linear subspaces and we can hope for a tangent plane characterization of
rectifiability. Now $Y$ is $l^{\infty}$, or more generally the dual of a
separable Banach space. In addition to Kirchheim’s metric differentiability
result Ambrosio and Kirchheim [AK00a] proved the weak star differentiability
for a Lipschitz map $f:\mathbb{R}^{m}\to Y$: for almost all
$x\in\mathbb{R}^{m}$ there exists a $w^{\ast}$-differential of $f$ at $x$,
that is, a linear map $wdf_{x}:\mathbb{R}^{m}\to Y$ such that

(8.5) $w^{\ast}-\lim_{y\to x}(f(y)-f(x)-wdf_{x}(y-x))/|y-x|)=0.$

If $E\subset X$ is $\mathcal{H}^{m}$ measurable and $m$-rectifiable, then by
Lemma 8.5 we can decompose almost all of it into sets $f_{i}(B_{i})$ where
$B_{i}\subset\mathbb{R}^{m}$ is a Borel set and $f_{i}$ is bi-Lipschitz on
$B_{i}$. Then for $\mathcal{H}^{m}$ almost all $a\in E$ we can define the
_weak approximate tangent plane_ of $E$ at $a=f_{i}(x)\in f_{i}(B_{i})$ as

$\operatorname{Tan}^{(m)}(E,a)=wdf_{ix}(\mathbb{R}^{m}).$

It further follows that if $\pi_{a}:Y\to\operatorname{Tan}^{(m)}(E,a)$ is a
weak star continuous projection ($\pi_{a}(x)=x$ for
$x\in\operatorname{Tan}^{(m)}(E,a)$), then $\|\pi_{a}(x)-a\|/\|x-a\|\to 1$ as
$x\to a,x\in E_{a}$, where $\Theta^{m}(E\setminus E_{a},a)=0$.

Ambrosio and Kirchheim also had a converse in [AK00a, Theorem 6.3]:

######  Theorem 8.7.

Let $E\subset Y$ be $\mathcal{H}^{m}$ measurable with
$\mathcal{H}^{m}(E)<\infty$. Then $E$ is $m$-rectifiable if and only if for
$\mathcal{H}^{m}$ almost all $a\in E$ there is a weak star continuous linear
map $\pi_{a}:Y\to Y,\dim\pi_{a}(Y)=m,$ such that for some $s>0$,

$\lim_{r\to 0}r^{-m}\mathcal{H}^{m}(\\{x\in E\cap
B(a,r):\|\pi_{a}(x-a)\|<s\|x-a\|\\})=0.$

Ambrosio and Kirchheim had this with positive lower density condition, but
using the argument from the Euclidean case one easily sees that this is not
needed, see [Mat95, Lemma 15.14].

Ambrosio and Kirchheim also proved in [AK00a] an area formula for Lipschitz
maps between rectifiable sets and a coarea formula for $\mathbb{R}^{k}$ valued
Lipschitz maps on rectifiable subsets of metric spaces.

###  8.5. Cheeger’s differentiability spaces and Alberti representations

As discussed above Kirchheim showed that Lipschitz maps from $\mathbb{R}^{m}$
to $X$ are differentiable in a sense. In [Che99] Cheeger introduced conditions
under which the differentiability of Lipschitz maps from $X$ to
$\mathbb{R}^{m}$ also makes sense and is true. This was further developed and
generalized by Keith [Kei04]. Suppose that $X$ is equipped with a Borel
measure $\mu$. We say that $(X,d,\mu)$ is an $m$-dimensional _Lipschitz
differentiability space_ , LDS, if there is a countable collection of local
Lipschitz charts $\phi_{i}:U_{i}\to\mathbb{R}^{m},U_{i}\subset
X,X=\bigcup_{i}U_{i}$, with respect to which every Lipschitz function
$f:X\to\mathbb{R}$ is differentiable at $\mu$ almost all $x\in U_{i}$ in the
sense that there is a unique linear function
$Df(x):\mathbb{R}^{m}\to\mathbb{R}$ such that

$f(y)-f(x)=Df(x)(\phi_{i}(y)-\phi_{i}(x))+o(d(x,y)),y\in U_{i}.$

It is clear that Euclidean spaces and Riemannian manifolds with the Lebesgue
measure are LDS. Not all, even compact, metric measure spaces are LDS, but
there are a lot of non-Euclidean ones that are, for example Heisenberg groups.

Recall the role of Alberti representations in Euclidean spaces from Section
5.8. _Alberti representation_ of a measure $\mu$ on $X$ is a Fubini-type
decomposition of $\mu$ into $1$-rectifiable measures $\mu_{\gamma}$:

(8.6) $\mu(B)=\int\mu_{\gamma}(B)\,dP\gamma,\ B\subset X\ \text{Borel}.$

Here $P$ is a probability measure on the space of curve fragments, that is,
bi-Lipschitz mappings $\gamma:C_{\gamma}\to X,C_{\gamma}\subset\mathbb{R}$
compact, and the measures $\mu_{\gamma}$ are absolutely continuous with
respect to
$\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}\gamma(C_{\gamma})$.

For $\phi:X\to\mathbb{R}^{m}$ the Alberti representations
$\gamma_{i},i=1,\dots,m$, are said to be $\phi$-independent if the derivatives
of $\phi\circ\gamma_{i},i=1,\dots,m$, are linearly independent and belong to
disjoint cones.

Bate characterised in [Bat15] the Lipschitz differentiability spaces via
Alberti representations. The following theorem tells us a great deal of
relations between Lipschitz differentiability spaces, Alberti representations
and rectifiablity:

######  Theorem 8.8.

Suppose that $\mathcal{H}^{m}(X)<\infty$ and $\Theta_{\ast}^{m}(X,x)>0$ for
$\mathcal{H}^{m}$ almost all $x\in X$. Then the following conditions are
equivalent.

  * (1)

$X$ is $m$-rectifiable.

  * (2)

There are Borel sets $U_{i}\subset X,i=1,2,\dots,$ with
$\mathcal{H}^{m}(X\setminus\bigcup_{i}U_{i})=0$ such that each
$(U_{i},d,\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}U_{i})$ is an $m$-dimensional LDS.

  * (3)

There are Borel sets $U_{i}\subset X,i=1,2,\dots,$ with
$\mathcal{H}^{m}(X\setminus\bigcup_{i}U_{i})=0$ and Lipschitz functions
$\phi_{i}:U_{i}\to\mathbb{R}^{m}$ such that each
$\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}U_{i}$ has $m$
$\phi_{i}$-independent Alberti representations.

That (1) implies (2) follows from Kirchheim’s work [Kir94]. He showed that in
the definition of rectifiable sets Lipschitz maps can be replaced by bi-
Lipschitz maps, recall Lemma 8.5. Then these can be used to find the required
charts. The implication from (2) to (3) follows from [Bat15]. Rectifiability
from independent Alberti representations was proved by Bate and Li in [BL17].
They used some of the ideas of David from [Dav15], who proved related weaker
results. A key feature in this technically very complicated argument is to
show that the maps $\phi_{i}$ are bi-Lipschitz on large subsets.

Notice that $m$ in the assumptions and in (2) is the same. This leaves out
many interesting LDSs. For example, the Hausdorff dimension of the Heisenberg
group $\mathbb{H}^{1}$ is 4 but it is a 2-dimensional LDS.

David and Kleiner [DK19] proved a general result for a measure $\mu$ in a
metric space without any density conditions: if at $\mu$ almost all points
$\mu$ is pointwise doubling, has two independent Alberti representations, and
pointed Gromov-Hausdorff limits (see Section 8.7) are homeomorphic to
$\mathbb{R}^{2}$, then $\mu$ is 2- rectifiable.

###  8.6. Projections as Lipschitz images

First some bad news. The Besicovitch-Federer projection theorem fails in every
infinite-dimensional separable Banach space $Y$: Bate, Csörnyei and Wilson
[BCW17] constructed a purely unrectifiable set $E\subset Y$ with
$\mathcal{H}^{1}(E)<\infty$ for which $\mathcal{L}^{1}(L(E))>0$ for every non-
zero continuous linear function $L:Y\to\mathbb{R}$. An earlier weaker result
was given by De Pauw in [Dep17].

However, instead of linear maps Bate [Bat20] considered Lipschitz maps from
the metric space $X$ to $\mathbb{R}^{n}$ and obtained a rectifiability
characterization in the spirit of the Besicovitch-Federer projection theorem.
There is no natural measure on the space of Lipschitz maps (at least for this
purpose), but Baire category gives a notion of typical maps. Related results
were proven by Pugh [Pug16] and Galeski [Gal18].

For the rest of this section we shall assume that $X$ is complete. To state
Bate’s result let us equip the space of bounded Lipschitz functions
$f:X\to\mathbb{R}^{n}$ such that $\text{Lip}(f)\leq 1$ with the supremum norm
to have the complete metric space $\text{Lip}_{1}(X,n)$. A subset of
$\text{Lip}_{1}(X,n)$ is _residual_ if it contains a countable intersection of
dense open sets. These are the complements of the ’small’ first category sets,
countable unions of nowhere dense sets. Bate proved the following

######  Theorem 8.9.

Let $0<m\leq n$. If $E\subset X$ is purely $m$-unrectifiable with
$\mathcal{H}^{m}(E)<\infty$ and $\Theta^{m}_{\ast}(E,x)>0$ for
$\mathcal{H}^{m}$ almost all $x\in E$, then the set of all
$f\in\text{Lip}_{1}(X,n)$ with $\mathcal{H}^{m}(f(E))=0$ is residual.

Conversely, if $F\subset X$ is $m$-rectifiable with $\mathcal{H}^{m}(F)>0$,
then the set of all $f\in\text{Lip}_{1}(X,n)$ with $\mathcal{H}^{m}(f(F))>0$
is residual.

Bate also showed that the positive lower density assumption is not needed if
$X=\mathbb{R}^{n}$ or $m=1$. It may be that it is never needed, but this
depends on an announced but unpublished result of Csörnyei and Jones.

The first statement of the theorem is the more essential one. To prove it one
needs to approximate well any $f\in\text{Lip}_{1}(X,n)$ with
$g\in\text{Lip}_{1}(X,n)$ for which $\mathcal{L}^{m}(g(E))=0$, or at least
arbitrarily small. There are several very interesting ingredients in this
argument.

The main tools consist of Alberti representations and weak tangent fields.
Recall from Section 5.8 that both play essential roles in the investigation of
differentiability of Lipschitz maps in [ACP05] and [ACP10]. According to
Definition 5.23 a set $E\subset\mathbb{R}^{n}$ has a weak $(m-1)$-dimensional
tangent field if, generically, the tangents of its 1-rectifiable subsets span
at most $(m-1)$-dimensional subspaces. In the metric space $X$ this spanning
condition is interpreted via Lipschitz maps $f:E\to\mathbb{R}^{n},0<m\leq n$.
A little more precisely, a Borel function $\tau:E\to G(n,m-1)$ is a weak
tangent field with respect to $f$ if for 1-rectifiable sets $\gamma\subset E$
at almost all points $x\in\gamma$ the tangent of $f(\gamma)$ at $f(x)$ lies in
$\tau(x)$.

If $E$ is purely $m$-unrectifiable, then by Theorem 8.8 $E$ can have at most
$m-1$ independent Alberti representations, which roughly means that one can
move along $E$ from the points of $E$ in at most to $m-1$ directions. This
leads to the existence of a weak tangent field. The next step in the proof of
Theorem 8.9 is to perturb $f$ slightly to a Lipschitz function $g$ with
$\mathcal{H}^{m}(g(E))$ small. This can be done contracting along directions
orthogonal to the planes $\tau(x)$. As these planes are $(m-1)$-dimensional,
the small measure is achieved.

As an application of Bate’s result David and Le Donne proved in [DL20] that if
a subset of a compact metric space has finite $\mathcal{H}^{m}$ measure,
positive lower density and topological dimension $m$, then it is not purely
$m$-unrectifiable. In Euclidean spaces this follows, without positive lower
density assumption, from the Besicovitch-Federer projection theorem, see
[Fed47b].

There exist easy examples of sets in Euclidean spaces with positive Hausdorff
$m$-measure which project to measure zero in _all_ $m$-planes. This is true in
the Lipschitz setting, too. Vitushkin, Ivanov and Melnikov [VIM63] constructed
a purely $1$-unrectifiable subset $E$ of the plane with $\mathcal{H}^{1}(E)=1$
such that $\mathcal{L}^{1}(f(E))=0$ for every Lipschitz function
$f:E\to\mathbb{R}$, see [Kel95] for a simplification. Modifying an idea of
Konyagin, Ambrosio and Kirchheim [AK00a] showed that for any $m>0$, not
necessarily an integer, there exists a compact metric space $X$ such that
$\mathcal{H}^{m}(X)=1$ and $\mathcal{H}^{m}(f(X))=0$ for every Lipschitz map
$f$ from $X$ into a Euclidean space.

Kun, Maleva and Máthé [KMM05] proved that an analytic subset
$A\subset\mathbb{R}^{n}$ is purely 1-unrectifiable if and only if for any
compact subset $F$ of $A$, $\mathcal{L}^{1}(f(F))=0$ for every local Lipschitz
quotient map $f:F\to\mathbb{R}$. A Lipschitz function $f:F\to\mathbb{R}$ is a
local Lipschitz quotient if there is $c>0$ such that $f(F)\cap
B(f(x),cr)\subset f(B(x,r))$ for all $x\in F,r>0.$

###  8.7. Metric tangents

In Theorem 8.7 we had a characterization of rectifiability in terms of
approximate tangent planes, which are linear subspaces of the bigger Banach
space. Bate [Bat21] proved corresponding results approximating in the Gromov-
Hausdorff distance by $m$-dimensional Banach spaces. Even more, he proved an
analogue of Theorem 5.9 in metric spaces.

First we have to define the relevant concepts. The _Gromov–Hausdorff distance_
$d_{GH}(X_{1},X_{2})$ between metric spaces $(X_{i},d_{i}),i=1,2,$ is the
infimum of those $\varepsilon>0$ for which there exists a metric space $Z$ and
isometric embedddings $Z_{1}$ and $Z_{2}$ of $X_{1}$ and $X_{2}$ into $Z$ such
that the Hausdorff distance $d_{H}(Z_{1},Z_{2})<\varepsilon$. Then
$d_{GH}(X_{1},X_{2})=0$ if and only if the completions of $(X_{1},d_{1})$ and
$(X_{2},d_{2})$ are isometric. If $(X_{i},d_{i}),i=0,1,2,\dots$, are metric
spaces and $x_{i}\in X_{i}$, we say that the sequence $(X_{i},d_{i},x_{i})$ of
pointed metric spaces converges to $(X_{0},d_{0},x_{0})$ if for any $r>0$
there is a sequence $r_{i}\to r$ such that
$d_{GH}(B_{d_{i}}(x_{i},r_{i}),B_{d_{0}}(x_{0},r))\to 0$ as $i\to\infty$. A
_pointed metric measure space_ $(X,d,\mu,x)$ consists of a metric space
$(X,d)$, a locally finite (bounded sets have finite measure) Borel measure
$\mu$ on $X$ and a distinguished point $x\in\operatorname{spt}\mu$.

Let $(X,d)$ be a complete metric space. For $K\geq 1$ let
$\operatorname{biLip}(K)$ be the set of metric spaces
$Y=(\mathbb{R}^{n},\rho)$ such that $\rho$ is $K$-bi-Lipschitz equivalent to
the Euclidean norm. Denote by $\operatorname{biLip}(K)^{\ast}$ the set of
pointed metric measure spaces $(X,d,\mu,x)$ with
$(X,d)\in\operatorname{biLip}(K)$. First a geometric version from [Bat21]:

######  Theorem 8.10.

Let $E\subset$ be $\mathcal{H}^{m}$ measurable with
$\mathcal{H}^{m}(E)<\infty$. Suppose that for $\mathcal{H}^{m}$ almost all
$x\in E,\ \Theta_{\ast}^{m}(E,x)>0$ and there exists $K_{x}\geq 1$ such that
for each $r>0$ there exist $Y_{r}\in$ $\operatorname{biLip}(K_{x})$ and
$E_{r}\subset E\cap B(x,r)$ for which

$\lim_{r\to 0}r^{-m}\mathcal{H}^{m}(E\cap B(x,r)\setminus E_{r})=0$

and

$\lim_{r\to 0}r^{-1}d_{GH}(Y_{r}\cap B(0,r),E_{r})=0.$

Then $E$ is $m$-rectifiable.

This result is new also in Euclidean spaces, because to conclude
rectifiability it allows much more general approximating sets than planes.

Next we define the tangent measure spaces. For a pointed metric measure space
$(X,d,\mu,x)$ let $\operatorname{Tan}(X,d,\mu,x)$ be the set of all pointed
metric measure spaces $(Y,\rho,\nu,y)$ such that there exist $r_{i}>0$ for
which $r_{i}\to 0$ and

(8.7) $(X,r_{i}^{-1}d,\mu(B(x,r_{i}))^{-1}\mu,x)\to(Y,\rho,\nu,y).$

Finding a metric which is suitable to define this convergence is a bit of a
delicate matter, since for the standard metric measure space convergence there
may not be any tangent measure spaces. Standard meaning that one uses the
Hausdorff distance for the spaces in $Z$, as in the definition of $d_{GH}$,
and a distance metrizing the weak convergence for the push-forwards of the
measures. Instead, Bate defined a metric $d_{\ast}$ that only considers the
distance between the measures and disregards the Hausdorff distance between
the embedded metric spaces.

######  Theorem 8.11.

Let $E\subset X$ be $\mathcal{H}^{m}$ measurable with
$\mathcal{H}^{m}(E)<\infty$ and $\Theta_{\ast}^{m}(E,x)>0$ for
$\mathcal{H}^{m}$ almost all $x\in E$. Then the following are equivalent:

  * (1)

$E$ is $m$-rectifiable.

