#  $(1,p)$-Sobolev spaces based on strongly local Dirichlet forms

Kazuhiro Kuwae Department of Applied Mathematics, Fukuoka University, Fukuoka
814-0180, Japan (kuwae@fukuoka-u.ac.jp). 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

In the framework of quasi-regular strongly local Dirichlet form
$(\mathscr{E},D(\mathscr{E}))$ on $L^{2}(X;\mathfrak{m})$ admitting minimal
$\mathscr{E}$-dominant measure $\mu$, we construct a natural $p$-energy
functional $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$ on
$L^{p}(X;\mathfrak{m})$ and $(1,p)$-Sobolev space
$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ for $p\in]1,+\infty[$. In this paper, we
establish the Clarkson type inequality for $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$.
As a consequence, $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ is a uniformly convex
Banach space, hence it is reflexive. Based on the reflexivity of
$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we prove that (generalized) normal
contraction operates on $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$, which has
been shown in the case of various concrete settings, but has not been proved
for such general framework. Moreover, we prove that $(1,p)$-capacity ${\rm
Cap}_{1,p}(A)<\infty$ for open set $A$ admits an equilibrium potential
$e_{A}\in D(\mathscr{E}^{\,p})$ with $0\leq e_{A}\leq 1$ $\mathfrak{m}$-a.e.
and $e_{A}=1$ $\mathfrak{m}$-a.e. on $A$.

Keywords: Dirichlet form, strongly local, quasi-regular,
$\mathscr{E}$-dominant measure, minimal $\mathscr{E}$-dominant measure, normal
contraction, carré du champ operator, $(1,p)$-Sobolev space,
Mathematics Subject Classification (2020): Primary 31C25; Secondary 30L15,
53C21, 58J35, 35K05, 42B05, 47D08

##  1 Statement of Main Results

Let $(X,\tau)$ 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 $X$ with full topological support. Let
$(\mathscr{E},D(\mathscr{E}))$ be a quasi-regular symmetric strongly local
Dirichlet space on $L^{2}(X;\mathfrak{m})$ and $(P_{t})_{t\geq 0}$ the
associated symmetric sub-Markovian strongly continuous semigroup on
$L^{2}(X;\mathfrak{m})$ (see [31, Chapter IV, Definition 3] for the quasi-
regularity and see [30, Theorem 5.1(ii)] 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 $f\in
L^{2}(X;\mathfrak{m})\cap\mathscr{B}(X)$, $P_{t}f(x)=\mathbb{E}_{x}[f(X_{t})]$
$\mathfrak{m}$-a.e. $x\in X$ (see [31, Chapter IV, Section 3]).

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

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

Here $\tilde{f}$ denotes the $\mathscr{E}$-quasi-continuous
$\mathfrak{m}$-version of $f$ (see [17, Theorem 2.1.3], [31, Chapter IV,
Proposition 3.3(ii)]). We set $\mu_{\langle f\rangle}:=\mu_{\langle
f,f\rangle}$ for $f\in D(\mathscr{E})\cap L^{\infty}(X;\mathfrak{m})$.
Moreover, for $f,g\in D(\mathscr{E})$, there exists a signed finite measure
$\mu_{\langle f,g\rangle}$ on $X$ such that $\mathscr{E}(f,g)=\mu_{\langle
f,g\rangle}(X)$, hence $\mathscr{E}(f,f)=\mu_{\langle f\rangle}(X)$. A Borel
measure $\mu$ on $X$ is said to be _$\mathscr{E}$ -dominant_ if it is
$\sigma$-finite and $\mu_{\langle f\rangle}\ll\mu$ for all $f\in
D(\mathscr{E})$, and it is said to be _minimal $\mathscr{E}$-dominant_ if it
is $\mathscr{E}$-dominant and $\mu\ll\nu$ for every $\mathscr{E}$-dominant
Borel measure $\nu$ on $X$. Any two minimal $\mathscr{E}$-dominant measure are
mutually equivalent. The existence of minimal $\mathscr{E}$-dominant measure
$\mu$ is shown by Nakao [33, Lemma 2.2], which is given by a finite smooth
measure $\mu:=\sum_{k=1}^{\infty}\frac{\mu_{\langle
f_{k}\rangle}}{2^{k}(1+\mu_{\langle f_{k}\rangle}(X))}$ for some countable
$\mathscr{E}_{1}^{1/2}$-dense subcollection $\\{f_{k}\\}$ of $D(\mathscr{E})$.
The class of $f\in D(\mathscr{E})$ for which $\mu_{\langle f\rangle}$ is
minimal $\mathscr{E}$-dominant is dense in $D(\mathscr{E})$ (see [25,
Proposition 2.7]), and every minimal $\mathscr{E}$-dominant measure does not
charge $\mathscr{E}$-exceptional sets (see [12, Proposition 2.18]). Given any
$\mathscr{E}$-dominant $\mu$, there exists a unique symmetric bilinear form
$\Gamma_{\mu}:D(\mathscr{E})_{e}\times D(\mathscr{E})_{e}\to L^{1}(X;\mu)$
such that for every $f,g\in D(\mathscr{E})_{e}$,

$\displaystyle\mathscr{E}(f,g)=\int_{X}\Gamma_{\mu}(f,g)\mathrm{d}\mu$

(see [9, after Definition 1.13]). Hereafter, we fix a minimal
$\mathscr{E}$-dominant measure $\mu$. The calculus rules from [9, Proposition
1.12] transfer accordingly to $\Gamma_{\mu}$ at the $\mu$-a.e. level as in the
following: Take $u,v\in D(\mathscr{E})$. Then

  1. (1)

(Parallelogram law):
$\Gamma_{\mu}(u+v)+\Gamma_{\mu}(u-v)=2\Gamma_{\mu}(u)+2\Gamma_{\mu}(v)$
$\mu$-a.e.

  2. (2)

(Strong local property): ${\bf 1}_{G}\Gamma_{\mu}(u,v)=0$ $\mu$-a.e. for all
open sets $G\subset X$ on which $u$ is constant.

  3. (3)

(Cauchy-Schwarz inequality): We have
$|\Gamma_{\mu}(u,v)|\leq\Gamma_{\mu}(u)^{\frac{1}{2}}\Gamma_{\mu}(v)^{\frac{1}{2}}$
$\mu$-a.e. In particular,
$\Gamma_{\mu}(u+v)^{\frac{1}{2}}\leq\Gamma_{\mu}(u)^{\frac{1}{2}}+\Gamma_{\mu}(v)^{\frac{1}{2}}$
$\mu$-a.e. and
$|\Gamma_{\mu}(u)^{\frac{1}{2}}-\Gamma_{\mu}(v)^{\frac{1}{2}}|\leq\Gamma_{\mu}(u-v)^{\frac{1}{2}}$
$\mu$-a.e.

  4. (4)

(Truncation Property): $\Gamma_{\mu}(u\lor v)={\bf
1}_{\\{\tilde{u}\geq\tilde{v}\\}}\Gamma_{\mu}(u)+{\bf
1}_{\\{\tilde{u}<\tilde{v}\\}}\Gamma_{\mu}(v)$ $\mu$-a.e. and
$\Gamma_{\mu}(u\land v)={\bf
1}_{\\{\tilde{u}\geq\tilde{v}\\}}\Gamma_{\mu}(v)+{\bf
1}_{\\{\tilde{u}<\tilde{v}\\}}\Gamma_{\mu}(u)$ $\mu$-a.e.

  5. (5)

(Contraction property): $\Gamma_{\mu}(0\lor u\land 1)\leq\Gamma_{\mu}(u)$
$\mu$-a.e.

  6. (6)

(Leibniz rule):
$\Gamma_{\mu}(uv,w)=\tilde{u}\Gamma_{\mu}(v,w)+\tilde{v}\Gamma_{\mu}(u,w)$
$\mu$-a.e. for $u,v\in D(\mathscr{E})\cap L^{\infty}(X;\mu)$ and $w\in
D(\mathscr{E})$.

  7. (7)

(Chain rule): Let $\phi\in C^{1}(\mathbb{R})$ with $\phi(0)=0$ and bounded
derivative $\phi^{\prime}$. Then $\phi(u)\in D(\mathscr{E})$ and
$\Gamma_{\mu}(\phi(u),v)=\phi^{\prime}(\tilde{u})\Gamma_{\mu}(u,v)$ $\mu$-a.e.

We say that $(X,\mathscr{E},\mathfrak{m})$ or simply $\mathscr{E}$ admits a
carré-du-champ $\Gamma$ if $\mathfrak{m}$ is $\mathscr{E}$-dominant in which
case we abbreviate $\Gamma_{\mathfrak{m}}$ by $\Gamma$ and term it the _carré
du champ_ , then $\mathfrak{m}$ is already _minimal_ $\mathscr{E}$-dominant
according to the remark after [12, Definition 3.4].

Fix $p\in[1,+\infty[$. Let $(\mathscr{E},D(\mathscr{E}))$ be a quasi-regular
strongly local Dirichlet form on $L^{2}(X;\mathfrak{m})$ admitting a minimal
$\mathscr{E}$-dominant measure $\mu$. Set $\mathscr{D}_{1,p}:=\\{u\in
L^{p}(X;\mathfrak{m})\cap D(\mathscr{E})\mid\Gamma_{\mu}(u)^{\frac{1}{2}}\in
L^{p}(X;\mu)\\}$ and fix a subspace $\mathscr{D}$ of $\mathscr{D}_{1,p}$. For
$u\in\mathscr{D}$, we define the _$p$ -energy_
$\mathscr{E}^{\,p}(u):=\|\Gamma_{\mu}(u)^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$.

######  Assumption 1.

We assume the closability of $(\mathscr{E}^{\,p},\mathscr{D})$ on
$L^{p}(X;\mathfrak{m})$, i.e. for any $\mathscr{E}^{\,p}$-Cauchy sequence
$\\{u_{n}\\}\subset\mathscr{D}$ satisfying $u_{n}\to 0$ in
$L^{p}(X;\mathfrak{m})$ as $n\to\infty$, then $\mathscr{E}^{\,p}(u_{n})\to 0$
as $n\to\infty$.

######  Proposition 1.1.

Assumption 1 is satisfied if $\mathfrak{m}(X)+\mu(X)<\infty$ and $p\geq 2$.

