# Principal frequency of $p$-sub-Laplacians for general vector fields

Michael Ruzhansky  Michael Ruzhansky: Department of Mathematics: Analysis,
Logic and Discrete Mathematics Ghent University, Belgium and School of
Mathematical Sciences Queen Mary University of London United Kingdom E-mail
address<EMAIL_ADDRESS>,  Bolys Sabitbek  Bolys Sabitbek: School
of Mathematical Sciences Queen Mary University of London United Kingdom, and
Department of Mechanics and Mathematics Al-Farabi Kazakh National University
71 al-Farabi Ave., Almaty, 050040 Kazakhstan and Institute of Mathematics and
Mathematical Modeling 125 Pushkin Street., Almaty, 050010 Kazakhstan E-mail
address<EMAIL_ADDRESS>and  Durvudkhan Suragan  Durvudkhan Suragan:
Department of Mathematics Nazarbayev University 53 Kabanbay batyr Ave.,
Astana, 010000 Kazakhstan E-mail address<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove the uniqueness and simplicity of the principal
frequency (or the first eigenvalue) of the Dirichlet $p$-sub-Laplacian for
general vector fields. As a byproduct, we establish the Caccioppoli
inequalities and also discuss the particular cases on the Grushin plane and on
the Heisenberg group.

###### Key words and phrases:

p-sub-Laplacian, smooth manifold, principal frequency, Picone’s identity,
Caccioppoli inequality

###### 1991 Mathematics Subject Classification:

35P30, 35H99.

The first author was supported by FWO Odysseus 1 grant G.0H94.18N: Analysis
and Partial Differential Equations and EPSRC grant EP/R003025/1. The second
author was supported in parts by the MES RK grant AP08053051 and EPSRC grant
EP/R003025/1. The third author was supported by the Nazarbayev University
program 091019CRP2120 and the Nazarbayev University grant 240919FD3901.

##  1\. Introduction

Let $M$ be a smooth manifold of dimension $n$ with a volume form $dx$. Let
$\\{X_{k}\\}_{k=1}^{N}$ with $n\geq N$ be a family of vector fields defined on
$M$. We consider the operator

$\mathcal{L}:=\sum_{k=1}^{N}X^{*}_{k}X_{k}.$ (1.1)

It is is locally hypoelliptic if the commutators of the vector fields
$\\{X_{k}\\}_{k=1}^{N}$ generate the tangent space of $M$ as the Lie algebra.

In addition, we define the $p$-sub-Laplacian for general vector fields by the
formula

$\mathcal{L}_{p}f:=\nabla_{X}^{*}\cdot\left(|\nabla_{X}f|^{p-2}\nabla_{X}f\right),\,1<p<\infty,$
(1.2)

and the horizontal gradients

$\nabla_{X}:=(X_{1},\ldots,X_{N})\,\,\,\text{and}\,\,\,\nabla_{X}^{*}:=(X_{1}^{*},\ldots,X_{N}^{*}),$

where

$X_{k}=\sum_{j=1}^{n}a_{kj}(x)\frac{\partial}{\partial
x_{j}},\,\,k=1,\ldots,N,$

with the formal adjoint

$X_{k}^{*}=-\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}(a_{kj}(x)),\,\,k=1,\ldots,N.$

Let $\Omega\subset M$ be an open set. We define the functional spaces

$S^{1,p}(\Omega)=\\{u:\Omega\rightarrow\mathbb{R};u,|\nabla_{X}u|\in
L^{p}(\Omega)\\}.$ (1.3)

We also consider the following functional

$J_{p}(u)=\left(\int_{\Omega}|\nabla_{X}u|^{p}dx\right)^{\frac{1}{p}}.$ (1.4)

Now let us define the functional class $\mathring{S}^{1,p}(\Omega)$ to be the
completion of $C_{0}^{1}(\Omega)$ in the norm generated by $J_{p}$, see in
[3].