  * (2)

For $\mathcal{H}^{m}$ almost all $x\in E$ there exists an $m$-dimensional
Banach space $(\mathbb{R}^{m},\|\cdot\|_{x})$ such that

$\operatorname{Tan}(X,d,\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E,x)=\\{(\mathbb{R}^{m},\|\cdot\|_{x},\mathcal{L}^{m},0)\\}.$

  * (3)

For $\mathcal{H}^{m}$ almost all $x\in E$ there exists $K_{x}\geq 1$ such that

$\operatorname{Tan}(X,d,\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E,x)\subset\operatorname{biLip}(K_{x})^{\ast}.$

That (1) implies (2) uses Kirchheim’s results in [Kir94] discussed above. The
main and hardest arguments involve the proofs of Theorem 8.10 and the
implication (3) $\Rightarrow$ (1) in Theorem 8.11. They use Bate’s Lipschitz
projection Theorem 8.9. Recall that the proof of the Euclidean Theorem 5.9
also used projections. Another basic tool needed consists of approximation of
$E$ with continuous images of cubes in $\mathbb{R}^{m}$. Its formulation is
rather complicated, so I only give Bate’s own informal description from his
introduction: Roughly speaking, it shows that, under the hypotheses of Theorem
8.10, for $\mathcal{H}^{m}$ almost all $x\in E$, the following is true: for
any $\varepsilon>0$ and any sufficiently small $r>0$, there exists a metric
space $\tilde{E}$ containing $E$ and a continuous (in fact Hölder) map
$\iota:[0,r]^{m}\to\tilde{E}$ such that $\iota|\partial[0,r]^{m}$ is close to
having Lipschitz inverse and
$\mathcal{H}^{m}_{\infty}(\iota([0,r]^{m})\setminus E)<\varepsilon r^{m}$.
Here $\mathcal{H}^{m}_{\infty}$ denotes the $m$-dimensional Hausdorff content.

To finish the proofs Bate showed that as a consequence of Theorem 8.9 $E$ is
rectifiable if all of its subsets posses this approximation property. To get
some (not quite correct) idea how Theorem 8.9 is applied imagine that for some
$x\in E,r>0,$ with $\mathcal{H}^{m}(E\cap B(x,r))\sim r^{m}$ we would have
$\iota([0,r]^{m})=E\cap B(x,r)$ for $\iota$ as above. Extending, if possible,
$(\iota|\partial[0,r]^{m})^{-1}$ to a Lipschitz map $f$ from $E\cap B(x,r)$
onto $[0,r]^{m}$ we would then have $\mathcal{L}^{m}(f(E\cap B(x,r)))\sim
r^{m}$. By some topology the same would be true for small perturbations of
$f$. If $E$ were purely unrectifiable this would contradict Theorem 8.9.

In addition, for the proof of Theorem 8.11 Bate extensively developed the
properties of the metric $d_{\ast}$ and the metric tangent measure spaces.

###  8.8. Menger curvature

The curvature of a smooth plane curve $C$ at a point $p$ can be defined as the
limit of the reciprocals of the largest radii of the circles approaching and
only touching $C$ at $p$. The obstruction for not being able to make the radii
bigger comes from triples of points on the curve near $p$. So the curvature at
$p$ is also the limit of $c(x,y,z)$, where $x,y,z\in C$ go to $p$ and
$c(x,y,z)=1/R$ with $R$ equal to the radius of the circle passing through
$x,y$ and $z$. If we take three points in a metric space $X$ there is an
isometric triple in the plane. Thus we can define the _Menger curvature_
$c(x,y,z)$ for any $x,y,z\in X$. It is not necessary to pass through the
plane, because there is a formula which gives $c(x,y,z)$ in terms of the three
distances, see [Men30],[Hah05] or [Hah08].

Menger introduced his curvature in [Men30] as a part of extensive studies of
geometry of metric spaces. In particular, he characterized in terms of this
curvature the metric arcs which are isometric to Euclidean line segments. An
interested reader might also wish to have a look at the book [BK70].

We already discussed this topic in Section 4.4. In [Hah05] Hahlomaa proved a
variant of Jones’s theorem 4.16 in general metric spaces. Now $\beta(F)$ is
defined in terms of the Menger curvature. In [Hah08] he generalized the David-
Léger theorem 4.18 to metric spaces:

######  Theorem 8.12.

If $\mathcal{H}^{1}(X)<\infty$ and

$\int_{X}\int_{X}\int_{X}c(x,y,z)^{2}\,d\mathcal{H}^{1}x\,d\mathcal{H}^{1}y\,d\mathcal{H}^{1}z<\infty,$

then $X$ is 1-rectifiable

The proof follows similar lines as David’s original proof in the plane, but
much also has to be changed.

##  9\. Heisenberg and Carnot groups

For an introduction to Heisenberg and Carnot groups, see for example [CDPT07]
or [BLU07]. Nice surveys related to rectifiability are given by Serapioni
[Ser08] and by Serra Cassano [Ser16]. Both Euclidean spaces and Heisenberg
groups are special cases of Carnot groups. Except for the Euclidean case they
are non-Abelian and have a similar structure as Heisenberg groups but instead
of two levels there can be any finite number of levels with different dilation
exponents. I do not discuss them explicitly, but I make some comments on them
along the way.

###  9.1. The Heisenberg group $\mathbb{H}^{n}$

Heisenberg group $\mathbb{H}^{n}$ is $\mathbb{R}^{2n+1}$ as a set but with a
different metric and non-Abelian group structure. We denote the points of
$\mathbb{H}^{n}$ by
$p=(z,t)=(x,y,t),z=(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n},t\in\mathbb{R},$
and define the non-Abelian group operation by

$p\cdot p^{\prime}=(z+z^{\prime},t+t^{\prime}+\omega(p,p^{\prime})),$

where

$\omega(p,p^{\prime})=\omega(z,z^{\prime})=-2\sum_{i=1}^{n}(x_{i}y^{\prime}_{i}-y_{i}x^{\prime}_{i}).$

We shall use the Koranyi metric $d$ given by

(9.1)
$d(p,p^{\prime})=(|z-z^{\prime}|^{4}+(t-t^{\prime}-\omega(z,z^{\prime}))^{2})^{\frac{1}{4}}.$

Then the ball of radius $r$ at the origin

$B(0,r)=\\{p:(|z|^{4}+t^{2})^{\frac{1}{4}}\leq r\\}$

is like a cylinder of width $2r$ and height $2r^{2}$, so
$\mathcal{L}^{2n+1}(B(0,r))\sim r^{2n+2}$. The ball $B(p,r)$ is the image of
$B(0,r)$ under the left translation $\tau_{p};\tau_{p}(q)=p\cdot q.$ The
metric is left invariant, that is, $\tau_{p}$ is an isometry. Moreover,
$\mathcal{L}^{2n+1}=c(n)\mathcal{H}_{d}^{2n+2}$ is a Haar measure of the
group, so $\mathcal{L}^{2n+1}(B(p,r))=c^{\prime}(n)r^{2n+2}$. In particular,
we see that the Hausdorff dimension of $\mathbb{H}^{n}$ is $2n+2$. We also
define the dilations $\delta_{r},r>0$, by

$\delta_{r}(z,t)=(rz,r^{2}t).$

Then $d(\delta_{r}(p),\delta_{r}(q))=rd(p,q)$.

In $\mathbb{H}^{n}$ the metric on all _horizontal lines_
$L_{e}:=\\{te,0):t\in\mathbb{R}\\},e\in S^{2n-1}$, and their left translates
is Euclidean, so their subsets are $1$-rectifiable. There are many other
1-rectifiable sets; any two points of $\mathbb{H}^{n}$ can be joined with a
rectifiable curve. This fact also leads to the geodesic metric which is
equivalent to the one we have chosen.

When $n>1$ there are many $m$-rectifiable sets for $m=1,\dots,n$, in the same
way. Define the horizontal plane $\mathbb{H}=\\{t=0\\}$ identified with
$\mathbb{R}^{2n}$ and define the space of _horizontal subgroups_

$G_{h}(2n,m)=\\{V\in G(2n,m):\omega(p,q)=0\ \text{for all}\ p,q\in V\\}.$

The metric on each $V\in G_{h}(2n,m)$ is Euclidean, so they are nice
$m$-rectifiable sets. The unitary transformations act transitively on
$G_{h}(2n,m)$ and lead to an invariant measure $\mu_{n,m}$. Not all linear
subspaces of $\mathbb{H}$ of dimension $1<m\leq n$ are subgroups, and none of
dimensions bigger than $n$.

The vertical subgroups of linear dimension $m-1,1\leq m-1\leq 2n$, and
Hausdorff dimension $m$ are the vertical planes $V\times\mathbb{R},V\in
G(2n,m-2)$. They together with the horizontal subgroups are the only non-
trivial homogeneous subgroups of $\mathbb{H}^{n}$, that is, they are closed
and invariant under the dilations. But the narrower collection of
complementary subgroups will be more relevant for us.

The subgroups $V\in G_{h}(2n,m)$ and $W=V^{\perp}$ are complementary: $V\cap
W=\\{0\\}$ and they span $\mathbb{H}^{n}$ in the sense that $V\cdot
W=\mathbb{H}^{n}$. In particular, any $p\in\mathbb{H}^{n}$ has a unique
decomposition

(9.2) $p=p_{V}\cdot p_{W},p_{V}\in V,p_{W}\in W.$

Define $\mathcal{G}(\mathbb{H}^{n},m)$ as the set of $V\in G_{h}(2n,m)$, when
$1\leq m\leq n$, and $V\times\mathbb{R},V\in G(2n,m-2),V^{\perp}\in
G_{h}(2n,2n+2-m)$, when $n+2\leq m\leq 2n+1$. They are the homogeneous
subgroups of Hausdorff dimension $m$ which admit a complement in the sense of
(9.2).

Also the vertical subgroups in $\mathcal{G}(\mathbb{H}^{n},m)$ will be
$m$-rectifiable, but we have to change the definition, as we shall soon do.

###  9.2. Some analytic tools in Heisenberg and Carnot groups

The analytic structure of $\mathbb{H}^{n}$ is generated by the following
vector fields:

$X_{i}=\partial_{x_{i}}+2y_{i}\partial_{t},Y_{i}=\partial_{y_{i}}-2x_{i}\partial_{t},i=1,\dots
n,\nabla_{H}=(X_{1},\dots,X_{n},Y_{1},\dots,Y_{n}).$

These vector fields at the origin span the horizontal plane $\mathbb{H}$ and
at $p$ the plane $\tau_{p}(\mathbb{H})$. The tangent vectors of rectifiable
curves are spanned by them. We say that a continuous function $u$ on an open
subset $U$ of $\mathbb{H}^{n}$ belongs to $C^{1}_{H}$ if $\nabla_{H}u$ is
continuous.

There is a very general Rademacher theorem due to Pansu [Pan89]. It says that
Lipschitz maps $f$ between Carnot groups are almost everywhere differentiable.
Now the derivative $d_{H}f(p)$ is a homogeneous (it commutes with the
dilations) homomorphism between the groups such that

$\lim_{q\to p}\frac{d(f(p)^{-1}\cdot f(q),d_{H}f(p)(p^{-1}\cdot
q))}{d(p,q)}=0.$

Franchi, Serapioni and Serra Cassano proved an implicit function theorem for
real-valued functions on subsets of $\mathbb{H}^{n}$ and a Whitney extension
theorem for Euclidean valued (or values in the horizontal subbundle) functions
on subsets of $\mathbb{H}^{n}$, see [FSS01] and [FSS03]. But a Whitney
extension theorem for $\mathbb{H}^{n}$ valued functions is missing, and hence
also approximation of Lipschitz maps with differentiable maps.

There also are several versions of the area formula, see [JNV20] and [Vit20]
and the references given there.

###  9.3. Definitions of rectifiability

There are several natural ways to define rectifiable sets in Heisenberg and
Carnot groups. Some of them are known to be equivalent in some cases, for some
the relations are unknown. We begin with the definition we have already used
in metric spaces:

######  Definition 9.1.

Let $1\leq m\leq n$. A set $E\subset\mathbb{H}^{n}$ is $m$-rectifiable if
there are Lipschitz maps
$f_{i}:A_{i}\to\mathbb{H}^{n},A_{i}\subset\mathbb{R}^{m},i=1,2,\dots,$ such
that $\mathcal{H}_{d}^{m}(E\setminus\bigcup_{i=1}^{\infty}f_{i}(A_{i}))=0$.

The corresponding notion for measures is defined as in 5.2. I only gave this
definition for low dimensional sets for a good reason: there are no non-
trivial $m$-rectifiable subsets of $\mathbb{H}^{n}$ for $m>n$ by the following
result. In $\mathbb{H}^{1}$ it was proved by Ambrosio and Kirchheim in
[AK00a], a very general statement in Carnot groups is due to Magnani [Mag04].

######  Theorem 9.2.

If $m=n+1,\dots,2n+2$ and $f:A\to\mathbb{H}^{n},A\subset\mathbb{R}^{m},$ is
Lipschitz, then $\mathcal{H}_{d}^{m}(f(A))=0$. In particular, $\mathbb{H}^{n}$
is purely $(2n+2)$-unrectifiable.

To get an idea why this is true, suppose that $n=1$ and $f=(f_{1},f_{2})$ maps
$\mathbb{R}^{3}$ into the vertical plane $W=\\{y=0\\}$. The metric in $W$ is
given by the ’norm’ $(|x|^{4}+t^{2})^{1/4}$. Then
$|f_{2}(u)-f_{2}(v)|\lesssim|u-v|^{2}$, so $f_{2}$ is constant and
$\mathcal{H}_{d}^{3}(f(\mathbb{R}^{3}))=0$ follows. Of course $f$ need not map
into a vertical plane, but its Pansu differentials $d_{H}f(u)$ must, because
they are group homomorphisms and thus $d_{H}f(u)(\mathbb{R}^{3})$ is an
Abelian subgroup. From this one can argue similarly.

Are there any non-trivial $m$-rectifiable subsets when $m>n$? No with our
present definition, as we saw above. But there are many with an alternate
definition. Recall that Euclidean rectifiable sets can be defined using level
sets of regular functions. Based on this we first define _regular surfaces_
$S$. If $1\leq m\leq n$ this means that $S$ is locally the image of an open
subset of $\mathbb{R}^{m}$ under an injective continuously differentiable (in
Pansu’s sense) map with injective derivative. If $n+2\leq m\leq 2n+1$ we say
that $S$ is regular if for every $p\in S$ there are an open set $U$ with $p\in
U$ and a function $u:U\to\mathbb{R}^{2n+2-m},$ whose coordinate functions
belong to $C^{1}_{H}$ such that $S\cap U=\\{q\in U:u(q)=0\\}$ and for $q\in U$
the Pansu derivative $d_{H}u(q)$ is surjective. These surfaces are also
regular in the sense that they have tangent subgroups, in the first case via
Pansu derivative and in the second via the kernel of $d_{H}f(p)$. Notice that
when $m\geq n+2$, the topological dimension of $S$ is $m-1$.

######  Definition 9.3.

Let $m=1,\dots,n,n+2,\dots,2n+1$. A set $E\subset\mathbb{H}^{n}$ is
$(m,\mathbb{H})$-rectifiable if there are $m$-regular surfaces
$S_{i},i=1,2,\dots,$ such that
$\mathcal{H}_{d}^{m}(E\setminus\bigcup_{i=1}^{\infty}S_{i})=0$.

This definition in codimension 1 is due to Franchi, Serapioni and Serra
Cassano [FSS01]. They introduced this concept and used it to develop De
Giorgi’s theory of sets of finite perimeter in Heisenberg groups. We shall
return to this in Section 13.4. For general dimensions and Carnot groups, see
[Mag06] and [FSS07].

As an example, consider the vertical plane $W=\\{y=0\\}\subset\mathbb{H}^{1}$.
Then $u(x,y,t)=y$ belongs to $C^{1}_{H}$ with $Y_{1}u\not=0$ on $W$. Hence $W$
is a regular surface and a $(3,\mathbb{H})$-rectifiable set. As an another
example the horizontal plane $\mathbb{H}$ in $\mathbb{H}^{n}$ is not a regular
surface, because it has a singularity, a characteristic point, at $0$. But it
is regular outside $0$, and hence a $(2n+1,\mathbb{H})$-rectifiable set.

In fact, all $C^{1}$ smooth Euclidean $m$-dimensional, $n+1\leq m\leq 2n$,
surfaces are $(m+1,\mathbb{H})$-rectifiable. They have positive and locally
finite $\mathcal{H}^{m+1}_{d}$ measure, they are regular outside the set of
characteristic points, which has $\mathcal{H}_{d}^{m+1}$ measure zero by
results of Balogh [Bal03] for $m=2n$ and Magnani [Mag06] for general $m$. A
point $p$ is a characteristic point of a hypersurface $S$ if the tangent space
(in Heisenberg sense) at $p$ is spanned by the horizontal vector fields
$X_{i},Y_{j}$. More generally, the Euclidean $m$-rectifiable sets, $n+1\leq
m\leq 2n$, are $(m+1,\mathbb{H})$-rectifiable since $\mathcal{H}^{m}(A)=0$
implies $\mathcal{H}_{d}^{m+1}(A)=0$ for $A\subset\mathbb{H}^{n}$. The
converse is false, because $(2n+1,\mathbb{H})$-rectifiable sets can have
Euclidean Hausdorff dimension bigger than $2n$, see [KS04] for an example in
$\mathbb{H}^{1}$ of Hausdorff dimension is $2.5$. General comparisions of
Euclidean and Carnot Hausdorff measures can be found in [BTW09].

When $1\leq m\leq n$, clearly $(m,\mathbb{H})$-rectifiable sets are
$m$-rectifiable, but it is not known if the converse holds. Although Lipschitz
maps are almost everywhere differentiable by Pansu’s theorem, one would need
something like Whitney’s extension theorem to go to $C_{H}^{1}$ from
Lipschitz. This is not known for Heisenberg valued maps.

