We introduce a notion of distributional $k$-forms on $d$-dimensional manifolds which can be integrated against suitably regular $k$-submanifolds.
Our approach combines ideas from Whitney's geometric integration <cit.> with those of sewing approaches to rough integration <cit.>.

§ INTRODUCTION

In this work we introduce Banach spaces of generalised/rough differential forms (or cochains) which are analogous to Hölder distributions.
Following the approach of geometric integration, pioneered by Whitney <cit.> and further developed by Harrison <cit.>, an element $A$ of such a space is characterized by its evaluation on simplices $\sigma \mapsto A(\sigma)$.
We work in a rougher setting where these elements can be of “infinite variation”, that is we allow $\sup_{\mathcal{K} | \sigma} \sum_{\sigma' \in \mathcal{K}} |A(\sigma')| = \infty$ where the supremum is over subdivisions $\mathcal{K}$ of $\sigma$ into smaller simplices.
In particular, we can make sense of a class of
$k$-forms formally given by $f = \sum_{I} f_{I} \mathrm{d}x^I$  where the $(f_{I})_{I}$ are genuine distributions.

Our work here can be seen as formulating a geometric analogue of the integration theory of Young <cit.>, building on earlier work such as <cit.>.
It is also motivated by the study of random fields in probability theory where one is often interested in integrating rough random distributions on $\mathbb{R}^{d}$ over $k < d$ dimensional objects, for instance circle averages of the Gaussian Free Field <cit.> or Wilson loop observables for quantum gauge fields <cit.>.

§.§ Main results and past work

We introduce a space $\Omega^{k}_{(\alpha,\beta)}$ of generalised $k$-forms in $d$-dimensional ambient space
for $\alpha \in (0,1]$, $\beta \in [0,1]\cup \{\infty\}$.
Our results for this space are as follows:

* Elements of $\Omega^{k}_{(\alpha,\beta)}$ be integrated in the sense geometric integration theory over embedded $k$-manifolds.

* Under point-wise multiplication $\Omega^{k}_{(\alpha,\beta)}$ is $C^\gamma$–module for $\gamma\in (1-\alpha+\beta,1]$.

* Whenever $\alpha>1/2$, these forms can be pulled-back which allows them to be defined intrinsically on manifolds. This allows for a notion of integration by pullback for which a corresponding Stokes' theorem holds.

* For $\alpha \in (0,1)$,
there is a natural embedding $\Omega^{k}_{\alpha,\beta}(\mathbb{R}^d) \hookrightarrow \prod_{I}  \mathcal{C}^{\alpha - 1}(\mathbb{R}^{d})$ which extends the mapping
$f = \sum_{I} f_I(x) dx^{I}\mapsto \{f_I\}_{I}$ on smooth forms $f$.

* We give a Kolmogorov criterion for the spaces $\Omega^{k}_{(\alpha,\beta)}$, which is checked for a class of  (distributional) Gaussian fields.

Along the way, a useful tool that we formulate and prove is a “simplicial sewing lemma”, Proposition <ref>, which might be of independent interest as it provides a coordinate invariant formulation[See also Remark <ref>.] of the multidimensional sewing lemma of <cit.> and<cit.>.

This work also unifies several previous constructions, below we compare our spaces $\Omega_{(\alpha,\beta)}^k$ and results about them to past work.

* For $\alpha=\beta=1$ these spaces agree with the flat cochains of Whitney, <cit.>, and for $\alpha=1$, $0<\beta<1$ with the class of $(k-1+\beta)$- cochains introduced in <cit.>, see Remark <ref>.

* They contain the generalised $d$-forms $f\cdot dg_1\wedge...\wedge dg_d$ introduced in <cit.>  and recently studied in <cit.> for $d = 2$ by discrete approximation. We extend several results therein, see Remark <ref>, Corollary <ref>, and Theorem <ref>.

* In dimension $d=2$ and for $k=1$, they are very similar to the spaces in <cit.>, <cit.>, see Remark <ref>, Remark <ref>, Proposition <ref> and Section <ref>.

§.§ Outline of the article

In Section <ref> we introduce geometric notions, such as the set  $\mathfrak{X}^k$ of oriented $k$-simplices in $\mathbb{R}^d$ and a corresponding space of chains $\mathcal{X}^k=\mathbb{Z}[\mathfrak{X}^k]/{\sim}$ where $\mathbb{Z}[\mathfrak{X}^k]$ denotes the free $\mathbb{Z}$-module generated by $\mathfrak{X}^k$ and the equivalence relation $\sim$ identifies the operators of addition and negation on $\mathbb{Z}[\mathfrak{X}^k]$with the geometric operations of gluing simplices and inverting orientation.
The subset $\Omega^k$ of measurable elements of the algebraic dual of $\mathcal{X}^k$ provides us with a `universe' of differential forms/cochains.

The first novelty that appears is our notion of size/mass of a simplex $\mass^k_{\alpha}: \mathfrak{X}^k\to  \mathbb{R}$ for $\alpha\in [0,1]$, given by the maximum of $\Vol^{k-1} (F) \cdot h^{\alpha}_{F}$ over faces $F$ of the simplex, where $h_F$ is the height measured from $F$.

In Section <ref> we define spaces $\Omega^k_{(\alpha,\beta)}\subset \Omega^k $ of `distributional' forms by imposing for $ \sigma \in \mathfrak{X}^k$ and $\omega \in  \mathfrak{X}^{k+1}$ the inequalities

$$| A (\sigma)| \lesssim \mass_\alpha^k (\sigma), \qquad  | \partial A (\omega)| \lesssim \mass_\alpha^k (\omega) \ .$$

Here $\partial:\mathcal{X}^{k+1}\to \mathcal{X}^k$ is the linear extension of the boundary operator on oriented simplices, which by duality maps $\Omega^k$ to $\Omega^{k+1}$.
Lemma <ref> provides equivalent characterisations of the spaces $\Omega^k_{(\alpha,\beta)}$ by controlling $A(\sigma)$ either in terms of the diameter of $\sigma$, or in terms of the value of $A$ on cubes.
Subsection <ref> then introduces a norm $| \cdot |_{(\alpha,\beta)}$ on $\mathcal{X}^{k}$ which for $\alpha=\beta=1$ is equivalent to Whitneys flat norm, see <cit.>, and for $\alpha=1, \beta\in (0,1)$ equivalent to the $(k-1+\beta)$-norm of Harrison, <cit.>.

We then establish an integration theory in the spirit of geometric integration by showing that $\Omega^{k}_{(\alpha,\beta)}$ is contained in the
continuous dual of $\mathcal{B}_{\alpha,\beta}^k$, the completion of $(\mathcal{X}^k, | \cdot |_{(\alpha,\beta)})$,
which in particular canonically contains closed manifolds.

Section <ref> establishes the Young type multiplication theorem, Theorem <ref>, which in particular generalises  <cit.> from dimension $2$ to arbitrary dimension.
In Corollary <ref>, we generalise <cit.> and establish that spaces of forms considered there are contained in our spaces.

The main result of Section <ref> shows that the notion of pull-back on smooth forms can be extended to $\Omega_{(\alpha,\beta)}^k$. As a consequence one can canonically define spaces $\Omega_{(\alpha,\beta)}^k(M)$ on any $C^{1,\eta}$-manifold $M$ as soon as $\alpha>1/(1+\eta)$.
Furthermore, the Stokes' theorem extends to forms in $\Omega_{(\alpha,\beta)}^k$.

In Section <ref>,  analogously to <cit.>, we show that the spaces $\Omega_{(\alpha,\beta)}^k(\mathbb{R}^{d})$ embed into spaces of Hölder distributions and, for codimension $d - k = 0$, this embedding is an isomorphism.

Section <ref> states and proves the promised Kolmogorov criterion which is then checked for a class of Gaussian fields.

Lastly, Section <ref> contains a proof of the simplicial sewing lemma, Proposition <ref>.
We present the proof, which only requires notions from Section <ref>, at the end of the article in order to streamline exposition. For the proof we define the notion of a “strongly regular method of subdivision” (see Definition <ref>) – the existence of such a method is non-trivial and follows from <cit.>.

We shall often use for the notation $\lesssim$ to mean that an inequality holds up to multiplication by a constant which may change from line to line but is uniform over any stated quantities. We forthermore write $\lesssim_{C}$ to emphasise the dependance of that constant on $C$. We write
$a\vee b:=\max\{a,b\}$ and $a\wedge b:=\min\{a,b\}$
for $a,b\in \mathbb{R}$.

For $m\in \mathbb{N}$ we equip $\mathbb{R}^m$ with the Borel sigma algebra and a norm $|\cdot|$ which is fixed throughout the article. We write $\Vol^k$ for the $k$-dimensional Hausdorff measure.
For a function $f: \mathbb{R}^d \to \mathbb{R}^m$, we shall use the notation $Df=\{\partial_{i} f\}_{i=1}^d$ to denote its derivative, as well as multi-index notation $D^I f:= \partial_1^{i_1},...,\partial_d^{i_d} f$ for $I=(i_1,...,i_d)\in \mathbb{N}^{d}$. We extend differentiation to distributions in the usual way. For $\Omega \subset \mathbb{R}^d$ and a function $f:\Omega \to \mathbb{R}^m$ we write $\|f\|_{L^{\infty}(\Omega)}:= \sup_{x\in \Omega} |f|$.  For $\alpha\in (0,1]$, we denote by $C^{\alpha}(\Omega)$ the usual space of Hölder continuous functions equipped with the norm

$$\|f \|_{C^\alpha(\Omega)}:= \|f\|_{L^{\infty}(\Omega)} + \sup_{x,y\in \Omega} \frac{|f(x)-f(y)| }{|x-y|^\alpha\wedge 1}\;.$$

As standard, $C^{k,\alpha}(\Omega)$ denotes $k$-times differentiable function such that the $k$- derivatives is are elements of $C^{\alpha}$. For $\alpha<0$, we work with the standard distribution spaces $C^\alpha$ as in <cit.>.

HS would like to thank Máté Gerencsér for his hospitality during a stay at TU Wien and gratefully acknowledges financial support from the EPSRC via Ilya Chevyrev’s New Investigator Award EP/X015688/1.
AC gratefully acknowledges partial support by the EPSRC through the “Mathematics of Random Systems” CDT EP/S023925/1.

§ PRELIMINARY GEOMETRIC NOTIONS

In this section we recall standard notation and facts about simplices and chains, c.f. <cit.>.
Unless mentioned otherwise, we shall be working in the ambient space $\mathbb{R}^d$.

§.§ Simplices and chains

Vectors $v_0,v_1,...v_k\in \mathbb{R}^d$ are called affinely independent if they lie in a unique $k$-dimensional affine subspace of $\mathbb{R}^d$, or equivalently, if the vectors $\{v_i-v_0\}_{i=1}^k$ are linearly independent.

The (unoriented) $k$-simplex $\simplex$ spanned by affinely independent vectors $v_0,..,v_k\in \mathbb{R}^d$ is defined as

$$\simplex:=\llbracket v_0,..,v_k \rrbracket:=\left\{ \sum_{i=0}^k t_i v_i\in \mathbb{R}^d \ : \ t_i\in [0,1], \ \sum_{i=0}^k t_i=1 \right\} \ .$$

The points $v_i$ are called the vertices of $\simplex$ and we write $\VV(\simplex)$ for this set.
A simplex spanned by a non-empty subset of $\VV(\simplex)$ is called a face of $\simplex$.
A face which is a $k-1$-simplex is called a boundary face.

We say a collection $\simplex_1,\dots,\simplex_N$ of $k$-simplices are non-overlapping if $\Vol^k(\simplex_i \cap \simplex_j) = 0$ for every $1 \le i < j \le N$.

A subdivision of a simplex $\pmb{\sigma}$ consists of a  finite set of non-overlapping k-simplices $\mathcal{K}$
such that $\pmb{\sigma}=\bigcup_{\pmb{w}\in \mathcal{K}} \pmb{w}$.

The mesh of the subdivision $\mathcal{K}$ is defined as $\diam(\mathcal{K}):=\max_{\pmb{w} \in \mathcal{K} } \diam (\pmb{w})$.
We call a sequence of subdivisions $(\mcK_n)_{n\in \mathbb{N}}$ of a $k$-simplex $\simplex$ regular if, for any $\beta>k$,

$$\lim_{n\to \infty} \sum_{\pmb{w}\in \mathcal{K}_n}\diam(\pmb{w})^\beta \to 0\ .$$

An orientation of a simplex spanned by affinely independent vectors $v_0,..,v_k\in \mathbb{R}^d$ is an ordering of its vertices $(v_0,...v_k)$, where two orderings are declared equivalent if they differ by an even permutation.

An oriented simplex $\osimplex$ consists of an unoriented simplex $\simplex$ together with an orientation $o$ of its vertices, that is
$	\osimplex=(\simplex, o)$.
We shall use the notation $[v_0,..,v_k]$ to denote the simplex $\llbracket v_0,..,v_k \rrbracket$ with the orientation inherited from the ordering $(v_0,..,v_k)$.

We denote by $\mathfrak{X}^k$ the set of all oriented $k$-simplices.
Given $\mathfrak{K} \subset \mathbb{R}^{d}$, we define $\mathfrak{X}^{k}(\mathfrak{K})$ to be the set of oriented simplices $\osimplex \in \mathfrak{X}^{k}$ with $\simplex \subset \mathfrak{K}$.
For $R > 0$, we also write $\mathfrak{X}_{\le R}^{k}(\mathfrak{K})$ for the set of all $\osimplex \in \mathfrak{X}^{k}(\mathfrak{K})$ with $\diam(\osimplex) \leq R$.

Note that $0$-simplices only have one orientation. For $k \geq 1$ a $k$-simplex $\simplex$ can carry one of two distinct orientations and for a given an orientation $o$, we write $-o$ to denote the other orientation on $\simplex$. We observe that the orientation $o$ of a simplex $\simplex$ can be canonically identified with an orientation of the (unique) $k$-dimensional hyperplane containing $\simplex$ – we shall freely use this identification
and compare the orientation of two simplices contained in the same hyperplane.
We also extend the notion of non-overlapping to oriented simplices in the natural way.

