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

  • Xavier TolsaEmail author
Correction
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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. \(x\in \Gamma \).

3.4 The \(\alpha _\mu \) coefficients of the \(\Gamma \)-cubes

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. \(\square \)

Notes

References

  1. 1.
    Tolsa, X.: Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality. Proc. Lond. Math. Soc. 98(2), 393–426 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Tolsa, X.: Characterization of n-rectifiability in terms of Jones’ square function: part I. Calc. Var. PDE 25(4), 3643–3665 (2015)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.ICREABarcelonaSpain
  2. 2.Departament de Matemàtiques, and BGSMathUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain

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