# Local space-preserving decompositions for the bubble transform

Richard S. Falk  Department of Mathematics, Rutgers University, Piscataway, NJ
08854<EMAIL_ADDRESS>and  Ragnar Winther  Department of Mathematics,
University of Oslo, 0316 Oslo, Norway<EMAIL_ADDRESS>
(Date: December 20, 2023)

###### Abstract.

The bubble transform is a procedure to decompose differential forms, which are
piecewise smooth with respect to a given triangulation of the domain, into a
sum of local bubbles. In this paper, an improved version of a construction in
the setting of the de Rham complex previously proposed by the authors is
presented. The major improvement in the decomposition is that unlike the
previous results, in which the individual bubbles were rational functions with
the property that groups of local bubbles summed up to preserve piecewise
smoothness, the new decomposition is strictly space-preserving in the sense
that each local bubble preserves piecewise smoothness. An important property
of the transform is that the construction only depends on the given
triangulation of the domain and is independent of any finite element space. On
the other hand, all the standard piecewise polynomial spaces are invariant
under the transform. Other key properties of the transform are that it
commutes with the exterior derivative, is bounded in $L^{2}$, and satisfies
the stable decomposition property.

###### Key words and phrases:

simplicial mesh, commuting decomposition of $k$-forms, preservation of
piecewise polynomial spaces

###### 2020 Mathematics Subject Classification:

65N30, 52-08

The research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Programme
(FP7/2007-2013) / ERC grant agreement 339643.

##  1\. Introduction

The present paper is a continuation of the earlier papers [5] and [6] of the
authors. While the first paper was devoted to scalar valued functions, the
second paper, [6], develops a theory for decomposing differential forms into a
sum of functions, or bubbles, which have local support on domains defined by a
given simplicial mesh of the domain. The decomposition, which we refer to as
the bubble transform, commutes with the exterior derivative, and has the
additional property that all the standard piecewise polynomial spaces utilized
in the finite element exterior calculus (FEEC), cf. [1, 2, 3], are in some
sense invariant. However, the piecewise polynomial spaces are not strictly
invariant for the decomposition constructed in [6]. In general, each
individual bubble is a rational function, but with the property that groups of
the local bubbles sum up to preserve the desired polynomial structure. The
purpose of the theory presented here is to refine the earlier theory, so that
we obtain a transform which is strictly space-preserving in the sense that
each local bubble preserves piecewise smoothness and the standard piecewise
polynomial spaces of FEEC.

The development of the bubble transform is partly motivated by the $hp$-finite
element method, i.e., where both piecewise polynomials of arbitrary high
degree and arbitrary small mesh cells are allowed. While the analysis of
finite element methods based on mesh refinements and a fixed polynomial
degree, i.e., the $h$-method, is by now well understood, the corresponding
analysis for the $p$-method, where the polynomial degree is unbounded, is so
far less canonical. However, the bubble transform represents a theory where
the decomposition itself, and the associated operator bounds, are obtained
independently of any finite element space. The entire construction only
depends on a given triangulation of the domain. In fact, the decomposition is
also stable with respect to mesh refinements, and therefore the results will
apply to general $hp$-methods. As a consequence, the decomposition represents
a new tool for understanding $hp$-methods. As an example, consider the
analysis of overlapping Schwarz preconditioners. In [7], it is established how
to construct such preconditioners for second order elliptic problems in the
setting of $hp$-refinements. But so far, the corresponding verification for
more general Hodge-Laplace problems appears to be open. In fact, a key
obstacle for establishing such a bound is to verify a so-called stable
decomposition property, i.e., to establish the existence of a bounded
decomposition, cf. [9, Chapter 2] and references given therein. Such a bound
is simply a special case of the bounds we derive here. Although the discussion
in the present paper will be restricted to the basic theory of the bubble
transform, a more thorough motivation can be found in [6].

In order to describe the main results of this paper, it is first necessary to
introduce some basic notation. Throughout this paper, $\Omega$ will be a
bounded polyhedral domain in $\mathbb{R}^{n}$, and for $0\leq k\leq n$, the
space of smooth differential $k$-forms on $\Omega$ will be denoted
$\Lambda^{k}(\Omega)$. The construction of the bubble transform is based on a
simplicial mesh ${\mathcal{T}}$ of $\Omega$. The corresponding space,
$\Lambda^{k}({\mathcal{T}})$, is the space of $k$–forms on $\Omega$ which are
piecewise smooth with respect to ${\mathcal{T}}$. More precisely, the elements
of $\Lambda^{k}({\mathcal{T}})$ are smooth on the closed simplices $T$ in the
triangulation and have single-valued traces on each subsimplex of
${\mathcal{T}}$. For the piecewise polynomial subspaces of
$\Lambda^{k}({\mathcal{T}})$, we will adopt the standard notation of FEEC, cf.
[1, 2, 3], i.e., ${\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}})$ denotes the
space of piecewise polynomial forms of degree less than or equal to $r$, while
${\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}})$ is the corresponding space
of trimmed polynomials. The set of all subsimplices of ${\mathcal{T}}$ is
denoted $\Delta({\mathcal{T}})$, while $\Delta_{m}({\mathcal{T}})$ is the set
of simplices of dimension $m$. For each $f\in\Delta({\mathcal{T}})$, the
macroelement $\Omega_{f}$ consists of the union of all $n$–simplices in
$\Delta({\mathcal{T}})$ containing $f$ as a subsimplex. Furthermore,
${\mathcal{T}}_{f}$ is the restriction of the mesh ${\mathcal{T}}$ to the
macroelement $\Omega_{f}$, and $\mathaccent
23{\Lambda}^{k}({\mathcal{T}}_{f})$ is the subspace of
$\Lambda^{k}({\mathcal{T}})$ consisting of forms which have support on
$\Omega_{f}$, i.e., which vanish on $\Omega\setminus\Omega_{f}$.

For given $u\in\Lambda^{k}({\mathcal{T}})$, the bubble transform leads to a
decomposition of the form

(1.1)
$u=W^{k}u+\sum_{f\in\Delta({\mathcal{T}})}B_{f}^{k}u=W^{k}u+\sum_{m=0}^{n}\sum_{f\in\Delta_{m}({\mathcal{T}})}B_{f}^{k}u,$

where the bubbles $B_{f}^{k}u$ belong to $\mathaccent
23{\Lambda}^{k}({\mathcal{T}}_{f})$, and where $W^{k}u$ is a trimmed piecewise
linear $k$-form. More precisely, we will show how to construct linear
operators
$W^{k}:\Lambda^{k}({\mathcal{T}})\to{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$
and local operators $B_{f}^{k}:\Lambda^{k}({\mathcal{T}})\to\mathaccent
23{\Lambda}^{k}({\mathcal{T}}_{f})$ which commute with the exterior derivative
$d$, i,e.,

$dW^{k}=W^{k+1}d,\quad\text{and }dB_{f}^{k}=B_{f}^{k+1}d,\quad 0\leq k\leq
n-1.$

In fact, if we let ${\mathcal{B}}^{k}$ denote the collection of all the
operators $\\{B_{f}^{k}\\}_{f\in\Delta({\mathcal{T}})}$, such that we can view

${\mathcal{B}}^{k}:\Lambda^{k}({\mathcal{T}})\to\prod_{f\in\Delta({\mathcal{T}})}\mathaccent
23{\Lambda}^{k}({\mathcal{T}}_{f}):=\mathaccent
23{\Lambda}^{k}({\mathcal{T}},\Delta),$

we can summarize and state that the diagram

$\begin{CD}\Lambda^{k}({\mathcal{T}})@>{d}>{}>\Lambda^{k+1}({\mathcal{T}})\\\
@V{}V{(W^{k},{\mathcal{B}}^{k})}V@V{}V{(W^{k+1},{\mathcal{B}}^{k+1})}V\\\
{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})\times\mathaccent
23{\Lambda}^{k}({\mathcal{T}},\Delta)@>{d}>{}>{\mathcal{P}}_{1}^{-}\Lambda^{k+1}({\mathcal{T}})\times\mathaccent
23{\Lambda}^{k+1}({\mathcal{T}},\Delta)\end{CD}$

commutes. We will also show that the operators $W^{k}$ and $B_{f}^{k}$ can be
extended to bounded operators in $L^{2}$. Furthermore, the polynomial
preserving property of the bubble transform can simply be expressed by the
fact that

(1.2)
$B_{f}^{k}({\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}}))\subset\mathaccent
23{\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}}_{f})\quad\text{and
}B_{f}^{k}({\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}}))\subset\mathaccent
23{\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}}_{f})$

for all $f\in\Delta({\mathcal{T}})$ and $r\geq 1$, where $\mathaccent
23{\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}}_{f})={\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}})\cap\mathaccent
23{\Lambda}^{k}({\mathcal{T}}_{f})$ and with corresponding definition for the
${\mathcal{P}}_{r}^{-}$-spaces.

The individual bubbles, $B_{f}^{k}u$, introduced above, will not correspond to
the bubbles constructed in [6]. However, a key part of the analysis given in
[6] is the study of a family of trace preserving operators,
$C_{m}^{k}:\Lambda^{k}({\mathcal{T}})\to\Lambda^{k}({\mathcal{T}})$, where
$0\leq m\leq n-1$. These operators are explicitly given in formula (2.19)
below. A key property of these operators is that if
$f\in\Delta_{m}({\mathcal{T}})$, where $m\geq k$, then

(1.3) $\operatorname{tr}_{f}C_{m}^{k}u=\operatorname{tr}_{f}u,$

cf. [6, Lemma 2.2]. Here $\operatorname{tr}_{f}$ is the trace operator.
Furthermore, these operators commute with the exterior derivative and they
preserve piecewise smoothness and the standard spaces of piecewise
polynomials, cf. [6, Proposition 7.1]. It was also shown how the global
functions $C_{m}^{k}u$ can be decomposed into a sum of local bubbles, but
these bubbles were rational functions leading to the apparent defect of the
theory of [6]. However, in the analysis below, where we will overcome the
problems just mentioned, the operators $C_{m}^{k}$ will still play an
essential part. In fact, the new decomposition of
$u\in\Lambda^{k}({\mathcal{T}})$ will be initialized by expressing $u$ as

$u=(u-C_{n-1}^{k}u)+C_{n-1}^{k}u.$

It follows from (1.3) above that $u-C_{n-1}^{k}u$ is a global piecewise smooth
function which has zero trace on elements of $\Delta_{n-1}({\mathcal{T}})$. As
a consequence, we can decompose this function as
$u-C_{n-1}^{k}u=\sum_{f\in\Delta_{n}({\mathcal{T}})}B_{f}^{k}u$, where each
local bubble, $B_{f}^{k}u:=\operatorname{tr}_{f}(u-C_{n-1}^{k}u)$, is
piecewise smooth, and with support in $f=\Omega_{f}$, for
$f\in\Delta_{n}({\mathcal{T}})$. Furthermore, the operators
$\\{B_{f}^{k}\\}_{f\in\Delta_{n}({\mathcal{T}})}$ will commute with the
exterior derivative and preserve the piecewise polynomial spaces. We can
conclude that we have decomposed $u$ into

(1.4) $u=\sum_{f\in\Delta_{n}({\mathcal{T}})}B_{f}^{k}u+C_{n-1}^{k}u,\quad
B_{f}^{k}u=\operatorname{tr}_{f}(u-C_{n-1}^{k}u),$

where the first part is a sum of local bubbles. The task for the rest of the
construction will be to show that the second part, the function
$C_{n-1}^{k}u$, also admits such a decomposition.

The present paper is organized as follows. In the next section, we list some
assumptions we will make, recall some standard notation and properties of
differential forms and simplicial complexes, and define some of the basic
operators that we will use in the construction of our decomposition. We end
Section 2 with an outline of the construction we will present in the remainder
of the paper, and state the main results, cf. Theorem 2.3. To motivate the
general theory, we discuss the decomposition in the case of scalar-valued
functions in Section 3. In Section 4, we define various local functions
depending on the given mesh ${\mathcal{T}}$. The delicate recursive
construction of these mesh functions, which represents a completely new
approach as compared to the construction performed in [6], can be seen as the
main new tool utilized to obtain the improved results of this paper. In
particular, the new mesh functions are used to define families of order
reduction operators in Section 5, and these operators are then used to define
the local bubbles $B_{f}^{k}$, cf. Section 6. In Section 7, we focus on the
operator $C_{m}^{k}-C_{m-1}^{k}$, and show that this operator admits a
decomposition into local bubbles with desired properties. By utilizing a
telescoping series argument, we will then obtain a similar decomposition for
the operator $C_{n-1}^{k}$. In the final section, we show that all the
operators of the decomposition (1.1) are bounded in $L^{2}$.

##  2\. Preliminaries

In this section, we introduce the basic assumptions, notation, and concepts
that will be used in the construction below, and give an outline of the
complete theory.

###  2.1. Assumptions

We assume that $\Omega\subset\mathbb{R}^{n}$ is a bounded polyhedral Lipschitz
domain which is partitioned into a finite set of $n$ simplices,
$\Delta_{n}({\mathcal{T}})$. The simplicial mesh ${\mathcal{T}}$ is assumed to
be a simplicial decomposition of $\Omega$, i.e., the union of the simplices in
$\Delta_{n}({\mathcal{T}})$ is the closure of $\Omega$ and the intersection of
any two is either empty or a common subsimplex of each. The set of all such
simplices of dimension $m$ is denoted $\Delta_{m}({\mathcal{T}})$, while
$\Delta({\mathcal{T}})=\bigcup_{0\leq m\leq n}\Delta_{m}({\mathcal{T}})$.
Furthermore, below we will frequently write $\Delta$ instead of
$\Delta({\mathcal{T}})$ and $\Delta_{m}$ instead of
$\Delta_{m}({\mathcal{T}})$.

###  2.2. Notation

The space of smooth differential $k$-forms on $\Omega$ will be denoted
$\Lambda^{k}(\Omega)$. More precisely, for each $x\in\Omega$,
$u_{x}\in\operatorname{Alt}^{k}$, where $\operatorname{Alt}^{k}$ is the space
of alternating $k$ -linear maps
$\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}\to\mathbb{R}$. We recall that
a projection, $\operatorname{skw}$, mapping $k$-linear forms to
$\operatorname{Alt}^{k}$, is given by

$(\operatorname{skw}u)(v_{1},\ldots,v_{k})=\frac{1}{k!}\sum_{\sigma}\operatorname{sign}(\sigma)u(v_{\sigma(1)},\ldots,v_{\sigma(k)}),$

where the sum is over all permutations of $\\{1,\ldots,k\\}$. In our
discussion below, we will encounter both the tensor product, $\otimes$, and
the wedge product, $\wedge$, of differential forms. If
$u_{1}\in\Lambda^{j}(\Omega)$ and $u_{2}\in\Lambda^{k}(\Omega)$, then
$u_{1}\wedge u_{2}\in\Lambda^{j+k}(\Omega)$, and the identity

(2.1) $u_{1}\wedge u_{2}=\binom{k+j}{k}\operatorname{skw}(u_{1}\otimes u_{2})$

holds. The exterior derivative will be denoted
$d=d_{k}:\Lambda^{k}(\Omega)\to\Lambda^{k+1}(\Omega)$. Furthermore, if
$F:{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a smooth map between manifolds,
then the corresponding pullback $F^{*}$ is a map from
$\Lambda^{k}({\mathcal{M}}^{\prime})\to\Lambda^{k}({\mathcal{M}})$. In
particular, when ${\mathcal{M}}$ is a submanifold of ${\mathcal{M}}^{\prime}$,
then the pullback by the inclusion map,
$\Lambda^{k}({\mathcal{M}}^{\prime})\to\Lambda^{k}({\mathcal{M}})$, is the
trace map $\operatorname{tr}_{{\mathcal{M}}}$. The notation
$H\Lambda^{k}(\Omega)$ refers to the Sobolev space

$H\Lambda^{k}(\Omega)=\\{u\in L^{2}\Lambda^{k}(\Omega):du\in
L^{2}\Lambda^{k+1}(\Omega)\\},$

where $u\in L^{2}\Lambda^{k}(\Omega)$, if for all vectors
$v_{1},\ldots,v_{k}\in\mathbb{R}^{n}$, the function
$u_{x}(v_{1},\ldots,v_{k})\in L^{2}(\Omega)$ as a function of $x$. The
corresponding space of piecewise smooth $k$-forms with single valued traces
with respect to ${\mathcal{T}}$, $\Lambda^{k}({\mathcal{T}})$, will then be a
subspace of $H\Lambda^{k}(\Omega)$. Furthermore, the set of piecewise
polynomial $k$-forms of order $r$,
${\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}})$, and the corresponding space of
trimmed piecewise polynomials,
${\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}})$, will satisfy

