# On the Schwartz Estimate for Hodge Laplacians on semisimple Lie Groups

Zhicheng Han  George-August-Universität Göttingen, Mathematisches Institut,
Bunsenstraße 3-5, 37073 Göttingen, Germany<EMAIL_ADDRESS>gottingen.de

###### Abstract.

In this paper, we prove Schwartz estimates for Hodge Laplacian and Dirac
operators on semisimple Lie groups. Alongside, we gives a version of Kuga’s
lemma for its Lie algebra cohomology. This is a generalization of similar
results of Barbasch and Moscovici [BM83] on symmetric spaces. The main purpose
of such estimates is to study the heat problem not only in the scalar case,
but also for sections of vector bundles on homogeneous spaces using Fourier
analysis.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Lie algebra cohomology
  4. 4 Elliptic operators on Lie groups
  5. 5 Proof of the Schwartz estimate
  6. 6 Schwartz estimates on nilpotent groups

##  1\. Introduction

Let $G$ be a real, connected Lie group with infinite center. Denote its Lie
algebra as $\mathfrak{g}$. Furnish $G$ with a left-$G$-invariant Riemannian
metric. Let $\Lambda^{*}G$ be the bundle of differential forms over $G$, and
let $\Delta_{p}$ be the Hodge Laplacian acting on the $p$-forms of $G$. This
operators is well-known to be a second-order elliptic operator on $G$, which
admits a kernel of the heat exonerator $e^{-t\Delta_{p}}$, which we denote as
$k_{t}^{p}(g)$.

It is our intention to study the spectral properties of $\Delta_{p}$ on a
semisimple Lie group with infinite center as group manifold. We are
particularly interested in the spectral properties of $\Delta_{p}$ on the
$L^{2}$-spaces. The spectral properties of $\Delta_{p}$ are closely related to
the representation theory of $G$.

The main purpose of this paper is to prove that given the most invariant
metric on $G$, we can prove that the heat kernel $k^{p}_{t}(g)$ associated
with the corresponding Hodge Laplacian $\Delta_{p}$ is a Schwartz function,
defined in the sense of Herb and Wolf [HW86].

To be more precise, let $G$ be a connected reductive Lie group of class
$\widetilde{\mathcal{H}}$. This is a class defined by Wolf that contains all
semisimple Lie groups with infinite center [Wol74]. If we fix $\theta$ a
Cartan involution of $G$, and denotes its fixed point to be $K$, then there is
a canonical way to construct a left $G$-invariant metric on $G$ that is also
$K$-bi-invariant from the Killing form $B$ of $G$. Denote this metric to be
$B^{\theta}$. Our main result is then stated as the following:

######  Theorem A (Schwartz kernel).

Let $t>0$. Then for any connected reductive Lie group $G$ of class
$\widetilde{\mathcal{H}}$ and corresponding metric $B^{\theta}$, the kernels
of the Hodge Laplacian $k^{p}_{t}$ associated with the differential forms of
$G$ are $L^{q}$-Schwartz functions on $G$, in the sense of Harish-Chandra, for
all $q>0$.

The property of the function to be of Schwartz class is a vital condition for
many applications in sight, because it is the class of functions where the
most general version of Plancherel formula is established in current
literature. We will later deploy the Plancherel formula in another paper
[Han24, TOCHANGE], to study the spectral properties of $\Delta_{p}$ on groups
such as $\widetilde{SL_{2}()}$. The proof that $k_{t}^{p}$ is a Schwartz
function is a crucial step in such applications.

Such a class of Schwartz function denoted by $\mathcal{S}(G)$ was first
defined by Harish-Chandra [HC84, Section 6] on reductive Lie groups with
compact center and later extended by Herb and Wolf [HW86] to the wider class
of semisimple groups with infinite center. The latter class of Schwartz
functions was suitably defined to capture the growth of functions in the
$K$-direction, which is now noncompact.

The most general Plancherel formula known to the author was established by
Herb and Wolf in the same paper, extending the previous work of Harish-Chandra
[HC75, HC76a, HC76b]. The main result of this paper therefore validates the
application of the Plancherel formula and subsequently representation theory
of $G$ to studying the spectral problems of $\Delta_{p}$ on group manifolds
$G$.

The left $G$-invariant differential operators on $G$ can be identified with
elements of the universal enveloping algebra $U(\mathfrak{g}_{\mathbb{C}})$.
Following the same methodology, we express the Hodge Laplacians in terms of
actions of $U(\mathfrak{g}_{\mathbb{C}})$. This is a common strategy used in
the study of symmetric spaces, where the Hodge Laplacian is essentially the
Casimir element $\Omega_{G}$ in the center of $U(\mathfrak{g}_{\mathbb{C}})$.

We begin with an expression for the Hodge Laplacian associated with any Lie
group $G$ and any left $G$-invariant metric. In the case $G$ is reductive, we
fix the most invariant Riemannian metric in sight, that is the Killing form
$B$ twisted with the Cartan involution $\theta$:

(1) $B^{\theta}(X,Y)=B(X,\theta Y)$

for $X,Y\in\mathfrak{g}_{\mathbb{C}}$. This exploits the underlying symmetry
and gives a refined version of Kuga’s lemma on the Lie algebra cohomology of
$\mathfrak{g}$ twisted with any unitary $\mathfrak{g}$-module $V$ (see
Proposition 3.3 and Corollary 3.5):

######  Proposition B.

Given a unitary $\mathfrak{g}$-module $(\pi,V_{\pi})$, the Hodge Laplacian
$\Delta_{p}$ on $C^{\infty}(\mathfrak{g},V_{\pi})$ can be expressed
explicitly. If $G$ is further assumed to be reductive, then this expression
for the Hodge Laplacian associated with $B^{\theta}$ is given in terms of the
representation $\pi$ and the adjoint representation $\operatorname{ad}$ of
$\mathfrak{g}$.

This is also where we depart from the case of symmetric spaces. In both cases,
the differential operator $D$ in question can be written as

$D=-R(\sum_{i}X_{i}^{2})+B$

as the sum of the Laplacian $-\sum_{i}X_{i}^{2}$ on functions of $G$ and a
perturbation term $B$. In the case of symmetric spaces, $B$ is essentially a
scalar operator, whereas in our cases the perturbation term is typically a
first-order differential operator with variable coefficients. Moreover, the
fact that $K$ is noncompact raises yet further difficulties when we estimate
derivatives from both sides.

The main novelty in this paper is first to give a very explicit description of
the Hodge Laplacian, so that the perturbation term and corresponding
perturbation estimates can be made. Secondly, we incorporate the perturbation
theory into the Gaussian estimates of the heat kernel. This allows us to
estimate the kernel under derivatives from one side for a large class of
differential operators on homogeneous vector bundles, which in particular
contains the Hodge Laplacian on Lie groups. Lastly, we also extends an
argument of Harish-Chandra to estimate the kernel when $K$ is noncompact.

The paper is organized as follows. In Section 2 we collect some general facts
about groups of class $\widetilde{\mathcal{H}}$. We also introduce the notion
of Schwartz functions on such groups. In Section 3 we give an explicit
expression of the Hodge Laplacian on general Lie algebra cohomology
$C^{*}(\mathfrak{g};V)$. This allows us to distinguish the first-order
perturbation operator. In Section 4 we collect some results on perturbation
theory for the heat semigroup. This allows us to control the kernel estimates.
Eventually in Section 5 we assemble all the ingredients to prove the Schwartz
estimate for the Hodge Laplacian. Lastly in Section 6 we discuss how this
method can be readily applied to prove a similar result in nilpotent Lie
groups. This is an easy consequence derived from the estimates obtained in
Section 5 that will be useful in a subsequent paper.

In a subsequent paper [Han24, TOCHANGE] we will use the Schwartz estimate to
study the spectral properties of the Hodge Laplacian on semisimple Lie groups
such as $\widetilde{SL_{2}()}$. A similar method is also used to study the
spectral properties of the Dirac operator on such groups.

This paper is a part of the author’s PhD thesis at the University of Göttingen
[Han24]. The author would like to thank his advisor Thomas Schick for
suggesting the research question and for carefully reading the first draft.

##  2\. Preliminaries

In this section we collect some facts about the structure of general
semisimple Lie groups with infinite center. We also recall the notion of
Schwartz functions on such groups, the Plancherel formula of which was
established by Herb and Wolf [HW86].

Let $G$ be a reductive Lie group. This implies that its Lie algebra admits the
following decomposition:

$\mathfrak{g}=Z_{\mathfrak{g}}\oplus\mathfrak{g}_{0}\quad\text{where
$Z_{\mathfrak{g}}$ is central and
$\mathfrak{g}_{0}=[\mathfrak{g},\mathfrak{g}]$ semisimple }$

In [Wol74], Wolf defined a reductive Lie group $G$ to be of class
$\widetilde{\mathcal{H}}$ if it satisfies the following properties:

  1. (1)

If $g\in G$ then $\operatorname{ad}(g)$ is an inner automorphism of
$\mathfrak{g}_{\mathbb{C}}$;

  2. (2)

It contains a closed normal abelian subgroup $Z$ that centralizes the identity
component $G^{0}$ of $G$;

  3. (3)

$ZG^{0}$ has finite index in $G$;

  4. (4)

$Z\cap G^{0}$ is cocompact in the center $Z_{G^{0}}$ of $G^{0}$.

As discussed in [Wol74, §0], this is a convenient class of reductive groups
that contains every connected semisimple group and is stable under passage to
Levi components of cuspidal parabolic subgroups of $G$. In particular this
contains the original class of Harish-Chandra, which are groups where
$G/G^{0}$ and the center of $[G^{0},G^{0}]$ are compact. In addition it also
contains the class of connected semisimple groups with infinite center, such
as the universal covers of linear reductive groups like $SL(2,)$, $Sp(n,)$ and
$U(n,n)$.

We assume throughout this paper that $G$ is connected. This lightens some of
the exposition significantly. Note in such case $ZG^{0}=G^{0}=G$.

A Cartan involution on $G$ is an involutive automorphism $\Theta$ such that
the fixed point set $K=G^{\Theta}$, which we call the relative maximal compact
subgroup, is the full inverse image of a maximal compact subgroup of
$\operatorname{Ad}(G)$. Note that $K$ is only compact if the center $Z_{G}$ of
$G$ is compact. Fix a Cartan involution $\Theta$ of $G$ and let $K$ be the
corresponding relative maximal compact subgroup. Then $K$ has a unique maximal
compact subgroup $K_{1}$ and has a closed normal vector subgroup $V_{G}$ such
that [HW90, Proposition 2.1]:

  1. (1)

$K=K_{1}\times V_{G}$;

  2. (2)

$Z=Z_{G}\cap V_{G}$ is cocompact in both $V_{G}$ and $Z_{G}$.

The Cartan involution $\Theta$ differentiates to give a Cartan involution
$\theta$ on $\mathfrak{g}$. Denote $\mathfrak{k,p}$ to be the eigenspaces of
$\theta$ corresponding to the eigenvalues $+1$ and $-1$ respectively. Choose a
maximal abelian subspace $\mathfrak{a_{p}}$ of $\mathfrak{p}$ and form the
restricted root system $\Delta^{+}(\mathfrak{g};\mathfrak{a_{p}})$. By picking
the polarity $\Delta^{+}(\mathfrak{g};\mathfrak{a_{p}})$ on
$\Delta(\mathfrak{g};\mathfrak{a_{p}})$, we form an Iwasawa decomposition

(2) $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{a_{p}}\oplus\mathfrak{n_{p}}$

where $\mathfrak{n_{p}}$ is the sum of positive $\mathfrak{a_{p}}$-root
spaces. Denote $A$ and $N$ to be the exponential of $\mathfrak{a_{p}}$ and
$\mathfrak{n_{p}}$ respectively. At group level this gives a Iwasawa
decomposition $G=NAK$

(3) $x=n(x)\cdot\exp H(x)\cdot\kappa(x),$

where $n(x)\in N$, $H(x)\in\mathfrak{a}$ and $\kappa(x)\in K$. Denote

$\rho_{\mathfrak{p}}=\frac{1}{2}\sum_{\alpha\in\Delta^{+}(\mathfrak{g};\mathfrak{a_{p}})}\alpha$

the half sum of positive restricted roots, counted with multiplicity. Define
now the zonal spherical function in a similar way as in the classical case
(c.f. [Kna86, §VII.8]):

(4)
$\phi_{0}^{G/Z}(g):=\int_{K/Z}e^{-(\rho_{\mathfrak{p}})H(kg)}\mathop{}\\!\mathrm{d}{(kZ)}.$

Note this enjoys all the desired properties of the classical zonal spherical
functions: inverse symmetries, bi-$K$-invariance, and of moderate growth (c.f.
[HW86, (2.5)]), i.e.:

  1. (i)

For all $k_{1},k_{2}\in K,g\in G$,
$\phi_{0}^{G/Z}(k_{1}gk_{2})=\phi_{0}^{G/Z}(g)$ and
$\phi_{0}^{G/Z}(g)=\phi_{0}^{G/Z}(g^{-1})$;

  2. (ii)

For some $d\geq 0$ and for all $a\in\exp(\overline{\mathfrak{a}^{+}})$:

(5) $\phi_{0}^{G/Z}(a)\leq Ca^{-\rho_{\mathfrak{p}}}(1+\rho_{\mathfrak{p}}\log
a)^{d}$

In particular, $\phi^{G/Z}_{0}$ is tempered: $\phi_{0}^{G/Z}\in
L^{2+\epsilon}(G/Z)$ for any $\epsilon>0$.

