# On $L^{p}$ Liouville theorems for Dirichlet forms

Bobo Hua  School of Mathematical Sciences, LMNS, Fudan University, Shanghai
200433, China; Shanghai Center for Mathematical Sciences, Jiangwan Campus,
Fudan University, No. 2005 Songhu Road, Shanghai 200438, China.
<EMAIL_ADDRESS>,  Matthias Keller  Institut für Mathematik, Universität
Potsdam, 14476 Potsdam, Germany<EMAIL_ADDRESS>,  Daniel
Lenz  Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07743
Jena, Germany<EMAIL_ADDRESS>and  Marcel Schmidt  Mathematisches
Institut, Universität Leipzig, 04109 Leipzig, Germany.
<EMAIL_ADDRESS>
###### Abstract.

We study harmonic functions for general Dirichlet forms. First we review
consequences of Fukushima’s ergodic theorem for the harmonic functions in the
domain of the $L^{p}$ generator. Secondly we prove analogues of Yau’s and
Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly
harmonic functions which satisfy certain $L^{p}$ growth criteria must be
constant. As consequence we give an integral criterion for recurrence.

###### Contents

  1. 1 Introduction
  2. 2 Fukushima’s ergodic theorem
  3. 3 Regular Dirichlet forms and harmonic functions
    1. 3.1 Basic notions and intrinsic metrics
    2. 3.2 Harmonic functions
  4. 4 A Caccioppoli inequality
  5. 5 Proof of Yau’s and Karp’s theorem and recurrence
    1. 5.1 Proof of Yau’s and Karp’s theorem
    2. 5.2 Proof of the growth test for recurrence

##  1\. Introduction

Liouville theorems for harmonic functions have a long tradition and circle
around the idea that a harmonic function which satisfies certain boundedness
conditions must be constant. The first result of this type for analytic
functions was proven by Cauchy in 1844, see [1]. In 1847 Liouville presented
the result in a lecture and that is how the name arose. Since then numerous
results in various contexts were proven.

Our setting is the one of Dirichlet forms. This setting includes on the one
hand well studied objects such as Laplacians on (sub) Riemannian manifolds but
also non-local operators arising from jump processes.

We start by stating the results and refer for the definitions to Section 2 and
Section 3. For a general background on Dirichlet forms we refer to [6]. Let
$\mathcal{E}$ be a Dirichlet form on $L^{2}(m)$, where $m$ is a
$\sigma$-finite measure. The associated self-adjoint operator generates a
Markovian $C_{0}$-semigroup on $L^{2}(m)$, which for $p\in[1,\infty)$ extends
to a Markovian $C_{0}$-semigroup on $L^{p}(m)$.

We first present a consequence of Fukushima’s ergodic theorem [5] for harmonic
functions in the domain of the generator of the $L^{p}$-semigroup, which we
also refer to as the $L^{p}$-generator of the Dirichlet form.

######  Theorem 1.1 (Basic $L^{p}$-Liouville theorem).

For $p\in(1,\infty)$ any harmonic function in the domain of the
$L^{p}$-generator of an irreducible Dirichlet form is constant. In particular,
in case there is a non-trivial harmonic function in the domain of the
$L^{p}$-generator, then $m$ is a finite measure.

Next, we restrict ourselves to regular Dirichlet forms where one has an
intrinsic metric $\rho$ in the sense of [4] for which all distance balls are
precompact. The first result is an analogue of Yau’s $L^{p}$-Liouville type
theorem [16]. For some nonlocal operators on Euclidean space a related result
is contained in [12]. Speaking about _(sub)harmonic_ functions we refer here
to weakly (sub)harmonic functions, see Definition 3.4.

######  Theorem 1.2 (Yau’s $L^{p}$-Liouville theorem).

Let $\mathcal{E}$ be an irreducible regular Dirichlet form without killing
with an intrinsic metric for which all distance balls are precompact. Let
$p\in(1,\infty)$ and let $f\in L^{p}(m)$ be a non-negative subharmonic
function. Then $f$ is constant if one of the following additional conditions
is satisfied:

  1. (a)

$1<p\leq 2$.

  2. (b)

$m$ is finite.

  3. (c)

$p>2$ and $f\in L^{q}(m)$ for some $q\in[2p-2,\infty]$.

  4. (d)

$p>2$, the intrinsic metric has finite jump size and $f\in L^{2p-2}_{\rm
loc}(m)$.

The next result is an analogue to Karp’s $L^{p}$-Liouville theorem [10], which
is found for strongly local regular Dirichlet forms in [15] and for graphs in
[7, 8]. Here we additionally need that the intrinsic metric has finite jump
size, see Section 3. We denote the distance balls with radius $r\geq 0$ about
a fixed point $o\in X$ with respect to the intrinsic metric $\rho$ by
$B_{r}=\\{x\in X\mid\rho(x,o)\leq r\\}$.

######  Theorem 1.3 (Karp’s $L^{p}$-Liouville theorem).

Let $\mathcal{E}$ be an irreducible regular Dirichlet form without killing
with an intrinsic metric for which all distance balls are precompact and the
jump size is finite. Let $p\in(1,\infty)$ and let $q=\max\\{p,2p-2\\}$. Then
every non-negative subharmonic function $f\in L^{q}_{\rm loc}(m)$ satisfying

$\displaystyle\int_{r_{0}}^{\infty}\frac{r}{\|f1_{B_{r}}\|_{p}^{p}}dr=\infty$

for all $r_{0}>0$ is constant.

Other than in the above mentioned references in the case $p>2$ we need some
more integrability than $L^{p}$ for the analogue of Yau’s theorem or
$L^{p}_{\rm loc}$ for the analogue of Karp’s theorem. The reason for this is a
technical issue concerning the existence of certain integrals in our proof. In
many concrete applications $L^{\infty}_{\rm loc}$-integrability (which is
sufficient for our results) of (sub)harmonic functions is known. This follows
either from hypellipticity of the corresponding operators (which yields
smoothness of harmonic functions) or, more generally, from local Hölder
estimates for nonnegative subharmonic functions deduced by De Giorgi-Nash-
Moser iteration. In this sense, the additional integrability assumptions we
make can be seen as rather mild assumptions (at least when the jump size is
finite).

In many special situations the positive and negative part of harmonic
functions are positive subharmonic functions. This is for example the case for
strongly local Dirichlet forms or Dirichlet forms on discrete sets. In these
cases the results above direclty imply the corresponding results for harmonic
functions.

We can use Karp’s Liouville theorem to prove an integral criterion for
recurrence.

######  Theorem 1.4 (Recurrence).

Let $\mathcal{E}$ be an irreducible regular Dirichlet form without killing
with an intrinsic metric for which all distance balls are precompact and the
jump size is finite. If

$\displaystyle\int_{r_{0}}^{\infty}\frac{r}{m(B_{r})}dr=\infty$

for some $r_{0}>0$, then $\mathcal{E}$ is recurrent.

##  2\. Fukushima’s ergodic theorem

Let $m$ be a $\sigma$-finite measure on (a $\sigma$-algebra on) $X$. Let
$\mathcal{E}$ be a Dirichlet form on $L^{2}(m)$ with nonnegative generator $L$
and associated Markovian semigroup $(T_{t})=(e^{-tL})$. For $1\leq
p\leq\infty$, let $(T^{p}_{t})$ be the Markovian extension of the semigroup to
$L^{p}(m)$. For $1\leq p<\infty$, these are $C_{0}$-semigroups of contractions
and we denote by $L_{p}$ the corresponding generator, [2]. For $p=\infty$, it
is also a semigroup of contractions, which is only weak-*-continuous. These
extension are compatible in the sense that for $p\neq q$ they agree on
$L^{p}(m)\cap L^{q}(m)$. Hence, on the semigroup level we often omit the
superscript $p$.

In this section a function $f\in L^{p}(m)$ is called $L_{p}$-harmonic if it
belongs to the domain of $L_{p}$ and $L_{p}f=0$ holds. For $p=2$, we also
speak about $L$-harmonic functions instead of $L_{2}$-harmonic functions.
Clearly, a function $f$ is $L$-harmonic if and only if $\mathcal{E}(f)=0$.