We consider the following Dirichlet boundary value problem for
$\mathcal{L}_{p}$:

$\begin{cases}\mathcal{L}_{p}u=\lambda|u|^{p-2}u,\,\,&\text{in}\,\,\Omega,\\\
u=0,\,\,&\text{on}\,\,\partial\Omega.\end{cases}$ (1.5)

In the Euclidean setting, the first eigenvalue for the Dirichlet boundary
value problem for $\mathcal{L}_{p}$ was obtained by Lindqvist in [16]. In the
sub-elliptic setting, the study of boundary value problems started by the
pioneering works of Bony [2], Gaveau [9], Kohn and Nirenberg [15]. In 1981,
the Dirichlet problem for the Kohn Laplacian on the Heisenberg group was
studied by Jerison in [13, 14] as a part of his dissertation under the
supervision of E. Stein. Also, semilinear equations on the Heisenberg group
for sums of vector fields were studied in [25] and [19]. Those works attracted
significant attention to boundary value problems for the sub-elliptic
operators, see e.g. [4], [5], [6], [7], [8] and references therein.

As usual, a weak solution of equation (1.5) means a function
$u\in\mathring{S}^{1,p}(\Omega)$ such that

$\int_{\Omega}|\nabla_{X}u|^{p-2}\langle\nabla_{X}u,\nabla_{X}\phi\rangle
dx-\lambda\int_{\Omega}|u|^{p-2}u\phi dx=0,$ (1.6)

for all $\phi\in\mathring{S}^{1,p}(\Omega)$. Similarly, by the sup-solution
and sub-solution of equation (1.5) we mean a function
$u\in\mathring{S}^{1,p}(\Omega)$ such that

$\int_{\Omega}|\nabla_{X}u|^{p-2}\langle\nabla_{X}u,\nabla_{X}\phi\rangle
dx-\lambda\int_{\Omega}|u|^{p-2}u\phi dx\geq 0,$ (1.7)

and

$\int_{\Omega}|\nabla_{X}u|^{p-2}\langle\nabla_{X}u,\nabla_{X}\phi\rangle
dx-\lambda\int_{\Omega}|u|^{p-2}u\phi dx\leq 0,$ (1.8)

for all $\phi\in\mathring{S}^{1,p}(\Omega)$, respectively. Here
$\lambda\in\mathbb{R}$ and $u$ are called a Dirichlet eigenvalue and
eigenfunction of the operator $\mathcal{L}_{p}$ in $\Omega\subset M$,
respectively. If we put $u$ instead of $\phi$ in (1.6), we have

$\int_{\Omega}|\nabla_{X}u|^{p}dx=\lambda\int_{\Omega}|u|^{p}dx,$ (1.9)

which can be written for $u\neq 0$ as

$\lambda:=\frac{\int_{\Omega}|\nabla_{X}u|^{p}dx}{\int_{\Omega}|u|^{p}dx}.$

Thus, one can define the principal Dirichlet frequency $\lambda_{1}(\Omega)$
of the operator $\mathcal{L}_{p}$ as the minimum of the Rayleigh quotient
$\int_{\Omega}|\nabla_{X}u|^{p}dx/\int_{\Omega}|u|^{p}dx$ taken over all
nontrivial functions $u\in\mathring{S}^{1,p}(\Omega)$, that is,

$\lambda_{1}(\Omega):=\min_{\mathring{S}^{1,p}(\Omega)}\left\\{\int_{\Omega}|\nabla_{X}u|^{p}dx:\int_{\Omega}|u|^{p}dx=1\right\\}.$
(1.10)

The main aim of this paper is to prove uniqueness, simplicity, and a domain
monotonicity of the principal frequency (or the first eigenvalue) of the
Dirichlet $p$-sub-Laplacian for general vector fields (1.2). We also present
the Caccioppoli inequality for this operator in the form

$\int_{\Omega}\phi^{p}|\nabla_{X}v|^{p}dx\leq
p^{p}\int_{\Omega}v^{p}|\nabla_{X}\phi|^{p}dx,$

for every $0\leq\phi\in C_{0}^{\infty}(\Omega)$, where $v>0$ is a sub-solution
of (1.5) in $\Omega\subset M$.

