# Riesz transforms for Dirichlet spaces tamed by distributional curvature
lower bounds

Syota Esaki111Syota Esaki<EMAIL_ADDRESS>Department of Applied
Mathematics, Fukuoka University, Fukuoka 814-0180, Japan. Supported in part by
JSPS Grant-in-Aid for Scientific Research (C) (No. 23K03158) and fund fund
(No. 215001) from the Central Research Institute of Fukuoka University. Zi
Jian Xu222Zi Jian Xu<EMAIL_ADDRESS>Department of Applied
Mathematics, Fukuoka University, Fukuoka 814-0180, Japan.  and Kazuhiro
Kuwae333🖂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 [22] as a Dirichlet space having a weak form of Bakry-Émery curvature
lower bounds in distribution sense. After their work, Braun [12] 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 Littlewood-Paley-Stein
inequality for $1$-forms as an element of $L^{p}$-cotangent bundles and
boundedness of Riesz transforms, which partially solves the problem raised by
Kawabi-Miyokawa [35].

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, $L^{2}$-differential forms, $L^{2}$-vector fields, Littlewood-Paley-
Stein inequality, Riesz transform, Bochner Laplacian, Hodge-Kodaira Laplacian,
measure valued Bochner-Wietzenböck formula, intertwining property, Wiener
space, path space with Gibbs measure, RCD-space, weighted Riemannian manifold,
Riemmanian manifold with boundary, configuration space,

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 $(P_{t})_{t\geq 0}$ the
associated symmetric sub-Markovian strongly continuous semigroup on
$L^{2}(M;\mathfrak{m})$ (see [43, Chapter IV, Definition 3] for the quasi-
regularity and see [36, 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 $f\in
L^{2}(M;\mathfrak{m})\cap\mathscr{B}(M)$, $P_{t}f(x)=\mathbb{E}_{x}[f(X_{t})]$
$\mathfrak{m}$-a.e. $x\in M$ (see [43, Chapter IV, Section 3]). The
$\mathfrak{m}$-symmetry of ${\bf X}$, $(P_{t})_{t\geq 0}$ can be extended to a
strongly continuous contraction semigroup on $L^{p}(M;\mathfrak{m})$ for
$p\in[1,+\infty[$. Denote by $(D(\Delta_{p}),\Delta_{p})$ its generator on
$L^{p}(M;\mathfrak{m})$ defined by

$\displaystyle\left\\{\begin{array}[]{cl}D(\Delta_{p})&\displaystyle{:=\left\\{f\in
L^{p}(M;\mathfrak{m})\mid\exists\lim_{t\to 0}\frac{P_{t}f-f}{t}\text{ in
}L^{p}(M;\mathfrak{m})\right\\}},\\\ \Delta_{p}f&\displaystyle{:=\lim_{t\to
0}\frac{P_{t}f-f}{t}\text{ for }f\in D(\Delta_{p})}.\end{array}\right.$

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 [25, 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 [36]).

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 [25, Theorem 2.1.3], [43, 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 [36, 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 a smooth measure of Dynkin class ($\kappa^{+}\in S_{D}({\bf X})$ in short)
and $2\kappa^{-}$ is a smooth measure of extended Kato class ($2\kappa^{-}\in
S_{E\\!K}({\bf X})$ 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 [43, Chapter VI,
Definition 2.3], [25, 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

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

$\displaystyle
C^{-1}\mathscr{E}_{1}(f,f)\leq\mathscr{E}^{q\kappa}_{\alpha_{0}}(f,f)\leq
C\mathscr{E}_{1}(f,f)\quad\text{ for all }\quad f\in
D(\mathscr{E}^{q\kappa})=D(\mathscr{E})$ (1.2)

by (1.1) (see [14, (3.8)] and [15, 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 [15, 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 $C=C(\kappa)>0,C_{\kappa}\geq 0$ depending only on
$\kappa^{-}$ such that for every $t\geq 0$

$\displaystyle\|P_{t}^{\kappa}\|_{p,p}\leq Ce^{C_{\kappa}t}.$ (1.3)

Here $C=1$ under $\kappa^{-}=0$. $C_{\kappa}\geq 0$ can be taken to be $0$
under $\kappa^{-}=0$.

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

$\displaystyle\left\\{\begin{array}[]{ll}D(\Delta^{q\kappa})&:=\\{u\in
D(\mathscr{E})\mid\text{there exists }w\in L^{2}(M;\mathfrak{m})\text{ such
that }\\\ &\hskip
85.35826pt\mathscr{E}^{q\kappa}(u,v)=-\int_{M}wv\,\mathrm{d}\mathfrak{m}\quad\text{
for any }\quad v\in D(\mathscr{E})\\},\\\ \Delta^{q\kappa}u&:=w\quad\text{ for
}w\in L^{2}(M;\mathfrak{m})\quad\text{specified as above,}\end{array}\right.$
(1.7)

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 3.2 below), in particular, the
heat flow $(P_{t})_{t\geq 0}$ satisfies

$\displaystyle\sqrt{\Gamma(P_{t}f)}\leq
P_{t}^{\kappa}\sqrt{\Gamma(f)}\quad\mathfrak{m}\text{-a.e.~{}for any }f\in
D(\mathscr{E})\quad\text{ and }\quad t\geq 0$ (1.8)

(see [22, Definition 3.3 and Theorem 3.4]). The inequality (1.8) 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 [22] for the definition of tamed
space. The $\mathfrak{m}$-symmetric Markov process ${\bf X}$ treated in our
paper may not be conservative in general. Under Assumption 1, a sufficient
condition (intrinsic stochastic completeness called in [22]) for
conservativeness of ${\bf X}$ is discussed in [22, Section 3.3], in
particular, under $1\in D(\mathscr{E})$ and Assumption 1, we have the
conservativeness of ${\bf X}$.

Our main theorem under Assumption 1 is the following:

######  Theorem 1.2.

Let $p\in]1,+\infty[$ and $\alpha>C_{\kappa}$. For $f\in
L^{p}(M;\mathfrak{m})\cap L^{2}(M;\mathfrak{m})$, we define the Riesz operator
$R_{\alpha}(\Delta)$ by

$\displaystyle
R_{\alpha}(\Delta)f:=\Gamma((\alpha-\Delta)^{-\frac{1}{2}}f)^{\frac{1}{2}}.$

Suppose $\kappa^{-}\in S_{K}({\bf X})$ and $p\in[2,+\infty[$, or
$\kappa_{\ll}^{-}=-R\mathfrak{m}$ with $R\geq 0$, $\kappa_{\perp}^{-}=0$ and
$p\in]1,2]$. Then, $R_{\alpha}(\Delta)$ can be extended to a (non-linear)
bounded operator on $L^{p}(M;\mathfrak{m})$. The operator norm
$\|R_{\alpha}(\Delta)\|_{p,p}$ depends only on $\kappa,p$ and $C_{\kappa}$.

Fix $p\in]1,\infty[$. We define the $(1,p)$-Sobolev space
$(H^{1,p}(M),\|\cdot\|_{H^{1,p}})$ as follows:

######  Definition 1.3.

Let $(\mathscr{E},D(\mathscr{E}))$ be a quasi-regular strongly local Dirichlet
form on $L^{2}(M;\mathfrak{m})$ admitting carré du champ $\Gamma$. Set
$\mathscr{D}_{1,p}:=\\{u\in L^{p}(M;\mathfrak{m})\cap
D(\mathscr{E})\mid\Gamma(u)^{\frac{1}{2}}\in L^{p}(M;\mathfrak{m})\\}$. For
$u\in\mathscr{D}_{1,p}$, we define
$\|u\|_{H^{1,p}}^{p}:=\|u\|_{L^{p}(M;\mathfrak{m})}^{p}+\|\Gamma(u)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}^{p}$.
Then $\|\cdot\|_{H^{1,p}}$ forms a norm on $\mathscr{D}_{1,p}$. We now define
the Sobolev space $H^{1,p}(M)$ as the $\|\cdot\|_{H^{1,p}}$-completion of
$\mathscr{D}_{1,p}$:

$\displaystyle\left\\{\begin{array}[]{cl}H^{1,p}(M)&:=\overline{\mathscr{D}_{1,p}}^{\|\cdot\|_{H^{1,p}}},\\\
\|u\|_{H^{1,p}}&:=\left(\|u\|_{L^{p}(M;\mathfrak{m})}^{p}+\|\Gamma(u)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}^{p}\right)^{\frac{1}{p}}\quad\text{
for }\quad u\in H^{1,p}(M).\end{array}\right.$

More precisely, for $u\in H^{1,p}(M)$, we can extend the carré du champ
$\Gamma$ to $u$ with $\Gamma(u)^{\frac{1}{2}}\in L^{p}(M;\mathfrak{m})$. We
set $D(\mathscr{E}^{\,p}):=H^{1,p}(M)$ and
$\mathscr{E}^{\,p}(u):=\|\Gamma(u)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}^{p}$
for $u\in D(\mathscr{E}^{p})$, which is called the $p$-energy of $u\in
H^{1,p}(M)$.

It is proved in [38, Lemma 3.1] that $(H^{1,p}(M),\|\cdot\|_{H^{1,p}})$ is a
uniformly convex Banach space and consequently it is reflexive.

The following theorems are essentially shown in the framework of weighted
Riemannian manifolds $(M,g,e^{-\phi}v_{g})$ (without boundary) with Witten
Laplacian $L:=\Delta-\langle\nabla\phi,\cdot\rangle$. For the completeness, we
restate them in our framework under Assumption 1.

######  Theorem 1.4.

Fix $\alpha>C_{\kappa}$. The following statements are equivalent, and one of
them (hence all) holds under $\kappa^{-}\in S_{K}({\bf X})$ and
$p\in[2,+\infty[$, or under $\kappa_{\ll}^{-}=-R\mathfrak{m}$ with $R\geq 0$,
$\kappa_{\perp}^{-}=0$ and $p\in]1,2]$.

  1. (1)

The Riesz transform $R_{\alpha}(\Delta)$ is bounded in $L^{p}(M;\mathfrak{m})$
for all $p\in]1,+\infty[$. That is, for every $p\in]1,+\infty[$, there exists
a constant $\|R_{\alpha}(\Delta)\|_{p,p}\in]0,+\infty[$ such that

$\displaystyle\|R_{\alpha}(\Delta)f\|_{L^{p}(M;\mathfrak{m})}\leq\|R_{\alpha}(\Delta)\|_{p,p}\|f\|_{L^{p}(M;\mathfrak{m})},\quad
f\in{\rm Test}(M)_{fs},$

where ${\rm Test}(M)_{fs}$ is the family of test functions with finite
supports (see Definition 3.3 below).

  2. (2)

The Sobolev norms $\|\cdot\|_{H^{1,p}}$, $\|\cdot\|_{1,p}$ and
$\||\cdot|\|_{1,p}$ are equivalent, where

$\displaystyle\|f\|_{1,p}:$
$\displaystyle=\sqrt{\alpha}\|f\|_{L^{p}(M;\mathfrak{m})}+\|\Gamma(f)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})},$
$\displaystyle\||f|\|_{1,p}:$
$\displaystyle=\|(\alpha-\Delta_{p})^{\frac{1}{2}}f\|_{L^{p}(M;\mathfrak{m})}$

for $f\in H^{1,p}(M)$.

  3. (3)

The Sobolev space $H^{1,p}(M)$ coincides with the Sobolev space $W^{1,p}(M)$
defined by $W^{1,p}(M):=V_{\alpha}(L^{p}(M;\mathfrak{m}))$. Here $V_{\alpha}$
is the bounded linear operator on $L^{p}(M;\mathfrak{m})$ defined by
$V_{\alpha}f:=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}e^{-\alpha
t}t^{-\frac{1}{2}}P_{t}f\mathrm{d}t=(\alpha-\Delta_{p})^{-\frac{1}{2}}f\in
L^{p}(M;\mathfrak{m})$ satisfying
$\sqrt{\alpha}\|V_{\alpha}f\|_{L^{p}(M;\mathfrak{m})}\leq\|f\|_{L^{p}(M;\mathfrak{m})}$.