As usual we denote the dual pairing between $L^{p}(m)$ and $L^{q}(m)$ with
$1/q+1/p=1$ by $(\cdot,\cdot)$.

Recall that a Dirichlet form $\mathcal{E}$ is called conservative if
$T_{t}1=1$. In 1982 Fukushima [5] proved that if $\mathcal{E}$ is conservative
(and coming from an $m$-symmetric Markov transition function), then for
$p\in(1,\infty)$ and all $f\in L^{p}(m)$

$\displaystyle\lim_{t\to\infty}T_{t}f=g\quad m\text{-a.e.},$

where $g\in L^{p}(m)$ is a $(T_{t})$-invariant function. In particular, if
$\mathcal{E}$ is additionally irreducible, then $g$ is constant and equal to
$m(X)^{-1}\int_{X}fdm$ whenever $m(X)<\infty$ and equal to $0$ if
$m(X)=\infty$. Now, one can argue that $L_{p}$-harmonic functions are
invariant under the semigroup and therefore all harmonic functions must be
constant in the above setting.

Indeed, the setting of [5] starts from an $m$-symmetric, conservative Markov
transition function on a $\sigma$-finite measure space. The proof then relies
on Rota’s ergodic theorem [3]. Here, we give an analytic proof of a related
result under weaker assumptions. We only need a (symmetric) Dirichlet form but
only show strong convergence in $L^{p}(m)$ for $p\in(1,\infty)$. This is
however enough to establish Theorem 1.1 along the same lines as discussed
above.

######  Lemma 2.1.

If $\mathcal{E}$ is irreducible and $0$ is an eigenvalue of $L$, then any
corresponding eigenfunction must be constant.

###### Proof.

Let $\varphi$ be an eigenfunction to $0$. As $\mathcal{E}$ is a Dirichlet
form, we infer, for any normal contraction $C$,

$0\leq\mathcal{E}(C\varphi)\leq\mathcal{E}(\varphi)=0$

and, hence,

$\mathcal{E}(C\varphi)=0.$

Thus, $C\varphi$ is an eigenfunction to $0$ for any normal contraction $C$.
Now, as $\mathcal{E}$ is irreducible, we know that the eigenspace to $0$ is
spanned by a unique function of fixed sign. The preceding reasoning then shows
that the span of this function must be invariant under normal contractions.
This is only possible if the function is constant. ∎

######  Definition 2.2 (The ground state $\Phi$).

If $\mathcal{E}$ is irreducible, we define $\Phi$ to be zero if $0$ is not an
eigenvalue of $L$ and to be the unique positive constant eigenfunction to $0$
with $\lVert\Phi\rVert_{2}=1$ otherwise.

######  Remark.

  1. (a)

Let us emphasize that $\Phi$ is a constant function in $L^{2}(m)$ (in all
situations). Hence, $\Phi\neq 0$ can only occur if $m(X)<\infty$. In this
case, constant functions are eigenfunctions to $0$ if and only if
$\mathcal{E}$ is conservative. Indeed, if $m(X)<\infty$ and $\mathcal{E}$ is
conservative, we have by definition $T_{t}1=1$ for all $t>0$. This shows

$\lim_{h\to 0+}h^{-1}(T_{h}1-1)=0$

in $L^{2}(m)$, so that $1\in D(L)$ and $L1=0$. Conversely, if $1\in D(L)$ with
$L1=0$, we have $1\in D(\mathcal{E})$ and $\mathcal{E}(1)=0$. This implies
conservativeness, see [6, Theorem 1.6.6].

In summary we obtain

$\Phi=\begin{cases}\frac{1}{\sqrt{m(X)}}&\text{if }m(X)<\infty\text{ and
}\mathcal{E}\text{ is conservative},\\\ 0&\text{else}.\end{cases}$

  2. (b)

Since $\Phi\neq 0$ implies $m(X)<\infty$, the constant function $\Phi$ belongs
to $L^{p}(m)$ for any $p\in[1,\infty]$. Thus, in both cases $\Phi=0$ and
$\Phi\neq 0$, the map

$L^{p}(m)\to L^{p}(m),\quad f\mapsto(\Phi,f)\Phi$

is well defined and continuous. By our discussion above it is given by

$(\Phi,f)\Phi=\begin{cases}\frac{1}{m(X)}\int_{X}fdm&\text{if
}m(X)<\infty\text{ and }\mathcal{E}\text{ is conservative},\\\
0&\text{else}.\end{cases}$

######  Lemma 2.3.

If $\mathcal{E}$ is irreducible, then any $L$-harmonic function is a multiple
of $\Phi$.

###### Proof.

This is clear (as any $L$-harmonic function either vanishes or is an
eigenfunction to $0$). ∎

######  Lemma 2.4.

If $\mathcal{E}$ is irreducible, then for $p\in(1,\infty)$ and $f\in L^{p}(m)$

$\lim_{t\to\infty}T_{t}f=(\Phi,f)\Phi\qquad\text{ in }L^{p}(m).$

###### Proof.

On the $L^{2}(m)$-level, the spectral theorem implies that $T_{t}f$ converges
to the projection of $f$ to the kernel of $L$, as $t\to\infty$. In our
situation this reads

$T_{t}f\stackrel{{\scriptstyle L^{2}}}{{\to}}\langle\Phi,f\rangle\Phi,$

see e.g. [11, Theorem 1.1]. For $f\in L^{2}(m)\cap L^{\infty}(m)\cap
L^{1}(m)\subseteq L^{p}(m)$, we can estimate by Littlewood’s inequality for
$L^{p}$-spaces (i.e., Hölder inequality with a smart choice of parameters)

$\displaystyle\lVert T_{t}f-(\Phi,f)\Phi\rVert_{p}$ $\displaystyle\leq\lVert
T_{t}f-(\Phi,f)\Phi\rVert^{1-\theta}_{r}\lVert
T_{t}f-(\Phi,f)\Phi\rVert_{2}^{\theta}$ $\displaystyle\leq C_{r}\lVert
f\rVert_{r}^{1-\theta}\lVert
T_{t}f-\langle\Phi,f\rangle\Phi\rVert_{2}^{\theta},$

with $r=\infty$ and $\theta=2/p$ if $p\geq 2$, and $r=1$ and
$\theta=2(p-1)/p$, if $p\leq 2$. The second inequality follows from the fact
that $(T_{t})$ is a contraction on $L^{1}(m)$ and $L^{\infty}(m)$ and that the
semigroups agree on $L^{p}(m)\cap L^{2}(m).$

This estimate and our discussion on the $L^{2}$-case show that the desired
convergence holds on a dense subspace of $L^{p}(m)$. Since the semigroups are
uniformly bounded, it extends to all of $L^{p}(m)$. ∎

######  Remark.

Our discussion after the definition of $\Phi$ shows that this a version of
Fukushima’s ergodic theorem for semigroups associated with not necessarily
conservative Dirichlet forms but with the weaker statement on
$L^{p}$-convergence instead of $m$-a.e. convergence.

######  Lemma 2.5.

For $p\in[1,\infty)$, let $f\in L^{p}(m)$ be an $L_{p}$-harmonic function.
Then, $T_{p}f=f$ for any $t\geq 0$.

###### Proof.

By abstract theory for $C_{0}$-semigroups, for a given $g\in D(L_{p})$, the
function $[0,\infty)\to L^{p}(m)$, $t\mapsto T_{t}g$ is the unique solution in
$C^{1}([0,\infty);L^{p}(m))$ of the Cauchy problem

$\begin{cases}\dot{u}_{t}=L_{p}u_{t}\quad\text{ for }t>0\\\
u_{0}=g\end{cases}.$

Now, obviously, $[0,\infty)\to L^{p}(m),\,t\mapsto f$ is continuously
differentiable and solves the problem with initial value $f$ so that
$T_{t}f=f$ for $t>0$. ∎

Theorem 1.1 is a direct consequence of the following theorem.

