# Hardy and Rellich identities and inequalities for Grushin operators via
spherical vector fields and Bessel pairs

Debdip Ganguly  Debdip Ganguly: Department of Mathematics Indian Institute of
Technology Delhi IIT Campus, Hauz Khas, Delhi, New Delhi 110016 India E-mail
address<EMAIL_ADDRESS>,  K. Jotsaroop  Jotsaroop Kaur: Department of
Mathematics Indian Institute of Science Education and Research Mohali Mohali,
Punjab 140306 India E-mail address<EMAIL_ADDRESS>and  Prasun
Roychowdhury  Prasun Roychowdhury: Mathematics Division National Center for
Theoretical Sciences NTU, Cosmology Building, No. 1, Sec. 4, Roosevelt Rd.,
Taipei City 106 Taiwan E-mail address<EMAIL_ADDRESS>
###### Abstract.

For Grushin vector fields, we prove Hardy, Hardy-Rellich, and Rellich
identities and inequalities with sharp constants. Our explicit remainder terms
significantly improve those found in the literature. Our arguments build on
abstract Hardy-Rellich identities involving the Bessel pair along with the use
of spherical harmonics developed by Garofalo-Shen [19]. Furthermore, in the
spirit of Bez-Machihara-Ozawa [4], we construct spherical vector fields that
correspond to the Grushin vector fields and prove identities that, in turn,
establish optimal Rellich identities by comparing the Grushin operator with
its radial and spherical components. We give alternate proofs of Hardy
identities and inequalities with enhanced Hardy constants in some subspaces of
the Sobolev space, among other things. Additionally, we compute the deficit
involving $L^{2}$-norm of the Laplacian and radial components of the Laplacian
for the Grushin operator with an explicit remainder term, which leads to a
comparison of Laplacian with the radial components of the Laplacian. As a
consequence of the main results, new second-order Heisenberg-Pauli-Weyl
uncertainty principles and Hydrogen uncertainty principles are also derived.
Furthermore, we also derive certain symmetrization principles on the Grushin
space.

###### Key words and phrases:

Grushin space, Hardy identity, Hardy-Rellich identity, Rellich inequality,
spherical harmonics

###### 2020 Mathematics Subject Classification:

35H10, 26D10, 42B37, 46E35

##  1\. Introduction

The study of functional inequalities has become a fascinating branch of
analysis and PDEs in recent years. One of the central attractions of this type
of study is the Hardy inequality. The basic one-dimensional weighted Hardy
inequality [26, Theorem 330] reads as follows: let $f\in
C_{c}^{\infty}(0,\infty)$ and $\alpha\in(-1,\infty)$, then with the sharp
constant, there holds

$\int_{0}^{\infty}\frac{{f^{\prime}}^{2}(x)}{x^{\alpha}}\>{\rm
d}x\geq\frac{(\alpha+1)^{2}}{4}\int_{0}^{\infty}\frac{f^{2}(x)}{x^{\alpha+2}}\>{\rm
d}x.$

There has been a growing interest in studying functional inequalities and
identities for Hardy-Rellich type for sub-Laplacian as square sums of vector
fields on $\mathbb{R}^{n+m}.$ In particular, we consider the vector fields
given by

$X_{i}=\frac{\partial}{\partial x_{i}},\quad i\,=\,1,\ldots,n,\quad
Y_{j}=|x|^{\alpha}\frac{\partial}{\partial y_{j}},\ j\,=\,1,\ldots,m$

and the corresponding sub-Laplacian, well-known as the Baouendi-Grushin
operator:

$\mathcal{L}_{G}^{\alpha}\,=\,\Delta_{x}\,+\,|x|^{2\alpha}\Delta_{y}.$

When $\alpha=0,$ the operator $\mathcal{L}_{G}^{0}$ is reduced to the standard
Laplacian in $\mathbb{R}^{n+m}.$ If $\alpha$ is a positive even integer, i.e,
$\alpha=2k,$ with $k\in\mathbb{N},$ the vector fields $X_{i}^{\prime}s$ and
$Y_{j}^{\prime}s$ satisfy Hörmander’s finite rank condition [27]. But in the
general case Hörmander’s condition is incoherent since the vector fields are
not sufficiently smooth. Note that $\mathcal{L}_{G}^{\alpha}$ does not belong
to the class of sub-Laplacian associated with a homogeneous group. However, it
is known that $\mathcal{L}_{G}^{\alpha}$ is a subelliptic operator for
$\alpha>0$ (see [15, 17]). The analysis of the Grushin operator
$\mathcal{L}_{G}^{\alpha}$ is subtle and in particular, we shall focus on the
case $\alpha=1$ and $m=1.$

Let us first rewrite the vector fields and the Grushin operator for $\alpha=1$
and $m=1$ explicitly. Let $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$ with $n\geq
1$. For $i=1,\cdots,n$ and consider the vector fields

$\displaystyle X_{i}=\frac{\partial}{\partial x_{i}},\quad
Y=|x|\frac{\partial}{\partial t},$

where $|x|=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}$. Then $\mathbb{R}^{n+1}$
equipped with the above vector fields is called Grushin space. The Grushin
gradient is defined by

$\displaystyle\nabla_{G}=(X_{1},\cdots,X_{n},Y)=(\nabla_{x},|x|\partial_{t}),$

where $\nabla_{x}$ is the standard Euclidean gradient on $\mathbb{R}^{n}$. Now
recall the Grushin operator defined by

$\displaystyle\mathcal{L}_{G}=\Delta_{x}+|x|^{2}\frac{\partial^{2}}{\partial
t^{2}},$

where $\Delta_{x}$ is the standard Laplacian on $\mathbb{R}^{n}$. Also note
that, for some vector field $X$, if $\text{div}_{G}X=\nabla_{G}\cdot X$, then
$\mathcal{L}_{G}=\text{div}_{G}\circ\nabla_{G}$.

Let $\delta_{\lambda}(x,t):=(\lambda x,\lambda^{2}t)$ for $\lambda>0$. We can
compute that
$\mathcal{L}_{G}(f\circ\delta_{\lambda})=\lambda^{2}\left(\mathcal{L}_{G}f\right)\circ\delta_{\lambda}$.
Note that $|\delta_{\lambda}U|=\lambda^{Q}|U|$ for any $U$ Lebesgue measurable
set in $\mathbb{R}^{n+1}$ where $Q=n+2$. We define on Grushin space
$\mathbb{R}^{n+1}$, a distance function for each $(x,t)\in\mathbb{R}^{n+1}$ by

$\displaystyle\varrho(x,t)=\big{(}|x|^{4}+4t^{2}\big{)}^{\frac{1}{4}}$

compatible with the dilation $\delta_{\lambda}$ defined above. From the above
definition of $\varrho$, it satisfies the following:
$(x,t)\mapsto\varrho(x,t)$ is $C^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\}),$
positive definite, $\varrho(\delta_{\lambda}(x,t))=\lambda\varrho(x,t)$ for
$\lambda>0,$ and $o=(\vec{0},0).$ In fact, $\varrho$ is deeply connected to
the fundamental solution of $\mathcal{L}_{G}$ (see, [17, Proposition 2.1]) and
we will call $\varrho$ as a homogeneous norm on $\mathbb{R}^{n+1}$.

This paper’s major goal is to prove the Hardy and Hardy-Rellich inequalities
for the Grushin operator by a method that fully addresses the identity
framework. The second significant point we discuss is finding the radial and
spherical derivatives of the operator $\mathcal{L}_{G}$ and obtaining an
identity incorporating the radial $L^{2}$-norm and the spherical contribution
of the Grushin operator.

