\in\mathsf{Tree}_{0}$ such that $\ell(Q)>\ell(P)$ and $x\in 3B_{Q}$ we have $z\in 4B_{Q}\cap G$. Thus, summing (6.14) over all such $Q\subset R_{0}$, and noticing that for any fixed sidelength $\ell(Q_{0})>\ell(P)$ there are only boundedly many $Q$ with $\ell(Q)=\ell(Q_{0})$ and $3B_{Q}\ni x$, yields

[TABLE]

Since the estimate holds for arbitrary $P\in\mathsf{Tree}_{0}$ with $x\in 3B_{P}$, (6.11) follows.

To see (6.9), we integrate (6.11) over $x\in 3B_{R}$ to get

[TABLE]

The estimates for $\beta_{\mu,2}(3B_{Q})$ can be shown in the same way. ∎

Corollary 6.4**.**

We have

[TABLE]

Proof.

Let $Q\in\mathsf{Tree}_{0}$. Recall that by (6.4), (6.5), we have $\mu(2.5B_{Q})\approx_{A,\tau}\mu(3B_{Q})\approx_{A,\tau}\ell(Q)^{n}.$ Moreover, it follows easily by (6.4) and the smallness of $\alpha$ and $\beta$ numbers (6.11),(6.12), that the best approximating planes for $\beta_{\mu,2}(3B_{Q})$ and $\alpha_{\mu}(3B_{Q})$ intersect $2B_{Q}$.

Hence, by Lemma 3.4 applied to $B_{1}=2.5B_{Q}$ and $B_{2}=3B_{Q}$, and by Lemma 6.3, we get the desired estimates. ∎

Corollary 6.5**.**

For every $Q\in\mathsf{Tree}_{0}$

[TABLE]

Proof.

By (6.4), (6.5), we have $\mu(1.5B_{Q})\approx_{A,\tau}\mu(9B_{Q})\approx_{A,\tau}\ell(Q)^{n}.$ Together with the smallness of $\alpha_{\mu}(3B_{Q})$ (6.11), this implies that the best approximating plane for $\alpha_{\mu}(3B_{Q})$ intersects $2B_{Q}$. Thus, Lemma 3.3 yields

[TABLE]

∎

Lemma 6.6**.**

We have

[TABLE]

Proof.

We begin by using Chebyshev and Cauchy-Schwarz inequalities to obtain

[TABLE]

By Fubini, the right hand side is equal to

[TABLE]

We can estimate this using the smallness of $\beta$-numbers (6.10), and thus

[TABLE]

∎

6.2. Balanced balls

Lemma 6.7** ([AT15, Lemma 3.1, Remark 3.2]).**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$, and let $B\subset\mathbb{R}^{d}$ be some ball with radius $r>0$ such that $\mu(B)>0$. Let $0<\gamma<1$. Then there exist constants $\rho_{1}=\rho_{1}(\gamma)>0$ and $\rho_{2}=\rho_{2}(\gamma)>0$ such that one of the following alternatives holds:

• (a)

There are points $x_{0},\dots,x_{n}\in B$ such that

[TABLE]

and for any $y_{k}\in B(x_{k},\rho_{1}r),\ k=1,\dots,n$, if we denote by $L^{y}_{k}$ the $k$-plane passing through $y_{0},\dots,y_{k}$, then we have

[TABLE]

• (b)

There exists a family of balls $\{B_{i}\}_{i\in I_{B}},$ with radii $r(B_{i})=4\gamma r$, centered on $B$, so that the balls $\{10B_{i}\}_{i\in I_{B}}$ are pairwise disjoint,

[TABLE]

and

[TABLE]

We will say that a ball $B$ is $\gamma$-balanced if the alternative (a) holds.

Lemma 6.8**.**

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$, $B\subset\mathbb{R}^{d}$ be a ball such that $\mu(B)\approx\mu(1.1B)>0$. Suppose $L$ is the $n$-plane minimizing $\alpha_{\mu}(1.1B)$ and that $L$ intersects $0.9B$. There exist $C=C(n,d)<1,\,\gamma=\gamma(n,d)<1$ such that if $\alpha_{\mu}(1.1B)\leq C\gamma$, then $B$ is $\gamma$-balanced.

Proof.

Proof by contradiction. Suppose that $B$ is not $\gamma$-balanced, i.e. that the alternative (b) in Lemma 6.7 holds.

We will estimate $\alpha_{\mu}(1.1B)$ from below. Let $c$ be the constant minimizing $\alpha_{\mu}(1.1B)$, so that by (3.5)

[TABLE]

Let balls $\{B_{i}\}_{i\in I_{B}}$ be as in Lemma 6.7 (b), with $r(B_{i})=r_{i}=4\gamma r(B)$. Let $f\in\operatorname{\text{Lip}}_{1}(1.1B)$ be defined in such a way that $f\equiv r_{i}$ on each $B_{i}$ and $\operatorname{supp}f\subset\bigcup_{i\in I_{B}}2B_{i}\subset 1.1B$. Then,

[TABLE]

On the other hand,

[TABLE]

The two estimates above imply that for some dimensional constants $C_{1},C_{2}$

[TABLE]

if we take $\gamma$ and $C=C(C_{1},C_{2})$ small enough. We reach a contradiction with the assumption $\alpha_{\mu}(1.1B)\leq C\gamma$. ∎

Corollary 6.9**.**

Let $Q\in\mathsf{Tree}_{0}$. Then $2.5B_{Q}$ is $\gamma$-balanced, where $\gamma=\gamma(n,d)$.

Proof.

We know that $\mu(1.5B_{Q})\approx_{A,\tau}\mu(9B_{Q})$, and that

[TABLE]

which implies (for $\varepsilon_{0}$ small enough) that the best approximating plane for $3B_{Q}$ intersects $2B_{Q}$. Applying Lemma 6.8 to $B=2.5B_{Q}$ finishes the proof.

∎

6.3. Small measure of cubes from $\mathsf{BS}$ and $\mathsf{F}$

Lemma 6.10**.**

We have

[TABLE]

Proof.

We start by estimating the measure of cubes from $\mathsf{BS}$. We use the definition of $\mathsf{BS}$ (6.1) to get

[TABLE]

Concerning $\mathsf{F}$, we use the $5R$-covering lemma to get a countable family of pairwise disjoint balls $B_{i}\mathrel{\mathop{\ordinarycolon}}=3B_{Q_{i}}$, $Q_{i}\in\mathsf{F}$, such that $\bigcup_{i}5B_{i}\supset\bigcup_{Q\in\mathsf{F}}Q$. For every $i$ we have

[TABLE]

Then

[TABLE]

∎

7. Construction of the Lipschitz graph

In this section we construct the Lipschitz graph $\Gamma$. At the beginning of Subsection 7.2 we define also the good set $R_{G}\subset\Gamma\cap R_{0}$, and we show that ${\left.\kern-1.2pt\mu\right|_{R_{G}}}\ll\mathcal{H}^{n}$. We start by proving some auxiliary estimates.

7.1. Estimates involving best approximating planes

Lemma 7.1**.**

[AT15, Lemma 6.4]** Suppose $P_{1},P_{2}$ are $n$-planes in $\mathbb{R}^{d}$, $X=\{x_{0},\dots,x_{n}\}$ is a collection of $n$ points, and

[TABLE]

where $d_{2}<d_{1}/(2d)$. Then for $y\in P_{2}$

[TABLE]

Lemma 7.2**.**

Suppose $Q_{1},Q_{2}\in\mathsf{Tree}_{0}$ are such that $\operatorname{dist}(Q_{1},Q_{2})\lesssim\ell(Q_{1})\approx\ell(Q_{2})$. Let $P\in\mathsf{Tree}_{0}$ be the smallest cube such that $3B_{P}\supset 3B_{Q_{1}}\cup 3B_{Q_{2}}.$ Then $\ell(P)\approx\ell(Q_{1}),$ and for all $y\in L_{Q_{2}}$

[TABLE]

In particular,

[TABLE]

Proof.

Since $3B_{0}\supset 3B_{Q_{1}}\cup 3B_{Q_{2}}$ and $R_{0}\in\mathsf{Tree}_{0}$, the cube $P$ is well-defined. The comparability $\ell(P)\approx\ell(Q_{2})$ holds due to the assumption $\operatorname{dist}(Q_{1},Q_{2})\lesssim\ell(Q_{1})\approx\ell(Q_{2})$.

Since $Q_{1}\in\mathsf{Tree}_{0}$, Corollary 6.9 tells us that $2.5B_{Q_{1}}$ is $\gamma$-balanced. Let $x_{0},\dots,x_{n}\in 2.5B_{Q_{1}}$ be the points from alternative (a) in Lemma 6.7. Thus, we have a family of balls $\{B_{k}\coloneqq B(x_{k},\rho_{1}r(2.5B_{Q_{1}}))\}_{k=0,\dots,n}$, such that $\mu(B_{k}\cap 2.5B_{Q_{1}})\geq\rho_{2}\mu(2.5B_{Q_{1}})\approx_{A,\tau}\rho_{2}\ell(Q_{1}).$

Since $r(B_{k})=\rho_{1}r(2.5B_{Q_{1}})\approx\ell(P)$, and $B_{k}\subset 3B_{Q_{1}}\subset 3B_{P}$, it is clear that

[TABLE]

and

[TABLE]

Keeping in mind that $\rho_{2}$ is a dimensional constant, we will not signal dependence on it in further computations. We use the above estimates and the Chebyshev inequality to find points $y_{k}\in B_{k}$ such that

[TABLE]

We would like to apply Lemma 7.1 to $n$-planes $L_{Q_{1}},L_{P}$ and points $X=\{y_{0},\dots,y_{n}\}$. We have $d_{1}\gtrsim\gamma$ thanks to (6.19). Furthermore, due to estimate (6.12) we know that $\beta_{\mu,2}(3B_{P})\lesssim_{A,\tau}\varepsilon_{0}$, and so $\beta_{\mu,2}(3B_{P})\approx_{A,\tau}d_{2}<d_{1}/(2d)$ for $\varepsilon_{0}$ small enough. Thus,

[TABLE]

Since the assumptions about cubes $Q_{1}$ and $Q_{2}$ are identical, it turns out that the estimates above are also valid if we replace $Q_{1}$ with $Q_{2}$, i.e.

[TABLE]

Using the triangle inequality, estimates (7.1), (7.3), and the fact that $(\operatorname{dist}(y,Q_{1})+\ell(Q_{1}))\approx(\operatorname{dist}(y,Q_{2})+\ell(Q_{2}))$ we finally reach the desired inequality

[TABLE]

∎

Lemma 7.3**.**

Let $Q,P\in\mathsf{Tree}$ be such that $\ell(Q)\lesssim\ell(P)$ and $\operatorname{dist}(Q,P)\lesssim\ell(P)$. Then for any $x\in L_{Q}\cap CB_{Q}$ we have

[TABLE]

Proof.

Consider first the special case $Q\subset P$.