For a simplex $\osimplex=[v_0,...,v_k]$, we denote by $\sigma_{\setminus v_i}:= (-1)^i[v_0, ... ,\hat{v}_i, ... ,v_k]$ the (oriented) face which does not contain $v_i\in V(\sigma)$, where we have used standard notation which means leaving out the term with a hat.
We set $\Bd(\sigma):= \{ \sigma_{\setminus v} \ : \ v\in V(\sigma)\}$.

Given an oriented simplex $\osimplex$, we write $\simplex$ for the underlying non-oriented simplex.
We sometimes overload notation by allowing functions which take unoriented simplices $\osimplex$ as arguments to also take oriented simplices $\osimplex = (\simplex,o)$ as arguments by just disregarding the orientation $o$.

Given an oriented simplex $\osimplex \in \mathfrak{X}^{k}$, we say $\mathcal{K} \subset \mathfrak{X}^{k}$ is a subdivision of $\osimplex$ if every $\tau \in \mathcal{K}$ has the same orientation as $\osimplex$ and $\big\{ \pmb{\tau}: \tau \in \mathcal{K} \big\}$ is a subdivision of the non-oriented simplex $\pmb{\sigma}$ (Definition <ref>).

We write $\mathcal{K} | \sigma$ to denote that $\mathcal{K}$ is a subdivision of the oriented simplex $\sigma$.
Note that the notions of mesh and of regularity clearly extend to the subdivisions of oriented simplicies.

The set of $k$-chains is given by $\mathcal{X}^k= \mathbb{Z}(\mathfrak{X}^k)/ \sim$ where the equivalence relation $\sim$ for
$k=0$ is trivial and equivalence classes consist of singletons. For $k\geq 1$

* $-\osimplex \sim (\simplex,-o) \ $ for every $\osimplex = (\simplex,o) \in \mathfrak{X}^k$.

* $
\displaystyle\sum_{\tau \in \mathcal{K}}  \tau \sim \osimplex\;
$ for any $\osimplex \in \mathfrak{X}^{k}$ and subdivision $\mathcal{K} | \sigma$.

We recall the boundary operator

∂: 𝔛^k →𝒳^k-1,     σ↦∂σ:= ∑_v∈(σ) σ_∖v = ∑_F∈(σ) F  .

Note that this induces a map $\partial: \mathcal{X}^{k}\to \mathcal{X}^{k-1}$
which satisfies
$\partial\circ \partial=0$, c.f. <cit.>.

We call $\pmb{Q} \subset \mathbb{R}^{d}$ a (non-oriented) $k$-cube of side length $r>0$ if it is is the image of $[0,r]^k\subset \mathbb{R}^k$ under an isometric embedding.
An oriented cube is specified by $Q= (\pmb{Q}, o)$, where $o$ is an orientation of the $k$-hyperplane containing $Q$.
For a set $\mathfrak{K}\subset \mathbb{R}^d$, $R>0$, we write  $\mathfrak{Q}^k_{\leq R}(\mathfrak{K})$ for the set of oriented cubes $Q= (\pmb{Q}, o)$, where $\pmb{Q} \subset \mathfrak{K}$ has side–length bounded by $R$.

We view $Q = (\pmb{Q},o)$ as an element of $\mathcal{X}^{k}$ by setting $Q = \sum_{\pmb{\tau} \in \mathcal{K}} (\pmb{\tau},o)$ for any $\mathcal{K} \subset \mathfrak{X}^{k}$ consisting of non-overlapping simplices which satisfy $\bigcup_{\pmb{\tau} \in \mathcal{K}} \pmb{\tau} = \pmb{Q}$.

Note that the notion of non-overlapping clearly extends to cubes.

Working with cubes or simplices both offer distinct advantages.
While cubes are advantageous when working on fibered spaces, c.f. <cit.>,
simplices are combinatorially simpler.
For example the boundary faces of a simplex can easily by explicitly expressed and any polygon can be decomposed into finitely many simplices.
We recall here a special case of Whitney's covering theorem <cit.>, which will allow us to interchange between control over simplices and control over cubes, see Lemma <ref>.

Given any non-oriented $k$-simplex $\simplex$, there exists a family of non-overlapping $k$-cubes $\big( \pmb{Q}^{i}_{2^{-n}} : n \in \mathbb{N}, i \in I_{n} \big)$ where each $I_{n}$ is a finite set, $\pmb{Q}^{i}_{2^{-n}}$ is $k$-cube of side length $2^{-n}$, and

\[
\simplex
\overline{ \bigcup_{n=0}^{\infty} \bigcup_{i \in I_{n}} \pmb{Q}^{i}_{2^{-n}}}
\quad
\text{and}
\quad
\le
\mathrm{dist}(\pmb{Q}^i_{2^{-n}}, \mathbb{R}^{d} \setminus \simplex)
\le 4 \cdot 2^{-n} \sqrt{k}\;.
\]

Moreover, the Whitney decomposition can be chosen so that for any $\simplex$, we can find index sets
$(J_{N} : N \in \mathbb{N})$ and collections of non-overlapping simplices $\{\simplex_{j} : j \in J_{N}\} $ of diameter at most $2^{-N}$

such that

- ∑_n = 0^N ∑_ i∈I_n Q^i_2^-n
=∑_j ∈J_N  _j

≲_k  2^N(k-1) ,

where the first equality above is between elements in $\mathcal{X}^{k}$ and the second inequality above is uniform in $N$.

We extend the notion of Whitney decomposition to oriented simplices $\simplex$ in the obvious way, that is, one has a family of oriented cubes $\big( Q^{i}_{2^{-n}} : n \in \mathbb{N}, i \in I_{n} \big)$ where the $\pmb{Q}^{i}_{2^{-n}}$ form a Whitney decomposition of $\pmb{\sigma}$, and the $Q^{i}_{2^{-n}}$ are oriented as $\sigma$ is.

Lastly, we explain how to construct a smooth partition of unity subordinate to the Whitney decomposition of a $d$-simplex $\simplex\subset \mathbb{R}^d$.

For an axis parallel cube $\pmb{Q}\subset \mathbb{R}^d$ we denote by
$x_{\pmb{Q}}\in \mathbb{R}^d$ resp. $r_{\pmb{Q}}\in \mathbb{R}^d$ the center, respectively the side length of $\pmb{Q}$ which are characterised by
$\pmb{Q}= x_{\pmb{Q}}+ r_{\pmb{Q}} \cdot [-1/2,1/2]^d$.
Let $\phi\in C^\infty$ be compactly supported on $[-2/3,2/3]^d$ such that $\phi|_{[-1/2,1/2]^d}=1$,
we define

$$\phi_{\pmb{Q}}(x)= \phi \left(\frac{x-x_{\pmb{Q}}}{r_{\pmb{Q}}} \right)\;.$$

For a given Whitney decomposition  $\simplex
\overline{ \bigcup_{n=0}^{\infty} \bigcup_{i \in I_{n}} \pmb{Q}^{i}_{2^{-n}}} $ set

$$\phi_{n,i}(x) := \begin{cases}\frac{
\phi_{\pmb{Q}^{i}_{2^{-n}}}(x)
}{\sum_{n\in \mathbb{N},i\in I_n}  \phi_{\pmb{Q}^{i}_{2^{-n}} } (x)} & \text{if } x\in \text{Int}(\simplex)\\
0 & \text{else.}
\end{cases}

We call by $\{\phi_{n,i}\}_{n\in \mathbb{N}, i\in I_n}$ the partition of unity subordinate to the above Whitney decomposition constructed from $\phi\in C^\infty$. Finally, one notes that by construction

$$ \|D^k \phi_{n,i}\|_{L^\infty} \lesssim_k 2^{nk}$$

uniformly over $\simplex$, and $n\in \mathbb{N}, i\in I_n$.

§.§ Controls on simplices

For any non-oriented $j$-simplex $\mathbf{F}$, we denote by $\langle \pmb{F} \rangle \subset \mathbb{R}^{d}$ the $j$-hyperplane containing $\pmb{F}$.
We define for $\alpha\in (0,1]$ and any $k$-simplex $\pmb{\sigma}$,

h_σ:=min_ F ∈(σ) sup{d(x, ⟨F ⟩)  :  x∈σ }
_α^k (σ) := max_ F ∈(σ) ^k-1( F) h_σ^α .

We observe that

h_σ^1-α _α^k (σ) = 1/k ^k(σ),     ^k_α(σ)≤(σ)^k-1+α ∧^k(σ)^α .

We also set $\mass_\infty^k (\pmb{\sigma})=0$ and $\mass_0^k(\pmb{\sigma})=1$.

We observe that the left side of the inequality (<ref>) is close to sharp when either the simplex $\pmb{\sigma}$ is equilateral or when $\pmb{\sigma}$ is a cone of small height over an equilateral simplex.

Suppose that we have $k$-simplices $\simplex, \simplex_1,\dots,\simplex_{n}$ such that $\diam (\simplex) \leq 1$ and $\simplex= \bigcup_{i=1}^{n} \simplex_i$, then

$${\mass}^k_\alpha(\pmb{\sigma}) \leq \sum_{i=1}^{n} {\mass}^k_\alpha(\pmb{\sigma}_i)\;.$$

Using the first equality in (<ref>) we have

\[
{\mass}^k_\alpha(\pmb{\sigma}) = \frac{\Vol^k(\pmb{\sigma})}{k h^{1-\alpha}_{\pmb{\sigma}}}
\leq \sum_{i=1}^{n}  \frac{\Vol^k(\pmb{\sigma}_i)}{k h^{1-\alpha}_{\pmb{\sigma} }}
= \sum_{i=1}^{n}  {\mass}^k_\alpha(\pmb{\sigma}_i)   \frac{k h^{1-\alpha}_{\pmb{\sigma}_i}}{k h^{1-\alpha}_{\pmb{\sigma}}}
\leq \sum_{i=1}^{n}  {\mass}^k_\alpha(\pmb{\sigma}_i)
\]

where we used that
$h^{1-\alpha}_{\sigma_{i}} \leq h^{1-\alpha}_{\sigma}$ and that $\Vol^k(\cdot)$ is additive.

There exists a constant $C=C(k)$, such that
$${\mass}^k_\alpha(\pmb{\sigma}) \leq \inf
\Big\{  \sum_{j \in J}  \diam (\simplex_j)^{k-1+\alpha}:
\pmb{\sigma} = \bigcup_{j \in J} \simplex_j  \Big\}  \leq C  {\mass}^k_\alpha(\pmb{\sigma}) \ .$$

for all $k$-simplices $\simplex$ with $\diam(\pmb{\sigma})\leq 1$ and $\alpha\in (0,1]$.

The first inequality above follows by combining the inequality in (<ref>) and Lemma <ref>.
For the second inequality above, observe that one can write
$\simplex= \bigcup_{i\in I} \simplex_i$ where $\{ \simplex_{i} \}_{i \in I}$ are non-overlapping, $\diam (\simplex_i) \leq h_{\simplex}$, and $|I| \lesssim_{k} \frac{ \Vol^{k-1}(\pmb{F})}{h_{\simplex}^{k-1}}$ – here $\pmb{F} \in \Bd(\simplex)$ is where the maximum in the second definition of (<ref>) is achieved.[Such a subdivision of $\simplex$ can be generated by appropriately subdividing $\pmb{F}$ and taking simplicial cones. ]

Using this subdivision, we have

\sum_{i \in I}  \diam (\simplex_i)^{k-1+\alpha} \lesssim_{k} \Vol^{k-1}(\pmb{F}) h_{\simplex}^{\alpha} = \mass_\alpha (\simplex)\;.

§.§ Functions on chains

We denote by $\Omega^k\subset(\mathcal{X}^k)^* $ the subspace of the
algebraic dual of $\mathcal{X}^k$ consisting of all elements $A\in (\mathcal{X}^k)^*$
such that the map $(\mathbb{R}^d)^{k+1 }\ni (v_0,\dots,v_k) \mapsto A[v_0,...,v_k] \in \mathbb{R}$
is Borel-measurable.

Observe that any $A \in \Omega^{k}$ can canonically be identified with
a (measurable) map $A: \mathfrak{X}^k\to \mathbb{R}$ satisfying the following two properties

* If $k \ge 1$, then for any oriented $k$-simplex $\osimplex$ one has $A(-\osimplex)= -A(\osimplex)$.

* Given $\osimplex \in \mathfrak{X}^{k}$ and $\mathcal{K} | \osimplex$,

\[
A(\osimplex) = \sum_{\tau \in \mathcal{K}} A(\tau)\;.
\]

Note that there is a natural corresponding space of $k$-chains $\mathcal{X}^{k}(\mathfrak{K})$ and additive maps $\Omega^{k}(\mathfrak{K})$.

Let $\alpha \in (0,1]$. Then, for any $A \in \Omega^k$, the following three conditions are equivalent:

$|A(Q)|\lesssim \diam(Q)^{k-1 + \alpha} $ uniformly over  $Q\in \mathfrak{Q}_{\leq 1}^k$ and, for any Whitney decomposition $\big( {Q}^{i}_{2^{-n}} : n \in \mathbb{N}, i\in I_{n} \big)$ of $\sigma$, $\sum_{n\in \mathbb{N}} \sum_{ i\in I_n} A({Q}^{i}_{2^{-n}})$ converges absolutely to $A(\sigma)$.

$|A(\sigma)|\lesssim \mass_{\alpha}^{k}(\sigma) \ , $ uniformly over $\sigma\in \mathfrak{X}_{\leq 1}^{k}$.

*  $|A(\sigma)|\lesssim \diam(\sigma)^{k-1 + \alpha}$, uniformly over $\sigma\in \mathfrak{X}_{\leq 1}^{k}$.

Note that the additional assumption in Item <ref> on additivity over Whitney decompositions is needed due to the fact that one can in general not write a simplex as a union of finitely many cubes.

Below, we write $\eta = k - 1 + \alpha$.
We first prove Item <ref> implies Item <ref>.
We apply Lemma <ref> and write $( Q^{i}_{2^{-n}} : n \in \mathbb{N}, i \in I_{n})$ for a Whitney decomposition of $\simplex$.