###### Proof.

Suppose that $\\{f_{n}\\}\subset\mathscr{D}$ is an $\mathscr{E}^{\,p}$-Cauchy
sequence such that $f_{n}\to 0$ in $L^{p}(X;\mathfrak{m})$. Applying Hölder’s
inequality with $\mathfrak{m}(X)+\mu(X)<\infty$ and $p\geq 2$, we see that
$\\{f_{n}\\}\subset\mathscr{D}$ is an $\mathscr{E}$-Cauchy sequence satisfying
$f_{n}\to 0$ in $L^{2}(X;\mathfrak{m})$. Then $\mathscr{E}(f_{n},f_{n})\to 0$
as $n\to\infty$ by the closedness of $(\mathscr{E},D(\mathscr{E}))$ on
$L^{2}(X;\mathfrak{m})$. From this, we see $\Gamma_{\mu}(f_{n})\to 0$ in
$\mu$-measure as $n\to\infty$. On the other hand, there exists $v\in
L^{p}(X;\mu)_{+}$ such that $\Gamma_{\mu}(f_{n})^{\frac{1}{2}}\to v$ in
$L^{p}(X;\mu)$ as $n\to\infty$. This implies $v=0$ $\mu$-a.e., hence
$\mathscr{E}^{\,p}(f_{n})=\int_{X}\Gamma_{\mu}(f_{n})^{\frac{p}{2}}\mathrm{d}\mu\to\int_{X}v^{\,p}\mathrm{d}\mu=0$
as $n\to\infty$. ∎

######  Definition 1.2 ($(1,p)$-Sobolev space
$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$).

For $u\in\mathscr{D}$, we define
$\|u\|_{H^{1,p}}^{p}:=\|u\|_{L^{p}(X;\mathfrak{m})}^{p}+\mathscr{E}^{\,p}(u)$.
Then $\|\cdot\|_{H^{1,p}}$ forms a norm on $\mathscr{D}$. We now define the
Sobolev space $H^{1,p}(X)$ as the $\|\cdot\|_{H^{1,p}}$-completion of
$\mathscr{D}$:

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

More precisely, for $u\in H^{1,p}(X)$, we can uniquely extend $\Gamma_{\mu}$
to $u$ with $\Gamma_{\mu}(u)^{\frac{1}{2}}\in L^{p}(X;\mu)$ as the
$L^{p}$-limit of $\Gamma_{\mu}(u_{n})^{\frac{1}{2}}$ for its approximating
sequence $\\{u_{n}\\}\subset\mathscr{D}$ in view of the inequality
$|\sqrt{\Gamma_{\mu}(f)}-\sqrt{\Gamma_{\mu}(g)}|\leq\sqrt{\Gamma_{\mu}(f-g)}$
$\mu$-a.e. The quantity $\Gamma_{\mu}(u)^{\frac{1}{2}}$ does not depend on the
choice of approximating sequence $\\{u_{n}\\}\subset\mathscr{D}$ . We set
$D(\mathscr{E}^{\,p}):=H^{1,p}(X)$ and
$\mathscr{E}^{\,p}(u):=\|\Gamma_{\mu}(u)^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
for $u\in D(\mathscr{E}^{\,p})$, which is called the _$p$ -energy_ of $u\in
H^{1,p}(X)$. It is easy to see that for $u,v\in D(\mathscr{E}^{\,p})$, the
Parallelogram law (1) above remains valid. We can define $\Gamma_{\mu}(u,v)$
as

$\displaystyle\Gamma_{\mu}(u,v):=\Gamma_{\mu}\left(\frac{u+v}{2}\right)-\Gamma_{\mu}\left(\frac{u-v}{2}\right)\in
L^{\frac{p}{2}}(X;\mu),$

which forms a bilinear form $\Gamma_{\mu}(\cdot,\cdot):H^{1,p}(X)\times
H^{1,p}(X)\to L^{\frac{p}{2}}(X;\mu)$ under the Parallelogram law (1) above,
and Cauchy-Schwarz inequality (3) above also remains valid for $u,v\in
D(\mathscr{E}^{\,p})$.

For $u,v\in D(\mathscr{E}^{\,p})$ and $\alpha\geq 0$, we define

$\displaystyle\mathscr{E}^{\,p}(u,v):$
$\displaystyle=\int_{X}\Gamma_{\mu}(u)^{\frac{p-2}{2}}\Gamma_{\mu}(u,v)\mathrm{d}\mu,$
(1.1) $\displaystyle\mathscr{E}^{\,p}_{\alpha}(u,v):$
$\displaystyle=\mathscr{E}^{\,p}(u,v)+\alpha\int_{X}|u|^{p-2}uv\,\mathrm{d}\mathfrak{m}.$
(1.2)

Clearly we see $\mathscr{E}^{\,p}_{0}(u,v)=\mathscr{E}^{\,p}(u,v)$ and the
linearities of $v\mapsto\mathscr{E}^{\,p}(u,v)$ and
$v\mapsto\mathscr{E}^{\,p}_{\alpha}(u,v)$. Moreover, we have

$\displaystyle|\mathscr{E}^{\,p}(u,v)|$
$\displaystyle\leq\mathscr{E}^{\,p}(u)^{1-\frac{1}{p}}\mathscr{E}^{\,p}(v)^{\frac{1}{p}},\qquad|\mathscr{E}^{\,p}_{\alpha}(u,v)|\leq\mathscr{E}^{\,p}_{\alpha}(u)^{1-\frac{1}{p}}\mathscr{E}^{\,p}_{\alpha}(v)^{\frac{1}{p}}.$
(1.3)

Indeed, the first inequality in (1.3) and the following inequality

$\displaystyle\left|\alpha\int_{X}|u|^{p-2}uv\,\mathrm{d}\mathfrak{m}\right|\leq\left(\alpha\int_{X}|u|^{p}\mathrm{d}\mathfrak{m}\right)^{1-\frac{1}{p}}\left(\alpha\int_{X}|v|^{p}\mathrm{d}\mathfrak{m}\right)^{\frac{1}{p}}$
(1.4)

are easy consequences of Hölder’s inequality, and the second inquality in
(1.3) follows the first one in (1.3) and (1.4) by way of the usual Hölder’s
inequality in $\mathbb{R}^{2}$:
$|x_{1}y_{1}+x_{2}y_{2}|\leq(|x_{1}|^{p}+|x_{2}|^{p})^{1/p}(|y_{1}|^{q}+|y_{2}|^{q})^{1/q}$,
$1/p+1/q=1$. We call $\mathscr{E}^{\,p}_{\alpha}(u,v)$ the _$\alpha$ -order
mutual energy of $u$ and $v$_. Note that
$\mathscr{E}^{\,p}_{\alpha}(u,u)=\int_{X}\Gamma_{\mu}(u)^{\frac{p}{2}}\mathrm{d}\mu+\alpha\int_{X}|u|^{p}\mathrm{d}\mathfrak{m}$,
in particular,
$\mathscr{E}^{\,p}_{1}(u,u)=\mathscr{E}^{\,p}_{1}(u)=\|u\|_{H^{1,p}}^{p}$ and
$\mathscr{E}^{\,p}(u,u)=\mathscr{E}^{\,p}(u)$.

######  Remark 1.3.

The coenergy (1.1) and the inequality (1.3) are also noted in [5] under the
existence of curré du champ operator. These are considered for the $p$-energy
associated with $p$-Laplacian over Euclidean space (see also [4]).

######  Remark 1.4.

To well-define $(1,p)$-Sobolev space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we
need Assumption 1. Otherwise the unique continuous map
$\overline{\iota}:H^{1,p}(X)\to L^{p}(X;\mathfrak{m})$ extending the inclusion
map $\iota:\mathscr{D}\to L^{p}(X;\mathfrak{m})$ is not necessarily injective
(see [31, Chapter I Remark 3.2(ii)]).

######  Remark 1.5.

The normed vector space $(\mathscr{D},\|\cdot\|_{1,p})$ with the following
norm:

$\displaystyle\|u\|_{1,p}:=\|u\|_{L^{p}(X;\mathfrak{m})}+\|\Gamma_{\mu}(u)^{\frac{1}{2}}\|_{L^{p}(X;\mu)},\quad
u\in\mathscr{D}_{1,p}$

is defined in [8, Notation 6.2.1] for $p\in[2,+\infty[$ under the condition
that $\mu=\mathfrak{m}$ is a minimal $\mathscr{E}$-dominant measure, i.e.,
$(\mathscr{E},D(\mathscr{E}))$ admits a carré du champ
$\Gamma=\Gamma_{\mathfrak{m}}$. But, in [8], there is no consideration for its
completion.

######  Remark 1.6.

In [26, Theorem 6.1], there is a similar construction of $(1,p)$-Sobolev space
denoted by $(H^{1,p}_{0}(X),\|\cdot\|_{1,p})$ under $p>2$ and
$\mathfrak{m}(X)<\infty$ in the framework of strongly local regular Dirichlet
form over locally compact separable metric space. The first inequality in
(1.3) is also noted in [26, Remark 6.1].

######  Remark 1.7.

  1. (i)

There are various ways to define $(1,p)$-Sobolev space for metric measure
spaces based on curré du champ (see [3, 7, 20, 22]). But in the past
literature, there is no synthetic treatment for the construction of such
spaces based on the theory of strongly local quasi-regular Dirichlet forms
admitting carré du champ operator including the case of infinite dimensional
spaces, like Wiener space, configuration spaces, interacting particle systems
etc. Such an infinite dimensional space has been thought to be considered in
the framework of extended metric measure spaces (see also [2, 35]). So the
construction of $(1,p)$-Sobolev space based on the theory of strongly local
quasi-regular Dirichlet forms is quite applicable.

  2. (ii)

There is another way to define $(r,p)$-Sobolev spaces (including the case of
$r=1$) as the Bessel potential space (see [16, 27, 28]). Note that our Sobolev
space based on the curré du champ operator is equivalent to the
$(1,p)$-Sobolev spaces as the Bessel potential space provided the Riesz
operator satisfies the $L^{p}$-boundedness (see [15, Theorem 1.4(2)]).

######  Remark 1.8.

There is a complete different approach to define $(1,p)$-Sobolev space on
fractals (see [24, 10, 38, 32]). It is unclear the relation with our Sobolev
space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$.

Our first main theorem is the Clarkson type inequality of the $(1,p)$-Sobolev
space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$.

######  Theorem 1.9 (Clarkson’s inequality).

Suppose $p\in]1,+\infty[$. The energy functional $\mathscr{E}^{\,p}_{1}$
satisfies the following inequalities of Clarkson type: for $\alpha\geq 0$ and
$f,g\in D(\mathscr{E}^{\,p})=H^{1,p}(X)$

$\displaystyle\mathscr{E}^{\,p\\!}_{\alpha}\left(\frac{f+g}{2}\right)+\mathscr{E}^{\,p\\!}_{\alpha}\left(\frac{f-g}{2}\right)$
$\displaystyle\leq\frac{\mathscr{E}^{\,p}_{\alpha}(f)+\mathscr{E}^{\,p}_{\alpha}(g)}{2},\quad
p\in[2,+\infty[,$ (1.5)
$\displaystyle\mathscr{E}^{\,p\\!}_{\alpha}\left(\frac{f+g}{2}\right)^{\frac{1}{p-1}}+\mathscr{E}^{\,p\\!}_{\alpha}\left(\frac{f-g}{2}\right)^{\frac{1}{p-1}}$
$\displaystyle\leq\left(\frac{\mathscr{E}^{\,p}_{\alpha}(f)+\mathscr{E}^{\,p}_{\alpha}(g)}{2}\right)^{\frac{1}{p-1}},\quad
p\in]1,2].$ (1.6)

In particular, $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ is a uniformly convex Banach
space and consequently it is reflexive.

Thanks to the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we can
establish that the (generalized) normal contraction operates on
$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$. For this, we define such notion below.