By Corollary 6.9, there exist balls $B_{k}=B(x_{k},\rho_{1}r(Q)),\ k=0,\dots,n,$ such that $\mu(B_{k}\cap 2.5B_{Q})\geq\rho_{2}\mu(2.5B_{Q})$, and $\operatorname{dist}(y_{k},L^{y}_{k-1})\gtrsim\gamma\ell(Q)$ for $y_{k}\in B_{k}$ (see (6.19)).

It follows by (6.8) that, for $\varepsilon_{0}$ small enough, $B_{i}\setminus R_{\mathsf{Far}}\not=\varnothing$. Fix some $y_{i}\in B_{i}\setminus R_{\mathsf{Far}}$ for every $i=0,\dots,n$, so that

[TABLE]

Let $z_{i}$ be the orthogonal projection of $y_{i}$ onto $L_{Q}$. Since $\ell(Q)\lesssim\ell(P)$, the triangle inequality yields

[TABLE]

Furthermore, if $\varepsilon_{0}$ is small enough, $|y_{i}-z_{i}|\lesssim\sqrt{\varepsilon_{0}}\ell(Q)$ and $\operatorname{dist}(y_{k},L^{y}_{k-1})\gtrsim\ell(Q)$ imply that $\operatorname{dist}(z_{k},L^{z}_{k-1})\gtrsim\ell(Q)$, and that $z_{i}\in 3B_{Q}$. Since $L_{Q}=\text{span}(z_{0},\dots,z_{n})$, it follows by elementary geometry and (7.5) that for any $x\in L_{Q}\cap CB_{Q}$

[TABLE]

which concludes the proof in the case $Q\subset P$.

Now, the general case follows by the above and Lemma 7.2. Indeed, take a cube $R\in\mathsf{Tree}$ such that $R\supset Q$ and $\ell(R)=\ell(P)$. The assumption $\operatorname{dist}(Q,P)\lesssim\ell(P)$ gives us $\operatorname{dist}(R,P)\lesssim\ell(P)$, and so we can apply Lemma 7.2 to get

[TABLE]

On the other hand, since $Q\subset R$, we already know that for $x\in L_{Q}\cap CB_{Q}$ we have

[TABLE]

Putting together the two inequalities above yields the desired result. ∎

Lemma 7.4**.**

Suppose the cubes $Q_{1},Q_{2}\in\mathsf{Tree}_{0}$ satisfy $2.5B_{Q_{1}}\subset 2.5B_{Q_{2}},\ \ell(Q_{1})\approx\ell(Q_{2}).$ Then

[TABLE]

Proof.

Set $B_{i}=2.5B_{Q_{i}},r_{i}=r(B_{i}),z_{i}=z(B_{i}),c_{i}=c_{Q_{i}},L_{i}=L_{Q_{i}}$ for $i=1,2.$ Let $\phi(z)=(r_{1}-|z_{1}-z|)_{+}\in\operatorname{\text{Lip}}_{1}(B_{1}).$ Then

[TABLE]

The fact that the last term above can also be estimated by $\varepsilon_{0}r_{1}^{n}$ follows easily by the fact that $L_{1}$ and $L_{2}$ are close to each other, see Lemma 7.2. ∎

7.2. Lipschitz function $F$ corresponding to the good part of $R_{0}$

Consider an auxiliary function

[TABLE]

Let

[TABLE]

Observe that, by the definition of function $d$, we have $R_{0}\setminus\bigcup_{Q\in\mathsf{Stop}}Q\subset R_{G}$.

Lemma 7.5**.**

We have ${\left.\kern-1.2pt\mu\right|_{R_{G}}}\ll\mathcal{H}^{n}$, and for $x\in R_{G}$

[TABLE]

In consequence, $d{\left.\kern-1.2pt\mu\right|_{R_{G}}}=g\,d{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}$ with $g\approx_{A,\tau}1$.

Proof.

Let $x\in R_{G}$. Given some small $h>0$ we use the fact that $d(x)=0$ to find $Q\in\mathsf{Tree}$ such that $B(x,h)\subset 3B_{Q}$ and $\ell(Q)\approx h$. Then

[TABLE]

Now, let $P\in\mathsf{Tree}$ be such that $3B_{P}\subset B(x,h)$ and $\ell(P)\approx h$. Then

[TABLE]

Letting $h\rightarrow 0$ we get $1\lesssim_{\tau}\Theta^{n}_{*}(\mu,x)\leq\Theta^{*n}(\mu,x)\lesssim_{A}1$ for $x\in R_{G}$. The upper density estimate and [Mat95, Theorem 6.9 (1)] imply ${\left.\kern-1.2pt\mu\right|_{R_{G}}}\ll{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}$ and ${\left.\kern-1.2pt\mu\right|_{R_{G}}}(B)\lesssim_{A}{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}(B)$ for all $B\subset\mathbb{R}^{d}$ Borel. The lower density estimate together with [Mat95, Theorem 6.9 (2)] give ${\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}\ll{\left.\kern-1.2pt\mu\right|_{R_{G}}}$ and ${\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}(B)\lesssim_{\tau}{\left.\kern-1.2pt\mu\right|_{R_{G}}}(B)$ (in particular, ${\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}$ is a finite Radon measure). Putting it all together, we use Radon-Nikodym theorem to get $d{\left.\kern-1.2pt\mu\right|_{R_{G}}}=g\,d{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}$, with $g\approx_{A,\tau}1$. ∎

In this subsection we will define function $F(x)$ for $x\in\Pi_{0}(R_{G})\subset L_{0}.$

Lemma 7.6**.**

If $\varepsilon_{0}$ and $\theta$ are small enough, then for any $x_{1},x_{2}\in\mathbb{R}^{d}$

[TABLE]

Proof.

Fix some small $h>0$. Let $Q_{1},Q_{2}\in\mathsf{Tree}$ be such that

[TABLE]

Take any $y_{i}\in Q_{i}$. Note that $|x_{i}-y_{i}|\leq d(x_{i})+h$. The triangle inequality gives us

[TABLE]

and similarly

[TABLE]

Hence, if we show that

[TABLE]

use the two former inequalities, and let $h\rightarrow 0$, we will get (7.7).

Let $P_{i}\in\mathsf{Tree}$ be the smallest cubes such that $3B_{P_{i}}\supset B_{Q_{i}}$ and

[TABLE]

We also take the smallest cube $R\in\mathsf{Tree}$ such that $3B_{R}\supset 3B_{P_{1}}\cup 3B_{P_{2}}$ and

[TABLE]

We use the fact that $3B_{R}\supset 3B_{P_{1}}\cup 3B_{P_{2}}$, the estimates (6.4), (6.5), the smallness of $\beta$ numbers (6.12), and the bound $\varepsilon_{0}\ell(R)^{n}\lesssim\ell(P_{i})^{n}$, to get

[TABLE]

Hence, by Chebyshev’s inequality, there exist some $z_{i}\in 3B_{P_{i}}$ such that

[TABLE]

where $\pi$ denotes orthogonal projection onto $L_{R}$, and the second inequality is due to (7.9). Note also that, since $y_{i},z_{i}\in 3B_{P_{i}}$, we have

[TABLE]

Now, the triangle inequality and 1-Lipschitz property of $\Pi_{0}^{\perp}$ give us

[TABLE]

To estimate the first term from the right hand side we use the fact that projections onto $L_{R}$ and $L_{0}$ are close to each other (6.7), the triangle inequality, and 1-Lipschitz property of $\Pi$:

[TABLE]

Putting together the two estimates above, as well as (7.10), (7.11), yields

[TABLE]

Since $|y_{1}-y_{2}|\approx|\Pi_{0}(y_{1})-\Pi_{0}(y_{2})|+|\Pi_{0}^{\perp}(y_{1})-\Pi_{0}^{\perp}(y_{2})|$, we may take $\varepsilon_{0}=\varepsilon_{0}(A,\tau,\theta)$ so small that

[TABLE]

Then, for $\theta$ small enough, we obtain the desired inequality (7.8):

[TABLE]

∎

The lemma above gives us for any $x,y\in R_{G}$

[TABLE]

This allows us to define a function $F$ on $\Pi_{0}(R_{G})\subset L_{0}$ as

[TABLE]

with $\operatorname{\text{Lip}}(F)\lesssim\theta$. Note that the graph of such $F$ is precisely $R_{G}$.

7.3. Extension of $F$ to the whole $L_{0}$

For any $z\in L_{0}$ let us define

[TABLE]

For each $z\in L_{0}$ with $D(z)>0$, i.e. $z\in L_{0}\setminus\Pi_{0}(R_{G})$, we define $J_{z}$ as the largest dyadic cube from $L_{0}$ such that $z\in J_{z}$ and

[TABLE]

Let $J_{i},i\in I,$ be a relabeling of the set of all such cubes $J_{z}$, without repetition.

Lemma 7.7**.**

The cubes $\{J_{i}\}_{i\in I}$ are disjoint and satisfy the following:

• (a)

If $z\in 15J_{i}$, then $5\operatorname{diam}(J_{i})\leq D(z)\leq 50\operatorname{diam}(J_{i})$.

• (b)

If $15J_{i}\cap 15J_{i^{\prime}}\not=\varnothing$, then

[TABLE]

• (c)

For each interval $J_{i}$ there are at most $N$ intervals $J_{i^{\prime}}$ such that $15J_{i}\cap 15J_{i^{\prime}}\not=\varnothing$.

• (d)

$L_{0}\setminus\Pi_{0}(R_{G})=\bigcup_{i\in I}J_{i}=\bigcup_{i\in I}15J_{i}.$**

The proof is straightforward and follows directly from the definition of $J_{i}$, see [Tol14, Lemma 7.20].

Note that, since $\beta_{\mu,2}(3B_{0})$ is very small (6.12) and $R_{0}$ is doubling, we have $\operatorname{dist}(z_{0},L_{0})\leq 2r(R_{0})=\frac{1}{14}r_{0}$. It follows that

[TABLE]

We define the set of indices

[TABLE]

Lemma 7.8**.**

The following holds:

• (a)

If $i\in I_{0}$, then $\operatorname{diam}(J_{i})\leq 0.2r_{0}$, and $3J_{i}\subset L_{0}\cap 1.9B_{0}$.

• (b)

If $J_{i}\cap 1.4B_{0}=\varnothing$ (in particular if $i\not\in I_{0}$), then

[TABLE]

Proof.