${\mathcal{P}}_{r-1}\Lambda^{k}({\mathcal{T}})\subset{\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}})\subset{\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}})\subset\Lambda^{k}({\mathcal{T}})\subset
H\Lambda^{k}(\Omega).$

If $f\in\Delta_{m}({\mathcal{T}})$, then $f$ corresponds to an ordered subset
of the vertices, $\Delta_{0}({\mathcal{T}})$. We assume that all the vertices
are numbered by a set of integers
${\mathcal{I}}=\\{0,1,\ldots,N({\mathcal{T}})\\}$ such that

$\Delta_{0}({\mathcal{T}})=\\{x_{i}\,:\,i\in{\mathcal{I}}\,\\}.$

Any $f\in\Delta_{m}({\mathcal{T}})$ is of the form
$f=[x_{j_{0}},x_{j_{1}},\ldots x_{j_{m}}]$, where $j_{0}<j_{1}<\ldots<j_{m}$
and $I(f):=\\{j_{0},j_{1},\ldots,j_{m}\\}\subset{\mathcal{I}}$. Here we have
used the notation $[\cdot,\cdots,\cdot]$ to denote convex combinations.
Furthermore, the statement $g\in\Delta(f)$ means that $g$ is a subsimplex of
$f$, and with increasingly ordered vertices. We define the map
$\sigma_{f}:\Delta_{0}(f)\to\\{0,\ldots,m\\}$ by

$\sigma_{f}(x_{j_{i}})=i.$

In other words, $\sigma_{f}(y)$ gives the internal numbering of a vertex $y$
relative to the simplex $f$. The number of vertices in $f$ is denoted $|f|$,
i.e., $|f|=m+1$ if $f\in\Delta_{m}({\mathcal{T}})$. If
$e,f\in\Delta({\mathcal{T}})$, with a disjoint set of vertices, and such that
the union of the vertices defines a simplex, then although $e$ and $f$ are
increasingly ordered, the simplex $[e,f]$ will not necessarily be increasingly
ordered. We then denote by $\langle e,f\rangle$ the increasingly ordered
simplex composed of the vertices of $e$ and $f$. The set $\bar{\Delta}(f)$
contains the emptyset, $\emptyset$, in addition to the elements of
$\Delta(f)$, and $\emptyset$ is the single element of
$\Delta_{-1}({\mathcal{T}})$. Since the ordering of a simplex
$f=[x_{j_{0}},x_{j_{1}},\ldots x_{j_{m}}]\in\Delta_{m}({\mathcal{T}})$ is
inherited from the global numbering of the vertices, the various simplices are
not necessarily equally oriented. If $m=n$, we define the orientation of $f$,
$o(f)$, by

$o(f)=\operatorname{sign}\det\Big{(}x_{j_{1}}-x_{j_{0}},\ldots,x_{j_{m}}-x_{j_{0}}\Big{)},$

i.e., it is the sign of the determinant of a nonsingular $n\times n$ matrix.

In the analysis that follows, we will make use of some concepts from
simplicial complexes, e.g., see [8]. The $k$-chains defined by the mesh
${\mathcal{T}}$ is a vector space consisting of linear combinations of the
form $\sum_{f\in\Delta_{k}}c_{f}f$. We will let ${\mathscr{C}}_{k}$ denote the
corresponding space of vector representations, i.e., if
$c\in{\mathscr{C}}_{k}$ then $c=\\{c_{f}\\}_{f\in\Delta_{k}}$,
$c_{f}\in\mathbb{R}$. In fact, in the development of the theory below, we will
consider $k$-chains with values in a finite dimensional vector spaces $X$, by
which we mean spaces of the form ${\mathscr{C}}_{k}\otimes X$. In particular,
$X$ will be a subspace of ${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$.
The boundary operator is a chain map,

$f\mapsto\sum_{i\in I(f)}(-1)^{\sigma_{f}(x_{i})}f(\hat{x}_{i}),$

where the hat is used to indicate a suppressed argument. Using vector
representations, the corresponding operator
$\partial_{k}:{\mathscr{C}}_{k}\otimes X\to{\mathscr{C}}_{k-1}\otimes X$ can
be expressed as

$(\partial_{k}c)_{f}=\sum_{i\in{\mathcal{I}}}c_{[x_{i},f]}\equiv\sum_{i\in{\mathcal{I}}}(-1)^{\sigma_{\langle
x_{i},f\rangle}(x_{i})}c_{\langle x_{i},f\rangle},\quad
f\in\Delta_{k-1}({\mathcal{T}}),\,1\leq k\leq n,$

where $c_{\langle x_{i},e\rangle}=0$ if $\langle x_{i},e\rangle$ is not an
element of $\Delta_{k}({\mathcal{T}})$. We can also identify
${\mathscr{C}}_{-1}\otimes X$ as $X$, corresponding to the single element
$\emptyset$ of $\Delta_{-1}({\mathcal{T}})$, and
$\partial_{0}:{\mathscr{C}}_{0}\otimes X\to X$ by
$\partial_{0}c=\sum_{i\in{\mathcal{I}}}c_{x_{i}}$.

The corresponding coboundary operators are cochain maps
$\delta_{k}:{\mathscr{C}}_{k}\otimes X\to{\mathscr{C}}_{k+1}\otimes X$ given
by

$(\delta_{k}c)_{f}=\sum_{i\in
I(f)}(-1)^{\sigma_{f}(x_{i})}c_{f(\hat{x}_{i})},\quad
f\in\Delta_{k+1}({\mathcal{T}}),$

for $-1\leq k\leq n-1$. If $X$ and $Y$ are finite dimensional vector spaces,
then these definitions lead to the identity

(2.2)
$\sum_{f\in\Delta_{k-1}({\mathcal{T}})}(\partial_{k}c)_{f}\otimes\tilde{c}_{f}=\sum_{f\in\Delta_{k}({\mathcal{T}})}c_{f}\otimes(\delta_{k-1}\tilde{c})_{f},\quad
c\in{\mathscr{C}}_{k}\otimes X,\,\tilde{c}\in{\mathscr{C}}_{k-1}\otimes Y,$

i.e., $\partial$ is the adjoint of $\delta$ with respect to the inner product
on ${\mathscr{C}}_{k}$. Furthermore, the operators $\partial$ and $\delta$
will satisfy the complex properties $\partial^{2}=\delta^{2}=0$.

Associated to any vertex $x_{i}\in\Delta_{0}({\mathcal{T}})$, we denote by
$\lambda_{i}(x)$ the nonnegative piecewise linear function equal to one at
$x_{i}$ and zero at all other vertices. More generally, if $f$ is a ordered
subset of $\Delta_{0}({\mathcal{T}})$, but not necessarily increasingly
ordered, then $\phi_{f}$ will denote the Whitney form associated to $f$. More
precisely, if $f=[x_{j_{0}},x_{j_{1}},\ldots,x_{j_{m}}]$, then $\phi_{f}$ is
given by

$\phi_{f}=m!\sum_{i=0}^{m}(-1)^{i}\lambda_{j_{i}}d\lambda_{j_{0}}\wedge\ldots\wedge\widehat{d\lambda_{j_{i}}}\wedge\ldots\wedge
d\lambda_{j_{m}}.$

In particular, for $f=[x_{i}]$, $\phi_{f}=\lambda_{i}$. If $m>0$, then
$\int_{g}\phi_{f}$ is plus or minus one if $g=f$, and zero if
$g\in\Delta_{m}({\mathcal{T}})$, $g\neq f$. The forms
$\\{\phi_{f}\\}_{f\in\Delta_{m}({\mathcal{T}})}$ span the space
${\mathcal{P}}_{1}^{-}\Lambda^{m}({\mathcal{T}})$ and they are local with
support in $\Omega_{f}$. It can also be easily checked that

(2.3) $d\phi_{f}=(m+1)!d\lambda_{j_{0}}\wedge\ldots\wedge
d\lambda_{j_{m}}=\sum_{j\in{\mathcal{I}}}\phi_{[x_{j},f]}=(\partial_{m+1}\phi)_{f},$

where $\phi_{[x_{j},f]}=0$ if $[x_{j},f]$ does not correspond to a subsimplex
of ${\mathcal{T}}$. Here the operator $\partial_{m+1}$ has the interpretation
given above, where $\\{\phi_{f}\\}$ for $f\in\Delta_{m+1}({\mathcal{T}})$
should be seen as an element of
${\mathscr{C}}_{m+1}\otimes{\mathcal{P}}_{1}\Lambda^{m+1}({\mathcal{T}})$.
Furthermore, if $u\in{\mathcal{P}}_{1}^{-}\Lambda^{m}({\mathcal{T}})$ is
expanded, such that $u=\sum_{f\in\Delta_{m}({\mathcal{T}})}c_{f}\phi_{f}$,
then

(2.4)
$du=\sum_{f\in\Delta_{m}({\mathcal{T}})}c_{f}(\partial_{m+1}\phi)_{f}=\sum_{f\in\Delta_{m+1}({\mathcal{T}})}(\delta_{m}c)_{f}\phi_{f}.$

It is also a consequence of the definition of $\phi_{f}$ and (2.3) that

(2.5)
$\phi_{[x_{i},f]}=\Big{(}\lambda_{i}d-md\lambda_{i}\wedge\Big{)}\phi_{f},\quad
f\in\Delta_{m-1}({\mathcal{T}}).$

If $0\leq m\leq n$, then associated to the simplex
$f=[x_{j_{0}},x_{j_{1}},\ldots,x_{j_{m}}]$, we define the standard simplex
${\mathcal{S}}_{f}$ by

${\mathcal{S}}_{f}=\big{\\{}\,\lambda=(\lambda_{j_{0}},\lambda_{j_{1}},\ldots,\lambda_{j_{m}})\in\mathbb{R}^{m+1}\,:\,\sum_{i=0}^{m}\lambda_{j_{i}}=1,\quad\lambda_{j_{i}}\geq
0\,\big{\\}},$

while ${\mathcal{S}}_{f}^{c}$ is the set of all convex combinations between
${\mathcal{S}}_{f}$ and the origin, i.e.,
${\mathcal{S}}_{f}^{c}=[0,{\mathcal{S}}_{f}]$. Alternatively,

${\mathcal{S}}_{f}^{c}=\big{\\{}\,\lambda=(\lambda_{j_{0}},\lambda_{j_{1}},\ldots,\lambda_{j_{m}})\in\mathbb{R}^{m+1}\,:\,\sum_{i=0}^{m}\lambda_{j_{i}}\leq
1,\quad\lambda_{j_{i}}\geq 0\,\big{\\}}.$

If $f=[x_{j_{0}},x_{j_{1}},\ldots x_{j_{m}}]\in\Delta_{m}({\mathcal{T}})$,
then $L_{f}:\Omega\to{\mathcal{S}}_{f}^{c}$ will be the map

$L_{f}(x)=\big{(}\lambda_{j_{0}}(x),\lambda_{j_{1}}(x),\ldots,\lambda_{j_{m}}(x)\big{)}.$

In the special case $f=\emptyset$, we let
${\mathcal{S}}_{\emptyset}={\mathcal{S}}_{\emptyset}^{c}=\\{0\\}$ and
$L_{f}(\Omega)=0$. In the construction below, we will frequently use the
pullback $L_{f}^{*}$ mapping $\Lambda^{k}({\mathcal{S}}_{f}^{c})$ to
$\Lambda^{k}({\mathcal{T}})$, i.e., $L_{f}^{*}$ maps smooth forms on
${\mathcal{S}}_{f}^{c}$ to piecewise smooth forms on $\Omega$, and also
polynomial forms to piecewise polynomial forms. Another key tool we will
utilize below is the piecewise linear function $\rho_{f}$, defined by

$\rho_{f}(x)=1-\sum_{i\in I(f)}\lambda_{i}(x),$

which can be seen as a distance function between
$f\in\bar{\Delta}({\mathcal{T}})$ and $x\in\Omega$. Note that
$0\leq\rho_{f}(x)\leq 1$ and $\rho_{f}\equiv 1$ if $f=\emptyset$.
Alternatively, we have $\rho_{f}=L_{f}^{*}b$, where
$b=b_{f}:{\mathcal{S}}_{f}^{c}\to\mathbb{R}$ is the distance to the origin,
i.e., $b(\lambda)=1-\sum_{i=0}^{m}\lambda_{j_{i}}$.

###  2.3. The macroelements

We recall that for any $f\in\Delta({\mathcal{T}})$, we denote by $\Omega_{f}$
the union of the simplices in $\Delta_{n}({\mathcal{T}})$ containing $f$ as a
subsimplex, while ${\mathcal{T}}_{f}$ is the restriction of the mesh
${\mathcal{T}}$ to $\Omega_{f}$. We refer to $\Omega_{f}$ as the macroelement
of $f$, or alternatively, as the star of $f$. The domain $\Omega_{f}$ is
contractible with respect to any $x\in f$. If
$f\in\Delta_{m-1}({\mathcal{T}})$ and $T\in\Delta_{n}({\mathcal{T}}_{f})$, we
let $f^{*}(T)\in\Delta_{n-m}(T)$ be the subsimplex of $T$ opposite $f$. The
link of $f$, denoted by $f^{*}$, is given as
$f^{*}=\bigcup_{T\in\Delta_{n}({\mathcal{T}}_{f})}f^{*}(T)$ (e.g., see [4]).
More precisely,

$f^{*}=\\{x\in\Omega_{f}:\lambda_{i}(x)=0,i\in I(f)\\}.$

The link $f^{*}$ can be viewed as an $n-m$ dimensional oriented manifold
composed of the simplices $f^{*}(T)$, where $f^{*}(T)$ has an orientation,
$o(f^{*}(T),T)$, induced by the $n$-simplex $T$. More precisely, if
$f=[x_{j_{0}},\ldots,x_{j_{m-1}}]$, then

$o(f^{*}(T),T)=o(T)\prod_{i=0}^{m-1}(-1)^{\sigma_{T_{i}}(x_{j_{i}})},$

where $T_{i}=T(\hat{x}_{j_{0}},\ldots,\hat{x}_{j_{i-1}})\in\Delta_{n-i}(T)$.
Any $x\in\Omega_{f}$ can be written uniquely as a convex combination of the
points $x_{i},\,i\in I(f)$, and points in $f^{*}$, i.e.,
$\Omega_{f}=[f,f^{*}]$. In the special case when $m=n-1$, the manifold $f^{*}$
will be reduced to two vertices, or only one close to the boundary, while in
the special case $f=\emptyset$, we have $\Omega_{f}=f^{*}=\Omega$. Below we
will also encounter the extended macroelement, $\Omega_{f}^{E}$, defined by
$\Omega_{f}^{E}=\cup_{i\in I(f)}\Omega_{x_{i}}$.

$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{0}$$x_{1}$ Figure 2.1. The macroelement
$\Omega_{f}\subset\mathbb{R}^{3}$, where $f=[x_{0},x_{1}]$ and $f^{*}$ is the
closed curve connecting the vertices $x_{2},x_{3},x_{4}$, and $x_{5}$.