######  Theorem 2.6.

Let $\mathcal{E}$ be an irreducible Dirichlet form and let $f$ be an
$L_{p}$-harmonic function for some $p\in(1,\infty)$. Then, $f$ is constant,
and in fact

$f=(\Phi,f)\Phi=\begin{cases}\frac{1}{m(X)}\int_{X}fdm&\text{if
}m(X)<\infty\text{ and }\mathcal{E}\text{ is conservative}\\\
0&\text{else}\end{cases}.$

###### Proof.

For $p\in(1,\infty)$, we infer from the previous two lemmas

$f=T_{t}f\to(\Phi,f)\Phi,\text{ as }t\to\infty.$

Combined with our computation of $(\Phi,f)\Phi$ this yields the result. ∎

##  3\. Regular Dirichlet forms and harmonic functions

###  3.1. Basic notions and intrinsic metrics

We use the notation of the previous section. Moreover, from now on we
additionally assume that $X$ is a locally compact separable metric space, $m$
is a Radon measure of full support and $\mathcal{E}$ is a regular Dirichlet
form on $L^{2}(m)$, see [6].

By $C(X)$ we denote the space of continuous functions on $X$ and by $C_{c}(X)$
the space of continuous functions of compact support.

As for continuous functions, we write $D(\mathcal{E})_{c}$ for the functions
in $D(\mathcal{E})$ with compact support. We denote by $D(\mathcal{E})_{\rm
loc}$ the space of functions locally in $D(\mathcal{E})$, that is the set of
functions $f$ such that for every open precompact set $G$ there is a function
in $D(\mathcal{E})$ which coincides with $f$ on $G$.

By the Beurling-Deny formula [6, Theorems 3.2.1 and 5.2.1] we have for $f,g\in
D(\mathcal{E})$

$\mathcal{E}(f)=\int_{X}d{\Gamma}^{(c)}(f)+\int_{X\times X\setminus
d}(\tilde{f}(x)-\tilde{f}(y))^{2}dJ(x,y)+\int_{X}\tilde{f}(x)^{2}k(dx),$

where ${\Gamma}^{(c)}$ is a measure valued strongly local quadratic form on
$D(\mathcal{E})$ (i.e., ${\Gamma}^{(c)}(f,g)=0$ if $f$ is constant on a
neighborhood of the support of $g$), $J$ is a Radon measure on $X\times
X\setminus d=\\{(x,y)\in X\times X\mid x\neq y\\}$ and $k$ is a Radon measure
on $X$. Moreover, $\tilde{f}$ is a quasi-continuous representative of $f$.
Such a representative exists for $f\in D(\mathcal{E})$ by [6, Theorem 2.1.7]
and the argument given there extends directly to $f\in D(\mathcal{E})_{\rm
loc}$, see e.g. [4, Proposition 3.1]. To simplify notation, below we will
always choose a quasi-continuous representative and just write $f$ instead of
$\tilde{f}$. Since we exclusively deal with continuous functions or functions
from $D(\mathcal{E})_{\rm loc}$, this is always possible.

Assumption: From now on we assume that $\mathcal{E}$ has no killing, i.e.,
$k=0$.

Using the measure $J$ from the decomposition we obtain a finite Radon measure
valued quadratic form ${\Gamma}^{(j)}$ on $D(\mathcal{E})$, which is uniquely
determined by

$\displaystyle\int_{K}d{\Gamma}^{(j)}(f)=\int_{K\times X\setminus
d}(f(x)-f(y))^{2}dJ(x,y)$

for all compact $K\subseteq X$ and $f\in D(\mathcal{E})$.

Next, we discuss how to extend ${\Gamma}^{(c)}$ and ${\Gamma}^{(j)}$ to larger
classes of functions. We denote by $D(\mathcal{E})_{\rm loc}^{*}$ the space of
functions in $f\in D(\mathcal{E})_{\rm loc}$ such that for every compact
$K\subseteq X$

$\displaystyle\int_{K\times X\setminus d}(f(x)-f(y))^{2}dJ(x,y)<\infty.$

This space was introduced in [4]. For later purposes, we note that any $f\in
D(\mathcal{E})_{\rm loc}^{*}$ with compact support belongs to
$D(\mathcal{E})_{\rm c}$, see [4, Theorem 3.5].

The bilinear forms on $D(\mathcal{E})\times D(\mathcal{E})$ induced from
${\Gamma}^{(c)}$ and ${\Gamma}^{(j)}$ by polarization can be (partially)
extended to bilinear forms $D(\mathcal{E})_{\rm loc}^{*}\times
D(\mathcal{E})_{c}$ with values in the space of finite signed Radon measures.
More precisely, for $f\in D(\mathcal{E})_{\rm loc}^{*}$ and $\varphi\in
D(\mathcal{E})_{c}$ these extensions are characterized by

$\int_{K}d{\Gamma}^{(c)}(f,\varphi)=\int_{K}d{\Gamma}^{(c)}(f_{K},\varphi),$

and

$\int_{K}d{\Gamma}^{(j)}(f,\varphi)=\int_{K\times X\setminus
d}(f(x)-f(y))(\varphi(x)-\varphi(y))dJ(x,y),$

for all compact $K\subseteq X$. Here, $f_{K}\in D(\mathcal{E})$ is a function
with $f=f_{K}$ on a relatively compact open neighborhood of $K$. Due to the
strong locality ${\Gamma}^{(c)}(f,\varphi)$ is well definied (i.e.,
independent of the choice of $f_{K}$, see the remarks after the proof of [6,
Theorem 3.2.2]) and the definition of $D(\mathcal{E})_{\rm loc}^{*}$ ensures
that ${\Gamma}^{(j)}(f,\varphi)$ is well defined (i.e., it yields a finite
signed Radon measure, see [4, Theorem 3.4]). Similarly, the quadratic forms
${\Gamma}^{(c)}$ and ${\Gamma}^{(j)}$ can be extended to quadratic forms on
$D(\mathcal{E})_{\rm loc}^{*}$ taking values in the set of nonnegative (not
necessarily finite) Radon measures, see [4, Proposition 3.3]. We let
$\Gamma={\Gamma}^{(c)}+{\Gamma}^{(j)}$ denote both the quadratic form on
$D(\mathcal{E})_{\rm loc}^{*}$ and the induced bilinear form on
$D(\mathcal{E})_{\rm loc}^{*}\times D(\mathcal{E})_{c}$.

Since $\mathcal{E}$ is a Dirichlet form, for any normal contraction
$C\colon{\mathbb{R}}\to{\mathbb{R}}$ and $f\in D(\mathcal{E})_{\rm loc}^{*}$,
we have $C\circ f\in D(\mathcal{E})_{\rm loc}^{*}$ and

$\int_{X}d\Gamma(C\circ f)\leq\int_{X}d\Gamma(f).$

Since the killing $k$ vanishes, this inequality extends to $1$-Lipschitz
functions. Indeed, if $C\colon{\mathbb{R}}\to{\mathbb{R}}$ is $1$-Lipschitz,
then $C\circ f=\tilde{C}(f)+C(0)$ with the normal contraction
$\tilde{C}=C-C(0)$. Since the constant function $1$ belongs to
$D(\mathcal{E})_{\rm loc}^{*}$, see [4, Proposition 3.1] and satisfies
$\Gamma(1)=0$, this yields the claim.

The strongly local part satisfies a Leibniz rule. If $f,g\in
D(\mathcal{E})_{\rm loc}^{*}$ such that $fg\in D(\mathcal{E})_{\rm loc}^{*}$,
then

$d\Gamma^{(c)}(fg,\varphi)=fd\Gamma^{(c)}(g,\varphi)+gd\Gamma^{(c)}(f,\varphi)$

for all $\varphi\in D(\mathcal{E})_{c}$, see the proof of [4, Theorem 3.9].
Moreover, it satisfies a chain rule. If $C\in C^{1}({\mathbb{R}})$ with
bounded derivative, then for $f\in D(\mathcal{E})_{\rm loc}^{*}$, we have
$C\circ f\in D(\mathcal{E})_{\rm loc}^{*}$ and

${\Gamma}^{(c)}(C(f),\varphi)=C^{\prime}(f){\Gamma}^{(c)}\Gamma(f,\varphi)$

for all $\varphi\in D(\mathcal{E})_{c}$, see [6, Theorem 3.2.2].

