# Bounded Poincaré operators for twisted and BGG complexes

Andreas Čap and Kaibo Hu  Faculty of Mathematics, University of Vienna, Oskar-
Morgenstern-Platz 1, 1090 Wien, Austria<EMAIL_ADDRESS>Mathematical
Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory
Quarter, Oxford, OX2 6GG, UK<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

Nous construisons des opérateurs de Poincaré bornés pour les complexes tordus
et les complexes BGG pour une large classe d’espaces fonctionnels (par
exemple, les espaces de Sobolev) sur des domaines de Lipschitz bornés. Ces
opérateurs sont dérivés des versions de de Rham en utilisant les diagrammes
BGG et, pour une cohomologie triviale, satisfont à l’identité d’homotopie
$dP+Pd=I$ pour des degrés $>0$. Les opérateurs préservent les classes
polynomiales si les versions de Rham le font. Le cas d’une cohomologie non
triviale ainsi que la propriété de complexe $P\circ P=0$ peuvent être
incorporés. Nous présentons des applications aux suites exactes de polynômes.

###### Abstract.

We construct bounded Poincaré operators for twisted complexes and BGG
complexes with a wide class of function classes (e.g., Sobolev spaces) on
bounded Lipschitz domains. These operators are derived from the de Rham
versions using BGG diagrams and, for vanishing cohomology, satisfy the
homotopy identity $dP+Pd=I$ in degrees $>0$. The operators preserve polynomial
classes if the de Rham versions do so. Nontrivial cohomology and the complex
property $P\circ P=0$ can be incorporated. We present applications to
polynomial exact sequences.

Keywords. Bernstein-Gelfand-Gelfand construction, Hilbert complexes, Poincaré
operators, polynomial complexes

MSC Codes. 58J10, 65N30

##  1\. Introduction

Poincaré operators are a basic tool in the theory of complexes. In the
simplest case, they can be used to prove vanishing cohomology of a complex
$(V^{{\scriptscriptstyle\bullet}},D^{{\scriptscriptstyle\bullet}})$. For this
purpose, one defines a graded operator $P^{{\scriptscriptstyle\bullet}}$ such
that $P^{k}:D^{k}\to D^{k-1}$ and such that $D^{k-1}P^{k}+P^{k+1}D^{k}=I$ for
$k>0$ together with a simple condition in degree zero. In the case of non-
trivial cohomology, a Poincaré operator determines a subcomplex with
isomorphic cohomology. For this case, the above relation has to be modified to
$D^{k-1}P^{k}+P^{k+1}D^{k}=I-L^{k}$ for operators $L^{k}:D^{k}\to D^{k}$ with
favorable properties. Poincaré operators for the de Rham complex play a key
role in various applications. In manifold theory, the existence of Poincaré
operators on star-shaped domains immediately implies local exactness of the de
Rham complex. In PDE theories, the Poincaré operators serve as the homotopy
inverse of differential operators and thus imply the existence of solutions
for, e.g., the Stokes problem. See, e.g., [1, 23], for the inverse of the
divergence operator. For smooth de Rham complexes, the Poincaré operators also
form a complex $P^{k-1}\circ P^{k}=0$ and are polynomial-preserving. These
properties are important for constructing finite element spaces [5, 17, 18,
29, 30, 31, 32] and establishing $p$-robustness for high order methods (see,
e.g., [10, 25]).

Many of these applications require Poincaré operators that are bounded between
certain function spaces (e.g., Sobolev spaces). Costabel and McIntosh [22]
established such operators and showed that these operators are pseudo-
differential operators of order -1 and thus are bounded in a broad class of
function spaces. The regularized Poincaré and Bogovskiĭ operators are special
cases of this general construction. Consequences of the construction in [22]
include a number of analytic results, such as the Poincaré inequalities, the
existence of regular potential and compactness [8, Section 2]. From explicit
Poincaré operators, one may also estimate constants in inequalities. We refer
to [28] for work in this direction.

In addition to the de Rham complexes, other differential complexes encode
structures in various problems in continuum mechanics [2, 3, 33] and general
relativity [9, 35]. The Bernstein-Gelfand-Gelfand (BGG) machinery provides a
systematic way to construct such complexes [8, 11, 12, 24]. This includes the
elasticity complex as an example. Analytic results of Costabel-McIntosh’s type
were established in [8, 11] for these complexes. The results were directly
carried over from the de Rham complexes via BGG diagrams, which avoided the
use of Poincaré operators. Nevertheless, inspired by applications associated
with the de Rham complexes, explicit Poincaré operators are still in need for
these complexes, as they

  * •

establish exact sequences with polynomial coefficients, which are crucial for
finite element computation;

  * •

facilitate the construction and analysis of $p$-robust numerical methods
(algorithms that are uniformly robust with polynomial degree);

  * •

contain explicit constants, such that refined estimates of the constants are
possible;

  * •

provide tools for establishing mechanics models (e.g., applications of the
Cesàro-Volterra formula in intrinsic elasticity [20]).

For the elasticity complex with smooth functions on domains with trivial
topology, Poincaré operators were derived in [19]. The operator at index one
coincides with the Cesàro-Volterra formula in classical elasticity. However,
these operators are not bounded between function spaces involved in PDEs and
numerical analysis. This makes it impossible to directly generalize several of
the applications discussed for the de Rham complex above to tensor-valued
problems such as elasticity and relativity.

In this paper, we fill this gap by deriving bounded Poincaré operators for BGG
complexes. The basic idea is to follow the BGG diagrams and use complex maps
and a diagram chase to obtain the operators for the BGG complexes from the de
Rham versions. The strategy is a generalization of the special case in [19].
We show that the construction can be applied to the BGG complexes constructed
in [11] from representation theory input, where the two steps in the
construction need different regularity assumptions. In particular, the $K$
operators in the BGG diagram that were a crucial tool in [19] are not
available (as bounded operators) in the second step. Compared to [19],

  * •

The Poincaré operators are bounded and valid for all the BGG complexes in [8]
and [11]. Particularly, this works for not only BGG complexes derived from
diagrams with two rows (e.g., the elasticity complex), but also for those
derived from diagrams with several rows (e.g., the conformal deformation
complex).

  * •

For the examples from [11] coming from representation theory, we show that the
derived operators are polynomial-preserving.

  * •

We develop a general algebraic strategy to modify the derived operators such
that the complex property $\mathscr{P}^{k-1}\circ\mathscr{P}^{k}=0$ holds.

  * •

The work [19] assumes trivial cohomology, while in this paper, we discuss the
incorporation of nontrivial cohomology.

An intermediate step in the construction of BGG complexes is the twisted
complexes, which are important in their own right. In geometry, the twisted
complex can be viewed as providing a prolongation of the operators of
geometric origin that show up in the BGG complexes, (c.f. the interpretation
of the elasticity complex as the Riemannian deformation complex). In continuum
mechanics, the twisted complexes can be interpreted as encoding additional
degrees of freedom compared to the BGG complexes. More specifically, from one
to three space dimensions, Hodge-Laplacian problems of an appropriate twisted
de Rham complex correspond to the linearized Timoshenko beam, the Reissner-
Mindlin plate, and the Cosserat elasticity models, respectively; the resulting
BGG complexes correspond to the linearized Euler-Bernoulli beam, the
Kirchhoff-Love plate, and the standard elasticity, respectively. See [11,
Section 5] for more details. In this paper, we also derive bounded Poincaré
operators for the twisted complexes. Existing results for the elasticity
version of the twisted complex in the context of Cosserat continuum can be
found in [27].

The Cesáro-Volterra formula has been generalized to the case with “little
regularity” in [21]. The construction in [21] leads to an implicit form based
on duality, while the corresponding result in this paper has a more explicit
form, preserves polynomial classes and is valid for a wider range of function
spaces. As a further application of the new Poincaré operators, we construct
exact BGG complexes with polynomial coefficients.

We summarize the main results in this paper using a specific example, i.e.,
the elasticity complex in 3D, which has the form (see Section 2 for the
notation):

(1.1)
${0}$${H^{q}(\Omega)\otimes\mathbb{V}}$${H^{q-1}(\Omega)\otimes\mathbb{S}}$${H^{q-3}(\Omega)\otimes\mathbb{S}}$${H^{q-4}(\Omega)\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

Similar conclusions hold for other complexes, e.g., the strain and stress
complexes in 2D, and the Hessian, $\operatorname{div}\operatorname{div}$,
conformal Hessian, and conformal deformation complexes in 3D [8, 11].

######  Theorem 1.

Let $\Omega$ be a bounded Lipschitz domain. For any real number $q$, there
exist bounded operators $\mathscr{P}^{i},~{}i=1,2,3$ that fit in the following
sequence:

(1.2)
${0}$${H^{q}(\Omega)\otimes\mathbb{V}}$${H^{q-1}(\Omega)\otimes\mathbb{S}}$${H^{q-3}(\Omega)\otimes\mathbb{S}}$${H^{q-4}(\Omega)\otimes\mathbb{V}}$${0,}$$\scriptstyle{\mathscr{P}^{1}}$$\scriptstyle{\mathscr{P}^{2}}$$\scriptstyle{\mathscr{P}^{3}}$

such that the following homotopy relations hold:

$\displaystyle\mathscr{P}^{1}\operatorname{def}u$
$\displaystyle=(I-\mathscr{L}_{V}^{0})u,\quad\forall u\in
H^{q}\otimes\mathbb{V},$
$\displaystyle(\operatorname{def}\mathscr{P}^{1}+\mathscr{P}^{2}\operatorname{inc})e$
$\displaystyle=(I-\mathscr{L}_{V}^{1})e,\quad\forall e\in
H^{q-1}\otimes\mathbb{S},$
$\displaystyle(\operatorname{inc}\mathscr{P}^{2}+\mathscr{P}^{3}\operatorname{div})\sigma$
$\displaystyle=(I-\mathscr{L}_{V}^{2})\sigma,\quad\forall\sigma\in
H^{q-3}\otimes\mathbb{S},$ $\displaystyle\operatorname{div}\mathscr{P}^{3}v$
$\displaystyle=(I-\mathscr{L}_{V}^{3})v,\quad\forall v\in
H^{q-4}\otimes\mathbb{V},$

where $\mathscr{L}_{V}^{i},~{}i=0,1,2,3$ are induced by pseudodifferential
operators of order $-\infty$ and hence are smoothing operators.

If $\Omega$ is a contractible domain, then ${\mathscr{L}}_{V}^{k}=0$ for
$k>0$, while the cohomology in degree $0$ is the space
$\mathrm{RM}:=\\{a+b\wedge x:a,b\in\mathbb{R}^{3}\\}$ of infinitesimal rigid
body motions. If $\Omega$ is further starlike, one can choose the operators
$\mathscr{P}^{{\scriptscriptstyle\bullet}}$ such that the complex property
$\mathscr{P}\circ\mathscr{P}=0$ holds, i.e., (1.2) is a complex. The operators
$\mathscr{P}^{{\scriptscriptstyle\bullet}}$ preserve polynomial classes, i.e.,
the operators fit in the following sequence for any $r\geq 0$:

(1.3)
${0}$${\mathcal{H}_{r+4}\otimes\mathbb{V}}$${\mathcal{H}_{r+3}\otimes\mathbb{S}}$${\mathcal{H}_{r+1}\otimes\mathbb{S}}$${\mathcal{H}_{r}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\mathscr{P}^{1}}$$\scriptstyle{\mathscr{P}^{2}}$$\scriptstyle{\mathscr{P}^{3}}$

where $\mathcal{H}_{s}$ is the space of homogeneous polynomials of degree $s$.

The rest of the paper is organized as follows. In Section 2, we review basic
facts on Poincaré operators as well as the results on existence of bounded
Poincaré operators for de Rham complexes. In Section 3, we present the main
algebraic construction for the BGG versions in a general setting and assuming
trivial cohomology. In Section 4, we first show that our construction can be
applied to the examples from [11] derived from representation theory. We prove
that in these cases the resulting operators are polynomial preserving and that
the complex property can be achieved. Then we discuss several explicit
examples of the construction. In Section 5, we discuss the changes needed to
deal with non-vansihing cohomology. Finally, vector/matrix proxies and an
application to polynomial exact sequences are presented in Section 6.

##  2\. Review: Poincaré operators and de Rham complexes

In this section, we review Poincaré operators, in particular various versions
for de Rham complexes. The results will be a crucial ingredient in the
construction of the BGG version. Moreover, the applications will serve as a
motivation for the development and fix the notation used in the rest of this
paper.