We also extend the original norm on $\mathfrak{p}$ suitably to capture the
growth in the vector subgroup $V_{G}$ of $K$. Recall that the Killing form
$B(-,-)$ is positive definite on $\mathfrak{p}$ and negative definite on
$\mathfrak{k}$, and is $G$-bi-invariant. Given

(6) $x=v(x)\cdot\kappa_{1}(x)\cdot\exp X$

with respect to the Cartan decomposition $G=V_{G}K_{1}\exp\mathfrak{p}$, we
define

(7)
$|x|_{\mathfrak{\mathfrak{p}}}:=B(X,X)^{1/2},\quad|x|_{v}=-B(v(x),v(x))^{1/2}.$

This define two seminorms on $G$. Define now
$|\cdot|_{\mathfrak{pz}}:G\to_{\geq 0}$ by:

(8) $|x|_{\mathfrak{pz}}:=|x|_{v}+|x|_{\mathfrak{p}}$

Note that $|x|_{\mathfrak{pz}}$ grows at the same rate as $|x|_{\mathfrak{p}}$
if $Z_{G}$ is compact. Moreover, $|\cdot|_{\mathfrak{pz}}$ is
$\operatorname{Ad}(K)$-invariant, $K_{1}$-bi-invariant [HW90, (2.9)].

We now define the Schwartz functions on $G$ in a similar way as Harish-
Chandra, replacing $|\cdot|_{\mathfrak{p}}$ by $|\cdot|_{\mathfrak{pz}}$.
Denote $U(\mathfrak{g}_{\mathbb{C}})$ as the universal enveloping algebra of
$\mathfrak{g}$. It acts on functions of $G$ by differentiation from both left
and right. We denote the corresponding action as $L(D)$ and $R(D)$ for $D\in
U(\mathfrak{g}_{\mathbb{C}})$.

Given $D_{1},D_{2}\in U(\mathfrak{g}_{\mathbb{C}})$ and $r\in$, we define a
family of seminorms:

(9) $v^{p}_{D_{1},D_{2},r}(f)=\sup_{x\in G}\bigg{|}(1+\left\lvert
x\right\rvert_{\mathfrak{pz}})^{r}\phi_{0}^{G/Z}(x)^{-2/p}L(D_{1})R(D_{2})F(x)\bigg{|}$

for $f\in C^{\infty}(G)$. Then we define _Harish-Chandra $L^{p}$-Schwartz
space_ on $G$ to be

(10) $\mathcal{S}^{p}(G)=\left\\{f\in
C^{\infty}(G):v^{p}_{D_{1},D_{2},r}(F)<\infty\text{ for all }D_{1},D_{2}\in
U(\mathfrak{g}_{\mathbb{C}}),r\in\right\\}.$

We will prove later the kernel functions we care about lie in $S^{p}(G)$ for
all $p>0$.

As in the classical case, for all$f\in\mathcal{S}^{2}(G)$, the Plancherel
formula of groups of class $\widetilde{\mathcal{H}}$ is established [HW86,
Theorem 7.6]. Moreover, $\mathcal{S}^{2}(G)$ a dense subspace of $L^{2}(G)$,
and is a topological algebra under convolution [HW86, §6].

##  3\. Lie algebra cohomology

In this section we study the Lie algebra cohomology of $\mathfrak{g}$ on
arbitrary connected Lie group $\mathfrak{g}$. We first express the Hodge
Laplacian on the Lie algebra cohomology in a general form, and then we
specialize it to the case of $\mathfrak{g}$ is reductive. We will see how the
terms simplified significantly in this case.

First we recall some basics of Lie algebra cohomology. The details can be
found in [BW00] In particular, fix a $\mathfrak{g}$-module $(V,\tau)$ over the
field $F=$ or $\mathbb{C}$, where $\tau$ denotes the $\mathfrak{g}$-module
structure of $V$. The
$C^{q}=C^{q}(\mathfrak{g};V)=\operatorname{Hom}_{F}(\Lambda^{q}\mathfrak{g},V)$
together with differentials $d:C^{q}\to C^{q+1}$

(11)
$\begin{split}df(X_{0},\cdots,X_{q})=\sum_{i}&(-1)^{i}\tau(X_{i})f(X_{0},\cdots,\widehat{X_{i}},\cdots,X_{q})\\\
&+\sum_{i<j}(-1)^{i+j}f([X_{i},X_{j}],X_{0},\cdots,\widehat{X_{i}},\cdots,\widehat{X_{j}},\cdots,X_{q}).\end{split}$

As usual, the $\widehat{\cdot}$ stands for omission of the argument. Then we
denote $H^{*}(\mathfrak{g};V)$ the cohomology of the complex. We denote the
first sum as $d_{\circ}$ and the second sum as $d_{\wedge}$ respectively, that
is: $d=d_{\circ}+d_{\wedge}$.

Fix now a left $G$-invariant Riemannian metric on $G$. Denote
$\operatorname{coad}:\mathfrak{g}\to\operatorname{End}(\mathfrak{g}^{*}_{\mathbb{C}})$
the coadjoint representation of $G$ on $\wedge^{*}\mathfrak{g}^{*}$, defined
by

(12) $\operatorname{coad}X(l)(Y):=l([Y,X])=l(\operatorname{ad}_{-X}Y)$

for $X,Y\in\mathfrak{g}$ and $l\in\mathfrak{g}^{*}_{\mathbb{C}}$. The
Riemannian metric also induces a natural hermitian inner product on
$\bigwedge^{*}\mathfrak{g}^{*}_{\mathbb{C}}$. We further assume the
$\mathfrak{g}$-module $V$ is hermitian with hermitian product
$\langle-,-\rangle_{V}$, i.e.:

(13) $\langle\tau(X)u,v\rangle_{V}+\langle u,\tau(X)v\rangle_{V}=0$

for all $u,v\in V$ and all $X\in\mathfrak{g}$. Then
$C^{*}(\mathfrak{g};V)\cong V\otimes\wedge^{*}\mathfrak{g}^{*}_{\mathbb{C}}$
is furnished with a hermitian structure, which we denote as
$\langle-,-\rangle$.

Follow the convention of [BW00, §II.1.4], we define the adjoint operator
$\delta$ to $d$ with respect to $\langle-,-\rangle$: Fix an orthonormal basis
$X_{1},\dots,X_{n}$, Denote $\omega^{i}$ the dual basis of
$\mathfrak{g}^{*}_{\mathbb{C}}$ with respect to the perfect pairing
$\omega_{i}(X_{j})=\delta_{ij}$, and put

$\omega^{J}=\omega^{j_{1}}\wedge\omega^{j_{2}}\wedge\cdots\wedge\omega^{j_{q}}$

for a multi-index $J=\\{j_{1},\cdots,j_{q}\\}$ and $J_{m}=\\{1,\cdots,m\\}$.

Denote the structural constants associated with this Lie algebra as
$C^{\gamma}_{\alpha,\beta}$, i.e.:

(14)
$[X_{\alpha},X_{\beta}]=\sum_{\gamma}C_{\alpha,\beta}^{\gamma}X_{\gamma}.$

It is immediate from (12) that:

(15)
$\operatorname{coad}X_{\alpha}(\omega^{\beta})=\sum_{\gamma}C_{\gamma,\alpha}^{\beta}\omega^{\gamma}.$

We denote the index set of $\mathfrak{g}$ as $I_{\mathfrak{g}}$. Whenever the
index is unspecified, we meant to sum over the whole basis of $\mathfrak{g}$.

If $\eta\in C^{q}(\mathfrak{g};V)$ one writes
$\eta_{J}=\eta(X_{j_{1}},\cdots,X_{j_{q}})$ and
$\eta=\sum_{J}\eta_{J}\cdot\omega^{J}$. In this way (11) can be rewritten for
$\eta\in C^{q}(\mathfrak{g};V)$ as

$(d_{\circ}\eta)_{J}=\sum_{1\leq
u\leq{q+1}}(-1)^{u-1}\tau(X_{j_{u}})\cdot\eta_{J(u)}\qquad\text{for
}J\subseteq J_{m},|J|={q+1}$

where $J(u_{1},u_{2},\dots,u_{n})$ denotes the $J$ with the $u_{i}$th entries
removed for $i=1,\dots,n$. Meanwhile,

(16)
$\begin{split}(d_{\wedge}\eta)_{J}&=\sum_{1\leq\alpha<\beta\leq{q+1}}(-1)^{\alpha+\beta}\eta_{[j_{\alpha},j_{\beta}]\cup
J(\alpha,\beta)}\\\
&=\sum_{1\leq\alpha<\beta\leq{q+1}}\sum_{j}(-1)^{\alpha+\beta}C_{j_{\alpha},j_{\beta}}^{j}\eta_{j\cup
J(\alpha,\beta)}\end{split}$

######  Definition 3.1.

Let $\delta:C^{q}(\mathfrak{g};V)\to C^{q-1}(\mathfrak{g};V)$ be the linear
operator adjoint to $d$:

$\langle\delta\eta,\mu\rangle=\langle\eta,d\mu\rangle\quad\textrm{for all
}\eta\in C^{q}(\mathfrak{g};V),\mu\in C^{q-1}(\mathfrak{g};V)$

The summands $d_{\circ}$ and $d_{\wedge}$ admit formal adjoints
$\delta_{\circ}$ and $\delta_{\wedge}$ respectively. Their action can be
explicitly expressed on the basis as follows:

######  Proposition 3.2.

The operator $\delta_{\circ}$ satisfies

(17) $(\delta_{\circ}\eta)_{J}=\sum_{j}\tau(X_{j})^{*}\eta_{\\{j\\}\cup J},$

where $\tau(X)^{*}$ is the adjoint of $\tau(X)$ with respect to
$\langle-,-\rangle_{V}$. The operator $\delta_{\wedge}$ satisfies

(18) $(\delta_{\wedge}\eta)_{J}=\sum_{\alpha<\beta}\sum_{1\leq u\leq
q-1}(-1)^{u-1}C_{\alpha,\beta}^{j_{u}}\eta_{\\{\alpha,\beta\\}\cup J(u)}$

for $\eta\in C^{q}(\mathfrak{g};V)$ and $|J|=q-1$.

###### Proof.

The first statement is a direct adaptation of the argument in [BW00,
Proposition II.2.3]. As for the second, it suffices to verify the identity
$\langle\delta_{\wedge}\eta,\nu\rangle=\langle\eta,d_{\wedge}\nu\rangle$ for
$\eta\in C^{q}(\mathfrak{g};V)$ and $\nu\in C^{q-1}(\mathfrak{g};V)$. Note:

(19)
$\langle\eta,d_{\wedge}\nu\rangle=\sum_{|J|=q}\langle\eta_{J},(d_{\wedge}\nu)_{J}\rangle=\sum_{|J|=q}\langle\eta_{J},\sum_{1\leq\alpha<\beta\leq{q}}\sum_{j}(-1)^{\alpha+\beta}C_{j_{\alpha},j_{\beta}}^{j}\nu_{j\cup
J(\alpha,\beta)}\rangle,$

and similarly

(20)
$\langle\delta_{\wedge}\eta,\nu\rangle=\sum_{|J^{\prime}|=q-1}\langle\sum_{\alpha<\beta}\sum_{1\leq
u\leq
q-1}(-1)^{u-1}C_{\alpha,\beta}^{j^{\prime}_{u}}\eta_{\\{\alpha,\beta\\}\cup
J^{\prime}(u)},\nu_{J^{\prime}}\rangle.$

For fixed index set $J$ of $\eta$ in the second expression, the indices
$J^{\prime}$ of $\nu$ associated with $\delta_{\wedge}\eta$ are

$\bigcup_{\alpha,\beta}\left\\{J^{\prime}\in
I_{\mathfrak{g}}\>\middle|\>|J^{\prime}|=q-1\quad\\{\alpha,\beta\\}\cup
J^{\prime}(u)=J\quad\text{for some }1\leq u\leq q-1\right\\},$

whereas for the same $J$, the corresponding indices of $J^{\prime}$ associated
with $(\delta_{\wedge})_{J^{\prime}}$ are

$\begin{split}&\bigcup_{1\leq u\leq
q-1}\bigcup_{\alpha,\beta}\bigcup_{1\leq\alpha^{\prime}<\beta^{\prime}\leq
q}\left\\{J^{\prime}\>\middle|\>j^{\prime}_{\alpha^{\prime}}=\alpha,j^{\prime}_{\beta^{\prime}}=\beta,\\{\alpha,\beta\\}\cup
J^{\prime}(u)=J\right\\}\\\ =&\bigcup_{1\leq u\leq
q-1}\bigcup_{1\leq\alpha^{\prime}<\beta^{\prime}\leq
q}\left\\{J^{\prime}\>\middle|\>J^{\prime}(u)=J(\alpha^{\prime},\beta^{\prime})\right\\}\\\
=&\bigcup_{j}\bigcup_{1\leq\alpha,\beta\leq
q}\left\\{J^{\prime}\>\middle|\>J^{\prime}=j\cup
J(\alpha,\beta)\right\\}\end{split}$

which is precisely the index set in the first expression. Because
$C_{\alpha,\beta}^{j}=-C_{\beta,\alpha}^{j}$ and $C_{\alpha,\alpha}^{j}=0$ by
the antisymmetry of the Lie bracket, together with the following fact