###  1.1. State of the art and main novelties

Let us discuss briefly the current state of the art of the problems in the
hypoelliptic settings. First, begin with the Hardy inequality for the Grushin
operator. For $Q\geq 3$ and $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, then with sharp constant
there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t\geq\frac{(Q-2)^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm d}t,$ (1.1)

where $Q=n+2$ is the homogeneous dimension of the Grushin space. Much research
has been done on the Hardy inequalities and the best way to improve them in
the sub-Laplacian situation, which is to add a positive remainder term to the
inequality’s right side (1.1). The aforementioned inequality has been
demonstrated even within the broader context of Carnot groups, namely inside
the Heisenberg group. We refer to the seminal work of Garofalo-Lanconelli
[18], where a following variant of Hardy inequality on the Heisenberg group
$\mathbb{H}^{n}$ was derived

$\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}u|^{2}\,{\rm d}\xi\geq
n^{2}\int_{\mathbb{H}^{n}}\frac{|u|^{2}|\psi|^{2}}{\varrho^{2}}\,{\rm d}\xi,$

where $\varrho$ and $\psi$ are, respectively, a suitable distance from the
origin and a weight function with $0<\psi<1$ and the corresponding homogeneous
dimension $Q=2n+2.$ Moreover, depending on the weight function
$\frac{\psi^{2}}{\varrho^{2}},$ several other variants of the Hardy
inequalities on $\mathbb{H}^{n}$ were proved in [1, 7, 34, 37]. Later,
Ruzhansky and Suragan extended the above Hardy inequalities to homogeneous
groups, for a detailed description of these results we refer the interested
reader to [41] (also see [22]). Now coming back to the Grushin vector fields,
related Hardy inequalities of type (1.1) were proved by L. D’Ambrosio in [6]
and later it was extended by many authors in different directions, see e.g.,
[9, 24, 28, 30, 31, 45, 48, 52, 53, 54] and references therein. This is by no
means an exhaustive list. It is important to note that Ruzhansky and Suragan’s
most recent finding [40] has improved the inequality mentioned above in the
framework of equality, and hence obtained a sharp remainder term on the right-
hand of the inequality (1.1) for the homogeneous groups (also see [42] for
related results). They have exploited the group and dilation structures of the
homogeneous groups. We will use a unified strategy to propose numerous
improvements of Hardy equality (instead of inequality) using Bessel pairs and
spherical harmonics in this article. Specifically, we will re-prove all known
results with improved remainder terms found in the literature. We can compare
our findings with those of the Carnot groups in [11, 14, 24, 32].

The next topic we shall deal with here is the concern with the validity of
higher-order Hardy inequalities, namely, Rellich inequalities, which in the
Euclidean setting go back to [38], and thereafter several improvements have
been made to date. The natural extension to the sub-Laplacian setting is
relatively new starting from the work of L. D’Ambrosio in [7], where the
author proved Rellich inequality on the Heisenberg group of homogeneous
dimension $Q=2n+2,$ for $n>2$ and all $u\in C_{c}^{\infty}(\mathbb{H}^{n}),$
there holds

$\int_{\mathbb{H}^{n}}|\Delta_{\mathbb{H}^{n}}u|^{2}\,{\rm d}x\,{\rm d}t\geq
n^{2}(n-2)^{2}\int_{\mathbb{H}^{n}}\frac{u^{2}}{|x|^{4}}\,{\rm d}x\,{\rm d}t,$
(1.2)

where $\Delta_{\mathbb{H}^{n}}$ is the Kohn’s sub-Laplacian on the Heisenberg
group and $|x|$ is the standard Euclidean distance function in
$\mathbb{R}^{2n}.$ In particular, using (1.2), there holds