We define ${\mathscr{C}}_{k}(f^{*})$ as the space of vector representations of
$k$-chains defined on the manifold $f^{*}$, i.e., if
$c\in{\mathscr{C}}_{k}(f^{*})$ then $c=\\{c_{f}\\}_{f\in\Delta_{k}(f^{*})}$.
The corresponding boundary and coboundary operators, $\partial_{k}(f^{*})$ and
$\delta_{k}(f^{*})$, are defined as the operators $\partial_{k}$ and
$\delta_{k}$ above, but by restricting to the manifold $f^{*}$. In the
construction performed later in this paper, we will utilize the fact that for
any $f\in\Delta_{m-1}({\mathcal{T}})$, $1\leq m\leq n-1$, the cochain complex

(2.6)
$\setcounter{MaxMatrixCols}{11}\begin{CD}\mathbb{R}@>{}>{}>{\mathscr{C}}_{0}(f^{*})@>{\delta}>{}>{\mathscr{C}}_{1}(f^{*})@>{\delta}>{}>\cdots
@>{\delta}>{}>{\mathscr{C}}_{n-m}(f^{*})@>{\delta}>{}>\mathbb{R}\end{CD}$

is exact, where $\delta=\delta(f^{*})$, and where the special operator
$\delta_{n-m}(f^{*}):\ C_{n-m}(f^{*})\to\mathbb{R}$ will be defined below.
This exactness is a consequence of the corresponding property for the trimmed
linear forms restricted to $f^{*}$. Since $f^{*}$ is a piecewise flat
submanifold of the boundary of $\Omega_{f}$, defined by the mesh
${\mathcal{T}}$, we can consider the trace spaces
${\mathcal{P}}_{1}^{-}\Lambda^{k}(f^{*})$, defined as
$\operatorname{tr}_{f^{*}}{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$. If
$f$ is an interior simplex, such that $f^{*}$ is a manifold without a
boundary, the complex

(2.7)
$\begin{CD}\mathbb{R}@>{}>{}>{\mathcal{P}}_{1}^{-}\Lambda^{0}(f^{*})@>{d}>{}>{\mathcal{P}}_{1}^{-}\Lambda^{1}(f^{*})@>{d}>{}>\cdots{\mathcal{P}}_{1}^{-}\Lambda^{n-m}(f^{*})@>{}>{}>\mathbb{R}\end{CD}$

is exact, where the final arrow represents the integral over $f^{*}$. If $f$
is a boundary simplex, the manifold $f^{*}$ may have a boundary, and in this
case the last arrow in (2.7) is redundant. By expanding the elements of
${\mathcal{P}}_{1}^{-}\Lambda^{k}(f^{*})$ in the basis functions, and using
the identity (2.4), we obtain the equivalent complex (2.6). More specifically,
the map
$l_{k}:{\mathcal{P}}_{1}^{-}\Lambda^{k}(f^{*})\to{\mathscr{C}}_{k}(f^{*})$
given by $\sum_{e\in\Delta_{k}(f^{*})}c_{e}\phi_{e}\mapsto c_{e}$ gives the
commuting relation $\delta_{k}(f^{*})\circ l_{k}=l_{k+1}\circ d$. As a
consequence, the exactness of the complex (2.6) follows from the exactness of
(2.7). The special operator $\delta_{n-m}(f^{*})$ is defined by

(2.8)
$\delta_{n-m}(f^{*})c=\sum_{e\in\Delta_{n-m}(f^{*})}o(e,T_{e})c_{e},\quad
T_{e}=\langle e,f\rangle.$

Note that this operator is always well-defined, but that the last arrow in
(2.6) may be redundant if $f$ intersects the boundary of $\Omega$. Since
$\partial$ is the adjoint of $\delta$ with respect to the inner product of
${\mathscr{C}}(f^{*})$, the following result is a consequence of the exactness
of the complex (2.6).

######  Lemma 2.1.

Let $f\in\Delta_{m-1}({\mathcal{T}})$ and $0\leq k<n-m$. If
$c\in{\mathscr{C}}_{k}(f^{*})$ satisfies $\partial_{k}(f^{*})c=0$, then there
is a $\tilde{c}\in{\mathscr{C}}_{k+1}(f^{*})$ such that
$c=\partial_{k+1}(f^{*})\tilde{c}$. Furthermore, $\tilde{c}$ is uniquely
determined if we require $\delta_{k+1}(f^{*})\tilde{c}=0$.

Below we will use Lemma 2.1 in a generalized sense, where we consider
${\mathscr{C}}_{k}$ with values in a finite dimensional vector space $X$, cf.
Sections 3 and 4.1.

###  2.4. The average operators and their generalizations

A key tool for our construction below is a family of average operators,
$A_{f}^{k}$, where $f\in\Delta$, which map piecewise smooth $k$-forms on
$\Omega_{f}$ to smooth $k$-forms on ${\mathcal{S}}_{f}^{c}$. The operators
$A_{f}^{k}$ will be defined by a function
$G_{f}:\Omega_{f}\times{\mathcal{S}}_{f}^{c}\to\Omega_{f}$, given by

$G_{f}(y,\lambda)=\sum_{i\in{\mathcal{I}}(f)}\lambda_{i}x_{i}+b(\lambda)y,$

Note that if $x\in f$ then, since $b(L_{f}x)=0$, we have

(2.9) $G_{f}(y,L_{f}x)=x,\quad x\in f.$

For each fixed $y\in\Omega_{f}$, $G_{f}(y,\lambda)$ is linear with respect to
$\lambda$. The corresponding derivative with respect $\lambda$,
$DG_{f}(y,\cdot)$, is therefore an operator mapping tangent vectors of
${\mathcal{S}}_{f}^{c}$, $TS_{f}^{c}$, into $T\Omega_{f}$ which is independent
of $\lambda$. It is given by

$DG_{f}(y,\cdot)=\sum_{i\in I(f)}(x_{i}-y)d\lambda_{i}.$

Since for each $y$, the map $G_{f}(y,\cdot)$ maps ${\mathcal{S}}_{f}^{c}$ to
$\Omega_{f}$, the corresponding pullback, $G_{f}(y,\cdot)^{*}$, maps
$\Lambda^{k}(\Omega_{f})$ to $\Lambda^{k}({\mathcal{S}}_{f}^{c})$. As a
further consequence, an average of these maps over $\Omega_{f}$ with respect
to $y$ will also map $\Lambda^{k}(\Omega_{f})$ to
$\Lambda^{k}({\mathcal{S}}_{f}^{c})$. In order to define the averages we want,
we will introduce a family of piecewise constant $n$-forms,
$z_{f}\in{\mathcal{P}}_{1}^{-}\Lambda^{n}({\mathcal{T}}_{f})$, with the
property that $z_{f}$ has support in $\Omega_{f}$ and

(2.10) $\int_{\Omega_{f}}z_{f}=\int_{\Omega}z_{f}=1.$

The operator $A_{f}^{k}$ is then defined for $f\in\Delta_{m}$ by

$A_{f}^{k}u=\int_{\Omega}G_{f}(y,\cdot)^{*}u\wedge z_{f}.$

Since pullbacks commute with the exterior derivative, so do the operators
$A_{f}^{k}$, i.e., $dA_{f}^{k}=A_{f}^{k+1}d$, and from (2.9), we obtain that
for $f\in\Delta_{m}$,

$\operatorname{tr}_{f}L_{f}^{*}A_{f}^{k}=\operatorname{tr}_{f},\quad m\geq k.$

Of course, the properties of the operators $A_{f}^{k}$ will also depend on the
choice of the functions $z_{f}$. For $f\in\Delta_{n}$, these functions are
defined to be

(2.11) $z_{f}=\frac{\kappa_{f}}{|\Omega_{f}|}\mathsf{vol},$

where $\kappa_{f}$ is the characteristic function of $\Omega_{f}$, while for
$f\in\Delta_{m},\,0\leq m<n$, the functions $z_{f}$ are defined recursively by
the relation

(2.12) $z_{f}=\frac{1}{|f^{*}|}\sum_{i\in I(f^{*})}z_{\langle
x_{i},f\rangle}.$

We note that it follows by construction that all the functions $z_{f}$ have
support in $\Omega_{f}$ and satisfy relation (2.10). Furthermore, it is a
consequence of Lemma 2.1 of [6] that the operators $A_{f}^{k}$ map
$\Lambda^{k}({\mathcal{T}}_{f})$ to $\Lambda^{k}(S_{f}^{c})$ and also map
piecewise polynomial forms to polynomial forms. More precisely, we have for
$f\in\Delta$,

(2.13)
$A_{f}^{k}({\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}}_{f}))\subset{\mathcal{P}}_{r}\Lambda^{k}({\mathcal{S}}_{f}^{c}),\qquad
A_{f}^{k}({\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}}_{f}))\subset{\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{S}}_{f}^{c}).$

######  Remarks.

In the argument given in [6], it was assumed that the functions $z_{f}$ were
given by (2.11) for all $f$. However, the modification we need to cover the
more general average functions $z_{f}$ introduced above is straightforward.

Consider the case of scalar valued functions, i.e., the case $k=0$. Since all
the functions $z_{f}$ satisfy the identity (2.10), it follows that for
$f\in\Delta$, and $e\in\Delta_{1}(f)$,

$(\delta A^{0}u)_{e,f}:=\sum_{i\in
I(e)}(-1)^{\sigma_{e}(x_{i})}A_{f(\hat{x}_{i})}^{0}u$

will be zero when $u$ is a constant. Therefore, this expression only depends
on $du$. Below we will construct a corresponding operator $R^{1}$, mapping
one-forms to zero-forms, such that

(2.14) $R^{1}du=(\delta A^{0}u)_{e,f},\quad e\in\Delta_{1}(f).$

A natural choice seems to be to label this operator by the simplex pair
$(e,f)$. In fact, this was the choice used in [6]. However, for the theory
developed in this paper, it appears more appropriate to use the equivalent
label, $(e,f\cap e^{*})$, where $f\cap e^{*}$ belongs to $e^{*}$ and vice
versa.

More generally, we introduce the sets
$\Delta_{j,m}=\Delta_{j,m}({\mathcal{T}})$ of pairs of simplices, given by

$\Delta_{j,m}:=\\{(e,f)\,:\,f\in\Delta_{m}({\mathcal{T}}),\,e\in\Delta_{j}(f^{*})\,\\}$

for $-1\leq m\leq n$ and $0\leq j<n-m$. For $(e,f)\in\Delta_{j,m}$, $j\geq 0$,
we will define operators $R_{e,f}^{k}$ mapping $k$-forms to ($k-j$)-forms. We
will refer to these operators as order reduction operators. To define them, we
recall that the map $G_{f}$ maps $\Omega_{f}\times{\mathcal{S}}_{f}^{c}$ to
$\Omega_{f}$, and as a consequence, the corresponding pullback, $G_{f}^{*}$,
is a map

$G_{f}^{*}:\Lambda^{k}(\Omega_{f})\to\Lambda^{k}(\Omega_{f}\times{\mathcal{S}}_{f}^{c})$

and we can express

$\Lambda^{k}(\Omega_{f}\times{\mathcal{S}}_{f}^{c})=\sum_{j=0}^{k}\Lambda^{j}(\Omega_{f})\otimes\Lambda^{k-j}({\mathcal{S}}_{f}^{c}).$

Furthermore, for each $0\leq j\leq k$, there is a canonical map
$\Pi_{j}:\Lambda^{k}(\Omega_{f}\times{\mathcal{S}}_{f}^{c})\to\Lambda^{j}(\Omega_{f})\otimes\Lambda^{k-j}({\mathcal{S}}_{f}^{c})$
such that $\sum_{j=0}^{k}\Pi_{j}$ is the identity. We refer to [6, Section
5.1] for more details. As in [6], all the operators $R_{e,f}^{k}$ will be of
the form

(2.15)
$(R_{e,f}^{k}u)_{\lambda}=\int_{\Omega}(\Pi_{j}G_{f}^{*}u)_{\lambda}\wedge
z_{e,f},\quad\lambda\in{\mathcal{S}}_{f}^{c},$

where the functions $z_{e,f}$ are trimmed linear $n-j$ forms for
$(e,f)\in\Delta_{j,m}$, and with support in $\Omega_{f}\cap\Omega_{e}^{E}$.
The construction of the functions $\\{z_{e,f}\\}$, given in Section 4 below,
will deviate from the corresponding construction given in [6]. In fact, the
careful construction of these functions below represents the main tool for
obtaining the improved results we derive in this paper, as compared to the
results presented in [6].

For pairs $(e,f)\in\Delta_{0,m}$, i.e., when $e$ is a vertex, we define
$z_{e,f}=-z_{\langle e,f\rangle}$, where the $n$-forms $z_{\langle
e,f\rangle}$ are defined by (2.11) and (2.12). As a consequence, for any
$e\in\Delta_{0}(f^{*})$, we have $R_{e,f}^{k}=-A_{\langle e,f\rangle}^{k}$.
The functions $\\{z_{e,f}\\}$ will be constructed to satisfy the relation

(2.16) $dz_{e,f}=(-1)^{j+1}(\delta z)_{e,f},\quad(e,f)\in\Delta_{j,m},$

for $j\geq 1$, where the generalized coboundary operator defined for pairs of
simplices in $\\{\Delta_{j,m}\\}$ is given by

$(\delta z)_{e,f}=\sum_{i\in
I(e)}(-1)^{\sigma_{e}(x_{i})}z_{e(\hat{x}_{i}),f}.$

######  Remarks.

The identity, $R_{e,f}^{k}=-A_{\langle e,f\rangle}^{k}$ and the definition of
the $\delta$ operator appear to deviate from the corresponding relations in
[6]. However, as stated above, in the present paper it is more convenient to
use a different, but equivalent, labeling of the $z$ functions and $R$
operators as compared to the previous paper. More precisely, the operators
$R_{e,f}^{k}$ introduced above were labeled by the pair $(e,\langle
e,f\rangle)$ in [6], and this causes minor differences in the relations above,
and also at a few occurrences below.

As a consequence of the definition of the operators $R_{e,f}^{k}$, given by
(2.15) and (2.16), the operators $R_{e,f}^{k}$ will satisfy the relation

(2.17) $R_{e,f}^{k+1}du=(-1)^{j}dR_{e,f}^{k}u-(\delta
R^{k}u)_{e,f},\quad(e,f)\in\Delta_{j,m},\,0\leq j\leq k+1,$

where $\Delta_{j,m}$ is defined to be the emptyset if $j\geq n-m$.
Furthermore, $(\delta R^{k}u)_{e,f}$ is taken to be zero for $e\in\Delta_{0}$.
When $e\in\Delta_{k+1}(f^{*})$, $R_{e,f}^{k}u=0$ and (2.17) reduces to

(2.18) $R_{e,f}^{k+1}du=-(\delta R^{k}u)_{e,f},\quad(e,f)\in\Delta_{k+1,m},$

which is consistent with (2.14). We refer to Section 5.3 of [6] and Section 5
below for more details.

The following lemma generalizes the mapping properties (2.13) of the average
operators $A_{f}^{k}$ to similar results for the order reduction operators. In
fact, this result corresponds to Proposition 5.2 of [6].

######  Lemma 2.2.

Assume that $(e,f)\in\Delta_{j,m}({\mathcal{T}})$.

  * i)

If $u\in\Lambda^{k}({\mathcal{T}}_{f})$, then
$b^{-j}R_{e,f}^{k}u\in\Lambda^{k-j}({\mathcal{S}}_{f}^{c})$.

  * ii)

If $u\in{\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}}_{f})$, then
$b^{-j}R_{e,f}^{k}u\in{\mathcal{P}}_{r}\Lambda^{k-j}({\mathcal{S}}_{f}^{c})$.

  * iii)

If $u\in{\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}}_{f})$, then
$b^{-j}R_{e,f}^{k}u\in{\mathcal{P}}_{r}^{-}\Lambda^{k-j}({\mathcal{S}}_{f}^{c})$.

######  Remarks.

The proof of Proposition 5.2 of [6] carries over directly to the present
setting, even if the definition of the weight functions $z_{e,f}$ will be
modified below. Therefore, we omit the proof here.

###  2.5. An outline of the construction

In the present notation, the trace preserving operators $C_{m}^{k}$,
introduced in [6, Section 6], admit the representation

(2.19)
$C_{m}^{k}u=\sum_{f\in\Delta_{m}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}L_{g}^{*}A_{f}^{k}u\\\
+\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\ 0\leq j\leq
n-m\end{subarray}}(-1)^{j-1}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\phi_{e}}{\rho_{g}}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u.$

where $0\leq m\leq n-1$, and where we recall that $R_{e,f}^{k}=0$ if
$e\in\Delta_{j}$, $j>k$. The definition of the operator $C_{m}^{k}$ contains
the rational terms $\phi_{e}/\rho_{g}$. However, it is established in [6], cf.
Lemma 2.2 and Proposition 7.1 of that paper, that the operator $C_{m}^{k}$
commutes with the exterior derivative, preserves piecewise smoothness and the
piecewise polynomial spaces ${\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}})$ and
${\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}})$, and preserves the trace of
$u$ on all $f\in\Delta_{m}$ if $m\geq k$.

