# Extremal functions for the second-order Sobolev inequality on groups of
polynomial growth

Bobo Hua, Ruowei Li and Florentin Münch  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>Ruowei Li: School of
Mathematical Sciences, Fudan University, Shanghai 200433, China; Shanghai
Center for Mathematical Sciences, Jiangwan Campus, Fudan University, No. 2005
Songhu Road, Shanghai 200438, China<EMAIL_ADDRESS>Florentin Münch: MPI
MiS Leipzig, 04103 Leipzig, Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs
of groups of polynomial growth. We use the discrete Concentration-Compactness
principle to prove the existence of extremal functions for best constants in
supercritical cases. As applications, we get the existence of positive ground
state solutions to the $p$-biharmonic equations and the Lane-Emden systems.

##  1\. Introduction

Sobolev inequalities are important in the studies of partial differential
equations and Riemannian geometry etc. For $N,\,k,\,p\geq
1,\,kp<N,\frac{1}{\,p^{\star}}=\frac{1}{p}-\frac{k}{N}$, we have the classical
Sobolev inequality,

(1.1) ${\displaystyle\lVert u\rVert_{p^{\star}}\leq C_{p}\lVert
u\rVert_{D^{k,p}},\;\forall u\in
D_{0}^{k,p}\left(\mathbb{R}^{\mathit{N}}\right),}$

where $C_{p}$ is a constant depending on $N$ and $p$, we omit the dependence
of constants $N$ for convenience, and
$D_{0}^{k,p}\left(\mathbb{R}^{\mathit{N}}\right)$ denotes the completion of
$C_{0}^{\infty}\left(\mathbb{R}^{\mathit{N}}\right)$ in the norm $\lVert
u\rVert_{D^{k,p}}^{p}\coloneqq\sum\limits_{|\alpha|=k}\int\lvert
D^{\alpha}u\rvert^{p}\mathbf{\mathrm{d}}x$. Define

$D^{k,p}\left(\mathbb{R}^{\mathit{N}}\right)\coloneqq\left\\{u\in
L^{\frac{Np}{N-kp}}(\mathbb{R}^{\mathit{N}}):\lVert
u\rVert_{D^{k,p}}<\infty\right\\},$

and it is well-known that
$D_{0}^{k,p}\left(\mathbb{R}^{\mathit{N}}\right)=D^{k,p}\left(\mathbb{R}^{\mathit{N}}\right)$.
It is sufficient to establish the inequality (1.1) for $k=1$ as its validity
for higher $k$ can be obtained by induction on $k$ [2].

Whether the best constant is attained by some $u\in
D_{0}^{k,p}\left(\mathbb{R}^{\mathit{N}}\right)$, which is called the extremal
function, has been intensively investigated in the literature. When $k=1,$
$p=1$ the best constant is the isoperimetric constant and the extremal
function is the characteristic function of a ball [23, 24]. For $k=1,p>1$, the
best constants and extremal functions were obtained in [4, 70, 71, 61, 15].
For $k>1,$ P. L. Lions [44, 45, 46, 47] established the Concentration-
Compactness method, which provided a new idea for proving the existence of
extremal functions. The general idea is as follows. The best constant in the
Sobolev inequality (1.1) is given by

$K:=\inf_{\begin{subarray}{c}u\in D^{k,p}(\mathbb{R}^{N})\\\ \lVert
u\rVert_{p^{\star}}=1\end{subarray}}\lVert u\rVert_{D^{k,p}}^{p}>0.$

Take a minimizing sequence $\\{u_{n}\\}$ and regard
$\left\\{|u_{n}|^{p^{\star}}\mathrm{dx}\right\\}$ as a sequence of probability
measures. He proved in [44, 46] that there are three cases of the limit of the
sequence: compactness, vanishing and dichotomy. Vanishing and dichotomy are
ruled out by the rescaling trick and subadditivity inequality. Therefore, the
extremal function exists by the compactness. Since the Concentration-
Compactness principle requires weak convergence
$u_{n}\xrightharpoonup{}u\;\hphantom{}\text{in}\;D^{k,p}(\mathbb{R}^{\mathit{N}})$,
this method does not apply the case of $p=1$. And Lieb proved the existence of
extremal functions by a compactness technique [42, Lemma 2.7] which is induced
by the Brézis-Lieb lemma [10].

In recent years, people paid attention to the analysis on discrete groups.
Since the Sobolev inequalities are useful analytical tools, they have been
extended to the discrete setting [14, 58]. For finite graphs, Sobolev
inequalities and sharp constants have been obtained by [55, 56, 67, 69, 80,
81]. In this article, we generalize our previous results on the existence of
extremal functions for the first-order Sobolev inequality [33] to higher order
inequalities on Cayley graphs of groups of polynomial growth.

Let $(G,S)$ be a Cayley graph of a group $G$ with a finite symmetric
generating set $S$, i.e. $S=S^{-1}.$ There is a natural metric on $(G,S)$
called the word metric, denoted by $d^{S}$. Let $B_{p}^{S}(n):={\left\\{x\in
G\mid d^{S}(p,x)\leq n\right\\}}$ denote the closed ball of radius $n$
centered at $p\in G$ and denote $\mid B_{p}^{S}(n)\mid:=\sharp B_{p}^{S}(n)$
as the volume (i.e. cardinality) of the set $B_{p}^{S}(n)$. When $e$ is the
unit element of $G$, the volume $\beta_{S}(n):=\mid B_{e}^{S}(n)\mid$ of
$B_{e}^{S}(n)$ is called the growth function of the group, see [49, 50, 78,
28, 26, 27, 72]. A group $G$ is called of polynomial growth, or of polynomial
volume growth, if $\beta_{S}(n)\leq Cn^{A}$, for any $n\geq 1$ and some $A>0$,
which is independent of the choice of the generating set $S$ since the metrics
$d^{S}$ and $d^{S_{1}}$ are bi-Lipschitz equivalent for different finite
generating sets $S$ and $S_{1}$. By Gromov’s theorem and Bass’ volume growth
estimate of nilpotent groups [6], for any group $G$ of polynomial growth there
are constants $C_{1}(S)$, $C_{2}(S)$ depending on $S$ and $N\in\mathbb{N}$
such that for any $n\geq 1$,

$C_{1}(S)n^{N}\leq\beta_{S}(n)\leq C_{2}(S)n^{N},$

where the integer $N$ is called the homogeneous dimension or the growth degree
of $G$. Since $N$ is sort of dimensional constant of $G$, we always omit the
dependence of $N$ in various constants.

In this paper, we consider the Cayley graph $(G,S)$ of a group of polynomial
growth with the homogeneous dimension $N\geq 3$. We denote by $\ell^{p}(G)$
the $\ell^{p}$-summable functions on $G$ and by $D_{0}^{k,p}(G)$ $(k=1,2)$
(resp. $\widetilde{D}_{0}^{2,p}(G)$ ) the completion of finitely supported
functions in the $D^{k,p}$ (resp. $\widetilde{D}^{2,p}$ ) norm, where $\lVert
u\rVert_{D^{1,p}(G)}\coloneqq\lVert\nabla u\rVert_{\ell^{p}(G)}$, $\lVert
u\rVert_{D^{2,p}(G)}\coloneqq\lVert\Delta u\rVert_{\ell^{p}(G)}$ and $\lVert
u\rVert_{\widetilde{D}^{2,p}(G)}\coloneqq\lVert\nabla^{2}u\rVert_{\ell^{p}(G)}$,
see Section 2 for details. Using Dungey’s result [22, Theorem 1], we prove the
second-order norm defined by the Hessian and Laplace operators are equivalent
on a nilpotent group $G$, hence $D_{0}^{2,p}(G)=\widetilde{D}_{0}^{2,p}(G)$,
see Lemma 6. Analogous to the continuous setting, set

$D^{k,p}(G)\coloneqq\left\\{u\in\ell^{\frac{Np}{N-kp}}(G):\lVert
u\rVert_{D^{k,p}(G)}<\infty\right\\},$
$\widetilde{D}^{2,p}(G)\coloneqq\left\\{u\in\ell^{\frac{Np}{N-2p}}(G):\lVert
u\rVert_{\widetilde{D}^{2,p}(G)}<\infty\right\\}.$

For $\mathbb{R}^{N}$, the equivalence
$D_{0}^{k,p}(\mathbb{R}^{N})=D^{k,p}(\mathbb{R}^{N})$ follows from the
approximation of $f\in D^{k,p}(\mathbb{R}^{N})$ by compact supported functions
using nice cutoff functions. Inspired by the continuous setting, for a
concrete example, integer lattice graph $\mathbb{Z}^{N}$, the equivalence
$D_{0}^{k,p}(\mathbb{Z}^{N})=D^{k,p}(\mathbb{Z}^{N})$ can be proved by
constructing cutoff functions. We prove that any $D^{1,p}(\mathbb{Z}^{N})$
function can be approximated by a "linear" cutoff function of the logarithmic
function (Theorem 7) and Cosco, Nakajima and Schweiger give a more explicit
estimate in [17]. However, the "linear" cutoff function does not work for
$D^{2,p}(\mathbb{Z}^{N})$ since the Laplacian is "bad" when it comes to the
sharp corner. Different from the continuous setting which can be mollified
locally, we find a way to fix this by taking a "close to linear" third-order
polynomial function being smooth at the sharp corner. Moreover, the Euclidean
distance is necessary for cutoff functions since its Laplacian is decreasing
while the combinatorial distance is not. Then using the discrete chain rule
(Lemma 5) we can get a good estimate of the second-order norm, and finally
construct desired cut-off functions, see Theorem 11. Then we prove the
equivalence $D_{0}^{k,p}(G)=D^{k,p}(G)$ on general Cayley graphs $(G,S)$, see
Theorem 8 and Theorem 12. First $D_{0}^{k,p}(G)\subseteq D^{k,p}(G)$ follows
from Sobolev inequalities, then we prove the converse direction via the
parabolicity theory on graphs [63, Corollary 2.6] for $k=1$ and the functional
analysis method by checking the Laplace operator is an isometry from
$D_{0}^{2,p}(G)$ (also $D^{2,p}(G)$) to $\ell^{p}(G)$.

For special case $p=2$, we extend the results to the Cayley graph $(G,S)$
satisfying $\beta_{S}(n)\geq C(S)n^{N},\ \forall n\geq 1$. We prove the Hodge
decomposition theorem on $1$-forms on edges (Theorem 18) and use it to prove
$D_{0}^{1,2}=D^{1,2}$ on $G$ (Corollary 19). Moreover, we prove
$D_{0}^{2,2}=D^{2,2}$ on the groups satisfying the second-order Sobolev
inequality including polynomial growth groups and non-amenable groups.

Using the boundedness of Riesz transforms [22] and the functional calculus in
Banach spaces [30] we get the following discrete second-order Sobolev
inequality by induction on the first-order Sobolev inequality:

(1.2) $\lVert u\rVert_{\ell^{q}}\leq C_{p,q}\lVert u\rVert_{D^{2,p}},\;\forall
u\in D^{2,p}(G),$

where $N\geq 3,1<p<\dfrac{N}{2},q=p^{\ast\ast}\coloneqq\dfrac{Np}{N-2p}$, see
Theorem 10. Since $\ell^{p}(G)$ embeds into $\ell^{q}(G)$ for any $q>p$, one
verifies that the inequality (1.2) holds for $q\geq p^{\ast\ast}$. Recalling
the continuous setting, it is called subcritical for $q<p^{\ast\ast}$,
critical for $q=p^{\ast\ast}$ and supercritical for $q>p^{\ast\ast}$ for
Sobolev inequalities. Therefore, (1.2) holds in both critical and
supercritical cases.