For a given function $T:\mathbb{R}^{N}\to\mathbb{R}$, $T$ is said to be a
_(generalized) normal contraction_ if

$\displaystyle T(0)=0\quad\text{ and
}\quad|T(x)-T(y)|\leq\sum_{k=1}^{N}|x_{k}-y_{k}|$

for any $x=(x_{1},\cdots,x_{N})$, $y=(y_{1},\cdots,y_{N})\in\mathbb{R}^{N}$.
When $N=1$, we call such $T:\mathbb{R}\to\mathbb{R}$ a _normal contraction_.
Note that $\phi(t):=t^{\sharp}:=0\lor t\land 1$, $\phi(t):=|t|$,
$\phi(t):=t^{+}:=t\lor 0$ and $\phi(t):=t^{-}:=(-t)\lor 0$ are typical
examples of normal contractions. More generally, the real-valued function
$\phi_{{\varepsilon}}$ on $\mathbb{R}$ depending on ${\varepsilon}>0$ such
that $\phi_{{\varepsilon}}(t)=t$ for $t\in[0,1]$,
$-{\varepsilon}\leq\phi_{{\varepsilon}}(t)\leq 1+{\varepsilon}$ for all
$t\in\mathbb{R}$, and
$0\leq\phi_{{\varepsilon}}(t)-\phi_{{\varepsilon}}(s)\leq t-s$ whenever $s<t$,
is also a normal contraction.

######  Theorem 1.10.

Suppose $p\in]1,+\infty[$ and $u=(u_{1},\cdots,u_{N})\in H^{1,p}(X)^{N}$. Let
$T:\mathbb{R}^{N}\to\mathbb{R}$ be a (generalized) normal contraction. Then
$T(u)\in H^{1,p}(X)$ and
$\mathscr{E}^{\,p}(T(u))^{\frac{1}{p}}\leq\sum_{k=1}^{N}\mathscr{E}^{\,p}(u_{k})^{\frac{1}{p}}$,
hence $\|T(u)\|_{H^{1,p}}\leq
2^{1-\frac{1}{p}}\sum_{k=1}^{n}\|u_{k}\|_{H^{1,p}}$.

By Theorem 1.10, we can call $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$ a
_non-linear Dirichlet space_ on $L^{p}(X;\mathfrak{m})$.

######  Corollary 1.11.

Suppose $p\in]1,+\infty[$. Then we have the following:

  1. (1)

For any $u\in H^{1,p}(X)$, we have $u^{\sharp},|u|,u^{+},u^{-}\in H^{1,p}(X)$,
and $\mathscr{E}^{\,p}(u^{\sharp})\leq\mathscr{E}^{\,p}(u)$,
$\mathscr{E}^{\,p}(|u|)\leq\mathscr{E}^{\,p}(u)$,
$\mathscr{E}^{\,p}(u^{+})\leq\mathscr{E}^{\,p}(u)$ and
$\mathscr{E}^{\,p}(u^{-})\leq\mathscr{E}^{\,p}(u)$.

  2. (2)

For any $u,v\in H^{1,p}(X)$, $u\lor v,u\land v\in H^{1,p}(X)$ and
$\mathscr{E}^{\,p}(u\lor
v)^{\frac{1}{p}}\leq\mathscr{E}^{\,p}(u)^{\frac{1}{p}}+\mathscr{E}^{\,p}(v)^{\frac{1}{p}}$,
$\mathscr{E}^{\,p}(u\land
v)^{\frac{1}{p}}\leq\mathscr{E}^{\,p}(u)^{\frac{1}{p}}+\mathscr{E}^{\,p}(v)^{\frac{1}{p}}$.

  3. (3)

For any $T\in C^{1}(\mathbb{R}^{n})$ with $T(0)=0$, $|\nabla T|\in
C_{b}(\mathbb{R}^{N})$ and $u=(u_{1},\cdots,u_{N})\in(H^{1,p}(X))^{N}$, we
have $T(u)\in H^{1,p}(X)$ with
$\mathscr{E}^{\,p}(T(u))^{\frac{1}{p}}\leq\max\limits_{1\leq k\leq
N}\left\|\frac{\partial T}{\partial
x_{k}}\right\|_{\infty}\sum_{k=1}^{N}\mathscr{E}^{\,p}(u_{k})^{\frac{1}{p}}$.

######  Theorem 1.12.

Suppose $p\in]1,+\infty[$ and $u,v\in H^{1,p}(X)$. Then we have the following
strong subadditive of $p$-energy $\mathscr{E}^{\,p}$:

$\displaystyle\mathscr{E}^{\,p}(u\lor v)+\mathscr{E}^{\,p}(u\land
v)\leq\mathscr{E}^{\,p}(u)+\mathscr{E}^{\,p}(v).$ (1.7)

######  Definition 1.13.

Denote by $\mathcal{O}(X)$ the family of all open subsets of $X$. For
$A\in\mathcal{O}(X)$ we define

$\displaystyle\mathcal{L}_{A}:$ $\displaystyle=\\{u\in
D(\mathscr{E}^{\,p})\mid u\geq 1\;\mathfrak{m}\text{-a.e.~{}on }A\\},$ (1.8)
$\displaystyle{\rm Cap}_{1,p}(A):$
$\displaystyle=\left\\{\begin{array}[]{cc}\inf_{u\in\mathcal{L}_{A}}\mathscr{E}^{\,p}_{1}(u,u),&\mathcal{L}_{A}\neq\emptyset\\\
+\infty&\mathcal{L}_{A}=\emptyset,\end{array}\right.$ (1.11)

and for any set $A\subset X$ we let

$\displaystyle{\rm Cap}_{1,p}(A):=\inf_{B\in\mathcal{O}(X),A\subset B}{\rm
Cap}_{1,p}(B).$ (1.12)

We call this _$1$ -order $(1,p)$-capacity_ of $A$ or simply the _capacity_ of
$A$.

Let
$\mathcal{O}_{0}(X):=\\{A\in\mathcal{O}(X)\mid\mathcal{L}_{A}\neq\emptyset\\}$.

######  Theorem 1.14.

For $A,B\in\mathcal{O}_{0}(X)$, we have the following:

  1. (1)

There exists a unique element $e_{A}\in\mathcal{L}_{A}$ such that

$\displaystyle\mathscr{E}^{\,p}_{1}(e_{A},e_{A})={\rm Cap}_{1,p}(A).$ (1.13)

  2. (2)

$0\leq e_{A}\leq 1$ $\mathfrak{m}$-a.e. and $e_{A}=1$ $\mathfrak{m}$-a.e. on
$A$.

  3. (3)

$e_{A}$ is a unique element of $D(\mathscr{E}^{\,p})$ satisfying $e_{A}=1$
$\mathfrak{m}$-a.e. on $A$ and $\mathscr{E}^{\,p}_{1}(e_{A},v)\geq 0$ for all
$v\in D(\mathscr{E}^{\,p})$ with $v\geq 0$ $\mathfrak{m}$-a.e. on $A$.

  4. (4)

If $v\in D(\mathscr{E}^{\,p})$ satisfies $v=1$ $\mathfrak{m}$-a.e. on $A$,
then $\mathscr{E}^{\,p}_{1}(e_{A},v)={\rm Cap}_{1,p}(A)$.

  5. (5)

If $A\subset B$, then $e_{A}\leq e_{B}$ $\mathfrak{m}$-a.e.

######  Remark 1.15.

  1. (i)

The assertion of Theorem 1.14(1),(5) is proved in [6, Proposition 2.1] based
on the existence of lower semi continuous convex functional functional $\Phi$
on $L^{p}(X;\mathfrak{m})$ in the framework of locally compact separable
metric space $X$ having positive Radon measure $\mathfrak{m}$ with ${\rm
supp}[\mathfrak{m}]=X$ under several assumptions related to the existence of
uniformly convex Banach space in terms of $\Phi$ and the conclusion of Theorem
1.12. In [6, Proposition 2.1], they did not show Theorem 1.14(2) under the
condition $(H_{6})$ in [6]. The contents of Theorem 1.14(3),(4) are not shown
in [6, Proposition 2.2]. Theorem 1.14(1),(5) is not completely covered by [6,
Proposition 2.1], because our underlying space $X$ is not necessarily locally
compact.

  2. (ii)

The assertion of Theorem 1.14(1) is also proved in [18, Theorem 2.1] in the
framework of $(1,p)$-Sobolev space of pure jump type by proving Hanner type
inequality for Sobolev spaces.

######  Corollary 1.16.

The $(1,p)$-capacity ${\rm Cap}_{1,p}$ is a Choquet capacity, i.e.,

  1. (1)

If $A\subset B$, then ${\rm Cap}_{1,p}(A)\leq{\rm Cap}_{1,p}(B)$.

  2. (2)

If $\\{A_{n}\\}_{n\in\mathbb{N}}$ is an increasing sequence of subsets of $X$,
then

$\displaystyle{\rm
Cap}_{1,p}\left(\bigcup_{n=1}^{\infty}A_{n}\right)\leq\sup_{n\in\mathbb{N}}{\rm
Cap}_{1,p}(A_{n}).$ (1.14)

  3. (3)

If $\\{A_{n}\\}_{n\in\mathbb{N}}$ is a decreasing sequence of compact subsets
of $X$, then

$\displaystyle{\rm
Cap}_{1,p}\left(\bigcap_{n=1}^{\infty}A_{n}\right)\leq\inf_{n\in\mathbb{N}}{\rm
Cap}_{1,p}(A_{n}).$ (1.15)

Moreover, it holds that for any Borel set $A$

$\displaystyle{\rm Cap}_{1,p}(A)=\sup_{K:\text{compact},K\subset A}{\rm
Cap}_{1,p}(K),$ (1.16)

and for any $A,B\subset X$

$\displaystyle{\rm Cap}_{1,p}(A\cup B)+{\rm Cap}_{1,p}(A\cap B)\leq{\rm
Cap}_{1,p}(A)+{\rm Cap}_{1,p}(B).$ (1.17)

######  Remark 1.17.

  1. (i)

The assertion of Corollary 1.16 is proved by [6, Proposition 2.2] in their
framework noted in Remark 1.15. Theorem 1.16 is not completely covered by [6,
Proposition 2.2], because our underlying space $X$ is not necessarily locally
compact.

  2. (ii)

The assertion of Corollary 1.16 is also proved in [18, Corollary 2.2] in the
framework of $(1,p)$-Sobolev space of pure jump type.

######  Remark 1.18.

For $\alpha>0$, we can define the $\alpha$-order $(1,p)$-capacity ${\rm
Cap}_{\alpha,p}(A)$ of $A$ by using $\mathscr{E}^{\,p}_{\alpha}(u,v)$. Similar
statements of Theorem 1.16, Theorem 1.14 and Lemma 2.2 below remain valid for
${\rm Cap}_{\alpha,p}$. Moreover, for $0<\alpha<\beta$, we have
$\frac{\alpha}{\beta}{\rm Cap}_{\beta,p}(A)\leq{\rm Cap}_{\alpha,p}(A)\leq{\rm
Cap}_{\beta,p}(A)$.

