# On the Moduli of Lipschitz Homology Classes

Ilmari Kangasniemi  Department of Mathematics, Syracuse University, Syracuse,
NY 13244, USA<EMAIL_ADDRESS>and  Eden Prywes  Department of Mathematics,
Princeton University, Princeton, NJ 08544, USA<EMAIL_ADDRESS>
###### Abstract.

We define a type of modulus $\operatorname{dMod}_{p}$ for Lipschitz surfaces
based on $L^{p}$-integrable measurable differential forms, generalizing the
vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a
homological duality theorem, where for Hölder conjugate exponents
$p,q\in(1,\infty)$, every relative Lipschitz $k$-homology class $c$ has a
unique dual Lipschitz $(n-k)$-homology class $c^{\prime}$ such that
$\operatorname{dMod}_{p}^{1/p}(c)\operatorname{dMod}_{q}^{1/q}(c^{\prime})=1$
and the Poincaré dual of $c$ maps $c^{\prime}$ to 1. As
$\operatorname{dMod}_{p}$ is larger than the classical surface modulus
$\operatorname{Mod}_{p}$, we immediately recover a more general version of the
estimate
$\operatorname{Mod}_{p}^{1/p}(c)\operatorname{Mod}_{q}^{1/q}(c^{\prime})\leq
1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is
formulated in the general setting of Lipschitz Riemannian manifolds, though
our results appear new in the smooth setting as well. We also provide a
characterization of closed and exact Sobolev forms on Lipschitz manifolds
based on integration over Lipschitz $k$-chains.

###### Key words and phrases:

Surface Modulus, Duality, Lipschitz Manifolds, Lipschitz Homology, Extremal
Length of Vector Measures, Sobolev Differential Forms

###### 2020 Mathematics Subject Classification:

Primary 31B15; Secondary 28A75, 512F30

Eden Prywes was partially supported by the NSF grant RTG-DMS-1502424.

##  1\. Introduction

In [7], Fuglede defined the modulus for a family of measures. He used this to
define the modulus for a family of Lipschitz surfaces, extending the classical
definition for a family of paths. In this paper, we give a definition for a
new version of the modulus of a family of surfaces using differential forms,
which we denote $\operatorname{dMod}$. We are able to prove that our
definition possesses duality properties that Fuglede’s classical definition of
the modulus of surfaces does not satisfy. The main theorem of the paper is as
follows:

######  Theorem 1.1.

Let $M$ be a closed Lipschitz Riemannian manifold. Let $c\in
H_{k}^{\operatorname{lip}}(M;{\mathbb{Z}})$ be a non-torsion element of the
rank $k$ Lipschitz homology group. For all $p\in(1,\infty)$, there exists a
unique element $c^{\prime}\in H^{\operatorname{lip}}_{n-k}(M;{\mathbb{R}})$ so
that the Poincaré dual of $c$ maps $c^{\prime}$ to $1$ and

$\displaystyle\operatorname{dMod}_{p}(c)^{1/p}\operatorname{dMod}_{q}(c^{\prime})^{1/q}=1,$

where $p^{-1}+q^{-1}=1$.

###  1.1. Modulus and duality

The study of path modulus was initiated by Beurling and Ahlfors [1] in order
to study quasiconformal maps. If $\Gamma$ is a family of paths in
$\Omega\subset{\mathbb{C}}$, then the _$p$ -modulus_ is defined as

$\displaystyle\operatorname{Mod}_{p}(\Gamma)=\inf\bigg{\\{}\int_{\Omega}\rho^{p}:\rho\in
L^{p}(\Omega)\text{ and }\int_{\gamma}\rho\geq 1\text{ for
}\gamma\in\Gamma\bigg{\\}}.$

The $2$-modulus of path families is preserved by planar conformal maps and is
preserved up to a constant by planar quasiconformal maps. It is also connected
to capacity problems and the calculus of variations.

The $p$-modulus satisfies a duality property for topological quadrilaterals.
Let $Q$ be a topological quadrilateral with sides $E_{1},E_{2},E_{3}$ and
$E_{4}$ arranged cyclically. If $\Gamma_{1}$ is the family of paths connecting
$E_{1}$ to $E_{3}$ in $Q$ and $\Gamma_{2}$ is the family of paths connecting
$E_{2}$ to $E_{4}$ in $Q$, then

$\displaystyle\operatorname{Mod}_{p}(\Gamma_{1})^{1/p}\operatorname{Mod}_{q}(\Gamma_{2})^{1/q}=1,$

where $p^{-1}+q^{-1}=1$.

Fuglede [7] defined the modulus for a family of measures in an analogous way.
Let $\mathcal{M}$ be a family of measures on a set
$\Omega\subset{\mathbb{R}}^{n}$. The $p$-modulus of $\mathcal{M}$ is defined
as

$\displaystyle\operatorname{Mod}_{p}(\mathcal{M})=\inf\bigg{\\{}\int_{\Omega}\rho^{p}:\rho\in
L^{p}(\Omega)\text{ and }\int_{\Omega}\rho d\mu\geq 1\text{ for
}\mu\in\mathcal{M}\bigg{\\}}.$

Since Lipschitz surfaces always have corresponding surface measures, this
definition also gives a definition for a family of Lipschitz surfaces. Fuglede
was interested in duality results that resemble the duality for quadrilaterals
in the curves setting and was able to prove such a theorem for the $2$-modulus
of surfaces. In particular, he proved that the collection of all curves that
connect a compact set $K\subset\mathbb{R}^{n}$ to $\infty$ has dual
$2$-modulus to the collection of Lipschitz hypersurfaces that separate $K$
from $\infty$ [7, Theorem 9].

Gehring [8] and Ziemer [20] proved a similar duality results for the exponent
$p=n$. Gehring showed, under smoothness assumptions on the surfaces, that the
$n$-modulus of a family of curves connecting two compact continua is dual to
the $n/(n-1)$-modulus of the family of surfaces separating them. Ziemer
extended the result to remove the smoothness assumption on the surface family.

A general duality theory for families of surfaces does not hold. This was
observed by Freedman and He in [6]. However, they pose similar capacity
problems, which do satisfy duality properties in the cases they are interested
in, namely solid tori in $3$-dimensional space. They are also able to show
that for families of surfaces $A$ and $B$ which are dual in a homological
sense, one has

$\displaystyle\operatorname{Mod}_{p}(A)^{1/p}\operatorname{Mod}_{q}(B)^{1/q}\leq
1,$

where $p^{-1}+q^{-1}=1$.

In [13], Lohvansuu studied the modulus of surfaces in Lipschitz cubes. Let
$Q=h(I^{n})$, where $I=[0,1]$ and $h$ is a bilipschitz map. Let $A=h(\partial
I^{k}\times I^{n-k})$ and $B=h(I^{k}\times\partial I^{n-k})$. In this case,
the relative homology satisfies $H^{k}(Q,A;{\mathbb{Z}})\cong
H^{k}(Q,B;{\mathbb{Z}})\cong{\mathbb{Z}}$. If $\Gamma_{A}$ and $\Gamma_{B}$
are the generators of the homology groups, Lohvansuu proved that

$\displaystyle\operatorname{Mod}_{p}(\Gamma_{A})^{1/p}\operatorname{Mod}_{q}(\Gamma_{B})^{1/q}\leq
1,$

where $p^{-1}+q^{-1}=1$.

###  1.2. Modulus with differential forms

The above results indicate that finding a modulus theory that gives a duality
property is not trivial even in settings where the topology is not
complicated, like a cube or torus. For this reason, in this paper we focus on
a modulus that uses differential forms instead of densities. In [2], Aikawa
and Ohtsuka defined a vector modulus that assigned a modulus to a family of
vector valued measures. This is very similar to the modulus we define, however
they only consider families of curves. They are able to show duality
properties in this setting and relate their theory to capacity properties of
sets.

Let $\mathcal{S}$ be a family of Lipschitz $k$-chains in
$\Omega\subset{\mathbb{R}}^{n}$. We define the _$p$ -differential form
modulus_ as

$\displaystyle\operatorname{dMod}_{p}(\mathcal{S})=\inf\int_{\Omega}|\omega|^{p},$

where the infimum is taken over all $k$-forms $\omega\in L^{p}(\Omega)$ such
that

$\displaystyle\int_{\sigma}\omega\geq 1,$

for $\sigma\in\mathcal{S}$, outside a set of zero $p$-modulus in
$\mathcal{S}$. Excluding an exceptional set in $\mathcal{S}$ is necessary, as
noted in [2] and discussed in Section 8. The differential form modulus is
particularly suited to proving duality theorems.

We prove Theorem 1.1 by proving a more general theorem that includes manifolds
with boundary. Let $M$ be a closed $n$-dimensional Lipschitz submanifold of an
oriented boundaryless Lipschitz Riemannian reference $n$-manifold
$\mathcal{R}$. Additionally, suppose that $\partial M=D\cup E$, where $D$ and
$E$ are closed $(n-1)$-dimensional Lipschitz submanifolds of $\partial M$ with
$\partial D=\partial E$.

######  Theorem 1.2.

Let $p,q\in(1,\infty)$ satisfy $p^{-1}+q^{-1}=1$. If $c\in
H_{k}^{\operatorname{lip}}(M,D;{\mathbb{Z}})$ is a non-torsion element, then
there exists a unique element $c^{\prime}\in
H_{n-k}^{\operatorname{lip}}(M,E;{\mathbb{R}})$ such that the Poincaré dual of
$c$ maps $c^{\prime}$ to $1$ and

$\displaystyle\operatorname{dMod}_{p}(c)^{1/p}\operatorname{dMod}_{q}(c^{\prime})^{1/q}=1.$

Theorem 1.1 follows from Theorem 1.2 by letting $M=\mathcal{R}$ and having
$\partial M=D=E=\emptyset$. In the case where
$H_{k}^{\operatorname{lip}}(M,D;{\mathbb{Z}})\cong
H_{n-k}^{\operatorname{lip}}(M,E;{\mathbb{Z}})\cong{\mathbb{Z}}$, as in the
setting of [13], we can strengthen our theorem to ensure that $c^{\prime}\in
H_{n-k}^{\operatorname{lip}}(M,E;{\mathbb{Z}})$.

######  Theorem 1.3.

Let $p,q\in(1,\infty)$ satisfy $p^{-1}+q^{-1}=1$. If
$H_{k}^{\operatorname{lip}}(M,D;{\mathbb{Z}})\cong
H_{n-k}^{\operatorname{lip}}(M,E;{\mathbb{Z}})\cong{\mathbb{Z}}$ and $c\in
H_{k}^{\operatorname{lip}}(M,D;{\mathbb{Z}})$ and $c^{\prime}\in
H_{n-k}^{\operatorname{lip}}(M,E;{\mathbb{Z}})$ are the generators of the
homology groups, then

$\displaystyle\operatorname{dMod}_{p}(c)^{1/p}\operatorname{dMod}_{q}(c^{\prime})^{1/q}=1.$

An immediate consequence of Theorem 1.3 is the following corollary, which
generalizes the main result in [13].

######  Corollary 1.4.

Let $p,q\in(1,\infty)$ with $p^{-1}+q^{-1}=1$. Suppose that
$H_{k}^{\operatorname{lip}}(M,D;\mathbb{Z})\cong
H_{n-k}^{\operatorname{lip}}(M,E;\mathbb{Z})\cong\mathbb{Z}$ for some
$k\in\\{1,\dots,n-1\\}$. Let $c$ and $c^{\prime}$ be the generators of
$H_{k}^{\operatorname{lip}}(M;\mathbb{Z})$ and
$H_{n-k}^{\operatorname{lip}}(M;\mathbb{Z})$, respectively. Then

$\operatorname{Mod}_{p}(c)^{\frac{1}{p}}\operatorname{Mod}_{q}(c^{\prime})^{\frac{1}{q}}\leq
1.$

As mentioned above, a version of Theorem 1.3 does not in general hold for the
classical surface modulus defined by Fuglede, due to a counterexample given by
Freedman and He in [6]. This counterexample is a solid torus with a
$180^{\circ}$ twist, the slices of which are dumbbell-shaped surfaces
isometric to $S=B^{2}((1,0),2^{-1})\cup
B^{2}((-1,0),2^{-1})\cup([-1,1]\times[-\varepsilon,\varepsilon])$. See Figure
1 for an illustration. A flat version of this example can be defined by
$M=(S\times[0,1])/\sim$, with the equivalence relation $\sim$ given by
$(z,0)\sim(-z,1)$.

Figure 1. The dumbbell-shaped twisted solid torus of Freedman and He that acts
as a counterexample to a version of Theorem 1.3 for the classical modulus.

Indeed, in the context of Theorem 1.3, $M$ is an oriented Lipschitz Riemannian
manifold with $H_{1}(M,\emptyset;{\mathbb{Z}})\cong H_{2}(M,\partial
M;{\mathbb{Z}})\cong{\mathbb{Z}}$. If $c\in H_{1}(M,\emptyset;{\mathbb{Z}})$
is a generator, then the density $\rho$ which is $1$ in the handle
$[-2^{-1},2^{-1}]\times[-\varepsilon,\varepsilon]\times[0,1]$ and $0$
elsewhere will be admissible for $c$, as any loop winding around $M$ exactly
once that doesn’t stay in the handle has to cross it. Hence,
$\operatorname{Mod}_{p}(c)\leq\varepsilon$. However, for a generator
$c^{\prime}$ of $H_{2}(M,\partial M;{\mathbb{Z}})$, by taking a constant
density $\rho^{\prime}$ in the end-part $(B^{2}((1,0),2^{-1})\cup
B^{2}((-1,0),2^{-1}))\times[0,1]$ of the dumbbell, one finds an upper bound
for $\operatorname{Mod}_{q}(c^{\prime})$ independent of $\varepsilon$. Thus,
for small enough $\varepsilon$, one has
$\operatorname{Mod}_{p}(c)^{1/p}\operatorname{Mod}_{q}(c^{\prime})^{1/q}<1$.

Similarly to the planar case discussed earlier, the classical path modulus is
connected to the theory of quasiconformal maps in that quasiconformal maps on
${\mathbb{R}}^{n}$ preserve the $n$-modulus, up to a multiplicative constant.
For surface modulus, only partial results have been shown. For example, Rajala
in [15] showed a result for the image of sphere families by quasiregular maps.
It is an interesting question whether there exists a class of $k$-surfaces
that is invariant under quasiconformal transformations and has a reasonable
definition for the $n/k$-modulus that is quasi-preserved by quasiconformal
maps. A similar question can also be asked for the differential form modulus.

###  1.3. Ideas of the proofs

We now outline the proofs of the main theorems in this paper. Since our
setting is that of Lipschitz manifolds, there are technical difficulties in
defining the integration theory for surfaces. This is mainly due to the fact
that the Riemannian metric $g$ for a Lipschitz manifold $M$ is defined almost
everywhere. Thus, the surface measure of an arbitrary Lipschitz $k$-chain
$\sigma$ on $M$, which is used to define the $p$-modulus and consequently
$p$-exceptional sets, is not necessarily defined in an intrinsic way. We are
however able to define $p$-exceptional families of Lipschitz $k$-chains on $M$
without first defining the $p$-modulus of surfaces on $M$, essentially relying
on the general modulus of measures and the quasi-preservation of surface
measures under bilipschitz maps.

This definition of $p$-exceptional sets unlocks the requisite integration
theory of $L^{p}$-forms over $p$-almost every Lipschitz $k$-chain on $M$ and
moreover allows us to define the classical $p$-modulus on $M$ using the fact
that $p$-almost every Lipschitz $k$-chain has a surface measure. Following
this, we obtain versions of the de Rham theorem and Stokes’ theorem in the
setting of Sobolev forms and Lipschitz chains. Our arguments take advantage of
the integration theory of flat forms by Whitney [19] and Federer [5], which
also applies in our setting of Lipschitz manifolds.

We then prove a characterization of weak exactness and closedness for Sobolev
differential forms via integration over Lipschitz $k$-chains. This
characterization is similar in spirit to Fuglede’s original motivation for
defining $p$-modulus. In [7, Section 3], he showed that the $p$-modulus can be
used to detect when vector fields have zero divergence or curl. In particular,
our version generalizes Fuglede’s results to differential forms of arbitrary
degree, applies on Lipschitz manifolds and also includes the characterization
of weakly vanishing boundary values. The following theorem is a simplified
version of this characterization not involving boundary values; the full
statement can be found in Section 7. While the theorem does not use
differential form modulus, it is an essential ingredient in its study.

######  Theorem 1.5.

Let $M$ be an oriented Lipschitz Riemannian $n$-manifold without boundary. Let
$p\in(1,\infty)$ and $\omega\in L^{p}(\wedge^{k}T^{*}M)$. The form $\omega$ is
weakly closed if and only if

$\displaystyle\int_{\sigma}\omega=0,$

for all $\sigma\in\partial C_{k+1}^{\operatorname{lip}}(M;{\mathbb{Z}})$
outside a $p$-exceptional set. Additionally, $\omega$ is weakly exact if and
only if

$\displaystyle\int_{\sigma}\omega=0,$

for all $\sigma\in
C_{k}^{\operatorname{lip}}(M;{\mathbb{Z}})\cap\ker(\partial)$ outside a
$p$-exceptional set.

Following this, we define the modulus $\operatorname{dMod}_{p}$ of
differential forms and study admissible forms for a homology class $c$. An
$L^{p}$-differential form $\omega$ is _admissible_ for $c$ if

$\displaystyle\int_{\sigma}\omega\geq 1$

for $\sigma\in c$, outside family that is $p$-exceptional in terms of the
classical modulus of surfaces. The key observation is that in this case, we
can use the characterization theory outlined in Theorem 1.5 to show that
$\omega$ is in fact weakly closed, despite us having no regularity assumptions
on $\omega$ beyond $L^{p}$-integrability. This directly connects admissibility
of forms for homology classes to Sobolev-de Rham cohomology theory.

With these preliminaries we are finally able to show the duality theorems,
Theorems 1.1 and 1.2. The proof uses the connection between $p$-differential
form modulus and the capacity of differential forms, which was observed in
[6]. We first show that for a Lipschitz homology class $c$, the modulus
problem admits a unique minimizer $\omega$. In particular, we prove in Section
8 that this minimizer satisfies the following properties.

######  Theorem 1.6.

Let $M$ be a closed $n$-dimensional Lipschitz submanifold in an oriented
Lipschitz Riemannian reference $n$-manifold $\mathcal{R}$ and let $D$ be a
closed $(n-1)$-dimensional Lipschitz submanifold of $\partial M$. Let
$p\in(1,\infty)$ and let $c\in H^{\operatorname{lip}}_{k}(M,D;{\mathbb{Z}})$,
with $0<k<n$. If $c$ is not a torsion element, then there exists a unique
differential form $\omega$ such that $\omega$ is $L^{p}$-integrable, weakly
closed, $p$-harmonic, $\omega$ has weakly vanishing tangential part in $D$,
$|\omega|^{p-2}\omega$ has weakly vanishing normal part in $E=\partial
M\setminus D$ and

$\displaystyle\operatorname{dMod}_{p}(c)=\int_{M}|\omega|^{p}\operatorname{vol}_{M}=\inf_{\tau}\int_{M}|\omega+d\tau|^{p}\operatorname{vol}_{M},$

where the infimum is over all $L^{p}$-integrable $\tau$, with an
$L^{p}$-integrable weak exterior derivative $d\tau$ and with weakly vanishing
tangential part in $D$.

We then connect the form $\omega$ to Poincaré duality. In particular, if $M$
is compact, then the Poincaré dual of $c$ corresponds to a Sobolev-de Rham
cohomology class $[\zeta]$, where we may assume that $\zeta$ solves the
capacity problem,

$\displaystyle\int_{M}\left|{\zeta}\right|^{q}\operatorname{vol}_{M}=\inf_{\tau}\int_{M}\left|{\zeta+d\tau}\right|^{q}\operatorname{vol}_{M},$

with $p^{-1}+q^{-1}=1$ and with the infimum taken over Sobolev forms $\tau$.
We prove that $\zeta$ and $\omega$ are in fact related to each other via the
formula

$\displaystyle\omega=\frac{|\zeta|^{q-2}\mathtt{\star}\hskip
1.0pt\zeta}{\|\zeta\|_{q}^{q}}.$

This indicates the connection between capacity problems and the differential
form modulus. The dual homology class $c^{\prime}$ of $c$ is then found as the
Poincaré dual of $[\omega]$ up to a sign and we prove that $\zeta$ is a
$q$-modulus minimizing admissible form for $c^{\prime}$. The class
$c^{\prime}$ a priori has real coefficients, but in the case where
$H_{k}^{\operatorname{lip}}(M,D;\mathbb{Z})\cong
H_{n-k}^{\operatorname{lip}}(M,E;\mathbb{Z})\cong\mathbb{Z}$, we are able to
show that $c^{\prime}$ in fact corresponds to a generator of
$H_{n-k}^{\operatorname{lip}}(M,E;\mathbb{Z})$.