Intrinsic differentiable graphs and intrinsic Lipschitz graphs have recently
been investigated intensively. They provide another definition for
rectifiability. In Euclidean spaces cones were used to characterize
rectifiability in terms of the approximate tangent planes, and they are
directly connected to Lipschitz maps as in the argument preceding Theorem 4.3.
In Heisenberg groups the situation is more complicated but we can define a
class of Lipschitz maps geometrically in terms of cones. This was done by
Franchi, Serapioni and Serra Cassano in [FSS06] in Heisenberg groups and in
[FS16] in general Carnot groups.

For a homogeneous subgroup $G$ define the cone

(9.3) $\begin{split}X(p,G,s)&=\\{q\in\mathbb{H}^{n}:d(p^{-1}\cdot
q,G)<sd(p,q)\\}\\\ &=p\\{q\in\mathbb{H}^{n}:d(q,G)<sd(q,0)\\}.\end{split}$

Geometrically these cones look rather different from the Euclidean cones.

Let $V$ and $W$ be complementary subgroups of $\mathbb{H}^{n}$; $V\cap
W=\\{0\\}$ and $V\cdot W=\mathbb{H}^{n}$, with $V$ horizontal and $W$
vertical. For much that follows they could also be complementary homogeneous
subgroups of a general Carnot group. We say that $S\subset\mathbb{H}^{n}$ is a
(vertical) _intrinsic Lipschitz graph_ if there is $s>0$ such that for all
$p\in S$,

$S\cap X(p,V,s)=\\{p\\}.$

We say that a function $f:A\to V,A\subset W,$ is (vertical) _intrinsic
Lipschitz_ if $gr(f):=\\{p\cdot f(p):p\in A\\}$ is an intrinsic Lipschitz
graph. In Euclidean spaces this just means that $f$ is Lipschitz. But now the
intrinsic Lipschitz functions need not be Lipschitz in the metric sense, and
vice versa, see [AS09, Example 3.21].

Changing the roles of $V$ and $W$ we get horizontal intrinsic Lipschitz graphs
and functions. Arena and Serapioni proved in [AS09, Proposition 3.20] that a
horizontal function $f$ is intrinsic Lipschitz if and only if $p\mapsto p\cdot
f(p)$ is metric Lipschitz.

Intrinsic Lipschitz graphs over the whole planes have positive and locally
finite Hausdorff measure (in the appropriate dimension), they are even AD-
regular. This holds in general Carnot groups, see [FS16, Theorem 3.9].

######  Definition 9.4.

Let $m=1,\dots,n,n+2,\dots,2n+1$. A set $E\subset\mathbb{H}^{n}$ is
$(m,\mathcal{H}_{intL})$-rectifiable if there are $m$-dimensional (in terms of
Hausdorff dimension) intrinsic Lipschitz graphs $S_{i},i=1,2,\dots,$ such that
$\mathcal{H}_{d}^{m}(E\setminus\bigcup_{i=1}^{\infty}S_{i})=0$.

There is a slightly more complicated notion of intrinsic differentiable
functions and graphs, see [FSS06], [AS09]. Arena and Serapioni proved in
[AS09, Theorem 4.2] that regular surfaces and intrinsic differentiable graphs
are the same locally. So the rectifiable sets defined in terms of intrinsic
differentiable graphs are the same as $(m,\mathbb{H})$-rectifiable sets. To
get to intrinsic Lipschitz one would need a Rademacher type theorem for
intrinsic Lipschitz functions. Such a theorem was proved by Franchi, Serapioni
and Serra Cassano [FSS11] in the codimension 1 case $(m=2n+1)$ in Heisenberg
groups. For this they used their results on sets of finite perimeter. Franchi,
Marchi and Serapioni [FMS14] extended this to some Carnot groups. In [Vit20]
Vittone proved that intrinsic Lipschitz functions $f:A\to V,A\subset W,W$
vertical, are almost everywhere intrinsic differentiable. From this using
Whitney type arguments he further showed that intrinsic Lipschitz graphs can
be approximated in measure by regular surfaces. This leads to, see [Vit20,
Corollary 7.4],

######  Theorem 9.5.

Let $n+2\leq m\leq 2n+1$ and let $E\subset\mathbb{H}^{n}$ be
$\mathcal{H}_{d}^{m}$ measurable with $\mathcal{H}_{d}^{m}(E)<\infty$. Then
$E$ is $(m,\mathbb{H})$-rectifiable if and only if $E$ is
$(m,\mathcal{H}_{intL})$-rectifiable.

There are several ingredients of independent interest in Vittone’s proof. He
introduced an alternative equivalent definition. According to it $S$ is a
vertical intrinsic Lipschitz $W$-graph if and only if there exist $\delta>0$
and a Lipschitz map $g:\mathbb{H}^{n}\to W^{\perp}$ such that $g(x)=0$ on $S$
and $g$ satisfies the ellipticity type condition $(g(p\cdot v)-g(p))\cdot
v\geq\delta|v|^{2}$ for $v\in W^{\perp},p\in\mathbb{H}^{n}$ (the second
$\cdot$ is the inner product). In the main part of the argument he used
currents (in Heisenberg sense) to show that the blow-ups of $S$ converge to a
vertical plane locally uniformly, which is the content of Rademacher’s theorem
in this setting.

Counter-examples to Rademacher’s theorem for intrinsic Lipschitz graphs in
some Carnot groups were given by Julia, Nicolussi Golo and Vittone in
[JNV21a]. In [JNV21b] they proved the almost everywhere tangential
differentiability of Euclidean valued functions on $C^{1}_{\mathbb{H}}$
submanifolds of $\mathbb{H}^{n}$ yielding on $\mathbb{H}$-rectifiable sets a
Lusin type approximation of Lipschitz functions and a coarea formula.

As said before, there are no Lipschitz maps from Euclidean spaces to
parametrize higher dimensional non-trivial sets. But perhaps one could
consider Lipschitz maps from other spaces. For instance, in $\mathbb{H}^{1}$
could 3-dimensional rectifiable sets be defined using Lipschitz maps from a
fixed vertical plane? Although an intrinsic Lipschitz map need not be metric
Lipschitz, maybe some other Lipschitz map could be used to parametrize an
intrinsic Lipschitz graph? Such a definition of rectifiability and its
consequences, in the generality of Carnot groups, was studied by Pauls [Pau04]
and by Cole and Pauls [CP06]. Let $(G_{1},d_{1})$ and $(G_{2},d_{2})$ be
Carnot groups and $G$ a subgroup of $G_{1}$ with Hausdorff dimension $m$. Let
us say that $E\subset G_{2}$ is $G$-rectifiable if up to
$\mathcal{H}^{m}_{d_{2}}$ measure zero it can be covered with countably many
Lipschitz images of subsets of $G$. Except for the cases where $G_{1}$ is
Euclidean, not much is known about the relation of $G$-rectifiability to other
concepts of rectifiability, but some partial information exists.

Fix a vertical subgroup $W_{n}\in\mathcal{G}(\mathbb{H}^{n},2n+1)$. It does
not matter which, they all are isometric. Recall that $C^{1}$ Euclidean
hypersurfaces in $\mathbb{H}^{n}$ are $(2n+1,\mathbb{H})$-rectifiable. Cole
and Pauls proved in [CP06] that the $C^{1}$ hypersurfaces in $\mathbb{H}^{1}$
are $W_{1}$-rectifiable, too. Di Donato, Fässler and Orponen [DFO19] proved
that the $C^{1,\alpha},\alpha>0,$ hypersurfaces in $\mathbb{H}^{n}$ are
$W_{n}$-rectifiable, moreover, they have big pieces of bi-Lipschitz images of
$W_{n}$. Earlier the rectifiability was proved by Antonelli and Le Donne
[AL20] for $C^{\infty}$ surfaces.

Antonelli and Le Donne showed in [AL20] that there exists a Carnot group
containing a $C^{\infty}$ hypersurface without characteristic points which is
not $G$-rectifiable for any Carnot group $G$, but it is rectifiable in the
sense of Franchi, Serapioni and Serra Cassano.

The papers [DFO19] and [AL20] show more than rectifiability, instead of
Lipschitz maps they use bi-Lipschitz maps. Antonelli and Le Donne discuss more
generally definitions of rectifiability based on bi-Lipschitz maps. Bigolin
and Vittone [BV10] give a counter-example concerning bi-Lipschitz
parametrizations. Orponen [Orp20] shows that bi-Lipschitz images of $W_{1}$ in
$\mathbb{H}^{1}$ admit a corona decomposition by intrinsic bi-Lipschitz
graphs.

In [LY19] Le Donne and Young proved that a sub-Riemannian manifold which has a
Carnot group $G$ as a constant Gromov-Hausdorff limit, see Section 8.7, is
$G$-rectifiable with bi-Lipschitz parametrizations. The converse also holds by
Pansu’s Rademacher theorem.

A possibly weaker notion of rectifiability with cones was proposed by Don, Le
Donne, Moisala and Vittone in [DLMV19]. They applied it to finite perimeter
sets in general Carnot groups.

###  9.4. Rectifiable sets and tangent subgroups

Next we shall give a characterization of rectifiability in terms of
approximate tangent subgroups and tangent measures. They are defined as in the
Euclidean case but using the Heisenberg structure. Recall the definition of
the cone $X(p,G,s)$ from (9.3).

We say that $G\in\mathcal{G}(\mathbb{H}^{n},m)$ is an _approximate tangent
subgroup_ of $E\subset\mathbb{H}^{n}$ at a point $p\in\mathbb{H}^{n}$ if
$\Theta^{\ast m}(E,p)>0$ and for all $0<s<1$,

$\lim_{r\to 0}r^{-m}\mathcal{H}_{d}^{m}(E\cap B(p,r)\setminus X(p,G,s))=0.$

To define the tangent measures we now set

$T_{a,r}(p)=\delta_{1/r}(a^{-1}\cdot p),\ p,a\in\mathbb{H}^{n},r>0.$

Then, as before, if $\mu$ is a Radon measure on $\mathbb{H}^{n}$, a non-zero
Radon measure $\nu$ is called a _tangent measure_ of $\mu$ at
$a\in\mathbb{H}^{n}$ if there are sequences $(c_{i})$ and $(r_{i})$ of
positive numbers such that $r_{i}\to 0$ and $c_{i}T_{a,r_{i}\\#}\mu\to\nu$
weakly. We denote the set of tangent measures of $\mu$ at $a$ by
$\operatorname{Tan}(\mu,a)$. The following theorems were proven in [MSS10]:

######  Theorem 9.6.

Let $1\leq m\leq n$ and let $E\subset\mathbb{H}^{n}$ be $\mathcal{H}_{d}^{m}$
measurable with $\mathcal{H}_{d}^{m}(E)<\infty$. Then the following are
equivalent:

  * (1)

$E$ is $m$-rectifiable.

  * (2)

$E$ has an approximate tangent subgroup
$G_{p}\in\mathcal{G}(\mathbb{H}^{n},m)$ at $\mathcal{H}_{d}^{m}$ almost all
$p\in E$.

  * (3)

For $\mathcal{H}_{d}^{m}$ almost all $p\in E$ there is
$G_{p}\in\mathcal{G}(\mathbb{H}^{n},m)$ such that

$\operatorname{Tan}(\mathcal{H}_{d}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E,p)=\\{c\mathcal{H}_{d}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}G_{p}:0<c<\infty\\}.$

  * (4)

For $\mathcal{H}_{d}^{m}$ almost all $p\in E$
$\mathcal{H}_{d}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ has a unique tangent measure at
$p$.

######  Theorem 9.7.

Let $n+2\leq m\leq 2n+1$ and let $E\subset\mathbb{H}^{n}$ be
$\mathcal{H}_{d}^{m}$ measurable with $\mathcal{H}_{d}^{m}(E)<\infty$. Suppose
also that $\Theta_{\ast}^{m}(E,p)>0$ for $\mathcal{H}_{d}^{m}$ almost all
$p\in E$. Then the following are equivalent:

  * (1)

$E$ is $(m,\mathbb{H})$-rectifiable.

  * (2)

$E$ has an approximate tangent subgroup
$G_{p}\in\mathcal{G}(\mathbb{H}^{n},m)$ at $\mathcal{H}_{d}^{m}$ almost all
$p\in E$.

  * (3)

For $\mathcal{H}_{d}^{m}$ almost all $p\in E$ there is
$G_{p}\in\mathcal{G}(\mathbb{H}^{n},m)$ such that

$\operatorname{Tan}(\mathcal{H}_{d}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E,p)=\\{c\mathcal{H}_{d}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}G_{p}:0<c<\infty\\}.$

  * (4)

For $\mathcal{H}_{d}^{m}$ almost all $p\in E$
$\mathcal{H}_{d}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ has a unique tangent measure at
$p$.

That (4) implies (3) follows from Theorem 5.7. There the uniqueness means
uniqueness up to multiplication by positive constants.

It is not known if the positive lower density assumption in Theorem 9.7 is
needed. Also it is not known if we can replace $m$-rectifiable by
$(m,\mathbb{H})$-rectifiable in Theorem 9.6(1). As mentioned before, the
problem here is the lack of Whitney’s extension theorem for $\mathbb{H}^{n}$
valued functions.

A few words about the proofs. The case $m\leq n$ is proved by arguments
similar to those used in the Euclidean case. But the cones now are harder to
deal with if the lower density is 0. This is overcome by proving first that
the existence of tangent subgroups implies positive lower density. When $m\geq
n+2$ the key for proving that approximate tangents imply rectifiability is the
Whitney extension theorem for $\mathbb{R}^{k}$ valued maps of Franchi,
Serapioni and Serra Cassano, [FSS01].

Theorem 9.6 was generalized by Idu, Magnani and Maiale [IMM20] to general
homogeneous groups, which need not be Carnot groups.

###  9.5. Densities

For low dimensional sets ($m\leq n$) we can apply Kirchheim’s theorem 8.6 to
conclude that $m$-rectifiable subsets of $\mathbb{H}^{n}$ with finite measure
have density 1 almost everywhere. The converse is not known, except for $m=1$,
see Theorem 9.12.

Recently there has been a remarkable progress on densities by Merlo in the two
papers [Mer19] and [Mer20]. He proved Preiss’s density characterization of
rectifiability for codimension 1 subsets of $\mathbb{H}^{n}$:

######  Theorem 9.8.

Let $\mu\in\mathcal{M}(\mathbb{H}^{n})$ be such that the positive and finite
limit
$\lim_{r\to 0}r^{-2n-1}\mu(B(p,r))$ exists for $\mu$ almost all
$p\in\mathbb{H}^{n}$. Then $\mu$ is $(2n+1,\mathbb{H})$-rectifiable.

As a corollary we have

######  Theorem 9.9.

Let $E\subset\mathbb{H}^{n}$ be $\mathcal{H}_{d}^{2n+1}$ measurable with
$\mathcal{H}_{d}^{2n+1}(E)<\infty$. Then $E$ is
$(2n+1,\mathbb{H})$-rectifiable if and only if $\Theta^{2n+1}(E,p)$ exists for
$\mathcal{H}_{d}^{2n+1}$ almost all $p\in E$.

The fact that the density exists for rectifiable sets follows from [JNV20,
Corollary 3.6]. Julia, Nicolussi Golo and Vittone proved this for sets of
general dimensions in arbitrary Carnot groups. They derive the result from an
area formula. See also, [AM22].

Often the density of high dimensional Heisenberg rectifiable sets is strictly
less than $1$, but for many metrics the density of the spherical Hausdorff
measure is 1 almost everywhere, while for some others it is strictly less than
$1$, see [Mag17].

As in the Euclidean case, the proof of Theorem 9.8 splits into two parts. In
the first part [Mer19] Merlo proves that the tangent measures are flat:

######  Theorem 9.10.

Let $\mu\in\mathcal{M}(\mathbb{H}^{n})$ be such that the positive and finite
limit
$\lim_{r\to 0}r^{-2n-1}\mu(B(p,r))$ exists for $\mu$ almost all
$p\in\mathbb{H}^{n}$. Then for $\mu$ almost all $p\in\mathbb{H}^{n}$,

$\operatorname{Tan}(\mu,p)\subset\\{c\mathcal{H}^{2n+1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}V:V\in\mathcal{G}(\mathbb{H}^{n},2n+1),0<c<\infty\\}.$

Again an easy argument shows that the tangent measures $\nu$ are uniform;

$\nu(B(p,r))=cr^{2n+1}\ \text{for}\ p\in\operatorname{spt}\nu,\ r>0.$

Using moments in the spirit of [Pre87] and [KP87] Merlo proves that their
supports are contained in quadratic conical surfaces. Then disconnectedness is
verified saying that vertical flat uniform measures are separated from the
others. As in the Euclidean case this implies that typically only vertical
flat uniform tangent measures exist. Although many of the arguments run as in
the Euclidean case, many essential changes have to be made, in particular,
replacing algebraic computations by geometric arguments. I find it surprising
that Preiss’s proof can be followed at all, since it used heavily the fact
that the metric is given by an inner product. Merlo overcomes this by cleverly
applying the polarization

$V(p,q)=(\|p\|^{4}+\|q\|^{4}-\|p^{-1}\cdot q\|^{4})/2.$

It can be written, with a rather complicated formula, in terms of the inner
product in $\mathbb{R}^{2n}$ involving for $p=(w,s),q=(z,t)$ all the
coordinates $w,z,s,t$ in various combinations.

In the second paper [Mer20] Merlo proves that the flatness of the tangent
measures implies rectifiability. This is the case $m=2n+1$ in the following
theorem:

######  Theorem 9.11.

Let $m=1,\dots,n$ or $m=2n+1$ and let $\mu\in\mathcal{M}(\mathbb{H}^{n})$ be
such that for $\mu$ almost all $p\in\mathbb{H}^{n}$,
$0<\Theta_{\ast}^{m}(\mu,p)\leq\Theta^{\ast m}(\mu,p)<\infty$ and

$\operatorname{Tan}(\mu,p)\subset\\{c\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}V:V\in\mathcal{G}(\mathbb{H}^{n},m),0<c<\infty\\}.$

Then $\mu$ is $(m,\mathbb{H})$-rectifiable.