Let $\rho\colon X\times X\to[0,\infty)$ be a pseudo metric, that is a
symmetric map with zero diagonal satisfying the triangle inequality. For
measurable sets $A\subseteq X$, we write $\rho_{A}=\inf_{x\in
A}\rho(x,\cdot)$. The following definition is a slight modification of the one
given in [4].

######  Definition 3.1 (Intrinsic metric).

A pseudo metric $\rho$ on $X$ is called intrinsic (for the regular Dirichlet
form $\mathcal{E}$) if it is continuous with respect to the topology on $X$
and there are Radon measures $m^{(c)}$, $m^{(j)}$ with $m^{(c)}+m^{(j)}\leq m$
such that for every measurable $A\subseteq X$ the following holds:
$\rho_{A}\in D(\mathcal{E})_{\rm loc}^{*}$ and

$\displaystyle{\Gamma}^{(c)}(\rho_{A})\leq
m^{(c)}\quad\mbox{and}\quad{\Gamma}^{(j)}(\rho_{A})\leq m^{(j)}.$

######  Remark.

In [4], $\rho$ is allowed to take the value infinity. In this sense our
definition is a bit more restrictive.

######  Definition 3.2.

The _jump size_ $s$ of a continuous pseudo metric $\rho$ with respect to the
Dirichlet form $\mathcal{E}$ is given by

$\displaystyle s:=\inf\\{t\geq 0\mid J(\\{(x,y)\in X\times X\setminus
d\mid\rho(x,y)>t\\})=0\\}\in[0,\infty].$

If $\rho$ is a fixed pseudo-metric, for $U\subseteq X$ and $r\in\mathbb{R}$ we
write

$B_{r}(U)=\\{x\in X\mid\rho(x,y)\leq r\mbox{ for some }y\in U\\}.$

The relevance of intrinsic metrics comes from the fact that Lipschitz
functions with respect to intrinsic metrics induce good cut-off functions.
This is discussed next.

Fix $o\in X$ and let $0\leq r<R$. Below we will make use of

$\displaystyle\eta=\eta_{r,R}\colon
X\to{\mathbb{R}},\quad\eta(x)=1\wedge\left(\frac{R-\rho(x,o)}{R-r}\right)_{+}.$

It is $(R-r)^{-1}$-Lipschitz and satisfies $\eta=1$ on $B_{r}$ and $\eta=0$ on
$X\setminus B_{R}$.

######  Lemma 3.3.

Let $\rho$ be an intrinsic metric for $\mathcal{E}$ with jump size $s$. Then
$\eta\in D(\mathcal{E})_{\rm loc}^{*}$ and it satisfies

$\Gamma^{(c)}(\eta)\leq\frac{1}{(R-r)^{2}}1_{B_{R}\setminus
B_{r}}m^{(c)}\text{ and
}\Gamma^{(j)}(\eta)\leq\frac{1}{(R-r)^{2}}1_{B_{R+s}\setminus
B_{r-s}}m^{(j)}.$

If, moreover, balls with respect to $\rho$ are precompact, then $\eta\in
D(\mathcal{E})_{c}$.

###### Proof.

The first statement and the estimates are the content of [4, Theorem 4.9 and
Proposition 8.5]. The last statement follows from the fact that functions in
$D(\mathcal{E})_{\rm loc}^{*}$ with compact support belong to
$D(\mathcal{E})_{c}$. ∎

If $\rho$-balls are precompact, the previous lemma and the definitions of
${\Gamma}^{(c)}$ and ${\Gamma}^{(j)}$ yield for $f\in D(\mathcal{E})_{\rm
loc}^{*}$

$\int_{X}d{\Gamma}^{(c)}(f,\eta)=\int_{B_{R}\setminus
B_{r}}d{\Gamma}^{(c)}(f,\eta),$

and

$\int_{X\times X\setminus
d}(f(x)-f(y))(\eta(x)-\eta(y))dJ(x,y)=\int_{U_{r,R}}(f(x)-f(y))(\eta(x)-\eta(y))dJ(x,y)$

with $U_{r,R}=(B_{R+s}\setminus B_{r-s})\times(B_{R+s}\setminus
B_{r-s})\setminus d$, whenever the integral makes sense.

###  3.2. Harmonic functions

In this subsection we introduce (sub)harmonic functions and discuss their
basic properties.

######  Definition 3.4.

A function $f:X\to{\mathbb{R}}$ is called _harmonic_ (_subharmonic_) if $f\in
D(\mathcal{E})_{\rm loc}^{*}$ and

$\displaystyle\int_{X}d{\Gamma}(f,{\varphi})=0\quad\mbox{($\leq 0$),}$

for all $0\leq{\varphi}\in D(\mathcal{E})_{c}$.

######  Remark.

Since domains Dirichlet forms are lattices, a harmonic function $f$ statisfies
$\int_{X}d{\Gamma}(f,{\varphi})=0$ for all $\varphi\in D(\mathcal{E})_{c}$.

Below we will need the following extension of Lemma 2.1 to
$D(\mathcal{E})_{\rm loc}^{*}$ functions. It can also be viewed as a
Liouville-type theorem.

######  Lemma 3.5.

Let $\mathcal{E}$ be irreducible. Then any $f\in D(\mathcal{E})_{\rm loc}^{*}$
with $\Gamma(f)=0$ is constant.

###### Proof.

We show that for each $\alpha>0$ the set $A_{\alpha}=\\{f>\alpha\\}$ is
$\mathcal{E}$-invariant. Since $\mathcal{E}$ is irreducible, this is only
possible if $f$ is constant. To this end, we show
$\mathcal{E}(1_{A_{\alpha}}\varphi)\leq\mathcal{E}(\varphi)$ which implies the
$\mathcal{E}$-invariance, see e.g. [13, Lemma 2.32].

We have

$1_{A_{\alpha}}=\lim_{n\to\infty}(n(f-\alpha)_{+})\wedge 1$

pointwise $m$-a.e. Let $g_{n}=(n(f-\alpha)_{+})\wedge 1$. Up to a constant
$g_{n}$ is a contraction of $f$ and so we have

$\Gamma(g_{n})\leq n^{2}\Gamma(f)=0.$

Now let $\varphi\in D(\mathcal{E})\cap C_{c}(X)$ be given. Then $\varphi
g_{n}\in D(\mathcal{E})$ (here we use that $\varphi g_{n}$ has compact support
and belongs to $D(\mathcal{E})_{\rm loc}^{*}\cap L^{\infty}(m)$ by [4,
Proposition 3.10]). The Leibniz rule for ${\Gamma}^{(c)}$, the “discrete
Leibniz rule” (i.e., $f(x)g(x)-f(y)g(y)=f(x)(g(x)-g(y))+g(y)(f(x)-f(y))$ for
functions $f,g$ on $X$ and $x,y\in X$) and the Cauchy-Schwarz inequality yield

$\displaystyle\mathcal{E}(\varphi g_{n})=\int_{X}\varphi
d\Gamma(g_{n},g_{n}\varphi)+\int_{X}g_{n}d\Gamma(\varphi,g_{n}\varphi)$
$\displaystyle\leq\left(\int_{X}d\Gamma(g_{n})\right)^{1/2}\left(\int_{X}|\varphi|^{2}d\Gamma(\varphi
g_{n})\right)^{1/2}+\left(\int_{X}|g_{n}|^{2}d\Gamma(\varphi)\right)^{1/2}\left(\int_{X}d\Gamma(\varphi
g_{n})\right)^{1/2}$
$\displaystyle\leq\mathcal{E}(\varphi)^{1/2}\mathcal{E}(\varphi g_{n})^{1/2}.$

This shows $\mathcal{E}(\varphi g_{n})\leq\mathcal{E}(\varphi)$. Since by
dominated convergence $1_{A_{\alpha}}\varphi=\lim_{n\to\infty}\varphi g_{n}$
in $L^{2}(m)$, the lower semicontinuity of $\mathcal{E}$ and this inequality
yield $1_{A_{\alpha}}\varphi\in D(\mathcal{E})$ and

$\mathcal{E}(1_{A_{\alpha}}\varphi)\leq\mathcal{E}(\varphi).$

Since $D(\mathcal{E})\cap C_{c}(X)$ is dense in $D(\mathcal{E})$, this
inequality extends to all $\varphi\in D(\mathcal{E})$ (use lower
semicontinuity) and we obtain the $\mathcal{E}$-invariance of $A_{\alpha}$. ∎

##  4\. A Caccioppoli inequality

The following Caccioppoli-type inequality is the key to Yau’s
$L^{p}$-Liouville theorem. A somewhat more subtle estimate is needed for
Karp’s result. Throughout $\mathcal{E}$ is a regular Dirichlet form without
killing and $\rho$ is an intrinsic metric for $\mathcal{E}$.