We remark that key parts of our theory rely on a technical result that
Lipschitz homology classes on Lipschitz manifolds are never $p$-exceptional in
the sense of the classical modulus of surfaces. Our proof of this fact ended
up sufficiently involved to deserve its own section. Indeed, even though the
ideas behind this are clear locally, $p$-exceptionality is a global property,
so most standard local proofs appear to fail. Our approach uses a technique
based on mollification and leads to a working proof that is mostly local,
although the method becomes increasingly technical in the case involving
relative homology.

### Structure of this paper

In Section 2, we recall Fuglede’s modulus and present the relevant theorems.
In Section 3, we recall the necessary background information regarding
Lipschitz Riemannian manifolds and in Section 4 we discuss integration theory
on Lipschitz Riemannian manifolds. In Section 5, we recall the construction of
local Sobolev cohomology on Lipschitz manifolds, as well as the related de
Rham isomorphisms. In Section 6, we prove that Lipschitz homology classes are
not $p$-exceptional, a technical result we require for the duality theory. In
Section 7, we prove the characterization of weak exactness and closedness of
differential forms outlined in Theorem 1.5. In Section 8, we define the
differential form modulus and prove its basic properties. This includes a
proof for Theorem 1.6. Section 9 then includes the proofs of the main duality
theorems for differential form modulus.

### Acknowledgements

The authors would like to thank the AMS’s Mathematical Research Communities,
where this project was initiated in the Analysis on Metric Spaces session.
They would also like to thank the participants of the session for helpful
conversations on the topic, especially Mario Bonk and Sylvester Eriksson-
Bique. The first author would like to thank Princeton University for hosting
his visit, which was of significant use in completing this paper.

##  2\. Modulus of measures

We recall Fuglede’s definition for the modulus of a family of measures (see
[7]).

######  Definition 2.1.

Let $(X,\mu)$ be a complete measure space and let $\mathcal{M}_{\mu}$ be the
family of all measures $\nu$ on $X$ such that every $\mu$-measurable set is
$\nu$-measurable for all $\nu\in\mathcal{M}_{\mu}$. If
$\Sigma\subset\mathcal{M}_{\mu}$, then a measurable function $f\colon
X\to[0,\infty]$ is called _admissible for $\Sigma$_ if

(1) $\int_{X}f\mathop{}\\!\mathrm{d}\nu\geq 1$

for every $\nu\in\Sigma$. We denote the set of admissible functions for
$\Sigma$ by $\operatorname{adm}(\Sigma)$. The _$p$ -modulus_ of $\Sigma$ for
$p\in[1,\infty)$ is given by

$\operatorname{Mod}_{p}(\Sigma)=\inf_{f\in\operatorname{adm}(\Sigma)}\int_{X}\left|{f}\right|^{p}\mathop{}\\!\mathrm{d}\mu.$

A family of measures $\Sigma$ on $(X,\mu)$ is called _$p$ -exceptional_ if
$\operatorname{Mod}_{p}(\Sigma)=0$. A property of measures is said to hold for
_$p$ -almost every ($p$-a.e.) $\nu\in\mathcal{M}_{\mu}$_ if it holds outside a
$p$-exceptional subset of $\mathcal{M}_{\mu}$.

We recall some basic properties of the $p$-modulus (see [7, Chapter I]). Given
a measure $\nu$, we use $\overline{\nu}$ to denote its completion. Moreover,
if $\Sigma,\Sigma^{\prime}\subset\mathcal{M}_{\mu}$, then we denote
$\Sigma\geq\Sigma^{\prime}$ if, for every $\sigma\in\Sigma$, there exists
$\sigma^{\prime}\in\Sigma^{\prime}$ such that
$\sigma(A)\geq\sigma^{\prime}(A)$ for all $\mu$-measurable $A\subset X$. In
this case, we say that $\Sigma^{\prime}$ _minorizes_ $\Sigma$.

######  Lemma 2.2.

Let $(X,\mu)$ be a complete measure space and let $p\in[1,\infty)$.

  1. (1)

The modulus $\operatorname{Mod}_{p}$ is an outer measure on
$\mathcal{M}_{\mu}$.

  2. (2)

If $\Sigma,\Sigma^{\prime}\subset\mathcal{M}_{\mu}$ and
$\Sigma\geq\Sigma^{\prime}$, then
$\operatorname{Mod}_{p}(\Sigma)\leq\operatorname{Mod}_{p}(\Sigma^{\prime})$.

  3. (3)

If $E\subset X$ is such that $\mu(E)=0$, then $\nu(E)=0$ for $p$-a.e.
$\nu\in\mathcal{M}_{\mu}$.

  4. (4)

If $f\in L^{p}(X,\mu)$, then $f\in L^{1}(X,\overline{\nu})$ for $p$-a.e.
$\nu\in\mathcal{M}_{\mu}$.

  5. (5)

If $f\in L^{p}(X,\mu)$ and $f_{i}\to f$ in $L^{p}(X,\mu)$, then there exists a
subsequence $f_{i_{j}}$ of $f_{i}$ such that $f_{i_{j}}\to f$ in
$L^{1}(X,\overline{\nu})$ for $p$-a.e. $\nu\in\mathcal{M}_{\mu}$.

The following lemma gives a characterization for $p$-exceptional sets.

######  Lemma 2.3 ([7, Theorem 2]).

Let $(X,\mu)$ be a complete measure space, let
$\Sigma\subset\mathcal{M}_{\mu}$ and let $p\in[1,\infty)$. The set $\Sigma$ is
$p$-exceptional if and only if there exists a function $f\in L^{p}(X,\mu)$
such that $f\geq 0$ and

$\int_{X}f\mathop{}\\!\mathrm{d}\sigma=\infty$

for every $\sigma\in\Sigma$.

For two sets $\Sigma,\Sigma^{\prime}\subset\mathcal{M}_{\mu}$, we say that
$\Sigma^{\prime}$ _weakly minorizes_ $\Sigma$ if, for every $\sigma\in\Sigma$,
there exists $\sigma^{\prime}\in\Sigma^{\prime}$ and $c_{\sigma}\in(0,1)$ such
that $\sigma(A)\geq c_{\sigma}\sigma^{\prime}(A)$ for all $\mu$-measurable
$A\subset X$. In this case, we denote $\Sigma\gtrsim\Sigma^{\prime}$. Since
the constant $c_{\sigma}$ in the definition of weak minorization can depend on
$\sigma$, there is no estimate on $\operatorname{Mod}_{p}(\Sigma)$ in terms of
$\operatorname{Mod}_{p}(\Sigma^{\prime})$ if $\Sigma\gtrsim\Sigma^{\prime}$.
However, we do get the following result for exceptional sets as an immediate
corollary of Lemma 2.3.

######  Corollary 2.4.

Let $(X,\mu)$ be a complete measure space, let
$\Sigma,\Sigma^{\prime}\subset\mathcal{M}_{\mu}$ and let $p\in[1,\infty)$. If
$\Sigma^{\prime}$ is $p$-exceptional and $\Sigma\gtrsim\Sigma^{\prime}$, then
$\Sigma$ is $p$-exceptional.

A measurable function $f\colon X\to[0,\infty]$ is _$p$ -weakly admissible_ for
$\Sigma\subset\mathcal{M}_{\mu}$ if (1) holds for $p$-almost every
$\nu\in\mathcal{M}_{\mu}$. The set of such functions is denoted by
$\operatorname{ess\,adm}_{p}(\Sigma)$. It follows that

(2)
$\operatorname{Mod}_{p}(\Sigma)=\inf_{f\in\operatorname{ess\,adm}_{p}(\Sigma)}\int_{X}\left|{f}\right|^{p}\mathop{}\\!\mathrm{d}\mu.$

If $p>1$, the infimum in (2) is in fact a minimum with a unique minimizer (see
the discussion in [7, pp. 181–182]).

##  3\. Lipschitz manifolds

###  3.1. Lipschitz manifolds without boundary

Let $\mathcal{R}$ be a topological $n$-manifold without boundary, where we
assume all manifolds are Hausdorff and second-countable. An atlas
$\mathcal{A}$ of charts $\phi\colon\Omega_{\phi}\to U_{\phi}$ with
$\Omega_{\phi}\subset\mathbb{R}^{n}$ and $U_{\phi}\subset\mathcal{R}$ is a
_Lipschitz atlas_ if, for any two charts $\phi,\psi\in\mathcal{A}$, the
transition map $\phi^{-1}\circ\psi\colon\psi^{-1}(U_{\phi}\cap
U_{\psi})\to\phi^{-1}(U_{\phi}\cap U_{\psi})$ is bilipschitz. A topological
manifold $\mathcal{R}$ equipped with a Lipschitz atlas $\mathcal{A}$ is called
a _Lipschitz manifold_. The charts $\phi\in\mathcal{A}$ are called
_bilipschitz charts_.

A _Lipschitz Riemannian metric_ $g$ on $\mathcal{R}$ is a choice of a
measurable Riemannian metric $\phi^{*}g\in\Gamma(T^{*}\Omega_{\phi}\otimes
T^{*}\Omega_{\phi})$ for every chart $\phi\in\mathcal{A}$. . The metric
satisfies that for all $\phi,\psi\in\mathcal{A}$ and $v\in{\mathbb{R}}^{n}$,
there exists $C_{\phi}\in[1,\infty)$ so that
$C_{\phi}^{-1}\left|{v}\right|^{2}\leq\phi^{*}g(v,v)\leq
C_{\phi}\left|{v}\right|^{2}$ a.e. on $\Omega_{\phi}$ and
$(\phi^{-1}\circ\psi)^{*}(\phi^{*}g)=\psi^{*}g$ a.e. on
$\psi^{-1}(U_{\phi}\cap U_{\psi})$. A Lipschitz manifold $\mathcal{R}$
equipped with a Lipschitz Riemannian metric $g$ is a _Lipschitz Riemannian
manifold_.

With a Lipschitz structure, we may define the set of measurable differential
forms, $\Gamma(\wedge^{k}T^{*}M)$ in the usual way, for $M\subset\mathcal{R}$
measurable (see e.g. [16]). If $M$ is orientable, we may also define a volume
form, denoted $\operatorname{vol}_{g}$. If $f\colon\mathcal{R}\to[0,\infty]$
is a measurable function we can define the usual integral of $f$ against the
volume form using a continuous partition of unity. In particular,
$\operatorname{vol}_{g}$ induces a measure on $\mathcal{R}$. Moreover, if
$\omega,\omega^{\prime}\in\Gamma(\wedge^{k}T^{*}\mathcal{R})$, we obtain a
well-defined measurable inner product
$\left<\omega,\omega^{\prime}\right>\colon\mathcal{R}\to\mathbb{R}$. This
allows us to define point-wise norms
$\left|{\omega}\right|\colon\mathcal{R}\to[0,\infty)$ for measurable $k$-forms
a.e. on $\mathcal{R}$. We thus obtain the spaces $L^{p}(\wedge^{k}T^{*}D)$ and
$L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}D)$ for any open set
$D\subset\mathcal{R}$ in the usual way, where the $L^{p}$-norm is given by

$\lVert\omega\rVert_{L^{p}}=\left(\int_{\mathcal{R}}\left|{\omega}\right|^{p}\operatorname{vol}_{g}\right)^{\frac{1}{p}}.$

###  3.2. Sobolev spaces on Lipschitz manifolds

Let $\Omega$ be a domain in $\mathbb{R}^{n}$. We recall that if $\omega\in
L^{1}_{\mathrm{loc}}(\wedge^{k}T^{*}\Omega)$, then $d\omega\in
L^{1}_{\mathrm{loc}}(\wedge^{k+1}T^{*}\Omega)$ is a _weak differential of
$\omega$_ if

(3) $\int_{\Omega}d\omega\wedge\eta=(-1)^{k+1}\int_{\Omega}\omega\wedge d\eta$

for all $\eta\in C^{\infty}_{0}(\wedge^{n-k-1}T^{*}\Omega)$. The Sobolev space
of all $\omega\in L^{p}(\wedge^{k}T^{*}\Omega)$ with a weak differential
$d\omega\in L^{p}(\wedge^{k}T^{*}\Omega)$ is denoted by
$W^{d,p}(\wedge^{k}T^{*}\Omega)$. Locally integrable versions
$W^{d,p}_{\mathrm{loc}}(\wedge^{k}T^{*}\Omega)$ are defined accordingly.

If $\Upsilon\subset\mathbb{R}^{n}$ is another domain and
$f\colon\Omega\to\Upsilon$ is bilipschitz, then the Lusin $(N^{-1})$-property
of $f$ along with the almost everywhere differentiability of $f$ yields a
measurable pull-back $f^{*}\omega$ on $\Omega$ for any measurable $k$-form
$\omega$ on $\Upsilon$. We recall the following chain rule for such pull-
backs, sketching its proof for the convenience of the reader (See also [4,
Lemma 2.22] and [11, Lemma 3.6]).

######  Lemma 3.1.

Let $\Omega,\Upsilon\subset\mathbb{R}^{n}$ be connected open domains, let
$f\colon\Omega\to\Upsilon$ be bilipschitz and let $p\in[1,\infty]$. If
$\omega\in W^{d,p}(\wedge^{k}T^{*}\Upsilon)$, then $f^{*}\omega\in
W^{d,p}(\wedge^{k}T^{*}\Omega)$ and $df^{*}\omega=f^{*}d\omega$.

###### Proof.

Let $L$ be the bilipschitz constant of $f$. If $\omega\in
L^{p}(\wedge^{k}T^{*}\Upsilon)$, then $f^{*}\omega\in
L^{p}(\wedge^{k}T^{*}\Omega)$ for $p\in[1,\infty)$ by the following standard
change of variables argument:

(4)
$\int_{\Omega}\left|{f^{*}\omega}\right|^{p}\leq\int_{\Omega}(\left|{\omega}\right|^{p}\circ
f)\left|{Df}\right|^{kp}\leq
L^{n+kp}\int_{\Omega}(\left|{\omega}\right|^{p}\circ
f)J_{f}=L^{n+kp}\int_{f(\Omega)}\left|{\omega}\right|^{p}.$

The same holds for $p=\infty$ by using $\left|{f^{*}\omega}\right|\leq
L^{k}(\left|{\omega}\right|\circ f)$ and the Lusin properties of $f$.

Checking that $df^{*}\omega=f^{*}d\omega$ weakly is also a standard
approximation argument. Indeed, one selects $\Omega^{\prime}$ compactly
contained in $\Omega$, chooses a compactly supported $\omega^{\prime}\in
W^{d,p}_{0}(\wedge^{k}T^{*}\mathbb{R}^{n})$ that coincides with $\omega$ on
$f(\Omega^{\prime})$ and mollifies both $f$ and $\omega^{\prime}$ to obtain
$f_{i}\in C^{\infty}(\Omega^{\prime},\mathbb{R}^{n})$ and $\omega_{i}\in
C^{\infty}_{0}(\wedge^{k}T^{*}\mathbb{R}^{n})$. We have
$df_{i}^{*}\omega_{j}=f_{i}^{*}d\omega_{j}$. Since $\left|{f_{i}-f}\right|\to
0$ uniformly, $\left|{Df_{i}}\right|\leq L$ a.e. on $\Omega^{\prime}$,
$Df_{i}\to Df$ pointwise a.e. on $\Omega$ and each $\omega_{j}$ and
$d\omega_{j}$ is uniformly continuous, dominated convergence yields that
$f_{i}^{*}\omega_{j}\to f^{*}\omega_{j}$ and $f_{i}^{*}d\omega_{j}\to
f^{*}d\omega_{j}$ in the $L^{1}$-norm on $\Omega^{\prime}$. Hence, it follows
that $f^{*}d\omega_{j}=df^{*}\omega_{j}$. Then since
$\omega_{j}\to\omega^{\prime}$ and $d\omega_{j}\to d\omega^{\prime}$ in the
$L^{1}$-norm, (4) yields that $f^{*}\omega_{j}\to
f^{*}\omega^{\prime}=f^{*}\omega$ and $f^{*}d\omega_{j}\to
f^{*}d\omega^{\prime}=f^{*}d\omega$ in the $L^{1}$-norm on $\Omega^{\prime}$.
Hence, $df^{*}\omega=f^{*}d\omega$ weakly on $\Omega^{\prime}$. This
generalizes to $\Omega$ by the locality of weak differentials. ∎

For an open $U$ in an oriented Lipschitz Riemannian manifold, we define that
$d\omega\in L^{1}_{\mathrm{loc}}(\wedge^{k+1}T^{*}U)$ is a weak differential
of $\omega\in L^{1}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$ if $\phi^{*}d\omega$ is
a weak differential of $\phi^{*}\omega$, for every bilipschitz chart $\phi$
with $U_{\phi}\subset U$. This lets us define Sobolev spaces
$W^{d,p}(\wedge^{k}T^{*}U)$ and $W^{d,p}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$.
The space $W^{d,p}(\wedge^{k}T^{*}U)$ has the norm
$\lVert\omega\rVert_{W^{d,p}}=\lVert\omega\rVert_{L^{p}}+\lVert
d\omega\rVert_{L^{p}}$. It can be shown that $d\omega$ is a weak differential
of $\omega\in L^{1}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$ precisely if it
satisfies (3) for test forms $\eta\in W^{d,\infty}(\wedge^{n-k-1}T^{*}U)$ with
$\operatorname{spt}\eta\subset U$ compact.

We also state the following approximation results.

######  Lemma 3.2.

Let $U$ be an open set in an oriented Lipschitz Riemannian manifold
$\mathcal{R}$ and let $p\in[1,\infty)$. The space
$W^{d,\infty}(\wedge^{k}T^{*}U)$ is dense in both $L^{p}(\wedge^{k}T^{*}U)$
and $W^{d,p}(\wedge^{k}T^{*}U)$.

###### Proof.

We sketch the case $\omega\in W^{d,p}(\wedge^{k}T^{*}U)$, as the proof of the
other case is analogous. Let $\varepsilon>0$. We take a countable locally
finite cover of $U$ with charts $\phi_{i}\colon\Omega_{i}\to U_{i}$,
$i\in{\mathbb{N}}$ and use a subordinate $W^{d,\infty}$-partition of unity to
find $\omega_{i}\in W^{d,p}(\wedge^{k}T^{*}U)$ such that
$\omega=\sum_{i}\omega_{i}$ and $\operatorname{spt}\omega_{i}\subset U_{i}$.
We take a smooth convolution approximation $\eta_{i}$ of
$\phi_{i}^{*}\omega_{i}$ such that
$\lVert\eta_{i}-\phi_{i}^{*}\omega_{i}\rVert_{W^{d,p}}<C_{i}^{-(k+1)p-n}2^{-i}\varepsilon$,
where $C_{i}$ is the constant such that
$C_{i}^{-1}\left|{v}\right|\leq\sqrt{(\phi_{i}^{*}g)_{x}(v,v)}\leq
C_{i}\left|{v}\right|$ for every $v\in\mathbb{R}^{n}$ and a.e.
$x\in\Omega_{i}$. The extension of $(\phi_{i}^{-1})^{*}\eta_{i}$ by zero
defines a $W^{d,\infty}$-form on $U$ such that
$\lVert\omega_{i}-(\phi_{i}^{-1})^{*}\eta_{i}\rVert_{W^{d,p}}<2^{-i}\varepsilon$.
The sum of these forms gives the desired approximation of $\omega$. ∎

###  3.3. Lipschitz domains and submanifolds

A connected, closed region $M\subset\mathcal{R}$ in a Lipschitz $n$-manifold
$\mathcal{R}$ is called a _(weakly) Lipschitz domain_ if it satisfies the
following condition: For any $x\in\partial M$, there exists a bilipschitz
chart $\phi\in\mathcal{A}$ such that $x\in U_{\phi}$,
$\phi(\mathbb{H}^{n}_{+}\cap\Omega_{\phi})=\operatorname{int}(M)\cap
U_{\phi}$, $\phi(\mathbb{H}^{n}_{-}\cap\Omega_{\phi})=(\mathcal{R}\setminus
M)\cap U_{\phi}$ and
$\phi(\partial\mathbb{H}^{n}_{+}\cap\Omega_{\phi})=\partial M\cap U_{\phi}$.
Here, $\mathbb{H}^{n}_{+}=\\{x\in\mathbb{R}^{n}:x_{n}>0\\}$ and
$\mathbb{H}^{n}_{-}=\\{x\in\mathbb{R}^{n}:x_{n}<0\\}$ are half-spaces of
$\mathbb{R}^{n}$. A chart $\phi$ as above is denoted a _bilipschitz boundary
chart of $M$_. Note that we allow for a Lipschitz domain $M$ to be non-
compact. More generally, a _closed Lipschitz $n$-submanifold_ $M$ of
$\mathcal{R}$ is a locally finite disjoint union of Lipschitz domains in
$\mathcal{R}$; this is equivalent to the definition of a Lipschitz domain
without the connectedness assumption. If a closed Lipschitz $n$-submanifold is
compact, we refer to it as a _compact Lipschitz $n$-submanifold_.