  4. (4)

The $(1,p)$-capacities $c_{1,p}$ and $C_{1,p}$ are equivalent, where

$\displaystyle c_{1,p}(O):$ $\displaystyle=\inf\\{\|f\|_{1,p}^{p}\mid f\in
H^{1,p}(M),f\geq 0\;\mathfrak{m}\text{-a.e.~{}on }M,f\geq
1\;\mathfrak{m}\text{-a.e.~{}on }O\\},$ $\displaystyle C_{1,p}(O):$
$\displaystyle=\inf\\{\|f\|_{1,p}^{p}\mid f\in W^{1,p}(M),f\geq
0\;\mathfrak{m}\text{-a.e.~{}on }M,f\geq 1\;\mathfrak{m}\text{-a.e.~{}on
}O\\},$

for all open sets $O\subset X$, and for all $A\subset X$,

$\displaystyle c_{1,p}(A):$ $\displaystyle=\inf\\{c_{1,p}(O)\mid f\in
H^{1,p}(M),A\subset O\subset X,O\text{ is open}\\},$ $\displaystyle
C_{1,p}(A):$ $\displaystyle=\inf\\{C_{1,p}(O)\mid f\in W^{1,p}(M),A\subset
O\subset X,O\text{ is open}\\}.$

###### Proof.

The proof is quite similar with the proof of [40, Theorem 3.1]. We omit it. ∎

######  Theorem 1.5.

Let $p\in]1,+\infty[$. Suppose that the Riesz transform $R_{\alpha}(\Delta)$
is bounded in $L^{p}(M;\mathfrak{m})$ for some $\alpha\geq 0$. Then
$R_{\beta}(\Delta)$ is bounded in $L^{p}(M;\mathfrak{m})$ for all
$\beta>\alpha\wedge 0$. Moreover, there exists a constant $A_{p}>0$ such that

$\displaystyle\|R_{\beta}(\Delta)\|_{p,p}\leq
A_{p}(\sqrt{\alpha\beta^{-1}}\lor 1)\|R_{\alpha}(\Delta)\|_{p,p}.$

###### Proof.

The proof is quite similar with the proof of [40, Theorem 3.2]. We omit it. ∎

######  Theorem 1.6.

Let $p\in]1,+\infty[$. Suppose that the Riesz transform
$R_{\alpha}(\Delta):=\Gamma(\nabla(\alpha-\Delta_{p})f)^{\frac{1}{2}}$ is
bounded in $L^{p}(M;\mathfrak{m})$ for some $\alpha>0$ and the bottom of
spectrum of $-\Delta$ is strictly positive, i.e.,

$\displaystyle\lambda_{2}(-\Delta):=\inf_{f\in
D(\mathscr{E})\setminus\\{0\\}}\frac{\mathscr{E}(f,f)}{\;\quad\|f\|_{L^{2}(M;\mathfrak{m})}^{2}}.$

Then the Riesz $R_{0}(\Delta):=\Gamma(\nabla(-\Delta_{p})f)^{\frac{1}{2}}$ is
bounded in $L^{p}(M;\mathfrak{m})$.

###### Proof.

The proof is quite similar with the proof of [40, Theorem 3.3]. We omit it. ∎

Under Assumption 1, we have also the following:

######  Theorem 1.7.

Let $p\in]1,+\infty[$ and $\alpha>C_{\kappa}$. Suppose $\kappa^{-}\in
S_{K}({\bf X})$ under $p\in[2,+\infty[$, or $\kappa_{\ll}=-R\mathfrak{m}$,
$\kappa_{\perp}=0$ under $p\in]1,2]$. Then we have the following:

  1. (1)

There exists $A_{p}>0$ such that for any $t>0$ and $f\in
L^{p}(M;\mathfrak{m})$

$\displaystyle\|\Gamma(P_{t}f)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}\leq
A_{p}\|R_{\alpha}(\Delta)\|_{p,p}(\alpha^{1/2}+t^{-1/2})\|f\|_{L^{p}(M;\mathfrak{m})}.$
(1.9)

  2. (2)

There exists $A_{p}>0$ such that for any ${\varepsilon}\in]0,1[$, $t>0$ and
$f\in L^{p}(M;\mathfrak{m})$

$\displaystyle\|\Gamma(P_{t}f)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}\leq
A_{p}\frac{\|R_{\alpha}(\Delta)\|_{p,p}}{\sqrt{{\varepsilon}}}\cdot\frac{e^{\alpha{\varepsilon}t}}{\sqrt{t}}\|f\|_{L^{p}(M;\mathfrak{m})}.$
(1.10)

  3. (3)

If the heat semigroup $P_{t}$ is hypercontractive, i.e.,
$\|P_{t}\|_{p,q(t)}=m_{p,q}(t)$, then there exists $C_{p}>0$ such that for any
${\varepsilon}\in]0,1[$. $t>0$ and $f\in L^{p}(M;\mathfrak{m})$

$\displaystyle\|\Gamma(P_{t}f)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}\leq
C_{p}\frac{\|R_{\alpha}(\Delta)\|_{p,p}}{\sqrt{{\varepsilon}}}\cdot\frac{e^{\alpha{\varepsilon}t}m_{p,q}((1-{\varepsilon})t)}{\sqrt{t}}\|f\|_{L^{p}(M;\mathfrak{m})}.$
(1.11)

  4. (4)

There exists $A_{p}>0$ such that, for any ${\varepsilon}\in]0,1[$, $t>0$ and
$f\in L^{p}(M;\mathfrak{m})$

$\displaystyle\|\Gamma(Q_{t}^{(\alpha)}f)^{\frac{1}{2}}\|_{L^{p}(M;\mathfrak{m})}\leq
C_{p}\frac{\|R_{\alpha}(\Delta)\|_{p,p}}{{\varepsilon}}\cdot\frac{e^{-\sqrt{\alpha}{\varepsilon}t}}{t}\|f\|_{L^{p}(M;\mathfrak{m})},$
(1.12)

where $Q_{t}^{(\alpha)}f:=e^{-t\sqrt{\alpha-\Delta}}f$ is the Cauchy
semigroup.

###### Proof.

The proof is quite similar with the proof of [40, Theorem 3.4]. We omit it. ∎

##  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 [25, 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 [25, (5.14)]). A smooth measure $\nu$ 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
smooth measure $\nu$ 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
[54, 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,
$\|P_{t}^{-\nu}\|_{p,p}\leq Ce^{C_{\nu}t}$ for $p\in[2,+\infty]$. 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

$\displaystyle C_{\beta\nu}\in]0,C_{\nu}]\quad\text{ and }\quad
C(\nu)=C(\beta\nu).$ (2.2)

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

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

for some small $t_{0}\in]0,1]$ satisfying
$\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{\infty}<1$ (see [54, 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. $C_{\beta\kappa}$).

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. $1$).

The following lemma might be known for experts:

######  Lemma 2.1.

For $\nu\in S_{E\\!K}({\bf X})$ and $\alpha>C_{\nu}$,

$\displaystyle\left\|\mathbb{E}_{\cdot}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]\right\|_{L^{\infty}(M;\mathfrak{m})}<1.$

###### Proof.

Take any sufficiently small $t_{0}\in]0,T[$. By use of Markov property,

$\displaystyle\mathbb{E}_{x}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]$
$\displaystyle=\sum_{n=0}^{\infty}\mathbb{E}_{x}\left[\int_{nt_{0}}^{(n+1)t_{0}}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]$
$\displaystyle=\sum_{n=0}^{\infty}\mathbb{E}_{x}\left[\int_{0}^{t_{0}}e^{-\alpha(nt_{0}+t)}\mathrm{d}A_{t}^{\nu}\circ\theta_{nt_{0}}\right]$
$\displaystyle=\sum_{n=0}^{\infty}e^{-\alpha
nt_{0}}\mathbb{E}_{x}\left[\mathbb{E}_{X_{nt_{0}}}\left[\int_{0}^{t_{0}}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]\right]$
$\displaystyle\leq\sum_{n=0}^{\infty}e^{-\alpha nt_{0}}q\text{-}\sup_{x\in
M}\mathbb{E}_{x}\left[A_{t_{0}}^{\nu}\right]$
$\displaystyle=\frac{1}{1-e^{-\alpha
t_{0}}}\|\mathbb{E}_{\cdot}[A_{t_{0}}^{\nu}]\|_{L^{\infty}(M;\mathfrak{m})}$
$\displaystyle\leq\frac{1-e^{-C_{\nu}(t_{0})t_{0}}}{1-e^{-\alpha t_{0}}}.$

Then

$\displaystyle\left\|\mathbb{E}_{\cdot}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]\right\|_{L^{\infty}(M;\mathfrak{m})}\leq\frac{1-e^{-C_{\nu}(t_{0})t_{0}}}{1-e^{-\alpha
t_{0}}}<1$

under $\alpha>C_{\nu}(t_{0})$. ∎

######  Corollary 2.2.

For $\nu\in S_{E\\!K}({\bf X})$ and $\alpha>C_{\nu}$,

$\displaystyle(1-\|U_{\alpha}\nu\|_{\infty})\mathscr{E}(f,f)\leq\mathscr{E}^{-\nu}_{\alpha\|U_{\alpha}\nu\|_{\infty}}(f,f)\leq\mathscr{E}_{\alpha\|U_{\alpha}\nu\|_{\infty}}(f,f)\quad\text{
for }\quad f\in D(\mathscr{E}),$ (2.3)

where $U_{\alpha}\nu(x):=\mathbb{E}_{x}\left[\int_{0}^{\infty}e^{-\alpha
t}\mathrm{d}A_{t}^{\nu}\right]$. In particular, under $\kappa^{+}\in
S_{D}({\bf X})$ and $\kappa^{-}\in S_{E\\!K}({\bf X})$, for
$\alpha>C_{\kappa}$, we have

$\displaystyle(1-\|U_{\alpha}\kappa^{-}\|_{\infty})\mathscr{E}(f,f)\leq\mathscr{E}^{\kappa}_{\alpha\|U_{\alpha}\kappa^{-}\|_{\infty}}(f,f)\leq
C\mathscr{E}_{1}(f,f)\quad\text{ for }\quad f\in D(\mathscr{E}),$ (2.4)

where
$C:=\max\left\\{(1+\|U_{\alpha}\kappa^{+}\|_{\infty}),\alpha\|U_{\alpha}|\kappa|\|_{\infty}\right\\}$.

##  3 Test functions

The following lemmas hold.

######  Lemma 3.1 ([22, Proposition 6.7]).

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

$\displaystyle\Gamma(P_{t}f)\leq\frac{1}{2t}\|P_{t}^{-\kappa^{-}}\|_{\infty,\infty}\cdot\|f\|_{L^{\infty}(M;\mathfrak{m})}^{2}.$
(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 3.2 ([22, 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 (1.8).

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

######  Definition 3.3.

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 $p\in]1,+\infty[$.

######  Lemma 3.4 ([46, 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 3.1 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 [27, 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 [27, 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 [27,
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 [27, Corollary 1.2.18], while for $p=1$, the
implication from “reflexive as Banach space” to “reflexive as
$L^{\infty}$-module” still holds [27, 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 [27, 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 ([27, 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 [27, 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 [30, 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 [27, 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 $\|A\|_{\mathscr{M}_{1}\otimes\mathscr{M}_{2}}langle\infty$. Denote by
$A^{\top}\in\mathscr{M}^{\otimes 2}$ the transpose of
$A\in\mathscr{M}^{\otimes 2}$ as defined in [27, 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 [27, 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 [30, 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 [30, 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 [30, 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 [12, §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 [12, Lemma 2.3 and Theorem 2.4]:

######  Lemma 4.9 ([12, Lemma 2.3]).

The map from ${\rm SF}(M)\times{\rm Pcm}/\\!\\!\sim$ into ${\rm
Pcm}/\\!\\!\sim$ defined in (4.6) extends continuously and unniquely 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, [12, 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 [12, 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, [12, 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 [30, 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)

######  Definition 4.15 (Measure-valued divergence).

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

$\displaystyle D({\bf div}):$ $\displaystyle=\Bigl{\\{}X\in
L^{2}(TM)\,\Bigl{|}\,\text{ there exists a $\sigma$-finite signed Radon smooth
measure }\nu$ $\displaystyle\hskip 28.45274pt\text{ such that for every }h\in
D(\mathscr{E})_{c}\cap
L^{\infty}(M;\mathfrak{m}),\quad-\int_{M}h\,\mathrm{d}\nu=\int_{M}\mathrm{d}h(X)\mathrm{d}\mathfrak{m}\Bigr{\\}}.$
(4.12)

Such a $\sigma$-finite signed Radon smooth measure $\nu$ is unique, and it is
called the _measure-valued divergence_ of $X$ and denoted by ${\bf div}\,X$.
Moreover, we define the subclass $D_{f}({\bf div})$ of $D({\bf div})$ as
follows:

$\displaystyle D_{f}({\bf div}):$ $\displaystyle=\Bigl{\\{}X\in
L^{2}(TM)\,\Bigl{|}\,\text{ there exists a finite signed smooth measure }\nu$
$\displaystyle\hskip 14.22636pt\text{ such that for every }h\in
D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m}),\quad-\int_{M}h\,\mathrm{d}\nu=\int_{M}\mathrm{d}h(X)\mathrm{d}\mathfrak{m}\Bigr{\\}}.$
(4.13)

Such a finite signed smooth measure $\nu$ is unique, and it is also called the
_measure-valued divergence_ of $X$ and denoted by ${\bf div}\,X$.