$\int_{\mathbb{H}^{n}}|\Delta_{\mathbb{H}^{n}}u|^{2}\,{\rm d}x\,{\rm d}t\geq
b\int_{\mathbb{H}^{n}}u^{2}\frac{|x|^{4}}{\varrho^{8}}\,{\rm d}x\,{\rm d}t,$
(1.3)

where $\varrho$ is the homogeneous distance function on the Heisenberg group
with $b:=n^{2}(n-2)^{2}$. Moreover, the inequality (1.3) is valid for $n>1$
with $b:=(n^{2}-1)^{2}$ for all the radial functions in
$C_{c}^{\infty}(\mathbb{H}^{n}),$ and the constant is sharp. Attached to the
above Rellich inequality (1.3), there is an analogous with a slightly
different form of Rellich inequality established by Y. Qiaohua in [51]. In
context to the other sub-Laplacian settings, namely, in the Carnot group,
Rellich-type inequalities and their improvement have been quite extensively
studied in the recent past, see e.g., [23, 43, 46, 50] and references therein.

Subsequently, in [45, Theorem 2.4], the authors provided an improved version
of the Rellich inequality for the Grushin operator. For $Q\geq 5$ and $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, then with sharp constant
there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\frac{Q^{2}(Q-4)^{2}}{16}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{4}}\>{\rm
d}x\>{\rm d}t,$ (1.4)

where $Q$ is the homogeneous dimension of the Grushin space and the constant
$\frac{Q^{2}(Q-4)^{2}}{16}$ is sharp. Moreover, in the same article, a Hardy-
Rellich-type inequality was also derived, which states as follows, for $Q\geq
4$ and $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, then with
sharp constant there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\frac{Q^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm d}t,$

and again the constant here is sharp. Driven by the aforementioned outcomes
acquired within the framework of the sub-Laplacian, we will present many
noteworthy enhancements of Hardy-Rellich and Rellich-type inequalities for the
degenerate Grushin vector fields here. To demonstrate identities and
inequalities, we will critically utilise the spherical harmonics for the
Grushin operator in the present work. To the best of our knowledge, this
method is new and offers a novel approach to dealing with such issues. We will
now quickly go over this article’s primary goal. Specifically, the main
contributions of the present article are as follows :

  * •

In Theorem 1.1, we first establish an Abstract Hardy identity involving a
Bessel pair. This identity is comparable to [10, Theorem 3.2]. In fact, for
the Grushin Vector fields, this identity generalises all of the earlier, well-
known Hardy inequality. Additionally, we replicate every known result for the
weighted version of the Hardy inequality by selecting certain Bessel pairings.
Second, we may establish Hardy inequalities for specific subspaces of the
smooth, compactly supported functions by tweaking the proof of the abstract
identity.

  * •

We demonstrate an abstract form of the Hardy-Rellich identity, or more
broadly, an inequality, by utilising the aforementioned abstract Hardy
identity and the Grushin operator’s structure. Given that the argument makes
use of spherical harmonics, which Garofalo-Shen [19] designed for the Grushin
operator, it appears to be rather delicate. This is the first use of spherical
harmonics to the proof of Hardy-Rellich inequality for the Grushin operator
that we are aware of. Our obtained results include explicit remainder terms
and are crisp. Rellich-type inequality, however, has been inferred for
hypoelliptic operators, such as the Baouendi-Grushin operator, using the
Bessel pair in [44].

  * •

This paper’s ultimate objective is to calculate the radial and spherical
contributions of the Grushin operator independently by using an identity
involving the radial and spherical contributions of the Grushin Laplacian’s
$L^{2}$-norm. We establish many identities that are comparable to the findings
for Euclidean vector fields found in [3, 4, 35, 36]. Using a more geometric
approach, we prove an identity for the Rellich situation, even for non-radial
functions, as opposed to an inequality.

###  1.2. Main Results

We lay down our primary theorems. We establish the Hardy identity involving
Bessel pairs in the following theorem. We employ the spherical harmonics for
the Grushin vector fields in the proof. We next re-prove various well-known
Hardy inequalities for the Grushin operator in the literature with much-
improved remainder terms by inserting specific Bessel pairs. This reads as
follows.