We begin by proving (a). Suppose $i\in I_{0}$. Then $J_{i}\cap 1.5B_{0}\not=\varnothing$ and

[TABLE]

We need to estimate $\operatorname{diam}(J_{i})$. By the definition of $J_{i}$, we have

[TABLE]

Since $J_{i}\cap 1.5B_{0}\not=\varnothing$ we have $\inf_{u\in J_{i}}D(u)\leq\max_{u\in L_{0}\cap 1.5B_{0}}D(u)$, and so it suffices to estimate the latter quantity. Note that the definition of $d$ (7.6) gives for $x\in 1.5B_{0}$

[TABLE]

Hence, by the definition of $D$ (7.13)

[TABLE]

It follows that $\operatorname{diam}(J_{i})\leq\frac{7}{40}r_{0}$, and

[TABLE]

Now, let us prove (b). Suppose $J_{i}\cap 1.4B_{0}=\varnothing$ and $z\in J_{i}$. Clearly, $|z_{0}-z|\geq 1.4r_{0}$. Together with the definition of $D$ (7.13) this gives

[TABLE]

On the other hand, by (7.14) we have

[TABLE]

Putting together the two estimates above gives for $z\in J_{i}$

[TABLE]

Applying Lemma 7.7 (a) yields

[TABLE]

Moreover, since

[TABLE]

we finally obtain

[TABLE]

∎

Lemma 7.9**.**

Given $i\in I_{0}$, there exists a cube $Q_{i}\in\mathsf{Tree}$ such that

[TABLE]

Proof.

Let $i\in I_{0}$ and $z\in J_{i}$. We know by Lemma 7.7 (a) that $D(z)\approx\ell(J_{i})$. Thus, by the definition of $D$ (7.13) we may find $Q\in\mathsf{Tree}$ such that

[TABLE]

Clearly, $\ell(Q)\lesssim\ell(J_{i})$, and $\operatorname{dist}(J_{i},\Pi_{0}(Q))\lesssim\ell(J_{i})$. If $\ell(Q)\gtrsim\ell(J_{i})$, we set $Q_{i}=Q$ and we are done. If that is not the case, then we define $Q_{i}$ as the ancestor $P\supset Q$ satisfying $\ell(P)\gtrsim\ell(J_{i})$ (we can always do that because $\ell(J_{i})\lesssim\ell(R_{0})$ by Lemma 7.8 (a)). ∎

For all $i\in I_{0}$ we define $F_{i}\mathrel{\mathop{\ordinarycolon}}L_{0}\rightarrow L_{0}^{\perp}$ as the affine function whose graph is the $n$-plane $L_{Q_{i}}$. Since $\measuredangle(L_{Q_{i}},L_{0})\leq\theta$ by (6.7), we have $\operatorname{\text{Lip}}(F_{i})\lesssim\theta$. For $i\not\in I_{0}$ set $F_{i}\equiv 0$, so that the graph of $F_{i}$ is the plane $L_{0}$.

Lemma 7.10**.**

Suppose $10J_{i}\cap 10J_{i^{\prime}}\not=\varnothing$. We have:

• (a)

if $i,i^{\prime}\in I_{0}$, then

[TABLE]

• (b)

for $x\in 100J_{i}$

[TABLE]

• (c)

$\lVert\nabla F_{i}-\nabla F_{i^{\prime}}\rVert_{\infty}\lesssim\sqrt{\varepsilon_{0}}.$**

Proof.

Let us start with (a). We know by Lemma 7.7(b) and Lemma 7.9 that $\ell(Q_{i})\approx\ell(Q_{i^{\prime}})\approx\ell(J_{i})\approx\ell(J_{i^{\prime}})$. Let $z_{1}\in Q_{i},z_{2}\in Q_{i^{\prime}}$ be such that $|\Pi_{0}(z_{1})-\Pi_{0}(z_{2})|\approx\operatorname{dist}(\Pi_{0}(Q_{i}),\Pi_{0}(Q_{i^{\prime}}))$. Note that $d(z_{1})\lesssim\ell(Q_{i}),\ d(z_{2})\lesssim\ell(Q_{i^{\prime}})$. It follows that

[TABLE]

On the other hand, we have by Lemma 7.9

[TABLE]

The two estimates together give us (a).

Now, (b) and (c) for $i,\ i^{\prime}\in I_{0}$ follow immediately because we can apply Lemma 7.2 to $Q_{i}$ and $Q_{i^{\prime}}$. If $i,\ i^{\prime}\notin I_{0}$, then (b) and (c) are trivially true, since $F_{i}=F_{i^{\prime}}\equiv 0$. The only remaining case is $i\in I_{0},\ i^{\prime}\notin I_{0}$.

Since $10J_{i}\cap 10J_{i^{\prime}}\not=\varnothing$, we know by Lemma 7.7 (b) and Lemma 7.8 that $\ell(J_{i})\approx\ell(J_{i^{\prime}})\approx\ell(R_{0}).$ We apply Lemma 7.2 to $Q_{i}$ and $R_{0}$, and the result follows.. ∎

Now, to define function $F$ on $L_{0}\setminus\Pi_{0}(R_{G})$ we consider the following partition of unity: for each $i\in I$ let $\widetilde{\varphi}_{i}\in C^{\infty}(L_{0})$ be such that $\widetilde{\varphi}_{i}\equiv 1$ on $2J_{i}$, $\operatorname{supp}\widetilde{\varphi}_{i}\subset 3J_{i}$, and

[TABLE]

Now, we set

[TABLE]

Clearly, the family $\{\varphi_{i}\}_{i\in I}$ is a partition of unity subordinated to sets $\{3J_{i}\}_{i\in I}$. Moreover, the inequalities above together with Lemma 7.7 imply that each $\varphi_{i}$ satisfies

[TABLE]

Recall that in (7.12) we defined $F(z)$ for $z\in\Pi_{0}(R_{G})$. Concerning $L_{0}\setminus\Pi_{0}(R_{G})$, by Lemma 7.7 (d) we have $L_{0}\setminus\Pi_{0}(R_{G})=\bigcup_{i\in I}J_{i}=\bigcup_{i\in I}3J_{i}.$ Thus, for $z\in L_{0}\setminus\Pi_{0}(R_{G})$ we may set

[TABLE]

Using Lemmas 7.7–7.10, one may follow the proofs of [Tol14, Lemma 7.24, Remark 7.26, Lemma 7.27] to get the following.

Lemma 7.11**.**

The function $F\mathrel{\mathop{\ordinarycolon}}L_{0}\rightarrow L_{0}^{\perp}$ is supported on $L_{0}\cap 1.9B_{0}$ and is $C\theta$-Lipschitz, where $C>0$ is an absolute constant. Furthermore, for $z\in 15J_{i},i\in I,$

[TABLE]

and

[TABLE]

We denote the graph of $F$ as $\Gamma$, and we define a function $f\mathrel{\mathop{\ordinarycolon}}L_{0}\rightarrow\Gamma$ as

[TABLE]

We set also

[TABLE]

Lemma 7.12**.**

Let $i\in I_{0}$. Then $B(f(z_{J_{i}}),2\operatorname{diam}(J_{i}))\subset 2.3B_{0}$.

Proof.

By the definition of $I_{0}$ we have $J_{i}\cap 1.5B_{0}\neq\varnothing$. We know by Lemma 7.8 that $\operatorname{diam}(J_{i})\leq 0.2r_{0}$, and so $z_{J_{i}}\in 1.7B_{0}$. Moreover, since $F$ is supported on $L_{0}\cap 1.9B_{0}$ and is Lipschitz continuous with constant comparable to $\theta$, we have $\operatorname{dist}(f(z_{J_{i}}),z_{J_{i}})=|F(z_{J_{i}})|\lesssim\theta r_{0}$.

It follows easily that $B(f(z_{J_{i}}),2\operatorname{diam}(J_{i}))\subset 2.3B_{0}$. ∎

We have defined a Lipschitz graph $\Gamma$, and a set $R_{G}\subset\Gamma\cap R_{0}$ such that ${\left.\kern-1.2pt\mu\right|_{R_{G}}}\ll\mathcal{H}^{n}$. Clearly, measure ${\left.\kern-1.2pt\mu\right|_{R_{G}}}$ is $n$-rectifiable. What remains to be shown is that $\mu(R_{G})\geq 0.5\mu(R_{0})$. Since $R_{G}$ contains $R_{0}\setminus\bigcup_{Q\in\mathsf{Stop}}Q$, it is enough to estimate the measure of the stopping cubes – this is what we will do in the remaining part of the article.

8. Small measure of cubes from $\mathsf{LD}$

In this section we will bound the measure of low density cubes. First, let us prove some additional estimates.

8.1. $\Gamma$ lies close to $R_{0}$

Lemma 8.1**.**

There exists a constant $C_{1}$ such that for any $x\in 3B_{0}$

[TABLE]

Proof.

First, notice that if $x\in 3B_{0}\setminus 1.01B_{0}$, then $d(x)\gtrsim r_{0}$, and so the estimate $\operatorname{dist}(x,\Gamma)\leq C_{1}\,d(x)$ is trivial. Now, assume $x\in 1.01B_{0}$.

Let $\xi=\Pi_{0}(x)\in L_{0},\ y=(\xi,F(\xi))\in\Gamma$. Lemma 7.6 gives us

[TABLE]

If $\xi\in\Pi_{0}(R_{G})$, then $y\in R_{G}$, which means that $d(y)=0$ and we get $\operatorname{dist}(x,\Gamma)\lesssim d(x).$

Now suppose $\xi\not\in\Pi_{0}(R_{G})$. Let $i\in I$ be such that $\xi\in J_{i}$. Note that since $x\in 1.01B_{0}$, then by (7.14) $\xi\in 1.1B_{0}$, and so $J_{i}\cap 1.5B_{0}\not=\varnothing$. Hence, $i\in I_{0}$. Let $Q_{i}\in\mathsf{Tree}$ be the cube from Lemma 7.9 corresponding to $J_{i}$. It follows that

[TABLE]

Now we will estimate $\operatorname{dist}(y,Q_{i}).$ Let $z=(\xi,F_{i}(\xi))\in L_{Q_{i}}$. We have

[TABLE]

Since $\varphi_{j}(\xi)\not=0$ only for $j\in I_{0}$ such that $\xi\in 3J_{j}$, we get from Lemma 7.10 (b) that $\lvert F_{j}(\xi)-F_{i}(\xi)\rvert\lesssim\ell(J_{i})$. Hence,

[TABLE]

We use the smallness of $\beta_{\mu,2}(3B_{Q_{i}})$ and Chebyshev inequality to find $p\in 2B_{Q_{i}},\ q\in L_{Q_{i}}$ such that $|p-q|\lesssim\ell(J_{i})$. We know from Lemma 7.9 (b) that $|\Pi_{0}(p)-\xi|\lesssim\ell(J_{i})$, and so $|\Pi_{0}(q)-\xi|\lesssim\ell(J_{i}).$ Together with the fact that both $q$ and $z$ belong to $L_{Q_{i}}$, and that $\measuredangle(L_{0},L_{Q_{i}})\leq\theta$ by (6.7), this implies

[TABLE]

Thus,

[TABLE]

From this, (8.2), Lemma 7.7 (a), and the definition od $D$, we get

[TABLE]

The estimate above together with (8.1) conclude the proof. ∎

Corollary 8.2**.**

For every $Q\in\mathsf{Tree}$ we have

[TABLE]

Moreover, for $i\in I_{0}$ we have

[TABLE]

Proof.