The proof for $m=2n+1$ is very complicated. It is given in general Carnot
groups. As for Theorem 5.9 one of the ideas is to use big projections implied,
but not easily, by the assumptions. For this some ideas of David and Semmes
[DS93] in uniform rectifiability help. But now projections mean the splitting
projections $p\mapsto p_{W}$, recall (9.2), which are much more complicated to
handle than the Euclidean projections. They are not even Lipschitz. Big
projection on $W$ improves the information given by the assumption: the
approximating planes have to be rather close to $W$. Then one can proceed to
show that some intrinsic Lipschitz graph intersects $\operatorname{spt}\mu$ in
positive measure.

The proof for low dimensions was given by Antonelli and Merlo in [AM22]. It
follows similar patterns, but now the projections are Lipschitz, which helps.
On the other hand, serious difficulties are caused by the fact that they prove
this in quite general Carnot settings; it is assumed that the tangent
subgroups admit at least one normal complementary subgroup.

There is also interesting information about the uniform measures in
$\mathbb{H}^{1}$: they are just the flat measures on horizontal lines,
$t$-axis and vertical planes. The first two cases are due to Chousionis,
Magnani and Tyson [CMT20], the third to Merlo [Mer19], using also some results
from [CMT20]. It seems to be open whether there can be non-flat uniform
measures in higher dimensional Heisenberg groups.

Combining the case $m=1$ with Theorem 9.11 we get the Besicovitch-Preiss
theorem for one-dimensional measures in $\mathbb{H}^{1}$:

######  Theorem 9.12.

Let $\mu\in\mathcal{M}(\mathbb{H}^{1})$ be such that the positive and finite
limit
$\lim_{r\to 0}r^{-1}\mu(B(p,r))$ exists for $\mu$ almost all
$p\in\mathbb{H}^{n}$. Then $\mu$ is $1$-rectifiable.

In Theorems 9.8 and 9.12 it is important that we use the Koranyi metric $d$.
We have been using it all the time, but everywhere else any equivalent metric
would be fine. For other, essentially different, metrics the validity of this
theorem is unknown. Recall the discussion about the Euclidean $l^{\infty}$
norm in Section 5.4.

In [AM20], [AM21] and [AM22] Antonelli and Merlo base in arbitrary Carnot
groups the definition of the rectifiability of a measure on the condition that
almost everywhere the tangent measures are flat or that there is a unique flat
tangent measure. Flat means a constant multiple of the Hausdorff measure on a
homogeneous subgroup that has a complementary subgroup. Compare with Theorems
9.6, 9.7, 9.10 and 9.11. Antonelli and Merlo make a systematic study of such
measures with many interesting results generalizing a lot of earlier
rectifiability theory. In addition to the analogue of Theorem 9.11, they prove
a characterization with intrinsic Lipschitz and differentiable graphs, a very
general area formula, and rectifiability of level sets of Lipschitz maps.

Let us still have a look at the full group $\mathbb{H}^{n}$ from the point of
view of general metric spaces. We know that it has positive and locally finite
$\mathcal{H}^{2n+2}$ measure and it is purely unrectifiable. The density of
the spherical measure $\Theta^{2n+2}(\mathcal{S}^{2n+2},p)=1$ but
$\Theta^{2n+2}(\mathcal{H}^{2n+2},p)=c(n)<1$ for every $p\in\mathbb{H}^{n}$,
see [Mag17]. This difference comes from the fact that balls are not extremals
for the isodiametric inequality in $\mathbb{H}^{n}$, see [Rig11]. Recall that
it is open in general metric spaces if the Hausdorff density one implies
rectifiability. So the Heisenberg group does not give a counter-example.

###  9.6. Projections

Recall from (9.2) that if the subgroups $V\in G_{h}(2n,m)$ and $W=V^{\perp}$
are complementary, then any $p\in\mathbb{H}^{n}$ has a unique decomposition
$p=p_{V}\cdot p_{W},p_{V}\in V,p_{W}\in W$. So we have the projections
$P_{V}(p)=p_{V}$ and $Q_{W}(p)=p_{W}$. Here $V$ is horizontal and $P_{V}$ is
just the standard orthogonal projection onto $V$ but $Q_{W}$ is more awkward,
in particular, it is not Lipschitz. The effect of these projections on the
Hausdorff dimension (Marstrand type theorems) was studied in [BFMT12].

Very little is known about the relations between projections and
rectifiability. There is a bit of that in the proof of Theorem 9.11, as
briefly explained above. As mentioned in Section 5.5 Hovila, E. and M.
Järvenpää and Ledrappier [HJJL14] proved the Besicovitch-Federer projection
theorem for transversal families of linear maps
$\mathbb{R}^{n}\to\mathbb{R}^{m}$. In [Hov14] Hovila proved that the family
$\\{P_{V}:V\in G_{h}(2n,m)\\}$ is transversal. Thus the Besicovitch-Federer
projection theorem holds in $\mathbb{R}^{2n}$ for $G_{h}(2n,m)$, which is
strictly lower dimensional than the whole $G(2n,m)$ when $m>1$. This
immediately leads to the following result in $\mathbb{H}^{n}$. There
$\pi(z,t)=z$.

######  Theorem 9.13.

Let $1\leq m\leq n$ and let $E\subset\mathbb{H}^{n}$ be a Borel set with
$\mathcal{H}^{m}(\pi(E))<\infty$. Then $\mathcal{H}^{m}(P_{V}(E))=0$ for
$\mu_{n,m}$ almost all $V\in G_{h}(2n,m)$ if and only if $\pi(E)$ is purely
$m$-unrectifiable.

###  9.7. Uniform rectifiability

The above results on rectifiability could give a starting point for uniform
rectifiability. So far it is not nearly fully developed, but there are many
interesting results, however, as far as I know, only in the cases of dimension
1 and codimension 1. Some of the terminology here is analogous to that in
Chapter 6.

For one-dimensional sets the question again is about traveling salesman type
results. Define $\beta_{E}(x,r)$ as in (4.1) but the infimum is taken only
over the horizontal lines, that is, the left translates of the of the lines
through 0 in the horizontal plane.

######  Theorem 9.14.

Let $G$ be a step two Carnot group of Hausdorff dimension $Q$. If
$\Gamma\subset G$ is a rectifiable curve, then

$\int_{G}\int_{0}^{\infty}r^{-Q}\beta_{\Gamma}(x,r)^{4}\,dr\,dx\lesssim\mathcal{H}^{1}(\Gamma).$

######  Theorem 9.15.

If $p<4$ and $E\subset\mathbb{H}^{1}$ is compact and such that

$\beta_{p}(E):=\int_{\mathbb{H}^{1}}\int_{0}^{\infty}r^{-4}\beta_{E}(x,r)^{p}\,dr\,dx<\infty,$

then $E$ is contained in a rectifiable curve $\Gamma$ for which
$\mathcal{H}^{1}(\Gamma)\lesssim d(E)+\beta_{p}(E).$

Theorem 9.14 was first proved in $\mathbb{H}^{1}$ by Li and Schul [LS16a] and
then extended by Chousionis, Li and Zimmerman [CLZ19a], with a modified
satement, to arbitrary Carnot groups. Theorem 9.15 was proved by Li and Schul
in [LS16b]. The exponent $4$ of $\beta$ comes from the geometry of
$\mathbb{H}^{1}$.

It is not known if Theorem 9.15 holds with $p=4$. However, Li [Li19] produced
a Carnot group where there is a gap for the exponents. He then defined
modified $\beta$ numbers, based on the stratification of the group, and proved
that they give a necessary and sufficient condition in arbitrary Carnot
groups; the analogues of Theorems 9.14 and 9.15 hold with the exponent $2s$,
where $s$ is the step of the group.

As we have seen, intrinsic Lipschitz graphs play an important role in low
codimensional rectifiability, so they probably should be basic examples of
uniformly rectifiable sets. Here are some results in this direction for
codimension 1 subsets of $\mathbb{H}^{n}$. The $\beta$ numbers are defined as
before but restricting the approximation to vertical planes.

Chousionis, Fässler and Orponen proved in [CFO19a] that an AD-3-regular set in
$\mathbb{H}^{1}$ has big pieces of intrinsic Lipschitz graphs if and only if
it satisfies the weak geometric lemma and has big vertical projections. The
projections here are the group projections $Q_{W}$ onto vertical planes
defined above.

Let $E\subset\mathbb{H}^{n}$ be Lebesgue measurable such that $E$ and
$\mathbb{H}^{n}\setminus E$ are AD-$(2n+2)$-regular and $\partial E$ is
AD-$(2n+1)$-regular. Then Naor and Young [NY18] established a corona
decomposition with intrinsic Lipschitz graphs for $\partial E$. This lead to
an isoperimetric-type inequality, which was used to solve a fundamental
combinatorial problem. See also [NY20]. By [FOR20, Section 8] the set
$\partial E$ as above is a particular case of a Semmes surface, that is, a
closed set $F$ such that for every $p\in F$ and $0<r<r_{0}$ there are balls of
radius $cr$ in different components of $B(p,r)\setminus F$. Partially using
techniques of [NY18] Fässler, Orponen and Rigot proved in [FOR20] that Semmes
surfaces have big pieces of intrinsic Lipschitz graphs. David [Dav88] had
earlier proved the corresponding Euclidean result for which this paper gives a
new proof.

Di Donato, Fässler and Orponen proved in [DFO19] that $C^{1,\alpha},\alpha>0$,
codimension 1 intrinsic graphs have big pieces of bi-Lipschitz images of
vertical hyperplanes in $\mathbb{H}^{n}$.

Chousionis, Li and Young proved in [CLY20] that intrinsic Lipschitz graphs in
$\mathbb{H}^{n}$ satisfy the geometric lemma when $n\geq 2$. They used a
Dorronsoro inequality of Fässler and Orponen [FO20] and reduction to lower
dimensional groups by slicing. This slicing technique does not work when $n=1$
and the problem is open.

We shall return to Heisenberg groups in connection of singular integrals.

##  10\. Bounded analytic functions and the Cauchy transform

General references for this chapter are [Tol14], [Paj02], [Chr90a] [Dud10],
[Mat95] and [Gar72]. Verdera has a recent survey in [Ver21]. Many of the
topics are continued in the higher dimensional case in the next chapter.

###  10.1. Removable sets and Menger curvature

In 1888 Painlevé [Pai1888] proved that any compact subset $E$ of the plane
with one-dimensional Hausdorff measure zero is removable for bounded analytic
functions. This was before the existence of Hausdorff measures, but the
condition simply means that $E$ can be covered with finitely many discs the
sum of whose diameters is arbitrarily small.

The removability means that any bounded complex analytic function in
$U\setminus E$, where $U$ is an open set containing $E$, has an analytic
extension to $U$. It is easy to see by the Cauchy integral formula that it is
enough to consider $U=\mathbb{C}$ and then by Liouville’s theorem $E$ is
removable if and only if every bounded analytic function in the complement of
$E$ must be constant. The problem of finding a geometric characterization of
removability is called Painlevé’s problem.

In 1947 Ahlfors [Ahl47] characterized removable sets in terms of the _analytic
capacity_ $\gamma$:

$\gamma(E)=\sup\\{\lim_{|z|\to\infty}|z(f(z)-f(\infty))|:|f|\leq 1,f\
\text{analytic in}\ \mathbb{C}\setminus E\\}.$

Then $\gamma(E)=0$ if and only $E$ is removable. However, this is still a
complex analytic characterization.

Here is an easy proof of Painlevé’s theorem: Let $f:\mathbb{C}\setminus
E\to\mathbb{C}$ be analytic with $|f|\leq 1,f(\infty)=0$, and let
$z\in\mathbb{C}\setminus E$. Let $\varepsilon>0$ and cover $E$ with discs
$B_{j},j=1,\dots,k,$ such that $\sum_{j=1}^{k}d(B_{j})<\varepsilon$ and $z$ is
outside them. Let $\Gamma$ be the boundary of their union. Then by the Cauchy
integral formula

$f(z)=-\frac{1}{2\pi
i}\int_{\Gamma}\frac{f(\zeta)}{\zeta-z}\,d\mathcal{H}^{1}\zeta.$

For small $\varepsilon$ this is bounded by $C(z)\varepsilon$, so $f(z)=0$.

What can we say if we only have $\mathcal{H}^{1}(E)<\infty$? From the above
argument we still get that

$f(z)=-\int_{\Gamma_{j}}\frac{f(\zeta)}{\zeta-z}\,d\mathcal{H}^{1}\zeta,$

where the $\Gamma_{j}$ are surrounding $E$ with
$\mathcal{H}^{1}(\Gamma_{j})<\pi\mathcal{H}^{1}(E)+1$ and $d(\Gamma_{j},E)\to
0$. Some subsequence of
$f\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}\Gamma_{j}$ converges weakly to a
complex measure on $E$. Looking at this a bit more closely one finds that this
measure is absolutely continuous with respect to
$\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$. This leads to the
representation

$f(z)=C_{E}(\varphi)(z):=\int_{E}\frac{\varphi(\zeta)}{\zeta-z}\,d\mathcal{H}^{1}\zeta,\
z\in\mathbb{C}\setminus E,$

where $\varphi$ is a bounded complex valued Borel function on $E$ and
$C_{E}(\varphi)$ is the _Cauchy transform_ of $\varphi$.

So we can rephrase Painlevé’s problem, at least for sets with finite length,
in terms of the Cauchy transform: when can we put a bounded non-trivial Borel
function on $E$ whose Cauchy transform is bounded? How does this relate to
rectifiability? For $\mathcal{H}^{1}$ almost all points $z\in E$ with
$\varphi(z)\neq 0$ the integral
$\int_{E}|\zeta-z|^{-1}|\varphi(z)|\,d\mathcal{H}^{1}\zeta=\infty$, so
$C_{E}(\varphi)(z)$ is not defined on $E$. Thus $C_{E}(\varphi)(z)$ can be
bounded when $z$ is near $E$ only due to cancellation and the cancellation
comes from the symmetries of $E$: for many points $z\in E$, if $\zeta\in E$
the symmetric point $2z-\zeta$ should be close to $E$. Heuristically,
rectifiable sets have such symmetries and purely unrectifiable fail to have
them. We have a theorem of David [Dav98]:

######  Theorem 10.1.

Let $E\subset\mathbb{C}$ be compact with $\mathcal{H}^{1}(E)<\infty$. Then $E$
is removable for bounded analytic functions if and only if $E$ is purely
1-unrectifiable.

Although the above discussion may give some indication why this could be true,
completing the proof has been a long and difficult journey. It was finished
when David in [Dav98] showed that purely unrectifiable sets are removable. But
let us first look at the other direction. So we should show that if $E$ is
rectifiable with $0<\mathcal{H}^{1}(E)<\infty$ then there is a bounded non-
constant analytic function in its complement. We may assume that $E$ is a
subset of a $C^{1}$ graph $\Gamma$, even with a small Lipschitz constant. If
$E$ is a line segment, finding $\varphi$ is easy: if $\varphi$ is a smooth
function on $E$ vanishing at the end-points, then $C_{E}(\varphi)$ is bounded
by a direct computation. If $E$ is a subset of the graph $\Gamma$ of a
$C^{1,\alpha}$ function for some $\alpha>0$, then one can still construct
$\varphi$, see [Gar72, Theorem I.7.1]. But nobody knows how to construct it if
$\Gamma$ is only $C^{1}$. However, it is known to exist by duality methods
involving the Hahn-Banach theorem or some of its equivalent forms, see
[Chr90a], [Paj02] or [Tol14]. To do this one considers $C_{\Gamma}$ as a
singular integral operator on $\Gamma$. We shall discuss this and other
singular integrals more in the next chapter. The final step needed was
Calderón’s theorem, [Cal77], saying $C_{\Gamma}$ is bounded in
$L^{2}(\Gamma)$. More precisely,

$\int_{\Gamma}|C_{\Gamma,\varepsilon}(g)|^{2}\,d\mathcal{H}^{1}\lesssim\int_{\Gamma}|g|^{2}\,d\mathcal{H}^{1}\
\text{for}\ g\in L^{2}(\Gamma),$

with constant independent of $\varepsilon$, where

(10.1)
$C_{\Gamma,\varepsilon}(g)(z)=\int_{\\{\zeta\in\Gamma:|\zeta-z|>\varepsilon\\}}\frac{g(\zeta)}{\zeta-z}\,d\mathcal{H}^{1}\zeta,\
z\in\Gamma.$

This only gives us non-zero functions $g$ such that
$C_{\Gamma,\varepsilon}(g)\in L^{2}(\Gamma)$ uniformly in $\varepsilon$, but
then duality methods based on the Hahn-Banach theorem can be used to produce
$\varphi$ for which $C_{\Gamma}(\varphi)$ is bounded in $\mathbb{C}\setminus
E$.

Calderón [Cal77] proved the $L^{2}$ boundedness of the Cauchy transform on
Lipschitz graphs with small Lipschitz constant in 1977. In 1982 Coifman,
McIntosh and Meyer [CMM82] proved this for general Lipschitz graphs. Later
many people have given different proofs. David [Dav84] proved that the Cauchy
transform is bounded on a rectifiable curve $\Gamma$ if and only if $\Gamma$
is AD-1-regular, that is, uniformly 1-rectifiable.

To prove the converse statement of Theorem 10.1 we need to show that if $E$ is
not removable, then it contains a rectifiable subset of positive measure. We
can again start with a bounded complex valued Borel function $\varphi$ on $E$
such that $C_{E}(\varphi)$ is bounded in $\mathbb{C}\setminus E$. There are
three main steps in the proof:

(1) Modify
$\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ and $\varphi$ to a finite Borel
measure $\mu$ and a bounded Borel function $g$ such that
$\mu\sim\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ on some $F\subset E$ with
$\mu(F)>0,\mu(B(z,r))\leq r$ for $z\in\mathbb{C},r>0$, the real part of
$g\geq\delta>0$ on $\mathbb{C}$ and $C_{\mu}g\in BMO(\mu)$.