######  Remarks.

The fact that the operator $C_{m}^{k}$ preserves the trace of $u$ on all
$f\in\Delta_{m}$ if $m\geq k$, can be derived easily from the definition
above. In fact, by combining the first term in (2.19) and the second term with
$j=0$, we obtain the primal operator studied in Section 4 of [6]. This
operator can be rewritten as

(2.20)
$C_{m}^{k}(primal)=\sum_{f\in\Delta_{m}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\rho_{f}}{\rho_{g}}L_{g}^{*}A_{f}^{k}u.$

By arguing as in the proof of [6, Lemma 4.1], using the cancellation property
with respect to $g$, we can conclude that this operator preserves the trace on
all elements of $\Delta_{m}$ if $m\geq k$. Furthermore, for the second term in
(2.19) with a fixed $f\in\Delta_{m-1}$ and $e\in\Delta_{j}(f^{*})$, we can
argue in the same way that the resulting function has support in
$\Omega_{e}\cap\Omega_{f}=\Omega_{\langle e,f\rangle}$. As a consequence, if
$j>0$, such that the simplex $\langle e,f\rangle\in\Delta_{s}$ for $s>m$,
these functions have vanishing trace on all $m$-simplices. Hence, the trace
property of $C_{m}^{k}$ follows. The commuting property of the operator
$C_{m}^{k}$ can also be shown directly from the properties of the operators
$A_{f}^{k}$ and $R_{e,f}^{k}$, cf. Section 2.4 above, while the space
preserving properties require a deeper analysis, performed in [6, Section 7].
In fact, an alternative proof of the space preserving properties follows as a
corollary of the analysis given in this paper.

We will derive the desired decomposition (1.1) of functions
$u\in\Lambda^{k}({\mathcal{T}})$ from the telescoping identity

(2.21) $u=C_{0}^{k}u+\sum_{m=1}^{n}(C_{m}^{k}-C_{m-1}^{k})u,$

where $C_{n}^{k}$ is the identity operator. We have already observed, cf.
(1.4), that

$(C_{n}^{k}-C_{n-1}^{k})u=u-C_{n-1}^{k}u=\sum_{f\in\Delta_{n}}B_{f}^{k}u,$

where $B_{f}^{k}u=\operatorname{tr}_{f}(u-C_{n-1}^{k}u)$ is a sum of piecewise
smooth local bubbles. For the special case $m=0$, there are no rational
functions present in the definition of the operator $C_{0}^{k}$. In fact, by
utilizing that the range of the operator $L_{\emptyset}$ is a single point,
i.e., the origin, it follows that $L_{\emptyset}^{*}$ maps forms of order
greater than zero, to zero. As a consequence, we can represent the operator
$C_{0}^{k}$ as

(2.22)
$C_{0}^{k}u=\sum_{f\in\Delta_{0}}\Big{(}L_{f}^{*}A_{f}^{k}u-L_{\emptyset}^{*}A_{f}^{k}u\Big{)}+(-1)^{k-1}\sum_{e\in\Delta_{k}}\phi_{e}\wedge
L_{\emptyset}^{*}R_{e,\emptyset}^{k}u.$

In particular, from the definition of the operator $R_{e,\emptyset}^{k}$, it
follows that the second term in (2.22) can be expressed as

$W^{k}u:=(-1)^{k-1}\sum_{e\in\Delta_{k}}\phi_{e}\Big{(}\int_{\Omega}u\wedge
z_{e,\emptyset}\Big{)},$

which is an element of ${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$. In
other words, $W^{k}$ is an operator which maps piecewise smooth $k$-forms into
the simplest class of piecewise polynomial $k$ forms, i.e., into trimmed
linear forms. In particular, we recall that for $e\in\Delta_{k}$, the
functions $z_{e,\emptyset}$ are elements of
${\mathcal{P}}_{1}^{-}\Lambda^{n-k}({\mathcal{T}})$, with support in
$\Omega_{e}^{E}$. We define the local operators $K_{0,f}^{k}$ by

$K_{0,f}^{k}u=L_{f}^{*}A_{f}^{k}u-L_{\emptyset}^{*}A_{f}^{k}u.$

The functions $K_{0,f}^{k}u$ will have support on $\Omega_{f}$, and by using
these operators, the identity (2.22) can be written as

$C_{0}^{k}u=\sum_{f\in\Delta_{0}}K_{0,f}^{k}u+W^{k}u.$

As a consequence of the fact that the operators $A_{f}^{k}$ commute with the
exterior derivative, it follows that the operators $K_{0,f}^{k}$ will also
commute with $d$. Furthermore, the operator $W^{k}$ also commutes with the
exterior derivative. In fact, from (2.3), we have

$dW^{k}u=(-1)^{k-1}\sum_{e\in\Delta_{k}}(\partial\phi)_{e}\Big{(}\int_{\Omega}u\wedge
z_{e,\emptyset}\Big{)},$

and from (2.2), (2.4), (2.16), and the Leibniz rule, we obtain

$\sum_{e\in\Delta_{k}}(\partial\phi)_{e}\Big{(}\int_{\Omega}u\wedge
z_{e,\emptyset}\Big{)}=\sum_{e\in\Delta_{k+1}}\phi_{e}\Big{(}\int_{\Omega}u\wedge(\delta
z)_{e,\emptyset}\Big{)}\\\
=(-1)^{k}\sum_{e\in\Delta_{k+1}}\phi_{e}\Big{(}\int_{\Omega}u\wedge
dz_{e,\emptyset}\Big{)}=-\sum_{e\in\Delta_{k+1}}\phi_{e}\Big{(}\int_{\Omega}du\wedge
z_{e,\emptyset}\Big{)},$

which implies that $dW^{k}=W^{k+1}d$.

We will also show below in Section 8 that $W^{k}$ can be extended to a bounded
linear operator on $L^{2}$.

Since we have obtained desired decompositions of the operators
$C_{n}^{k}-C_{n-1}^{k}$ and $C_{0}^{k}$, it remains to decompose the functions
$(C_{m}^{k}-C_{m-1}^{k})u$ for $1\leq m\leq n-1$, cf. (2.21). In fact, the
main contribution of this paper is to show that these operators admit the
representation

(2.23) $(C_{m}^{k}-C_{m-1}^{k})u=\sum_{\begin{subarray}{c}f\in\Delta_{j}\\\
j=m,m-1\end{subarray}}K_{m,f}^{k}u,\quad 1\leq m\leq n-1,$

cf. Proposition 7.3 below, where the functions $K_{m,f}^{k}u$ have support in
$\Omega_{f}$. Furthermore, each operator $K_{m,f}^{k}$ commutes with the
exterior derivative and preserves piecewise smoothness and the piecewise
polynomial spaces ${\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}})$ and
${\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}})$. Our derivation below of the
identity (2.23) depends on a careful construction of the family of operators
$\\{R_{e,f}^{k}\\}$, or more precisely of the functions $\\{z_{e,f}\\}$
defining these operators, cf. (2.15). As a consequence of the identity (2.23),
we obtain the desired decomposition (1.1), since the function $u-W^{k}u$ can
be decomposed into local bubbles of the form

$u-W^{k}u=\sum_{f\in\Delta}B_{f}^{k}u,$

where each function $B_{f}^{k}u$ has support on $\Omega_{f}$. More precisely,
if we let $K_{n,f}^{k}=0$ for each $f\in\Delta_{n-1}$, then

(2.24) $B_{f}^{k}=K_{m,f}^{k}+K_{m+1,f}^{k},\quad f\in\Delta_{m},\,0\leq m\leq
n-1,$

while for $f\in\Delta_{n}$,

(2.25)
$B_{f}^{k}u=\operatorname{tr}_{f}(u-C_{n-1}^{k}u)=\operatorname{tr}_{f}\Big{(}u-(W^{k}u+\sum_{\begin{subarray}{c}g\in\Delta_{m}\\\
0\leq m\leq n-1\end{subarray}}B_{g}^{k}u)\Big{)}.$

We can summarize the main results we will obtain for the construction outlined
above in the following theorem.

######  Theorem 2.3.

For $0\leq k\leq n$, there exist operators
$W^{k}:\Lambda^{k}({\mathcal{T}})\to{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$
and for each $f\in\Delta_{m}({\mathcal{T}})$, $0\leq m\leq n$, local operators
$B_{f}^{k}:\Lambda^{k}({\mathcal{T}})\to\mathaccent
23{\Lambda}^{k}({\mathcal{T}}_{f})$ such that the decomposition (1.1) holds.
The operators $B_{f}^{k}$ can be extended to bounded operators from
$L^{2}\Lambda^{k}(\Omega_{f})$ to itself if $0\leq m<n$, and from
$L^{2}\Lambda^{k}(\Omega_{f}^{E})$ to $L^{2}\Lambda^{k}(\Omega_{f})$ when
$m=n$. Furthermore, all these operators commute with the exterior derivative,
and satisfy the invariant property (1.2). Finally, the decomposition given by
(1.1) satisfies the stable decomposition property, detailed in Proposition
8.1.

##  3\. The case of scalar-valued functions

To motivate the general theory developed later in this paper, we will discuss
the decomposition (1.1) in the case $k=0$. In fact, to derive the
decomposition (1.1), we only need to establish the identity (2.23) for $1\leq
m\leq n-1$. A key ingredient in the derivation of (2.23) is to rely on two
slightly different representations of the operator $C_{m}^{0}$. For $k=0$, the
expression (2.19) can be written as

(3.1)
$C_{m}^{0}u=\sum_{f\in\Delta_{m}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\Big{[}L_{g}^{*}A_{f}^{0}u-\sum_{i\in
I(f\cap g^{*})}\frac{\lambda_{i}}{\rho_{g}}L_{g}^{*}A_{f}^{0}u\Big{]},$

where we recall that the operator $R_{e,f}^{k}u=-A_{\langle e,f\rangle}^{k}u$
when $(e,f)\in\Delta_{0,m-1}$. Alternatively, we also have

(3.2)
$C_{m}^{0}u=\sum_{f\in\Delta_{m}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\rho_{f}}{\rho_{g}}L_{g}^{*}A_{f}^{0}u,$

cf. (2.20). Motivated by the first term in (3.1), we will define the operators
$K_{m,f}^{0}$ by

(3.3)
$K_{m,f}^{0}u=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}L_{g}^{*}A_{f}^{0}u,\quad
f\in\Delta_{m}({\mathcal{T}}).$

From the properties of the average operators, cf. (2.13), it follows that the
operators $K_{m,f}^{0}u$ have domain of dependence $\Omega_{f}$, preserve
piecewise smoothness and the piecewise polynomial spaces, and have support on
$\Omega_{f}$.. The latter property follows from a standard cancellation
argument. To see this, consider an index $i\in I(f)$. For each $g\in\Delta(f)$
such that $i\notin I(g)$, we have that the two terms

$L_{g}^{*}A_{f}^{0}u,\quad\text{and }L_{\langle x_{i},g\rangle}^{*}A_{f}^{0}u$

cancel, when $\lambda_{i}(x)=0$. By repeating this argument, we see that
$K_{m,f}^{0}u\equiv 0$ for all $x$ such that $\lambda_{i}(x)=0$ for all $i\in
I(f)$. However, this means that the function $K_{m,f}^{0}u$ has support on
$\Omega_{f}$. From (3.1), it follows that

$\displaystyle C_{m}^{0}u-\sum_{f\in\Delta_{m}}K_{m,f}^{0}u$
$\displaystyle=-\sum_{f\in\Delta_{m}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\sum_{i\in
I(f\cap g^{*})}\frac{\lambda_{i}}{\rho_{g}}L_{g}^{*}A_{f}^{0}u$
$\displaystyle=\sum_{f^{\prime}\in\Delta_{m-1}}\sum_{g\in\bar{\Delta}(f^{\prime})}(-1)^{|f^{\prime}|-|g|}\sum_{i\in
I((f^{\prime})^{*})}\frac{\lambda_{i}}{\rho_{g}}L_{g}^{*}A_{\langle
x_{i},f^{\prime}\rangle}^{0}u,$

where we have introduced $f^{\prime}=f(\hat{x}_{i})$. Furthermore, from (2.12)
and (3.2), we also obtain

$C_{m-1}^{0}u=\sum_{f\in\Delta_{m-1}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\rho_{f}}{|f^{*}|\rho_{g}}\sum_{i\in
I(f^{*})}L_{g}^{*}A_{\langle x_{i},f\rangle}^{0}u.$

By combining these representations of $C_{m}^{0}$ and $C_{m-1}^{0}$, we obtain
the identity

(3.4) $(C_{m}^{0}-C_{m-1}^{0})u-\sum_{f\in\Delta_{m}}K_{m,f}^{0}u\\\
=\sum_{f\in\Delta_{m-1}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\rho_{g}^{-1}\sum_{i\in
I(f^{*})}\Big{[}\Big{(}\lambda_{i}-\frac{\rho_{f}}{|f^{*}|}\Big{)}L_{g}^{*}A_{\langle
x_{i},f\rangle}^{0}u\Big{]}.$

Therefore, if we define the operators $K_{m,f}^{0}$ by

(3.5)
$K_{m,f}^{0}u=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\rho_{g}^{-1}\sum_{i\in
I(f^{*})}\Big{[}\Big{(}\lambda_{i}-\frac{\rho_{f}}{|f^{*}|}\Big{)}L_{g}^{*}A_{\langle
x_{i},f\rangle}^{0}u\Big{]},$

for $f\in\Delta_{m-1}$, we obtain that (3.4) simply reads

(3.6) $(C_{m}^{0}-C_{m-1}^{0})u=\sum_{\begin{subarray}{c}f\in\Delta_{j}\\\
j=m,m-1\end{subarray}}K_{m,f}^{0}u,$

which is the desired identity (2.23) for $k=0$.

It remains to see that the operators $K_{m,f}^{0}$, $f\in\Delta_{m-1}$, have
the desired properties. Again, by the properties of the map $A_{f}^{0}$, the
operators $L_{g}^{*}A_{\langle x_{i},f\rangle}^{0}$ and hence $K_{m.f}^{0}$
will have domain of dependence $\Omega_{f}$. In addition, we need to show that
the operators $K_{m,f}^{0}$, given by (3.5), preserve piecewise smoothness and
piecewise polynomials, and that the target function has local support. In
fact, the property that the functions $K_{m,f}^{0}u$ are supported on the
macroelements $\Omega_{f}$ follows by a cancellation argument similar to the
one given above. However, formula (3.5) contains rational functions, and
therefore, at first glance, it seems unlikely that the corresponding operators
$K_{m,f}^{0}$ will preserve piecewise smoothness. On the other hand, since
$\operatorname{supp}K_{m,f}^{0}u\subset\Omega_{f}$, we can restrict the
analysis of the functions $K_{m,f}^{0}u$ to the domain $\Omega_{f}$, and on
this domain we will rely on an alternative representation of the operators.

To derive the alternative representation of $K_{m,f}^{0}$, we recall that when
$f\in\Delta_{m-1}$, the manifold $f^{*}$ is of dimension $n-m$, and since we
assume that $m\leq n-1$, the manifold $f^{*}$ is of dimension greater or equal
to one. For $x_{i}\in f^{*}$, we define a corresponding piecewise linear
function, $\beta_{x_{i}}=\beta_{x_{i}}(f)$ by

$\beta_{x_{i}}=\lambda_{i}-\frac{\rho_{f}}{|f^{*}|}.$

The collection $\\{\beta_{x_{i}}(f)\\}_{i\in I(f^{*})}$ can be seen as an
element in ${\mathscr{C}}_{0}(f^{*})\otimes{\mathcal{P}}_{1}({\mathcal{T}})$.
Furthermore, on the domain $\Omega_{f}$, we have that

$\partial_{0}\beta=\sum_{i\in I(f^{*})}\beta_{x_{i}}=\Big{(}\sum_{i\in
I(f^{*})}\lambda_{i}\Big{)}-\rho_{f}=0,$

where $\partial_{0}=\partial_{0}(f^{*})$. From Lemma 2.1, we therefore
conclude that there is a unique element
$\mu=\mu(f)=\\{\mu_{e}(f)\\}_{e\in\Delta_{1}(f^{*})}\in{\mathscr{C}}_{1}(f^{*})\otimes{\mathcal{P}}_{1}({\mathcal{T}})$,
such that the identities

(3.7)
$(\partial\mu)_{x_{i}}=\lambda_{i}-\frac{\rho_{f}}{|f^{*}|}=\beta_{x_{i}},\quad
i\in I(f^{*}),\quad\text{and }\delta\mu=0,$

hold on $\Omega_{f}$, where $\partial=\partial_{1}(f^{*})$ and
$\delta=\delta_{1}(f^{*})$. Note that when $n-m=1$, there is no space
${\mathscr{C}}_{2}(f^{*})$ (see Figure 2.1) and hence we define
$\delta_{1}(f^{*})$ by (2.8).