Let $V^{i},i=0,1,\cdots,n$ be vector spaces and $d^{i}:V^{i}\to
V^{i+1},i=0,1,\cdots,n-1$ be linear operators. Then a sequence

${\cdots}$${V^{i-1}}$${V^{i}}$${V^{i+1}}$${\cdots}$$\scriptstyle{d^{i-1}}$$\scriptstyle{d^{i}}$

is called a complex if $d^{i+1}\circ d^{i}=0,~{}\forall i$. A consequence of
the complex property is $\mathcal{R}(d^{i-1})\subset\mathcal{N}(d^{i})$.
Cohomology is the quotient space
$\mathscr{H}^{i}:=\mathcal{N}(d^{i})/\mathcal{R}(d^{i-1})$. Suppose that there
exists linear maps $P^{k}:V^{k}\to V^{k-1},k=1,2,\cdots,n$, such that

(2.1) $d^{k-1}P^{k}+P^{k+1}d^{k}=I_{V^{k}}\text{\ for\ }k>0\text{\ and\
}d^{0}P^{1}d^{0}=d^{0},$

then we call $P^{{\scriptscriptstyle\bullet}}$ an (algebraic) Poincaré
operator. We will call (2.1) a null-homotopy relation. Note that (2.1) for
$k>0$ immediately implies the exactness of the complex and hence vanishing
cohomology. Indeed, for any $u\in V^{k}$ with $du=0$, we get $u=d(Pu)$, so the
class of $u$ in $\mathscr{H}^{k}$ vanishes. The condition in degree $0$
readily implies that for $u\in V^{0}$, $u-Pdu$ lies in the kernel of $d$,
which coincides with the cohomology in degree zero. To allow for nontrivial
cohomology, one can weaken the relation (2.1) to the form

(2.2) $d^{k-1}P^{k}+P^{k+1}d^{k}=I_{V^{k}}-L^{k},$

for a family $L^{\scriptscriptstyle\bullet}$ of linear maps $L^{k}:V^{k}\to
V^{k}$. Observe that the the relation (2.2) readily implies that
$d^{k}L^{k}=d^{k}P^{k+1}d^{k}-d^{k}=L^{k+1}d^{k}$. Hence defining
$W^{k}:=\mathcal{R}(L^{k})\subset V^{k}$, we conclude that
$d^{k}(W^{k})\subset W^{k+1}$, so this defines a subcomplex
$(W^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$ of
$(V^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$. The
inclusions $W^{k}\to V^{k}$ induce a map between the cohomologies of the two
complexes and the maps $L^{k}:V^{k}\to W^{k}$ define a map in the opposite
direction. The relation (2.2) easily implies that these maps induce
isomorphisms in cohomology which are inverse to each other.

There are different choices for $P^{{\scriptscriptstyle\bullet}}$ and
$L^{{\scriptscriptstyle\bullet}}$ such that (2.2) holds (actually, once we
choose $P^{{\scriptscriptstyle\bullet}}$, (2.2) defines
$L^{{\scriptscriptstyle\bullet}}$). We should specify what properties we want
$L^{{\scriptscriptstyle\bullet}}$ to satisfy. Following [22], in this paper we
are mainly concerned with the case where $L^{{\scriptscriptstyle\bullet}}$ are
smoothing operators. Before presenting details of this choice, we mention
another possibility. In fact, the most efficient version of such a relation is
when the resulting differentials on the complex
$W^{\scriptscriptstyle\bullet}$ are the zero maps, i.e. if $d^{k}L^{k}=0$ for
any $k$. Then the cohomology of
$(V^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$ is simply
given by $\mathscr{H}^{i}=\mathcal{R}(L^{i})$. Observe also, that starting
from (2.2) the additional conditions in (2.1) are equivalent to $dL^{0}=0$ and
$L^{k}=0$ for $k>0$.

We have to remark that in the setting of the current article we will rarely
meet the situation of bounded Poincaré operators which lead to $L$-operators
that satisfy the strict relation $d^{k}L^{k}=0$. However, this case is very
important in different situations, whence we decided to also discuss results
on this case. The prime example for this situation comes from Hodge theory on
smooth differential forms on compact Riemannian manifolds. Under this
assumption, one in addition to the exterior derivative
$d:\Omega^{k}(M)\to\Omega^{k+1}(M)$ obtains the codifferential
$\delta:\Omega^{k}(M)\to\Omega^{k-1}(M)$ and hence the Hodge-Laplace operator
$\Delta:=d\delta+\delta d$. Now Hodge theory states that for compact $M$ and
each degree $k$ the kernel of $\Delta$ on $\Omega^{k}(M)$ equals
$\mathcal{N}(d)\cap\mathcal{N}(\delta)$ and isomorphically projects onto the
cohomology in degree $k$. The key towards proving this is the construction of
a bounded operator (called Green’s operator) $G:\Omega^{k}(M)\to\Omega^{k}(M)$
for each $k$ such that $\mathcal{N}(G)=\mathcal{N}(\Delta)$, $Gd=dG$ and
$\delta G=G\delta$ and such that $G\Delta=I-L$, where $L$ is a projection onto
$\mathcal{N}(\Delta)$. In fact, $G$ can be defined by solving a Hodge-
Laplacian problem

$(d\delta+\delta d)G(u):=P_{\mathfrak{H}^{\perp}}u,$

where $P_{\mathfrak{H}^{\perp}}$ is the projection to the ($L^{2}$) orthogonal
complement of harmonic forms. This shows that defining $P:=\delta G$, one
obtains $dP+Pd=G\Delta=I-L$ and of course $dL=0$.

The main facts from Hodge theory can be generalized to the setting of
complexes of densely defined unbounded operators between Hilbert spaces, see
[4, 6, 7]. The BGG machinery can also be applied in that setup and then the
above discussion should apply to the twisted complex and BGG complexes
obtained in that setting.

For the case of the de Rham complex, we consider a domain
$\Omega\subset\mathbb{R}^{n}$ (conditions on $\Omega$ will be imposed later).
Then there is the smooth de Rham complex

${0}$${C^{\infty}\Lambda^{0}}$${C^{\infty}\Lambda^{1}}$${\cdots}$${C^{\infty}\Lambda^{n}}$${0.}$$\scriptstyle{d^{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{n-1}}$

Here $V^{k}:=C^{\infty}\Lambda^{k}$ is the space of smooth differential
$k$-forms and $d^{k}$ is the $k$-th exterior derivatives. Assuming that the
domain $\Omega$ is star shaped around a point $x_{0}\in\Omega$, one can
explicitly define a Poincaré operator. For $t\in[0,1]$, one defines a map
$F_{t}:\Omega\to\Omega$ by $F_{t}(x)=tx+(1-t)x_{0}$. Hence $F_{1}$ is identity
and $F_{0}$ is the constant map $x_{0}$. Now for a $k$-form $u$ on $\Omega$,
define

(2.3)
$(\mathfrak{p}u)_{x}(\xi_{2},\ldots,\xi_{k})=\int_{0}^{1}u_{F_{t}(x)}(\partial_{t}F_{t}(x),DF_{t}(x)\xi_{2},\cdots,DF_{t}(x)\xi_{k})\,dt.$

Then one verifies that $d\mathfrak{p}+\mathfrak{p}d=I$ for $k\geq 1$ and
$\mathfrak{p}du(x)=u(x)-u(x_{0})$ for $k=0$ [13, 17, 34]. The existence of the
$u(x_{0})$ term reflects the fact that the de Rham complex is not exact at
index zero ($d^{0}$ has constants as its kernel). This Poincaré operator is
the standard tool for proving the Poincaré lemma (local exactness of the de
Rham complex) [34]. Of course, if $\Omega$ has nontrivial cohomology then a
Poincaré operator that satisfies (2.1) cannot exist.

For many applications, versions of the de Rham complex with non-smooth
differential forms are needed. To fix the notation, let $\Omega$ be a bounded
Lipschitz domain. Then we let $H^{q}(\Omega)$ be the Sobolev space with all
derivatives of order $\leq q$ in $L^{2}(\Omega)$, and $H_{0}^{q}(\Omega)$ be
the closure of $C^{\infty}_{0}(\Omega)$ with the $H^{q}$ norm. Moreover,
$H^{q}\Lambda^{k}(\Omega)$ is the space of $k$-forms with coefficients in
$H^{q}$. Similarly, one can define $H^{q}_{0}\Lambda^{k}(\Omega)$. We also
omit the domain $\Omega$ when there is no danger of confusion. In the sequel,
we will use $\|\cdot\|_{q}$ to denote the Sobolev $H^{q}$ norm, and
$\|\cdot\|$ for the $L^{2}$ norm.

The following de Rham complex with Sobolev spaces plays an important role in
the analysis of fluid and electromagnetic problems:

(2.4)
${0}$${H^{q}\Lambda^{0}}$${H^{q-1}\Lambda^{1}}$${\cdots}$${H^{q-n}\Lambda^{n}}$${0.}$$\scriptstyle{d^{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{n-1}}$

For the analytical applications, it is desirable to find a Poincaré operator
such that each of the operators $P^{i}$ is bounded. The operators (2.3) cannot
be used for (2.4) as they are not bounded between the Sobolev spaces. Costabel
and McIntosh [22] derived a bounded Poincaré operator for (2.4) on bounded
Lipschitz domains. This starts from the case of a domain $\Omega$ that is
starlike with respect to all points in a ball $B$. Then one chooses a bump
function (a non-negative function whose integral is $1$) $\theta$ with support
in $B$ and averages the operators coming from (2.3) for the points $x_{0}\in
B$ with a weight $\theta(x_{0})$. It turns out that the resulting operators
$P^{\scriptscriptstyle\bullet}$ still satisfy the homotopy relation (2.1), but
in addition, they are pseudodifferential operators of order $-1$ on
$\mathbb{R}^{n}$. This implies favorable analytical properties in a wide range
of spaces, e.g., $W^{k,p}$, Besov spaces and Triebel-Lizorkin spaces. For
simplicity, we focus on Sobolev spaces $H^{q}$, for which obtains bounded
operators $P^{i}:H^{q-i}\Lambda^{i}\to H^{q-(i-1)}\Lambda^{i-1}$. Moreover,
the operators turn out to not only map smooth forms to smooth forms but also
preserve spaces of forms with appropriate polynomial coefficients.

A bounded Lipschitz domain $\Omega\subset\mathbb{R}^{n}$ then admits a finite
covering by domains that are starlike with respect to all points in some open
ball. One then glues together the Poincaré operators for these starlike
domains using a partition of unity. It turns out that the resulting operators
$P^{\scriptscriptstyle\bullet}$ still are pseudodifferential operators of
order $-1$, so analytical properties like boundedness properties are still
available. They satisfy homotopy relations of the form (2.2), but the
properties of the resulting operators $L^{\scriptscriptstyle\bullet}$, which
are caused by the glueing with a partition of unity, are slightly unexpected.
They are not finite rank operators (so their images are much larger than the
cohomology of the de Rham complex) but they are pseudodifferential operators
of order $-\infty$ and thus smoothing operators. So additional work is needed
to complete the analysis of the Sobolev de Rham complex (2.4). This is based
on the fact that, since $L^{\scriptscriptstyle\bullet}$ is a smoothing
operator, $I-L$ is invertible modulo compact operators. We remark that in [22]
also the compactly supported version, i.e. the Hodge duals of Poincaré
operators are studied under the name generalized Bogovskiĭ operators.

The Poincaré operators for (2.4) have several consequences and applications.

  * •

The complex (2.4) has a uniform representation of cohomology for any real
number $q$, i.e.,

$\mathcal{N}(d^{k},H^{q-k}\Lambda^{k})=d^{k-1}H^{q-(k-1)}\Lambda^{k-1}\oplus\mathscr{H}^{k},$

where $\mathscr{H}^{k}$ is a linear subspace of $C^{\infty}\Lambda^{k}$ that
is independent of $q$, see [22]. Moreover, $\dim\mathscr{H}^{k}=\beta_{k}$,
the $k$-th Betti number. The uniform representation of cohomology further
implies analytical results, e.g., the Poincaré inequalities, regular
decomposition and compactness [8].

  * •

The special case of $P^{n}$ implies that for any $q\in L^{2}$ satisfying
$\int_{\Omega}q\,dx=0$, there exists $v\in[H_{0}^{1}]^{n}$ such that
$\operatorname{div}v=q$, and $\|v\|_{1}\leq\|q\|$. This establishes the well-
posedness of the Stokes problem (see, e.g., [26, 36]): given $f\in H^{-1}$,
find $u\in[H_{0}^{1}]^{n}$ and $p\in L^{2}/\mathbb{R}$, such that

$\displaystyle-\Delta u+\nabla p$ $\displaystyle=$ $\displaystyle f,$
$\displaystyle\operatorname{div}u$ $\displaystyle=$ $\displaystyle 0.$

  * •

The Poincaré operators can be used to construct polynomial complexes and show
their exactness, which are needed for finite element methods. Examples include

(2.5)
${\cdots}$${\mathcal{P}_{r-(k-1)}\Lambda^{k-1}}$${\mathcal{P}_{r-k}\Lambda^{k}}$${\mathcal{P}_{r-(k+1)}\Lambda^{k+1}}$${\cdots,}$$\scriptstyle{d^{k-1}}$$\scriptstyle{d^{k}}$

and

(2.6)
${\cdots}$${\mathcal{P}_{r}\Lambda^{k-1}+\mathfrak{p}^{k}\mathcal{P}_{r}\Lambda^{k}}$${\mathcal{P}_{r}\Lambda^{k}+\mathfrak{p}^{k+1}\mathcal{P}_{r}\Lambda^{k+1}}$${\mathcal{P}_{r}\Lambda^{k}+\mathfrak{p}^{k+1}\mathcal{P}_{r}\Lambda^{k+1}}$${\cdots,}$$\scriptstyle{d^{k-1}}$$\scriptstyle{d^{k}}$

where $\mathcal{P}_{r}\Lambda^{k}$ denotes the space of $k$-forms with
polynomial coefficients of degree less than or equal to $r$ [4, 6, 29]. See
also, e.g., [17, 31, 32], for applications to the construction of finite
elements for Stokes problems. The Poincaré operators (2.3) for the smooth de
Rham complex are sufficient for this purpose.

