# Hess-Schrader-Uhlenbrock inequality for the heat semigroup on differential
forms over Dirichlet spaces tamed by distributional curvature lower bounds

Kazuhiro Kuwae111🖂Kazuhiro Kuwae<EMAIL_ADDRESS>Department of Applied
Mathematics, Fukuoka University, Fukuoka 814-0180, Japan. Supported in part by
JSPS Grant-in-Aid for Scientific Research (S) (No. 22H04942) and fund (No.
215001) from the Central Research Institute of Fukuoka University.

###### Abstract

The notion of tamed Dirichlet space was proposed by Erbar, Rigoni, Sturm and
Tamanini [27] as a Dirichlet space having a weak form of Bakry-Émery curvature
lower bounds in distribution sense. After their work, Braun [14] established a
vector calculus for it, in particular, the space of $L^{2}$-normed
$L^{\infty}$-module describing vector fields, $1$-forms, Hessian in
$L^{2}$-sense. In this framework, we establish the Hess-Schrader-Uhlenbrock
inequality for $1$-forms as an element of $L^{2}$-cotangent module (an
$L^{2}$-normed $L^{\infty}$-module), which extends the Hess-Schrader-
Uhlenbrock inequality by Braun [14] under an additional condition.

Keywords: Strongly local Dirichlet space, tamed space, Bakry-Émery condition,
smooth measures of (extended) Kato class, smooth measures of Dynkin class,
$L^{p}$-normed $L^{\infty}$-module, Hilbert module, tensor product of Hilbert
module, exterior product of $L^{\infty}$-module, $L^{2}$-differential forms,
$L^{2}$-vector fields, Littlewood-Paley-Stein inequality, Riesz transform,
Bochner Laplacian, Hodge-Kodaira Laplacian, Hess-Schrader-Uhlenbrock
inequality, intertwining property.

Mathematics Subject Classification (2020): Primary 31C25, 60H15, 60J60 ;
Secondary 30L15, 53C21, 58J35, 35K05, 42B05, 47D08

##  1 Statement of Main Theorem

###  1.1 Framework

Let $(M,\tau)$ be a topological Lusin space, i.e., a continuous injective
image of a Polish space, endowed with a $\sigma$-finite Borel measure
$\mathfrak{m}$ on $M$ with full topological support. Let
$(\mathscr{E},D(\mathscr{E}))$ be a quasi-regular symmetric strongly local
Dirichlet space on $L^{2}(M;\mathfrak{m})$ and $(P_{t})_{t\geq 0}$ (resp.
$(G_{\alpha})_{\alpha>0}$) the associated symmetric sub-Markovian strongly
continuous semigroup (resp. resolvent) on $L^{2}(M;\mathfrak{m})$ with the
relation $G_{\alpha}f:=\int_{0}^{\infty}e^{-\alpha t}P_{t}f\mathrm{d}t$ for
$f\in L^{2}(M;\mathfrak{m})$ (see [50, Chapter IV, Definition 3] for the
quasi-regularity and see [44, Theorem 5.1(i)] for the strong locality). Then
there exists an $\mathfrak{m}$-symmetric special standard process ${\bf
X}=(\Omega,X_{t},\mathbb{P}_{x})$ associated with
$(\mathscr{E},D(\mathscr{E}))$, i.e. for $f\in
L^{2}(M;\mathfrak{m})\cap\mathscr{B}(M)$, $P_{t}f(x)=p_{t}f(x)$
$\mathfrak{m}$-a.e. $x\in M$ (see [50, Chapter IV, Section 3]). Here
$p_{t}f(x):=\mathbb{E}_{x}[f(X_{t})]$ and $\mathscr{B}(M)$ denotes the family
of Borel measurable functions on $M$ (the symbol $\mathscr{B}(M)$ is also used
for the family of Borel measurable subsets of $M$ in some context). It is
known that $(P_{t})_{t\geq 0}$ (resp. $(G_{\alpha})_{\alpha>0}$) under
$p\in[1,+\infty]$ can be extended to bounded contractive operators (still
denoted by the same symbol) on $L^{p}(M;\mathfrak{m})$ and is strongly
continuous on $L^{p}(M;\mathfrak{m})$ under $p\in[1,+\infty[$ in the sense
that $\lim_{t\to 0}\|P_{t}f-f\|_{L^{p}(M;\mathfrak{m})}=0$ (resp.
$\lim_{\alpha\to\infty}\|\alpha G_{\alpha}f-f\|_{L^{p}(M;\mathfrak{m})}=0$),
and weakly* continuous on $L^{\infty}(M;\mathfrak{m})$ in the sense that
$\lim_{t\to 0}\int_{M}(P_{t}f-f)g\mathrm{d}\mathfrak{m}=0$ (resp.
$\lim_{\alpha\to\infty}\int_{M}(\alpha
G_{\alpha}f-f)g\mathrm{d}\mathfrak{m}=0$) for $f\in
L^{\infty}(M;\mathfrak{m})$ and $g\in L^{1}(M;\mathfrak{m})$. An increasing
sequence $\\{F_{n}\\}$ of closed subsets is called an _$\mathscr{E}$ -nest_ if
$\bigcup_{n=1}^{\infty}D(\mathscr{E})_{F_{n}}$ is $\mathscr{E}_{1}$-dense in
$D(\mathscr{E})$, where $D(\mathscr{E})_{F_{n}}:=\\{u\in D(\mathscr{E})\mid
u=0\;\mathfrak{m}\text{-a.e.~{}on }M\setminus F_{n}\\}$. A subset $N$ is said
to be _$\mathscr{E}$ -exceptional_ or _$\mathscr{E}$ -polar_ if there exists
an $\mathscr{E}$-nest $\\{F_{n}\\}$ such that
$N\subset\bigcap_{n=1}^{\infty}(M\setminus F_{n})$. For two subsets $A,B$ of
$M$, we write $A\subset B$ $\mathscr{E}$-q.e. if $A\setminus B$ is
$\mathscr{E}$-exceptional, hence we write $A=B$ $\mathscr{E}$-q.e. if
$A\setminus B$ $\mathscr{E}$-q.e. and $B\setminus A$ $\mathscr{E}$-q.e. hold.
It is known that any $\mathscr{E}$-exceptional set is
$\mathfrak{m}$-negligible and for any $\mathfrak{m}$-negligible
$\mathscr{E}$-quasi-open set is $\mathscr{E}$-exceptional, in particular, for
an $\mathscr{E}$-quasi continuous function $u$, $u\geq 0$ $\mathfrak{m}$-a.e.
implies $u\geq 0$ $\mathscr{E}$-q.e. For a statement $P(x)$ with respect to
$x\in M$, we say that $P(x)$ holds $\mathscr{E}$-q.e. $x\in M$ (simply $P$
holds $\mathscr{E}$-q.e.) if the set $\\{x\in M\mid P(x)\text{ holds }\\}$ is
$\mathscr{E}$-exceptional. A subset $G$ of $M$ is called an _$\mathscr{E}$
-quasi-open set_ if there exists an $\mathscr{E}$-nest $\\{F_{n}\\}$ such that
$G\cap F_{n}$ is an open set of $F_{n}$ with respect to the relative topology
on $F_{n}$ for each $n\in\mathbb{N}$. A function $u$ on $M$ is said to be
_$\mathscr{E}$ -quasi continuous_ if there exists an $\mathscr{E}$-nest
$\\{F_{n}\\}$ such that $u|_{F_{n}}$ is continuous on $F_{n}$ for each
$n\in\mathbb{N}$.

Denote by $(\mathcal{E},D(\mathcal{E})_{e})$ the extended Dirichlet space of
$(\mathscr{E},D(\mathscr{E}))$ defined by

$\displaystyle\left\\{\begin{array}[]{rl}D(\mathcal{E})_{e}&=\\{u\in
L^{0}(M;\mathfrak{m})\mid\exists\\{u_{n}\\}\subset
D(\mathscr{E}):\mathscr{E}\text{-Cauchy sequence}\\\ &\hskip 170.71652pt\text{
such that }\lim_{n\to\infty}u_{n}=u\;\mathfrak{m}\text{-a.e.}\\},\\\
\mathscr{E}(u,u)&=\lim_{n\to\infty}\mathscr{E}(u_{n},u_{n}),\end{array}\right.$

(see [30, p. 40] for details on extended Dirichlet space). Denote by
$\dot{D}(\mathscr{E})_{{\rm loc}}$, the space of functions locally in
$D(\mathscr{E})$ in the broad sense defined by

$\displaystyle\dot{D}(\mathscr{E})_{{\rm loc}}$ $\displaystyle=\\{u\in
L^{0}(M;\mathfrak{m})\mid\exists\\{u_{n}\\}\subset D(\mathscr{E})\text{ and
}\exists\\{G_{n}\\}:\mathscr{E}\text{-nest of }\mathscr{E}\text{-quasi-open
sets}$ $\displaystyle\hskip 227.62204pt\text{ such that
}u=u_{n}\;\mathfrak{m}\text{-a.e.~{}on }G_{n}\\}.$

Here an increasing sequence $\\{G_{n}\\}$ of $\mathscr{E}$-quasi open sets is
called an _$\mathscr{E}$ -nest_ if $M=\bigcup_{n=1}^{\infty}G_{n}$
$\mathscr{E}$-q.e. For $u\in\dot{D}(\mathscr{E})_{{\rm loc}}$ and an
$\mathscr{E}$-nest $\\{G_{n}\\}$ of $\mathscr{E}$-quasi-open sets satisfying
$u|_{G_{n}}\in D(\mathscr{E})|_{G_{n}}$, we write $\\{G_{n}\\}\in\Xi(u)$. It
is known that $D(\mathscr{E})\subset
D(\mathscr{E})_{e}\subset\dot{D}(\mathscr{E})_{{\rm loc}}$ and each
$u\in\dot{D}(\mathscr{E})_{{\rm loc}}$ admits an $\mathscr{E}$-quasi
continuous $\mathfrak{m}$-version (see [44]).

It is known that for $u,v\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})$
there exists a unique signed finite Borel measure $\mu_{\langle u,v\rangle}$
on $M$ such that

$\displaystyle 2\int_{M}\tilde{f}\mathrm{d}\mu_{\langle
u,v\rangle}=\mathscr{E}(uf,v)+\mathscr{E}(vf,u)-\mathscr{E}(uv,f)\quad\text{
for }\quad u,v\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m}).$

Here $\tilde{f}$ denotes the $\mathscr{E}$-quasi-continuous
$\mathfrak{m}$-version of $f$ (see [30, Theorem 2.1.3], [50, Chapter IV,
Proposition 3.3(ii)]). We set $\mu_{\langle f\rangle}:=\mu_{\langle
f,f\rangle}$ for $f\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})$.
Moreover, for $f,g\in D(\mathscr{E})$, there exists a signed finite measure
$\mu_{\langle f,g\rangle}$ on $M$ such that $\mathscr{E}(f,g)=\mu_{\langle
f,g\rangle}(M)$, hence $\mathscr{E}(f,f)=\mu_{\langle f\rangle}(M)$. We assume
$(\mathscr{E},D(\mathscr{E}))$ admits a carré-du-champ $\Gamma$, i.e.
$\mu_{\langle f\rangle}\ll\mathfrak{m}$ for all $f\in D(\mathscr{E})$ and set
$\Gamma(f):=\mathrm{d}\mu_{\langle f\rangle}/\mathrm{d}\mathfrak{m}$. Then
$\mu_{\langle f,g\rangle}\ll\mathfrak{m}$ for all $f,g\in D(\mathscr{E})$ and
$\Gamma(f,g):=\mathrm{d}\mu_{\langle f,g\rangle}/{\mathrm{d}\mathfrak{m}}\in
L^{1}(M;\mathfrak{m})$ is expressed by
$\Gamma(f,g)=\frac{1}{4}(\Gamma(f+g)-\Gamma(f-g))$ for $f,g\in
D(\mathscr{E})$.

We can extend the singed smooth $\mu_{\langle f,g\rangle}$ or carré-du-champ
$\Gamma(f,g)$ from $f,g\in D(\mathscr{E})$ to
$f,g\in\dot{D}(\mathscr{E})_{{\rm loc}}$ by polarization (see [44, Lemmas 5.2
and 5.3]). In this case, $\mu_{\langle f,g\rangle}$ (resp. $\Gamma(f,g)$) is
no longer a finite signed measure (resp. an $\mathfrak{m}$-integrable
function). For $u\in\dot{D}(\mathscr{E})_{{\rm loc}}$ and an
$\mathscr{E}$-nest $\\{G_{n}\\}$ of $\mathscr{E}$-quasi-open sets satisfying
$u|_{G_{n}}\in D(\mathscr{E})|_{G_{n}}$, then we can define $\mathscr{E}(u,v)$
for $v\in\bigcup_{n=1}^{\infty}D(\mathscr{E})_{G_{n}}$ by

$\displaystyle\mathscr{E}(u,v)=\mu_{\langle u,v\rangle}(M).$

Here $D(\mathscr{E})_{G_{n}}:=\\{u\in
D(\mathscr{E})\mid\tilde{u}=0\;\mathscr{E}\text{-q.e.~{}on }G_{n}^{c}\\}$.

Fix $q\in\\{1,2\\}$. Let $\kappa$ be a signed smooth measure with its Jordan-
Hahn decomposition $\kappa=\kappa^{+}-\kappa^{-}$. We assume that $\kappa^{+}$
is of Dynkin class smooth measure ($\kappa^{+}\in S_{D}({\bf X})$ in short)
and $2\kappa^{-}$ is of extended Kato class smooth measure ($2\kappa^{-}\in
S_{E\\!K}({\bf X})$ in short). More precisely, $\nu\in S_{D}({\bf X})$ (resp.
$\nu\in S_{E\\!K}({\bf X})$) if and only if $\nu\in S({\bf X})$ and
$\mathfrak{m}\text{-}\sup_{x\in M}\mathbb{E}_{x}[A_{t}^{\nu}]<\infty$ for
any/some $t>0$ (resp. $\lim_{t\to 0}\mathfrak{m}\text{-}\sup_{x\in
M}\mathbb{E}_{x}[A_{t}^{\nu}]<1$) (see [2]). Here $S({\bf X})$ denotes the
family of smooth measures with respect to ${\bf X}$ (see [50, Chapter VI,
Definition 2.3], [30, p. 83] for the definition of smooth measures) and
$\mathfrak{m}$-$\sup_{x\in M}f(x)$ denotes the $\mathfrak{m}$-essentially
supremum for a function $f$ on $M$. For $\nu\in S({\bf X})$ and set
$U_{\alpha}\nu(x):=\mathbb{E}_{x}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]$ with its $\mathfrak{m}$-essentially supremum
$\|U_{\alpha}\nu\|_{\infty}:=\mathfrak{m}$-$\sup_{x\in M}U_{\alpha}\nu(x)$,
$\|U_{\alpha}\nu\|_{\infty}<\infty$ for some/any $\alpha>0$ (resp.
$\lim_{\alpha\to\infty}\|U_{\alpha}\nu\|_{\infty}=0$,
$\lim_{\alpha\to\infty}\|U_{\alpha}\nu\|_{\infty}<1$) if and only if $\nu\in
S_{D}({\bf X})$ (resp. $\nu\in S_{E\\!K}({\bf X})$, $\nu\in S_{K}({\bf X})$).
For $\nu\in S_{D}({\bf X})$, the following inequality holds

$\displaystyle\int_{M}\tilde{f}^{2}\mathrm{d}\nu\leq\|U_{\alpha}\nu\|_{\infty}\mathscr{E}_{\alpha}(f,f),$
(1.1)

which is called the Stollmann-Voigt’s inequality. Here
$\mathscr{E}_{\alpha}(f,f):=\mathscr{E}_{\alpha}(f,f)+\alpha\|f\|_{L^{2}(M;\mathfrak{m})}^{2}$.

Then, for $q\in\\{1,2\\}$, the quadratic form

$\displaystyle\mathscr{E}^{q\kappa}(f):=\mathscr{E}(f)+\langle
2\kappa,\tilde{f}^{2}\rangle$

with finiteness domain $D(\mathscr{E}^{q\kappa})=D(\mathscr{E})$ is closed,
lower semi-bounded, moreover, there exists $\alpha_{0}>0$ and $C>0$ such that

$\displaystyle
C^{-1}\mathscr{E}_{1}(f,f)\leq\mathscr{E}^{q\kappa}_{\alpha_{0}}(f,f)\leq
C\mathscr{E}_{1}(f,f)\quad\text{ for all }\quad f\in
D(\mathscr{E}^{q\kappa})=D(\mathscr{E})$

by (1.1) (see [16, (3.3)] and [17, Assumption of Theorem 1.1]). The Feynman-
Kac semigroup $(p_{t}^{q\kappa})_{t\geq 0}$ defined by

$\displaystyle
p_{t}^{q\kappa}f(x)=\mathbb{E}_{x}[e^{-qA_{t}^{\kappa}}f(X_{t})],\quad
f\in\mathscr{B}_{b}(M)$

is $\mathfrak{m}$-symmetric, i.e.

$\displaystyle\int_{M}p_{t}^{q\kappa}f(x)g(x)\mathfrak{m}(\mathrm{d}x)=\int_{M}g(x)p_{t}^{q\kappa}g(x)\mathfrak{m}(\mathrm{d}x)\quad\text{
for all }\quad f,g\in\mathscr{B}_{+}(M)$

and coincides with the strongly continuous semigroup $(P_{t}^{q\kappa})_{t\geq
0}$ on $L^{2}(M;\mathfrak{m})$ associated with
$(\mathscr{E}^{q\kappa},D(\mathscr{E}^{q\kappa}))$ (see [17, Theorem 1.1]).
Here $A_{t}^{q\kappa}$ is a continuous additive functional (CAF in short)
associated with the signed smooth measure $q\kappa$ under Revuz
correspondence. Under $\kappa^{-}\in S_{K}({\bf X})$ and $p\in[1,+\infty]$,
$(p_{t}^{\kappa})_{t\geq 0}$ can be extended to be a bounded operator on
$L^{p}(M;\mathfrak{m})$ denoted by $P_{t}^{\kappa}$ such that there exist
finite constants $C(\kappa)>0,C_{\kappa}\geq 0$ depending only on $\kappa^{-}$
such that for every $t\geq 0$

$\displaystyle\|P_{t}^{\kappa}\|_{p,p}\leq C(\kappa)e^{C_{\kappa}t}.$ (1.2)

Here $C(\kappa)=1$ under $\kappa^{-}=0$. $C_{\kappa}\geq 0$ can be taken to be
$0$ under $\kappa^{-}=0$. We define $R_{\alpha}^{\kappa}$ on
$L^{p}(M;\mathfrak{m})$ for $\alpha>C_{\kappa}$ by

$\displaystyle R_{\alpha}^{\kappa}f:=\int_{0}^{\infty}e^{-\alpha
t}P_{t}^{\kappa}f\,\mathrm{d}t.$

Let $\Delta^{q\kappa}$ be the $L^{2}$-generator associated with
$(\mathscr{E}^{q\kappa},D(\mathscr{E}))$ defined by

$\displaystyle\left\\{\begin{array}[]{ll}D(\Delta^{q\kappa})&:=\\{u\in
D(\mathscr{E})\mid\text{there exists }w\in L^{2}(M;\mathfrak{m})\text{ such
that }\\\ &\hskip
85.35826pt\mathscr{E}^{q\kappa}(u,v)=-\int_{M}wv\,\mathrm{d}\mathfrak{m}\quad\text{
for any }\quad v\in D(\mathscr{E})\\},\\\ \Delta^{q\kappa}u&:=w\quad\text{ for
}w\in L^{2}(M;\mathfrak{m})\quad\text{specified as above,}\end{array}\right.$
(1.6)

called the Schrödinger operator with potential $q\kappa$. Formally,
$\Delta^{q\kappa}$ can be understood as “$\Delta^{q\kappa}=\Delta-q\kappa$”,
where $\Delta$ is the $L^{2}$-generator associated with
$(\mathscr{E},D(\mathscr{E}))$.

######  Definition 1.1 ($q$-Bakry-Émery condition).

Suppose that $q\in\\{1,2\\}$, $\kappa^{+}\in S_{D}({\bf X})$, $2\kappa^{-}\in
S_{E\\!K}({\bf X})$ and $N\in[1,+\infty]$. We say that
$(M,\mathscr{E},\mathfrak{m})$ or simply $M$ satisfies the $q$-Bakry-Émery
condition, briefly ${\sf BE}_{q}(\kappa,N)$, if for every $f\in D(\Delta)$
with $\Delta f\in D(\mathscr{E})$ and every nonnegative $\phi\in
D(\Delta^{q\kappa})$ with $\Delta^{q\kappa}\phi\in L^{\infty}(M;\mathfrak{m})$
(we further impose $\phi\in L^{\infty}(M;\mathfrak{m})$ for $q=2$), we have

$\displaystyle\frac{1}{q}\int_{M}\Delta^{q\kappa}\phi\Gamma(f)^{\frac{q}{2}}\mathrm{d}\mathfrak{m}-\int_{M}\phi\Gamma(f)^{\frac{q-2}{2}}\Gamma(f,\Delta
f)\mathrm{d}\mathfrak{m}\geq\frac{1}{N}\int_{M}\phi\Gamma(f)^{\frac{q-2}{2}}(\Delta
f)^{2}\mathrm{d}\mathfrak{m}.$

The latter term is understood as $0$ if $N=\infty$.

######  Assumption 1.

We assume that $M$ satisfies ${\sf BE}_{2}(\kappa,N)$ condition for a given
signed smooth measure $\kappa$ with $\kappa^{+}\in S_{D}({\bf X})$ and
$2\kappa^{-}\in S_{E\\!K}({\bf X})$ and $N\in[1,\infty]$.

Under Assumption 1, we say that $(M,\mathscr{E},\mathfrak{m})$ or simply $M$
is tamed. In fact, under $\kappa^{+}\in S_{D}({\bf X})$ and $2\kappa^{-}\in
S_{E\\!K}({\bf X})$, the condition ${\sf BE}_{2}(\kappa,\infty)$ is equivalent
to ${\sf BE}_{1}(\kappa,\infty)$ (see Lemma 3.3 below), in particular, the
heat flow $(P_{t})_{t\geq 0}$ satisfies

$\displaystyle\sqrt{\Gamma(P_{t}f)}\leq
P_{t}^{\kappa}\sqrt{\Gamma(f)}\quad\mathfrak{m}\text{-a.e.~{}for any }f\in
D(\mathscr{E})\quad\text{ and }\quad t\geq 0$ (1.7)

(see [27, Definition 3.3 and Theorem 3.4]). The inequality (1.7) plays a
crucial role in our paper. Note that our condition $\kappa^{+}\in S_{D}({\bf
X})$, $\kappa^{-}\in S_{E\\!K}({\bf X})$ (resp. $\kappa^{+}\in S_{D}({\bf
X})$, $2\kappa^{-}\in S_{E\\!K}({\bf X})$) is stronger than the $1$-moderate
(resp. $2$-moderate) condition treated in [27] for the definition of tamed
space.