We can use the piecewise linear functions,
$\\{\mu_{e}(f)\\}_{e\in\Delta_{1}(f^{*})}$, to obtain an alternative
representation of the operators $K_{m,f}^{0}$. In fact, we have, using (2.2)
and (2.18), that

$\sum_{i\in I(f^{*})}(\partial\mu)_{x_{i}}\wedge L_{g}^{*}A_{\langle
x_{i},f\rangle}^{0}u=-\sum_{i\in I(f^{*})}(\partial\mu)_{x_{i}}\wedge
L_{g}^{*}R_{x_{i},f}^{0}u\\\ =-\sum_{e\in\Delta_{1}(f^{*})}\mu_{e}\wedge
L_{g}^{*}(\delta R^{0}u)_{e,f}=\sum_{e\in\Delta_{1}(f^{*})}\mu_{e}\wedge
L_{g}^{*}R_{e,f}^{1}du.$

As a consequence, we obtain that the alternative representation,

$K_{m,f}^{0}u=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\sum_{e\in\Delta_{1}(f^{*})}\Big{[}\mu_{e}(f)\wedge
L_{g}^{*}b^{-1}R_{e,f}^{1}du\Big{]},$

holds on $\Omega_{f}$. However, from this alternative representation, the
mapping properties of the operators $K_{m,f}^{0}$ follow directly from the
corresponding properties of the operators $R_{e,f}^{1}$, given in Lemma 2.2.

We close the discussion given in this section by summarizing the results we
have obtained for the operators $K_{m,f}^{0}$.

######  Lemma 3.1.

Let $1\leq m\leq n-1$ and $f\in\Delta_{j}({\mathcal{T}})$, where $j=m,m-1$.
Assume that the corresponding operators $K_{m,f}^{0}$ are defined by (3.3) and
(3.5). Then the identity (3.6) holds,
$\operatorname{supp}K_{m,f}^{0}u\subset\Omega_{f}$, and the operators
$K_{m,f}^{0}$ have the mapping properties

$K_{m,f}^{0}(\Lambda^{0}({\mathcal{T}}_{f}))\subset\Lambda^{0}({\mathcal{T}}_{f}),\qquad
K_{m,f}^{0}({\mathcal{P}}_{r}({\mathcal{T}}_{f}))\subset{\mathcal{P}}_{r}({\mathcal{T}}_{f}).$

##  4\. The local structure of the mesh

Key tools for decomposing the operators $C_{m}^{k}-C_{m-1}^{k}$ into a sum of
local operators with local target space will be various local functions
derived from the given mesh, ${\mathcal{T}}$. To describe these functions, we
introduce the space ${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})$ as
the subset of ${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$ corresponding
to degrees of freedom on $f^{*}$. More precisely,

${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})=\operatorname{span}_{e\in\Delta_{k}(f^{*})}\phi_{e}.$

###  4.1. The functions $\mu_{e}(f)$

Let $m$ be an index, $1\leq m\leq n$, and $f\in\Delta_{m-1}$, such that the
associated manifold, $f^{*}$, will be of dimension $n-m$. In the special case
when $m=n$, the manifold $f^{*}$ will be of dimension zero, and consist of one
or two points depending on the location of $f$ relative to the boundary of
$\Omega$. Since this case is special, we will first assume that $m\leq n-1$
such that the manifold $f^{*}$ is a least one dimensional.

In the previous section, we constructed functions
$\\{\mu_{e}(f)\\}_{e\in\Delta_{1}(f^{*})}$ such that the identity (3.7) holds,
where we can consider the collection $\\{\mu_{e}(f)\\}$ as an element of
${\mathscr{C}}_{1}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{0}({\mathcal{T}},f^{*})$.
In fact, we will construct the collection
$\\{\mu_{e}(f)\\}_{e\in\Delta_{j}(f^{*})}\in{\mathscr{C}}_{j}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{j-1}({\mathcal{T}},f^{*})$
for $1\leq j\leq n$. This will be done by an inductive process with respect to
$j$. For $e\in\Delta_{0}(f^{*})$, we define $\mu_{e}(f)$ to be the constant
$-1/|f^{*}|$, such that $\\{\mu_{e}(f)\\}_{e\in\Delta_{0}(f^{*})}$ can be
viewed as an element of
${\mathscr{C}}_{0}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{-1}({\mathcal{T}},f^{*})$,
where ${\mathcal{P}}_{1}^{-}\Lambda^{-1}({\mathcal{T}},f^{*})$ is identified
as $\mathbb{R}$. Below we will apply the difference operators,
$\partial(f^{*})$ and $\delta(f^{*})$, to elements of
${\mathscr{C}}_{j}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})$.
This is done with respect to ${\mathscr{C}}_{j}(f^{*})$, while the polynomial
space ${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})$ is considered
fixed. For example, the operator $\partial(f^{*})$ maps elements of
${\mathscr{C}}_{j}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})$
to
${\mathscr{C}}_{j-1}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})$.
On the other hand, the exterior derivative, $d$, will map elements of
${\mathscr{C}}_{j}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}},f^{*})$
to
${\mathscr{C}}_{j}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{k+1}({\mathcal{T}},f^{*})$.

In addition to the collection of functions $\\{\mu_{e}(f)\\}$, we will
introduce the associated collection of functions
$\\{\beta_{e}\\}=\\{\beta_{e}(f)\\}_{e\in\Delta_{j}(f^{*})}$ as elements of
${\mathscr{C}}_{j}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}})$
for $0\leq j\leq n-m$. The function $\beta_{e}(f)$ is defined from the
corresponding function $\mu_{e}(f)$ by

(4.1)
$\beta_{e}:=\rho_{f}^{j+1}d\Big{(}\frac{\mu_{e}}{\rho_{f}^{j}}\Big{)}+(-1)^{j}\phi_{e},\quad
e\in\Delta_{j}(f^{*}),$

if $1\leq j\leq n-m$. In fact, if we let the exterior derivative $d_{-1}$ be
the inclusion map from constants to
${\mathcal{P}}_{1}\Lambda^{0}({\mathcal{T}},f^{*})$, then the definition (4.1)
also holds when $j=0$, cf. (3.7). We observe that if we restrict to the domain
$\Omega_{f}$, such that $\rho_{f}=\sum_{i\in I(f^{*})}\lambda_{i}$, then
$\beta_{e}(f)$ also admits the representation

(4.2) $\beta_{e}=\sum_{i\in
I(f^{*})}(\lambda_{i}d-jd\lambda_{i}\wedge)\mu_{e}+(-1)^{j}\phi_{e},\quad
e\in\Delta_{j}(f^{*}).$

It then follows that

$\mu_{e}\in{\mathcal{P}}_{1}^{-}\Lambda^{j-1}({\mathcal{T}},f^{*})\,\Longrightarrow\,\operatorname{tr}_{\Omega_{f}}\beta_{e}\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}}_{f},f^{*}),$

where
${\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}}_{f},f^{*})=\operatorname{tr}_{\Omega_{f}}{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})$.
Furthermore, the relation (3.7) can be rephrased as
$\beta_{e}=(\partial(f^{*})\mu)_{e}$ on $\Omega_{f}$ for
$e\in\Delta_{0}(f^{*})$.

######  Lemma 4.1.

Let $f\in\Delta_{m-1}({\mathcal{T}})$, where $1\leq m\leq n-1$, and $j$ an
index such that $1\leq j\leq n-m$. Assume that
$\\{\mu_{e}\\}=\\{\mu_{e}(f)\\}\in{\mathscr{C}}_{s}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{s-1}({\mathcal{T}},f^{*})$
has been defined for $s=j-1,j$ such that the identity

(4.3) $\beta_{e}(f)=(\partial(f^{*})\mu)_{e},$

holds on $\Omega_{f}$ for $e\in\Delta_{j-1}(f^{*})$. Then
$(\partial(f^{*})\beta)_{e}=0$ for all $e\in\Delta_{j-1}(f^{*})$, and if
$j<n-m$, there exist
$\\{\mu_{e}\\}=\\{\mu_{e}(f)\\}\in{\mathscr{C}}_{j+1}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})$
such that (4.3) holds on $\Omega_{f}$ for all $e\in\Delta_{j}(f^{*})$.
Furthermore, $\\{\mu_{e}\\}\in
C_{j+1}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})$ is
uniquely determined by the condition $\delta(f^{*})\mu=0$.

###### Proof.

Throughout the proof, all the identities should be considered to hold on the
domain $\Omega_{f}$, and the operators $\partial$ and $\delta$ are defined
with respect to $f^{*}$. By assumption, we have

$\beta_{e}=\rho^{j}d\Big{(}\frac{\mu_{e}}{\rho^{j-1}}\Big{)}+(-1)^{j-1}\phi_{e}=(\partial_{j}\mu)_{e},\quad
e\in\Delta_{j-1}(f^{*}),$

where $\rho=\rho_{f}$. Since $d^{2}=0$ and $d$ commutes with $\partial_{j}$,
this gives

$(\partial_{j}d\Big{(}\frac{\mu}{\rho^{j}}\Big{)})_{e}=(-1)^{j-1}d\Big{(}\frac{\phi_{e}}{\rho^{j}}\Big{)},\quad
e\in\Delta_{j-1}(f^{*}).$

Hence, it follows from (4.1) that for $e\in\Delta_{j-1}(f^{*})$

$(\partial_{j}\beta)_{e}=\rho^{j+1}(\partial_{j}d\Big{(}\frac{\mu}{\rho^{j}}\Big{)})_{e}+(-1)^{j}(\partial_{j}\phi)_{e}=(-1)^{j-1}[\rho^{j+1}d\Big{(}\frac{\phi_{e}}{\rho^{j}}\Big{)}-(\partial_{j}\phi)_{e}].$

However, a direct computation, using (2.5) and $\rho_{f}=\sum_{i\in
I(f^{*})}\lambda_{i}$ on $\Omega_{f}$, shows

$\rho^{(j+1)}d\Big{(}\frac{\phi_{e}}{\rho^{j}}\Big{)}=\sum_{i\in I(f^{*}\cap
e^{*})}(\lambda_{i}d-jd\lambda_{i}\wedge)\phi_{e}=\sum_{i\in I(f^{*}\cap
e^{*})}\phi_{[x_{i},e]}=(\partial_{j}\phi)_{e}.$

As a consequence, $(\partial_{j}\beta)_{e}=0$ for $e\in\Delta_{j-1}(f^{*})$.
If $j<n-m$, it follows from Lemma 2.1 that there exist a uniquely determined
$\\{\mu_{e}\\}\in{\mathscr{C}}_{j+1}(f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}}_{f},f^{*})$
such that $\beta_{e}=(\partial_{j+1}\mu)_{e}$ on $\Omega_{f}$, for all
$e\in\Delta_{j}(f^{*})$ and $\delta_{j+1}\mu=0$. Furthermore, each function
$\mu_{e}\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}}_{f},f^{*})$ can be
uniquely extended to a function in
${\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})$. ∎

It remains to discuss the case $m=n$. In this case, $f\in\Delta_{n-1}$, so
$f^{*}$ will only consist of one or two points, and there is no element in
$\Delta_{1}(f^{*})$. In fact, we will have $|f^{*}|=1$ if $f$ is a boundary
simplex, and otherwise $|f^{*}|=2$. On the other hand, by adopting the
interpretation above of $d_{-1}$ as the inclusion operator and
$\mu_{e}(f)=-1/|f^{*}|$ for $e\in\Delta_{0}(f^{*})$, we obtain

$\beta_{e}(f):=-\frac{\rho_{f}}{|f^{*}|}+\phi_{e},\quad
e\in\Delta_{0}(f^{*}),$

and $\partial_{0}(f^{*})\beta=0$.

From Lemma 4.1 and an induction argument with respect to $j$, we obtain the
following result.

######  Corollary 4.2.

Let $f\in\Delta_{m-1}({\mathcal{T}})$, where $1\leq m\leq n$. For all
$e\in\Delta_{j}(f^{*})$, $0\leq j\leq n-m$, there exists functions
$\mu_{e}(f)\in{\mathcal{P}}_{1}^{-}\Lambda^{j-1}({\mathcal{T}},f^{*})$ and
$\beta_{e}(f)\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}})$, uniquely
defined by the inductive procedure above, satisfying
$\partial_{j}(f^{*})\beta=0$ on $\Omega_{f}$. Furthermore, if $j<n-m$, then
the identity (4.3) holds on $\Omega_{f}$ for all $e\in\Delta_{j}(f^{*})$.

######  Remarks.

An alternative view of the construction of the functions $\\{\mu_{e}(f)\\}$
given above can be given by expanding the function
$\mu_{e}\in{\mathcal{P}}_{1}^{-}\Lambda^{j-1}({\mathcal{T}},f^{*})$ in the
form

(4.4)
$\mu_{e}=\sum_{e^{\prime}\in\Delta_{j-1}(f^{*})}a_{e,e^{\prime}}\phi_{e^{\prime}},$

where the real coefficients $\\{a_{e,e^{\prime}}\\}$ can be identified with an
element $a_{e}\in{\mathscr{C}}_{j-1}$. It follows from (2.3), (2.5), and
(4.2), that if we restrict to $\Omega_{f}$, then the function
$\beta_{e}=\beta_{e}(f)$ admits the representation

(4.5)
$\beta_{e}-(-1)^{j}\phi_{e}=\sum_{\begin{subarray}{c}e^{\prime}\in\Delta_{j}(f^{*})\\\
i\in
I(e^{\prime})\end{subarray}}(-1)^{\sigma_{e^{\prime}}(x_{i})}a_{e,e^{\prime}(\hat{x}_{i})}\phi_{e^{\prime}}=\sum_{e^{\prime}\in\Delta_{j}(f^{*})}(\delta
a_{e,\cdot})_{e^{\prime}}\phi_{e^{\prime}},$

for $e\in\Delta_{j}(f^{*})$. As a consequence, the equation (4.3) for
$e\in\Delta_{j}(f^{*})$ can be represented by the algebraic system

(4.6)
$(\partial_{j+1}a_{\cdot,e^{\prime}})_{e}=(\delta_{j-1}a_{e,\cdot})_{e^{\prime}}+(-1)^{j}1_{e,e^{\prime}},\quad
e^{\prime}\in\Delta_{j}(f^{*}),$

where $\partial=\partial(f^{*})$, and $1_{e,e^{\prime}}=1$ if $e^{\prime}=e$
and equals zero otherwise. Furthermore, the condition
$\delta_{j+1}(f^{*})\mu=0$ is equivalent to

(4.7) $\delta_{j+1}a_{\cdot,e^{\prime}}=0,\quad
e^{\prime}\in\Delta_{j}(f^{*}).$

If $(e,f)\in\Delta_{j,m-1}$ and $g\in\bar{\Delta}(f)$, then
$\rho_{g}-\rho_{f}=\sum_{i\in I(f\cap g^{*})}\lambda_{i}$, which leads to

$\rho_{g}^{j+1}d\Big{(}\frac{\mu_{e}(f)}{\rho_{g}^{j}}\Big{)}=\rho_{f}^{j+1}d\Big{(}\frac{\mu_{e}(f)}{\rho_{f}^{j}}\Big{)}+\sum_{i\in
I(f\cap g^{*})}\big{(}\lambda_{i}d-jd\lambda_{i}\wedge\big{)}\mu_{e}(f).$

In the analysis below, the functions
$\psi_{e,g}(f)\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},g^{*})$,
defined by

(4.8) $\psi_{e,g}(f)=(-1)^{j-1}\sum_{i\in I(f\cap
g^{*})}\big{(}\lambda_{i}d-jd\lambda_{i}\wedge\big{)}\mu_{e}(f),$

will be useful. We observe that $\psi_{e,f}(f)=0$, and it follows from (4.1),
that