The boundary $\partial M$ of a closed Lipschitz $n$-submanifold $M$ is a
Lipschitz $(n-1)$-manifold, as the restrictions of $\phi\in\mathcal{A}$ to
$\partial\mathbb{H}^{n}_{+}\cong\mathbb{R}^{n-1}$ provide a Lipschitz atlas
$\mathcal{A}_{\partial M}$. For subsets $A\subset\partial M$, we use
$\operatorname{int}(A)$ and $\partial A$ to refer to the interior and
boundary, respectively, in $\partial M$.

We can define $L^{p}$ and $L^{p}_{\mathrm{loc}}$-spaces of differential forms
on a closed Lipschitz $n$-submanifold $M$ in the usual way. We consequently
define Sobolev spaces $W^{d,p}(\wedge^{k}T^{*}M)$ by requiring that
$d\omega\in L^{p}(\wedge^{k+1}T^{*}M)$ is a weak differential of $\omega\in
L^{p}(\wedge^{k}T^{*}M)$ in $\operatorname{int}(M)$ and also
$W^{d,p}_{\mathrm{loc}}$-spaces with a similar definition. For the rest of
this section, we let $M$ be a closed Lipschitz $n$-submanifold in
$\mathcal{R}$, $U^{\prime}$ be an open subset of $\mathcal{R}$ and
$U=U^{\prime}\cap M$.

Suppose that $D$ is a closed Lipschitz $(n-1)$-submanifold in $\partial M$.
Let $\omega\in W^{d,1}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$. _The tangential part
$\omega_{T}$ of $\omega$ vanishes weakly on $D$_ if the extension of $d\omega$
by zero to $(U^{\prime}\setminus\partial M)\cup(U\cap\operatorname{int}(D))$
is the weak differential of the extension of $\omega$ by zero to
$(U^{\prime}\setminus\partial M)\cup(U\cap\operatorname{int}(D))$. This is
only dependent on $U$, $M$ and $D$ and not the choice of $U^{\prime}$. The
space of $\omega\in W^{d,p}(\wedge^{k}T^{*}U)$ with a weakly vanishing
$\omega_{T}$ on $D$ is denoted $W^{d,p}_{T(D)}(\wedge^{k}T^{*}U)$ and a
similar notation $W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}U)$ is used for
the locally integrable version.

The Lipschitz Riemannian metric $g$ on $\mathcal{R}$ defines a Hodge star
operator $\mathtt{\star}\hskip 1.0pt_{g}$ on $\mathcal{R}$. The weak
codifferential is defined by $d^{*}\omega=(-1)^{n(k-1)+1}\mathtt{\star}\hskip
1.0pt_{g}d\mathtt{\star}\hskip 1.0pt_{g}\omega$, whenever $\omega\in
L^{1}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$ is such that $\mathtt{\star}\hskip
1.0pt_{g}\omega$ is weakly differentiable. The space of all forms $\omega\in
L^{p}(\wedge^{k}T^{*}U)$ with a weak codifferential $d^{*}\omega\in
L^{p}(\wedge^{k-1}T^{*}U)$ is denoted $W^{d^{*},p}(\wedge^{k}T^{*}U)$ and
locally integrable versions of the spaces are defined accordingly.

For $\omega\in W^{d^{*},1}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$, the _normal part
$\omega_{N}$ of $\omega$ vanishes weakly on $D$_ if the tangential part
$(\mathtt{\star}\hskip 1.0pt_{g}\omega)_{T}$ of $\mathtt{\star}\hskip
1.0pt_{g}\omega$ vanishes weakly on $D$. In particular, this occurs if the
extension of $d^{*}\omega$ by zero to $(U^{\prime}\setminus\partial
M)\cup(U\cap\operatorname{int}(D))$ is the weak codifferential of the
extension of $\omega$ by zero to $(U^{\prime}\setminus\partial
M)\cup(U\cap\operatorname{int}(D))$. We denote the space of Sobolev $k$-forms
$\omega\in W^{d^{*},p}(\wedge^{k}T^{*}U)$ with weakly vanishing normal part on
$D$ by $W^{d^{*},p}_{N(D)}(\wedge^{k}T^{*}U)$ and a corresponding locally
integrable version $W^{d^{*},p}_{N(D),\mathrm{loc}}(\wedge^{k}T^{*}U)$ is also
defined.

We note that the spaces $W^{d,p}_{T(D)}(\wedge^{k}T^{*}U)$ and
$W^{d^{*},p}_{N(D)}(\wedge^{k}T^{*}U)$ are complete. Indeed, this is a direct
consequence of the completeness of $W^{d,p}(\wedge^{k}T^{*}U)$ and
$W^{d^{*},p}(\wedge^{k}T^{*}U)$, as taking extensions by zero past
$\operatorname{int}(D)$ has no effect on $L^{p}$-convergence.

We prove a version of Lemma 3.2 for $W^{d,p}_{T(D)}(\wedge^{k}T^{*}M)$. For
smooth domains $M$ in smooth manifolds $\mathcal{R}$, the case $D=\partial M$
can be found in [12, Corollary 3.8]. Our proof is somewhat similar to theirs,
except the handling of charts that meet $\partial D$ requires some technical
modifications.

######  Lemma 3.3.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let $U\subset M$ be
open in $M$ and let $D$ be a closed Lipschitz $(n-1)$-submanifold in $\partial
M$. If $p\in[1,\infty)$, then every $\omega\in
W^{d,p}_{T(D)}(\wedge^{k}T^{*}U)$ can be approximated in
$\lVert\cdot\rVert_{W^{d,p}}$-norm by $\omega_{j}\in
W^{d,\infty}_{T(D)}(\wedge^{k}T^{*}U)$ with
$\operatorname{spt}\omega_{j}\subset U\setminus D$.

Before we begin the proof, we recall the following reflection property of
$W^{d,p}$-forms in the Euclidean setting. It is discussed in [12, p.48, 50].

######  Lemma 3.4.

Let $B$ be the unit ball $B^{n}(0,1)$ in $\mathbb{R}^{n}$ and suppose that
$\omega\in W^{d,p}(\wedge^{k}T^{*}(B\cap\mathbb{H}^{n}_{+}))$ for some
$p\in[1,\infty]$. Extend $\omega$ to $B\cap\mathbb{H}^{n}_{-}$ by
$\Psi^{*}(\omega)=-\omega$, where $\Psi\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$
is the reflection map defined by
$\Psi(x_{1},\dots,x_{n})=\Psi(x_{1},\dots,x_{n-1},-x_{n})$. Then the extended
$\omega$ is in $W^{d,p}(\wedge^{k}T^{*}B)$.

###### Proof of Lemma 3.3.

Let $B$ be the unit ball $B^{n}(0,1)$ in $\mathbb{R}^{n}$. We can take a
locally finite cover of $U$ with bilipschitz charts which are of the following
three types:

  1. (1)

$\phi_{i}\colon B\to U_{i}$, where $U_{i}\subset\operatorname{int}(M)$;

  2. (2)

$\phi_{i}\colon B\to U_{i}$, where
$\phi_{i}(B\cap\mathbb{H}^{n}_{+})\subset\operatorname{int}M$,
$\phi_{i}(B\cap\mathbb{H}^{n}_{-})\subset\mathcal{R}\setminus M$ and
$\phi_{i}(B\cap\partial\mathbb{H}^{n}_{+})\subset\operatorname{int}D$;

  3. (3)

$\phi_{i}\colon B\to U_{i}$, where
$\phi_{i}(B\cap\mathbb{H}^{n}_{+})\subset\operatorname{int}M$,
$\phi_{i}(B\cap\mathbb{H}^{n}_{-})\subset\mathcal{R}\setminus M$ and
$\phi_{i}(B\cap\partial\mathbb{H}^{n}_{+})\subset\partial M\setminus D$;

  4. (4)

$\phi_{i}\colon B\to U_{i}$, where

     * •

$\phi_{i}(B\cap\mathbb{H}^{n}_{+})\subset\operatorname{int}M$,

     * •

$\phi_{i}(B\cap\mathbb{H}^{n}_{-})\subset\mathcal{R}\setminus M$,

     * •

$\phi_{i}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=0,x_{1}>0\\})\subset\operatorname{int}D$,

     * •

$\phi_{i}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=0,x_{1}<0\\})\subset\partial
M\setminus D$,

     * •

and $\phi_{i}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=0,x_{1}=0\\})\subset\partial
D$.

Similarly as in the proof of Lemma 3.2, we can use a $W^{d,\infty}$ partition
of unity to reduce the question into the case where $\operatorname{spt}\omega$
is compact and contained in a single chart neighborhood $U_{i}$. We only
present the last case (4), as it demonstrates all the required techniques and
the other cases can be shown by leaving out parts of the argument.

Suppose we are in the last case. We use another bilipschitz chart to further
modify our domain. Define $\phi_{i}^{\prime}\colon B\to U_{i}$ by

  * •

$\phi_{i}^{\prime}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}\geq\max(0,x_{1})\\})=M\cap
U_{i}$,

  * •

$\phi_{i}^{\prime}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=0,x_{1}<0\\})=U_{i}\cap(\partial
M\setminus D)$,

  * •

$\phi_{i}^{\prime}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=0,x_{1}=0\\})=U_{i}\cap\partial
D$,

  * •

$\phi_{i}^{\prime}(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=x_{1},x_{1}>0\\})=U_{i}\cap\operatorname{int}D$.

We extend $\omega$ by zero outside $M$ and let
$\omega^{\prime}=(\phi_{i}^{\prime})^{*}\omega$ on $B\cap\mathbb{H}^{n}_{+}$.
By the weak vanishing of $\omega_{T}$ on $D$, we have that this extension by
zero is in $W^{d,p}(\wedge^{k}T^{*}\phi_{i}(B\cap\mathbb{H}^{n}_{+}))$.
Consequently, $d\omega^{\prime}=(\phi_{i}^{\prime})^{*}d\omega$ weakly,
$\omega^{\prime}\in W^{d,p}(\wedge^{k}T^{*}(B\cap\mathbb{H}^{n}_{+}))$ and
$\omega^{\prime}\equiv 0$ on $\\{x\in B\cap\mathbb{H}^{n}_{+}:x_{n}<x_{1}\\}$.

We then extend $\omega^{\prime}$ to $B$ via the reflection Lemma 3.4. The form
$\omega^{\prime}$ is compactly supported in $W^{d,p}(\wedge^{k}T^{*}B)$ and
$\omega^{\prime}$ vanishes on the set of points $\\{x\in
B:\left|{x_{n}}\right|<x_{1}\\}$. Let
$\tau_{t}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote the translation
$x\mapsto(x_{1}-t,x_{2},x_{3},\dots,x_{n})$. If $j>0$, then for small enough
$t$, $\tau_{t}^{*}\omega^{\prime}$ and $\tau_{t}^{*}d\omega^{\prime}$ are
well-defined compactly supported forms in $B^{n}(0,1)$ and

$\int_{B^{n}(0,1)}(\left|{\omega^{\prime}-\tau_{t}^{*}\omega^{\prime}}\right|^{p}+\left|{d\omega^{\prime}-d\tau_{t}^{*}\omega^{\prime}}\right|^{p})<\varepsilon_{j}.$

Moreover, the translated form $\tau_{t}^{*}\omega^{\prime}$ vanishes in a
neighborhood of the set $B\cap\\{x:\left|{x_{n}}\right|\leq x_{1}\\}$. Hence,
this remains true for convolutions with a small enough mollifying kernel and
we may use a convolution to pick a smooth approximation
$\omega_{j}^{\prime}\in
C^{\infty}_{0}(\wedge^{k}T^{*}(B\setminus\\{x:\left|{x_{n}}\right|\leq
x_{1}\\}))$ with
$\lVert\omega_{j}^{\prime}-\tau_{t}^{*}\omega^{\prime}\rVert_{p}+\lVert
d\omega_{j}^{\prime}-d\tau_{t}^{*}\omega^{\prime}\rVert_{p}<\varepsilon_{j}$.
Then $((\phi_{i}^{\prime})^{-1})^{*}\omega_{j}^{\prime}$ yields the desired
approximation in the chart $U_{i}$, concluding the proof of this case. ∎

######  Remark 3.5.

Note that the same argument as above can be used in the smooth manifold
setting to prove a smooth approximation version of Lemma 3.3. The only change
one has to do is to replace the condition $x_{n}\geq x_{1}$ in the definition
of $\phi_{i}^{\prime}$ by a condition of the form $x_{n}\geq f(x_{1})$, where
$f$ is a smooth non-decreasing function such that $f(t)=0$ if $t\leq 0$ and
$f(t)>0$ if $t>0$.

We end this section with a lemma about wedge products and vanishing tangential
parts, the proof of which takes advantage of Lemma 3.3.

######  Lemma 3.6.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let
$p,q,r\in[1,\infty]$ with $p^{-1}+q^{-1}\geq r^{-1}$ and let $U\subset M$ be
open in $M$. Let $D\subset\partial M$ be a closed Lipschitz
$(n-1)$-submanifold and let $E=\partial M\setminus\operatorname{int}D$. If
$\omega\in W^{d,p}_{T(D)}(\wedge^{k}T^{*}U)$ and $\tau\in
W^{d,q}_{T(E)}(\wedge^{l}T^{*}U)$, where $0\leq l+k\leq n$, then
$\omega\wedge\tau\in W^{d,r}_{T(\partial M)}(\wedge^{l+k}T^{*}U)$.

###### Proof.

By standard properties of wedge products of weakly differentiable forms, we
have $\omega\wedge\tau\in W^{d,r}(\wedge^{k+l}T^{*}U)$ with
$d(\omega\wedge\tau)=d\omega\wedge\tau+(-1)^{k}\omega\wedge d\tau$. By Lemma
3.3, we may approximate $\omega$ by $\omega_{i}\in
W^{d,\infty}_{T(D)}(\wedge^{k}T^{*}U)$ such that
$\operatorname{spt}\omega_{i}\subset M\setminus D$ and $\tau$ by $\tau_{i}\in
W^{d,\infty}_{T(E)}(\wedge^{l}T^{*}U)$ such that
$\operatorname{spt}\tau_{i}\subset M\setminus E$. Consequently,
$\operatorname{spt}(\omega_{i}\wedge\tau_{i})$ does not meet $\partial M$ and
hence $\omega_{i}\wedge\tau_{i}\in W^{d,\infty}_{T(\partial
M)}(\wedge^{k+l}T^{*}U)$. Since also
$\omega_{i}\wedge\tau_{i}\to\omega\wedge\tau$ in the $W^{d,r}$-norm on $U$,
the claim follows by completeness of $W^{d,r}_{T(\partial
M)}(\wedge^{k+l}T^{*}U)$. ∎

##  4\. Lipschitz chains and integration

Let $\mathcal{R}$ be a Lipschitz $n$-manifold without boundary. A map
$\sigma\colon\Delta_{k}\to\mathcal{R}$ that is Lipschitz, where
$\Delta_{k}\subset\mathbb{R}^{k}$ is the (closed) standard $k$-simplex, is
called a _singular Lipschitz $k$-simplex_. Given a field of coefficients
$\mathbb{K}$, a _Lipschitz $k$-chain on $\mathcal{R}$ with coefficients in
$\mathbb{K}$_ is a formal sum $\sigma=k_{1}\sigma_{1}+\dots+k_{m}\sigma_{m}$
of singular Lipschitz $k$-simplices $\sigma_{i}$, where $k_{i}\in\mathbb{K}$.
We denote the space of all such Lipschitz $k$-chains by
$C^{\operatorname{lip}}_{k}(\mathcal{R};\mathbb{K})$. If $U$ is a subset of
$\mathcal{R}$, then $C^{\operatorname{lip}}_{k}(U;\mathbb{K})$ is the space of
all $\sigma=k_{1}\sigma_{1}+\dots+k_{m}\sigma_{m}\in
C^{\operatorname{lip}}_{k}(\mathcal{R};\mathbb{K})$ such that the image of
each $\sigma_{i}$ is contained in $U$. We also obtain a boundary map
$\partial\colon C^{\operatorname{lip}}_{k+1}(U;\mathbb{K})\to
C^{\operatorname{lip}}_{k}(U;\mathbb{K})$ in the usual way.

###  4.1. Lipschitz chains in Euclidean domains

We first go over the key properties of the space
$C^{\operatorname{lip}}_{k}(\Omega;\mathbb{R})$, where
$\Omega\subset\mathbb{R}^{n}$ is a connected open domain.

Suppose that $\sigma\colon\Delta_{k}\to\Omega$ is a singular Lipschitz
$k$-simplex. Rademacher’s theorem yields a derivative
$D\sigma(x)\colon\mathbb{R}^{k}\to\mathbb{R}^{n}$ for almost every
$x\in\Delta_{k}$. We may define a corresponding Borel measure $\nu_{\sigma}$
for $\sigma$ on $\Omega$ by

(5)
$\int_{\Omega}\eta\mathop{}\\!\mathrm{d}\nu_{\sigma}=\int_{\Delta_{k}}(\eta\circ\sigma)\left|{J_{\sigma}}\right|\operatorname{vol}_{k}.$

for any Borel measurable function $\eta\colon\Omega\to\mathbb{R}$. Here, the
absolute value of the Jacobian $\left|{J_{\sigma}}\right|$ is defined a.e. on
$\Delta_{k}$ by

(6) $\left|{J_{\sigma}}\right|(x)=\sqrt{\det D^{T}\sigma(x)D\sigma(x)}.$

Moreover, we can extend this definition to Lipschitz $k$-chains
$\sigma=k_{1}\sigma_{1}+\dots+k_{m}\sigma_{m}$ by
$\nu_{\sigma}=k_{1}\nu_{\sigma_{1}}+\dots+k_{m}\nu_{\sigma_{m}}$. Using the
measure $\nu_{\sigma}$, we define the modulus of a family $\mathcal{S}$ of
Lipschitz $k$-chains, denoted

$\operatorname{Mod}_{p}(\mathcal{S})=\operatorname{Mod}_{p}\left(\left\\{\overline{\nu_{\sigma}}:\sigma\in\mathcal{S}\right\\}\right).$

Let $\sigma\colon\Delta_{k}\to\Omega$ be a singular Lipschitz $k$-simplex in
$\Omega$ and let $\omega$ be a Borel differential $k$-form on $\Omega$ such
that $\left|{\omega}\right|\in L^{1}(\nu_{\sigma})$. We may define the
integral of $\omega$ over the surface $\sigma$. The pull-back
$\sigma^{*}\omega$ is a well-defined measurable form since $\omega$ is Borel.
Moreover, since $\sigma^{*}\omega$ is a top-dimensional form in $\Delta_{k}$,
we have

$\displaystyle\left|{\sigma^{*}\omega_{x}}\right|$
$\displaystyle\leq\left|{\omega_{\sigma(x)}}\right|\left|{J_{\sigma}}\right|$

and

(7)
$\int_{\Delta_{k}}\left|{\sigma^{*}\omega}\right|\operatorname{vol}_{k}\leq\int_{\Delta_{k}}(\left|{\omega}\right|\circ\sigma)\left|{J_{\sigma}}\right|\operatorname{vol}_{k}=\int_{\Omega}\left|{\omega}\right|\mathop{}\\!\mathrm{d}\nu_{S}<\infty.$

Thus, $\sigma^{*}\omega$ is integrable. So

(8) $\int_{\sigma}\omega:=\int_{\Delta_{k}}\sigma^{*}\omega.$

The definition extends to $\sigma\in
C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$ linearly.

We crucially obtain a version of the Fuglede lemma for differential forms.