We define w

_n σ
{ p∈σ  :  2^-n-1√(k)
≤dist (p, ℝ^d ∖)
≤4 ·2^-n+1 √(k) } ,

Next we note that  $2^{-n}>h_{\pmb{\sigma}} \Rightarrow  I_n = \emptyset$ and for uniform in $n \in \mathbb{N}$,  $|I_n|\lesssim_{k}  \Vol^{k}( \mfd_{n} \pmb{\sigma} ) 2^{nk}$.
It follows that

|A(σ)|    ≤∑_n = 0^∞ ∑_i∈I_n |A(Q_2^-n^i)|
≤∑_n=0^∞ |I_n| 2^-nη
≲∑_n ≥-log_2(h_σ)  ^k(_n σ)    2^-n(η- k)

≲^k(σ) max_ n ≥-log_2(h_σ)2^-n(η-k)
≤^k(σ)  h_σ^η-k
^k(σ)/h_σ  h_σ^η-k-1
≤_η-(k-1)^k(σ) ,

where in the first inequality on the second line above we used that $\sum_{n = 0}^{\infty} \Vol^{k}(\mfd_{n} \pmb{\sigma}) \leq 4 \cdot \Vol^{k}(\pmb{\sigma})$.
This concludes showing Item <ref> implies Item <ref>.

The fact that Item <ref> implies Item <ref> follows from the last inequality in <ref>.

Finally, to argue that Item <ref>  follows from Item <ref> we first observe that we can realize any $Q\in \mathfrak{Q}_{\leq 1}^k$ of diameter $r$  as a sum of $k!$ oriented simplices of diameter at most $r$ and thus the estimate on $A(Q)$ in Item <ref> follows directly.

For the rest of  Item <ref>, given a Whitney decomposition of $\sigma$ we note that the absolute convergence of $\sum_{n \in \mathbb{N}} \sum_{i \in I_{n}} A(Q^{i}_{2^{-n}})$ follows from the estimate from the first part of Item <ref> and proceeding as in (<ref>).
To show convergence to $A(\sigma)$, using the notation of (<ref>) we have

\[
\Big|A(\sigma) - \sum_{n = 0}^{N} \sum_{ i\in I_n} Q^{i}_{2^{-n}}  A({Q}^{i}_{2^{-n}}) \Big|\leq \sum_{j\in J_N}  |A(\sigma_j )| \lesssim
2^{ -N(k-1 + \alpha)} 2^{N(k-1)} \xrightarrow[N \rightarrow \infty]{} 0\;.
\]

§.§ Simplicial sewing lemma

In this subsection we state the promised simplicial sewing lemma – Proposition <ref>.
This tool will be used frequently in our analysis, but as mentioned earlier the proof is deferred to Section <ref>.

For a function $\Xi: \mathfrak{X}^k(\mathfrak{K})  \to \mathbb{R}$, a simplex $\sigma\in \mathfrak{X}^k$ and a subdivision $\mathcal{K} | \sigma$ we introduce the following quantity which measures to what degree $\Xi$ fails to be additive over $\mathcal{K}$:

δ_;σ Ξ:= Ξ(σ)-∑_σ' ∈ Ξ(σ') .

We now introduce a space of “almost additive” $\Xi$ which will be the codomain of our sewing map.

Let $C_{2,k}^{{\eta,\gamma}}(\mathfrak{K})$ consist of all functions
$\Xi: \mathfrak{X}^k(\mathfrak{K})  \to \mathbb{R}$ satisfying  $\Xi(\sigma)=- \Xi(-\sigma)$ and

\[
\llbracket \Xi\rrbracket_{(\eta,\gamma);\mathfrak{K} }:= \llbracket \Xi\rrbracket_{\eta,\mathfrak{K}} +\llbracket \delta\Xi\rrbracket_{\gamma,\mathfrak{K}} <+ \infty\;,
\]

\[
\llbracket  \Xi\rrbracket_{\eta,\mathfrak{K}}:= \sup_{\sigma \in \mathfrak{X}^k(\mathfrak{K})} \frac{|A(\sigma)|}{\diam(\sigma)^{\eta} }\;,
\qquad
\llbracket  \delta\Xi\rrbracket_{\gamma,\mathfrak{K}}  := \sup_{\sigma \in \mathfrak{X}^k(\mathfrak{K})} \sup_{\mathcal{K}|\sigma} \frac{| \delta_{\mcK;\sigma} \Xi  |}{|\mathcal{K}| \diam(\sigma)^{\gamma}}\;,
\]

and the supremum over $\mathcal{K} | \sigma$ in the second expression runs over all subdivisions $\mathcal{K}$ of $\sigma$.

Let $0<\eta\leq k <\gamma$.
There exists a unique linear map
$\mathcal{I}: C_{2,k}^{\eta, \gamma}(\mathfrak{K}) \to \Omega^k(\mathfrak{K}),$

|ℐ Ξ(σ)- Ξ(σ)|≲δΞ_γ,𝔎 (σ)^γ,     |ℐ Ξ(σ) | ≲Ξ_(η,γ);   (σ)^η ,
uniformly in $\sigma \in \mathfrak{X}_{\le 1}^{k}(\mathfrak{K})$.
Furthermore, for any regular sequence of subdivisions $(\mathcal{K}_n)_{n \in \mathbb{N}}$ of $\sigma\in \mathfrak{X}^{k}(\mathfrak{K})$,

$$\mathcal{I}\Xi(\sigma)= \lim_{n\to \infty} \sum_{\sigma'\in \mathcal{K}_n} \Xi(\sigma')\ .$$

Note that the condition $\Xi(\sigma)=- \Xi(-\sigma)$ in Definition <ref> could be replaced by $|\Xi(\sigma)+ \Xi(-\sigma)|=o(\diam(\sigma)^k)$.
If one did not impose any relation between $\Xi(\sigma)$ and $\Xi(-\sigma)$ an analogue of Proposition <ref> still holds, but $\mathcal{I}\Xi$ would only be additive but not an element of $\Omega^k$ in general.

Strictly speaking, our sewing lemma is less general than the one in <cit.> where one is allowed to scale different directions differently.
However, we believe our results could be formulated on $\mathbb{R}^d$ is equipped with a non-trivial scaling $\fraks$ c.f. <cit.>, but we refrain from doing so as it would distract from the main ideas.

We state the following corollary of Proposition <ref>.

In the setting of Proposition <ref>, if $ \alpha:=\eta-k+1\in (0,1]$,

$$ \big|\mathcal{I} \Xi(\sigma) \big| \lesssim \llbracket \Xi\rrbracket_{(\eta,\gamma); \mfK}  \mass_{{\alpha}}^k (\sigma)\ .$$

This follows immediately from Proposition <ref> and Lemma <ref>

In <cit.>, the authors write It is worth emphasizing that this result might seem counter-intuitive...It is therefore not at all clear how one can expect to be nonatomic a limit of sums of germs which are essentially not so.
The proof of the corollary above clarifies this point.

We also note that the threshold $\eta > k-1$ above  is expected  to be sharp,  c.f. Proposition <ref>.

§ DISTRIBUTIONAL $K$-FORMS

For $\alpha\in (0,1]$ and $\mathfrak{K} \subset \mathbb{R}^{d}$, we define the following norm on $\Omega^k(\mathfrak{K})$

$$\|A\|_{\alpha; \mathfrak{K}}
\sup_{\osimplex\in \mathfrak{X}_{\le 1}^k(\mathfrak{K})}
\frac{|A(\osimplex)|}{\mass^k_\alpha(\osimplex)} \ \qquad
\text{ and }
\qquad \|A\|_{\infty; \mathfrak{K}} = \begin{cases} +\infty &\text{ if } A\neq 0, \\
0 &\text{ else.}
\end{cases}

For $\alpha,\beta \in [0,1]\cup \{\infty\}$ set $\|A\|_{\alpha,\beta; \mathfrak{K}}:= \|A\|_{\alpha; \mathfrak{K}} + \|\partial A\|_{\beta; \mathfrak{K}}$ and

\[
\Omega^{k}_{(\alpha,\beta)} (\mathfrak{K}):= \Big\{ A \in \Omega^k \ : \  \|A\|_{\alpha,\beta; \mathfrak{K}}< +\infty  \Big\}
%	\|A\|_{\alpha,\beta; \mathfrak{K}}
%	=
%	\sup_{\osimplex\in \mathfrak{X}_{\le 1}^k(\mathfrak{K})}
%	\frac{|A(\osimplex)|}{\mass_\alpha(\osimplex)}
%	+
%	\sup_{\osimplex\in \mathfrak{X}_{\le 1}^{k+1}(\mathfrak{K})}
%	\frac{|A(\partial\osimplex)|}{\mass_\beta(\osimplex)}
\;.
\]

Note that $\Omega^{0}_{\alpha,\beta}=\Omega^{0}_{0,\beta} $ is canonically identified with $\beta$-Hölder functions whenever $\beta \in (0,1]$.
Also observe the inclusions $\Omega^k_{\alpha', \beta'} \subset \Omega^k_{\alpha, \beta} $ whenever $\alpha'\geq \alpha$, $\beta' \geq \beta$.

By construction the linear map

∂: Ω^k_(α,β)→Ω^k+1_(β,∞) .

is bounded and
the spaces $A\in\Omega^k_{(\alpha,\infty)}$ are a natural analogue of closed forms.

When $\mathfrak{K} = \mathbb{R}^{d}$ we often suppress it from the notation, for instance writing $\|A\|_{\alpha}$ for $\|A\|_{\alpha, \mathbb{R}^{d}}$.
We also state most of our estimates below in terms of global norms $\| \cdot \|_{\alpha}$ to lighten notation,
but it will be clear how they can be localized to compact $\mathfrak{K} \subset \mathbb{R}^{d}$.

When taking $k=1$ and $n=2$, the first term in the above definition corresponds to the growth norm $| \cdot |_{\gr\alpha}$ as in <cit.>, and the second term serves a similar purpose as $| \cdot |_{\tri\alpha}$ in <cit.>.

§.§ Estimates on chains and a generalised flat norm

In order to do analysis on $\mathcal{X}^k$, we introduce, for $X\in \mathcal{X}^k$,

| X |_{\alpha;k}:= \inf
\Big\{  \sum_{j \in J} |c_j|  \cdot \text{\textbf{m}}_\alpha^k(\simplex_j):
X= \sum_{j \in J} c_j \osimplex_j ,\ \osimplex_j \in \mathfrak{X}_{\le 1}^{k} \Big\}\;.

Clearly $| \cdot |_{\alpha;k}$ satisfies the triangle inequality.

Using the above control, we now define a “Hölder flat-norm” on $\mathcal{X}^{k}(\mathfrak{K})$.

For $\alpha,\beta \in [0,1]\cup\{\infty\}$ define the $(\alpha,\beta)$-flat norm on $\mathcal{X}^{k}(\mathfrak{K})$ as

inf_Z ∈𝒳^k+1(𝔎)
|X- ∂Z|_α;k
| Z |_β;k+1
} .

The definition (<ref>) for $\alpha=\beta=1$ is  equivalent to the “flat-norm” introduced in <cit.>. For $\alpha=1$, $0<\beta< 1$ it is seen to be equivalent to the $(k-1+\beta)$-flat norm introduced in <cit.> using Lemma <ref>.

It is clear $|\cdot|_{(\alpha,\beta);\mathfrak{K}}$ satisfies the triangle inequality, that is for  $X,Y\in \mathcal{X}^{k}(\mathfrak{K})$

|X+Y|_(α,β);𝔎≤|X|_(α,β);𝔎 + |X|_(α,β);𝔎 .

Similarly, for $Z \in  \mathcal{X}^{k+1}(\mathfrak{K})$,

|X|_(α,β);𝔎≤|X+∂Z|_(α,β);𝔎 + |Z|_β, k+1 .

For $A\in \Omega^k$ we have that

$$\sup_{X\in \mathcal{X}^{k}(\mathfrak{K}) }\frac{|A(X)|}{|X|_{(\alpha, \beta);\mathfrak{K}}} \leq \|A\|_{(\alpha,\beta); \mathfrak{K}} \ . $$

For $X\in \mathcal{X}^k$ and any decomposition $X= \sum_i c_i \sigma_i$

$$|A(X)|\leq \sum_i |c_i| |A(\sigma_i)| \leq \|A\|_{\alpha} \sum_i |c_i| \mass_\alpha^k (\sigma_i) $$

and thus taking an infimum over all decompositions gives $|A(X)|\leq \|A\|_{\alpha} |X|_\alpha$.
Similarly, for any $Z\in \mathcal{X}^{k+1}$

$$|A(\partial Z)| \lesssim \|\partial A\|_{\beta} | Z |_{\beta}\ .$$

$$|A(X)|\leq |A(X + \partial Z)| + |A(\partial Z)| \leq  \|A\|_{\alpha} |X+ \partial Z|_\alpha  + \|\partial A\|_{\beta} | Z |_{\beta} \leq \|A\|_{(\alpha,\beta)}  \left(  |X+ \partial Z|_\alpha  +| Z |_{\beta} \right)\ .  $$

Taking the infimum over $Z \in \mathcal{X}^{k+1}(\mathfrak{K})$ concludes the claim.

$|\cdot |_{(\alpha,\beta)}: \mathcal{X}^k \to \mathbb{R}$ is a norm.

Consider $0\neq X\in \mathcal{X}^k$. Without loss of generality we can write for some $n\in \mathbb{N}$

$$X= \sum_{i=0}^n c_i \sigma_i$$
where $\Vol^{k}(\sigma_i\cap \sigma_j)=0$ whenever $i\neq j$.
Note that smooth differential forms are contained in $\Omega^{k}_{(1,1)}\subset \Omega^{k}_{(\alpha,\beta)}$, thus one easily constructs $A\in  \Omega^{k}_{(\alpha,\beta)}$ such that $|A(X)|>0$. In view of Lemma <ref>, this implies $|X|_{(\alpha,\beta)}>0.$

Let $\mathcal{B}^{k}_{(\alpha,\beta)}$ be the completion of the normed $Z$-module $(\mathcal{X}^k, |\cdot|_{(\alpha,\beta)})$.

The pairing
$\mathcal{X}^k \times \Omega^k_{(\alpha,\beta)}\to \mathbb{R}$ continuously extends to a pairing $\mathcal{B}_{\alpha,\beta}^{k} \times \Omega^k_{(\alpha,\beta)}\to \mathbb{R}$.

Lemma (<ref>) shows that elements of $\Omega^k_{(\alpha,\beta)}$ are elements of the continuous dual of
$(\mathcal{X}^k, |\cdot|_{(\alpha,\beta)})$.