  * •

The bounded Poincaré operators [22] for (2.4) preserve certain function
classes, particularly polynomials. This fact is used to show the
$p$-robustness (uniformity with respect to polynomial degrees) of high order
finite element methods. For this application, it is not enough to know the
exactness (cohomology) of the Sobolev de Rham complexes, but the explicit form
of the Poincaré operators is important.

For $n\leq 3$, the de Rham complex can be written in terms of vector-matrix
proxies on $\mathbb{R}^{n}$. This means that we can choose coordinates and
identify differential forms with the (function) coefficients in front of the
form bases (see [4, 6] for more details). In particular, the proxy of the de
Rham complex in three dimensions involves $\operatorname{grad}$,
$\operatorname{curl}$ and $\operatorname{div}$ operators:

${0}$${C^{\infty}}$${C^{\infty}\otimes\mathbb{R}^{3}}$${C^{\infty}\otimes\mathbb{R}^{3}}$${C^{\infty}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$

Correspondingly, the Poincaré operators have the following vector-matrix
proxies (without loss of generality, we have chosen 0 as the base point and
the path $\gamma(t)=tx$ connecting $0$ and $x$; see, e.g., [29, (7)-(8)], [17,
(34)]):

(2.7) $\displaystyle P^{1}u(x)=\int_{0}^{1}u(tx)\cdot x\,dt,\quad
P^{2}v(x)=\int_{0}^{1}tv(tx)\wedge xdt,\quad
P^{3}w(x)=\int_{0}^{1}t^{2}w(tx)x\,dx.$

In the proxies below, we use the following convention of notation:

(2.8) $\int_{0}^{x}u(y)\cdot dy:=\int_{\gamma}u(y)\,dy=\int_{0}^{1}u(tx)\cdot
x\,dt,$

where $\gamma(t):=tx$ for $t\in[0,1]$ is as above. Similarly, for smooth
functions $v=(v^{1},v^{2},v^{3})$ and $w$,

(2.9) $\int_{0}^{x}dy\wedge v(y)=-\int_{0}^{x}v(y)\wedge
dy:=\left(\begin{array}[]{c}\int_{0}^{1}tv_{3}(tx)x^{2}\,dt-\int_{0}^{1}tv_{2}(tx)x^{3}\,dt\\\
\int_{0}^{1}tv_{1}(tx)x^{3}\,dt-\int_{0}^{1}tv_{3}(tx)x^{1}\,dt\\\
\int_{0}^{1}tv_{2}(tx)x^{1}\,dt-\int_{0}^{1}tv_{1}(tx)x^{2}\,dt\end{array}\right),$
(2.10) $\int_{0}^{x}w(y)\otimes dy=\int_{0}^{x}dy\otimes
w(y):=\int_{0}^{1}t^{2}w(tx)x\,dt.$

In this notation, (2.7) becomes

(2.11) $\displaystyle P^{1}u(x)=\int_{0}^{x}u(y)\cdot dy,\quad
P^{2}v(x)=\int_{0}^{x}v(y)\wedge dy,\quad P^{3}w(x)=\int_{0}^{x}w(y)\otimes
dy.$

These operators are not bounded between the desired Sobolev spaces because the
integral operators involve fixed base points.

In elasticity, the Cesàro-Volterra integral operator provides an explicit
potential for strain tensors satisfying the Saint-Venant compatibility
condition. In terms of the elasticity complex, this can be viewed as a
Poincaré operator in the elasticity complex at index one

(2.12)
${0}$${C^{\infty}\otimes\mathbb{V}}$${C^{\infty}\otimes\mathbb{S}}$${C^{\infty}\otimes\mathbb{S}}$${C^{\infty}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

where $\operatorname{def}u:=\frac{1}{2}(\nabla u+\nabla^{T}u)$ is the
symmetric gradient,
$\operatorname{inc}u:=\operatorname{curl}\circ\mathrm{T}\circ\operatorname{curl}u$
is the incompatibility operator (the Saint-Venant operator, or the linearized
Einstein tensor up to a constant coefficient), and $\operatorname{div}$ is the
column-wise divergence. Here and in the sequel, $\mathbb{V}:=\mathbb{R}^{n}$
denotes vectors and $\mathbb{M}$ is the space of all $n\times n$-matrices. Let
further $\mathbb{S}$ and $\mathbb{T}$ be the subspaces of matrices that are
symmetric and trace-free, respectively.

##  3\. Poincaré operators for BGG complexes with trivial cohomology

In this section, we derive bounded Poincaré operators for the BGG complexes in
the general (abstract) setting used in [11]. For different parts of the
construction in that reference different assumptions are needed, so we have to
connect to the right point in the development.

Let $Z^{i,j},~{}0\leq i\leq n,0\leq j\leq N$ be Hilbert spaces and
$d^{i,j}:Z^{i,j}\to Z^{i+1,j}$ be bounded linear operators such that for each
$j$ we obtain a complex
$(Z^{{\scriptscriptstyle\bullet},j},d^{{\scriptscriptstyle\bullet},j})$, i.e.
$d^{i+1,j}\circ d^{i,j}=0$ for all $i,j$. Assume further that we have given
bounded linear operators $S^{i,j}:Z^{i,j}\to Z^{i+1,j-1}$ satisfying
$d^{i+1,j-1}S^{i,j}=-S^{i+1,j}d^{i,j}$, $S^{i+1,j-1}\circ S^{i,j}=0,\forall
i,j$ (or, in short, $dS=-Sd$ and $S^{2}=0$). At this stage, we don’t need or
use the operators called $K^{i,j}$ in [11].

Let $Y^{i}:=Z^{i,0}\times\cdots\times Z^{i,N}$. Following the convention in
[11], we use $S^{i}:Y^{i}\to Y^{i+1}$ to denote the vectorized operator, i.e.,
for any $u^{i,j}\in Z^{i,j}$,

$S^{i}(u^{i,0},u^{i,1},\cdots,u^{i,N}):=\left(\begin{array}[]{ccccc}0&S^{i,1}&&&\\\
&0&S^{i,2}&&\\\ &&\ddots&\ddots&\\\ &&&0&S^{i,n}\\\
&&&&0\end{array}\right)\left(\begin{array}[]{c}u^{i,0}\\\ u^{i,1}\\\ \vdots\\\
u^{i,N-1}\\\
u^{i,N}\end{array}\right)=(S^{i,1}u^{i,1},\cdots,S^{i,N}u^{i,N},0).$

These ingredients are sufficient to define the twisted complex: Defining the
twisted operator $d_{V}:=d-S$, the twisted complex is

(3.1)
${\cdots}$${Y^{i-1}}$${Y^{i}}$${Y^{i+1}}$${\cdots,}$$\scriptstyle{d_{V}^{i-1}}$$\scriptstyle{d_{V}^{i}}$

Also, the construction of the BGG complex from the twisted complex [11] can be
carried out with the ingredients we have at hand so far, the $K$ operators are
not needed in this construction.

Now we assume in addition that for each of the initial complexes
$(Z^{{\scriptscriptstyle\bullet},j},d^{{\scriptscriptstyle\bullet},j})$, we
have given bounded Poincaré operators. This means that for each $i,j$, we have
given a bounded operator $P^{i,j}:Z^{i,j}\to Z^{i-1,j},i=1,2,\cdots,n$, such
that

(3.2) $d^{i-1,j}P^{i,j}+P^{i+1,j}d^{i,j}=I_{Z^{i,j}}\text{\ for\ }i>0\text{\
and\ }d^{0,j}P^{1,j}d^{0,j}=d^{0,j}.$

As noted in Section 2 already, this implies that the cohomologies of all the
complexes are trivial. In typical examples, each of the complexes
$(Z^{{\scriptscriptstyle\bullet},j},d^{{\scriptscriptstyle\bullet},j})$ is a
vector-valued de Rham complex on a star shaped domain, and appropriate
Poincaré operators are provided by [22]. As above, we define $P^{i}:Y^{i}\to
Y^{i-1}$ as the vectorized component-wise version of $P^{i,j}$.

Our setup is summarized in the following diagram:

(3.3)
${0}$${Z^{0,0}}$${Z^{1,0}}$${\cdots}$${Z^{n,0}}$${0}$${0}$${Z^{0,1}}$${Z^{1,1}}$${\cdots}$${Z^{n,1}}$${0}$${\cdots}$${\cdots}$${\cdots}$${\cdots}$${0}$${Z^{0,N}}$${Z^{1,N}}$${\cdots}$${Z^{n,N}}$${0.}$$\scriptstyle{d^{0,0}}$$\scriptstyle{P^{1,0}}$$\scriptstyle{d^{1,0}}$$\scriptstyle{P^{2,0}}$$\scriptstyle{d^{n-1,0}}$$\scriptstyle{P^{n,0}}$$\scriptstyle{d^{0,1}}$$\scriptstyle{S^{0,1}}$$\scriptstyle{P^{1,1}}$$\scriptstyle{d^{1,1}}$$\scriptstyle{S^{1,1}}$$\scriptstyle{P^{2,1}}$$\scriptstyle{d^{n-1,1}}$$\scriptstyle{S^{n-1,1}}$$\scriptstyle{P^{n,1}}$$\scriptstyle{S^{0,2}}$$\scriptstyle{S^{1,2}}$$\scriptstyle{S^{n-1,2}}$$\scriptstyle{d^{0,N}}$$\scriptstyle{S^{0,N}}$$\scriptstyle{P^{1,N}}$$\scriptstyle{d^{1,N}}$$\scriptstyle{S^{1,N}}$$\scriptstyle{P^{2,N}}$$\scriptstyle{d^{n-1,N}}$$\scriptstyle{S^{n-1,N}}$$\scriptstyle{P^{n,N}}$

######  Remark 1.

In the following developments, the composition $PS$ will play an important
role. The components of this are $P^{i+1,j-1}\circ S^{i,j}:Z^{i,j}\to
Z^{i,j-1}$, so the map between the same spaces as the operators $K^{i,j}$ used
in [11]. As we shall see in the proof of Lemma 1, the analogy goes further,
since $d(PS)-(PS)d=S$. Hence it is tempting to view $PS$ as a “replacement”
for the $K$ operators. This is even more tempting in the situation of diagrams
with two rows from [8] since in that case $e^{-K}=1-K$ which looks even close
to what we are doing here. One has to be aware of the fact that there are
substantial and important differences between the two situations, in
particular in the examples derived from de Rham complexes. In this situation
the $K$ operators from [11] are local operators defined on all of
$\mathbb{R}^{n}$ and can be used on any domain in $\mathbb{R}^{n}$ simply by
restriction. They induce isomorphisms between the sum of the initial complexes
and the twisted complex on any domain, regardless of whether the topology is
trivial or not. In contrast, the version of Poincaré operators we need here is
only available for star-shaped domains and depends on the global behavior on
the domain. Also the isomorphisms between the sum of the original complexes
and the twisted complex is constructed in an essentially different way here.

We will derive the Poincaré operators for the BGG complex in two steps. The
first step is to construct bounded Poincaré operators for the twisted complex
$(Y^{{\scriptscriptstyle\bullet}},d_{V}^{{\scriptscriptstyle\bullet}})$. The
second step is then to pass to the BGG complex
$(\Upsilon^{{\scriptscriptstyle\bullet}},\mathscr{D}^{{\scriptscriptstyle\bullet}})$.
The derivation is sketched in the following diagram:

(3.4)
${Y^{i}}$${Y^{i+1}}$${Y^{i}}$${Y^{i+1}}$${\Upsilon^{i}}$${\Upsilon^{i+1}}$$\scriptstyle{F}$$\scriptstyle{d^{i}}$$\scriptstyle{P^{i+1}}$$\scriptstyle{F}$$\scriptstyle{B^{i}}$$\scriptstyle{d_{V}^{i}}$$\scriptstyle{P_{V}^{i+1}}$$\scriptstyle{B^{i+1}}$$\scriptstyle{\mathscr{D}^{i}}$$\scriptstyle{A^{i}}$$\scriptstyle{\mathscr{P}^{i+1}}$$\scriptstyle{A^{i+1}}$

Here $F$ is an isomorphism of complexes. On the other hand, $A$ and $B$ are
morphisms of complexes which induce isomorphisms in cohomology.

Details will be explained in the rest of this section.

###  3.1. Poincaré operators for the twisted complex

We first use the Poincaré operators for the initial complexes to construct an
isomorphism between the sum of these complexes and the twisted complex.

######  Lemma 1.

We have $d(I-PS)=(I-PS)d_{V}$.