The following Leibniz rule is an improvement of [12, Lemma 3.13] and is
useful.

######  Lemma 4.16.

For every $X\in D({\bf div})$ (resp. $X\in D_{f}({\bf div})$) and every $f\in
D(\mathscr{E})_{e}\cap L^{\infty}(M;\mathfrak{m})$, we have $fX\in D({\bf
div})$ (resp. $fX\in D_{f}({\bf div})$) with

$\displaystyle{\bf div}(fX)=\tilde{f}{\bf div}\,X+\mathrm{d}f(X)\mathfrak{m}.$

###### Proof.

We only show the proof for the case $X\in D({\bf div})$. The proof for the
case $X\in D_{f}({\bf div})$ is similar. It is easy to see $fX\in L^{2}(TM)$.
Moreover $(\mathrm{d}f)(X)=\langle\nabla f,X\rangle\in L^{1}(M;\mathfrak{m})$,
since $X\in D({\bf div})$ and $f\in D(\mathscr{E})_{e}\cap
L^{\infty}(M;\mathfrak{m})$. Then $(\mathrm{d}f)(X)\mathfrak{m}$ is a finite
signed smooth measure, hence a $\sigma$-finite signed Radon smooth measure.
Then the measure $\nu$ defined by

$\displaystyle\nu:=\tilde{f}{\bf div}\,X+\mathrm{d}f(X)\mathfrak{m}$

is also a signed Radon smooth measure, because $\tilde{f}$ is bounded. For
$h\in D(\mathscr{E})_{c}\cap L^{\infty}(M;\mathfrak{m})$,

$\displaystyle-\int_{M}\tilde{h}\,\mathrm{d}\nu$
$\displaystyle=-\int_{M}\tilde{h}\tilde{f}\,\mathrm{d}{\bf
div}\,X-\int_{M}h\,\mathrm{d}f(X)\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}\mathrm{d}(hf)(X)\mathrm{d}\mathfrak{m}-\int_{M}h\mathrm{d}f(X)\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}(\mathrm{d}(hf)-h\mathrm{d}f)(X)\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}f\mathrm{d}h(X)\mathrm{d}\mathfrak{m}=\int_{M}\mathrm{d}h(fX)\,\mathrm{d}\mathfrak{m},$

which implies $\nu={\bf div}\,fX$. Here we use the Leibniz rule (see [12,
Proposition 2.11]): For every $f,g\in D(\mathscr{E})_{e}\cap
L^{\infty}(M;\mathfrak{m})$,

$\displaystyle\mathrm{d}(fg)=\tilde{f}\,\mathrm{d}g+\tilde{g}\,\mathrm{d}f.$

∎

###  4.7 Measure-valued Laplacian

######  Definition 4.17 (Measure-valued Schrödinger operator).

We define $D({\text{\boldmath$\Delta$}}^{2\kappa})$ to consist of all $u\in
D(\mathscr{E})$ for which there exists a $\sigma$-finite Radon signed smooth
measure $\nu$ such that for $D(\mathscr{E})\subset L^{1}(M;|\nu|)$ and for
every $h\in D(\mathscr{E})$, we have

$\displaystyle-\int_{M}\tilde{h}\mathrm{d}\nu=\mathscr{E}^{2\kappa}(h,u).$

In case of existence, $\nu$ is unique, denoted by
${\text{\boldmath$\Delta$}}^{2\kappa}u$ and shall be called the _measure-
valued Schrödinger operator_ with potential $2\kappa$. We define

$\displaystyle
D_{f}({\text{\boldmath$\Delta$}}^{2\kappa}):=\\{u\in\text{$D(\text{\boldmath$\Delta$}^{2\kappa})$}\mid{\text{\boldmath$\Delta$}}^{2\kappa}u\text{
is a finite signed smooth measure }\\}.$

######  Definition 4.18 (Measure-valued Laplacian).

We define $D({\text{\boldmath$\Delta$}})$ to consist of all $u\in
D(\mathscr{E})$ for which there exists a $\sigma$-finite Radon signed smooth
measure $\nu$ such that for every $h\in D(\mathscr{E})_{c}\cap
L^{\infty}(M;\mathfrak{m})$, we have

$\displaystyle-\int_{M}\tilde{h}\mathrm{d}\nu=\mathscr{E}(h,u).$

In case of existence, $\nu$ is unique, denoted by
${\text{\boldmath$\Delta$}}u$ and shall be called the _measure-valued
Laplacian_. In another word, $D({\text{\boldmath$\Delta$}})$ consists of all
$u\in D(\mathscr{E})$ for which $\nabla u\in D({\bf div})$, and
${\text{\boldmath$\Delta$}}u:={\bf div}\nabla u$. The subspace
$D_{f}({\text{\boldmath$\Delta$}})(\subset\text{$D(\text{\boldmath$\Delta$})$})$
such that the ${\text{\boldmath$\Delta$}}u$ forms a finite signed smooth
measure for $u\in D_{f}({\text{\boldmath$\Delta$}})$ can be similarly defined
by replacing the test functions $h\in D(\mathscr{E})_{c}\cap
L^{\infty}(M;\mathfrak{m})$ with $h\in D(\mathscr{E})\cap
L^{\infty}(M;\mathfrak{m})$. In particular,
$D_{f}({\text{\boldmath$\Delta$}})$ consists of all $u\in D(\mathscr{E})$ for
which $\nabla u\in D_{f}({\bf div})$. For $u\in
D({\text{\boldmath$\Delta$}})$, denote by ${\text{\boldmath$\Delta$}}_{\ll}u$
(resp. ${\text{\boldmath$\Delta$}}_{\perp}u$) the absolutely continuous (resp.
singular) part of ${\text{\boldmath$\Delta$}}u$ with respect to $\mathfrak{m}$
and set

$\displaystyle\Delta_{\ll}u:=\frac{\mathrm{d}{\text{\boldmath$\Delta$}}_{\ll}u}{\mathrm{d}\mathfrak{m}}.$

######  Lemma 4.19 (cf. [12, Lemma 8.12]).

The signed Borel measure $\tilde{u}^{2}\kappa$ for $u\in D(\mathscr{E})$ has
finite total variation. For $u\in D(\mathscr{E})$ with $u^{2}\in
D({\text{\boldmath$\Delta$}}^{2\kappa})$ (resp. $u^{2}\in
D_{f}({\text{\boldmath$\Delta$}}^{2\kappa})$), we have $u^{2}\in
D({\text{\boldmath$\Delta$}})$ (resp. $u^{2}\in
D_{f}({\text{\boldmath$\Delta$}})$) with

$\displaystyle{\text{\boldmath$\Delta$}}^{2\kappa}u^{2}={\text{\boldmath$\Delta$}}u^{2}-2\tilde{u}^{2}\kappa.$

###### Proof.

Since $|\kappa|\in S_{D}({\bf X})$, we see that $\tilde{u}^{2}|\kappa|$ and
$\tilde{u}^{2}\kappa^{\pm}$ are finite smooth measures in view of Stollmann-
Voigt’s inequality (1.1). Since $(M,\tau)$ is a Lusin topological space, any
finite measure is a Radon smooth measure. Thus, $\tilde{u}^{2}\kappa$ is a
signed Radon smooth measure. For $u\in D(\mathscr{E})$ with $u^{2}\in
D({\text{\boldmath$\Delta$}}^{2\kappa})$ (resp. $u^{2}\in
D_{f}({\text{\boldmath$\Delta$}}^{2\kappa})$), if we set
$\nu:={\text{\boldmath$\Delta$}}^{2\kappa}u^{2}+2\tilde{u}^{2}\kappa$, then
$\nu$ is a signed Radon (resp. finite) smooth measure, and for $h\in
D(\mathscr{E})_{c}\cap L^{\infty}(M;\mathfrak{m})$ (resp. $h\in
D(\mathscr{E})\cap L^{\infty}(M;\mathfrak{m})$), we have

$\displaystyle-\int_{M}\tilde{h}\,\mathrm{d}\nu$
$\displaystyle=-\int_{M}\tilde{h}\,\mathrm{d}{\text{\boldmath$\Delta$}}^{2\kappa}u^{2}-2\int_{M}\tilde{h}\tilde{u}^{2}\mathrm{d}\kappa$
$\displaystyle=\mathscr{E}^{2\kappa}(h,u^{2})-\langle
2\kappa,\tilde{h}\tilde{u}^{2}\rangle$ $\displaystyle=\mathscr{E}(h,u^{2}),$

hence $u^{2}\in D({\text{\boldmath$\Delta$}})$ (resp. $u^{2}\in
D_{f}({\text{\boldmath$\Delta$}})$) and $\nu={\text{\boldmath$\Delta$}}u^{2}$.
∎

######  Lemma 4.20.

Take $f\in D({\text{\boldmath$\Delta$}})$. Let $\Phi\in C^{2}(\mathbb{R})$
satisfy $\Phi(0)=0$, $\Phi^{\prime},\Phi^{\prime\prime}\in C_{b}(\mathbb{R})$.
Then $\Phi(f)\in D({\text{\boldmath$\Delta$}})$ with

$\displaystyle{\text{\boldmath$\Delta$}}\Phi(f)$
$\displaystyle=\Phi^{\prime\prime}(f)|\nabla
f|^{2}\mathfrak{m}+\Phi^{\prime}(f){\text{\boldmath$\Delta$}}f,$ (4.14)
$\displaystyle\Delta_{\ll}\Phi(f)$
$\displaystyle=\Phi^{\prime\prime}(f)|\nabla
f|^{2}+\Phi^{\prime}(f)\Delta_{\ll}f.$ (4.15)

If further $f\in D_{f}({\text{\boldmath$\Delta$}})$, then $\Phi(f)\in
D_{f}({\text{\boldmath$\Delta$}})$ with (4.14) and (4.15).

###### Proof.

It is easy to see $\Phi(f)\in D(\mathscr{E})$ by $\Phi(0)=0$. Take $h\in
D(\mathscr{E})_{c}\cap L^{\infty}(M;\mathfrak{m})$. Then, by way of the
Leibniz rule for measure-valued divergence (see Lemma 4.16),

$\displaystyle-\mathscr{E}(h,\Phi(f))$ $\displaystyle=-\int_{M}\langle\nabla
h,\nabla\Phi(f)\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle=-\int_{M}\langle\nabla h,\Phi^{\prime}(f)\nabla
f\rangle\mathrm{d}\mathfrak{m}$
$\displaystyle=\int_{M}\tilde{h}\,\mathrm{d}\,{\bf div}(\Phi^{\prime}(f)\nabla
f)$ $\displaystyle=\int_{M}\tilde{h}\,\mathrm{d}\left(\Phi^{\prime}(f){\bf
div}(\nabla f)+\mathrm{d}\Phi^{\prime}(f)(\nabla
f)\right)\quad(\text{Lemma~{}\ref{lem:LeibnizRule}})$
$\displaystyle=\int_{M}\tilde{h}\,\mathrm{d}\left(\Phi^{\prime}(f){\text{\boldmath$\Delta$}}f+\Phi^{\prime\prime}(f)|\nabla
f|^{2}\mathfrak{m}\right),$

which implies $\Phi(f)\in D({\text{\boldmath$\Delta$}})$ and
${\text{\boldmath$\Delta$}}\Phi(f)=\Phi^{\prime}(f){\text{\boldmath$\Delta$}}f+\Phi^{\prime\prime}(f)|\nabla
f|^{2}\mathfrak{m}$. The proof of the latter assertion is similar. ∎

######  Definition 4.21.

We define $\dot{D}({\text{\boldmath$\Delta$}})_{{\rm loc}}$ to consist of all
$u\in\dot{D}(\mathscr{E})_{{\rm loc}}$ for which there exists a
$\sigma$-finite signed Radon smooth measure $\nu$ such that for every
$h\in\bigcup_{k=1}^{\infty}D(\mathscr{E})_{G_{k}}\cap
L^{\infty}(M;\mathfrak{m})$ with $\\{G_{k}\\}\in\Xi(u)$, we have

$\displaystyle-\int_{M}\tilde{h}\,\mathrm{d}\nu=\mathscr{E}(h,u).$

In case of existence, $\nu$ is unique, denoted by
${\text{\boldmath$\Delta$}}u$ and shall be called the _measure-valued
Laplacian_. We define the subspace
$\dot{D}_{f}({\text{\boldmath$\Delta$}})_{{\rm loc}}$ of
$\dot{D}({\text{\boldmath$\Delta$}})_{{\rm loc}}$ by

$\displaystyle\dot{D}_{f}({\text{\boldmath$\Delta$}})_{{\rm
loc}}:=\\{u\in\dot{D}({\text{\boldmath$\Delta$}})_{{\rm
loc}}\mid{\text{\boldmath$\Delta$}}u\text{ is a finite signed smooth
measure}\\}.$

It is easy to see
$D({\text{\boldmath$\Delta$}})\subset\dot{D}({\text{\boldmath$\Delta$}})_{{\rm
loc}}$,
$D_{f}({\text{\boldmath$\Delta$}})\subset\dot{D}_{f}({\text{\boldmath$\Delta$}})_{{\rm
loc}}$ and $1\in\dot{D}_{f}({\text{\boldmath$\Delta$}})_{{\rm loc}}$ with
${\text{\boldmath$\Delta$}}1=0$.