Denote by $L^{2}(T^{*}\\!M)$ the space of $L^{2}$-cotangent module defined as
an $L^{2}$-normed $L^{\infty}$-module in Definition 4.8 below, and by
$W^{1,2}(T^{*}\\!M)$ the Sobolev space of $L^{2}$-$1$-forms defined by
$W^{1,2}(T^{*}\\!M):=D({\rm d})\cap D({\rm d}_{*})$, where $D({\rm d})$ (resp.
$D({\rm d}_{*})$) is the domain of exterior differential (resp.
codifferential) defined in Definition 6.1 (resp. Definition 6.2) below. We
endow $W^{1,2}(T^{*}\\!M)$ with the norm $\|\cdot\|_{W^{1,2}(T^{*}\\!M)}$
given by

$\displaystyle\|\omega\|_{W^{1,2}(T^{*}\\!M)}^{2}:=\|\omega\|_{L^{2}(T^{*}\\!M)}^{2}+\|\mathrm{d}\omega\|_{L^{2}(\Lambda^{2}T^{*}\\!M)}^{2}+\|\mathrm{d}_{*}\omega\|_{L^{2}(M;\mathfrak{m})}^{2}.$

Here $L^{2}(\Lambda^{2}T^{*}\\!M):=\Lambda^{2}L^{2}(T^{*}\\!M)$ denotes the
space of $2$-fold exterior products of $L^{2}(T^{*}\\!M)$ defined in
Definition 4.6. We define the _contravariant_ functional $\mathscr{E}_{\rm
con}:L^{2}(T^{*}\\!M)\to[0,+\infty]$ by

$\displaystyle\mathscr{E}_{\rm
con}(\omega):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}\left[|\mathrm{d}\omega|^{2}+|\mathrm{d}_{*}\omega|^{2}\right]\mathrm{d}\mathfrak{m}}&\text{
if }\omega\in W^{1,2}(T^{*}\\!M),\\\
\infty&\text{otherwise.}\end{array}\right.$

We define the subspace $H^{1,2}(T^{*}\\!M)\subset W^{1,2}(T^{*}\\!M)$ by the
closure of ${\rm Reg}(T^{*}\\!M)$ with respect to
$\|\cdot\|_{W^{1,2}(T^{*}\\!M)}$:

$\displaystyle H^{1,2}(T^{*}\\!M):=\overline{{\rm
Reg}(T^{*}\\!M)}^{\|\cdot\|_{W^{1,2}(T^{*}\\!M)}}.$

Here ${\rm Reg}(T^{*}\\!M)$ is the space of regular $1$-forms defined in
Definition 4.16 below. The space $D({\Delta\Delta}^{\rm\\!HK})$ is defined to
consist of all $\omega\in H^{1,2}(T^{*}\\!M)$ for which there exists
$\alpha\in L^{2}(T^{*}\\!M)$ such that for every $\eta\in H^{1,2}(T^{*}\\!M)$,

$\displaystyle\int_{M}\langle\alpha,\eta\rangle\mathrm{d}\mathfrak{m}=-\int_{M}\left[\langle\mathrm{d}\omega,\mathrm{d}\eta\rangle+\langle\mathrm{d}_{*}\omega,\mathrm{d}_{*}\eta\rangle\right]\mathrm{d}\mathfrak{m}.$

In case of existence, the element $\alpha$ is unique, denoted by
${\Delta\Delta}^{\rm\\!HK}\omega$ and called the _Hodge Laplacian_ , _Hodge-
Kodaira Laplacian_ or _Hodge-de Rham Laplacian_ of $\omega$. Formally
${\Delta\Delta}^{\rm\\!HK}\omega$ can be written
“${\Delta\Delta}^{\rm\\!HK}\omega=-(\mathrm{d}\mathrm{d}_{*}+\mathrm{d}_{*}\mathrm{d})\omega$”.

Let $P_{t}^{\rm HK}$ be the heat flow on $1$-forms associated to the
functional $\widetilde{\mathscr{E}}_{\rm con}:L^{2}(T^{*}\\!M)\to[0,+\infty]$
with

$\displaystyle\widetilde{\mathscr{E}}_{\rm
con}(\omega):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}\left[|\mathrm{d}\omega|^{2}+|\mathrm{d}_{*}\omega|^{2}\right]\mathrm{d}\mathfrak{m}}&\text{
if }\omega\in H^{1,2}(T^{*}\\!M),\\\ \infty&\text{
otherwise.}\end{array}\right.$

We can write $\mathscr{E}^{\rm HK}$ instead of $\widetilde{\mathscr{E}}_{\rm
con}$. Let $(P_{t}^{\rm HK})_{t\geq 0}$ be the heat semigroup of bounded
linear and self-adjoint operator on $L^{2}(T^{*}\\!M)$ formally written by

$\displaystyle\text{``$P_{t}^{\rm HK}:=e^{t\,{\Delta\Delta}^{\rm\\!HK}}$''}.$

We define $R_{\alpha}^{\rm HK}\omega$ for $\omega\in L^{2}(T^{*}\\!M)$ by

$\displaystyle R_{\alpha}^{\rm HK}\omega:=\int_{0}^{\infty}e^{-\alpha
t}P_{t}^{\rm HK}\omega\,\mathrm{d}t.$

###  1.2 Main Results

Our main theorem under Assumption 1 is the following:

######  Theorem 1.2 (Hess-Schrader-Uhlenbrock inequality).

We have the following:

  1. (i)

For every $\omega\in L^{2}(T^{*}\\!M)$ and $\alpha>C_{\kappa}$,

$\displaystyle|R_{\alpha}^{\rm HK}\omega|\leq
R_{\alpha}^{\kappa}|\omega|\quad\mathfrak{m}\text{-a.e.}$ (1.8)

  2. (ii)

For every $\omega\in L^{2}(T^{*}\\!M)$ and every $t\geq 0$,

$\displaystyle|P_{t}^{\rm HK}\omega|\leq
P_{t}^{\kappa}|\omega|\quad\mathfrak{m}\text{-a.e.}$ (1.9)

The assertion of Theorem 1.2(ii) is known as Hess-Schrader-Uhlenbrock
inequality for $L^{2}$-differential forms, which was firstly established in
the case that $M$ is a compact Riemannian manifold without boundary by Hess-
Schrader-Uhlenbrock [38, 39] and by Simon [60]. This inequality was extended
to the case of compact Riemannian manifold with convex boundary by Ouhabaz
[53] and by Shigekawa [58, 59]. For general compact Riemannian manifolds with
boundary, Hess-Schrader-Uhlenbrock inequality was proved by Hsu [41] in
probabilistic way. For non-compact Riemannian manifolds without boundary, it
was proved by Güneysu [37] in analytic way, and by Driver-Thalmaier [23] and
Elworthy-Le Jan-Li [24, 49] in stochastic way. $L^{p}$-properties of
$(P_{t}^{\rm HK})_{t\geq 0}$ and related heat kernel estimates on Riemannian
manifolds have been studied by Magniez-Ouhabaz [51] under the Ricci lower
bounds having Kato class condition. Very recently, Theorem 1.2(ii) is proved
by Braun [13, Theorem A] in the framework of RCD$(K,\infty)$-space for
$K\in\mathbb{R}$, and also proved by Braun [14, Theorem 8.41] in the present
framework with an extra assumption [14, Assumption 8.37]. Our Theorem 1.2(ii)
extends [14, Theorem 8.41] completely. Theorem 1.2(ii) also plays a crucial
role to prove the $L^{p}$-boundedness of Riesz operator for $p\in[2,+\infty[$
(see [29]).

As a corollary of Theorem 1.2, we can establish the strong continuity of the
semigroup $(P_{t}^{\rm HK})_{t\geq 0}$ on $L^{p}(T^{*}\\!M)$ under Kato class
Ricci lower bounds.

######  Corollary 1.3 ($C_{0}$-property of $(P_{t}^{\rm HK})_{t\geq 0}$ on
$L^{p}(M;\mathfrak{m})$).

Suppose $p\in[2,+\infty]$, or $\kappa^{-}\in S_{K}({\bf X})$ and
$p\in[1,+\infty]$. Then the heat flow $(P_{t}^{\rm HK})_{t\geq 0}$ can be
extended to a semigroup on $L^{p}(T^{*}\\!M)$ and and for each $t>0$

$\displaystyle\|P_{t}^{\rm HK}\omega\|_{L^{p}(T^{*}\\!M)}\leq
C(\kappa)e^{C_{\kappa}t}\|\omega\|_{L^{p}(T^{*}\\!M)},\quad\omega\in
L^{p}(T^{*}\\!M).$ (1.10)

Moreover, if $\kappa^{-}\in S_{K}({\bf X})$ and $p\in[1,+\infty[$, then
$(P_{t}^{\rm HK})_{t\geq 0}$ is strongly continuous on $L^{p}(T^{*}\\!M)$,
i.e., $(P_{t}^{\rm HK})_{t\geq 0}$ is a $C_{0}$-semigroup on
$L^{p}(T^{*}\\!M)$, and further $(P_{t}^{\rm HK})_{t\geq 0}$ is weakly*
continuous on $L^{\infty}(T^{*}\\!M)$.

##  2 Smooth measures of extended Kato class

In this section, we summarize results on smooth measures of extended Kato
class. Denote by $S({\bf X})$ the family of (non-negative) smooth measures on
$(M,\mathscr{B}(M))$ (see [30, p. 83] for the precise definition of smooth
measures). Then there exists a PCAF $A^{\nu}$ admitting exceptional set
satisfying

$\displaystyle\mathbb{E}_{h\mathfrak{m}}\left[\int_{0}^{\infty}e^{-\alpha
t}f(X_{t})\mathrm{d}A_{t}^{\nu}\right]=\mathbb{E}_{f\nu}\left[\int_{0}^{\infty}e^{-\alpha
t}f(X_{t})\mathrm{d}t\right],\quad f,h\in\mathscr{B}_{+}(M)$

(see [30, (5.14)]). A measure $\nu\in S({\bf X})$ is said to be of _Dynkin
class_ (resp.  _Kato class_) if
$\|\mathbb{E}_{\cdot}[A_{t}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}<\infty$ for
some/all $t>0$ (resp.
$\|\mathbb{E}_{\cdot}[A_{t}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}=0$). A
measure $\nu\in S({\bf X})$ is said to be of _extended Kato class_ if
$\lim_{t\to 0}\|\mathbb{E}_{x}[A_{t}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}<1$.
Denote by $S_{D}({\bf X})$ (resp. $S_{K}({\bf X})$, $S_{E\\!K}({\bf X})$) the
family of (non-negative) smooth measures of Dynkin class (resp. Kato class,
extended Kato class). For $\nu\in S_{D}({\bf X})$,
$x\mapsto\mathbb{E}_{x}[A_{t}^{\nu}]$ is $\mathscr{E}$-quasi-continuous, hence
$\|\mathbb{E}_{\cdot}[A_{t}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}$ actually
coincides with a quasi-essentially supremum:

$\displaystyle\|\mathbb{E}_{\cdot}[A_{t}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}=q\text{-}\sup_{x\in
M}\mathbb{E}_{x}[A_{t}^{\nu}]:=\inf_{N:\mathscr{E}\text{-exceptional}}\sup_{x\in
N^{c}}\mathbb{E}_{x}[A_{t}^{\nu}].$

For $\nu\in S_{E\\!K}({\bf X})$, there exist positive constants $C=C(\nu)$ and
$C_{\nu}>0$ such that

$\displaystyle\|\mathbb{E}_{\cdot}[e^{A_{t}^{\nu}}]\|_{L^{\infty}(M;\mathfrak{m})}\leq
Ce^{C_{\nu}t}.$ (2.1)

Indeed, we can choose
$C(\nu):=(1-\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})})^{-1}$
and $C_{\nu}:=\frac{1}{t_{0}}\log C(\nu)$. for some sufficiently small
$t_{0}\in]0,1[$ satisfying
$\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}<1$ (see
[62, Proof of Theorem 2.2]). Denote by $P_{t}^{-\nu}f(x)$ the Feynman-Kac
semigroup defined by
$P_{t}^{-\nu}f(x):=\mathbb{E}_{x}[e^{A_{t}^{\nu}}f(X_{t})]$,
$f\in\mathscr{B}_{+}(M)$. (2.1) is nothing but
$\|P_{t}^{-\nu}\|_{\infty,\infty}\leq Ce^{C_{\nu}t}$. Moreover, for the
conjugate exponent $q:=p/(p-1)\in[1,+\infty]$ we can deduce

$\displaystyle\|P_{t}^{-\nu}\|_{p,p}\leq
C(q\nu)e^{\frac{C_{q\nu}}{q}t}\quad\text{ for }\quad p\in]1,+\infty].$ (2.2)

Here
$\|P_{t}^{-\nu}\|_{p,p}:=\sup\\{\|P_{t}^{-\nu}f\|_{L^{p}(M;\mathfrak{m})}\mid
f\in L^{p}(M;\mathfrak{m}),\|f\|_{L^{p}(M;\mathfrak{m})}=1\\}$ stands for the
operator norm from $L^{p}(M;\mathfrak{m})$ to $L^{p}(M;\mathfrak{m})$. For
each $\beta\geq 1$ satisfying $\beta\nu\in S_{E\\!K}({\bf X})$, we can choose

$\displaystyle C_{\beta\nu}\in]0,C_{\nu}]\quad\text{ and }\quad
C(\nu)=C(\beta\nu).$ (2.3)

Indeed $C_{\nu}$ and $C(\nu)$ can be given by

$\displaystyle
C_{\nu}=\frac{1}{t_{0}}\log\left(1-\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{\infty}\right)^{-1},\quad
C(\nu)=\left(1-\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{\infty}\right)^{-1}$
(2.4)

for some small $t_{0}\in]0,1]$ satisfying
$\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{\infty}<1$ (see [62, Proof of
Theorem 2.2]). If we choose $s_{0}\in]0,t_{0}]$ satisfying
$\|\mathbb{E}_{\cdot}[A_{s_{0}}^{\nu}]\|_{\infty}=\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\beta\nu}]\|_{\infty}<1$,
then we see $C_{\beta\nu}\leq C_{\nu}$ and $C(\nu)=C(\beta\nu)$ for such
selection of $s_{0}$ (resp. $t_{0}$) to $C_{\nu}$ (resp. $C_{\beta\kappa}$).
If $\beta\nu\in S_{E\\!K}({\bf X})$ for $\beta\geq 1$, then we can deduce from
(2.2) and (2.3) that for $p\in[\beta/(\beta-1),+\infty]$

$\displaystyle\|P_{t}^{-\nu}\|_{p,p}\leq C(\nu)e^{C_{\nu}t}.$ (2.5)

In particular, if $\nu\in S_{K}({\bf X})$, then (2.5) holds for any
$p\in[1,+\infty]$ (the proof for $p=1$ can be directly deduced). When
$\nu=-R\mathfrak{m}$ for a constant $R\in\mathbb{R}$, $C_{\nu}$ (resp.
$C(\nu)$) can be taken to be $R\lor 0$ (resp. $1$).

It is known that $\nu\in S_{D}({\bf X})$ (resp. $\nu\in S_{K}({\bf X})$) if
and only if $\|\mathbb{E}_{\cdot}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]\|_{L^{\infty}(M;\mathfrak{m})}<\infty$ for
some/all $\alpha>0$ (resp.
$\lim_{\alpha\to\infty}\|\mathbb{E}_{\cdot}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]\|_{L^{\infty}(M;\mathfrak{m})}=0$), and $\nu\in
S_{E\\!K}({\bf X})$ if and only if
$\lim_{\alpha\to\infty}\|\mathbb{E}_{\cdot}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]\|_{L^{\infty}(M;\mathfrak{m})}<1$.

##  3 Test functions

Under Assumption 1, we now introduce ${\rm Test}(M)$ the set of test
functions:

######  Definition 3.1.

Let $(M,\mathscr{E},\mathfrak{m})$ be a tamed space. Let us define the set of
test functions by

$\displaystyle{\rm Test}(M):$ $\displaystyle=\\{f\in D(\Delta)\cap
L^{\infty}(M;\mathfrak{m})\mid\Gamma(f)\in L^{\infty}(M;\mathfrak{m}),\Delta
f\in D(\mathscr{E})\\},$ $\displaystyle{\rm Test}(M)_{fs}:$
$\displaystyle=\\{f\in{\rm Test}(M)\mid\mathfrak{m}({\rm supp}[f])<\infty\\}.$

It is easy to see that ${\rm Test}(M)_{fs}\subset\mathscr{D}_{1,p}\subset
H^{1,p}(M)$ for any $p\in]1,+\infty[$.

The following lemmas hold.

######  Lemma 3.2 ([27, Proposition 6.8]).

Under Assumption 1, ${\sf BE}_{2}(-\kappa^{-},+\infty)$ holds. Moreover, for
every $f\in L^{2}(M;\mathfrak{m})\cap L^{\infty}(M;\mathfrak{m})$ and $t>0$,
it holds

$\displaystyle\Gamma(P_{t}f)\leq\frac{1}{2t}\|P_{t}^{-2\kappa^{-}}\|_{\infty,\infty}\cdot\|f\|_{L^{\infty}(M;\mathfrak{m})}^{2}.$
(3.1)

In particular, for $f\in L^{2}(M;\mathfrak{m})\cap
L^{\infty}(M;\mathfrak{m})$, then $P_{t}f\in{\rm Test}(M)$.

######  Lemma 3.3 ([27, Theorems 3.4 and 3.6, Proposition 3.7 and Theorem
6.10]).

Under $\kappa^{+}\in S_{D}({\bf X})$ and $2\kappa^{-}\in S_{E\\!K}({\bf X})$.
the condition ${\sf BE}_{2}(\kappa,\infty)$ is equivalent to ${\sf
BE}_{1}(\kappa,\infty)$. In particular, we have (1.7).

######  Lemma 3.4 ([54, Lemma 3.2]).

Under Assumption 1, for every $f\in{\rm Test}(M)$, we have $\Gamma(f)\in
D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})$ and there exists
$\mu=\mu^{+}-\mu^{-}$ with $\mu^{\pm}\in D(\mathscr{E})^{*}$ such that

$\displaystyle-\mathscr{E}^{2\kappa}(u,\varphi)=\int_{M}\tilde{\varphi}\,\mathrm{d}\mu\quad\text{
for all }\quad\varphi\in D(\mathscr{E}).$ (3.2)

Moreover, ${\rm Test}(M)$ is an algebra, i.e., for $f,g\in{\rm Test}(M)$,
$fg\in{\rm Test}(M)$, if further $f$$\in{\rm Test}(M)^{n}$, then
$\Phi($$f$$)\in{\rm Test}(M)$ for every smooth function
$\Phi:\mathbb{R}^{n}\to\mathbb{R}$ with $\Phi(0)=0$.

######  Lemma 3.5.

${\rm Test}(M)\cap L^{p}(M;\mathfrak{m})$ is dense in
$L^{p}(M;\mathfrak{m})\cap L^{2}(M;\mathfrak{m})$ both in $L^{p}$-norm and in
$L^{2}$-norm. In particular, ${\rm Test}(M)$ is dense in
$(\mathscr{E},D(\mathscr{E}))$.

###### Proof.

Take $f\in L^{p}(M;\mathfrak{m})\cap L^{2}(M;\mathfrak{m})$. We may assume
$f\in L^{\infty}(M;\mathfrak{m})$, because $f$ is $L^{p}$(and also
$L^{2}$)-approximated by a sequence $\\{f^{k}\\}$ of
$L^{p}(M;\mathfrak{m})\cap L^{2}(M;\mathfrak{m})\cap
L^{\infty}(M;\mathfrak{m})$-functions defined by $f^{k}:=(-k)\lor f\land k$.
If $f\in L^{p}(M;\mathfrak{m})\cap L^{2}(M;\mathfrak{m})\cap
L^{\infty}(M;\mathfrak{m})$, $P_{t}f\in{\rm Test}(M)$ by Lemma 3.2 and
$\\{P_{t}f\\}\subset{\rm Test}(M)\cap L^{p}(M;\mathfrak{m})$ converges to $f$
in $L^{p}$ and in $L^{2}$ as $t\to 0$. If $f\in D(\mathscr{E}))$, then $f$ can
be approximated by $\\{P_{t}f^{k}\\}$ in $(\mathscr{E},D(\mathscr{E}))$. This
shows the last statement. ∎

######  Remark 3.6.

As proved above, ${\rm Test}(M)$ forms an algebra and dense in
$(\mathscr{E},D(\mathscr{E}))$ under Assumption 1. However, ${\rm Test}(M)$ is
not necessarily a subspace of $C_{b}(M)$. When the tamed space comes from
${\rm RCD}$-space, the Sobolev-to-Lipschitz property of RCD-spaces ensures
${\rm Test}(M)\subset C_{b}(M)$.

##  4 Vector space calculus for tamed space

###  4.1 $L^{\infty}$-module

We need the notion of $L^{p}$-normed $L^{\infty}(M;\mathfrak{m})$-modules.

######  Definition 4.1 ($L^{\infty}$-module).

Given $p\in[1,+\infty]$, a real Banach space
$(\mathscr{M},\|\cdot\|_{\mathscr{M}})$, or simply, $\mathscr{M}$ is called an
_$L^{p}$ -normed $L^{\infty}$-module_ (over $(M,\mathfrak{m})$) if it
satisfies

  1. (a)

a bilinear map $\cdot$ :
$L^{\infty}(M;\mathfrak{m})\times\mathscr{M}\to\mathscr{M}$ satisfying

$\displaystyle(fg)\cdot v$ $\displaystyle=f\cdot(gv),$ $\displaystyle{\bf
1}_{M}\cdot v$ $\displaystyle=v,$

  2. (b)

a nonnegatively valued map $|\cdot|_{\mathfrak{m}}:\mathscr{M}\to
L^{p}(M;\mathfrak{m})$ such that

$\displaystyle|f\cdot v|_{\mathfrak{m}}$
$\displaystyle=|f||v|_{\mathfrak{m}}\quad\mathfrak{m}\text{-a.e.},$
$\displaystyle\|v\|_{\mathscr{M}}$
$\displaystyle=\||v|_{\mathfrak{m}}\|_{L^{p}(M;\mathfrak{m})},$

for every $f,g\in L^{\infty}(M;\mathfrak{m})$ and $v\in\mathscr{M}$. If only
(a) is satisfied, we call $(\mathscr{M},\|\cdot\|_{\mathscr{M}})$ or simply
$\mathscr{M}$ an $L^{\infty}(M;\mathfrak{m})$-module.

Throughout this paper, we always assume that for every $v\in\mathscr{M}$ its
point-wise norm $|v|_{\mathfrak{m}}$ is Borel measurable. We call
$\mathscr{M}$ an _$L^{\infty}$ -module_ if it is $L^{p}$-normed for some
$p\in[1,+\infty]$. $\mathscr{M}$ is called _separable_ if it is a separable
Banach space. We call $v\in\mathscr{M}$ is ($\mathfrak{m}$-essentially)
bounded if $|v|_{\mathfrak{m}}\in L^{\infty}(M;\mathfrak{m})$. $\mathscr{M}$
is called _Hilbert module_ if is an $L^{2}$-normed $L^{\infty}$-module, in
this case, the point-wise norm $|\cdot|_{\mathfrak{m}}$ satisfies a point-wise
$\mathfrak{m}$-a.e. parallelogram identity, hence it induces a _point-wise
scalar product_
$\langle\cdot,\cdot\rangle_{\mathfrak{m}}:\mathscr{M}\times\mathscr{M}\to
L^{1}(M;\mathfrak{m})$ which is $L^{\infty}$-bilinear, $\mathfrak{m}$-a.e.
nonnegative definite, local in both components, satisfies the point-wise
$\mathfrak{m}$-a.e. Cauchy-Schwarz inequality, and reproduces the Hilbertian
scalar product on $\mathscr{M}$ by integration with respect to $\mathfrak{m}$.