######  Lemma 4.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a connected open domain and let
$p\in[1,\infty)$. If $\omega\in L^{p}(\wedge^{k}T^{*}\Omega)$ and
$\omega_{i}\to\omega$ in $L^{p}(\wedge^{k}T^{*}\Omega)$, then there exists a
subsequence $\omega_{i_{j}}$ of $\omega_{i}$ such that

$\int_{\sigma}\omega_{i_{j}}\to\int_{\sigma}\omega$

for $p$-a.e. $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$.

###### Proof.

The forms $\omega$ and $\omega_{i}$ are integrable over $\sigma$ for $p$-a.e.
$\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$. Lemma 2.2, (5)
yields a subsequence $\omega_{i_{j}}$ such that

$\int_{\Omega}\left|{\omega_{i_{j}}-\omega}\right|\mathop{}\\!\mathrm{d}\nu_{\sigma}\to
0$

for $p$-a.e. $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$. It
follows using (7) and (8) that

$\int_{\sigma}(\omega_{i_{j}}-\omega)\to 0$

for $p$-a.e. $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$. ∎

###  4.2. Wolfe’s theorem

We have a reasonable integral for $\omega\in L^{p}(\wedge^{k}T^{*}\Omega)$
over $p$-a.e. Lipschitz $k$-chain $\sigma\in
C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$. However, a much stronger
result can be achieved for $\omega\in W^{d,\infty}(\wedge^{k}T^{*}\Omega)$.
According to _Wolfe’s theorem_ , there is a definition of an integral of
$\omega\in W^{d,\infty}(\wedge^{k}T^{*}\Omega)$ over _all_ Lipschitz
$k$-chains $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$, despite
the fact that $\omega$ is defined only up to a set of measure zero. Standard
references for this can be found in the books of Whitney [19, Chapter IX] and
Federer [5, 4.1.19]. See also the work of Petit, Rajala and Wenger [14], which
uses methods closer to the setting of this paper.

We state the precise properties of the integral we require. The following
Proposition 4.2 and Lemma 4.3 are reasonably immediate consequences from the
results in [5] and [19]. However, for the reader unfamiliar with this theory,
we have given a more detailed account of the proof with precise literary
references in the appendix of this paper.

######  Proposition 4.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a connected open domain. There exists a
unique bilinear map

$W^{d,\infty}(\wedge^{k}T^{*}\Omega)\times
C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})\to\mathbb{R},\quad(\omega,\sigma)\mapsto\int_{\sigma}\omega$

with the following properties.

  1. (1)

If $\omega\in W^{d,\infty}(\wedge^{k}T^{*}\Omega)$ is smooth, then

$\int_{\sigma}\omega=\int_{\Delta_{k}}\sigma^{*}\omega.$

for all singular Lipschitz $k$-simplices $\sigma\colon\Delta_{k}\to\Omega$.

  2. (2)

If $\omega\in W^{d,\infty}(\wedge^{k}T^{*}\Omega)$ is Borel, then for every
$p\in[1,\infty)$,

$\int_{\sigma}\omega=\int_{\Delta_{k}}\sigma^{*}\omega,$

for $p$-a.e. singular Lipschitz $k$-simplices
$\sigma\colon\Delta_{k}\to\Omega$.

  3. (3)

For all $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$ and
$\omega\in W^{d,\infty}(\wedge^{k-1}T^{*}\Omega)$,

$\int_{\partial\sigma}\omega=\int_{\sigma}d\omega.$

  4. (4)

For all $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$ and
$\omega,\omega_{j}\in W^{d,\infty}(\wedge^{k}T^{*}\Omega)$ where
$j\in\mathbb{N}$, if $\omega_{j}\to\omega$ almost everywhere pointwise and
$\sup_{j}\lVert\omega_{j}\rVert_{W^{d,\infty}}<\infty$, then

$\lim_{j\to\infty}\int_{\sigma}\omega_{j}=\int_{\sigma}\omega.$

  5. (5)

If $\sigma\in C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$, $\tau\in
C_{k+1}^{\operatorname{lip}}(\Omega;\mathbb{R})$ and $\omega\in
W^{d,\infty}(\wedge^{k}T^{*}\Omega)$, then

$\left|{\int_{\sigma+\partial\tau}\omega}\right|\leq(\lVert\omega\rVert_{L^{\infty}}+\lVert
d\omega\rVert_{L^{\infty}})(\nu_{\sigma}(\Omega)+\nu_{\tau}(\Omega)).$

  6. (6)

If $\Omega^{\prime}\subset\mathbb{R}^{n}$ is another domain, $\omega\in
W^{d,\infty}(\wedge^{k}T^{*}\Omega)$ and $\omega^{\prime}\in
W^{d,\infty}(\wedge^{k}T^{*}\Omega^{\prime})$ are such that
$\omega|_{\Omega\cap\Omega^{\prime}}=\omega^{\prime}|_{\Omega\cap\Omega^{\prime}}$
and $\sigma\in
C^{\operatorname{lip}}_{k}(\Omega\cap\Omega^{\prime};\mathbb{R})$, then

$\int_{\sigma}\omega=\int_{\sigma}\omega^{\prime}.$

Suppose that $f\colon\Omega\to\Upsilon$ is a Lipschitz map between two
Euclidean domains. The push-forward $f_{*}\sigma\in
C_{k}^{\operatorname{lip}}(\Upsilon;\mathbb{R})$ for every $\sigma\in
C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$ is given by
$f_{*}\sigma=\sigma\circ f$ for any Lipschitz $k$-simplex $\sigma$. The
definition is extended to chains linearly.

######  Lemma 4.3.

Let $\Omega,\Upsilon\subset\mathbb{R}^{n}$ be connected open domains, let
$f\colon\Omega\to\Upsilon$ be bilipschitz and let $p\in[1,\infty]$. If
$\omega\in W^{d,p}(\wedge^{k}T^{*}\Upsilon)$, then

$\int_{\sigma}f^{*}\omega=\int_{f_{*}\sigma}\omega.$

for $p$-a.e. $\sigma\in C^{\operatorname{lip}}_{k}(\Omega;\mathbb{R})$ and for
all $\sigma\in C^{\operatorname{lip}}_{k}(\Omega;\mathbb{R})$ when $p=\infty$.

Finally, in our applications, we will also have to consider integration of
$W^{d,\infty}$-forms over Lipschitz $k$-chains that meet the boundary of a
Lipschitz domain $D$. The following property of the integral will play a
significant role in making that boundary part of the theory work.

######  Lemma 4.4.

Let $\Omega\subset\mathbb{R}^{n}$ be open and let $\omega\in
W^{d,\infty}(\wedge^{k}T^{*}\Omega)$. If $\omega\equiv 0$ on
$\Omega\cap\mathbb{H}_{+}^{n}$, then the integral of $\omega$ over every
$\sigma\in
C_{k}^{\operatorname{lip}}(\Omega\cap\overline{\mathbb{H}^{n}_{+}};\mathbb{R})$
vanishes.

###### Proof.

The claim is clear for all $\sigma\in
C_{k}^{\operatorname{lip}}(\Omega\cap\mathbb{H}^{n}_{+};\mathbb{R})$ by
Proposition 4.2 part (6), since $\omega$ vanishes in a neighborhood of the
total image of $\sigma$. The part that requires attention is proving the
result for a Lipschitz $k$-simplex
$\sigma\colon\Delta_{k}\to\Omega\cap\overline{\mathbb{H}^{n}_{+}}$ that meets
$\partial\mathbb{H}^{n}_{+}$. Let $\sigma$ be such a $k$-simplex.

The proof is via a standard flat approximation argument. For every
$\varepsilon>0$, we define a map
$H_{\varepsilon}\colon\Delta_{k}\times[0,\varepsilon]\to\Omega\cap\overline{\mathbb{H}^{n}_{+}}$
by $H_{\varepsilon}(x,t)=\sigma(x)+(0,0,\dots,0,t)$; for small enough
$\varepsilon>0$ the image is indeed in
$\Omega\cap\overline{\mathbb{H}^{n}_{+}}$. Let
$\sigma_{\varepsilon}(x)=H_{\varepsilon}(x,\varepsilon)$. The map
$H_{\varepsilon}$ can be understood as a Lipschitz $(k+1)$-chain; its total
measure is $\nu_{H_{\varepsilon}}(\Omega)=\varepsilon\nu_{\sigma}(\Omega)$ and
its boundary consists of $\sigma-\sigma_{\varepsilon}$ and a part with total
measure $\nu_{\partial
H_{\varepsilon}-(\sigma-\sigma_{\varepsilon})}(\Omega)=\varepsilon\nu_{\partial\sigma}(\Omega)$.
By Proposition 4.2 part (5), we get

$\left|{\int_{\sigma-\sigma_{\varepsilon}}\omega}\right|\leq(\lVert\omega\rVert_{L^{\infty}}+\lVert
d\omega\rVert_{L^{\infty}})(\varepsilon\nu_{\sigma}(\Omega)+\varepsilon\nu_{\partial\sigma}(\Omega))\xrightarrow[\varepsilon\to
0]{}0.$

On the other hand, $\omega$ vanishes in a neighborhood of
$\sigma_{\varepsilon}$, so

$\int_{\sigma-\sigma_{\varepsilon}}\omega=\int_{\sigma}\omega,$

completing the proof. ∎

###  4.3. Integration of $W^{d,\infty}$-forms in Lipschitz manifolds

We now have the necessary preliminaries to discuss integration over Lipschitz
$k$-chains on Lipschitz manifolds. Suppose that $\mathcal{R}$ is a Lipschitz
Riemannian manifold without boundary with metric $g$ and let
$A\subset\mathcal{R}$ be any subset of $\mathcal{R}$. Since we have given a
definition for Lipschitz maps $\sigma\colon\Delta_{k}\to\mathcal{R}$ in
Section 3.1, we obtain the spaces $C^{\operatorname{lip}}_{k}(A;\mathbb{K})$
of Lipschitz chains on $A$ with $\mathbb{K}\in\\{\mathbb{R},\mathbb{Z}\\}$.

Suppose that $U\subset\mathcal{R}$ is an open subset of $\mathcal{R}$ and that
$\omega\in W^{d,\infty}(U)$. If $\sigma\in
C^{\operatorname{lip}}_{k}(U;\mathbb{K})$, then we may subdivide $\sigma$ into
a $\sigma^{\prime}=k_{1}\sigma_{1}+\dots+k_{i}\sigma_{i}$ such that every
$\sigma_{j}$ is a Lipschitz $k$-simplex contained in a single chart
neighborhood $U_{\phi_{j}}\subset U$. Then

$\int_{\sigma}\omega=\sum_{j=1}^{i}k_{j}\int_{\phi_{j}^{-1}\circ\sigma_{j}}\phi_{j}^{*}\,\omega,$

where we use Wolfe’s canonical integral on the right hand side. By Lemma 4.3,
the right hand side integrals are independent of the choice of chart
$\phi_{j}$. By considering a common refinement of two subdivisions, we see
that the integral depends only on $\sigma$.

Next, we consider integration on Lipschitz submanifolds. Let $M$ be a closed
Lipschitz $n$-submanifold in $\mathcal{R}$, let $U\subset M$ be open in $M$
and let $\omega\in W^{d,\infty}(\wedge^{k}T^{*}U)$. Our previous definition
lets us integrate $\omega$ over all $\sigma\in
C_{k}^{\operatorname{lip}}(U\cap\operatorname{int}M;\mathbb{R})$, but we wish
to extend this to all $\sigma\in C_{k}^{\operatorname{lip}}(U;\mathbb{R})$. By
subdivision, it is again enough to consider integration over a single
Lipschitz $k$-simplex $\sigma\colon\Delta_{k}\to U$ contained in a chart
neighborhood $U_{\phi}$, where the remaining non-trivial case is that
$\phi\colon B^{n}(0,1)\to U_{\phi}$ maps
$B^{n}(0,1)\cap\overline{\mathbb{H}^{n}_{+}}$ into $M$ and
$B^{n}(0,1)\setminus\overline{\mathbb{H}^{n}_{+}}$ outside $M$.

The form $\phi^{*}\omega$ is in
$W^{d,\infty}(\wedge^{k}T^{*}(B^{n}(0,1)\cap\mathbb{H}^{n}_{+}))$. Hence, we
can use the reflection Lemma 3.4 to extend $\phi^{*}\omega$ to a form
$\alpha\in W^{d,\infty}(\wedge^{k}T^{*}B^{n}(0,1))$. The integral of $\omega$
over $\sigma$ is the integral of $\alpha$ over $\phi^{-1}\circ\sigma$. In
order to see that this is independent of our choices, we let $\psi$ be another
such chart and let $\alpha^{\prime}$ be the extension of $\psi^{*}\omega$ by
reflection. Then $\alpha-(\psi^{-1}\circ\phi)^{*}\alpha^{\prime}$ vanishes in
the subset of $\mathbb{H}^{n}_{+}$ where it is defined. Lemma 4.4 shows that
the integral of $\alpha-(\psi^{-1}\circ\phi)^{*}\alpha^{\prime}$ over
$\phi^{-1}\circ\sigma$ vanishes and the same integral is obtained by using
$\psi$ and $\alpha^{\prime}$ instead of $\phi$ and $\alpha$. So the integral
of $\omega\in W^{d,\infty}(\wedge^{k}T^{*}U)$ is well-defined over every
$\sigma\in C_{k}^{\operatorname{lip}}(U;\mathbb{R})$, when $U$ is an open
subset of a closed Lipschitz $n$-submanifold $M$.

We also note that the definition extends to forms $\omega\in
W^{d,\infty}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$. This is since there exists an
increasing sequence of sets $U_{i}\subset U$ such that $U_{i}$ are open in
$M$, $\omega\in W^{d,\infty}(\wedge^{k}T^{*}U_{i})$ and $U=\bigcup_{i}U_{i}$.
Every Lipschitz $k$-simplex $\sigma\colon\Delta_{k}\to U$ is then contained in
one of the sets $U_{i}$ by compactness.

By using bilipschitz charts, subdivision of Lipschitz $k$-simplices and
Proposition 4.2 part (3), we immediately have the following version of Stokes’
theorem for the integral of $W^{d,\infty}_{\mathrm{loc}}$-forms.

######  Proposition 4.5.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$, let $U\subset M$ be open in $M$,
$\sigma\in C_{k}^{\operatorname{lip}}(U;\mathbb{R})$ be a singular Lipschitz
$k$-chain and $\omega\in W^{d,\infty}_{\mathrm{loc}}(\wedge^{k-1}T^{*}U)$.
Then

$\int_{\partial\sigma}\omega=\int_{\sigma}d\omega.$

We also require the following result for the canonical integral regarding
boundary simplices.

######  Lemma 4.6.

Suppose that $M$ is a closed Lipschitz $n$-submanifold in an oriented
Lipschitz Riemannian $n$-manifold $\mathcal{R}$ without boundary and that
$U\subset M$ is an open subset of $M$. Let $D\neq\emptyset$ be a closed
Lipschitz $(n-1)$-submanifold in $\partial M$ and let $\omega\in
W^{d,\infty}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}U)$. Then

$\int_{\sigma}\omega=0$

for every $\sigma\in C^{\operatorname{lip}}_{k}(D\cap U;\mathbb{R})$.

###### Proof.

Let $B=B^{n}(0,1)$. By subdivision, we may assume that $\sigma$ is a Lipschitz
$k$-simplex $\sigma\colon\Delta_{k}\to D$ that is contained in the image of a
boundary chart $\phi\colon B\to V$ with $\phi(B\cap\mathbb{H}^{n}_{+})\subset
U$ and $\phi(B\cap\mathbb{H}^{n}_{-})\subset\mathcal{R}\setminus M$. We wish
to show that the integral of $\phi^{*}\omega$ over
$\sigma^{\prime}=\phi^{-1}\circ\sigma$ vanishes. The case where $V\cap\partial
M\subset D$ is simple, as we may just extend $\phi^{*}\omega$ by zero into
$B\cap\mathbb{H}^{n}_{-}$ and use Lemma 4.4.

The other case to consider is when $V$ meets $\partial D$, in which case we
can set $\phi$ up so that $V\cap D=\phi(\\{x\in B:x_{n}=0,x_{1}>0\\})$. Let
$\tau_{\varepsilon}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the translation
map by $(\varepsilon,0,\dots,0)$. For small enough $\varepsilon>0$,
$(\tau_{\varepsilon})_{*}\sigma^{\prime}$ remains in $B$ and the integral of
$\phi^{*}\omega$ over $(\tau_{\varepsilon})_{*}\sigma^{\prime}$ again vanishes
by a use of Lemma 4.4. Similarly as in the proof of Lemma 4.4, we can define
$H_{\varepsilon}\colon\Delta_{k}\times[0,\varepsilon]\to B$ by
$H_{\varepsilon}(x,t)=(\tau_{t})_{*}\sigma^{\prime}(x)$ and we have by
Proposition 4.2 part (5) that

$\displaystyle\left|{\int_{\sigma^{\prime}-(\tau_{\varepsilon})_{*}\sigma^{\prime}}\phi^{*}\omega}\right|$
$\displaystyle\leq(\lVert\phi^{*}\omega\rVert_{L^{\infty}}+\lVert
d\phi^{*}\omega\rVert_{L^{\infty}})(\nu_{H_{\varepsilon}}(B)+\nu_{\partial
H_{\varepsilon}-(\sigma^{\prime}-(\tau_{\varepsilon})_{*}\sigma^{\prime})}(B))$
$\displaystyle\leq(\lVert\phi^{*}\omega\rVert_{L^{\infty}}+\lVert
d\phi^{*}\omega\rVert_{L^{\infty}})(\varepsilon\nu_{\sigma^{\prime}}(B)+\varepsilon\nu_{\partial\sigma^{\prime}}(B))\xrightarrow[\varepsilon\to
0]{}0.$

We conclude that the integral of $\phi^{*}\omega$ over $\sigma^{\prime}$
vanishes. ∎

###  4.4. Integration of $L^{p}$-forms in Lipschitz manifolds

Next, we consider integration of $\omega\in
L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}\mathcal{R})$ over Lipschitz $k$-chains on
a Lipschitz Riemannian manifold $\mathcal{R}$. For this, we need to first
define $p$-exceptional families of surfaces. The main technical complication
is that you cannot use (6) to define $\left|{J_{\sigma}}\right|$ for every
Lipschitz $\sigma\colon\Delta_{k}\to\mathcal{R}$ due to $g$ only being defined
almost everywhere.

We approach this as follows. Let $\sigma\colon\Delta_{k}\to\mathcal{R}$ be
Lipschitz. We can choose a subdivision of $\Delta_{k}$ into finitely many
$\Delta_{k,i}\subset\Delta_{k}$ such that $\sigma(\Delta_{k,i})$ is contained
in the image $U_{i}$ of a bilipschitz chart $\phi_{i}\colon\Omega_{i}\to
U_{i}$ for every $i$. Now, for every $i$, the map
$\sigma_{i}=\phi_{i}^{-1}\circ\sigma|_{\Delta_{k,i}}$ is Lipschitz and we can
define a measure $\nu_{\sigma_{i}}$ on $\Omega_{i}$ with respect to the
Euclidean metric. We then define

(9) $\mu_{\sigma}=\sum_{i}(\phi_{i})_{*}\nu_{\sigma_{i}}.$

Once we have fixed a measure $\mu_{\sigma}$ for every Lipschitz $k$-simplex,
we can similarly define measures $\mu_{\sigma}$ for every formal linear
combination $\sigma=c_{1}\sigma_{1}+\dots+c_{k}\sigma_{k}\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$ of Lipschitz $k$-simplices
$\sigma_{i}$. We note that the measures $\mu_{\sigma}$ depend on the choices
of subdivisions $\Delta_{k,i}$ and charts $\phi_{i}$ for each $\sigma$.
However, we still have the following.

######  Lemma 4.7.

Suppose $\mathcal{R}$ is a Lipschitz Riemannian manifold and that
$\sigma\colon\Delta_{k}\to\mathcal{R}$ is a Lipschitz $k$-simplex. Suppose
that $\mu_{\sigma}$ and $\mu_{\sigma}^{\prime}$ are both defined using (9),
with different subdivisions and bilipschitz charts. There exists
$C\in(0,\infty)$ such that $C^{-1}\mu_{\sigma}^{\prime}\leq\mu_{\sigma}\leq
C\mu_{\sigma}^{\prime}$. Moreover, if $\mathcal{R}$ is a smooth Riemannian
manifold and if we define $\nu_{\sigma}$ using (5) and (6) where
$D^{T}\sigma(x)$ is the adjoint satisfying
$g(D\sigma(x)v,w)=g_{\text{eucl}}(v,D^{T}\sigma(x)w)$, then there exists a
constant $C\in(0,\infty)$ such that $C^{-1}\nu_{\sigma}\leq\mu_{\sigma}\leq
C\nu_{\sigma}$.