(4.9)
$\rho_{g}^{j+1}d\Big{(}\frac{\mu_{e}(f)}{\rho_{g}^{j}}\Big{)}+(-1)^{j}[\phi_{e}+\psi_{e,g}(f)]=\beta_{e}(f).$

Also observe that in the special case when $e\in\Delta_{0}(f^{*})$, then

(4.10) $\psi_{e,g}(f)=\frac{1}{|f^{*}|}\sum_{i\in I(f\cap
g^{*})}\lambda_{i}=\frac{1}{|f^{*}|}(\rho_{g}-\rho_{f}).$

###  4.2. Construction of the weight functions $z_{e,f}$

We recall from Section 2.4 that the weight functions $z_{e,f}$ are an
essential ingredient for the construction of the order reduction operators
$R_{e,f}^{k}$. This section is devoted to an inductive process for
constructing the functions $z_{e,f}$. In fact, as a preliminary step, we will
first construct a family of local functions $w_{e,f}$, and then the functions
$z_{e,f}$ will be constructed as

(4.11) $z_{e,f}=(\delta^{+}w)_{e,f}.$

Here the operator $\delta^{+}$ is a variant of the coboundary operator defined
for pairs of simplices, given by

$(\delta^{+}w)_{e,f}=\sum_{i\in
I(e)}(-1)^{\sigma_{e}(x_{i})}w_{e(\hat{x}_{i}),\langle
x_{i},f\rangle},\quad(e,f)\in\Delta_{j,m}.$

This operator will satisfy the complex property, $(\delta^{+})^{2}=0$, and
from [6, Lemma 5.2], we recall that
$\delta\circ\delta^{+}=-\delta^{+}\circ\delta$. For the construction below, we
will utilize exactness of the complex of trimmed linear forms with support on
$\Omega_{f}$. Recall that in Section 1, we introduced the space $\mathaccent
23{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}}_{f})$ as the subspace of
${\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}})$ consisting of functions
which vanish on $\Omega\setminus\Omega_{f}$. However, if $f$ is a boundary
simplex, then functions in this space will in general not have vanishing trace
on the boundary of $\Omega_{f}$. Therefore, we introduce the notation
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to denote the subspace of $\mathaccent
23{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}}_{f})$ with vanishing trace on
$\partial\Omega_{f}$. The two spaces are equal if $f$ is an interior simplex,
but in general $\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{k}({\mathcal{T}}_{f})\subset\mathaccent
23{\mathcal{P}}_{1}^{-}\Lambda^{k}({\mathcal{T}}_{f})$. We also recall from
(2.4) that if
$w=\sum_{g\in\Delta_{j}}c_{g}\phi_{g}\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}})$,
then $dw=\sum_{g\in\Delta_{j+1}}(\delta_{j}c)_{g}\phi_{g}$. In particular, if
$w$ has support on $\Omega_{f}$ for $f\in\Delta$, then the sum can be
restricted to all simplices $g$ such that $g\supset f$. Motivated by this, we
define $d_{f}^{*}:\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{j}({\mathcal{T}}_{f})\to\accentset{~{}{\leavevmode\hbox
to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{j-1}({\mathcal{T}}_{f})$,
by

(4.12) $d_{f}^{*}w=\sum_{\begin{subarray}{c}g\in\Delta_{j-1}\\\ g\supset
f\end{subarray}}(\partial_{j}c)_{g}\phi_{g}.$

Below we will utilize exactness of the complex
$(\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda({\mathcal{T}}_{f}),d)$
to conclude that a function $w\in\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{j}({\mathcal{T}}_{f})$
is uniquely determined by $dw$ and $d_{f}^{*}w$.

The functions $w_{e,f}$ will be defined inductively with respect to $m$ for
all pairs $(e,f)\in\Delta_{j,m}$, for $0\leq m\leq n$ and $-1\leq j<n-m$ as
functions in $\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}}_{f})$.
We start the induction process with $m=n$, and hence only $j=-1$ is allowed.
The set $\Delta_{-1,n}$ consists of pairs of the form $(\emptyset,f)$, where
$f\in\Delta_{n}$. In this case, we define
$w_{\emptyset,f}=-z_{f}=-(\kappa_{f}/|\Omega_{f}|)\mathsf{vol}$. For the
general case, when $0\leq m<n$, we will use a variational approach utilizing
tensor product spaces of the form
${\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}})$,
i.e., we consider products of elements in the two functions spaces and with
independent spatial variables. We assume, as an induction hypothesis, that
$w_{e,f}\in\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}}_{f})$
for all $(e,f)\in\Delta_{j,m}$ and $-1\leq j<n-m$, have already been
constructed. Furthermore, we assume that these functions satisfy

(4.13)
$(\delta^{+}dw)_{e,f}\in\text{range}(\delta),\quad(e,f)\in\Delta_{n-m,m-1}(f^{*}).$

In the case $m=n$, the functions of the form $w_{\emptyset,\langle
x_{i},f\rangle}$, involved in (4.13), have support in $\Omega_{\langle
x_{i},f\rangle}\subset\Omega_{f}$. Since we will utilize exactness of the
complex consisting of trimmed differential forms with boundary conditions, the
exterior derivative in (4.13), in the case $m=n$, should be interpreted as the
integral, and for $(x_{i},f)\in\Delta_{0,n-1}$, we have

$(\delta^{+}dw)_{x_{i},f}=d(\delta^{+}w)_{x_{i},f}=\int_{\Omega}w_{\emptyset,\langle
x_{i},f\rangle}=-1=\int_{\Omega}w_{\emptyset,f}=(\delta dw)_{x_{i},f}.$

Therefore, property (4.13) holds initially.

For a fixed $f\in\Delta_{m-1}$ and $-1\leq j<n-m$,
$w_{e,f}\in{\mathcal{P}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}})$ is defined by

(4.14) $\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\otimes
w_{e,f}=(-1)^{j}\sum_{e\in\Delta_{j+1}(f^{*})}\mu_{e}(f)\otimes(\delta^{+}w)_{e,f},$

where we observe that all functions on the right hand side are already
constructed. Since
$\mu_{e}(f)\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})$ for
$e\in\Delta_{j+1}(f^{*})$, we can view the right hand side of (4.14) is an
element of
${\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}})$,
i.e, it is a trimmed linear $j$-form with values in
${\mathcal{P}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}})$. Therefore, the
coefficients $w_{e,f}$ of the left hand side are uniquely determined as
functions in ${\mathcal{P}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}})$.
Furthermore, since all the domains of the form $\\{\Omega_{x_{i},f}\\}$, $i\in
I(f^{*})$ are contained in $\Omega_{f}$, we can conclude from the induction
hypothesis that $(\delta^{+}w)_{e,f}\in\accentset{~{}{\leavevmode\hbox
to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}}_{f})$
and hence the function $w_{e,f}\in\accentset{~{}{\leavevmode\hbox
to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}}_{f})$
for $(e,f)\in\Delta_{j,m-1}$. In particular, since $\mu_{e}(f)=-1/|f^{*}|$ for
$e\in\Delta_{0}(f^{*})$, we obtain that the function $w_{\emptyset,f}$,
$f\in\Delta_{m-1}$ satisfies the recurrence relation

$w_{\emptyset,f}=\frac{1}{|f^{*}|}\sum_{i\in I(f^{*})}w_{\emptyset,\langle
x_{i},f\rangle}.$

Let $0\leq j\leq n-m$. For the discussion below, it will be useful to observe
that the function just defined satisfies

$\displaystyle\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\otimes(\delta w)_{e,f}$
$\displaystyle=\sum_{e\in\Delta_{j-1}(f^{*})}(\partial\phi)_{e}\otimes
w_{e,f}$ $\displaystyle=\sum_{i\in
I(f^{*})}(\lambda_{i}d-jd\lambda_{i}\wedge)\sum_{e\in\Delta_{j-1}(f^{*})}\phi_{e}\otimes
w_{e,f}$ $\displaystyle=(-1)^{j-1}\sum_{i\in
I(f^{*})}(\lambda_{i}d-jd\lambda_{i}\wedge)\sum_{e\in\Delta_{j}(f^{*})}\mu_{e}\otimes(\delta^{+}w)_{e,f},$

where we have used (2.2) and (2.5), in addition to (4.14). Alternatively, from
(4.2) we have

(4.15) $\sum_{e\in\Delta_{j}(f^{*})}\Big{[}\phi_{e}\otimes(\delta
w)_{e,f}-\Big{(}\phi_{e}+(-1)^{j-1}\beta_{e}\Big{)}\otimes(\delta^{+}w)_{e,f}\Big{]}=0,$

where $f\in\Delta_{m-1}$ and $0\leq j\leq n-m$, and where the spatial variable
for $\phi_{e}$ and $\beta_{e}$ is restricted to $\Omega_{f}$.

The set up above defines the functions $w_{e,f}$ for all
$(e,f)\in\Delta_{j,m-1}$, where $-1\leq j<n-m$, from the corresponding
functions defined for elements in $\Delta_{j,m}$. However, we will also need
the functions $w_{e,f}$ for $(e,f)\in\Delta_{n-m,m-1}$. In this case, the
right hand side of (4.14) is not well defined since there are no elements in
$\Delta_{n-m,m}$. Instead, for $f\in\Delta_{m-1}({\mathcal{T}})$ and $j=n-m$,
we will require $w_{e,f}$ to satisfy

(4.16) $\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\otimes
dw_{e,f}=\sum_{e\in\Delta_{j}(f^{*})}\beta_{e}(f)\otimes(\delta^{+}w)_{e,f},$

and $d_{f}^{*}w_{e,f}=0$. In (4.16), the spatial variables are restricted to
$\Omega_{f}\times\Omega$, i.e., $\phi_{e}$ and $\beta_{e}(f)$ mean the
restrictions of these quantities to $\Omega_{f}$. It follows from the
induction hypothesis and the fact that
$\operatorname{tr}_{\Omega_{f}}\beta_{e}(f)\in{\mathcal{P}}_{1}^{-}\Lambda^{n-m}({\mathcal{T}}_{f},f^{*})$,
that the right hand side of (4.16) can be viewed as an element of
${\mathcal{P}}_{1}^{-}\Lambda^{n-m}({\mathcal{T}}_{f},f^{*})\otimes{\mathcal{P}}_{1}^{-}\Lambda^{m}({\mathcal{T}})$.
Furthermore, from the hypothesis (4.13), combined with (2.2), we obtain that

$\sum_{e\in\Delta_{j}(f^{*})}\beta_{e}(f)\otimes(\delta^{+}dw)_{e,f}=0,$

since $\partial\beta(f)=0$. As a consequence, from the exactness of the
complex $(\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda({\mathcal{T}}_{f}),d)$,
we obtain that there exist elements $w_{e,f}\in\accentset{~{}{\leavevmode\hbox
to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{m-1}({\mathcal{T}}_{f})$
such that (4.16) holds. Finally, since we require $d_{f}^{*}w_{e,f}=0$, the
functions $w_{e,f}$ are uniquely determined.

To complete the induction argument, we need to verify that the assumption
(4.13) is preserved by the induction step, i.e., that the identity (4.13)
holds with $m$ replaced by $m-1$. However, by combining (4.15) and (4.16), we
obtain that the identity

(4.17) $dw_{e,f}=(-1)^{j+1}[(\delta w)_{e,f}-(\delta^{+}w)_{e,f}]$

holds for $(e,f)\in\Delta_{j,m-1}$, $j=n-m$. As a further consequence of the
complex property of $\delta^{+}$, and the identity
$\delta^{+}\circ\delta=-\delta\circ\delta^{+}$, we then obtain

$(\delta^{+}dw)_{e,f}=(-1)^{n-m+1}(\delta\circ\delta^{+}w)_{e,f},$

for $(e,f)\in\Delta_{n-m+1,m-2}$. Hence, property (4.13) at level $m-1$ is
verified.

We summarize the properties of the construction above.

######  Lemma 4.3.

The inductive procedure above uniquely specifies the functions
$w_{e,f}\in\accentset{~{}{\leavevmode\hbox to1.8pt{\vbox
to1.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.9pt\lower-0.9pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to1.8pt{\vbox to1.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.9pt\lower-0.9pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.8pt}{0.0pt}\pgfsys@curveto{0.8pt}{0.44183pt}{0.44183pt}{0.8pt}{0.0pt}{0.8pt}\pgfsys@curveto{-0.44183pt}{0.8pt}{-0.8pt}{0.44183pt}{-0.8pt}{0.0pt}\pgfsys@curveto{-0.8pt}{-0.44183pt}{-0.44183pt}{-0.8pt}{0.0pt}{-0.8pt}\pgfsys@curveto{0.44183pt}{-0.8pt}{0.8pt}{-0.44183pt}{0.8pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}}{{\mathcal{P}}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}}_{f})$
for all $(e,f)\in\Delta_{j,m}$, where $0\leq m\leq n$ and $-1\leq j<n-m$.

Next we will establish that the functions $w_{e,f}$, introduced above, satisfy
the identity (4.17) more generally, i.e., not only for
$(e,f)\in\Delta_{n-m,m-1}$.

######  Lemma 4.4.

The functions $w_{e,f}$ satisfy the identity (4.17) for all
$(e,f)\in\Delta_{j,m}$, where $0\leq m\leq n-1$ and $0\leq j<n-m$.

###### Proof.

The proof will be done by induction with respect to $m$. For $m=n-1$, the only
possible value of $j$ is $j=0$. In this case, the desired identity has already
been verified above. Next, we assume that the identity holds for all
$(e,f)\in\Delta_{j,m}$, where $0\leq j<n-m$. Note that the case
$(e,f)\in\Delta_{n-m,m-1}$ is already established above. Therefore, we can
assume that $j<n-m$, and from the support property of the functions $w_{e,f}$,
it is enough to show this identity on $\Omega_{f}$. It follows from (4.15)
that the desired identity will follow if we can show that

(4.18)
$\sum_{e\in\Delta_{j}(f^{*})}\beta_{e}\otimes(\delta^{+}w)_{e,f}=\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\otimes
dw_{e,f},\quad f\in\Delta_{m-1},$

when $\phi_{e}$ and $\beta_{e}$ are restricted to $\Omega_{f}$. If $0\leq
j<m-n$, then from Corollary 4.2 we have the identity
$\beta_{e}=(\partial\mu)_{e}$ which gives

$\sum_{e\in\Delta_{j}(f^{*})}\beta_{e}\otimes(\delta^{+}w)_{e,f}=\sum_{e\in\Delta_{j}(f^{*})}(\partial\mu)_{e}\otimes(\delta^{+}w)_{e,f}\\\
=-\sum_{e\in\Delta_{j+1}(f^{*})}\mu_{e}\otimes(\delta^{+}\circ\delta
w)_{e,f}=(-1)^{j}\sum_{e\in\Delta_{j+1}(f^{*})}\mu_{e}\otimes(\delta^{+}dw)_{e,f}\\\
=\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\otimes dw_{e,f},$

where we have used the fact that
$\delta\circ\delta^{+}=-\delta^{+}\circ\delta$, (4.14), and the induction
hypothesis (4.17) with $f^{\prime}=\\{x_{i},f\\}\in\Delta_{m}$ and
$e^{\prime}=e(\hat{x}_{i})\in\Delta_{j}((f^{\prime})^{*})$. This establishes
the identity (4.18), and therefore the proof is completed. ∎

The desired weight functions $z_{e,f}$ are defined from the corresponding $w$
functions by the relation (4.11). More precisely, the functions $z_{e,f}$ are
defined by (4.11) for $(e,f)\in\Delta_{j,m}$ for $-1\leq m<n$ and $0\leq
j<n-m$. In particular, for $(e,f)\in\Delta_{0,n-1}$, we have
$z_{e,f}=-z_{\langle e,f\rangle}$. Recall also that a key property of these
functions is that they satisfy the identity (2.16), i.e., $dz=(-1)^{j+1}\delta
z$ for $(e,f)\in\Delta_{j,m}$.

######  Lemma 4.5.

The functions $z_{e,f}$, where $(e,f)\in\Delta_{j,m}$ for $-1\leq m\leq n-1$
and $0\leq j<n-m$ belong to
${\mathcal{P}}_{1}^{-}\Lambda^{n-j}({\mathcal{T}})$, and with support in
$\Omega_{f}\cap\Omega_{e}^{E}$. In addition, $z_{e,f}$ has vanishing trace on
the boundary of $\Omega$. Furthermore, for $j>0$, the identities (2.16) and
$(\delta^{+}z)_{e,f}=0$ hold.