######  Definition 4.2 (Dual module).

Let $\mathscr{M}$ and $\mathscr{N}$ be $L^{p}$-normed $L^{\infty}$-module.
Denote both point-wise norms by $|\cdot|_{\mathfrak{m}}$. A map
$T:\mathscr{M}\to\mathscr{N}$ is called _module morphism_ if it is a bounded
linear map in the sense of functional analysis and

$\displaystyle T(f\,v)=f\,T(v)$ (4.1)

for every $v\in\mathscr{M}$ and every $f\in L^{\infty}(M;\mathfrak{m})$. The
set of all module morphisms is written ${\rm Hom}(\mathscr{M},\mathscr{N})$
and equipped with the usual operator norm
$\|\cdot\|_{\mathscr{M},\mathscr{N}}$. We call $\mathscr{M}$ and $\mathscr{N}$
_isomorphic_ (as $L^{\infty}$-module) if there exists $T\in{\rm
Hom}(\mathscr{M},\mathscr{N})$ and $S\in{\rm Hom}(\mathscr{N},\mathscr{M})$
such that $T\circ S={\rm Id}_{\mathscr{N}}$ and $S\circ T={\rm
Id}_{\mathscr{M}}$. Any such $T$ is called _module isomorphism_. If in
addition, such a $T$ is a norm isometry, it is called _module isometric
isomorphism_. In fact, by (4.1) every module isometric isomorphism $T$
preserves pointwise norms $\mathfrak{m}$-a.e., i.e. for every
$v\in\mathscr{M}$,

$\displaystyle|T(v)|_{\mathfrak{m}}=|v|_{\mathfrak{m}}\quad\mathfrak{m}\text{-a.e.}$

The dual module $\mathscr{M}^{*}$ is defined by

$\displaystyle\mathscr{M}^{*}:={\rm Hom}(\mathscr{M},L^{1}(M;\mathfrak{m}))$

and will be endowed with the usual operator norm. The point-wise paring
between $v\in\mathscr{M}$ and $L\in\mathscr{M}^{*}$ is denoted by $L(v)\in
L^{1}(M;\mathfrak{m})$. If $\mathscr{M}$ is $L^{p}$-normed, then
$\mathscr{M}^{*}$ is an $L^{q}$-normed $L^{\infty}$-module, where
$p,q\in[1,+\infty]$ with $1/p+1/q=1$ (see [32, Proposition 1.2.14]) with
naturally defined multiplications and, and by a slight abuse of notation,
point-wise norm is given by

$\displaystyle|L|_{\mathfrak{m}}:=\mathfrak{m}\text{\rm-esssup}\\{|L(v)|\mid
v\in\mathscr{M},|v|_{\mathfrak{m}}\leq 1\;\mathfrak{m}\text{-a.e.}\\}.$ (4.2)

By [32, Corollary 1.2.16], if $p<\infty$,

$\displaystyle|v|_{\mathfrak{m}}=\mathfrak{m}\text{\rm-esssup}\\{|L(v)|\mid
L\in\mathscr{M},|L|_{\mathfrak{m}}\leq 1\;\mathfrak{m}\text{-a.e.}\\}$

for every $v\in\mathscr{M}$. Moreover, under $p<\infty$, in the sense of
functional analysis $\mathscr{M}^{*}$ and the dual Banach space
$\mathscr{M}^{\prime}$ of $\mathscr{M}$ are isometrically isomorphic [32,
Proposition 1.2.13]. In this case, the natural point-wise paring map
$\mathscr{I}:\mathscr{M}\to\mathscr{M}^{**}$, where $\mathscr{M}^{**}:={\rm
Hom}(\mathscr{M}^{*},L^{1}(M;\mathfrak{m}))$, belongs to ${\rm
Hom}(\mathscr{M},\mathscr{M}^{**})$ and constitutes a norm isometry is
$L^{p}$-normed for $p\in]1,+\infty[$, this is equivalent to $\mathscr{M}$
being reflexive as Banach space [32, Corollary 1.2.18], while for $p=1$, the
implication from “reflexive as Banach space” to “reflexive as
$L^{\infty}$-module” still holds [32, Propositions 1.2.13 and 1.2.17]. In
particular all Hilbert modules are reflexive in both senses.

######  Definition 4.3 ($L^{0}$-module).

Fix an $L^{\infty}$-module $\mathscr{M}$. Let $(B_{i})_{i\in\mathbb{N}}$ a
Borel partition of $M$ such that $\mathfrak{m}(B_{i})\in]0,+\infty[$ for each
$i\in\mathbb{N}$. Denote by $\mathscr{M}^{0}$ the completion of $\mathscr{M}$
with respect to $\mathscr{M}$ the distance ${\sf
d}_{\mathscr{M}^{0}}:\mathscr{M}\times\mathscr{M}\to[0,+\infty[$ defined by

$\displaystyle{\sf
d}_{\mathscr{M}^{0}}(v,w):=\sum_{i\in\mathbb{N}}\frac{1}{2^{i}\mathfrak{m}(B_{i})}\int_{B_{i}}(|v-w|_{\mathfrak{m}}\land
1)\mathrm{d}\mathfrak{m}.$

We call $\mathscr{M}^{0}$ as the _$L^{0}$ -module_ associated with
$\mathscr{M}$. The induced topology on $\mathscr{M}^{0}$ does not depend on
the choice of $(B_{i})_{i\in\mathbb{N}}$ (see [32, p. 31]). In addition,
scalar and functional multiplication, and the point-wise norm
$|\cdot|_{\mathfrak{m}}$ extend continuously to $\mathscr{M}^{0}$, so that all
$\mathfrak{m}$-a.e. properties mentioned for $L^{\infty}$-module hold for
general elements in $\mathscr{M}^{0}$ and $L^{0}(M;\mathfrak{m})$ in place of
$\mathscr{M}$ and $L^{\infty}(M;\mathfrak{m})$. The point-wise paring of
$\mathscr{M}$ and $\mathscr{M}^{*}$extends uniquely and continuously to a
bilinear map on $\mathscr{M}\times(\mathscr{M}^{*})^{0}$ with values in
$L^{0}(M;\mathfrak{m})$ such that for every $v\in\mathscr{M}^{0}$ and every
$L\in(\mathscr{M}^{*})^{0}$,

$\displaystyle|L(v)|\leq|L|_{\mathfrak{m}}|v|_{\mathfrak{m}}\quad\mathfrak{m}\text{-a.e.}$

We have the following characterization of elements in $(\mathscr{M}^{*})^{0}$:

######  Proposition 4.4 ([32, Proposition 1.3.2]).

Let $T:\mathscr{M}^{0}\to L^{0}(M;\mathfrak{m})$ be a linear map for which
there exists $f\in L^{0}(M;\mathfrak{m})$ such that for every
$v\in\mathscr{M}$,

$\displaystyle|T(v)|\leq f|v|_{\mathfrak{m}}\quad\mathfrak{m}\text{-a.e.}$

Then there exists a unique $L\in(\mathscr{M}^{*})^{0}$ such that for every
$v\in\mathscr{M}$,

$\displaystyle L(v)=T(v)\quad\mathfrak{m}\text{-a.e.,}$

and we furthermore have

$\displaystyle|L|_{\mathfrak{m}}\leq f\quad\mathfrak{m}\text{-a.e.}$

###  4.2 Tensor products

######  Definition 4.5 (Tensor products).

Let $\mathscr{M}_{1}$ and $\mathscr{M}_{2}$ be two Hilbert module. By a slight
abuse of notation, we denote both point-wise scalar products by
$\langle\cdot,\cdot\rangle_{\mathfrak{m}}$. We consider the $L^{0}$-module
$\mathscr{M}_{i}^{0}$ induced from $\mathscr{M}_{i}$, $i=1,2$. Let
$\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0}$ be the “algebraic tensor
product” consisting of all finite linear combinations of formal elements
$v\otimes w$, $v\in\mathscr{M}_{1}^{0}$ and $w\in\mathscr{M}_{2}^{0}$,
obtained by factorizing appropriate vector spaces (see [32, Section 1.5]). It
naturally comes with a multiplications
$\cdot\,:L^{0}(M;\mathfrak{m})\times\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0}\to\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0}$
defined through

$\displaystyle f(v\otimes w):=(fv)\otimes w=v\otimes(fw)$

and a point-wise scalar product
$\langle\cdot\,|\,\cdot\rangle_{\mathfrak{m}}\,:(\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0})^{2}\to
L^{0}(M;\mathfrak{m})$ given by

$\displaystyle\langle(v_{1}\otimes w_{1})\,|\,(v_{2}\otimes
w_{2})\rangle:=\langle v_{1},v_{2}\rangle_{\mathfrak{m}}\langle
w_{1},w_{2}\rangle_{\mathfrak{m}},$

both extended to $\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0}$ by bi-
linearity. Then $\langle\cdot\,|\,\cdot\rangle_{\mathfrak{m}}$ is bilinear,
$\mathfrak{m}$-a.e. non-negative definite, symmetric, and local in both
components (see [35, Lemma 3.2.19]).

The point-wise _Hilbert-Schmidt norm_ $|\cdot|_{{\rm
HS}}:\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0}\to L^{0}(M;\mathfrak{m})$ is
given by

$\displaystyle|A|_{{\rm HS}}:=\sqrt{\langle A\,|\,A\rangle_{\mathfrak{m}}}.$

This map satisfies the $\mathfrak{m}$-a.e. triangle inequality and is
$1$-homogenous with respect to multiplication with
$L^{0}(M;\mathfrak{m})$-functions (see [32, p. 44]). Consequently, the map
$\|\cdot\|_{\mathscr{M}_{1}\otimes\mathscr{M}_{2}}:\mathscr{M}_{1}^{0}\odot\,\mathscr{M}_{2}^{0}\to[0,+\infty]$
defined through

$\displaystyle\|A\|_{\mathscr{M}_{1}\otimes\mathscr{M}_{2}}:=\||A|_{{\rm
HS},\mathfrak{m}}\|_{L^{2}(M;\mathfrak{m})}$

has all properties of a norm except that it might take the value $+\infty$.

Now we define the _tensor product_ $\mathscr{M}_{1}\otimes\mathscr{M}_{2}$ the
$\|\cdot\|_{\mathscr{M}_{1}\otimes\mathscr{M}_{2}}$-completion of the subspace
that consists of all $A\in\mathscr{M}_{1}^{0}\odot\mathscr{M}_{2}^{0}$ such
that $\|A\|_{\mathscr{M}_{1}\otimes\mathscr{M}_{2}}<\infty$. Denote by
$A^{\top}\in\mathscr{M}^{\otimes 2}$ the transpose of
$A\in\mathscr{M}^{\otimes 2}$ as defined in [32, Section 1.5]. For instance,
for bounded $v,w\in\mathscr{M}$ we have

$\displaystyle(v\otimes w)^{\top}=w\otimes v.$ (4.3)

###  4.3 Exterior products

Let $\mathscr{M}$ be a Hilbert module and $k\in\mathbb{N}\cup\\{0\\}$. Set
$\Lambda^{0}\\!\mathscr{M}^{0}:=L^{0}(M;\mathfrak{m})$ and, for $k\geq 1$, let
$\Lambda^{k}\\!\mathscr{M}^{0}$ be the “exterior product”constructed by
suitable factorizing $(\mathscr{M}^{0})^{\odot k}$ (see [32, Section 1.5]).
The representative of $v_{1}\odot\cdots\odot
v_{k},v_{1},\cdots,v_{k}\in\mathscr{M}^{0}$ in $\Lambda^{k}\\!\mathscr{M}^{0}$
is written $v_{1}\wedge\cdots\wedge v_{k}$. $\Lambda^{k}\\!\mathscr{M}^{0}$
naturally comes with a multiplication
$\cdot:L^{0}(M;\mathfrak{m})\times\Lambda^{k}\\!\mathscr{M}^{0}$ via

$\displaystyle f(v_{1}\wedge\cdots\wedge v_{k}):=(fv_{1})\wedge\cdots\wedge
v_{k}=\cdots=v_{1}\wedge\cdots\wedge(fv_{k})$

and a pointwise scalar product
$\langle\cdot,\cdot\rangle_{\mathfrak{m}}:(\Lambda^{k}\\!\mathscr{M}^{0})^{2}\to
L^{0}(M;\mathfrak{m})$ defined by

$\displaystyle\langle v_{1}\wedge\cdots\wedge v_{k},w_{1}\wedge\cdots\wedge
w_{k}\rangle_{\mathfrak{m}}:={\rm det}\left[\langle
v_{i},w_{j}\rangle_{\mathfrak{m}}\right]_{i,j\in\\{1,2,\cdots,k\\}}$ (4.4)

up to a factor $k!$, both extended to $\Lambda^{k}\\!\mathscr{M}^{0}$ by
(bi-)linearity. Then $\langle\cdot,\cdot\rangle_{\mathfrak{m}}$ is bilinear,
$\mathfrak{m}$-a.e. non-negative definite, symmetric, and local in both
components.

Given any $k,k^{\prime}\in\mathbb{N}\cup\\{0\\}$, the map assigning to
$v_{1}\wedge\cdots\wedge v_{k}\in\Lambda^{k}\\!\mathscr{M}^{0}$ and
$w_{1}\wedge\cdots\wedge
w_{k^{\prime}}\in\Lambda^{k^{\prime}}\\!\mathscr{M}^{0}$ the element
$v_{1}\wedge\cdots\wedge v_{k}\wedge w_{1}\wedge\cdots\wedge
w_{k^{\prime}}\in\Lambda^{k+k^{\prime}}\\!\mathscr{M}^{0}$ can and will be
uniquely extended by bilinearity and continuity to a bilinear map
$\bigwedge:\Lambda^{k}\\!\mathscr{M}^{0}\times\Lambda^{k^{\prime}}\\!\mathscr{M}^{0}$
termed _wedge product_ (see [35, p. 47]). If $k=0$ or $k^{\prime}=0$, it
simply corresponds to multiplication of elements of
$\Lambda^{k^{\prime}}\\!\mathscr{M}^{0}$ or $\Lambda^{k}\\!\mathscr{M}^{0}$,
respectively, with function in $L^{0}(M;\mathfrak{m})$ according to (4.4).

By a slight abuse of notation, define the map
$|\cdot|_{\mathfrak{m}}:\Lambda^{k}\\!\mathscr{M}^{0}\to
L^{0}(M;\mathfrak{m})$ by

$\displaystyle|\omega|_{\mathfrak{m}}:=\sqrt{\langle\omega,\omega\rangle_{\mathfrak{m}}}.$

It obeys the $\mathfrak{m}$-a.e. triangle inequality and is homogenous with
respect to multiplication with $L^{0}(M;\mathfrak{m})$-functions (see [35, p.
47]).

It follows that the map
$\|\cdot\|_{\Lambda^{k}\\!\mathscr{M}}:=\||\cdot|_{\mathfrak{m}}\|_{L^{2}(M;\mathfrak{m})}$
has all properties of a norm except that $\|w\|_{\Lambda^{k}\\!\mathscr{M}}$
might be infinite.

######  Definition 4.6.

The ($k$-fold) exterior product $\Lambda^{k}\\!\mathscr{M}$ is defined as the
completion with respect to $\|\cdot\|_{\Lambda^{k}\\!\mathscr{M}}$ of the
subspace consisting of all $\omega\in\Lambda^{k}\\!\mathscr{M}^{0}$ such that
$\|\omega\|_{\Lambda^{k}\\!\mathscr{M}}$.

The space $\Lambda^{k}\\!\mathscr{M}$ naturally becomes a Hilbert module and,
if $\mathscr{M}$ is separable, is separable as wel (see [35, p. 47]).

###  4.4 Cotangent module

Let $(\mathscr{E},D(\mathscr{E}))$ be a quasi-regular strongly local Dirichlet
form on $L^{2}(M;\mathfrak{m})$. We define the _cotangent module_
$L^{2}(T^{*}\\!M)$, i.e., the space of differential $1$-forms that are
$L^{2}$-integrable in a certain “universal” sense.

######  Definition 4.7 (Pre-cotangent module).

We define the _pre-cotangent module_ ${\rm Pcm}$ by

$\displaystyle{\rm Pcm}:$
$\displaystyle=\Biggl{\\{}\left.(f_{i},A_{i})_{i\in\mathbb{N}}\;\Biggr{|}\,(A_{i})_{i\in\mathbb{N}}\text{
Borel partition of }X,\right.$ $\displaystyle\hskip
85.35826pt\left.(f_{i})_{i\in\mathbb{N}}\subset
D(\mathscr{E})_{e},\;\sum_{i\in\mathbb{N}}\int_{A_{i}}\Gamma(f_{i})\mathrm{d}\mathfrak{m}<\infty\right\\}$

Moreover, we define a relation $\sim$ on ${\rm Pcm}$ by
$(f_{i},A_{i})_{i\in\mathbb{N}}\sim(g_{j},B_{j})_{j\in\mathbb{N}}$ if and only
if $\int_{A_{i}\cap B_{j}}\Gamma(f_{i}-g_{j})\mathrm{d}\mathfrak{m}=0$ for
every $i,j\in\mathbb{N}$. The relation, in fact forms an equivalence relation
by [14, §2.1]. The equivalence class of an element
$(f_{i},A_{i})_{i\in\mathbb{N}}\in{\rm Pcm}$ with respect to $\sim$ is denoted
by $[f_{i},A_{i}]$. The space ${\rm Pcm}/\\!\\!\sim$ of equivalence classes
becomes a vector space via the well-defined operations

$\displaystyle[f_{i},A_{i}]+[g_{j},B_{j}]:=[f_{i}+g_{j},A_{i}\cap
B_{j}],\qquad\lambda[f_{i},A_{i}]:=[\lambda f_{i},A_{i}]$ (4.5)

for every $[f_{i},A_{i}],[g_{j},B_{j}]\in{\rm Pcm}/\\!\\!\sim$ and
$\lambda\in\mathbb{R}$.

Now we define the space ${\rm SF}(M;\mathfrak{m})\subset
L^{\infty}(M;\mathfrak{m})$ of simple functions, i.e., each element $h\in{\rm
SF}(M;\mathfrak{m})$ attains only a finite number values. For
$[f_{i},A_{i}]\in{\rm Pcm}/\\!\\!\sim$ and $h=\sum_{j=1}^{\ell}a_{j}{\bf
1}_{B_{j}}\in{\rm SF}(M;\mathfrak{m})$ with a Borel partition $(B_{j})$ of
$M$, we define the product $h[f_{i},A_{i}]\in{\rm Pcm}/\\!\\!\sim$ as

$\displaystyle h[f_{i},A_{i}]:=[a_{j}f_{i},A_{i}\cap B_{j}],$ (4.6)

where we set $B_{j}:=\emptyset$ and $a_{j}:=0$ for every $j>{\ell}$. It is
readily verified that this definition is well-posed and that the resulting
multiplication is a bilinear map from ${\rm SF}(M;\mathfrak{m})\times{\rm
Pcm}/\\!\\!\sim$ into ${\rm Pcm}/\\!\\!\sim$ such that for every
$[f_{i},A_{i}]\in{\rm Pcm}/\\!\\!\sim$ and every $h,k\in{\rm
SF}(M;\mathfrak{m})$

$\displaystyle(hk)[f_{i},A_{i}]=h(k[f_{i},A_{i}]),\qquad{\bf
1}[f_{i},A_{i}]=[f_{i},A_{i}].$ (4.7)

Moreover, the map $\|\cdot\|_{L^{2}(T^{*}\\!M)}:{\rm
Pcm}/\\!\\!\sim\,\to[0,+\infty[$ given by

$\displaystyle\|[f_{i},A_{i}]\|_{L^{2}(T^{*}\\!M)}^{2}:=\sum_{i\in\mathbb{N}}\int_{A_{i}}\Gamma(f_{i})\mathrm{d}\mathfrak{m}<\infty$

constitutes a norm on ${\rm Pcm}/\\!\\!\sim$.

######  Definition 4.8 (Cotangent module).

We define the Banach space $(L^{2}(T^{*}\\!M),\|\cdot\|_{L^{2}(T^{*}\\!M)})$
as the completion of $({\rm Pcm}/\\!\\!\sim,\|\cdot\|_{L^{2}(T^{*}\\!M)})$.
The pair $(L^{2}(T^{*}\\!M),\|\cdot\|_{L^{2}(T^{*}\\!M)})$ or simply
$L^{2}(T^{*}\\!M)$ is called _cotangent module_ , and the elements of
$L^{2}(T^{*}\\!M)$ are called _cotangent vector fields_ or (_differential_)
_$1$ -forms_.

The following are shown in [14, Lemma 2.3 and Theorem 2.4]:

######  Lemma 4.9 ([14, Lemma 2.3]).

The map from ${\rm SF}(M)\times{\rm Pcm}/\\!\\!\sim$ into ${\rm
Pcm}/\\!\\!\sim$ defined in (4.6) extends continuously and uniquely to a
bilinear map from $L^{\infty}(M;\mathfrak{m})\times L^{2}(T^{*}\\!M)$ into
$L^{2}(T^{*}\\!M)$ satisfying, for every $f,g\in L^{\infty}(M;\mathfrak{m})$
and every $\omega\in L^{2}(T^{*}\\!M)$,

$\displaystyle(fg)\omega=f(g\omega),\quad{\bf
1}_{M}\omega=\omega,\quad\|f\omega\|_{L^{2}(T^{*}\\!M)}\leq\|f\|_{L^{\infty}(M;\mathfrak{m})}\cdot\|\omega\|_{L^{2}(T^{*}\\!M)}.$

######  Theorem 4.10 (Module property, [14, Theorem 2.4]).

The cotangent module $L^{2}(T^{*}\\!M)$ is an $L^{2}$-normed
$L^{\infty}$-module over $M$ with respect to $\mathfrak{m}$ whose point-wise
norm $|\cdot|_{\mathfrak{m}}$ satisfies, for every $[f_{i},A_{i}]\in{\rm
Pcm}/\\!\\!\sim$,

$\displaystyle|[f_{i},A_{i}]|_{\mathfrak{m}}=\sum_{i\in\mathbb{N}}{\bf
1}_{A_{i}}\Gamma(f_{i})^{\frac{1}{2}}\quad\mathfrak{m}\text{-a.e.}$ (4.8)

In particular, $L^{2}(T^{*}\\!M)$ is a Hilbert module with respect to
$\mathfrak{m}$.

######  Definition 4.11 ($L^{2}$-differential).

The $L^{2}$-differential $\mathrm{d}f$ of any function $f\in
D(\mathscr{E})_{e}$ is defined by

$\displaystyle{\mathrm{d}f}:=[f,X]\in L^{2}(T^{*}\\!M),$

where $[f,X]\in{\rm Pcm}/\\!\\!\sim\;\subset L^{2}(T^{*}\\!M)$ is the
representative of the sequence $(f_{i},A_{i})_{i\in\mathbb{N}}$ given by
$f_{i}:=f$, $A_{1}:=X$, $f_{i}:=0$ and $A_{i}:=\emptyset$ for every $i\geq 2$.

As usual, we call a $1$-form $\omega\in L^{2}(T^{*}\\!M)$ _exact_ if, for some
$f\in D(\mathscr{E})_{e}$,

$\displaystyle\omega=\mathrm{d}f.$

The $L^{2}$-differential $\mathrm{d}$ is a linear operator on
$D(\mathscr{E})_{e}$. By (4.8), the $L^{\infty}$-module structure induced by
$\mathfrak{m}$ according to Theorem 4.10,

$\displaystyle|\mathrm{d}f|_{\mathfrak{m}}=\Gamma(f)^{\frac{1}{2}}\quad\mathfrak{m}\text{-a.e.}$

holds for every $f\in D(\mathscr{E})_{e}$.

###  4.5 Tangent module

In this section, let $(M,{\sf d},\mathfrak{m})$ be a metric measure space and
assume the infinitesimally Hilbertian condition for it. In particular, we are
given a strongly local Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ on
$L^{2}(M;\mathfrak{m})$. We define the notion of _tangent module_.