Picone’s identity for general vector fields plays an important role in some of
our computations. Recall Picone’s identity for differentiable functions $u$
and $v$ with $v(x)\neq 0$ in $\Omega\subset\mathbb{R}^{n}$:

$|\nabla u|^{2}-\left\langle\nabla
v,\nabla\left(\frac{|u|^{2}}{v}\right)\right\rangle=|\nabla
u|^{2}+\frac{|u|^{2}}{|v|^{2}}|\nabla v|^{2}-2\frac{u}{v}\langle\nabla
v,\nabla u\rangle\geq 0,$ (1.11)

where $\nabla$ and $\langle\cdot,\cdot\rangle$ are the standard gradient and
the inner product in $\mathbb{R}^{n}$, respectively. The Picone identity is
one of the important tools in the theory of partial differential equations
(see, e.g. [24]). As consequences of Picone’s identity, one can obtain, for
instance, the simplicity of principal eigenvalues, Barta’s inequalities,
nonexistence of positive solutions, Hardy’s inequalities, and Sturmian
comparison. In the Euclidean setting, the Picone identity was investigated by
many authors and generalised in different directions. For example, in [1]
Allegato and Huang extended the Picone identity to the case of any $p>1$.
Recently, in [11] and [12] Jaros presented the Picone identity for the Finsler
$p$-Laplacian and obtained the Caccioppoli inequality, which has the form

$\int_{\Omega}|\phi\nabla v|^{2}dx\leq 4\int_{\Omega}|v\nabla\phi|^{2}dx,$
(1.12)

where $v>0$ is a weak solution of $\Delta v=0$ in
$\Omega\subset\mathbb{R}^{n}$ and $0\leq\phi\in C^{\infty}_{0}(\Omega)$ is a
test function. The present paper is motivated by Jaros’ results. The
inequality (1.12) can be derived from (1.11) by integrating over $\Omega$ with
$u=v\phi$, then applying the Cauchy-Schwartz inequality and the Young
inequality. The $L^{p}$-version of inequality (1.12) was obtained in [10],
[17], and [20]. Note that the authors in [21] have obtained the weighted
anisotropic Hardy and Rellich inequalities making use of the (first and second
order) anisotropic Picone identities. We also refer to a recent open access
book [22] for further discussions in this direction. Here, we establish a
version of Picone’s identity for general vector fields:

######  Lemma 1.1.

Let $\Omega\subset M$ be an open set. Let $p>1$. Let $u$ and $v$ be
differentiable in a given set $\Omega$ with $v(x)\neq 0$ in $\Omega$. Define

$\displaystyle
L(u,v):=|\nabla_{X}u|^{p}-p\frac{|u|^{p-2}u}{|v|^{p-2}v}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}u\rangle+(p-1)\frac{|u|^{p}}{|v|^{p}}|\nabla_{X}v|^{p},$
(1.13) $\displaystyle
R(u,v):=|\nabla_{X}u|^{p}-|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\left(\frac{|u|^{p}}{|v|^{p-2}v}\right)\rangle.$
(1.14)

Then we have

$L(u,v)=R(u,v)\geq 0.$ (1.15)

Moreover, we have $L(u,v)=0$ a.e. in $\Omega$ if and only if $u=cv$ a.e. in
$\Omega$ with a positive constant $c$.

A Carnot group (stratified group) version of the Picone identity was obtained
in [23] for a general case. Also, Niu, Zhang, and Wang in [18] obtained the
Picone identity on the Heisenberg group and remarked that it could be extended
to general vector fields satisfying Hörmander’s condition. This idea was later
extended in [22, Section 11.6]. Indeed, all the extensions are a direct
reworking of the proof [18, Lemma 2.1]. Therefore, here we omit the proof of
Lemma 1.1.

The paper is organised in the following way: In Section 2, we prove the
uniqueness and simplicity of the principal frequency (or the first eigenvalue)
of the Dirichlet $p$-sub-Laplacian for general vector fields. In Section 3, we
obtain the Caccioppoli inequalities for general vector fields making use of
Picone’s identity. Then, we present some examples on the Grushin plane and the
Heisenberg group.

##  2\. Principal frequency of $p$-sub-Laplacians

######  Lemma 2.1.

Assume that there exists a strictly positive sup-solution of (1.5). Then we
have

$\int_{\Omega}|\nabla_{X}u|^{p}dx\geq\lambda\int_{\Omega}|u|^{p}dx,$ (2.1)

for all functions $u\in\mathring{S}^{1,p}(\Omega)$.

We recall that a sup-solution $v$ of (1.5) has the following form

$\int_{\Omega}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\phi\rangle
dx-\lambda\int_{\Omega}|v|^{p-2}v\phi dx\geq 0,$

for all $\phi\in\mathring{S}^{1,p}(\Omega)$.