##  2 Proofs of Theorems 1.9, 1.10, Corollary 1.11, Theorem 1.12 and Corollary
1.16

###### Proof of Theorem 1.9.

We first prove the assertion for $\alpha=0$. For $u,v\in D(\mathscr{E})$, we
know

$\displaystyle\Gamma_{\mu}\left(\frac{u+v}{2}\right)+\Gamma_{\mu}\left(\frac{u-v}{2}\right)=\frac{\Gamma_{\mu}(u)+\Gamma_{\mu}(v)}{2}\quad\mu\text{-a.e.~{}on
}X.$ (2.1)

Since any $u\in D(\mathscr{E}^{\,p})$ can be approximated by a sequence
$\\{u_{n}\\}\subset\mathscr{D}_{1,p}$ in $H^{1,p}$-norm, we see the identity
(2.1) for $u,v\in D(\mathscr{E}^{\,p})$. Suppose $p\in[2,+\infty[$. From the
inequality $\|x\|_{p}\leq\|x\|_{2}$ for $x\in\mathbb{R}^{2}$ and (2.1),

$\displaystyle\Gamma_{\mu}\left(\frac{u+v}{2}\right)^{\frac{p}{2}}+\Gamma_{\mu}\left(\frac{u-v}{2}\right)^{\frac{p}{2}}$
$\displaystyle\leq\left(\Gamma_{\mu}\left(\frac{u+v}{2}\right)+\Gamma_{\mu}\left(\frac{u-v}{2}\right)\right)^{\frac{p}{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:curre-
DuChampIdentity}}}{{=}}\left(\frac{\Gamma_{\mu}(u)+\Gamma_{\mu}(v)}{2}\right)^{\frac{p}{2}}$
$\displaystyle\leq\frac{\Gamma_{\mu}(u)^{\frac{p}{2}}+\Gamma_{\mu}(v)^{\frac{p}{2}}}{2},\qquad\mu\text{-a.e.}$
(2.2)

where we use the fact that $\mathbb{R}_{+}\ni t\mapsto t^{\frac{p}{2}}$ is
convex in the last step. Integrating this we deduce (1.5) under $\alpha=0$.
Next we suppose $p\in]1,2]$. Obviously $p\leq 2\leq q:=p/(p-1)$ and thus for
any $u,v\in D(\mathscr{E}^{\,p})$ we have

$\displaystyle\Gamma_{\mu}\left(\frac{u+v}{2}\right)^{\frac{q}{2}}+\Gamma_{\mu}\left(\frac{u-v}{2}\right)^{\frac{q}{2}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Clarkson*}}}{{\leq}}\frac{\Gamma_{\mu}(u)^{\frac{q}{2}}+\Gamma_{\mu}(v)^{\frac{q}{2}}}{2}$
$\displaystyle\leq\left(\frac{\Gamma_{\mu}(u)^{\frac{p}{2}}+\Gamma_{\mu}(v)^{\frac{p}{2}}}{2}\right)^{\frac{q}{p}},\quad\mu\text{-a.e.}$
(2.3)

where in the last inequality we used the fact that $\|x\|_{2}\leq\|x\|_{p}$
for any $x\in\mathbb{R}^{2}$. Thanks to the reverse triangle inequality in
$L^{r}(X;\mu)$ for $r\in]0,1[$ proved in [22, Proposition 4.4], for any non-
negative Borel functions $f,g$, it holds

$\displaystyle\left(\int_{X}f^{r}\mathrm{d}\mu\right)^{\frac{1}{r}}+\left(\int_{X}g^{r}\mathrm{d}\mu\right)^{\frac{1}{r}}\leq\left(\int_{X}(f+g)^{r}\mathrm{d}\mu\right)^{\frac{1}{r}}.$
(2.4)

We apply this with $r:=\frac{p}{q}$,
$f:=\Gamma_{\mu}\left(\frac{u+v}{2}\right)^{\frac{q}{2}}$ and
$g:=\Gamma_{\mu}\left(\frac{u-v}{2}\right)^{\frac{q}{2}}$ to obtain

$\displaystyle\left(\left(\int_{X}\Gamma_{\mu}\left(\frac{u+v}{2}\right)^{\frac{p}{2}}\mathrm{d}\mu\right)^{\frac{q}{p}}+\left(\int_{X}\Gamma_{\mu}\left(\frac{u-v}{2}\right)^{\frac{p}{2}}\mathrm{d}\mu\right)^{\frac{q}{p}}\right)^{\frac{p}{q}}$
$\displaystyle\hskip
113.81102pt\leq\int_{X}\left(\Gamma_{\mu}\left(\frac{u+v}{2}\right)^{\frac{q}{2}}+\Gamma_{\mu}\left(\frac{u-v}{2}\right)^{\frac{q}{2}}\right)^{\frac{p}{q}}\mathrm{d}\mu.$

This and (2.3) give (1.6) under $\alpha=0$.

Next we prove the case for $\alpha>0$. In this case, the proof for
$p\in[2,+\infty[$ is easy, because it is done by adding (1.5) under $\alpha=0$
to the usual Clarkson’s inequality over $L^{p}(X;\mathfrak{m})$. Next we
assume $p\in]1,2]$. We have two inequalities of Clarkson type below:

$\displaystyle\mathscr{E}^{\,p\\!}\left(\frac{f+g}{2}\right)^{\frac{1}{p-1}}+\mathscr{E}^{\,p\\!}\left(\frac{f-g}{2}\right)^{\frac{1}{p-1}}$
$\displaystyle\leq\left(\frac{\mathscr{E}^{\,p}(f)+\mathscr{E}^{\,p}(g)}{2}\right)^{\frac{1}{p-1}},$
(2.5)
$\displaystyle\left\|\alpha\cdot\frac{f+g}{2}\right\|_{L^{p}(X;\mathfrak{m})}^{\frac{p}{p-1}}+\left\|\alpha\cdot\frac{f-g}{2}\right\|_{L^{p}(X;\mathfrak{m})}^{\frac{p}{p-1}}$
$\displaystyle\leq\left(\alpha\cdot\frac{\|f\|_{L^{p}(X;\mathfrak{m})}^{p}+\|g\|_{L^{p}(X;\mathfrak{m})}^{p}}{2}\right)^{\frac{p}{p-1}}.$
(2.6)

Then we can deduce (1.6) from (2.5) and (2.6) by using the closed convex body
$\overline{B}_{1}^{\,p}(0):=\\{(x,y)\in\mathbb{R}^{2}\mid|x|^{\frac{p}{p-1}}+|y|^{\frac{p}{p-1}}\leq
1\\}$ under $p\in]1,2]$. ∎

######  Remark 2.1.

The content of Theorem 1.9 is proved in [22, Proposition 4.4] in the framework
of infinitesimal Hilbertian metric measure space with $p$-independence of weak
upper gradients in the weak sense. Our construction of $(1,p)$-Sobolev space
is completely different from the Sobolev space $W^{1,p}(X)$ constructed in
[22] and does not need to impose the $p$-independence of weak upper gradient.

###### Proof of Theorem 1.10.

First we assume $T\in C^{\infty}(\mathbb{R}^{N})$. Then the generalized normal
contraction property implies $|\partial T/\partial x_{k}|\leq 1$ on
$\mathbb{R}^{N}$ for each $k\in\\{1,\cdots,N\\}$. By [8, Corollary 6.1.3] or
[30, Lemma 5.2], under
$u\in(\mathscr{D}_{1,p})^{N}\subset(D(\mathscr{E}))^{N}$, we have

$\displaystyle\Gamma_{\mu}(T(u))$
$\displaystyle=\left|\Gamma_{\mu}(T(u))\right|$
$\displaystyle=\left|\sum_{i,j=1}^{N}\frac{\partial T}{\partial
x_{i}}(u)\frac{\partial T}{\partial x_{j}}(u)\Gamma_{\mu}(u_{i},u_{j})\right|$
$\displaystyle\leq\sum_{i,j=1}^{N}\left|\frac{\partial T}{\partial
x_{i}}(u)\right|\left|\frac{\partial T}{\partial
x_{j}}(u)\right|\left|\Gamma_{\mu}(u_{i},u_{j})\right|$
$\displaystyle\leq\sum_{i,j=1}^{N}|\Gamma_{\mu}(u_{i},u_{j})|$ (2.7)
$\displaystyle\leq\sum_{i,j=1}^{N}\Gamma_{\mu}(u_{i})^{\frac{1}{2}}\Gamma_{\mu}(u_{j})^{\frac{1}{2}}$
$\displaystyle=\left(\sum_{i=1}^{N}\Gamma_{\mu}(u_{i})^{\frac{1}{2}}\right)^{2}\quad\mu\text{-a.e.}$

From this, we can easily confirm $T(u)\in\mathscr{D}_{1,p}$ for
$u=(u_{1},\cdots,u_{N})\in(\mathscr{D}_{1,p})^{N}$ and
$\mathscr{E}^{\,p}(T(u))^{\frac{1}{p}}\leq\sum_{k=1}^{N}\mathscr{E}^{\,p}(u_{k})^{\frac{1}{p}}$.
Next we take $u\in H^{1,p}(X)^{N}$. Then there exists an
$\|\cdot\|_{1,p}$-approximating sequence
$\\{u^{n}\\}\subset(\mathscr{D}_{1,p})^{N}$ to $u$ in the sense that
$u_{k}^{n}\to u_{k}$ in $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ as $n\to\infty$ for
each $k\in\\{1,\cdots,N\\}$. In particular, $\\{u^{n}\\}$ is
$L^{p}(X;\mathfrak{m})$-convergent to $u$ in the sense that $u_{k}^{n}\to
u_{k}$ in $L^{p}(X;\mathfrak{m})$ as $n\to\infty$ for each
$k\in\\{1,\cdots,N\\}$. Moreover, $\\{T(u^{n})\\}$ is also
$L^{p}(X;\mathfrak{m})$-convergent to $T(u)$. Indeed, since
$|T(u^{n})|\leq\sum_{k=1}^{N}|u^{n}_{k}|$, we see that $\\{T(u^{n})\\}$ is
uniformly integrable and it converges to $T(u)$ in $\mathfrak{m}$-measure.
This implies the $L^{p}$-convergence of $\\{T(u^{n})\\}$ to $T(u)$ (see
Vitali’s convergence theorem in [36, Theorem 16.6]).