######  Definition 4.12 (Tangent module).

The tangent module $(L^{2}(TM),\|\cdot\|_{L^{2}(TM)})$ or simply $L^{2}(TM)$
is

$\displaystyle L^{2}(TM):=L^{2}(T^{*}\\!M)^{*}$

and it is endowed with the norm $\|\cdot\|_{L^{2}(TM)}$ induced by (4.2). The
elements of $L^{2}(TM)$ will be called _vector fields_.

As in Subsection 4.1, the point-wise pairing between $\omega\in
L^{2}(T^{*}\\!M)$ and $X\in L^{2}(TM)$ is denoted by $\omega(X)\in
L^{1}(M;\mathfrak{m})$, and, by a slight abuse of notation, $|X|\in
L^{2}(M;\mathfrak{m})$ denotes the point-wise norm of $M$. By [14, Lemma 2.7
and Proposition 1.24], $L^{2}(TM)$ is a separable Hilbert module. Furthermore,
in terms of the point-wise scalar product $\langle\cdot,\cdot\rangle$ on
$L^{2}(T^{*}\\!M)$ and $L^{2}(TM)$, respectively, [14, Proposition 1.24]
allows us to define the _(Riesz) musical isomorphisms_
$\sharp:L^{2}(T^{*}\\!M)\to L^{2}(TM)$ and $\flat:=\sharp^{-1}$ defined by

$\displaystyle\langle\omega^{\sharp},X\rangle:=\omega(X)=:\langle
X^{\flat},\omega\rangle\quad\mathfrak{m}\text{-a.e.}$ (4.9)

######  Definition 4.13 ($L^{2}$-gradient).

The _$L^{2}$ -gradient_ $\nabla f$ of a function $f\in D(\mathscr{E})_{e}$ is
defined by

$\displaystyle\nabla f:=(\mathrm{d}f)^{\sharp}.$

Observe from (4.9) that $f\in D(\mathscr{E})_{e}$, is characterized as the
unique element $X\in L^{2}(TM)$ which satisfies

$\displaystyle\mathrm{d}f(X)=|\mathrm{d}f|^{2}=|X|^{2}\quad\mathfrak{m}\text{-a.e.}$

###  4.6 Divergences

######  Definition 4.14 ($L^{2}$-divergence).

We define the space $D({\rm div})\subset L^{2}(TM)$ by

$\displaystyle D({\rm div}):$ $\displaystyle=\Bigl{\\{}X\in
L^{2}(TM)\,\Bigl{|}\,\text{ there exists a function }f\in
L^{2}(M;\mathfrak{m})$ $\displaystyle\hskip 56.9055pt\text{ such that for
every }h\in
D(\mathscr{E}),\quad-\int_{M}h\,f\,\mathrm{d}\mathfrak{m}=\int_{M}\mathrm{d}h(X)\mathrm{d}\mathfrak{m}\Bigr{\\}}.$

Such $f\in L^{2}(M;\mathfrak{m})$ is unique, and it is called the _(
$L^{2}$-)divergence of_ $M$ and denoted by ${\rm div}\,X$.

The uniqueness of ${\rm div}\,X$ comes from the denseness of $D(\mathscr{E})$
in $L^{2}(M;\mathfrak{m})$. Note that the map ${\rm div}:D({\rm div})\to
L^{2}(M;\mathfrak{m})$ is linear, hence $D({\rm div})$ is a vector space. By
definition of $D(\Delta)$, $\nabla D(\Delta)\subset D({\rm div})$ and

$\displaystyle{\rm div}\nabla f=\Delta f\quad\mathfrak{m}\text{-a.e.}$ (4.10)

for every $f\in D(\Delta)$. Moreover, using the Leibniz rule in [35, Theorem
4.4], one can verify that for every $X\in D({\rm div})$ and $f\in
D(\mathscr{E})_{e}\cap L^{\infty}(M;\mathfrak{m})$ with $|\mathrm{d}f|\in
L^{\infty}(M;\mathfrak{m})$, we have $fX\in D({\rm div})$ and

$\displaystyle{\rm div}(fX)=f\,{\rm
div}(X)+\mathrm{d}f(X)\quad\mathfrak{m}\text{-a.e.}$ (4.11)

###  4.7 The Lebesgue spaces $L^{p}(T^{*}\\!M)$ and $L^{p}(TM)$

Let $L^{0}(T^{*}\\!M)$ and $L^{0}(TM)$ be the $L^{0}$-modules as in Definition
4.3 associated to $L^{2}(T^{*}\\!M)$ and $L^{2}(TM)$, i.e.,

$\displaystyle L^{0}(T^{*}\\!M):=L^{2}(T^{*}\\!M)^{0},\quad
L^{0}(TM):=L^{2}(TM)^{0}.$

The characterization of Cauchy sequences in these spaces grants that the
point-wise norms $|\cdot|_{\mathfrak{m}}:L^{2}(T^{*}\\!M)\to
L^{2}(M;\mathfrak{m})$ and $|\cdot|_{\mathfrak{m}}:L^{2}(TM)\to
L^{2}(M;\mathfrak{m})$ as well as the (Riesz) musical isomorphisms
$\flat:L^{2}(TM)\to L^{2}(T^{*}\\!M)$ and $\sharp:L^{2}(T^{*}\\!M)\to
L^{2}(TM)$ uniquely extend to (non-relabeled) continuous map
$\flat:L^{0}(TM)\to L^{0}(T^{*}\\!M)$ and $\sharp:L^{0}(T^{*}\\!M)\to
L^{0}(TM)$.

For $p\in[1,+\infty]$, let $L^{p}(T^{*}\\!M)$ and $L^{p}(TM)$ be the Banach
spaces consisting of all $\omega\in L^{0}(T^{*}\\!M)$ and $X\in L^{0}(TM)$
such that $|\omega|\in L^{p}(M;\mathfrak{m})$ and $|X|\in
L^{p}(M;\mathfrak{m})$, respectively, endowed with the norms

$\displaystyle\|\omega\|_{L^{p}(T^{*}\\!M)}:=\||\omega|_{\mathfrak{m}}\|_{L^{p}(M;\mathfrak{m})},\qquad\|X\|_{L^{p}(TM)}:=\||X|_{\mathfrak{m}}\|_{L^{p}(M;\mathfrak{m})}.$

Since by [14, Lemma 2.7], $L^{2}(T^{*}\\!M)$ is separable, and so is
$L^{2}(TM)$ by [14, Proposition 1.24]. Then one easily derives that if
$p\neq+\infty$, the spaces $L^{p}(T^{*}\\!M)$ and $L^{p}(TM)$ are separable as
well. Since $L^{2}(T^{*}\\!M)$ and $L^{2}(TM)$ are reflexive as Hilbert
spaces, by the discussion in Definition 4.2 it follows that $L^{p}(T^{*}\\!M)$
and $L^{p}(TM)$ are reflexive for every $p\in]1,+\infty[$. For
$q\in[1,+\infty[$ such that $1/p+1/q=1$ in the sense of $L^{\infty}$-modules
we have the duality

$\displaystyle L^{p}(T^{*}\\!M)^{*}=L^{q}(TM).$

The point-wise norms $|\cdot|_{\mathfrak{m}}:L^{p}(T^{*}\\!M)\to
L^{p}(M;\mathfrak{m})$ and $|\cdot|_{\mathfrak{m}}:L^{p}(TM)\to
L^{p}(M;\mathfrak{m})$ as well as the (Riesz) musical isomorphisms
$\flat:L^{p}(TM)\to L^{p}(T^{*}\\!M)$ and $\sharp:L^{p}(T^{*}\\!M)\to
L^{p}(TM)$ are defined by

$\displaystyle{}_{L^{p}(TM)}\langle\omega^{\sharp},X\rangle_{L^{q}(TM)}:=\omega(X)=:{}_{L^{q}(T^{*}\\!M)}\langle
X^{\flat},\omega\rangle_{L^{p}(T^{*}\\!M)}\quad\mathfrak{m}\text{-a.e.}$
(4.12)

for $X\in L^{q}(TM)$ and $\omega\in L^{p}(T^{*}\\!M)$, and we write (4.12) by

$\displaystyle\langle\omega^{\sharp},X\rangle:=\omega(X)=:\langle
X^{\flat},\omega\rangle\quad\mathfrak{m}\text{-a.e.}$ (4.13)

for simplicity.

###  4.8 Test and regular objects

######  Definition 4.15 (${\rm Test}(TM)$ and ${\rm Reg}(TM)$).

We define the subclass of $L^{2}(TM)$ consisting of _test vector fields_ or
_regular vector fields_ , respectively:

$\displaystyle{\rm Test}(TM)$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}g_{i}\nabla
f_{i}\,\right|\,n\in\mathbb{N},f_{i},g_{i}\in{\rm Test}(M)\right\\},$
$\displaystyle{\rm Reg}(TM)$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}g_{i}\nabla
f_{i}\,\right|\,n\in\mathbb{N},f_{i}\in{\rm Test}(M),g_{i}\in{\rm
Test}(M)\cup\mathbb{R}{\bf 1}_{M}\right\\}.$

######  Definition 4.16 (${\rm Test}(T^{*}\\!M)$ and ${\rm Reg}(T^{*}\\!M)$).

We define the subclass of $L^{2}(T^{*}\\!M)$ consisting of _test $1$-forms_ or
_regular $1$-forms_, respectively:

$\displaystyle{\rm Test}(T^{*}\\!M)$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}g_{i}\mathrm{d}f_{i}\,\right|\,n\in\mathbb{N},f_{i},g_{i}\in{\rm
Test}(M)\right\\},$ $\displaystyle{\rm Reg}(T^{*}\\!M)$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}g_{i}\mathrm{d}f_{i}\,\right|\,n\in\mathbb{N},f_{i}\in{\rm
Test}(M),g_{i}\in{\rm Test}(M)\cup\mathbb{R}{\bf 1}_{M}\right\\}.$

${\rm Test}(TM)$ (resp. ${\rm Test}(T^{*}\\!M)$) is a dense subspace of
$L^{p}(TM)$ (resp. $L^{p}(T^{*}\\!M)$) for $p\in[1,+\infty[$, hence ${\rm
Reg}(TM)\cap L^{p}(TM)$ (resp. ${\rm Reg}(T^{*}\\!M)\cap L^{p}(T^{*}\\!M)$) is
dense in $L^{p}(TM)$ (resp. $L^{p}(T^{*}\\!M)$) (see Section 9). From this, we
have that $L^{2}(TM)\cap L^{p}(TM)$ (resp. $L^{2}(T^{*}\\!M)\cap
L^{p}(T^{*}\\!M)$) is dense in $L^{p}(TM)$ (resp. $L^{p}(T^{*}\\!M)$) for
$p\in[1,+\infty[$.

###  4.9 Lebesgue spaces on tensor products

Denote the two-fold tensor products of $L^{2}(T^{*}\\!M)$ and $L^{2}(TM)$,
respectively, in the sense of Definition 4.5 by

$\displaystyle L^{2}((T^{*})^{\otimes 2}X):=L^{2}(T^{*}\\!M)\otimes
L^{2}(T^{*}\\!M),\quad L^{2}((T)^{\otimes 2}X):=L^{2}(TM)\otimes L^{2}(TM).$

By the discussion from Subsection 4.2,Theorem 4.10 and [14, Proposition 1.24],
both are separable modules. They are point-wise isometrically module
isomorphic: the respective pairing is initially defined by

$\displaystyle(\omega_{1}\otimes\omega_{2})(X_{1}\otimes
X_{2}):=\omega_{1}(X_{1})\omega_{2}(X_{2})\quad\mathfrak{m}\text{-a.e.}$

for $\omega_{1},\omega_{2}\in L^{2}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$ and
$X_{1},X_{2}\in L^{2}(TM)\cap L^{\infty}(TM)$, and is extended by linearity
and continuity to $L^{2}((T^{*})^{\otimes 2}X)$ and $L^{2}((T)^{\otimes 2}X)$,
respectively. By a slight abuse of notation, this pairing, with [14,
Proposition 1.24], induces the (Riesz) musical isomorphisms
$\flat:L^{2}((T)^{\otimes 2}X)\to L^{2}((T^{*})^{\otimes 2}X)$ and
$\sharp:=\flat^{-1}$ given by

$\displaystyle\langle A^{\sharp}\,|\,T\rangle_{\mathfrak{m}}:=A(T)=:\langle
A\,|\,T^{\flat}\rangle_{\mathfrak{m}}\quad\mathfrak{m}\text{-a.e.}$ (4.14)

and write $|A|_{\rm HS}:=\sqrt{\langle A\,|\,A\rangle_{\mathfrak{m}}}$ and
$|T|_{\rm HS}:=\sqrt{\langle T\,|\,T\rangle_{\mathfrak{m}}}$ for $A\in
L^{2}((T^{*})^{\otimes 2}X)$ and $T\in L^{2}((T)^{\otimes 2}X)$.

We let $L^{p}((T^{*})^{\otimes 2}X)$ and $L^{p}(T^{\otimes 2}X)$,
$p\in\\{0\\}\cup[1,+\infty]$, can be defined similarly as in Subsection 4.7.
For $p\in[1,+\infty]$, these spaces naturally become Banach space which, if
$p<\infty$, are separable. The following coincidences also hold

$\displaystyle L^{p}((T^{*})^{\otimes 2}X)=L^{p}(T^{*}\\!M)\otimes
L^{p}(T^{*}\\!M),\quad L^{q}((T)^{\otimes 2}X):=L^{q}(TM)\otimes L^{q}(TM)$

and the (Riesz) musical isomorphisms $\flat:L^{q}((T)^{\otimes 2}X)\to
L^{p}((T^{*})^{\otimes 2}X)$ and $\sharp:=\flat^{-1}$ can be defined by (4.14)
for $A\in L^{p}((T^{*})^{\otimes 2}X)$ and $T\in L^{q}((T)^{\otimes 2}X)$.

We define the $L^{p}$-dense sets, $p\in[1,+\infty]$, intended strongly if
$p<\infty$ and weakly* if $p=\infty$, reminiscent of Subsection 4.2.

$\displaystyle{\rm Test}((T^{*})^{\otimes 2}X):={\rm Test}(T^{*}\\!M)^{\odot
2},$ $\displaystyle{\rm Test}(T^{\otimes 2}X):={\rm Test}(TM)^{\odot 2},$
$\displaystyle{\rm Reg}((T^{*})^{\otimes 2}X):={\rm Reg}(T^{*}\\!M)^{\odot
2},$ $\displaystyle{\rm Reg}(T^{\otimes 2}X):={\rm Reg}(TM)^{\odot 2}.$

But the denseness in $L^{p}((T^{*})^{\otimes 2}X)$ (resp. $L^{p}(TM)^{\otimes
2}$) for ${\rm Reg}((T^{*})^{\otimes 2}X)$ (resp. ${\rm Reg}(T^{\otimes 2}X)$)
should be understood as for ${\rm Reg}((T^{*})^{\otimes 2}X)\cap
L^{p}((T^{*})^{\otimes 2}X)$ (resp. ${\rm Reg}(T^{\otimes 2}X)\cap
L^{p}(T^{\otimes 2}X)$).

###  4.10 Lebesgue spaces on exterior products

Given any $k\in\mathbb{N}\cup\\{0\\}$, we set

$\displaystyle L^{2}(\Lambda^{k}T^{*}\\!M):$
$\displaystyle=\Lambda^{k}L^{2}(T^{*}\\!M),$ $\displaystyle
L^{2}(\Lambda^{k}TM):$ $\displaystyle=\Lambda^{k}L^{2}(TM),$

where the exterior products are defined in Subsection 4.3. For
$k\in\\{0,1\\}$, we see

$\displaystyle L^{2}(\Lambda^{1}T^{*}\\!M)$ $\displaystyle=L^{2}(T^{*}\\!M),$
$\displaystyle L^{2}(\Lambda^{1}TM)$ $\displaystyle=L^{2}(TM),$ $\displaystyle
L^{2}(\Lambda^{0}T^{*}\\!M)$
$\displaystyle=L^{2}(\Lambda^{0}TM)=L^{2}(M;\mathfrak{m}).$

By Subsection 4.3, these are naturally Hilbert modules. As in Subsection 4.9,
$L^{2}(\Lambda^{k}T^{*}\\!M)$ and $L^{2}(\Lambda^{k}TM)$ are pointwise
isometrically module isomorphic. For brevity, the induced pointwise pairing
between $\omega\in L^{2}(\Lambda^{k}T^{*}\\!M)$ and $X_{1}\land\cdots\land
X_{k}\in L^{2}(\Lambda^{k}TM)$ with $X_{1},\cdots,X_{k}\in L^{2}(TM)\cap
L^{\infty}(TM)$, is written by

$\displaystyle\omega(X_{1},\cdots,X_{k}):=\omega(X_{1}\land\cdots\land
X_{k}).$

We let $L^{p}(\Lambda^{k}T^{*}\\!M)$ and $L^{p}(\Lambda^{k}TM)$,
$p\in\\{0\\}\cup[1,+\infty]$, be as in Subsection 4.7. For $p\in[1,+\infty]$,
these spaces are Banach and, if $p<\infty$, additionally separable. We define
the formal $k$-th exterior products, $k\in\mathbb{N}\cup\\{0\\}$, of the
classes from Subsection 4.8 as follows:

$\displaystyle{\rm Test}(\Lambda^{k}T^{*}\\!M):$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}f_{i}^{0}\mathrm{d}f_{i}^{1}\land\cdots\land\mathrm{d}f_{i}^{k}\;\right|\;n\in\mathbb{N},f_{i}^{j}\in{\rm
Test}(M)\text{ for }0\leq j\leq k\right\\},$ $\displaystyle{\rm
Test}(\Lambda^{k}TM):$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}f_{i}^{0}\nabla
f_{i}^{1}\land\cdots\land\nabla
f_{i}^{k}\;\right|\;n\in\mathbb{N},f_{i}^{j}\in{\rm Test}(M)\text{ for }0\leq
j\leq k\right\\},$ $\displaystyle{\rm Reg}(\Lambda^{k}T^{*}\\!M):$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}f_{i}^{0}\mathrm{d}f_{i}^{1}\land\cdots\land\mathrm{d}f_{i}^{k}\;\right|\;n\in\mathbb{N},f_{i}^{j}\in{\rm
Test}(M)\text{ for }1\leq j\leq k,\right.$ $\displaystyle\hskip
256.0748ptf_{i}^{0}\in{\rm Test}(M)\cup\mathbb{R}{\bf 1}_{M}\Biggr{\\}},$
$\displaystyle{\rm Reg}(\Lambda^{k}TM):$
$\displaystyle=\left\\{\left.\sum_{i=1}^{n}f_{i}^{0}\mathrm{d}f_{i}^{1}\land\cdots\land\mathrm{d}f_{i}^{k}\;\right|\;n\in\mathbb{N},f_{i}^{j}\in{\rm
Test}(M)\text{ for }1\leq j\leq k,\right.$ $\displaystyle\hskip
256.0748ptf_{i}^{0}\in{\rm Test}(M)\cup\mathbb{R}{\bf 1}_{M}\Biggr{\\}}.$

We employ the evident interpretations for $k=1$, while the respective spaces
for $k=0$ are identified with those spaces to which their generic elements’s
zeroth order terms belong to. These classes are dense in their respective
$L^{p}$-spaces, $p\in[1,+\infty]$, intended strongly if $p<\infty$ and weakly*
if $p=\infty$. But the denseness in $L^{p}(\Lambda^{k}T^{*}\\!M)$ (resp.
$L^{p}(\Lambda^{k}TM)$) for ${\rm Reg}(\Lambda^{k}T^{*}\\!M)$ (resp. ${\rm
Reg}(\Lambda^{k}TM)$) should be modified as noted before.

##  5 Covariant derivative

###  5.1 The Sobolev spaces $W^{1,2}(TM)$ and $H^{1,2}(TM)$

We construct the tangent module $L^{2}(TM)$ in Subsection 4.5. Now we define
the $(1,2)$-Sobolev space $W^{1,2}(TM)$ based on the notion of
$L^{2}$-Hessian.

######  Definition 5.1 ($(1,2)$-Sobolev space $W^{1,2}(TM)$).

The space $W^{1,2}(TM)$ is defined to consist of all $X\in L^{2}(TM)$ for
which there exists a $T\in L^{2}(T^{\otimes 2}X)$ such that for every
$g_{1},g_{2},h\in{\rm Test}(M)$,

$\displaystyle\int_{M}h\langle T\,|\,$ $\displaystyle\nabla g_{1}\otimes\nabla
g_{2}\rangle\mathrm{d}\mathfrak{m}$ $\displaystyle=-\int_{M}\langle X,\nabla
g_{2}\rangle{\rm div}(h\nabla g_{1})\mathrm{d}\mathfrak{m}-\int_{M}\,h\,{\rm
Hess}\,g_{2}(X,\nabla g_{1})\mathrm{d}\mathfrak{m}.$

Here ${\rm Hess}\,g_{2}\in L^{2}((T^{*})^{\otimes 2}X)$ is the _Hessian_
defined for $g_{2}\in{\rm Test}(M)$ (see [14, Definition 5.2]). In case of
existence, the element $T$ is unique, denoted by $\nabla X$ and called the
_covariant derivative_ of $M$.

Arguing as in [14, after Definition 5.2], the uniqueness statement in
Definition 5.1 is derived. In particular, $W^{1,2}(TM)$ constitutes a vector
space and the covariant derivative $\nabla$ is a linear operator on it. The
space $W^{1,2}(TM)$ is endowed with the norm $\|\cdot\|_{W^{1,2}(M)}$ given by

$\displaystyle\|X\|_{W^{1,2}(TM)}^{2}:=\|X\|_{L^{2}(TM)}^{2}+\|\nabla
X\|_{L^{2}(T^{\otimes 2}X)}^{2}.$

We also define the _covariant functional_ $\mathscr{E}_{\rm
cov}:L^{2}(TM)\to[0,+\infty[$ by

$\displaystyle\mathscr{E}_{\rm
cov}(X):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}|\nabla X|_{\rm
HS}^{2}\mathrm{d}\mathfrak{m}}&\text{ if }X\in W^{1,2}(TM),\\\ \infty&\text{
otherwise. }\end{array}\right.$ (5.3)

It is proved in [14, Theorem 6.3] that $(W^{1,2}(TM),\|\cdot\|_{W^{1,2}(TM)})$
is a separable Hilbert space, $\nabla$ is a closed operator, ${\rm
Reg}(TM)\subset W^{1,2}(TM)$, $\|\cdot\|_{L^{2}(TM)}$-denseness of
$W^{1,2}(TM)$ in $L^{2}(TM)$, and the lower semi continuity of
$\mathscr{E}_{\rm cov}:L^{2}(TM)\to[0,+\infty[$.

######  Definition 5.2 ($(1,2)$-Sobolev space $H^{1,2}(TM)$).

We define the space $H^{1,2}(TM)\subset W^{1,2}(TM)$ as the
$\|\cdot\|_{W^{12,}(TM)}$-closure of ${\rm Reg}(TM)$:

$\displaystyle H^{1,2}(TM):=\overline{{\rm
Reg}(TM)}^{\|\cdot\|_{W^{1,2}(TM)}}.$

$H^{1,2}(TM)$ is in general a strict subset of $W^{1,2}(TM)$.

The following lemma is a version of what is known as _Kato’s inequality_ (for
the Bochner Laplacian) in the smooth case (see [39, Chapter 2], [36, Lemma
3.5]).

######  Lemma 5.3 (Kato’s inequality, [14, Lemma 6.12]).

For every $X\in H^{1,2}(TM)$, $|X|\in D(\mathscr{E})$ and

$\displaystyle|\nabla|X||\leq|\nabla X|_{\rm HS}\quad\mathfrak{m}\text{-a.e.}$

In particular, if $X\in H^{1,2}(TM)\cap L^{\infty}(M;\mathfrak{m})$, then
$|X|^{2}\in D(\mathscr{E})$.