###### Proof of Lemma 2.1.

Suppose that $v$ is a strictly positive sup-solution of (1.5) in $\Omega$ and
assume that $u\in C_{0}^{\infty}(\Omega)$. Then for a given small
$\varepsilon>0$ in (1.7) we set

$\phi:=\frac{|u|^{p}}{(v+\varepsilon)^{p-1}}.$

A straightforward computation gives that

$\displaystyle\int_{\Omega}\lambda
v^{p-1}\frac{|u|^{p}}{(v+\varepsilon)^{p-1}}dx$
$\displaystyle\leq\int_{\Omega}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\left(\frac{|u|^{p}}{(v+\varepsilon)^{p-1}}\right)\rangle
dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla_{X}u|^{p}dx-\int_{\Omega}|\nabla_{X}u|^{p}-|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\left(\frac{|u|^{p}}{(v+\varepsilon)^{p-1}}\right)\rangle
dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla_{X}u|^{p}dx-\int_{\Omega}L(u,v+\varepsilon)dx.$

In the last line, we have used the Picone identity. Now by taking the limit as
$\varepsilon\rightarrow 0^{+}$ and making use of Fatou’s lemma on the left-
hand side and the Lebesgue dominated convergence theorem on the right-hand
side of above expression, we arrive at

$\int_{\Omega}|\nabla_{X}u|^{p}dx-\lambda\int_{\Omega}|u|^{p}dx-\int_{\Omega}L(u,v)dx\geq
0.$ (2.2)

Since $L(u,v)\geq 0$ a.e. in $\Omega$, it yields

$\int_{\Omega}|\nabla_{X}u|^{p}dx-\lambda\int_{\Omega}|u|^{p}dx\geq 0.$

By a density argument, we assert that the claim is valid for
$\mathring{S}^{1,p}(\Omega)$. ∎

######  Theorem 2.2.

Let $\Omega\subset M$ be a bounded open set. If there exists $\lambda$ and a
strictly positive $v\in\mathring{S}^{1,p}(\Omega)$ such that

$\int_{\Omega}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\phi\rangle
dx\geq\lambda\int_{\Omega}|v|^{p-2}v\phi dx,$ (2.3)

for every nonnegative function $\phi\in\mathring{S}^{1,p}(\Omega)$, then we
have

$\lambda_{1}(\Omega)\geq\lambda,$ (2.4)

and

$\int_{\Omega}|\nabla_{X}u|^{p}dx\geq\lambda\int_{\Omega}|u|^{p}dx,$ (2.5)

for all $u\in\mathring{S}^{1,p}(\Omega)$.

Note that here and after we denote by $\lambda_{1}$ the principal frequency of
$\mathcal{L}_{p}$ and by $u_{1}$ the corresponding positive eigenfunction
without the existence argument, that is, we assume that

$\int_{\Omega}|\nabla_{X}u_{1}|^{p-2}\langle\nabla_{X}u_{1},\nabla_{X}\phi\rangle
dx=\lambda_{1}(\Omega)\int_{\Omega}|u_{1}|^{p-2}u_{1}\phi dx,$ (2.6)

for every $\phi\in\mathring{S}^{1,p}(\Omega)$.

###### Proof of Theorem 2.2.

Note that inequality (2.5) was proved in Lemma 2.1. So now we need to prove
(2.4).

Let $u_{1}\in\mathring{S}^{1,p}(\Omega)$ be the eigenfunction associated to
the principal frequency $\lambda_{1}(\Omega)$. By choosing $\varepsilon>0$, we
integrate $L(u_{1},v+\varepsilon)$ over $\Omega$, then using (2.3) and (2.6)
with $\phi=u_{1}$, we have

$\displaystyle 0\leq$ $\displaystyle\int_{\Omega}L(u_{1},v+\varepsilon)dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla_{X}u_{1}|^{p}dx-\int_{\Omega}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\left(\frac{|u_{1}|^{p}}{(v+\varepsilon)^{p-1}}\right)\rangle
dx$ $\displaystyle\leq$
$\displaystyle\,\lambda_{1}(\Omega)\int_{\Omega}|u_{1}|^{p}dx-\lambda\int_{\Omega}v^{p-1}\frac{|u_{1}|^{p}}{(v+\varepsilon)^{p-1}}dx.$