######  Corollary 4.22.

Take ${\varepsilon}>0$ and $p\in[1,2]$. If $u\in
D({\text{\boldmath$\Delta$}})_{+}$ (resp. $u\in
D_{f}({\text{\boldmath$\Delta$}})_{+}$), then
$(u+{\varepsilon}^{2})^{\frac{p}{2}}\in\dot{D}({\text{\boldmath$\Delta$}})_{{\rm
loc}}$ (resp.
$(u+{\varepsilon}^{2})^{\frac{p}{2}}\in\dot{D}_{f}({\text{\boldmath$\Delta$}})_{{\rm
loc}}$) and

$\displaystyle{\text{\boldmath$\Delta$}}(u+{\varepsilon}^{2})^{\frac{p}{2}}$
$\displaystyle=\frac{p}{2}\left(\frac{p}{2}-1\right)\left(u+{\varepsilon}^{2}\right)^{\frac{p}{2}-2}|\nabla
u|^{2}\mathfrak{m}+\frac{p}{2}(\tilde{u}+{\varepsilon}^{2})^{\frac{p}{2}-1}{\text{\boldmath$\Delta$}}u,$
(4.16) $\displaystyle\Delta_{\ll}(u+{\varepsilon}^{2})^{\frac{p}{2}}$
$\displaystyle=\frac{p}{2}\left(\frac{p}{2}-1\right)\left(u+{\varepsilon}^{2}\right)^{\frac{p}{2}-2}|\nabla
u|^{2}+\frac{p}{2}(\tilde{u}+{\varepsilon}^{2})^{\frac{p}{2}-1}\Delta_{\ll}u\quad\mathfrak{m}\text{-a.e.}$
(4.17)

###### Proof.

Take $\Psi(r):=(r+{\varepsilon}^{2})^{\frac{p}{2}}-{\varepsilon}^{p}$ for
$r\geq 0$. $\Psi$ can be extended to a $C^{2}$-function on $\mathbb{R}$ with
$\Psi(0)=0$, $\Psi^{\prime},\Psi^{\prime\prime}\in C_{b}(\mathbb{R})$. Then we
have $\Psi(u)\in D({\text{\boldmath$\Delta$}})$ and

$\displaystyle{\text{\boldmath$\Delta$}}\Psi(u)=\frac{p}{2}\left(\frac{p}{2}-1\right)\left(u+{\varepsilon}^{2}\right)^{\frac{p}{2}-2}|\nabla
u|^{2}\mathfrak{m}+\frac{p}{2}(\tilde{u}+{\varepsilon}^{2})^{\frac{p}{2}-1}{\text{\boldmath$\Delta$}}u.$

In particular,
$(u+{\varepsilon}^{2})^{\frac{p}{2}}=\Psi(u)+{\varepsilon}^{p}\in\dot{D}({\text{\boldmath$\Delta$}})_{{\rm
loc}}$ and
${\text{\boldmath$\Delta$}}(u+{\varepsilon}^{2})^{\frac{p}{2}}={\text{\boldmath$\Delta$}}(\Psi(u)+{\varepsilon}^{p})={\text{\boldmath$\Delta$}}\Psi(u)$.
∎

###  4.8 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 [12, Lemma 2.7], $L^{2}(T^{*}\\!M)$ is separable, and so is
$L^{2}(TM)$ by [12, 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

$\displaystyle{}_{L^{p}(TM)}\langle\omega^{\sharp},X\rangle_{L^{q}(TM)}:=\omega(X)=:{}_{L^{q}(T^{*}\\!M)}\langle
X^{\flat},\omega\rangle_{L^{p}(T^{*}\\!M)}\quad\mathfrak{m}\text{-a.e.}$
(4.18)

for $X\in L^{q}(TM)$ and $\omega\in L^{p}(T^{*}\\!M)$, and we write (4.18) by

$\displaystyle\langle\omega^{\sharp},X\rangle:=\omega(X)=:\langle
X^{\flat},\omega\rangle\quad\mathfrak{m}\text{-a.e.}$ (4.19)

for simplicity.

###  4.9 Test and regular objects

######  Definition 4.23 (${\rm Test}(TM)$ 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 4.24 (${\rm Test}(T^{*}\\!M)$ 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
$L^{p}(TM)$ for $p\in[1,+\infty[$ (resp. $L^{p}(T^{*}\\!M)$) (see [39]), 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. $L^{p}(T^{*}\\!M)$). 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
$p\in[1,+\infty[$.

###  4.10 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 [12, 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 [12,
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

$\displaystyle\langle A^{\sharp}\,|\,T\rangle_{\mathfrak{m}}:=A(T)=:\langle
A\,|\,T^{\flat}\rangle_{\mathfrak{m}}\quad\mathfrak{m}\text{-a.e.}$ (4.20)

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 4.8.
For $p\in[1,+\infty]$, these spaces naturally become Banach space which, if
$p<\infty$, are separable. The following coincidence also holds

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

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 (4.20)
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. ${\rm Reg}(T^{\otimes 2}X)\cap
L^{p}(T^{\otimes 2}X)$).

###  4.11 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 4.10,
$L^{2}(\Lambda^{k}T^{*}\\!M)$ 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 4.8. 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 4.9 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 [12, Definition 5.2]). In case of
existence, the element $T$ is unique, denoted by $\nabla X$ and called the
_covariant derivative_ of $X$.

Arguing as in [12, 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 [12, Theorem 6.3] that $(W^{1,2}(TM),\|\cdot\|_{W^{1,2}(TM)})$
is a separable Hilbert space, $\nabla$ is a closed operator, ${\rm
Reg}(TM)\subset W^{1,2}(TM)$, $\|\cdot\|_{W^{1,2}(TM)}$-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 [32, Chapter 2], [31, Lemma
3.5]).

######  Lemma 5.3 (Kato’s inequality, [12, 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
$|X|^{2}\in D(\mathscr{E})$.

##  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 4.8. 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 [12, 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 [12, 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 [12, 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 [12, 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 [12, 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 [12, 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 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 write $\mathscr{E}\mathscr{E}^{\rm HK}$ 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 ([12, 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.

The following theorem is known as _Hess-Schrader-Uhlenbrock inequality_. This
is proved in [12, Theorem 8.41] under an extra assumption [12, Assumption
8.37] in the framework of tamed Dirichlet space (Assumption 1). In [39], we
improve [12, Theorem 8.41] without this assumption. Theorem 6.8 also plays a
crucial role to prove the $L^{p}$-boundedness of Riesz operator.

######  Theorem 6.8 (Hess-Schrader-Uhlenbrock inequality, [39, Theorem 1.2]).

We have the following:

  1. (1)

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

$\displaystyle|R_{\alpha}^{\rm HK}\omega|\leq
R_{\alpha}^{\kappa}|\omega|\quad\mathfrak{m}\text{-a.e.}$ (6.5)

  2. (2)

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

$\displaystyle|P_{t}^{\rm HK}\omega|\leq
P_{t}^{\kappa}|\omega|\quad\mathfrak{m}\text{-a.e.}$ (6.6)

Theorem 6.8(ii) is proved in [11, Theorem A] in the framework of
RCD$(K,\infty)$-space for $K\in\mathbb{R}$, and also proved in [12, Theorem
8.41] in the present framework with an extra assumption [12, Assumption 8.37].

######  Corollary 6.9 ($C_{0}$-property of $(P_{t}^{\rm HK})_{t\geq 0}$ on
$L^{p}(M;\mathfrak{m})$, [39, Corollary 1.3]).

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 for each $t>0$

$\displaystyle\|P_{t}^{\rm HK}\omega\|_{L^{p}(T^{*}\\!M)}\leq
C(\kappa)e^{C_{\kappa}t}\|\omega\|_{L^{p}(T^{*}\\!M)},\quad\omega\in
L^{p}(T^{*}\\!M).$ (6.7)

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 $L^{\infty}(T^{*}\\!M)$.

##  7 Ricci curvature measures

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

The following are shown in [12, Lemmas 8.1 and 8.2]:

######  Lemma 7.1 ([12, 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 ([12, 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 [12, Lemma 8.8]:

######  Lemma 7.4 ([12, 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 $X\in
H^{1,2}_{\sharp}(TM)$,

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

Then, we can state the main result of [12]:

######  Theorem 7.5 ([12, 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$}}^{2\kappa}\frac{\langle
X,Y\rangle}{2}-\left[\frac{1}{2}\langle
X,({\Delta\Delta}^{\rm\\!HK}Y^{\flat})^{\sharp}\rangle+\frac{1}{2}\langle
Y,({\Delta\Delta}^{\rm\\!HK}X^{\flat})^{\sharp}\rangle+\langle\nabla
X\,|\,\nabla Y\rangle\right]\mathfrak{m}$ (7.1)

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

$\displaystyle{\bf Ric}^{\kappa}(X,X)$ $\displaystyle\geq 0,$ (7.2)
$\displaystyle{\bf Ric}^{\kappa}(X,Y)(M)$
$\displaystyle=\int_{M}\left[\langle\mathrm{d}X^{\flat},\mathrm{d}Y^{\flat}\rangle+\langle\mathrm{d}_{*}X^{\flat},\mathrm{d}_{*}Y^{\flat}\rangle\right]\mathrm{d}\mathfrak{m}$
$\displaystyle\hskip 56.9055pt-\int_{M}\langle\nabla X\,|\,\nabla
Y\rangle\mathrm{d}\mathfrak{m}-\langle\kappa,\widetilde{\langle
X,Y\rangle}\rangle,$ (7.3) $\displaystyle\|{\bf Ric}^{\kappa}(X,Y)\|_{\rm
TV}^{2}$ $\displaystyle\leq\left[\mathscr{E}_{\rm
con}(X^{\flat})-\langle\kappa,|X|^{2}\rangle\right]\left[\mathscr{E}_{\rm
con}(Y^{\flat})-\langle\kappa,|Y|^{2}\rangle\right].$ (7.4)

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

$\displaystyle{\bf Ric}(X,Y):={\bf
Ric}^{\kappa}(X,Y)+\langle\kappa,\widetilde{\langle X,Y\rangle}\rangle.$ (7.5)

This map is well-defined, symmetric and $\mathbb{R}$-bilinear. ${\bf Ric}$ is
also jointly continuous, which follows from polarization, [12, Corollary 6.14]
and Lemma 7.4. Lastly, by Theorem 7.5, Lemma 7.2 and [12, Lemma 8.12], for
every $X,Y\in{\rm Reg}(TM)$, we have a similar expression with (7.1):

$\displaystyle{\bf Ric}(X,Y)={\text{\boldmath$\Delta$}}\frac{\langle
X,Y\rangle}{2}-\left[\frac{1}{2}\langle
X,({\Delta\Delta}^{\rm\\!HK}Y^{\flat})^{\sharp}\rangle+\frac{1}{2}\langle
Y,({\Delta\Delta}^{\rm\\!HK}X^{\flat})^{\sharp}\rangle+\langle\nabla
X\,|\,\nabla Y\rangle\right]\mathfrak{m}$ (7.6)

Denote by ${\bf Ric}_{\ll}^{\kappa}(X,Y)$ (resp. ${\bf
Ric}_{\perp}^{\kappa}(X,Y)$) the absolutely continuous (resp. singular) part
of ${\bf Ric}^{\kappa}(X,Y)$ for $X,Y\in H^{1,2}_{\sharp}(TM)$ with respect to
$\mathfrak{m}$. Then we see ${\bf Ric}_{\ll}^{\kappa}(X,X)\geq 0$ and ${\bf
Ric}_{\perp}^{\kappa}(X,X)\geq 0$ for $X\in H^{1,2}_{\sharp}(TM)$, which
yields