###### Proof.

We have

$(I-PS)d_{V}=(I-PS)(d-S)=d-S+PdS=d-dPS=d(I-PS),$

where we have used $dP+Pd=I$, $S^{2}=0$ and $dS=-Sd$. This implies the desired
result. ∎

Since the bounded operator $PS$ maps each of the spaces $Z^{i,j}$ to
$Z^{i,j-1}$ it is evidently nilpotent and hence $I-PS$ is invertible. Define
$F^{i}:=(I-P^{i+1}S^{i})^{-1}=\sum_{l=0}^{\infty}(P^{i+1}S^{i})^{l}$, noting
that the sum is finite since $PS$ is nilpotent. Lemma 1 implies that the
following diagram commutes:

(3.5)
${\cdots}$${Y^{i}}$${Y^{i+1}}$${\cdots}$${\cdots}$${Y^{i}}$${Y^{i+1}}$${\cdots.}$$\scriptstyle{F^{i}}$$\scriptstyle{d}$$\scriptstyle{F^{i+1}}$$\scriptstyle{d_{V}}$

From the commuting diagram (3.5), we immediately get bounded Poincaré
operators for the twisted complex.

######  Theorem 2.

Define

(3.6)
$P_{V}^{i}:=F^{i-1}P^{i}(F^{i})^{-1}=\sum_{l=0}^{\infty}(P^{i}S^{i-1})^{l}P^{i}(I-P^{i+1}S^{i}).$

We have

(3.7) $d^{i-1}_{V}P^{i}_{V}+P^{i+1}_{V}d^{i}_{V}=I\text{\ for\ }i>0\text{\
and\ }d^{0}_{V}P^{1}_{V}d^{0}_{V}=d^{0}_{V}.$

###### Proof.

Exert $F$ on the left and $F^{-1}$ on the right of (3.2) and use Lemma 1. ∎

In the special case where the diagram has two rows ($N=2$), $F^{i}$ has the
form

$F^{i}=\left(\begin{array}[]{cc}I&P^{i+1}S^{i}\\\ 0&I\end{array}\right).$

Now the definition (3.6) of $P_{V}$ boils down to $P_{V}=(I+PS)P(I-PS)$. If
$P\circ P=0$, we further have $P_{V}=P+PSP$, which has the matrix form

(3.8) $P_{V}=\left(\begin{array}[]{cc}I&PS\\\
0&I\end{array}\right)\left(\begin{array}[]{cc}P&0\\\
0&P\end{array}\right)\left(\begin{array}[]{cc}I&-PS\\\
0&I\end{array}\right)=\left(\begin{array}[]{cc}P&-P(PS-SP)\\\
0&P\end{array}\right).$

###  3.2. BGG complexes

For this step, we have to make an additional assumption, namely that for each
$i,j$, the operator $S^{i,j}:Z^{i,j}\to Z^{i+1,j-1}$ has closed range. This of
course implies that also $S^{i}:Y^{i}\to Y^{i+1}$ has closed range. The spaces
showing up in the BGG complex associated to the twisted complex are then
closed linear subspaces of the spaces $Y^{i}$ that form the twisted complex.
One defines
$\Upsilon^{i}:=\mathcal{N}(S^{i})\cap\mathcal{R}(S^{i-1})^{\perp}$, or more
explicitly,

$\Upsilon^{i}:=\\{(u_{0},\cdots u_{N})\in
Y^{i}:u_{0}\in\mathcal{R}(S^{i-1,1})^{\perp},~{}u_{\ell}\in\mathcal{R}(S^{i-1,\ell+1})^{\perp}\cap\mathcal{N}(S^{i,\ell}),\ell=2,\cdots,N-1,~{}u_{N}\in\mathcal{N}(S^{i,N})\\}.$

Intuitively, $\Upsilon^{{\scriptscriptstyle\bullet}}$ comes from eliminating
(as much as possible) subspaces that are connected isomorphically by the $S$
operators from (3.3). One then defines bounded operators

(3.9)
${\cdots}$${\Upsilon^{i-1}}$${\Upsilon^{i}}$${\Upsilon^{i+1}\to\cdots}$$\scriptstyle{\mathscr{D}^{i-1}}$$\scriptstyle{\mathscr{D}^{i}}$

as follows. Since $S^{i}\circ S^{i-1}=0$, we conclude that
$\mathcal{R}(S^{i-1})$ is a closed linear subspace of $\mathcal{N}(S^{i})$ and
indeed $\mathcal{N}(S^{i})=\mathcal{R}(S^{i-1})\oplus\Upsilon^{i}$. Let us
denote by $\Pi_{\mathcal{N}(S^{i})}$, $\Pi_{\mathcal{R}(S^{i})^{\perp}}$ and
$\Pi_{\Upsilon^{i}}$ be the (orthogonal) projections onto
$\mathcal{N}(S^{i})$, $\mathcal{R}(S^{i})^{\perp}$ and $\Upsilon^{i}$,
respectively. For simplicity of notation, we will simply write
$\Pi_{\mathcal{N}}$, $\Pi_{\mathcal{R}^{\perp}}$ and $\Pi_{\Upsilon}$, when
there is no danger of confusion. Recall from [11] that we define
$T:=(S|_{\mathcal{N}(S)^{\perp}})^{-1}\Pi_{\mathcal{R}(S)}$ and denote its
components by $T^{i,j}:Z^{i,j}\to Z^{i-1,j+1}$, so
$T^{i,j}:=(S^{i-1,j+1}|_{\mathcal{N}(S)^{\perp}})^{-1}\circ\Pi_{\mathcal{R}(S^{i-1,j+1})}$.
Then $G^{i}:Z^{i}\to Z^{i-1}$ is defined by
$G^{i}:=-\sum_{k=0}^{\infty}(T^{i}d^{i-1})^{k}T^{i}$. Next, one defines
$A^{i}:\Upsilon^{i}\to Y^{i}$ by $A^{i}:=I-G^{i+1}d_{V}^{i}$ and the operators
$\mathscr{D}^{{\scriptscriptstyle\bullet}}$ in (3.9) are given by
$\mathscr{D}^{i}:=P_{\Upsilon^{i+1}}d_{V}^{i}A^{i}$. It then turns out that
$d^{i}_{V}A^{i}=A^{i+1}\mathscr{D}^{i}$ so $A$ defines a cochain map from the
sequence (3.9) to the twisted complex. Via this, the fact that
$d^{i+1}_{V}\circ d^{i}_{V}=0$ easily implies that also (3.9) is a complex,
and hence $A$ induces a map in cohomology. Finally,
$B^{i}:Y^{i}\to\Upsilon^{i}$ defined by
$B^{i}:=\Pi_{\Upsilon^{i}}(I-d_{V}^{i-1}G^{i})$ provides a cochain map in the
other direction, such that $BA=I$. One also shows that $AB$ is chain homotopic
to the identity, whence these two chain maps induced isomorphisms in
cohomology, which are inverse to each other.

Recalling the diagram (3.4), we define the composition of chain maps
$\pi^{i}:=B^{i}F^{i}$ and $\pi_{\dagger}^{i}:=(F^{i})^{-1}A^{i}$. Now we
define Poincaré operators for the BGG complex by

(3.10)
$\displaystyle\mathscr{P}^{i}:=B^{i-1}P^{i}_{V}A^{i}=\pi^{i-1}P^{i}\pi_{\dagger}^{i}.$

######  Theorem 3.

The Poincaré operators satisfy

$\mathscr{D}^{i-1}\mathscr{P}^{i}+\mathscr{P}^{i+1}\mathscr{D}^{i}=I{\text{\
for\ }i>0\text{\ and\
}\mathscr{D}^{0}\mathscr{P}^{1}\mathscr{D}^{0}=\mathscr{D}^{0}.}$

###### Proof.

By definition, we have

$\mathscr{D}\mathscr{P}+\mathscr{P}\mathscr{D}=\mathscr{D}BP_{V}A+BP_{V}A\mathscr{D}.$

Now $A^{{\scriptscriptstyle\bullet}}$ and $B^{{\scriptscriptstyle\bullet}}$
are cochain maps, meaning that $\mathscr{D}B=Bd_{V}$ and
$A\mathscr{D}=d_{V}A$. Moreover, $BA=I$ [11, p.9]. This immediately implies
the claim. ∎

In the case where the diagram has two rows ($N=2$),

(3.11) $A=\left(\begin{array}[]{cc}I&0\\\
Td&\Pi_{\mathcal{N}}\end{array}\right),\quad
B=\left(\begin{array}[]{cc}\Pi_{\mathcal{R}^{\perp}}&0\\\
\Pi_{\mathcal{N}}dT&\Pi_{\mathcal{N}}\end{array}\right),$

and

$\displaystyle\mathscr{P}=BP_{V}A$
$\displaystyle=\left(\begin{array}[]{cc}\Pi_{\mathcal{R}^{\perp}}&0\\\
\Pi_{\mathcal{N}}dT&\Pi_{\mathcal{N}}\end{array}\right)\left(\begin{array}[]{cc}P&-P(PS-
SP)\\\ 0&P\end{array}\right)\left(\begin{array}[]{cc}I&0\\\
Td&\Pi_{\mathcal{N}}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\Pi_{\mathcal{R}^{\perp}}(P-P(PS-
SP)Td)&-\Pi_{\mathcal{R}^{\perp}}P(PS-SP)\Pi_{\mathcal{N}}\\\
\Pi_{\mathcal{N}}dT(P-P(PS-
SP)Td)+\Pi_{\mathcal{N}}PTd&-\Pi_{\mathcal{N}}dTP(PS-
SP)\Pi_{\mathcal{N}}+\Pi_{\mathcal{N}}P\Pi_{\mathcal{N}}\end{array}\right).$

The explicit formula has several terms, but in examples, many of the $S$
operators are injective or surjective. This will simplify the formulas. More
precisely, if $S^{i,j}:Z^{i,j}\to Z^{i+1,j-1}$ is injective, then $Z^{i,j}$ is
removed from the BGG complex
$(\Upsilon^{{\scriptscriptstyle\bullet}},\mathscr{D}^{{\scriptscriptstyle\bullet}})$,
i.e., the $i$-th component of $\Upsilon^{j}$ is zero. Similarly, if
$S^{i,j}:Z^{i,j}\to Z^{i+1,j-1}$ is surjective, then $Z^{i+1,j-1}$ is removed.

##  4\. Examples

In this section, we present the construction of Poincaré operators for the
examples derived from representation theory in [11]. In particular, we verify
that the assumptions in the previous section are satisfied. Moreover, in this
setup, we discuss additional properties such as the polynomial-preservation
and the complex property.

###  4.1. The setting of [11]

Let us first describe explicitly how to apply the general scheme derived in
Section 3 to the BGG complexes constructed using representation theory in
[11]. We will also deduce that in this setting the resulting operators
preserve appropriately chosen subspaces of polynomial maps which is important
in applications to finite element methods. The basic input for the
construction in [11] (in dimension $n$) is the choice of a representation
$\mathbb{V}$ of $O(n)$ which decomposes as
$\mathbb{V}=\oplus_{i=0}^{N}\mathbb{V}_{i}$ into $O(n)$-invariant subspaces.
For appropriate choices of $\mathbb{V}$, representation theory provides us
with linear maps
$\partial^{i.j}:\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}\to\Lambda^{i+1}\mathbb{R}^{n*}\otimes\mathbb{V}_{j-1}$
such that $\partial^{i+1,j-1}\circ\partial^{i.j}=0$. Then one fixes a number
$q\in\mathbb{R}$ and a bounded domain $\Omega\subset\mathbb{R}^{n}$ and
defines
$Z^{i,j}:=H^{q-i-j}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j})$,
which can be viewed as the Sobolev space of differential $i$-forms on $\Omega$
with values in the vector space $\mathbb{V}_{j}$ and Sobolev regularity
$q-i-j$. In particular, for each $i,j$, the exterior derivative defines a
bounded linear operator $d^{i,j}:Z^{i,j}\to Z^{i+1,j}$, because the Sobolev
regularity in the target is one lower than in the source. On the other hand,
the Sobolev regularities in $Z^{i,j}$ and $Z^{i+1,j-1}$ are the same, so
$\partial^{i.j}$ induces a bounded linear operator $S^{i,j}:Z^{i,j}\to
Z^{i+1,j-1}$ with closed range. It is shown in [11] that all the assumptions
we made in Section 3 are satisfied and that the cohomology of
$(Y^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet}_{V})$ is
isomorphic to the direct sum of the cohomologies of the complexes
$(Z^{{\scriptscriptstyle\bullet},j},d^{{\scriptscriptstyle\bullet},j})$.

In this situation, appropriate Poincaré operators for the de Rham complex are
constructed in the work [22] of Costabel and McIntosh. For a domain
$\Omega\subset\mathbb{R}^{n}$ which is starlike with respect to all points in
some open ball, they construct integral operators $R_{\ell}$ on compactly
supported $\ell$-forms on $\mathbb{R}^{n}$ which satisfy appropriate null
homotopy relations and by Theorem 3.2 of that reference, these are
pseudodifferential operators of order $-1$ with symbol in an appropriate
Hörmander class. In particular, for bounded domain which is starlike to some
ball, they induce bounded linear operators
$H^{s}(\Omega,\Lambda^{\ell}\mathbb{R}^{n*})\to
H^{s+1}(\Omega,\Lambda^{\ell-1}\mathbb{R}^{n*})$ that satisfy appropriate
homotopy relations. Now of course we can also identify $Z^{i,j}$ with
$H^{q-i-j}(\Omega,\Lambda^{i}\mathbb{R}^{n*})\otimes\mathbb{V}_{j}$, so
tensorizing with the identity on $\mathbb{V}_{j}$, the operator $R_{i}$ gives
rise to bounded linear operators $P^{i,j}:Z^{i,j}\to Z^{i-1,j}$ for each $i$
and $j$. Hence the construction in Section 3 applies and we get

######  Theorem 4.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain which is starlike with
respect to an open ball. Then for any representation $\mathbb{V}$ as discussed
in Section 3 of [11] there is a bounded Poincaré operator for both the twisted
complex and the BGG complex on $\Omega$ constructed from $\mathbb{V}$ as in
that reference.

