# Correction to: Characterization of n-rectifiability in terms of Jones’ square function: part I

• Xavier Tolsa
Correction

## 1 Correction to: Calc. Var. (2015) 54:3643–3665  https://doi.org/10.1007/s00526-015-0917-z

Abstract. In the author’s paper “Characterization of n-rectifiability in terms of Jones’ square function: Part I”, Calc. Var. PDE. (2015), no. 4, 3643–3665, there is a gap in the proof of Main Lemma 2.1. In this note, this gap is corrected.

## 2 Introduction

In [2], it is shown that if $$\mu$$ is an n-rectifiable measure in $$\mathbb {R}^d$$, then
\begin{aligned} \int _0^\infty \beta _{2,\mu }^n(x,r)^2\,\frac{dr}{r}<\infty \quad \text{ for }~ \mu \text{-a.e. }~ {x\in \mathbb {R}^d}. \end{aligned}
(for the notation, see the aforementioned paper). An essential step for the proof is the following result (Main Lemma 2.1 in [2]):

### Main Lemma

Let $$\mu$$ be a finite Borel measure on $$\mathbb {R}^d$$ and let $$\Gamma \subset \mathbb {R}^d$$ be an n-dimensional Lipschitz graph in $$\mathbb {R}^d$$. Then
\begin{aligned} \int _0^\infty \alpha _\mu ^n(x,r)^2\,\frac{dr}{r}<\infty \quad \text{ for }~ \mathcal {H}^n\text{-a.e. }~ {x\in \Gamma .} \end{aligned}

The proof of this result in [2] has a gap because of an incorrect statement just above (2.22) in that paper. In this note, this gap is corrected. The new arguments are along the same lines as in [2], but they require many changes of technical type.

## 3 The proof

### 3.1 Preliminaries

Given two finite Borel measures $$\sigma$$, $$\mu$$ on $$\mathbb {R}^d$$ and a closed ball $$B\subset \mathbb {R}^d$$, we set
\begin{aligned} \mathrm{dist}_B(\sigma ,\mu ):= \sup \left\{ \left| {\textstyle \int f\,d\sigma - \int f\,d\mu }\right| :\,\mathrm{Lip}(f) \le 1,\,{\text {supp}}(f)\subset B\right\} , \end{aligned}
where $$\mathrm{Lip}(f)$$ stands for the Lipschitz constant of f. We also set
\begin{aligned} \alpha _\mu ^n(B)= \frac{1}{r(B)^{n+1}}\,\inf _{a\ge 0,L} \,\mathrm{dist}_{B}\left( \mu ,\,a\mathcal {H}^n_{L}\right) , \end{aligned}
where the infimum is taken over all the constants $$a\ge 0$$ and all the n-planes L which intersect B. If the n-plane L is fixed, we write
\begin{aligned} \alpha _{\mu ,L}^n(B)= \frac{1}{r(B)^{n+1}}\,\inf _{a\ge 0} \,\mathrm{dist}_{B}\left( \mu ,\,a\mathcal {H}^n_{L}\right) . \end{aligned}
We will drop the exponent n, and we will write $$\alpha _\mu (x,r)$$ and $$\alpha _{\mu ,L}(x,r)$$ instead of $$\alpha _\mu ^n({\bar{B}}(x,r))$$ and $$\alpha _{\mu ,L}^n({\bar{B}}(x,r))$$ to shorten the notation.

### 3.2 The case when $$\mu$$ is supported on $$\Gamma$$

We will prove first the following partial result, which can be deduced by stopping time arguments analogous to some of the ones in [2]. However, for completeness, we show here the detailed proof.

### Lemma 2.1

Let $$\Gamma \subset \mathbb {R}^d$$ be an n-dimensional Lipschitz graph in $$\mathbb {R}^d$$, and let $$\mu$$ be a finite Borel measure supported on $$\Gamma$$. Then
\begin{aligned} \int _0^\infty \alpha _\mu (x,r)^2\,\frac{dr}{r}<\infty \quad \text{ for }~ \mathcal {H}^n\text{-a.e. }~ {x\in \Gamma .} \end{aligned}

To prove the preceding result, we need to introduce the following “$$\Gamma$$-cubes” associated with $$\Gamma$$. We assume that $$\Gamma$$ equals the graph of a Lipschitz function $$A:\mathbb {R}^n\rightarrow \mathbb {R}^{d-n}$$. We say that $$Q\subset \Gamma$$ is a $$\Gamma$$-cube if it is a subset of the form $$Q=\Gamma \cap (Q_0\times \mathbb {R}^{d-n})$$, where $$Q_0\subset \mathbb {R}^n$$ is an n-dimensional cube. We denote $$\ell (Q):=\ell (Q_0)$$. We say that Q is a dyadic $$\Gamma$$-cube if $$Q_0$$ is a dyadic cube. The center of Q is the point $$x_Q=(x_{Q_0},A(x_{Q_0}))$$, where $$x_{Q_0}$$ is the center of $$Q_0$$. The collection of dyadic $$\Gamma$$-cubes Q with $$\ell (Q)=2^{-j}$$ is denoted by $$\mathcal {D}_{\Gamma ,j}$$. Also, we set $$\mathcal {D}_{\Gamma }=\bigcup _{j\in \mathbb {Z}}\mathcal {D}_{\Gamma ,j}$$ and $$\mathcal {D}_{\Gamma }^k=\bigcup _{j\ge k}\mathcal {D}_{\Gamma ,j}$$.