The optimal constant in the Sobolev inequality (1.2) is given by

(1.3) $K:=\inf_{\begin{subarray}{c}u\in D^{2,p}(G)\\\ \lVert
u\rVert_{q}=1\end{subarray}}\lVert u\rVert_{D^{2,p}}^{p}.$

In order to prove that the infimum is achieved, we consider a minimizing
sequence $\\{u_{n}\\}\subset D^{2,p}(G)$ satisfying

(1.4) $\lVert u_{n}\rVert_{q}=1,\lVert u_{n}\rVert_{D^{2,p}}^{p}\to
K,n\to\infty.$

We want to prove $u_{n}\to u$ strongly in $D^{2,p}(G)$, which yields that $u$
is a minimizer.

We prove the following main results.

######  Theorem 1.

For $N\geq 3,1<p<\frac{N}{2}$, $q>p^{\ast\ast}=\frac{Np}{N-2p}$, let
$\left\\{u_{n}\right\\}\subset D^{2,p}(G)$ be a minimizing sequence satisfying
(1.4). Then there exists a sequence $\\{x_{n}\\}\subset G$ and $v\in
D^{2,p}(G)$ such that the sequence after translation
$\left\\{v_{n}(x):=u_{n}(x_{n}x)\right\\}$ contains a convergent subsequence
that converges to $v$ in $D^{2,p}(G)$. And $v$ is a minimizer for $K$.

######  Remark.

(1) This result implies that best constant can be obtained in the
supercritical case.

(2) If we define the second-order norm by the Hessian operator, i.e.
$\tilde{D}^{2,p}$ (see Section 2), then the second-order Sobolev inequalities
also hold, and the strong convergence and existence of minimizer in
$\tilde{D}_{0}^{2,p}(G)$ follows from a similar argument.

(3) In particular, for $p=1$, we can get the same results in
$\tilde{D}^{2,1}(\mathbb{Z}^{N})$, see Section 7.

We will provide two proofs for Theorem 1. In the continuous setting, Lions
proved the existence of extremal functions by Concentration-Compactness
principle [46, Lemma I.1.] and a rescaling trick [46, Theorem I.1, (17)]. And
Lieb in [42] used a compactness technique and the rearrangement inequalities.
Following Lions, the main idea of proof I is to prove a discrete analog of
Concentration-Compactness principle, see Lemma 15. However, we don’t know
proper notion of rescaling and rearrangement tricks on graphs to exclude the
vanishing case of the limit function. Inspired by [35], for the supercritical
case, we prove that the translation sequence has a uniform positive lower
bound at the unit element, see Lemma 16, which excludes the vanishing case.
The idea of proof II is based on a compactness technique by Lieb [42, Lemma
2.7] and the nonvanishing of the limit of translation sequence.

As applications, we get the existence of positive ground state solutions for
the discrete nonlinear $p$-biharmonic equations and the Lane–Emden systems in
supercritical cases. They have certain physical backgrounds, such as traveling
waves in a suspension bridge [21] and the static deflection of an elastic
plate [1], and have been well studied in continuous setting, see for example
[43, 13, 74, 64, 66, 65, 12, 59, 52, 51, 68, 39, 48, 25, 19, 73, 75, 31, 41,
54, 53, 57, 29, 36, 37, 38] and references therein. For discrete setting,
there are few results of the fourth order nonlinear equations [32, 79].

First by Theorem 1, we can get the existence of positive solutions to Euler-
Lagrange equation of (1.2) as follows.

######  Corollary 2.

For $N\geq 3,1<p<\frac{N}{2},q>p^{\ast\ast}$, there is a positive ground state
solution of the nonlinear $p$-biharmonic equation

(1.5) $\Delta\left(\mid\Delta u\mid^{p-2}\Delta u\right)-\mid
u\mid^{q-2}u=0,\;\forall u\in D^{2,p}(G),$

where $\Delta$ is the Laplace on graphs defined as $\Delta
u(x)\coloneqq\sum\limits_{y\sim x}(u(y)-u(x))$.

Consider the Lane-Emden system

(1.6) $\begin{cases}\begin{array}[]{cc}-\Delta u=\lvert
v\rvert^{p^{\prime}-2}v\\\ -\Delta v=\lvert u\rvert^{q-2}u\\\ u\in
D^{2,p}(G),v\in D^{2,q^{\prime}}(G),\end{array}\end{cases}$

where $N\geq 3$, $p,q>1$, $p^{\prime}=\frac{p}{p-1}$,
$q^{\prime}=\frac{q}{q-1}$, $(p^{\prime},q)$ lies on the critical hyperbola,
that is,

$\frac{1}{p^{\prime}}+\frac{1}{q}=\frac{N-2}{N}.$

Indeed, $(u,v)$ is a solution of (1.6) if and only if $u$ is a solution of
(1.5) with $v\coloneqq-\mid\Delta u\mid^{p-2}\Delta u$ inB critical cases.
This relation is sometimes called reduction-by-inversion, see [16, Lemma 2.1]
[9, 20, 62] for continuous setting. We can also prove the equivalence in
supercritical cases for discrete setting, see Lemma 17. Hence, by Corollary 2
we have a positive solution of the Lane-Emden system.

######  Corollary 3.

For $N\geq 3$, $p,q>1$, $\frac{1}{p^{\prime}}+\frac{1}{q}<\frac{N-2}{N}$,
there is a pair of positive solution $(u,v)$ for the Lane-Emden system (1.6).

######  Conjecture 4.

According to the results in continuous cases [46, Corollary I.2] [43, 13, 74,
64, 12, 59, 52, 68, 19, 54, 75, 73], we conjecture that (1.5) and (1.6) have
positive solutions in critical cases and the non-negative solutions in
subcritical cases are trivial.

The paper is organized as follows. In Section 2, we recall some basic facts
and prove some useful lemmas. In Section 3, we study the important equivalence
of Sobolev spaces and prove the discrete second-order Sobolev inequality using
the boundedness of Riesz transforms and the functional calculus. In Section 4,
we introduce the Brézis-Lieb lemma and prove the Concentration-Compactness
principle on $G$. In Section 5, we prove a key lemma to exclude the vanishing
case and give two proofs for Theorem 1. As applications, we get the existence
results for $p$-biharmonic equations and Lane-Emden systems in Section 6\. In
Section 7, we prove the Hodge decomposition theorem on $1$-forms on edges for
$p=2$ and study the existence of extremal functions for $p=1$.

##  2\. Preliminaries

Let $G$ be a countable group. It is called finitely generated if it has a
finite generating set $S$. We always assume that the generating set $S$ is
symmetric, i.e. $S=S^{-1}$. The Cayley graph of $(G,S)$ is a graph structure
$(V,E)$ with the set of vertices $V=G$ and the set of edges $E$ where for any
$x,y\text{$\in$}G,xy\in E$ (also denoted by $x\sim y$) if $x=ys$ for some
$s\in S$. The Cayley graph of $(G,S)$ is endowed with a natural metric, called
the word metric [11]: For any $x,y\in G$, the distance between them is defined
as the length of the shortest path connecting $x$ and $y$ by assigning each
edge of length one,

$d^{S}(x,y)=\text{inf}{\left\\{k:x=x_{0}\sim\cdot\cdot\cdot\sim
x_{k}=y\right\\}.}$

One easily sees that for two generating sets $S$ and $S_{1}$ the metrics
$d^{S}$ and $d^{S_{1}}$ are bi-Lipschitz equivalent, i.e. there exist two
constants $C_{1}(S,S_{1}),C_{2}(S,S_{1})$ such that for any $x,y\in G$

$C_{1}(S,S_{1})d^{S_{1}}(x,y)\leq d^{S}(x,y)\leq
C_{2}(S,S_{1})d^{S_{1}}(x,y).$

Let $B_{p}^{S}(n):={\left\\{x\in G\mid d^{S}(p,x)\leq n\right\\}}$ denote the
closed ball of radius $n$ centered at $p\in G$. By the group structure, it is
obvious that $\mid B_{p}^{S}(n)\mid=\mid B_{q}^{S}(n)\mid$, for any $p,q\in
G$. The growth function of $(G,S)$ is defined as $\beta_{S}(n):=\mid
B_{e}^{S}(n)\mid$ where $e$ is the unit element of $G$. A group $G$ is called
of polynomial growth if there exists a finite generating set $S$ such that
$\beta_{S}(n)\leq Cn^{A}$ for some $C,A>0$ and any $n\geq 1$. One checks that
this definition is independent of the choice of the generating set $S$. Thus,
the polynomial growth is indeed a property of the group $G$. In this paper, we
consider the Cayley graph $(G,S)$ of a group of polynomial growth

$C_{1}(S)n^{N}\leq\beta_{S}(n)\leq C_{2}(S)n^{N},$

for some $N\in\mathbb{N}$ and any $n\geq 1$, where $N$ is called the
homogenous dimension of $G$.

We denote by $C(G)$ the space of functions on $G$. The support of $u\in C(G)$
is defined as $\textrm{supp}(u)\coloneqq\\{x\in G:u(x)\neq 0\\}$. Let
$C_{0}(G)$ be the set of all functions with finite support. For any $u\in
C(G)$, the $\ell^{p}$ norm of $u$ is defined as

$\lVert u\rVert_{\ell^{p}(G)}\coloneqq\begin{cases}\left(\underset{x\in
G}{\sum}\lvert u(x)\rvert^{p}\right)^{1/p}&\text{$0<p<\infty,$}\\\
\underset{x\in G}{\sup}\lvert u(x)\rvert&p=\infty,\end{cases}$

and we shall write $\lVert u\rVert_{{\ell^{p}(G)}}$ as $\lVert u\rVert_{{p}}$
for convenience. The $\ell^{p}(G)$ space is defined as

$\ell^{p}(G)\coloneqq\left\\{u\in C(G):\lVert
u\rVert_{{\ell^{p}(G)}}<\infty\right\\}.$

For any $u\in C(G)$, the difference operator is defined as for any $x\sim y$

$\nabla_{xy}u=u(y)-u(x).$

In particular, for the Cayley graph $(G,S)$ with symmetric generating set
$S\coloneqq\left\\{s_{1},s_{2},\cdots,s_{m}\right\\}$, we can define the
difference in $i$-th direction as

$\partial_{i}u(x)\coloneqq u(xs_{i})-u(x).$

Let

$\lvert\nabla u(x)\rvert_{p}\coloneqq\left(\sum\limits_{y\sim
x}\lvert\nabla_{xy}u\rvert^{p}\right)^{1/p}$

be the $p$-norm of the gradient of $u$ at $x$.

We define the Laplace operator as

$\Delta u(x)\coloneqq\sum\limits_{y\sim x}(u(y)-u(x)).$

We write $p$-norm of the Hessian of $u$ as

$\lVert\nabla^{2}u\rVert_{p}\coloneqq\left(\stackrel{{\scriptstyle[}}{{i}},j=1]{m}{\sum}\underset{x\in
G}{\sum}\lvert\partial_{j}\partial_{i}u(x)\lvert^{p}\right)^{1/p}=\left(\stackrel{{\scriptstyle[}}{{i}},j=1]{m}{\sum}\lVert\partial_{j}\partial_{i}u(x)\lVert_{p}^{p}\right)^{1/p}.$

The $D^{k,p}$ ($k=1,2$) norms of $u$ are given by

$\lVert u\rVert_{D^{1,p}(G)}\coloneqq\lVert\lvert\nabla
u\rvert_{p}^{p}\rVert_{\ell^{1}(G)}^{1/p}=\left(\sum\limits_{x\in
G}\sum\limits_{y\sim x}\lvert\nabla_{xy}u\rvert^{p}\right)^{1/p},$ $\lVert
u\rVert_{D^{2,p}(G)}\coloneqq\lVert\Delta
u\rVert_{\ell^{p}(G)}=\left(\sum\limits_{x\in G}\lvert\sum\limits_{y\sim
x}\nabla_{xy}u\rvert^{p}\right)^{1/p}.$

We use the Hessian operator to define $\widetilde{D}^{2,p}$ norm as

$\lVert u\rVert_{\widetilde{D}^{2,p}(G)}\coloneqq\lVert\nabla^{2}u\rVert_{p}.$

Since $G$ is a regular graph, that is, the degree
$d_{x}\coloneqq\sharp\left\\{y:y\sim x\right\\}$ is constant for $x\in G$, we
have

$\lVert\nabla u\lVert_{p}\coloneqq\lVert\lvert\nabla
u(x)\rvert_{1}\lVert_{p}\sim\lVert\lvert\nabla
u(x)\rvert_{p}^{p}\lVert_{1}^{1/p}=\lVert u\rVert_{D^{1,p}}.$

In this paper, we use $a\lesssim b$ to denote $a\leq Cb$ for some $C>0$ and
$a\sim b$ to denote $a\lesssim b\lesssim a$.