It is easy to see that $\\{T(u^{n})\\}$ is a $\|\cdot\|_{H^{1,p}}$-bounded
sequence of $\mathscr{D}_{1,p}$. In view of the reflexivity of
$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, we can apply Banach-Alaoglu Theorem to the
topological dual space
$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})=(H^{1,p}(X)^{**},\|\cdot\|_{H^{1,p}(X)^{**}})$
of $(H^{1,p}(X)^{*},\|\cdot\|_{H^{1,p}(X)^{*}})$ so that there exists a
subsequence $\\{T(u^{n_{i}})\\}\subset\mathscr{D}_{1,p}$ and $v\in H^{1,p}(X)$
such that $\\{T(u^{n_{i}})\\}$ $H^{1,p}(X)$-weakly converges to $v$, in
particular, it converges to $v$ $L^{p}$-weakly, because any $\ell\in
L^{p}(X;\mu)^{*}=L^{q}(X;\mu)$ can be regarded as $\ell|_{H^{1,p}(X)^{*}}\in
H^{1,p}(X)^{*}$. Since $\\{T(u^{n})\\}$ converges to $T(u)$ in
$L^{p}(X;\mathfrak{m})$, we see $v=T(u)$. From this, we have

$\displaystyle\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\|\Gamma_{\mu}(T(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle=\|T(u)\|_{H^{1,p}}^{p}$
$\displaystyle\leq\varliminf_{i\to\infty}\|T(u^{n_{i}})\|_{H^{1,p}}^{p}$
$\displaystyle\leq\varliminf_{i\to\infty}\left(\|T(u^{n_{i}})\|_{L^{p}(X;\mathfrak{m})}^{p}+\|\Gamma_{\mu}(T(u^{n_{i}}))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}\right)$
$\displaystyle=\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\varliminf_{i\to\infty}\|\Gamma_{\mu}(T(u^{n_{i}}))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle\leq\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\varliminf_{i\to\infty}\left(\sum_{k=1}^{N}\|\Gamma_{\mu}(u^{n_{i}}_{k})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}\right)^{p}$
$\displaystyle=\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\left(\sum_{k=1}^{N}\|\Gamma_{\mu}(u_{k})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}\right)^{p}.$

In the last inequality above, we use the fact that
$u^{n_{i}}\in(\mathscr{D}_{1,p})^{N}$ implies
$T(u^{n_{i}})\in\mathscr{D}_{1,p}$ and
$\Gamma_{\mu}(T(u^{n_{i}}))^{\frac{1}{2}}\leq\sum_{k=1}^{N}\Gamma_{\mu}(u^{n_{i}}_{k})^{\frac{1}{2}}$
$\mu$-a.e. on $X$ shown in (2.7). Consequently, we obtain that

$\displaystyle\mathscr{E}^{\,p}(T(u))^{\frac{1}{p}}=\|\Gamma_{\mu}(T(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}\leq\sum_{k=1}^{N}\|\Gamma_{\mu}(u_{k})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}=\sum_{k=1}^{N}\mathscr{E}^{\,p}(u_{k})^{\frac{1}{p}}$

and

$\displaystyle\|T(u)\|_{H^{1,p}(X)}\leq
2^{1-\frac{1}{p}}\sum_{k=1}^{N}\|u_{k}\|_{H^{1,p}(X)}.$

Finally, we consider the general case. Since $x_{k}\mapsto
T(x_{1},\cdots,x_{k},\cdots,x_{N})$ is a $1$-Lipschitz function on
$\mathbb{R}$, it is differential almost everywhere on $\mathbb{R}$ in view of
Rademacher Theorem and its $k$-th derivative $\frac{\partial T}{\partial
x_{k}}$ exists for a.e. $x_{k}$ and $|\frac{\partial T}{\partial x_{k}}|\leq
1$ a.e. $x_{k}$ for each $k\in\\{1,\cdots,x_{N}\\}$. In particular,
$x_{k}\mapsto\frac{\partial T}{\partial x_{k}}$ is locally integrable on
$\mathbb{R}$.

Take a function $j$ defined by $j(x):=e^{-\frac{1}{1-|x|^{2}}}$ for $|x|\leq
1$ and $j(x):=0$ for $|x|>1$. Set $j_{n}(x):=n^{N}j(nx)$ and
$T_{n}(x):=j_{n}*T(x)-j_{n}*T(0):=\int_{\mathbb{R}^{N}}(j_{n}(x-y)-j_{n}(y))T(y)\mathrm{d}y$.
Then $T_{n}\in C^{\infty}(\mathbb{R}^{N})$, $T_{n}(0)=0$ and
$\lim_{n\to\infty}T_{n}(x)=T(x)$ for each $x\in\mathbb{R}^{N}$. We easily see

$\displaystyle\frac{\partial T_{n}}{\partial x_{k}}(x)$
$\displaystyle=\int_{\mathbb{R}^{N}}\frac{\partial}{\partial
x_{k}}j_{n}(x-y)T(y)\mathrm{d}y$
$\displaystyle=-\int_{\mathbb{R}^{N}}\frac{\partial}{\partial
y_{k}}j_{n}(x-y)T(y)\mathrm{d}y=\int_{\mathbb{R}^{N}}j_{n}(x-y)\frac{\partial
T}{\partial y_{k}}(y)\mathrm{d}y,$ (2.8)

because

$\displaystyle-\int_{\mathbb{R}}\frac{\partial}{\partial
y_{k}}j_{n}(x-y)T(y)\mathrm{d}y_{k}$
$\displaystyle=-\lim_{-a,b\to\infty}\left[j_{n}(x-y)T(y)\right]_{y_{k}=a}^{y_{k}=b}+\int_{\mathbb{R}}j_{n}(x-y)\frac{\partial
T}{\partial y_{k}}(y)\mathrm{d}y_{k}$
$\displaystyle=\int_{\mathbb{R}}j_{n}(x-y)\frac{\partial T}{\partial
y_{k}}(y)\mathrm{d}y_{k}.$

From (2.8), we see $|\frac{\partial T_{n}}{\partial x_{k}}|\leq 1$ on
$\mathbb{R}^{N}$ for all $k\in\\{1,\cdots,N\\}$ and $T_{n}(0)=0$, hence
$T_{n}$ is a generalized normal contraction on $\mathbb{R}^{N}$. Now take
$u\in H^{1,p}(X)^{N}$. Then we already know $T_{n}(u)\in H^{1,p}(X)$ and
$\|\Gamma_{\mu}(T_{n}(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}\leq\sum_{k=1}^{N}\|\Gamma_{\mu}(u_{k})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}$,
and $\|T_{n}(u)\|_{H^{1,p}}\leq
2^{1-\frac{1}{p}}\sum_{k=1}^{N}\|u_{k}\|_{H^{1,p}}$.

Since $|T_{n}(u)|\leq\sum_{k=1}^{N}|u_{k}|$ and $T_{n}(u)\to T(u)$ as
$n\to\infty$ $\mathfrak{m}$-a.e., $\\{T_{n}(u)\\}$
$L^{p}(X;\mathfrak{m})$-converges to $T(u)$ as $n\to\infty$. Moreover
$\Gamma_{\mu}(T_{n}(u))^{\frac{1}{2}}\leq\sum_{k=1}^{N}\Gamma_{\mu}(u_{k})^{\frac{1}{2}}$
$\mu$-a.e. yields the $\|\cdot\|_{H^{1,p}}$-boundedness of $\\{T_{n}(u)\\}$.
In view of the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, there exists
an $H^{1,p}$-weakly convergent subsequence $\\{T_{n_{i}}(u)\\}$ to $T(u)$.
Therefore, we can deduce

$\displaystyle\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\|\Gamma_{\mu}(T(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle=\|T(u)\|_{H^{1,p}}^{p}$
$\displaystyle\leq\varliminf_{i\to\infty}\|T_{n_{i}}(u)\|_{H^{1,p}}^{p}$
$\displaystyle\leq\varliminf_{i\to\infty}\left(\|T_{n_{i}}(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\|\Gamma_{\mu}(T_{n_{i}}(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}\right)$
$\displaystyle=\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\varliminf_{i\to\infty}\|\Gamma_{\mu}(T_{n_{i}}(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle\leq\|T(u)\|_{L^{p}(X;\mathfrak{m})}^{p}+\left(\sum_{k=1}^{N}\|\Gamma_{\mu}(u_{k})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}\right)^{p}.$

Therefore, we obtain

$\displaystyle\mathscr{E}^{\,p}(T(u))^{\frac{1}{p}}=\|\Gamma_{\mu}(T(u))^{\frac{1}{2}}\|_{L^{p}(X;\mu)}\leq\sum_{k=1}^{N}\|\Gamma_{\mu}(u_{k})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}=\sum_{k=1}^{N}\mathscr{E}^{\,p}(u_{k})^{\frac{1}{p}}$

and

$\displaystyle\|T(u)\|_{H^{1,p}}\leq
2^{1-\frac{1}{p}}\sum_{k=1}^{N}\|u_{k}\|_{H^{1,p}}.$

∎

###### Proof of Corollary 1.11.

The assertion is an easy consequence of Theorem 1.10. We omit it. ∎

###### Proof of Theorem 1.12.

First we prove the assertion for $u,v\in\mathscr{D}_{1,p}$. In this case, by
way of truncation property of carré du champ $\Gamma_{\mu}$, we have

$\displaystyle\Gamma_{\mu}(u\lor v)^{\frac{p}{2}}+\Gamma_{\mu}(u\land
v)^{\frac{p}{2}}$ $\displaystyle={\bf 1}_{\\{u\geq
v\\}}\Gamma_{\mu}(u)^{\frac{p}{2}}+{\bf
1}_{\\{u<v\\}}\Gamma_{\mu}(v)^{\frac{p}{2}}+{\bf 1}_{\\{u\geq
v\\}}\Gamma_{\mu}(v)^{\frac{p}{2}}+{\bf
1}_{\\{u<v\\}}\Gamma_{\mu}(u)^{\frac{p}{2}}$
$\displaystyle=\Gamma_{\mu}(u)^{\frac{p}{2}}+\Gamma_{\mu}(v)^{\frac{p}{2}},\quad\mu\text{-a.e.}$

Integrating this, we obtain the conclusion. Next, we consider general $u,v\in
H^{1,p}(X)$. Then there exist sequences
$\\{u_{n}\\},\\{v_{n}\\}\subset\mathscr{D}_{1,p}$ converging to $u,v$ in
$H^{1,p}(X)$, respectively. By Corollary 1.11(2), $\\{u_{n}\lor v_{n}\\}$ and
$\\{u_{n}\land v_{n}\\}$ are $\|\cdot\|_{H^{1,p}}$-bounded. Then, by way of
the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$, there exists a common
subsequence $\\{n_{i}\\}$ such that $\\{u_{n_{i}}\lor v_{n_{i}}\\}$ and
$\\{u_{n_{i}}\land v_{n_{i}}\\}$ $H^{1,p}$-weakly converges to $u\lor v$ and
$u\land v$, respectively. Therefore,

$\displaystyle\mathscr{E}^{\,p}(u\lor v)+\mathscr{E}^{\,p}(u\land v)$
$\displaystyle=\|\Gamma_{\mu}(u\lor
v)^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}+\|\Gamma_{\mu}(u\land
v)^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle\leq\varliminf_{i\to\infty}\|\Gamma_{\mu}(u_{n_{i}}\lor
v_{n_{i}})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}+\varliminf_{i\to\infty}\|\Gamma_{\mu}(u_{n_{i}}\land
v_{n_{i}})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle\leq\varliminf_{i\to\infty}\left(\|\Gamma_{\mu}(u_{n_{i}}\lor
v_{n_{i}})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}+\|\Gamma_{\mu}(u_{n_{i}}\land
v_{n_{i}})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}\right)$
$\displaystyle=\varliminf_{i\to\infty}\left(\|\Gamma_{\mu}(u_{n_{i}})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}+\|\Gamma_{\mu}(v_{n_{i}})^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}\right)$
$\displaystyle=\|\Gamma_{\mu}(u)^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}+\|\Gamma_{\mu}(v)^{\frac{1}{2}}\|_{L^{p}(X;\mu)}^{p}$
$\displaystyle=\mathscr{E}^{\,p}(u)+\mathscr{E}^{\,p}(v).$

∎

###### Proof of Theorem 1.14.

First we prove (1) under $p\in[2,+\infty[$. The proof under $p\in]1,2]$ is
similar. Let $\\{u_{n}\\}\subset\mathcal{L}_{A}$ be a minimizing sequence to
${\rm Cap}_{1,p}(A)=\inf_{u\in\mathcal{L}_{A}}\mathcal{E}^{\,p}_{1}(u_{n})$.
By (1.5), we can see that any minimizing sequence
$\\{u_{n}\\}\subset\mathcal{L}_{A}$
($\lim_{n\to\infty}\mathscr{E}^{\,p}_{1}(u_{n})={\rm Cap}_{1,p}(A)$) is
$H^{1,p}$-convergent to an element $e_{A}\in\mathcal{L}_{A}$ satisfying (1.13)
and that such an $e_{A}$ is unique.