Since $Q\subset R_{0}\subset B_{0}$, the first inequality follows immediately by Lemma 8.1 and the definition of function $d$.

The second inequality is implied by the first one, the fact that $\operatorname{dist}(\Pi_{0}(Q_{i}),J_{i})\lesssim\ell(Q_{i})$ by Lemma 7.9, and that $\Gamma$ is a Lipschitz graph with a small Lipschitz constant. ∎

Lemma 8.3**.**

Let $C>0$. If $\varepsilon_{0}$ is chosen small enough, then for each $Q\in\mathsf{Tree}$ and $x\in\Gamma\cap CB_{Q}$

[TABLE]

Proof.

There are three cases to consider.

Case 1. $x\in R_{G}$, i.e. $d(x)=0$.

Fix some small $h>0$. Let $P\in\mathsf{Tree}$ be such that $(\operatorname{dist}(x,P)+\operatorname{diam}(B_{P}))\leq h\ll\ell(Q)$. Since $x\in\Gamma\cap CB_{Q}$, we have $\operatorname{dist}(P,Q)\lesssim\ell(Q)$. Setting $y=\Pi_{L_{P}}(x)$, we clearly have $|x-y|\lesssim h$, and in consequence $y\in L_{P}\cap C^{\prime}B_{Q}$ with $C^{\prime}\approx C$. Thus, we may apply Lemma 7.3 to get

[TABLE]

Thus, $\operatorname{dist}(x,L_{Q})\lesssim_{A,\tau,C}\sqrt{\varepsilon_{0}}\ell(Q)+h$. Letting $h\rightarrow 0$ ends the proof in this case.

Case 2. $x=(\zeta,F(\zeta))$ for $\zeta\in L_{0}\setminus\Pi_{0}(R_{G})$, and

[TABLE]

Since $F(\zeta)=\sum_{i}\varphi_{i}(\zeta)F_{i}(\zeta)$, we get that $x$ is a convex combination of points $\{(\zeta,F_{i}(\zeta))\}_{i\in I_{1}}$, where $I_{1}\subset I_{0}$ consists of indices $i$ such that $\varphi_{i}(\zeta)\not=0$. Thus, it suffices to show that for each $i\in I_{1}$

[TABLE]

First, note that since $x\in CB_{Q}$,

[TABLE]

Let $J_{i^{\prime}}$ be the dyadic cube containing $\zeta$, $i^{\prime}\in I_{1}$. Then

[TABLE]

Moreover, as each $\varphi_{i}$ is supported in $3J_{i}$, we necessarily have $3J_{i}\cap J_{i^{\prime}}\not=\varnothing$ for $i\in I_{1}$. Thus, by Lemma 7.7 (b) and by Lemma 7.9,

[TABLE]

Furthermore, Lemma 7.10 (a) implies

[TABLE]

Taking into account Lemma 7.9 and the fact that $\zeta\in J_{i^{\prime}}\cap\Pi_{0}(CB_{Q})$ we obtain

[TABLE]

The three estimates above yield

[TABLE]

Applying Lemma 7.6 to any $y_{1}\in Q_{i},y_{2}\in Q$ gives us

[TABLE]

Note that $(\zeta,F_{i}(\zeta))\in L_{Q_{i}}\cap C^{\prime}B_{Q_{i}}$ for some $C^{\prime}=C^{\prime}(n,d)>0$. Indeed: $(\zeta,F_{i}(\zeta))\in L_{Q_{i}}$ by the definition of $F_{i}$; to see that $(\zeta,F_{i}(\zeta))\in C^{\prime}B_{Q_{i}}$ observe that $\varphi_{i}(\zeta)\not=0$, and so $\zeta\in 3J_{i}$, which together with Lemma 7.9 gives $(\zeta,F_{i}(\zeta))\in C^{\prime}B_{Q_{i}}$.

Due to the observation above and (8.5), (8.6), we can use Lemma 7.3 to get the desired inequality:

[TABLE]

Case 3. $x=(\zeta,F(\zeta))$ for $\zeta\in L_{0}\setminus\Pi_{0}(R_{G})$, and

[TABLE]

It follows that there exists some $k\not\in I_{0}$ such that $\zeta\in 3J_{k}$. Hence, by Lemma 7.8 (b)

[TABLE]

Furthermore, if $J_{i^{\prime}}$ is the cube containing $\zeta=\Pi_{0}(x)$, then using the definition of functions $d$ and $D$ yields

[TABLE]

Since $J_{i^{\prime}}\cap 3J_{k}\not=\varnothing$, Lemma 7.7 (b) gives us $\ell(J_{i^{\prime}})\approx\ell(J_{k}).$ Thus,

[TABLE]

and again using Lemma 7.7 (b) we get that $\ell(J_{i})\approx\ell(R_{0})$ for all $i\in I_{1}$, where $I_{1}\subset I_{0}$ are indices such that $\zeta\in 3J_{i}$. By the definition of cubes $Q_{i}$ in Lemma 7.9, we also have $\ell(Q_{i})\approx\ell(R_{0})$.

It is clear that $\operatorname{dist}(Q_{i},R_{0})=0$, and so the assumptions of Lemma 7.2 are satisfied for $Q_{i}$ and $R_{0}$. Since $\operatorname{dist}((\zeta,F_{i}(\zeta)),Q_{i})\lesssim\ell(R_{0})\approx\ell(Q_{i})$, we get that

[TABLE]

for $i\in I_{1}$. Hence,

[TABLE]

At the same time, the planes $L_{Q}$ and $L_{0}$ are close to each other due to Lemma 7.2, and so

[TABLE]

∎

Corollary 8.4**.**

Let $\theta$ and $\varepsilon_{0}$ be small enough. Suppose $Q\in\mathsf{Tree}$ satisfies $10B_{Q}\cap\Gamma\not=\varnothing$. Then for $y\in L_{Q}\cap 10B_{Q}$

[TABLE]

Proof.

Let $\widetilde{F}\mathrel{\mathop{\ordinarycolon}}L_{Q}\rightarrow L_{Q}^{\perp}$ be defined in such a way that $\Gamma$ is the graph of $\widetilde{F}$. This definition makes sense because $\measuredangle(L_{Q},L_{0})\leq\theta$. Moreover, $\operatorname{\text{Lip}}(F)\lesssim\theta$ implies that $\operatorname{\text{Lip}}(\widetilde{F})\lesssim\theta$.

Let $x\in L_{Q}$ be such that $(x,\widetilde{F}(x))\in 10B_{Q}\cap\Gamma$. By the triangle inequality and Lemma 8.3 we have for $y\in L_{Q}\cap 10B_{Q}$

[TABLE]

Thus, for $\theta$ and $\varepsilon_{0}$ small enough, we have $(y,\widetilde{F}(y))\in 11B_{Q}\cap\Gamma$ and we may use Lemma 8.3 once again to conclude that

[TABLE]

∎

Recall that

[TABLE]

Lemma 8.5**.**

For all $x\in 3B_{0}\setminus R_{\mathsf{Far}}$

[TABLE]

Proof.

If $d(x)=0$, then $x\in R_{G}\subset\Gamma$ and we are done. Suppose that $d(x)>0$. Let $Q\in\mathsf{Tree}$ be such that

[TABLE]

Fix some $z\in Q$ and note that $|z-x|\leq 2d(x)$.

Let $C_{1}$ be the constant from Lemma 8.1. If we have $B(z,2(C_{1}+2)d(x))\subset 3B_{0}$, then let $P\in\mathsf{Tree}$ be the smallest cube satisfying $B(z,2(C_{1}+2)d(x))\subset 3B_{P}$; otherwise, set $P=R_{0}$.

In both cases we have $\ell(P)\approx d(x)$, as well as $x\in 3B_{P}$. Moreover, we know from Lemma 8.1 that

[TABLE]

Hence, $3B_{P}\cap\Gamma\neq\varnothing$ (for $P=R_{0}$ this is obvious, and for $P\subsetneq R_{0}$ it follows from the fact that $B(z,2(C+2)d(x))\subset 3B_{P}$).

The assumption $x\not\in R_{\mathsf{Far}}$ gives us

[TABLE]

and so $\Pi_{L_{P}}(x)\in 4B_{P}\cap L_{P}$. We apply Corollary 8.4 to $\Pi_{L_{P}}(x)$ to get

[TABLE]

The two inequalities above and the fact that $\ell(P)\approx d(x)$ imply

[TABLE]

∎

Lemma 8.6**.**

For every $x\in\Gamma$ we have $D(\Pi_{0}(x))\leq d(x)\lesssim D(\Pi_{0}(x)).$

Proof.

The inequality $D(\Pi_{0}(x))\leq d(x)$ follows directly from the definition of $D$ (7.13).

To see that $d(x)\lesssim D(\Pi_{0}(x))$, let $Q\in\mathsf{Tree}$ be such that

[TABLE]

for some small $h>0$. Take any $y\in 3B_{Q}\setminus R_{\mathsf{Far}}$, then by Lemma 8.5 we have some $z\in\Gamma$ such that

[TABLE]

Using the fact that $x,z\in\Gamma$, that $y\in 3B_{Q}$, and the inequality above, we have

[TABLE]

and so

[TABLE]

It follows that

[TABLE]

Letting $h\rightarrow 0$ ends the proof. ∎

8.2. Estimating the measure of $\mathsf{LD}$

Lemma 8.7**.**

If $\varepsilon_{0}$ and $\tau$ are small enough, with $\varepsilon_{0}=\varepsilon_{0}(\tau)\ll\tau$, then

[TABLE]

Proof.

Recall that by Lemma 6.6 we have $\mu(R_{\mathsf{Far}})\lesssim_{A,\tau}\sqrt{\varepsilon_{0}}\mu(R_{0}).$ Hence, for $\varepsilon_{0}$ small enough we get $\mu(R_{\mathsf{Far}})\leq\tau\mu(R_{0}),$ and so to show (8.8) it suffices to prove

[TABLE]

where $R_{\mathsf{LD}}=\bigcup_{Q\in\mathsf{LD}}Q\setminus R_{\mathsf{Far}}$.

We use Besicovitch covering theorem to find a countable collection of points $x_{i}\in R_{\mathsf{LD}}$ such that $x_{i}\in Q_{i}\setminus R_{\mathsf{Far}},\ Q_{i}\in\mathsf{LD},$ and

[TABLE]

where $N$ is a dimensional constant.