We define $D_{0}^{k,p}(G)$ ($k=1,2$) and $\widetilde{D}_{0}^{2,p}(G)$ as the
completion of $C_{0}\left(G\right)$ in $D^{k,p}$ norm and
$\widetilde{D}^{2,p}$ norm respectively. And define

$D^{k,p}(G)\coloneqq\left\\{u\in\ell^{\frac{Np}{N-kp}}(G):\lVert
u\rVert_{D^{k,p}(G)}<\infty\right\\},$
$\widetilde{D}^{2,p}(G)\coloneqq\left\\{u\in\ell^{\frac{Np}{N-2p}}(G):\lVert
u\rVert_{\widetilde{D}^{2,p}(G)}<\infty\right\\}.$

Let $c_{0}(G)$ be the completion of $C_{0}\left(G\right)$ in $\ell^{\infty}$
norm. It is well-known that $\ell^{1}(G)=(c_{0}(G))^{\ast}$. We set

$\lVert\mu\rVert\coloneqq\sup\limits_{u\in c_{0}(G),\lVert
u\rVert_{\infty}=1}\langle\mu,u\rangle,\qquad\forall\mu\in\ell^{1}(G).$

By definition,

$\mu_{n}\xrightharpoonup{w^{\ast}}\mu\;\text{in}\;\ell^{1}(G)\;\text{if and
only if}\;\langle\mu_{n},u\rangle\longrightarrow\langle\mu,u\rangle,\forall
u\in c_{0}(G).$

In the proof, we will use the following results, see [18].

######  Fact.

(a)Every bounded sequence of $\ell^{1}(G)$ contains a $w^{\ast}$-convergent
subsequence.
(b)If $\mu_{n}\xrightharpoonup{w^{\ast}}\mu$ in $\ell^{1}(G)$, then
${\mu_{n}}$ is bounded and

$\lVert\mu\rVert\leq\varliminf\limits_{n\to\infty}\lVert\mu_{n}\rVert.$

(c)If $\mu\in\ell^{1+}(G):=\left\\{\mu\in\ell^{1}(G):\mu\geq 0\right\\}$, then

$\lVert\mu\rVert=\langle\mu,1\rangle.$

Next we give a discrete chain rule.

######  Lemma 5.

For a weighted graph $G=(V,w,m)$, where $w_{xy}$, $m(x)$ denote the weights
assigned to the edge $xy\in E$ and to the vertex $v$, respectively. Let
$\phi\in C^{2}(\mathbb{R})$, $f\in C(V)$, then for some $\xi\in[m,M]$,

$\Delta\phi(f)(x)=\phi^{\prime}(f(x))\Delta
f(x)+\phi^{\prime\prime}(\xi)\Gamma f(x),$

where $\Delta f(x)\coloneqq\underset{y\sim
x}{\sum}\frac{w_{xy}}{m(x)}\left(f\left(y\right)-f(x)\right)$, $\Gamma
f(x)\coloneqq\frac{1}{2}\underset{y\sim x}{\sum}\frac{w_{xy}}{m(x)}\mid
f(y)-f(x)\mid^{2}$ and $m\coloneqq\underset{y\in B_{x}(1)}{\textrm{min}}f(y)$,
$M\coloneqq\underset{y\in B_{x}(1)}{\text{max}}f(y)$.

###### Proof.

With $s=f(x)$ and $t_{y}=f(y)-f(x)$, Taylor’s theorem gives

$\phi(s+t_{y})=\phi(s)+t_{y}\phi^{\prime}(s)+\frac{1}{2}t_{y}^{2}\phi^{\prime\prime}(\xi_{y})$

for some $\xi_{y}\in[s\wedge(s+t_{y}),s\vee(s+t_{y})]$. Summing up gives

$\displaystyle\Delta\phi(f)(x)$ $\displaystyle=\underset{y\sim
x}{\sum}\frac{w_{xy}}{m(x)}\left(\phi\left(s+t_{y}\right)-\phi(s)\right)$
$\displaystyle=\underset{y\sim
x}{\sum}\frac{w_{xy}}{m(x)}\left(t_{y}\phi^{\prime}\left(s\right)+\frac{1}{2}t_{y}^{2}\phi^{\prime\prime}(\xi_{y})\right)$
$\displaystyle=\underset{y\sim
x}{\sum}\frac{w_{xy}}{m(x)}\left(t_{y}\phi^{\prime}\left(s\right)+\frac{1}{2}t_{y}^{2}\phi^{\prime\prime}(\xi)\right)$
$\displaystyle=\phi^{\prime}(f(x))\Delta f(x)+\phi^{\prime\prime}(\xi)\Gamma
f(x),$

where in the third identity, we can choose some $\xi$ between minimum and
maximum of the $\xi_{y}$ by continuity of $\phi^{\prime\prime}$, that is,

$m\leq\xi\leq M.$

This finishes the proof. ∎

##  3\. Sobolev spaces and second-order Sobolev inequalities

In this section, inspired by the results in the continuous setting, we study
some important properties of Sobolev spaces defined in Section 2\. By
induction on the discrete first-order Sobolev inequalities, we prove the
second-order Sobolev inequalities using the boundedness of Riesz transforms
and the functional calculus.

First, we introduce a key lemma of norm equivalence using Dungey’s result [22,
Theorem 1].

######  Lemma 6.

Let Cayley graph $(G,S)$ is a nilpotent group of polynomial growth with
symmetric generating set
$S\coloneqq\left\\{s_{1},s_{2},\cdots,s_{m}\right\\}$, for $p>1,$ if $u\in
C_{0}(G)$, then

$\lVert u\rVert_{\widetilde{D}^{2,p}(G)}\sim\lVert u\rVert_{D^{2,p}(G)}.$

Moreover, $D_{0}^{2,p}(G)=$$\widetilde{D}_{0}^{2,p}(G)$.

###### Proof.

Since $-\Delta
u(x)=\frac{1}{2}\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}\partial_{s_{i}^{-1}}\partial_{s_{i}}u(x),$
then

$\displaystyle\lVert\Delta u\rVert_{p}$
$\displaystyle=\left(\underset{x}{\sum}\mid\frac{1}{2}\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}\partial_{s_{i}^{-1}}\partial_{s_{i}}u(x)\mid^{p}\right)^{1/p}$
$\displaystyle\lesssim\left(\underset{x}{\sum}\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}\mid\partial_{s_{i}^{-1}}\partial_{s_{i}}u(x)\mid^{p}\right)^{1/p}$
$\displaystyle\leq\left(\underset{x}{\sum}\stackrel{{\scriptstyle[}}{{i}},j=1]{m}{\sum}\mid\partial_{s_{j}}\partial_{s_{i}}u(x)\mid^{p}\right)^{1/p}=\lVert\nabla^{2}u\rVert_{p}.$

Using Dungey’s result in [22, Theorem 1], we know for any $i,j$,

(3.1)
$\lVert\partial_{s_{j}}\partial_{s_{i}}u(x)\rVert_{p}\lesssim\lVert\Delta
u\rVert_{p},$

which implies

$\lVert\nabla^{2}u\rVert_{p}\lesssim\lVert\Delta u\rVert_{p}.$

Hence the norms defined by the Laplace and Hessian operator are equivalent. So
that the spaces defined by the completion of finitely supported functions in
the equivalent norms coincide. ∎

######  Remark.

The assumption of nilpotent group is necessary, if $G$ is not nilpotent then,
(3.1) may fail, see [3, Section 1].

For a Cayley graph it is well known that if $\beta_{S}(n)\geq C(S)n^{N},\
\forall n\geq 1$ for $N\geq 3,$ then the first-order Sobolev inequality holds
[34, Theorem 3.6],

(3.2) ${\displaystyle\lVert u\rVert_{\ell^{q}}\leq C_{p,q}\lVert
u\rVert_{D^{1,p}},\;\forall u\in D_{0}^{1,p}(G)}$

where $1\leq p<N,q\geq p^{\ast}\coloneqq\dfrac{Np}{N-p}.$ In fact, this
follows from a standard trick and the isoperimetric estimate [77, Theorem
4.18].

We consider a concrete example of Cayley graphs of polynomial growth with the
homogeneous dimension $N$, the integer lattice graph $\mathbb{Z}^{N}$, which
serve as the discrete counterpart of $\mathbb{R}^{N}$. It consists of the set
of vertices $V=\mathbb{Z}^{N}$ and the set of edges

$E=\left\\{\left\\{x,y\right\\}:x,y\in\mathbb{Z}^{N},\mathop{\sum\limits_{i=1}^{N}\lvert
x_{i}-y_{i}\rvert=1}\right\\}.$

Analogous the continuous setting, we construct cutoff functions to prove the
results on $\mathbb{Z}^{N}$.

######  Theorem 7.

If $N\geq 3$, $1<p<N$, then
$D_{0}^{1,p}(\mathbb{Z}^{N})=D^{1,p}(\mathbb{Z}^{N}).$

###### Proof.

For any $u\in D_{0}^{1,p}(\mathbb{Z}^{N})$, there exists a sequence
$\left\\{u_{n}\right\\}\subset C_{0}\left(\mathbb{Z}^{N}\right)$ such that

$u_{n}\stackrel{{\scriptstyle}}{{\longrightarrow}}u\text{ in }D^{1,p}.$

And $u\in\ell^{\frac{Np}{N-p}}(\mathbb{Z}^{N})$ by the Sobolev inequality
(3.2). Hence, $D_{0}^{1,p}(\mathbb{Z}^{N})\subseteq D^{1,p}(\mathbb{Z}^{N})$.

In the other direction, the key is to find suitable cutoff functions
$\eta_{n}(x)\in C_{0}\left(\mathbb{Z}^{N}\right)$. For any $u\in
D^{1,p}(\mathbb{Z}^{N})$, set $u_{n}\coloneqq u\eta_{n}\in
C_{0}\left(\mathbb{Z}^{N}\right)$, then by H$\ddot{\text{o}}$lder inequalities

$\displaystyle\lVert u_{n}-u\rVert_{D^{1,p}(\mathbb{Z}^{N})}^{p}$
$\displaystyle=\sum\limits_{x\in\mathbb{Z}^{N}}\sum\limits_{y\sim
x}\lvert\nabla_{xy}(u\eta_{n})-\nabla_{xy}u\rvert^{p}$ (3.3)
$\displaystyle=\sum\limits_{x\in\mathbb{Z}^{N}}\sum\limits_{y\sim
x}\lvert\nabla_{xy}u\eta_{n}(y)+\nabla_{xy}\eta_{n}u(x)-\nabla_{xy}u\rvert^{p}$
$\displaystyle\lesssim\sum\limits_{x\in\mathbb{Z}^{N}}\lvert\nabla
u(x)\mid_{p}^{p}\underset{y\sim
x}{\text{max}}\mid\eta_{n}(y)-1\mid^{p}+\lVert\nabla\eta_{n}\rVert_{\ell^{N}}^{p}\lVert
u\rVert_{\ell^{p^{\ast}}}^{p},$

Hence, if the cutoff functions satisfy

$\eta_{n}\text{ is uniformly bounded,}$ (3.4) $\eta_{n}\to 1\text{ pointwise
on $\mathbb{Z}^{N}$}$

and

$\lVert\nabla\eta_{n}\rVert_{\ell^{N}(\mathbb{Z}^{N})}\rightarrow 0,$

then we can prove the other direction by the dominated convergence theorem.