For later use, we observe that the Poincaré operators for the twisted complex
have additional analytic properties. Indeed, since the Poincaré operators for
the de Rham complex we start with are restrictions of pseudodifferential
operators of order $-1$ on $\mathbb{R}^{n}$, the same is true for the
operators $PS$ and hence its powers are such restrictions, too. Consequently,
also the Poincaré operators for the twisted complex obtained in Theorem 4 are
restrictions of pseudodifferential operators of order $-1$. They also have the
same support properties as the Poincaré operators for the de Rham complex
constructed in [22].

###  4.2. Polynomial-preservation

A remarkable property of the constructions in [22] is that the operators
preserve subspaces of differential forms with polynomial coefficients under
rather weak and general assumptions. We can carry this over to the BGG
complexes constructed in [11] under appropriate assumptions. Following [22],
we take a kind of axiomatic approach to this, i.e., we consider subspaces
$\Pi^{i,j}\subset
C^{\infty}(\mathbb{R}^{n},\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j})$ of
polynomial maps and impose conditions on these subspaces that are needed in
order to make sure that they are preserved by the Poincaré operator. The first
two conditions are the exact analog of the conditions in [22]:

  * (a)

For any $t\in\mathbb{R}$, $a\in\mathbb{R}^{n}$ and $u\in\Pi^{i,j}$, also
$x\mapsto u(tx+a)$ lies in $\Pi^{i,j}$.

  * (b)

For any $u\in\Pi^{i,j}$ and the Euler vector field $E(x)=x$, we get
$i_{E}u\in\Pi^{i-1,j}$.

The following condition, which would make sense in the setting of [22] is not
explicitly stated or used there. It is very natural, however, for constructing
subcomplexes of polynomial forms and it is needed to pass to BGG complexes.

  * (c)

For any $i,j$, $d(\Pi^{i,j})\subset\Pi^{i+1,j}$.

Finally, we need two conditions relating the subspace to the operators that
occur in the BGG construction, namely:

  * (d)

For any $i<n$ and $j>0$ we have $S(\Pi^{i,j})\subset\Pi^{i+1,j-1}$.

  * (e)

For any $i>0$ and $j<N$ we have $T(\Pi^{i,j})\subset\Pi^{i-1,j+1}$.

Now on any bounded domain $\Omega\subset\mathbb{R}^{n}$, the restriction of
any form with polynomial coefficients lie in any Sobolev space. Consequently,
the spaces $\Pi^{i,j}$ are always contained in the spaces $Z^{i,j}$ used above
and we get

######  Corollary 1.

In the setting of Theorem 4 above, suppose that we have selected spaces
$\Pi^{i,j}$ of polynomial $i$-forms with coefficients in $\mathbb{V}_{j}$ such
that the assumptions (a)–(e) from above are all satisfied and put
$\Pi^{i}:=\oplus_{j}\Pi^{i,j}$. Then the Poincaré operators for the twisted
complex constructed in Theorem 4 map each $\Pi^{i}$ to $\Pi^{i-1}$, while the
Poincaré operators for the BGG complex map each $\Upsilon^{i}\cap\Pi^{i}$ to
$\Upsilon^{i-1}\cap\Pi^{i-1}$.

###### Proof.

By [22], assumptions (a) and (b) imply that the Poincaré operator $P$ for the
de Rham complex have the property that $P^{i,j}(\Pi^{i,j})\subset\Pi^{i-1,J}$
and together with (c), we conclude that
$(I-P^{i+1}S^{i})(\Pi^{i})\subset\Pi^{i}$. Similarly, all powers of
$P^{i}S^{i-1}$ map $\Pi^{i-1}$ to itself, so the claim on the twisted complex
follows readily from Theorem 2.

To pass to the BGG complex, we just have to observe that by definition, we get
$G^{i}(\Pi^{i})\subset\Pi^{i-1}$ and $A^{i}(\Pi^{i})\subset\Pi^{i}$. For the
map $B^{i}$, we just have to observe that the projection onto $\Upsilon^{i}$
can be written as $I-S^{i-1}T^{i}-T^{i+1}S^{i}$, so this also maps $\Pi^{i}$
to itself. Hence also the claim on the BGG complex follows directly from the
construction. ∎

Observe that the conditions (a)–(e) are not difficult to satisfy. For example,
we can fix some homogeneity $m>n+N$, denote by $\mathcal{H}^{r}$ the space of
homogeneous polynomials of degree $r\in\mathbb{N}$ and then put
$\Pi^{i,j}:=\mathcal{H}^{m-i-j}\otimes\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$.
The conclusion also holds for
$\Pi^{i,j}:=\mathcal{P}_{m-i-j}\otimes\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$,
where $\mathcal{P}_{r}=(\oplus_{\ell\leq r}\mathcal{H}^{r})$ is the space of
polynomials of degree at most $r$.

###  4.3. Complex property

Another favorable property of a Poincaré operator is the complex property,
i.e. that $P^{i-1}\circ P^{i}=0$ for any $i$. This property is obviously
preserved under the passage to the twisted complex discussed in Section 3.1.
The situation is not so simple for the passage to the BGG complex. From the
definition, it follows directly that

$\mathscr{P}^{i-1}\mathscr{P}^{i}=B^{i-2}P^{i-1}_{V}A^{i-1}B^{i-1}P^{i}_{V}A^{i},$

and $A^{i-1}B^{i-1}$ is far from being the identity. Fortunately, there is a
universal solution to this problem, which does not destroy boundedness or the
property of being a pseudodifferential operator.

######  Theorem 5.

Let $(V^{{\scriptscriptstyle\bullet}},D^{{\scriptscriptstyle\bullet}})$ be a
complex and $P^{{\scriptscriptstyle\bullet}}$ be a Poincaré operator
satisfying the null homotopy relation $D^{i-1}P^{i}+P^{i+1}D^{i}=I$. Define
$\tilde{P^{i}}:=P^{i}-{D^{i-2}}P^{i-1}P^{i}$. Then
$\tilde{P}^{\scriptscriptstyle\bullet}$ satisfies the null homotopy relation
$D^{i-1}\tilde{P}^{i}+\tilde{P}^{i+1}D^{i}=I$ and the complex property
$\tilde{P}^{i-1}\circ\tilde{P}^{i}=0$.

If the spaces $V^{i}$ are Banach spaces and the operators $D^{i}$ and $P^{i}$
all are bounded, then also the modified operators $\tilde{P}^{i}$ are bounded.

###### Proof.

The last statement is obvious from the definitions. For the relations, we
compute using $D^{2}=0$

$D\tilde{P}+\tilde{P}D=DP+PD-DPPD.$

Using the null homotopy relation twice, we get

$DPPD=-PDPD+PD=PDDP=0,$

so $\tilde{P}^{\scriptscriptstyle\bullet}$ satisfies the null homotopy
relation, too. Finally,

$\tilde{P}^{2}=P^{2}-DP^{3}-PDP^{2}+DP^{2}DP^{2}=P^{2}-(DP+PD)P^{2}+DP^{2}DP^{2}=DP^{2}DP^{2},$

and this vanishes since $DP^{2}D=0$. ∎

######  Remark 2.

By the null-homotopy relation, we observe that
$\tilde{P^{i}}:=P^{i}-{D^{i-2}}P^{i-1}P^{i}={P^{i}}D^{i-1}P^{i}$.

##  5\. The case of nontrivial cohomology

We next discuss modifications that are needed to treat cases with non-trivial
cohomology, so the null homotopy relation (2.1) gets replaced by the more
general version (2.2). While the basic ideas are not so different from the
case of vanishing cohomology, there is the question of which properties one
should assume about $L$ and how to prove that they are preserved by the
construction. This is particularly relevant for the examples related to de
Rham complexes where the form of these properties is rather unexpected.
Consequently, we will discuss two different situations here. One basic case
that was mentioned in Section 2 already, is that the image of $L$ coincides
with the cohomology, or equivalently that $dL=Ld=0$. We can deal with this
case in a general abstract setting. The second case is specifically tailored
to the examples coming from de Rham complexes and representation theory as
discussed in Section 4.1. Here the treatment is based on a variant of the
construction for the de Rham complex in [22].

###  5.1. $L$ operators representing cohomology

We discuss the first case, where we require that $dL=0$ and hence
$\mathcal{R}(L)$ coincides with the cohomology. Our goal is to construct the
corresponding operators $P_{V}$, $L_{V}$ for the twisted complex and
$\mathscr{P}$, $\mathscr{L}$ for the BGG complex, such that analogous
properties hold. To this end, we will follow the derivation of the twisted and
BGG complexes, and carry over properties of the input operators $P$, $L$ to
these cases by complex maps $F$ and $A$, $B$. The main difficulty is that in
the setup in Section 3, the complex maps $F^{{\scriptscriptstyle\bullet}}$
from the input to the twisted complex involves Poincaré operators
$P^{{\scriptscriptstyle\bullet}}$ of the input complexes and the null-homotopy
identity $dP+Pd=I$ plays an important role in verifying the properties. Hence
in the case of nontrivial cohomology, a different argument is needed.

One option is to return to the original setting of [11] and make different
assumptions for the two steps. For the first step, we also need the
$K$-operators from the first step there. So in addition to the assumptions in
the beginning of Section 3, we assume that we have given bounded operators
$K^{i,j}:Z^{i,j}\to Z^{i,j-1}$ such that
$S^{i,j}=d^{i,j-1}K^{i,j}-K^{i+1,j}d^{i,j}$ and such that
$S^{i,j-1}K^{i,j}=K^{i+1,j-1}S^{i,j}$. Then it was shown in Theorem 1 of [11]
that viewing our operators as $K^{i}:Y^{i}\to Y^{i}$, the exponentials
$F^{i}=\exp(K^{i})$ (which are just polynomials, since each $K^{i}$ is
nilpotent) are bounded operators that fit together to define an isomorphism
$F^{\scriptscriptstyle\bullet}:(Y^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})\to(Y^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$
of differential complexes. So in particular, we have $F^{i+1}\circ
d^{i}=d^{i}_{V}\circ F^{i}$. Using this, the passage to the twisted complex is
fairly trivial. Given a Poincaré operator $P^{\scriptscriptstyle\bullet}$ for
$(Y^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$ we define
$\tilde{P}^{i}_{V}:Y^{i}\to Y^{i-1}$ by $\tilde{P}^{i}_{V}=F^{i}\circ
P^{i}\circ(F^{i-1})^{-1}$.

######  Proposition 1.

Suppose that $P^{\scriptscriptstyle\bullet}$ is a Poincaré operator for
$(Y^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$ such that
$dP+Pd=I-L$. Then $\tilde{P}^{\scriptscriptstyle\bullet}$ is a Poincaré
operator for
$(Y^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$ with
$d_{V}\tilde{P}_{V}+\tilde{P}_{V}d_{V}=I-L_{V}$, where
$L^{i}_{V}=F^{i}L^{i}(F^{i})^{-1}$. If each $P^{i}$ is bounded then each
$\tilde{P}_{V}^{i}$ is bounded and if $P^{\scriptscriptstyle\bullet}$ has the
complex property then $\tilde{P}^{\scriptscriptstyle\bullet}$ has the complex
property, too. Finally, if $L^{i}$ has finite rank then the same is true for
$L^{i}_{V}$ and if $dL=Ld=0$, then also $d_{V}L_{V}=L_{V}d_{V}=0$.

###### Proof.

Since $Fd=dF$, we immediately get $d_{V}\tilde{P}_{V}=FdPF^{-1}$ and
$\tilde{P}_{V}d_{V}=FPdF^{-1}$ which immediately implies that homotopy
relation with the right form of $L^{i}_{V}$. The remaining statements then are
obvious, they just carry over through conjugation by $F$. ∎

For the passage to the BGG complex, we don’t need the $K$-operators any more.
So we are back to the setting from the beginning of Section 3 and in addition,
we assume that we have given a Poincaré operator
$P_{V}^{\scriptscriptstyle\bullet}$ for the twisted complex
$(Y^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$. As in
Section 3.2 we define the operators
$\mathscr{P}^{i}:\Upsilon^{i}\to\Upsilon^{i-1}$ as
$\mathscr{P}^{i}:=B^{i-1}P^{i}_{V}A^{i}$.