###### Proof.

That the functions $z_{e,f}$ belong to
${\mathcal{P}}_{1}^{-}\Lambda^{n-j}({\mathcal{T}})$ follows from the fact that
$w_{e,f}$ are elements of
${\mathcal{P}}_{1}^{-}\Lambda^{n-j-1}({\mathcal{T}})$. Furthermore, since
$\delta^{+}$ satisfies the complex property, we obtain that $\delta^{+}z=0$,
and from (4.17) we have

$(dz)_{e,f}=(\delta^{+}dw)_{e,f}=(-1)^{j}(\delta^{+}\circ\delta w)_{e,f}\\\
=(-1)^{j+1}(\delta\circ\delta^{+}w)_{e,f}=(-1)^{j+1}(\delta z)_{e,f},$

where, as above, we have used the fact that
$\delta\circ\delta^{+}=-\delta^{+}\circ\delta$.

From the support property of the $w$ functions given in Lemma 4.3, we have for
$(e,f)\in\Delta_{j,m}$ and $i\in I(e)$ that

$\operatorname{supp}w_{e(\hat{x}_{i}),\langle
x_{i},f\rangle}\subset\Omega_{\langle
x_{i},f\rangle}=\Omega_{f}\cap\Omega_{x_{i}},$

which implies that

$\operatorname{supp}z_{e,f}\subset\bigcup_{i\in
I(e)}\big{(}\Omega_{f}\cap\Omega_{x_{i}}\big{)}=\Omega_{f}\cap\Omega_{e}^{E}.$

Finally, since all functions $\\{w_{e,f}\\}$ have vanishing trace on the
boundary of $\Omega$, this property will also hold for all the functions
$\\{z_{e,f}\\}$. ∎

Recall that the functions $\psi_{e,g}(f)$, defined by (4.8), where
$(e,f)\in\Delta_{j,m-1}$ and $g\in\bar{\Delta}(f)$, satisfy the relation
(4.9). The following relation between the functions $\psi_{e,g}(f)$ and the
weight functions $z_{e,f}$ will be crucial in our analysis below.

######  Lemma 4.6.

Let $0\leq m\leq n-1$ and $0\leq j<n-m$. For any
$g\in\Delta_{s}({\mathcal{T}})$, where $-1\leq s\leq m-1$, the identity

(4.19) $\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m}\\\ f\supset
g\end{subarray}}\psi_{e,g}(f)\otimes
z_{e,f}=\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\ f\supset
g\end{subarray}}\phi_{e}\otimes z_{e,f},$

holds.

###### Proof.

Since $z=\delta^{+}w$, we can reformulate the right hand side of the identity
as

$\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\ f\supset
g\end{subarray}}\sum_{i\in I(e)}(-1)^{\sigma_{e}(x_{i})}\phi_{e}\otimes
w_{e(\hat{x}_{i}),\langle
x_{i},f\rangle}=\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j-1,m}\\\ f\supset
g\end{subarray}}\sum_{i\in I(f\cap g^{*})}\phi_{[x_{i},e]}\otimes w_{e,f}.$

On the other hand, using the definition of $\psi_{e,g}(f)$, (2.5), and (4.14),
for $e\in\Delta_{j-1}$, the left hand side can be rewritten as

$(-1)^{j-1}\sum_{\begin{subarray}{c}f\in\Delta_{m}\\\ f\supset
g\end{subarray}}\sum_{i\in I(f\cap
g^{*})}(\lambda_{i}d-jd\lambda_{i}\wedge)\sum_{e\in\Delta_{j}(f^{*})}\mu_{e}\otimes(\delta^{+}w)_{e,f}\\\
=\sum_{\begin{subarray}{c}f\in\Delta_{m}\\\ f\supset
g\end{subarray}}\sum_{i\in I(f\cap
g^{*})}(\lambda_{i}d-jd\lambda_{i}\wedge)\sum_{e\in\Delta_{j-1}(f^{*})}\phi_{e}\otimes
w_{e,f}\\\ =\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j-1,m}\\\ f\supset
g\end{subarray}}\sum_{i\in I(f\cap g^{*})}\phi_{[x_{i},e]}\otimes w_{e,f},$

and hence the desired identity is verified. ∎

##  5\. The order reduction operators

We recall from Section 2.4 above that the order reduction operators,
$R_{e,f}^{k}$, are defined for $(e,f)\in\Delta_{j,m}$ from corresponding
functions $z_{e,f}$ by

$(R_{e,f}^{k}u)_{\lambda}=\int_{\Omega}(\Pi_{j}G_{f}^{*}u)_{\lambda}\wedge
z_{e,f},\quad\lambda\in{\mathcal{S}}_{f}^{c},$

cf. (2.15). The functions $z_{e,f}$ will be taken to be the weight functions
constructed in the previous section. In particular,
$z_{e,f}\in{\mathcal{P}}_{1}^{-}\Lambda^{n-j}({\mathcal{T}}_{f})$ for
$(e,f)\in\Delta_{j,m}$, and with support in $\Omega_{f}\cap\Omega_{e}^{E}$,
while the function $G_{f}$, defined above, maps the product
$\Omega_{f}\times{\mathcal{S}}_{f}^{c}$ to $\Omega_{f}$. Since $G_{f}$ is
defined on a product space, the target space for the pullback, $G_{f}^{*}$,
can be represented as the sum of tensor products, and $\Pi_{j}$ is the
canonical map of $\Lambda^{k}(\Omega_{f}\times{\mathcal{S}}_{f}^{c})$ to
$\Lambda^{j}(\Omega_{f})\otimes\Lambda^{k-j}({\mathcal{S}}_{f}^{c})$. By
construction, the operators $R_{e,f}^{k}$ map $k$-forms to $(k-j)$-forms for
$(e,f)\in\Delta_{j,m}$, $0\leq j\leq k$. Furthermore, $R_{e,f}^{k}\equiv 0$
for $j>k$.

The commuting property $dG_{f}^{*}u=G_{f}^{*}du$, where $u$ is $k$-form on
$\Omega_{f}$, can in the present setting be expressed by

(5.1)
$d_{\Omega}\Pi_{j-1}G_{f}^{*}u+(-1)^{j}d_{S}\Pi_{j}G_{f}^{*}u=\Pi_{j}dG_{f}^{*}u=\Pi_{j}G_{f}^{*}du,\quad
j=1,\ldots,k,$

where $d_{\Omega}$ and $d_{S}$ denote the exterior derivative with respect to
the spaces $\Omega$ and ${\mathcal{S}}_{f}^{c}$, respectively. By combining
this with property (2.16), cf. Lemma 4.5, we derive, exactly as in the proof
of [6, Proposition 5.4], that the operators $R_{e,f}^{k}$ satisfy the
fundamental relation (2.17). Furthermore, from the fact that $\delta^{+}z=0$,
cf. Lemma 4.5, we obtain that $(\delta^{+}R^{k})_{e,f}=0$.

Next we will use Lemma 4.6 to obtain the following property of the operators
$R_{e,f}^{k}$.

######  Lemma 5.1.

Let $0\leq m\leq n-1$ and $0\leq j<n-m$. For any
$g\in\Delta_{s}({\mathcal{T}})$, where $-1\leq s\leq m-1$, the identity

(5.2) $\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m}\\\ f\supset
g\end{subarray}}\psi_{e,g}(f)\wedge
L_{g}^{*}R_{e,f}^{k}u=\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\
f\supset g\end{subarray}}\phi_{e}\wedge L_{g}^{*}R_{e,f}^{k}u,$

holds.

###### Proof.

We consider the simplex $g$ to be fixed. Since
$\psi_{e,g}(f)\in{\mathcal{P}}_{1}^{-}\Lambda^{j}({\mathcal{T}},g^{*})$, it
follows that there exist constants $\\{a_{e,e^{\prime}}(f)\\}$ such that
$\psi_{e,g}(f)=\sum_{e^{\prime}\in\Delta_{j}(g^{*})}a_{e,e^{\prime}}(f)\phi_{e^{\prime}}$.
As a consequence, the identity (4.19) can be expressed as

$\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\ f\supset
g\end{subarray}}\phi_{e}\otimes
z_{e,f}=\sum_{e\in\Delta_{j}(g^{*})}\phi_{e}\otimes\sum_{\begin{subarray}{c}(e^{\prime},f)\in\Delta_{j,m}\\\
f\supset g\end{subarray}}a_{e^{\prime},e}(f)z_{e^{\prime},f},$

which implies that

$\sum_{\begin{subarray}{c}f\in\Delta_{m-1}(e^{*})\\\ f\supset
g\end{subarray}}z_{e,f}=\sum_{\begin{subarray}{c}(e^{\prime},f)\in\Delta_{j,m}\\\
f\supset g\end{subarray}}a_{e^{\prime},e}(f)z_{e^{\prime},f},\quad
e\in\Delta_{j}(g^{*}).$

From the definition of the order reduction operators $R_{e,f}^{k}$, we then
obtain the corresponding identity

$\sum_{\begin{subarray}{c}f\in\Delta_{m-1}(e^{*})\\\ f\supset
g\end{subarray}}R_{e,f}^{k}=\sum_{\begin{subarray}{c}(e^{\prime},f)\in\Delta_{j,m}\\\
f\supset g\end{subarray}}a_{e^{\prime},e}(f)R_{e^{\prime},f}^{k},\quad
e\in\Delta_{j}(g^{*}).$

By applying $L_{g}^{*}$ and then reversing the steps above, we obtain the
desired identity. ∎

In addition to the operators $R_{e,f}^{k}$, we will also use the operators
$Q_{e,f}^{k}$, defined from the functions $w_{e,f}$. More precisely,

$(Q_{e,f}^{k}u)_{\lambda}=\int_{\Omega}(\Pi_{j+1}G_{f}^{*}u)_{\lambda}\wedge
w_{e,f},\quad(e,f)\in\Delta_{j,m},$

where $0\leq m\leq n$ and $-1\leq j<n-m$. We recall that for
$e\in\Delta_{j}(f^{*})$, the functions $w_{e,f}$ are trimmed linear
$(n-j-1)$-forms with support in $\Omega_{f}$, and as a consequence, the
operator $Q_{e,f}^{k}$ maps $k$-forms to $(k-j-1)$-forms. In particular, if
$k<j+1$, then $Q_{e,f}^{k}\equiv 0$. Since $z_{e,f}=(\delta^{+}w)_{e,f}$, we
also have ,

(5.3) $(\delta^{+}Q^{k}u)_{e,f}=R_{e,f}^{k}u.$

Since the operators $Q_{e,f}^{k}$ are constructed from a similar procedure as
the operators $R_{e,f}^{k}$, the new operators will also preserve piecewise
smoothness and the piecewise polynomial spaces. In particular, we have for
$(e,f)\in\Delta_{j,m}$ that

(5.4)
$b^{-(j+1)}Q_{e,f}^{k}({\mathcal{P}}\Lambda^{k}({\mathcal{T}}_{f}))\subset{\mathcal{P}}\Lambda^{k}({\mathcal{S}}_{f}^{c}),$

where ${\mathcal{P}}$ can either be ${\mathcal{P}}_{r}$ or
${\mathcal{P}}_{r}^{-}$. Here, the extra factor $b^{-1}$, as compared to the
mapping properties of the corresponding operators $R_{e,f}^{k}$ given in Lemma
2.2, is due to the fact that the order of the forms $w_{e,f}$ are reduced by
one as compared to the forms $z_{e,f}$. We refer to the proof of Proposition
5.2 of [6] for further details.

######  Lemma 5.2.

Let $1\leq m\leq n$ and $j=n-m$. Assume that $f\in\Delta_{m-1}({\mathcal{T}})$
and $g\in\bar{\Delta}(f)$. The identity

$\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\wedge
L_{g}^{*}\Big{(}(-1)^{j+1}Q_{e,f}^{k+1}du-
dQ_{e,f}^{k}u\Big{)}=\sum_{e\in\Delta_{j}(f^{*})}\beta_{e}(f)\wedge
L_{g}^{*}R_{e,f}^{k}u,$

holds on $\Omega_{f}$.

###### Proof.

For a fixed $e\in\Delta_{j}(f^{*})$, we have

$(-1)^{j+1}Q_{e,f}^{k+1}du-
dQ_{e,f}^{k}u=\int_{\Omega}\Big{(}(-1)^{j+1}\Pi_{j+1}G_{f}^{*}du-
d_{S}\Pi_{j+1}G_{f}^{*}u\Big{)}\wedge w_{e,f}\\\
=(-1)^{j+1}\int_{\Omega}d_{\Omega}\Pi_{j}G_{f}^{*}u\wedge
w_{e,f}=\int_{\Omega}\Pi_{j}G_{f}^{*}u\wedge dw_{e,f},$

where we have used the identity (5.1), and the local support of $w_{e,f}$.
However, from (4.16) we have that

$\sum_{e\in\Delta_{j}(f^{*})}\phi_{e}\otimes
L_{g}^{*}\int_{\Omega}\Pi_{j}G_{f}^{*}u\wedge
dw_{e,f}=\sum_{e\in\Delta_{j}(f^{*})}\beta_{e}(f)\otimes
L_{g}^{*}R_{e,f}^{k}u$

on $\Omega_{f}$, and hence the desired identity follows from (2.1). ∎

##  6\. The local operators $K_{m,f}^{k}$

We recall that the operators $\\{B_{f}^{k}\\}_{f\in\Delta}$, appearing in the
decomposition (1.1), will be defined by the operator $W^{k}$, mapping into the
space of trimmed linear $k$-forms, and from the local operators
$\\{K_{m,f}^{k}\\}_{f\in\Delta_{j}}$, $j=m,m-1$, by formulas (2.24)–(2.25).
The purpose of this section is to define the operators $\\{K_{m,f}^{k}\\}$.
For each value of $m$, $1\leq m\leq n-1$, we define the operators
$K_{m,f}^{k}$, for $f\in\Delta_{m}$, by

(6.1)
$K_{m,f}^{k}u=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}L_{g}^{*}A_{f}^{k}u.$

This is the obvious generalization of the corresponding operators defined for
$k=0$ in Section 3 above. From the properties of the operators $A_{f}^{k}$,
cf. Section 2.4, it follows that the operator $L_{g}^{*}A_{f}^{k}$, and hence
$K_{m,f}^{k}$, has domain of dependence $\Omega_{f}$. Furthermore, the
operator (6.1) commutes with the exterior derivative, preserves piecewise
smoothness and the piecewise polynomial spaces, and by the cancellation
argument, cf. Section 3, we obtain that the functions $K_{m,f}^{k}u$ have
support on $\Omega_{f}$.