Let $r>1$, and $R\gg r$ be large enough. Define

$\eta(x)\coloneqq 1\wedge\frac{\textrm{log$R$}-\text{log$\mid
x\mid$}}{\textrm{log$R$}-\text{log$r$}}\vee 0,$

where $\mid x\mid$ stands for the Euclidean distance. Then

$\displaystyle\lVert\nabla\eta\rVert_{\ell^{N}(\mathbb{Z}^{N})}^{N}$
$\displaystyle=\sum\limits_{r\leq\mid x\mid\leq R}\sum\limits_{y\sim
x}\lvert\nabla_{xy}\eta\rvert^{N}$
$\displaystyle\lesssim\left(\textrm{log$\frac{R}{r}$}\right)^{-N}\sum\limits_{r-1\leq\mid
x\mid\leq R+1}\sum\limits_{y\sim x}\lvert\textrm{log$\mid
x\mid$}-\text{log$\mid y\mid$}\rvert^{N}$
$\displaystyle\lesssim\left(\textrm{log$\frac{R}{r}$}\right)^{-N}\sum\limits_{r-1\leq\mid
x\mid\leq R+1}\lvert x\rvert^{-N},$

where the second inequality follows from the mean value theorem. In the
following, we can estimate the summation on $\mathbb{Z}^{N}$ by the integral
on $\mathbb{R}^{\mathit{N}}$,

$\displaystyle\sum\limits_{r-1\leq\mid x\mid\leq R+1}\lvert x\rvert^{-N}$
$\displaystyle\lesssim\sum\limits_{r-1\leq\mid x\mid\leq
R+1}\int\limits_{S_{x}(\frac{1}{2})}\lvert x\rvert^{-N}\text{d$t$}$
$\displaystyle\lesssim\sum\limits_{r-1\leq\mid x\mid\leq
R+1}\int\limits_{S_{x}(\frac{1}{2})}\lvert t\rvert^{-N}\text{d$t$}$
$\displaystyle\lesssim\int\limits_{\widetilde{B}(R+2)\setminus\widetilde{B}(r-2)}\lvert
t\rvert^{-N}\text{d$t$},$

where $S_{x}(\frac{1}{2})\coloneqq\left\\{t\in\mathbb{R}^{\mathit{N}}:\lvert
t_{i}-x_{i}\rvert<\frac{1}{2},1\leq i\leq N\right\\}$ is the Euclidean cube,
and $\widetilde{B}(r)$ is the Euclidean ball in $\mathbb{R}^{\mathit{N}}$ with
radius of $r$ and centered at the origin. Hence,

$\lVert\nabla\eta\rVert_{\ell^{N}(\mathbb{Z}^{N})}^{N}\lesssim\left(\textrm{log$\frac{R}{r}$}\right)^{-N}\text{log}\frac{R}{r}=O\left(\textrm{$\left(\text{log}\frac{R}{r}\right)$}^{1-N}\right).$

For fixed $r$, letting $R\to\infty$ when $N\geq 3$, we have that
$\lVert\nabla\eta\rVert_{N}\to 0$. Since $\eta(x)$ is uniformly bounded and
tends to $1$ pointwise, we prove $u\in D_{0}^{1,p}(\mathbb{Z}^{N})$ by (3.3).
∎

######  Remark.

For $D^{1,p}(\mathbb{Z}^{N})$, one can replace the Euclidean distance
$\mid\cdot\mid$ in the definition of cutoff function $\eta$ by any norm
$\parallel\cdot\parallel_{r}$ with $r>0$, and Cosco, Nakajima and Schweiger
give a more explicit estimate of the $p$-capacity [17].

For a Cayley graph $(G,S)$ of polynomial growth with the homogeneous dimension
$N$, it is difficult to construct desired cutoff functions. We find an
alternative method to prove the result by the parabolicity theory [63].

######  Theorem 8.

If $N\geq 3$, $1<p<N$, then $D_{0}^{1,p}(G)=D^{1,p}(G)$.

###### Proof.

By the same argument as Theorem 7, $D_{0}^{1,p}(G)\subseteq D^{1,p}(G)$
follows from the Sobolev inequalities (3.2).

In the other direction, by the parabolicity theory on graphs [63, Corollary
2.6] we know that $G$ is $N$-parabolic which implies the $N$-capacity is zero.
Hence, we can get a uniformly bounded sequence $\eta_{n}(x)$ with finite
support satisfying $\eta_{n}(x)\to 1$ pointwise and
$\lVert\nabla\eta_{n}\rVert_{\ell^{N}(G)}\rightarrow 0$. Set $u_{n}\coloneqq
u\eta_{n}\in C_{0}(G)$, then by H$\ddot{\text{o}}$lder inequalities

$\displaystyle\lVert u_{n}-u\rVert_{D^{1,p}(G)}^{p}$
$\displaystyle=\sum\limits_{x\in G}\sum\limits_{y\sim
x}\lvert\nabla_{xy}u\eta_{n}(y)+\nabla_{xy}\eta_{n}u(x)-\nabla_{xy}u\rvert^{p}$
$\displaystyle\lesssim\sum\limits_{x\in G}\lvert\nabla
u(x)\mid_{p}^{p}\underset{y\sim
x}{\text{max}}\mid\eta_{n}(y)-1\mid^{p}+\lVert\nabla\eta_{n}\rVert_{\ell^{N}}^{p}\lVert
u\rVert_{\ell^{p^{\ast}}}^{p}\to 0.$

That is, $D^{1,p}(G)\subseteq D_{0}^{1,p}(G)$. ∎

######  Remark.

For $N\geq 3,$ if the Cayley graph $(G,S)$ satisfies $\beta_{S}(n)\geq
C(S)n^{N},\ \forall n\geq 1$, then we can prove $D_{0}^{1,2}(G)=D^{1,2}(G)$
using the Hodge decomposition theorem on $1$-forms on edges, see Theorem 18
and Corollary 19.

By the properties of semigroup [5, Chapter 5], we have the following lemma.

######  Lemma 9.

For any $u\in\ell^{p}(G)$, $p\in(1,\infty)$,

$Mu\coloneqq\sideset{\frac{1}{\mid\varGamma(-\frac{1}{2})\mid}}{{}_{0}^{\infty}}{\int}\left(e^{t\Delta}u-u\right)t^{-\frac{3}{2}}\text{d}t,$

where $e^{t\Delta}$ is the semigroup of $\Delta$. Then $M$ is a bounded linear
operator in $\ell^{p}(G)$, and

$L^{\frac{1}{2}}u\coloneqq(-\Delta)^{\frac{1}{2}}u=Mu,\forall
u\in\ell^{p}(G).$

###### Proof.

Obviously, $M$ is well-defined and linear. And for any $u\in C_{0}(G),$

$Mu\sim\int_{0}^{1}\left(\int_{0}^{t}e^{s\Delta}\Delta u\text{
d$s$}\right)t^{-\frac{3}{2}}\text{d}t+\int_{1}^{\infty}\left(e^{t\Delta}u-u\right)t^{-\frac{3}{2}}\text{d}t.$

Then,

$\displaystyle\lVert Mu\lVert_{p}$
$\displaystyle\lesssim\int_{0}^{1}\left(\int_{0}^{t}\lVert
e^{s\Delta}\lVert_{p\rightarrow p}\lVert\Delta u\lVert_{p}\text{
d$s$}\right)t^{-\frac{3}{2}}\text{d}t+\int_{1}^{\infty}\left(\lVert
e^{t\Delta}\lVert_{p\rightarrow p}\lVert u\lVert_{p}+\lVert
u\lVert_{p}\right)t^{-\frac{3}{2}}\text{d}t$ $\displaystyle\lesssim\lVert
u\lVert_{p}\int_{0}^{1}t^{-\frac{1}{2}}\text{d}t+\lVert
u\lVert_{p}\int_{1}^{\infty}t^{-\frac{3}{2}}\text{d}t\lesssim\lVert
u\lVert_{p}.$

Hence, $M$ is a bounded operator in $\ell^{p}(G)$. And these definitions are
consistent for different $p$, i.e. two of them agree on their common domain
since the extensions of semigroup $e^{t\Delta}$ are consistent in different
$\ell^{p}(G)$, see [40].

By the spectral mapping theorem in Banach spaces [30, Proposition 3.1.1], we
know that $L^{\frac{1}{2}}$ is a bounded operator in $\ell^{p}(G)$. For $p=2$,
using the functional calculus in Hilbert spaces, for any $u\in\ell^{2}(G),$ we
have

$L^{\frac{1}{2}}u=Mu,$

which is also true for any $u\in C_{0}(G)$. Hence by the Bounded Linear
Transformation theorem [60, Theorem I.7], we get

$L^{\frac{1}{2}}u=Mu,\>\forall u\in\ell^{p}(G).$

∎

Now we are ready to prove the discrete second-order Sobolev inequality.

######  Theorem 10.

For $N\geq 3,1<p<\dfrac{N}{2},p^{\ast\ast}\coloneqq\dfrac{Np}{N-2p}$, we have
the second-order Sobolev inequalities

(3.5) $\lVert u\rVert_{\ell^{p^{\ast\ast}}}\leq C_{p}\lVert
u\rVert_{D^{2,p}},\;\forall u\in D_{0}^{2,p}(G),$

and for $1\leq p<\dfrac{N}{2}$,

(3.6) $\lVert u\rVert_{\ell^{p^{\ast\ast}}}\leq C_{p}\lVert
u\rVert_{\widetilde{D}^{2,p}},\;\forall u\in\widetilde{D}_{0}^{2,p}(G).$

###### Proof.

Using the completion trick, it suffices to prove that the second-order Sobolev
inequalities (3.5) and (3.6) hold for any $u\in C_{0}(G)$. By the first-order
Sobolev inequality (3.2) we have

(3.7) $\lVert u\rVert_{\ell^{p^{\ast\ast}}}\leq C_{p}\lVert\nabla
u\lVert_{p^{\ast}}.$

By the boundedness of Riesz transforms [22], we know that

(3.8) $\lVert\nabla u\lVert_{p^{\ast}}\lesssim\lVert
L^{\frac{1}{2}}u\lVert_{p^{\ast}}.$

And $L^{\frac{1}{2}}u\in\ell^{p}(G)$ since $L^{\frac{1}{2}}$ is a bounded
linear operator in $\ell^{p}(G)$ by Lemma 9. Then $L^{\frac{1}{2}}u\in
D_{0}^{1,p}(G)$ follows from $\ell^{p}(G)$ embeds into $D^{1,p}(G)$ and
$D_{0}^{1,p}(G)=D^{1,p}(G)$. Hence by the first-order Sobolev inequality again
we know

(3.9) $\lVert L^{\frac{1}{2}}u\lVert_{p^{\ast}}\leq C_{p}\lVert\nabla
L^{\frac{1}{2}}u\lVert_{p}.$

By the boundedness of Riesz transforms, we get

(3.10) $\lVert\nabla L^{\frac{1}{2}}u\lVert_{p}\lesssim\lVert
L^{\frac{1}{2}}L^{\frac{1}{2}}u\lVert_{p}=\lVert u\rVert_{D^{2,p}}.$

The inequality (3.5) is proved by (3.7)-(3.10).