$\displaystyle\left\\{\begin{array}[]{rl}{\bf Ric}_{\ll}(X,X)&={\bf
Ric}_{\ll}^{\kappa}(X,X)+\widetilde{|X|^{2}}\kappa_{\ll}\geq\widetilde{|X|^{2}}\kappa_{\ll},\\\
{\bf Ric}_{\perp}(X,X)&={\bf
Ric}_{\perp}^{\kappa}(X,X)+\widetilde{|X|^{2}}\kappa_{\perp}\geq\widetilde{|X|^{2}}\kappa_{\perp}\end{array}\right.$
(7.9)

for $X\in H^{1,2}_{\sharp}(TM)$. Here $\kappa_{\ll}$ (resp. $\kappa_{\perp}$)
is the absolutely continuous (resp. singular) part of $\kappa$ with respect to
$\mathfrak{m}$. We define the Ricci tensor $\mathfrak{Ric}(X,Y)$ by

$\displaystyle\mathfrak{Ric}(X,Y):=\frac{\mathrm{d}{\bf
Ric}_{\ll}(X,Y)}{\mathrm{d}\mathfrak{m}}\quad\text{ for }\quad X,Y\in
H^{1,2}_{\sharp}(TM).$ (7.10)

In particular, by (7.9)

$\displaystyle\mathfrak{Ric}(X,X)\geq|X|^{2}\frac{\mathrm{d}\kappa_{\ll}}{\mathrm{d}\mathfrak{m}}\quad\text{
for }\quad X\in H^{1,2}_{\sharp}(TM).$ (7.11)

##  8 Littlewood-Paley $G$-functions

Recall the heat flow $(P_{t}^{\rm HK})_{t\geq 0}$ of Hodge-Kodaira Laplacian
${\Delta\Delta}^{\rm\\!HK}$ acting on $L^{2}(T^{*}\\!M)$. By Corollary 6.9,
$(P_{t}^{\rm HK})_{t\geq 0}$ can be regarded as a bounded semigroup on
$L^{p}(T^{*}\\!M)$. Let $(Q_{t}^{(\alpha),{\rm HK}})_{t\geq 0}$ be the
$\alpha$-order Cauchy semigroup acting on $L^{p}(T^{*}\\!M)$ defined by

$\displaystyle Q_{t}^{(\alpha),{\rm HK}}\theta:=\int_{0}^{\infty}e^{-\alpha
s}P_{s}^{\rm HK}\theta\lambda_{t}(\mathrm{d}s),\quad\theta\in
L^{p}(T^{*}\\!M),$

where $\lambda_{t}$ with $t\geq 0$ is the probability measure on
$[0,+\infty[$, whose Laplace transform is given by

$\displaystyle\int_{0}^{\infty}e^{-\gamma
s}\lambda_{t}(\mathrm{d}s)=e^{-\sqrt{\gamma}t},\quad\gamma\geq 0.$

For $t>0$, $\lambda_{t}$ has the following expression

$\displaystyle\lambda_{t}(\mathrm{d}s)=\frac{t}{2\sqrt{\pi}}e^{-t^{2}/4s}s^{-3/2}\mathrm{d}s.$

For $\theta\in L^{2}(T^{*}\\!M)$, $Q_{t}^{(\alpha),{\rm HK}}\theta\in
D({\Delta\Delta}^{\rm\\!HK})$ and

$\displaystyle\|{\Delta\Delta}^{\rm\\!HK}Q_{t}^{(\alpha),{\rm
HK}}\theta\|_{L^{2}(T^{*}\\!M)}\leq\left(\frac{1}{\sqrt{2}}\int_{0}^{\infty}e^{-\alpha
s}\frac{1}{s}\lambda_{t}(\mathrm{d}s)\right)\|\theta\|_{L^{2}T^{*}\\!M)}<\infty.$

$Q_{t}^{(\alpha),{\rm HK}}\theta$ can be formally written by

$\displaystyle Q_{t}^{(\alpha),{\rm
HK}}\theta(x)=\text{``$e^{-\sqrt{\alpha-{\Delta\Delta}^{\rm\\!HK}}\,t}\theta(x)$''.}$

By Corollary 6.9 and $\kappa^{-}\in S_{K}({\bf X})$, we see that

$\displaystyle\|Q_{t}^{(\alpha),{\rm HK}}\theta\|_{L^{p}(T^{*}\\!M)}\leq
Ce^{-\sqrt{\alpha-C_{\kappa}}t}\|\theta\|_{L^{p}(T^{*}\\!M)}$ (8.1)

holds for any $p\in[1,+\infty]$ and $(Q_{t}^{(\alpha),{\rm HK}})_{t\geq 0}$ is
a strongly continuous semigroup on $L^{p}(T^{*}\\!M)$ for $p\in[1,+\infty[$
and $\alpha\geq C_{\kappa}$ and weakly* continuous on $L^{\infty}(T^{*}\\!M)$
in the same way of the proof of [39, Corollary 1.3]. The infinitesimal
generator of $(Q_{t}^{(\alpha),{\rm HK}})_{t\geq 0}$ on $L^{p}(T^{*}\\!M)$ is
denoted by $-\sqrt{\alpha-{\Delta\Delta}^{\rm\\!HK}_{p}}$, where
${\Delta\Delta}^{\rm\\!HK}_{p}$ is the infinitesimal generator of $(P_{t}^{\rm
HK})_{t\geq 0}$ on $L^{p}(T^{*}\\!M)$. We may omit the subscript $p$ for
simplicity.

For $\theta\in L^{2}(T^{*}\\!M)\cap L^{p}(T^{*}\\!M)$ and $\alpha\geq
C_{\kappa}$, we define the Littlewood-Paley’s $G$-functions by

$\displaystyle{g_{\stackrel{{\scriptstyle\rightarrow}}{{\theta}}}}(x,t)$
$\displaystyle:=\left|\frac{\partial}{\partial t}(Q_{t}^{(\alpha),\rm
HK}\theta)(x)\right|,\qquad\qquad\quad{G_{\stackrel{{\scriptstyle\rightarrow}}{{\theta}}}}(x):=\left(\int_{0}^{\infty}t{g_{\stackrel{{\scriptstyle\rightarrow}}{{\theta}}}}(x,t)^{2}\mathrm{d}t\right)^{1/2},$
$\displaystyle{g_{\theta}^{\uparrow}}(x,t)$
$\displaystyle:=\left|\nabla(Q_{t}^{(\alpha),\rm
HK}\theta)^{\sharp}(x)\right|_{\rm
HS},\qquad\quad\quad{G_{\theta}^{\uparrow}}(x):=\left(\int_{0}^{\infty}t{g_{\theta}^{\uparrow}}(x,t)^{2}\mathrm{d}t\right)^{1/2},$
$\displaystyle g_{\theta}(x,t)$
$\displaystyle:=\sqrt{{g_{\stackrel{{\scriptstyle\rightarrow}}{{\theta}}}}(x,t)^{2}+{g_{\theta}^{\uparrow}}(x,t)^{2}},\qquad\quad
G_{\theta}(x):=\left(\int_{0}^{\infty}tg_{\theta}(x,t)^{2}\mathrm{d}t\right)^{1/2}.$

######  Lemma 8.1 (Maximal ergodic inequality, [50, Theorem 3.3]).

Suppose $p\in]1,+\infty[$. For $f\in L^{p}(M;\mathfrak{m})$,

$\displaystyle\left\|\sup_{t\geq
0}P_{t}|f|\right\|_{L^{p}(M;\mathfrak{m})}\leq\frac{p}{p-1}\|f\|_{L^{p}(M;m)}.$
(8.2)

######  Proposition 8.2.

Suppose that $\kappa^{-}\in S_{K}({\bf X})$ and $p\in]1,+\infty[$. Then there
exists $C_{p}>0$ such that for any $\alpha\geq C_{\kappa}$,

$\displaystyle\int_{M}\sup_{t\geq 0}\left|Q_{t}^{(\alpha),{\rm
HK}}\theta(x)\right|^{p}\mathfrak{m}(\mathrm{d}x)\leq
C_{p}\|\theta\|_{L^{p}(T^{*}\\!M)}^{p},\quad\theta\in L^{p}(T^{*}\\!M).$

###### Proof.

By Theorem 6.8,

$\displaystyle\left|Q_{t}^{(\alpha),{\rm HK}}\theta(x)\right|$
$\displaystyle=\left|\int_{0}^{\infty}e^{-\alpha s}P_{s}^{\rm
HK}\theta(x)\lambda_{t}(\mathrm{d}s)\right|$
$\displaystyle\leq\int_{0}^{\infty}e^{-\alpha s}\left|P_{s}^{\rm
HK}\theta(x)\right|\lambda_{t}(\mathrm{d}s)$
$\displaystyle\leq\int_{0}^{\infty}e^{-\alpha
s}P_{s}^{\kappa}|\theta|(x)\lambda_{t}(\mathrm{d}s)$
$\displaystyle=\int_{0}^{\infty}\mathbb{E}_{x}\left[e^{-\alpha
s-A_{s}^{\kappa}}|\theta|(X_{s})\right]\lambda_{t}(\mathrm{d}s).$

Take any $\beta>p/(p-1)>1$ and $\beta^{\prime}:=\beta/(\beta-1)\in]1,p[$. Then

$\displaystyle\sup_{t\geq 0}\left|Q_{t}^{(\alpha),{\rm HK}}\theta(x)\right|$
$\displaystyle\leq\sup_{s\geq 0}\mathbb{E}_{x}\left[e^{-\alpha
s-A_{s}^{\kappa}}|\theta|(X_{s})\right]$ $\displaystyle\leq\sup_{s\geq
0}\mathbb{E}_{x}\left[e^{-\beta(\alpha
s+A_{s}^{\kappa})}\right]^{\frac{1}{\beta}}\mathbb{E}_{x}\left[|\theta|^{\beta^{\prime}}(x)\right]^{\frac{1}{\beta^{\prime}}}$
$\displaystyle\leq\sup_{s\geq 0,y\in
M}\left(P_{s}^{\beta(\alpha\mathfrak{m}+\kappa)}1(y)\right)^{\frac{1}{\beta}}\cdot\sup_{s\geq
0}\left(P_{s}|\theta|^{\beta^{\prime}}(x)\right)^{\frac{1}{\beta^{\prime}}}$
$\displaystyle\leq\sup_{s\geq 0,y\in
M}\left(P_{s}^{\alpha\mathfrak{m}-\beta\kappa^{-}}1(y)\right)^{\frac{1}{\beta}}\cdot\sup_{s\geq
0}\left(P_{s}|\theta|^{\beta^{\prime}}(x)\right)^{\frac{1}{\beta^{\prime}}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:KatoCoincidence}}}{{\leq}}\sup_{s\geq
0}\left(Ce^{-\alpha s+C_{\kappa}s}\right)^{\frac{1}{\beta}}\cdot\sup_{s\geq
0}\left(P_{s}|\theta|^{\beta^{\prime}}(x)\right)^{\frac{1}{\beta^{\prime}}}$
$\displaystyle\leq C^{\frac{1}{\beta}}\sup_{s\geq
0}\left(P_{s}|\theta|^{\beta^{\prime}}(x)\right)^{\frac{1}{\beta^{\prime}}},\quad(\alpha\geq
C_{\kappa}).$

Applying Lemma 8.1, we obtain

$\displaystyle\int_{M}\sup_{t\geq 0}\left|Q_{t}^{(\alpha),{\rm
HK}}\theta(x)\right|^{p}\mathfrak{m}(\mathrm{d}x)$
$\displaystyle=C^{\frac{p}{\beta}}\int_{M}\left|\sup_{s\geq
0}P_{s}|\theta|^{\beta^{\prime}}(x)\right|^{\frac{p}{\beta^{\prime}}}\mathfrak{m}(\mathrm{d}x)$
$\displaystyle\leq
C^{\frac{p}{\beta}}\left(\frac{\frac{p}{\beta^{\prime}}}{\frac{p}{\beta^{\prime}}-1}\right)^{\frac{p}{\beta^{\prime}}}\int_{M}|\theta(x)|^{\beta^{\prime}\frac{p}{\beta^{\prime}}}\mathfrak{m}(\mathrm{d}x)$
$\displaystyle=C^{\frac{p}{\beta}}\left(\frac{p}{p-\beta^{\prime}}\right)^{\frac{p}{\beta^{\prime}}}\|\theta\|_{L^{p}(T^{*}\\!M)}^{p}$
$\displaystyle=C^{\frac{p}{\beta}}\left(\frac{p(\beta-1)}{(p-1)\beta-p}\right)^{\frac{p(\beta-1)}{\beta}}\|\theta\|_{L^{p}(T^{*}\\!M)}^{p}.$

∎

Now we set $\eta=\eta(x,t):=Q_{t}^{(\alpha),{\rm HK}}\theta(x)$. Let
$\overline{\nabla}:=\left(\nabla,\frac{\partial}{\partial t}\right)$ be the
covariant derivative over $X\times\mathbb{R}$ and $\nabla$ the covariant
derivative acting on $W^{1,2}(TM)$:

$\displaystyle|\overline{\nabla}\eta(x,t)|^{2}:=\left|\nabla(Q_{t}^{(\alpha),{\rm
HK}}\theta)^{\sharp}(x)\right|_{\rm HS}^{2}+\left|\frac{\partial}{\partial
t}Q_{t}^{(\alpha),{\rm
HK}}\theta(x)\right|^{2}=|\nabla\eta^{\sharp}(x,t)|_{\rm
HS}^{2}+\left|\frac{\partial}{\partial t}\eta(x,t)\right|^{2}.$

##  9 Auxiliary Brownian motion

Let $(B_{t},\mathbb{P}_{\stackrel{{\scriptstyle\rightarrow}}{{a}}})$ be one-
dimensional Brownian motion starting at $a\in\mathbb{R}$ with the generator
$\frac{\partial^{2}}{\partial a^{2}}$. We set
$\widehat{M}:=M\times\mathbb{R}$, $\hat{x}:=(x,a)\in\widehat{M}$,
$\widehat{X}_{t}:=(X_{t},B_{t})$, $t\geq 0$,
$\widehat{\mathfrak{m}}:=\mathfrak{m}\otimes m$ and
$\mathbb{P}_{\hat{x}}:=\mathbb{P}_{x}\otimes\mathbb{P}_{\stackrel{{\scriptstyle\rightarrow}}{{a}}}$.
Then $\widehat{\bf X}:=(\widehat{X}_{t},\mathbb{P}_{\hat{x}})$ is an
$\widehat{\mathfrak{m}}$-symmetric diffusion process on $\widehat{M}$ with the
(formal) generator $\Delta+\frac{\partial^{2}}{\partial a^{2}}$, where $m$ is
one-dimensional Lebesgue measure. Let $\tau:=\inf\\{t>0\mid B_{t}=0\\}$ be the
first hitting time of $B_{t}$ to $\\{0\\}$. We denote the semigroup on
$L^{p}(\widehat{M};\widehat{\mathfrak{m}})$ associated with the diffusion
process $(\widehat{X}_{t})_{t\geq 0}$ by $(\widehat{P}_{t})_{t\geq 0}$ and its
generator by $\widehat{\Delta}_{p}$. We also denote the Dirichlet form on
$L^{2}(\widehat{M};\widehat{\mathfrak{m}})$ associated with
$\widehat{\Delta}_{2}$ by $(\widehat{\mathscr{E}},D(\widehat{\mathscr{E}}))$.
That is,

$\displaystyle D(\widehat{\mathscr{E}}):$ $\displaystyle=\left\\{u\in
L^{2}(\widehat{M};\widehat{\mathfrak{m}})\;\left|\;\lim_{t\to
0}\frac{1}{t}(u-\widehat{P}_{t}u,u)_{L^{2}(\widehat{M};\widehat{\mathfrak{m}})}<+\infty\right.\right\\},$
$\displaystyle\widehat{\mathscr{E}}(u,v):$ $\displaystyle=\lim_{t\to
0}\frac{1}{t}(u-\widehat{P}_{t}u,v)_{L^{2}(\widehat{M};\widehat{\mathfrak{m}})}\qquad\text{
for }\quad u,v\in D(\widehat{\mathscr{E}}).$

Since $(\widehat{\mathscr{E}},D(\widehat{\mathscr{E}}))$ is associated to the
$\widehat{\mathfrak{m}}$-symmetric Borel right process $\widehat{\bf X}$, it
is quasi-regular by Fitzsimmons [24]. We need the following lemma:

######  Lemma 9.1.

We have the following:

  1. (1)

Let $N$ be an $\widehat{\mathscr{E}}$-exceptional set. Then, for $m$-a.e.
$a\in\mathbb{R}$, $N_{a}:=\\{x\in M\mid(x,a)\in N\\}$ is an
$\mathscr{E}$-exceptional set.

  2. (2)

Let $(x,a)\mapsto u(x,a)$ be an $\widehat{\mathscr{E}}$-quasi continuous
function. Then, for $m$-a.e. $a\in\mathbb{R}$, $x\mapsto u(x,a)$ is an
$\mathscr{E}$-quasi continuous function.

###### Proof.

(1) is proved in $\widehat{\rm O}$kura [45, Theorem 4.1(4)]. We prove (2).
Since the $\mathscr{E}$-quasi continuity is equivalent to the
$\mathscr{E}$-q.e. fine continuity by [25, Theorem 4.2.2 and Theorem 4.6.1],
it suffices to prove that the fine continuity of $(x,a)\mapsto u(x,a)$ with
respect to $\widehat{\bf X}$ implies the fine continuity of $x\mapsto u(x,a)$
for each $a\in\mathbb{R}$ with respect to ${\bf X}$. This can be confirmed by
that for each $a\in\mathbb{R}$

$\displaystyle t\mapsto u(\widehat{X}_{t})=u(X_{t},a)\;\text{ is right
continuous }\;\mathbb{P}_{(x,a)}\text{-a.s.~{}equivalently
}\;\mathbb{P}_{x}\text{-a.s.~{}for }\;x\in M$

in view of [25, Theorem A.2.7]. ∎

Throughout this subsection, we assume $\kappa^{+}\in S_{D}({\bf X})$ and
$\kappa^{-}\in S_{E\\!K}({\bf X})$. We define
$\widehat{\kappa}^{\,\pm}:=\kappa^{\pm}\otimes m$ and
$\widehat{\kappa}:=\widehat{\,\kappa}^{+}-\widehat{\,\kappa}^{-}$. The
associated positive continuous additive functional (PCAF in short)
$\widehat{A}_{t}^{\;\widehat{\,\kappa}^{\pm}}$ in Revuz correspondence under
$\widehat{\bf X}$ is given by
$\widehat{A}_{t}^{\;\widehat{\kappa}^{\pm}}=A_{t}^{\kappa^{\pm}}$, in
particular, $\widehat{\kappa}^{\,+}$ (resp. $\widehat{\kappa}^{\,-}$) is a
smooth measure of Dynkin (resp. extended Kato) class with respect to
$\widehat{\bf X}$, i.e. $\widehat{\kappa}^{\,+}\in S_{D}(\widehat{\bf X})$
(resp. $\widehat{\kappa}^{\,-}\in S_{E\\!K}(\widehat{\bf X})$). Then we can
define the following quadratic form
$(\widehat{\mathscr{E}}^{\;\widehat{\kappa}},D(\widehat{\mathscr{E}}^{\;\widehat{\kappa}}))$
on $L^{2}(\widehat{M};\widehat{\mathfrak{m}})$:

$\displaystyle
D(\widehat{\mathscr{E}}^{\;\widehat{\kappa}}):=D(\widehat{\mathscr{E}}),\qquad\widehat{\mathscr{E}}^{\;\widehat{\kappa}}(u,v):=\widehat{\mathscr{E}}(u,v)+\langle\widehat{\kappa},\tilde{u}\tilde{v}\rangle\quad\text{
for }\quad u,v\in D(\widehat{\mathscr{E}}^{\;\widehat{\kappa}}).$ (9.1)

Here $\tilde{u}$ denotes the $\widehat{\mathscr{E}}$-quasi-continuous
$\widehat{\mathfrak{m}}$-version of $u$ with respect to
$(\widehat{\mathscr{E}},D(\widehat{\mathscr{E}}))$. The strongly continuous
semigroup $(\widehat{P}_{t}^{\;\widehat{\kappa}})_{t\geq 0}$ on
$L^{2}(\widehat{M};\widehat{\mathfrak{m}})$ associated with the quadratic form
$(\widehat{\mathscr{E}}^{\;\widehat{\kappa}},D(\widehat{\mathscr{E}}^{\;\widehat{\kappa}}))$
is given by

$\displaystyle\widehat{P}_{t}^{\widehat{\;\kappa}}u(\hat{x})=\mathbb{E}_{\hat{x}}[e^{-A_{t}^{\kappa}}u(\widehat{X}_{t})]\quad\text{
for }\quad u\in
L^{2}(\widehat{M};\widehat{\mathfrak{m}})\cap\mathscr{B}(\widehat{M}).$ (9.2)

We denote by $\widehat{\mathscr{C}}:={\rm Test}(M)\odot
C_{c}^{\infty}(\mathbb{R})$ the totality of all finite linear combinations of
$f\otimes\varphi$, $f\in{\rm Test}(M)$, $\varphi\in
C_{c}^{\infty}(\mathbb{R})$, where $(f\otimes\varphi)(y):=f(x)\varphi(a)$.
Meanwhile, the space $L^{2}(M;\mathfrak{m})\otimes L^{2}(\mathbb{R})$ and
$D(\mathscr{E})\otimes H^{1,2}(\mathbb{R})$ are usual tensor products of
Hilbert spaces, where $H^{1,2}(\mathbb{R})$ is the Sobolev space which
consists of all functions $\varphi\in L^{2}(\mathbb{R})$ such that the weak
derivative $\dot{\varphi}$ exists and belongs to $L^{2}(\mathbb{R})$. Then we
have

######  Lemma 9.2 (Lemma 3.1 in [23]).

$\widehat{\mathscr{C}}$ is dense in $D(\widehat{\mathscr{E}})$. Moreover, for
$u,v\in D(\mathscr{E})\otimes H^{1,2}(\mathbb{R})$, we have

$\displaystyle\widehat{\mathscr{E}}(u,v)$
$\displaystyle=\int_{\mathbb{R}}\mathscr{E}(u(\cdot,a),v(\cdot,a))m(\mathrm{d}a)+\int_{M}\mathfrak{m}(\mathrm{d}x)\int_{\mathbb{R}}\frac{\partial
u}{\partial a}(x,a)\frac{\partial v}{\partial a}(x,a)m(\mathrm{d}a).$ (9.3)

The following lemma is proved in [50, Proposition 3.9]:

######  Lemma 9.3 ([50, Proposition 3.9]).

Let $h$ be a non-negative measurable function on $[0,+\infty[$. Then

$\displaystyle\mathbb{E}_{\stackrel{{\scriptstyle\rightarrow}}{{a}}}\left[\int_{0}^{\tau}h(B_{s})\mathrm{d}s\right]=\int_{0}^{\infty}(a\land
t)h(t)\mathrm{d}t.$ (9.4)

Based on Lemma 9.3, we can obtain the following generalization of [49,
Proposition 6.1]

######  Proposition 9.4 ([49, Proposition 6.1]).

Let $\nu$ be a non-negative smooth measure on $M$ associated with a PCAF
$A^{\nu}$ in Revuz correspondence under ${\bf X}$ and $j$ be a non-negative
measurable function on $X\times[0,+\infty[$. Denote by
$\widehat{A}^{\,\widehat{\nu}}$ the PCAF of $\widehat{\bf X}$ associated with
the smooth measure $\widehat{\nu}:=\nu\otimes m$ with respect to $\widehat{\bf
X}$. Then for all $a\in]0,+\infty[$ and Borel measurable non-negative function
$j$ on $X\times\mathbb{R}$

$\displaystyle\mathbb{E}_{\mathfrak{m}\otimes\delta_{a}}\left[\int_{0}^{\tau}j(\widehat{X}_{s})\mathrm{d}\widehat{A}_{s}^{\,\widehat{\nu}}\right]=\int_{M}\int_{0}^{\infty}(a\land
t)j(x,t)\nu(\mathrm{d}x)\mathrm{d}t.$ (9.5)

Let ${\rm Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R})$ be the algebraic
tensor product of ${\rm Reg}(T^{*}\\!M)$ and $C_{c}^{\infty}(\mathbb{R})$,
i.e., any element of ${\rm Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R})$
forms a finite linear combination of $\theta\otimes\varphi$ with
$\theta\in{\rm Reg}(T^{*}\\!M)$ and $\varphi\in C_{c}^{\infty}(\mathbb{R})$,
which is a dense subspace of the tensor product $L^{2}(T^{*}\\!M)\otimes
L^{2}(\mathbb{R})$ of Hilbert spaces $L^{2}(T^{*}\\!M)$ and
$L^{2}(\mathbb{R})$. We now define the quadratic form
$\widehat{\mathscr{E}}^{\,\rm HK}$ on ${\rm Reg}(T^{*}\\!M)\odot
C_{c}^{\infty}(\mathbb{R})$ by

$\displaystyle\widehat{\mathscr{E}}^{\rm
HK}(\theta,\eta):=\int_{\mathbb{R}}\mathscr{E}^{\rm
HK}(\theta(\cdot,t),\omega(\cdot,t))\mathrm{d}t+\int_{\mathbb{R}}\langle\dot{\theta}(\cdot,t),\dot{\eta}(\cdot,t)\rangle_{L^{2}(T^{*}\\!M)}\mathrm{d}t,\quad\theta,\eta\in{\rm
Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R}).$

######  Lemma 9.5.

The quadratic form $(\widehat{\mathscr{E}}^{\,\rm HK},{\rm
Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R}))$ is closable on
$L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$. Denote by
$(\widehat{\mathscr{E}}^{\,\rm HK},D(\widehat{\mathscr{E}}^{\,\rm HK}))$ its
closure on $L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$. Then,
$(\widehat{\mathscr{E}}^{\,\rm HK},D(\widehat{\mathscr{E}}^{\,\rm HK}))$ is a
closed bilinear form on $L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$.