$\eta_{J}=(-1)^{\alpha-1+\beta-2}\eta_{j_{\alpha},j_{\beta},j_{1},\cdots,\widehat{j_{\alpha}},\cdots,\widehat{j_{\beta}},\cdots,j_{q}}=(-1)^{\alpha+\beta-1}\eta_{\\{j_{\alpha},j_{\beta}\\}\cup
J(j_{\alpha},j_{\beta})},$

and the claim is proven. ∎

Next we express the Hodge Laplacian $\Delta:=\delta\circ d+d\circ\delta$ as
sum of four parts:

(21)
$\Delta=(d_{\circ}+d_{\wedge})(\delta_{\circ}+\delta_{\wedge})+(\delta_{\circ}+\delta_{\wedge})(d_{\circ}+d_{\wedge})=\Delta_{\circ}+\Delta_{\wedge}+\Delta_{\circ,\wedge}+\Delta_{\wedge.\circ}$

where:

(22) $\displaystyle\Delta_{\circ}$
$\displaystyle=d_{\circ}\delta_{\circ}+\delta_{\circ}d_{\circ}\qquad$
$\displaystyle\Delta_{\wedge}=d_{\wedge}\delta_{\wedge}+\delta_{\wedge}d_{\wedge}$
(23) $\displaystyle\Delta_{\circ,\wedge}$
$\displaystyle=d_{\circ}\delta_{\wedge}+\delta_{\wedge}d_{\circ}\qquad$
$\displaystyle\Delta_{\wedge,\circ}=d_{\wedge}\delta_{\circ}+\delta_{\circ}d_{\wedge}$

For convenience, we choose $X_{i}$ to be an orthonormal basis of
$\mathfrak{g}$ with respect to the metric, and denote

$\bar{\Omega}_{G}=\sum_{i}X_{i}^{2}\in U(\mathfrak{g}_{\mathbb{C}}).$

We also note that $X_{i}$ acts via coadjoint representation on the basis of
$\wedge^{*}\mathfrak{g}^{*}_{\mathbb{C}}$ as

(24) $\begin{split}\operatorname{coad}(X_{i})(\eta)_{J}=\sum_{1\leq u\leq
q}\sum_{j\in I_{\mathfrak{g}}}(-1)^{u-1}C^{j_{u}}_{j,i}\eta_{j\cup
J(u)},\end{split}$

This identity comes from the expression (15). Define the coadjoint*
representation as the dual representation of the coadjoint representation of
$\mathfrak{g}$, i.e.: For all $X\in\mathfrak{g}$:

(25)
$((\operatorname{coad}^{*}X)l,l^{\prime})_{\mathfrak{g}^{*}_{\mathbb{C}}}=(l,(\operatorname{coad}X)l^{\prime})_{\mathfrak{g}^{*}_{\mathbb{C}}}$

This immediately extends to an action of $\mathfrak{g}$ on
$\wedge^{*}\mathfrak{g}^{*}$, which we also denote as
$\operatorname{coad}^{*}$. Note $\operatorname{coad}^{*}$ and
$-\operatorname{ad}$ are isomorphic representations by the following diagram:

${\mathfrak{g}^{*}}$${\mathfrak{g}}$${\mathfrak{g}^{*}}$${\mathfrak{g}}$$\scriptstyle{\flat,\cong}$$\scriptstyle{\sharp,\cong}$$\scriptstyle{-\operatorname{coad}_{X}^{*}}$$\scriptstyle{\operatorname{ad}_{X}}$

for all $X\in\mathfrak{g}$. Here $\flat$ and $\sharp$ are the musical
isomorphisms.

Next we form a general expression for each component of the Hodge Laplacian.
The following proposition holds for an arbitrary Lie algebra $\mathfrak{g}$:

######  Proposition 3.3.

Given a hermitian $\mathfrak{g}$-module $(V,\tau)$ and form
$C^{q}(\mathfrak{g};V)$ as above. Then the Laplacian admits the following
expressions:

  1. (A)

$\Delta_{\circ}$ acts on $C^{q}(\mathfrak{g};V)$ as

(26)
$(\Delta_{\circ}\eta)_{J}=-\sum_{j}\tau(X_{j})^{*}\tau(X_{j})\eta_{J}+\sum_{a}\tau(X_{a})\sum_{1\leq
u\leq q}(-1)^{u}C^{a}_{j,j_{u}}\eta_{\\{j\\}\cup J(u)};$

  2. (B)

$\Delta_{\wedge}$ acts on $C^{q}(\mathfrak{g};V)$ as

(27)
$\begin{split}(\Delta_{\wedge}\eta)_{J}=\sum_{j}\sum_{\alpha<\beta}C^{j}_{\alpha,\beta}\Biggl{(}&\sum_{1\leq\gamma\leq
q}(-1)^{\gamma}C^{j_{\gamma}}_{\alpha,\beta}\eta_{j\cup J(\gamma)}\\\
+&\sum_{1\leq u<v\leq
q}(-1)^{u+v}C^{j}_{j_{u},j_{v}}\eta_{\\{\alpha,\beta\\}\cup
J(u,v)}\Biggr{)};\end{split}$

  3. (C)

$\Delta_{\circ,\wedge}$ acts on $C^{q}(\mathfrak{g};V)$ as
$\sum_{k}\tau(X_{k})^{*}\operatorname{coad}(X_{k})$;

  4. (D)

$\Delta_{\wedge,\circ}$ acts on $C^{q}(\mathfrak{g};V)$ as
$-\sum_{k}\tau(X_{k})\operatorname{coad}^{*}(X_{k})$;

In particular, if we take $V$ to be unitary, then the $0$-th Laplacian on
$C^{0}(\mathfrak{g};V)$ is seen to take the form:

(28) $\Delta_{0}=-\tau(\bar{\Omega}_{G}).$

###### Proof.

We prove the formula for $\Delta_{\circ,\wedge}$ as an example. The rest can
be found in [Han24, Proposition 1.5].

$\begin{split}(\delta_{\wedge}d_{\circ}\eta)_{J}&=\sum_{\alpha<\beta}\sum_{1\leq
u\leq
q-1}(-1)^{u-1}C^{j_{u}}_{\alpha,\beta}(d_{\circ}\eta)_{\\{\alpha,\beta\\}\cup
J(u)}\\\ &=\sum_{\alpha<\beta}\sum_{1\leq u\leq
q-1}(-1)^{u-1}C_{\alpha,\beta}^{j_{u}}\Bigg{(}\tau(X_{\alpha})\eta_{\beta\cup
J(u)}-\tau(X_{\beta})\eta_{\alpha\cup J(u)}\\\
&\qquad\qquad-\sum_{\begin{subarray}{c}1\leq v\leq q\\\ v\neq
u\end{subarray}}(-1)^{v[u]}\tau(X_{j_{v}})\eta_{\\{\alpha,\beta\\}\cup
J(u,v)}\Bigg{)}.\end{split}$

In the last step, the notation $v[u]$ is an abbreviation of notation, defined
by

(29) $v[u]=\begin{cases}v&\textnormal{ if }v<u\\\ v-1&\textnormal{ if
}u<v.\end{cases}$

On the other hand,

$\displaystyle(d_{\circ}\delta_{\wedge}\eta)_{J}$ $\displaystyle=\sum_{1\leq
v\leq q+1}(-1)^{v-1}\tau(X_{j_{v}})(\delta_{\wedge}\eta)_{J(v)}$
$\displaystyle=\sum_{1\leq v\leq
q+1}\sum_{\alpha<\beta}\sum_{\begin{subarray}{c}1\leq u\leq q\\\ u\neq
v\end{subarray}}(-1)^{v-1}\tau(X_{j_{v}})C^{j_{u}}_{\alpha,\beta}\cdot(-1)^{u[v]}\eta_{\\{\alpha,\beta\\}\cup
J(u,v)}$

We see the last summand in $d_{\circ}\delta_{\wedge}\eta$ cancels with
$\delta_{\wedge}d_{\circ}$ as

$(-1)^{v+u[v]}=-(-1)^{u+v[u]}$

for all $u\neq v$. Hence

(30) $\begin{split}(\Delta_{\circ,\wedge}\eta)_{J}&=\sum_{1\leq u\leq
q}\sum_{\alpha<\beta}(-1)^{u-1}C^{j_{u}}_{\alpha,\beta}\left(\tau(X_{\alpha})\eta_{\beta\cup
J(u)}-\tau(X_{\beta})\eta_{\alpha\cup J(u)}\right)\\\
&=\sum_{\gamma}(\tau(X_{\gamma})\operatorname{coad}(X_{\gamma}))(\eta_{J}),\end{split}$

where the last identity comes from our convention (24). Hence the third
statement is proven. The rest is proved similarly. ∎

######  Remark 3.4.

For later purposes, we note that the operator

$\begin{split}(\square_{\circ}\eta)_{J}:=(\Delta_{\circ}\eta+\tau(\bar{\Omega}_{G})\eta)_{J}=\sum_{a}\tau(X_{a})\sum_{\begin{subarray}{c}j\\\
1\leq u\leq q\end{subarray}}(-1)^{u}C^{a}_{j,j_{u}}\eta_{\\{j\\}\cup
J(u)}\end{split}$

acts on the $C^{q}(\mathfrak{g};V)\cong
V\otimes\wedge^{p}\mathfrak{g}^{*}_{\mathbb{C}}$ as derivations on the
exterior algebra part, in the following sense:

(31)
$\square_{\circ}(\eta_{J}\omega^{J})=\square_{\circ}(\eta_{J}\omega^{J_{1}})\wedge\omega^{J_{2}}+\omega^{J_{1}}\wedge\square_{\circ}(\eta_{J}\omega^{J_{2}})$

for any $\omega^{J}=\omega^{J_{1}}\wedge\omega^{J_{2}}$. This can be proved by
comparing the expressions on both sides:

$\begin{split}&\square_{\circ}(\eta_{J}\omega^{J_{1}})\wedge\omega^{J_{2}}+\omega^{J_{1}}\wedge\square_{\circ}(\eta_{J}\omega^{J_{2}})\\\
=&\sum_{a}\tau(X_{a})(\eta_{J})\biggl{(}\sum_{\begin{subarray}{c}j\\\ 1\leq
u\leq|J_{1}|\end{subarray}}(-1)^{u}C^{a}_{j,j_{u}}\omega^{j}\wedge\omega^{J_{1}(u)}\wedge\omega^{J_{2}}\\\
&\ \ +\sum_{\begin{subarray}{c}j\\\ |J_{1}|+1\leq
u\leq|J_{2}|\end{subarray}}(-1)^{u-|J_{1}|}C^{a}_{j,j_{u}}\omega^{J_{1}}\wedge\omega^{j}\wedge\omega^{J_{2}(u-|J_{1}|)}\biggr{)}\\\
=&\square_{\circ}(\eta_{J}\omega^{J})\end{split}$

where the last identity follows from
$(-1)^{u-|J_{1}|}\omega^{J_{1}}\wedge\omega^{j}=(-1)^{u}\omega^{j}\wedge\omega^{J_{1}}$.
Hence the claim is proven.

If we assume further that $\mathfrak{g}$ is real reductive, one can fix a non-
degenerate $G$-invariant bilinear form $B(-,-)$ on $\mathfrak{g}_{\mathbb{C}}$
that restricts to the Killing form on $[\mathfrak{g,g}]$. If we fix a Cartan
involution $\theta$ as in the previous section, we note $B(-,-)$ is negative
definite on $\mathfrak{k}$ and positive definite on $\mathfrak{p}$. This gives
us a positive definite $\operatorname{Ad}(K)$-invariant hermitian inner
product, denoted as $B^{\theta}$, such that

(32) $B^{\theta}(X,Y):=B(X,\theta\bar{Y})$

for all $X,Y\in\mathfrak{g}_{\mathbb{C}}$, where $\bar{\cdot}$ denotes the
complex conjugation of $\mathfrak{g}_{\mathbb{C}}$ over $\mathfrak{g}$. In
this case the previous $\bar{\Omega}_{G}$ admits a nicer expression:

(33) $\bar{\Omega}_{G}=\Omega_{G}-2\Omega_{K}\in U(\mathfrak{g}_{\mathbb{C}})$

Recall $\Omega_{G}$ is the Casimir element in the center
$Z(\mathfrak{g}_{\mathbb{C}})$ of $U(\mathfrak{g}_{\mathbb{C}})$ induced by
the $G$-bi-invariant bilinear form $B$:

$\Omega_{G}=\sum_{i}X_{i}X^{i}$

for a basis $\\{X_{i}\\}$ of $\mathfrak{g}$ with the dual basis $\\{X^{i}\\}$
with respect to $B$. Here $\Omega_{K}$ is defined in a similar way by
replacing the basis of $\mathfrak{g}$ by that of $\mathfrak{k}$. If we take
$\\{X_{i}\\}$ to be an orthonormal basis of $\mathfrak{g}$ with respect to
$B^{\theta}$, then $\bar{\Omega}_{G}=\sum_{i}X_{i}^{2}$ is just the sum of
squares.