######  Theorem 4.1 (Caccioppoli-type inequality).

Assume the distance balls of $\rho$ are precompact and $p\in(1,\infty)$. Then,
there is $C>0$ such that for every non-negative subharmonic function $f$ and
all $0<r<R$ such that $f1_{B_{R+s}\setminus B_{r-s}}\in L^{q}(m)$ for some
$q\in[2p-2,\infty]$ we have

$\displaystyle\int_{B_{r}}f^{p-2}d{\Gamma}^{(c)}(f)+\int_{X\times
B_{r}\setminus d}(f(x)\vee f(y))^{p-2}$ $\displaystyle(f(x)-f(y))^{2}dJ(x,y)$
$\displaystyle\leq\frac{C}{(R-r)^{2}}\lVert f1_{B_{R+s}\setminus
B_{r-s}}\rVert_{p}^{p},$

where $s$ is the jump size of $\rho$.

######  Remark.

  1. (a)

Note that $p\leq 2$ if and only if $2p-2\leq p$. Hence, in this case,
$f1_{B_{R+s}\setminus B_{r-s}}\in L^{q}(m)$ for some $q\geq 2p-2$ is already
satisfied if the right side of the Caccioppoli-type inequality is finite.

  2. (b)

If the jump-size of $\rho$ is finite, the set $B_{R+s}$ is precompact and
hence has finite measure. In this case, for $p>2$ the assumption on the
integrability on $f$ is satisfied for all $0\leq r<R$ if and only if $f\in
L^{2p-2}_{\rm loc}(m)$.

######  Remark.

For $p\geq 2$ the inequality above yields (with a larger constant $C$ and
under suitable conditions on $f$)

$\displaystyle\int_{X}f^{p-2}d{\Gamma}(f)$
$\displaystyle\leq\frac{C}{(R-r)^{2}}\lVert f1_{B_{R+s}\setminus
B_{r-s}}\rVert_{p}^{p}.$

Let $\mathrm{Lip}_{c}(X)$ be the space of Lipshitz continuous functions with
compact support with respect to the intrinsic metric $\rho$.

######  Lemma 4.2 (The key estimate).

Let $p\in(1,\infty)$ and let $n\in{\mathbb{N}}$. For every non-negative
subharmonic function $f$ and ${\varphi}\in\mathrm{Lip}_{c}(X)$, we have

$\displaystyle\int_{A_{n}}f^{p-2}{\varphi}^{2}d{\Gamma}^{(c)}(f)+\int_{X\times
X\setminus d}(f_{n}(x)\vee
f_{n}(y))^{p-2}{\varphi}(y)^{2}(f_{n}(x)-f_{n}(y))^{2}dJ(x,y)$
$\displaystyle\leq-C\int_{X}f_{n}^{p-1}{\varphi}d{\Gamma}^{(c)}(f,{\varphi})$
$\displaystyle\qquad-C\int_{X\times X\setminus
d}f_{n}^{p-1}(x){\varphi}(y)(f(x)-f(y))({\varphi}(x)-{\varphi}(y))dJ(x,y)<\infty,$

where $f_{n}=f\wedge n$, $A_{n}=\\{f<n\\}$ and $C=2/((p-1)\wedge 1)$.

###### Proof.

For $j\in{\mathbb{N}}$, we let $g_{j}=f_{n}\vee j^{-1}$. We show the
inequality with $f_{n}$ replaced by $g_{j}$ and $A_{n}$ replaced by
$\\{j^{-1}<f<n\\}$. Then the statement follows from Fatou’s lemma and
Lebegue’s dominated convergence theorem after letting $j\to\infty$. Note that
the integral on the right hand side of the equation exists due to the
boundedness of $f_{n}$ and the definition of $D(\mathcal{E})_{\rm loc}^{*}$.

Since $g_{j}\in D(\mathcal{E})_{\rm loc}^{*}$ is bounded from below by $1/j$
and from above by $n$, we have $g_{j}^{p-1}\in D(\mathcal{E})_{\rm loc}^{*}$
(here we use that $[j^{-1},n]\to{\mathbb{R}},t\mapsto t^{p-1}$ is Lipschitz).
Using the boundedness of $g_{j}^{p-1}$, for
${\varphi}\in\mathrm{Lip}_{c}(X)\subseteq D(\mathcal{E})_{c}\cap
L^{\infty}(m)$ we obtain ${\varphi}^{2}g_{j}^{p-1}\in D(\mathcal{E})_{c}$ as
$D(\mathcal{E})_{c}\cap L^{\infty}(m)$ is an ideal in the algebra
$D(\mathcal{E})\cap L^{\infty}(m)$. The subharmonicity and non-negativity of
$f$ yields

$\displaystyle
0\geq\int_{X}d{\Gamma}^{(c)}(f,{\varphi}^{2}g^{p-1}_{j})+\int_{X}d{\Gamma}^{(j)}(f,{\varphi}^{2}g^{p-1}_{j}).$

To shorten notation, we write $\nabla_{xy}g=g(x)-g(y)$ for $x,y\in X$ and
functions $g$. We apply the “discrete Leibniz rules”
$\nabla_{xy}(fg)=g(x)\nabla_{xy}f+f(y)\nabla_{xy}g$ and
$\nabla_{xy}f^{2}=2f(y)\nabla_{xy}f+(\nabla_{xy}f)^{2}$ to the jump term

$\displaystyle\int_{X}d{\Gamma}^{(j)}(f,{\varphi}^{2}g^{p-1}_{j})$
$\displaystyle=$ $\displaystyle\int_{X\times X\setminus
d}\nabla_{xy}f\cdot\big{(}{\varphi}^{2}(y)\nabla_{xy}g^{p-1}_{j}+g^{p-1}_{j}(x)\nabla_{xy}{\varphi}^{2}\big{)}dJ(x,y)$
$\displaystyle=$ $\displaystyle\int_{X\times X\setminus
d}\nabla_{xy}f\cdot\big{(}{\varphi}^{2}(y)\nabla_{xy}g^{p-1}_{j}+2g^{p-1}_{j}(x){\varphi}(y)\nabla_{xy}{\varphi}+g^{p-1}_{j}(x)(\nabla_{xy}{\varphi})^{2}\big{)}dJ(x,y).$

We start by estimating the first term, where we apply [8, Lemma 2.9], which
states that if $\nabla_{xy}g\geq 0$ for a non-negative function $g$, then

$\displaystyle\nabla_{xy}g^{p-1}\geq C(g(x)\vee g(y))^{p-2}\nabla_{xy}g$

for the constant $C=(p-1)\wedge 1$. Since $\nabla_{xy}f\geq 0$ implies
$\nabla_{xy}g^{p-1}_{j}\geq 0$,
$\nabla_{xy}f\nabla_{xy}g_{j}^{p-1}=\nabla_{yx}f\nabla_{yx}g_{j}^{p-1}\geq 0$
and
$\nabla_{xy}f\nabla_{xy}g_{j}=\nabla_{yx}f\nabla_{yx}g_{j}\geq|\nabla_{xy}g_{j}|^{2}$,
this gives