Definition <ref> together with Corollary <ref> gives a
geometric integration theory similar to <cit.>, see also more generally <cit.>. For $\alpha=1$, $\beta\in (0,1]$ it agrees with the one in <cit.>.

By covering a closed manifold by images of (say) squares and approximating as in <cit.> one sees that closed, oriented, $C^{1,\eta}$-manifolds are canonically contained in $\mathcal{B}_{\alpha,\beta}^{k}$ as soon as $\eta>0$.
Similarly to the proof of Lemma <ref>, one can check that a compact oriented $C^{1,\eta}$-manifold with boundary is in $\mathcal{B}_{\alpha,\beta}^{k}$ as soon as $\alpha>\frac{1}{1+\eta}$, see also Theorem <ref> and Remark <ref>.

§.§ Further $(\alpha,\beta)$-flat norm estimates

We collect some Lemmata about the $(\alpha,\beta)$-flat norm which will be useful later on.

Let $v_0,...v_k\in \mathbb{R}^d$ and $v'_0\in \mathbb{R}^d$ all be contained in a ball of radius $r>0$. Then,

\Big|[v_0,v_1,...,v_k]-[v'_0,v_1,...,v_k] \Big|_{(\alpha,\beta) } \lesssim_k r^{k-1} d(v_0,v_0')^\alpha + r^{k}  d(v_0,v_0')^\beta \ .$$

Consider $Z= [v'_0,v_0,v_1,...,v_k]$ and write $X= [v_0,v_1,...,v_k]-[v'_0,v_1,...,v_k]$.

\begin{align*}
|X|_{(\alpha,\beta) } &\leq |X- \partial  Z|_{\alpha;k}
| Z |_{\beta;k+1}  \\
&\leq \sum_{j=1}^k \big| [v'_0,v_0,v_1,...,\hat{v}_j,...,v_k] \big|_{\alpha;k}  + | Z |_{\beta;k+1} \\
&\leq k   r^{k-1} d(v'_0,v_0)^\alpha +  r^{k}d(v'_0,v_0)^\beta  \ .
\end{align*}

Consider simplices $\sigma= [v_0,...,v_k]$, $\sigma'= [v'_0,...,v'_k]$ contained in a ball of radius $r>0$. Let
$\mu=\max \big\{|v_i-v_i'| \ \in \mathbb{R} : i=0,...,k \big\}$.

$$|\sigma-\sigma'|_{(\alpha,\beta)} \lesssim r^{k-1} \mu^\alpha   + r^{k}\mu^\beta . $$

We define $\sigma_i= [v_0,...,v_{i-1},v'_i, ...,v'_k]$ and note that $\sigma=\sigma_{k+1}$ and $\sigma'=\sigma_{0}$.
The statement follows from the using (<ref>) to write
$|\sigma-\sigma'|_{(\alpha,\beta) }\leq \sum_{i=0}^{k} |\sigma_i-\sigma_{i+1}|_{(\alpha,\beta)} $ and then applying Lemma <ref>.

Fix a family of maps $(\pi_{n})_{n \in \mathbb{N}}$, with  $\pi_{n}: \mathbb{R}^{d}\to 2^{-n} \mathbb{Z}^{d} \subset \mathbb{R}^{d}$, such that, uniform  in $n \in \mathbb{N}$ and $x \in \mathbb{R}^{d}$, we have

$$ d(x,\pi_{n}x) \lesssim 2^{-n}\ ,$$

uniformly in $n\in \mathbb{N}$ and $x\in \mathbb{R}^{d}$.
We extend this map to a map on simplices by setting

$$\pi_{n} [v_0,v_1,...,v_k]= [\pi_{n}v_0,\pi_{n}v_1,...,\pi_{n}v_k] \;. $$

$\pi_{n}$ naturally extends to $\mathcal{X}^{k}$.

The following is a direct corollary of Lemma <ref>.

For any  $\sigma \in \mathfrak{X}^{k}$ and $m,n \in \mathbb{N}$,

$$|\pi_{m}\sigma-\pi_{n}\sigma|_{(\alpha,\beta)} \lesssim \diam(\sigma)^{k-1} 2^{- \alpha(m \wedge n)} + \diam(\sigma)^{k} 2^{- \beta(m \wedge n)}\ .$$

§ MULTIPLICATION WITH REGULAR FUNCTIONS

For a smooth function $f: \mathbb{R}^{d}\to \mathbb{R}$ and a smooth differential $k$-form $A$, one can easily see that the pointwise product $f\cdot A$ is  characterised by

∫_σ f·A =
lim_n→∞ ∑_σ'∈𝒦_n  ( 1/k+1 ∑_i=0^k f(v_i)  ·  ∫_σ' A )

where $(\mathcal{K}_n)_{n \in \mathbb{N}}$ is any regular sequence of subdivisions of $\sigma$.
In order to define a product

$$C^\beta \times \Omega^k_\alpha \to \Omega^{k}_\alpha, \qquad (f, A)\mapsto f\cdot A $$

we mimic the right hand side of (<ref>), but allow for slightly more general approximations.[
Note that $\Xi^{f\cdot A} [v_0,...,v_k]=\frac{1}{k+1} \sum_{i=0}^k f(v_i) A[v_0,...,v_k]$ is clearly a special case of (<ref>).]
For this consider a family $\mu=\{\mu_{\sigma}\}_{\sigma\in \mathfrak{X}^k}$ of probability measures
where each $\mu_{\sigma}$ is supported on $\pmb{\sigma}$ and independent of the orientation of $\sigma$, and set

Ξ^f·A_μ(σ):= μ_σ(f)  ·A(σ)  .

Let $\alpha,\gamma \in (0,1]$ such that $\alpha+\gamma>1$ and $\mu=\{\mu_{\sigma}\}_{\sigma\in \mathfrak{X}^k}$ as above.  Then, $\Xi^{f\cdot A}_\mu\in C^{\alpha,\gamma+ k-1 +\alpha}_{2,k}$ for every $f\in C^\gamma$, $A \in \Omega^{k}_{\alpha,\beta}$.

It follows that, with the map $\mathcal{I}$ as in Proposition <ref>,

$$f\cdot A:=\mathcal{I}\Xi^{f\cdot A}_\mu  \in \Omega^k_{\alpha, (\alpha+ \gamma-1) \wedge \beta}\;,$$

is in fact independent of the choice of $\mu$ and satisfies

f·A_α ≲ f_C^γ A_α,      ∂(f·A) _(α+ γ-1) ∧β<  f_C^γA_(α,β)  .

To verify that $\Xi^{f\cdot A}\in C^{\alpha,\gamma+ k-1+ \alpha}_{2,k}$, observe that

\[
|\Xi^{f\cdot A}_\mu(\sigma)|\leq \|f\|_{L^\infty} |A(\sigma)| \leq \|f\|_{L^\infty} \|A\|_{\alpha} \mass_\alpha^k(\sigma)  \leq \|f\|_{L^\infty} \|A\|_{\alpha} \diam(\sigma)^{k-1+\alpha}\;,
\]

\begin{align*}
|\delta_{\mcK;\sigma} \Xi^{f\cdot A}(\sigma)| &= \Big|\sum_{\sigma'\in \mathcal{K}} \left( \mu_{\sigma}(f) -\mu_{\sigma'}(f)   \right) A(\sigma') \Big|\\
& \leq |\mathcal{K}| \cdot \|f\|_{C^\gamma} \diam(\sigma)^\gamma \|A\|_{\alpha} \mass_\alpha^k(\sigma) \\
&\leq|\mathcal{K}| \cdot \|f\|_{C^\gamma}  \|A\|_{\alpha} \diam(\sigma)^{\gamma+ k-1+ \alpha}\ .
\end{align*}

The first estimate of (<ref>)  then follows from the first estimate of (<ref>).

The fact that $f \cdot A$ does not depend on the particular choice of $\mu$, follows from the uniqueness in Proposition <ref> and the fact and observing that given a second family $\mu'=\{\mu'_\sigma\}_{\sigma\in \mathfrak{X}^k}$ of measures,

$$|\Xi^{f\cdot A}_\mu(\sigma)- \Xi^{f\cdot A}_{\mu'}(\sigma)|\leq \|f\|_{C^\gamma} \diam(\sigma)^\gamma \|A\|_{\alpha} \mass_\alpha^k(\sigma)\;.$$

We now turn to the second inequality of (<ref>).
For a family of measures $\mu = \{ \mu_{w} \}_{w \in \mathfrak{X}^{k}}$ as before we define
$\Xi_{\mu}^{\partial (f\cdot A)}(\sigma) :=\sum_{F\in \Bd(\sigma)} \Xi_{\mu}^{f\cdot A}( F)$ for $\sigma \in \mathfrak{X}^{k+1}$.

Given another family of measures on $k+1$ simplices
$\bar{\mu}= \{\bar{\mu}_\sigma\}_{\sigma\in \mathfrak{X}^{k+1} }$,
we set, for $\sigma \in  \mathfrak{X}^{k+1}$.

$$\Xi^{ (\partial f)\wedge A}_{\mu, \bar{\mu}}(\sigma):= \sum_{F\in \Bd(\sigma)} \left(\mu_F(f)-\bar{\mu}_{\sigma}(f)\right) A(F)\ .$$

$\Xi^{\partial (f\cdot A)}_{\mu}= \Xi^{ (\partial f)\wedge A}_{\mu, \bar{\mu}} + \Xi^{f\cdot \partial A}_{\bar \mu}$,

\begin{align*}
|\Xi_\mu^{\partial (f\cdot A)}(\sigma)|&\leq \|f\|_{C^\gamma} \|A\|_{\alpha} \diam(\sigma)^{\gamma+k-1+ \alpha} + \|f\|_{L^\infty} \|\partial A\|_{\beta} \diam(\sigma)^{k+\beta} \\
&\leq \|f\|_{C^\gamma}\|A\|_{(\alpha,\beta)} \diam(\sigma)^{k + ( \gamma+ \alpha  -1 ) \wedge \beta}\;,\;
\end{align*}

and thus

\begin{align*}
&|\partial (f\cdot A) (\sigma )| \leq |\Xi_\mu^{\partial (f\cdot A)}(\sigma) - \partial (f \cdot A) (\sigma )| + |\Xi_\mu^{\partial (f\cdot A)}(\sigma)|\\
(k+1) \llbracket \delta \Xi_{\mu}^{f\cdot A} \rrbracket_{\gamma+k-1+\alpha} \diam({\sigma})^{\gamma+k-1+\alpha}  + \|f\|_{C^\gamma}\|A\|_{(\alpha,\beta)} \diam(\sigma)^{k + ( \gamma+ \alpha  -1 ) \wedge \beta}
\\
% & \leq
%{k+1 \choose 2} \| \Xi^{f\cdot A}\|_{\gamma+k-1+\alpha} \diam({\sigma})^{\gamma+k-1+\alpha}
%+  \|f\|_{C^\gamma} \|A\|_{\alpha} \diam^{\gamma+k-1+ \alpha} + |f|_{l^\infty} \|\partial A\|_{\beta} \diam^{k+\beta}\\
&\lesssim \|f\|_{C^\gamma}\|A\|_{(\alpha,\beta)} \diam(\sigma)^{k + ( \gamma+ \alpha  -1 ) \wedge \beta }   \ ,
\end{align*}

which is the second inequality of (<ref>).
In the first inequality above, we used that $|\Xi_\mu^{\partial (f\cdot A)}(\sigma) - \partial (f \cdot A) (\sigma)|
\le \sum_{F \in \Bd(\sigma)} |\Xi_\mu^{f\cdot A}(F) - (f\cdot A)(F)|$ and we have already estimates the terms on the right hand side.

The theorem above allows us to extend a wedge product on differential forms to our rough setting.

Let  $\alpha, \beta, \gamma \in (0,1]$ with $\tilde{\alpha}:=\alpha+\gamma-1>0$ and $\tilde{\beta}:=\beta+\gamma-1>0$.
Then the identity $df\wedge A:= d(f\cdot A) -f\cdot dA$ for smooth forms extends to a continuous bilinear map

C^γ×Ω_α,β →Ω^k+1_α̃ ∧β, β̃,     (f, A)↦df∧A:= d(f·A)

This follows from Theorem <ref> which gives
$d(f\cdot A) \in \Omega^{k+1}_{(\alpha+\gamma -1)\wedge \beta, \infty}$ and $f\cdot dA \in  \Omega^{k+1}_{ \beta, \beta+\gamma-1}$.

We state another corollary, which in the view of Remark <ref> and the preceding corollary, reproduces <cit.> when specialized to $n=k-1$.

Let $n\leq k$.
Fix $\alpha,\gamma_0,...,\gamma_n\in (0,1]$ and $ \beta\in (0,1]\cup\{\infty\}$ such that

\[
\alpha+\sum_{i=0}^{n} \gamma_i > n  \enskip \text{ and } \enskip \beta+ \sum_{i=0}^{n} \gamma_i >n-1\;.
\]

Then, writing $\tilde\alpha=  \alpha+\sum_{i=1}^{n} \gamma_i -(n-1)$, $\tilde{\beta}=\big(\alpha + \sum_{i=0}^{n} \gamma_i-n \big)\wedge \big(\beta+\sum_{i=1}^{n} \gamma_i -(n-1) \big)$, one has that the mapping

$$ C^{\gamma_0} \times \Big( \prod_{i=1}^{n} C^{\gamma_i} \Big) \times\Omega^{k-n}_{\alpha,\beta} \to \Omega^{k}_{\tilde \alpha,\tilde \beta}
,\qquad (g_0,...,g_n, A) \mapsto g_0 \cdot dg_1 \wedge ...\wedge d g_n \wedge A $$

is a bounded multi-linear map.

We write $\alpha_{k-n} = \alpha, \beta_{k-n}= \beta$ and $A_{k-n}= A$.
Then, for $l\in \{1,...,k\}$, inductively define $\alpha_{k-n+l} = (\alpha_{k-n+l-1} +\gamma_{k-l} -1)  \wedge  \beta_{k-n+l-1}$ and
$ \beta_{k-n+l}= \beta_{k-n+l-1} +\gamma_{k-l} -1$. Note that $\alpha_{k-n+l}, \beta_{k-n+l}>0$ by assumption and thus
$A_{k-n+l}:= dg_n \wedge A_{k-n+l-1}\in \Omega^{k-n+l}_{\alpha_k-n+l  ,\beta_k-n+l} $ by (<ref>).
For $l = n$, we have $A_{k}= dg_1 \wedge ...\wedge d g_n \wedge A$ and $\alpha_k=\tilde\alpha$ and  $\beta_k= \beta+\sum_{i=1}^{n} \gamma_i -(n-1)$.
Applying Theorem <ref> once more to control $g_0 \cdot A_{k}$ finishes the proof.