###### Proof.

For $\theta\in{\rm Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R})$, it is
easy to see $\widehat{\Delta\Delta}^{\rm
HK}\theta:=\left(\frac{\partial^{2}}{\partial
t^{2}}+{\Delta\Delta}^{\rm\\!HK}\right)\theta\in L^{2}(T^{*}\\!M)\otimes
L^{2}(\mathbb{R})$. Then

$\displaystyle\widehat{\mathscr{E}}^{\,\rm
HK}(\theta,\eta)=\left(-\widehat{\Delta\Delta}^{\rm
HK}\theta,\eta\right)_{L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})}\quad\text{
for }\quad\theta,\eta\in{\rm Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R}).$

Then the closability of $(\widehat{\mathscr{E}}^{\,\rm
HK},D(\widehat{\mathscr{E}}^{\,\rm HK}))$ on $L^{2}(T^{*}\\!M)\otimes
L^{2}(\mathbb{R})$ follows [43, Chapter I, Proposition 3.3]. ∎

######  Remark 9.6.

The norm $\|\cdot\|_{H^{1,2}(T^{*}\\!M)\otimes H^{1,2}(\mathbb{R})}$ of the
tensor product $H^{1,2}(T^{*}\\!M)\otimes H^{1,2}(\mathbb{R})$ of Hilbert
spaces $H^{1,2}(T^{*}\\!M)$ and $H^{1,2}(\mathbb{R})$ does not coincide with
the norm $\|\cdot\|_{\widehat{\mathscr{E}}_{1}^{\,\rm HK}}$ derived from
$(\widehat{\mathscr{E}}^{\,\rm HK},D(\widehat{\mathscr{E}}^{\,\rm HK}))$ on
$L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$. Indeed, for $\theta\in{\rm
Reg}(T^{*}\\!M)$ and $\varphi\in C_{c}^{\infty}(\mathbb{R})$.

$\displaystyle\|\theta\otimes\varphi\|_{H^{1,2}(T^{*}\\!M)\otimes
H^{1,2}(\mathbb{R})}^{2}$
$\displaystyle=\|\theta\|_{H^{1,2}(T^{*}\\!M)}^{2}\|\varphi\|_{H^{1,2}(\mathbb{R})}^{2}$
$\displaystyle=\left(\mathscr{E}^{\rm
HK}(\theta,\theta)+\|\theta\|_{L^{2}(T^{*}\\!M)}^{2}\right)\left(\|\dot{\varphi}\|_{L^{2}(\mathbb{R})}^{2}+\|\varphi\|_{L^{2}(\mathbb{R})}^{2}\right)$
$\displaystyle\|\theta\otimes\varphi\|_{\widehat{\mathscr{E}}_{1}^{\,\rm
HK}}^{2}$ $\displaystyle=\widehat{\mathscr{E}}^{\,\rm
HK}_{1}(\theta\otimes\varphi,\theta\otimes\varphi)$
$\displaystyle=\mathscr{E}^{\rm
HK}(\theta,\theta)\|\varphi\|_{L^{2}(\mathbb{R})}^{2}+\|\theta\|_{L^{2}(T^{*}\\!M)}^{2}\|\dot{\varphi}\|_{L^{2}(\mathbb{R})}^{2}+\|\theta\|_{L^{2}(T^{*}\\!M)}^{2}\|\varphi\|_{L^{2}(\mathbb{R})}^{2}.$

######  Lemma 9.7.

We have the inclusion $H^{1,2}(T^{*}\\!M)\odot H^{1,2}(\mathbb{R})\subset
D(\widehat{\mathscr{E}}^{\rm HK})$. The map $H^{1,2}(T^{*}\\!M)\times
H^{1,2}(\mathbb{R})\to H^{1,2}(T^{*}\\!M)\odot H^{1,2}(\mathbb{R})$ is
continuous.

###### Proof.

It suffices to show that $\theta\otimes\varphi\in D(\widehat{\mathscr{E}}^{\rm
HK})$ for $\theta\in H^{1,2}(T^{*}\\!M)$ and $\varphi H^{1,2}(\mathbb{R})$.
Take an approximating sequence $\\{\theta_{n}\\}\subset{\rm Reg}(T^{*}\\!M)$
(resp. $\\{\varphi_{n}\\}\subset C_{c}^{\infty}(\mathbb{R})$) in
$H^{1,2}(T^{*}\\!M)$ (resp. in $H^{1,2}(\mathbb{R})$) to $\theta\in
H^{1,2}(T^{*}\\!M)$ (resp. $\varphi\in H^{1,2}(\mathbb{R})$). Then it is easy
to see that $\\{\theta_{n}\otimes\varphi_{n}\\}\subset H^{1,2}(T^{*}\\!M)\odot
H^{1,2}(\mathbb{R})$ is an $\widehat{\mathscr{E}}^{\,\rm HK}_{1}$-Cauchy
sequence. This yields the conclusion and the following identity: for
$\theta,\eta\in H^{1,2}(T^{*}\\!M)$ and $\varphi,\psi\in H^{1,2}(\mathbb{R})$,

$\displaystyle\widehat{\mathscr{E}}^{\,\rm
HK}(\theta\otimes\varphi,\eta\otimes\psi)=\mathscr{E}^{\rm
HK}(\theta,\eta)(\varphi,\psi)_{L^{2}(\mathbb{R})}+\langle\theta,\eta\rangle_{L^{2}(T^{*}\\!M)}(\dot{\varphi},\dot{\psi})_{L^{2}(\mathbb{R})},$

which implies the desired continuity. ∎

######  Lemma 9.8.

Suppose $\eta\in L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$. Then the
following are equivalent to each other:

  1. (1)

$\eta\in D(\widehat{\mathscr{E}}^{\,\rm HK})$.

  2. (2)

For a.e. $t\in\mathbb{R}$, we have $\eta(\cdot,t)\in D({\mathscr{E}}^{\,\rm
HK})$ and $\dot{\eta}(\cdot,t)\in L^{2}(T^{*}\\!M)$. Moreover,
$\int_{\mathbb{R}}{\mathscr{E}}^{\,\rm
HK}_{1}(\eta(\cdot,t),\eta(\cdot,t))\mathrm{d}t<\infty$ and
$\int_{\mathbb{R}}\|\dot{\eta}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t<\infty$.

Here $\dot{\eta}(\cdot,t)$ denotes the $L^{2}(T^{*}\\!M)$-valued
distributional derivative of $\eta(\cdot,t)$ with respect to $t\in\mathbb{R}$

###### Proof.

(1) $\Longrightarrow$ (2): Suppose $\eta\in D(\widehat{\mathscr{E}}^{\,\rm
HK})$. Let $\eta_{n}\in{\rm Reg}(T^{*}\\!M)\odot C_{c}^{\infty}(\mathbb{R})$
be an $\widehat{\mathscr{E}}^{\,\rm HK}_{1}$-approximating sequence to $\eta$.
By taking a subsequence if necessary we may assume

$\displaystyle\sum_{k=1}^{\infty}2^{k}\widehat{\mathscr{E}}^{\,\rm
HK}_{1}(\eta_{k+1}-\eta_{k},\eta_{k+1}-\eta_{k})<\infty.$ (9.6)

Then, for each $T\in]0,+\infty[$,

$\displaystyle\int_{0}^{T}$
$\displaystyle\sum_{k=1}^{\infty}\mathscr{E}^{\,\rm
HK}_{1}(\eta_{k+1}(\cdot,t)-\eta_{k}(\cdot,t),\eta_{k+1}(\cdot,t)-\eta_{k}(\cdot,t))^{\frac{1}{2}}\mathrm{d}t$
$\displaystyle\leq\sum_{k=1}^{\infty}\sqrt{T}2^{-\frac{k}{2}}\left(2^{k}\int_{0}^{T}{\mathscr{E}}^{\,\rm
HK}_{1}(\eta_{k+1}(\cdot,t)-\eta_{k}(\cdot,t),\eta_{k+1}(\cdot,t)-\eta_{k}(\cdot,t))\mathrm{d}t\right)^{\frac{1}{2}}$
$\displaystyle=\sqrt{T}\left(\sum_{k=1}^{\infty}2^{-k}\right)^{\frac{1}{2}}\left(\sum_{k=1}^{\infty}2^{k}\widehat{\mathscr{E}}^{\,\rm
HK}_{1}(\eta_{k+1}-\eta_{k},\eta_{k+1}-\eta_{k})\right)^{\frac{1}{2}}<\infty.$

Thus,

$\displaystyle\sum_{k=1}^{\infty}\mathscr{E}^{\rm
HK}_{1}(\eta_{k+1}(\cdot,t)-\eta_{k}(\cdot,t),\eta_{k+1}(\cdot,t)-\eta_{k}(\cdot,t))^{\frac{1}{2}}<\infty\quad\text{
for\;\; a.e. }\quad t\in\mathbb{R}.$

This means that $\\{\eta_{n}(\cdot,t)\\}$ is an $\mathscr{E}^{\,\rm
HK}_{1}$-Cauchy sequence for a.e. $t\in\mathbb{R}$. On the other hand, by
taking a further subsequence, $\\{\eta_{n}(\cdot,t)\\}$ converges to
$\eta(\cdot,t)$ in $L^{2}(T^{*}\\!M)$ for a.e. $t\in\mathbb{R}$. Therefore,
$\eta(\cdot,t)\in H^{1,2}(T^{*}\\!M)=D(\mathscr{E}^{\,\rm HK})$ and
$\eta_{n}(\cdot,t)\to\eta(\cdot,t)$ in $H^{1,2}(T^{*}\\!M)$ for a.e.
$t\in\mathbb{R}$. Moreover,

$\displaystyle\int_{\mathbb{R}}\mathscr{E}^{\rm
HK}_{1}(\eta(\cdot,t),\eta(\cdot,t))\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}}\lim_{n\to\infty}\mathscr{E}^{\rm
HK}_{1}(\eta_{n}(\cdot,t),\eta_{n}(\cdot,t))\mathrm{d}t$
$\displaystyle\leq\varliminf_{n\to\infty}\int_{\mathbb{R}}\mathscr{E}^{\rm
HK}_{1}(\eta_{n}(\cdot,t),\eta_{n}(\cdot,t))\mathrm{d}t$
$\displaystyle=\lim_{n\to\infty}\widehat{\mathscr{E}}^{\,\rm
HK}_{1}(\eta_{n},\eta_{n})=\widehat{\mathscr{E}}^{\,\rm
HK}_{1}(\eta,\eta)<\infty.$

From (9.6), we can similarly deduce

$\displaystyle\sum_{k=1}^{\infty}\|\dot{\eta}_{k+1}(\cdot,t)-\dot{\eta}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}<\infty\quad\text{
for\;\; a.e. }\quad t\in\mathbb{R}$

as above. Hence $\\{\dot{\eta}_{n}(\cdot,t)\\}$ is an
$L^{2}(T^{*}\\!M)$-Cauchy sequence for a.e. $t\in\mathbb{R}$. Let $v(\cdot,t)$
be its limit in $L^{2}(T^{*}\\!M)$. By using an integration by parts, we see
for $\theta\in{\rm Reg}(T^{*}\\!M)$ and $\varphi\in
C_{c}^{\infty}(\mathbb{R})$