Fix an orthonormal basis of $\mathfrak{g}$

$\\{X_{1},\cdots,X_{m},Y_{1},\cdots,Y_{n-m}\\}$

with respect to the Killing form $B^{\theta}$, such that $\\{X_{i}\\}$ is an
the orthonormal basis of $\mathfrak{k}$ and $\\{Y_{\alpha}\\}$ that of
$\mathfrak{p}$. We denote the index sets of $\mathfrak{k}$ and $\mathfrak{p}$
as $I_{\mathfrak{k}}$ and $I_{\mathfrak{p}}$ respectively, and again
$I_{\mathfrak{g}}=I_{\mathfrak{k}}\sqcup I_{\mathfrak{p}}$ the index set of
$\mathfrak{g}$. Before the proof of the corollary, we remark that the
coadjoint and coadjoint* representations satisfy the following relations when
restricted to $\mathfrak{k}$ and $\mathfrak{p}$ respectively:

(34)
$\operatorname{coad}|_{\mathfrak{k}}=\operatorname{coad}^{*}|_{\mathfrak{k}}\qquad\operatorname{coad}|_{\mathfrak{p}}=-\operatorname{coad}^{*}|_{\mathfrak{p}}.$

######  Corollary 3.5 (Generalized Kuga’s Lemma).

Assume $\mathfrak{g}$ is reductive with a Cartan decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ as above. Let $V$ be an
hermitian $\mathfrak{g}$-module. Define a bigrading on
$C^{k}(\mathfrak{g};V)=\sum_{p+q=k}C^{p,q}(\mathfrak{g};V)$ with

$C^{p,q}(\mathfrak{g};V):=\operatorname{Hom}_{\mathbb{C}}(\wedge^{p}\mathfrak{k}\otimes\wedge^{q}{\mathfrak{p}},V)\cong
V\otimes\wedge^{p}\mathfrak{k}^{*}\otimes\wedge^{q}\mathfrak{p}^{*}.$

Then $\Delta_{\circ}$ and $\Delta_{\wedge}$ take the following simplified
form:

  1. (I)

$\Delta_{\circ}=\tau(\bar{\Omega}_{G})+\square_{\circ}$ acts on
$C^{p,0}(\mathfrak{g};V)$ and $C^{0,q}(\mathfrak{g};V)$ by

(35)
$\begin{split}\square_{\circ}\Big{|}_{C^{p,0}(\mathfrak{g};V)}&\cong+\sum_{j\in
I_{\mathfrak{g}}}\tau(X_{j})\operatorname{coad}^{*}(X_{j})\\\
\square_{\circ}\Big{|}_{C^{0,q}(\mathfrak{g};V)}&\cong-\sum_{j\in
I_{\mathfrak{g}}}\tau(X_{j})\operatorname{coad}^{*}(X_{j})\end{split}$

This extends to an action on $C^{p,q}(\mathfrak{g};V)$ by derivation, as a
consequence of Remark 3.4.

  2. (II)

$\Delta_{\wedge}$ acts on $C^{n}(\mathfrak{g};V)$ by
$\sum_{j}\operatorname{coad}^{*}(X_{j})\operatorname{coad}(X_{j})$.

In particular, if we abbreviate the action by:

(36)
$\begin{split}A_{\mathfrak{g}}&=-\tau(\bar{\Omega}_{G}):C^{p,q}(\mathfrak{g};V)\longrightarrow
C^{p,q}(\mathfrak{g};V)\\\ B_{\mathfrak{k}}&=\sum_{k\in
I_{\mathfrak{k}}}\tau(X_{k})\operatorname{coad}^{*}(X_{k}):C^{p,q}(\mathfrak{g};V)\longrightarrow
C^{p,q}(\mathfrak{g};V)\\\ C_{\mathfrak{p}}&=\sum_{\alpha\in
I_{\mathfrak{p}}}\tau(Y_{\alpha})\operatorname{coad}^{*}(Y_{\alpha}):C^{p,q}(\mathfrak{g};V)\longrightarrow
C^{p-1,q+1}(\mathfrak{g};V)\oplus C^{p+1,q-1}(\mathfrak{g};V)\\\
D_{\mathfrak{g}}&=\frac{1}{2}\operatorname{coad}^{*}(\Omega_{G}):C^{p,q}(\mathfrak{g};V)\longrightarrow
C^{p,q}(\mathfrak{g};V)\end{split}$

then each component of $\Delta_{1}$ acts on $1$-forms
$C^{1}(\mathfrak{g};V)\cong(V\otimes\mathfrak{k}^{*})\oplus(V\otimes\mathfrak{p}^{*})$
via the following operator-valued matrices:

(37)
$\begin{split}\Delta_{\circ}|_{C^{1}(\mathfrak{g};V)}=\begin{pmatrix}A_{\mathfrak{g}}+B_{\mathfrak{k}}&-C_{\mathfrak{p}}\\\
C_{\mathfrak{p}}&A_{\mathfrak{g}}-B_{\mathfrak{k}}\end{pmatrix}&\qquad\Delta_{\wedge}|_{C^{1}(\mathfrak{g};V)}=\begin{pmatrix}D_{\mathfrak{g}}&\\\
&D_{\mathfrak{g}}\end{pmatrix}\\\
\Delta_{\circ,\wedge}|_{C^{1}(\mathfrak{g};V)}=\begin{pmatrix}-B_{\mathfrak{k}}&C_{\mathfrak{p}}\\\
C_{\mathfrak{p}}&-B_{\mathfrak{k}}\end{pmatrix}&\qquad\Delta_{\wedge,\circ}|_{C^{1}(\mathfrak{g};V)}=\begin{pmatrix}-B_{\mathfrak{k}}&-C_{\mathfrak{p}}\\\
-C_{\mathfrak{p}}&-B_{\mathfrak{k}}\end{pmatrix}\end{split}$

Summing all terms up, the $1$-th Laplacian on $C^{1}(\mathfrak{g};V)$ is seen
to take the form:

(38)
$\Delta_{1}=\begin{pmatrix}A_{\mathfrak{g}}-B_{\mathfrak{k}}+D_{\mathfrak{g}}&-C_{\mathfrak{p}}\\\
C_{\mathfrak{p}}&A_{\mathfrak{g}}-3B_{\mathfrak{k}}+D_{\mathfrak{g}}\end{pmatrix}$

###### Proof.

The details of the proof can be found in [Han24, Corollary 1.7]. The main
ingredient is to refine the results from Proposition 3.3 and by exploiting the
extra symmetries in the structural constants: Given
$X_{i},X_{j},X_{k}\in\mathfrak{k}$ and $Y_{\alpha},Y_{\beta}\in\mathfrak{p}$
some basis vectors, one readily verify the following identity:

$C^{\beta}_{\alpha,i}=C^{\alpha}_{i,\beta}=-C_{\beta,\alpha}^{i}\qquad
C^{k}_{i,j}=C^{i}_{j,k}$

whereas all structural constants $C^{i}_{\alpha,j}$ vanish by orthogonality.

Assuming $J\subseteq I_{\mathfrak{k}}$, the coadjoint action for fixed $a$ can
be rewritten as:

$\displaystyle(\square_{\circ}\eta)_{J}$ $\displaystyle=\sum_{a\in
I_{\mathfrak{g}}}\tau(X_{a})\sum_{\begin{subarray}{c}j\\\ 1\leq u\leq
q\end{subarray}}(-1)^{u-1}C^{a}_{j,j_{u}}\eta_{j\cup J(u)}$
$\displaystyle=-\sum_{a\in
I_{\mathfrak{k}}}\biggl{(}\tau(X_{a})(-1)^{u}\sum_{\begin{subarray}{c}1\leq
u\leq q\\\ j\in I_{\mathfrak{k}}\end{subarray}}C^{a}_{j,j_{u}}\eta_{j\cup
J(u)}\biggr{)}$ $\displaystyle\qquad\qquad-\sum_{\alpha\in
I_{\mathfrak{p}}}\biggl{(}\tau(X_{\alpha})(-1)^{u}\sum_{\begin{subarray}{c}1\leq
u\leq q\\\ \beta\in
I_{\mathfrak{p}}\end{subarray}}C^{\alpha}_{\beta,j_{u}}\eta_{j\cup
J(u)}\biggr{)}$ $\displaystyle=\sum_{a\in
I_{\mathfrak{k}}}\biggl{(}\tau(X_{a})(-1)^{u}\sum_{\begin{subarray}{c}1\leq
u\leq q\\\ j\in I_{\mathfrak{k}}\end{subarray}}C^{j}_{a,j_{u}}\eta_{j\cup
J(u)}\biggr{)}$ $\displaystyle\qquad\qquad+\sum_{\alpha\in
I_{\mathfrak{p}}}\biggl{(}\tau(X_{\alpha})(-1)^{u}\sum_{\begin{subarray}{c}1\leq
u\leq q\\\ \beta\in
I_{\mathfrak{p}}\end{subarray}}C^{\beta}_{\alpha,j_{u}}\eta_{j\cup
J(u)}\biggr{)}$ $\displaystyle=\sum_{a\in
I_{\mathfrak{g}}}\tau(X_{a})\operatorname{coad}(X_{a})(\eta_{J}),$

where in the last identity we used
$\operatorname{coad}X_{\alpha}(\omega^{\beta})=\sum_{\gamma}C_{\gamma,\alpha}^{\beta}\omega^{\gamma}$,
as well as our convention (24), and then by exploiting the identities of the
structural constants above. The rest of the identities are derived in a
similar way. ∎

##  4\. Elliptic operators on Lie groups

In this section we review and prove some classical results about kernel
estimates of the heat semigroup. The strategy is to first develop a Gaussian
estimate for the second order differential operator $\bar{\Omega}_{G}$.

We first derive a Gaussian upper bound for strongly elliptic operators with
constant coefficients. Given a continuous representation $(\pi,H_{\pi})$ of
$G$, i.e, a homomorphism $\pi$ from $G$ to the group of linear transformation
on $H_{\pi}$ such that the map

$G\times H_{\pi}\to H_{\pi}\qquad(g,v)\mapsto\pi(g)v$

is continuous. We can choose a basis $\\{X_{i}\\}$ of $\mathfrak{g}$ and
define _differential operator with constant coefficients_ as a polynomial in
$D_{i}=\pi(X_{i})$, where:

(39) $D=\sum_{I}c_{I}D^{I}$

with $D^{I}=D_{i_{1}}D_{i_{2}}\dots D_{i_{n}}$ for $I=(i_{1},\cdots,i_{n})$,
and $c_{I}$ are complex coefficients. We say the differential operator is
_strongly elliptic_ if its associated principal symbol satisfies the following
inequality

$\operatorname{Re}\left((-1)^{\frac{m}{2}}\sum_{|I|=m}c_{I}\xi^{I}\right)>0$

for all $\xi\in^{d}\backslash\\{0\\}$, where $m$ is the order of the
differential operator. Furthermore, we denote the $C^{n}$-vectors in the
representation $H_{\pi}$ as

(40) $C^{n}(H_{\pi})=\bigcap_{1\leq i_{1},\dots,i_{n}\leq
d}\operatorname{Dom}(\pi(X_{i_{1}}\dots X_{i_{n}})),$

which defines a subspace of the smooth vectors of $H_{\pi}$ independent of the
choice of basis [Rob91, Page 8]. The $C^{n}$-norm of $v\in H_{\pi}$ is defined
as

(41) $\left\lVert v\right\rVert_{C^{n}(H_{\pi})}=\sup_{|I|\leq
n}\left\lVert\pi(X^{I})v\right\rVert_{H_{\pi}}$

and the analytic vectors of $H_{\pi}$ are defined to be those $v\in H_{\pi}$
such that

$\sum_{n\geq 1}\frac{t^{n}}{n!}\left\lVert
v\right\rVert_{C^{n}(H_{\pi})}<\infty$

for some $t>0$. It is a classical result proved independently by Langlands
[Lan60] and Nelson [Nel59] that the analytic vectors are dense in the
continuous representation $H_{\pi}$ of $G$. We collect some results of
Langlands’ thesis which we will use in establishing the Gaussian estimate. The
details can be found in [Rob91].

Let $(\pi,H_{\pi})$ be weakly* or strongly continuous representation, and let
$D$ be a strongly elliptic operator associated with $\pi$. Then $D$ is
closable and its closure $\bar{D}$ generates a holomorphic semigroup, denoted
as $e^{-t\bar{D}}$ on respective representations satisfying the property:

$\bar{D}v=\lim_{t\to 0}\frac{v-e^{-t\bar{D}}v}{t}$

for all $v\in\mathrm{Dom}(D)$. Moreover, the semigroup satisfies the following
the property [Rob91, Theorem 1.5.1]:

  1. I.

$e^{-t\pi(\bar{D})}H_{\pi}\subseteq H_{\pi}^{\infty}$ maps the space into its
smooth vectors $H_{\pi}^{\infty}$ for all $t>0$;

  2. II.

The map $z\mapsto e^{-z\pi(\bar{D})}$ defines a holomorphic family of bounded
operators on $H_{\pi}$ in the sector $\\{z\in\mathbb{C}:|\arg z|<C\\}$ for
some $C\in(0,\pi/2]$ depending on the elliptic operator. Moreover,
$e^{-t\pi(\bar{D})}H_{\pi}\subseteq\operatorname{Dom}(\bar{D})$ for $t>0$ and

(42) $\left\lVert\pi(\bar{D})e^{-t\pi(\bar{D})}v\right\rVert_{H_{\pi}}\leq
Mt^{-1}\left\lVert v\right\rVert_{H_{\pi}}$

for some $M\geq 1$.