Next by the first-order Sobolev inequality (3.2),

$\displaystyle\lVert u\rVert_{\ell^{p^{\ast\ast}}}$
$\displaystyle\lesssim\lVert\nabla
u\rVert_{\ell^{p^{\ast}}}=\left(\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}\lVert\partial_{i}u\rVert_{\ell^{p^{\ast}}}^{p^{\ast}}\right)^{1/p^{\ast}}$
$\displaystyle\lesssim\left(\stackrel{{\scriptstyle[}}{{i}}=1]{m}{\sum}\lVert\nabla\partial_{i}u\rVert_{\ell^{p}}^{p^{\ast}}\right)^{1/p^{\ast}}$
$\displaystyle\lesssim\left(\left(\stackrel{{\scriptstyle[}}{{i}},j=1]{m}{\sum}\lVert\partial_{j}\partial_{i}u\rVert_{\ell^{p}}\right)^{p^{\ast}}\right)^{1/p^{\ast}}\sim\lVert\nabla^{2}u\rVert_{\ell^{p}}.$

Hence the Sobolev inequality (3.6) holds. ∎

######  Remark.

(1) Since $\ell^{p}(G)$ embeds into $\ell^{q}(G)$ for any $q>p$, see [35,
Lemma 2.1], we get the second-order Sobolev inequalities (1.2) and (3.6) in
supercritical cases $q>p^{\ast\ast}$.

(2) The second-order Sobolev inequality for $u\in D_{0}^{2,1}(G)$ can not be
proved directly since the Riesz transforms is weak type $(1,1)$ for the
nilpotent group $G$ [22, Theorem 1], and we don’t know whether the inequality

$\lVert u\rVert_{\ell^{1^{\ast}}}\lesssim\lVert\nabla
u\rVert_{\ell^{1,\infty}}\coloneqq\underset{t>0}{\text{sup
}}t\left(\sharp\left\\{x\in G:\mid\nabla u(x)\mid_{1}>t\right\\}\right)$

holds or not.

Then we prove the equivalence of higher-order Sobolev spaces.

######  Theorem 11.

If $N\geq 3$, $1<p<\dfrac{N}{2}$, then

(3.11) $D_{0}^{2,p}(\mathbb{Z}^{N})=D^{2,p}(\mathbb{Z}^{N}),\text{}$

and for $1\leq p<\dfrac{N}{2}$,

$\widetilde{D}_{0}^{2,p}(\mathbb{Z}^{N})=\widetilde{D}^{2,p}(\mathbb{Z}^{N}).$

###### Proof.

By the same argument as Theorem 7, $D_{0}^{2,p}(G)\subseteq D^{2,p}(G)$ and
$\widetilde{D}_{0}^{2,p}(\mathbb{Z}^{N})\subseteq\widetilde{D}^{2,p}(\mathbb{Z}^{N})$
follow from the Sobolev inequalities (3.5) and (3.6).

In the other direction, the key is to find suitable cutoff functions
$\eta_{n}(x)\in C_{0}\left(\mathbb{Z}^{N}\right)$. For any $u\in
D^{2,p}(\mathbb{Z}^{N})$(resp. $\widetilde{D}^{2,p}(\mathbb{Z}^{N})$), set
$u_{n}\coloneqq u\eta_{n}\in C_{0}\left(\mathbb{Z}^{N}\right)$, then by
H$\ddot{\text{o}}$lder inequalities

(3.12) $\lVert
u_{n}-u\rVert_{D^{2,p}(\mathbb{Z}^{N})}^{p}\lesssim\sum\limits_{x\in\mathbb{Z}^{N}}\lvert\Delta
u(x)\mid^{p}\underset{y\sim
x}{\text{max}}\mid\eta_{n}(y)-1\mid^{p}+\lVert\Delta\eta_{n}\rVert_{\ell^{\frac{N}{2}}}^{p}\lVert
u\rVert_{\ell^{p^{\ast\ast}}}^{p}$

and

$\displaystyle\lVert u_{n}-u\rVert_{\tilde{D}^{2,p}(\mathbb{Z}^{N})}^{p}$
$\displaystyle\lesssim\sum\limits_{x\in\mathbb{Z}^{N}}\stackrel{{\scriptstyle[}}{{i}},j=1]{m}{\sum}\\{\lvert\partial_{j}\partial_{i}u(x)\mid^{p}\mid\eta_{n}(xs_{i}s_{j})-1\mid^{p}+\lvert\partial_{j}u(x)\mid^{p}\mid\partial_{i}\eta_{n}(xs_{j})\mid^{p}$
(3.13)
$\displaystyle+\lvert\partial_{i}u(x)\mid^{p}\mid\partial_{j}\eta_{n}(xs_{i})\mid^{p}+\lvert
u(x)\mid^{p}\mid\partial_{j}\partial_{i}\eta_{n}(x)\mid^{p}\\}$
$\displaystyle\lesssim\sum\limits_{x\in\mathbb{Z}^{N}}\stackrel{{\scriptstyle[}}{{i}},j=1]{m}{\sum}\lvert\partial_{j}\partial_{i}u(x)\mid^{p}\underset{i,j}{\text{max}}\mid\eta_{n}(xs_{i}s_{j})-1\mid^{p}+\lVert\nabla\eta_{n}\rVert_{\ell^{N}}^{p}\lVert\nabla
u\rVert_{\ell^{p^{\ast}}}^{p}$
$\displaystyle+\lVert\nabla^{2}\eta_{n}\rVert_{\ell^{\frac{N}{2}}}^{p}\lVert
u\rVert_{\ell^{p^{\ast\ast}}}^{p}.$

Hence, if the cutoff functions satisfy

$\eta_{n}\text{ is uniformly bounded,}$ (3.14) $\eta_{n}\to 1\text{ pointwise
on $\mathbb{Z}^{N}$}$

and

$\begin{cases}\begin{array}[]{cc}\lVert\Delta\eta_{n}\rVert_{\ell^{\frac{N}{2}}(\mathbb{Z}^{N})}\rightarrow
0,\>\text{for }D^{2,p}(\mathbb{Z}^{N}),\\\
\lVert\nabla\eta_{n}\rVert_{\ell^{N}(\mathbb{Z}^{N})},\>\lVert\nabla^{2}\eta_{n}\rVert_{\ell^{\frac{N}{2}}(\mathbb{Z}^{N})}\rightarrow
0,\>\text{for }\widetilde{D}^{2,p}(\mathbb{Z}^{N}),\end{array}\end{cases}$

then we can prove the other direction by the dominated convergence theorem. We
distinguish two cases and construct cutoff functions respectively:

Figure 3.1. The figure shows the cutoff functions constructed for
$D^{1,p}(\mathbb{Z}^{N})$ (solid line) and $D^{2,p}(\mathbb{Z}^{N})$ (dotted
line), respectively.

Case 1. For $D^{2,p}(\mathbb{Z}^{N})$, define
$f(x)\coloneqq\sum\limits_{i}x_{i}^{2}$ and

$\phi(s)\coloneqq
1\wedge\frac{\textrm{log$(1-(1-\frac{s}{R})^{3})$}}{\textrm{log$(1-(1-\frac{r}{R})^{3})$}}\vee
0.$

Let $\eta(x)\coloneqq\phi(f(x))$,
$X\coloneqq\left\\{x\in\mathbb{Z}^{N}:f(y)>r\text{ for all $y\sim
x$}\right\\}$. Then $\Delta f=2N$, $\Gamma f=4f+N$. The numbers $r$ and $R$
rather stand for the square of the distance as $f$ is the square of the
Euclidean distance. By definition $\phi\in C^{2}\left(r,\infty\right)$ and

$-(\text{log}(\frac{R}{c_{1}r}))^{-1}\frac{c_{2}}{s}\leq\phi^{\prime}(s)\leq
0\leq\phi^{\prime\prime}(s)\leq(\text{log}(\frac{R}{c_{1}r}))^{-1}\frac{c_{3}}{s^{2}},\;\forall
s\in(r,R].$

By the discrete chain rule (Lemma 5) on $X$, we have

$\Delta\eta(x)=\Delta\phi(f)(x)=2N\phi^{\prime}\left(f(x)\right)+(4f(x)+N)\phi^{\prime\prime}(\xi^{2})$

with $f(x)-2\sqrt{f(x)}+1\leq\xi\leq f(x)+2\sqrt{f(x)}+1$. And thus if
$100<r<f(y)$ for all $y\sim x$, we have $2\xi^{2}\geq f^{2}(x)$ giving

$-(\text{log}(\frac{R}{c_{1}r}))^{-1}2N\frac{c_{2}}{f(x)}\leq\Delta\eta(x)\leq(\text{log}(\frac{R}{c_{1}r}))^{-1}(4f(x)+N)\frac{2c_{3}}{f^{2}(x)}.$

Then taking $c\coloneqq\text{max}\left\\{2c_{2},9c_{3}\right\\}$, we get

$\mid\Delta\eta(x)\mid\leq\frac{cN}{f(x)\text{log}(\frac{R}{c_{1}r})}\sim\frac{1}{f(x)\text{log}\frac{R}{r}}.$

Thus,

(3.15) $\displaystyle\sum\limits_{x\in X}\mid\Delta\eta(x)\mid^{\frac{N}{2}}$
$\displaystyle\lesssim\left(\frac{1}{\text{log}\frac{R}{r}}\right)^{\frac{N}{2}}\sum\limits_{r<f(x)\leq
R}f(x)^{-\frac{N}{2}}$
$\displaystyle\lesssim\left(\frac{1}{\text{log}\frac{R}{r}}\right)^{\frac{N}{2}}\int\limits_{\widetilde{B}_{\sqrt{R}+1}\setminus\widetilde{B}_{\sqrt{r}-1}}\mid
t\mid^{-N}\text{d$t$}$ $\displaystyle\lesssim
O\left(\textrm{$\left(\text{log}\frac{R}{r}\right)$}^{1-\frac{N}{2}}\right).$

This goes to $0$ as $R\to\infty$ for $N\geq 3$. The boundary at $\sqrt{r}$
does not matter as we can keep $r$ constant and let $R\to\infty$. At the
neighbourhood of boundary $\partial\sqrt{r}$, i.e.
$Y\coloneqq\left\\{x\in\mathbb{Z}^{N}:\sqrt{r}-2\leq\mid
x\mid\leq\sqrt{r}+2\right\\}$, by the mean value theorem we have

$\mid\Delta\eta(x)\mid=\mid\underset{y\sim
x}{\sum}\left(\phi(f(y))-\phi(f(x))\right)\mid\lesssim\mid\phi^{\prime}(r)\mid\mid
f(y)-f(x)\mid\lesssim\frac{1}{\sqrt{r}\text{log}\frac{R}{r}},$

Since the cardinality of $Y$ is finite, then

(3.16) $\sum\limits_{x\in
Y}\mid\Delta\eta(x)\mid^{\frac{N}{2}}\lesssim\sum\limits_{x\in
Y}\left(\frac{1}{\sqrt{r}\text{log}\frac{R}{r}}\right)^{\frac{N}{2}}\to
0\text{ as }R\to\infty.$

Hence, by (3.15) and (3.16) we know

$\lVert\Delta\eta(x)\rVert_{\ell^{\frac{N}{2}}(\mathbb{Z}^{N})}\to 0\text{ as
}R\to\infty,$

and we prove $u\in D_{0}^{2,p}(\mathbb{Z}^{N})$ by (3.12).