######  Theorem 6.

Suppose that the Poincaré operator $P_{V}^{\scriptscriptstyle\bullet}$
satisfies a homotopy relation $d_{V}P_{V}+P_{V}d_{V}=I-L_{V}$ such that
$d_{V}L_{V}=L_{V}d_{V}=0$. Then the induced Poincaré operator
$\mathscr{P}^{\scriptscriptstyle\bullet}$ on the BGG complex satisfies a
homotopy relation of the form
$\mathscr{D}\mathscr{P}+\mathscr{P}\mathscr{D}=I-\mathscr{L}$, with
$\mathscr{L}\mathscr{D}=\mathscr{D}\mathscr{L}=0$. Moreover, if
$L_{V}^{2}=L_{V}$ then also $\mathscr{L}^{2}=\mathscr{L}$. Finally, if $P_{V}$
has the complex property then $L_{V}^{2}=L_{V}$ and we can modify
$\mathscr{P}^{\scriptscriptstyle\bullet}$ in such a way that in addition to
all proceeding properties, it has the complex property, too.

###### Proof.

As before, we see (without assumptions on $L_{V}$) that
$\mathscr{P}^{\scriptscriptstyle\bullet}$ satisfies the homotopy relation with
$\mathscr{L}^{i}=B^{i}L_{V}^{i}A^{i}$, so the question is whether the nice
properties are preserved. But then $A^{i}\mathscr{D}^{i-1}=d^{i}_{V}A^{i}$ and
$\mathscr{D}^{i-1}B^{i-1}=B^{i}d_{V}^{i-1}$ together with
$d_{V}L_{V}=L_{V}d_{V}=0$ implies that
$\mathscr{L}\mathscr{D}=\mathscr{D}\mathscr{L}=0$. By construction, we also
get $(\mathscr{L}^{i})^{2}=B^{i}L^{i}_{V}A^{i}B^{i}L^{i}_{V}A^{i}$. Next,
equation (25) of [11] shows that
$A^{i}B^{i}=I-d^{i-1}_{V}G^{i}-G^{i+1}d^{i}_{V}$ and hence we conclude that
$L^{i}_{V}A^{i}B^{i}L^{i}_{V}=(L^{i}_{V})^{2}=L^{i}_{V}$ and hence
$(\mathscr{L}^{i})^{2}=\mathscr{L}^{i}$.

To prove the last statement, assume that $P^{i-1}_{V}P^{i}_{V}=0$. Then the
homotopy relation readily implies that
$L_{V}P_{V}=P_{V}-P_{V}d_{V}P_{V}=P_{V}L_{V}$. Using this and
$L_{V}d_{V}=d_{V}L_{V}=0$, we see that composing the homotopy relation with
$L_{V}$ gives $0=L_{V}-(L_{V})^{2}$, so $(L_{V})^{2}=L_{V}$ follows. Now we
first define
$\widehat{\mathscr{P}}^{i}:=\mathscr{P}^{i}-\mathscr{P}^{i}\mathscr{L}^{i}-\mathscr{L}^{i-1}\mathscr{P}^{i}+\mathscr{L}^{i-1}\mathscr{P}^{i}\mathscr{L}^{i}$.
This implies that $\mathscr{D}\widehat{\mathscr{P}}=\mathscr{D}\mathscr{P}$
and $\widehat{\mathscr{P}}\mathscr{D}=\mathscr{P}\mathscr{D}$, so the homotopy
relation holds for $\widehat{\mathscr{P}}$ in the same form as for
$\mathscr{P}$. But then $\mathscr{L}^{2}=\mathscr{L}$ and the definition of
$\widehat{\mathscr{P}}$ readily imply that
$\mathscr{L}\widehat{\mathscr{P}}=\widehat{\mathscr{P}}\mathscr{L}=0$.

This brings us into a situation in which the trick used in the proof of
Theorem 5 works. The homotopy relation for $\widehat{\mathscr{P}}$ together
with $\mathscr{D}\mathscr{L}=0$ implies that
$\mathscr{D}\widehat{\mathscr{P}}\mathscr{D}=\mathscr{D}$ and in turn
$\mathscr{D}\widehat{\mathscr{P}}^{2}\mathscr{D}=0$. Then we can define
$\widetilde{\mathscr{P}}^{i}:=\widehat{\mathscr{P}}^{i}-\mathscr{D}^{i-1}\widehat{\mathscr{P}}^{i-1}\widehat{\mathscr{P}}^{i}$
for each $i$. This definition implies that
$\mathscr{D}\widetilde{\mathscr{P}}=\mathscr{D}\widehat{\mathscr{P}}$ and
similar for the other composition. Thus we still have the homotopy relation
with the same map $\mathscr{L}$. The proof that
$\widetilde{\mathscr{P}}^{2}=0$ is exactly as the last computation in the
proof of Theorem 5. ∎

######  Remark 3.

As we assume that there exist bounded $K$ operators, the above construction
only applies to BGG complexes with smooth functions and twisted complexes with
smooth and certain Sobolev functions (e.g., the rows in the input complex have
equal Sobolev regularity). This does not apply to BGG complexes with Sobolev
spaces, as they will require input complexes with different kinds of
regularity, for which the $K$ operators are rarely known.

###  5.2. Bounded Lipschitz domains

We now discuss the second setting for non-trivial cohomology, so we use the
setup of Section 4.1 but for the case of a bounded Lipschitz domain $\Omega$.
So as there, we start with a representation
$\mathbb{V}=\oplus_{i=0}^{N}\mathbb{V}_{i}$ and for a fixed $q\in\mathbb{R}$
define
$Z^{i,j}:=H^{q-i-j}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j})$.
We also have the bounded operators $d^{i,j}:Z^{i,j}\to Z^{i+1,j}$ and
$S^{i,j}:Z^{i,j}\to Z^{i+1,j-1}$ induced by the exterior derivative and the
linear maps $\partial^{i,j}$ as there. This gives us the twisted complex
$(Y^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet}_{V})$. Following
the developments in Section 4.3 of [22], we find a finite covering of
$\overline{\Omega}$ by open sets $U_{1},\dots,U_{m}$ such that for each $i$,
the domain $\Omega_{i}:=U_{i}\cap\Omega$ is starlike with respect to some open
ball. Moreover, via a partition of unity, we get smooth functions $\chi_{i}\in
C^{\infty}(\mathbb{R}^{n},\mathbb{R})$ with values in $[0,1]$ such that
$\operatorname{supp}(\chi_{i})\subset U_{i}$ and such that
$\sum_{i}\chi_{i}\equiv 1$ on some open neighborhood of $\overline{\Omega}$.

Now for each $i$, the Poincaré operator for the twisted complex constructed in
Theorem 4 makes sense on smooth forms, so we get operators that we denote by
$P^{i}_{\ell}:C^{\infty}(\Omega_{\ell},\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V})\to
C^{\infty}(\Omega_{\ell},\Lambda^{i-1}\mathbb{R}^{n*}\otimes\mathbb{V})$ such
that $d^{i-1}_{V}P^{i}_{\ell}+P^{i+1}_{\ell}d^{i}_{V}=I$ for each $i>0$. Hence
for $u\in C^{\infty}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V})$ we
can restrict to $\Omega_{\ell}$ and then consider $\chi_{\ell}P^{i}_{\ell}u\in
C^{\infty}(\Omega_{\ell},\Lambda^{i-1}\mathbb{R}^{n*}\otimes\mathbb{V})$. But
this vanishes identically on the open subset
$\Omega\setminus(\Omega\cap\operatorname{supp}(\chi_{i}))$ which together with
$\Omega_{i}$ covers $\Omega$, so it can be extended by zero to a form in
$C^{\infty}(\Omega,\Lambda^{i-1}\mathbb{R}^{n*}\otimes\mathbb{V})$. So in
particular, we can define

(5.1) $\tilde{P}^{i}(u):=\sum_{\ell}\chi_{\ell}P^{i}_{\ell}u\in
C^{\infty}(\Omega,\Lambda^{i-1}\mathbb{R}^{n*}\otimes\mathbb{V})$

By construction, this is agian the restriction of a pseudodifferential
operator of order $-1$ and hence it defines bounded operators
$\tilde{P}^{i}:Y^{i}\to Y^{i-1}$, where as before $Y^{i}=\oplus_{j}Z^{i,j}$.
Moreover, we get

$(d^{i}_{V}\tilde{P}^{i}+\tilde{P}^{i}d^{i}_{V})(u)=\textstyle\sum_{\ell}\chi_{\ell}(d^{i}_{V}P^{i}_{\ell}u+P^{i}_{\ell}d^{i}_{V}u)+\sum_{\ell}(d^{i}_{V}(\chi_{\ell}P^{i}_{\ell}u)-\chi_{\ell}d^{i}_{V}(P^{i}_{\ell}u)).$

The homotopy relations for the operators $P^{i}_{\ell}$ show that the first
sum gives $\sum_{\ell}\chi_{\ell}u|_{\Omega_{\ell}}=u$ while the second sum
can be written in the language of differential forms as
$\sum_{\ell}d\chi_{i}\wedge P^{i}_{\ell}u$. Here we use the wedge product of a
real-valued one-form and a $\mathbb{V}$-valued $(i-1)$-form, which is a
$\mathbb{V}$-valued $i$-form. Now each summand in the latter sum is a form on
$\Omega_{\ell}$ which vanishes identically on the complement of
$\operatorname{supp}(\chi_{i})$, so as above, this defines

(5.2) $\tilde{L}^{i}u:=-\sum_{\ell}d\chi_{\ell}\wedge P^{i}_{\ell}u\in
C^{\infty}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}).$

As above, $\tilde{L}^{i}$ is the restriction of a pseudodifferential operator
of order $-1$ on $\mathbb{R}^{n}$ so again it defines a bounded operator
$Y^{i}\to Y^{i}$ (which even improves Sobolev regularity). Now by definition,
we get the homotopy relation

(5.3) $d^{i-1}_{V}\tilde{P}^{i}+\tilde{P}^{i+1}d_{V}^{i}=I-\tilde{L}^{i}$

and hitting this with $d_{V}$ from the left and from the right, we conclude
that $d^{i}_{V}\tilde{L}^{i}=\tilde{L}^{i+1}d^{i}_{V}$.

Still following [22], we can improve things further. This is based on the
observation that the functions $\chi_{\ell}$ coming from a partition of unity
can often be chosen to be constant on large subsets. Now suppose that
$x\in\Omega$ is a point at which the family $\\{\chi_{\ell}\\}$ is flat in the
sense that each $\chi_{i}$ is constant on some neighborhood of $x$. Then
evidently $d\chi_{\ell}(x)=0$ for each $\ell$ and hence $\tilde{L}^{i}(u)$
vanishes at $x$ for any $u\in
C^{\infty}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V})$. Now a
compactness argument shows that given $\Omega$, we can find finitely many
coverings $\\{U^{(s)}_{\ell}\\}$ of $\overline{\Omega}$ and associated
families $\\{\chi_{\ell}^{(s)}\\}$ of functions such that each
$U^{(s)}_{\ell}\cap\Omega$ is starlike with respect to some ball and for each
$x_{0}\in\mathbb{R}^{n}$ at least one of the families
$\\{\chi_{\ell}^{(s)}\\}$ is flat at $x_{0}$, see Lemma 4.5 of [22]. Now we
denote the operators for the covering $\\{U^{(s)}_{\ell}\\}$ by
$\tilde{P}^{i}_{(s)}$ and $\tilde{L}^{i}_{(s)}$ for $s=1,\dots,m$ and the put

(5.4) $\displaystyle
P^{i}_{V}:=\tilde{P}^{i}_{(1)}+\tilde{L}^{i-1}_{(1)}\tilde{P}^{i}_{(2)}+\tilde{L}^{i-1}_{(1)}\tilde{L}^{i-1}_{(2)}\tilde{P}^{i}_{(3)}+\dots+\tilde{L}^{i-1}_{(1)}\dots\tilde{L}^{i-1}_{(m-1)}\tilde{P}^{i}_{(m)}.$
(5.5) $\displaystyle L^{i}_{V}:=\tilde{L}^{i}_{(1)}\dots\tilde{L}^{i}_{(m)}$

######  Theorem 7.

(1) The operators
$P^{i}_{V}:C^{\infty}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V})\to
C^{\infty}(\Omega,\Lambda^{i-1}\mathbb{R}^{n*}\otimes\mathbb{V})$ are
restrictions of pseudodifferential operators of order $-1$ on $\mathbb{R}^{n}$
and hence induce bounded operators $Y^{i}\to Y^{i-1}$ that we denote by the
same symbol. These satisfy the relations
$d^{i-1}_{V}P^{i}_{V}+P^{i+1}_{V}d^{i}_{V}=I-L^{i}_{V}$ and we get
$d^{i}_{V}L^{i}_{V}=L^{i+1}_{V}d_{V}^{i}$.

(2) The operators
$L^{i}_{V}:C^{\infty}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V})\to
C^{\infty}(\Omega,\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V})$ are
restrictions of pseudodifferential operators of order $-\infty$ on
$\mathbb{R}^{n}$ and hence they are smoothing operators.