###  5.2 Bochner Laplacian

We have a preliminary choice to make, i.e. either to define the Bochner
Laplacian $\square^{B}$ on $W^{1,2}(TM)$ or on the _strictly small space_
$H^{1,2}(TM)$. We choose the latter one since the calculation rules is more
powerful.

######  Definition 5.4 (Bochner Laplacian).

We define $D(\square^{B})$ to consist of all $X\in H^{1,2}(TM)$ for which
there exists $Z\in L^{2}(TM)$ such that for every $Y\in H^{1,2}(TM)$,

$\displaystyle\int_{M}\langle
Y,Z\rangle\mathrm{d}\mathfrak{m}=-\int_{M}\langle\nabla Y\,|\,\nabla
X\rangle\mathrm{d}\mathfrak{m}.$

In case of existence, $Z$ is uniquely determined, denoted by $\square^{B}X$
and called the _Bochner Laplacian_ (or _connection Laplacian_ or _horizontal
Laplacian_) of $M$.

observe that $D(\square^{B})$ is a vector space, and that
$\square^{B}:D(\square^{B})\to L^{2}(TM)$ is a linear operator. Both are easy
to see from the linearity of the covariant derivative.

We modify the functional from (5.3) with the domain $W^{1,2}(TM)$ by
introducing the “augmented” covariant energy functional
$\widetilde{\mathscr{E}}_{\rm cov}:L^{2}(TM)\to[0,+\infty]$ with

$\displaystyle\widetilde{\mathscr{E}}_{\rm
cov}(X):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}|\nabla X|_{\rm
HS}^{2}\mathrm{d}\mathfrak{m}}&\text{ if }X\in H^{1,2}(TM),\\\ \infty&\text{
otherwise. }\end{array}\right.$ (5.6)

Clearly, its (non-relabeled) polarization $\widetilde{\mathscr{E}}_{\rm
cov}:H^{1,2}(TM)^{2}\to\mathbb{R}$ is a closed, symmetric form, and
$\square^{B}$ is the non-positive, self-adjoint generator uniquely associated
to it according to [30, Theorem 3.3]. We can write $\mathscr{E}^{B}$ instead
of $\widetilde{\mathscr{E}}_{\rm cov}$. Let $(P_{t}^{\rm B})_{t\geq 0}$ be the
heat semigroup of bounded linear and self-adjoint operator on $L^{2}(TM)$
formally written by

$\displaystyle\text{``$P_{t}^{\rm B}:=e^{t\,\square^{\rm B}}$''}.$

For $\alpha>0$ and $X\in L^{2}(TM)$, we set $R_{\alpha}^{\rm
B}X:=\int_{0}^{\infty}e^{-\alpha t}P_{t}^{\rm B}X\mathrm{d}t$.

######  Lemma 5.5 ([14, Corollary 6.21, Theorem 6.26]).

We have the following:

  1. (1)

$0\leq\inf\sigma(-\Delta)\leq\inf\sigma(-\square^{\rm B})$.

  2. (2)

For every $X\in L^{2}(TM)$ and every $t\geq 0$,

$\displaystyle|P_{t}^{\rm B}X|\leq P_{t}|X|\quad\mathfrak{m}\text{-a.e.}$
(5.7)

######  Corollary 5.6 ([14, Corollary 6.29]).

Suppose $p\in[1,+\infty[$. Then the heat flow $(P_{t}^{\rm B})_{t\geq 0}$ can
be extended to a contractive semigroup on $L^{p}(TM)$, which is strongly
continuous on $L^{p}(TM)$ under $p\in[1,+\infty[$ and weakly* continuous on
$L^{\infty}(TM)$.

##  6 Exterior derivative

Throughout this section, we fix $k\in\mathbb{N}\cup\\{0\\}$.

###  6.1 The Sobolev space $D(\mathrm{d}^{k})$ and $D(\mathrm{d}_{*}^{k})$

Given $\omega\in L^{0}(\Lambda^{k}T^{*}\\!M)$ and $X_{0},\cdots,X_{k},Y\in
L^{0}(TM)$, we shall use the standard abbreviations: for $1\leq i<j\leq k$

$\displaystyle\omega(\widehat{X}_{i}):$
$\displaystyle=\omega(X_{0},\cdots,\widehat{X}_{i},\cdots,X_{k}),$
$\displaystyle:$ $\displaystyle=\omega(X_{0}\land\cdots\land X_{i-1}\land
X_{i+1}\land\cdots\land X_{k}),$
$\displaystyle\omega(Y,\widehat{X}_{i},\widehat{Y}_{j}):$
$\displaystyle=\omega(Y,X_{0},\cdots,\widehat{X}_{i},\cdots,\widehat{X}_{j},\cdots,X_{k}),$
$\displaystyle:$ $\displaystyle=\omega(Y\land X_{0}\land\cdots\land
X_{i-1}\land X_{i+1}\land\cdots\land Y_{j-1}\land Y_{j+1}\land\cdots\land
X_{k}).$

######  Definition 6.1 (Sobolev space $D(\mathrm{d}^{k})$).

We define $D(\mathrm{d}^{k})$ to consist of all $\omega\in
L^{2}(\Lambda^{k}T^{*}\\!M)$ for which there exists $\eta\in
L^{2}(\Lambda^{k+1}T^{*}\\!M)$ such that for every $X_{0},\cdots,X_{k}\in{\rm
Test}(M)$,

$\displaystyle\int_{M}\eta(X_{0},\cdots,X_{k})\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}\sum_{i=0}^{k}(-1)^{i+1}\omega(\widehat{X}_{i}){\rm
div}\,X_{i}\mathrm{d}\mathfrak{m}$ $\displaystyle\hskip
56.9055pt+\int_{M}\sum_{i=0}^{k}\sum_{j=i+1}^{k}(-1)^{i+j}\omega([X_{i},X_{j}],\widehat{X}_{i},\widehat{X}_{j})\mathrm{d}\mathfrak{m}.$

In case of existence, the element $\eta$ is unique, denoted by
$\mathrm{d}\omega$ and called the _exterior derivative_ (or _exterior
differential_) of $\omega$.

The uniqueness follows by density of ${\rm Test}(\Lambda^{k+1}T^{*}\\!M)$ in
$L^{2}(\Lambda^{k+1}T^{*}\\!M)$ as discussed in Section 4.7. It is then clear
that $D(\mathrm{d}^{k})$ is a real vector space and that $\mathrm{d}$ is a
linear operator on it.

We always endow $D(\mathrm{d}^{k})$ with the norm
$\|\cdot\|_{D(\mathrm{d}^{k})}$ given by

$\displaystyle\|\omega\|_{D(\mathrm{d}^{k})}^{2}:=\|\omega\|_{L^{2}(\Lambda^{k}T^{*}\\!M)}^{2}+\|\mathrm{d}\omega\|_{L^{2}(\Lambda^{k}T^{*}\\!M)}^{2}.$

We introduce the functional
$\mathscr{E}_{\mathrm{d}}:L^{2}(\Lambda^{k}T^{*}\\!M)\to[0,+\infty]$ with

$\displaystyle\mathscr{E}_{\mathrm{d}}(\omega):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}|\mathrm{d}\omega|^{2}\mathrm{d}\mathfrak{m}}&\text{
if }\omega\in D(\mathrm{d}^{k}),\\\ \infty&\text{
otherwise.}\end{array}\right.$

We do not make explicit the dependency of $\mathscr{E}_{\mathrm{d}}$ on the
degree $k$. It will always be clear from the context which one is intended. It
is proved in [14, Theorem 7.5] that
$(D(\mathrm{d}^{k}),\|\cdot\|_{D(\mathrm{d}^{k})})$ is a separable Hilbert
space, the exterior differential $\mathrm{d}$ is a closed operator, ${\rm
Reg}(\Lambda^{k}T^{*}\\!M)\subset D(\mathrm{d}^{k})$, $D(\mathrm{d}^{k})$ is
dense in $L^{2}(\Lambda^{k}T^{*}\\!M)$, and the functional
$\mathscr{E}_{\mathrm{d}}:L^{2}(\Lambda^{k}T^{*}\\!M)\to[0,+\infty]$ is lower
semi continuous.

######  Definition 6.2 (The space $D_{\rm reg}(\mathrm{d}^{k})$).

We define the space $D_{\rm reg}(\mathrm{d}^{k})\subset D(\mathrm{d}^{k})$ by
the closure of ${\rm Reg}(\Lambda^{k}T^{*}\\!M)$ with respect to the norm
$\|\cdot\|_{D(\mathrm{d}^{k})}$:

$\displaystyle D_{\rm reg}(\mathrm{d}^{k}):=\overline{{\rm
Reg}(\Lambda^{k}T^{*}\\!M)}^{\|\cdot\|_{D(\mathrm{d}^{k})}}.$

It is proved in [14, Theorem 7.5] that for every $\omega\in D_{\rm
reg}(\mathrm{d}^{k})$, we have $\mathrm{d}\omega\in D_{\rm
reg}(\mathrm{d}^{k+1})$ with $\mathrm{d}(\mathrm{d}\omega)=0$.

######  Definition 6.3 (The space $D(\mathrm{d}_{*}^{k})$).

Given any $k\in\mathbb{N}$, the space $D(\mathrm{d}_{*}^{k})$ is defined to
consist of all $\omega\in L^{2}(\Lambda^{k}T^{*}\\!M)$ for which there exists
$\rho\in L^{2}(\Lambda^{k-1}T^{*}\\!M)$ such that for every $\eta\in{\rm
Test}(\Lambda^{k-1}T^{*}\\!M)$, we have

$\displaystyle\int_{M}\langle\rho,\eta\rangle\,\mathrm{d}\mathfrak{m}=\int_{M}\langle\omega,\mathrm{d}\eta\rangle\,\mathrm{d}\mathfrak{m}.$

If it exists, $\rho$ is unique, denoted by $\mathrm{d}_{*}\omega$ and called
the _codifferential_ of $\omega$. We simply define
$D(\mathrm{d}_{*}^{0}):=L^{0}(M;\mathfrak{m})$ and $\mathrm{d}_{*}:=0$ on this
space.

By the density of ${\rm Test}(\Lambda^{k-1}T^{*}\\!M)$ in
$L^{2}(\Lambda^{k-1}T^{*}\\!M)$, the uniqueness statement is indeed true.
Furthermore, $\mathrm{d}_{*}$ is a closed operator, i.e., the image of the
assignment ${\rm Id}\times\mathrm{d}_{*}:D(\mathrm{d}_{*}^{k})\to
L^{2}(\Lambda^{k}T^{*}\\!M)\times L^{2}(\Lambda^{k-1}T^{*}\\!M)$ is closed in
$L^{2}(\Lambda^{k}T^{*}\\!M)\times L^{2}(\Lambda^{k-1}T^{*}\\!M)$.

By way of [14, Theorem 7.5 and Lemma 7.15], we have the following: For $f\in
D(\mathscr{E})_{e}\cap L^{\infty}(M;\mathfrak{m})$ and $\omega\in
H^{1,2}(T^{*}\\!M)$ satisfying $\mathrm{d}f\in L^{\infty}(M;\mathfrak{m})$ or
$\omega\in L^{\infty}(T^{*}\\!M)$, $f\omega\in H^{1,2}(T^{*}\\!M)$ with

$\displaystyle\mathrm{d}(f\omega)$
$\displaystyle=f\mathrm{d}\omega+\mathrm{d}f\land\omega,$ (6.1)
$\displaystyle\mathrm{d}_{*}(f\omega)$
$\displaystyle=f\mathrm{d}_{*}\omega-\langle\mathrm{d}f,\omega\rangle.$ (6.2)

###  6.2 The Sobolev spaces $W^{1,p}(\Lambda^{k}T^{*}\\!M)$ and
$H^{1,p}(\Lambda^{k+1}T^{*}\\!M)$

######  Definition 6.4 (The space $W^{1,2}(\Lambda^{k}T^{*}\\!M)$).

We define the space $W^{1,2}(\Lambda^{k}T^{*}\\!M)$ by

$\displaystyle W^{1,2}(\Lambda^{k}T^{*}\\!M):=D(\mathrm{d}^{k})\cap
D(\mathrm{d}_{*}^{k}).$

By [14, Theorem 7.5 and Lemma 7.15], we already know that
$W^{1,2}(\Lambda^{k}T^{*}\\!M)$ is a dense subspace of
$L^{2}(\Lambda^{k}T^{*}\\!M)$.

We endow $W^{1,2}(\Lambda^{k}T^{*}\\!M)$ with the norm
$\|\cdot\|_{W^{1,2}(\Lambda^{k}T^{*}\\!M)}$ given by

$\displaystyle\|\omega\|_{W^{1,2}(\Lambda^{k}T^{*}\\!M)}^{2}:=\|\omega\|_{L^{2}(\Lambda^{k}T^{*}\\!M)}^{2}+\|\mathrm{d}\omega\|_{L^{2}(\Lambda^{k+1}T^{*}\\!M)}^{2}+\|\mathrm{d}_{*}\omega\|_{L^{2}(\Lambda^{k-1}T^{*}\\!M)}^{2}$

and we define the _contravariant_ functional $\mathscr{E}_{\rm
con}:L^{2}(\Lambda^{k}T^{*}\\!M)\to[0,+\infty]$ by

$\displaystyle\mathscr{E}_{\rm
con}(\omega):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}\left[|\mathrm{d}\omega|^{2}+|\mathrm{d}_{*}\omega|^{2}\right]\mathrm{d}\mathfrak{m}}&\text{
if }\omega\in W^{1,2}(\Lambda^{k}T^{*}\\!M)\\\
\infty&\text{otherwise.}\end{array}\right.$

Arguing as for [14, Theorems 5.3, 6.3 and 7.5],
$W^{1,2}(\Lambda^{k}T^{*}\\!M)$ becomes a separable Hilbert space with respect
to $\|\cdot\|_{W^{1,2}(\Lambda^{k}T^{*}\\!M)}$. moreover, the functional
$\mathscr{E}_{\rm con}:L^{2}(\Lambda^{k}T^{*}\\!M)\to[0,+\infty]$ is clearly
lower semi continuous.

Again by [14, Theorem 7.5 and Lemma 7.15], we have ${\rm
Reg}(\Lambda^{k}T^{*}\\!M)\subset W^{1,2}(\Lambda^{k}T^{*}\\!M)$, so that the
following definition makes sense.

######  Definition 6.5 (The space $H^{1,2}(\Lambda^{k}T^{*}\\!M)$).

The space $H^{1,2}(\Lambda^{k}T^{*}\\!M)\subset W^{1,2}(\Lambda^{k}T^{*}\\!M)$
is defined by the closure of ${\rm Reg}(\Lambda^{k}T^{*}\\!M)$ with respect to
$\|\cdot\|_{W^{1,2}(\Lambda^{k}T^{*}\\!M)}$:

$\displaystyle H^{1,2}(\Lambda^{k}T^{*}\\!M):=\overline{{\rm
Reg}(\Lambda^{k}T^{*}\\!M)}^{\|\cdot\|_{W^{1,2}(\Lambda^{k}T^{*}\\!M)}}.$

###  6.3 Hodge-Kodaira Laplacian

We now define the Hosdge-Kodaira Laplacian in L2-sense:

######  Definition 6.6 ($L^{2}$-Hodge-Kodaira Laplacian
${\Delta\Delta}^{\rm\\!HK}_{k}$).

The space $D({\Delta\Delta}^{\rm\\!HK}_{k})$ is defined to consist of all
$\omega\in H^{1,2}(\Lambda^{k}T^{*}\\!M)$ for which there exists $\alpha\in
L^{2}(\Lambda^{k}T^{*}\\!M)$ such that for every $\eta\in
H^{1,2}(\Lambda^{k}T^{*}\\!M)$,

$\displaystyle\int_{M}\langle\alpha,\eta\rangle\mathrm{d}\mathfrak{m}=-\int_{M}\left[\langle\mathrm{d}\omega,\mathrm{d}\eta\rangle+\langle\mathrm{d}_{*}\omega,\mathrm{d}_{*}\eta\rangle\right]\mathrm{d}\mathfrak{m}.$

In case of existence, the element $\alpha$ is unique, denoted by
${\Delta\Delta}^{\rm\\!HK}_{k}\omega$ and called the _Hodge Laplacian_ ,
_Hodge-Kodaira Laplacian_ or _Hodge-de Rham Laplacian_ of $\omega$. Formally
${\Delta\Delta}^{\rm\\!HK}_{k}\omega$ can be written
“${\Delta\Delta}^{\rm\\!HK}_{k}\omega=-(\mathrm{d}\mathrm{d}_{*}+\mathrm{d}_{*}\mathrm{d})\omega$”.

For the most important case $k=1$, we write ${\Delta\Delta}^{\rm\\!HK}$
instead of ${\Delta\Delta}^{\rm\\!HK}_{1}$. We see
${\Delta\Delta}^{\rm\\!HK}_{0}=\Delta$ the usual $L^{2}$-generator associated
to the given quasi-regular strongly local Dirichlet form
$(\mathscr{E},D(\mathscr{E}))$. Moreover, the Hodge-Kodaira Laplacian
${\Delta\Delta}^{\rm\\!HK}_{k}$ is a closed operator.

###  6.4 Heat flow of $1$-forms associated with ${\Delta\Delta}^{\rm\\!HK}$

We define the heat flow $P_{t}^{\rm HK}$ on $1$-forms associated to the
functional $\widetilde{\mathscr{E}}_{\rm con}:L^{2}(T^{*}\\!M)\to[0,+\infty]$
with

$\displaystyle\widetilde{\mathscr{E}}_{\rm
con}(\omega):=\left\\{\begin{array}[]{cc}\displaystyle{\int_{M}\left[|\mathrm{d}\omega|^{2}+|\mathrm{d}_{*}\omega|^{2}\right]\mathrm{d}\mathfrak{m}}&\text{
if }\omega\in H^{1,2}(T^{*}\\!M),\\\ \infty&\text{
otherwise.}\end{array}\right.$

We can write $\mathscr{E}^{\rm HK}$ instead of $\widetilde{\mathscr{E}}_{\rm
con}$. Let $(P_{t}^{\rm HK})_{t\geq 0}$ be the heat semigroup of bounded
linear and self-adjoint operator on $L^{2}(T^{*}\\!M)$ formally written by

$\displaystyle\text{``$P_{t}^{\rm HK}:=e^{t\,{\Delta\Delta}^{\rm\\!HK}}$''}.$

The following are important:

######  Lemma 6.7 ([14, Lemma 8.32, Corollaries 8.35 and 8.36]).

We have the following:

  1. (1)

For every $f\in D(\mathscr{E})$ and every $t>0$, $\mathrm{d}P_{t}f\in
D({\Delta\Delta}^{\rm\\!HK})$ and

$\displaystyle P_{t}^{\rm HK}\mathrm{d}f=\mathrm{d}P_{t}f.$ (6.3)

  2. (2)

If $\omega\in D(\mathrm{d}_{*})$ and $t>0$, then $P_{t}^{\rm HK}\omega\in
D(\mathrm{d}_{*})$ and

$\displaystyle\mathrm{d}_{*}P_{t}^{\rm HK}\omega=P_{t}\mathrm{d}_{*}\omega.$
(6.4)

  3. (3)

$\inf\sigma(-\Delta^{\kappa})\leq\inf\sigma(-{\Delta\Delta}^{\rm\\!HK})$.

The formulas (6.3) and (6.4) are called _intertwining properties_ , which play
a crucial role to prove the $L^{p}$-boundedness of Riesz operator.

##  7 Ricci curvature measures

To state the notion of Ricci curvature measure, we need preliminary
preparation.

The following are shown in [14, Lemmas 8.1 and 8.2]:

######  Lemma 7.1 ([14, Lemma 8.1]).

For every $g\in{\rm Test}(M)\cup\mathbb{R}{\bf 1}_{M}$ and every $f\in{\rm
Test}(M)$. we have $g\,\mathrm{d}f\in D({\Delta\Delta}^{\rm\\!HK})$ with

$\displaystyle{\Delta\Delta}^{\rm\\!HK}(g\,\mathrm{d}f)=g\,\mathrm{d}\Delta
f+\Delta g\,\mathrm{d}f+2{\rm Hess}\,f(\nabla g,\cdot),$

with the usual interpretations $\nabla{\bf 1}_{M}=0$ and $\Delta{\bf
1}_{M}=0$. More generally, for every $X\in{\rm Reg}(TM)$ and every $h\in{\rm
Test}(M)\cup\mathbb{R}{\bf 1}_{M}$, we have $hX^{\flat}\in
D({\Delta\Delta}^{\rm\\!HK})$ with

$\displaystyle{\Delta\Delta}^{\rm\\!HK}(hX^{\flat})=h{\Delta\Delta}^{\rm\\!HK}X^{\flat}-{\Delta\Delta}^{\rm\\!HK}hX^{\flat}+2(\nabla_{\nabla
h}X)^{\flat}.$

######  Lemma 7.2 ([14, Lemma 8.2]).

For every $X\in{\rm Reg}(TM)$, we have $|X|^{2}\in
D({\text{\boldmath$\Delta$}}^{2\kappa})$ with

$\displaystyle{\text{\boldmath$\Delta$}}^{2\kappa}\frac{|X|^{2}}{2}\geq\left[|\nabla
X|_{\rm HS}^{2}+\langle
X,({\Delta\Delta}^{\rm\\!HK}X^{\flat})^{\sharp}\rangle\right]\mathfrak{m}.$

######  Definition 7.3 (The space $H^{1,2}_{\sharp}(TM)$).

We define the space $H^{1,2}_{\sharp}(TM)$ as the image of
$H^{1,2}(T^{*}\\!M)$ under the map $\sharp$, endowed with the norm

$\displaystyle\|X\|_{H^{1,2}_{\sharp}(TM)}:=\|X^{\flat}\|_{H^{1,2}(T^{*}\\!M)}.$

The following is proved in [14, Lemma 8.8]:

######  Lemma 7.4 ([14, Lemma 8.8]).

$H^{1,2}_{\sharp}(TM)$ is a subspace of $H^{1,2}(TM)$. The aforementioned
natural inclusion is continuous. Additionally, for every $X\in
H^{1,2}_{\sharp}(TM)$,

$\displaystyle\mathscr{E}_{\rm cov}(X)\leq\mathscr{E}_{\rm
con}(X^{\flat})-\langle\kappa,\widetilde{|X|}^{2}\rangle.$ (7.1)

Then, we can state the main result of [14]:

######  Theorem 7.5 ([14, Theorem 8.9]).

There exists a unique continuous mapping ${\bf
Ric}^{\kappa}:H^{1,2}_{\sharp}(TM)^{2}\to\mathfrak{M}_{\rm
f}^{\pm}(M)_{\mathscr{E}}$ satisfying the identity

$\displaystyle{\bf
Ric}^{\kappa}(X,Y)={\text{\boldmath$\Delta$}}^{2\kappa}\frac{\langle
X,Y\rangle}{2}-\left[\frac{1}{2}\langle
X,({\Delta\Delta}^{\rm\\!HK}Y^{\flat})^{\sharp}\rangle+\frac{1}{2}\langle
Y,({\Delta\Delta}^{\rm\\!HK}X^{\flat})^{\sharp}\rangle+\langle\nabla
X\,|\,\nabla Y\rangle\right]\mathfrak{m}$ (7.2)

for every $X,Y\in{\rm Reg}(TM)$. Here $\mathfrak{M}_{\rm
f}^{\pm}(M)_{\mathscr{E}}$ denotes the family of finite signed smooth
measures. The map ${\bf Ric}^{\kappa}$ is symmetric and $\mathbb{R}$-linear.
Furthermore, for every $X,Y\in H^{1,2}_{\sharp}(TM)$, it obeys

$\displaystyle{\bf Ric}^{\kappa}(X,X)$ $\displaystyle\geq 0,$ (7.3)
$\displaystyle{\bf Ric}^{\kappa}(X,Y)(M)$
$\displaystyle=\int_{M}\left[\langle\mathrm{d}X^{\flat},\mathrm{d}Y^{\flat}\rangle+\langle\mathrm{d}_{*}X^{\flat},\mathrm{d}_{*}Y^{\flat}\rangle\right]\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip 56.9055pt-\int_{M}\langle\nabla X\,|\,\nabla
Y\rangle\mathrm{d}\mathfrak{m}-\langle\kappa,\widetilde{\langle
X,Y\rangle}\rangle,$ (7.4) $\displaystyle\|{\bf Ric}^{\kappa}(X,Y)\|_{\rm
TV}^{2}$ $\displaystyle\leq\left[\mathscr{E}_{\rm
con}(X^{\flat})-\langle\kappa,\widetilde{|X|}^{2}\rangle\right]\left[\mathscr{E}_{\rm
con}(Y^{\flat})-\langle\kappa,\widetilde{|Y|}^{2}\rangle\right].$ (7.5)

Keeping in mind Theorem 7.5, we define the map ${\bf
Ric}:H^{1,2}_{\sharp}(TM)^{2}\to\mathfrak{M}_{\rm f}^{\pm}(M)_{\mathscr{E}}$
by

$\displaystyle{\bf Ric}(X,Y):={\bf
Ric}^{\kappa}(X,Y)+\langle\kappa,\widetilde{\langle X,Y\rangle}\rangle.$ (7.6)

This map is well-defined, symmetric and $\mathbb{R}$-bilinear. ${\bf Ric}$ is
also jointly continuous, which follows from polarization, [14, Corollary 6.14]
and Lemma 7.4. Lastly, by Theorem 7.5, Lemma 7.2 and [14, Lemma 8.12], for
every $X,Y\in{\rm Reg}(TM)$, we have a similar expression with (7.2):

$\displaystyle{\bf Ric}(X,Y)={\text{\boldmath$\Delta$}}\frac{\langle
X,Y\rangle}{2}-\left[\frac{1}{2}\langle
X,({\Delta\Delta}^{\rm\\!HK}Y^{\flat})^{\sharp}\rangle+\frac{1}{2}\langle
Y,({\Delta\Delta}^{\rm\\!HK}X^{\flat})^{\sharp}\rangle+\langle\nabla
X\,|\,\nabla Y\rangle\right]\mathfrak{m}$ (7.7)

######  Lemma 7.6 ([14, Lemma 8.15]).

For every $X,Y\in H_{\sharp}^{1,2}(TM)$ and $f\in{\rm Test}(M)$, we have

$\displaystyle{\bf Ric}(fX,Y)=\tilde{f}\,{\bf Ric}(X,Y),$ (7.8)

in particular,

$\displaystyle{\bf Ric}^{\kappa}(fX,Y)=\tilde{f}\,{\bf Ric}^{\kappa}(X,Y).$
(7.9)

##  8 Proof of Theorem 1.2

First note that the following condition ${\bf(R_{3})}$ in [12, Corollary
4.2.3] is satisfied for tamed Dirichlet space.