(2) By Theorem 1.10, $u:=(0\lor e_{A})\land 1\in\mathcal{L}_{A}$ and
$\mathscr{E}^{\,p}_{1}(u,u)\leq\mathscr{E}^{\,p}_{1}(e_{A},e_{A})={\rm
Cap}_{1,p}(A)$. Hence $u=e_{A}$.

(3) If $v$ has the stated property, then
$e_{A}+{\varepsilon}v\in\mathcal{L}_{A}$ and
$\mathscr{E}^{\,p}_{1}(e_{A}+{\varepsilon}v,e_{A}+{\varepsilon}v)\geq\mathscr{E}^{\,p}_{1}(e_{A},e_{A})$
for any ${\varepsilon}\geq 0$. This means that

$\displaystyle 0$
$\displaystyle\leq\left.\frac{\mathrm{d}}{\mathrm{d}{\varepsilon}}\mathscr{E}^{\,p}_{1}(e_{A}+{\varepsilon}v,e_{A}+{\varepsilon}v)\right|_{{\varepsilon}=0}=p\,\mathscr{E}^{\,p}_{1}(e_{A},v)\quad\text{
for }v\in D(\mathscr{E}^{\,p})\text{ with }v\geq
0\;\mathfrak{m}\text{-a.e.~{}on }A.$

Indeed,

$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}{\varepsilon}}\mathscr{E}^{\,p}_{1}(e_{A}+{\varepsilon}v,e_{A}+{\varepsilon}v)\right|_{{\varepsilon}=0}$
$\displaystyle=\left.\frac{p}{2}\int_{X}\Gamma_{\mu}(e_{A}+{\varepsilon}v)^{\frac{p-2}{2}}\left(2{\varepsilon}\Gamma_{\mu}(v,v)+2\Gamma_{\mu}(e_{A},v)\right)\mathrm{d}\mu\right|_{{\varepsilon}=0}$
$\displaystyle\hskip
28.45274pt+\left.\frac{p}{2}\int_{X}|e_{A}+{\varepsilon}v|^{p-2}(2{\varepsilon}|v|^{2}+2e_{A}v)\mathrm{d}\mathfrak{m}\right|_{{\varepsilon}=0}$
$\displaystyle=p\int_{X}\Gamma_{\mu}(e_{A})^{\frac{p}{2}-1}\Gamma_{\mu}(e_{A},v)\mathrm{d}\mu+p\int_{X}e_{A}^{\,p-1}v\mathrm{d}\mathfrak{m}$
$\displaystyle=p\,\mathcal{E}^{\,p}_{1}(e_{A},v).$

Convsersely, suppose $u\in D(\mathscr{E}^{\,p})$ satisfies the conditions in
(3), then $u\in\mathcal{L}_{A}$ and $w-u\geq 0$ $\mathfrak{m}$-a.e. on $A$ for
any $w\in\mathcal{L}_{A}$. In particular, $\mathscr{E}^{\,p}_{1}(u,w-u)\geq
0$. Hence

$\displaystyle\mathscr{E}^{\,p}_{1}(u,u)\leq\mathscr{E}^{\,p}_{1}(u,w)\leq\mathscr{E}^{\,p}_{1}(u,u)^{1-\frac{1}{p}}\mathscr{E}^{\,p}_{1}(w,w)^{\frac{1}{p}},$

which implies $\mathscr{E}^{\,p}_{1}(u,u)\leq\mathscr{E}^{\,p}_{1}(w,w)$,
proving $u=e_{A}$.

(4) follows immediately from (3).

(5) Since $A\subset B$, $e_{A}=e_{A}\land e_{B}$ $\mathfrak{m}$-a.e. on $A$.
This implies
$\mathscr{E}^{\,p}_{1}(e_{A},e_{A})=\mathscr{E}^{\,p}_{1}(e_{A},e_{A}\land
e_{B})$, because $\mathscr{E}^{\,p}_{1}(e_{A},v)=0$ if $v\in
D(\mathscr{E}^{\,p})$ satisfies $v=0$ $\mathfrak{m}$-a.e. on $A$. Therefore,

$\displaystyle\mathscr{E}^{\,p}_{1}(e_{A},e_{A})$
$\displaystyle=\mathscr{E}^{\,p}_{1}(e_{A},e_{A}\land e_{B})$
$\displaystyle\leq\mathscr{E}^{\,p}_{1}(e_{A},e_{A})^{1-\frac{1}{p}}\mathscr{E}^{\,p}_{1}(e_{A}\land
e_{B},e_{A}\land e_{B})^{\frac{1}{p}},$

which implies
$\mathscr{E}^{\,p}_{1}(e_{A},e_{A})\leq\mathscr{E}^{\,p}_{1}(e_{A}\land
e_{B},e_{A}\land e_{B})$, proving $e_{A}=e_{A}\land e_{B}$. ∎

To prove Corollary 1.16, we need the following lemma.

######  Lemma 2.2.

We have the following:

  1. (1)

If $A,B\in\mathcal{O}(X)$ with $A\subset B$, then ${\rm Cap}_{1,p}(A)\leq{\rm
Cap}_{1,p}(B)$.

  2. (2)

(1.17) holds for $A,B\in\mathcal{O}(X)$.

  3. (3)

Let $\\{A_{n}\\}$ be an increasing sequence of $A_{n}\in\mathcal{O}(X)$. Then
(1.14) holds.

###### Proof.

(1) is clear. (2) For $A,B\in\mathcal{O}_{0}(X)$, by Theorem 1.12,

$\displaystyle{\rm Cap}_{1,p}(A\cup B)+{\rm Cap}_{1,p}(A\cap B)$
$\displaystyle\leq\mathscr{E}^{\,p}_{1}(e_{A}\lor e_{B},e_{A}\lor
e_{B})+\mathscr{E}^{\,p}_{1}(e_{A}\land e_{B},e_{A}\lor e_{B})$
$\displaystyle\leq\mathscr{E}^{\,p}_{1}(e_{A},e_{A})+\mathscr{E}^{\,p}_{1}(e_{B},e_{B})$
$\displaystyle={\rm Cap}_{1,p}(A)+{\rm Cap}_{1,p}(B).$

(3) Suppose $\sup_{n\in\mathbb{N}}{\rm Cap}_{1,p}(A_{n})<\infty$. We prove the
assertion for the case $p\in[2,+\infty[$. The proof for the case $p\in]1,2]$
is similar based on (1.6). Suppose $n>m$. Then
$\frac{e_{A_{n}}+e_{A_{m}}}{2}\in\mathcal{L}_{A_{m}}$. According to the
inequality (1.5) of Clarkson type,

$\displaystyle\mathscr{E}^{\,p}_{1}\left(\frac{e_{A_{n}}-e_{A_{m}}}{2}\right)+{\rm
Cap}_{1,p}(A_{m})$
$\displaystyle\leq\mathscr{E}^{\,p}_{1}\left(\frac{e_{A_{n}}-e_{A_{m}}}{2}\right)+\mathscr{E}^{\,p}_{1}\left(\frac{e_{A_{n}}+e_{A_{m}}}{2}\right)$
$\displaystyle\leq\frac{\mathscr{E}^{\,p}_{1}(e_{A_{n}})+\mathscr{E}^{\,p}_{1}(e_{A_{m}})}{2}$
$\displaystyle=\frac{1}{2}\left({\rm Cap}_{1,p}(A_{n})+{\rm
Cap}_{1,p}(A_{m})\right),$

which implies that $\\{e_{A_{n}}\\}_{n\in\mathbb{N}}$ converges to some $u\in
D(\mathscr{E}^{\,p})$ in $H^{1,p}$-norm. Clearly $u=1$ $\mathfrak{m}$-a.e. on
$A:=\bigcup_{n=1}^{\infty}A_{n}$. If $v\in D(\mathscr{E}^{\,p})$ is non-
negative $\mathfrak{m}$-a.e. on $A$, then
$\mathscr{E}^{\,p}_{1}(u,v)=\lim_{n\to\infty}\mathscr{E}^{\,p}_{1}(e_{A_{n}},v)\geq
0$. Hence $u=e_{A}$ by Lemma 1.14(3) and $\sup_{n\in\mathbb{N}}{\rm
Cap}_{1,p}(A_{n})=\lim_{n\to\infty}\mathscr{E}^{\,p}_{1}(e_{A_{n}},e_{A_{n}})=\mathscr{E}^{\,p}_{1}(u,u)={\rm
Cap}_{1,p}(A)$. ∎

###### Proof of Corollary 1.16.

The conclusion follows Lemma 2.2 and [17, Theorem A.1.2]. (1.16) is a
consequence of [17, Theorem A.1.1]. (1.17) directly follows from Lemma 2.2(3).
∎

##  3 Examples

######  Example 3.1 (Abstract Wiener space).

Let $(B,H,\mu)$ be an abstract Wiener space, i.e. $B$ is a real separable
Banach space and $H$ is a real separable Hilbert space that is densely and
continuously imbedded in $B$ and $\mu$ is a Gaussian measure on $B$ satisfying

$\displaystyle\int_{B}\exp\left(\sqrt{-1}{}_{B}\langle
w,\varphi\rangle_{B^{*}}\right)\mathrm{d}\mu=\exp\left(-\frac{1}{2}|\varphi|_{H^{*}}^{2}\right)\quad\text{
for }\quad\varphi\in B^{*}\subset H^{*}.$ (3.1)

(see Shigekawa [37, Definition 1.1]). Let $(T_{t}^{\rm OU})_{t\geq 0}$ be the
Ornstein-Uhlenbeck semigroup defined by

$\displaystyle T_{t}^{\rm
OU}f(x):=\int_{B}f(e^{-t}x+\sqrt{1-e^{-2t}}y)\mu(\mathrm{d}y)\quad\text{ for
}\quad f\in\mathscr{B}(B)_{+}.$ (3.2)

It is known that $T_{t}^{\rm OU}$ is $\mu$-symmetric and $(T_{t}^{\rm
OU})_{t\geq 0}$ can be extended to a strongly continuous contraction semigroup
in $L^{p}(B;\mu)$ for $p\geq 1$ (see Shigekawa [37, Proposition 2.4]). Let
$(\mathscr{E}^{\rm OU},D(\mathscr{E}^{\rm OU}))$ be the Dirichlet form on
$L^{2}(B;\mu)$ associated to $(T_{t}^{\rm OU})_{t\geq 0}$. It is known that
$(\mathscr{E}^{\rm OU},D(\mathscr{E}^{\rm OU}))$ is associated to a diffusion
process ${\bf X}^{\rm OU}$ called _Ornstein-Uhlenbeck process_. A function
$f:B\to\mathbb{R}$ is called $H$-differentiable at $x\in B$ if there exists an
$h^{*}\in H^{*}$ such that
$\left.\frac{\mathrm{d}}{\mathrm{d}t}f(x+th)\right|_{t=0}={}_{H}\langle
h,h^{*}\rangle_{H^{*}}$ $h\in H$. Then $h^{*}$ is called the $H$-derivative of
$f$ at $x$ denoted by $D_{H}f(x)$. Let