This was done in [DM00] except that we only derived $L^{2}$ estimates for
$C_{\mu}g$, the BMO estimates were then (later, although the papers appeared
in different order) proved in [Dav98]. For the modifications we needed to
construct generalized dyadic cubes for non-doubling measures similar to those
mentioned in Section 6.5. Then $BMO(\mu)$ refers to BMO defined with these
cubes. The construction of $\mu$ and $g$ is done by stopping time arguments
similar to those used by Christ in [Chr90b] in the AD-regular case.

Before explaining the step (2) let us define that $C_{\mu}$ is bounded in
$L^{2}(\mu)$ if the truncated _Cauchy transforms_ $C_{\mu,\varepsilon}$,

$C_{\mu,\varepsilon}g(z)=\int_{|\zeta-z|>\varepsilon}\frac{g(\zeta)}{\zeta-z}\,d\mu\zeta,\varepsilon>0,$

are uniformly bounded in $L^{2}(\mu)$. Furthermore, we say that $C_{E}$ is
bounded in $L^{2}(E)$ if this holds for
$\mu=\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$.

(2) Use a $T(b)$-theorem to prove that the singular integral operator
$C_{\mu}$ is bounded in $L^{2}(\mu)$. Again, a suitable $T(b)$-theorem in the
AD-regular case was proved in [Chr90b]. But in the present general non-
doubling situation no such result was known and David proved it in [Dav98].
Roughly speaking it says that the existence of $b=g$ as in step (1) implies
the $L^{2}$-boundedness of $C_{\mu}$.

(3) Recall from (4.2) the Menger curvature $c(z_{1},z_{2},z_{3})=1/R$, where
$R$ is the radius of the circle passing through $z_{1},z_{2}$ and $z_{3}$. The
final step (which was completed first) is based on its relation to the Cauchy
kernel $1/z$, which is provided by the following remarkable formula of
Melnikov [Mel95]:

(10.2)
$\sum_{\sigma}\frac{1}{(z_{\sigma(1)}-z_{\sigma(3)})\overline{(z_{\sigma(2)}-z_{\sigma(3)})}}=c(z_{1},z_{2},z_{3})^{2}$

for distinct points $z_{1},z_{2},z_{3}\in\mathbb{C}$, where $\sigma$ runs
through the six permutations of $\\{1,2,3\\}$. See [Tol14, Section 3.2], for
the easy elementary proof. Define the curvature of $\mu$ by

$c^{2}(\mu)=\iiint c(z_{1},z_{2},z_{3})^{2}\,d\mu z_{1}\,d\mu z_{2}\,d\mu
z_{3}.$

Melnikov and Verdera [MV96] integrated (10.2) with respect to $\mu$, wrote
$|C_{\mu,\varepsilon}1(z)|^{2}$ as a double integral and used Fubini’s theorem
six times to get

(10.3)
$\int|C_{\mu,\varepsilon}1|^{2}\,d\mu=\frac{1}{6}c^{2}_{\varepsilon}(\mu)+O(\mu(\mathbb{C})),$

where in $c^{2}_{\varepsilon}(\mu)$ the integration is performed over the
triples with mutual distances bigger than $\varepsilon$. Since the left hand
side is bounded by the step (2), so is the right hand side, whence
$c^{2}(\mu)<\infty$, and the proof is completed by the David-Legér theorem
4.18.

The formula (10.2) is remarkable firstly because it relates to the Cauchy
kernel a non-negative quantity which vanishes on lines, and only on lines.
This alone would be very useful. Secondly this quantity has a concrete
geometric meaning.

The above argument gives

######  Theorem 10.2.

Let $E\subset\mathbb{C}$ be $\mathcal{H}^{1}$ measurable with
$\mathcal{H}^{1}(E)<\infty$. If $C_{E}$ is bounded in $L^{2}(E)$, then $E$ is
$1$-rectifiable.

Nazarov, Treil and Volberg, see [NV99] or [NTV02], proved a little later than
David a $T(b)$ theorem which also completes the proof of Theorem 10.1. Their
method was quite different. They used random translations of the standard
dyadic squares finding them in a good position with big probability. This
method has turned out to be very useful in many later developments.

The formula (10.3) was used by Melnikov and Verdera in [MV96] to give yet
another proof for the $L^{2}$-boundedness of the Cauchy transform on Lipschitz
graphs.

If in the above sketch of the proof we can start with a real-valued $\varphi$,
then the modified function $g$ will be positive and we can immediately go from
step (1) to step (3). This was done in [DM00] and it resulted to the analogue
of Theorem 10.1 for Lipschitz harmonic functions. We shall discuss them in
higher dimensions in the next chapter.

It is easy to see that sets of Hausdorff dimension bigger than 1 are not
removable. Hence after David’s theorem only the case with dimension 1 and
infinite measure remained open. The complete solution of Painlevé’s problem
was given by Tolsa in [Tol03]:

######  Theorem 10.3.

Let $E\subset\mathbb{C}$ be compact. Then $E$ is not removable if and only
there is $\mu\in\mathcal{M}(E)$ such that $\mu(B(z,r))\leq r$ for
$z\in\mathbb{C}$ and $r>0$ and $c^{2}(\mu)<\infty$.

Notice that when $\mathcal{H}^{1}(E)<\infty$ Tolsa’s criterion is equivalent
to rectifiablity by Theorems 4.18 and 6.2, the latter applied to Lipschitz
graphs.

By results of Melnikov [Mel95] and Tolsa [Tol05] we also have a quantitative
version of Theorem 10.3 in terms of the analytic capacity, see also Theorems
4.14 and 6.1 in [Tol14].

######  Theorem 10.4.

Let $E\subset\mathbb{C}$ be compact. Then

$\gamma(E)\sim\sup\\{\mu(E):\mu\in\mathcal{M}(E),c^{2}(\mu)\leq\mu(E),\mu(B(z,r))\leq
r\ \text{for}\ z\in\mathbb{C},r>0\\}.$

For AD-regular sets we have

######  Theorem 10.5.

If $E\subset\mathbb{R}^{2}$ is closed and AD-$1$-regular, then $E$ is
uniformly $1$-rectifiable if and only $C_{E}$ is bounded in $L^{2}(E)$.

As mentioned above the boundedness on uniformly rectifiable sets is due to
David [Dav84]. The converse was proved in [MMV96] using (10.3), the above
mentioned results of Christ and Theorem 6.4, with $p=2$, of David and Semmes.
It also gave Theorem 10.1 for AD-regular sets.

Jaye and Nazarov gave in [JN14] a different proof for Theorem 10.5 using
reflectionless measures, see some comments on them at the end of Section 11.2.
Their proof does not use Menger curvature but relies on other special
properties of the Cauchy kernel.

For general measures $\mu$ on $\mathbb{C}$ with linear growth;
$\mu(B(z,r))\leq r$ for $z\in\mathbb{C},r>0$, the boundedness of $C_{\mu}$ in
$L^{2}(\mu)$ is equivalent to
$c^{2}(\mu\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}B)\lesssim\mu(2B)$ for all discs
$B$, see [Tol14, Theorem 3.5]. Hence, recalling Theorem 6.1, the following
result of Azzam and Tolsa [AT15] extends Theorem 10.5:

######  Theorem 10.6.

Let $\mu\in\mathcal{M}(\mathbb{C})$ with $\mu(B(z,r))\leq r$ for
$z\in\mathbb{C},r>0$. Then

(10.4)
$c^{2}(\mu)+\mu(\mathbb{C})\sim\int_{0}^{\infty}\int\beta^{1,2}_{\mu}(x,r)^{2}\frac{\mu(B(x,r))}{r}\,d\mu
x\frac{1}{r}\,dr+\mu(\mathbb{C}).$

Recall their closely related rectifiability characterization in Theorem 7.2.
For the corresponding result for the Riesz kernels $R_{n-1}$ in
$\mathbb{R}^{n}$, see Theorem 11.5.

Combining Theorem 10.6 with Theorems 10.3 and 10.4 we have a $\beta$
characterization of removable singularities of bounded analytic functions and
of analytic capacity.

###  10.2. Projections

The validity of Theorem 10.1 was conjectured by Vitushkin in [Vit67]. Actually
he formulated his question for general compact sets $E$: is $E$ removable if
and only if $\mathcal{H}^{1}(P_{L}(E))=0$ for almost all lines $L$ through the
origin? By the Besicovitch projection theorem 4.13 this is equivalent to pure
unrectifiability when $E$ has finite measure. In general the answer is no. I
showed in [Mat86b] that the projection condition is not conformally invariant
whereas the removability obviously is. This did not tell us which of the
implications is false, but soon afterwards Jones and Murai constructed in
[JM88] a non-removable set with zero projections. An easier construction with
Menger curvature was done by Joyce and Mörters in [JM00]. The other direction
is still open. Because of Tolsa’s Theorem 10.3 this can be stated purely as a
geometric measure theory problem: if $\mathcal{H}^{1}(P_{L}(E))>0$ for
positively many lines $L$, is it then possible to construct
$\mu\in\mathcal{M}(E)$ such that $\mu(B(z,r))\leq r$ for $z\in\mathbb{C}$ and
$r>0$ and $c^{2}(\mu)<\infty$? Chang and Tolsa proved in [CT20] a partial
result in this direction.

Dabrowski and Villa [DV22] proved the quantitative estimate $\gamma(E)\sim
d(E)$ provided the compact set $E\subset\mathbb{C}$ satisfies the projection
condition (6.6). For this they used Theorems 10.4 and 10.6.

###  10.3. Principal values

From Theorem 10.2 we get a characterization of rectifiability in terms of the
boundedness of the Cauchy transform. Now we give a characterization in terms
of the convergence properties of the Cauchy transform.

######  Theorem 10.7.

Let $E\subset\mathbb{C}$ be $\mathcal{H}^{1}$ measurable with
$\mathcal{H}^{1}(E)<\infty$. Then $E$ is 1-rectifiable if and only the finite
limit

$\lim_{\varepsilon\to 0}\int_{\\{\zeta\in
E:|\zeta-z|>\varepsilon\\}}\frac{1}{\zeta-z}\,d\mathcal{H}^{1}\zeta$

exists for $\mathcal{H}^{1}$ almost all $z\in E$.

The convergence for rectifiable sets was proved in [MM94], in [Ver92] Verdera
gave a different Hahn-Banach proof. The other, more difficult, direction was
proved by Tolsa in [Tol00]. Under the assumption of positive lower density it
was proved in [Mat95a]. Both papers contain more general results for measures.
We say that $\mu\in\mathcal{M}(\mathbb{C})$ has a _principal value_ at $z$ if
the finite limit

$C\mu(z):=\lim_{\varepsilon\to
0}\int_{\\{\zeta:|\zeta-z|>\varepsilon\\}}\frac{1}{\zeta-z}\,d\mu\zeta$

exists. Define the maximal transform

$C^{\ast}\mu(z):=\sup_{\varepsilon>0}\left|\int_{\\{\zeta:|\zeta-z|>\varepsilon\\}}\frac{1}{\zeta-z}\,d\mu\zeta\right|.$

Proving and using a variant of a $T(b)$ theorem of Nazarov, Treil and Volberg,
see [NV99] or [NTV02], Tolsa proved first that if $\Theta^{\ast
1}(\mu,z)<\infty$ and $C^{\ast}\mu(z)<\infty$ for $\mu$ almost all
$z\in\mathbb{C}$, then $C_{\mu}$ is $L^{2}$ bounded on a set of large $\mu$
measure. Then he concluded that by (10.3) there are compact sets $E_{i}$ such
that $\mu(\mathbb{C}\setminus\cup_{j=1}^{\infty}E_{j})=0$ and
$c^{2}(\mu\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E_{j})<\infty.$ Applying this and
Theorem 4.18 to
$\mu=\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$, Theorem 10.7 follows, and even
with the existence of principal values replaced by the finiteness of the
maximal function.

In [Mat95a] I proved

######  Theorem 10.8.

Let $\mu\in\mathcal{M}(\mathbb{C})$. If $\Theta^{1}_{\ast}(\mu,z)>0$ and
$C\mu(z)$ exists for $\mu$ almost all $z\in\mathbb{C}$, then $\mu$ is
1-rectifiable.

The proof uses tangent measures, recall Section 5.3. The assumptions imply
that for $\mu$ almost all $z\in\mathbb{C}$ every
$\nu\in\operatorname{Tan}(\mu,z)$ is _symmetric_ which means that

$\int_{B(z,r)}(\zeta-z)\,d\nu\zeta=0\ \text{for}\
z\in\operatorname{spt}\nu,r>0.$

Then up to discrete measures the symmetric measures were characterized. Every
non-discrete symmetric measure is either a constant multiple of the two-
dimensional Lebesgue measure, or a constant multiple of the one-dimensional
Lebesgue measure on a line, or a countable sum of constant multiples of the
one-dimensional Lebesgue measures on parallel lines. Theorem 10.8 was deduced
from this.

Tolsa extended also Theorem 10.8 in [Tol07]:

######  Theorem 10.9.

Let $\mu\in\mathcal{M}(\mathbb{C})$. If $\Theta^{\ast 1}(\mu,z)>0$ and
$C^{\ast}\mu(z)<\infty$ for $\mu$ almost all $z\in\mathbb{C}$, then $\mu$ is
1-rectifiable.

The proof involves similar ingredients as [Tol00] but considerable extra
difficulties are caused by the weaker density assumptions. The paper contains
more detailed information about measures with finite Cauchy maximal transform.

For $\mu\in\mathcal{M}(\mathbb{R}^{n}),x\in\mathbb{R}^{n}$ and $r>0$, define

$C^{m}_{\mu}(x,r)=r^{-m-1}\int_{B(x,r)}(y-x)\,d\mu y.$

It vanishes on $\operatorname{spt}\mu$ for all $r>0$ if and only if $\mu$ is
symmetric. The finiteness in the following theorem is a kind of approximate
symmetry condition.

######  Theorem 10.10.

Let $\mu\in\mathcal{M}(\mathbb{R}^{n})$ with $0<\Theta^{\ast m}(\mu,x)<\infty$
for $\mu$ almost all $x\in\mathbb{R}^{n}$. Then $\mu$ is $m$-rectifiable if
and only if $\int_{0}^{\infty}|C^{m}_{\mu}(x,r)|^{2}/r\,dr<\infty$ for $\mu$
almost all $x\in\mathbb{R}^{n}$.

The sufficiency of this condition for rectifiability follows from the work of
Mayboroda and Volberg [MV09b] while Villa proved the necessity in [Vil19b].
Villa also proved an analogous result for uniform rectifiability and
considered more general kernels.

###  10.4. Square functions

In [DS93] David and Semmes proved the following

######  Theorem 10.11.

Let $E\subset\mathbb{C}$ be AD-1-regular. For $f\in L^{2}(E)$ and
$z\in\mathbb{C}\setminus E$ define

$F(z)=\int_{E}\frac{f(w)}{z-w}\,d\mathcal{H}^{1}w.$

Then $E$ is uniformly rectifiable if and only if

$\int_{\mathbb{C}}|F^{\prime}(z)|^{2}d(z,E)\,dz\lesssim\int_{E}|f|^{2}\,d\mathcal{H}^{1}\
\text{for all}\ f\in L^{2}(E).$

See Theorem I.2.41 in [DS93], where also an $(n-1)$-dimensional version in
$\mathbb{R}^{n}$ is given with the Cauchy transform replaced by the Riesz
transform $R_{n-1}$.

Here is a related result with harmonic functions

######  Theorem 10.12.

Let $\Omega\subset\mathbb{R}^{n}$ be a corkscrew domain (see Section 12.1)
with AD-$(n-1)$-regular boundary. Then $\partial\Omega$ is uniformly
rectifiable if and only if

$\int_{\Omega\cap B(a,r)}|\nabla u(x)|^{2}d(x,\partial\Omega)\,dx\lesssim
r^{n-1}|\sup\\{u(x):x\in\Omega\\}|^{2}$

for all $a\in\partial\Omega,r>0$ and for every bounded harmonic function $u$
in $\Omega$.

The ’only if’ direction was proved by Hofmann, Martell and Mayboroda [HMM16]
and the ’if’ direction by Garnett, Mourgoglou and Tolsa [GMT18]. These papers
show that this estimate is also equivalent to an approximation property of
harmonic functions. Related results were proven by Hofmann and Tapiola [HT20]
and Bortz and Tapiola [BT19].

###  10.5. Other related kernels

The Menger curvature trick (10.2) is particular to the Cauchy kernel $1/z$. We
shall now discuss some positive results and counterexamples with other
kernels.

For $\Omega:S^{1}\to\mathbb{C}$, define

$k_{\Omega}(z)=\Omega(z/|z|)/|z|,\ z\neq 0.$

Huovinen proved in [Huo97] that Theorem 10.8 remains valid for
$k_{\Omega_{k}}$ with $\Omega_{k}(z)=z^{k}/|z|^{k}$, where $k$ is an odd
positive integer. Assuming additionally that the lower density is finite, it
holds for finite linear combinations of such kernels. Now the tangent measures
satisfy the $\Omega$-symmetricity $\int_{B(z,r)}\Omega(\zeta-z)\,d\nu\zeta=0\
\text{for}\ z\in\operatorname{spt}\nu,r>0.$ Huovinen characterized the
supports of such non-discrete measures. They are either unions of lines or the
whole plane $\mathbb{C}$. Observe that the cancellation for $z^{k}$ does not
only come from $-z$, but from $k-1$ other points, too. Thus, when $k\geq 3$,
in addition to flat tangent measures also certain finite sums of up to $k$
flat measures on lines through the origin, are possible tangent measures.
These are called spike measures by Jaye and Merchán and they cause new
problems as compared to the case $k=1$. Anyway, Huovinen was able to show that
under positive lower density and existence of principal values only flat
measures occur as tangent measures almost everywhere.