###### Proof.

By a finite subdivision and use of bilipschitz charts, the claims reduce to
the fact that if
$\sigma^{\prime}\colon\Delta_{k}\to\Omega\subset\mathbb{R}^{n}$ is Lipschitz
and
$f\colon(\Omega,g_{\text{eucl}})\to(\Omega^{\prime},g)\subset(\mathbb{R}^{n},g)$
is bilipschitz, where the Riemannian metric $g$ in the target is smooth, then
$C^{-1}\left|{J_{f\circ\sigma^{\prime}}}\right|\leq\left|{J_{\sigma^{\prime}}}\right|\leq
C\left|{J_{f\circ\sigma^{\prime}}}\right|$ a.e. in $\Delta_{k}$. For a simple
proof of this fact, let $x$ be a point in the interior of $\Delta_{k}$ such
that $f\circ\sigma^{\prime}$ and $\sigma^{\prime}$ are differentiable at $x$
and use the singular value decomposition of matrices to select an orthonormal
basis $v_{1},\dots,v_{k}$ of $\mathbb{R}^{k}$ such that the image vectors
$w_{i}=D(f\circ\sigma^{\prime})(x)v_{i}$ are $g$-orthogonal. Now, the
bilipschitz condition of $f$ and Hadamard’s inequality yield that

$\displaystyle\left|{J_{f\circ\sigma^{\prime}}}\right|$
$\displaystyle=\prod_{i=1}^{k}\left|{w_{i}}\right|_{g}$
$\displaystyle=\prod_{i=1}^{k}\lim_{t\to
0}\frac{d_{g}(f\circ\sigma^{\prime}(x+tv_{i}),f\circ\sigma^{\prime}(x))}{t}$
$\displaystyle\geq\prod_{i=1}^{k}C^{-1}\lim_{t\to
0}\frac{\left|{\sigma^{\prime}(x+tv_{i})-\sigma^{\prime}(x)}\right|}{t}$
$\displaystyle=C^{-k}\prod_{i=1}^{k}\left|{D\sigma^{\prime}(x)v_{i}}\right|$
$\displaystyle\geq C^{-k}\left|{J_{\sigma^{\prime}}(x)}\right|.$

The other direction of the estimate is analogously obtained by using the
singular value decomposition theorem on $D\sigma^{\prime}(x)$ instead. ∎

In particular, Lemma 4.7 states that different choices in the construction of
the measures $\mu_{\sigma}$ lead to families
$\mathcal{M}=\\{\mu_{\sigma}:\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})\\}$ and
$\mathcal{M}^{\prime}=\\{\mu_{\sigma}^{\prime}:\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})\\}$ such that we have weak
minorizations
$\mathcal{M}^{\prime}\lesssim\mathcal{M}\lesssim\mathcal{M}^{\prime}$.
Therefore, these families have the same $p$-exceptional sets by Corollary 2.4.
These exceptional sets are the same as the exceptional sets of
$\\{\nu_{\sigma}:\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})\\}$ in the smooth setting.
Hence, we can define that a subset $\Sigma\subset
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$ is _$p$ -exceptional_ if
the family of measures $\\{\mu_{\sigma}:\sigma\in\Sigma\\}$ is
$p$-exceptional.

Before we proceed to define integration for $\omega\in
L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}\mathcal{R})$, we prove one more lemma
that will be repeatedly useful in following considerations. Let $\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$, let $\phi\colon\Omega\to
U$ be a chart and let $\sigma^{\prime}\in
C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$. We say that _$\sigma$ covers
$\sigma^{\prime}$ via $\phi$_ if there exists a constant $c>0$ such that
$\mu_{\sigma}\geq c\cdot\phi_{*}\nu_{\sigma^{\prime}}$. By Lemma 4.7, this is
independent on the choice of $\mu_{\sigma}$.

######  Lemma 4.8.

Suppose $\mathcal{R}$ is a Lipschitz Riemannian manifold and
$\phi\colon\Omega\to U$ is a chart.

  1. (1)

Let $\sigma\in C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$ and
$\sigma^{\prime}\colon\Delta_{k}\to\Omega$ be a Lipschitz $k$-simplex in
$\Omega$. If $a(\phi\circ\sigma^{\prime})$ (or any bilipschitz
reparameterization) occurs in a subdivision of $\sigma$ with
$a\in\mathbb{R}\setminus\\{0\\}$, then $\sigma$ covers $\sigma^{\prime}$ via
$\phi$.

  2. (2)

Let $\Sigma^{\prime}\subset C_{k}^{\operatorname{lip}}(\Omega;\mathbb{R})$ and

$\Sigma=\left\\{\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R}):\sigma\text{ covers
}\sigma^{\prime}\text{ via }\phi\text{ for some
}\sigma^{\prime}\in\Sigma^{\prime}\right\\}.$

If $\Sigma^{\prime}$ is $p$-exceptional, then $\Sigma$ is $p$-exceptional.

###### Proof.

If $a(\phi\circ\sigma^{\prime})$ occurs in a subdivision of $\sigma$, then by
further subdividing one can make it so that
$\left|{a}\right|\phi_{*}\nu_{\sigma^{\prime}}$ is one of the measures added
together to form $\mu_{\sigma}$. This gives the first property.

For the second, since $\phi^{*}\operatorname{vol}_{g}$ is a.e. uniformly
comparable with the Lebesgue volume, it follows that the family of measures
$\\{\phi_{*}\nu_{\sigma^{\prime}}:\sigma^{\prime}\in\Sigma^{\prime}\\}$ is
$p$-exceptional in $(\mathcal{R},\operatorname{vol}_{g})$. Therefore, the
family $\\{\mu_{\sigma}:\sigma\in\Sigma\\}$ is also $p$-exceptional by
Corollary 2.4. ∎

Suppose that $\omega\in L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}\mathcal{R})$,
with the aim of defining the integral of $\omega$ over $p$-a.e. $\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$. We select a countable
collection of bounded bilipschitz charts $\phi_{i}\colon\Omega_{i}\to U_{i}$
such that the sets $U_{i}$ cover $\mathcal{R}$ and fix a Borel representative
of $\phi_{i}^{*}\omega\in L^{p}(\wedge^{k}T^{*}\Omega_{i})$ for each $i$. We
choose the measures $\mu_{\sigma}$ for all $\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$ using only the charts
$\phi_{i}$.

For every chart $\phi_{i}$, let $\Sigma_{i}^{\prime}\subset
C_{k}^{\operatorname{lip}}(\Omega_{i};\mathbb{R})$ be the collection of all
Lipschitz $k$-chains $\sigma$ such that the integral of $\phi_{i}^{*}\omega$
over $\sigma$ is not defined. Similarly, for any two charts $\phi_{i}$ and
$\phi_{j}$ such that $U_{i}\cap U_{j}\neq\emptyset$, we denote
$\Omega_{ij}=\phi_{i}^{-1}(U_{i}\cap U_{j})$ and let $\Sigma_{ij}^{\prime}$ be
the family of $\sigma\in C_{k}^{\operatorname{lip}}(\Omega_{ij};\mathbb{R})$
such that the integral of $\phi_{i}^{*}\omega$ over $\sigma$ and the integral
of $\phi_{j}^{*}\omega$ over $(\phi_{j}^{-1}\circ\phi_{i})_{*}\sigma$ are both
well-defined, but different. The families $\Sigma^{\prime}_{i}$ and
$\Sigma^{\prime}_{ij}$ essentially encompass all the simplices where
integration behaves poorly and are both $p$-exceptional in $\Omega_{i}$, since
$\phi_{i}^{*}\omega\in L^{p}(\wedge^{k}T^{*}\Omega_{i})$.

Let $\Sigma$ be the family of all $\sigma\in
C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$ such that $\sigma$ covers
a $\sigma^{\prime}\in\Sigma^{\prime}_{i}\cup\bigcup_{j}\Sigma^{\prime}_{ij}$
via $\phi_{i}$ for some $i$. Since there are only a countable amount of
charts, $\Sigma$ is $p$-exceptional by Lemma 4.8. Moreover, for all $k$-chains
$\sigma\in C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})\setminus\Sigma$,
Lemma 4.8 also yields that no subdivision contains a simplex where the
integral behaves poorly. So integration of $\omega$ over $\sigma$ is defined
as usual via subdivision and charts and the result is independent of the exact
subdivision used.

Overall, the set $\Sigma$ and the values of the integral of $\omega$ over
$\sigma\in C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})\setminus\Sigma$
depend on the choices of charts $\phi_{i}$ and Borel representatives
$\phi_{i}^{*}\omega$. However, it is reasonably straightforward to see that a
different choice of $\phi_{i}$ and $\phi_{i}^{*}\omega$ will only change the
integral of $\omega$ in a $p$-exceptional family.

The definition extends to $\omega\in L^{p}(\wedge^{k}T^{*}U)$ with
$U\subset\mathcal{R}$ by extending $\omega$ as zero outside $U$. If instead
$U\subset M$ is an open subset of a closed Lipschitz $n$-submanifold $M$ in
$\mathcal{R}$, then by a similar extension by zero to
$U^{\prime}\subset\mathcal{R}$ with $U^{\prime}\cap M=U$, the definition also
extends to $\omega\in L^{p}(\wedge^{k}T^{*}U)$ and $\omega\in
L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}U)$.

We next prove Lipschitz manifold versions of two Euclidean results. The first
is a version of the differential forms Fuglede Lemma 4.1.

######  Lemma 4.9.

Let $\mathcal{R}$ be a Lipschitz Riemannian manifold without boundary and let
$p\in[1,\infty)$. If $\omega\in L^{p}(\wedge^{k}T^{*}\mathcal{R})$ and
$\omega_{i}\to\omega$ in $L^{p}(\wedge^{k}T^{*}\mathcal{R})$, then there
exists a subsequence $\omega_{i_{j}}$ of $\omega_{i}$ such that

$\int_{\sigma}\omega_{i_{j}}\to\int_{\sigma}\omega$

for $p$-a.e. $\sigma\in C^{\operatorname{lip}}_{k}(\mathcal{R};\mathbb{R})$.

###### Proof.

Let $\\{\phi_{l}:l=1,\dots\\}$ be a countable cover of $\mathcal{R}$ by
bilipschitz charts. By the Euclidean result from Lemma 4.1 and a standard
diagonal subsequence argument, we may select $i_{j}$ such that

$\int_{\sigma^{\prime}}\phi_{l}^{*}\omega_{i_{j}}\to\int_{\sigma^{\prime}}\phi_{l}^{*}\omega$

for every $l$ and $p$-a.e. $\sigma^{\prime}\in
C_{k}^{\operatorname{lip}}(\Omega_{l};\mathbb{R})$. We denote the set of
$\sigma^{\prime}\in C_{k}^{\operatorname{lip}}(\Omega_{l};\mathbb{R})$ for
which this convergence does not hold by $\Sigma^{\prime}_{l}$. The claim then
follows by Lemma 4.8, which shows the $p$-exceptionality of the family of all
$\sigma\in C_{k}^{\operatorname{lip}}(\mathcal{R};\mathbb{R})$ which cover an
element $\sigma^{\prime}\in\Sigma^{\prime}_{l}$ via $\phi_{l}$ for some
$l\in{\mathbb{N}}$. ∎

Second, we give a Lipschitz version of Proposition 4.2 part (2), which
connects our two integrals.

######  Lemma 4.10.

Let $\mathcal{R}$ be a Lipschitz Riemannian manifold without boundary, $M$ be
a closed Lipschitz $n$-submanifold in $\mathcal{R}$, $U\subset M$ be open in
$M$ and $\omega\in W^{d,\infty}(\wedge^{k}T^{*}U)$. For every
$p\in[1,\infty)$, outside a $p$-exceptional subset of
$C^{\operatorname{lip}}_{k}(U;\mathbb{R})$, the integral of $\omega$ as an
$L^{p}$-form over $p$-a.e. $\sigma\in
C^{\operatorname{lip}}_{k}(U;\mathbb{R})$ coincides with the integral of
$\omega$ as a $W^{d,\infty}$-form over all $\sigma\in
C^{\operatorname{lip}}_{k}(U;\mathbb{R})$.

###### Proof.

Suppose that $\phi$ is a chart such that $U_{\phi}\subset
U\cap\operatorname{int}(M)$. If we fix a Borel representative of
$\phi^{*}\omega$ and let $\Sigma^{\prime}$ be the set of $\sigma\in
C^{\operatorname{lip}}_{k}(\Omega_{i};\mathbb{R})$ for which
$\phi_{i}^{*}\omega$ does not satisfy the result, then $\Sigma^{\prime}$ is
$p$-exceptional by Proposition 4.2 part (2).

On the other hand, suppose that $\phi\colon\Omega_{\phi}\to U_{\phi}$ is a
chart that takes $\Omega_{\phi}\cap\overline{\mathbb{H}^{n}_{+}}$ into $U$ and
$\Omega_{\phi}\cap\partial\mathbb{H}^{n}_{+}$ into $U\cap\partial M$. Then the
$W^{d,\infty}$-integral in $U_{\phi}$ is defined using the reflection
extension of $\phi^{*}\omega$ outlined in Lemma 3.4 and the $L^{p}$-integral
is defined using the extension of $\phi^{*}\omega$ by zero. These two
extensions have Borel representatives $\alpha$ and $\beta$ that coincide in
$\Omega_{\phi}\cap\overline{\mathbb{H}^{n}_{+}}$. In this case, $\alpha$ and
$\beta$ have the exact same $L^{p}$-integral over every $\sigma\in
C^{\operatorname{lip}}_{k}(\Omega_{\phi}\cap\overline{\mathbb{H}^{n}_{+}};\mathbb{R})$.
If $\Sigma^{\prime}$ is the set of $\sigma\in
C^{\operatorname{lip}}_{k}(\Omega_{\phi}\cap\overline{\mathbb{H}^{n}_{+}};\mathbb{R})$
for which $\phi_{i}^{*}\omega$ does not satisfy the result, then the
$L^{p}$-integral of $\alpha$ over every $\sigma\in\Sigma^{\prime}$ differs
from the corresponding $W^{d,\infty}$-integral. Similarly as before,
$\Sigma^{\prime}$ is $p$-exceptional by Proposition 4.2 part (2).

The claim again follows by taking a countable cover of $U$ with such charts
and by using Lemma 4.8 on the corresponding sets $\Sigma^{\prime}$ as in the
previous lemma. ∎

To end this section, we comment on the measures $\nu_{\sigma}$. As mentioned
earlier, if the Lipschitz Riemannian metric is only measurable, then we cannot
define $\nu_{\sigma}$ canonically for every $\sigma\in
C^{\operatorname{lip}}_{k}(M;\mathbb{R})$. However, outside a $p$-exceptional
family of Lipschitz $k$-chains with $p\in[1,\infty)$, the measure
$\nu_{\sigma}$ can be defined.

Indeed, in the following Lemma, we prove that one can define $\nu_{\sigma}$
outside a $p$-exceptional family of chains in a domain of $\mathbb{R}^{n}$
equipped with a bounded measurable Riemannian metric $g^{\prime}$. Once this
is done, if $(M,g)$ is a Lipschitz Riemannian manifold and
$\phi\colon\Omega\to U$ is a chart, we can then define $\nu_{\sigma}$ for
Lipschitz $k$-simplexes $\sigma\colon\Delta_{k}\to U$ by defining
$\nu_{\phi^{-1}\circ\sigma}$ with respect to $\phi^{*}g$ and applying push-
forward of measures with $\phi$. The definition for $p$-a.e. $\sigma\in
C^{\operatorname{lip}}_{k}(M;\mathbb{R})$ follows by a similar countable
covering and subdivision argument as was used to define integration of
$k$-forms over $p$-a.e. Lipschitz $k$-chain, including the use of Lemma 4.8.
The measures $\nu_{\sigma}$ again depend on the choices made in the
construction, but a different set of choices changes them only in a
$p$-exceptional family.

######  Lemma 4.11.

Let $U\subset{\mathbb{R}}^{n}$ be an open set and let $g$ be a Lipschitz
Riemannian metric on $U$. There exists a collection of Lipschitz $k$-simplices
$\mathcal{S}$ so that $\nu_{\sigma}$ is defined by (5) and (6) with respect to
$g$ and whose complement in the set of all Lipschitz $k$-simplices is
$p$-exceptional.

###### Proof.

Let $g_{n}$ be a sequence of smooth metrics that converge to $g$ pointwise
almost everywhere and in $L^{p}$. Let $\mu_{\sigma}$ be the surface measure of
$\sigma$ with respect to the Euclidean metric on $U$. By Lemma 2.2, outside a
$p$-exceptional set of Lipschitz $k$-simplices,

$\displaystyle\lim_{n\to\infty}\int_{\sigma}\|g_{n}-g\|_{\text{op}}d\mu_{\sigma}=0.$

Therefore, on $\sigma$, $g$ is a Lipschitz Riemannian metric outside a set of
$0$ measure with respect to $\mu_{\sigma}$. This allows us to define
$\nu_{\sigma}$ with respect to $g$. ∎

##  5\. Sobolev cohomology theories

We next discuss the relevant parts of Sobolev cohomology theory on Lipschitz
manifolds.

###  5.1. Local Sobolev cohomology

Let $M$ be a closed Lipschitz $n$submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let $D$ be a closed
Lipschitz $(n-1)$-submanifold in $\partial M$ and let $p\in[1,\infty]$. We
note that the map $U\mapsto W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}U)$ is
a sheaf on $M$.

In order to define a cohomology theory in the Sobolev setting the only
analytic tool needed is a local exactness result. This is provided by a
Sobolev variant of the Poincaré lemma. A version in the Euclidean setting with
$D=\emptyset$ can be found in [10, Section 4]. We give the version of the
result that we require, relying heavily on the exposition in [10, Section 4]
for the proof.

######  Lemma 5.1.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let $p\in[1,\infty]$
and let $D$ be a closed Lipschitz $(n-1)$-submanifold in $\partial M$. Let
$x\in M$, $k\geq 1$, $U$ a neighborhood of $x$ in $M$ and suppose that
$\omega\in W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}U)$ is such that
$d\omega=0$. There exists a neighborhood $V\subset U$ of $x$ and a form
$\tau\in W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k-1}T^{*}V)$ such that
$\omega|_{V}=d\tau$.

###### Proof.

We first recall an explicit construction for $\tau$ in the Euclidean setting.
Let $B=B^{n}(0,1)$. Fix $\eta\in C^{\infty}_{0}(B)$ such that $\eta\geq 0$ and
$\int_{B}\eta=1$ and define a linear operator $T\colon
C^{\infty}(\wedge^{k}T^{*}B)\to C^{\infty}(\wedge^{k-1}T^{*}B)$ by

(10)
$(T\omega)_{x}(v_{1},\dots,v_{k-1})=\int_{B}\eta(y)\int_{0}^{1}t^{k-1}\omega_{tx+(1-t)y}(x-y,v_{1},\dots,v_{k-1})\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}y.$

Then $\omega=dT(\omega)+T(d\omega)$ and $\lVert T\omega\rVert_{L^{p}}\leq
C_{\eta}\lVert\omega\rVert_{L^{p}}$ for all $p\in[1,\infty]$ and all
$\omega\in C^{\infty}(\wedge^{k}T^{*}B)$ (see [10, (4.6) and (4.15)]). If
$p\in[1,\infty)$, then $C^{\infty}(\wedge^{k}T^{*}B)$ is dense in
$L^{p}(\wedge^{k}T^{*}B)$ and hence $T$ extends to a bounded linear map
$T\colon L^{p}(\wedge^{k}T^{*}B)\to L^{p}(\wedge^{k-1}T^{*}B)$. Moreover, a
similar extension remains valid for $p=\infty$ even though
$C^{\infty}(\wedge^{k}T^{*}B)$ is not dense in $L^{\infty}(\wedge^{k}T^{*}B)$.
Indeed, we may smoothly approximate $\omega\in L^{\infty}(\wedge^{k}T^{*}B)$
with $\omega_{j}\in C^{\infty}(\wedge^{k}T^{*}B)$ such that
$\lVert\omega_{j}\rVert_{L^{\infty}}\leq\lVert\omega\rVert_{L^{\infty}}$ and
$\lVert\omega_{j}-\omega\rVert_{L^{1}}\to 0$. In which case, $T(\omega_{j})$
converges pointwise to a unique $T(\omega)\in L^{1}(\wedge^{k}T^{*}B)$ and the
uniform bound $\lVert T(\omega_{j})\rVert_{L^{\infty}}\leq
C_{\eta}\lVert\omega\rVert_{L^{\infty}}$ yields that $\lVert
T(\omega)\rVert_{L^{\infty}}\leq C_{\eta}\lVert\omega\rVert_{L^{\infty}}$.