Given a $$\Gamma$$-cube Q, we denote by $$B_Q$$ a closed ball concentric with Q with $$r(B_Q)=3{\text {diam}}(Q)$$. Note that $$B_Q$$ contains Q and is centered on $$\Gamma$$. We set
\begin{aligned} \alpha _\mu (Q) :=\alpha _\mu (B_Q). \end{aligned}
By standard methods, it follows that Lemma 2.1 is an immediate consequence of the following more precise result:

### Lemma 2.2

Let $$\Gamma$$ be an n-dimensional Lipschitz graph in $$\mathbb {R}^d$$, let $$\mu$$ be a finite Borel measure supported on $$\Gamma$$, and let $$\mathcal {D}_\Gamma$$ be the lattice of $$\Gamma$$-cubes introduced above. Denoting by $$L_Q$$ the n-plane that minimizes $$\alpha _{\mathcal {H}^n|_\Gamma }(Q)$$, we have
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q}\alpha _{\mu ,L_Q}(Q)^2<\infty \quad \text{ for }~ {\mathcal {H}^n}\text{-a.e. }~ {x\in \Gamma .} \end{aligned}

### Proof

First note that if $$\mu =g\, \mathcal {H}^n|_\Gamma$$ with $$g\in L^\infty (\mathcal {H}^n|_\Gamma )$$, then the conclusion of the lemma is true. Indeed, it turns out that the measure $$\sigma := (1 + g)\,\mathcal {H}^n|_\Gamma$$ is n-AD-regular and thus
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q,\ell (Q)\le 1}\alpha _{\sigma ,L_Q}(Q)^2<\infty \quad \text{ for }~ {\mathcal {H}^n}\text{-a.e. }~ {x\in \Gamma ,} \end{aligned}
by the results from [1]. Then, for any $$Q\in \mathcal {D}_\Gamma$$, we have
\begin{aligned} \alpha _{\mu ,L_Q}(Q) \le \alpha _{\mathcal {H}^n|_\Gamma ,L_Q}(Q) + \alpha _{\sigma ,L_Q}(Q), \end{aligned}
and thus
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q,\ell (Q)\le 1}\alpha _{\mu ,L_Q}(Q)^2&\lesssim \sum _{Q\in \mathcal {D}_\Gamma :x\in Q,\ell (Q)\le 1}\alpha _{\mathcal {H}^n|_\Gamma ,L_Q}(Q)^2\\&\quad +\sum _{Q\in \mathcal {D}_\Gamma :x\in Q,\ell (Q)\le 1}\alpha _{\sigma ,L_Q}(Q)^2<\infty \quad \text{ for } \mathcal {H}^n\text{-a.e. }~ {x\in \Gamma ,} \end{aligned}
On the other hand, using just that $$\mu$$ is a finite measure, it follows easily that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q,\ell (Q)\ge 1}\alpha _{\mu ,L_Q}(Q)^2 <\infty \quad \text{ for } \text{ all }~ x\in \Gamma , \end{aligned}
and so we have
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q}\alpha _{\mu ,L_Q}(Q)^2 <\infty \quad \text{ for } \mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma . \end{aligned}
In the case when $$\mu$$ is an arbitrary finite Borel measure supported on $$\Gamma$$, we consider a suitable Calderón-Zygmund decomposition of $$\mu$$ with respect to $$\mathcal {H}^n|_\Gamma$$. So for some big $$\lambda >0$$, we let $${\mathcal B}$$ be the maximal family of $$\Gamma$$-cubes P such that $$\mu (P)>\lambda \,\mathcal {H}^n(P)$$, and then we have
\begin{aligned} \mu = g\,\mathcal {H}^n|_\Gamma + \nu , \end{aligned}
where
\begin{aligned} g\,\mathcal {H}^n|_\Gamma = \mu |_G + \sum _{P\in {\mathcal B}} \frac{\mu (P)}{\mathcal {H}^n(P)}\,\mathcal {H}^n|_P, \end{aligned}
with $$G= \mathbb {R}^d {\setminus } \bigcup _{P\in {\mathcal B}} P$$, and also
\begin{aligned} \nu = \sum _{P\in {\mathcal B}} \nu _P,\qquad \nu _P = \mu |_P - \frac{\mu (P)}{\mathcal {H}^n(P)}\,\mathcal {H}^n|_P. \end{aligned}
By standard arguments, it is clear that $$\Vert g\Vert _{L^\infty (\mu )}\lesssim \lambda$$, and thus if we denote $$\sigma = g\,d\mathcal {H}^n|_\Gamma$$, then
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q}\alpha _{\sigma ,L_Q}(Q)^2<\infty \quad \text{ for } \mathcal {H}^n\hbox {-a.e. } x\in \Gamma . \end{aligned}
(2.1)
Let $${\widetilde{G}}= \Gamma {\setminus } \bigcup _{P\in {\mathcal B}} 3B_{P}$$, and observe that
\begin{aligned} \mathcal {H}^n(\Gamma {\setminus } {\widetilde{G}}) = \sum _{P\in {\mathcal B}} \mathcal {H}^n(3B_{P})\le c\sum _{P\in {\mathcal B}} \mathcal {H}^n(P)\le \frac{c}{\lambda }\sum _{P\in {\mathcal B}} \mu (P) \le \frac{c}{\lambda }\,\Vert \mu \Vert . \end{aligned}