Observe that $B(x_{i},r(Q_{i}))\subset 1.5B_{Q_{i}}$. It follows that

[TABLE]

where the last inequality was obtained using the fact that $Q_{i}\in\mathsf{LD}$. Furthermore, since $x_{i}\not\in R_{\mathsf{Far}}$ we may use Lemma 8.5 to get $\operatorname{dist}(x_{i},\Gamma)\lesssim_{A,\tau}\sqrt{\varepsilon_{0}}\,d(x_{i})$. Note also that $d(x_{i})\lesssim r(Q_{i})$. Hence,

[TABLE]

So, if $\varepsilon_{0}$ is small enough, $\Gamma$ passes close to the center of $B(x_{i},r(Q_{i}))$. Since $\Gamma$ is a Lipschitz graph with small Lipschitz constant we get

[TABLE]

Thus,

[TABLE]

We have $\ell(R_{0})^{n}\approx\mu(3B_{0})\approx\mu(R_{0})$ because $\Theta_{\mu}(3B_{0})=1$, see Remark 5.5, and $R_{0}$ is doubling. Hence,

[TABLE]

∎

9. Approximating measure $\nu$

In order to estimate the measure of high density cubes, we need to introduce a measure $\nu$ supported on $\Gamma$ which will approximate $\mu$.

9.1. Definition and properties of $\nu$

Let $\eta<1/1000$ be a small dimensional constant which will be fixed in the proof of Lemma 9.1 (c). For every $i\in I$ (the set of indices from Section 7.3) consider a finite collection of points $\{z^{\prime}_{k}\}_{k\in K_{i}}\subset J_{i},\,\#K_{i}\lesssim_{n}1,$ such that the balls $B(z^{\prime}_{k},0.5\eta\ell(J_{i}))$ cover the whole $J_{i}$. We set $K=\bigcup_{i}K_{i}$, $K_{0}=\bigcup_{i\in I_{0}}K_{i}$.

For $k\in K_{i}$ we define

[TABLE]

The following lemma collects basic properties of $B_{k}$.

Lemma 9.1**.**

We have the following:

• (a)

For $k\in K_{i}$

[TABLE]

• (b)

For $k\in K$ there exist at most $C=C(n)$ indices $k^{\prime}\in K$ such that $\Pi_{0}(3B_{k})\cap\Pi_{0}(3B_{k^{\prime}})\not=\varnothing$ (in particular, there are at most $C$ indices $k^{\prime}\in K$ such that $3B_{k}\cap 3B_{k^{\prime}}\not=\varnothing$). Moreover, for all such $k^{\prime}$ we have

[TABLE]

• (c)

For $k\in K$ and $x\in 3B_{k}$ we have

[TABLE]

• (d)

For $k\in K_{0}$

[TABLE]

• (e)

For $k\not\in K_{0}$

[TABLE]

If additionally $3B_{k}\cap 3B_{0}\not=\varnothing$, then

[TABLE]

• (f)

Finally,

[TABLE]

Proof.

(a) follows immediately by the definition of $B_{k}$.

Concerning (b), suppose $k\in K_{i}$ and $\Pi_{0}(3B_{k})\cap\Pi_{0}(3B_{k^{\prime}})\not=\varnothing$ for some $k^{\prime}\in K_{i^{\prime}}$. By (a) we know that $2J_{i}\cap 2J_{i^{\prime}}\not=\varnothing$, and there are at most $N$ such indices $i^{\prime}$, see Lemma 7.7) (c). Since $\#K_{i}\lesssim_{n}1$ by the definition, we get that there are at most $C(n,N)$ indices $k^{\prime}$ satisfying $\Pi_{0}(3B_{k})\cap\Pi_{0}(3B_{k^{\prime}})\not=\varnothing$. The estimate $r_{k}\approx r_{k^{\prime}}$ follows by Lemma 7.7 (b).

To prove (c), recall that $z_{k}$ is the center of $B_{k}$. By the definition, $\Pi_{0}(z_{k})\in J_{i}$ for $i\in I$ such that $r_{k}=\eta\ell(J_{i})$. Lemma 7.7 (a) gives us $D(\Pi_{0}(z_{k}))\approx_{n}\ell(J_{i})$. Hence, by Lemma 8.6 we get

[TABLE]

Now, for an arbitrary $x\in 3B_{k}$ we have by the 1-Lipschitz property of function $d$ that

[TABLE]

Since $d(z_{k})\approx\eta^{-1}r_{k},$ choosing $\eta$ small enough we arrive at $d(x)\approx\eta^{-1}r_{k},$ and so for $\eta$ small enough $r_{k}\leq d(x)\leq\eta^{-3/2}r_{k}$.

Concerning (d), let $i\in I_{0}$ be such that $\Pi_{0}(z_{k})\in J_{i}$. We know by Lemma 7.12 that $B(f(z_{J_{i}}),2\operatorname{diam}(J_{i}))\subset 2.3B_{0}$. Since $3B_{k}\subset B(f(z_{J_{i}}),2\operatorname{diam}(J_{i})),$ we get $3B_{k}\subset 2.3B_{0}$.

To show (e), let $k\in K\setminus K_{0}$. Let $i\in I\setminus I_{0}$ be such that $k\in K_{i}$, i.e. $\Pi_{0}(z_{k})\in J_{i}$. By (c) and Lemma 8.6 we have $d(z_{k})\approx D(\Pi_{0}(z_{k}))\approx r_{k}$. At the same time, $|\Pi_{0}(z_{k})-z_{0}|\approx\ell(J_{i})\gtrsim\ell(R_{0})$ by Lemma 7.8 (b). Recall also that $\lVert F\rVert_{\infty}\lesssim\theta\ell(R_{0})$ due to Lipschitz continuity and the fact that $\operatorname{supp}(F)\subset 1.9B_{0}$, see Lemma 7.11. It follows that

[TABLE]

and on the other hand

[TABLE]

for $\theta$ small enough. Hence, $|z_{k}-z_{0}|\approx\ell(J_{i})\approx r_{k}\gtrsim\ell(R_{0})$.

Now, assume also $3B_{k}\cap 3B_{0}\not=\varnothing$, and suppose $x\in 3B_{0}\cap 3B_{k}$. We have $\Pi_{0}(x)\in 2J_{i}$ by (9.1). Clearly, $D(\Pi_{0}(x))\leq d(x)\lesssim\ell(R_{0})$, and so $r_{k}\approx\ell(J_{i})\approx D(\Pi_{0}(x))\lesssim\ell(R_{0})$ by Lemma 7.7 (a).

Finally, to see (9.7) note that by the definition of $B_{k}$ and by (a) we have

[TABLE]

Together with Lemma 7.7 (d) this implies (9.7). ∎

Since $\eta$ is a dimensional constant, we will usually not mention dependence on it in our further estimates.

Due to bounded superposition of $3B_{k}$ (Lemma 9.1 (b)) we may define a partition of unity $\{h_{k}\}_{k\in K}$ such that $0\leq h_{k}\leq 1,\ \operatorname{supp}h_{k}\subset 3B_{k},\ \operatorname{\text{Lip}}(h_{k})\approx\ell(J_{i})^{-1},$ and

[TABLE]

Again, by the bounded superposition of $3B_{k}$ we may assume

[TABLE]

Recall that $\sigma={\left.\kern-1.2pt\mathcal{H}^{n}\right|_{\Gamma}},$ and that $c_{0}$ is a constant minimizing $\alpha_{\mu}(3B_{0})$. We set

[TABLE]

We define the approximating measure as

[TABLE]

Note that, since ${\left.\kern-1.2pt\mu\right|_{R_{G}}}\ll\sigma$ by Lemma 7.5, we also have $\nu\ll\sigma$. To simplify the notation, we introduce

[TABLE]

Note that by Lemma 9.1 (d), (9.7), and the fact that $R_{G}\subset B_{0}$, we get

[TABLE]

and so by the definition of $\nu$ we have

[TABLE]

Lemma 9.2**.**

For each $k\in K_{0}$ there exists $P_{k}\in\mathsf{Tree}$ such that $3B_{k}\subset 2.5B_{P_{k}}$, and $\ell(P_{k})\approx r_{k}$.

Proof.

We know by (9.4) that $3B_{k}\subset 2.3B_{0}$. Thus, we may define $P_{k}$ as the smallest cube in $\mathsf{Tree}$ such that $3B_{k}\subset 2.5B_{P_{k}}$. We have $\ell(P_{k})\approx r_{k}$ due to (9.3). ∎

We will write for $k\in K_{0}$

[TABLE]

and for $k\not\in K_{0}$ set $\widetilde{B}_{k}=2.5B_{0},\ \widetilde{c}_{k}=c_{0}$, and $L_{k}=L_{0}$.

Note that for every $k\in K$

[TABLE]

Indeed, for $k\in K_{0}$, it follows by Lemma 8.3 applied to $z_{k}$ and $P_{k}$. For $k\not\in K_{0}$, but such that $z_{k}\in 1.9B_{0}$, again it follows by Lemma 8.3 applied to $z_{k}$ and $R_{0}$. Finally, for $k\not\in K_{0}$ such that $z_{k}\not\in 1.9B_{0}$ this is trivially true because $\Gamma\setminus(1.9B_{0})=L_{0}\setminus(1.9B_{0})$, and so $\operatorname{dist}(z_{k},L_{0})=0$.

Lemma 9.3**.**

For $k\in K$ the set $\Gamma\cap 3B_{k}$ is a Lipschitz graph over $L_{k}$, with a Lipschitz constant at most $C\sqrt{\varepsilon_{0}}$.

Proof.

Suppose $k\in K_{0}$, i.e. that $k\in K_{i}$ for some $i\in I_{0}$. We know by (7.17) that $f(15J_{i})$ is a $C\sqrt{\varepsilon_{0}}$-Lipschitz graph over $L_{Q_{i}}$ (recall that $F_{i}$ is an affine function whose graph is $L_{Q_{i}}$).

At the same time, since $P_{k}$ satisfies $\operatorname{dist}(P_{k},Q_{i})\lesssim\ell(Q_{i})$ (see (8.3) and the definition of $P_{k}$) and $\ell(P_{k})\approx r_{k}\approx\ell(Q_{i}),$ we can apply Lemma 7.2 to get $\measuredangle(L_{Q_{i}},L_{k})\lesssim_{A,\tau}\varepsilon_{0}$. It follows that $f(15J_{i})$ is a $C\sqrt{\varepsilon_{0}}$-Lipschitz graph over $L_{k}$. The same is true for $k\not\in K_{0}$: since $L_{k}=L_{0}=\operatorname{graph}({F_{i}})$, it follows immediately by (7.17). We conclude by noting that

[TABLE]

∎

Lemma 9.3 and (9.14) imply that for every $k\in K$

[TABLE]

Furthermore, by $\eqref{eq:sum of F containing x estimate}$ we have

[TABLE]

Lemma 9.4**.**

For $k\in K$ we have

[TABLE]

Proof.

For $k\not\in K_{0}$ we have $c_{k}=c_{0}=\widetilde{c}_{k}$, so the claim is trivially true. Suppose $k\in K_{0}$. Recall that $h_{k}\approx 1$ in $B_{k}$ (9.9), $\operatorname{\text{Lip}}(h_{k})\approx r_{k}^{-1}$, and $\widetilde{c}_{k}\approx_{A,\tau}1$ by (6.17). It follows that

[TABLE]

∎

An immediate corollary of (6.17) and the lemma above is that for $k\in K$

[TABLE]

Lemma 9.5**.**

The measure $\nu$ is $n$-AD-regular, that is, for $x\in\Gamma,\ r>0$

[TABLE]

Proof.