For $f\in\Delta_{m-1}$, the generalization of the operators $K_{m,f}^{0}$,
introduced in Section 3, to the case of $k$-forms is less obvious. In this
case, we define $K_{m,f}^{k}$ by an alternating sum of the form

(6.2) $K_{m,f}^{k}=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}K_{m,f,g}^{k},$

where

(6.3)
$K_{m,f,g}^{k}u=-L_{g}^{*}A_{f}^{k}u+\sum_{j=0}^{n-m}\frac{(-1)^{j-1}}{\rho_{g}^{j+1}}\sum_{e\in\Delta_{j}(f^{*})}\Big{(}\phi_{e}+\psi_{e,g}(f)\Big{)}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\ +\sum_{\begin{subarray}{c}e\in\Delta_{n-m}(f^{*})\\\
j=n-m\end{subarray}}(-1)^{j}\Big{(}d\frac{\phi_{e}}{\rho_{g}^{j+1}}\Big{)}\wedge
L_{g}^{*}Q_{e,f}^{k}u.$

These operators reduce to the corresponding operators $K_{m,f}^{0}$ if $k=0$.
To see this, observe that all the operators $Q_{e,f}^{0}$ are identically
zero. Similarly, $R_{e,f}^{0}=0$ if $(e,f)\in\Delta_{j,m}$, $j>0$, and if
$j=0$ then $R_{e,f}^{0}=-A_{\langle e,f\rangle}^{0}$. As a consequence, we
obtain from (6.2) and (6.3) that

$K_{m,f}^{0}u=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\Big{[}-L_{g}^{*}A_{f}^{0}u+\rho_{g}^{-1}\sum_{i\in
I(f^{*})}\Big{(}\lambda_{i}+\psi_{x_{i},g}(f)\Big{)}\wedge L_{g}^{*}A_{\langle
x_{i},f\rangle}^{0}u\Big{]}\\\
=\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\rho_{g}^{-1}\Big{[}\sum_{i\in
I(f^{*})}\Big{(}\lambda_{i}-\frac{\rho_{f}}{|f^{*}|}\Big{)}\wedge
L_{g}^{*}A_{\langle x_{i},f\rangle}^{0}u\Big{]},$

where we have used (2.12) and (4.10). Hence, we can conclude that the
definition above agrees with the definition given in Section 3 when $k=0$, cf.
formula (3.5). We note that the domain of dependence of all the operators in
(6.3) is $\Omega_{f}$ and hence the domain of dependence of $K_{m,f}^{k}$ is
$\Omega_{f}$. By the cancellation argument introduced in Section 3, it is also
easy to see that the functions $K_{m,f}^{k}u$ have support on $\Omega_{f}$. If
$g\in\bar{\Delta}(f)$ and $i\in I(f\cap g^{*})$, then

$\operatorname{tr}_{\lambda_{i}=0}\Big{(}K_{m,f,\\{x_{i},g\\}}^{k}u-K_{m,f,g}^{k}u\Big{)}=0,$

which shows that $K_{m,f}^{k}u$ has support on $\Omega_{x_{i}}$. Furthermore,
since $i\in I(f)$ is arbitrary, the support of $K_{m,f}^{k}u$ must be limited
to

$\Omega_{f}=\bigcap_{i\in I(f)}\Omega_{x_{i}}.$

A key step to show that the operator $K_{m,f}^{k}$ commutes with the exterior
derivative and preserves piecewise smoothness is the following alternative
expression of the first part of the operator $K_{m,f,g}^{k}$.

######  Lemma 6.1.

Assume that $1\leq m\leq n-1$. For each $f\in\Delta_{m-1}({\mathcal{T}})$ and
$g\in\bar{\Delta}(f)$, the identity

(6.4)
$-Lg^{*}A_{f}^{k}u+\sum_{j=0}^{n-m}\frac{(-1)^{j-1}}{\rho_{g}^{j+1}}\sum_{e\in\Delta_{j}(f^{*})}\Big{(}\phi_{e}+\psi_{e,g}\Big{)}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\
=\sum_{j=1}^{n-m}\sum_{e\in\Delta_{j}(f^{*})}\Big{[}d(\mu_{e}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u)+\mu_{e}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k+1}du\Big{]}-\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\frac{\beta_{e}}{\rho_{g}^{j+1}}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\ $

holds on $\Omega_{f}$. Here, the functions $\mu_{e},\,\beta_{e},\,\psi_{e,g}$
are all associated to the simplex $f$.

The proof of the lemma above is partly technical. Therefore, we delay the
proof, and we will first use the identity to prove the following key result.

######  Lemma 6.2.

Let $1\leq m\leq n-1$ and assume $f\in\Delta_{j}({\mathcal{T}})$, $j=m,m-1$.
The function $K_{m,f}^{k}u$ has domain of dependence $\Omega_{f}$ and support
on $\Omega_{f}$. Furthermore, the operator $K_{m,f}^{k}$ commutes with the
exterior derivative and maps the spaces $\Lambda^{k}({\mathcal{T}}_{f})$,
${\mathcal{P}}_{r}^{-}\Lambda^{k}({\mathcal{T}}_{f})$, and
${\mathcal{P}}_{r}\Lambda^{k}({\mathcal{T}}_{f})$ to themselves.

###### Proof.

Following the discussion above, it remains to show that the operator defined
by (6.2) and (6.3) preserves piecewise smoothness and, in particular, the
piecewise polynomial spaces, and that this operator commutes with the exterior
derivative. In fact, we will show that each of the operators $K_{m,f,g}^{k}$
has these properties, where $g\in\bar{\Delta}(f)$. Furthermore, due to the
support property of the functions $K_{m,f}^{k}u$ already verified, it is
enough to establish these properties on the domain $\Omega_{f}$.

It is a consequence of the identity of Lemma 6.1 that the operator
$K_{m,f,g}^{k}$ admits the alternative representation

$K_{m,f,g}^{k}u=\sum_{j=1}^{n-m}\sum_{e\in\Delta_{j}(f^{*})}\Big{(}d(\mu_{e}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u)+\mu_{e}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k+1}du\Big{)}\\\
+\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\Big{(}(-1)^{j}\Big{(}d\frac{\phi_{e}}{\rho_{g}^{j+1}}\Big{)}\wedge
L_{g}^{*}Q_{e,f}^{k}u-\frac{\beta_{e}}{\rho_{g}^{j+1}}\wedge
L_{g}^{*}R_{e,f}^{k}u\Big{)}.$

However, by Lemma 5.2, the two last terms can be rewritten in the form

$\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\Big{[}(-1)^{j}\Big{(}d\frac{\phi_{e}}{\rho_{g}^{j+1}}\Big{)}\wedge
L_{g}^{*}Q_{e,f}^{k}u-\frac{\phi_{e}}{\rho_{g}^{j+1}}\wedge
L_{g}^{*}\Big{(}(-1)^{j+1}Q_{e,f}^{k+1}du-dQ_{e,f}^{k}u\Big{)}\Big{]}\\\
=\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}(-1)^{j}\Big{(}d\Big{(}\frac{\phi_{e}}{\rho_{g}^{j+1}}\wedge
L_{g}^{*}Q_{e,f}^{k}u\Big{)}+\frac{\phi_{e}}{\rho_{g}^{j+1}}\wedge
L_{g}^{*}Q_{e,f}^{k+1}du\Big{)}.$

As a consequence, it follows that the operator $K_{m,f,g}^{k}$ can be
expressed as

$K_{m,f,g}^{k}u=\sum_{j=1}^{n-m}\sum_{e\in\Delta_{j}(f^{*})}\Big{(}d(\mu_{e}(f)\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u)+\mu_{e}(f)\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k+1}du\Big{)}\\\
+\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}(-1)^{j}\Big{(}d\Big{(}\phi_{e}\wedge
L_{g}^{*}b^{-(j+1)}Q_{e,f}^{k}u\Big{)}+\phi_{e}\wedge
L_{g}^{*}b^{-(j+1)}Q_{e,f}^{k+1}du\Big{)}.$

From this representation, commuting with the exterior derivative is obvious,
and the space preserving properties follow directly from the corresponding
properties of the operators $R_{e,f}^{k}$ and $Q_{e,f}^{k}$ given in Lemma 2.2
and (5.4). ∎

Next, we will establish Lemma 6.1.

###### Proof.

(of Lemma 6.1) From the identity (4.9), we obtain

$\sum_{j=0}^{n-m}\sum_{e\in\Delta_{j}(f^{*})}\frac{(-1)^{j-1}}{\rho_{g}^{j+1}}\Big{(}\phi_{e}+\psi_{e,g}\Big{)}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\
=\sum_{j=0}^{n-m}\sum_{e\in\Delta_{j}(f^{*})}\Big{[}d\Big{(}\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}-\frac{\beta_{e}}{\rho_{g}^{j+1}}\Big{]}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\ =\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\Big{[}d\Big{(}\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}-\frac{\beta_{e}}{\rho_{g}^{j+1}}\Big{]}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\
+\sum_{j=0}^{n-m-1}\sum_{e\in\Delta_{j}(f^{*})}\Big{[}d\Big{(}\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}-\frac{(\partial\mu)_{e}}{\rho_{g}^{j+1}}\Big{]}\wedge
L_{g}^{*}R_{e,f}^{k}u.$

where we have used the identity $\beta_{e}=(\partial\mu)_{e}$ from Lemma 4.1,
and where $\partial=\partial(f^{*})$. Recall also that for
$e\in\Delta_{0}(f^{*})$, we obtain from (2.12) that

$\sum_{e\in\Delta_{0}(f^{*})}d\mu_{e}\wedge
L_{g}^{*}R_{e,f}^{k}u=L_{g}^{*}A_{f}^{k}u,$

since $d\mu_{e}=-1/|f^{*}|$ and $R_{e,f}^{k}u=-A_{\langle e,f\rangle}^{k}$.
Hence, the left hand side of (6.4) is given by

(6.5) $\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\Big{[}d\Big{(}\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}-\frac{\beta_{e}}{\rho_{g}^{j+1}}\Big{]}\wedge
L_{g}^{*}R_{e,f}^{k}u-\sum_{e\in\Delta_{0}(f^{*})}\frac{(\partial\mu)_{e}}{\rho_{g}}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\
+\sum_{j=1}^{n-m-1}\sum_{e\in\Delta_{j}(f^{*})}\Big{[}d\Big{(}\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}-\frac{(\partial\mu)_{e}}{\rho_{g}^{j+1}}\Big{]}\wedge
L_{g}^{*}R_{e,f}^{k}u.$

On the other hand, it follows from (2.17) and the Leibniz rule for the wedge
product, that

$d(\mu_{e}\wedge L_{g}^{*}b^{-j}R_{e,f}^{k}u)+\mu_{e}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k+1}du\\\
=\Big{(}d\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}\wedge
L_{g}^{*}R_{e,f}^{k}u+\mu_{e}\wedge
L_{g}^{*}b^{-j}((-1)^{j-1}dR_{e,f}^{k}u+R_{e,f}^{k+1}du)\\\
=\Big{(}d\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}\wedge
L_{g}^{*}R_{e,f}^{k}u-\mu_{e}\wedge L_{g}^{*}b^{-j}(\delta R^{k}u)_{e,f},$

for $e\in\Delta_{j}(f^{*})$, $0\leq j\leq n-m$. As a consequence, the right
hand side of (6.4) is given by

$=\sum_{j=1}^{n-m-1}\sum_{e\in\Delta_{j}(f^{*})}\Big{(}d\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}\wedge
L_{g}^{*}R_{e,f}^{k}u+\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\Big{[}\Big{(}d\frac{\mu_{e}}{\rho_{g}^{j}}\Big{)}-\frac{\beta_{e}}{\rho_{g}^{j+1}}\Big{]}\wedge
L_{g}^{*}R_{e,f}^{k}u\\\
-\sum_{j=0}^{n-m-1}\sum_{e\in\Delta_{j}(f^{*})}\partial\mu_{e}\wedge
L_{g}^{*}b^{-(j+1)}(R^{k}u)_{e,f},$

where we have used (2.2) to rewrite the last term. If we combine terms and
compare this with (6.5), we obtain (6.4). ∎

The definitions of the operators $\\{K_{m,f}^{k}\\}$, given above, combined
with the operators $\\{W^{k}\\}$ introduced in Section 2.5, complete the
construction of all the operators required for the decomposition (1.1), cf.
(2.24) and (2.25).

######  Proposition 6.3.

The operators $\\{B_{f}^{k}\\}$, defined by (2.24) and (2.25), have domain of
dependence $\Omega_{f}$ if $f\in\Delta_{m}$, $m<n$, and $\Omega_{f}^{E}$ if
$m=n$. Furthermore, the functions ${\mathcal{B}}_{f}^{k}u$ have support in
$\Omega_{f}$.

###### Proof.

All these properties follow directly from the corresponding properties of the
operators $\\{K_{m,f}\\}$ given in Lemma 6.2, except for the domain of
dependence property in the case $f\in\Delta_{n}$. However, it is a consequence
(2.25) that in this case the operator $B_{f}^{k}$ has a domain of dependence
included in

$\big{(}\bigcup_{e\in\Delta_{k}(f)}\Omega_{e}^{E}\big{)}\cup\big{(}\bigcup_{g\in\Delta(f)}\Omega_{g}\big{)}\subset\Omega_{f}^{E}.$

∎

To complete the proof of the main theorem of the paper, Theorem 2.3, it
remains to verify the identity (2.23) and to establish the desired operator
bounds. This will be done in the two next sections.

##  7\. Verifying the fundamental identity

The purpose of this section is to establish the fundamental identity (2.23),
i.e.,

$C_{m}^{k}u-\sum_{\begin{subarray}{c}f\in\Delta_{j}\\\
j=m,m-1\end{subarray}}K_{m,f}^{k}u=C_{m-1}^{k}u,\quad 1\leq m\leq n-1.$

Therefore, we have to study sums of the operators $K_{m,f}^{k}$ introduced in
the previous section. As a preliminary step, we study sums of expressions
corresponding to a part of the definition (6.3).

######  Lemma 7.1.

Assume that $1\leq m\leq n-1$. Then

(7.1) $\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j.m-1}\\\ 0\leq j\leq
n-m\end{subarray}}(-1)^{j-1}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\psi_{e,g}(f)}{\rho_{g}}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u\\\
=-\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j.m-2}\\\ 0\leq j\leq
n-m\end{subarray}}(-1)^{j-1}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\phi_{e}}{\rho_{g}}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u.$

###### Proof.

Recall from (4.8) that $\psi_{e,g}(f)=0$ if $g=f$. Therefore, by changing the
order of summation and then using (5.2), the left hand side of (7.1) can be
rewritten as

$\sum_{\begin{subarray}{c}g\in\Delta_{s}\\\ -1\leq s\leq
m-2\end{subarray}}(-1)^{m-|g|}\sum_{j=0}^{n-m}\frac{(-1)^{j-1}}{\rho_{g}^{j+1}}\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\
f\supset g\end{subarray}}\psi_{e,g}(f)\wedge L_{g}^{*}R_{e,f}^{k}u\\\
=\sum_{\begin{subarray}{c}g\in\Delta_{s}\\\ -1\leq s\leq
m-2\end{subarray}}(-1)^{m-|g|}\sum_{j=0}^{n-m}(-1)^{j-1}\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-2}\\\
f\supset g\end{subarray}}\frac{\phi_{e}}{\rho_{g}}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u,$

where the right hand side corresponds exactly to the right hand side of (7.1).
∎

Next, we consider a corresponding sum of the last term of the definition
(6.3).

######  Lemma 7.2.

Assume that $1\leq m\leq n-1$. Then

(7.2) $\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-1}\\\
j=n-m\end{subarray}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\Big{(}d\frac{\phi_{e}}{\rho_{g}^{j+1}}\Big{)}\wedge
L_{g}^{*}Q^{k}_{e,f}u\\\ =-\sum_{\begin{subarray}{c}(e,f)\in\Delta_{j,m-2}\\\
j=n-m+1\end{subarray}}\sum_{g\in\bar{\Delta}(f)}(-1)^{|f|-|g|}\frac{\phi_{e}}{\rho_{g}}\wedge
L_{g}^{*}b^{-j}R_{e,f}^{k}u.$

###### Proof.

Let $f\in\Delta_{m-1}$ and $g\in\bar{\Delta}(f)$ be fixed. Since
$\Omega_{f}\subset\Omega_{g}$, it follows that $\rho_{g}=\sum_{i\in
I(g^{*})}\lambda_{i}$ on $\Omega_{f}$. Since $\dim f^{*}=n-m$ and
$e\in\Delta_{n-m}(f^{*})$, $\phi_{[x_{i},e]}=0$ unless $i\in I(f)$ and so we
have

$d\Big{(}\frac{\phi_{e}}{\rho_{g}^{n-m+1}}\Big{)}=\frac{1}{\rho_{g}^{n-m+2}}\sum_{i\in
I(f\cap g^{*})}\phi_{[x_{i},e]},\quad\text{on }\Omega_{f}.$

As a consequence, when we restrict to $\Omega_{f}$, we can conclude that

$\sum_{\begin{subarray}{c}e\in\Delta_{j}(f^{*})\\\
j=n-m\end{subarray}}\Big{(}d\frac{\phi_{e}}{\rho_{g}^{j+1}}\Big{)}\wedge
L_{g}^{*}Q_{e,f}^{k}u\\\ =\sum_{i\in I(f\cap
g^{*})}\sum_{\begin{subarray}{c}e\in\Delta_{j}(f(\hat{x}_{i})^{*})\\\ e\supset
x_{i}\\\
j=n-m+1\end{subarray}}(-1)^{\sigma_{e(x_{i})}}\frac{\phi_{e}}{\rho_{g}}\wedge
L_{g}^{*}b^{-j}Q_{e(\hat{x}_{i}),f}^{k}u.$

Next, if we sum over all $g\in\bar{\Delta}(f)$ we obtain

(7.3)