Since $$\lambda$$ can be taken arbitrarily big, to prove the lemma, it suffices to show that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :x\in Q}\alpha _{\mu ,L_Q}(Q)^2<\infty \quad \text{ for } \mathcal {H}^n\hbox {-a.e. }x\in {\widetilde{G}}. \end{aligned}
(2.2)
Denote by $${\mathcal G}$$ the family of those $$\Gamma$$-cubes which are not contained in $$\bigcup _{P\in {\mathcal B}} 3B_{P}$$. Notice that, for $$x\in {\widetilde{G}}$$, all the $$\Gamma$$-cubes in the sum in (2.2) are from $${\mathcal G}$$. For a given $$Q\in {\mathcal G}$$, we have
\begin{aligned} \alpha _{\mu ,L_Q}(Q) \le \alpha _{\sigma ,L_Q}(Q) + \frac{1}{\ell (Q)^{n+1}}\,\sup _f \left| \int f\,d\nu \right| , \end{aligned}
where the supremum is taken over all functions f which are 1-Lipschitz and supported on $$B_Q$$. To estimate the last integral on the right hand side, we write
\begin{aligned} \left| \int f\,d\nu \right| \le \sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing }\left| \int f\,d\nu _P\right| . \end{aligned}
Since $$\int d\nu _P=0$$ for all $$P\in {\mathcal B}$$, we have
\begin{aligned} \left| \int f\,d\nu _P\right| \le \int |f-f(x_P)|\,d|\nu _P|\lesssim \ell (P) \,\Vert \nu _P\Vert \lesssim \ell (P)\,\mu (P). \end{aligned}
Thus,
\begin{aligned} \alpha _{\mu ,L_Q}(Q) \lesssim \alpha _{\sigma ,L_Q}(Q) + \sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}}. \end{aligned}
So, because of (2.1), to prove the lemma, it suffices to show that
\begin{aligned} \sum _{Q\in {\mathcal G}:x\in Q}\left( \sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}}\right) ^2<\infty \quad \text{ for } \mathcal {H}^n\hbox {-a.e. }x\in {\widetilde{G}}. \end{aligned}
(2.3)
To prove (2.3), first we take into account that if $$P\in {\mathcal B}$$ is such that $$P\cap B_Q\ne \varnothing$$, then
\begin{aligned} r(B_P)\le r(B_Q), \end{aligned}
(2.4)
because otherwise $$Q\subset B_Q\subset 3B_P,$$ which contradicts the fact that $$Q\in {\mathcal G}$$. Now, from (2.4), we deduce that $$P\subset B_P\subset 3B_Q$$, and taking also into account that $$\mu (P)\lesssim \lambda \,\ell (P)^n$$, we get
\begin{aligned} \sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}} \lesssim \lambda \sum _{P\in {\mathcal B}:P\subset 3B_Q} \frac{\ell (P)^{n+1}}{\ell (Q)^{n+1}} \lesssim \lambda . \end{aligned}
Therefore,
\begin{aligned} \sum _{Q\in {\mathcal G}:x\in Q}\left( \sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}}\right) ^2\lesssim \lambda \sum _{Q\in {\mathcal G}:x\in Q}\sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}}. \end{aligned}
Now we can show that the last sum is finite for $$\mathcal {H}^n$$-a.e. $$x\in {\widetilde{G}}$$ by integrating this with respect to $$\mathcal {H}^n|_\Gamma$$. Then we obtain
\begin{aligned} \int _\Gamma \sum _{Q\in {\mathcal G}:x\in Q}\sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}}\,d\mathcal {H}^n(x)&\approx \sum _{Q\in {\mathcal G}}\,\sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)}\\&= \sum _{P\in {\mathcal B}}\mu (P) \sum _{Q\in {\mathcal G}:P\cap B_Q\ne \varnothing }\frac{\ell (P)}{\ell (Q)}. \end{aligned}
Recalling that $$r(B_Q)\ge r(B_P)$$ for every Q in the last sum, we get $$\sum _{Q\in {\mathcal G}:P\cap B_Q\ne \varnothing }\frac{\ell (P)}{\ell (Q)}\lesssim 1$$, and so we deduce that
\begin{aligned} \int _\Gamma \sum _{Q\in {\mathcal G}:x\in Q}\sum _{P\in {\mathcal B}:P\cap B_Q\ne \varnothing } \frac{\ell (P)\,\mu (P)}{\ell (Q)^{n+1}}\,d\mathcal {H}^n(x) \lesssim \sum _{P\in {\mathcal B}}\mu (P)\le \Vert \mu \Vert , \end{aligned}
which implies (2.3) and completes the proof of the lemma. $$\square$$