(3) Using the operators $P^{i}_{V}:Y^{i}\to Y^{i-1}$ as the starting point of
the construction discussed in Section 3.2, we obtain a bounded Poincare
operator $\mathscr{P}^{\scriptscriptstyle\bullet}$ for the BGG complex
determined by $\mathbb{V}$ which satisfies a homotopy relation
$\mathscr{D}\mathscr{P}+\mathscr{P}\mathscr{D}=I-\mathscr{L}$ where
$\mathscr{L}$ is the restriction of a pseudodifferential operator of order
$-\infty$ on $\mathbb{R}^{n}$ and hence a smoothing operator and invertible
module compact operators.

###### Proof.

(1) The fact that $P^{i}_{V}$ is the restriction of a pseudodifferential
operator follows readily from its definition as a composition of such
operators. Moreover, the homotopy relations for the operators $P^{i}_{(s)}$
together with $d_{V}^{i-1}\tilde{L}^{i}_{(s)}=\tilde{L}^{i+1}_{(s)}d_{V}^{i}$
recursively imply that

$d^{i-1}_{V}\tilde{L}^{i-1}_{(1)}\dots\tilde{L}^{i-1}_{(s)}\tilde{P}^{i}_{(s+1)}+\tilde{L}^{i}_{(1)}\dots\tilde{L}^{i}_{(s)}\tilde{P}^{i+1}_{(s+1)}d_{V}^{i}=\tilde{L}^{i}_{(1)}\dots\tilde{L}^{i}_{(s)}-\tilde{L}^{i}_{(1)}\dots\tilde{L}^{i}_{(s+1)}$

which immediately gives the required homotopy relation, while
$d^{i}_{V}L^{i}_{V}=L^{i+1}_{V}d_{V}^{i}$ is obvious from the definition.

(2) If the family $\\{\chi_{\ell}^{(s)}\\}$ is flat on a neighborhood of
$x_{0}$ then for any distribution $u$, the distribution $\tilde{L}^{i}_{(s)}u$
is smooth on a neighborhood of $x_{0}$. But since all the other operators
$\tilde{L}^{i}_{(r)}$ are pseudolocal, also the composition $L^{i}_{V}$ has
the property that $L^{i}_{V}u$ is smooth on a neighborhood of $x_{0}$. But by
construction, this works for any $x_{0}\in\mathbb{R}^{n}$ and the claim
follows.

(3) We get $\mathscr{P}^{i}=B^{i-1}P^{i}_{V}A^{i}$ for the operators $A$ and
$B$ from section 3.2. Since $\mathscr{D}B=Bd_{V}$ and $A\mathscr{D}=d_{V}A$,
we get the claimed homotopy relation with
$\mathscr{L}^{i}=B^{i-1}L^{i}_{V}A^{i}$. But in our setting, the operators $A$
and $B$ both are differential operators that are defined on smooth maps on all
of $\mathbb{R}^{n}$ so the representation as a composition implies that
$\mathscr{L}^{i}$ is the restriction of a pseudodifferential operator of order
$-\infty$. ∎

##  6\. Proxies and Applications

In this section, we express the Poincaré operators derived in the previous
sections in terms of vector/matrix proxies. First, we discuss the case of
dimension one using proxies and explicit forms of spaces and operators. The
result is trivial, as one can obtain the inverse of a second-order derivative
by simply integrating twice, but carrying out this simple example may
illustrate the general construction. Then we discuss the case of the
elasticity complex in 3D and compare the result at index one to the Cesàro-
Volterra integral and generalizations [21]. We also present explicit results
for the elasticity version of the twisted complex, inspired by their
applications in continuum mechanics.

Finally, we present an application of the newly derived Poincaré operators on
a systematic construction of polynomial versions of BGG complexes. The
homotopy relation and the polynomial-preserving property imply their
exactness.

###  6.1. Poincaré operators in one dimension.

The geometric version of the BGG machinery usually is not discussed in
dimension one, since the corresponding geometric structures are only available
in higher dimensions. Still one can follow the representation theory setup
discussed in Section 3 of [11] for the Lie algebra
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ with $\mathfrak{g}_{0}=\mathbb{R}$
and use the corresponding maps to setup BGG diagrams. The simplest case
corresponds to the standard representation $\mathbb{R}^{2}$ of $\mathfrak{g}$.
This leads to the following BGG diagram in one space dimension for any real
number $q$.

(6.1)
${H^{q}}$${H^{q-1}}$${H^{q-1}}$${H^{q-2},}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$$\scriptstyle{I}$

where $\partial:=\frac{d}{dx}$. The BGG complex is obtained by simply
connecting the two rows:

(6.2) ${H^{q}}$${H^{q-2}.}$$\scriptstyle{\partial^{2}}$

Suppose that for the two rows of (6.1) we have Poincaré operators
$P_{\sharp}:H^{q-1}\to H^{q}$ and $P_{\flat}:H^{q-2}\to H^{q-1}$,
respectively. That is, $\partial P_{\sharp}=I_{H^{q-1}}$,
$P_{\sharp}\partial=I_{H^{q}}+c$, where $c$ is a constant. Similar identities
hold for $P_{\flat}$. Then we want to derive Poincaré operators for the BGG
complex (6.2) satisfying $\partial^{2}\mathscr{P}=I_{H^{q-2}}$ and
$\mathscr{P}\partial^{2}=I_{H^{q}}+g$, where $g\in\mathcal{N}(\partial^{2})$
is a linear function. In fact, one can verify that composition of the de Rham
Poincaré operators $\mathscr{P}:=P_{\sharp}\circ P_{\flat}$ gives the desired
properties. Nevertheless, we go through our construction to illustrate the
theory.

The first step is to derive the Poincaré operators for the twisted complex by
the following diagram:

(6.3) ${\left(\begin{array}[]{c}H^{q}\\\
H^{q-1}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q-1}\\\
H^{q-2}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q}\\\
H^{q-1}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q-1}\\\
H^{q-2}\end{array}\right),}$$\scriptstyle{F^{0}}$$\scriptstyle{\left(\begin{array}[]{cc}\partial&\\\
&\partial\end{array}\right)}$$\scriptstyle{F^{1}}$$\scriptstyle{\left(\begin{array}[]{cc}\partial&-I\\\
0&\partial\end{array}\right)}$

where

$F^{0}:=\left(\begin{array}[]{cc}I&P_{\sharp}\\\ 0&I\end{array}\right),\quad
F^{1}:=\left(\begin{array}[]{cc}I&0\\\ 0&I\end{array}\right).$

Using the commuting diagram, we define

$P_{V}:=F^{0}\circ P\circ(F^{1})^{-1}=\left(\begin{array}[]{cc}I&P_{\sharp}\\\
0&I\end{array}\right)\left(\begin{array}[]{cc}P_{\sharp}&0\\\
0&P_{\flat}\end{array}\right)\left(\begin{array}[]{cc}I&0\\\
0&I\end{array}\right)=\left(\begin{array}[]{cc}P_{\sharp}&P_{\sharp}P_{\flat}\\\
0&P_{\flat}\end{array}\right).$

The operators in (3.11) now boil down to

$A^{0}:=\left(\begin{array}[]{cc}I&0\\\ \partial&0\end{array}\right),\quad
A^{1}:=\left(\begin{array}[]{cc}I&0\\\ 0&I\end{array}\right)\quad
B^{0}:=\left(\begin{array}[]{cc}I&0\\\ 0&0\end{array}\right),\quad
B^{1}:=\left(\begin{array}[]{cc}0&0\\\ \partial&I\end{array}\right),$

This leads to the following commuting diagram:

(6.4) ${\left(\begin{array}[]{c}H^{q}\\\
H^{q-1}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q-1}\\\
H^{q-2}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q}\\\
0\end{array}\right)}$${\left(\begin{array}[]{c}0\\\
H^{q-2}\end{array}\right),}$$\scriptstyle{B^{0}}$$\scriptstyle{\left(\begin{array}[]{cc}\partial&-I\\\
0&\partial\end{array}\right)}$$\scriptstyle{B^{1}}$$\scriptstyle{\left(\begin{array}[]{cc}0&0\\\
\partial^{2}&0\end{array}\right)}$

Then we obtain Poincaré operators for the bottom row of (6.4):

${\mathscr{P}}=B^{0}\circ P_{V}\circ A^{1}=\left(\begin{array}[]{cc}I&0\\\
0&0\end{array}\right)\left(\begin{array}[]{cc}P_{\sharp}&P_{\sharp}P_{\flat}\\\
0&P_{\flat}\end{array}\right)\left(\begin{array}[]{cc}I&0\\\
0&I\end{array}\right)=\left(\begin{array}[]{cc}P_{\sharp}&P_{\sharp}P_{\flat}\\\
0&0\end{array}\right).$

Note that the first component of $\Upsilon^{1}$ is zero because the $S$
operator is bijective ($\mathcal{R}(S)^{\perp}=0$). Therefore

$\mathscr{P}\left(\begin{array}[]{c}0\\\
u\end{array}\right)=\left({\begin{array}[]{c}P_{\sharp}P_{\flat}u\\\
0\end{array}}\right)$

gives the Poincaré operator $\mathscr{P}:=P_{\sharp}P_{\flat}$ for the
nontrivial component.

###  6.2. Vector and matrix proxies

In higher dimensions, the diagrams are more delicate and more structures come
in. Following [8, 11], we summarize our notations for vector/matrix proxies in
the table above.

$\mathbb{V}$ | $\mathbb{R}^{n}$
---|---
$\mathbb{M}$ | the space of all $n\times n$-matrices
$\mathbb{S}$ | symmetric matrices
$\mathbb{K}$ | skew symmetric matrices
$\mathbb{T}$ | trace-free matrices
$\operatorname{skw}:\mathbb{M}\to\mathbb{K}$ | skew symmetric part of a matrix
$\operatorname{sym}:\mathbb{M}\to{\mathbb{S}}$ | symmetric part of a matrix
$\operatorname{tr}:\mathbb{M}\to\mathbb{R}$ | matrix trace
$\iota:\mathbb{R}\to\mathbb{M}$ | the map $\iota u:=uI$ identifying a scalar with a scalar matrix
$\mathrm{dev}:\mathbb{M}\to\mathbb{T}$ | deviator (trace-free part of a matrix) given by $\mathrm{dev}u:=u-1/n\operatorname{tr}(u)I$

Moreover, in three space dimensions, we can identify a skew symmetric matrix
with a vector,

$\operatorname{mskw}\left(\begin{array}[]{c}v_{1}\\\ v_{2}\\\
v_{3}\end{array}\right):=\left(\begin{array}[]{ccc}0&-v_{3}&v_{2}\\\
v_{3}&0&-v_{1}\\\ -v_{2}&v_{1}&0\end{array}\right).$

Consequently, we have $\operatorname{mskw}(v)w=v\times w$ for
$v,w\in{\mathbb{V}}$. We also define
$\operatorname{vskw}=\operatorname{mskw}^{-1}\circ\operatorname{skw}:\mathbb{M}\to{\mathbb{V}}$.
We further define the Hessian operator
$\operatorname{hess}:=\operatorname{grad}\circ\operatorname{grad}$ and for any
matrix-valued function $u$, $\mathcal{S}u:=u^{T}-\operatorname{tr}(u)I$. In
our convention, the proxies of exterior derivatives ($\operatorname{grad}$,
$\operatorname{curl}$, $\operatorname{div}$, etc.) act column-wise, and so do
the Poincaré operators for the de Rham complexes.

Some examples in three dimensions can be demonstrated by the following diagram
in vector/matrix forms:

(6.5)
${0}$${H^{q}\otimes\mathbb{R}}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{V}}$${H^{q-3}\otimes\mathbb{R}}$${0}$${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{M}}$${H^{q-3}\otimes\mathbb{M}}$${H^{q-4}\otimes\mathbb{V}}$${0}$${0}$${H^{q-2}\otimes\mathbb{V}}$${H^{q-3}\otimes\mathbb{M}}$${H^{q-4}\otimes\mathbb{M}}$${H^{q-5}\otimes\mathbb{V}}$${0}$${0}$${H^{q-3}\otimes\mathbb{R}}$${H^{q-4}\otimes\mathbb{V}}$${H^{q-5}\otimes\mathbb{V}}$${H^{q-6}\otimes\mathbb{R}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{I}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{2\operatorname{vskw}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{2\operatorname{vskw}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\iota}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{I}$

This diagram should not be confused with (3.3), as in (6.5), we only read two
rows at one time. From the first two rows, we get the Hessian complex; the
middle two rows give the elasticity complex; and the last two rows give the
$\operatorname{div}\operatorname{div}$ complex. See [8] for more details.

The elasticity complex: In three space dimensions, the diagram for deriving
the elasticity complex and the Poincaré operators is the two middle rows in
(6.5). We first discuss the case with smooth functions. The benefit is that we
get $P\circ P=0$ if we use the standard Poincaré operators for the de Rham
complex (which may not hold for the regularized version [22]).

In the diagram, $S^{0}=-\operatorname{mskw}$ and $S^{1}=\mathcal{S}$ are
injective. Therefore the second components of $Y^{0}$ and $Y^{1}$ are zero (in
the kernel of $S^{0}$ and $S^{1}$, respectively). Then the operator
$\mathscr{P}^{1}$ at index one essentially maps between the first components
with the form $P-P(PS-SP)Td$, which further simplifies to $P+PSPTd$ if $P\circ
P=0$.