$\displaystyle{\varphi}^{2}(y)\nabla_{xy}f\cdot\nabla_{xy}g^{p-1}_{j}\geq
C{\varphi}^{2}(y)(g_{j}(x)\vee g_{j}(y))^{p-2}|\nabla_{xy}g_{j}|^{2}$

for the first term in the estimate above. We leave the second term as it is
for now. For third term we obtain with having
$\nabla_{xy}f\nabla_{xy}g_{j}\geq 0$ in mind

$\displaystyle 0$ $\displaystyle\leq\int_{X\times X\setminus
d}\nabla_{xy}g^{p-1}_{j}\nabla_{xy}f(\nabla_{xy}{\varphi})^{2}dJ(x,y)$
$\displaystyle=2\int_{X\times X\setminus
d}g^{p-1}_{j}(x)\nabla_{xy}f(\nabla_{xy}{\varphi})^{2}dJ(x,y),$

where the equality holds as the term on the right hand side converges
absolutely by Hölder’s inequality. Hence, we obtain

$\displaystyle\int_{X}d{\Gamma}^{(j)}(f,{\varphi}^{2}g^{p-1}_{j})\geq$
$\displaystyle C\int_{X\times X\setminus d}{\varphi}^{2}(y)(g_{j}(x)\vee
g_{j}(y))^{p-2}|\nabla_{xy}g_{j}|^{2}dJ(x,y)$
$\displaystyle\quad+2\int_{X\times X\setminus
d}g^{p-1}_{j}(x){\varphi}(y)\nabla_{xy}f\nabla_{xy}{\varphi}dJ(x,y).$

Applying the Leibniz rule and the chain rule to the strongly local term we get
immediately

$\displaystyle\int_{X}d{\Gamma}^{(c)}(f,{\varphi}^{2}g^{p-1}_{j})$
$\displaystyle=(p-1)\int_{X}{\varphi}^{2}g^{p-2}_{j}d{\Gamma}^{(c)}(f,g_{j})+2\int_{X}g^{p-1}_{j}{\varphi}d{\Gamma}^{(c)}(f,{\varphi})$
$\displaystyle=(p-1)\int_{\\{m^{-1}<f<n\\}}{\varphi}^{2}f^{p-2}d{\Gamma}^{(c)}(f)+2\int_{X}g^{p-1}_{j}{\varphi}d{\Gamma}^{(c)}(f,{\varphi}).$

For the last equality we used the truncation property of $\Gamma^{(c)}$
discussed in [15, Appendix 4.1]. Putting the two estimates for the terms
$\int_{X}d{\Gamma}^{(j)}(f,{\varphi}^{2}g^{p-1}_{j})$ and
$\int_{X}d{\Gamma}^{(c)}(f,{\varphi}^{2}g^{p-1}_{j})$ into the inequality at
the beginning of the proof yields the statement. ∎

###### Proof of the Caccioppoli-type inequality, Theorem 4.1.

Let $\eta=\eta_{r,R}$ be the cut-off functions discussed in Lemma 3.3. We use
$\varphi=\eta=\eta_{r,R}$ in our key estimate and deal with the terms on the
right-hand side separately. Without loss of generality we can assume
$f1_{B_{R+s}\setminus B_{r-s}}\in L^{p}(m)$ for otherwise there is nothing to
show. We further assume $q<\infty$ as it will become clear in the course of
the proof that the case $q=\infty$ is simpler and follows along the same
lines.

We use the inequality $|Q(u,v)|\leq\varepsilon
Q(u,u)^{2}+\frac{1}{4\varepsilon}Q(v,v)^{2}$, which holds for nonnegative
bilinear forms $Q$ and $\varepsilon>0$, and $\eta\leq 1$ to estimate

$\displaystyle C\left|\int_{A_{n}}f_{n}^{p-1}\eta
d{\Gamma}^{(c)}(f,\eta)\right|$
$\displaystyle\leq\frac{1}{2}\int_{A_{n}}f_{n}^{p-2}\eta^{2}d\Gamma^{(c)}(f)+C^{\prime}\int_{A_{n}}f^{p}d\Gamma^{(c)}(\eta)$
$\displaystyle\leq\frac{1}{2}\int_{A_{n}}f_{n}^{p-2}\eta^{2}d\Gamma^{(c)}(f)+\frac{C^{\prime}}{(R-r)^{2}}\lVert
f^{p}1_{B_{R}\setminus B_{r}}\rVert_{p}^{p},$

where $f_{n}=f\wedge n$ and $A_{n}=\\{f<n\\}$. Moreover, $f_{n}^{p-1}=n^{p-1}$
on $X\setminus A_{n}$, Cauchy-Schwarz inequality, the cut-off properties of
$\eta$ and Chebyshev’s inequality yield

$\left|\int_{X\setminus A_{n}}f_{n}^{p-1}\eta
d{\Gamma}^{(c)}(f,\eta)\right|\leq n^{p-1}\left(\int_{X\setminus
A_{n}}d{\Gamma}^{(c)}(\eta)\right)^{1/2}\left(\int_{X\setminus
A_{n}}\eta^{2}d{\Gamma}^{(c)}(f)\right)^{1/2}\\\
\leq\frac{n^{p-1}}{(R-r)}m(\\{f\geq n\\}\cap B_{R}\setminus
B_{r})^{1/2}\left(\int_{X\setminus
A_{n}}\eta^{2}d{\Gamma}^{(c)}(f)\right)^{1/2}\\\
\leq\frac{n^{p-1-q/2}}{(R-r)}\lVert f1_{B_{R}\setminus
B_{r}}\rVert^{q/2}\left(\int_{X\setminus
A_{n}}\eta^{2}d{\Gamma}^{(c)}(f)\right)^{1/2}.$

Since $\infty>q\geq 2p-2$ and $f1_{B_{R}\setminus B_{r}}\in L^{q}(m)$, the
right side of this inequality converges to $0$, as $n\to\infty$. A similar
reasoning yields

$\displaystyle C\left|\int_{A_{n}\times X\setminus
d}f_{n}^{p-1}(x)\eta(y)(f(x)-f(y))(\eta(x)-\eta(y))dJ(x,y)\right|$
$\displaystyle\leq C\int_{A_{n}\times X\setminus d}(f_{n}(x)\vee
f_{n}(y))^{p-1}\eta(y)|f(x)-f(y)||\eta(x)-\eta(y)|dJ(x,y)$
$\displaystyle\leq\frac{1}{2}\int_{A_{n}\times X\setminus d}(f_{n}(x)\vee
f_{n}(y))^{p-2}\eta^{2}(y)(f(x)-f(y))^{2}dJ(x,y)$
$\displaystyle\quad+C^{\prime}\int_{A_{n}\times X\setminus d}(f_{n}(x)\vee
f_{n}(y))^{p}(\eta(x)-\eta(y))^{2}dJ(x,y)$
$\displaystyle\leq\frac{1}{2}\int_{A_{n}\times X\setminus d}(f_{n}(x)\vee
f_{n}(y))^{p-2}\eta^{2}(y)(f(x)-f(y))^{2}dJ(x,y)$
$\displaystyle\quad+\frac{C^{\prime\prime}}{(R-r)^{2}}\lVert
f^{p}1_{B_{R+s}\setminus B_{r-s}}\rVert_{p}^{p}.$

For the last inequality we used the properties of $\eta$ and $(a\wedge
b)^{p}\leq a^{p}+b^{p}$ as well as the symmetry of $J$. Moreover, as for the
strongly local part we obtain using the cut-off properties of $\eta$

$\displaystyle\int_{(X\setminus A_{n})\times X\setminus
d}f_{n}(x)^{p-1}\eta(y)|f(x)-f(y)||\eta(x)-\eta(y)|dJ(x,y)$
$\displaystyle\leq\frac{n^{p-1-q/2}}{R-r}\lVert f1_{B_{R+s}\setminus
B_{r-s}}\rVert^{q/2}\left(\int_{X\setminus
A_{n}}\eta^{2}d{\Gamma}^{(j)}(f)\right)^{1/2},$

with the right side converging to $0$, as $n\to\infty$.