Moreover, there exists $k,l>0$ such that we have the following small time
estimate [Rob91, Theorem II.2.2]:

(43) $\left\lVert e^{-t\pi(\bar{D})}v\right\rVert_{C^{n}(H_{\pi})}\leq
kl^{n}n!t^{-\frac{n}{m}}\left\lVert v\right\rVert_{H_{\pi}}$

for all $v\in H_{\pi}$ and $t\in(0,1]$. On the other hand, using the fact that
$e^{-t\bar{D}}$ is a holomorphic family of bounded operators for $t>0$, and
general semigroup theory [Rob91, p.96], we choose the lower bound of the
operator norm

$\omega=\inf_{t>0}\frac{\log\|e^{-t\bar{D}}\|}{t}>-\infty.$

Here $\left\lVert\cdot\right\rVert$ denotes the operator norm on $H_{\pi}$.
Consequently there is a $M\geq 1$, such that

(44) $\left\lVert e^{-t\bar{D}}\phi\right\rVert_{H_{\pi}}\leq Me^{\omega
t}\left\lVert\phi\right\rVert_{H_{\pi}}$

for all $t>0$. We call this the continuity bound.

Before moving on to establish the Gaussian estimate, we introduce the last
bound which is a Sobolev inequality in the Lie group setting. If we choose the
representations $(\pi,H_{\pi})=(L,L^{p}(G))$ with $L$ the left translation of
$G$ on $L^{p}(G)$, then we can define the Sobolev spaces

$W^{k,p}(G):=C^{k}(L^{p}(G))$

as the $C^{k}$-vectors of $L^{p}(G)$ with $C^{k}$-norm in $L^{p}(G)$ defined
as in (41), and correspondingly the $C^{k}(G)$ as the $C^{k}$-vectors of
$C_{0}(G)$, the continuous functions on $G$ that vanish at infinity with:

$C^{k}(G):=C^{k}(C_{0}(G))\quad\left\lVert\varphi\right\rVert_{C^{m}}=\sup_{|I|\leq
m}\left\lVert L(X^{I})\varphi\right\rVert_{L^{\infty}(G)}.$

Let $U$ be a bounded open subset of the identity component of $G$. Denote
$d:=\dim G$, and the Sobolev inequality is proved essentially the same way as
the real case [Rob91, Lemma III.2.3]:

(45) $\left\lVert\varphi\right\rVert_{C^{m}(U)}\leq
C\left\lVert\varphi\right\rVert_{W^{k,p}(U)}$

for all $0\leq m<k-\frac{d}{p}$ and for all $\varphi\in W^{k,p}(U)$, with $C$
being the implied constant.

Next we consider the heat kernel associated with $D$. Fix a left $G$-invariant
metric on $G$. This defines a distance function $|g|:=d(g,e_{G})$ on $G$, as
well as a left Haar measure on $G$. Then there is integrable function
$k_{t}:_{+}\times G\to\mathbb{C}$ such that for every strongly continuous and
weakly* continuous representation $(H_{\pi},\pi)$ of $G$, the corresponding
heat semigroup $e^{-t\pi(\bar{D})}$ defined above satisfies

(46) $e^{-t\pi(\bar{D})}=\int_{G}\pi(g)k_{t}(g)\mathop{}\\!\mathrm{d}{g}.$

with the kernel $(t,g)\mapsto k_{t}(g)$ defines an analytic function on
${}_{+}\times G$ [Rob91, Theorem III.2.1].

For $1\leq p\leq\infty$, we define the weighted $L^{p}$-space $L^{p}_{\rho}$
for any $\rho\geq 0$ as

$L^{p}_{\rho}:=L^{p}(G;e^{\rho|g|}\mathop{}\\!\mathrm{d}{g})\qquad\left\lVert\varphi\right\rVert_{L^{p}_{\rho}}:=\left(\int\Big{|}\varphi(g)e^{\rho|g|}\Big{|}^{p}\mathop{}\\!\mathrm{d}{g}\right)^{1/p}$

and let $L^{\rho}$ be the left regular representation of $G$ on it. All
weighted $L^{p}$-spaces are Banach algebras under convolution [KMB12, Section
I]. Similarly, define the weighted $L^{\infty}$-space $L^{\infty}_{\rho}(G)$
with weighted $L^{\infty}$-norm as

(47)
$\left\lVert\varphi\right\rVert_{L^{\infty}_{\rho}}:=\operatorname*{ess.sup}_{g\in
G}e^{\rho|g|}|\varphi(g)|.$

We now derive a pointwise Gaussian bound from the inequalities we listed
above. The representation in question is $(L^{\rho},L^{p}_{\rho}(G))$ for
suitably chosen $\rho$. The following lemma is stated in [Rob91, Corollary
III.2.5]. We include the proof to complete the discussion.

######  Lemma 4.1.

Let $G$ be a Lie group, and $X_{1},\dots,X_{d}$ a basis of the Lie algebra
$\mathfrak{g}$. Given a $G$-invariant strongly elliptic operator $D$ of order
$m$ on $G$ expressed in $X_{i}$, and form the kernel $k_{t}$ for $t>0$ as
above. Then for each $\rho\geq 0$ there exist $a,b,c>0$ and $\omega\geq 0$
such that we have the following pointwise Gaussian estimate:

(48) $\left\lvert L(X^{\alpha})\partial_{t}^{\ell}k_{t}(g)\right\rvert\leq
ab^{|\alpha|}c^{\ell}|\alpha|!\ell!(1+t^{-(\ell+\frac{|\alpha|+d+1}{m})})e^{\omega
t}e^{-\rho|g|}$

for all $g>G$ and $t>0$. Here again $L(X^{\alpha})$ denotes the left regular
representation of $G$ and $\partial_{t}$ denotes the differential in the time
direction.

###### Proof.

For simplicity we prove the statement only for unimodular $G$ and comment on
the other case in the end of the proof. We consider the representation to be
the left-regular translation $L^{\rho}$ of $G$ on the weighted $L^{1}$-space
$L_{\rho}^{1}(G)$.

Let $e^{-tL^{\rho}(\bar{D})}$ be the corresponding heat semigroup and:

(49)
$L^{\rho}(X^{\alpha})D^{\ell}e^{-tL^{\rho}(\bar{D})}\varphi(e_{G})=\int_{G}\left(L(X^{\alpha})(-\partial_{t})^{\ell}k_{t}\right)(g)e^{\rho|g|}\cdot(\varphi(g^{-1})e^{-\rho|g|})\mathop{}\\!\mathrm{d}{g}.$

Here we use the fact that $Dk_{t}+\partial_{t}k_{t}=0$, and we can interchange
the order of the derivative and the integral via the following equality

$D(k_{t}*\varphi)=(Dk_{t})*\varphi,$

again using Langlands’ result that $k_{t}\in C^{\infty}(G)\cap L^{1}(G)$. Next
by the duality between $L^{1}$ and $L^{\infty}$, we rewrite the integral as

$\sup_{g\in
G}|(\partial_{g}^{\alpha}\partial_{t}^{\ell}k_{t})(g)|e^{\rho|g|}=\sup_{\left\lVert\varphi\right\rVert_{L^{1}_{\rho}}\leq
1}\left\\{|L^{\rho}(X^{\alpha})D^{\ell}e^{-tL^{\rho}(\bar{D})}\varphi(e_{G})|\right\\}.$

Now consider the $C^{k}$-vectors of the weighted $L^{1}$-space

$W^{k,1}_{\rho}(G):=C^{k}(L^{1}_{\rho}(G)),$

with weighted $C^{k}$-norm $\left\lVert\cdot\right\rVert_{C^{k}_{\rho}}$
defined as in (41) by replacing $H_{\pi}$ by $L^{1}_{\rho}(G)$. Consider their
restriction to a bounded open neighborhood $U$ of $e_{G}$, where the
$L^{1}$-norm and the weighted $L^{1}$-norm are equivalent. Applying now the
Sobolev embedding (45) by taking $m,k,p=0,d+1,1$ we get a constant
$c_{\rho,U}>0$ such that:

$\begin{split}|X^{\alpha}D^{\ell}e^{-tL^{\rho}(\bar{D})}\varphi(e_{G})|&\leq\left\lVert
X^{\alpha}D^{\ell}e^{-tL^{\rho}(\bar{D})\varphi(e_{G})}\right\rVert_{C^{0}(U)}\\\
&\leq c_{\rho,U}\left\lVert
X^{\alpha}D^{\ell}e^{-tL^{\rho}(\bar{D})}\varphi\right\rVert_{W^{d+1,1}_{\rho}(U)}\\\
&\leq c_{\rho,U}\left\lVert
D^{\ell}e^{-tL^{\rho}(\bar{D})}\varphi\right\rVert_{W^{|\alpha|+d+1,1}_{\rho}(U)}.\end{split}$

Hence $|\partial_{g}^{\alpha}\partial_{t}^{\ell}k_{t}(g)|$ remains bounded for
all $g\in G$ and $t>0$ by the operator norm

$|\partial_{g}^{\alpha}\partial_{t}^{\ell}k_{t}(g)|\leq
c_{\rho,U}e^{-\rho|g|}\left\lVert
D^{\ell}e^{-tL^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to
W^{|\alpha|+d+1,1}_{\rho}(U)}.$

with $\left\lVert\cdot\right\rVert_{X\to Y}$ denotes the operator norm between
Banach spaces $X$ and $Y$. Again from the fact that
$\phi=e^{-tL^{\rho}(\bar{D})}\varphi$ solves the equation
$\partial_{t}\phi+\bar{D}\phi=0$, we obtain the following identity:

$D^{\ell}e^{-tL^{\rho}(\bar{D})}=e^{-t_{1}L^{\rho}(\bar{D})}(De^{-\frac{t_{2}}{\ell}L^{\rho}(\bar{D})})^{\ell}e^{-t_{3}L^{\rho}(\bar{D})}$

for every triplet $t_{1},t_{2},t_{3}>0$ with $t_{1}+t_{2}+t_{3}=t$. Rewrite
$\beta:=|\alpha|+d+1$, we therefore want to estimate the following:

$\begin{split}&e^{\rho|g|}\left\lvert(\partial_{g}^{\alpha}\partial_{t}^{\ell}k_{t}(g))\right\rvert\\\
\leq&c_{\rho,U}\left\lVert
e^{-t_{1}L^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to
W^{\beta,1}}\left(\left\lVert
De^{-\frac{t_{2}}{\ell}L^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to
L^{1}_{\rho}}\right)^{\ell}\left\lVert
e^{-t_{3}L^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to
L^{1}_{\rho}}.\end{split}$

The norms in three time intervals are bounded via different methods:

  1. (I)

For $t_{1}\in(0,1]$, we use the small time estimate (43). Applying to
$L^{1}_{\rho}$-norm to estimate the $W^{\beta,1}$-norm:

(50) $\left\lVert e^{-t_{1}L^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to
W^{\beta,1}}\leq c^{\prime}c^{\beta}\beta!t^{-\frac{\beta}{m}}$

for some $c^{\prime},c>0$. This contributes the factor
$a^{\prime}b^{|\alpha|}|\alpha|!t^{-\beta/m}$ to the final expression;

  2. (II)

For $t_{2}\in(0,\ell]$, recall that $e^{-tL^{\rho}(\bar{D})}$ defines a
holomorphic family of bounded operator for $t$ in the real line, and
$e^{-tL^{\rho}(\bar{D})}L^{1}_{\rho}\subseteq\operatorname{Dom}(\bar{D})$.
Hence we have the following bound:

(51) $\left\lVert
De^{-\frac{t_{2}}{\ell}L^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to
L^{1}_{\rho}}\leq c_{2}{\frac{\ell}{t_{2}}}$

for some $c_{2}>0$, by changing the variable $t_{2}$ to $t_{2}/\ell$ and
recalling the bound (42). Hence the second factor contributes a term
$\ell^{\ell}c_{2}^{\ell}t_{2}^{-\ell}\sim\ell!c_{2}^{\ell}t_{2}^{-\ell}$ to
the final expression by Stirling’s approximation;

  3. (III)

Lastly the continuity bound (44) gives

$\left\lVert e^{-t_{3}L^{\rho}(\bar{D})}\varphi\right\rVert_{L^{1}_{\rho}}\leq
C^{\prime\prime}e^{\omega
t_{3}}\left\lVert\varphi\right\rVert_{L^{1}_{\rho}},$

and consequently $\left\lVert
e^{-t_{3}L^{\rho}(\bar{D})}\right\rVert_{L^{1}_{\rho}\to L^{1}_{\rho}}\leq
C^{\prime\prime}e^{\omega t_{3}}$ for some $C^{\prime\prime}>0$. Hence the
third factor contributes a term $e^{\omega t_{3}}$ to the final expression.

Summing up all the contributions we see the upper bound of the derivative is
indeed as claimed.