Jaye and Merchán [JM21] proved Huovinen’s $\Omega_{k}$ result assuming
positive and finite upper density almost everywhere. New methods are required;
for $k\geq 3$ the Menger curvature is not available and tangent measures do
not seem to work either. To deal with this, Jaye and Merchán combined Tolsa’s
and Huovinen’s methods. They introduced $\alpha$ numbers in the spirit of
(6.4), but now minimizing the distance to spike measures. They showed that
positive and finite upper density and existence of principal values imply that
these $\alpha$ numbers tend to zero. This alone does not imply rectifiability,
but combined with $L^{2}$-boundedness it does. The needed $L^{2}$-boundedness
again follows by $T(b)$ theorems. The proof is also based on the earlier work
[JM20a] and [JM20b] of these authors.

In [Vil21] Villa proved the rectifiability result for odd bi-Lipschitz
functions $\Omega:S^{1}\to S^{1}$ with constant close to 1 under positive
lower density and finite upper density. Then the AD-1-regular
$\Omega$-symmetric measures are 1-flat.

In [Huo01] Huovinen considered kernels satisfying standard Calderón-Zygmund
conditions and some additional cancellation conditions. Let
$K_{t}(z)=Re(z)/|z|^{2}-tRe(z)^{3}/|z|^{4},t\in\mathbb{R}$. When $t=1$, he
proved that there exist compact purely 1-unrectifiable AD-regular sets such
that the principal values exist almost everywhere and the operator is $L^{2}$
bounded on some subset of positive measure. This phenomenon is caused by the
cancellation coming from the coordinate axis. Jaye and Nazarov [JN18] found
for $t=3/4$ a compact purely 1-unrectifiable set with positive and finite
1-measure for which the operator is $L^{2}$ bounded. This set is not AD-
regular but it has the interesting property that, despite the $L^{2}$
boundedness, the principal values do not exist. In fact, their kernel was much
simpler, $\bar{z}/z^{2}$, but its real part is $4K_{t}$. Mateu and Prat [MP21]
gave an example in higher dimensions.

Chousionis, Mateu, Prat and Tolsa [CMPT12] considered the kernels
$Re(z)^{k}/|z|^{k+1}$ for positive odd integers $k$. They proved for them the
analogues of Theorems 10.2 and 10.5. They did it by relating to these kernels
a sum of permutations as in (10.2) and showed that it is non-negative and has
properties similar to the Menger curvature. This allowed them to prove the
analogue of the David-Legér theorem 4.18. Chunaev [Chu17] did the same for a
larger class of kernels including $K_{t}$ as above for certain parameters $t$
for which the permutation sum is non-negative. But for some $t$ this sum takes
both positive and negative values. Even for a range of such $t$ Chunaev,
Mateu, and Tolsa [CMT19] managed to prove analogous results.

David [Dav01] and Chousionis [Cho09] considered some self-similar fractals and
constructed Calderón-Zygmund kernels for which the operators are bounded on
these fractals. The kernels are defined to fit the self-similarities of the
sets.

##  11\. Singular integrals

###  11.1. A few words in general

The classical Calderón-Zygmund theory of singular integrals deals with the
Lebesgue measure on $\mathbb{R}^{m}$ and operators $T_{K}$;

(11.1) $T_{K}f(x)=\int K(x-y)f(y)\,dy.$

Here $K$ usually is nice except that it has a singularity of order $|x|^{-m}$
at the origin. Because of this the integral in (11.1) often does not exist.
But $K$ is also assumed to possess cancellation, being odd or something less.
Then the principal values

(11.2) $T_{K}f(x)=\lim_{\varepsilon\to
0}\int_{|x-y|>\varepsilon}K(x-y)f(y)\,dy$

exist if $f$ is sufficiently nice, usually Lipschitz is enough. This is easily
checked for instance for the Riesz kernel $|x|^{-m-1}x$.

The core of the theory is the $L^{2}$-boundedness; when is $T_{K}:L^{2}\to
L^{2}$ bounded? Very general regularity assumptions on $K$, saying that the
singularity at $0$ is not too bad, suffice. For instance the condition that
$|x|^{m+j}|\nabla^{j}K(x)|$ is bounded for $j=0,1,2,\dots$, that will appear
later, is much more than enough. The $L^{2}$-boundedness implies, and often is
equivalent to, many other fundamental properties ($L^{p},1<p<\infty,$ BMO,
weak $L^{1}$, $T(b)$, etc.). In particular it implies that the convergence of
(11.2) takes place for all $f\in L^{1}$ for almost all $x\in\mathbb{R}^{m}$.
This is because nice functions are dense in $L^{1}$.

Our interest is in the case where the Lebesgue measure is replaced by in some
sense $m$-dimensional measure on $\mathbb{R}^{n}$. To define the
$L^{2}$-boundedness without having the pointwise formula (11.2), we define the
truncated operators $T_{K,\mu,\varepsilon},\varepsilon>0$:

$T_{K,\mu,\varepsilon}g(x)=\int_{|x-y|>\varepsilon}K(x-y)g(y)\,d\mu y.$

We say that $T_{K,\mu}$ is bounded in $L^{2}(\mu)$ if the
$T_{K,\mu,\varepsilon},\varepsilon>0$, are uniformly bounded in $L^{2}(\mu)$.
Often this is equivalent to saying that the maximal transform
$T_{K,\mu}^{\ast}$,

$T_{K,\mu}^{\ast}g(x)=\sup_{\varepsilon>0}|\int_{|x-y|>\varepsilon}K(x-y)g(y)\,d\mu
y|,$

is bounded in $L^{2}(\mu)$. In case
$\mu=\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ we say that $T_{K,E}$ is
bounded in $L^{2}(E)$.

Coifman and Weiss extended in [CW71] most of the basic theory from the
Lebesgue measure in $\mathbb{R}^{m}$ to doubling measures $\mu$ in metric
spaces $E$, in particular to AD-$m$-regular sets in $\mathbb{R}^{n}$. Again a
basic question is: when is $T_{K}:L^{2}(\mu)\to L^{2}(\mu)$ bounded? But now
it is not only about the kernel, but also, and usually mainly, about $E$ and
$\mu$. Still $L^{2}$-boundedness implies a lot of other things, but not any
more automatically the existence of principal values. The cancellation
properties of the kernel do not help even for constant functions if the
measure does not have symmetry properties. But we showed with Verdera in
[MV09a] that under very general conditions $L^{2}$-boundedness implies weak
convergence of the truncated operators.

Is the doubling condition necessary for the general theory? It seems that this
question was seriously considered only when it was needed in connection of the
Cauchy integral and analytic capacity, recall the $T(b)$-theorems of David and
of Nazarov, Treil and Volberg from the previous chapter. In addition to
$T(b)$, Nazarov, Treil and Volberg, and later also Tolsa, developed the non-
homogeneous (i.e., non-doubling) Calderón-Zygmund theory with surprising
success in many papers.

###  11.2. $L^{2}$-boundedness and uniform rectifiability

This topic has its origins in David’s work in the 1980s, see [Dav84] and
[Dav88]. Most of the basics was developed by David and Semmes in [DS91] and
[DS93]. The main problem related to rectifiability is the following conjecture
of David and Semmes:

######  Conjecture 11.1.

Let $0<m<n$ be integers and let $E\subset\mathbb{R}^{n}$ be AD-$m$-regular.
Then the _Riesz transform_ $R_{E}^{m}$ is bounded in $L^{2}(E)$ if and only if
$E$ is uniformly $m$-rectifiable.

The kernel of $R_{E}^{m}$ is the _Riesz kernel_
$R_{m}(x)=|x|^{-m-1}x,x\in\mathbb{R}^{n}\setminus\\{0\\}$, so
$R_{E}^{m}=T_{R_{m},E}$ according to the above notation. For a measure $\mu$
we shall also set $R_{\mu}^{m}=T_{R_{m},\mu}$. The $L^{2}$-boundedness again
means that the truncated operators $R^{m}_{E,\varepsilon},\varepsilon>0$,

$R_{E,\varepsilon}^{m}g(x)=\int_{\\{y\in
E:|x-y|>\varepsilon\\}}R_{m}(x-y)g(y)\,d\mathcal{H}^{m}y,$

are uniformly bounded in $L^{2}(E)$.

The boundedness of much more general singular integral operators on uniformly
rectifiable sets was proved in [DS91]. We shall return to this soon. The
problem is the converse. It is known for $m=1,n-1$, and only then:

######  Theorem 11.2.

Let $m=1$ or $m=n-1$ and let $E\subset\mathbb{R}^{n}$ be AD-$m$-regular. Then
the Riesz transform $R_{E}^{m}$ is bounded in $L^{2}(E)$ if and only if $E$ is
uniformly $m$-rectifiable.

For $m=1$ the proof of Theorem 10.5 gives this, too. So it is based on
symmetrization as in (10.2). For $m>1$ this method does not work because the
corresponding sum takes both positive and negative values as was shown, more
generally, by Farag in [Far99] and [Far00b].

The case $m=n-1$ is due to Nazarov, Tolsa and Volberg in [NTV14a]. Their proof
is very long and complicated and contains many brilliant ideas. Before saying
a few words about it, let us look at a much weaker and simpler result which
holds for all $m$, see [Mat02]

######  Proposition 11.3.

Let $0<m<n$ and let $\mu\in\mathcal{M}(\mathbb{R}^{n})$ be AD-$m$-regular. If
the Riesz transform $R_{\mu}^{m}$ is bounded in $L^{2}(\mu)$, then for $\mu$
almost all $a\in\mathbb{R}^{n}$ $\mu$ has some $m$-flat tangent measures at
$a$.

The proof is easy. A duality argument, recall the discussion around (10.1),
gives a bounded function $g$ such that the maximal function $R_{\mu}^{m\ast}g$
is bounded. Then we can find an AD-$m$-regular tangent measure $\nu$ such that

(11.3) $|\int_{r<|x-y|<R}|x-y|^{-m-1}(x-y)\,d\nu y|\leq C\ \text{for}\
x\in\operatorname{spt}\nu,0<r<R.$

Since $\operatorname{spt}\nu$ cannot be the whole space we can find an open
ball $U$ disjoint from $\operatorname{spt}\nu$ such that there is
$x\in\operatorname{spt}\nu\cap\partial U$. Any tangent measure $\pi$ of $\nu$
at $x$ is again AD-$m$-regular and satisfies (11.3). It has support in a half-
space $H$ with $0\in\operatorname{spt}\pi\cap\partial H$, so due to (11.3)
$\operatorname{spt}\pi$ must be tangential to $\partial H$ at $0$. The next
tangent measure will have support in $\partial H$, perhaps with a non-constant
density, but the last tangent measure will be flat, and it is also a tangent
measure of $\mu$ by Preiss’s ’tangent measures to tangent measures are tangent
measures’ principle, see [Mat95, Theorem 14.16].

Proposition 11.3 gives approximation by planes at _some_ arbitrarily small
scales, but even for ordinary rectifiability we would need it at _all_
arbitrarily small scales. As such it is useless for the proof of Theorem 11.2,
but in [NTV14a] Nazarov, Tolsa and Volberg used tangent measures to prove a
quantitative form of this: sufficiently good pointwise boundedness of the
Riesz transform at a range of scales gives a good approximation at some cube
in the same range. Then the counter-assumption that the set in Theorem 6.8 is
not Carleson allows to build a Cantor structure based on the generalized
dyadic cubes, mentioned in Section 6.5, where the cubes of nearly flatness of
$\operatorname{spt}\mu$ (or rather of $\mu$) and cubes of non-flatness are
alternating. The proof ends with clever applications of an extremal problem
and a maximum principle. These have their origins in [ENV14]. The maximum
principle requires harmonicity of the kernel and this is the main reason (or
maybe the only reason) why the proof does not extend to $m<n-1$. The end
result is that the $L^{2}$-boundedness implies the property of approximation
with unions of planes as in Theorem 6.8 and hence uniform rectifiability. This
is not a sketch of the proof, only a few tiles from a magnificent structure.

For sets of finite measure we have the analogue of Theorem 10.2:

######  Theorem 11.4.

Let $E\subset\mathbb{R}^{n}$ be $\mathcal{H}^{n-1}$ measurable with
$\mathcal{H}^{n-1}(E)<\infty$. If $R^{n-1}_{E}$ is bounded in $L^{2}(E)$, then
$E$ is $(n-1)$-rectifiable.

This was proved by Nazarov, Tolsa and Volberg in [NTV14b]. They used a result
of Eiderman, Nazarov and Volberg from [ENV14] according to which $R^{n-1}_{E}$
is unbounded in $L^{2}(E)$ if $E$ has zero lower density. Thus they could
assume that $\Theta_{\ast}^{n-1}(E,x)>0$ for $\mathcal{H}^{n-1}$ almost all
$x\in E$. Then they used an argument of Pajot from [Paj96] to find
AD-$(n-1)$-regular measures $\mu_{j}$ such that $R_{\mu_{j}}^{n-1}$ is bounded
in $L^{2}(\mu_{j})$ and
$\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E\leq\sum_{j}\mu_{j}$. This
allowed them to conclude the proof by Theorem 11.2.

As in 10.6 for $m=1$, there is a generalization of Theorem 11.2:

######  Theorem 11.5.

Let $\mu\in\mathcal{M}(\mathbb{R}^{n})$ with $\mu(B(x,r))\leq r^{n-1}$ for
$x\in\mathbb{R}^{n},r>0$. Then

(11.4)
$\|R^{n-1}_{\mu}\|_{L^{2}(\mu)}^{2}+\mu(\mathbb{R}^{n})\sim\int_{0}^{\infty}\int\beta^{n-1,2}_{\mu}(x,r)^{2}\frac{\mu(B(x,r))}{r^{n-1}}\,d\mu
x\frac{1}{r}\,dr+\mu(\mathbb{R}^{n}).$

The estimate ${}^{\prime}\lesssim^{\prime}$ was proved by Girela-Sarrión
[Gir19] for general $m$ and general kernels. The converse was proved by Tolsa
[Tol21] based on his paper with Dabrowski [DT21a].

The validity of ${}^{\prime}\gtrsim^{\prime}$ in Theorem 11.5 for $1<m<n-1$ is
an open question, as it should be, since the David-Semmes conjecture is then
open. However, Jaye, Nazarov and Tolsa proved in [JNT18] that the
$L^{2}$-boundedness for all radial-type kernels
$K\in\mathcal{K}_{m}(\mathbb{R}^{n})$, see below, implies that the right hand
side of (11.4) is finite.

Prat, Puliatti and Tolsa [PPT21] extended Theorems 11.2 and 11.4 to kernels
which are gradients of the fundamental solutions of more general elliptic
equations; $cR_{n-1}$ is the gradient of the fundamental solution of the
Laplace equation. Let $A(x),x\in\mathbb{R}^{n},$ be an $n\times n$ matrix with
Hölder continuous entries satisfying the ellipticity conditions

(11.5) $|\xi|^{2}\lesssim A(x)\xi\cdot\xi\ \text{for all}\
\xi,x\in\mathbb{R}^{n},$ (11.6) $A(x)\xi\cdot\eta\lesssim|\xi||\eta|\
\text{for all}\ \xi,\eta,x\in\mathbb{R}^{n}.$

The equation

(11.7) $L_{A}u(x):=-\text{div}(A\nabla u)(x)=0,$

has a fundamental solution $\Gamma_{A}(x,y)$. The kernel then is its gradient,
$K_{A}(x,y)=\nabla_{x}\Gamma_{A}(x,y)$. Technically the situation now is more
complicated than for the Riesz kernel, but the authors of [PPT21] managed with
several modifications to follow the same strategy to prove the analogues of
Theorems 11.2 and 11.4.

Mas and Tolsa proved in [MT14] a characterization of uniform rectifiability in
terms of the $L^{2}$-boundedness of variations of the Riesz transform. They
are defined by maximizing
$\sum_{m\in\mathbb{Z}}|R^{m}_{\mu,\varepsilon_{m+1}}f(x)-R^{m}_{\mu,\varepsilon_{m}}f(x)|^{2}$
over decreasing sequences of positive numbers $\varepsilon_{m}$.

Except for the Riesz kernels and some other particular cases, recall the
discussion in Section 10.5, the question for which kernels Theorem 11.2 holds
is pretty much open. However, if we consider a large class of kernels the
characterization of uniform rectifiability in all dimensions was already
obtained by David and Semmes in [DS91]. Let us denote by
$\mathcal{K}_{m}(\mathbb{R}^{n})$ the set of smooth real valued odd functions
$K$ on $\mathbb{R}^{n}\setminus\\{0\\}$ such that $|x|^{m+j}|\nabla^{j}K(x)|$
is bounded for $j=0,1,2,\dots$.

######  Theorem 11.6.

Let $E$ be AD-$m$-regular. Then $E$ is uniformly rectifiable if and only
$T_{K,E}$ is bounded in $L^{2}(E)$ for all kernels
$K\in\mathcal{K}_{m}(\mathbb{R}^{n})$.

Already the boundedness with the kernels $K\in\mathcal{K}_{m}(\mathbb{R}^{n})$
which are of the form $K(x)=x_{j}k(x)$, where $k$ is radial, is enough for
uniform rectifiability, see [DS93, Theorem I.2.59], for rectifiability this
was proved in [MP95b]. Related results are in [Tol09].

Jaye and Nazarov studied _reflectionless_ measures in several papers. They
have some resemblance to symmetric measures. A measure $\mu$ is reflectionless
with respect to a kernel $K$ if $T_{K,\mu}1$ vanishes, in a weak sense, on the
support of $\mu$. In [JN19] they showed that if $K$ is a kernel with respect
to which the AD-$m$-regular reflectionless measures are flat, then the
$L^{2}$-boundedness on any AD-$m$-regular set $E$ implies the uniform
rectifiablity of $E$. This condition is known to hold for the Cauchy kernel
and $R_{1}$ in the plane but it is unknown for higher dimensional Riesz
kernels. With this property of the Cauchy kernel Jaye and Nazarov gave in
[JN14] the earlier mentioned new proof of Theorem 10.5. The example of [JN18]
mentioned near the end of Section 10.5 relies on the fact that the Lebesgue
measure on $\mathbb{C}$ is reflectionless with respect to the kernel
$\bar{z}/z^{2}$.