### 3.3 The approximating measure $$\sigma$$ for the general case

To prove the main lemma in full generality, we consider a Whitney decomposition of $$\mathbb {R}^d{\setminus } \Gamma$$ as in Section 2.3 of [2]. That is, we have a family $${\mathcal W}$$ of dyadic cubes in $$\mathbb {R}^d$$ with disjoint interiors such that
\begin{aligned} \bigcup _{Q\in {\mathcal W}} Q = \mathbb {R}^d{\setminus } \Gamma , \end{aligned}
and moreover, there are some constants $$R>20$$ and $$D_0\ge 1$$ such the following holds for every $$Q\in {\mathcal W}$$:
1. (i)

$$10Q \subset \mathbb {R}^d{\setminus } \Gamma$$;

2. (ii)

$$R Q \cap \Gamma \ne \varnothing$$;

3. (iii)

there are at most $$D_0$$ cubes $$Q'\in {\mathcal W}$$ such that $$10Q \cap 10Q' \ne \varnothing$$. Further, for such cubes $$Q'$$, we have $$\ell (Q')\approx \ell (Q)$$.

From properties (i) and (ii), it is clear that $$\mathrm{dist}(Q,\Gamma )\approx \ell (Q)$$. We assume that the Whitney cubes are small enough so that
\begin{aligned} {\text {diam}}(Q)< \mathrm{dist}(Q,\Gamma ). \end{aligned}
(2.5)
This can be achieved by replacing each cube $$Q\in {\mathcal W}$$ by its descendants $$P\in \mathcal {D}_k(Q)$$, for some fixed $$k\ge 1$$, if necessary. From (2.5), we infer that if $$Q\in {\mathcal W}$$ intersects some ball B(yr) with $$y\in \Gamma$$, then
\begin{aligned} {\text {diam}}(Q)\le r, \end{aligned}
(2.6)
and thus
\begin{aligned} Q\subset B(y,3r). \end{aligned}
(2.7)
For a given Borel measure $$\mu$$, we denote
\begin{aligned} M_n\mu (x) = \sup _{r>0}\frac{\mu (B(x,r))}{r^n}. \end{aligned}
From the growth condition $$\mathcal {H}^n(\Gamma \cap B(x,r))\le c\,r^n$$ for all x and $$r>0$$, it follows easily that the maximal operator $$M_n$$ is bounded from the space of finite signed Radon measures $$M(\mathbb {R}^d)$$ into $$L^{1,\infty }(\mathcal {H}^n|_\Gamma )$$. As a consequence, for any arbitrary finite Borel measure $$\mu$$ in $$\mathbb {R}^d$$, $$M_n\mu (x)<\infty$$ for $$\mathcal {H}^n$$-a.e $$x\in \Gamma .$$

Denote by $$\Pi _\Gamma$$ the projection on $$\Gamma$$ given by $$\Pi _\Gamma (x) = (x_1,\ldots ,x_n,A(x_1,\ldots ,x_n))$$. To each cube $$Q\in {\mathcal W}$$ we associate a ball $${\widetilde{B}}_Q$$ with radius $$r({\widetilde{B}}_Q)=\frac{1}{4} \ell (Q)$$, centered in $${\widetilde{x}}_Q:=\Pi _\Gamma (x_Q)$$, where $$x_Q$$ is the center of Q. In particular, we have $$\mathrm{dist}(Q,{\widetilde{B}}_Q)\approx \ell (Q)\approx r({\widetilde{B}}_Q)$$.