$\displaystyle{\bf(R_{3})}:\text{There exists }\mathscr{C}\subset
D(\Delta)\text{ dense in }D(\mathscr{E})\text{ such that for all
}f\in\mathscr{C},f^{2}\in D(\Delta).$

In fact, $\mathscr{C}={\rm Test}(M)$ satisfies ${\bf(R_{3})}$. It is proved in
[12, Corollary 4.2.3] that ${\bf(R_{3})}$ implies the following ${\bf(R_{2})}$
and ${\bf(R_{2})^{\prime}}$.

$\displaystyle{\bf(R_{2})}:$ $\displaystyle\text{ For all }f\in
D(\Delta),\,f^{2}\in D(\Delta_{1}).$ $\displaystyle{\bf(R_{2})^{\prime}}:$
$\displaystyle\text{ There exists }\mathscr{C}\subset D(\Delta)\text{ dense in
}D(\mathscr{E})\text{ such that for all }f\in\mathscr{C},\,f^{2}\in
D(\Delta_{1}).$

The proof of Theorem 1.2 is split in several lemmas.

######  Lemma 8.1.

$X\in H^{1,2}(TM)$ implies $|X|\in D(\mathscr{E})$ and

$\displaystyle\mathscr{E}(|X|,|X|)\leq\widetilde{\mathscr{E}}_{\rm
cov}(X,X)<\infty.$ (8.1)

Moreover, for $f\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})_{+}$ with
$fX\in H^{1,2}(TM)$, we have $f|X|\in D(\mathscr{E})$ and

$\displaystyle\mathscr{E}(|X|,f|X|)\leq\widetilde{\mathscr{E}}_{\rm
cov}(X,fX).$ (8.2)

###### Proof.

The proof of (8.1) can be directly deduced from Lemma 5.3. Next we show (8.2).
Assume $fX\in H^{1,2}(TM)$ for $f\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})_{+}$. By (8.1), we have $f|X|\in D(\mathscr{E})$.
Moreover, by Lemma 5.5(2),

$\displaystyle((I-P_{t})|X|,f|X|)_{L^{2}(M;\mathfrak{m})}\leq((I-P_{t}^{\rm
B})X,fX)_{L^{2}(TM)}.$

Divided by $t>0$ and letting $t\to 0$, we obtain (8.2). ∎

######  Lemma 8.2.

Take $\omega\in H^{1,2}(T^{*}\\!M)(=D(\mathscr{E}^{\rm HK}))$. Then
$|\omega|\in D(\mathscr{E})$ and

$\displaystyle\mathscr{E}^{\kappa}(|\omega|,|\omega|)\leq\mathscr{E}^{\rm
HK}(\omega,\omega).$ (8.3)

###### Proof.

$\omega\in H^{1,2}(T^{*}\\!M)$ implies $\omega^{\sharp}\in
H^{1,2}_{\sharp}(TM)\subset H^{1,2}(TM)$ and $|\omega|=|\omega^{\sharp}|\in
D(\mathscr{E})$. Then

$\displaystyle\mathscr{E}(|\omega|,|\omega|)$
$\displaystyle=\mathscr{E}(|\omega^{\sharp}|,|\omega^{\sharp}|)\stackrel{{\scriptstyle\eqref{eq:ContractionBochner1}}}{{\leq}}\widetilde{\mathscr{E}}_{\rm
cov}(\omega^{\sharp},\omega^{\sharp})\stackrel{{\scriptstyle\eqref{eq:ContBochHK}}}{{\leq}}\mathscr{E}^{\rm
HK}(\omega,\omega)-\langle\kappa,|\omega|^{2}\rangle,$

which implies the conclusion. ∎

######  Lemma 8.3.

If $\omega\in H^{1,2}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$ (resp. $\omega\in
H^{1,2}(T^{*}\\!M)$) and $f\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})$
(resp. $f\in{\rm Test}(M)$), then $f\omega\in H^{1,2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$ (resp. $f\omega\in H^{1,2}(T^{*}\\!M)$).

###### Proof.

By [14, Remark 6.10], we have that $f\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})$ (resp. $f\in{\rm Test}(M)$) and $X\in
H^{1,2}(TM)\cap L^{\infty}(TM)$ (resp. $X\in H^{1,2}(TM)$) imply $fX\in
H^{1,2}(TM)\cap L^{\infty}(TM)$ (resp. $fX\in H^{1,2}(TM)$) and
$\nabla(fX)=\nabla f\otimes X+f\nabla X$. Therefore, $f\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})$ (resp. $f\in\text{\rm Test}(M)$) and $\omega\in
H^{1,2}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$ (resp. $\omega\in
H^{1,2}(T^{*}\\!M)$) imply $f\omega\in H^{1,2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$ (resp. $f\omega\in H^{1,2}(T^{*}\\!M)$), because
$(f\omega)^{\sharp}=f\omega^{\sharp}$ holds in view of that for any $\eta\in
H^{1,2}(T^{*}\\!M)$

$\displaystyle\langle(f\omega^{\sharp})^{\flat},\eta\rangle=\eta(f\omega^{\sharp})=\langle\eta^{\sharp},f\omega^{\sharp}\rangle$
$\displaystyle=f\langle\omega,\eta\rangle=\langle f\omega,\eta\rangle.$

. ∎

######  Lemma 8.4.

Take $\omega\in H^{1,2}(T^{*}\\!M)$ and $f\in{\rm Test}(M)_{+}$. Then
$f|\omega|\in D(\mathscr{E}^{\kappa})=D(\mathscr{E})$ and

$\displaystyle\mathscr{E}^{\kappa}(|\omega|,f|\omega|)\leq\mathscr{E}^{\rm
HK}(\omega,f\omega).$ (8.4)

###### Proof.

Suppose $\omega\in H^{1,2}(T^{*}\\!M)$ and $f\in{\rm Test}(M)_{+}$. By Lemma
8.3, we have $f\omega\in H^{1,2}(T^{*}\\!M)$, hence $f|\omega|\in
D(\mathscr{E})$ by Lemma 8.3. On the other hand, by (7.4), we have

$\displaystyle{\bf Ric}(\omega^{\sharp},f\omega^{\sharp})(M)=\mathscr{E}^{\rm
HK}(\omega,f\omega)-\widetilde{\mathscr{E}}_{\rm
cov}(\omega^{\sharp},f\omega^{\sharp}).$ (8.5)

By Lemma 8.1, we then have

$\displaystyle\mathscr{E}^{\kappa}(|\omega|,f|\omega|)$
$\displaystyle=\mathscr{E}(|\omega|,f|\omega|)+\langle\kappa,f|\omega|^{2}\rangle$
$\displaystyle\leq\widetilde{\mathscr{E}}_{\rm
cov}(\omega^{\sharp},f\omega^{\sharp})+\langle\kappa,f|\omega|^{2}\rangle$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:RicciIdentity}}}{{=}}\mathscr{E}^{\rm
HK}(\omega,f\omega)-{\bf Ric}^{\kappa}(\omega^{\sharp},f\omega^{\sharp})(M)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:RicciDensityKappa}}}{{=}}\mathscr{E}^{\rm
HK}(\omega,f\omega)-\int_{M}\widetilde{f}{\rm d}{\bf
Ric}^{\kappa}(\omega^{\sharp},\omega^{\sharp})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:NonKappaRicci}}}{{\leq}}\mathscr{E}^{\rm
HK}(\omega,f\omega).$

∎

######  Corollary 8.5.

Take $\omega\in D({\Delta\Delta}^{\rm\\!HK})\cap L^{\infty}(T^{*}\\!M)$ and
$f\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})_{+}$. Then $f\omega\in
H^{1,2}(T^{*}\\!M)$, $f|\omega|\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})$, and (8.4) hold.

###### Proof.

We already know $f\omega\in H^{1,2}(T^{*}\\!M)$ by Lemma 8.3. By Lemma 8.4, we
have $(P_{t}f)\omega\in H^{1,2}(T^{*}\\!M)$ and $(P_{t}f)|\omega|\in
D(\mathscr{E})$, and

$\displaystyle\mathscr{E}^{\kappa}(|\omega|,(P_{t}f)|\omega|)\leq\left(-{\Delta\Delta}^{\rm\\!HK}\omega,(P_{t}f)\omega\right)_{L^{2}(T^{*}\\!M)},$

because $P_{t}f\in{\rm Test}(M)_{+}$. Since $\\{P_{t}f\\}_{t\geq 0}$ converges
to $f$ in $D(\mathscr{E})$ as $t\to 0$ and weakly* converges to $f$ in
$L^{\infty}(M;\mathfrak{m})$ as $t\to 0$, we can deduce (8.4). ∎

######  Lemma 8.6.

Take $\omega\in D({\Delta\Delta}^{\rm\\!HK})\cap L^{\infty}(T^{*}\\!M)$. Then

$\displaystyle\left(-{\Delta\Delta}^{\rm\\!HK}\omega,g\frac{\omega}{|\omega|}\right)_{L^{2}(T^{*}\\!M)}\geq\mathscr{E}^{\kappa}(|\omega|,g)\quad\text{
for all }\quad g\in{\rm Test}(M)_{+}.$ (8.6)

Here we set $\omega/|\omega|:=0$ if $\omega=0$.

###### Proof.

By Lemma 8.2, we see $|\omega|\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})$. For each ${\varepsilon}>0$, we set
$|\omega|_{{\varepsilon}}:=\sqrt{|\omega|^{2}+{\varepsilon}^{2}}$. Then we see
$\frac{|\omega|}{|\omega|_{{\varepsilon}}}\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})$ (cf. [50, Chapter I, Theorem 4.12 and Exercise
4.16]). We may assume $|\omega|$ and $|\omega|_{{\varepsilon}}$ are
$\mathscr{E}$-quasi continuous.

For $g\in{\rm Test}(M)_{+}$, we consider $f:=g/|\omega|_{{\varepsilon}}\in
D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})_{+}$. We apply Corollary 8.5 for
$f\in D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})_{+}$ so that $f\omega\in
H^{1,2}(T^{*}\\!M)$, $f|\omega|\in D(\mathscr{E})$ and

$\displaystyle\left(-{\Delta\Delta}^{\rm\\!HK}\omega,g\frac{\omega}{|\omega|_{\varepsilon}}\right)_{L^{2}(T^{*}\\!M)}$
$\displaystyle=\left(-{\Delta\Delta}^{\rm\\!HK}\omega,f\omega\right)_{L^{2}(T^{*}\\!M)}$
$\displaystyle=\mathscr{E}^{\rm HK}(\omega,f\omega)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:KEnergyCont**}}}{{\geq}}\mathscr{E}^{\kappa}(|\omega|,f|\omega|)$
$\displaystyle=\mathscr{E}^{\kappa}\left(|\omega|,g\frac{|\omega|}{|\omega|_{{\varepsilon}}}\right)$
$\displaystyle=\int_{M}\frac{g}{|\omega|_{{\varepsilon}}^{3}}(|\omega|_{{\varepsilon}}^{2}-|\omega|^{2})\Gamma(|\omega|,|\omega|)\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
28.45274pt+\int_{M}\frac{|\omega|}{|\omega|_{{\varepsilon}}}{\Gamma(|\omega|,g)\mathrm{d}\mathfrak{m}+\left\langle\kappa,g\frac{|\omega|^{2}}{|\omega|_{{\varepsilon}}}\right\rangle}$
$\displaystyle\geq\int_{M}\frac{|\omega|}{|\omega|_{{\varepsilon}}}{\Gamma(|\omega|,g)\mathrm{d}\mathfrak{m}+\left\langle\kappa,g\frac{|\omega|^{2}}{|\omega|_{{\varepsilon}}}\right\rangle}.$

Now letting ${\varepsilon}\to 0$, we have

$\displaystyle\int_{M}\frac{|\omega|}{|\omega|_{{\varepsilon}}}\Gamma(|\omega|,g)\mathrm{d}\mathfrak{m}\to\int_{M}{\bf
1}_{\\{\omega\neq 0\\}}\Gamma(|\omega|,g)\mathrm{d}\mathfrak{m}.$

On the other hand,

$\displaystyle\Gamma(|\omega|^{1+{\varepsilon}},|\omega|^{1+{\varepsilon}})=(1+{\varepsilon})^{2}|\omega|^{2{\varepsilon}}\Gamma(|\omega|,|\omega|).$

Hence

$\displaystyle{\bf
1}_{\\{\omega=0\\}}\Gamma(|\omega|^{1+{\varepsilon}},|\omega|^{1+{\varepsilon}})=0.$

Letting ${\varepsilon}\to 0$, we see

$\displaystyle{\bf 1}_{\\{\omega=0\\}}\Gamma(|\omega|,|\omega|)=0,$

in particular,

$\displaystyle{\bf 1}_{\\{\omega=0\\}}\Gamma(|\omega|,g)=0.$

Now we eventually have

$\displaystyle\left(-{\Delta\Delta}^{\rm\\!HK}\omega,g\frac{\omega}{|\omega|}\right)_{L^{2}(T^{*}\\!M)}$
$\displaystyle\geq\int_{M}\Gamma(|\omega|,g)\mathrm{d}\mathfrak{m}+\langle\kappa,g|\omega|\rangle$
$\displaystyle=\mathscr{E}(|\omega|,g)+\langle\kappa,g|\omega|\rangle$
$\displaystyle=\mathscr{E}^{\kappa}(|\omega|,g)$

for $g\in{\rm Test}(M)_{+}$. ∎

######  Lemma 8.7.

Suppose $\omega\in H^{1,2}(T^{*}\\!M)$. Then

$\displaystyle\int_{0}^{t}\int_{M}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}\mathrm{d}s<\infty.$ (8.7)

###### Proof.

Under $\omega\in H^{1,2}(T^{*}\\!M)$, (8.7) follows from the continuity of

$\displaystyle[0,t]\ni s\mapsto\int_{M}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}<\infty.$ (8.8)

The continuity (8.8) can be confirmed by combining [14, Corollary 6.14, (8.5)
and Theorem 8.31(v)]. ∎

######  Lemma 8.8.

For every $\omega\in L^{2}(T^{*}\\!M)$ and $t>0$, we have

$\displaystyle|P_{t}^{\rm HK}\omega|^{2}\leq
P_{t}^{-2\kappa^{-}}|\omega|^{2}\quad\mathfrak{m}\text{-a.e.}$ (8.9)

In particular, for $\omega\in L^{2}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$ and
$\alpha>C_{\kappa}$,

$\displaystyle\|R_{\alpha}^{\rm
HK}\omega\|_{L^{\infty}(T^{*}\\!M)}\leq\frac{\sqrt{C(\kappa)}}{\alpha-
C_{\kappa}}\|\omega\|_{L^{\infty}(T^{*}\\!M)}.$ (8.10)

Here $C(\kappa)$ and $C_{\kappa}$ are the positive constants appeared in
(1.2).

###### Proof.

We may assume $\kappa^{+}=0$, because ${\sf BE}_{2}(-\kappa^{-},\infty)$ is
satisfied. Since $H^{1,2}(T^{*}\\!M)$ is dense in $L^{2}(T^{*}\\!M)$ and
$\|P_{t}^{\rm HK}\omega\|_{L^{2}(T^{*}\\!M)}\leq\|\omega\|_{L^{2}(T^{*}\\!M)}$
for $\omega\in L^{2}(T^{*}\\!M)$ (see [14, Theorem 8.31(iii)]), if (8.9) holds
for $\omega\in H^{1,2}(T^{*}\\!M)$, then it holds for $\omega\in
L^{2}(T^{*}\\!M)$. So we may assume $\omega\in H^{1,2}(T^{*}\\!M)$. Take $g\in
L^{2}(M;\mathfrak{m})\cap L^{\infty}(M;\mathfrak{m})\cap\mathscr{B}_{+}(M)$
and set
$g_{\alpha}:=R_{\alpha}g=\mathbb{E}_{\cdot}[\int_{0}^{\infty}e^{-\alpha
s}g(X_{s})\mathrm{d}s]$. Note here that $(G_{\alpha})_{\alpha\geq 0}$ defined
by $G_{\alpha}f:=\int_{0}^{\infty}e^{-\alpha t}P_{t}f\mathrm{d}t$ for $f\in
L^{1}(M;\mathfrak{m})$ forms a strongly continuous resolvent on
$L^{1}(M;\mathfrak{m})$. Since
$R_{\alpha}f(x):=\mathbb{E}_{x}\left[\int_{0}^{\infty}e^{-\alpha
t}f(X_{t})\mathrm{d}t\right]$ is $\alpha$-excessive for $f\in
L^{1}(M;\mathfrak{m})\cap\mathscr{B}(M)_{+}$, it is an $\mathscr{E}$-q.e.
finite and $\mathscr{E}$-quasi continuous function for $f\in
L^{1}(M;\mathfrak{m})\cap\mathscr{B}(M)_{+}$ in view of [45, Lemma 4.4] and
[30, Theorem 4.6.1]. The content of [30, Theorem 4.6.1] remains valid for
quasi-regular Dirichlet forms. We may assume $\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\in L^{1}(M;\mathfrak{m})\cap\mathscr{B}(M)_{+}$, where
$\widetilde{|P_{s}^{\rm HK}\omega|}$ is an $\mathscr{E}$-quasi continuous
$\mathfrak{m}$-version of $|P_{s}^{\rm HK}\omega|\in D(\mathscr{E})$.