If $p\in[1,\infty)$, it follows by smooth approximation that $T(\omega)\in
W^{d,p}(\wedge^{k-1}T^{*}B)$ and $\omega=dT(\omega)+T(d\omega)$ if $\omega\in
W^{d,p}(\wedge^{k}T^{*}B)$ (see [10, Lemma 4.2]). The same is also true for
$\omega\in W^{d,\infty}(\wedge^{k}T^{*}B)$, since
$W^{d,\infty}(\wedge^{k}T^{*}B)\subset W^{d,1}(\wedge^{k}T^{*}B)$ implies that
$dT(\omega)=\omega-T(d\omega)$ weakly and hence $\lVert
dT(\omega)\rVert_{L^{\infty}}\leq\lVert\omega\rVert_{L^{\infty}}+\lVert
T(d\omega)\rVert_{L^{\infty}}<\infty$. In the Euclidean setting, if $\omega\in
W^{d,p}(\wedge^{k}T^{*}B)$ weakly, then a choice of $\tau=T(\omega)$ yields
$\tau\in W^{d,p}(\wedge^{k-1}T^{*}B)$ and $d\tau=\omega$ weakly.

We return to the Lipschitz manifold setting. Suppose first that
$x\in\operatorname{int}(M)$. We can choose a bilipschitz chart $\phi\colon
B\to V$ such that $x\in V\subset U\cap\operatorname{int}(M)$. The Euclidean
result implies that $\phi^{*}\omega\in W^{d,p}(\wedge^{k}T^{*}B)$ and the
desired $\tau\in W^{d,p}(\wedge^{k-1}T^{*}V)$ is given by
$\phi^{*}\tau=T(\phi^{*}\omega)$.

The next case we consider is when $x\in\partial M\setminus D$. There exists a
bilipschitz chart $\phi\colon B\to V$ such that $x\in V$, $V\cap M\subset U$,
$V\cap D=\emptyset$, $\phi(B\cap\overline{\mathbb{H}^{n}_{+}})=V\cap M$ and
$\phi(B\setminus\overline{\mathbb{H}^{n}_{+}})=V\setminus M$. In this case,
the reflection Lemma 3.4 extends $\phi^{*}\omega$ to $B$. A valid choice of
$\tau$ is given by $\phi^{*}\tau=T(\phi^{*}\omega)$ on
$B\cap\mathbb{H}^{n}_{+}$ and $\phi^{*}\tau=0$ on $B\cap\mathbb{H}^{n}_{-}$.

Next, we consider the case $x\in\operatorname{int}D$. There exists a
bilipschitz chart $\phi\colon B\to V$ such that $x\in V$, $V\cap M\subset U$,
$V\cap\partial M\subset\operatorname{int}D$,
$\phi(B\cap\overline{\mathbb{H}^{n}_{+}})=V\cap M$ and
$\phi(B\setminus\overline{\mathbb{H}^{n}_{+}})=V\setminus M$. If we extend
$\phi^{*}\omega$ by zero to $B$, the extension is in
$W^{d,p}(\wedge^{k}T^{*}B)$ with
$\operatorname{spt}(\phi^{*}\omega)\subset\overline{\mathbb{H}^{n}_{+}}$. By
selecting the $\eta$ in the definition of $T$ to satisfy
$\operatorname{spt}\eta\subset B\cap\mathbb{H}^{n}_{-}$ and since
$B\cap\mathbb{H}^{n}_{-}$ is convex, it follows from (10) that
$T(\phi^{*}\omega)=0$ on $B\cap\mathbb{H}^{n}_{-}$ (see also the comment
before [10, (4.14)]). Thus, $\phi^{*}\tau=T(\phi^{*}\omega)$ remains a valid
choice that vanishes weakly on $D$.

We have one more remaining case, which is $x\in\partial D$. For this case, we
select the bilipschitz chart $\phi\colon B\to V$, where
$\phi(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=0,x_{1}<0\\})=V\cap(\partial M\setminus
D)$ and
$\phi(B\cap\\{x\in\mathbb{R}^{n}:x_{n}=x_{1},x_{1}>0\\})=V\cap\operatorname{int}D$.
By extending $\phi^{*}\omega$ by zero to $B\cap\mathbb{H}^{n}_{+}$ and then
further extending to $B$ by Lemma 3.4, we obtain an extension
$\phi^{*}\omega\in W^{d,p}(\wedge^{k}T^{*}B)$ that vanishes in
$H=B\cap\\{x\in\mathbb{R}^{n}:\left|{x_{n}}\right|<x_{1}\\}$. Choosing $\eta$
such that $\operatorname{spt}\eta\subset H$, $T(\phi^{*}\omega)$ vanishes on
$H$ due to convexity of $H$. The desired $\tau$ is given by
$\phi^{*}\tau=T(\phi^{*}\omega)$ in $B\cap\mathbb{H}^{n}_{+}$ and
$\phi^{*}\tau=0$ in $B\cap\mathbb{H}^{n}_{-}$. ∎

Let $H_{p}^{k}(M,D)$ denote the degree $k$ cohomology space of the complex
$W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{*}T^{*}M)$. We call $H_{p}^{k}(M,D)$ the
degree $k$ _relative $L^{p}_{\mathrm{loc}}$-cohomology of the pair $(M,D)$_.
Due to the Poincaré lemma shown in Lemma 5.1, the methods of sheaf cohomology
now apply to the spaces $H_{p}^{k}(M,D)$. We begin with the following
corollary of the Poincaré lemma. The proof assumes familiarity with sheaf
theory and sheaf cohomology; for standard reference works on the topic, we
refer the reader to the comprehensive treatment in [3], as well as the more
focused introductions in [17, Chapter 5] and [18, Chapter II].

######  Corollary 5.2.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let $p\in[1,\infty)$
and let $D$ be a closed Lipschitz $(n-1)$-submanifold in $\partial M$. The
relative $L^{p}_{\mathrm{loc}}$-cohomology spaces $H_{p}^{k}(M,D)$ and
$H_{\infty}^{k}(M,D)$ are isomorphic as vector spaces, with the isomorphism
induced by the inclusion maps
$W^{d,\infty}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}M)\hookrightarrow
W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{*}T^{k}M)$.

###### Proof.

Based on the Poincaré Lemma 5.1, we obtain that the graded differential
sheaves $\mathcal{W}_{p,D}^{*}\colon U\mapsto
W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{*}T^{*}U)$ and
$\mathcal{W}_{\infty,D}^{*}\colon U\mapsto
W^{d,\infty}_{T(D),\mathrm{loc}}(\wedge^{*}T^{*}U)$ on $M$ each form a fine
torsionless resolution of the sheaf $\mathcal{R}_{D}$ of locally constant
functions on $M$ which vanish on $D$. Since the inclusion maps
$W^{d,\infty}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}M)\hookrightarrow
W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{*}T^{k}M)$ form a sheaf homomorphism of
resolutions, they must induce an isomorphism in cohomology by [3, II.4.2]. ∎

###  5.2. Lipschitz homology and the de Rham theorem

On smooth manifolds $M$, the de Rham theorem shows that the cohomology of the
complex $C^{\infty}(\wedge^{*}T^{*}M)$ is dual to singular homology and the
duality map is explicitly given by integration over smooth simplices. Since we
can integrate $W^{d,\infty}_{T(D),\mathrm{loc}}(\wedge^{*}T^{*}M)$-forms over
Lipschitz simplices, a similar explicit duality relation between Lipschitz
homology and $L^{\infty}_{\mathrm{loc}}$-cohomology exists.

######  Proposition 5.3.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let $p\in[1,\infty]$
and let $D$ be a closed Lipschitz $(n-1)$-submanifold in $\partial M$. There
is a canonical isomorphism

$H^{k}_{\infty}(M,D)\cong\hom(H_{k}^{\operatorname{lip}}(M,D;\mathbb{Z}),\mathbb{R})\cong\hom(H_{k}^{\operatorname{lip}}(M,D;\mathbb{R}),\mathbb{R}),$

which, for $[\omega]\in H^{k}_{\infty}(M,D)$ and $[\sigma]\in
H_{k}^{\operatorname{lip}}(M,D;\mathbb{K})$ with
$\mathbb{K}\in\\{\mathbb{Z},\mathbb{R}\\}$, is given by

(11) $[\omega]([\sigma])=\int_{\sigma}\omega.$

Moreover, this isomorphism maps the wedge product to the cup product on the
level of cohomology classes: if $E=\partial M\setminus D$, then
$[\omega]\cup[\omega^{\prime}]=[\omega\wedge\omega^{\prime}]$ for all
$[\omega]\in H^{k}_{\infty}(M,D),[\omega^{\prime}]\in H^{l}_{\infty}(M,E)$,
where $[\omega\wedge\omega^{\prime}]\in H^{k+l}_{\infty}(M,\partial M)$.

The argument is essentially the standard sheaf-theoretic proof of de Rham
-type theorems. For the convenience of the reader we provide a proof in the
appendix. It is also notable that (11) also defines a chain map from
$W^{d,\infty}_{T(D),\mathrm{loc}}(\wedge^{*}T^{*}M)$ to
$\hom(C^{\operatorname{lip}}_{*}(M;\mathbb{Z}),\mathbb{R})$, but this map does
not preserve products: the equivalence of $\cup$ and $\wedge$ applies only for
cohomology classes.

##  6\. $p$-exceptional homology classes

In this section we show that homology classes are not $p$-exceptional

######  Proposition 6.1.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary, let $p\in(1,\infty)$
and $k\in\\{0,\dots,\infty\\}$ and let $D$ be a closed Lipschitz
$(n-1)$-submanifold in $\partial M$. If $[\sigma]\in
H_{k}^{\operatorname{lip}}(M,D;\mathbb{K})$ for
$\mathbb{K}\in\\{\mathbb{Z},\mathbb{R}\\}$, then $[\sigma]$ is not
$p$-exceptional.

We present the argument in increasingly general cases, building up to
Proposition 6.1. The case involving relative homology presents significant
technical difficulties, which make the originally simple argument far more
difficult to follow. We start with the case where $M$ has no boundary, where
the proof is straightforward.

######  Lemma 6.2.

Let $M$ be an oriented Lipschitz Riemannian manifold without boundary and let
$p\in(1,\infty)$, $k\in\\{0,\dots,\infty\\}$. If $[\sigma]\in
H_{k}^{\operatorname{lip}}(M;\mathbb{K})$, for
$\mathbb{K}\in\\{\mathbb{Z},\mathbb{R}\\}$, then $[\sigma]$ is not
$p$-exceptional.

###### Idea of proof.

Suppose that $c\in H_{k}^{\operatorname{lip}}(M;\mathbb{K})$ is not
$p$-exceptional. Then there is a function $\rho\in L^{p}(M)$ such that the
integral of $\rho$ over the measure of every $\sigma\in c$ is infinite. The
idea is to select an arbitrary chain $\sigma\in c$ and to construct a function
$\rho^{\prime}$ by smoothing $\rho$ in a neighborhood of the total image of
$\sigma$. By the averaging properties of the convolution, the property that
the integral of $\rho^{\prime}$ over the measure of $\sigma$ is infinite
remains true. However, the resulting $\rho^{\prime}$ will be continuous and
finite-valued on the compact total image of $\sigma$ and $\rho^{\prime}$ will
therefore have a maximal value on it. This is a contradiction. ∎

###### Proof of Lemma 6.2.

Suppose towards contradiction that $c\in
H_{k}^{\operatorname{lip}}(M;\mathbb{K})$ is $p$-exceptional. By Lemma 2.3
there exists a Borel function $\rho\colon M\to[0,\infty]$ such that
$\lVert\rho\rVert_{L^{p}}<\infty$, but $\rho$ has infinite integral over every
$\mu_{\sigma}$, $\sigma\in c$. We fix a representative $\sigma\in c$ and note
that the total image $K$ of $\sigma$ is compact.

We select bilipschitz charts $\phi_{i}\colon B^{n}(0,2)\to M$,
$i\in\\{1,\dots,m\\}$, such that $U_{i}=\phi_{i}(B^{n}(0,1))$ cover $K$. We
proceed by induction. Suppose that $\rho_{i-1}\in L^{p}(M)$ is such that
$\rho_{i-1}$ is continuous and finite-valued on $U_{1}\cup\dots\cup U_{i-1}$
and that $\rho_{i-1}$ has infinite integral over every $\mu_{\sigma}$,
$\sigma\in c$. We construct $\rho_{i}$ that is continuous and finite-valued on
$U_{1}\cup\dots\cup U_{i}$ and has otherwise the same properties. Note that we
take $\rho_{0}=\rho$. Once these properties are shown for all $\rho_{i}$, we
can pick $\rho^{\prime}=\rho_{m}$ and the proof is complete.

We fix $\varepsilon>0$ and define a function $\eta\colon
B^{n}(0,2)\to[0,\varepsilon]$ by

$\eta(x)=\max\left(0,(1-\left|{x}\right|)\varepsilon\right)$

For every vector $v\in B^{n}(0,1)$, we define a homeomorphism
$G_{v}:B^{n}(0,2)\to B^{n}(0,2)$ by

$G_{v}(x)=x+\eta(x)v.$

With a suitable choice of $\varepsilon>0$ dependent only on $n$, this is
bilipschitz. By setting $F_{i,v}=\phi_{i}\circ G_{v}\circ\phi_{i}^{-1}$ in
$U_{i}$ and extending to the rest of $M$ as identity, we hence obtain a
bilipschitz automorphism $F_{i,v}\colon M\to M$. We next define
$\rho_{i}\colon M\to[0,\infty]$ in the following way:

$\rho_{i}(x)=\int_{B^{n}(0,1)}\rho_{i-1}\circ
F_{i,v}(x)\mathop{}\\!\mathrm{d}v.$

We first show that the integral of $\rho_{i}$ with respect to $\mu_{\sigma}$
is still infinite for every $\sigma\in c$. For this, note that
$(F_{i,v})_{*}\sigma$ is Lipschitz homotopic to $\sigma$ via
$H(z,t)=((F_{i,tv})_{*}\sigma)(z)$ and thus $(F_{i,v})_{*}\sigma\in c$. Hence,

(12)
$\displaystyle\begin{split}\int_{M}\rho_{i}(x)\mathop{}\\!\mathrm{d}\mu_{\sigma}(x)&=\int_{M}\int_{B^{n}(0,1)}\rho_{i-1}\circ
F_{i,v}(x)\mathop{}\\!\mathrm{d}v\mathop{}\\!\mathrm{d}\mu_{\sigma}(x)\\\
&=\int_{B^{n}(0,1)}\int_{M}\rho_{i-1}(z)\mathop{}\\!\mathrm{d}(F_{i,v})_{*}\mu_{\sigma}(z)\mathop{}\\!\mathrm{d}v\\\
&\geq
C^{-1}\int_{B^{n}(0,1)}\int_{M}\rho_{i-1}(z)\mathop{}\\!\mathrm{d}\mu_{(F_{i,v})_{*}\sigma}(z)\mathop{}\\!\mathrm{d}v\\\
&=C^{-1}\int_{B^{n}(0,1)}\infty\mathop{}\\!\mathrm{d}v=\infty.\end{split}$

Note that the inequality here is a change of variables that uses the fact that
$F_{i,v}$ is bilipschitz.

Next, we show that $\rho_{i}$ is still in $L^{p}(M)$. It is enough to check
that $\rho_{i}\circ\phi_{i}\in L^{p}(B^{n}(0,1))$. However, if $x\in
B^{n}(0,1)$,

$\rho_{i}\circ\phi_{i}(x)=\int_{B^{n}(0,1)}\rho_{i-1}\circ\phi_{i}(x+\eta(x)v)\mathop{}\\!\mathrm{d}v=\frac{1}{\eta^{n}(x)}\int_{B^{n}(x,\eta(x))}\rho_{i-1}\circ\phi_{i}(w)\mathop{}\\!\mathrm{d}w.$

Notably, $\rho_{i}\circ\phi_{i}(x)$ is bounded from above up to a constant by
the maximal function
$\operatorname{{\mathbf{M}}}(\rho_{i-1}\circ\phi_{i})(x)$. By the Hardy-
Littlewood maximal inequality for $p>1$, we have $\rho_{i}\circ\phi_{i}\in
L^{p}(B^{n}(0,1))$ and hence $\rho_{i}\in L^{p}(M)$.

It remains to prove continuity and finiteness of $\rho_{i}$ on
$U_{1}\cup\dots\cup U_{i}$. Finiteness on $U_{i}$ can be immediately seen from
the previously stated expression for $\rho_{i}\circ\phi_{i}$, since
$\eta(x)>0$ and $\rho_{i-1}\circ\phi_{i}$ is in $L^{1}(B^{n}(x,\eta(x)))$.
Suppose then that $x_{k},x\in B^{n}(0,1)$ for $k\in\mathbb{Z}_{>0}$ and that
$x_{k}\to x$ as $k\to\infty$. Then, denoting $B_{x}=B^{n}(x,\eta(x))$ and
$B_{x,k}=B^{n}(x_{k},\eta(x_{k}))$, we have

$\displaystyle\left|{\rho_{i}\circ\phi_{i}(x)-\rho_{i}\circ\phi_{i}(x_{k})}\right|$
$\displaystyle\qquad\leq\frac{1}{\eta^{n}(x)}\int_{(B_{x}\cup
B_{x,k})\setminus(B_{x}\cap
B_{x,k})}\rho_{i-1}\circ\phi_{i}+\left|{\frac{1}{\eta^{n}(x_{k})}-\frac{1}{\eta^{n}(x)}}\right|\int_{B^{n}(0,R)}\rho_{i-1}\circ\phi_{i},$

where both right hand side terms tend to 0 since $\eta$ is continuous and
$\rho_{i-1}\circ\phi_{i}$ is $L^{1}(B^{n}(0,1))$. We conclude that $\rho_{i}$
is continuous at $\phi_{i}(x)$.

Moreover, since $\rho_{i}=\rho_{i-1}$ outside $U_{i}$, we have that $\rho_{i}$
is finite-valued in $U_{1}\cup\dots\cup U_{i-1}$ and continuous in
$(U_{1}\cup\dots\cup U_{i-1})\setminus\overline{U_{i}}$. Finally, suppose that
$x\in\partial B^{n}(0,1)$ and $\phi_{i}(x)\in U_{1}\cup\dots\cup U_{i-1}$.
Then $\rho_{i-1}\circ\phi_{i}$ is continuous at $x$ and hence if $x_{k}\in
B^{n}(0,1)$ are such that $x_{k}\to x$, we get

$\displaystyle\left|{\rho_{i}\circ\phi_{i}(x)-\rho_{i}\circ\phi_{i}(x_{k})}\right|$
$\displaystyle\leq\int_{B^{n}(0,1)}\left|{\rho_{i-1}\circ\phi_{i}(x_{k}+\eta(x_{k})v)-\rho_{i-1}\circ\phi_{i}(x)}\right|$
$\displaystyle\leq\operatorname{vol}_{n}(B^{n}(0,1))\max_{w\in
B^{n}(x,\left|{x-x_{k}}\right|+\eta(x_{k}))}\left|{\rho_{i-1}\circ\phi_{i}(w)-\rho_{i-1}\circ\phi_{i}(x)}\right|$
$\displaystyle\xrightarrow{k\to\infty}0.$

From this, it follows that $\rho_{i}$ is continuous at
$\phi_{i}(x)\in(U_{1}\cup\dots\cup U_{i-1})\cap\partial U_{i}$, completing the
proof that $\rho_{i}$ is continuous in $U_{1}\cup\dots\cup U_{i}$. ∎

The non-relative case where $M$ has boundary is a reduction to the previous
proof.

######  Lemma 6.3.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$ without boundary and let
$p\in(1,\infty)$, $k\in\\{0,\dots,\infty\\}$. Then no homology class
$[\sigma]\in H_{k}^{\operatorname{lip}}(M;\mathbb{K})$ is $p$-exceptional,
where $\mathbb{K}\in\\{\mathbb{Z},\mathbb{R}\\}$.