$\displaystyle\mathcal{P}:$ $\displaystyle=\\{F:B\to\mathbb{R}\mid\text{ there
exists }n\in\mathbb{N},\varphi_{1},\cdots,\varphi_{n}\in B^{*},$
$\displaystyle\hskip 85.35826ptf\in C^{\infty}(\mathbb{R}^{n})\text{ is a
polynomial on }\mathbb{R}^{n}\text{ such that }$ $\displaystyle\hskip
199.16928ptF(x)=f(\langle x,\varphi_{1}\rangle,\cdots,\langle
x,\varphi_{n}\rangle)\\}$

be the family of polynomial functions on $B$. For $f\in\mathcal{P}$, its
$H^{1,p}$-norm $\|f\|_{H^{1,p}}$ is given by

$\displaystyle\|f\|_{H^{1,p}}^{p}:=\int_{B}|f|^{p}\mathrm{d}\mu+\int_{B}|D_{H}f|_{H^{*}}^{p}\mathrm{d}\mu<\infty.$

When $p=2$, the $H^{1,2}$-completion of $(\mathcal{P},\|\cdot\|_{H^{1,2}})$ is
nothing but the Dirichlet form $(\mathscr{E}^{\rm OU},D(\mathscr{E}^{\rm
OU}))$ and it admits carré du champ $\Gamma$ given by
$\Gamma(f)=|D_{H}f|_{H^{*}}^{2}$, where $D_{H}:D(\mathscr{E}^{\rm OU})\to
L^{2}(B,H^{*};\mu)$ is a natural extension of $D_{H}$. It is shown in [37,
Theorem 4.4] that the norm $\|\cdot\|_{H^{1,p}}$ on $\mathcal{P}$ is
equivalent to another norm $\|\cdot\|_{W^{1,p}}$ on $\mathcal{P}$ based on the
Ornstein-Uhlenbeck operator $L^{\rm OU}$ associated to $(T_{t}^{\rm
OU})_{t\geq 0}$. In particular, our Assumption 1 for $\mathscr{D}=\mathcal{P}$
is satisfied. Then our $(1,p)$-Sobolev space
$(H^{1,p}(B),\|\cdot\|_{H^{1,p}})$ coincides with the $H^{1,p}$-completion of
$\mathcal{P}(\subset L^{p}(B;\mu))$.

######  Example 3.2 (${\sf RCD}(K,\infty)$-space).

A metric measure space is a triple $(X,{\sf d},\mathfrak{m})$ such that

$\displaystyle(X,{\sf d}):\quad$ $\displaystyle\text{ is a complete separable
metric space},$ $\displaystyle\mathfrak{m}\neq 0:\quad$ $\displaystyle\text{
is non-negative and boundedly finite Borel measure}.$

Any metric open ball is denoted by $B_{r}(x):=\\{y\in X\mid{\sf d}(x,y)<r\\}$
for $r>0$ and $x\in X$. A subset $B$ of $X$ is said to be bounded if it is
included in a metric open ball. Denote by $C([0,1],X)$ the space of continuous
curve defined on the unit interval $[0,1]$ equipped the distance ${\sf
d}_{\infty}(\gamma,\eta):=\sup_{t\in[0,1]}{\sf d}(\gamma_{t},\eta_{t})$ for
every $\gamma,\eta\in C([0,1],X)$. This turn $C([0,1],X)$ into complete
separable metric space. Next we consider the set of $q$-absolutely continuous
curves, denoted by $AC^{q}([0,1],X)$, is the subset of $\gamma\in C([0,1],X)$
so that there exists $g\in L^{q}(0,1)$ satisfying

$\displaystyle{\sf
d}(\gamma_{t},\gamma_{s})\leq\int_{s}^{t}g(r)\mathrm{d}r,\quad s<t\text{ in
}[0,1].$

Recall that for any $\gamma\in AC^{q}([0,1],X)$, there exists a minimal a.e.
function $g\in L^{q}(0,1)$ satisfying the above, called metric speed denoted
by $|\dot{\gamma}_{t}|$, which is defined as
$|\dot{\gamma}_{t}|:=\lim_{h\downarrow 0}{\sf d}(\gamma_{t+h},\gamma_{t})/h$
for $\gamma\in AC^{q}([0,1],X)$, $|\dot{\gamma}_{t}|:=+\infty$ otherwise. We
define the kinetic energy functional $C([0,1],X)\ni\gamma\mapsto{\sf
Ke}_{t}(\gamma):=\int_{0}^{1}|\dot{\gamma}_{t}|^{q}\mathrm{d}t$, if $\gamma\in
AC^{q}([0,1],X)$, ${\sf Ke}_{t}(\gamma):=+\infty$ otherwise. For every
$t\in[0,1]$, we define the evaluation map at time $t$ as follows: ${\sf
e}_{t}:C([0,1],X)\to X$, ${\sf e}_{t}(\gamma):=\gamma_{t}$ for $\gamma\in
C([0,1]\to X)$. We easily see that ${\sf e}_{t}$ is a $1$-Lipschitz map.

######  Definition 3.3 ($q$-test plan).

Let $(X,{\sf d},\mathfrak{m})$ be a metric measure space and
$q\in]1,+\infty[$. A measure $\pi$$\in\mathscr{P}(C([0,1],X))$ is said to be a
$q$-test plan, provided

  1. (i)

there exists $C>0$ so that $({\sf e}_{t})_{\sharp}$$\pi$$\leq C\mathfrak{m}$
for every $t\in[0,1]$;

  2. (ii)

we have $\int_{C([0,1],X)}{\sf
Ke}_{q}(\gamma)$$\pi$$(\mathrm{d}\gamma)<\infty$.

Moreover, we say that $\pi$ is an $\infty$-test plan if, instead of (ii), we
have ${\rm Lip}(\gamma)\leq L$ for $\pi$-a.e. $\gamma$ (and thus for every
$\gamma$ in the support of $\pi$ by the lower semi continuity of the global
Lipschitz constant with respect to uniform convergence).

######  Definition 3.4 (Sobolev spaces $S^{\hskip 0.85355ptp}(X)$ and
$W^{1,p}(X)$).

Let $(X,{\sf d},\mathfrak{m})$ be a metric measure space and
$p\in]1,+\infty[$. A Borel function $f$ belongs to $S^{\hskip 0.85355ptp}(X)$,
provided there exists a $G\in L^{p}(X;\mathfrak{m})$, called $p$-weak upper
gradient of $f$ so that

$\displaystyle\int_{C([0,1],X)}|f(\gamma_{1})-f(\gamma_{0})|\text{\boldmath$\pi$}(\mathrm{d}\gamma)\leq\int_{C([0,1],X)}\int_{0}^{1}G(\gamma_{t})|\dot{\gamma}_{t}|\mathrm{d}t\text{\boldmath$\pi$}(\mathrm{d}\gamma),\quad\text{$\forall${\boldmath$\pi$}
$q$-test plan}.$ (3.3)

the set of $p$-weak upper gradient of a given Borel function $f$ is a closed
convex subset of $L^{p}(X;\mathfrak{m})$. The minimal $p$-weak upper gradient,
denoted by $|Df|_{p}$ is then the element of minimal $L^{p}$-norm in this
class. Also, by making use of the lattice property of the set of $p$-weak
upper gradient, such a minimality is also in the $\mathfrak{m}$-a.e. sense
(see [11, Proposition 2.17 and Theorem 2.18]). It should be noted that
$S^{\hskip 0.85355ptp}(X)\cap L^{\infty}(X;\mathfrak{m})$ is an algebra. The
Sobolev space, denoted by $W^{1,p}(X)$, is $L^{p}(X;\mathfrak{m})\cap
S^{\,p}(X)$ as a set, equipped with the norm

$\displaystyle\|f\|_{W^{1,p}(X)}:=\left(\|f\|^{p}_{L^{p}(X;\mathfrak{m})}+\||Df|_{p}\|^{p}_{L^{p}(X;\mathfrak{m})}\right)^{\frac{1}{p}},\quad
f\in W^{1,p}(X).$

######  Remark 3.5.

The definition of $W^{1,p}(X)$ is based on the mass transport theory. There
are various way to define $(1,p)$-Sobolev spaces by Cheeger, Shanmugalingam
and etc. For general $p\in]1,+\infty[$, they are equivalent to each other (see
[3, Theorem 7.4] for general $p\in]1,+\infty[$, [23, Remark 2.27 and Theorem
2.28] for the case $p=2$).

It is a standard fact that $(W^{1,p}(X),\|\cdot\|_{W^{1,p}(X)})$ is a Banach
space in view of the weak lower semi continuity of weak upper gradient (see
[11, Theorem 2.7]). It should be noted that $W^{1,p}(X)\cap
L^{\infty}(X;\mathfrak{m})$ is an algebra. It is in general false that
$(W^{1,p}(X),\|\cdot\|_{W^{1,p}(X)})$ is reflexive and $p=2$ does not imply
that $(W^{1,2}(X),\|\cdot\|_{W^{1,2}(X)})$ is a Hilbert space. When the latter
situation occurs, we say that $(X,{\sf d},\mathfrak{m})$ is infinitesimally
Hilbertian (see [19]). Equivalently, we call $(X,{\sf d},\mathfrak{m})$
infinitesimally Hilbertian provided the following parallelogram identity
holds:

$\displaystyle
2|Df|_{2}^{2}+2|Dg|_{2}^{2}=|D(f+g)|_{2}^{2}+|D(f-g)|_{2}^{2},\quad\mathfrak{m}\text{-a.e.}\quad\forall
f,g\in W^{1,2}(X).$ (3.4)

Under the infinitesimally Hilbertian condition for $(X,{\sf d},\mathfrak{m})$,
$(W^{1,p}(X),\|\cdot\|_{W^{1,p}(X)})$ becomes a uniformly convex Banach space
for $p\in]1,+\infty[$, hence it is reflexive (see [22, Proposition 4.4]).
Denote by ${\rm Lip}(X)$ the family of Lipschitz functions on $(X,{\sf d})$
and let ${\rm lip}(f)(x):=\varlimsup_{y\to x}\frac{|f(y)-f(x)|}{{\sf d}(x,y)}$
if $x$ is an accumulation point. When $x$ is an isolated point, we set ${\rm
lip}(f)(x):=0$. It is easy to see that $\\{f\in{\rm Lip}(X)\mid{\rm lip}(f)\in
L^{p}(X;\mathfrak{m})\\}\subset S^{\,p}(X)$, consequently, $\\{f\in{\rm
Lip}(X)\cap L^{p}(X;\mathfrak{m})\mid{\rm lip}(f)\in
L^{p}(X;\mathfrak{m})\\}\subset W^{1,p}(X)$, because

$\displaystyle|f(\gamma_{1})-f(\gamma_{0})|\leq\int_{0}^{1}{\rm
lip}(f)(\gamma_{t})|\dot{\gamma}_{t}|\mathrm{d}t$

for any $\gamma\in C([0,1],X)$ (see [23, (2.17)]. In particular, ${\rm
Lip}(X)_{bs}:=\\{f\in{\rm Lip}(X)\mid{\rm supp}[f]\text{ is bounded
}\\}\subset W^{1,p}(X)$.