###  11.3. Principal values

We have the analogue of Theorem 10.7 for the Riesz kernels:

######  Theorem 11.7.

Let $E\subset\mathbb{R}^{n}$ be $\mathcal{H}^{m}$ measurable with
$\mathcal{H}^{m}(E)<\infty$. Then $E$ is $m$-rectifiable if and only if the
finite limit

$\lim_{\varepsilon\to 0}\int_{\\{y\in
E:|y-x|>e\\}}R_{m}(y-x)\,d\mathcal{H}^{m}y$

exists for $\mathcal{H}^{m}$ almost all $x\in E$.

Verdera’s Hahn-Banach proof [Ver92] for the Cauchy transform generalizes to
give that the principal values exist for rectifiable sets. The converse was
proved by Tolsa in [Tol08]. His proof for Theorem 10.7 does not work now as he
used Menger curvature for that. It is kind of replaced by $L^{2}$ estimates on
Lipschitz graphs. That is, Tolsa showed that the $L^{2}$ norm of the Riesz
transform on Lipschitz graphs is quantitatively bounded by the $L^{2}$ norm of
the gradient of the function, not only from above but also from below. These
estimates are then used to construct Lipschitz graphs containing a positive
measure of $E$. Very roughly, the existence of principal values implies some
approximation with a graph of a Lipschitz function, whose gradient cannot have
too big $L^{2}$ norm by the bounds on the Riesz transform. Arguments similar
to those of Legér in the proof of Theorem 4.18 are essential.

The analogue of Theorem 10.8 was proved in [MP95b] but in addition to positive
lower density we had to assume that it also is finite:

######  Theorem 11.8.

Let $\mu\in\mathcal{M}(\mathbb{R}^{n})$. If
$0<\Theta^{m}_{\ast}(\mu,x)<\infty$ and the finite limit $\lim_{\varepsilon\to
0}\int_{\\{|y-x|>\varepsilon\\}}R_{m}(y-x)\,d\mu y$ exists for $\mu$ almost
all $x\in\mathbb{R}^{n}$, then $\mu$ is $m$-rectifiable.

Again we showed that tangent measures $\nu$ are symmetric:

$\int_{B(x,r)}(y-x)\,d\nu y=0\ \text{for}\ x\in\operatorname{spt}\nu,r>0.$

We could not say very much about them, except that the AD-$m$-regular
symmetric measures are flat. But we could show that if almost everywhere $\mu$
has positive lower density and the tangent measures are symmetric, then they
are flat.

Combining theorems 11.4 and 11.7 we see that if $m=1$ or $m=n-1$,
$E\subset\mathbb{R}^{n}$ is $\mathcal{H}^{m}$ measurable with
$\mathcal{H}^{m}(E)<\infty$ and the Riesz transform is bounded in $L^{2}(E)$,
then the principal values exist $\mathcal{H}^{m}$ almost everywhere in $E$.
But we only know this going through rectifiability. It would be interesting to
have a direct proof, or any proof when $1<m<n-1$. Combining with Theorem 11.7
this would give a proof that the $L^{2}$-boundedness implies rectifiability.

The more general question for which kernels $L^{2}$-boundedness implies almost
everywhere convergence is open, even for AD-regular sets. We know from Section
10.5 that there are reasonable kernels for which this fails, but the Jaye-
Nazarov example is not AD-regular. In [MV09a] we proved with Verdera under
very general conditions that the $L^{2}$-boundedness together with zero lower
density implies the existence of principal values. Jaye and Merchán [JM20a]
strengthened this by replacing zero density by the condition that
modifications of Tolsa’s $\alpha$s (recall Section 7.4) tend to zero. See also
[JM20b] for related results and recall the discussion in 10.5 on [JM21].

Let $\Omega\subset\mathbb{R}^{n}$ be a domain with compact AD-$(n-1)$-regular
(and a little more) boundary. D. and M. Mitrea and Verdera [MMV16] proved that
then $\Omega$ is a $C^{1+\alpha},0<\alpha<1,$ domain if and only if the Riesz
transform $R^{n-1}_{\partial\Omega}$ maps $C^{\alpha}(\partial\Omega)$ into
itself. The second half actually means several equivalent conditions, some of
them involving uniform rectifiability, corresponding to different definitions
of the Riesz transform, for example principal value and distributional
definitions. The proof uses Clifford algebras and an interesting formula
expressing the unit normal in terms of the Riesz transform and the Cauchy-
Clifford transform.

###  11.4. Lipschitz harmonic functions

The kernel $R_{n-1}$ is a constant multiple of the gradient of the fundamental
solution of the Laplacian in $\mathbb{R}^{n}$, which is $c_{n}|x|^{2-n}$ for
$n\geq 3$, $c_{2}\log|x|$ for $n=2$. Hence the codimension 1 Riesz transform
is related to the removability of Lipschitz harmonic functions in the same way
as the Cauchy transform is related to the removable sets of bounded analytic
functions.

Let us say that a compact set $E\subset\mathbb{R}^{n}$ is _removable for
Lipschitz harmonic functions_ , abbreviated RLH, if whenever $U$ is an open
set containing $E$ every Lipschitz function $u:U\to\mathbb{R}$ which is
harmonic in $U\setminus E$ is harmonic in $U$. Since Lipschitz functions
always have Lipschitz extensions, we can start with $u$ defined in all of $U$.

######  Theorem 11.9.

Let $E\subset\mathbb{R}^{n}$ be compact with $\mathcal{H}^{n-1}(E)<\infty$.
Then $E$ is RHL if and only if $E$ is purely $(n-1)$-unrectifiable.

The non-removability of rectifiable sets was proved in [MP95a] by methods
similar to those used for the analytic functions and discussed in the previous
chapter. The converse was proved for $n=2$ in [DM00] based on Menger curvature
(recall the discussion after Theorem 10.2) and the David-Léger theorem 4.18.
The case of general $n$ is due to Nazarov, Tolsa and Volberg in [NTV14b]. The
proof is reduced to Theorem 11.2 via a $T(b)$-theorem from [Vol03] in a
similar manner as was argued for analytic functions.

In the plane Tolsa’s characterization, Theorem 10.3, is valid also for
Lipschitz harmonic functions. Hence

######  Theorem 11.10.

Let $E\subset\mathbb{C}$ be compact. Then the following are equivalent:

  * (1)

$E$ is removable for bounded analytic functions.

  * (2)

$E$ is removable for Lipschitz harmonic functions.

  * (3)

If $\mu\in\mathcal{M}(E)$ is such that $\mu(B(z,r))\leq r$ for
$z\in\mathbb{C}$ and $r>0$, then $c^{2}(\mu)=\infty$.

The equivalence of (1) and (2) is only known passing through (3), and hence by
a very complicated proof. It is easy to see that (1) implies (2): if $u$ is
Lipschitz harmonic, the $\partial_{\overline{z}}\partial_{z}u=\Delta u=0$, so
$f=\partial_{z}u$ is bounded analytic. The converse, to get $u$ from $f$,
would require some kind of integration, which is possible in some special
cases but maybe not always.

Tolsa [Tol21] characterized also the removable sets for Lipschitz harmonic
functions in all dimensions:

######  Theorem 11.11.

Let $E\subset\mathbb{R}^{n}$ be compact. Then $E$ is not removable for
Lipschitz harmonic functions if and only if there exists
$\mu\in\mathcal{M}(E)$ such that $\mu(B(x,r))\leq r^{n-1}$ for
$x\in\mathbb{R}^{n}$ and $r>0$ and

$\int_{0}^{\infty}\int\beta^{n-1,2}_{\mu}(x,r)^{2}\frac{\mu(B(x,r))}{r^{n-1}}\,d\mu
x\frac{1}{r}\,dr<\infty.$

This is a consequence of Theorem 11.5.

###  11.5. Parabolic singular integrals

In [BHHLN20], [BHHLN21b] it is shown that a large class of parabolic singular
integral operators are $L^{2}$-bounded on parabolic uniformly rectifiable
sets. Recall Section 6.6, as mentioned there the motivation for this theory
comes from the heat equation $\Delta_{x}u(x,t)=\partial_{t}u(x,t).$ Its
fundamental solution $W$ is given for $x\in\mathbb{R}^{n},t>0$, by

$W(x,t)=ct^{-n/2}e^{-|x|^{2}/(4t)}.$

The kernel $K$, which now replaces the Riesz kernel $R_{n-1}$, is

$K(x,t)=\nabla_{x}W(x,t)=-(c/2)t^{-n/2-1}xe^{-|x|^{2}/(4t)},$

with $K(x,t)=0$, when $t\leq 0$. Notice that it is antisymmetric only in the
$x$ variable, but this is good enough for the $L^{2}$-boundedness on uniformly
rectifiable sets by [BHHLN20]. The converse, the analogues of Theorems 11.2
and 11.4, are unknown. These would be needed to get a removability result such
as Theorem 11.9 for parabolic regular Lipschitz (again BMO in $t$ variable,
recall Section 6.6) solutions of the heat equation. Mateu, Prat and Tolsa have
done in [MPT20] some preliminary work in this direction. For instance, they
showed that positive measure subsets of parabolic regular Lipschitz graphs are
not removable. They also constructed Cantor sets with positive measure which
are removable.

We shall discuss harmonic measure, induced by the Laplace equation, in the
next chapter. In the same way the heat equation leads to the caloric measure.
To get something like Theorem 12.2 in the parabolic case would also seem to
require information about the consequences of the $L^{2}$-boundedness of
$T_{K}$.

###  11.6. Heisenberg groups

As compared to the Euclidean theories, rather little is known about the
singular integrals

$T_{K}f(p)=\int K(p^{-1}\cdot q)f(q)\,d\mu q$

on lower than full dimensional subsets of the Heisenberg groups. As was seen
in Chapter 9 rectifiability is much better understood than uniform
rectifability. Anyway intrinsic Lipschitz graphs are good candidates for basic
uniformly rectifiable sets. So it makes sense to ask: For what kernels are the
singular integral operators $L^{2}$-bounded on intrinsic Lipschitz graphs?
When does $L^{2}$-boundedness or existence of principal values imply
rectifiability?

What kernels should replace the Riesz kernels? If we just look at the
expression and the scaling property of $R_{m}$ in $\mathbb{R}^{n}$ a similar
kernel in $\mathbb{H}^{n}$ is $R_{m}=(R_{m,1},R_{m,2})$ where for
$p=(z,t)\in\mathbb{H}^{n}$ and $\|p\|=(|z|^{4}+t^{2})^{1/4}$,

$R_{m,1}(p)=\|p\|^{-m-1}z\ \text{and}\ R_{m,2}(p)=\|p\|^{-m-2}t.$

In [CM11] an analogue of Proposition 11.3 was proved for these kernels.
However, if we want a connection to harmonicity, we should start from a
Laplacian. Let $X_{i},Y_{i},i=1,\dots,n,$ be the vector fields as is Section
9.2. The sub-Riemannian or Kohn Laplacian in $\mathbb{H}^{n}$ is defined by

$\Delta_{H}=\sum_{i=1}^{n}(X_{i}^{2}+Y_{i}^{2}).$

For the potential theory related to it, see [BLU07]. The fundamental solution
of $\Delta_{H}u=0$ is $\Gamma(p)=c\|p\|^{-2n}$. Note that
$2n=\dim\mathbb{H}^{n}-2$, so it has the same form as in $\mathbb{R}^{n}$. The
kernel related to the Lipschitz harmonic functions is
$K=\nabla_{H}\Gamma:\mathbb{H}^{n}\to\mathbb{R}^{2n}$. It looks a bit
complicated, since its coordinate functions are for $i=1,\dots,n$, and
$z=(x,y)\in\mathbb{R}^{2n}$,

$K_{i}(z,t)=\frac{x_{i}|z|^{2}+y_{i}t}{\|p\|^{2n+4}},\
K_{i+n}(z,t)=\frac{y_{i}|z|^{2}-x_{i}t}{\|p\|^{2n+4}}.$

However, it is a reasonable Calderón-Zygmund kernel. It is not odd, but it is
horizontally antisymmetric: $K(z,t)=-K(-z,t)$. Due to this Chousionis, Fässler
and Orponen [CFO19b] were able to prove in $\mathbb{H}^{1}$ that $T_{K}$, and
more general singular integrals, are $L^{2}$-bounded on $C^{1,\alpha}$
intrinsic graphs, see also [FO18] for related results.

Orponen [Orp18] proved that for AD-3-regular subsets of $\mathbb{H}^{1}$ the
$L^{2}$-boundedness of the 3-dimensional singular integrals with horizontally
antisymmetric kernels implies local symmetry of the type in Theorem 6.9 and
that this implies weak geometric lemma.

In [CM14] a class of self-similar purely $(2n+1,\mathbb{H})$-unrectifiable
subsets of $\mathbb{H}^{n}$ with positive $\mathcal{H}^{2n+1}$ measure was
introduced on which $T_{K}$ is $L^{2}$ unbounded and which are removable for
Lipschitz harmonic functions. Further results in this direction were proven by
Chousionis and Urbanski in [CU15].

Chousionis and Li [CL17] introduced a class of non-negative 1-homogeneous
kernels in $\mathbb{H}^{1}$ which vanish on the vertical axis $\\{z=0\\}$. For
some of them the operator is $L^{2}$-bounded on regular curves and for some
the $L^{2}$-boundedness on an AD-1-regular set implies that it is contained in
a regular curve. Extension of the first statement to general Carnot groups is
given by Chousionis, Li and Zimmerman [CLZ19b].

Fässler and Orponen [FO21] proved in $\mathbb{H}^{1}$ the $L^{2}$-boundedness
of many singular integral operators on AD-regular curves and on a class of
vertical graphs called Lipschitz flags.

##  12\. Harmonic measure and elliptic measures

For the harmonic measure in the plane, see [GM05]. Toro’s survey [Tor19]
discusses relations between geometric measure theory and harmonic measure both
in the plane and in higher dimensions.

###  12.1. Harmonic measure

Let $\Omega\neq\mathbb{R}^{n}$ be an open connected subset of
$\mathbb{R}^{n},n\geq 2$. We assume that $\partial\Omega$ is not too small; it
is not a polar set. This is true if $\mathcal{H}^{n-1}(\partial\Omega)>0$. By
classical potential theory, see, for example, [AG01], for any continuous
function $f$ on $\partial\Omega$ one can solve the Dirichlet problem to find a
harmonic function $u_{f}$ in $\Omega$ with boundary values $f$, in a
generalized sense if $\partial\Omega$ is not sufficiently nice. Fixing
$p\in\Omega$ for a while the Riesz representation theorem and the maximum
principle can be used to show that there is a probability measure
$\omega_{\Omega}^{p}\in\mathcal{M}(\partial\Omega)$ such that

(12.1) $u_{f}(p)=\int f\,d\omega_{\Omega}^{p}.$

Then $u(p)=\int u\,d\omega_{\Omega}^{p}$ if $u$ is continuous in
$\overline{\Omega}$ and harmonic in $\Omega$. The measure
$\omega_{\Omega}^{p}$ is called the _harmonic measure_ of $\Omega$ at $p$. It
depends on $p$, but by the Harnack inequality for any two points
$p,q\in\Omega$ the measures $\omega_{\Omega}^{p}$ and $\omega_{\Omega}^{q}$
are comparable.

According to Kakutani’s probabilistic characterization
$\omega_{\Omega}^{p}(A)$ is the probability that the Brownian traveller
starting from $p$ hits $A$ before hitting any other part of the boundary. This
helps to visualize the fact that in the case of complicated boundaries
$\omega_{\Omega}^{p}$ lives on parts of the boundary which are more easily
accessible from $\Omega$. A more precise statement in the plane is a result of
Wolff [Wol93] saying that harmonic measure lives on a set of sigma-finite
$\mathcal{H}^{1}$ measure. The corresponding statement in $\mathbb{R}^{3}$ is
false by his example in [Wol95]. However, Bourgain [Bou87] proved that in
$\mathbb{R}^{n}$ it lives on a set of dimension at most $n-\varepsilon(n)$,
where $\varepsilon(n)$ is a small positive constant, whose best value is
unknown.

From the point of view of this survey the main question is: what are the
relations between harmonic measure and the geometry of the boundary? More
precisely, for $E\subset\partial\Omega$ does rectifiability of $E$ imply
something on the harmonic measure on $E$, and conversely, do some properties
of the harmonic measure lead to rectifiability? The first general result was
proved already in 1916 by the brothers Riesz: if $\Omega\subset\mathbb{R}^{2}$
is simply connected and $\mathcal{H}^{1}(\partial\Omega)<\infty$, then
$\omega_{\Omega}^{p}$ and
$\mathcal{H}^{1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}\partial\Omega$ are mutually
absolutely continuous. Here the whole boundary is rectifiable but Bishop and
Jones extended this in [BJ90] by showing that if $\Omega\subset\mathbb{R}^{2}$
is simply connected and $E\subset\partial\Omega\cap\Gamma$, where $\Gamma$ is
a rectifiable curve, then $\omega_{\Omega}^{p}(E)=0$ if and only if
$\mathcal{H}^{1}(E)=0$. They used Jones’s traveling salesman theorem 4.16 for
this.

In the plane complex analytic tools are very effective, but in higher
dimensions quite different methods are required and the situation is in many
ways different. First, the analogue of the Riesz brothers theorem fails.
Ziemer constructed in [Zie74] a domain in $\mathbb{R}^{3}$ whose boundary is
2-rectifiable, it even has an ordinary tangent plane a each point, and it has
finite $\mathcal{H}^{2}$ measure, but $\mathcal{H}^{2}$ is not absolutely
continuous with respect to harmonic measure. To the other direction, Wu showed
in [Wu86] that there exists a domain $\Omega\subset\mathbb{R}^{3}$ and
$E\subset\partial\Omega\cap V$, where $V$ is a plane, with positive harmonic
measure and $\mathcal{H}^{2}(E)=0$, even $\dim E=1$.