We know by (9.7), the definition of $h$ (9.8), and (9.18) that

[TABLE]

Together with Lemma 7.5 this gives

[TABLE]

∎

Lemma 9.6**.**

If $k,j\in K$ satisfy $3B_{k}\cap 3B_{j}\not=\varnothing$, then

[TABLE]

Proof.

If $3B_{k}\cap 3B_{j}\not=\varnothing$, then by (9.1) and Lemma 7.7 (b) it follows that

[TABLE]

Now, since $3B_{k}\cap 3B_{j}\not=\varnothing$ and $r_{k}\approx r_{j}$, we get that there exists $R\in\mathsf{Tree}$ such that $2.5B_{R}\supset\widetilde{B}_{k}\cup\widetilde{B}_{j}$ and $\ell(R)\approx r_{k}$. Hence, we may use Lemma 7.4 and Lemma 9.4 to obtain

[TABLE]

∎

9.2. $\nu$ approximates $\mu$ well

Lemma 9.7**.**

We have

[TABLE]

In consequence, for every $x\in 3B_{0}\setminus(R_{G}\cup R_{\mathsf{Far}})$ we have $h(x)=1.$

Proof.

Let $x\in 3B_{0}\setminus(R_{G}\cup R_{\mathsf{Far}})$. We will find $k\in K$ such that $x\in 2B_{k}$.

By Lemma 8.5 we have $y\in\Gamma$ such that

[TABLE]

Since $x\not\in R_{G}$, we have $d(x)>0$. Moreover, since $d(x)\leq d(y)+|x-y|\leq d(y)+0.5d(x)$, we get that $0<d(x)\leq 2d(y)$. In particular, $y\not\in R_{G}$ and by (9.7) there exists $k\in K$ such that $y\in B_{k}\cap\Gamma$. It follows by Lemma 9.1 (c) that

[TABLE]

Together with (9.19) this gives $|x-y|\leq r_{k}/2$, for $\varepsilon_{0}$ small enough. Since $y\in B_{k}$, we get that $x\in 2B_{k}$.

∎

Lemma 9.8**.**

Suppose that $x\in 2.5B_{0}$, $r\geq Cd(x)$ for some $C>0$, and that $B(x,r)\subset 3B_{0}$. Then,

[TABLE]

Proof.

Since $B(x,r)\subset 3B_{0}$, and $r\geq Cd(x)$, there exists a cube $Q\in\mathsf{Tree}$ such that $B(x,r)\subset 3B_{Q}$ and $\ell(Q)\approx_{C}r$. In consequence, using the properties of $\mathsf{Tree}$ yields

[TABLE]

Thus, given any $\phi\in\operatorname{\text{Lip}}_{1}(B(x,r))$ we have

[TABLE]

and so $F_{B(x,r)}(\mu_{B},{\left.\kern-1.2pt\mu_{B}\right|_{(R_{\mathsf{Far}})^{c}}})\lesssim_{A,C}\varepsilon_{0}^{1/4}r^{n+1}.$ Similarly, $F_{B(x,r)}(h\mu,h{\left.\kern-1.2pt\mu\right|_{(R_{\mathsf{Far}})^{c}}})\lesssim_{A,C}\varepsilon_{0}^{1/4}r^{n+1}.$

Now, observe that $h\mu=h\mu_{B}$ by the definition of $h$. Moreover, inside $B(x,r)$ we have

[TABLE]

because $h\equiv 1$ on $3B_{0}\setminus(R_{G}\cup R_{\mathsf{Far}})$ by Lemma 9.7. Thus, the triangle inequality yields

[TABLE]

∎

Lemma 9.9**.**

If $x\in 2.5B_{0}$ and $r>0$ satisfy $B(x,r)\subset 2.5B_{0}$, then

[TABLE]

Proof.

Since $\nu_{B}=\sum_{k}c_{k}h_{k}\sigma$, our aim is estimating $F_{B(x,r)}(\sum_{k}c_{k}h_{k}\sigma,h\mu)$. Set

[TABLE]

First, we will deal with $k\in K(x,r)\setminus K_{0}$. For such $k$ by (9.6) we have

[TABLE]

In particular, $r\lesssim r_{k},$ and so given $\phi\in\operatorname{\text{Lip}}_{1}(B(x,r))$ we have $\operatorname{\text{Lip}}(\phi h_{k})\lesssim 1,\ \operatorname{supp}(\phi h_{k})\subset B(x,r)\cap 3B_{k}\subset 2.5B_{0}$.

Moreover, recall that

[TABLE]

In consequence, since $c_{k}=c_{0}$ by (9.10), we have for any $\phi\in\operatorname{\text{Lip}}_{1}(B(x,r))$

[TABLE]

Now, we turn our attention to $k\in K_{0}(x,r)=K(x,r)\cap K_{0}$. For any $\phi\in\operatorname{\text{Lip}}_{1}(B(x,r))$ we have

[TABLE]

We start by estimating $I_{1}$. Observe that setting $\Phi_{k}=(\phi-\phi(z_{k}))h_{k}$ we have $\operatorname{\text{Lip}}(\Phi_{k})\lesssim 1$ and $\operatorname{supp}\Phi_{k}\subset 3B_{k}$. Hence,

[TABLE]

Concerning $I_{2}$, note that for $k\in K_{0}(x,r)$ we have by the definition of $c_{k}$ (9.10)

[TABLE]

and so

[TABLE]

Putting together the estimates for $k\in K(x,r)\setminus K_{0}$ and for $k\in K_{0}(x,r)$, and taking supremum over $\phi\in\operatorname{\text{Lip}}_{1}(B(x,r))$, we finally get

[TABLE]

∎

The previous two lemmas, and the fact that $F_{B}(\nu,\mu)=F_{B}(\nu_{B},\mu_{B})$, imply the following:

Lemma 9.10**.**

For $x\in 2.5B_{0}$ and $r\gtrsim d(x)$ such that $B(x,r)\subset 2.5B_{0}$ we have

[TABLE]

In particular, we have

[TABLE]

Lemma 9.11**.**

For $x\in\Gamma$ and $r\gtrsim\ell(R_{0})$ such that $B(x,r)\cap 3B_{0}\not=\varnothing$ we have

[TABLE]

Proof.

Recall that by (9.12) we have

[TABLE]

To take advantage of this equality, we define an auxiliary function $\psi$ such that $\psi\equiv 1$ on $2.3B_{0}$, $\operatorname{supp}(\psi)\subset 2.5B_{0}$, and $\operatorname{\text{Lip}}(\psi)\lesssim\ell(R_{0})^{-1}$. Then,

[TABLE]

Recall that $z_{0}=z_{R_{0}}$. It follows that for $\phi\in\operatorname{\text{Lip}}_{1}(B(x,r))$ we have

[TABLE]

∎

10. Small measure of cubes from $\mathsf{HD}$

For brevity of notation let us denote by $\Pi_{*}\nu$ the image measure of $\nu$ by $\Pi_{0}$, that is the measure such that $\Pi_{*}\nu(A)=\nu(\Pi_{0}^{-1}(A))$. Set

[TABLE]

The key estimate necessary to bound the measure of high density cubes is the following.

Lemma 10.1**.**

We have

[TABLE]

We postpone the proof of the above lemma to the next subsection. Let us show now how we can use it to estimate the measure of cubes in $\mathsf{HD}$.

Lemma 10.2**.**

We have

[TABLE]

Proof.

Recall that by Lemma 6.6 we have $\mu(R_{\mathsf{Far}})\lesssim_{A,\tau}\sqrt{\varepsilon_{0}}\mu(R_{0}).$ Thus, to show (10.2) it suffices to prove

[TABLE]

where $R_{\mathsf{HD}}=\bigcup_{Q\in\mathsf{HD}}Q\setminus R_{\mathsf{Far}}$.

For every $x\in R_{\mathsf{HD}}$ we define $B_{x}=B(x,r(Q_{x})/100)$, where $Q_{x}\in\mathsf{HD}$ is such that $x\in Q_{x}$. We use the $5r$-covering theorem to choose $\{x_{j}\}_{j\in J}$ such that all $B_{x_{j}}$ are pairwise disjoint and $\bigcup_{j}5B_{x_{j}}$ covers $\bigcup_{x\in R_{\mathsf{HD}}}B_{x}$. Observe that $5B_{x_{j}}\subset 3B_{Q_{x_{j}}}$, and so by (6.5)

[TABLE]

For every $j$ set $B_{j}=\frac{1}{2}B_{x_{j}},Q_{j}=Q_{x_{j}}$, and let $P_{j}\in\mathsf{Tree}$ be the parent of $Q_{j}$. We have $\ell(P_{j})\approx\ell(Q_{j})\approx r(B_{j}).$ Since $x_{j}\not\in R_{\mathsf{Far}}$, we can use Lemma 8.5 to obtain

[TABLE]

Since $2B_{j}$ are disjoint, the centers of $B_{j}$ are close to $\Gamma$, and $\Gamma$ is a graph of function $F$ with $\operatorname{\text{Lip}}(F)\lesssim\theta\ll 1$, it follows that $\Pi_{0}(B_{j})$ are disjoint as well.

We use the above to get

[TABLE]

We claim that

[TABLE]

where

[TABLE]

and $\mathcal{M}$ is the Hardy-Littlewood maximal function on $L_{0}$. $\mathcal{BM}$ stands for “big $\mathcal{M}$”. Before we prove (10.5), note that due to the weak type $(2,2)$ estimate for $\mathcal{M}$ we have

[TABLE]

Putting this together with (10.4), (10.5), and our key estimate from Lemma 10.1, we get that

[TABLE]

Therefore, all that remains is to show (10.5).

Let $j\in J,\ y\in\Pi_{0}(B_{j})$. Since $|y-\Pi_{0}(x_{j})|\leq r(B_{j})\leq r(B_{Q_{j}})$ and $\Pi_{0}(x_{j})\in\Pi_{0}(B_{Q_{j}})$, we have $B(y,25r(B_{Q_{j}}))\supset\Pi_{0}(10B_{Q_{j}})$. Clearly, for some $C=C(n)>0$

[TABLE]

Recall that by (3.5) and Remark 5.5 we have

[TABLE]

Thus, if we show that $\nu(10B_{Q_{j}})\gtrsim Ar(B_{Q_{j}})^{n},$ for $A$ big enough we will have $\mathcal{M}(f-c_{0})(y)>1$, and so we will be done.