Let $e\in C^{\infty}\otimes\mathbb{S}$. Then $\mathscr{P}^{1}(e)\in
C^{\infty}\otimes\mathbb{V}$ is given by

$\displaystyle\mathscr{P}^{1}(e)$
$\displaystyle=\int_{0}^{x}e(y)\cdot\,dy-\int_{0}^{x}(\underbrace{\operatorname{mskw}\int_{0}^{y}\underbrace{(e({z})\times\nabla)^{T}}_{\text{$Td(e)$}}\cdot
dz)}_{\text{$SPTd(e)$}}\cdot dy.$

Here we have used the de Rham Poincaré operators with smooth forms (2.3) for
the definition of $P$. We can verify one of the null homotopy identities by
straightforward calculations. Indeed, if $e=\operatorname{def}(w)$ for some
vector-valued function $w$, then

$\displaystyle\mathscr{P}^{1}(e)$
$\displaystyle=\int_{0}^{x}e(y)\,dy-\frac{1}{2}\int_{0}^{x}\operatorname{mskw}(\int_{0}^{y}(\nabla\times
w\nabla)\cdot\,dz)\cdot\,dy$
$\displaystyle=\int_{0}^{x}e(y)\,dy-\frac{1}{2}\int_{0}^{x}\operatorname{mskw}(\nabla\times
w(y)-\nabla\times w(0))\cdot\,dy$

Note that

$\displaystyle\int_{0}^{x}\operatorname{def}(w)\,dy$
$\displaystyle=\int_{0}^{x}(w\nabla)\,dy+\frac{1}{2}\int_{0}^{x}(\nabla
w-w\nabla)\cdot\,dy$
$\displaystyle=w(x)-w(0)+\frac{1}{2}\int_{0}^{x}\operatorname{mskw}\nabla\times
w(y)\cdot\,dy.$

This gives

$\mathscr{P}^{1}(\operatorname{def}(w))=w(x)-w(0)+\frac{1}{2}x\wedge(\nabla\times
w(0)).$

Since the right hand side is an infinitesimal rigid motion, it lies in the
kernel of $\operatorname{def}$, so the right null homotopy relation is
satisfied in degree zero. This reflects the fact that the original sum of de
Rham complexes has a six dimensional space of constants as its cohomology in
degree zero.

In the case of general Sobolev spaces, the above smooth version generalized to
$\mathscr{P}^{1}=P(PS-SP)Td$, where the de Rham Poincaré operators
$P^{{\scriptscriptstyle\bullet}}$ are replaced by the regularized version in
[22].

At index two, the formula simplifies to
$\mathscr{P}^{2}=-{\Pi_{\mathcal{R}^{\perp}}}P(PS-SP)$, which maps the second
component of $Y^{2}$ to the first component of $Y^{1}$ as $S^{1}$ is
bijective. At index three, we get $\mathscr{P}^{3}=-{\Pi_{\mathcal{N}}}dTP(PS-
SP)+{\Pi_{\mathcal{N}}}P$. If $P\circ P=0$, these formulas further simplify to
$\mathscr{P}^{2}={\Pi_{\mathcal{R}^{\perp}}}PSP$ and
$\mathscr{P}^{3}={\Pi_{\mathcal{N}}}dTPSP+{\Pi_{\mathcal{N}}}P$, respectively.

Compared to the Cesàro-Volterra formula with little regularity [21], the new
formula $\mathscr{P}^{1}$ is polynomial-preserving (when the input de Rham
$P^{{\scriptscriptstyle\bullet}}$ operators are so) and has a more explicit
form (whereas the formula in [21] is given by duality), and is valid for a
wide range of spaces, following the results in [22]. To see that the two
results are different, we note that the formula in [21] boils down to the
classical Cesàro-Volterra formula if the fields it acts on are smooth enough.
Contrarily, this is not the case for the $\mathscr{P}^{1}$ operator derived
above as the input $P^{{\scriptscriptstyle\bullet}}$ involves smoothing
(averaging) even for smooth functions.

######  Remark 4.

In the proxies above, for the convenience of notation, we have used the
convention that differential operators act row-wise on a matrix-valued
function. For example, if $u$ is a vector-valued 0-form, then $du$ corresponds
to $(du)_{ij}:=\partial_{j}u_{i}$, where $(du)_{ij}$ denotes the $(i,j)$-th
entry of the matrix proxy of $du$. Alternatively, in another notation, $du$
corresponds to $u\nabla$, where $\nabla$ acts “from the right” (row-wise).
Correspondingly, the de Rham Poincaré operators act row-wise. For example, if
$v$ is a matrix-valued function, then $\int_{0}^{x}dy\wedge v(y)$ extends
(2.9) to each row of $v$. This is consistent with the convention in [19], but
different from [8, 11].

The twisted complex: The vector/matrix form of the elasticity version of the
twisted complex (obtained from the two middle rows in (6.5)) is the following:

(6.6) ${\left(\begin{array}[]{c}{C^{\infty}\otimes\mathbb{V}}\\\
{C^{\infty}\otimes\mathbb{V}}\end{array}\right)}$${\left(\begin{array}[]{c}{C^{\infty}\otimes\mathbb{M}}\\\
{C^{\infty}\otimes\mathbb{M}}\end{array}\right)}$${\left(\begin{array}[]{c}{C^{\infty}\otimes\mathbb{M}}\\\
{C^{\infty}\otimes\mathbb{M}}\end{array}\right)}$${\left(\begin{array}[]{c}{C^{\infty}\otimes\mathbb{V}}\\\
{C^{\infty}\otimes\mathbb{V}}\end{array}\right).}$$\scriptstyle{\left(\begin{array}[]{cc}\operatorname{grad}&\operatorname{mskw}\\\
0&\operatorname{grad}\end{array}\right)}$$\scriptstyle{\left(\begin{array}[]{cc}\operatorname{curl}&-\mathcal{S}\\\
0&\operatorname{curl}\end{array}\right)}$$\scriptstyle{\left(\begin{array}[]{cc}\operatorname{div}&-2\operatorname{vskw}\\\
0&\operatorname{div}\end{array}\right)}$

The Poincaré operators have the general form (see (3.8))

$P_{V}=\left(\begin{array}[]{cc}P&-P(PS-SP)\\\ 0&P\end{array}\right),$

which further simplifies to

$\left(\begin{array}[]{cc}P&-PSP\\\ 0&P\end{array}\right),$

if the complex property $P\circ P=0$ holds for the input.

For the complex with smooth fields (for which the classical de Rham Poincaré
operators (2.3) can be used), vector-matrix proxies have the form (see
(2.8)-(2.10) for the notation):

$P_{V}^{1}\left(\begin{array}[]{c}u\\\
w\end{array}\right)=\left(\begin{array}[]{c}\int_{0}^{x}u(y)\cdot\,dy+\int_{0}^{x}(-\operatorname{mskw}\int_{0}^{y}w(z)\cdot\,dz)\cdot\,dy\\\
\int_{0}^{x}w(y)\cdot\,dy\end{array}\right)=\left(\begin{array}[]{c}\int_{0}^{x}u(y)\cdot\,dy+\int_{0}^{x}(\int_{0}^{y}w(z)\cdot\,dz)\wedge
dy\\\ \int_{0}^{x}w(y)\cdot\,dy\end{array}\right),$
$P_{V}^{2}\left(\begin{array}[]{c}u\\\
w\end{array}\right)=\left(\begin{array}[]{c}\int_{0}^{x}u(y)\wedge
dy+\int_{0}^{x}(\mathcal{S}\int_{0}^{y}w(z)\wedge dz)\wedge dy\\\
\int_{0}^{x}w(y)\wedge dy\end{array}\right),$

and

$P_{V}^{3}\left(\begin{array}[]{c}u\\\
w\end{array}\right)=\left(\begin{array}[]{c}\int_{0}^{x}u(y)\otimes
dy+\int_{0}^{x}(2\operatorname{vskw}\int_{0}^{y}w(z)\otimes dz)\otimes dy\\\
\int_{0}^{x}w(y)\otimes dy\end{array}\right),$

respectively.

###  6.3. Polynomial complexes

The examples in [11] have vector-valued de Rham complexes as input (each row
in (3.3)). Then all the operators $d^{{\scriptscriptstyle\bullet}}$,
$P^{{\scriptscriptstyle\bullet}}$ (regularized Poincaré operators for the de
Rham complex) and $S^{{\scriptscriptstyle\bullet}}$ are polynomial-preserving.
Following the discussions in Section 4.2, we obtain polynomial exact
sequences. Examples include the elasticity complex,

(6.7)
${0}$${\mathcal{P}_{r}\otimes\mathbb{V}}$${\mathcal{P}_{r-1}\otimes\mathbb{S}}$${\mathcal{P}_{r-3}\otimes\mathbb{S}}$${\mathcal{P}_{r-4}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

the conformal Hessian complex

(6.8)
${0}$${\mathcal{P}_{r}}$${\mathcal{P}_{r-2}\otimes(\mathbb{S}\cap\mathbb{T})}$${\mathcal{P}_{r-3}\otimes(\mathbb{S}\cap\mathbb{T})}$${\mathcal{P}_{r-5}}$${0,}$$\scriptstyle{\mathrm{dev}\operatorname{hess}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$

and the conformal deformation complex

(6.9)
${0}$${\mathcal{P}_{r}\otimes\mathbb{V}}$${\mathcal{P}_{r-1}\otimes(\mathbb{S}\cap\mathbb{T})}$${\mathcal{P}_{r-4}\otimes(\mathbb{S}\cap\mathbb{T})}$${\mathcal{P}_{r-5}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\mathrm{dev}\operatorname{def}}$$\scriptstyle{\cot}$$\scriptstyle{\operatorname{div}}$

######  Theorem 8.

The polynomial complexes (6.7), (6.8) and (6.9) are exact in degrees $>0$
while the cohomologies in dimension $0$ are the $6$-dimensional space of rigid
infinitesimal motions, the $5$-dimensional space
$\mathcal{N}(\mathrm{dev}\operatorname{hess})$ and the $10$ dimensional space
$\mathcal{N}(\mathrm{dev}\operatorname{def})$, respectively.

###### Proof.

On $\mathbb{R}^{n}$, we get the Poincaré operators on the smooth complexes as
discussed above and they are all polynomial preserving. Thus we get the
claimed information on the cohomology. ∎

For the three examples derived from (6.5) (the elasticity, Hessian and
$\operatorname{div}\operatorname{div}$ complexes), polynomial complexes and
their exactness were discussed in [14, 16, 15, 19].

The above complexes are analogues of (2.5) with full polynomials. Another
family of polynomial complexes, aimed at an analogue of (2.6), can be derived
from a sequence of polynomial spaces of equal degree, e.g.,

(6.10)
${0}$${\mathcal{P}_{r}\otimes\mathbb{V}}$${\mathcal{P}_{r}\otimes\mathbb{S}}$${\mathcal{P}_{r}\otimes\mathbb{S}}$${\mathcal{P}_{r}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

Here we used the elasticity complex as an example, but the discussions also
hold for other complexes. Clearly, (6.10) is a complex, but we do not get
Poincaré operators here, since they do not preserve the degree of polynomials
and hence we cannot make conclusions on the cohomology. To fix this, we can
modify the spaces in the complex by including a
$\mathscr{P}^{\scriptscriptstyle\bullet}$-component, i.e.,

(6.11)
${0}$${\mathcal{P}_{r}\otimes\mathbb{V}+\mathscr{P}^{1}[\mathcal{P}_{r}\otimes\mathbb{S}]}$${\mathcal{P}_{r}\otimes\mathbb{S}+\mathscr{P}^{2}[\mathcal{P}_{r}\otimes\mathbb{S}]}$${\mathcal{P}_{r}\otimes\mathbb{S}+\mathscr{P}^{3}[\mathcal{P}_{r}\otimes\mathbb{V}]}$${\mathcal{P}_{r}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\operatorname{div}}$

The null homotopy relations for the operators $\mathscr{P}^{i}$ easily imply
that the operators in (6.11) indeed map each space in the sequence to the next
space. In general, it is still not clear that we get a Poincaré operator for
this complex, since the spaces are not necessarily mapped to each other by the
operators $\mathscr{P}^{i}$. This is true here, however, since we can use the
Poincaré operators derived from the smooth de Rham version (2.3) which satisfy
the complex property. Then (6.11) is also a complex with
$\mathscr{P}^{{\scriptscriptstyle\bullet}}$ and we get the conclusions on the
cohomology.

This construction above is a general procedure. We can start with any complex
of $d^{{\scriptscriptstyle\bullet}}$ and complete it by adding
$\mathscr{P}^{{\scriptscriptstyle\bullet}}$ components such that the sequence
is also a complex of $\mathscr{P}^{{\scriptscriptstyle\bullet}}$. One may
further investigate the construction of finite elements based on polynomial
exact sequences, but this is beyond the scope of this paper.

## Acknowledgement

A.Č. is supported by the Austrian Science Fund (FWF): P33559-N. This
article/publication is based upon work from COST Action CaLISTA CA21109
supported by COST (European Cooperation in Science and Technology).
www.cost.eu. K.H. is supported by a Royal Society University Research
Fellowship (URF$\backslash$R1$\backslash$221398).

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