Plugging these estimates into our key estimate in Lemma 4.2, using $f=f_{n}$
on $A_{n}$ and $\eta=1$ on $B_{r}$, yields a constant $C>0$ such that for all
$n\in{\mathbb{N}}$

$\displaystyle\int_{A_{n}\cap
B_{r}}f^{p-2}d{\Gamma}^{(c)}(f)+\int_{A_{n}\times(A_{n}\cap B_{r})\setminus
d}(f(x)\vee f(y))^{p-2}(f(x)-f(y))^{2}dJ(x,y)$
$\displaystyle\quad\leq\frac{C}{(R-r)^{2}}\lVert f1_{B_{R+s}\setminus
B_{r-s}}\rVert_{p}^{p}+E_{n},$

with $E_{n}\to 0$, as $n\to\infty$. With this at hand the statement follows
after letting $n\to\infty$.

In the case $q=\infty$ we can use the same estimates as above without invoking
the limit $n\to\infty$, as in this case $A_{n}=X$ for some $n\in{\mathbb{N}}$
and $E_{n}=0$ for some $n\in{\mathbb{N}}$. ∎

For proving Yau’s theorem the Caccioppoli-type inequality is sufficient. For
Karp’s theorem we need one further estimate, which follows from our key
estimate and the Caccioppoli-type inequality. Here, $\eta=\eta_{r,R}$ denotes
the cut-off function discussed in Lemma 3.3.

######  Lemma 4.3.

Assume the distance balls of $\rho$ are precompact and the jump-size $s$ is
finite. For $p\in(1,\infty)$ there is $C>0$ such that for every non-negative
subharmonic function $f\in L^{q}_{\rm loc}(m)$ with $q=\max\\{p,2p-2\\}$ and
all $0<r<R$ we have

$\left(\int_{X}f^{p-2}\eta^{2}d{\Gamma}^{(c)}(f)+\int_{X\times X\setminus
d}(f(x)\vee f(y))^{p-2}\eta^{2}(y)(f(x)-f(y))^{2}dJ(x,y)\right)^{2}\\\
\leq\frac{C}{(R-r)^{2}}\lVert f1_{B_{R+s}\setminus
B_{r-s}}\rVert_{p}^{p}\Biggl{(}\int_{B_{R}\setminus
B_{r}}f^{p-2}\eta^{2}d{\Gamma}^{(c)}(f)\\\ \qquad+\int_{U_{R+s}\setminus
U_{r-s}}(f(x)\vee f(y))^{p-2}\eta^{2}(y)(f(x)-f(y))^{2}dJ(x,y)\Biggr{)},$

with $\eta=\eta_{r,R}$ and $U_{r}=B_{r}\times B_{r}\setminus d$.

###### Proof.

We apply our key estimate Lemma 4.2 to $f$ with the cut-off function
$\eta=\eta_{r,R}$. Then, the Cauchy-Schwarz inequality and the cut-off
properties of $\eta$ yield

$\left(\int_{A_{n}}f^{p-2}\eta^{2}d{\Gamma}^{(c)}(f)+\int_{X\times X\setminus
d}(f_{n}(x)\vee
f_{n}(y))^{p-2}\eta(y)^{2}(f_{n}(x)-f_{n}(y))^{2}dJ(x,y)\right)^{2}\\\
\leq\frac{C}{(R-r)^{2}}\lVert f_{n}1_{B_{R+s}\setminus
B_{r-s}}\rVert_{p}^{p}\Biggl{(}\int_{B_{R}\setminus
B_{r}}f_{n}^{p-2}\eta^{2}d{\Gamma}^{(c)}(f)\\\ \qquad+\int_{U_{R+s}\setminus
U_{r-s}}(f_{n}(x)\vee
f_{n}(y))^{p-2}\eta^{2}(y)(f(x)-f(y))^{2}dJ(x,y)\Biggr{)}.$

By our Caccioppoli-type inequality Theorem 4.1 all of the integrals on the
right side exist when $f_{n}$ is replaced by $f$ (here we use that $B_{R+s}$
is precompact, $0\leq\eta\leq 1$ with $\eta=0$ on $X\setminus B_{R}$ and the
integrability assumption on $f$). Hence, we can take the limit $n\to\infty$ to
obtain the statement. ∎

##  5\. Proof of Yau’s and Karp’s theorem and recurrence

In this section we use the results from the previous sections to prove Yau’s
and Karp’s theorem, Theorem 1.2 and 1.3. Later in this section we prove the
growth test for recurrence, Theorem 1.4.

###  5.1. Proof of Yau’s and Karp’s theorem

The proofs of our Liouville theorems are based on the following observation.

######  Lemma 5.1.

Let $\mathcal{E}$ be irreducible and let $f\in D(\mathcal{E})_{\rm loc}^{*}$
be non-negative. If for some $p\in(1,\infty)$

$\int_{X}f^{p-2}d{\Gamma}^{(c)}(f)+\int_{X\times X\setminus d}(f(x)\vee
f(y))^{p-2}(f(x)-f(y))^{2}dJ(x,y)=0,$

then $f$ is constant.

###### Proof.

By the truncation property of the strongly local measure ${\Gamma}^{(c)}$ and
$f\geq 0$ we have

${\Gamma}^{(c)}(f)={\Gamma}^{(c)}(f_{+})=1_{\\{f>0\\}}{\Gamma}^{(c)}(f),$

see [15, Appendix 4.1]. Hence, $\int_{X}f^{p-2}d{\Gamma}^{(c)}(f)=0$ implies
${\Gamma}^{(c)}(f)=0$. Similarly, $\int_{X\times X\setminus d}(f(x)\vee
f(y))^{p-2}(f(x)-f(y))^{2}dJ(x,y)=0$ implies $\int_{X\times X\setminus
d}(f(x)-f(y))^{2}dJ(x,y)=0$. These two observations and Lemma 3.5 show the
claim. ∎

###### Proof of Yau’s $L^{p}$-Liouville theorem, Theorem 1.2.

The assumptions we made guarantee that we can apply the Caccioppoli-type
inequality Theorem 4.1 for all $0<r<R$. In this inequality letting first
$R\to\infty$ and then $r\to\infty$ yields

$\int_{X}f^{p-2}d{\Gamma}^{(c)}(f)+\int_{X\times X\setminus d}(f(x)\vee
f(y))^{p-2}(f(x)-f(y))^{2}dJ(x,y)=0.$

Hence, the previous lemma implies that $f$ is constant. ∎

###### Proof of Karp’s theorem, Theorem 1.3.

With Lemma 4.3 at hand we basically follow [15]. Suppose $f\neq 0$. Let $R\geq
4s$ be such that $f1_{B_{R}}\neq 0$. For $n\in{\mathbb{N}}$, we define
$R_{n}=2^{n}R$, $\eta_{n}=\eta_{R_{n-1}+s,R_{n}-s}$, $v_{n}=\lVert
f1_{B_{R_{n}}\setminus B_{R_{n-1}}}\rVert_{p}^{p}$ and

$Q_{n}=\int_{B_{R_{n}}}f^{p-2}\eta_{n}^{2}d{\Gamma}^{(c)}(f)+\int_{U_{R_{n}}}(f(x)\vee
f(y))^{p-2}\eta_{n}^{2}(y)(f(x)-f(y))^{2}dJ(x,y),$

with $U_{r}=B_{r}\times B_{r}\setminus d$. The assumption
$\int^{\infty}r/\lVert f1_{B_{r}}\rVert_{p}^{p}dr=\infty$ implies

$\sum_{n=1}^{\infty}\frac{R_{n}^{2}}{v_{n}}=\infty.$

Once we show $Q_{1}=0$, the assertion follows from Lemma 5.1 after letting
$R\to\infty$.