Case 2. For $\widetilde{D}^{2,p}(\mathbb{Z}^{N})$, set
$\eta(x)\coloneqq\phi(f(x))$ as in Case 1. Then we can check that

$\lVert\nabla\eta\rVert_{\ell^{N}(\mathbb{Z}^{N})}^{N}\lesssim
O\left(\textrm{$\left(\text{log}\frac{R}{r}\right)$}^{1-N}\right).$

By the norm equivalence (Lemma 6) and the result in Case 1, we know

$\lVert\nabla^{2}\eta\rVert_{\ell^{\frac{N}{2}}(\mathbb{Z}^{N})}^{\frac{N}{2}}\sim\lVert\Delta\eta\rVert_{\ell^{\frac{N}{2}}(\mathbb{Z}^{N})}^{\frac{N}{2}}\lesssim
O\left(\textrm{$\left(\text{log}\frac{R}{r}\right)$}^{1-\frac{N}{2}}\right).$

Hence,
$\widetilde{D}^{2,p}(\mathbb{Z}^{N})\subseteq\widetilde{D}_{0}^{2,p}(\mathbb{Z}^{N})$
by (3.13). This finishes the proof. ∎

######  Remark.

(1) The cutoff function $\eta$ defined for $D^{1,p}(\mathbb{Z}^{N})$ can not
work for $D^{2,p}(\mathbb{Z}^{N})$ directly since the Laplacian is "bad" when
it comes to the sharp corner $R$. We find a way to fix this by taking a "close
to linear" function $\phi$ being smooth at the boundary $R$. Now the Euclidean
distance is necessary for constructing the cutoff function since its Laplacian
is decreasing while the combinatorial distance is not. Then using the chain
rule we can get a good estimate.

(2) For $1<p<\dfrac{N}{2},$ by Lemma 6 and Theorem 11, we know
$D^{2,p}(\mathbb{Z}^{N})=D_{0}^{2,p}(\mathbb{Z}^{N})=\widetilde{D}_{0}^{2,p}(\mathbb{Z}^{N})=$$\widetilde{D}^{2,p}(\mathbb{Z}^{N})$.

Then we prove the higher-order equivalence on a Cayley graph $G$ using the
functional analysis.

######  Theorem 12.

If $N\geq 3$, $1<p<\dfrac{N}{2}$, then $D_{0}^{2,p}(G)=D^{2,p}(G)$.

###### Proof.

First $D_{0}^{2,p}(G)\subseteq D^{2,p}(G)$ by the Sobolev inequalities (3.5).
The other direction can be proved by checking the Laplace operator $\Delta$ is
an isometry from $D_{0}^{2,p}(G)$ to $\ell^{p}(G)$. The injection follows from
the $\ell^{p}-$Liouville property of graphs [5, Section 1.7]. The range of
$\Delta$, denoted by $\text{Ran}(\Delta)$, is closed by the completeness of
$D_{0}^{2,p}(G)$. For any $h\in\ell^{q}(G)$, the dual space of $\ell^{p}(G)$,

$0=h(f),\>\forall f\in\text{Ran}(\Delta),$

which implies

$0=\langle h,\Delta u\rangle=\langle\Delta h,u\rangle,\>\forall u\in
C_{0}\left(G\right).$

Hence $\Delta h=0$ and $h=0$ by the $\ell^{p}-$Liouville property, then the
surjection follows from the closed range theorem. Similarly, $\Delta$ is also
an isometry from $D^{2,p}(G)$ to $\ell^{p}(G)$. Then by the fact
$D_{0}^{2,p}(G)\subseteq D^{2,p}(G)$ we know $D_{0}^{2,p}(G)=D^{2,p}(G)$. ∎

######  Remark.

For the special case $p=2$, $D_{0}^{2,2}=D^{2,2}$ on general groups satisfying
the second-order Sobolev inequality, for example, polynomial growth groups and
non-amenable groups.

##  4\. The Concentration-Compactness principle

In this section, we prove the discrete Concentration-Compactness principle. We
first introduce a key lemma as in [10, Theorem 1].

Consider a measure space $(\Omega,\Sigma,\mu)$, which consists of a set
$\Omega$ equipped with a $\sigma$-algebra $\Sigma$ and a Borel measure
$\mu:\Sigma\to\left[0,\infty\right]$.

######  Lemma 13.

(Brézis-Lieb lemma) Let $(\Omega,\Sigma,\mu)$ be a measure space,
$\\{u_{n}\\}\subset L^{p}(\Omega,\Sigma,\mu)$, and $0<p<\infty.$ If
(a)$\;\\{u_{n}\\}$ is uniformly bounded in $L^{p}$, and
(b)$\;u_{n}\to u,n\to\infty$ $\mu$-almost everywhere in $\Omega$, then

(4.1) $\lim\limits_{n\to\infty}(\lVert u_{n}\rVert_{L^{p}}^{p}-\lVert
u_{n}-u\rVert_{L^{p}}^{p})=\lVert u\rVert_{L^{p}}^{p}.$

######  Remark.

(1) The preceding lemma is a refinement of Fatou’s Lemma.

(2) Since $\;\\{u_{n}\\}$ is uniformly bounded in $L^{p}$, passing to a
subsequence if necessary, we have

$\lim\limits_{n\to\infty}\lVert
u_{n}\rVert_{p}^{p}=\lim\limits_{n\to\infty}\lVert
u_{n}-u\rVert_{p}^{p}+\lVert u\rVert_{p}^{p}.$

(3) If $\Omega$ is countable and $\mu$ is a positive measure defined on
$\Omega$, then we get a discrete version of Lemma 13.

######  Corollary 14.

Let $\Omega\subset G$, ${\left\\{u_{n}\right\\}}\subset D^{2,p}(G)$, and
$1<p<\infty$. If
(a)′ $\left\\{{u_{n}}\right\\}$ is uniformly bounded in $D^{2,p}(G)$, and
(b)′ $u_{n}\to u,n\to\infty$ pointwise on $G$, then

(4.2) $\lim\limits_{n\to\infty}\left(\sum\limits_{x\in\Omega}\lvert\Delta
u_{n}(x)\rvert^{p}-\sum\limits_{x\in\Omega}\lvert\Delta(u_{n}-u)(x)\rvert^{p}\right)=\sum\limits_{x\in\Omega}\lvert\Delta
u(x)\rvert^{p}.$

###### Proof.

Since $\left\\{{\Delta u_{n}}\right\\}$ is uniformly bounded in $\ell^{p}(G)$
and $\Delta u_{n}\rightarrow\Delta u,n\rightarrow\infty$ pointwise on $G$, by
Lemma 13 we get

$\lim\limits_{n\to\infty}(\lVert\Delta
u_{n}\rVert_{\ell^{p}(\Omega)}^{p}-\lVert\Delta\left(u_{n}-u\right)\rVert_{\ell^{p}(\Omega)}^{p})=\lVert\Delta
u\rVert_{\ell^{p}(\Omega)}^{p},$

which is equivalent to the equation (4.2). ∎

Next, we establish the Concentration-Compactness principle on $G$.

######  Lemma 15.

(Discrete Concentration-Compactness lemma) For $N\geq 3,1<p<\frac{N}{2},q\geq
p^{\ast\ast}$, if $\\{u_{n}\\}$ is uniformly bounded in $D^{2,p}(G)$. Then
passing to a subsequence, still denoted by $\\{u_{n}\\}$, we have

(4.3) $u_{n}\longrightarrow u\quad\textrm{pointwise\;on}\;G,$ (4.4)
$\lvert\Delta u_{n}\rvert^{p}\xrightharpoonup{w^{\ast}}\lvert\Delta
u\rvert^{p}\quad\textrm{in}\;\ell^{1}(G).$

And the following limits

$\lim\limits_{R\to\infty}\lim\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
u_{n}\left(x\right)\rvert^{p}\coloneqq\mu_{\infty},\>\lim\limits_{R\to\infty}\lim\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert
u_{n}(x)\rvert^{q}\coloneqq\nu_{\infty},$

exist. For the above $\left\\{u_{n}\right\\}$, we have

(4.5)
$\lvert\Delta(u_{n}-u)\rvert^{p}\xrightharpoonup{w^{\ast}}0\quad\text{in}\;\ell^{1}(G),$
(4.6) $\lvert
u_{n}-u\rvert^{q}\xrightharpoonup{w^{\ast}}0\quad\text{in}\;\ell^{1}(G),$
(4.7) $\nu_{\infty}^{p/q}\leq K^{-1}\mu_{\infty},$ (4.8)
$\lim\limits_{n\to\infty}\lVert u_{n}\rVert_{D^{2,p}}^{p}=\lVert
u\rVert_{D^{2,p}}^{p}+\mu_{\infty},$ (4.9) $\lim\limits_{n\to\infty}\lVert
u_{n}\rVert_{q}^{q}=\lVert u\rVert_{q}^{q}+\nu_{\infty}.$

###### Proof.

Since $\\{u_{n}\\}$ is uniformly bounded in $\ell^{q}(G)$, they are bounded in
$\ell^{\infty}(G)$. By diagonal principle, passing to a subsequence we get
(4.3). Since $\left\\{\lvert\Delta u_{n}\rvert^{p}\right\\}$ is uniformly
bounded in $\ell^{1}(G)$, we get (4.4) by the Banach-Alaoglu theorem and
(4.3). For every $R\geq 1$, passing to a subsequence if necessary,

$\lim\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
u_{n}\left(x\right)\rvert^{p},\qquad\lim\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert
u_{n}(x)\rvert^{q}.$

exist. Then we can define $\mu_{\infty}$, $\nu_{\infty}$ by the monotonicity
in $R$.

Let $v_{n}:=u_{n}-u$, then $v_{n}\to 0$ pointwise on $G$ and
$\left\\{\lvert\Delta v_{n}\rvert^{p}\right\\}$ is uniformly bounded in
$\ell^{1}(G)$. Then any subsequence of $\left\\{\lvert\Delta
v_{n}\rvert^{p}\right\\}$ contains a subsequence, still denoted by
$\left\\{\lvert\Delta v_{n}\rvert^{p}\right\\}$, that $w^{\ast}$-converges to
0 in $\ell^{1}(G)$, which follows from

$\sum h\lvert\Delta v_{n}\rvert^{p}\to 0,\text{ }\forall h\in C_{0}(G).$

Hence we obtain (4.5). Similarly, we get (4.6).

For $R\geq 1$, let $\Psi_{R}\in C(G)$ such that $\Psi_{R}(x)=1$ for
$d^{S}(x,e)\geq R+1$, $\Psi_{R}(x)=0$ for $d^{S}(x,e)\leq R$. By the discrete
Sobolev inequality (1.2), we have

$(\sum\lvert\Psi_{R}v_{n}\rvert^{q})^{p/q}\leq
K^{-1}\sum\lvert\Delta(\Psi_{R}v_{n})\rvert^{p}=K^{-1}\underset{x}{\sum}\lvert\underset{y\sim
x}{\sum}\left(\nabla_{xy}\Psi_{R}v_{n}(y)+\Psi_{R}(x)\nabla_{xy}v_{n}\right)\rvert^{p}.$

For any $\varepsilon>0$, there exists $C_{\varepsilon}>0$ such that

$\lvert\underset{y\sim x}{\sum}\nabla_{xy}\Psi_{R}v_{n}(y)+\underset{y\sim
x}{\sum}\Psi_{R}(x)\nabla_{xy}v_{n}\rvert^{p}\leq
C_{\varepsilon}\lvert\underset{y\sim
x}{\sum}\nabla_{xy}\Psi_{R}v_{n}(y)\lvert^{p}+(1+\varepsilon)\lvert\Delta
v_{n}\lvert^{p}\lvert\Psi_{R}(x)\lvert^{p}.$

Since $v_{n}\to 0$ pointwise on $G$, by $\varepsilon\to 0^{+}$ we obtain

(4.10)
$\varlimsup\limits_{n\to\infty}(\sum\lvert\Psi_{R}v_{n}\rvert^{q})^{p/q}\leq
K^{-1}\varlimsup\limits_{n\to\infty}\sum\lvert\Delta
v_{n}\rvert^{p}\Psi_{R}^{p}.$