Let $${\varphi }:\mathbb {R}^d\rightarrow \mathbb {R}$$ be a smooth non-negative radial function supported in B(0, 1) which equals 1 in B(0, 1/2). For each $$Q\in {\mathcal W}$$, we consider the function $${\varphi }_Q:\mathbb {R}^d\rightarrow \mathbb {R}$$ defined by
\begin{aligned} {\varphi }_Q(x) ={\varphi }\left( \frac{x-{\widetilde{x}}_Q}{r({\widetilde{B}}_Q)}\right) \end{aligned}
and then we set
\begin{aligned} g_Q(x) = \frac{\mu (Q)}{\Vert {\varphi }_Q\Vert _{L^1(\mathcal {H}^n|_\Gamma )}}\,{\varphi }_Q(x), \end{aligned}
(2.8)
and also
\begin{aligned} g(x) = \sum _{Q\in {\mathcal W}} g_Q(x). \end{aligned}
Notice that $$\int g_Q\,d\mathcal {H}^n|_\Gamma = \mu (Q)$$ and also that
\begin{aligned} \Vert g_Q\Vert _\infty \lesssim \frac{\mu (Q)}{\ell (Q)^n},\qquad \Vert \nabla g_Q\Vert _\infty \lesssim \frac{\mu (Q)}{\ell (Q)^{n+1}}. \end{aligned}
Then we consider the following measure $$\sigma$$ on $$\Gamma$$, which approximates $$\mu |_{\Gamma ^c}$$, in a sense:
\begin{aligned} \sigma = g\,d\mathcal {H}^n|_\Gamma = \sum _{Q\in {\mathcal W}}g_Q\,d\mathcal {H}^n|_\Gamma . \end{aligned}
(2.9)
Next, for some constant $$A>10$$ to be chosen below, we also consider the auxiliary function
\begin{aligned} G_A(x) = \sum _{Q\in {\mathcal W}} \frac{\mu (Q)}{\ell (Q)^n}\,\chi _{A{\widetilde{B}}_Q}. \end{aligned}
Notice that, for some absolute constant C,
\begin{aligned} g(x) \le C\,G_A(x)\quad \text{ for } \text{ all } x\in \Gamma . \end{aligned}
Also,
\begin{aligned}\Vert G_A\Vert _{L^1(\mathcal {H}^n|_\Gamma )}\le C\,\sum _{Q\in {\mathcal W}}\frac{\mu (Q)\,A^n\,r({\widetilde{B}}_Q)^n}{\ell (Q)^n} \le C\,A^n\,\Vert \mu \Vert ,\end{aligned}
and thus $$G_A(x)<\infty$$ for $$\mathcal {H}^n$$-a.e. $$x\in \Gamma$$.

### Lemma 2.3

Assume A big enough. Then, for each $$Q\in \mathcal {D}_\Gamma$$, we have
\begin{aligned} \alpha _\mu (Q)&\lesssim \alpha _{\mu |_\Gamma ,L_Q}(Q) + \alpha _{\sigma ,L_Q}(Q) + \int _{6B_{Q}} \frac{\mathrm{dist}(x,\Gamma )}{\ell (Q)^{n+1}}\, d\mu (x)\\&\quad + \inf _{x\in Q}G_A(x) \,\alpha _{\mathcal {H}^n|_\Gamma }(Q) + \sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}, \end{aligned}
where $$L_Q$$ is the n-plane that minimizes $$\alpha _{\mathcal {H}^n|_\Gamma }(Q)$$.