Let us define

[TABLE]

Note that $\lambda$ is $1$-Lipschitz and that $\operatorname{supp}(\lambda)\subset 10B_{Q_{j}}\subset 2.5B_{0}$. Moreover,

[TABLE]

Note that $r(B_{Q_{j}})\gtrsim d(z_{Q_{j}})$. We get that

[TABLE]

Note that for all $k$ such that $3B_{k}\cap 10B_{Q_{j}}\not=\varnothing$ we have $r_{k}\lesssim_{A,\tau}r(B_{Q_{j}})$. Indeed, for $x\in 10B_{Q_{j}}$ it holds that $d(x)\lesssim_{A,\tau}r(B_{Q_{j}})$, and for $x\in 3B_{k}$ we have $r_{k}\leq d(x)$ by Lemma 9.1 (c). Moreover, since the balls $\Pi_{0}(3B_{k})$ are of bounded intersection by Lemma 9.1 (b), we get

[TABLE]

Hence, using the above and the fact that $Q_{j}\in\mathsf{HD}$

[TABLE]

for $\varepsilon_{0}$ small enough. Thus, $\nu(10B_{Q_{j}})\gtrsim Ar({B_{Q_{j}}})^{n}$ and by (10.6) we get $\mathcal{M}(f-c_{0})(y)>1$. ∎

10.1. $\Lambda$-estimates

The aim of this section is to prove the crucial estimate from Lemma 10.1, i.e.

[TABLE]

From now on we will denote by $\phi\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{d}\rightarrow\mathbb{R}$ a radial $C^{\infty}$ function such that $\phi\equiv 1$ on $B(0,1/2)$, $\operatorname{supp}(\phi)\subset B(0,1)$, and

[TABLE]

We also set

[TABLE]

A classical result of harmonic analysis (see [Ste93, Sections I.6.3, I.8.23]) states that

[TABLE]

and so we will work with the latter expression.

For $r>0$ and $x\in\mathbb{R}^{d}$ let us define

[TABLE]

Given a measure $\lambda$ on $\mathbb{R}^{d}$ we set

[TABLE]

Lemma 10.3**.**

We have for all $x\in\Gamma$

[TABLE]

Proof.

By the definition of $\Lambda_{\nu}$, it suffices to show that for all $x,y\in\Gamma$ we have

[TABLE]

Hence, by the definition of $\widetilde{\psi}_{r}$, we need to check that $\phi((5r)^{-1}(x-y))=1$ whenever $\psi_{r}(\Pi_{0}(x)-\Pi_{0}(y))\not=0$.

Since $\operatorname{supp}(\psi_{r})\subset B(0,2r)$, we get that $|\Pi_{0}(x)-\Pi_{0}(y)|\leq 2r$. Thus, due to the fact that $\Gamma$ is a $C\theta$-Lipschitz graph, we have

[TABLE]

Hence, $y\in B(x,5r/2)$, which gives $\phi((5r)^{-1}(x-y))=1$. ∎

The lemma above and the fact that $\Pi_{0}$ is bilipschitz between $\Gamma$ and $L_{0}$ imply that

[TABLE]

In consequence of (10.8) and the above, to prove Lemma 10.1 it suffices to show that

[TABLE]

We start with the following simple calculation.

Lemma 10.4**.**

For $x\in\Gamma$ we have

[TABLE]

Moreover, for $x\in\Gamma\cap 2.5B_{0}$ and $d(x)\lesssim r<\eta r_{0}$ we have

[TABLE]

for some $Q\in\mathsf{Tree}$ such that $B(x,5r)\subset 3B_{Q}$ and $r\approx\ell(Q)$.

Proof.

First, we will prove (10.11). Let $B=B(x,2r)$, and $L_{B},c_{B}$ be the minimizing plane and constant for $\alpha_{\nu}(B)$. Using the fact that $\Lambda_{\nu}(x,r)=|\psi_{r}\ast\Pi_{*}\nu(\Pi_{0}(x))|$ we get

[TABLE]

Hence, by $n$-AD-regularity of $\nu$ we arrive at

[TABLE]

Now, let us look at (10.12). Since $x\in\Gamma\cap 2.5B_{0}$ and $d(x)\lesssim r<\eta r_{0}$, we may find $Q\in\mathsf{Tree}$ such that $B(x,5r)\subset 3B_{Q}$ and $\ell(Q)\approx_{A,\tau}r$. We use the fact that $|\nabla\widetilde{\psi}_{r}|\lesssim r^{-n-1}$ and $\operatorname{supp}\widetilde{\psi}_{r}\subset B(x,5r)$ to get

[TABLE]

We claim that the last integral above is equal to [math]. To prove this, it suffices to show that for $x\in\Gamma$ and $y\in L_{Q}$ we have $\widetilde{\psi}_{r}(x-y)=\psi_{r}(\Pi_{0}(x)-\Pi_{0}(y))$, because

[TABLE]

Since $\widetilde{\psi}_{r}(x-y)=\psi_{r}(\Pi_{0}(y)-\Pi_{0}(x))\phi((5r)^{-1}(x-y))$, we only have to check that $\phi((5r)^{-1}(x-y))=1$ for $\Pi_{0}(y)-\Pi_{0}(x)\in\operatorname{supp}\psi_{r}$. In other words, knowing that $|\Pi_{0}(y)-\Pi_{0}(x)|\leq 2r$, we expect that $|x-y|\leq\frac{5}{2}r$.

Indeed, the fact that $\Gamma$ is a $C\theta$-Lipschitz graph, that $\measuredangle(L_{Q},L_{0})\leq\theta$, $|\Pi_{0}(y)-\Pi_{0}(x)|\leq 2r$, and Lemma 8.3, imply

[TABLE]

Hence,

[TABLE]

as expected. ∎

Before we proceed, let us state the following auxiliary result. Recall that given a ball $B$, $z(B)$ denotes the center of $B$.

Lemma 10.5** ([ATT18, Lemma 6.11]).**

Let $B$ be a ball centered on an $\varepsilon$-Lipschitz graph $\Gamma$, and $f$ a function such that

[TABLE]

and $f(x)\approx 1$ uniformly for $x\in 3B\cap\Gamma$. Then

[TABLE]

where $\sigma$ denotes the surface measure on $\Gamma$.

We split the area of integration from (10.10) into several pieces. We will estimate each of them separately.

Lemma 10.6**.**

For every $k\in K$ we have

[TABLE]

Proof.

By Lemma 9.1 (c) we know that for $x\in B_{k}$ we have $\eta^{2}d(x)\leq\eta^{1/2}r_{k}$. Hence,

[TABLE]

Let $g(x)=\sum_{j\in K}c_{j}h_{j}(x)$. Note that for $x\in 3B_{k}\cap\Gamma$ we have $h(x)=1$, due to (9.7) and the definition of $h$ (9.8). Thus, by Lemma 9.6,

[TABLE]

Hence, by (9.18), $g(x)\approx_{A,\tau}1$. Since ${\left.\kern-1.2pt\nu\right|_{3B_{k}}}=g{\left.\kern-1.2pt\sigma\right|_{3B_{k}}}$, and $\Gamma\cap 3B_{k}$ is a $C\sqrt{\varepsilon_{0}}$-Lipschitz graph by Lemma 9.3, we can apply Lemma 10.5 and get

[TABLE]

∎

Let $M(\mathbb{R}^{d})$ denote the space of finite Borel measures on $\mathbb{R}^{d}$.

Lemma 10.7** ([ATT18, Lemma 8.2]).**

For $\lambda\in M(\mathbb{R}^{d})$ we define

[TABLE]

and for $f\in L^{2}(\sigma)$ set $T_{\sigma}f=T(f\sigma)$. Then $T_{\sigma}$ is bounded in $L^{p}(\sigma)$ for $1<p<\infty$, and T is bounded from $M(\mathbb{R}^{d})$ to $L^{1,\infty}(\sigma)$. Furthermore, the norms $\lVert T_{\sigma}\rVert_{L^{p}(\sigma)\rightarrow L^{p}(\sigma)}$ and $\lVert T\rVert_{M(\mathbb{R}^{d})\rightarrow L^{1,\infty}(\sigma)}$ are bounded above by some absolute constants depending only on $p,n$ and $d$.

Lemma 10.8**.**

We have

[TABLE]

Proof.

Since $\nu=(\nu_{B}-h\mu)+(h\mu-\mu_{B})+\mu$, for each $x\in\Gamma\cap 2.4B_{0}$ we split

[TABLE]

Let

[TABLE]

We divide our area of integration into two parts:

[TABLE]

In order to estimate $I_{1}$, note that for $x\in 2.4B_{0}$ and $r<\eta r_{0}$ we have $B(x,5r)\subset 2.5B_{0}$. Since $\operatorname{supp}\widetilde{\psi}_{r}\subset 5B(0,r)$ we get $\Lambda_{\mu_{B}-h\mu}(x,r)=\Lambda_{{\left.\kern-1.2pt(\mu_{B}-h\mu)\right|_{2.5B_{0}}}}(x,r)$. Hence, by Lemma 10.7 applied to $\lambda={\left.\kern-1.2pt(\mu_{B}-h\mu)\right|_{2.5B_{0}}}$

[TABLE]

Since $h=1$ on $3B_{0}\setminus(R_{\mathsf{Far}}\cup R_{G})$ by Lemma 9.7, $\mu_{B}(R_{G})=(h\mu)(R_{G})=0$ by their definition and (9.7), and $\mu(R_{\mathsf{Far}})$ is small by Lemma 6.6, we have

[TABLE]

Thus, for $\varepsilon_{0}$ small enough

[TABLE]

Now, consider the density $q=\frac{d{\left.\kern-1.2pt\nu\right|_{2.5B_{0}}}}{d\sigma}$. Arguing as before we see that for $x\in 2.4B_{0}$ and $r<\eta r_{0}$ we have $\Lambda_{\nu}(x,r)=\Lambda_{q\sigma}(x,r)$. By $n$-AD-regularity of $\nu$ (Lemma 9.5) we get $\lVert q\rVert_{L^{4}(\sigma)}^{4}\lesssim_{A,\tau}\sigma(2.5B_{0})\approx\ell(R_{0})^{n}$. Using the $L^{4}(\sigma)$ boundedness of $T_{\sigma}$ yields

[TABLE]

We move on to estimating $I_{2}$. Observe that by the definition of $H$ we have

[TABLE]

Thus, by (10.14),

[TABLE]

To handle $I_{22}$, we use (10.12) to get for $x\in\Gamma\cap 2.4B_{0}$.