We end the proof with a comment on generalizing the proof to non-unimodular
cases. All the key ingredients are the same, but one needs to take into
account the difference between the right and the left-invariant Haar measure,
and also to measure the growth of the modular function $\Delta(g)$ resulting
from changing one measure to the other. The modular function is an analytic
homomorphism and hence we can bound it pointwise by $e^{\omega^{\prime}|g|}$
for some $\omega^{\prime}>0$ and $C\geq 1$. Replacing $\rho$ in the argument
by $\rho+\omega^{\prime}$ one then gets the correct estimate. ∎

######  Remark 4.2.

The Gaussian bound we obtained in Lemma 4.1 is in fact crude and can be
refined via techniques such as logarithmic Sobolev inequalities or Nash
inequalities (see [Rob91] for an extensive survey). But for our purpose, the
current estimate is sufficient.

##  5\. Proof of the Schwartz estimate

In this section we prove A, the major result of this paper. First we recall
the theory of bounded perturbation of heat semigroups. The following theorem
is from [HP74, Section 13.4].

######  Theorem 5.1.

Let $X$ be a Banach space. Let $e^{-tA}$ a continuous semigroup of bounded
operators on $X$ with $A$ be the infinitesimal generator of a semigroup, and
$B$ a linear operator such that
$\operatorname{Dom}(B)\supseteq\operatorname{Dom}(A)$, and $Be^{-tA}$ defines
a $t$-measurable family of bounded operators on $X$. Moreover, we assume that
both $e^{-tA}$ and $Be^{-tA}$ satisfy the following estimates:

  * •

For some $M,\omega>0$, $\left\lVert e^{-tA}\right\rVert\leq Me^{t\omega}$ for
all $t>0$.

  * •

For some $\alpha<1$ and $c>0$, $\left\lVert Be^{-tA}\right\rVert\leq
ct^{-\alpha}$ for all $t\in(0,1]$.

Here $\left\lVert\cdot\right\rVert$ denotes the operator norm on $X$. Then
$A+B$ also generates a strongly continuous semigroup $e^{-t(A+B)}$ which
admits the _Dyson-Phillips expansion_ , which is an absolutely convergent
series for all $t>0$:

(52) $e^{-t(A+B)}f=\sum_{k=0}^{\infty}B\operatorname{Per}^{k}(e^{-tA}f)$

for all $f\in X$. Here $\operatorname{Per}^{i}(u(t))$ is defined recursively
as:

(53)
$\operatorname{Per}^{0}(u(t))=u\quad\operatorname{Per}^{k}(u(t))=\int_{0}^{t}e^{-(t-s)A}B\operatorname{Per}^{k-1}(u(s))ds.$

The proof is modelled after a more general argument from [HP74, Chapter 13].
We include a proof here, as the main ingredients of the proof will be used
again in proving Proposition 5.3. We begin with proving that the Dyson-
Phillips expansion defines an absolutely convergent series, hence
$\\{e^{-t(A+B)}:t>0\\}$ indeed forms a strongly continuous semigroup. This can
be done by majorizing the series by the following two functions

(54) $\phi(t):=\left\lVert e^{-tA}\right\rVert,\qquad\psi(t):=\left\lVert
Be^{-tA}\right\rVert.$

Both functions are non-negative and measurable functions in $t$, and they
satisfy two conditions:

  1. Property 1:

$\int_{0}^{1}\phi(t)+\psi(t)\mathop{}\\!\mathrm{d}{t}<\infty$;

  2. Property 2:

By the semigroup property of $e^{-tA}$, $\psi(t)$ satisfies the inequality:
For all $t,s>0$,

(55) $\psi(t+s)\leq\psi(t)\phi(s).$

These two properties alone give some quantitative bounds on $\phi$ and $\psi$:

  1. (I)

The finite sub-multiplicative function $\phi(t)$ is bounded on each interval
of the form $(\epsilon,1/\epsilon)$ by [HP74, Theorem 7.4.1]. Together with
Property 2, we see $\psi(t)$ is also bounded there;

  2. (II)

$\lim_{t\to\infty}t^{-1}\log\phi(t)$ exists, by the fact that $\phi$ is sub-
multiplicative [HP74, Theorem 7.6.1]. Denote this limit by $\omega_{0}$. This
also gives an upper bound on $\psi$ by Property 2:

(56) $\limsup_{t\to\infty}t^{-1}\log\psi(t)\leq\omega_{0}$

  3. (III)

For any $\omega>\omega_{0}$ we have
$\int_{0}^{\infty}e^{-t\omega}(\phi(t)+\psi(t))\mathop{}\\!\mathrm{d}{t}=:M_{\omega}<\infty$.
This is immediate from the first property and (56). Using the property 2 with
the growth estimate of $\phi(t)$,

$0\leq
4e^{-t\omega}\psi(t)\leq(e^{-\omega(t-t_{1})}\phi(t-t_{1})+e^{-t_{1}\omega}\psi(t_{1}))^{2}$

by the sub-additivity inequality. Hence

$\displaystyle t(e^{-t\omega}\psi(t))^{1/2}$
$\displaystyle=2\int_{0}^{t/2}(e^{-t\omega}\psi(t))^{1/2}\mathop{}\\!\mathrm{d}{t_{1}}$
$\displaystyle\leq\int_{0}^{t/2}e^{-\omega(t-t_{1})}\psi(t-t_{1})+e^{-t_{1}\omega}\phi(t_{1})\mathop{}\\!\mathrm{d}{t_{1}}\leq
M_{\omega},$

and we obtain the upper bound

(57) $\psi(t)\leq e^{t\omega}t^{-2}M_{\omega}^{2}$

for all $t>0$.

The goal is to bound $\operatorname{Per}^{k}(e^{-tA}f)$ in the Dyson-Phillips
series by the convolution product $\phi*\psi^{*k}$, where $\psi^{*k}$ denotes
the $k$-th convolution product of $\psi$. The series of convolution products
can be estimated as follows [HP74, Lemma 13.4.3]:

######  Lemma 5.2.

Suppose $\psi_{0}$ and $\psi_{1}$ are two nonnegative measurable functions
satisfying property 1 and 2 as stated above. Then the series
$\theta(t):=\sum_{k=0}^{\infty}(\psi_{0}*\psi_{1}^{*k})(t)$ converges
uniformly with respect to $t$ in the interval of $(\varepsilon,1/\varepsilon)$
for $\varepsilon\in(0,1)$. Moreover, if $\omega>\omega_{0}>0$ is such that
$\int_{0}^{\infty}e^{-t\omega}\psi_{1}(t)\mathop{}\\!\mathrm{d}{t}\leq 1$,
then $\int_{0}^{\infty}e^{-t\omega}\theta(t)\mathop{}\\!\mathrm{d}{t}<\infty$.

###### Proof.

By the first quantitative bound above we see that both $\phi(t)$ and $\psi(t)$
are bounded on each interval of the form $(\epsilon,1/\epsilon)$. Choose
$\omega_{1}>\omega_{0}$ so that:

$\int_{0}^{\infty}e^{-\omega_{1}\xi}(\phi(t)+\psi_{0}(t))\mathop{}\\!\mathrm{d}{t}\leq
1;\quad\int_{0}^{\infty}e^{-\omega_{1}\xi}(\phi(t)+\psi_{1}(t))\mathop{}\\!\mathrm{d}{t}\leq\frac{1}{16}.$

Now from (57) we get $\psi_{0}(t)\leq t^{-2}e^{t\omega_{1}}$ and
$\psi_{1}(t)\leq\frac{1}{16}t^{-2}e^{t\omega_{1}}$. By induction we will
establish:

(58) $\psi_{0}*\psi_{1}^{*n}(t)\leq 2^{-n}t^{-2}e^{t\omega_{1}}$

It suffices to estimate the constant for the induction step using the
quantitative bounds we established before the lemma:

$\displaystyle(\psi_{0}*\psi_{1}^{*n})(t)$ $\displaystyle=$ $\displaystyle
e^{t\omega_{1}}\left(\int_{0}^{t/2}+\int_{t/2}^{t}\left(e^{-(t-s)\omega_{1}}\psi_{0}*\psi_{1}^{*(n-1)}(t-s)\right)\cdot\left(e^{-s\omega_{1}}\psi_{1}(s)\right)\mathop{}\\!\mathrm{d}{s}\right)$
$\displaystyle\leq$ $\displaystyle
e^{t\omega_{1}}\left(2^{-(n-1)}2^{2}t^{-2}\int^{t/2}_{0}e^{-s\omega_{1}}\psi_{1}(s)\mathop{}\\!\mathrm{d}{s}+\frac{2^{2}}{16}t^{-2}\int_{0}^{t/2}e^{-s\omega_{1}}(\psi_{0}*\psi_{1}^{*(n-1)})(s)\mathop{}\\!\mathrm{d}{s}\right)$
$\displaystyle\leq$ $\displaystyle 2^{-n}t^{-k}e^{-t\omega_{1}},$

with the two inequalities coming from the induction hypothesis. Having
established (58) we see $\theta_{t}$ is an absolutely convergent series, as it
is majorized by the uniformly convergent series
$\sum_{n}2^{-n}t^{-2}e^{t\omega_{1}}$. This proves the first claim. For the
second claim, we note the following:

$\int_{0}^{\infty}e^{-t\omega}(\psi_{0}*\psi_{1}^{*n})(t)\mathop{}\\!\mathrm{d}{t}=\left(\int_{0}^{\infty}e^{-t\omega}\psi_{0}(t)\mathop{}\\!\mathrm{d}{t}\right)\cdot\left(\int_{0}^{\infty}e^{-t\omega}\psi_{1}(t)\mathop{}\\!\mathrm{d}{t}\right)^{n}$

Hence we can write
$\int_{0}^{\infty}e^{-t\omega}\theta(t)\mathop{}\\!\mathrm{d}{t}=\left(\int_{0}^{\infty}e^{-t\omega}\psi_{0}(t)\mathop{}\\!\mathrm{d}{t}\right)\cdot(1-\int_{0}^{\infty}e^{-t\omega}\psi_{1}(t)\mathop{}\\!\mathrm{d}{t})^{-1}$,
which is finite by our assumption. ∎

Now the proof of the theorem is achieved by combining all these estimates:

###### Proof of Theorem 5.1.

We begin by proving that the Dyson-Phillips series converges uniformly in the
strong operator topology for $t>0$ by bounding it using $\phi$ and $\psi$:

$\left\lVert\operatorname{Per}^{n}(e^{-tA}f)\right\rVert\leq\phi*\psi^{*n}(t)\quad\left\lVert
B\operatorname{Per}^{n}(e^{-tA}f)\right\rVert\leq\psi^{*n}(t)$

First note that the functions $\phi=\psi_{0}$ and $\psi=\psi_{1}$ satisfy the
estimates in Lemma 5.2, by the assumption we made in Theorem 5.1. Also by our
assumption on $e^{-tA}$, the case $n=0$ is trivially true. The case for
general $n$ is shown by an easy induction with the following inequality:

$\left\lVert\operatorname{Per}^{k+1}(e^{-tA})\right\rVert\leq\int_{0}^{t}\left\lVert
e^{-(t-s)A}\right\rVert\cdot\left\lVert
B\operatorname{Per}^{k}(e^{-sA})\right\rVert\mathop{}\\!\mathrm{d}{s}\leq\phi*\psi^{*(k+1)}(t)$

and similarly for $B\operatorname{Per}^{k+1}(e^{-tA})$:

$\left\lVert
B\operatorname{Per}^{k+1}(e^{-tA}f)\right\rVert\leq\int_{0}^{t}\left\lVert
Be^{-(t-s)A}\right\rVert\cdot\left\lVert
B\operatorname{Per}^{k}(e^{-sA})\right\rVert\mathop{}\\!\mathrm{d}{s}\leq\psi^{*(k+2)}(t)$

Now
$\sum_{k=0}^{\infty}\left\lVert\operatorname{Per}^{k}(e^{-tA})\right\rVert$ is
bounded by $\theta(t)$, a fact we have proven in the above lemma to be
uniformly convergent in $t$ on any interval $(\varepsilon,1/\varepsilon)$ for
$0<\varepsilon<1$. These show $e^{-t(A+B)}$ defines a strongly continuous heat
semigroup. ∎

Fix now a finite-dimensional $G$-representation $(\tau,V_{\tau})$, together
with a compact subgroup $L\subseteq G$, such that $V_{\tau}|L$ is a unitary
representation of $L$. Form now a $G$-invariant strongly elliptic operator
$\mathcal{D}$ on the homogeneous vector bundle
$E_{\tau}:=G\times_{\tau}V_{\tau}\to G/L$ like [BM83, §1.1]. Note the smooth
sections $\Gamma(E)$ can be identified with the smooth vectors of the induced
module $\operatorname{ind}_{L}^{G}(\tau)$. Explicitly,

(59) $C^{\infty}(G;\tau):=\\{F:G\to V_{\tau}\mid f\in
C^{\infty},F(gl)=\tau(l^{-1})F(g)\text{ for all }l\in L\\},$

where $G$ acts on $C^{\infty}(G;\tau)$ via right regular representation. It is
isomorphic with $\Gamma(E)$ via the canonical $G$-equivariant isomorphism

(60) $\mathcal{A}:\Gamma(E)\longrightarrow C^{\infty}(G;\tau)\qquad
F(g):=(\mathcal{A}f)(g)=\tau(g^{-1})(f(gL)).$

Fix a complex basis $X_{i}$ of $\mathfrak{g}_{\mathbb{C}}$. Note that every
$G$-invariant operator $\mathcal{D}$ can be identified via $\mathcal{A}$ with
an operator of the following form

$D_{\tau}=\sum_{I}X^{I}\otimes
C_{I}\in[U(\mathfrak{g}_{\mathbb{C}})\otimes\operatorname{End}(V)]^{L},$

where $L$ acts on $U(\mathfrak{g}_{\mathbb{C}})\otimes\operatorname{End}(V)$
by:

(61) $l\circ(\sum_{i}D_{i}\otimes
C_{i}):=\sum_{i}\operatorname{Ad}_{l}(D_{i})\otimes(\tau(l)\circ
C_{i}\circ\tau(l)^{-1}).$

We further assume that $\mathcal{D}$ is $G$-invariant, i.e., it commutes with
the natural action of $G$ on the $E_{\tau}$. Then the heat kernel associated
with the semigroup $e^{-tD_{\tau}}$ can be treated as a function
$k_{t}^{\tau}:G\to\operatorname{End}(V)$ which satisfies the following
covariance property:

(62) $k^{\tau}_{t}(g)=\tau(a)k^{\tau}_{t}(a^{-1}gb)\tau(b)^{-1}\qquad\text{
for }x\in G,a,b\in L.$

We note that $D_{\tau}$ is often not an elliptic operator with constant
coefficients because of the covariance in the direction of fiber of
$E_{\tau}$. Nonetheless we can still estimate the growth via bounded
perturbations on part of the operator with constant coefficients.