We now set a function $F_{n}:[0,t]\to\mathbb{R}$ defined by

$\displaystyle
F_{n}(s):=\int_{M}P_{t-s}^{2\kappa}g_{\alpha}\,nP_{\frac{1}{n}}G_{n}|P_{s}^{\rm
HK}\omega|^{2}\mathrm{d}\mathfrak{m}.$

Note that $|P_{s}^{\rm HK}\omega|^{2}\in\mathscr{G}_{\rm
reg}\subset\mathscr{G}$ by [14, Proposition 6.11], where $\mathscr{G}$ and
$\mathscr{G}_{\rm reg}$ are the first order Sobolev spaces defined in [14,
Definitions 6.18 and 5.19]. Hence $P_{\frac{1}{n}}G_{n}|P_{s}^{\rm
HK}\omega|^{2}\in L^{1}(M;\mathfrak{m})$, and
$\Delta^{2\kappa}P_{t-s}^{2\kappa}g_{\alpha}\in L^{\infty}(M;\mathfrak{m})$.
Since $(P_{t}^{2\kappa})_{t\geq 0}$ is weakly* continuous semigroup on
$L^{\infty}(M;\mathfrak{m})$, we can deduce

$\displaystyle\frac{1}{n}F^{\prime}_{n}(s)$
$\displaystyle=\int_{M}\left(-\Delta^{2\kappa}P_{t-s}^{2\kappa}g_{\alpha}\,P_{\frac{1}{n}}G_{n}|P_{s}^{\rm
HK}\omega|^{2}+2(P_{t-s}^{2\kappa}g_{\alpha})P_{\frac{1}{n}}G_{n}\langle
P_{s}^{\rm HK}\omega,{\Delta\Delta}^{\rm\\!HK}P_{s}^{\rm
HK}\omega\rangle\right)\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}\left(-\Delta^{2\kappa}P_{t-s}^{2\kappa}g_{\alpha}\right)P_{\frac{1}{n}}G_{n}|P_{s}^{\rm
HK}\omega|^{2}\mathrm{d}\mathfrak{m}+2\int_{M}\langle(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha})P_{s}^{\rm
HK}\omega,{\Delta\Delta}^{\rm\\!HK}P_{s}^{\rm
HK}\omega\rangle\mathrm{d}\mathfrak{m}.$

The first term of the right hand side in the last line is

$\displaystyle\int_{M}$
$\displaystyle\left(-\Delta^{2\kappa}P_{t-s}^{2\kappa}g_{\alpha}\right)\,P_{\frac{1}{n}}G_{n}|P_{s}^{\rm
HK}\omega|^{2}\mathrm{d}\mathfrak{m}$
$\displaystyle=\lim_{k\to\infty}\int_{M}\left(-\Delta^{2\kappa}P_{t-s}^{2\kappa}g_{\alpha}\right)\,P_{\frac{1}{n}}G_{n}\left(|P_{s}^{\rm
HK}\omega|^{2}\land k^{2}\right)\mathrm{d}\mathfrak{m}$
$\displaystyle=\lim_{k\to\infty}\mathscr{E}^{2\kappa}(P_{t-s}^{2\kappa}g_{\alpha},P_{\frac{1}{n}}G_{n}(|P_{s}^{\rm
HK}\omega|^{2}\land k^{2}))$
$\displaystyle=\lim_{k\to\infty}\left\\{\mathscr{E}\left(P_{t-s}^{2\kappa}g_{\alpha},P_{\frac{1}{n}}G_{n}(|P_{s}^{\rm
HK}\omega|^{2}\land
k^{2})\right)-2\int_{M}p_{t-s}^{2\kappa}g_{\alpha}\;p_{\frac{1}{n}}R_{n}(\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\land k^{2})\mathrm{d}\kappa^{-}\right\\}$
$\displaystyle=\lim_{k\to\infty}\int_{M}\langle\nabla
P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha},\nabla(|P_{s}^{\rm
HK}\omega|^{2}\land
k^{2})\rangle\mathrm{d}\mathfrak{m}-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}$ $\displaystyle=\int_{M}\langle\nabla
P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha},\nabla|P_{s}^{\rm
HK}\omega|^{2}\rangle\mathrm{d}\mathfrak{m}-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-},$

where we use the fact that $\nabla(|P_{s}^{\rm HK}\omega|^{2}\land
k^{2})\to\nabla|P_{s}^{\rm HK}\omega|^{2}$ as $k\to\infty$ in $L^{1}(TM)$ and
$P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g\in{\rm Test}(M)_{+}$ by Lemma 3.2,
hence $|\nabla P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g|\in
L^{\infty}(M;\mathfrak{m})$. Moreover, $p_{t-s}^{2\kappa}g_{\alpha}$ is an
$\mathscr{E}$-quasi continuous $\mathfrak{m}$-version of
$P_{t-s}^{2\kappa}g_{\alpha}$ (see [46, Theorem 7.3]). Therefore, we have

$\displaystyle\frac{1}{n}F^{\prime}_{n}(s)$
$\displaystyle=\int_{M}\langle\nabla
P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha},\nabla|P_{s}^{\rm
HK}\omega|^{2}\rangle\mathrm{d}\mathfrak{m}-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}$ $\displaystyle\hskip
56.9055pt+2\int_{M}\langle
P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}\,P_{s}^{\rm
HK}\omega,{\Delta\Delta}^{\rm\\!HK}P_{s}^{\rm
HK}\omega\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle=2\int_{M}\langle\nabla(P_{s}^{\rm
HK}\omega)^{\sharp}\,|\,\nabla
P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}\otimes(P_{s}^{\rm
HK}\omega)^{\sharp}\rangle_{\rm HS}\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
56.9055pt-2\int_{M}\langle\mathrm{d}(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}P_{s}^{\rm
HK}\omega),\mathrm{d}P_{s}^{\rm HK}\omega\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
113.81102pt-2\int_{M}\langle\mathrm{d}_{*}(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}P_{s}^{\rm
HK}\omega),\mathrm{d}_{*}P_{s}^{\rm HK}\omega\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
170.71652pt-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}.$

Since

$\displaystyle\langle\nabla(P_{s}^{\rm HK}\omega)^{\sharp}\,$
$\displaystyle|\,\nabla
P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}\otimes(P_{s}^{\rm
HK}\omega)^{\sharp}\rangle_{\rm HS}$ $\displaystyle=\langle\nabla(P_{s}^{\rm
HK}\omega)^{\sharp}\,|\,\nabla(p_{\frac{1}{n}}R_{n}P_{t-s}^{2\kappa}g_{\alpha})P_{s}^{\rm
HK}\omega)^{\sharp}\rangle_{\rm
HS}-P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}|\nabla(P_{s}^{\rm
HK}\omega)^{\sharp}|_{\rm HS}^{2}$ $\displaystyle\leq\langle\nabla(P_{s}^{\rm
HK}\omega)^{\sharp}\,|\,\nabla(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha})P_{s}^{\rm
HK}\omega)^{\sharp}\rangle_{\rm HS},$

we have

$\displaystyle\frac{1}{n}F^{\prime}_{n}(s)$ $\displaystyle\leq
2\int_{M}\langle\nabla(P_{s}^{\rm
HK}\omega)^{\sharp}\,|\,\nabla(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha})P_{s}^{\rm
HK}\omega)^{\sharp}\rangle_{\rm HS}\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
56.9055pt-2\int_{M}\langle\mathrm{d}(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}P_{s}^{\rm
HK}\omega),\mathrm{d}P_{s}^{\rm HK}\omega\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
113.81102pt-2\int_{M}\langle\mathrm{d}_{*}(P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}P_{s}^{\rm
HK}\omega),\mathrm{d}_{*}P_{s}^{\rm HK}\omega\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip
170.71652pt-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}$
$\displaystyle=-2\int_{M}p_{\frac{1}{n}}R_{n}p_{t-s}^{2\kappa}g_{\alpha}\,\mathrm{d}{\bf
Ric}((P_{s}^{\rm HK}\omega)^{\sharp},(P_{s}^{\rm
HK}\omega)^{\sharp})-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}$
$\displaystyle=-2\int_{M}p_{\frac{1}{n}}R_{n}p_{t-s}^{2\kappa}g_{\alpha}\,\mathrm{d}{\bf
Ric}^{\kappa}((P_{s}^{\rm HK}\omega)^{\sharp},(P_{s}^{\rm
HK}\omega)^{\sharp})$ $\displaystyle\hskip
56.9055pt+2\int_{M}p_{\frac{1}{n}}R_{n}p_{t-s}^{2\kappa}g_{\alpha}\,\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:NonKappaRicci}}}{{\leq}}2\int_{M}p_{\frac{1}{n}}R_{n}p_{t-s}^{2\kappa}g_{\alpha}\,\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}-2\int_{M}(p_{t-s}^{2\kappa}g_{\alpha})\,p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}.$ (8.11)

In the first equality above, we use
$P_{\frac{1}{n}}G_{n}P_{t-s}^{2\kappa}g_{\alpha}\in{\rm Test}(M)_{+}$ again
and [14, Lemma 8.14]. Thanks to the right continuity of $s\mapsto
g_{\alpha}(X_{s})$ under the law $\mathbb{P}_{x}$ and the Lebesgue’s dominated
convergence theorem, we can deduce that
$np_{\frac{1}{n}}R_{n}p_{t-s}^{2\kappa}g_{\alpha}\to
p_{t-s}^{2\kappa}g_{\alpha}$ as $n\to\infty$ $\mathscr{E}$-q.e. From this,
(8.11) and the boundedness of $P_{t-s}^{2\kappa}g_{\alpha}$, we have

$\displaystyle\varlimsup_{n\to\infty}F^{\prime}_{n}(s)$ $\displaystyle\leq
2\int_{M}p_{t-s}^{2\kappa}g_{\alpha}\,\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}-\varliminf_{n\to\infty}2\int_{M}p_{t-s}^{2\kappa}g_{\alpha}\;np_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}\leq 0,$ (8.12)

where we use the Fatou’s lemma and dominated convergence theorem for

$\displaystyle\int_{M}p_{t-s}^{2\kappa}g_{\alpha}\,\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}=\lim_{n\to\infty}\int_{M}np_{\frac{1}{n}}R_{n}P_{t-s}^{2\kappa}g_{\alpha}\,\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}$

under (8.8). Recall that $p_{\frac{1}{n}}R_{n}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}=R_{n}p_{\frac{1}{n}}\widetilde{|P_{s}^{\rm HK}\omega|}^{2}\in
L^{1}(M;\mathfrak{m})$ is $\mathscr{E}$-q.e. finite and $\mathscr{E}$-quasi
continuous. Integrating (8.12) from $0$ to $t$ and multiplying $(-1)$, we have

$\displaystyle\int_{0}^{t}\sup_{n\geq 1}f_{n}(s)\mathrm{d}s\geq 0,$

where $f_{n}(s):=\inf_{\ell\geq n}\left(-F_{\ell}^{\prime}(s)\right)$. We see
$f_{1}(s)\leq f_{n}(s)\leq f_{n+1}(s)$ for all $s\in[0,t]$ and
$n\in\mathbb{N}$. We can deduce $\int_{0}^{t}f_{1}(s)\mathrm{d}s>-\infty$.
Indeed,

$\displaystyle-\int_{0}^{t}f_{1}(s)\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\sup_{\ell\geq
1}F^{\prime}_{\ell}(s)\mathrm{d}s$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Dominates}}}{{\leq}}\int_{0}^{t}\sup_{\ell\geq
1}\left(\int_{M}\ell
R_{\ell}p_{\frac{1}{\ell}}p_{t-s}^{2\kappa}g_{\alpha}\;\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}-\int_{M}p_{t-s}^{2\kappa}g_{\alpha}\;\ell
R_{\ell}p_{\frac{1}{\ell}}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}\right)\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|P_{t-s}^{2\kappa}g_{\alpha}\|_{\infty}\int_{M}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}\mathrm{d}s$ $\displaystyle\leq
C(2\kappa)e^{C_{2\kappa}t}\|g_{\alpha}\|_{\infty}\int_{0}^{t}\int_{M}\widetilde{|P_{s}^{\rm
HK}\omega|}^{2}\mathrm{d}\kappa^{-}\mathrm{d}s<\infty$

by (8.7) under $\omega\in H^{1,2}(T^{*}\\!M)$. By a variant of Beppo Levi’s
monotone convergence theorem under $\int_{0}^{t}f_{1}(s)\mathrm{d}s>-\infty$,
we get

$\displaystyle-\varlimsup_{n\to\infty}\int_{0}^{t}F_{n}^{\prime}(s)\mathrm{d}s\geq\lim_{n\to\infty}\int_{0}^{t}f_{n}(s)\mathrm{d}s=\int_{0}^{t}\sup_{n\geq
1}f_{n}(s)\mathrm{d}s\geq 0.$

Thus,

$\displaystyle\varlimsup_{n\to\infty}\left(\int_{M}g_{\alpha}\,nP_{\frac{1}{n}}G_{n}|P_{t}^{\rm
HK}\omega|^{2}\mathrm{d}\mathfrak{m}-\int_{M}(P_{t-s}^{2\kappa}g_{\alpha})\,nP_{\frac{1}{n}}G_{n}|\omega|^{2}\mathrm{d}\mathfrak{m}\right)$
$\displaystyle\leq\varlimsup_{n\to\infty}\int_{0}^{t}F_{n}^{\prime}(s)\mathrm{d}s\leq
0.$

Since
$\lim_{n\to\infty}\|P_{\frac{1}{n}}nG_{n}f-f\|_{L^{1}(M;\mathfrak{m})}=0$ for
$f\in L^{1}(M;\mathfrak{m})$, we have

$\displaystyle\int_{M}g_{\alpha}|P_{t}^{\rm
HK}\omega|^{2}\mathrm{d}\mathfrak{m}\leq\int_{M}(P_{t}^{2\kappa}g_{\alpha})|\omega|^{2}\mathrm{d}\mathfrak{m}=\int_{M}g_{\alpha}P_{t}^{2\kappa}|\omega|^{2}\mathrm{d}\mathfrak{m}.$

Since $\alpha g_{\alpha}=\alpha R_{\alpha}g\in L^{\infty}(M;\mathfrak{m})$
weakly* converges to $g$ in $L^{\infty}(M;\mathfrak{m})$ as $\alpha\to\infty$
and $g\in L^{2}(M;\mathfrak{m})\cap
L^{\infty}(M;\mathfrak{m})\cap\mathscr{B}_{+}(M)$ is arbitrary, we obtain
(8.9).

Next we prove (8.10). Under Assumption 1, from (1.2), we can get

$\displaystyle\|P_{t}^{2\kappa}\|_{\infty,\infty}\leq
C(\kappa)e^{2C_{\kappa}t}.$ (8.13)

That is, under $2\kappa^{-}\in S_{E\\!K}({\bf X})$, we have (8.13), because
$C_{2\kappa}\leq C_{\kappa}\leq 2C_{\kappa}$ and $C(2\kappa)=C(\kappa)$ from
(2.3). Then

$\displaystyle\|P_{t}^{\rm HK}\omega\|_{L^{\infty}(T^{*}\\!M)}$
$\displaystyle\leq\sqrt{P_{t}^{2\kappa}1}\|\omega\|_{L^{\infty}(T^{*}\\!M)}$
$\displaystyle\leq\sqrt{C(\kappa)e^{2C_{\kappa}t}}\|\omega\|_{L^{\infty}(T^{*}\\!M)}$
$\displaystyle=\sqrt{C(\kappa)}e^{C_{\kappa}t}\|\omega\|_{L^{\infty}(T^{*}\\!M)}$

for $\omega\in L^{2}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$. Therefore we
obtain (8.10). ∎

###### Proof of Theorem 1.2.

Take $g\in{\rm Test}(M)_{+}$. Then, for $\omega\in
D({\Delta\Delta}^{\rm\\!HK})\cap L^{\infty}(T^{*}\\!M)$

$\displaystyle\int_{M}(\Delta g-\alpha
g)|\omega|\mathrm{d}\mathfrak{m}-\int_{M}g|\omega|\mathrm{d}\kappa$
$\displaystyle\geq\int_{M}g\left(\left\langle{\Delta\Delta}^{\rm\\!HK}\omega,\frac{\omega}{|\omega|}\right\rangle-\alpha|\omega|\right)\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}g\left\langle{\Delta\Delta}^{\rm\\!HK}\omega-\alpha\omega,\frac{\omega}{|\omega|}\right\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle\geq-\int_{M}g\left|{\Delta\Delta}^{\rm\\!HK}\omega-\alpha\omega\right|\mathrm{d}\mathfrak{m}$

by Lemma 8.6, hence

$\displaystyle\mathscr{E}_{\alpha}^{\kappa}(|\omega|,g)\leq\int_{M}g\left|(\alpha-{\Delta\Delta}^{\rm\\!HK})\omega\right|\mathrm{d}\mathfrak{m}.$
(8.14)

Since ${\rm Test}(M)_{+}$ is dense in $D(\mathscr{E})_{+}$, (8.14) holds for
any $g\in D(\mathscr{E})_{+}$. By Lemma 8.8, for $\alpha>C_{\kappa}$, we can
set $\omega:=R_{\alpha}^{\rm HK}\eta\in D({\Delta\Delta}^{\rm\\!HK})\cap
L^{\infty}(T^{*}\\!M)$ for $\eta\in L^{2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$ and $g:=R_{\alpha}^{\kappa}\psi$ with $\psi\in
L^{2}(M;\mathfrak{m})_{+}$. Then we see

$\displaystyle(\psi,|R_{\alpha}^{\rm
HK}\eta|)_{\mathfrak{m}}\leq(\psi,R_{\alpha}^{\kappa}|\eta|)_{\mathfrak{m}}\quad\text{
for any }\quad\psi\in L^{2}(M;\mathfrak{m})_{+}.$

This implies that for $\alpha>C_{\kappa}$ and $\eta\in L^{2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$

$\displaystyle|R_{\alpha}^{\rm HK}\eta|\leq
R_{\alpha}^{\kappa}|\eta|\quad\mathfrak{m}\text{-a.e.}$ (8.15)

By Lemma 9.1 below, for any $\eta\in L^{2}(T^{*}\\!M)$, there exists
$\\{\eta_{n}\\}\subset L^{2}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$ such that
$\\{\eta_{n}\\}$ converges to $\eta$ in $L^{2}(T^{*}\\!M)$ and (8.15) holds by
replacing $\eta$ with $\eta_{n}$. By taking a subsequence if necessary, we can
deduce that (8.15) holds for general $\eta\in L^{2}(T^{*}\\!M)$.

From (8.15), we can obtain that $|P_{t}^{\rm HK}\eta|\leq
P_{t}^{\kappa}|\eta|$ $\mathfrak{m}$-a.e. for each $t>0$ in view of the
following observation:

$\displaystyle P_{t}^{\kappa}f$
$\displaystyle=\lim_{n\to\infty}\left(\frac{n}{t}\right)^{n}(R_{\frac{n}{t}}^{\kappa})^{n}f\quad\text{
in }L^{2}(M;\mathfrak{m})\quad\text{ for }\quad f\in L^{2}(M;\mathfrak{m}),$
$\displaystyle P_{t}^{\rm HK}\theta$
$\displaystyle=\lim_{n\to\infty}\left(\frac{n}{t}\right)^{n}(R_{\frac{n}{t}}^{\rm
HK})^{n}\theta\quad\text{ in }L^{2}(T^{*}\\!M)\quad\text{ for }\quad\theta\in
L^{2}(T^{*}\\!M)$

(cf. [58, Theorem 2.4]). ∎

##  9 Proof of Corollary 1.3

The following lemma is stated in [14] without proof. We provide its proof for
readers’ convenience. First note that ${\rm Test}(T^{*}\\!M)\subset
L^{p}(T^{*}\\!M)$ and ${\rm Test}(TM)\subset L^{p}(TM)$ for $p\in[1,+\infty]$.

######  Lemma 9.1.

Let $p\in[1,+\infty[$. Then ${\rm Test}(T^{*}\\!M)$ (resp. ${\rm Test}(TM)$)
is dense in $L^{p}(T^{*}\\!M)$ (resp. $L^{p}(TM)$). Moreover, ${\rm
Test}(T^{*}\\!M)$ (resp. ${\rm Test}(TM)$) is weakly* dense in
$L^{\infty}(T^{*}\\!M)$ (resp. $L^{\infty}(TM)$).

###### Proof.

Assume $p\in[1,+\infty[$. In this case, we take $X\in
L^{p}(T^{*}\\!M)^{*}=L^{q}(TM)$ with $q:=p/(p-1)\in]1,+\infty]$. Suppose
$X\perp{\rm Test}(T^{*}\\!M)$, i.e.,

$\displaystyle\left(\sum_{i=1}^{n}g_{i}\mathrm{d}f_{i}\right)(X)=0\quad\text{
for any }n\in\mathbb{N},\quad g_{i},f_{i}\in{\rm Test}(M)\;\;(1\leq i\leq n).$
(9.1)

In particular,

$\displaystyle g\langle X,\nabla f\rangle=\langle X,g\nabla
f\rangle=\left(g\mathrm{d}f\right)(X)=0\quad\text{ for any }\quad g,f\in{\rm
Test}(M).$

Owing to Lemma 3.2, we see $(P_{1/n}h)\langle X,\nabla f\rangle=0$ for $h\in
L^{2}(M;\mathfrak{m})\cap L^{\infty}(M;\mathfrak{m})$. By taking a subsequence
if necessary, we may assume $h=\lim_{n\to\infty}P_{1/n}h$ $\mathfrak{m}$-a.e.,
hence $h\langle X,\nabla f\rangle=0$ for such $h$. Let $\\{M_{n}\\}$ be an
increasing sequence of Borel sets with $M=\bigcup_{n=1}^{\infty}M_{n}$ and
$\mathfrak{m}(M_{n})<\infty$ for each $n\in\mathbb{N}$. Then for $h_{n}:={\bf
1}_{\\{|X|\leq n\\}\cap M_{n}}\in L^{2}(M;\mathfrak{m})\cap
L^{\infty}(M;\mathfrak{m})$, we see $h_{n}X\in L^{2}(TM)$. Since $L^{2}(TM)$
is generated by $\nabla D(\mathscr{E})$ in the sense of $L^{\infty}$-modules
(see [14, Lemma 2.7]), we have $h_{n}X=0$. Therefore $X=0$, consequently ${\rm
Test}(T^{*}\\!M)$ is dense in $L^{p}(T^{*}\\!M)$ by [19, Chapter III,
Corollary 6.14]. In view of the musical isomorphisms, ${\rm Test}(TM)$ is also
dense in $L^{p}(TM)$ for $p\in[1,+\infty[$.

Next we assume $p=\infty$. Take $X\in L^{1}(TM)$ and suppose $X\perp{\rm
Test}(T^{*}\\!M)$, i.e., (9.1) holds. In the same way as above, we can deduce
$X=0$. This means that ${\rm Test}(T^{*}\\!M)$ is dense in
$L^{\infty}(T^{*}\\!M)=L^{1}(TM)^{*}$ with respect to the weak* topology
$\sigma(L^{1}(T\\!M)^{*},L^{1}(TM))$ by [19, Chapter IV, Example 1.8,
Corollary 3.14]. In view of the musical isomorphisms, ${\rm Test}(TM)$ is also
weakly* dense in $L^{\infty}(TM)$. ∎

######  Corollary 9.2.

Suppose $p\in[1,+\infty[$. Then $L^{p}(T^{*}\\!M)\cap H^{1,2}(T^{*}\\!M)$
(resp. $L^{p}(TM)\cap H^{1,2}(TM)$) is dense in $L^{p}(T^{*}\\!M)$ (resp.
$L^{p}(TM)$). Moreover, $L^{\infty}(T^{*}\\!M)\cap H^{1,2}(T^{*}\\!M)$ (resp.
$L^{\infty}(TM)\cap H^{1,2}(TM)$) is weakly* dense in $L^{\infty}(T^{*}\\!M)$
(resp. $L^{\infty}(TM)$).

###### Proof.

The assertion easily follows from Lemma 9.1, because ${\rm
Test}(T^{*}\\!M)\subset H^{1,2}(T^{*}\\!M)$ and ${\rm Test}(TM)\subset
H^{1,2}(TM)$. ∎

######  Corollary 9.3.

Suppose $p\in[1,+\infty[$. Then ${\rm Reg}(T^{*}\\!M)\cap L^{p}(T^{*}\\!M)$ is
dense in $L^{p}(T^{*}\\!M)$ and ${\rm Reg}(TM)\cap L^{p}(TM)$ is dense in
$L^{p}(TM)$. Moreover, ${\rm Reg}(T^{*}\\!M)\cap L^{\infty}(T^{*}\\!M)$ is
weakly* dense in $L^{\infty}(T^{*}\\!M)$ and ${\rm Reg}(TM)\cap
L^{\infty}(TM)$ is dense in $L^{\infty}(TM)$.