From the definition of $\Psi_{R}$,

(4.11) $\varlimsup\limits_{n\to\infty}(\sum\limits_{d^{S}(x,e)>R}\lvert
v_{n}\rvert^{q})^{p/q}\leq
K^{-1}\varlimsup\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
v_{n}\rvert^{p}.$

By Lemma 13 and Corollary 14, we have

$\lim\limits_{n\to\infty}\left(\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
u_{n}(x)\rvert^{p}-\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
v_{n}(x)\rvert^{p}\right)=\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
u(x)\rvert^{p},$
$\lim\limits_{n\to\infty}\left(\sum\limits_{d^{S}(x,e)>R}\lvert
u_{n}(x)\rvert^{q}-\sum\limits_{d^{S}(x,e)>R}\lvert
v_{n}(x)\rvert^{q}\right)=\sum\limits_{d^{S}(x,e)>R}\lvert u(x)\rvert^{q}.$

Hence by letting $R\to\infty$, we get

(4.12)
$\lim\limits_{R\to\infty}\lim\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert\Delta
v_{n}(x)\rvert^{p}=\mu_{\infty},$ (4.13)
$\lim\limits_{R\to\infty}\lim\limits_{n\to\infty}\sum\limits_{d^{S}(x,e)>R}\lvert
v_{n}(x)\rvert^{q}=\nu_{\infty}.$

Combining the equations (4.11)-(4.13), we get

$\nu_{\infty}^{p/q}\leq K^{-1}\mu_{\infty}.$

Since $u_{n}\to u$ pointwise on $G$, then for every $R\geq 1$,

$\displaystyle\varlimsup\limits_{n\to\infty}\sum\lvert\Delta u_{n}\rvert^{p}$
$\displaystyle=\varlimsup\limits_{n\to\infty}\left(\sum\Psi_{R}\lvert\Delta
u_{n}\rvert^{p}+\sum(1-\Psi_{R})\lvert\Delta u_{n}\rvert^{p}\right)$
$\displaystyle=\varlimsup\limits_{n\to\infty}\sum\Psi_{R}\lvert\Delta
u_{n}\rvert^{p}+\sum(1-\Psi_{R})\lvert\Delta u\rvert^{p},$

and

$\displaystyle\varlimsup\limits_{n\to\infty}\sum\lvert u_{n}\rvert^{q}$
$\displaystyle=\varlimsup\limits_{n\to\infty}\left(\sum\Psi_{R}\lvert
u_{n}\rvert^{q}+\sum(1-\Psi_{R})\lvert u_{n}\rvert^{q}\right)$
$\displaystyle=\varlimsup\limits_{n\to\infty}\sum\Psi_{R}\lvert
u_{n}\rvert^{q}+\sum(1-\Psi_{R})\lvert u\rvert^{q}.$

Letting $R\to\infty$, we obtain

$\lim\limits_{n\to\infty}\sum\lvert\Delta
u_{n}\rvert^{p}=\mu_{\infty}+\sum\lvert\Delta u\rvert^{p}=\mu_{\infty}+\lVert
u\rVert_{D^{2,p}}^{p},$ $\lim\limits_{n\to\infty}\sum\lvert
u_{n}\rvert^{q}=\nu_{\infty}+\sum\lvert u\rvert^{q}=\nu_{\infty}+\lVert
u\rVert_{q}^{q}.$

∎

######  Remark.

(1) In the continuous setting, P. L. Lions [46], Bianchi et al.[8] and Ben-
Naoum et al.[7] proved that the limit of the minimizing sequence norm can be
divided into three parts, i.e. the norm of the limit function, the norm of the
limit of the difference between the sequence and the limit function, and the
norm of the sequence at infinity. The corresponding parts still satisfy the
Sobolev inequality, see [76, Lemma 1.40].

(2) The difference between the sequence and the limit function
$w^{\ast}$-converges to $0$ in $\ell^{1}(G)$, i.e. (4.5) and (4.6), which is
not true in continuous setting. For example, consider the sequence of
probability measures $\left\\{\delta_{n}\right\\}$ in $[0,1]$, where
$\delta_{n}(x)\coloneqq n\chi_{[0,\frac{1}{n}]}\textrm{d}x$, then
$\delta_{n}\to 0$ almost everywhere in $[0,1]$. However,
$\delta_{n}\xrightharpoonup{w^{\ast}}\delta_{0}$ in
$\left(C[0,1]\right)^{\ast}$ and the Dirac measure $\delta_{0}$ is non-zero.
This is the advantage of the discrete setting.

##  5\. Proof of Theorem 1

In this section, we will prove the existence of the extremal function for the
discrete second-order Sobolev inequality (1.2). Firstly, we prove that the
minimizing sequence after translation has a uniform positive lower bound at
the unit element $e\in G$. This is crucial to rule out the vanishing case of
the limit function.

######  Lemma 16.

For $N\geq 3,1<p<\frac{N}{2},q>p^{\ast\ast}$, let
$\left\\{u_{n}\right\\}\subset D^{2,p}(G)$ be a minimizing sequence satisfying
(1.2). Then $\varliminf\limits_{n\to\infty}\lVert
u_{n}\rVert_{\ell^{\infty}}>0$.

###### Proof.

Choosing $q^{\prime}$ such that $p^{\ast\ast}<q^{\prime}<q<\infty$, by
interpolation inequality we have

$1=\lVert u_{n}\rVert_{q}^{q}\leq\lVert
u_{n}\rVert_{q^{\prime}}^{q^{\prime}}\lVert
u_{n}\rVert_{\infty}^{q-q^{\prime}}\leq C_{q^{\prime},p}^{q^{\prime}}\lVert
u_{n}\rVert_{D^{2,p}}^{q^{\prime}}\lVert u_{n}\rVert_{\infty}^{q-q^{\prime}},$

where $C_{q^{\prime},p}$ is the constant in the Sobolev inequality (1.2).

By taking the limit, we obtain

$1\leq
C_{q^{\prime},p}^{q^{\prime}}K^{\frac{q^{\prime}}{p}}\varliminf\limits_{n\to\infty}\lVert
u_{n}\rVert_{\infty}^{q-q^{\prime}}.$

This proves the lemma. ∎

######  Remark.

The maximum of $\lvert u_{n}\rvert$ is attainable since $\lVert
u_{n}\rVert_{q}=1$. Define $v_{n}\left(x\right):=u_{n}(x_{n}x)$, where $\lvert
u_{n}(x_{n})\rvert=\max\limits_{x}\lvert u_{n}(x)\rvert$. Then the translation
sequence $\left\\{v_{n}\right\\}$ is uniformly bounded in $D^{2,p}(G)$,
$\lVert v_{n}\rVert_{q}=1$ and $\lvert v_{n}(e)\rvert=\lVert
u_{n}\rVert_{\infty}$, where $e$ is the unit element of $G$. By Lemma 16,
passing to a subsequence if necessary, we have

$v_{n}\longrightarrow v\quad\;\textrm{pointwise\;on}\;G,$ (5.1) $\lvert
v(e)\rvert=\varliminf\limits_{n\to\infty}\lVert
u_{n}\rVert_{\ell^{\infty}}>0.$

Next, we give the proof I of Theorem 1.

###### Proof I of Theorem 1.

Let $\left\\{u_{n}\right\\}\subset D^{2,p}(G)$ be a minimizing sequence
satisfying (1.4). And the translation sequence $\left\\{v_{n}\right\\}$ is
defined in the Remark after Lemma 16.

By equalities (4.9) and (4.8) in Lemma 15, passing to a subsequence, we get

$K=\lim\limits_{n\to\infty}\lVert v_{n}\rVert_{D^{2,p}}^{p}=\lVert
v\rVert_{D^{2,p}}^{p}+\mu_{\infty},$ $1=\lim\limits_{n\to\infty}\lVert
v_{n}\rVert_{q}^{q}=\lVert v\rVert_{q}^{q}+\nu_{\infty}.$

From the Sobolev inequality (1.2), (4.7) and the inequality

(5.2) $\left(a^{q}+b^{q}\right)^{p/q}\leq a^{p}+b^{p}\text{,}\;\forall a,b\geq
0,$

we get

$\displaystyle K$ $\displaystyle=\lVert v\rVert_{D^{2,p}}^{p}+\mu_{\infty}$
$\displaystyle\geq K((\lVert v\rVert_{q}^{q})^{p/q}+\nu_{\infty}^{p/q})$
$\displaystyle\geq K(\lVert v\rVert_{q}^{q}+\nu_{\infty})^{p/q}=K.$

Since $\left(a^{q}+b^{q}\right)^{p/q}<a^{p}+b^{p}$ unless $a=0$ or $b=0$, we
deduce from (5.1) that $\lVert v\rVert_{q}^{q}=1$. By

$\lVert v\rVert_{D^{2,p}}^{p}\geq K\lVert v\rVert_{q}^{p},$

we get

$\lVert v\rVert_{D^{2,p}}^{p}=K=\lim\limits_{n\to\infty}\lVert
v_{n}\rVert_{D^{2,p}}^{p}.$

That is, $v$ is a minimizer. ∎

Then we give another proof for Theorem 1 using the discrete Brézis-Lieb lemma.

###### Proof II of Theorem 1.

Using Lemma 16, by the translation and taking a subsequence if necessary as
before, we can get a minimizing sequence $\left\\{u_{n}\right\\}$ satisfying
(1.4), $u_{n}\to u$ pointwise on $G$, and $\lvert u(e)\rvert>0$.

By Lemma 13, the inequality (5.2) and the Sobolev inequality, passing to a
subsequence, we have

(5.3) $\displaystyle K=\underset{n\rightarrow\infty}{\lim}\lVert
u_{n}\rVert_{D^{2,p}}^{p}$
$\displaystyle=\underset{n\rightarrow\infty}{\lim}\frac{\lVert
u_{n}\rVert_{D^{2,p}}^{p}}{\lVert
u_{n}\rVert_{q}^{p}}=\underset{n\rightarrow\infty}{\overline{\lim}}\frac{\lVert
u_{n}-u\rVert_{D^{2,p}}^{p}+\lVert u\rVert_{D^{2,p}}^{p}}{\left(\lVert
u_{n}-u\rVert_{q}^{q}+\lVert u\rVert_{q}^{q}\right)^{p/q}}$
$\displaystyle\geq\underset{n\rightarrow\infty}{\overline{\lim}}\frac{\lVert
u_{n}-u\rVert_{D^{2,p}}^{p}+\lVert u\rVert_{D^{2,p}}^{p}}{\lVert
u_{n}-u\rVert_{q}^{p}+\lVert u\rVert_{q}^{p}}$
$\displaystyle\geq\underset{n\rightarrow\infty}{\overline{\lim}}\frac{K\lVert
u_{n}-u\rVert_{q}^{p}+\lVert u\rVert_{D^{2,p}}^{p}}{\lVert
u_{n}-u\rVert_{q}^{p}+\lVert u\rVert_{q}^{p}}.$

Since $u\text{$\not\equiv$}0$, by the Sobolev inequality we have that

$\lVert u\rVert_{D^{2,p}}^{p}\leq K\lVert u\rVert_{q}^{p},$

which implies

$\lVert u\rVert_{D^{2,p}}^{p}=K\lVert u\rVert_{q}^{p}.$

By (5.3), passing to a subsequence, we get

$\underset{n\rightarrow\infty}{\lim}\lVert
u_{n}-u\rVert_{D^{2,p}}^{p}=K\underset{n\rightarrow\infty}{\lim}\lVert
u_{n}-u\rVert_{q}^{p}.$

Since $0<\lVert u\rVert_{q}\leq\underset{n\rightarrow\infty}{\lim}\lVert
u_{n}\rVert_{q}=1$, it suffices to show that $\lVert u\rVert_{q}=1$. Suppose
that it is not true, i.e. $0<\lVert u\rVert_{q}=D<1$, then by Lemma 13,

$\underset{n\rightarrow\infty}{\lim}\lVert
u_{n}-u\rVert_{q}^{q}=\underset{n\rightarrow\infty}{\lim}\lVert
u_{n}\rVert_{q}^{q}-\lVert u\rVert_{q}^{q}=1-D^{q}>0.$

However, $\left(a^{q}+b^{q}\right)^{p/q}<a^{p}+b^{p}$ if $a,$ $b>0$. This
yields a contradiction by (5.3).