### Proof

Note first that $$\alpha _\mu (Q)\le \alpha _{\mu |_\Gamma ,L_Q}(Q) + \alpha _{\mu |_{\Gamma ^c,L_Q}}(Q)$$. So we just have to deal with $$\alpha _{\mu |_{\Gamma ^c},L_Q}(Q)$$. To estimate this coefficient, we consider a 1-Lipschitz function f supported on $$B_Q$$, and we write
\begin{aligned} \int f\,d\mu |_{\Gamma ^c} = \sum _{P\in {\mathcal W}}\int f\,d(\mu |_P - g_P \,\mathcal {H}^n|_\Gamma ) + \int f\,d\sigma , \end{aligned}
where $$g_P$$ is the function defined in (2.8) and $$\sigma$$ the measure in (2.9).
So, for any constant $$a_Q\in \mathbb {R}$$, we get
\begin{aligned} \alpha _{\mu |_{\Gamma ^c},L_Q}(Q)\le \alpha _{\sigma ,L_Q}(Q) + \frac{1}{\ell (Q)^{n+1}}\sup _f\left| \sum _{P\in {\mathcal W}}\int f\,d(\mu |_P - g_P \,\mathcal {H}^n|_\Gamma ) + a_Q\int f\,d\mathcal {H}^n|_{L_Q}\right| , \end{aligned}
where the supremum is taken over all 1-Lipschitz functions supported on $$B_Q$$.
We denote by $$I_a$$ the subfamily of the cubes from $${\mathcal W}$$ which intersect $$2B_Q$$, and $$I_b={\mathcal W}{\setminus } I_a$$, and we split the last sum above as follows:
\begin{aligned}&\left| \sum _{P\in {\mathcal W}}\int f\,d(\mu |_P - g_P \,\mathcal {H}^n|_\Gamma ) + a_Q\int f\,d\mathcal {H}^n|_{L_Q}\right| \\&\qquad \le \sum _{P\in I_a}\left| \int f\,d(\mu |_P - g_P \,\mathcal {H}^n|_\Gamma ) \right| + \left| \sum _{P\in I_b}\int f\,d(\mu |_P - g_P \,\mathcal {H}^n|_\Gamma ) + a_Q\int f\,d\mathcal {H}^n|_{L_Q}\right| \\&\qquad =: S_a + S_b. \end{aligned}
First we deal with the sum $$S_a$$. For each $$P\in I_a$$, since $$\int g_P\,d\mathcal {H}^n|_{\Gamma } = \mu (P)$$, we deduce that
\begin{aligned} \left| \int f\,d(\mu |_P- g_P\,\mathcal {H}^n|_\Gamma )\right|&\le \left| \int _P(f(x) - f(x_P))\,d\mu (x)\right| \\&\quad + \left| \int (f(x_P) - f(x))\, g_P(x)\,\mathcal {H}^n|_\Gamma (x))\right| . \end{aligned}
To deal with the first integral on the right-hand side, we take into account that for $$x\in P$$ we have
\begin{aligned} |f(x) - f(x_P)|\le \Vert \nabla f\Vert _\infty \,|x-x_P|\le c\,\ell (P). \end{aligned}
(2.10)
Concerning the second integral, recall that $${\text {supp}}g_P\subset \Gamma \cap {\bar{B}}(x_P,c\,\ell (P))$$, and thus we also have $$|x-x_P|\le c\,\ell (P)$$ in the domain of integration, so that (2.10) holds in this case too. Therefore,
\begin{aligned} \left| \int f\,d(\mu |_P- g_P\,\mathcal {H}^n|_\Gamma )\right| \le c\,\ell (P)\,\mu (P)\approx \int _P \mathrm{dist}(x,\Gamma )\,d\mu (x), \end{aligned}
where we took into account that $$\mathrm{dist}(x,\Gamma )\approx \ell (P)$$ for every $$x\in P$$. Recall that $${\text {supp}}f\subset B_Q$$ and since $$P\in I_a$$, then $$P\subset 6B_Q$$ by the argument in (2.7). Thus,
\begin{aligned} S_a = \sum _{P\in I_a}\left| \int f\,d(\mu |_P- g_P\,\mathcal {H}^n|_\Gamma )\right| \le c\,\int _{6B_Q} \mathrm{dist}(x,\Gamma )\,d\mu (x). \end{aligned}