§ PULLBACK AND STOKES' THEOREM

For $F : \mathbb{R}^{m}\to \mathbb{R}^{d}$ and a simplex $\sigma= [v_0,...,v_k] \in \mathfrak{X}^k(\mathbb{R}^m)$, write $F_*\sigma= [F(v_0),...,F(v_k)]$.
For $A\in \Omega^k_{\alpha,\beta}$, we shall (under appropriate regularity assumptions) define

(F^*A)(σ):=lim_n→∞ ∑_σ'∈𝒦_n  A(F_*σ')

along any regular sequence of subdivisions $\mathcal{K}_n$ of $\sigma$.
This is easily seen to agree with the classical definition whenever $F$ and $A$ are smooth.

Let $\alpha\in ( \frac{1}{1+\eta} ,1]$, $\beta>0$.
Then, for $A \in \Omega_{\alpha,\beta}$ and $F\in C^{1,\eta}$,
the limit in (<ref>) is well defined and for $\tilde{\beta}=  \big(\alpha(1+\eta)-1\big)\wedge \beta$ the linear map

$$ F^{*}: \Omega^k_{\alpha,\beta} \to \Omega^k_{\alpha,\tilde \beta }, \qquad A\mapsto   F^*A \ $$
is bounded.

Furthermore, for $ \bar F \in C^{1,\eta}$, $F^*(\bar{F}^*A)=( \bar{F}\circ F)^* A$,  and, for $\phi \in C^{\gamma}$ with $\gamma\in (1-\alpha,1]$, $F^{*}(\phi \cdot A)= (F^*\phi)\cdot F^{*} A$.

Finally, for $\tilde{\alpha}< \alpha$, $\tilde{\beta}<  \big(\alpha(1+\eta)-1\big)\wedge \beta$ the following map is continuous

$$  C^{1,\eta}\times \Omega^k_{\alpha,\beta}\to \Omega^k_{\tilde{\alpha},\tilde \beta}, \qquad (F,A)\mapsto   F^*A\ .$$

Combine Lemma <ref>, Lemma <ref> and Proposition <ref>.

Note that this implies that for $\eta >0$, $\alpha>\frac{1}{1+\eta} , \beta <  \alpha(1+\eta)-1$, and an oriented $C^{1,\eta}$-manifold $M$  (possibly with boundary) we can define a space
$\Omega_{(\alpha,\beta)}^k (M)$ in the usual way.
In particular, this provides a notion of integration by pull-back and mimicking standard proofs of Stokes' theorem, c.f. <cit.>, one obtains the following theorem.

Let $M$ be a
compact $(k+1)$–dimensional, oriented, $C^{1,\eta}$–manifold with boundary.
For any $\alpha> \frac{1}{1+\eta}$, $\beta>0$, and $A\in \Omega^{k}_{(\alpha,\beta)}(M)$, it holds that

$$ \int_M dA = \int_{\partial M} A \;. $$

Contrast the stronger regularity assumptions required to build integration by pull-back, Theorem <ref>,  compared to Remark <ref>.
It is not hard to see that when both notions apply, the resulting integrals are equal.

In contrast to <cit.> we do not work with a control, since it is unclear how to extend Definition 3.16 therein to $k>1$.
For $k=1$ (and $d\geq 2$) one can still define a control by replacing $|P;\bar{P}|^{\alpha/2}$ therein by the $(\alpha', \beta)$-flat norm for some $\alpha', \beta\in (0,1]$, but we shall not pursue this further since our focus is on generic $k\geq 1$.

§.§ Simplicial approximation

For $F: \mathbb{R}^m\to \mathbb{R}^{d}$ and  $\osimplex=[v_0,...,v_k]\in \mathfrak{X}^k(\mathbb{R}^m)$, let $F^{\sigma} : \mathbb{R}^m \to \mathbb{R}^d$
to be the unique map supported on $\pmb{\sigma}$ such that
$F^{\sigma}(\sum_{i} t_i v_i)= \sum_{i} t_i F(v_i)$ whenever $t_i\in [0,1]$ and $\sum_{i} t_i=1$.

For a simplex $\osimplex\in \mathfrak{X}^k(\mathbb{R}^m)$ let

|F|_(α,β);    :=sup_𝒦 | σ max_σ'∈𝒦 |F_*σ'- F^σ_*σ'|_(α,β)  ,

|F; F̅|_(α,β);    := sup_𝒦 | σ max_σ'∈𝒦 |F_*σ' -F̅_* σ' - F^σ_*σ'+ F̅^σ_*σ'|_(α,β) .

where the supremum runs over all subdivisions $\mathcal{K}$ of ${\osimplex}$. Let

F;F̅ _(α,β);η; k;  :   = sup_σ∈𝔛^k_≤1( ) ( |F_*σ- F̅_*σ|_(α, β)/(σ)^η   ) ,

F; F̅ _(α,β);γ; k; :   =  sup_σ∈𝔛^k_≤1( ) (  |F; F̅ |_(α,β);/(σ)^γ   ) ,

as well as $\|F\|_{(\alpha,\beta),\gamma; k ; \mfK}:= \|F;0 \|_{(\alpha,\beta),\gamma; k; \mfK}$ and $  \vertiii{F}_{(\alpha,\beta);k; \mfK} :=\vertiii{F; 0 }_{(\alpha,\beta);k; \mfK} $.
Note that it follows directly from
that $\|\ \cdot\ ;\ \cdot\ \|_{(\alpha,\beta);\eta; k; \mfK}$ and $\vertiii{\ \cdot\ ; \ \cdot\ }_{(\alpha,\beta);\gamma; k; \mfK}$ satisfy the triangle inequality.

Whenever $F$ is Lipschitz continuous, for every $R>0$ it holds that

\|F\|_{(\alpha,\beta); k-1+\alpha; k; \mfK} \lesssim_{R} \| DF\|_{L^{\infty}}\ ,$$

uniformly over $F$ satisfying $\| DF\|_{L^{\infty}}\leq R$.
Furthermore, for $\eta<k-1+\alpha$ there exists $\kappa= \kappa(\alpha,\eta)>0$ such that for every $R>0$

F; F̅_(α,β); η; k;  ≲_R F-F̅^κ_L^∞

uniformly over Lipschitz continuous $F,\bar F$ such that $\| F\|_{L^{\infty}},\ \| \bar F\|_{L^{\infty}}, \ \| D F\|_{L^{\infty}} ,\  \| D\bar F\|_{L^{\infty}}\leq R$.

The former estimate is seen directly. Next, note that on the one hand

$$|F; \bar{F}  |_{(\alpha, \beta);\sigma} \leq |F |_{(\alpha, \beta);\sigma}+| \bar{F}  |_{(\alpha, \beta);\sigma}  \lesssim  {\diam(\sigma)^{k-1+\alpha}} $$

and on the other hand by Lemma <ref>

$$|F_*\sigma- \bar{F}_*\sigma  |_{(\alpha, \beta)}\lesssim \|F-\bar{F}\|_{L^\infty}^{\alpha}$$

and (<ref>) follows by interpolation.

Let $\alpha\in ( \frac{1}{1+\eta}, 1]$, $\beta\in( 0,1]$, then $ \bar{\gamma}:= \left(k-1+ \alpha(1+\eta) \right) \wedge\left( k+ \beta(1+\eta)\right)>k$.
For every $R>0$ and $\gamma\leq \bar{\gamma}$ the bound
$\vertiii{F }_{(\alpha,\beta),\gamma;k }\lesssim_{R} \|F\|_{C^{1,\eta}}$ holds
uniformly over $F$ satisfying $\|F\|_{C^{1,\eta}}<R$.

Furthermore, if $\gamma< \bar{\gamma}$,
there exists $\kappa=\kappa(\alpha,\beta,\gamma,\eta)>0$ such that
$\vertiii{F; \bar F }_{(\alpha,\beta),\gamma}\lesssim_{R} \|F-\bar F\|^{\kappa}_{L^\infty}$ uniformly over $F,\bar F$ satisfying $\|F\|_{C^{1,\eta}}, \|\bar F\|_{C^{1,\eta}}<R$.

Let $\sigma' \subset \sigma$ and observe that each vertex $v'_i\in \VV(\sigma')$ can uniquely be written as a convex combination
$ v'_i= \sum_{j=0}^k \lambda^i_j v_j$ where $v_j\in\VV(\sigma)$.

Thus we observe that for $r= \diam (\sigma)$,

∑_j λ^i_j F(v_j) =    ∑_j λ^i_j ( F (v'_i) + ∇F (v'_i) (v_j -v'_i)  +𝒪(D^1F_C^η(v_j -v'_i)^1+η) )

= F (v'_i) + ∇F (v'_i)   ∑_j λ^i_j (v_j -v'_i)  +𝒪(D^1F_C^ηr^1+η)

= F (v'_i)+ 𝒪(D^1F_C^ηr^1+η)  .

Since both $F^\sigma (\sigma')$ and  $F (\sigma')$ are contained in a ball with radius $\lesssim \|DF\|_{L^\infty}r$,

we conclude  by Lemma <ref> that

$$|F^{\sigma}(\sigma')- F(\sigma')|_{(\alpha,\beta)}
\lesssim (\|DF\|_{L^\infty}r )^{k-1} (\|DF\|_{C^\eta}r)^{\alpha (1+\eta)}+(\|DF\|_{L^\infty}r)^{k}  (\|DF\|_{C^\eta}r)^{\beta(1+\eta)}\ , $$

which remains bounded by $r^{\gamma}$
whenever $\gamma\leq\bar{\gamma}$.

To see the second claim of the lemma, note that as above

$$ |F^{\sigma}(\sigma')- F(\sigma')-\bar{F}^{\sigma}(\sigma')+ \bar{F}(\sigma')|_{(\alpha,\beta)}
\leq  |F^{\sigma}(\sigma')-\bar{F}^{\sigma}(\sigma')  |_{(\alpha,\beta)} + | F(\sigma') -\bar{F}(\sigma')|_{(\alpha,\beta)} \lesssim_R \|F-\bar{F}\|_{L^\infty}\ .

On the other hand by the first part of this proof

$$\vertiii{F; \bar F }_{(\alpha,\beta); \bar \gamma;k}\leq \vertiii{F }_{(\alpha,\beta);\bar \gamma;k} + \vertiii{ \bar F }_{(\alpha,\beta);\bar \gamma;k}< \infty$$
and we

conclude by interpolation.

Note that the threshold $\alpha>1/2$ in Theorem <ref> is explained by the observation that even if $F$ is smooth, generically
$\vertiii{F}_{(\alpha,\beta); k;k}<\infty$ only if $\alpha \in [ \frac{1}{2}, 1]$. This can be seen by for example choosing
$F$ to be a smooth parametrisation of the closure of an open subset of an embedded $k$-sphere which makes  (<ref>) sharp.

In order to make sense of the limit in (<ref>) we define the increments

$$\Xi^{F^*A}(\sigma):= A(F_*\sigma), \qquad \Xi^{F^*A ; \bar F^*A }(\sigma):= A(F_*\sigma)- A(\bar F_* \sigma)\;.$$

Let $\eta>k-1$, $\gamma>k$. For $A \in \Omega^k_{\alpha,\beta}$ and $F,\bar{F}$ continuous such that
$\| F\|_{( \alpha, \beta);\eta;k}, \| \bar F\|_{( \alpha, \beta);\eta;k}, \vertiii{F}_{(\alpha,\beta); \gamma;k} , \vertiii{\bar F}_{(\alpha,\beta); \gamma;k}  <\infty$, one finds that
$\Xi^{F^*A}, \Xi^{F^*A ; \bar F^*A }\in C_{2,k}^{\eta, \gamma}$
and that $F^*A:= \mathcal{I}\Xi^{{F^*A}} \in \Omega^k$ satisfies

F^*A_η-(k-1)    ≲A_(α,β) (  F_( α, β);η;k  ∨F_(α,β);γ;k )  ,

F^*A- F̅^*A _η-(k-1)     ≲A_(α,β)  (  F;F̅_( α, β);η;k  ∨F;F̅_(α,β);γ;k ) .

If furthermore $\vertiii{F}_{(\beta,\infty);\tilde{\gamma}; k+1}, \vertiii{F}_{(\beta,\infty);\tilde{\gamma}; k+1}<\infty$ for $\tilde{\gamma}>k$, then
$F^*A\in \Omega_{\alpha, (\gamma\wedge \tilde \gamma)-k}\ $

∂(F^*A  -F̅^*A)_(γ∧γ̃)- k ≲ F;F̅_( α, β);γ;k  ∨F;F̅_(β,∞);γ̃; k+1 .

Finally, $F^*(\bar{F}^*A)=( \bar{F}\circ F)^* A$ and $F^{*}(\phi \cdot A)= F^*\phi\cdot F^{*} A$ for $\phi \in C^\rho$ whenever $\rho \in (1-\alpha,1]$.

First note that by Lemma <ref>

Ξ^F^*A _η≤A_ α  F_( α, β);η;k  ,     Ξ^F^*A ; F̅^*A_η≤A_ (α, β) F;F̅_( α, β);η;k    .

To see

δΞ^F^*A_γ ≤A_(α,β)  F_( α, β);γ;k    ,
δ_σ, 𝒦 Ξ^F^*A ; F̅^*A _γ≤A_(α, β)  F; F̅_(α,β);γ;k
note that for $\mathcal{K}|\sigma$

\begin{align*}
|\delta_{\sigma, \mathcal{K}} \Xi^F|&= \left|
A(F_* (\sigma)) - \sum_{\sigma'\in \mathcal{K} } A(F_* (\sigma')) \right| \leq \sum_{\sigma'\in \mathcal{K} } |A (F^{\sigma}_*\sigma'- F_*\sigma')| \\
&\leq  \|A\|_{(\alpha, {\beta}) }  \sum_{\sigma'\in \mathcal{K} }  |F^{\sigma}_*\sigma'- F_*\sigma'|_{( \alpha, \beta)} \\
&\leq  \|A\|_{\bar{\alpha}, \bar{\beta} }   |\mathcal{K}| \vertiii{F}_{( \alpha, \beta) ;\gamma; k} \diam(\sigma)^\gamma \ .
\end{align*}

The second inequality of (<ref>) follows similarly. Thus, we conclude the first part of the proposition by Corollary <ref>.