######  Proposition 5.3.

Let $D_{\tau}$ be a $G$-invariant differential operator on $E_{\tau}$ as given
above. Suppose $D_{\tau}=D+B$, with $D\in U(\mathfrak{g}_{\mathbb{C}})$ a
$G$-invariant strongly elliptic operator with constant coefficients of order
$m$; and $B$ a differential operator of order $\ell<m$ of the following form:

(63) $B=\sum_{|I|\leq\ell}D^{I}\otimes C_{I}\in
U(\mathfrak{g}_{\mathbb{C}})\otimes\operatorname{End}(V).$

Then the perturbed semigroup $e^{-t(\overline{D+B})}$ is strongly continuous
in $t$, with $k_{t}^{D+B}$ its kernel. The derivatives of $k^{D+B}_{t}$
satisfy the following estimate: For each $\rho>0$ and every derivative $X^{J}$
of order $|J|<\ell$,

(64) $\int_{G}e^{\rho|g|}\left\lVert
L(X^{J})k_{t}^{D+B}(g)\right\rVert_{\operatorname{End}(V)}\mathop{}\\!\mathrm{d}{g}<\infty$

for all fixed $t>0$.

###### Proof.

For clarity of exposition we abbreviate $L^{p}_{\tau}:=L^{p}(G;V_{\tau})$ the
$L^{p}$-spaces with coefficients in $V_{\tau}$ for $1\leq p\leq\infty$, and
the weighted $L^{p}$-spaces in a similar way. Recall that
$L^{\infty}_{\rho,\tau}(G)$ is the weighted $L^{\infty}$-space with norm

$\left\lVert\varphi\right\rVert_{L^{\infty}_{\rho,\tau}}=\sup_{g\in
G}e^{-\rho|g|}\left\lVert\varphi(g)\right\rVert_{V}.$

Denote $C_{\rho,\tau,\infty}^{k}=C^{k}(L_{\rho,\tau}^{\infty}(G))$ its $C^{k}$
vectors, and define its $C^{k}$-norms as (41), replacing $H_{\pi}$ by
$L_{\rho,\tau}^{\infty}(G)$.

The strong continuity of the perturbed heat semigroup $e^{-t(\overline{D+B})}$
is straightforward from Theorem 5.1. Derivatives of the perturbed heat kernel
can be estimated by bounding its weighted $L^{\infty}$-norm in a similar way
as in the proof of Lemma 4.1:

(65) $\begin{split}\int_{G}\left\lVert
L(X^{J})k_{t}^{D+B}(g)\right\rVert_{\operatorname{End}(V)}\mathop{}\\!\mathrm{d}{g}&=\sup_{\left\lVert\varphi\right\rVert_{L^{\infty}_{\rho,\tau}}\leq
1}\left\\{\left\lVert
L(X^{J})e^{-tL^{\rho}(\overline{D+B})}\varphi(e_{G})\right\rVert_{V}\right\\}\\\
&\leq\sup_{\left\lVert\varphi\right\rVert_{L^{\infty}_{\rho,\tau}}\leq
1}\left\\{\left\lVert
e^{-tL^{\rho}(\overline{D+B})}\varphi\right\rVert_{C_{\rho,\tau,\infty}^{\ell}}\right\\}.\end{split}$

As the Dyson-Phillips series (52) is absolutely convergent, it suffices to
estimate the $C^{k}$-norm of each term
$\operatorname{Per}^{k}(e^{-t\bar{D}}\varphi)$. Apply now Theorem 5.1 further
to the $C^{\ell}$-norms of each term by verifying the norm assumptions:

  * •

As $D$ acts on $L^{\infty}_{\rho,\tau}$ by $L^{\rho}\otimes\mathrm{Id}_{V}$,

$\left\lVert e^{-tL^{\rho}(\bar{D})}\right\rVert_{L^{\infty}_{\rho,\tau}\to
C^{\ell}_{\rho,\tau,\infty}}=\left\lVert
e^{-tL^{\rho}(\bar{D})}\right\rVert_{L^{\infty}_{\rho}\to
C^{\ell}_{\rho,\infty}}\leq Me^{t\omega}$

for some $M,\omega>0$, by the continuity bound (44);

  * •

For small $t\in(0,1]$ the small time estimate (43) gives:

(66) $\begin{split}\left\lVert
Be^{-tL^{\rho}(\bar{D})}\right\rVert_{L^{\infty}_{\rho,\tau}\to
L^{\infty}_{\rho,\tau}}&\leq\sum_{I}\left\lVert
B_{I}\right\rVert_{\operatorname{End}(V)}\left\lVert
D^{I}e^{-tL^{\rho}(\bar{D})}\right\rVert_{L^{\infty}_{\rho}\to
L^{\infty}_{\rho}}\\\ &\leq a\sum_{I}\left\lVert
e^{-tL^{\rho}(\bar{D})}\right\rVert_{L^{\infty}_{\rho}\to
C^{\ell}_{\rho,\infty}}\\\ &\leq a^{\prime}b^{\ell}\ell!t^{-\ell/m}\sim
C_{\ell}t^{-\ell/m}\end{split}$

with $\ell/m<1$ by our assumption.

Now repeating the notations and arguments in Lemma 5.2 we see that the
$C^{\ell}_{\rho,\infty}$-norm is indeed finite. For $\varphi=\psi\otimes v\in
L^{2}(G)\otimes V_{\tau}^{*}\cong L^{2}(G,V_{\tau})$ of unit norm,

(67) $\left\lVert
e^{-tL^{\rho}(\overline{D+B})}\varphi\right\rVert_{C_{\rho,\tau,\infty}^{\ell}}\leq\sum_{k=0}^{\infty}\left\lVert\operatorname{Per}^{k}(e^{-t\bar{D}}\varphi)\right\rVert_{C_{\rho,\infty}^{\ell}}\leq\theta_{B,A}(t)$

with the absolutely convergent series $\theta_{B,A}(t)$, expressed in the sum
of convolution products of functions in $t$, is finite:

(68) $\theta_{B,A}(t)=\sum_{k=0}^{\infty}\left\lVert
e^{-t\bar{D}}\right\rVert_{L^{\infty}_{\rho}\to
C_{\rho,\infty}^{\ell}}*\left\lVert
Be^{-t\bar{D}}\right\rVert_{L^{\infty}_{\rho,\tau}\to
L^{\infty}_{\rho,\tau}}^{k}(t)<\infty.$

As Lemma 5.2 proved, the series $\theta_{B,A}(t)$ is majorized by
$\sum_{n}2^{-n}t^{-2}e^{-t\omega_{1}}$ for some positive
$\omega_{1}>\omega>0$. This proves that
$e^{\rho|g|}\left\lVert\partial^{I}_{g}k_{t}^{D+B}\right\rVert_{\operatorname{End}(V)}\in
L^{1}(G)$ for all $t>0$. ∎

This estimate shows that $e^{\rho|g|}L_{X^{I}}k_{t}^{D+B}\in
L^{1}(G,\operatorname{End}(V))$ for any $\rho>0$ and any differentiation
$X^{I}\in U(\mathfrak{g}_{\mathbb{C}})$ from the left. To establish Schwartz
estimate, we only need the following variant of the Sobolev lemma adapted to
Lie groups [Pou72, Lemma 5.1], which follows directly from the Sobolev lemma
in n by choosing local coordinates:

######  Lemma 5.4.

Fix an integer $s>\dim G$. Then for each compact neighborhood
$\mathcal{B}(e_{G})$ of $e_{G}$ there exists a constant $C$ such that:

(69) $f(e_{G})\leq C\sum_{|I|\leq s}\int_{\mathcal{B}(e_{G})}\left\lvert
L_{X^{I}}f(y)\right\rvert\mathop{}\\!\mathrm{d}{y}$

for all $f\in C^{\infty}(G)$.

We have now prepared for the proof of the main result A. Recall the definition
of Schwartz spaces (10).

###### Proof of A.

First we dominate the growth of $\phi_{0}^{G/Z}(x)$ by choosing appropriate
$e^{\rho|g|}$. From (5) we see that:

(70) $-\log\phi_{0}^{G/Z}(x)\leq\gamma d(e_{G},x)\quad\text{ for all }x\in G$

for some constant $\gamma>0$. Hence if we choose the constant $\rho>\gamma>0$
in Proposition 5.3 this indeed implies
$\phi_{0}^{G}(x)^{-2/p}L_{X^{I}}k_{t}^{D+B}\in L^{1}(G,\operatorname{End}(V))$
for all $t>0$ and $p>0$. Apply Sobolev inequality (69) now:

(71)
$\begin{split}&\left\lVert\phi_{0}^{G}(x)^{-2/p}L_{X^{I}}k_{t}^{D+B}(x)\right\rVert_{\operatorname{End}(V)}\\\
\leq&C\sum_{|J|\leq s}\int_{B}\left\lVert
L_{X^{I}}\phi_{0}^{G}(yx)^{-2/p}L_{X^{J}}k_{t}^{D+B}(yx)\right\rVert_{\operatorname{End}(V)}\mathop{}\\!\mathrm{d}{y}\\\
\leq&C\sum_{|J|\leq s}\left\lVert
L_{X^{\alpha}}\phi_{0}^{G}(yx)^{-2/p}L_{X^{J}}k_{t}^{D+B}\right\rVert_{L^{1}(G;\operatorname{End}(V))}<\infty.\end{split}$

So we have proved that $\nu_{D_{1},-,r}(F)$ is finite for all $D_{1}\in
U(\mathfrak{g}_{\mathbb{C}})$ and $r>0$. Next we establish a relation between
$L_{X^{I}}R_{X^{J}}k_{t}$ and $L_{X^{I^{\prime}}}k_{t}$. To prove this we use
the fact that $k_{t}$ is $\operatorname{Ad}(K)$-invariant, i.e.:

$k_{t}(x^{-1}gx)=k_{t}(g)$

for all $x\in K$. This follows readily from the fact that $K$ acts on $V$ by
unitary operators, and acts on $\mathfrak{g}$ by isometries with respect to
the metric $B^{\theta}$.

The following lemma is modelled after an argument of Harish-Chandra [HC84,
§11, Lemma 17]:

######  Lemma 5.5.

Let $V$ be a unitary $K$-bimodule and $f:G\to V$ be an
$\operatorname{Ad}(K)$-invariant $C^{\infty}$-function. Then for each pair
$(D,D^{\prime})\in U(\mathfrak{g}_{\mathbb{C}})\times
U(\mathfrak{g}_{\mathbb{C}})$, one can choose a finite number of $D_{i}\in
U(\mathfrak{g}_{\mathbb{C}})(1\leq i\leq p)$ such that:

(72) $\left\lVert
L_{D}R_{D^{\prime}}f(g)\right\rVert_{V}\leq\sum_{j=1}^{p}\left\lVert
L_{D_{j}}f(g)\right\rVert_{V}\quad\left\lVert
L_{D}R_{D^{\prime}}f(g)\right\rVert_{V};\leq\sum_{j=1}^{p}\left\lVert
R_{D_{j}}f(g)\right\rVert_{V}.$

###### Proof.

Decompose the Lie algebra $\mathfrak{g=k\oplus a\oplus n_{p}}$ via the Iwasawa
decomposition as in (2). Recall
$\mathfrak{n_{p}}=\sum_{i=1}^{l}\mathfrak{g}_{\alpha_{i}}$ is the space
containing all the simple positive $(\mathfrak{g};\mathfrak{a})$-roots
$\\{\alpha_{i}\\}_{i=1}^{l}$. Applying the Poincaré-Birkhoff-Witt theorem, we
decompose the universal enveloping algebra as:

(73)
$U(\mathfrak{g}_{\mathbb{C}})=U(\mathfrak{k}_{\mathbb{C}})U(\mathfrak{a}_{\mathbb{C}})U(\mathfrak{n}_{\mathbb{C}}).$

Here $U(\mathfrak{l}_{\mathbb{C}})$ are the corresponding universal enveloping
algebra generated by corresponding algebra $\mathfrak{l}$.