###### Proof.

There exists a locally Lipschitz _collar neighborhood_ of $\partial M$ in $M$.
That is, there exists a locally Lipschitz homeomorphism $F_{\partial
M}\colon(\partial M)\times[0,2\varepsilon]\to U\subset M$ such that
$F_{\partial M}(x,0)=x$. We refer to [9, Prop. 3.42] for the general idea of
such constructions. For every $\delta\in[0,2\varepsilon]$, we define the
projection map $P^{\delta}\colon M\to M$ where
$P^{\delta}=\operatorname{id}_{M}$ outside $F_{\partial M}(\partial
M\times[0,\delta])$ and $P^{\delta}(F_{\partial M}(x,t))=F_{\partial
M}(x,\delta)$ for $t\in[0,\delta]$. Now, if $\sigma\in c\in
H_{k}^{\operatorname{lip}}(M;\mathbb{K})$, then
$\sigma^{\prime}=P^{\varepsilon}_{*}\sigma$ is Lipschitz homotopic to $\sigma$
by $H(x,t)=P^{t\varepsilon}_{*}(\sigma(x))$. Hence, $\sigma^{\prime}\in c$ and
since $\sigma^{\prime}$ does not meet the boundary of $M$, we can repeat the
proof of Lemma 6.2 on it with no significant changes. ∎

The final most general case is the full relative case of Proposition 6.1.
Here, the extra technical hurdles caused by relative homology take far more
effort to overcome.

###### Idea of the proof of Proposition 6.1.

We use a locally Lipschitz collar neighborhood to find a smaller submanifold
$M_{\varepsilon}\approx M$ inside $M$ such that
$M\setminus\operatorname{int}M_{\varepsilon}$ is essentially $\partial
M\times[0,\varepsilon]$. If $H_{k}^{\operatorname{lip}}(M,D;\mathbb{K})$ has a
$p$-exceptional class $c$, then
$H_{k}^{\operatorname{lip}}(M_{\varepsilon},D_{\varepsilon};\mathbb{K})$ also
has a $p$-exceptional class $c_{\varepsilon}$, where $D_{\varepsilon}$ is
$D\times\\{\varepsilon\\}$. We pick $\rho\in L^{p}(M_{\varepsilon})$ such that
the integral of $\rho$ over the measures of every chain
$\sigma_{\varepsilon}\in c_{\varepsilon}$ is infinite and extend $\rho$ by
zero to all of $M$. Moreover, we fix an element $\sigma_{\varepsilon}\in
c_{\varepsilon}$ and turn it into an element $\sigma\in c$ by adding a chain
that is essentially “$\partial\sigma_{\varepsilon}\times[0,\varepsilon]$”.

The strategy is again to construct a $\rho^{\prime}$ by smoothing out $\rho$
in a neighborhood of $\sigma_{\varepsilon}$. Notably, our convolutions will
now only translate the underlying space inwards in any neighborhoods meeting
$\partial M_{\varepsilon}$. Since $\rho$ was zero in $M\setminus
M_{\varepsilon}$, $\rho^{\prime}$ will be continuous and finite in a
neighborhood of the total image of $\sigma$ and it then remains to show that
the integral of $\rho^{\prime}$ over the measure of $\sigma$ remains infinite.

The idea for this is that after any of the translations used to perform the
convolution, the part of $\sigma$ that gets translated inside
$M_{\varepsilon}$ will be an element of $c_{\varepsilon}$ and hence $\rho$
will have infinite integral over it. A major technical issue, however, is that
the intersection of a Lipschitz chain and a Lipschitz subdomain is not always
a Lipschitz chain: a simple example is the graph of a Lipschitz function
$f\colon\mathbb{R}\to\mathbb{R}$ crossing the $x$-axis at infinitely many
points, in which case the part in the upper half-space is not a singular chain
due to having infinitely many components. Regardless, using the fact that our
chosen chain $\sigma$ is ”straight” in the region
$M\setminus\operatorname{int}M_{\varepsilon}\approx\partial\sigma_{\varepsilon}\times[0,\varepsilon]$
and the fact that any boundary translations in our convolution are inwards
into $M_{\varepsilon}$, we are able to verify that the required intersection
is Lipschitz if the translations we use in the convolution are small enough. ∎

###### Full proof of Proposition 6.1.

We define $F_{\partial M}\colon(\partial M)\times[0,2\varepsilon]\to U\subset
M$ and $P^{\delta}$ for $\delta\in[0,2\varepsilon]$ as in the proof of Lemma
6.3. For $\delta\in[0,2\varepsilon]$, we denote $M_{\delta}=P^{\delta}(M)$,
$\partial M_{\delta}=P^{\delta}(\partial M)$, $D_{\delta}=P^{\delta}(D)$ and
$\partial D_{\delta}=P^{\delta}(\partial D)$. We also define an alternate map
$Q^{\delta}\colon M\to M_{\delta}$ for $\delta\in[0,2\varepsilon)$, where
$Q^{\delta}=\operatorname{id}_{M}$ outside $U$ and $Q^{\delta}(F_{\partial
M}(x,t))=F_{\partial M}(x,\delta+(1-\delta/(2\varepsilon))t)$. In particular,
$Q^{\delta}$ is a locally bilipschitz homeomorphism $M\to M_{\delta}$.

Suppose towards contradiction that $c\in
H_{k}^{\operatorname{lip}}(M,D;\mathbb{K})$ is $p$-exceptional. We denote
$c_{\varepsilon}=Q^{\varepsilon}_{*}c\in
H^{k}_{\operatorname{lip}}(M_{\varepsilon},D_{\varepsilon};\mathbb{K})$. Since
$Q^{\varepsilon}$ is locally bilipschitz, it follows that $c_{\varepsilon}$ is
also $p$-exceptional. We consequently find a $\rho\in L^{p}(M_{\varepsilon})$
such that for every $\sigma\in c_{\varepsilon}$, the integral of $\rho$ over
$\mu_{\sigma}$ is infinite. We extend $\rho$ to $M$ by setting $\rho=0$ on
$M\setminus M_{\varepsilon}$.

We fix a Lipschitz $k$-chain $\sigma_{\varepsilon}\in c_{\varepsilon}$, in
which case $\partial\sigma_{\varepsilon}\in
C_{k-1}^{\operatorname{lip}}(D_{\varepsilon};\mathbb{K})$. Let $K$ be the
total image of $\sigma_{\varepsilon}$. We may assume that $K\cap(\partial
M_{\varepsilon}\setminus\operatorname{int}D_{\varepsilon})=\emptyset$. Indeed,
this can be achieved by similar methods as in the proof of Lemma 6.3, by using
a locally Lipschitz map similar to $P^{\delta}$ that takes $\partial
M_{\varepsilon}\setminus D_{\varepsilon}$ inside $M_{\varepsilon}$ and
$D_{\varepsilon}$ inside $\operatorname{int}D_{\varepsilon}$.

The boundary $\partial\sigma_{\varepsilon}$ is in
$C^{\operatorname{lip}}_{k-1}(\partial M_{\varepsilon},\mathbb{K})$. We use
$\partial\sigma_{\varepsilon}$ to define a chain $\sigma_{c}\in
C^{\operatorname{lip}}_{k}(U;\mathbb{K})$ as follows: for every simplex
$\gamma\colon\Delta_{k-1}\to\partial M_{\varepsilon}$ in the sum
representation of $\partial\sigma_{\varepsilon}$, we define
$\gamma_{c}\colon\Delta_{k-1}\times[0,\varepsilon]\to U$ such that
$F_{\partial M}^{-1}\circ\gamma_{c}(x,t)=((F_{\partial
M}^{-1}\circ\gamma(x))_{\partial M},t)$. Every $\gamma_{c}$ can be understood
as a Lipschitz $k$-chain following a suitable subdivision of the domain. Due
to this, we treat $\gamma_{c}$ like singular Lipschitz simplices during the
rest of the proof. We define $\sigma_{c}$ to be the sum of these $\gamma_{c}$
with the corresponding coefficients from $\partial\sigma_{\varepsilon}$. We
then fix $\sigma=\sigma_{\varepsilon}+\sigma_{c}$. Since $\sigma_{c}$ is
essentially a straight Lipschitz homotopy from $\partial\sigma_{\varepsilon}$
to $(Q^{\varepsilon})^{-1}_{*}\partial\sigma_{\varepsilon}$, it follows that
$\partial\sigma\in\operatorname{int}D$.

We next pick charts $\phi_{i}\colon(-2,2)^{n}\to M$, $i\in\\{1,\dots,m\\}$,
such that the sets $U_{i}=\phi_{i}((-1,1)^{n})$ cover $K$. Because of our
assumption that $K\cap(\partial
M_{\varepsilon}\setminus\operatorname{int}D_{\varepsilon})=\emptyset$, we can
restrict our attention to charts of two types. The first type consists of
charts inside the interior of $M_{\varepsilon}$, i.e. $U_{i}\subset
M_{\varepsilon}$. The second type consists of rectangular boundary charts
around points of $D_{\varepsilon}$. We may choose these to be such that
$U_{i}=F_{\partial M}(V_{i}\times(0,2\varepsilon))$ and $\phi_{i}=F_{\partial
M}\circ(\phi_{i}^{\prime}\times((\varepsilon/2)\operatorname{id}_{\mathbb{R}}+\varepsilon))$
where $V_{i}\subset\operatorname{int}D$ and
$\phi_{i}^{\prime}\colon(-2,2)^{n-1}\to V_{i}$ is a bilipschitz chart.

For every $i\in\\{1,\dots,m\\}$ we let $\varepsilon_{i}>0$ and define maps
$\eta_{i}\colon(-2,2)^{n}\to[0,\varepsilon_{i}]$ by
$\eta_{i}(x)=\max(0,1-\left|{x}\right|_{\ell^{\infty}})\varepsilon_{i}$. That
is, $\eta_{i}$ are similar as in Lemma 6.2, except adjusted to use
$\ell^{\infty}$-norms due to our domain being a square. Further following the
proof of Lemma 6.2, we also define for every $v\in B^{n}(0,1)$ a map
$G_{i,v}\colon(-2,2)^{n}\to(-2,2)^{n}$ by $G_{i,v}(x)=x+\eta_{i}(x)v$, assume
$\varepsilon_{i}$ are small enough that $G_{i,v}$ are bilipschitz and set
$F_{i,v}=\phi_{i}\circ G_{i,v}\circ\phi_{i}^{-1}$ in $U_{i}$ and
$F_{i,v}=\operatorname{id}_{M}$ elsewhere. We again set $\rho_{0}=\rho$ and
define inductively that

$\rho_{i}(x)=\int_{B^{n}(0,1)\cap\mathbb{H}^{n}_{+}}\rho_{i-1}\circ
F_{i,v}(x)\mathop{}\\!\mathrm{d}v.$

Note the difference to Lemma 6.2 that the averaging occurs over the upper half
ball.

By repeating the corresponding parts of the proof of Lemma 6.2 with trivial
changes, we are able to conclude that $\rho_{m}$ is in $L^{p}(M)$ and that
$\rho_{m}$ is continuous and finite-valued on $U_{1}\cup\dots\cup U_{m}$.
Moreover, since the $\rho$ we started with is identically zero in $M\setminus
M_{\varepsilon}$ and hence continuous there, it in fact follows that
$\rho_{m}$ is continuous and finite-valued in $U_{1}\cup\dots\cup
U_{m}\cup(M\setminus M_{\varepsilon})$. Since the total image of $\sigma$ lies
in $U_{1}\cup\dots\cup U_{m}\cup(M\setminus M_{\varepsilon})$, it only remains
to show that the integral of $\rho_{m}$ over $\mu_{\sigma}$ is infinite, in
which case a contradiction follows.

For this, fix $v_{1},\dots,v_{m}\in B^{n}(0,1)\cap\mathbb{H}^{n}_{+}$ and
denote
$\sigma_{v_{1},\dots,v_{m}}=(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\sigma$.
Due to our use of $v_{i}$ in the upper half space and to how our boundary
charts are set up, the total image of
$(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\sigma_{\varepsilon}$
lies inside $\operatorname{int}M_{\varepsilon}$. Therefore any intersections
of $\sigma_{v_{1},\dots,v_{m}}$ with $\partial M_{\varepsilon}$ will occur in
the $(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\sigma_{c}$-part.
The Lipschitz constants of $G_{i,sv_{i}}$ have an upper bound independent of
the choice of $s\in[0,1]$ and $v_{i}\in B^{n}(0,1)$. Hence, we may fix a
constant $L\in(1,\infty)$ independent of $s$ and $v_{i}$ (but dependent on the
charts $\phi_{i}$) such that the maps
$\phi_{i}^{-1}\circ\Phi_{i^{\prime}}\circ\phi_{i^{\prime\prime}}$ are all
$L$-Lipschitz in their domains of definition.

For every $i\in\\{0,\dots,m\\}$, we define
$\Phi_{i}\colon\partial_{M}\times[0,\varepsilon]\times[0,1]\to M$ by

$\displaystyle\Phi_{i}(x,t,s)=F_{i,sv_{i}}\circ
F_{i-1,sv_{i-1}}\circ\dots\circ F_{1,sv_{1}}\circ F_{\partial_{M}}(x,t).$

We fix a chart
$\phi=\phi_{j}^{\prime}\times\operatorname{id}_{[0,\varepsilon]}\times\operatorname{id}_{[0,1]}$
where $\phi_{j}^{\prime}\colon(-2,2)^{n-1}\to V_{j}$ is from the definition of
one of the charts $\phi_{i}$ that meet the boundary of $M_{\varepsilon}$. For
every $i\in\\{0,m\\}$ and $(x,s)\in V_{j}\times[0,1]$, we denote by
$t_{i}(x,s)$ the largest $t\in[0,\varepsilon]$ such that
$\Phi_{i}(x,t,s)\in\partial M_{\varepsilon}$; note that
$t_{0}(x,s)\equiv\varepsilon$. We claim that for every $i\in\\{1,m\\}$, if we
select $\varepsilon_{i}$ small enough, then the function $t_{i}$ is Lipschitz
in $V_{j}$. First we will show that for all $x,x^{\prime}\in V_{j}$,
$s_{1},s_{2}\in[0,1]$, $t_{1}\in[0,t_{i-1}(x,s_{1})]$ and
$t_{2}\in[0,t_{i-1}(x,s_{2})]$ with $t_{1}<t_{2}$, we have

(13)
$\displaystyle\begin{split}2^{-i}(t_{1}-t_{2})-(1-2^{-i})&\left(\left|{(\phi_{j}^{\prime})^{-1}(x)-(\phi_{j}^{\prime})^{-1}(x^{\prime})}\right|+\left|{s_{1}-s_{2}}\right|\right)\\\
&\leq(F_{\partial
M}^{-1}(\Phi_{i}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\\\
&\leq(2-2^{-i})(t_{1}-t_{2})+(1-2^{-i})\left(\left|{(\phi_{j}^{\prime})^{-1}(x)-(\phi_{j}^{\prime})^{-1}(x^{\prime})}\right|+\left|{s_{1}-s_{2}}\right|\right).\end{split}$

In fact, $t_{i}(x,s_{1})$ is the only $t\in[0,\varepsilon]$ such that
$\Phi_{i}(x,t,s_{1})\in\partial M_{\varepsilon}$.

We prove (13) by induction. Suppose that the claim holds for $i-1$, where the
case of $i-1=0$ is clear since $(F_{\partial
M}^{-1}(\Phi_{0}(x,t,s)))_{[0,2\varepsilon]}=t$. If $\Phi_{i-1}(x,t,s)\in
M\setminus M_{\varepsilon}$, then $\Phi_{i}(x,t,s)\in M\setminus
M_{2\varepsilon}$ and so (13) is well defined. Next, supposing that
$(x,s,t)\in V_{j}\times[0,\varepsilon]\times[0,1]$, if we have
$\Phi_{i-1}(x,t,s)\in U_{i}$, we can compute that

$\displaystyle(F_{\partial M}^{-1}(\Phi_{i}(x,t,s)))_{[0,2\varepsilon]}$
$\displaystyle=(F_{\partial M}^{-1}\circ\phi_{i}\circ
G_{i,sv_{i}}\circ\phi_{i}^{-1}\circ(\Phi_{i-1}(x,t,s)))_{[0,2\varepsilon]}$
$\displaystyle=(\varepsilon/2)(G_{i,sv_{i}}\circ\phi_{i}^{-1}(\Phi_{i-1}(x,t,s)))_{n}+\varepsilon$
$\displaystyle=(\varepsilon/2)(\phi_{i}^{-1}(\Phi_{i-1}(x,t,s)))_{n}+\varepsilon+(\varepsilon/2)\varepsilon_{i}(sv_{i})_{n}(1-\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t,s))\rvert_{\ell^{\infty}})$
$\displaystyle=(F_{\partial
M}^{-1}(\Phi_{i-1}(x,t,s)))_{[0,\varepsilon]}+(\varepsilon/2)\varepsilon_{i}s(v_{i})_{n}(1-\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t,s))\rvert_{\ell^{\infty}}).$

On the other hand, if $\Phi_{i-1}(x,t,s)\notin U_{i}$, then $(F_{\partial
M}^{-1}(\Phi_{i}(x,t,s)))_{[0,\varepsilon]}=(F_{\partial
M}^{-1}(\Phi_{i-1}(x,t,s)))_{[0,\varepsilon]}$. It follows that if
$\Phi_{i-1}(x,t_{1},s_{1})$ and $\Phi_{i-1}(x^{\prime},t_{2},s_{2})$ are both
in $U_{i}$, we may estimate using $s_{1},s_{2}\leq 1$ and
$\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))\rvert_{\ell^{\infty}}\leq 1$
that

(14) $\displaystyle\Big{\lvert}\bigl{(}(F_{\partial
M}^{-1}(\Phi_{i}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\bigr{)}$
$\displaystyle\qquad\qquad-\bigl{(}(F_{\partial
M}^{-1}(\Phi_{i-1}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i-1}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\bigr{)}\Big{\rvert}$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}\left(\left|{s_{1}-s_{2}}\right|+\left|{s_{1}\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))\rvert_{\ell^{\infty}}-s_{2}\lvert\phi_{i}^{-1}(\Phi_{i-1}(x^{\prime},t_{2},s_{2}))\rvert_{\ell^{\infty}}}\right|\right)$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}\left(2\left|{s_{1}-s_{2}}\right|+\left|{\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))\rvert_{\ell^{\infty}}-\lvert\phi_{i}^{-1}(\Phi_{i-1}(x^{\prime},t_{2},s_{2}))\rvert_{\ell^{\infty}}}\right|\right)$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}\left(2\left|{s_{1}-s_{2}}\right|+\left|{\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))-\phi_{i}^{-1}(\Phi_{i-1}(x^{\prime},t_{2},s_{2}))}\right|\right)$
$\displaystyle\qquad\leq\varepsilon\varepsilon_{i}(1+L)\left((t_{1}-t_{2})+\left|{(\phi_{j}^{\prime})^{-1}(x)-(\phi_{j}^{\prime})^{-1}(x^{\prime})}\right|+\left|{s-s^{\prime}}\right|\right).$

If on the other hand $\Phi_{i-1}(x,t_{1},s_{1})\in U_{i}$ and
$\Phi_{i-1}(x^{\prime},t_{2},s_{2})\notin U_{i}$, then there exists a convex
combination
$(x^{\prime\prime},t_{3},s_{3})=\phi(\lambda\phi^{-1}(x,t_{1},s_{1})+(1-\lambda)\phi^{-1}(x^{\prime},t_{2},s_{2}))\in
V_{i}\times[0,\varepsilon]\times[0,1]$, $\lambda\in(0,1]$, such that
$\Phi_{i-1}(x^{\prime\prime},t_{3},s_{3})\in(\phi_{i}(-2,2)^{n})\setminus
U_{i}$. Hence, we obtain that
$\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))\rvert_{\ell^{\infty}}<1\leq\lvert\phi_{i}^{-1}(\Phi_{i-1}(x^{\prime\prime},t_{3},s_{3}))\rvert_{\ell^{\infty}}\leq
2$. We instead estimate

(15) $\displaystyle\Big{\lvert}\bigl{(}(F_{\partial
M}^{-1}(\Phi_{i}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\bigr{)}$
$\displaystyle\qquad\qquad-\bigl{(}(F_{\partial
M}^{-1}(\Phi_{i-1}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i-1}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\bigr{)}\Big{\rvert}$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}\left|{1-\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))\rvert_{\ell^{\infty}}}\right|$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}\left|{\lvert\phi_{i}^{-1}(\Phi_{i-1}(x,t_{1},s_{1}))\rvert_{\ell^{\infty}}-\lvert\phi_{i}^{-1}(\Phi_{i-1}(x^{\prime\prime},t_{3},s_{3}))\rvert_{\ell^{\infty}}}\right|$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}L\left((t_{1}-t_{3})+\left|{(\phi_{j}^{\prime})^{-1}(x)-(\phi_{j}^{\prime})^{-1}(x^{\prime\prime})}\right|+\left|{s_{1}-s_{3}}\right|\right)$
$\displaystyle\qquad\leq(\varepsilon/2)\varepsilon_{i}L\left((t_{1}-t_{2})+\left|{(\phi_{j}^{\prime})^{-1}(x)-(\phi_{j}^{\prime})^{-1}(x^{\prime})}\right|+\left|{s_{1}-s_{2}}\right|\right).$

The case $\Phi_{i-1}(x,t_{1},s_{1})\notin
U_{i},\Phi_{i-1}(x^{\prime},t_{2},s_{2})\in U_{i}$ is identical to the
previous case and in the final case $\Phi_{i-1}(x,t_{1},s_{1})\notin
U_{i},\Phi_{i-1}(x^{\prime},t_{2},s_{2})\notin U_{i}$, we have

(16) $\displaystyle\Big{\lvert}\bigl{(}(F_{\partial
M}^{-1}(\Phi_{i}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\bigr{)}$
$\displaystyle\qquad\qquad-\bigl{(}(F_{\partial
M}^{-1}(\Phi_{i-1}(x,t_{1},s_{1})))_{[0,2\varepsilon]}-(F_{\partial
M}^{-1}(\Phi_{i-1}(x^{\prime},t_{2},s_{2})))_{[0,2\varepsilon]}\bigr{)}\Big{\rvert}=0.$

Thus, by choosing $\varepsilon_{i}<\varepsilon^{-1}(1+L)^{-1}2^{-i}$ and by
combining (14)-(16) with the case $i-1$ of (13), we get the case $i$ of (13).