To see (<ref>), note that

$$ |A ( \partial F_* \sigma) |  \leq \| \partial A \|_{\beta} \|F\|_{(\beta,\infty); \tilde{\gamma}; k+1 }\diam(\sigma)^{\tilde{\gamma}}$$

$$|F^*A( \partial \sigma) - A ( \partial F_* \sigma) |\leq  \sum_{B\in \Bd(\sigma)}  | \Xi^{F^*A} (B) - F^* A(B) |   \lesssim \llbracket \delta \Xi^{F^*A}\rrbracket_{\gamma} \diam(\sigma)^{\tilde{\gamma}} \ .  $$

Therefore by (<ref>)

\begin{align*}
|F^*A( \partial \sigma) |&\leq  |A ( \partial F_* \sigma) | + |F^*A( \partial \sigma) - A ( \partial F_* \sigma) | \\
&\lesssim  \|A\|_{(\alpha,\beta)} (\vertiii{ F}_{( \alpha, \beta);\gamma;k} \vee \|F\|_{(\beta,\infty); \tilde{\gamma}; k+1 })\diam(\sigma)^{\tilde{\gamma}\wedge\gamma} \ .
\end{align*}

Thus, Lemma <ref>

$$\|\partial F^*A \|_{(\gamma \wedge \tilde{\gamma})- k }\lesssim  \|A\|_{(\alpha,\beta)} (\| F\|_{( \alpha, \beta);\gamma} \vee \|F\|_{(\beta,\infty); \tilde{\gamma}; k+1; }) \ $$

and (<ref>) follows similarly.

The first identity in the final part follows from the fact that $G_* (F_*(\sigma))= (G\circ F)_* \sigma$.
For the second identity, note that $\Xi_{\mu}^{(\phi\circ F) \cdot F^*A }(\sigma)= \mu_\sigma (F^* \phi) \cdot (F^*A )(\sigma)$,

\Xi^{F^{*}(\phi \cdot A)}(\sigma)= (\phi\cdot A)(F_*\sigma),

and that

\Xi_{\mu}^{\phi \cdot A} (F_*\sigma) =\mu_\sigma (F^* \phi)  A(F_*\sigma)= \mu_\sigma (F^* \phi)  \Xi^{ F^*A}(\sigma)\ .

\begin{align*}
|\Xi^{F^{*}(\phi \cdot A)} (\sigma)- \Xi_{\mu}^{(F^* \phi) \cdot F^*A }(\sigma)|
&\leq |\Xi^{F^{*}(\phi \cdot A)} (\sigma)- \Xi_{\mu}^{\phi \cdot A} (F_*\sigma)| +|\Xi_{\mu}^{\phi \cdot A} (F_*\sigma)- \Xi_{\mu}^{(F^* \phi) \cdot F^*A }(\sigma)| \\
&= |(\phi \cdot A) (F_{*}\sigma)- \Xi_{\mu}^{\phi \cdot A} (F_*\sigma)| +|\mu_\sigma (F^* \phi) \big( \Xi^{ F^*A}- F^*A\big)(\sigma)| \\
%&\lesssim \diam(\sigma)^\gamma,
\end{align*}

and thee claim follows by (<ref>) together with Theorem <ref>.

§ EMBEDDINGS INTO DISTRIBUTION SPACES

For $1\leq k\leq d$, we write $\mathfrak{C}^d_{k}:= \{J\subset \{1,...,d\}\ : \ |J|=k \}$.
Given $J\in \mathfrak{C}^d_{k}$ we also write $J^{c}= \{1,..,d\}\setminus J$ and denote by
$E^J$ the hyperplane spanned by $(e_j : j\in J)$ equipped with its canonical orientation.
Given  $v,w\in \mathbb{R}^d$ we write $\llbracket w, v \rrbracket :=[w_1,v_1]\times...\times [w_d,v_d]$.

Given $A\in \Omega^k$, $J\in \mathfrak{C}^d_{k}$ and $\psi \in C^\infty_c(\mathbb{R}^d)$ we define

⟨π_J A, ψ⟩:= (-1)^|J| ∫_E^J^c ∫_E^J A(w,w+v ) D^J ψ(v+w) dv dw .

The following is a higher dimensional analogue of <cit.>.

In the setting above, the map characterised by (<ref>) restricts to a bounded linear map
$\pi_J: \Omega^k_{\alpha, 0}  \to C^{\alpha-1}$. Furthermore the map

$$\pi  :  \Omega^k_{\alpha, 0} \to \left( C^{\alpha-1}\right)^{\mathfrak{C}^d_k}, \qquad A \mapsto \pi(A)= \big( \pi_J(A) : J \in \mathfrak{C}^d_k \big)$$

is injective.

Recall that $\mass_0(\sigma)=1$ whenever $\sigma\neq 0$ and thus $\Omega^k_{\alpha, 0}= \{A \in \Omega^k \ : \  \|A\|_{\alpha}<+\infty\},$

Note that this suggests notation $A= \sum_{J \in \mathfrak{C}^d_{k}} (\pi_J A)(x) dx^J$ which is clearly an identity if $A$ arose from a smooth $k$-form.

Noting that for any simplex $\sigma\in \mathfrak{X}^k$ one has
$\int_{E^{J^c}} \int_{E^J} A(\sigma) D^{J} \psi(v+w) dv dw =0 $ and thus

$$  \int_{E^{J^c}} \int_{E^J} A(\llbracket w,w+v \rrbracket) D^{J} \psi(v+w) dv dw = \int_{E^{J^c}} \int_{E^J} A(\llbracket w +x,w+x+v \rrbracket) D^{J} \psi(v+w) dv dw\ .

Let $\psi\in C^{k+1}_{c}(B_{1/2}(0))$ satisfy $|\psi|_{C^{k+1}}\leq 1$ and write $\psi^{\lambda}_{x}:= \lambda^{-d}\psi\big( (\cdot- x)/\lambda \big)$, then

$$ |\langle\pi_{J} A, \psi^\lambda_x \rangle| \leq \lambda^{-k} \sup_{v\in E^J, w\in E^{J^c}, v+w \in \supp (\psi^{\lambda}_{x})  } |A(\llbracket w +x,w+x+v \rrbracket)| \lesssim \| A\|_{\alpha} \lambda^{-k} \lambda^{k-1+\alpha}\ .$$

It remains to check the map is injective.
Indeed if $\pi_{J} A=0$ for all $J \in  \mathfrak{C}^d_{k}$, this implies by the fundamental Lemma of calculus of variations that for almost every $w$ the function $E^J\ni v\mapsto A(\llbracket w,w+v\rrbracket)\in \mathbb{R}$ is the constant function. Since $A[w,w]=0$ this concludes the proof.

In the special case $k=d$ the map $\pi$ is surjective with bounded inverse.

Fix an orientation $o$ on $\mathbb{R}^d$.
Given any oriented $d$-simplex $\sigma=(\pmb{\sigma},o_{\sigma})$ we define $\mathrm{sign}(\sigma) = 1$ if $o_{\sigma} = o$ and $\mathrm{sign}(\sigma) = - 1$ otherwise.
We also set $\mathbf{1}_{\sigma} := \mathrm{sign}(\sigma) \mathbf{1}_{\pmb{\sigma}}$
Furthermore fix $\phi$ as in Remark <ref> and denote by $\big(\phi^{\simplex}_{n,i}\big)_{n\in \mathbb{N}, i\in I_n}$ the partition of unity constructed from $\phi\in C^\infty$ subordinate to a Whitney decomposition of $\simplex$.

Then, for $F\in C^{\alpha-1}$ one can set[One can check that the definition is independent of the specific choice of $\phi$.]

$$A_F(\sigma):= F(  \mathbf{1}_\sigma): =  \mathrm{sign}(\sigma)  \sum_{n\in \mathbb{N},i\in I_n} F(\phi^{\simplex}_{n,i})\;.$$
$\eta=d-1+\alpha$, one then finds that

\begin{align*}
\sum_{n\in \mathbb{N},i\in I_n} |F(\phi^{\simplex}_{n,i})| &\lesssim \sum_{n\in \mathbb{N},i\in I_n}  \|F\|_{C^{\alpha-1}} 2^{-nd}  2^{-n(\alpha-1)}\lesssim \|F\|_{C^{\alpha-1}} \sum_{n\in \mathbb{N}: 2^{-n}<\diam(\sigma) } |I_n|  2^{-n\eta } \\
&\lesssim \|F\|_{C^{\alpha-1}}  \diam(\sigma)^{\eta},
\end{align*}

where in the last step one argues ad verbatim like in (<ref>). Thus, the following linear map is bounded

$$\iota: C^{\alpha-1}\to \Omega_{\alpha,0}, \qquad F\mapsto A_F\ .$$

When $F$ is smooth, it is easily checked that

$$\big( \pi \circ \iota \big) (F) (w)= \pi A_F (w) = D^{J}_{v} F(\mathbf{1}_{\llbracket w, w+v \rrbracket} )= F(w)\ .$$

Thus, by continuity of $\iota$, it is a right inverse to $\pi$ which implies that $\pi$ is surjective.

§ APPLICATION TO RANDOM FIELDS

§.§ A Kolmogorov criterion

Fix a compact set $\mathfrak{K} \subset \mathbb{R}^{d}$. Let  $A=\big( A_{\sigma}: \sigma \in \mathfrak{X}^{k}(\mathfrak{K}) \big)$ be a $\mathfrak{X}^{k}(\mathfrak{K})$-indexed stochastic process such that the map $S\ni \sigma \mapsto A(\sigma) = A_{\sigma}$ satisfies the relations in Remark <ref> for any finite subset $S\subset\mathfrak{X}^{k}(\mathfrak{K})$ almost surely.

Assume furthermore that for some $\tilde{\alpha}, \tilde{\beta} \in (0,1]$ and $q \in \mathbb{N}$,

M_q = sup_∈𝔛_≤1^k(𝔎)   [|A()|^q]/ ^k_α̃()^q  +sup_∈𝔛_≤1^k+1(𝔎) [|A(∂)|^q]/ ^k+1_β̃()^q  < +∞

Then, for any $(\alpha,\beta) \in (0,  \tilde{\alpha} \wedge \tilde{\beta} -  d(k+1)/q) \times (0,  \tilde{\beta} -  d(k+2)/q)$, there exists a modification $\hat{A} \in \Omega^{k}_{(\alpha,\beta)} (\mathfrak{K})$ with

[ Â_α,β; 𝔎^q ] ≤M_q .

As is standard, outside of the proof below we just write $A$ to refer to the modification $\hat{A}$.

Note that the conclusion of Proposition <ref> remains true when (<ref>) is replaced by

M^_q = sup_∈𝔛_≤1^k(𝔎)   [|A()|^q]/ (σ)^q(k-1+α̃)  +sup_∈𝔛_≤1^k+1(𝔎) [|A(∂)|^q]/ (σ)^q(k+β̃)  < +∞ .

Additionally, the statement also holds if, for $j \in \{0,1\}$, one replaces the suprema over $\mathfrak{X}_{\le 1}^{k+j}(\mathfrak{K})$ in (<ref>) with suprema over $\mathfrak{Q}_{\le 1}^{k+j}(\mathfrak{K})$.

We write $(\tilde{\alpha}_{k}, \tilde{\alpha}_{k+1}) = (\tilde{\alpha}, \tilde{\beta})$, $(\alpha_k,\alpha_{k+1}) = (\alpha, \beta)$.
For  $n \in \mathbb{N}$ and $j \in \{k,k+1\}$ we write

\[
\mathfrak{Y}^{j}_{n}
\big\{ \sigma \in \mathfrak{X}_{\le 1}^j(\mathfrak{K}): \mass_{\alpha_j}^{j}(\sigma) \in (2^{-(n-1)\alpha_j},2^{-n \alpha_j}] \big\}\;.
\]

Note that $( \mathfrak{Y}^{j}_{n}: n \in \mathbb{N})$ is a partition of $\mathfrak{X}_{\le 1}^j(\mathfrak{K})$.
For any $\sigma \in  \mathfrak{X}_{\le 1}^{j}(\mathfrak{K})$, we define $n_{\sigma}$ so that $\sigma \in \mathfrak{Y}^{j}_{n_\sigma}$.

We shall show that

∑_m=0^∞ 2^mq α
|A(π_mσ)|^q + ∑_n>m |A(π_nσ)-A(π_n-1σ)|^q
]    < ∞ .

This will imply that, with probability $1$, for any $\sigma \in \mathfrak{X}_{\le 1}^{k}(\mathfrak{K})$, the limit
\hat{A}(\sigma) :=
\lim_{n \rightarrow \infty}

and one can extend $\hat{A}$ to $\mathcal{X}^k(\mathfrak{K})$.
Since measurability is preserved in this limit, we have $\hat{A} \in \Omega^{k}$.
In particular, for $\sigma \in \mathfrak{X}^{k+1}(\mathfrak{K})$, $\lim_{n \rightarrow \infty} A(\pi_{n}\partial\sigma) = \hat{A}(\partial \sigma )$, which together with showing

∑_m=0^∞ 2^mq α
|A(π_m ∂σ)|^q + ∑_n>m |A(π_n ∂σ)-A(π_n-1 ∂σ)|^q
] < ∞ ,

will prove that $\hat{A}$ satisfies the estimate (<ref>).

We use the parameter $j \in \{k,k+1\}$ so we can write discuss estimates (<ref>) and (<ref>) at the same time.
We claim that, uniformly in $m$, $\sigma \in \mathfrak{Y}^{j}_{m}$ and $n \ge m$, we have

[ |A(π_mσ_j)|^q ]
2^-q m α̂_j
2^-nq α̂_j    ,

where, $\sigma_k = \sigma$, $\hat{\alpha}_{k} =  \tilde{\alpha} \wedge \tilde{\beta}$, and $\sigma_{k+1} = \partial \sigma$, $\hat{\alpha}_{k+1} = \tilde{\beta}$.