Fix now an integer $d\geq 0$ such that $D,D^{\prime}\in
U_{d}(\mathfrak{g}_{\mathbb{C}})$. Denote
$U_{d}(\mathfrak{g}_{\mathbb{C}})\subseteq U(\mathfrak{g}_{\mathbb{C}})$ the
subset containing all elements of order $\leq d$. We can choose a basis
$\\{D_{\mathfrak{k}}D_{\mathfrak{a}}D_{\mathfrak{n}}\\}$ of
$U_{d}(\mathfrak{g}_{\mathbb{C}})$ where $D_{\mathfrak{k}}\in
U(\mathfrak{k}_{\mathbb{C}})$, $D_{\mathfrak{a}}\in
U(\mathfrak{a}_{\mathbb{C}})$ and $D_{\mathfrak{n}}\in
U(\mathfrak{n}_{\mathbb{C}})$. Denote this basis as $\mathcal{B}_{d}$.
Moreover, recall that the natural action of $G$ on
$U(\mathfrak{g}_{\mathbb{C}})$ extends the adjoint action on
$\mathfrak{g}_{\mathbb{C}}\subseteq U_{1}(\mathfrak{g}_{\mathbb{C}})$. Then:

(74) $\operatorname{Ad}(a)D_{\mathfrak{n}}=\exp\left(\sum_{1\leq i\leq
l}m_{i}\alpha_{i}(\log a)\right)D_{\mathfrak{n}}$

for all $a\in A=\exp\mathfrak{a}$. This equality is obtained by extending the
map $\operatorname{Ad}(a)\mathfrak{g}_{\alpha_{i}}=e^{\alpha(\log
a)}\mathfrak{g}_{\alpha_{i}}$. Moreover, as $U(\mathfrak{n}_{\mathbb{C}})$ is
generated by $g_{\alpha_{i}}$ as an algebra, we see that $m_{i}$ are non-
negative integers. For the same reason we can expand the
$\operatorname{Ad}(K)$-action based on its action on the basis:

(75)
$\operatorname{Ad}(k)D=\sum_{b\in\mathcal{B}_{d}}a_{b}(k)b\qquad\operatorname{Ad}(k)D^{\prime}=\sum_{b\in\mathcal{B}_{d}}a^{\prime}_{b}(k)b$

for $k\in K$. Observe that $U_{d}(\mathfrak{g}_{\mathbb{C}})$ is a continuous
representation of $K$, hence $a_{b}$ and $a_{b}^{\prime}$ are continuous
functions in $K$.

Next by the $KAK$-decomposition [Her92, Lemma 2.5], we write $G=KA^{+}K$ where
$A^{+}=\exp(\overline{\mathfrak{a^{+}_{p}}})$ where $\mathfrak{a}^{+}$
contains all $H\in\mathfrak{a}$ such that $\alpha_{i}(H)\geq 0$ for all
restricted roots $\alpha_{i}$. Denote:

(76) $c=\sup_{k\in K}\max_{b\in B}(|a_{b}(k)|,|a^{\prime}_{b}(k)|)$

This constant is clearly finite if $K$ is compact. In the case $K$ is
noncompact, we can appeal to (6) and then write $a_{b}$ and $a_{b}^{\prime}$
as continuous functions that are $Z$-invariant instead [Her92, Lemma 2.7]. In
all cases, $c$ is finite. Next we estimate growths in the $A$-direction under
derivations. Write $g=k_{1}ak_{2}\in KA^{+}K$, then:

(77) $\|L_{D}R_{D^{\prime}}f(g)\|_{V}\leq\left\lVert
L_{(\operatorname{Ad}_{k_{1}^{-1}}D)}R_{(\operatorname{Ad}_{k_{2}}D^{\prime})}f(a)\right\rVert_{V}\leq\sum_{b,b^{\prime}\in\mathcal{B}_{d}}c^{2}\left\lVert
L_{b}R_{b^{\prime}}f(a)\right\rVert_{V}.$

Here the first inequality uses the fact that $V$ is a unitary $K$-bimodule

$\|k_{1}vk_{2}\|_{V}=\|v\|_{V}$

for all $k_{1},k_{2}\in K$ and $v\in V$. Now it suffices to estimate the norm
for each $L_{b}R_{b^{\prime}}f(a)$. Write each $b\in\mathcal{B}_{d}$ in the
form of

$b=D_{\mathfrak{k}}^{b}D_{\mathfrak{a}}^{b}D_{\mathfrak{n}}^{b}$

with respect to the decomposition (73). Using the fact $f$ is
$\operatorname{Ad}(K)$-invariant, we see
$L_{D_{\mathfrak{k}}}f=R_{D_{\mathfrak{k}}}f$ for any $D_{\mathfrak{k}}\in
U(\mathfrak{k}_{\mathbb{C}})$. Now combining the fact that $\mathfrak{a}$ is
abelian and the identity (74), one shifts the actions on the left to the right
side one by one:

(78)
$L_{b}R_{b^{\prime}}f(a)=L_{D_{\mathfrak{k}}^{b}D_{\mathfrak{a}}^{b}D_{\mathfrak{n}}^{b}}R_{b^{\prime}}f(a)=R_{D^{b}_{\mathfrak{k}}D^{b}_{\mathfrak{a}}\operatorname{Ad}_{a^{-1}}(D_{\mathfrak{n}}^{b})b^{\prime}}f(a)$

Where $R_{D_{1}D_{2}}f=R_{D_{2}}(R_{D_{1}}f)$. Next,

(79) $\left\lVert
R_{D^{b}_{\mathfrak{k}}D^{b}_{\mathfrak{a}}\operatorname{Ad}_{a^{-1}}(D_{\mathfrak{n}}^{b})b^{\prime}}f(a)\right\rVert_{V}\leq\left\lVert
R_{D^{b}_{\mathfrak{k}}D^{b}_{\mathfrak{a}}D^{b}_{\mathfrak{n}}b^{\prime}}f(a)\right\rVert_{V}$

by combining (74) and the fact $\alpha_{i}(\log a)\geq 0$ since $\alpha_{i}$
is a positive root. At last one notices that the following finite set

$\\{D^{b}_{\mathfrak{k}}D^{b}_{\mathfrak{a}}D^{b}_{\mathfrak{n}}b^{\prime}\mid
b,b^{\prime}\in\mathcal{B}_{d}\\}$

spans a finite-dimensional subspace in $U(\mathfrak{g}_{\mathbb{C}})$. We
denote a basis of this subspace as $\\{l_{j}:1\leq j\leq p\\}$. We can choose
a uniform bound $C^{\prime}>0$ such that each element in the above set can be
written as sums $\sum_{1\leq j\leq p}\gamma_{j}l_{j}$ with $|\gamma_{j}|\leq
C^{\prime}$. We can now estimate the derivatives from both sides by combining
(77) and (79):

$\left\lVert
L_{D}R_{D^{\prime}}f(g)\right\rVert_{V}\leq\sum_{b,b^{\prime}\in\mathcal{B}_{d}}c^{2}\left\lVert
R_{D^{b}_{\mathfrak{k}}D^{b}_{\mathfrak{a}}D^{b}_{\mathfrak{n}}b^{\prime}}f(a)\right\rVert_{V}\leq\sum_{1\leq
j\leq p}|\mathcal{B}_{d}|C^{\prime}c^{2}\left\lVert
R_{l_{j}}f(g)\right\rVert_{V}.$

Note we have again exploited the fact that $f$ is
$\operatorname{Ad}(K)$-invariant and the fact that $K$ acts on $V$ by isometry
to deduce the last inequality. This concludes the proof. ∎

By inspecting Corollary 3.5, we see that the Hodge Laplacian $\Delta_{p}$
associated with the Riemannian metric induced by $B^{\theta}$ can be written
in the following form, by taking $\tau=R$ the right regular representation:

$\Delta_{p}=R(\Delta_{0})\otimes\mathrm{Id}_{\wedge^{p}\mathfrak{g}^{*}_{\mathbb{C}}}+\text{terms
with lower order}$

where $R(\Delta_{0})=-R(\bar{\Omega}_{G})$ is a strongly elliptic operator,
and the remainder terms can be expressed in the form of (63). Now combining
Lemma 5.5 with the Schwartz estimate from one side (71), we see that the
kernel of $\Delta_{p}$ is indeed a Schwartz function. ∎

##  6\. Schwartz estimates on nilpotent groups

The above Schwartz estimates can be extended to cases beyond the Hodge
Laplacian and reductive groups. In a subsequent paper we discuss how the
spinor Laplacian associated with the same Riemannian metric can be proven to
be Schwartz class in the sense of (10) following the same proof.

As an direct consequence of our estimate (71), we can also prove that the
Hodge Laplacian $\Delta_{p}$ on a nilpotent Lie group $N$ are of Schwartz
class. We briefly recall the definition of Schwartz functions on $N$ from
[CG90, §A.2], and then prove the result as a corollary that will be useful in
[Han24, TOCHANGE].

Let $N$ be a simply connected real nilpotent Lie group. If $N$ is given
polynomial coordinates $\gamma:^{n}\to N$, that is,

$\log\circ\gamma:^{n}\to\mathfrak{n}$

defines a polynomial map with polynomial inverse. Though it is not dependent
on the choice of basis, typically if we choose $\\{X_{1},\dots,X_{n}\\}$ to be
a weak Malcev basis [CG90, Theorem 1.1.13], then

$\gamma(x_{1},\dots,x_{n})=\exp(x_{1}X_{1})\cdots\exp(x_{n}X_{n})$

defines a polynomial coordinate. Now we can define $\mathcal{S}(N)$ as the
pullback of the Schwartz functions on n. More explicitly, a function $f$ on
$N$ is Schwartz if it is finite under all seminorms:

$\left\lVert
x^{\alpha}D^{\beta}(f\circ\gamma)\right\rVert_{L^{\infty}(^{n})}<\infty$

for all
$D^{\beta}=\partial_{x_{1}}^{\beta_{1}}\cdots\partial_{x_{n}}^{\beta_{n}}$ and
$x^{\alpha}=x^{\alpha_{1}}\cdots x_{\alpha_{n}}$ a monomial. Note this is
equivalent to defining it as the space of smooth functions $f\in
C^{\infty}(N)$ such that

(80) $\left\lVert p_{\alpha}L(X^{\alpha})f\right\rVert_{\infty}<\infty$

for all $p_{\alpha}\in\mathbb{C}[G]$ polynomials on $G$, and $X_{\alpha}\in
U(\mathfrak{g}_{\mathbb{C}})$ under any chosen basis of $\mathfrak{g}$ [CG90,
Corollary A.2.3]. Note it suffices to consider left derivatives in this case,
as the Lie brackets of nilpotent Lie groups can be expressed by polynomial
functions.

######  Corollary 6.1.

Let $N$ be a simply connected nilpotent Lie group, and fix on $N$ a left
$N$-invariant Riemannian metric. Then the heat kernel $k_{t}^{p}$ of the Hodge
Laplacian $\Delta_{p}$ on $N$ is a Schwartz function.

###### Proof.

Recall first that the content of Proposition 3.3 holds for any Lie group $G$.
If we fix $V$ to be the $L^{2}(N)$ with right-regular action, then we see

$\Delta_{p}=-R(\sum_{i}X_{i}^{2})+\sum_{i}R(X_{i})\otimes C_{i}$

with $C_{i}\in\operatorname{End}(\wedge^{*}\mathfrak{n}^{*})$. In particular,
$\Delta_{p}$ satisfies the assumptions of Proposition 5.3. Hence the kernel of
$\Delta_{p}$ is a Schwartz function. Consequently,

$\int_{N}e^{\rho|g|}\left\lVert
L(X^{J})k_{t}^{p}(g)\right\rVert_{\operatorname{End}(\wedge^{*}\mathfrak{n}^{*})}\mathop{}\\!\mathrm{d}{g}<\infty.$

Because the polynomial grows slower than $e^{\rho|g|}$, we have

$p_{I}L_{X^{I}}k_{t}^{p}\in
L^{1}(N,\operatorname{End}(\wedge^{p}\mathfrak{n}^{*})).$

In fact, $L_{X^{J}}p_{I}$ remains polynomial therefore bounded for any
$|J|\geq 0$, therefore,

(81) $L_{X^{J}}(p_{I}L_{X^{I}}k_{t}^{p})\in
L^{1}(N,\operatorname{End}(\wedge^{p}\mathfrak{n}^{*})).$

Now Lemma 5.4 and the bound (71) applies similarly to the nilpotent case.

(82) $\begin{split}\left\lVert
p_{I}L_{X^{I}}k_{t}^{D+B}(x)\right\rVert_{\operatorname{End}(V)}\leq
C\sum_{|J|\leq s}\left\lVert
L_{X^{J}}p_{I}L_{X^{I}}k_{t}^{D+B}\right\rVert_{L^{1}(G;\operatorname{End}(V))}<\infty,\end{split}$

and the claim is proven. ∎

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