By using (13) with $x=x^{\prime}$ and $s_{1}=s_{2}=s$, it follows that
$t\mapsto(F_{\partial M}^{-1}(H_{c,i}(x,t,s)))_{[0,2\varepsilon]}$ is
increasing on $[0,t_{i-1}(x,s)]$. Moreover, since $F_{i,sv_{i}}$ translate
inwards whenever $U_{i}$ meets $\partial M_{\varepsilon}$, we have
$F_{\partial M}^{-1}(H_{c,i}(x,t_{i-1}(x,s),s))\geq\varepsilon$. Hence, for
every $(x,s)\in V_{i}\times[0,1]$ there in fact exists exactly one
$t_{i}(x,s)\in[0,\varepsilon]$ such that
$\Phi_{i}(x,t_{i}(x,s),s)\in\operatorname{int}D_{\varepsilon}$, that is,
$(F_{\partial
M}^{-1}(\Phi_{i}(x,t_{i}(x,s),s)))_{[0,2\varepsilon]}=\varepsilon$. Moreover,
by applying (13) with $t_{1}=t_{i}(x,s_{1})$ and
$t_{2}=t_{i}(x^{\prime},s_{2})$, it follows that

$\left|{t_{i}(x,s_{1})-t_{i}(x^{\prime},s_{2})}\right|\leq
2^{i}(1-2^{-i})\left(\left|{(\phi_{j}^{\prime})^{-1}(x)-(\phi_{j}^{\prime})^{-1}(x^{\prime})}\right|+\left|{s_{1}-s_{2}}\right|\right),$

which implies that $t_{i}\circ\phi$ is Lipschitz. The induction part is hence
complete.

With the map $t_{m}$, we may define a map $P^{\prime}\colon\partial
M\times[0,\varepsilon]\times[0,1]\to\partial M\times[0,\varepsilon]$ by

$P^{\prime}(x,t,s)=(x,\max(t,t_{m}(x,s))).$

With the help of this map, we then define $P\colon M\times[0,1]\to M$ by
setting $P(x,s)=F_{\partial M}(P^{\prime}(F_{\partial M}^{-1}(x),s)$ on
$M\setminus M_{\varepsilon}$ and $P(x,s)=x$ on $M_{\varepsilon}$. Finally, we
define $\Theta\colon M\times[0,1]\to M$ by

$\Theta(x,s)=F_{m,sv_{m}}\circ\dots\circ F_{1,sv_{1}}\circ P(x,s).$

We note that the image of $\Theta$ lies in $M_{\varepsilon}$. By our proof
that $t_{m}$ is Lipschitz on any of our chosen boundary neighborhoods, we get
that $\Theta$ is Lipschitz in a neighborhood of $K\times[0,1]$. We denote
$\Theta^{s}=\Theta(\cdot,s)$ for $s\in[0,1]$.

The map $\Theta$ hence provides a Lipschitz homotopy between
$\Theta^{1}_{*}\sigma$ and $\Theta^{0}_{*}\sigma$. Recalling that
$\sigma=\sigma_{\varepsilon}+\sigma_{c}$ where $\sigma_{c}$ is a ”straight”
extension of $\partial\sigma_{\varepsilon}$ to $M\setminus M_{\varepsilon}$,
let $\gamma_{c}\colon\Delta_{k-1}\times[0,\varepsilon]\to M$ be one of the
terms in the sum representation of $\sigma_{c}$. Since
$t_{m}(\cdot,0)\equiv\varepsilon$, we have
$\Theta^{0}_{*}\gamma_{c}(x,t)=\gamma_{c}(x,\varepsilon)=\gamma(x)$ for all
$(x,t)\in\Delta_{k-1}\times[0,\varepsilon]$, where $\gamma$ again is the
simplex of $\partial\sigma_{\varepsilon}$ that $\gamma_{c}$ is a straight
extension of. Since also
$\Theta^{0}_{*}\sigma_{\varepsilon}=\sigma_{\varepsilon}$, we conclude that
$\Theta^{0}_{*}\sigma$ is in the same Lipschitz homology class
$c_{\varepsilon}$ as $\sigma_{\varepsilon}$. Moreover, the total image of the
boundary $\partial\sigma$ is in $K\cap\partial M$, which every $\Theta_{s}$
maps into $D_{\varepsilon}$. Because of this,
$\Theta^{1}_{*}\sigma-\Theta^{0}_{*}\sigma\in\partial
C^{\operatorname{lip}}_{k+1}(M_{\varepsilon};\mathbb{K})+C^{\operatorname{lip}}_{k}(D_{\varepsilon};\mathbb{K})$,
implying that $\Theta^{1}_{*}\sigma\in c_{\varepsilon}$.

Finally, we return to the $k$-chain
$\sigma_{v_{1},\dots,v_{m}}=(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\sigma$.
Clearly we have
$\Theta^{1}_{*}\sigma_{\varepsilon}=(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\sigma_{\varepsilon}$.
Moreover, if $\gamma_{c}$ is a component of the sum representation of
$\sigma_{c}$, we have
$\Theta^{1}_{*}\gamma_{c}(x,t)=(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\gamma_{c}(x,t)$
when $t\geq t_{m}((Q^{\varepsilon})^{-1}_{*}\gamma(x),1)$ and
$\Theta^{1}_{*}\gamma_{c}(x,t)=(F_{m,v_{m}})_{*}\dots(F_{2,v_{2}})_{*}(F_{1,v_{1}})_{*}\gamma_{c}(x,t_{m}((Q^{\varepsilon})^{-1}_{*}\gamma(x),1))$
otherwise. Notably, for $t\leq t_{m}((Q^{\varepsilon})^{-1}_{*}\gamma(x),1)$,
$\Theta^{1}_{*}\gamma_{c}(x,t)$ is independent of $t$ and this part will hence
yield a zero-dimensional Jacobian if $\Theta^{1}_{*}\gamma_{c}$ is post-
composed with a chart. It follows that $\mu_{\sigma_{v_{1},\dots,v_{m}}}$ is
bounded from below by a multiple of $\mu_{\Theta^{1}_{*}\sigma}$. Since
$\Theta^{1}_{*}\sigma\in c_{\varepsilon}$, the integral of $\rho$ over
$\mu_{\Theta^{1}_{*}\sigma}$ is infinite by the definition of $\rho$ and
therefore the integral of $\rho$ over $\mu_{\sigma_{v_{1},\dots,v_{m}}}$ is
also infinite. Since this holds for all choices of $v_{1},\dots,v_{m}$, it
follows by a computation as in (12) that the integral of $\rho_{m}$ over
$\mu_{\sigma}$ is infinite. The proof is hence complete.

∎

##  7\. Stokes-type theorems for $L^{p}$-forms

In this section, we prove Stokes-type results for $L^{p}$-forms in the setting
of Lipschitz manifolds. In particular, our aim is to provide a generalization
of the theory discussed in [7, Chapter III], where instead of vector fields in
Euclidean space, we consider Sobolev forms on Lipschitz domains in a Lipschitz
manifold.

We start by pointing out that, by combining Proposition 4.5 with the Fuglede
lemma for forms given in Lemma 4.9, we obtain the following $L^{p}$-version of
the Lipschitz Stokes theorem.

######  Corollary 7.1.

Let $M$ be a closed Lipschitz $n$-submanifold in an oriented Lipschitz
Riemannian $n$-manifold $\mathcal{R}$. Suppose that $\omega\in
W^{d,p}(\wedge^{k-1}T^{*}M)$. There exists a $p$-exceptional $E_{1}\subset
C_{k}^{\operatorname{lip}}(M;\mathbb{Z})$ and $E_{2}\subset
C_{k}^{\operatorname{lip}}(M;\mathbb{Z})$ for which $\partial E_{2}\subset
C_{k-1}^{\operatorname{lip}}(M;\mathbb{Z})$ is $p$-exceptional, such that

$\int_{\partial\sigma}\omega=\int_{\sigma}d\omega.$

for all $\sigma\in C_{k}^{\operatorname{lip}}(M;\mathbb{R})\setminus(E_{1}\cup
E_{2})$.

###### Proof.

By Lemma 3.2, we can approximate $\omega$ by $W^{d,\infty}$-forms
$\omega_{i}$. By Proposition 4.5, every $\omega_{i}$ satisfies the desired
Stokes’ theorem for every $\sigma\in
C_{k}^{\operatorname{lip}}(M;\mathbb{R})$. Then we extend $\omega$,
$\omega_{i}$, $d\omega$ and $d\omega_{i}$ by zero to $\mathcal{R}$. By Lemmas
4.9 and 4.10 and after moving to a subsequence,
$\int_{\sigma}d\omega_{j}\to\int_{\sigma}d\omega$ outside a $p$-exceptional
set of $\sigma\in C_{k}^{\operatorname{lip}}(M;\mathbb{Z})$. Additionally,
$\int_{\partial\sigma}\omega_{j}\to\int_{\partial\sigma}\omega$ outside a
$p$-exceptional set of $\partial\sigma\in\partial
C_{k-1}^{\operatorname{lip}}(M;\mathbb{Z})$. ∎

Having two exceptional sets in Corollary 7.1 is necessary. For the necessity
of $E_{2}$, we can take a smooth $\omega\in
C^{\infty}_{0}(\wedge^{n-1}\mathbb{R}^{n})$ and redefine it on the boundary of
a single $n$-simplex $\sigma$, making the formula fail on every simplex
$\sigma^{\prime}$ with $\partial\sigma^{\prime}=\partial\sigma$. The singleton
$\\{\partial\sigma\\}$ is $p$-exceptional, but the family of $n$-simplices
$\sigma^{\prime}$ with $\partial\sigma^{\prime}=\partial\sigma$ is not.

For the necessity of $E_{1}$, we redefine a smooth $d\omega\in
C^{\infty}_{0}(\wedge^{k}\mathbb{R}^{n})$ with $k<n$ on a single $k$-chain
$\sigma$ with $\partial\sigma=0$. The formula fails for all $k$-chains
$\sigma+\sigma^{\prime}$ where the image of $\sigma^{\prime}$ is disjoint from
the image of $\sigma$. The family of such $k$-chains is exceptional, but the
family of their boundaries is not.

###  7.1. The Fuglede-type characterization

The main result of this section characterizes closedness and exactness for
Sobolev forms purely by integration over $k$-chains. Moreover, it also
includes a similar characterization of the weak vanishing of $\omega_{T}$ in
such cases.

######  Theorem 7.2.

Suppose that $M$ is a closed Lipschitz $n$-submanifold in an oriented
Lipschitz Riemannian $n$-manifold $\mathcal{R}$ without boundary and $D$ is a
closed Lipschitz $(n-1)$-submanifold in $\partial M$. Let $p\in(1,\infty)$ and
$\omega\in L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}M)$. The following are
equivalent.

  * •

$\omega\in W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}M)$ and $d\omega=0$.

  * •

For $p$-a.e. $\sigma\in C_{k}^{\operatorname{lip}}(D;\mathbb{Z})+\partial
C_{k+1}^{\operatorname{lip}}(M;\mathbb{Z})\subset
C_{k}^{\operatorname{lip}}(M;\mathbb{Z})$,

$\int_{\sigma}\omega=0.$

The following are also equivalent.

  * •

$\omega=d\tau$ for some $\tau\in
W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k-1}T^{*}M)$.

  * •

For $p$-a.e. $\sigma\in C_{k+1}^{\operatorname{lip}}(M;\mathbb{Z})$ such that
$\partial\sigma\in C_{k}^{\operatorname{lip}}(D;\mathbb{Z})$,

$\int_{\sigma}\omega=0.$

The theorem immediately implies Theorem 1.5 by considering $M=\mathcal{R}$ and
$\partial M=\emptyset$. Theorem 7.2 consists of four individual implications.
Two of them can be proven locally and we begin with those statements. The
remaining two require global information, which in our proof is provided by
the use of $L^{p}$-cohomology. Note that the proofs of almost all of the
implications also apply for $p=1$, except for one proof which uses Proposition
6.1.

###  7.2. The local implications

We start by showing the following part of Theorem 7.2.

######  Lemma 7.3.

Suppose that $M$ is a closed Lipschitz $n$-submanifold in an oriented
Lipschitz Riemannian $n$-manifold $\mathcal{R}$ without boundary and $D$ is a
closed Lipschitz $(n-1)$-submanifold in $\partial M$. Let $p\in[1,\infty)$ and
$\omega\in L^{p}_{\mathrm{loc}}(\wedge^{k}T^{*}M)$. If

$\int_{\sigma}\omega=0,$

for $p$-a.e. $\sigma\in C_{k}^{\operatorname{lip}}(D;\mathbb{Z})+\partial
C_{k+1}^{\operatorname{lip}}(M;\mathbb{Z})$, then $\omega\in
W^{d,p}_{T(D),\mathrm{loc}}(\wedge^{k}T^{*}M)$ and $d\omega=0$.

We first prove Euclidean versions of the result, starting without boundary
conditions.

######  Lemma 7.4.

For $k\in\\{1,\dots,n\\}$, let $\mathcal{B}^{n}_{k}\subset
C_{k}^{\operatorname{lip}}(\mathbb{R}^{n};\mathbb{Z})$ denote the collection
of all isometrically embedded $k$-balls in $\mathbb{R}^{n}$. Let $k\geq 0$ and
$\omega\in L^{p}_{\mathrm{loc}}(\wedge^{k}\mathbb{R}^{n})$, where
$p\in[1,\infty)$. If

$\int_{\partial\sigma}\omega=0,$

for $p$-a.e. $\partial\sigma\in\partial\mathcal{B}^{n}_{k+1}$, then
$d\omega=0$ weakly.

###### Proof.

We first show that if $\sigma\in\mathcal{B}^{n}_{k+1}$ and
$A\subset\mathbb{R}^{n}$ is a measurable set such that the family
$\mathcal{S}_{A}:=\\{\partial\sigma-y:y\in A\\}$ is $p$-exceptional, then
$m_{n}(A)=0$. By subadditivity of $m_{n}$ and $\operatorname{Mod}_{p}$, we may
assume that $A$ is bounded and in particular $m_{n}(A)<\infty$. Since
$\operatorname{spt}\nu_{\partial\sigma}$ is also bounded, we may select a
bounded $B\subset\mathbb{R}^{n}$ such that $x-y\in B$ whenever $y\in A$ and
$x\in\operatorname{spt}\nu_{\partial\sigma}$. Suppose towards contradiction
that $m_{n}(A)>0$. Lemma 2.3 yields a function $\rho\in L^{p}(\mathbb{R}^{n})$
such that the integral of $\rho$ over every $\partial\sigma-y$ with $y\in A$
is infinite. By Tonelli’s theorem and our counterassumption that $m_{n}(A)>0$,
we have

$\displaystyle\int_{\partial\sigma}\int_{A}\rho(x-y)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle=\int_{A}\int_{\partial\sigma}\rho(x-y)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
$\displaystyle=\int_{A}\int_{\partial\sigma-y}\rho(x^{\prime})\mathop{}\\!\mathrm{d}x^{\prime}\mathop{}\\!\mathrm{d}y=\infty.$

Hence, the function

$x\mapsto\int_{A}\rho(x-y)dy$

must be unbounded over $\operatorname{spt}\nu_{\partial\sigma}$. However,
since $\rho\in L^{p}({\mathbb{R}}^{n})$ and $B\subset\mathbb{R}^{n}$ is
bounded, we have $\rho\in L^{1}(B)$. It follows that

$\displaystyle\int_{A}\rho(x-y)dy\leq\int_{B}\rho<\infty,$

for all $x\in\operatorname{spt}\nu_{\partial\sigma}$, resulting in a
contradiction. We conclude that indeed $m_{n}(A)=0$.

Suppose then that

$\int_{\partial\sigma}\omega=0$

for $p$-a.e. $\partial\sigma\in\partial\mathcal{B}^{n}_{k+1}$. We take the
convolution $\omega_{j}=\eta_{j}\ast\omega$ for a sequence of mollifying
kernels $\eta_{j}\in C^{\infty}_{0}(\mathbb{R}^{n})$. Fix
$\partial\sigma\in\partial\mathcal{B}^{n}_{k+1}$. By Fubini’s theorem and a
change of variables,

$\displaystyle\int_{\partial\sigma}\omega_{j}$
$\displaystyle=\int_{\partial\sigma}\int_{\mathbb{R}^{n}}\omega_{x-y}\eta_{j}(y)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{n}}\eta_{j}(y)\int_{\partial\sigma}\omega_{x-y}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
$\displaystyle=\int_{\mathbb{R}^{n}}\eta_{j}(y)\left(\int_{\partial\sigma-y}\omega\right)\mathop{}\\!\mathrm{d}y.$

For $p$-a.e. element of $\\{\partial\sigma-y:y\in\mathbb{R}^{n}\\}$, we have
by assumption that

$\int_{\partial\sigma-y}\omega=0.$

The $p$-exceptional family where this does not hold is of the form
$\mathcal{S}_{A}$ for some $A\subset\mathbb{R}^{n}$. We can place $A$ between
two measurable sets $A^{-}\subset A\subset A^{+}$ with the same
$m_{n}$-measure. By the statement shown in the beginning of the proof, we have
$m_{n}(A^{-})=0$, which also yields $m_{n}(A^{+})=0$. So

$\int_{\mathbb{R}^{n}}\eta(y)\left(\int_{\partial\sigma-y}\omega\right)\mathop{}\\!\mathrm{d}y=\int_{\mathbb{R}^{n}\setminus
A^{+}}0\mathop{}\\!\mathrm{d}y=0.$

In conclusion, the integral of $\omega_{j}$ over every
$\partial\sigma\in\partial\mathcal{B}^{n}_{k+1}$ is zero.

Since $\omega_{j}$ are smooth, it follows by the standard smooth Stokes’
theorem that the integral of $d\omega_{j}$ over every
$\sigma\in\mathcal{B}^{n}_{k+1}$ is zero. This is only possible if
$d\omega_{j}$ is identically zero. Let $V\subset\mathbb{R}^{n}$ be a bounded
domain, in which case $\omega_{j}\to\omega$ in $L^{p}(\wedge^{k}V)$. Since
$d\omega_{j}\to 0$ in $L^{p}(\wedge^{k}V)$, we obtain that $d\omega=0$ weakly
in $V$. By the locality properties of the weak differential, it follows that
$d\omega=0$ weakly in $\mathbb{R}^{n}$. ∎

Next, we give a Euclidean version with boundary.

######  Lemma 7.5.

Let $\mathbb{H}^{n}$ denote the upper half-space of $\mathbb{R}^{n}$. For
$k\in\\{1,\dots,n\\}$, let $\mathcal{B}^{n}_{k}\subset
C_{k}^{\operatorname{lip}}(\mathbb{R}^{n};\mathbb{Z})$ denote the collection