To prove the inequality (<ref>) we argue as in Corollary <ref>.
For any $\sigma  \in \mathfrak{X}_{\le 1}^{j}$ and $n,n' \in \mathbb{Z}$, there exists $Z^j = Z^j(\sigma,n,n') \in \mathcal{X}^{j+1}(\mathfrak{X})$ such that, uniform in such $\sigma,n,n'$,

\[
\Big|
\pi_{n}(\sigma) - \pi_{n'}(\sigma) -  \partial Z^j
\Big|_{\tilde{\alpha}_j;j}
\lesssim
2^{-(n \wedge n') \tilde{\alpha}_j}\;,
\]

and, for $j=k$, we also have

\[
\Big| Z^k \Big|_{\tilde{\beta};{k+1}} \lesssim 2^{-(n \wedge n') \tilde{\beta}}\;.
\]

Continuing in the case $j=k$, we have

|A (
π_n(σ) - π_n'(σ)
|^q ]
≲[ |A ( π_n(σ) - π_n'(σ) -  ∂Z^k)|^q]
[ |A (∂Z^k)|^q ]

≲M_q ( |
π_n(σ) - π_n'(σ) -  ∂Z^k |_α̃;k^q + |Z^k |^q_β̃;k+1  )

≲M_q ( 2^-α̃(n ∧n' )q +  2^-β̃(n ∧n' )q ) = M_q
2^-nq α̂_k   .

For $j=k+1$, since $\partial \partial Z^{k+1} = 0$, we have

( ∂π_n(σ) - ∂π_n'(σ))|^q
( ∂(π_n(σ) -  π_n'(σ) - ∂Z^k+1  ))

≲M_q |  π_n(σ) -  π_n'(σ) - ∂Z^k+1 |_β̃;k+1^q
≲M_q 2^-β̃(n ∧n')q = M_q 2^-α̂_k (n ∧n')q  .

Now that (<ref>) is proved, let $\mathfrak{Y}^{j}_{m,n} = \big\{\pi_{n} \sigma: \sigma \in \mathfrak{Y}^{j}_{m,n} \big\}$, and note that  $|\mathfrak{Y}^{j}_{m,n}| \le \diam(\mathfrak{K})^{d} 2^{n(j+1)}$ since each such simplex is determined by $j+1$ vertices in $\mathfrak{K} \cap (2^{-n} \mathbb{Z}^{d})$.
We then have

∑_m = 0^∞ 2^mq α_j
|A(π_mσ_j)|^q + ∑_n>m |A(π_nσ_j)-A(π_n-1σ_j)|^q

∑_m = 0^∞ 2^mq α_j
∑_σ̃ ∈𝔜^k_m,m
|A(σ̃_j)|^q + ∑_n>m ∑_σ̃ ∈𝔜^k_m,n
|A(σ̃_j)-A(π_n-1 σ̃_j)|^q

∑_m = 0^∞
2^mq α_j
2^-q mα̂_j  2^(j+1)dm
∑_n>m 2^-nq α̂_j  2^(j+1)dn

≲M_q ∑_m = 0^∞  2^q m ( α_j -  α̂_j) + (j+1)dm ≲M_q ,

where in  the third inequality we used that $\tilde{\alpha}_j  > (j+1)d/q$ and in the final inequality we used that $\alpha_j < \hat{\alpha}_j - d(j+1)/q$.

§.§ Rough Gaussian $k$-forms

In this section we exhibit a criterion for fractional Gaussian fields, c.f. <cit.>, to belong to the spaces $\Omega_{\alpha,\beta}^k$.

We start with the following generalisation of <cit.>.

Let $\pmb{Q} \subset \mathbb{R}^{d}$ be a $k$-cube of side length $r \in (0,1]$.
We associate to $\pmb{Q}$ the distribution $\delta_{\pmb{Q}}$ on $\mathbb{R}^{d}$ by setting, for any smooth $f$ on $\mathbb{R}^{d}$,

\[
\langle \delta_{\pmb{Q}},f\rangle = \int_{[0,r]^{k}} f(\gamma(x)) \mathrm{d}^{k}x\;,
\]

where $\gamma: [0,r]^{k} \rightarrow  \pmb{Q}$ is an isometric embedding and $\mathrm{d}^{k}x$ the Lebesgue measure. There exists $C>0$ independent of $r$ such that
\| \delta_{\pmb{Q}} \|_{H^{-\theta}(\mathbb{R}^{d})}
\leq C r^{ (k + (2 \theta - d + k) \wedge k)/2}
for any $\theta > (d-k)/2$.

Without loss of generality, we take $\pmb{Q} = [0,r]^{k} \times \{0\}^{d-k} \subset \mathbb{R}^{d}$.
We then have

\[
\widehat{\delta_{\pmb{Q}}}(p_1,\dots,p_{n})
\prod_{j=1}^{k} \frac{e^{2\pi i p_j r} - 1}{2 \pi i p_{j}}\;.
\]

δ_Q _H^-θ(ℝ^d)^2
∫_ℝ^d (1 + ∑_j=1^d |p_j| )^-2 θ | δ_Q(p) |^2   d^dp
≲∫_ℝ^d (1 + ∑_j=1^d |p_j| )^-2 θ ∏_j=1^k( r^2 ∧p_j^-2)   d^dp

≲∫_ℝ^k (1 + ∑_j=1^k |p_j| )^(d-k)-2 θ ∏_j=1^k( r^2 ∧p_j^-2)   d^kp

The estimate then follows from noting that for any $0 \le m \le k$,

∫_ℝ^k (1 + ∑_j=1^k |p_j| )^(d-k)-2 θ ∏_j=1^m ( p_j^-2 1 { |p_j| ≥r^-1 } )
∏_j=m+1^k ( r^2  1 { |p_j| < r^-1 }  ) d^kp
≲r^k +  (2θ- d + k) ∧k  .

Fix $k < d$, $(d-k)/2 < \theta $.
Let $(A_{I}: I\in \mathfrak{C}^d_{k})$ be a collection of centred, jointly Gaussian, random fields on $H^{-\theta}(\mathbb{R}^{d})$ for which there exists $C>0$ such that

\[
\EE[ A_{I}(f)^2] \le C \|f\|_{H^{ - \theta}}^2\qquad \text{uniformly over $I$ and }f \in H^{-\theta}(\mathbb{R}^{d})\;.
\]

Let $\bar{\alpha}= (\theta-d/2+1)\wedge 1$ and $\bar \beta = (\theta-d/2) \wedge 1$. Then, there exists
$A\in \Omega^{k}_{\alpha,\beta}(\mathfrak{K}) $
such that $\pi(A)$ is a modification of $(A_{I}: I\in \mathfrak{C}^d_{k})$ and
\EE \big[ \llbracket A\rrbracket_{(\alpha,\beta); \mathfrak{K}}^2 \big]
\lesssim_{\mathfrak{K}} C\;
for every $\alpha\in (0,\bar{\alpha}]$, $\beta\in (0, \bar{\beta}]$ and compact $\mathfrak{K} \subset \mathbb{R}^{d}$.

By Proposition <ref> and Remark <ref> with the choice $\bar{\alpha} = \theta/2 - d/2 + 1$ and $\bar{\beta} = \zeta$ and combined with equivalence of moments for Gaussian random variables, it suffices to prove the estimate

sup_ Q ∈𝔔_≤1^k(𝔎)   [|A(Q)|^2]/ (Q)^2(k-1 + α̅)  +sup_Q ∈𝔔_≤1^k+1(𝔎) [|A(∂Q)|^2]/ (Q)^2(k +β̅)
≲C .

For $Q \in \mathfrak{Q}^{k}(\mathfrak{K})$ of side length $r$, let
A(Q) = \frac{1}{r^k} \sum_{I} \mathrm{d}x^{I}(Q) \delta_{\pmb{Q}}(A_{I})\;.
To estimate the first term on the left hand side of (<ref>) we note that by Lemma <ref> we have, uniform in $Q \in  \mathfrak{Q}_{\le 1}^{k}(\mathfrak{K})$,

\[
\EE[
\delta_{\pmb{Q}}(A_{I})^2]
\le C \| \delta_{\pmb{Q}} \|_{H^{-\theta}}^2
\lesssim C \diam(Q)^{k + (2 \theta - d + k)\wedge k }= C \diam(Q)^{2(k  + \theta - d/2) \wedge 2k}\;.
\]

We turn to estimating the second term of (<ref>).
The desired estimate follows from applying Stokes' Theorem estimate to write  $|A(\partial Q)| = |(\partial A)(Q)|$ and then observing that, for any $I$ and $j \in [d] \setminus I$, and uniform in  $Q \in  \mathfrak{Q}_{\le 1}^{k+1}(\mathfrak{K})$,

\[
\mathbb{E}[\delta_{\pmb{Q}}(\partial_{j} A_{I})^2]
\lesssim C\| \partial_{j} \delta_{\pmb{Q}}\|_{H^{-\theta}}^2
\lesssim C\| \delta_{\pmb{Q}}\|_{H^{-\theta + 1}}^2
\lesssim C \diam(Q)^{k + 1 + (2 (\theta - 1) - d + k+1)\wedge(k+1) }   \;.
\]

§ METHODS OF SUBDIVISION

We define the eccentricity $\mathfrak{e}(\sigma)$ of $\sigma \in \mathfrak{X}^k$ as

(σ)^k/^k(σ) .

We call a family of maps $\mathcal{M}= \{\mathcal{M}_{\ell}\}_{\ell\in \mathbb{N}}$ where for each $\ell \in \mathbb{N}$,

$$\mathcal{M}_{\ell}: \mathfrak{X}^k \mapsto 2^{\mathfrak{X}^k}, $$
a method of subdivision (for $k$ simplices), if for each $\sigma \in \mathfrak{X}^k$, $\mathcal{M}_\ell(\sigma)$ is a subdivision of $\sigma$ and

ℳ_ℓ+ℓ'(σ)= ⋃_w∈ℳ_ℓ'(σ) ℳ_ℓ(w) ,     for every ℓ,ℓ'∈ .

We call $\mathrm{card}(\mathcal{M}) := \sup_{\sigma}|\mathcal{M}_1(\sigma)| $ the cardinality of $\mathcal{M}$ and write

\[
\| \mathcal{M} \|
\sup_{\sigma \in \mathfrak{X}^k}
\sup_{\ell \in \mathbb{N}}
\sup_{w \in\mathcal{M}_\ell(\sigma) }
\frac{\mathfrak{e}(w)}{\mathfrak{e}(\sigma)}
\quad
\text{and}
\quad
\mathfrak{c}_{\mathcal{M}}
\sup_{\sigma \in \mathfrak{X}^k}
\max_{w \in\mathcal{M}_1(\sigma) }
\Big( \frac{\diam(w)}{\diam(\sigma)} \Big)\; \]

as well as

ℳ := inf_μ>0sup_l∈ℕ ℓ^-μ  sup{
^k(w')/^k(w):  σ∈𝔛^k, w,w'∈_ℓ(σ) }  .

We call $\mathcal{M}$ strongly regular if  $\mathrm{card}(\mathcal{M}) \vee  \| \mathcal{M} \|\vee \vertiii{ \mathcal{M}} < \infty$ and  $\mathfrak{c}_{\mathcal{M}} < 1$.

Enforcing $\| \mathcal{M} \| < \infty$ rules out methods of iterative subdivision that would would be problematic for our analysis, such as slicing a triangle into thinner and thinner strips without significantly reducing the length of the strips.

The main theorem of <cit.> states that there exists a strongly regular method of subdivision with $\vertiii{ \mathcal{M}}=1$ .

Another natural method of subdivision is longest edge bisection, but the question of whether $\| \mathcal{M} \| < \infty$ for this method appears to be an open problem, see <cit.>.

For any strongly regular method of subdivision $\mathcal{M}$, $\sigma\in \mathfrak{X}^k$, and $n \in \mathbb{N}$,

$$\sum_{\sigma'\in \mathcal{M}_n(\sigma) }  \diam(\sigma')^{k}
\leq
% \max_{{\sigma'\in \mathcal{M}_n(\sigma) }} \frac{\diam(\sigma')^k}{\Vol^k(\sigma')} \Vol^k(\sigma)
\Big(
\max_{{\sigma'\in \mathcal{M}_n(\sigma) }} \mathfrak{e}(\sigma')
\Big) \Vol^k(\sigma)
\leq \| \mcM\| \mathfrak{e}(\sigma)\Vol^k(\sigma)
\leq  \| \mcM\| \diam(\sigma)^k\;.$$

In particular,  then sequence $\big(\mathcal{M}_{l}(\sigma) \big)_{l\in \mathbb{N}}$ is a regular sequence of subdivisions of $\sigma$ in the sense of Definition <ref> since, for $\gamma>k$,

∑_σ'∈_n (σ)  (σ')^γ
≤max_σ'∈_n(σ)   (σ')^γ-k
∑_σ'∈ℳ_n(σ)   (σ')^k

≤  ^k (σ) max_σ'∈𝒦^(n) _σ   (σ')^γ-k

≤c^n(γ-k)_  ^γ (σ)   .

Given two subdivisions $\mathcal{K} | \sigma$ and $\mathcal{K}' | \sigma$, we say $\mathcal{K}'$ is a refinement of $\mathcal{K}$ if, for every $\tau \in \mathcal{K}$, there exists $\mathcal{K}'' \subset \mathcal{K}'$ with $\mathcal{K}'' | \tau$.

Consider two strongly regular methods of subdivision $\mathcal{M} , \mathcal{M}'$. Then, there exists $\mu= \mu(\mcM)>0$, $C=C(\mcM)$
such that, for any $\sigma$ and $n\in \mathbb{N}$, there exists a common refinement $\mathcal{K}_n= \mathcal{K}(\mathcal{M},\mathcal{M}',\sigma,n) $ of both $\mathcal{M}_n(\sigma)$ and $\mathcal{M}_n(\sigma)$ satisfying

$$\max_{w'\in \mcM'_n(\sigma)}
\big|
\{ \tilde{w}\in \mcK_n : \tilde{w}\subset w' \} \big|
\lesssim_{k } C(\mcM)  \mathfrak{e}(\sigma) n^{\mu(\mcM)} \left(1+\frac{{\diam \big(\mcM'_n(\sigma) \big)}}{\diam\big(\mathcal{M}_n(\sigma)\big)} \right)^k\ .$$

Note that by symmetry one can reverse the roles of $\mcM$ and $\mcM'$ in the estimate above.

$$\mathcal{D}_{n;\sigma}^{\mathcal{M} ,\mathcal{M}'}(w') :=
\big|
\{w\in \mathcal{M}_n(\sigma): \Vol^k(w\cap w') \neq 0\}
\big|. $$

The lemma follows by combining the following claim with the fact that for any two simplices, their intersection can be written as a union of a bounded ($k$-dependent) number of simplices.