###### Proof.

By Lemma 9.1, the assertion clearly follow from ${\rm
Test}(T^{*}\\!M)\subset{\rm Reg}(T^{*}\\!M)\cap L^{p}(T^{*}\\!M)$ and ${\rm
Test}(TM)\subset{\rm Reg}(TM)\cap L^{p}(TM)$. ∎

###### Proof of Corollary 1.3.

First we suppose $p\in[2,+\infty[$. By Theorem 1.2 and (2.5) with
$\nu=\kappa^{-}$, we easily get

$\displaystyle\|P_{t}^{\rm
HK}\omega\|_{L^{p}(T^{*}\\!M)}\leq\|P_{t}^{\kappa}|\omega|\|_{L^{p}(M;\mathfrak{m})}\leq
C(\kappa)e^{C_{\kappa}t}\|\omega\|_{L^{p}(T^{*}\\!M)}$ (9.2)

holds for $\omega\in L^{p}(T^{*}\\!M)\cap L^{2}(T^{*}\\!M)$. For general
$\omega\in L^{p}(T^{*}\\!M)$, there exists $\\{\omega_{n}\\}\subset
L^{p}(T^{*}\\!M)\cap H^{1,2}(T^{*}\\!M)$ such that

$\displaystyle\lim_{n\to\infty}\|\omega_{n}-\omega\|_{L^{p}(T^{*}\\!M)}=0$

by Corollary 9.2. Then $\\{P_{t}^{\rm HK}\omega_{n}\\}$ forms an
$L^{p}(T^{*}\\!M)$-Cauchy sequence by (9.2). Denote by $P_{t}^{\rm HK}\omega$
its limit. Since $P_{t}^{\kappa}\omega_{n}$ converges to
$P_{t}^{\kappa}\omega$ in $L^{p}(M;\mathfrak{m})$, we can deduce $|P_{t}^{\rm
HK}\omega|\leq P_{t}^{\kappa}|\omega|$ $\mathfrak{m}$-a.e. Hence $P_{t}^{\rm
HK}$ can be extended to $L^{p}(T^{*}\\!M)$ and (9.2) holds for $\omega\in
L^{p}(T^{*}\\!M)$. In case $p=\infty$, we use another approximation. For
$\omega\in L^{\infty}(T^{*}\\!M)$, we set $\omega_{n}:={\bf 1}_{M_{n}}\omega$,
where $\\{M_{n}\\}$ is an increasing sequence of Borel sets with
$M=\bigcup_{n=1}^{\infty}M_{n}$ and $\mathfrak{m}(M_{n})<\infty$ for each
$n\in\mathbb{N}$. Then $\omega_{n}\in L^{2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$ for each $n\in\mathbb{N}$, hence (9.2) holds for
$\omega_{n}$ instead of $\omega$. Now, for a Borel partition
$(B_{i})_{i\in\mathbb{N}}$ with $\mathfrak{m}(B_{i})\in]0,+\infty[$, we
consider the distance ${\sf d}_{L^{0}(T^{*}\\!M)}$ on $L^{0}(T^{*}\\!M)$:

$\displaystyle{\sf
d}_{L^{0}(T^{*}\\!M)}(\eta_{1},\eta_{2}):=\sum_{i\in\mathbb{N}}\frac{2^{-i}}{\mathfrak{m}(B_{i})}\int_{B_{i}}\min\\{|\eta_{1}-\eta_{2}|,1\\}\mathrm{d}\mathfrak{m},\quad\eta_{1},\eta_{2}\in
L^{0}(T^{*}\\!M).$

This distance does not depend on the choice of such Borel partition
$(B_{i})_{i\in\mathbb{N}}$ (see [14, 1.4.3]). Since $P_{t}^{\kappa}{\bf
1}_{B_{i}}\in L^{2}(M;\mathfrak{m})$ under $2\kappa^{-}\in S_{E\\!K}({\bf
X})$, we see from Theorem 1.2 that

$\displaystyle\int_{B_{i}}|P_{t}^{\rm
HK}(\omega_{n}-\omega_{m})|\mathrm{d}\mathfrak{m}$
$\displaystyle\leq\int_{M}P_{t}^{\kappa}{\bf
1}_{B_{i}}|\omega_{n}-\omega_{m}|\mathrm{d}\mathfrak{m}$
$\displaystyle\leq\|\omega\|_{L^{\infty}(T^{*}\\!M)}\int_{M_{m}\setminus
M_{n}}P_{t}^{\kappa}{\bf 1}_{B_{i}}\mathrm{d}\mathfrak{m}\to 0\quad\text{ as
}\quad n,m\to\infty.$

This means that $(P_{t}^{\rm HK}\omega_{n})_{n\in\mathbb{N}}$ forms a ${\sf
d}_{L^{0}(T^{*}\\!M)}$-Cauchy sequence. Denote its limit by $P_{t}^{\rm
HK}\omega\in L^{0}(T^{*}\\!M)$. Then we see $P_{t}^{\rm HK}\omega\in
L^{\infty}(T^{*}\\!M)$ and (9.2) holds for $\omega$.

Next we suppose $\kappa^{-}\in S_{K}({\bf X})$ and $p\in[1,+\infty[$. Then
(2.5) with $\nu=\kappa^{-}$ yields

$\displaystyle\|P_{t}^{\kappa}\|_{p,p}\leq C(\kappa)e^{C_{\kappa}t}.$ (9.3)

Thus Theorem 1.2 and (9.3) yield the estimate (9.2) for $\omega\in
L^{p}(T^{*}\\!M)\cap L^{2}(T^{*}\\!M)$, consequently (9.2) holds for
$\omega\in L^{p}(T^{*}\\!M)$ as proved above.

Finally we prove the strong continuity of $(P_{t}^{\rm HK})_{t\geq 0}$ under
$\kappa^{-}\in S_{K}({\bf X})$ and $p\in[1,+\infty[$. Suppose first $\omega\in
L^{p}(T^{*}\\!M)\cap H^{1,2}(T^{*}\\!M)$. We then know

$\displaystyle\lim_{t\to 0}\|P_{t}^{\rm
HK}\omega-\omega\|_{H^{1,2}(T^{*}\\!M)}=0$ (9.4)

by [14, Theorem 8.31(v)]. In particular, $|P_{t}^{\rm HK}\omega-\omega|\to 0$
in $\mathfrak{m}$-measure as $t\to 0$. Take any subsequence $\\{t_{n}\\}$
tending to $0$ decreasingly as $n\to\infty$ such that $|P_{t_{n}}^{\rm
HK}\omega|\to|\omega|$ $n\to\infty$ $\mathfrak{m}$-a.e. Applying Fatou’s
lemma,

$\displaystyle\int_{M}|\omega|^{p}\mathrm{d}\mathfrak{m}\leq\varliminf_{n\to\infty}\int_{M}|P_{t_{n}}^{\rm
HK}\omega|^{p}\mathrm{d}\mathfrak{m}.$ (9.5)

On the other hand, we already know that (9.2) holds for $\omega\in
L^{p}(T^{*}\\!M)$. This implies

$\displaystyle\varlimsup_{n\to\infty}\|P_{t_{n}}^{\rm
HK}\omega\|_{L^{p}(T^{*}\\!M)}\leq C(\kappa)\|\omega\|_{L^{p}(T^{*}\\!M)}.$

Since the coefficient $C(\kappa)$ has the following expression for some
sufficiently small $t_{0}>0$

$\displaystyle
C(\kappa)=\frac{1}{1-\left\|\mathbb{E}_{\cdot}\left[A_{t_{0}}^{\kappa^{-}}\right]\right\|_{\infty}}$

by (2.4) (see [62, Proof of Theorem 2.2]), it can be taken to be arbitrarily
close to $1$ in view of the fact that $\kappa^{-}$ is a smooth measure of Kato
class. Then, we have

$\displaystyle\varlimsup_{n\to\infty}\|P_{t_{n}}^{\rm
HK}\omega\|_{L^{p}(T^{*}\\!M)}\leq\|\omega\|_{L^{p}(T^{*}\\!M)}.$

Combining this and (9.5), we have $\lim_{n\to\infty}\|P_{t_{n}}^{\rm
HK}\omega\|_{L^{p}(T^{*}\\!M)}=\|\omega\|_{L^{p}(T^{*}\\!M)}$. Applying [55,
Theorem 16.6], we obtain

$\displaystyle\lim_{n\to\infty}\|P_{t_{n}}^{\rm
HK}\omega-\omega\|_{L^{p}(T^{*}\\!M)}=0.$

Thus

$\displaystyle\lim_{t\to 0}\|P_{t}^{\rm
HK}\omega-\omega\|_{L^{p}(T^{*}\\!M)}=0.$

Now we consider general $\omega\in L^{p}(T^{*}\\!M)$. By Corollary 9.2, there
exists $\\{\omega_{n}\\}\subset L^{p}(T^{*}\\!M)\cap H^{1,2}(T^{*}\\!M)$ such
that $\lim_{n\to\infty}\|\omega_{n}-\omega\|_{L^{p}(T^{*}\\!M)}=0$. By use of
triangle inequality,

$\displaystyle\|P_{t}^{\rm
HK}\omega-\omega\|_{L^{p}(T^{*}\\!M)}\leq\|P_{t}^{\rm
HK}(\omega-\omega_{n})\|_{L^{p}(T^{*}\\!M)}+\|P_{t}^{\rm
HK}\omega_{n}-\omega_{n}\|_{L^{p}(T^{*}\\!M)}+\|\omega_{n}-\omega\|_{L^{p}(T^{*}\\!M)}.$

By (9.2) and the conclusion for $\omega_{n}\in L^{p}(T^{*}\\!M)\cap
L^{2}(T^{*}\\!M)$,

$\displaystyle\varlimsup_{t\to 0}\|P_{t}^{\rm
HK}\omega-\omega\|_{L^{p}(T^{*}\\!M)}\leq\left(C(\kappa)+1\right)\|\omega_{n}-\omega\|_{L^{p}(T^{*}\\!M)}\to
0\quad\text{ as }\quad n\to\infty.$

Therefore

$\displaystyle\lim_{t\to 0}\|P_{t}^{\rm
HK}\omega-\omega\|_{L^{p}(T^{*}\\!M)}=0\quad\text{ for }\quad\omega\in
L^{p}(T^{*}\\!M).$

Finally, we prove the weak* continuity of $(P_{t}^{\rm HK})_{t\geq 0}$ on
$L^{\infty}(T^{*}\\!M)$ under $\kappa^{-}\in S_{K}({\bf X})$. Take $X\in
L^{1}(TM)$. First we assume $\omega\in L^{2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$. Consider $X_{n}:={\bf 1}_{\\{|X|\leq n\\}\cap M_{n}}X$
as defined above. Then we see

$\displaystyle|\langle X_{n},P_{t}^{\rm
HK}\omega-\omega\rangle|\leq\|X_{n}\|_{L^{1}(TM)}\|P_{t}^{\rm
HK}\omega-\omega\|_{L^{2}(T^{*}\\!M)}\to 0\quad\text{ as }\quad t\to 0.$

From this,

$\displaystyle\varlimsup_{t\to 0}|\langle X,P_{t}^{\rm
HK}\omega-\omega\rangle|$ $\displaystyle\leq\varlimsup_{t\to
0}\|X-X_{n}\|_{L^{1}(TM)}\|P_{t}^{\rm
HK}\omega-\omega\|_{L^{\infty}(T^{*}\\!M)}+\varlimsup_{t\to 0}|\langle
X_{n},P_{t}^{\rm HK}\omega-\omega\rangle|$
$\displaystyle\leq\|X-X_{n}\|_{L^{1}(TM)}(1+C(\kappa)))\|\omega\|_{L^{\infty}(T^{*}\\!M)}\to
0\quad\text{ as }\quad n\to\infty.$

Then, we have $\lim_{t\to 0}\langle X_{n},P_{t}^{\rm
HK}\omega-\omega\rangle=0$ for $\omega\in L^{2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$. Next we take general $\omega\in
L^{\infty}(T^{*}\\!M)$. Consider $\omega_{n}:={\bf 1}_{M_{n}}\omega$ as above.
Then we know $P_{t}^{\rm HK}\omega\in L^{2}(T^{*}\\!M)\cap
L^{\infty}(T^{*}\\!M)$. Then we have

$\displaystyle|\langle X,\omega_{n}-\omega\rangle|$
$\displaystyle\leq\int_{M}|\langle X,{\bf
1}_{M_{n}^{c}}\omega\rangle_{\mathfrak{m}}|\mathrm{d}\mathfrak{m}\leq\|\omega\|_{L^{\infty}(T^{*}\\!M)}\int_{M_{n}^{c}}|X|_{\mathfrak{m}}\mathrm{d}\mathfrak{m}\to
0\quad\text{ as }\quad n\to\infty,$ $\displaystyle|\langle X,P_{t}^{\rm
HK}(\omega_{n}-\omega)\rangle|$ $\displaystyle\leq\int_{M}|\langle
X,P_{t}^{\rm HK}{\bf
1}_{M_{n}^{c}}\omega\rangle_{\mathfrak{m}}|\mathrm{d}\mathfrak{m}$
$\displaystyle\leq\|\omega\|_{L^{\infty}(T^{*}\\!M)}\int_{M_{n}^{c}}P_{t}^{\kappa}|X|_{\mathfrak{m}}\mathrm{d}\mathfrak{m}\to
0\quad\text{ as }\quad n\to\infty,$

where we use the $\mathfrak{m}$-symmetry of $P_{t}^{\kappa}$ and
$P_{t}^{\kappa}|X|\in L^{1}(M;\mathfrak{m})$ under $\kappa^{-}\in S_{K}({\bf
X})$. Combining this inequality and

$\displaystyle|\langle X,P_{t}^{\rm HK}\omega-\omega\rangle|$
$\displaystyle\leq|\langle X,P_{t}^{\rm
HK}(\omega-\omega_{n})\rangle|+|\langle X,P_{t}^{\rm
HK}\omega_{n}-\omega_{n})\rangle|+|\langle X,\omega_{n}-\omega)\rangle|,$

we obtain

$\displaystyle\varlimsup_{t\to 0}|\langle X,P_{t}^{\rm
HK}\omega-\omega\rangle|\leq\|\omega\|_{L^{\infty}(TM)}\int_{M_{n}^{c}}(|X|_{\mathfrak{m}}+P_{t}^{\kappa}|X|_{\mathfrak{m}})\mathrm{d}\mathfrak{m}\to
0\quad\text{ as }\quad n\to\infty.$

Therefore, $\lim_{t\to 0}\langle X,P_{t}^{\rm HK}\omega-\omega\rangle=0$ for
$X\in L^{1}(TM)$ and $\omega\in L^{\infty}(T^{*}\\!M)$. This implies the weak*
continuity of $(P_{t}^{\rm HK})_{t\geq 0}$ on
$L^{\infty}(T^{*}\\!M)=L^{1}(TM)^{*}$. ∎

##  10 Examples

######  Example 10.1 (Riemannian manifolds with boundary).

Let $(M,g)$ be a smooth Riemannian manifold with boundary $\partial M$. Denote
by $\mathfrak{v}:={\rm vol}_{g}$ the Riemannian volume measure induced by $g$,
and by $\mathfrak{s}$ the surface measure on $\partial M$ (see [14, §1.2]). If
$\partial M\neq\emptyset$, then $\partial M$ is a smooth co-dimension $1$
submanifold of $M$ and it becomes Riemannian when endowed with the pulback
metric

$\displaystyle\langle\cdot,\cdot\rangle_{j}:=j^{*}\langle\cdot,\cdot\rangle,\qquad\langle
u,v:=g(u,v)\quad\text{ for }\quad u,v\in TM.$

under the natural inclusion $j:\partial M\to M$. The map $j$ induces a natural
inclusion $d_{j}:T\partial M\to TM|_{\partial M}$ which is not surjective. In
particular, the vector bundles $T\partial M$ and $TM|_{\partial M}$ do not
coincide.

Let $\mathfrak{m}$ be a Borel measure on $M$ which is locally equivalent to
$\mathfrak{v}$. Let $D(\mathscr{E}):=W^{1,2}(M^{\circ})$ be the Sobolev space
with respect to $\mathfrak{m}$ defined in the usual sense on
$M^{\circ}:=M\setminus\partial M$. Define
$\mathscr{E}:W^{1,2}(M^{\circ})\to[0,+\infty[$ by

$\displaystyle\mathscr{E}(f):=\int_{M^{\circ}}|\nabla
f|^{2}\mathrm{d}\mathfrak{m}$

and the quantity $\mathscr{E}(f,g)$, $f,g\in W^{1,2}(M^{\circ})$, by
polarization. Then $(\mathscr{E},D(\mathscr{E}))$ becomes a strongly local
regular Dirichlet form on $L^{2}(M;\mathfrak{m})$, since $C_{c}^{\infty}(M)$
is a dense set of $D(\mathscr{E})$. Let $k:M^{\circ}\to\mathbb{R}$ and
$\ell:\partial M\to\mathbb{R}$ be continuous functions providing lower bounds
on the Ricci curvature and the second fundamental form of $\partial M$,
respectively. Suppose that $M$ is compact and $\mathfrak{m}=\mathfrak{v}$.
Then $(M,\mathscr{E},\mathfrak{m})$ is tamed by

$\displaystyle\kappa:=k\mathfrak{v}+\ell\mathfrak{s},$

because $\mathfrak{v},\mathfrak{s}\in S_{K}({\bf X})$ (see [27, Theorem 2.36]
and [40, Theorem 5.1]). Then one can apply Theorem 1.2 and Corollary 1.3 to
$(M,\mathscr{E},\mathfrak{m})$.

More generally, if $M$ is regularly exhaustible, i.e., there exists an
increasing sequence $(M_{n})_{n\in\mathbb{N}}$ of domains $M_{n}\subset
M^{\circ}$ with smooth boundary $\partial M_{n}$ such that $g$ is smooth on
$M_{n}$ and the following properties hold:

  1. (1)

The closed sets $(\overline{M}_{n})_{n\in\mathbb{N}}$ constitute an
$\mathscr{E}$-nest for $(\mathscr{E},W^{1,2}(M))$.

  2. (2)

For all compact sets $K\subset M^{\circ}$ there exists $N\in\mathbb{N}$ such
tht $K\subset M_{n}$ for all $n\geq N$.

  3. (3)

There are lower bounds $\ell_{n}:\partial M_{n}\to\mathbb{R}$ for the
curvature of $\partial M_{n}$ with $\ell_{n}=\ell$ on $\partial M\cap\partial
M_{n}$ such that the distributions
$\kappa_{n}=k\mathfrak{v}_{n}+\ell_{n}\mathfrak{s}_{n}$ are uniformly
$2$-moderate in the sense that

$\displaystyle\sup_{n\in\mathbb{N}}\sup_{t\in[0,1]}\sup_{x\in
M_{n}}\mathbb{E}^{(n)}\left[e^{-2A_{t}^{\kappa_{n}}}\right]<\infty,$ (10.1)

where $\mathfrak{v}_{n}$ is the volume measure of $M_{n}$ and
$\mathfrak{s}_{n}$ is the surface measure of $\partial M_{n}$.

Suppose $\mathfrak{m}=\mathfrak{v}$. Then $(M,\mathscr{E},\mathfrak{m})$ is
tamed by $\kappa=k\mathfrak{v}+\ell\mathfrak{s}$ (see [27, Theorem 4.5]). In
this case, Theorem 1.2 and Corollary 1.3 hold for
$(M,\mathscr{E},\mathfrak{m})$ provided $\kappa^{+}\in S_{D}({\bf X})$,
$\kappa^{-}\in S_{K}({\bf X})$.

Let $Y$ be the domain defined by

$\displaystyle Y:=\\{(x,y,z)\in\mathbb{R}^{3}\mid
z>\phi(\sqrt{x^{2}+y^{2}})\\},$

where $\phi:[0,+\infty[\to[0,+\infty[$ is $C^{2}$ on $]0,+\infty[$ with
$\phi(r):=r-r^{2-\alpha}$, $\alpha\in]0,1[$ for $r\in[0,1]$, $\phi$ is a
constant for $r\geq 2$ and $\phi^{\prime\prime}(r)\leq 0$ for
$r\in[0,+\infty[$. Let $\mathfrak{m}_{Y}$ be the $3$-dimensional Lebesgue
measure restricted to $Y$ and $\sigma_{\partial Y}$ the $2$-dimensional
Hausdorff measure on $\partial Y$. Denote by $\mathscr{E}_{Y}$ the Dirichlet
form on $L^{2}(Y;\mathfrak{m})$ with Neumann boundary conditions. The smallest
eigenvalue of the second fundamental form of $\partial Y$ can be given by

$\displaystyle\ell(r,\phi)=\frac{\phi^{\prime\prime}(r)}{(1+|\phi^{\prime}(r)|^{2})^{3/2}}(\leq
0),$

for $r\leq 1$ and $\ell=0$ for $r\geq 2$. It is proved in [27, Theorem 4.6]
that the Dirichlet space $(Y,\mathscr{E}_{Y},\mathfrak{m}_{Y})$ is tamed by

$\displaystyle\kappa=\ell\sigma_{\partial Y}.$

In this case, the measure-valued Ricci lower bound $\kappa$ satisfies
$|\kappa|=|\ell|\sigma_{\partial Y}\in S_{K}({\bf X})$ (see [27, Lemma 2.34,
Theorem 2.36, Proof of Theorem 4.6]). Then Theorem 1.2 and Corollay 1.3 hold
for $(Y,\mathscr{E}_{Y},\mathfrak{m}_{Y})$.

######  Example 10.2 (Configuration space over metric measure spaces).

Let $(M,g)$ be a complete smooth Riemannian manifold without boundary. The
configuration space $\Upsilon$ over $M$ is the space of all locally finite
point measures, that is,

$\displaystyle\Upsilon:=\\{\gamma\in\mathcal{M}(M)\mid\gamma(K)\in\mathbb{N}\cup\\{0\\}\quad\text{
for all compact sets}\quad K\subset M\\}.$

In the seminal paper Albeverio-Kondrachev-Röckner [1] identified a natural
geometry on $\Upsilon$ by lifting the geometry of $M$ to $\Upsilon$. In
particular, there exists a natural gradient $\nabla^{\Upsilon}$, divergence
${\rm div}^{\Upsilon}$ and Laplace operator $\Delta^{\Upsilon}$ on $\Upsilon$.
It is shown in [1] that the Poisson point measure $\pi$ on $\Upsilon$ is the
unique (up to intensity) measure on $\Upsilon$ under which the gradient and
divergence become dual operator in $L^{2}(\Upsilon;\pi)$. Hence, the Poisson
measure $\pi$ is the natural volume measure on $\Upsilon$ and $\Upsilon$ can
be seen as an infinite dimensional Riemannian manifold. The canonical
Dirichlet form

$\displaystyle\mathscr{E}(F)=\int_{\Upsilon}|\nabla^{\Upsilon}F|_{\gamma}^{2}\pi(\mathrm{d}\gamma)$

constructed in [1, Theorem 6.1] is quasi-regular and strongly local and it
induces the heat semigroup $T_{t}^{\Upsilon}$ and a Brownian motion ${\bf
X}^{\Upsilon}$ on $\Upsilon$ which can be identified with the independent
infinite particle process. If ${\rm Ric}_{g}\geq K$ on $M$ with
$K\in\mathbb{R}$, then $(\Upsilon,\mathscr{E}^{\Upsilon},\pi)$ is tamed by
$\kappa:=K\pi$ with $|\kappa|\in S_{K}({\bf X}^{\Upsilon})$ (see [26, Theorem
4.7] and [27, Theorem 3.6]). Then one can apply Theorem 1.2 and Corollary 1.3
for $(\Upsilon,\mathscr{E}^{\Upsilon},\pi)$.

More generally, in Dello Schiavo-Suzuki [21], configuration space $\Upsilon$
over proper complete and separable metric space $(M,{\sf d})$ is considered.
The configuration space $\Upsilon$ is endowed with the _vage topology_
$\tau_{V}$, induced by duality with continuous compactly supported functions
on $M$, and with a reference Borel probability measure $\mu$ satisfying [21,
Assumption 2.17], commonly understood as the law of a proper point process on
$M$. In [21], they constructed the strongly local Dirichlet form
$\mathscr{E}^{\Upsilon}$ defined to be the $L^{2}(\Upsilon;\mu)$-closure of a
certain pre-Dirichlet form on a class of certain cylinder functions and prove
its quasi-regularity for a wide class of measures $\mu$ and base spaces (see
[21, Proposition 3.9 and Theorem 3.45]). Moreover, in Dello Schiavo-Suzuki
[22], for any fixed $K\in\mathbb{R}$ they prove that a Dirichlet form
$(\mathscr{E},D(\mathscr{E}))$ with its carré-du-champ $\Gamma$ satisfies
${\sf BE}_{2}(K,\infty)$ if and only if the Dirichlet form
$(\mathscr{E}^{\Upsilon},D(\mathscr{E}^{\Upsilon}))$ on $L^{2}(\Upsilon;\mu)$
with its carré-du-champ $\Gamma^{\Upsilon}$ satisfies ${\sf
BE}_{2}(K,\infty)$. Hence, if $(M,\mathscr{E},\mathfrak{m})$ is tamed by
$K\mathfrak{m}$ with $|K|\mathfrak{m}\in S_{K}({\bf X})$, then
$(\Upsilon,\mathscr{E}^{\Upsilon},\mu)$ is tamed by $\kappa:=K\mu$ with
$|\kappa|\in S_{K}({\bf X}^{\Upsilon})$. Then Theorem 1.2 and Corollary 1.3
hold for $(\Upsilon,\mathscr{E}^{\Upsilon},\mu)$ under the suitable class of
measures $\mu$ defined in [21, Assumption 2.17].