Now we take taking the limit as $\varepsilon\rightarrow 0^{+}$, we arrive at

$0\leq\lambda_{1}(\Omega)-\lambda,$

which proves Theorem 2.2. ∎

######  Corollary 2.3 (Uniqueness).

If there exists $\lambda$ and a strictly positive
$v\in\mathring{S}^{1,p}(\Omega)$ such that

$\int_{\Omega}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}\phi\rangle
dx=\lambda\int_{\Omega}|v|^{p-2}v\phi dx,$ (2.7)

for every nonnegative function $\phi\in\mathring{S}^{1,p}(\Omega)$, then we
have

$\lambda_{1}(\Omega)=\lambda.$ (2.8)

###### Proof of Corollary 2.3.

Let $u_{1}\in\mathring{S}^{1,p}(\Omega)$ be the eigenfunction associated to
the principal frequency $\lambda_{1}(\Omega)$. By choosing $\varepsilon>0$, we
integrate $L(u_{1},v+\varepsilon)$ over $\Omega$, then using (2.6) and (2.7)
with $\phi=v$, we have

$\displaystyle 0\leq$ $\displaystyle\int_{\Omega}L(v,u_{1}+\varepsilon)dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla_{X}v|^{p}dx-\int_{\Omega}|\nabla_{X}u_{1}|^{p-2}\langle\nabla_{X}u_{1},\nabla_{X}\left(\frac{|v|^{p}}{(u_{1}+\varepsilon)^{p-1}}\right)\rangle
dx$ $\displaystyle=$
$\displaystyle\lambda\int_{\Omega}|v|^{p}dx-\lambda_{1}(\Omega)\int_{\Omega}u_{1}^{p-1}\frac{|v|^{p}}{(u_{1}+\varepsilon)^{p-1}}dx.$

Now we take the limit as $\varepsilon\rightarrow 0^{+}$, so we get
$\lambda_{1}(\Omega)\leq\lambda$. On the other hand, by Theorem 2.2 we have
the fact that $\lambda_{1}(\Omega)\geq\lambda$, that is, we arrive at

$\lambda_{1}(\Omega)=\lambda,$

completing the proof. ∎

######  Theorem 2.4 (Simplicity).

Any eigenfunction $u\in\mathring{S}^{1,p}(\Omega)$ of $\mathcal{L}_{p}$
associated to the principal frequency $\lambda_{1}(\Omega)$ is a constant
multiple of $u_{1}$.

###### Proof of Theorem 2.4.

Let both $u$ and $u_{1}$ be eigenfunctions associated to the principal
frequency $\lambda_{1}$. Let $L(u,u_{1})\geq 0$ in $\Omega$. For
$\varepsilon>0$, we integrate $L(u,u_{1}+\varepsilon)$ over $\Omega$ with
$v=u_{1}+\varepsilon$, we get

$\displaystyle\int_{\Omega}L(u,u_{1}+\varepsilon)dx=\int_{\Omega}|\nabla_{X}u|^{p}dx-\lambda_{1}(\Omega)\int_{\Omega}u_{1}^{p-1}\frac{|u|^{p}}{(u_{1}+\varepsilon)^{p-1}}dx.$

Taking the limit as $\varepsilon\rightarrow 0^{+}$ and using the Lebesgue
dominated convergence theorem and Fatou’s lemma, we arrive at

$\int_{\Omega}L(u,u_{1})dx\leq\int_{\Omega}|\nabla_{X}u|^{p}dx-\lambda_{1}(\Omega)\int_{\Omega}|u|^{p}dx=0.$

Since $L(u,u_{1})\geq$ in $\Omega$, it follows that $L(u,u_{1})=0$ a.e. in
$\Omega$. So it follows from Lemma 1.1 that $u=Cu_{1}$ with $C>0$. ∎

Then, by using the Picone identity, we show monotonicity of the principal
frequency $\lambda_{1}(\Omega)$ in Theorem 2.2 as the function of the set
$\Omega$.