There exists $ \mu(\mcM), C(\mcM)>0$
such that

$$\mathcal{D}_{n;\sigma}^{\mathcal{M} ,\mathcal{M}'}(w')\leq C(\mcM) \mathfrak{e}(\sigma) n^{\mu} \left(1+\frac{{\diam (w')}}{\diam(\mathcal{M}_n(\sigma))} \right)^k\ $$

uniformly in $n\in\mathbb{N}$

We now prove this claim.
Write $\mathcal{K}_{n}(w'):=\{w\in \mathcal{M}_n(\sigma): \Vol^k(w\cap w') \neq 0\}$ and let $\bar w\in \mcM(\sigma)$ be such that $\diam (\mcM_n(\sigma))=\diam(\bar w)$. Then,

\begin{align*}
|\mathcal{K}_{n}(w')| \diam(\bar w)^{k} & \leq  |\mathcal{K}_{n}(w')| \mathfrak{e}(\bar w)  \Vol^k(\bar{w})\leq
\mathfrak{e}(\bar w)\vertiii{\mathcal{M}}  n^{\mu(\mcM)} \sum_{w\in \mathcal{K}_{n}(w')} \Vol^k(w)
%\vertiii{\mathcal{M}} \mathfrak{e}(w') n^{\mu(\mcM)}\sum_{w\in \mathcal{K}_{n,w'}}  \Vol(w)\\
\\
& \leq \| \mcM\| \cdot \vertiii{\mathcal{M}} \mathfrak{e}(\sigma) n^{\mu(\mcM)} \sum_{w\in \mathcal{K}_{n}(w')} \Vol(w) \\
&\leq \| \mcM\|  \cdot \vertiii{\mathcal{M}} \mathfrak{e}(\sigma) n^{\mu(\mcM)} (\diam(\bar{w})+{\diam (w')})^k\
\end{align*}

and therefore,
$|\mathcal{K}_{n}(w')|\leq \| \mcM\|  \vertiii{\mathcal{M}}  \mathfrak{e}(\sigma) n^{\mu(\mcM)} \left(1+\frac{{\diam (w')}}{\diam(\bar w )} \right)^k\ .$

§.§ Proof of Proposition <ref>

We shall suppress $\mathfrak{K}\subset \mathbb{R}^d$ throughout the proof.

In the setting of Proposition <ref>, for any strongly regular method of subdivision $\mathcal{M}$, the map

$$\mathcal{I}_{\mathcal{M} } \Xi: \mathfrak{X}^k \to \mathbb{R},\qquad \sigma \mapsto  \mathcal{I}_{\mathcal{M} } \Xi (\sigma):= \lim_{n\to \infty } \sum_{\sigma'\in \mcM_n(\sigma)} \Xi(\sigma')$$

$$ |\mathcal{I}_{\mathcal{M} } \Xi (\sigma)-\Xi (\sigma) | \leq \frac{\| \mcM\|}{1-\mathfrak{c}^{(\gamma-k)}_\mcM}  \llbracket \delta\Xi\rrbracket_{\gamma,\mathfrak{K}}
\diam(\sigma)^{\gamma} \ .
as well as
$\mathcal{I}_{\mcM } \Xi (\sigma) = \sum_{\sigma'\in \mathcal{M}_n(\sigma) } \mathcal{I}_{\mathcal{M} }\Xi (\sigma')$.

Using (<ref>) in the last line we find that

\begin{align*}
\Bigg|\sum_{\sigma'\in \mcM_{n+1}(\sigma)} \Xi(\sigma')-\sum_{\sigma'\in \mcM_n(\sigma)} \Xi(\sigma') \Bigg|
&\leq \sum_{\sigma'\in \mcM_n(\sigma)}   \Big(\Xi(\sigma')- \sum_{\sigma''\in \mcM_1(\sigma')}  \Xi(\sigma'')\Big) \\
&\leq \llbracket \delta\Xi\rrbracket_{\gamma,\mathfrak{K}}  \sum_{\sigma'\in \mcM_n(\sigma)}   \diam (\sigma')^\gamma \\
& \leq  \mathfrak{c}^{n(\gamma-k)}_\mcM \| \mcM\| \llbracket \delta\Xi\rrbracket_{\gamma,\mathfrak{K}} \diam (\sigma)^{\gamma} \ .
\end{align*}

Since $c_\mcM<1$ and $\gamma>k$, this is summable in $n$. The remaining claim follows from (<ref>).

We define the following equivalence relation between methods of subdivisions: we say that
$\mcM\sim\mcM'$, whenever there exits $C>0$ such that $\frac{1}{C}\leq \sup_{n}\frac{\diam(\mcM_{n} (\sigma))}{\diam( \mcM'_{n} (\sigma))} \leq C\ $ uniformly over $\sigma\in \mathfrak{X}^k_{\leq 1}$, $n\in \mathbb{N}$.

For two strongly regular methods of subdivision $\mcM$ and $\mcM'$ the maps $\mathcal{I}_{\mathcal{M} }\Xi(\sigma)$ and $\mathcal{I}_{\mcM'}\Xi(\sigma)$ agree whenever $\mcM\sim\mcM'$.

For any two strongly regular methods of subdivision $\mcM$, $\mcM'$. By Lemma <ref> there exists $\mu=\mu(\mcM,\mcM')$ and a common refinement $\mathcal{K}_n$ of $\mcM_n(\sigma), \mcM'_n(\sigma)$ such that, if we write $\mathcal{K}_n(w):= \left\{ \tilde{ w} \in \mathcal{K}_n\ : \ \tilde w\subset w \right\} $  for $w\in \mcM_n(\sigma) \cup \mcM'_n(\sigma)$,

$$|\mathcal{K}_n(w)| \lesssim_{\mathcal{M},\tilde {\mathcal{M}},\sigma} n^{\mu}  .$$

\begin{align*}
\mathcal{I}_{\mathcal{M} } \Xi(\sigma) - \mathcal{I}_{\mathcal{M} } \Xi (\sigma) &=
\sum_{w\in  \mathcal{M}_n(\sigma)} \mathcal{I}_{\mathcal{M} } \Xi(w) -  \sum_{w\in \mathcal{M}_n'(\sigma)} \mathcal{I}_{\mathcal{M}' } \Xi(w) \\
&= \sum_{w\in  \mathcal{M}_n(\sigma)} \left( \mathcal{I}_{\mathcal{M} } \Xi(w)- \Xi(w) \right)  - \sum_{w'\in \mathcal{M}_n'(\sigma)} \left( \mathcal{I}_{\mathcal{M}' } \Xi(w')- \Xi(w') \right)\\
& \quad + \sum_{w\in  \mathcal{M}_n(\sigma)} \Big( \Xi(w) -\sum_{\tilde w\in \mathcal{K}_n(w) }  \Xi(\tilde w)   \Big)
- \sum_{w'\in \mathcal{M}_n'(\sigma)} \Big( \Xi(w') 		\sum_{\tilde w\in \mathcal{K}_n(w') } 	\Xi(\tilde w)	\Big)\ .
\end{align*}

The first two sums in the last expression above converge to $0$ as $n\to \infty$ by definition, and for the third sum we have
\begin{align*}
\Big| \Xi(w) -\sum_{\tilde w\in \mathcal{K}^{M,\tilde{M}}_n(\sigma); \tilde w\subset w }  \Xi(w)   \Big|
&= | \delta_{w,\mathcal{K}_n(w)} \Xi| \leq |\mathcal{K}_n(w)| \llbracket\delta\Xi\rrbracket_\gamma \diam(w)^\gamma\\
&\lesssim n^\mu  \llbracket \delta\Xi\rrbracket_\gamma \diam (w)^\gamma
\end{align*}

Similarly, one estimates the summands in the fourth sum. We conclude by (<ref>).

We leave the following observation as an exercise to the reader: Given a strongly regular method of subdivision $\mcM$, a simplex $\sigma$ and $\mathcal{K}$ a subdivision of $\sigma$. Then there exists a strongly regular method $\mcM' \sim \mcM$ such that $\mcK= \mcM_1(\sigma)$.

For any strongly regular method of subdivision
$\mathcal{I}_{\mcM} \Xi$ is an element of $\Omega^{k}$.

Note that the first condition of Definition <ref> follows directly, and it remains to check that for any $\sigma \in \mathfrak{X}^{k}$ and subdivision $\mcK | \sigma$, one has

$$\mathcal{I}_{\mcM}\Xi(\sigma) = \sum_{\sigma' \in \mcK} \mathcal{I}_{\mcM}\Xi(\sigma')\ .$$

The flexibility in Definition <ref> allows us to choose a strongly regular method of subdivision $\mcM'$ as in Remark <ref>.
It follows by Lemma <ref>  that

ℐ_ℳΞ(σ)     = ℐ_ℳ'Ξ(σ) = lim_n →∞  ∑_σ'∈'_n+1(σ) Ξ(σ') = ∑_σ'∈'_1(σ)  lim_n →∞  ∑_σ” ∈ℳ'_n(σ') ℐ_ℳ' Ξ(σ”)

=  ∑_σ'∈ ℐ_ Ξ(σ') .

We first note that
any map
$\mathcal{I}: C_{2,k}^{\eta, \gamma}(\mathfrak{K}) \to \Omega^k(\mathfrak{K}),$

\[
|\mathcal{I} \Xi (\sigma)- \Xi (\sigma)|\lesssim \llbracket \delta\Xi\rrbracket_{\gamma,\mathfrak{K}} \diam(\sigma)^{\gamma}
\]

also satisfies

$$|\mathcal{I}\Xi(\sigma)  |
\leq  |\Xi(\sigma) | +    |\Xi(\sigma)-\mathcal{I}\Xi(\sigma) |\lesssim \llbracket\Xi\rrbracket_{(\eta,\gamma)} \diam(\sigma)^\eta \ .$$

Considering a regular sequence of subdivisions $\{\mcK_n\}_{n\in \mathbb{N}}$ of $\sigma$ any such map also satisfies

|\mathcal{I} \Xi (\sigma)- \sum_{\sigma'\in \mcK_n}\Xi (\sigma')|\lesssim \llbracket \delta\Xi\rrbracket_{\gamma,\mathfrak{K}} \sum_{\sigma'\in \mcK_n}  \diam(\sigma')^{\gamma} \to 0\;.

Thus, in view of the preceding lemmas we have constructed for any strongly regular method of subdivision a map $\mathcal{I}$ satisfying the properties in Proposition <ref>.
It remains to prove uniqueness of the map $\mathcal{I}$. Assume $\mathcal{I}'$ is a second such map and let $\mcM$ be a strongly regular method of subdivision. Then, by additivity

\begin{align*}
|\mathcal{I}\Xi(\sigma) - \mathcal{I}'\Xi(\sigma)  |  &\leq \sum_{\sigma' \in \mcM_n} |\mathcal{I}\Xi(\sigma') - \mathcal{I}'\Xi(\sigma')  |\\
&\leq \sum_{\sigma' \in \mcM_n} \left( |\mathcal{I}\Xi(\sigma') -\Xi(\sigma') | +| \mathcal{I}'\Xi(\sigma')- \Xi(\sigma')  | \right)\\
&\lesssim \sum_{\sigma' \in \mcM_n} \diam(\sigma)^{\gamma}
\end{align*}

which converges to $0$ as $n\to \infty$ by (<ref>).

§ SYMBOLIC INDEX

Object     Meaning     Ref.

$\Vol^k$     $k$-dimensional Hausdorf measure     Page sec:preliminaries

$\pmb{\sigma}$    non-oriented simplex     Def. <ref>

$\sigma$    oriented simplex     Def. <ref>

$\VV(\sigma)$     vertices of $\sigma$      Def. <ref>

$\diam$    diameter of a simplex or cube and mesh of a subdivision      Def. <ref>

$\mathcal{K}|\sigma$    $\mathcal{K}$ is a subdivision of $\sigma$     Def. <ref>

$\mathfrak{X}^k$      set of oriented simplices     Def. <ref>

$\mathcal{X}^k $     $\mathbb{Z}$ module of chains     Def. <ref>

${\mass^k_\alpha }$     $\alpha$-mass     Eq. <ref>

$\Omega^k$     universe of (measurable) cochains     Def. <ref>

$\Omega_{\alpha,\beta}^{k}$    elements $A\in \Omega^k$ such that $\|A\|_{\alpha,\beta}<+\infty$      Def. <ref>

$\mathfrak{Q}^k$      set of oriented cubes     Def. <ref>

$\Bd(\sigma)$     oriented faces of $\sigma$     Def. <ref>

$\partial$     boundary operator $\delta \sigma = \sum_{F\in \Bd(\sigma)} F$     Eq. <ref>

$\mathcal{B}_{\alpha,\beta}$    completion of $(\mathcal{X}^k,  |\cdot |_{(\alpha,\beta)}) $     Def. <ref>

$C_2^{\eta,\gamma}$     certain functions on simplices/germs     Def. <ref>

$\Xi$    element of $C_2^{\eta,\gamma}$      Def. <ref>

$\mu_{\sigma}$    family of probability measure supported on $\pmb{\sigma}$     Page thm:multiplication

$\delta\Xi$    generalised increment of $\Xi$     Def. <ref>

$\mathcal{I}$     sewing/reconstruction operator $C_2^{\eta,\gamma}\to \Omega^k$     Prop. <ref>

$|\cdot |_{(\alpha,\beta)} $     $(\alpha,\beta)$-flat norm     Def. <ref>

$\delta_{\pmb Q}$    element of $D'$ which integrates functions over $\pmb{Q}$     Def. <ref>

$\mathcal{M}$    method of subdivision     Def. <ref>

$\mathfrak{e}$    eccentricity of a simplex $\mathfrak{e} (\sigma)= \frac{\diam(\sigma)^k}{\Vol^k(\sigma)}$     Eq. <ref>