[TABLE]

Hence,

[TABLE]

Finally, we deal with the integral $I_{21}$. Observe that, since $\Lambda_{\nu_{B}}(x,r)-\Lambda_{h\mu}(x,r)=\widetilde{\psi}_{r}\ast\nu_{B}(x)-\widetilde{\psi}_{r}\ast h\mu(x)$, and $|\nabla\widetilde{\psi}_{r}|\lesssim r^{-n-1}$, we have

[TABLE]

Note that for $k\in K$ such that $3B_{k}\cap B(x,5r)\not=\varnothing,$ for $\eta^{2}d(x)<r<\eta r_{0}$, and for any $y\in 3B_{k}\cap B(x,5r)$, we have

[TABLE]

Thus, $r_{k}\leq\eta^{-3}r$, and for some big $C^{\prime}=C^{\prime}(A,\tau)$ we have $B_{k}\subset C^{\prime}B_{0}$. It follows by the Cauchy-Schwarz inequality, the fact that $B_{k}$ are centered on $\Gamma$, and that they are of bounded intersection, that

[TABLE]

Together with the fact that $r_{k}\leq\eta^{-3}r$ this implies

[TABLE]

Now, note that if $B(x,5r)\cap 3B_{k}\not=\varnothing$, then

[TABLE]

Hence,

[TABLE]

Together with (10.15), (10.16), (10.17), and (10.18), this concludes the proof. ∎

We are finally ready to complete the proof of (10.10). Let us split the area of integration into four subsets:

[TABLE]

we also set

[TABLE]

Since ${\left.\kern-1.2pt\nu\right|_{(2.3B_{0})^{c}}}=c_{0}{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{L_{0}\cap(2.3B_{0})^{c}}}$ by (9.12), for $(x,r)\in A_{1}$ we have

[TABLE]

and so $I_{1}=0$.

Now let $(x,r)\in A_{2}$. Since $B(x,2r)\cap 2.3B_{0}\not=\varnothing,\ r>\eta r_{0}$, we have

[TABLE]

so that $r\geq\max(\eta r_{0},\,\eta^{2}|x-z_{0}|)$. It follows that

[TABLE]

where we used in the last line that $\Gamma\setminus 1.9B_{0}=L_{0}\setminus 1.9B_{0}$, see Lemma 7.11.

Concerning $(x,r)\in A_{3}$, note that necessarily $x\in 2.4B_{0}$, and so by Lemma 10.8

[TABLE]

Finally, for $(x,r)\in A_{4}$, we only need to consider $x$ such that $d(x)>0$ and $x\in 2.4B_{0}\cap\Gamma$, and since all such $x$ are contained in some $B_{k}$ we get

[TABLE]

Putting together all the estimates above finishes the proof of (10.10) and Lemma 10.1.

11. Small measure of cubes from $\mathsf{BA}$

We know by Lemma 6.6 that $\mu(R_{\mathsf{Far}})\lesssim_{A,\tau}\sqrt{\varepsilon_{0}}\mu(R_{0}).$ Thus, in order to estimate the measure of $\bigcup_{Q\in\mathsf{BA}}Q$, it suffices to bound the measure of

[TABLE]

Lemma 11.1**.**

We have

[TABLE]

Proof.

For every $x\in R_{\mathsf{BA}}$ we define $B_{x}=B(x,r(Q_{x})/100)$, where $Q_{x}\in\mathsf{BA}$ is such that $x\in Q_{x}$. We use the $5r$-covering theorem to choose $\{x_{i}\}_{i\in J}$ such that all $B_{x_{i}}$ are pairwise disjoint and $\bigcup_{i}5B_{x_{i}}$ covers $\bigcup_{x\in R_{\mathsf{BA}}}B_{x}$. Observe that

[TABLE]

Set $B_{i}=\frac{1}{2}B_{x_{i}},Q_{i}=Q_{x_{i}}$, and let $P_{i}\in\mathsf{Tree}$ be the parent of $Q_{i}$. We have $\ell(P_{i})\approx\ell(Q_{i})\approx r(B_{i}).$ Since $x_{i}\not\in R_{\mathsf{Far}}$, we can use Lemma 8.5 to obtain

[TABLE]

Hence, for small $\varepsilon_{0}$, we get that $\frac{1}{4}B_{i}\cap\Gamma\not=\varnothing$. It follows that for each $i\in J$ we can choose balls $B_{i,1},\,B_{i,2}\subset B_{i}$ centered at $\Gamma$, with $r(B_{i,1})\approx r(B_{i,2})\approx r(B_{i})$, and such that $\operatorname{dist}(B_{i,1},B_{i,2})\gtrsim r(B_{i})$. Then, for any points $y_{k}\in B_{i,k}\cap\Gamma,\,k=1,2,$ we have

[TABLE]

Since $y_{1},y_{2}\in\Gamma\cap B_{i}\subset\Gamma\cap B_{P_{i}},$ we have by Lemma 8.3

[TABLE]

Let $w_{k}=\Pi_{L_{P_{i}}}(y_{k})$. By the estimate above we have $|y_{k}-w_{k}|\lesssim_{A,\tau}\sqrt{\varepsilon_{0}}\ell(P_{i}).$ Moreover, it is easy to see that $w_{k}\in B_{P_{i}}$.

Since $\measuredangle(L_{Q_{i}},L_{0})>\theta$ and $Q_{i}\in\mathsf{Tree}_{0}$ by the definition of $\mathsf{BA}$ (6.2), $\ell(Q_{i})\approx\ell(P_{i})$, and $\operatorname{dist}(Q_{i},P_{i})=0,$ we may use Lemma 7.2 with $Q_{i},P_{i}$ to get

[TABLE]

Thus,

[TABLE]

for $\varepsilon_{0}$ small enough.

Now, denoting by $m_{i}$ the mean of $F$ over the ball $\Pi_{0}(B_{i})$, we have

[TABLE]

Hence, the estimates above give us for some $k\in\{1,2\}$

[TABLE]

Since the estimate above holds for all points $y_{k}\in B_{i,k}\cap\Gamma$, and $\Pi_{0}(B_{i,k}\cap\Gamma)\approx r(B_{i})^{n}$, we can use Poincaré’s inequality to get

[TABLE]

for all $i\in J$.

We claim that the $n$-dimensional balls $\{\Pi_{0}(B_{i})\}_{i\in J}$ are pairwise disjoint. This follows easily by the fact that $2B_{i}=B_{x_{i}}$ are pairwise disjoint, $\frac{1}{4}B_{i}\cap\Gamma\not=\varnothing$, and $\Gamma$ is a graph of a Lipschitz function with a small Lipschitz constant.

Hence, we may sum the inequality above over all $i\in J$ to finally get

[TABLE]

∎

To estimate $\lVert\nabla F\rVert_{L^{2}}$ we will use a well-known theorem due to Dorronsoro. We reformulate it slightly for the sake of convenience. In what follows, $\mathcal{D}_{\Gamma}$ denotes the dyadic grid on $\Gamma$, i.e. the image of the standard dyadic grid on $L_{0}$ under $f(x)=(x,F(x)).$ For $Q\in\mathcal{D}_{\Gamma}$ we set $B_{Q}=B(z_{Q},\operatorname{diam}(Q))$ and $r_{Q}=\operatorname{diam}(Q)\approx\ell(Q)$.

Theorem 11.2** ([Dor85, Theorem 2]).**

Let $F\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\rightarrow\mathbb{R}^{d-n}$ be an $L$-Lipschitz function, with $L$ sufficiently small, and let $\Gamma\subset\mathbb{R}^{d}$ be the graph of $F$, and $\sigma={\left.\kern-1.2pt\mathcal{H}^{n}\right|_{\Gamma}}$. Then

[TABLE]

To estimate the integral above we split the area of integration into four subfamilies:

[TABLE]

we also set

[TABLE]

Firstly, note that for $(x,r)\in A_{1}$ we have $B(x,r)\cap\Gamma=B(x,r)\cap L_{0}$ because $\operatorname{supp}(F)\subset 1.9B_{0}$, and so

[TABLE]

Lemma 11.3**.**

We have

[TABLE]

Proof.

Let $(x,r)\in A_{2}$. Observe that since $\sigma\approx_{A,\tau}\nu$, we have

[TABLE]

Note that if $B(x,r)\cap 1.9B_{0}\not=\varnothing$, then necessarily $x\in B(z_{0},1.9r_{0}+r)\subset B(z_{0},20r)$. Hence,

[TABLE]

∎

Lemma 11.4**.**

We have

[TABLE]

Proof.

Let $(x,r)\in A_{3}$. Since $B(x,r)\cap 1.9B_{0}\not=\varnothing$ and $\eta^{2}d(x)\leq r<0.1r_{0}$, it is clear that $B(x,2r)\subset 2.1B_{0}$ and we may find a cube $P=P(x,r)\in\mathsf{Tree}$ such that $B(x,2r)\subset 3B_{P}$ and $r\approx\ell(P)$. We will estimate the average distance of $B(x,r)\cap\Gamma$ to $L_{P}$.

Bounding the part corresponding to $B(x,r)\cap R_{G}\subset 3B_{P}\cap R_{G}$ is straightforward: Lemma 7.5 states that $d{\left.\kern-1.2pt\mu\right|_{R_{G}}}=g\,d{\left.\kern-1.2pt\mathcal{H}^{n}\right|_{R_{G}}}$ with $g\approx_{A,\tau}1$, and so

[TABLE]

Dealing with the part outside of $R_{G}$ is a bit more delicate. By (9.7) and the definition of functions $h_{k}$ (9.8),

[TABLE]

Consider the 1-Lipschitz function $\Phi(y)=\psi(y)\operatorname{dist}(y,L_{P}),$ where $\psi$ is $r^{-1}$-Lipschitz, $\psi\equiv 1$ on $B(x,r)$, $|\psi|\leq 1$, and $\operatorname{supp}(\psi)\subset B(x,2r)$.

[TABLE]

Since $|\psi|,|h|\leq 1$, the first term on the right hand side above can be bounded by $\beta_{\mu,2}(3B_{P})\ell(P)^{n}$, just as in (11.5). Concerning the second term,

[TABLE]

Gathering all the calculations above we get that

[TABLE]

Integrating the first term over $A_{3}$, since each $P(x,r)$ has sidelength comparable to $r$ and $\operatorname{dist}(P(x,r),x)\lesssim_{A,\tau}r$, it is easy to see that

[TABLE]

Moving on to the second term from (11.6), note that if $y\in 3B_{k}\cap B(x,2r)\not=\varnothing$, then by (9.3) we have $r_{k}\leq d(y)\leq 2r+d(x)\leq(2+\eta^{2})r.$ Thus, following calculations from the proof of Lemma 10.8 (more precisely (10.19) and onwards), we get that

[TABLE]

Hence, $I_{3}\lesssim_{A,\tau}\varepsilon_{0}\ell(R_{0})^{n}$. ∎

Lemma 11.5**.**

We have

[TABLE]

Proof.

Let $(x,r)\in A_{4}$, so that $\eta^{2}d(x)\geq r>0$. It follows by (9.7) that $x\in B_{k}$ for some $k\in K$. Then,

[TABLE]

Note also that $x\in 2B_{0}$. Thus,

[TABLE]

∎

Putting together the estimates for $I_{1},\ I_{2},\ I_{3}$ and $I_{4}$, we get that

[TABLE]

Thus, Lemma 11.1 and Theorem 11.2 give us

[TABLE]

Taking into account the estimates for other stopping cubes, we arrive at

[TABLE]

Thus, $\mu(R_{G})\geq 0.5\mu(R_{0})$, and since $R_{G}$ is a subset of the Lipschitz graph $\Gamma$ and ${\left.\kern-1.2pt\mu\right|_{R_{G}}}$ is $n$-rectifiable, the proof of Lemma 5.1 is finished.