(2.11)
Next we consider the sum $$S_b$$. For each $$P\in I_b$$, we have $$P\cap 2B_Q=\varnothing$$, and so $$\int f\,d\mu |_P=0$$. Therefore,
\begin{aligned} S_b&= \left| \sum _{P\in I_b}\int f\,g_P \,d\mathcal {H}^n|_\Gamma - a_Q\int f\,d\mathcal {H}^n|_{L_Q}\right| \\&\le \left| \sum _{P\in I_b}\int f\,g_P \,d(\mathcal {H}^n|_\Gamma - c_{Q}\,\mathcal {H}^n|_{L_Q})\right| + \int |f|\,\left| \sum _{P\in I_b} c_Q\,g_P -a_Q\right| \,d\mathcal {H}^n|_{L_Q}\\&=: S_{b,1} + S_{b,2}, \end{aligned}
where $$c_Q$$ is the constant minimizing $$\alpha _{\mathcal {H}^n|_\Gamma ,L_Q}(Q)$$. To deal with $$S_{b,1}$$ we take into account that $$\sum _{P\in I_b} f\,g_P$$ is a Lipschitz function supported in $$B_Q$$, and $$f\,g_P$$ vanishes unless $${\widetilde{B}}_P\cap B_Q\ne \varnothing$$. To shorten notation, we write $$P\sim Q$$ if $$P\in I_b$$ is such that $${\widetilde{B}}_P\cap B_Q\ne \varnothing$$. It is easy to check that $$P\sim Q$$ implies that $$\ell (P)\gtrsim \ell (Q)$$. Then we have
\begin{aligned} \left\| \nabla \left( \sum _{P\in I_b} f\,g_P\right) \right\| _\infty&\le \sum _{P\sim Q} \left( \Vert g_P\Vert _\infty + c\,\ell (Q) \Vert \nabla g_P\Vert _\infty \right) \\&\lesssim \sum _{P\sim Q}\left( \frac{\mu (P)}{\ell (P)^n} + \frac{\ell (Q)\,\mu (P)}{\ell (P)^{n+1}}\right) \lesssim \sum _{P\sim Q} \frac{\mu (P)}{\ell (P)^n}. \end{aligned}
From the fact that $$\ell (P)\gtrsim \ell (Q)$$, it also follows that, for A big enough, we have
\begin{aligned} \sum _{P\sim Q} \frac{\mu (P)}{\ell (P)^n}\lesssim \inf _{x\in Q}G_A(x). \end{aligned}
(2.12)
Therefore,
\begin{aligned} S_{b,1}\lesssim \alpha _{\mathcal {H}^n|_\Gamma }(Q)\,\ell (Q)^{n+1}\,\inf _{x\in Q}G_A(x). \end{aligned}
Finally, we turn our attention to the term $$S_{b,2}$$. Choosing $$a_Q = c_Q \sum _{P\sim Q}g_P(x_Q)$$ and using that $$|c_Q|\lesssim 1$$, we obtain
\begin{aligned}S_{b,2}\lesssim \int \left| f\sum _{P\sim Q} \bigl [g_P - g_P(x_Q)\bigr ]\right| \,d\mathcal {H}^n|_{L_Q}. \end{aligned}
Note now that, for $$x\in B_Q$$,
\begin{aligned}\sum _{P\sim Q} \bigl |g_P(x) - g_P(x_Q)\bigr | \lesssim \sum _{P\sim Q} \Vert \nabla g_P\Vert _\infty \,\ell (Q)\lesssim \sum _{P\sim Q} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}, \end{aligned}
and so, using also that $$|f|\le C\,\ell (Q)\,\chi _{B_Q}$$,
\begin{aligned} S_{b,2}\lesssim \sum _{P\sim Q} \frac{\mu (P)\,\ell (Q)^{n+2}}{\ell (P)^{n+1}}. \end{aligned}
Gathering all the estimates above, we obtain
\begin{aligned}&\alpha _{\mu |_{\Gamma ^c},L_Q}(Q)\lesssim \alpha _{\sigma ,L_Q}(Q) + \int _{6B_{Q}} \frac{\mathrm{dist}(x,\Gamma )}{\ell (Q)^{n+1}}\, d\mu (x) + \inf _{x\in Q}G_A(x) \,\alpha _{\mathcal {H}^n|_\Gamma }(Q) \\&\quad + \sum _{P\sim Q} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}, \end{aligned}
which completes the proof of the lemma. $$\square$$