######  Theorem 2.5.

Let $\Omega$ and $\widetilde{\Omega}$ be subsets of $M$. If
$\Omega\subset\widetilde{\Omega}$, then we have

$\lambda_{1}(\Omega)\geq\lambda_{1}(\widetilde{\Omega}).$ (2.9)

###### Proof of Theorem 2.5.

We have $\Omega\subset\widetilde{\Omega}$, and let $u_{1}$ and
$\widetilde{u}_{1}$ be positive eigenfunctions associated to
$\lambda_{1}(\Omega)$ and $\lambda_{1}(\widetilde{\Omega})$, respectively (see
the assumption (2.6)). For $\varepsilon>0$ we consider
$L(u_{1},\widetilde{u}_{1}+\varepsilon)$ where $u=u_{1}$ and
$v=\widetilde{u}_{1}+\varepsilon$. By using inequality (2.3), we get

$\displaystyle\int_{\Omega}L(u_{1},\widetilde{u}_{1}+\varepsilon)dx$
$\displaystyle=\int_{\Omega}|\nabla_{X}u_{1}|^{p}dx-\int_{\Omega}|\nabla_{X}\widetilde{u}_{1}|^{p-2}\langle\nabla_{X}\widetilde{u}_{1},\nabla_{X}\left(\frac{u_{1}^{p}}{(\widetilde{u}_{1}+\varepsilon)^{p-1}}\right)\rangle
dx$
$\displaystyle\leq\lambda_{1}(\Omega)\int_{\Omega}u_{1}^{p}dx-\lambda_{1}(\widetilde{\Omega})\int_{\Omega}\widetilde{u}_{1}^{p-2}\frac{u_{1}^{p}}{(\widetilde{u}_{1}+\varepsilon)^{p-1}}dx.$

Now by taking the limit as $\varepsilon\rightarrow 0^{+}$ and making use of
Fatou’s lemma and the Lebesgue dominated convergence theorem, we obtain

$(\lambda_{1}(\Omega)-\lambda_{1}(\widetilde{\Omega}))\int_{\Omega}u_{1}^{p}dx\geq\int_{\Omega}L(u_{1},\widetilde{u}_{1})dx\geq
0,$ (2.10)

which yields

$\lambda_{1}(\Omega)\geq\lambda_{1}(\widetilde{\Omega}).$ (2.11)

This proves Theorem 2.5. ∎

##  3\. Caccioppoli inequalities for general vector fields

Here we give the Caccioppoli inequalities for general vector fields by making
use of Picone’s identity, and we discuss some of their corollaries on the
Grushin plane and the Heisenberg group.

######  Theorem 3.1 (Caccioppoli inequality).

Let $v$ be a positive sub-solution of (1.5) in $\Omega\subset M$. Then for
every fixed $q>p-1$, $1<p<\infty$, and $\lambda\in\mathbb{R}$ we have

$\int_{\Omega}v^{q-p}\phi^{p}|\nabla_{X}v|^{p}dx\leq\left(\frac{p}{q-p+1}\right)^{p}\int_{\Omega}v^{q}|\nabla_{X}\phi|^{p}dx+\frac{\lambda
p}{q-p+1}\int_{\Omega}v^{q}\phi^{p}dx,$ (3.1)

for all nonnegative functions $\phi\in C^{\infty}_{0}(\Omega)$.

Note that in the case when $q=p$ and $\lambda=0$ in (3.1) we have

$\int_{\Omega}\phi^{p}|\nabla_{X}v|^{p}dx\leq
p^{p}\int_{\Omega}v^{p}|\nabla_{X}\phi|^{p}dx.$ (3.2)

###### Proof of Theorem 3.1.

Setting $u:=v^{q/p}\phi$ in $L(u,v)$ we have

$\displaystyle\int_{\Omega}L(v^{q/p}\phi,v)dx=$
$\displaystyle\int_{\Omega}|\nabla_{X}(v^{q/p}\phi)|^{p}dx+(p-1)\int_{\Omega}v^{q-p}|\phi\nabla_{X}v|^{p}dx$
$\displaystyle-p\int_{\Omega}|v^{\frac{q-p}{p}}\phi|^{p-1}|\nabla_{X}v|^{p-2}\langle\nabla_{X}v,\nabla_{X}(v^{q/p}\phi)\rangle
dx$ $\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla_{X}(v^{q/p}\phi)|^{p}dx-(q-p+1)\int_{\Omega}v^{q-p}|\phi\nabla_{X}v|^{p}dx$
$\displaystyle-p\int_{\Omega}|v^{\frac{q-p}{p}}\phi|^{p-1}|\nabla_{X}v|^{p-2}v^{q/p}\langle\nabla_{X}v,\nabla_{X}\phi\rangle
dx.$