$\displaystyle\langle v,\theta\otimes\varphi\rangle_{L^{2}(T^{*}\\!M)\otimes
L^{2}(\mathbb{R})}$
$\displaystyle=\int_{\mathbb{R}}\lim_{n\to\infty}\langle\dot{\eta}_{n}(\cdot,t),\theta\rangle_{L^{2}(T^{*}\\!M)}\varphi(t)\mathrm{d}t$
$\displaystyle=-\lim_{n\to\infty}\int_{\mathbb{R}}\langle\eta_{n}(\cdot,t),\theta\rangle_{L^{2}(T^{*}\\!M)}\dot{\varphi}(t)\mathrm{d}t$
$\displaystyle=-\int_{\mathbb{R}}\langle\eta(\cdot,t),\theta\rangle_{L^{2}(T^{*}\\!M)}\dot{\varphi}(t)\mathrm{d}t$
$\displaystyle=\left\langle-\int_{\mathbb{R}}\eta(\cdot,t)\dot{\varphi}(t)\mathrm{d}t,\theta\right\rangle_{L^{2}(T^{*}\\!M)}=\left\langle\int_{\mathbb{R}}\dot{\eta}(\cdot,t){\varphi}(t)\mathrm{d}t,\theta\right\rangle_{L^{2}(T^{*}\\!M)}$
$\displaystyle=\langle\dot{\eta},\theta\otimes\varphi\rangle_{L^{2}(T^{*}\\!M)\otimes
L^{2}(\mathbb{R})},$

hence $v(\cdot,t)=\dot{\eta}(\cdot,t)$ for a.e. $t\in\mathbb{R}$. Therefore

$\displaystyle\int_{\mathbb{R}}(\|\dot{\eta}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}+\|{\eta}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2})\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}}\lim_{n\to\infty}(\|\dot{\eta}_{n}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}+\|{\eta}_{n}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2})\mathrm{d}t$
$\displaystyle\leq\varliminf_{n\to\infty}\int_{\mathbb{R}}(\|\dot{\eta}_{n}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}+\|{\eta}_{n}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2})\mathrm{d}t$
$\displaystyle\leq\lim_{n\to\infty}\widehat{\mathscr{E}}^{\,\rm
HK}(\eta_{n},\eta_{n})=\widehat{\mathscr{E}}^{\,\rm HK}(\eta,\eta)<\infty.$

(2) $\Longrightarrow$ (1): Suppose the conditions in (2) for $\eta\in
L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$. Let $(\widehat{P}_{t}^{\,\rm HK})$
be the semigroup on $L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$ associated to
the quadratic form $(\widehat{\mathscr{E}}^{\,\rm
HK},D(\widehat{\mathscr{E}}^{\,\rm HK}))$ on $L^{2}(T^{*}\\!M)\otimes
L^{2}(\mathbb{R})$. Then we see

$\displaystyle\widehat{P}_{t}^{\,\rm HK}\eta(x,s)={P}_{t}^{\,\rm
HK}\left(P_{t}^{w}\eta(\cdot,s)\right)(x)\quad\text{ for
}\quad\widehat{\mathfrak{m}}\text{-a.e.~{}}(x,s),$ (9.7)

where

$\displaystyle P_{t}^{w}\eta(\cdot,s):=\frac{1}{\sqrt{4\pi
t}}\int_{\mathbb{R}}e^{-\frac{(u-s)^{2}}{4\pi t}}\eta(\cdot,u)\mathrm{d}u$

is defined to be a Bochner integral on $L^{2}(T^{*}\\!M)$. Similarly, we can
define $P_{t}^{w}\mathrm{d}\eta(\cdot,s)$ and
$P_{t}^{w}\mathrm{d}^{*}\eta(\cdot,s)$. Then

$\displaystyle\widehat{\mathscr{E}}^{\,\rm HK}$
$\displaystyle(\widehat{P}_{t}^{\,\rm HK}\eta,\widehat{P}_{t}^{\,\rm HK}\eta)$
$\displaystyle=\int_{\mathbb{R}}{\mathscr{E}}^{\rm HK}({P}_{t}^{\,\rm
HK}P_{t}^{w}\eta(\cdot,s),{P}_{t}^{\,\rm
HK}P_{t}^{w}\eta(\cdot,s))\mathrm{d}s+\int_{\mathbb{R}}\left|P_{t}^{\rm
HK}\frac{\partial}{\partial
s}P_{t}^{w}\eta(\cdot,s)\right|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}s$
$\displaystyle\leq\int_{\mathbb{R}}{\mathscr{E}}^{\rm
HK}(P_{t}^{w}\eta(\cdot,s),P_{t}^{w}\eta(\cdot,s))\mathrm{d}s+\int_{\mathbb{R}}\left|\frac{\partial}{\partial
s}P_{t}^{w}\eta(\cdot,s)\right|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}s$
$\displaystyle\leq\int_{\mathbb{R}}{\mathscr{E}}^{\rm
HK}(\eta(\cdot,s),\eta(\cdot,s))\mathrm{d}s+\int_{\mathbb{R}}\left|\frac{\partial}{\partial
s}\eta(\cdot,s)\right|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}s<\infty.$

In the last inequality, we use $|P_{t}^{w}\mathrm{d}\eta|\leq
P_{t}^{w}|\mathrm{d}\eta|$, $|P_{t}^{w}\mathrm{d}^{*}\eta|\leq
P_{t}^{w}|\mathrm{d}^{*}\eta|$, and $\frac{\partial}{\partial
s}P_{t}^{w}\eta(\cdot,s)=P_{t}^{w}\frac{\partial}{\partial s}\eta(\cdot,s)$
for $\widehat{\mathfrak{m}}$-a.e. $(x,s)$. Thus we have
$(\widehat{P}_{t}^{\,\rm HK}\eta)_{t\geq 0}$ is $\widehat{\mathscr{E}}^{\,\rm
HK}_{1}$-bounded. Therfore, $\eta\in D(\widehat{\mathscr{E}}^{\,\rm HK})$. ∎

######  Theorem 9.9.

Take $\omega=\omega(x,t)\in D(\widehat{\mathscr{E}}^{\,\rm HK})$. Then
$|\omega|\in D(\widehat{\mathscr{E}})$ and

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

###### Proof.

Applying the Fatou’s lemma for Dirichlet form on $L^{2}(\mathbb{R})$
associated to the one-dimensional Brownian motion $(B_{t})_{t\geq 0}$ (see
[47] for Fatou’s lemma for Dirichlet forms),

$\displaystyle\int_{\mathbb{R}}\left|\frac{\partial|\omega|}{\partial
t}\right|^{2}\mathrm{d}t$
$\displaystyle\leq\varliminf_{n\to\infty}\int_{\mathbb{R}}\left|\frac{\partial(|\omega|^{2}+\frac{1}{n^{2}})^{\frac{1}{2}}}{\partial
t}\right|^{2}\mathrm{d}t$
$\displaystyle\leq\varliminf_{n\to\infty}\int_{\mathbb{R}}\frac{\langle\omega,\dot{\omega}\rangle^{2}}{|\omega|^{2}+\frac{1}{n^{2}}}\mathrm{d}t\leq\varliminf_{n\to\infty}\int_{\mathbb{R}}\frac{|\omega|^{2}|\dot{\omega}|^{2}}{|\omega|^{2}+\frac{1}{n^{2}}}\mathrm{d}t$
$\displaystyle\leq\int_{\mathbb{R}}|\dot{\omega}|^{2}\mathrm{d}t.$ (9.8)

On the other hand, Hess-Schrader-Uhlenbrock inequality (6.6) tells us the
following (see [39])

$\displaystyle\mathscr{E}^{\kappa}(|\omega|(\cdot,t),|\omega|(\cdot,t))\leq\mathscr{E}^{\rm
HK}(\omega(\cdot,t),\omega(\cdot,t)).$ (9.9)

Combining (9.8) and (9.9), we have

$\displaystyle\widehat{\mathscr{E}}^{\,\widehat{\kappa}}(|\omega|,|\omega|)$
$\displaystyle=\int_{\mathbb{R}}\mathscr{E}^{\kappa}(|\omega|(\cdot,t),|\omega|(\cdot,t))\mathrm{d}t+\int_{M}\int_{\mathbb{R}}\left|\frac{\partial|\omega|}{\partial
t}\right|^{2}\mathrm{d}\mathfrak{m}\mathrm{d}t$
$\displaystyle\leq\int_{\mathbb{R}}\mathscr{E}^{\rm
HK}(\omega(\cdot,t),\omega(\cdot,t))\mathrm{d}t+\int_{M}\int_{\mathbb{R}}\left|\dot{\omega}\right|^{2}\mathfrak{m}(\mathrm{d}x)\mathrm{d}t\mathrm{d}\mathfrak{m}=\widehat{\mathscr{E}}^{\rm
HK}(\omega,\omega).$

∎

######  Corollary 9.10.

Let $\eta=\eta(x,t):=Q_{|t|}^{(\alpha),{\rm HK}}\theta(x)$ with $\theta\in
D({\Delta\Delta}^{\rm\\!HK})\cap L^{\infty}(T^{*}\\!M)$ and
$\alpha>C_{\kappa}$. Then $\eta\in D(\widehat{\mathscr{E}}^{\,\rm HK})$, hence
$|\eta|\in D(\widehat{\mathscr{E}}^{\,\widehat{\kappa}})\cap
L^{\infty}(\widehat{M};\widehat{\mathfrak{m}})=D(\widehat{\mathscr{E}})\cap
L^{\infty}(\widehat{M};\widehat{\mathfrak{m}})$. In particular, $|\eta|^{2}\in
D(\widehat{\mathscr{E}})\cap L^{\infty}(\widehat{M};\widehat{\mathfrak{m}})$.
Moreover, for $p\in]0,+\infty[$ and $\varepsilon>0$,
$\eta_{{\varepsilon}}:=(|\eta|^{2}+{\varepsilon}^{2})^{\frac{p}{2}}-{\varepsilon}^{p}$
satisfies $\eta_{{\varepsilon}}\in D(\widehat{\mathscr{E}})\cap
L^{\infty}(\widehat{M};\widehat{\mathfrak{m}})$.

###### Proof.

From (8.1), we easily see

$\displaystyle\int_{\mathbb{R}}\|\eta(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t\leq
C^{2}\|\theta\|_{L^{2}(T^{*}\\!M)}^{2}\int_{\mathbb{R}}e^{-2\sqrt{\alpha-
C_{\kappa}}|t|}\mathrm{d}t<\infty,$

hence $\eta\in L^{2}(T^{*}\\!M)\otimes L^{2}(\mathbb{R})$. For a fixed $t>0$,
we see

$\displaystyle\mathscr{E}^{\rm HK}_{\alpha}(\eta(\cdot,t),\eta(\cdot,t))$
$\displaystyle=((\alpha-{\Delta\Delta}^{\rm\\!HK})\eta(\cdot,t),\eta(\cdot,t))_{L^{2}(T^{*}\\!M)}$
$\displaystyle=\|\sqrt{\alpha-{\Delta\Delta}^{\rm\\!HK}}\eta(\cdot,t)\|_{L^{2}(T^{*}\\!M)}$
$\displaystyle=\int_{M}|\dot{\eta}(\cdot,t)|^{2}\mathrm{d}\mathfrak{m},$

where we use $\dot{\eta}=\sqrt{\alpha-{\Delta\Delta}^{\rm\\!HK}}\eta$ in
$L^{2}(T^{*}\\!M)$. Applying (8.1) again,

$\displaystyle\int_{\mathbb{R}}\|(\alpha-{\Delta\Delta}^{\rm\\!HK})\eta(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}}\|Q_{|t|}^{(\alpha),{\rm
HK}}(\alpha-{\Delta\Delta}^{\rm\\!HK})\theta\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t$
$\displaystyle\leq\|(\alpha-{\Delta\Delta}^{\rm\\!HK})\theta\|_{L^{2}(T^{*}\\!M)}^{2}C^{2}\int_{\mathbb{R}}e^{-2\sqrt{\alpha-
C_{\kappa}}|t|}\mathrm{d}t<\infty.$

Therefore, we obtain

$\displaystyle\widehat{\mathscr{E}}^{\,\rm HK}_{\alpha}(\eta,\eta)$
$\displaystyle=\int_{\mathbb{R}}{\mathscr{E}}^{\,\rm
HK}_{\alpha}(\eta(\cdot,t),\eta(\cdot,t))\mathrm{d}t+\int_{\mathbb{R}}\|\dot{\eta}(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t$
$\displaystyle=2\int_{\mathbb{R}}{\mathscr{E}}^{\,\rm
HK}_{\alpha}(\eta(\cdot,t),\eta(\cdot,t))\mathrm{d}t$ $\displaystyle\leq
2\sqrt{\int_{\mathbb{R}}\|(\alpha-{\Delta\Delta}^{\rm\\!HK})\eta(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t}\sqrt{\int_{\mathbb{R}}\|\eta(\cdot,t)\|_{L^{2}(T^{*}\\!M)}^{2}\mathrm{d}t}<\infty.$

By Lemma 9.8, this inequality shows $\eta\in D(\widehat{\mathscr{E}}^{\,\rm
HK})$. Theorem 9.9 also shows $|\eta|\in D(\widehat{\mathscr{E}})\cap
L^{\infty}(\widehat{M};\widehat{\mathfrak{m}})$ . The final assertion follows
from the fact that
$T(x):=\Psi(|x|)=(|x|+{\varepsilon}^{2})^{\frac{p}{2}}-{\varepsilon}^{p}$
satisfies $T(0)=0$ and $|T(x)-T(y)|\leq\frac{p}{2}{\varepsilon}^{p-2}|x-y|$