### 3.5 Proof of the Main Lemma

It is immediate to check that the statement in the main lemma is equivalent to the fact that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\alpha _\mu (Q)^2\,\chi _Q(x)<\infty \quad \text{ for } \mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma . \end{aligned}
(2.13)
To prove this estimate, we intend to use Lemma 2.3. To this end, observe that, by Lemma 2.1, we know that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma } \left( \alpha _{\mu |_\Gamma ,L_Q}(Q)^2 + \alpha _{\sigma ,L_Q}(Q)^2\right) \,\chi _Q(x)<\infty \quad \text{ for }~ \mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma . \end{aligned}
Also, for every $$x\in \Gamma$$,
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma :Q\ni x}\inf _{y\in Q}G_A(y)^2 \,\alpha _{\mathcal {H}^n|_\Gamma }(Q)^2 \le G_A(x)^2\,\sum _{Q\in \mathcal {D}_\Gamma :Q\ni x}\alpha _{\mathcal {H}^n|_\Gamma }(Q)^2 <\infty \quad \text{ for }~ \mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma , \end{aligned}
since $$G_A(x)<\infty$$ and $$\sum _{Q\in \mathcal {D}_\Gamma :Q\ni x}\alpha _{\mathcal {H}^n|_\Gamma }(Q)^2 <\infty$$ for $$\mathcal {H}^n$$-a.e. $$x\in \Gamma$$.
Next we show that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\left( \,\sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\right) ^2\,\chi _Q(x)<\infty \quad \text{ for } ~\mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma . \end{aligned}
(2.14)
To this end, note that, as in (2.12),
\begin{aligned} \sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\,\chi _Q(x) \lesssim \sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)}{\ell (P)^{n}}\,\chi _Q(x) \lesssim G_A(x). \end{aligned}
Hence,
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\left( \sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\right) ^2\,\chi _Q(x) \lesssim G_A(x) \sum _{Q\in \mathcal {D}_\Gamma }\sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\chi _Q(x). \end{aligned}
Thus to prove (2.14), it suffices to show that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\chi _Q(x)<\infty \quad \text{ for }~ \mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma . \end{aligned}
(2.15)
By Fubini, we have
\begin{aligned}&\int \sum _{Q\in \mathcal {D}_\Gamma }\sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\,\chi _Q(x)\,d\mathcal {H}^n|_\Gamma (x)\\ {}&\qquad \approx \sum _{Q\in \mathcal {D}_\Gamma } \sum _{\begin{array}{c} P\in {\mathcal W}:P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\mu (P)\,\ell (Q)}{\ell (P)^{n+1}}\,\ell (Q)^n\\&\qquad \approx \sum _{\begin{array}{c} P\in {\mathcal W} \end{array}}\mu (P) \sum _{\begin{array}{c} Q\in \mathcal {D}_\Gamma :P\cap 2B_Q=\varnothing ,\\ {\widetilde{B}}_P\cap B_Q\ne \varnothing \end{array}} \frac{\ell (Q)^{n+1}}{\ell (P)^{n+1}} \lesssim \sum _{\begin{array}{c} P\in {\mathcal W} \end{array}}\mu (P) = \Vert \mu \Vert , \end{aligned}
where in the last inequality we took into account that the $$\Gamma$$-cubes $$Q\in \mathcal {D}_\Gamma$$ such that $$P\cap 2B_Q=\varnothing$$ and $${\widetilde{B}}_P\cap B_Q\ne \varnothing$$ satisfy $$Q\subset c{\widetilde{B}}_P$$ for some $$c>1$$, because $$\ell (Q)\lesssim \ell (P)$$. So (2.15) is proved.
Finally, to complete the proof of this lemma, by Lemma 2.3, it remains to show that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\left( \int _{6B_{Q}} \frac{\mathrm{dist}(y,\Gamma )}{\ell (Q)^{n+1}}\, d\mu (y)\right) ^2\,\chi _Q(x)<\infty \quad \text{ for }~ \mathcal {H}^n\hbox {-a.e.}~ x\in \Gamma . \end{aligned}
(2.16)
The arguments for this are almost the same as the ones in [2]. We repeat them here for the reader’s convenience. By Cauchy–Schwarz, we get
\begin{aligned} \left( \int _{6B_{Q}} \frac{\mathrm{dist}(y,\Gamma )}{\ell (Q)^{n+1}}\, d\mu (y)\right) ^2\le \mu (6B_Q) \int _{6B_{Q}} \left( \frac{\mathrm{dist}(y,\Gamma )}{\ell (Q)^{n+1}}\right) ^2\, d\mu (y). \end{aligned}
Since $$\frac{\mu (6B_Q)}{\ell (Q)^n}\lesssim M_n \mu (x)$$ for all $$x\in Q$$, and $$M_n\mu (x)<\infty$$ for $$\mathcal {H}^n$$-a.e. $$x\in \Gamma$$, it turns that, to prove (2.16), it suffices to show that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\int _{6B_{Q}} \frac{\mathrm{dist}(y,\Gamma )^2}{\ell (Q)^{n+2}}\, d\mu (y)\,\chi _Q(x)<\infty \quad \text{ for } \mathcal {H}^n\hbox {-a.e. } x\in \Gamma . \end{aligned}
(2.17)
The integral of the left-hand side of (2.17) with respect to $$\mathcal {H}^n|_\Gamma$$ does not exceed
\begin{aligned} c\sum _{Q\in \mathcal {D}_\Gamma } \int _{6B_{Q}} \frac{\mathrm{dist}(y,\Gamma )^2}{\ell (Q)^{n+2}}\, d\mu (y)\,\ell (Q)^n = c\sum _{Q\in \mathcal {D}_\Gamma } \int _{6B_{Q}} \frac{\mathrm{dist}(y,\Gamma )^2}{\ell (Q)^2}\, d\mu (y). \end{aligned}
By Fubini, this equals
\begin{aligned} c\int \mathrm{dist}(y,\Gamma )^2\sum _{Q\in \mathcal {D}_\Gamma }\chi _{6B_{Q}}(y)\, \frac{1}{\ell (Q)^{2}} \, d\mu (y). \end{aligned}
Notice now that
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma }\chi _{6B_{Q}}(y)\, \frac{1}{\ell (Q)^2} = \sum _{Q\in \mathcal {D}_\Gamma : y\in 6B_{Q}} \frac{1}{\ell (Q)^2} \lesssim \frac{1}{\mathrm{dist}(y, \Gamma )^2}, \end{aligned}
because the condition $$y\in 6B_{Q}$$ implies that $$\mathrm{dist}(y,\Gamma )\le r(B_Q)\approx \ell (Q)$$. Thus,
\begin{aligned} \sum _{Q\in \mathcal {D}_\Gamma } \int _{6B_{Q}} \frac{\mathrm{dist}(y,\Gamma )^2}{\ell (Q)^{2}}\, d\mu (y)&\lesssim \int \frac{\mathrm{dist}(y,\Gamma )^2}{\mathrm{dist}(y,\Gamma )^2} \, d\mu (y)= \Vert \mu \Vert . \end{aligned}
Hence, (2.16) follows and we are done. $$\square$$

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