In the last line, we have used the equality
$\nabla_{X}(v^{q/p}\phi)=\frac{q}{p}v^{\frac{q-p}{p}}\phi\nabla_{X}v+v^{q/p}\nabla_{X}\phi$.
Noting that

$\displaystyle\langle\nabla_{X}v,\nabla_{X}\phi\rangle$
$\displaystyle=\sum_{i=1}^{N}X_{i}vX_{i}\phi\leq\sum_{i=1}^{N}|X_{i}v||X_{i}\phi|$
$\displaystyle\leq\left(\sum_{i=1}^{N}(X_{i}v)^{2}\right)^{1/2}\left(\sum_{i=1}^{N}(X_{i}\phi)^{2}\right)^{1/2}=|\nabla_{X}v||\nabla_{X}\phi|,$

and recalling Young’s inequality in the form

$ab^{p-1}\leq\frac{a^{p}}{ps^{p-1}}+\frac{p-1}{p}sb^{p},\,\,a,b\geq
0\,\text{and}\,\,s>0,$

with $a=v^{q/p}|\nabla_{X}\phi|$ and $b=v^{\frac{q-p}{p}}\phi|\nabla_{X}v|$,
we obtain

$\displaystyle 0\leq\int_{\Omega}L(v^{q/p}\phi,v)dx=$
$\displaystyle\int_{\Omega}|\nabla_{X}(v^{q/p}\phi)|^{p}dx+s^{1-p}\int_{\Omega}v^{q}|\nabla_{X}\phi|^{p}dx$
$\displaystyle-(q-p+1-s(p-1))\int_{\Omega}v^{q-p}|\phi\nabla_{X}v|^{p}dx$
$\displaystyle\leq$
$\displaystyle\lambda\int_{\Omega}|v^{q/p}\phi|^{p}dx+s^{1-p}\int_{\Omega}v^{q}|\nabla_{X}\phi|^{p}dx$
$\displaystyle-(q-p+1-s(p-1))\int_{\Omega}v^{q-p}|\phi\nabla_{X}v|^{p}dx.$

Here we have used
$\int_{\Omega}|\nabla_{X}u|^{p}dx\leq\int_{\Omega}\lambda|u|^{p}dx$ for the
sub-solution of (1.5). Thus, we arrive at

$\displaystyle\int_{\Omega}v^{q-p}\phi^{p}|\nabla_{X}v|^{p}dx\leq$
$\displaystyle\frac{s^{1-p}}{q-p+1-s(p-1)}\int_{\Omega}v^{q}|\nabla_{X}\phi|^{p}dx$
(3.3)
$\displaystyle+\frac{\lambda}{q-p+1-s(p-1)}\int_{\Omega}v^{q}\phi^{p}dx.$

Now we choose the suitable constant as $s=\frac{q-p+1}{p}$ which leads to

$\int_{\Omega}v^{q-p}\phi^{p}|\nabla_{X}v|^{p}dx\leq\left(\frac{p}{q-p+1}\right)^{p}\int_{\Omega}v^{q}|\nabla_{X}\phi|^{p}dx+\frac{\lambda
p}{q-p+1}\int_{\Omega}v^{q}\phi^{p}dx.$ (3.4)

This proves Theorem 3.1. ∎

Let us discuss some particular cases of the Caccioppoli inequalities on the
Grushin plane and the Heisenberg group. One of the important examples of a
sub-Riemannian manifold is the Grushin plane. Recall that the Grushin plane
$G$ is the space $\mathbb{R}^{2}$ with vector fields

$X_{1}=\frac{\partial}{\partial
x_{1}},\,\,\text{and}\,\,X_{2}=x_{1}\frac{\partial}{\partial x_{2}},$

and the gradient

$\nabla_{G}=(X_{1},X_{2})$ (3.5)

for $x:=(x_{1},x_{2})\in\mathbb{R}^{2}$.