Thus, $\lVert u\rVert_{q}=1$ and $u$ is a minimizer. ∎

##  6\. Proofs for corollaries

As applications, we prove corollaries in the introduction. The following lemma
establishes the equivalence between solutions to the higher-order quasilinear
problem (1.5) and to the system (1.6).

######  Lemma 17.

For $N\geq 3$, $p,q>1$, $\frac{1}{p^{\prime}}+\frac{1}{q}<\frac{N-2}{N}$,
$p^{\prime}=\frac{p}{p-1}$, $q^{\prime}=\frac{q}{q-1}$, $(u,v)$ is a solution
of (1.5) if and only if $u$ is a solution of (1.6) with $v\coloneqq-\mid\Delta
u\mid^{p-2}\Delta u$.

###### Proof.

Since $\frac{1}{p^{\prime}}+\frac{1}{p^{\ast\ast}}=\frac{N-2}{N},$
$\frac{1}{p^{\prime}}+\frac{1}{q}<\frac{N-2}{N}$ if and only if
$q>p^{\ast\ast}$. Suppose that $u\in D^{2,p}(G)$ is a solution of (1.5). Set
$v\coloneqq-\mid\Delta u\mid^{p-2}\Delta u$. Then $v\in\ell^{p^{\prime}}(G)$
and $u\in\ell^{p^{\ast\ast}}(G)$ by the Sobolev inequality. And
$u\in\ell^{q}(G)$, since $\ell^{p^{\ast\ast}}(G)$ embeds into $\ell^{q}(G)$.
Let $w$ be a solution of

$-\Delta w=\lvert u\rvert^{q-2}u.$

Hence, $\Delta w\in\ell^{q^{\prime}}$ and $w\in D^{2,q^{\prime}}(G)$ by the
fact that Laplace operator $\Delta$ is an isometry from $D^{2,q^{\prime}}(G)$
to $\ell^{q^{\prime}}(G)$, see Theorem 12. By the uniqueness theorem for
harmonic functions we know that

$v=w\in D^{2,q^{\prime}}(G).$

Note that $\lvert v\rvert^{p^{\prime}-2}v=-\Delta u$. That is, $(u,v)$ is a
solution of (1.6). On the other hand, if $(u,v)$ is a pair of solution for
(1.6), we have $-\mid\Delta u\mid^{p-2}\Delta u=v$. This proves the lemma. ∎

By Theorem 1, we can prove Corollary 2.

###### Proof of Corollary 2.

By Theorem 1 there exists a minimizer $u$ for the problem (1.3). Let $v$ be a
solution of

$-\Delta v=\mid-\Delta u\mid.$

Then $\lVert v\rVert_{q}\lesssim\lVert\Delta u\rVert_{p}$ since Laplace
operator $\Delta$ is an isometry from $D_{0}^{2,p}(G)$ to $\ell^{p}(G)$, and
$D_{0}^{2,p}(G)$ embeds into $\ell^{q}(G)$ for $q\geq p^{\ast\ast}$. Note that
$u\leq v$ by the maximum principle. Replacing $u$ by $-u$, we get $-u\leq v$
similarly. Hence,

$0\leq\mid u\mid\leq v.$

In particular we have $1=\lVert u\rVert_{q}\leq\lVert v\rVert_{q}$ and
$\lVert\Delta u\rVert_{p}=\lVert\Delta v\rVert_{p}$. Therefore, we know that
$\frac{v}{\lVert v\rVert_{q}}$ is a non-negative minimizer. It follows from
the Lagrange multiplier that $\frac{v}{\lVert v\rVert_{q}}$ is a non-negative
solution of (1.5). The maximum principle yields that it is positive. ∎

Finally, we can prove Corollary 3 by Corollary 2 and Lemma 17.

###### Proof of Corollary 3.

By Corollary 2 there exists a positive solution $u$ of equation (1.5). Set
$v\coloneqq-\mid\Delta u\mid^{p-2}\Delta u$ as Lemma 17 and we know

$-\Delta v=\lvert u\rvert^{q-2}u>0.$

Hence $v$ is positive by the maximum principle. ∎

##  7\. The cases of $p=2$ and $p=1$

In this section, we study the special cases of $p=2$ and $p=1$. First, for
$p=2$, consider a Cayley graph $(G,S)$ satisfying $\beta_{S}(n)\geq
C(S)n^{N},\ \forall n\geq 1$. By the first-order Sobolev inequality (3.2) we
know $D_{0}^{1,2}(G)\subseteq D^{1,2}(G)$. We fix an orientation for edge set
$E$. Then $D_{0}^{1,2}(G)$ and $D^{1,2}(G)$ are Hilbert spaces equipped with
the inner product

$\langle\nabla_{e}u,\nabla_{e}v\rangle_{E}\coloneqq\underset{e\in
E}{\sum}\left(u(e_{+})-u(e_{-})\right)\left(v(e_{+})-v(e_{-})\right),$

where $e_{-}$ and $e_{+}$ are the initial and terminal endpoints of $e$
respectively. For any $\alpha\in C(E)$,

$\text{div}\,\alpha(x)\coloneqq\underset{e=(y,x)\in
E}{\sum}\alpha(e)-\underset{\tilde{e}=(x,y)\in
E}{\sum}\alpha(\tilde{e}),\;\;x\in V.$

Then we can prove the Hodge decomposition theorem on $1$-forms on edges on the
Cayley graph.

######  Theorem 18.

If the Cayley graph $(G,S)$ satisfies $\beta_{S}(n)\geq C(S)n^{N},\ \forall
n\geq 1$, then we have decompositions

$\ell^{2}(E)=\nabla D_{0}^{1,2}(G)\oplus H$

and

$\ell^{2}(E)=\nabla D^{1,2}(G)\oplus H,$

where $H\coloneqq\left\\{u\in\ell^{2}(E):\text{div}\,u=0\right\\}$.

###### Proof.

For any $u\in\ell^{2}(E)$, we define a bounded linear operator on
$D_{0}^{1,2}(G)$ as

$L(v)\coloneqq-\langle\text{div}\,u,v\rangle.$

A nondegerate bilinear functional $B:D_{0}^{1,2}(G)\times
D_{0}^{1,2}(G)\to\mathbb{R}$ is defined as

$B(v,w)\coloneqq\langle\nabla_{e}v,\nabla_{e}w\rangle_{E}.$

By the Lax–Milgram theorem, there exists $f\in D_{0}^{1,2}(G)$ such that

$B(f,v)=L(v),\>\forall v\in D_{0}^{1,2}(G).$

Hence, for any $v\in C_{0}(G)$,

$\langle\Delta
f,v\rangle=-\langle\nabla_{e}f,\nabla_{e}v\rangle_{E}=\langle\text{div}\,u,v\rangle,$

which implies that $\Delta f=\text{div}\,u$.

Since $\lVert\nabla f\rVert_{\ell^{2}(E)}=\frac{1}{2}\lVert
f\rVert_{D^{1,2}(G)}$, $\nabla f\in\ell^{2}(E)$. Let $h=u-\nabla
f\in\ell^{2}(E)$. Then

$\text{div}\,h=\text{div}\,u-\text{div}\,\nabla f=\text{div}\,u-\Delta f=0.$

That is $h\in H$. The decomposition is proved. If $u\in\nabla
D_{0}^{1,2}(G)\cap H$, then there exists $v\in D_{0}^{1,2}(G)$ such that
$u=\nabla v$ and $\text{div}\,u=\Delta v=0$. Hence $v=0$ and $u=0$. The
property of the direct sum is proved. And we can get the decomposition
$\ell^{2}(E)=\nabla D^{1,2}(G)\oplus H$ by the same argument. ∎

######  Corollary 19.

If the Cayley graph $(G,S)$ satisfies $\beta_{S}(n)\geq C(S)n^{N},\ \forall
n\geq 1$, then $D_{0}^{1,2}(G)=D^{1,2}(G)$.

###### Proof.

By the Sobolev inequality we know $D_{0}^{1,2}(G)\subseteq D^{1,2}(G)$. Then
for any $u\in D^{1,2}(G)$, by the Hodge decomposition theorem we get

$\nabla u=\nabla\tilde{u}+h,\,\tilde{u}\in D_{0}^{1,2}(G),h\in H.$

And $h=\nabla u-\nabla\tilde{u}\in\nabla D^{1,2}(G)$, hence $h=0$, that is
$\nabla\left(u-\tilde{u}\right)=0$, which implies that $u=\tilde{u}\in
D_{0}^{1,2}(G)$. Hence $D^{1,2}(G)\subseteq D_{0}^{1,2}(G)$. ∎

######  Remark.

For the groups satisfying the second-order Sobolev inequality, for example,
polynomial growth groups and non-amenable groups, we can prove
$D_{0}^{2,2}=D^{2,2}$ by checking the Laplace operator is an isometry from
$D_{0}^{2,2}(G)$ (also $D^{2,2}(G)$) to $\ell^{2}(G)$ as Theorem 12.

For the special case $p=1$, since
$\widetilde{D}_{0}^{2,1}(\mathbb{Z}^{N})=\widetilde{D}^{2,1}(\mathbb{Z}^{N})$
by Lemma 11, we can consider the optimal constant $\widetilde{K}$ of Sobolev
inequality (3.6) in the supercritical case $q>1^{\ast\ast}=\frac{N}{N-2}$ on
$\mathbb{Z}^{N}$:

(7.1)
$\widetilde{K}:=\inf_{\begin{subarray}{c}u\in\widetilde{D}^{2,1}(\mathbb{Z}^{N})\\\
\lVert u\rVert_{q}=1\end{subarray}}\lVert u\rVert_{\widetilde{D}^{2,1}}.$

In order to prove that the infimum is achieved, consider a minimizing sequence
$\\{u_{n}\\}\subset\widetilde{D}^{2,1}(\mathbb{Z}^{N})$ satisfying

(7.2) $\lVert u_{n}\rVert_{q}=1,\lVert
u_{n}\rVert_{\widetilde{D}^{2,1}(\mathbb{Z}^{N})}\to\widetilde{K},n\to\infty.$

Then by the same argument as the proof of Theorem 1 we can get the following
result for $\widetilde{D}^{2,1}(\mathbb{Z}^{N})$.

######  Theorem 20.

For $N\geq 3,$ $q>1^{\ast\ast}=\frac{N}{N-2}$, let
$\left\\{u_{n}\right\\}\subset\widetilde{D}^{2,1}(\mathbb{Z}^{N})$ be a
minimizing sequence satisfying (7.2). Then there exists a sequence
$\\{x_{n}\\}\subset\mathbb{Z}^{N}$ and $v\in\widetilde{D}^{2,1}$ such that the
sequence after translation $\left\\{v_{n}(x):=u_{n}(x_{n}+x)\right\\}$
contains a convergent subsequence that converges to $v$ in
$\widetilde{D}^{2,1}$. And $v$ is a minimizer for $\widetilde{K}$.

######  Remark.

This result implies that the best constant can be obtained in the
supercritical case.

$\mathbf{\boldsymbol{\mathbf{Acknowledgements}\mathbf{}\text{:}}}$ The authors
would like to thank Xueping Huang, Genggeng Huang and Fengwen Han for helpful
discussions and suggestions. B.H. is supported by NSFC, no.11831004.

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