Under (3.4), we can give a bilinear form $\langle
D\cdot,D\cdot\rangle:W^{1,2}(X)\times W^{1,2}(X)\to L^{1}(X;\mathfrak{m})$
which is defined by

$\displaystyle\langle
Df,Dg\rangle:=\frac{1}{4}|D(f+g)|_{2}^{2}-\frac{1}{4}|D(f-g)|_{2}^{2},\quad
f,g\in W^{1,2}(X).$

Moreover, under the infinitesimally Hilbertian condition, the bilinear form
$(\mathscr{E},D(\mathscr{E}))$ defined by

$\displaystyle
D(\mathscr{E}):=W^{1,2}(X),\quad\mathscr{E}(f,g):=\frac{1}{2}\int_{X}\langle
Df,Dg\rangle\mathrm{d}\mathfrak{m}$

is a strongly local Dirichlet form on $L^{2}(X;\mathfrak{m})$ admitting carré
du champ.

Denote by $(P_{t})_{t\geq 0}$ the semigroup on $L^{2}(X;\mathfrak{m})$
associated with $(\mathscr{E},D(\mathscr{E}))$. We call $(P_{t})_{t\geq 0}$
_heat flow_. Under (3.4), let $(\Delta,D(\Delta))$ be the $L^{2}$-generator
associated with $(\mathscr{E},D(\mathscr{E}))$ defined by

$\displaystyle\left\\{\begin{array}[]{rl}D(\Delta)&:=\\{u\in
D(\mathscr{E})\mid\text{there exists }w\in L^{2}(X;\mathfrak{m})\text{ such
that }\\\ &\hskip
85.35826pt\mathscr{E}(u,v)=-\int_{X}wv\,\mathrm{d}\mathfrak{m}\quad\text{ for
any }\quad v\in D(\mathscr{E})\\},\\\ \Delta u&:=w\quad\text{ for }w\in
L^{2}(X;\mathfrak{m})\quad\text{specified as above,}\end{array}\right.$ (3.8)

######  Definition 3.6 (${\sf RCD}(K,\infty)$-spaces).

A metric measure space $(X,{\sf d},\mathfrak{m})$ is said to be an
RCD$(K,\infty)$-space if it satisfies the following conditions:

  1. (1)

$(X,{\sf d},\mathfrak{m})$ is infinitesimally Hilbertian.

  2. (2)

There exist $x_{0}\in X$ and constants $c,C>0$ such that
$\mathfrak{m}(B_{r}(x_{0}))\leq C\mathrm{e}^{cr^{2}}$.

  3. (3)

If $f\in W^{1,2}(X)$ satisfies $|Df|_{2}\leq 1$ $\mathfrak{m}$-a.e., then $f$
has a $1$-Lipschitz representative.

  4. (4)

For any $f\in D(\Delta)$ with $\Delta f\in W^{1,2}(X)$ and $g\in D(\Delta)\cap
L^{\infty}(X;\mathfrak{m})$ with $g\geq 0$ and $\Delta g\in
L^{\infty}(X;\mathfrak{m})$,

$\displaystyle\frac{1}{2}\int_{X}|Df|_{2}^{2}\Delta
g\,\mathrm{d}\mathfrak{m}-\int_{X}\langle Df,D\Delta f\rangle
g\,\mathrm{d}\mathfrak{m}\geq K\int_{X}|Df|_{2}^{2}g\,\mathrm{d}\mathfrak{m}$

Let ${\rm Test}(X)$ be the family of _test functions_ defined by

$\displaystyle{\rm Test}(X):=\\{u\in D(\Delta)\cap
L^{\infty}(X;\mathfrak{m})\mid\Gamma(u)^{\frac{1}{2}}=|Du|_{2}\in
L^{\infty}(X;\mathfrak{m})\text{ and }\Delta u\in D(\mathscr{E})\\}.$

It is shown in [34, Lemma 3.2] that ${\rm Test}(X)$ forms an algebra under
${\sf RCD}(K,\infty)$-condition for $(X,{\sf d},\mathfrak{m})$.

######  Theorem 3.7.

Let $K\in\mathbb{R}$ and $p\in]1,+\infty[$. Suppose that $(X,{\sf
d},\mathfrak{m})$ is an RCD$(K,\infty)$-space. Then
$(\mathscr{D}_{1,p},\|\cdot\|_{H^{1,p}})$ is closable on
$L^{p}(X;\mathfrak{m})$ and its closure $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ on
$L^{p}(X;\mathfrak{m})$ coincides with $(W^{1,p}(X),\|\cdot\|_{W^{1,p}})$.

###### Proof.

It is easy to see
$(H^{1,2}(X),\|\cdot\|_{H^{1,2}})=(W^{1,2}(X),\|\cdot\|_{W^{1,2}})$. Suppose
that RCD$(K,\infty)$-condition holds for $(X,{\sf d},\mathfrak{m})$. Then, by
[21, Theorem 3.4], for any given $p,q\in]1,+\infty[$, $u\in W^{1,p}(X)$ such
that $u,|Du|_{p}\in L^{q}(X;\mathfrak{m})$ implies $u\in W^{1,q}(X)$ and
$|Du|_{p}=|Du|_{q}$ $\mathfrak{m}$-a.e. From this, we can conclude that
$\mathscr{D}_{1,p}\subset W^{1,p}(X)$ with $\Gamma(u)^{\frac{1}{2}}=|Du|_{p}$
$\mathfrak{m}$-a.e., hence $\|u\|_{H^{1,p}}=\|u\|_{W^{1,p}}$ for any
$u\in\mathscr{D}_{1,p}$. Since ($W^{1,p}(X),\|\cdot\|_{W^{1,p}})$ is a Banach
space, the closability of $(\mathscr{D}_{1,p},\|\cdot\|_{H^{1,p}})$ on
$L^{p}(X;\mathfrak{m})$ is clear. Thus, we have the inclusion
$H^{1,p}(X)\subset W^{1,p}(X)$ with $\|u\|_{H^{1,p}}=\|u\|_{W^{1,p}}$ for
$u\in H^{1,p}(X)$.

We claim that ${\rm Test}(X)\cap W^{1,p}(X)$ is $W^{1,p}$-dense in
$W^{1,p}(X)$. For this, we show that any $u\in W^{1,p}(X)$ can be
$\|\cdot\|_{W^{1,p}}$-approximated by a sequence of $W^{1,p}(X)\cap
L^{\infty}(X;\mathfrak{m})\cap L^{2}(X;\mathfrak{m})$. By using the distance
function ${\sf d}:X^{2}\to[0,+\infty[$, we can construct functions
$\rho_{n}\in{\rm Lip}(X)$ such that $0\leq\rho_{n}\leq 1$ on $X$, $\rho_{n}=1$
on $B_{n}(o)$, $\rho_{n}=0$ on $X\setminus B_{n+1}(o)$ and ${\rm
lip}(\rho_{n})=|D\rho_{n}|_{p}=|D\rho_{n}|_{2}\leq 1$ $\mathfrak{m}$-a.e. on
$X$. Then $u_{n}:=(-n)\lor(u\rho_{n})\land n\in W^{1,p}(X)\cap
L^{\infty}(X;\mathfrak{m})\cap L^{2}(X;\mathfrak{m})$ forms an
$\|\cdot\|_{W^{1,p}}$-approximation to $u$. For any $u\in W^{1,p}(X)\cap
L^{\infty}(X;\mathfrak{m})\cap L^{2}(X;\mathfrak{m})$, we see $P_{t}u\in{\rm
Test}(X)\cap W^{1,p}(X)$ in view of $|DP_{t}u|_{p}^{p}\leq
e^{-pKt}P_{t}|Du|_{p}^{p}$ by [21, Proposition 3.1] and
$|DP_{t}u|_{2}^{2}\leq\frac{e^{K^{-}t}}{2t}\|u\|_{L^{\infty}(X;\mathfrak{m})}$
by [14, Proposition 6.8]. From this, $\\{P_{t}u\\}$ is a
$\|\cdot\|_{W^{1,p}}$-bounded sequence in ${\rm Test}(X)\cap W^{1,p}(X)$.
Thanks to the uniform convexity of $(W^{1,p}(X),\|\cdot\|_{W^{1,p}})$ and
Banach-Saks type theorem by Kakutani [29], any $u\in W^{1,p}(X)\cap
L^{\infty}(X;\mathfrak{m})\cap L^{2}(X;\mathfrak{m})$ can be
$\|\cdot\|_{W^{1,p}}$-approximated by a sequence in ${\rm Test}(X)\cap
W^{1,p}(X)$. Thus the claim is confirmed.

Applying [21, Theorem 3.4] again, we see the inclusion ${\rm Test}(X)\cap
W^{1,p}(X)\subset\mathscr{D}_{1,p}\subset H^{1,p}(X)$. Therefore we obtain the
inclusion $W^{1,p}(X)\subset H^{1,p}(X)$ with
$\|u\|_{H^{1,p}}=\|u\|_{W^{1,p}}$ for $u\in W^{1,p}(X)$. ∎

######  Example 3.8 (Configuration space over metric measure spaces).

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

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

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

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

constructed in [1, Theorem 6.1] is quasi-regular and strongly local and it
induces the heat semigroup $T_{t}^{\Upsilon}$ and a Brownian motion ${\bf
X}^{\Upsilon}$ on $\Upsilon$ which can be identified with the independent
infinite particle process. By Proposition 1.1, one can construct the
$(1,p)$-Sobolev space $(H^{1,p}(\Upsilon),\|\cdot\|_{H^{1,p}})$ for
$p\in[2,+\infty[$ based on the quasi regular strongly local Dirichlet form
$(\mathscr{E}^{\Upsilon},D(\mathscr{E}^{\Upsilon}))$ on $L^{2}(\Upsilon;\mu)$.

More generally, in Dello Schiavo-Suzuki [13], configuration space $\Upsilon$
over proper complete and separable metric space $(X,{\sf d})$ is considered.
The configuration space $\Upsilon$ is endowed with the _vage topology_
$\tau_{V}$, induced by duality with continuous compactly supported functions
on $X$, and with a reference Borel probability measure $\mu$ satisfying [13,
Assumption 2.17], commonly understood as the law of a proper point process on
$X$. In [13], they constructed the strongly local Dirichlet form
$\mathscr{E}^{\Upsilon}$ defined to be the $L^{2}(\Upsilon;\mu)$-closure of a
certain pre-Dirichlet form on a class of certain cylindrical functions and
prove its quasi-regularity for a wide class of measures $\mu$ and base spaces
(see [13, Proposition 3.9 and Theorem 3.45]).

By Proposition 1.1, one can construct the $(1,p)$-Sobolev space
$(H^{1,p}(\Upsilon),\|\cdot\|_{H^{1,p}})$ for $p\in[2,+\infty[$ based on the
quasi regular strongly local Dirichlet form
$(\mathscr{E}^{\Upsilon},D(\mathscr{E}^{\Upsilon}))$ on $L^{2}(\Upsilon;\mu)$.

Acknowledgment. The author would like to thank Dr. Ryosuke Shimizu for showing
the references [6] and [26] during the preparation of this paper. He also
thanks to Professor Lucian Beznea for showing his recent result [5] and
related literatures [4, 27, 28].

Conflict of interest. The authors have no conflicts of interest to declare
that are relevant to the content of this article.

Data Availability Statement. Data sharing is not applicable to this article as
no datasets were generated or analyzed during the current study.

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