######  Corollary 3.2.

Let $\Omega\subset G$. Let $v$ be a positive sub-solution of

$\begin{cases}-\nabla_{G}\cdot(|\nabla_{G}u|^{p-2}\nabla_{G}u)=\lambda|u|^{p-2}u,\,\,&\text{in}\,\,\Omega,\\\
u=0,\,\,&\text{on}\,\,\partial\Omega.\end{cases}$ (3.6)

Then for every fixed $q>p-1$ and $1<p<\infty$ we have

$\int_{\Omega}v^{q-p}\phi^{p}|\nabla_{G}v|^{p}dx\leq\left(\frac{p}{q-p+1}\right)^{p}\int_{\Omega}v^{q}|\nabla_{G}\phi|^{p}dx+\frac{\lambda
p}{q-p+1}\int_{\Omega}v^{q}\phi^{p}dx,$ (3.7)

and for $q=p$ and $\lambda=0$ we have

$\int_{\Omega}\phi^{p}|\nabla_{G}v|^{p}dx\leq
p^{p}\int_{\Omega}v^{p}|\nabla_{G}\phi|^{p}dx,$ (3.8)

for all nonnegative functions $\phi\in C^{\infty}_{0}(\Omega)$.

Let $\mathbb{H}^{n}$ be the Heisenberg group, that is, $\mathbb{R}^{2n+1}$
equipped with the group law

$\xi\circ\xi^{\prime}:=(x+x^{\prime},y+y^{\prime},t+t^{\prime}+2(x\cdot
y^{\prime}-x^{\prime}\cdot y)),$

where $\xi:=(x,y,t)\in\mathbb{R}^{2n+1}$ with $x\in\mathbb{R}^{n}$,
$y\in\mathbb{R}^{n}$, $t\in\mathbb{R}$. The dilation operation on the
Heisenberg group with respect to the group law has the form

$\delta_{\lambda}(\xi):=(\lambda x,\lambda
y,\lambda^{2}t)\,\,\text{for}\,\,\lambda>0.$

The Heisenberg group is a basic example of step 2 stratified Lie groups
(Carnot groups).

The Lie algebra $\mathfrak{h}$ of the left-invariant vector fields on the
Heisenberg group $\mathbb{H}^{n}$ is spanned by

$X_{j}:=\frac{\partial}{\partial x_{j}}+2y_{j}\frac{\partial}{\partial t},$
$Y_{j}:=\frac{\partial}{\partial y_{j}}-2x_{j}\frac{\partial}{\partial t},$

with their (non-zero) commutator

$T=[X_{j},Y_{j}]=-4\frac{\partial}{\partial t}.$

The horizontal gradient on $\mathbb{H}^{n}$ is given by

$\nabla_{H}:=(X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}),$

so the sub-Laplacian on $\mathbb{H}^{n}$ is given by

$\mathcal{L}:=\sum_{j=1}^{n}X_{j}^{2}+Y_{j}^{2}.$

Note that in the case of the general stratified Lie groups $\mathbb{G}$
(Carnot groups) we have $X_{k}=-X^{*}_{k}$, so the formula (1.2) can be
rewritten as

$\mathcal{L}_{p}f:=\nabla_{H}\cdot(|\nabla_{H}f|^{p-2}\nabla_{H}f),$ (3.9)

which is called the $p$-sub-Laplacian.

######  Corollary 3.3.

Let $\Omega\subset\mathbb{H}^{n}$. Let $v$ be a positive sub-solution of

$\begin{cases}-\nabla_{H}\cdot(|\nabla_{H}u|^{p-2}\nabla_{H}u)=\lambda|u|^{p-2}u,\,\,&\text{in}\,\,\Omega,\\\
u=0,\,\,&\text{on}\,\,\partial\Omega.\end{cases}$ (3.10)

Then for every fixed $q>p-1$ and $1<p<\infty$ we have

$\int_{\Omega}v^{q-p}\phi^{p}|\nabla_{H}v|^{p}dx\leq\left(\frac{p}{q-p+1}\right)^{p}\int_{\Omega}v^{q}|\nabla_{H}\phi|^{p}dx+\frac{\lambda
p}{q-p+1}\int_{\Omega}v^{q}\phi^{p}dx,$ (3.11)

and for $q=p$ and $\lambda=0$ we have

$\int_{\Omega}\phi^{p}|\nabla_{H}v|^{p}dx\leq
p^{p}\int_{\Omega}v^{p}|\nabla_{H}\phi|^{p}dx,$ (3.12)

for all nonnegative functions $\phi\in C^{\infty}_{0}(\Omega)$.

Here the Caccioppoli inequality on the Heisenberg group seems to be also new.

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