Starting from Dahlberg [Dah77] in 1977 and Lipschitz boundaries and followed
by David, Jerison, Kenig and Semmes and more general boundaries in the 1980s a
great number of people have produced results in the spirit that various
geometric properties of the boundary imply absolute continuity of harmonic
measure, often with quantitative estimates such as being an $A_{\infty}$
weight. Uniform rectifiability of the boundary alone is not sufficient to get
such results by an example of Bishop and Jones in [BJ90]. But starting with
some natural geometric conditions of the boundary, uniform rectifiability
often comes into play leading to, or even characterizing, quantitative
properties of the harmonic measure. Commonly used conditions are corkscrew and
Harnack chain conditions. Roughly speaking the former says that every ball
centered in the boundary contains a ball of comparable size inside the domain
and the latter that any two points in the domain can be joined with a chain of
balls whose size is comparable to the distance to the boundary.

The proof of the following theorem was completed by Azzam, Hofmann, Martell,
Mourgoglou and Tolsa in [AHMMT20]. The authors explain by several examples
that the result is in many ways optimal. It is a culmination of a long process
involving many other people and articles, see [AHMMT20] and [HLMN17] for the
history, references and other related results. In particular, that
$A_{\infty}$ implies uniform rectifiability was proved earlier by Hofmann and
Martell, see [HLMN17] where this is obtained even for the non-linear
$p$-harmonic equation.

######  Theorem 12.1.

Let $\Omega$ satisfy the corkscrew condition and have AD-$(n-1)$-regular
boundary. Then $\partial\Omega$ is uniformly rectifiable and satisfies the
weak local John condition if and only if $\omega_{\Omega}^{p}$ is locally in
weak $A_{\infty}$.

The weak local John condition is a quantitative connectivity condition saying,
roughly, that each point of $\Omega$ can be connected to many points in the
boundary by rectifiable curves in $\Omega$ staying away from the boundary.
That $\omega_{\Omega}^{p}$ is locally in weak $A_{\infty}$ means that there is
$s>0$ such that for every $x\in\partial\Omega$ and $0<r<d(\partial\Omega)/4$,

$\omega_{\Omega}^{p}(A)\lesssim\left(\frac{\mathcal{H}^{n-1}(A)}{\mathcal{H}^{n-1}(\partial\Omega\cap
B(x,r))}\right)^{s}\omega_{\Omega}^{p}(\partial\Omega\cap B(x,2r))$

for all $p\in\Omega\setminus B(x,4r)$ and for all Borel sets
$A\subset\partial\Omega\cap B(x,r)$.

The local weak $A_{\infty}$ condition is known to be equivalent to the
quantitative solvability of the $L^{p}$ Dirichlet problem.

Let us now look at a qualitative rectifiability criterion. After many partial
results by several people Azzam, Hofmann, Martell, Mayboroda, Mourgoglou,
Tolsa and Volberg proved in [AHMMMTV16]

######  Theorem 12.2.

Let $p\in\Omega$ and $E\subset\partial\Omega$ with
$\mathcal{H}^{n-1}(E)<\infty$.

  * (1)

If
$\omega_{\Omega}^{p}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E\ll\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$, then
$\omega_{\Omega}^{p}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ is $(n-1)$-rectifiable.

  * (2)

If $\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E\ll\omega_{\Omega}^{p}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$, then $E$ is
$(n-1)$-rectifiable.

By the Radon-Nikodym theorem it is easy to see that these statements are
equivalent. For example, if (1) holds and
$\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E\ll\omega_{\Omega}^{p}$, then
$\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E=g\omega_{\Omega}^{p}$ for some
non-negative function $g$ on $E$ for which $g(x)>0$ for $\mathcal{H}^{n-1}$
almost all $x\in E$. Then
$\omega_{\Omega}^{p}\ll\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}\\{x\in E:g(x)>0\\}$, and it
follows from (1) that $E$ is $(n-1)$-rectifiable.

The key to the proof of Theorem 12.2 is the relation between the Green
function and the Riesz transform and Theorem 11.4 of Nazarov, Tolsa and
Volberg. In classical potential theory the _Green function_
$G_{\Omega}:\Omega\times\Omega\setminus\\{(p,x):p=x\\}\to\mathbb{R}$ is a
basic tool to study harmonic measure. It is harmonic in both variables
separately. For a fixed $p\in\Omega$ $G(p,\cdot)$ has zero boundary values. At
$x=p$ it has the same singularity as the fundamental solution $\Gamma$ of the
Laplacian, which is a constant multiple of $|x|^{2-n}$, if $n\geq 3$, and of
$\log|x|$, if $n=2$. More precisely, $\Gamma(p-x)-G_{\Omega}(p,x)$ is a
harmonic function of $x$ in $\Omega$. For nice domains the harmonic measure is
absolutely continuous with respect to the surface measure and its density is
given by the normal derivative of the Green function. By (12.1) the Green
function can be written as

$G_{\Omega}(p,x)=\Gamma(p-x)-\int\Gamma(p-x)\,d\omega_{\Omega}^{p}x,\
p,x\in\Omega,x\neq p.$

Since $\nabla\Gamma=cR_{n-1}$ we have

(12.2) $\nabla_{x}G_{\Omega}(p,x)=cR_{n-1}(p-x)-c\int
R_{n-1}(p-x)\,d\omega_{\Omega}^{p}x,\ p,x\in\Omega,x\neq p.$

As in (1) of Theorem 12.2 suppose that
$\omega_{\Omega}^{p}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E\ll\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$, so that
$\omega_{\Omega}^{p}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}}
\end{picture}E=g\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ for some non-negative $g$.
Given $M>0$ it is enough to prove that $E_{M}:=\\{x\in E:1/M<g(x)<M\\}$ is
$(n-1)$-rectifiable. One could then hope that, similarly to the case of nice
boundaries, the left hand side of (12.2) would have enough boundedness to give
boundedness for the Riesz transform when $x\in E_{M}$. In this very general
case this is not clear at all, but the authors of [AHMMMTV16] managed to show
something like this. A bit more precisely, they again used generalized dyadic
cubes, now from [DM00] since there is no doubling, and they showed using
(12.2) that there are enough cubes $Q$ to cover large part of $E_{M}$ such
that the truncated Riesz transform $R^{n-1}_{\omega_{\Omega}^{p},r(Q)}(x)$
with a suitable $r(Q)$ is bounded for $x\in Q$, with a quantitative bound.
This allows to apply a $T(b)$-theorem of Nazarov, Treil and Volberg to get the
$L^{2}(\omega_{\Omega}^{p})$-boundedness of the Riesz transform on a subset of
$E_{M}$ with positive measure. From this an application of Theorem 11.4 yields
rectifiability.

There are also results on two-phase problems involving rectifiability. The
following was proved by Azzam, Mourgoglou, Tolsa and Volberg in [AMTV19], the
paper [AMT17] contains a quantitative version:

######  Theorem 12.3.

Let $\Omega_{1}$ and $\Omega_{2}$ be disjoint open connected subsets of
$\mathbb{R}^{n}$ and $E\subset\partial\Omega_{1}\cap\partial\Omega_{2}$ a
Borel set such that $\omega_{\Omega_{1}}^{p_{1}}$ and
$\omega_{\Omega_{2}}^{p_{2}},p_{i}\in\Omega_{i},$ are mutually absolutely
continuous on $E$. Then $E$ contains an $(n-1)$-rectifiable subset $F$ such
that $\omega_{\Omega_{i}}^{p_{i}}(E\setminus F)=0$ and
$\omega_{\Omega_{1}}^{p_{1}}$ and $\omega_{\Omega_{2}}^{p_{2}}$ are mutually
absolutely continuous with respect to
$\mathcal{H}^{n-1}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}F$.

The proof uses an interesting blow-up argument involving tangent measures and
the Alt-Caffarelli-Friedman monotonicity formula for pairs of subharmonic
functions applied to the Green functions. This method was introduced by Kenig,
Preiss and Toro in [KPT09], where a partial result and deep information about
harmonic measures was obtained. The proof also relies on Theorem 12.2, and so
on the Nazarov-Tolsa-Volberg Riesz transform theorems 11.2 and 11.4, and on
[GT18].

###  12.2. Elliptic measures in codimension 1

The previous section dealt with the Laplace equation $\Delta u=0$. It is
natural to expect that the results would have analogues for more general
elliptic equations, and this indeed is the case. Assuming that $\Omega$ is a
uniform domain, Hofmann, Martell, Mayboroda, Toro and Zhao [HMMTZ20]
characterized the $A_{\infty}$ property with uniform rectifiability for
elliptic measures corresponding to equations with optimal conditions for the
coefficients. Prat, Puliatti and Tolsa proved in [PPT21] an analogue of
Theorem 12.2 for such elliptic measures with Hölder continuous coefficients
using their singular integral results mentioned in Section 11.2. Then the same
arguments as for the Laplace equation work.

Cao, Hidalgo-Palencia and Martell [CHM22] investigated corona decompositions
associated to quite general elliptic measures. They showed, among other
things, that these are equivalent to square function estimates as in Theorem
10.12, also to a weaker form of them, which are equivalent to uniform
rectifiability. The boundary of the domain is assumed to be AD-regular and to
satisfy the cork-screw condition.

###  12.3. Elliptic measures in codimension bigger than one

If $E\subset\mathbb{R}^{n}$ is a closed set with
$\mathcal{H}^{n-2}(E)<\infty$, then it is polar. In particular, if
$E\subset\partial\Omega$, then $\omega_{\Omega}^{p}(E)=0$ and the properties
of the harmonic measure are in no way related to the geometric properties of
$E$. The same is true for the elliptic equations as above. But considering
suitable degenerate elliptic equations a rich theory can be developed. This
was realized and done by David, Feneuil and Mayboroda and their co-authors in
many papers, only some of which are listed in the references.

Let $E\subset\mathbb{R}^{n}$ be AD-$m$-regular for some integer $0<m<n-1$. The
standard ellipticity conditions (11.5) and (11.6) for a measurable $n\times n$
matrix valued function $A$ are now replaced by

(12.3) $|\xi|^{2}\lesssim d(E,x)^{n-m-1}A(x)\xi\cdot\xi\ \text{for all}\
\xi,x\in\mathbb{R}^{n},$

(12.4) $d(E,x)^{n-m-1}A(x)\xi\cdot\zeta\lesssim|\xi||\zeta|\ \text{for all}\
\xi,\zeta,x\in\mathbb{R}^{n}.$

The authors developed in [DFM17] a comprehensive theory for the degenerate
elliptic operators $L=-\operatorname{div}(A\nabla)$ analogous to the classical
theory including among others solutions of the Dirichlet problem with
continuous boundary data, the corresponding elliptic measure $\omega^{p}_{L}$
and its basic properties, and the existence and properties of the Green
function and its relations to the elliptic measure. As remarked before, in the
classical theory one often needs some conditions for the boundary to prevent
it from being too massive. Here they are not needed. The AD-$m$-regularity
with $m<n-1$ gives automatically the corkscrew and Harnack chain conditions.

To get absolute continuity of the elliptic measure on Lipschitz graphs or more
general sets one needs stronger assumptions on $A$, also in the classical
theory. I now restrict to a particular operator, which is the authors’
replacement of the Laplacian. Again there is a weight like $d(E,x)^{n-m-1}$,
but this seems to be too rough and it is replaced by a regularized distance
$D_{\alpha,\mu}$, where $\alpha>0$ is a parameter and
$\mu=\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$ or some other AD-$m$-regular
measure with support $E$:

$D_{\alpha,\mu}(x)=\left(\int|x-y|^{-m-\alpha}\,d\mu y\right)^{-1/\alpha}.$

AD-regularity easily implies that $D_{\alpha,\mu}(x)\sim d(E,x)$.

The elliptic operator attached to $\mu$ is given by

(12.5) $L_{\alpha,\mu}=-\operatorname{div}(D_{\alpha,\mu}^{m+1-n}\nabla).$

Denote now simply by $\omega^{p}$ the elliptic measure related to
$L_{\alpha,\mu}$ and some base point $p\in\mathbb{R}^{n}\setminus E$. In this
setting David and Mayboroda proved in [DM20a]

######  Theorem 12.4.

If $E$ is uniformly $m$-rectifiable, $m\leq n-2$, then $\omega^{p}\ll\mu$.
Moreover $\omega^{p}\in A_{\infty}(\mu)$, which means that for every
$\varepsilon>0$ there is $\delta>0$ such that if $x\in E,r>0$ and $F\subset
E\cap B(x,r)$ is a Borel set, then

(12.6) $\frac{\omega^{p}(F)}{\omega^{p}(E\cap
B(x,r))}<\delta\implies\frac{\mu(F)}{\mu(E\cap B(x,r))}<\varepsilon,$

where $p\in\mathbb{R}^{n}\setminus E$ is such that $d(E,p)\sim|p-x|\sim r$.

In [Fen20] Feneuil gave a different simpler proof.

The converse of Theorem 12.4 fails completely for $\alpha=n-m-2$, then
$m<n-2$. In this case $\omega\in A_{\infty}(\mu)$ for any AD-$m$-regular
measure $\mu$, and $m$ need not even be an integer. I shall say a bit more
about this soon. The above authors believe that this value of $\alpha$ is a
unique exception and the converse should be true for the other values.

In addition to extending from codimension one to general dimensions the work
of David, Feneuil, Mayboroda and their co-authors contains several interesting
new results and phenomena also for codimension one. I present here two new
characterizations of uniform rectifiability. Now $m$ can be any integer with
$0<m<n$.

In [DEM18] David, Engelstein and Mayboroda characterized uniform rectifiablity
in terms of the distance $D_{\alpha,\mu}$ in the spirit of Theorems 10.11 and
10.12.

######  Theorem 12.5.

Let $E\subset\mathbb{R}^{n}$ be AD-$m$-regular. Then $E$ is uniformly
rectifiable if and only if

(12.7) $d(x,E)^{m+2-n}|\nabla(|\nabla D_{\alpha,\mu}|^{2})(x)|^{2}dx$

is a Carleson measure on $\mathbb{R}^{n}\setminus E$.

The ’only if’ direction uses Tolsa’s $\alpha$s, recall Theorem 6.10; (12.7)
vanishes if $\mu$ is a flat measure and the uniform rectifiability leads to
approximation of $\mu$ with flat measures. For the converse direction the
starting idea is that if (12.7) vanishes then $\nabla D_{\alpha,\mu}$ is
constant, from which it follows, but with a lot of work, that $\mu$ is
$m$-flat. They also characterized the rectifiability of $E$ in terms of the
non-tangential limits of $\nabla|D_{\alpha,\mu}|$.

Another characterization is in terms of the Green functions. Let
$G_{\alpha,\mu}$ be the Green function corresponding to $L_{\alpha,\mu}$ with
the pole at $\infty$. If $E$ is an $m$-plane, then
$G_{\alpha,\mu}(x)=cd(E,x)$, and the elliptic measure is a constant multiple
of $\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$. Uniform rectifiability means
that $E$ is well approximated by $m$-planes and this turns out to be
equivalent that $G_{\alpha,\mu}$ is well approximated by distances to
$m$-planes:

######  Theorem 12.6.

Let $E\subset\mathbb{R}^{n}$ be an unbounded AD-$m$-regular set. When $m=n-1$
assume also the corkscrew and Harnack chain conditions. Then $E$ is uniformly
rectifiable if and only if for every $\varepsilon>0$ and $M>1$ the set
$E\times(0,\infty)\setminus\mathcal{G}(\varepsilon,M)$ is a Carleson set,
where $\mathcal{G}(\varepsilon,M)$ is the set of $(x,r)\in E\times(0,\infty)$
such that there are an affine $m$-plane $V$ and $c>0$ for which

$|d(y,V)-cG_{\alpha,\mu}(y)|\leq\varepsilon r\ \text{for}\
y\in(\mathbb{R}^{n}\setminus E)\cap B(x,Mr).$

This is a special case of the results proved by David and Mayboroda in
[DM20b]. Again the proof of the ’only if’ direction uses Tolsa’s $\alpha$s.
For the converse direction we have $G_{\alpha,\mu}(y)=0$ when $y\in E$, which
immediately gives some approximation of $E$ by planes. So it is a good
starting point, but it only gives weak geometric lemma and many more arguments
are needed to get the bilateral approximation, recall Theorem 6.7.

The result in [DM20b] is more general in that the authors considered much more
general degenerate elliptic operators. They also have similar results with the
regularized distance for $m=n-1$ where $d(y,V)-cG_{\alpha,\mu}(y)$ is replaced
by $D_{\beta,\mu}(y)-cG_{a,\mu}(y),\beta>0$. Then they can start with any
$n-2<m<n$ and the Carleson estimates force $m$ to be an integer.

For $m<n-2$ and $\alpha=n-m-2$ the smallness of $|D_{\alpha,\mu}-cG_{a,\mu}|$
or the absolute continuity of the elliptic measure do not imply any kind of
rectifiability, as was already stated after Theorem 12.4. The reason is that
then by direct computation
$L_{\alpha,\mu}D_{\alpha,\mu}=\tfrac{1}{\alpha}\Delta R_{\alpha,\mu}$ where,
when $\alpha=n-m-2$, $R_{\alpha,\mu}(x)=\int|x-y|^{2-n}\,d\mu y$ is harmonic,
so $L_{\alpha,\mu}D_{\alpha,\mu}=0$, from which it follows by the uniqueness
of the Green function that $G_{\alpha,\mu}$ is a constant multiple of
$D_{\alpha,\mu}$. This leads to $\omega_{L_{\alpha,\mu}}\sim\mu$. Further, if
$E$ is $m$-rectifiable, using basic properties of rectifiable sets it follows
that
$\omega_{L_{\alpha,\mu}}=c\mathcal{H}^{m}\begin{picture}(12.0,12.0)\put(2.0,0.0){\line(1,0){8.0}}
\put(2.0,0.0){\line(0,1){8.0}} \end{picture}E$. The details can be found in
[DEM18].

##  13\. Sets of finite perimeter and functions of bounded variation

This topic is well documented in many books, see [Fed69], Section 4.5,