Lemma 4.3 (applied to $r=R_{n-1}+s$ and $R=R_{n}+s$) and
$\eta_{n-1}\leq\eta_{n}$ yield

$\displaystyle Q_{n-1}Q_{n}\leq
Q_{n}^{2}\leq\frac{Cv_{n}}{(R_{n}-R_{n-1}-2s)^{2}}\left(Q_{n}-Q_{n-1}\right)\leq\frac{16Cv_{n}}{R_{n}^{2}}\left(Q_{n}-Q_{n-1}\right).$

For the last inequality we used $R\geq 4s$ and $R_{n}=2^{n}R$. If $Q_{1}>0$,
this would imply

$\frac{R_{n}^{2}}{v_{n}}\leq
16C\left(\frac{1}{Q_{n-1}}-\frac{1}{Q_{n}}\right)$

and hence

$\sum_{n=2}^{\infty}\frac{R_{n}^{2}}{v_{n}}\leq\frac{16C}{Q_{1}}<\infty,$

a contradiction. ∎

###  5.2. Proof of the growth test for recurrence

In this subsection we discuss how our version of Karp’s theorem can be used to
deduce the volume growth test for recurrence Theorem 1.4. As in the previous
sections we assume that $\mathcal{E}$ is a regular Dirichlet form without
killing.

In order to prove the theorem we employ some abstract results from [9], to
which we also refer to for the following facts about extended Dirichlet
spaces. Recall that the extended Dirichlet space $D(\mathcal{E}_{\rm e})$ of
$\mathcal{E}$ is defined by

$D(\mathcal{E}_{\rm e})=\\{f\in L^{0}(m)\mid\text{ex. $\mathcal{E}$-Cauchy
sequence }(f_{n})\text{ in }D(\mathcal{E})\text{ s.t. }f_{n}\to
f\,m\text{-a.e.}\\}.$

A sequence $(f_{n})$ as in the definition above is called approximating
sequence for $f$. Let $f\in D(\mathcal{E}_{\rm e})$ and let $(f_{n})$ be an
approximating sequence. We define

$\mathcal{E}_{\rm e}(f)=\lim_{n\to\infty}\mathcal{E}(f_{n}).$

This is indeed independent of the choice of the sequence $(f_{n})$ such that
$\mathcal{E}_{\rm e}$ is an extension of $\mathcal{E}$. Moreover, it holds
that $D(\mathcal{E})=D(\mathcal{E}_{\rm e})\cap L^{2}(m)$. We need the
following observation, which shows that our extension of the form via the
measure-valued quadratic form $\Gamma$ is compatible with the extended
Dirichlet space.

######  Lemma 5.2.

Let $f\in D(\mathcal{E}_{\rm e})\cap L^{\infty}(m)$. Then $f\in
D(\mathcal{E})_{\rm loc}^{*}$ and

$\mathcal{E}_{\rm e}(f)=\int_{X}d\Gamma(f).$

###### Proof.

Let $f\in D(\mathcal{E}_{\rm e})\cap L^{\infty}(m)$. We first show $f\in
D(\mathcal{E})_{\rm loc}$. For a compact $K\subseteq X$ we choose $\varphi\in
D(\mathcal{E})\cap C_{c}(X)$ with $0\leq\varphi\leq 1$ and $\varphi=1$ on $K$.
According to [6, Exercise 1.4.1] such a function always exists. Then $\varphi
f=f$ on $K$ and $\varphi f\in L^{2}(m)$. Since $D(\mathcal{E}_{\rm e})\cap
L^{\infty}(m)$ is an algebra and $D(\mathcal{E}_{\rm e})\cap
L^{2}(m)=D(\mathcal{E})$, we obtain $\varphi f\in D(\mathcal{E})$. This shows
$f\in D(\mathcal{E})_{\rm loc}$.

Now let $(f_{n})$ be an approximating sequence for $f$. Without loss of
generality $\lVert f_{n}\rVert_{\infty}\leq\lVert f\rVert_{\infty}$ and
according to [6, Theorem 2.1.7] we can additionally assume $f_{n}\to f$ q.e.
(recall that we always choose quasi-continuous representatives). Since the
jump measure does not charge sets of capacity $0$, Fatou’s lemma yields

$\displaystyle\int_{X\times X\setminus d}(f(x)-f(y))^{2}dJ(x,y)$
$\displaystyle\leq\liminf_{n\to\infty}\int_{X\times X\setminus
d}(f_{n}(x)-f_{n}(y))^{2}dJ(x,y)$
$\displaystyle\leq\liminf_{n\to\infty}\mathcal{E}(f_{n})$
$\displaystyle=\mathcal{E}_{\rm e}(f)<\infty.$

In particular, this implies $f\in D(\mathcal{E})_{\rm loc}^{*}$.

We will show $\int_{X}d\Gamma(f_{n})\to\int_{X}d\Gamma(f)$ as this yields

$\int_{X}d\Gamma(f)=\lim_{n\to\infty}\int_{X}d\Gamma(f_{n})=\lim_{n\to\infty}\mathcal{E}(f_{n})=\mathcal{E}_{\rm
e}(f).$

According to [14, Theorem 3.1], for $\varphi\in D(\mathcal{E})\cap C_{c}(X)$
with $0\leq\varphi\leq 1$ and $g\in D(\mathcal{E}_{\rm e})\cap L^{\infty}(m)$
we have (using $\varphi g,\varphi g^{2}\in D(\mathcal{E})\cap L^{\infty}(m)$,
see first part of the proof)

$\mathcal{E}(\varphi g)-\mathcal{E}(\varphi g^{2},\varphi)\leq\mathcal{E}_{\rm
e}(g).$

Using the Leibniz rule for $\Gamma^{(c)}$ and the discrete Leibniz rule for
the integration with respect to $J$ we obtain

$\int_{X}\varphi^{2}d\Gamma^{(c)}(g)+\int_{X\times X\setminus
d}\varphi(x)\varphi(y)(g(x)-g(y))^{2}dJ(x,y)=\mathcal{E}(\varphi
g)-\mathcal{E}(\varphi g^{2},\varphi).$

Letting $\varphi\nearrow 1$ this implies

$\int_{X}d\Gamma(g)\leq\mathcal{E}_{\rm e}(g).$

In particular, since the LHS is a quadratic form in $g$, for $(g_{n})$ in
$D(\mathcal{E}_{\rm e})\cap L^{\infty}(m)$ the convergence $g_{n}\to g$ with
respect to $\mathcal{E}_{\rm e}$ implies
$\int_{X}d\Gamma(g_{n})\to\int_{X}d\Gamma(g)$.

Since $(f_{n})$ is an approximating sequence for $f$, the lower semicontinuity
of $\mathcal{E}_{\rm e}$ with respect to $m$-a.e. convergence, see e.g. [14,
Lemma 2.3], yields

$\mathcal{E}_{\rm e}(f-f_{n})\leq\liminf_{m\to\infty}\mathcal{E}_{\rm
e}(f_{m}-f_{n})=\liminf_{m\to\infty}\mathcal{E}(f_{m}-f_{n})\to 0,\text{ as
}n\to\infty.$

Hence, we obtain the claimed convergence of $\Gamma(f_{n})$ to $\Gamma(f)$. ∎

###### Proof of the volume growth test for recurrence, Theorem 1.4.

According to [9, Theorem 1 and Proposition 3] it suffices to show that any
non-negative $h\in D(\mathcal{E}_{\rm e})\cap L^{\infty}(m)$ with
$\mathcal{E}_{\rm e}(h,\varphi)\geq 0$ for all nonnegative $\varphi\in
D(\mathcal{E})$ is constant (such functions are called excessive). By Lemma
5.2 such a function $h$ satisfies $h\in D(\mathcal{E})_{\rm loc}^{*}$ and

$\int_{X}d\Gamma(h,\varphi)\geq 0$

for all non-negative $\varphi\in D(\mathcal{E})_{c}$. Now assume without loss
of generality $\lVert h\rVert_{\infty}\leq 1$. Since $\Gamma(1)=0$, the
function $f=1-h$ is nonnegative and subharmonic in our sense. Moreover,
$\lVert f1_{B_{r}}\rVert_{p}^{p}\leq\lVert f\rVert_{\infty}^{p}m(B_{r}).$
Hence, the assumption on $m(B_{r})$ and Theorem 1.3 yield that $f$ is
constant. ∎

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