######  Theorem 1.1.

Let $G$ be the Grushin space with dimension $Q$ with $Q\geq 3$ and $B_{R}(o)$
denotes the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. If
$(V,W)$ is a $Q$-dimensional Bessel pair on $(0,R)$ with positive solution
$f$, then for $u\in C_{c}^{\infty}(B_{R}(o)\setminus\\{o\\})$ there holds

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm d}t$
$\displaystyle-\int_{B_{R}(o)}W(\varrho)u^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{G}\bigg{(}\frac{u}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm d}t,$

and

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{\varrho,G}u|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle-\int_{B_{R}(o)}W(\varrho)u^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{\varrho,G}\bigg{(}\frac{u}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm d}t.$

######  Remark 1.1.

An independent proof of [14, Theorem 3.2], which took into account more
general Hardy identities on the Carnot groups, is provided by Theorem 1.1.
Nevertheless, their methodology is dependent on certain factorization lemma.
While our method is quite flexible in applying the above theorem to specific
subspaces $\mathcal{S}_{j}$ (see Section 2 for the definition of
$\mathcal{S}_{j}$), where the Hardy identities with an improved constant are
derived (see Theorem 3.1 below), it necessitates the presence of spherical
harmonics for the operator and symmetric structure of the ambient space.

Next, we want to apply Theorem 1.1 to obtain the identities and inequalities
of Hardy-Rellich-type. Handling the second-order identities indeed requires
considerable care. After breaking down the Grushin operator on the radial
function, we obtain a Rellich identity involving Bessel pairs (for radial
functions) by applying the integration by parts formula found in the
previously mentioned Hardy identities on radial functions. The outcome is as
follows:

######  Theorem 1.2.

Let $G$ be the Grushin space with dimension $Q$ with $Q\geq 3$ and $B_{R}(o)$
denotes the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. Let
$\left(V,W\right)$ be a $Q$-dimensional Bessel pair on $(0,R)$ with positive
solution $f$ on $(0,R)$. Then for all radial function $u\in
C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$ there holds:

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$ $\displaystyle=\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+(Q-1)\int_{B_{R}(o)}\left(\frac{V(\varrho)}{\varrho^{2}}-\frac{V_{\varrho}(\varrho)}{\varrho}\right)\left|\nabla_{G}u\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}\left(\frac{u_{\varrho}}{f(\varrho)}\right)\right|^{2}f^{2}(\varrho)\>{\rm
d}x\>{\rm d}t,$

where the $\varrho$ in suffix means the derivative with respect to the radial
part $\varrho$.

######  Remark 1.2.

It is important to note that the aforementioned theorem refers to identity
rather than inequality. We may now deduce Hardy-Rellich identities with sharp
constant and precise remainder terms by selecting appropriate Bessel pairs
(see Corollary 1.2). In the Euclidean setting, the aforementioned can also be
compared with [8, Theorem 1.5].

We now demonstrate the Hardy-Rellich inequality for functions, which need not
be radial. We employ the spherical harmonics to address the non-radial
component of the Grushin operator in order to extend Theorem 1.2 to the non-
radial function. Restricted to the Grushin sphere, the proof contains a few
interesting stages that can be handled using the spectral information of the
Grushin operator. This brings us to the subsequent theorem:

######  Theorem 1.3.

Let $G$ be the Grushin space with dimension $Q$ with $Q\geq 3$ and $B_{R}(o)$
denotes the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. Let
$\left(V,W\right)$ be a $Q$-dimensional Bessel pair on $(0,R)$ with positive
solution $f$ on $(0,R)$ with $V\geq 0$. Then for all $u\in
C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$ there holds:

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle\geq\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t$
$\displaystyle+(Q-1)\int_{B_{R}(o)}\left(\frac{V(\varrho)}{\varrho^{2}}-\frac{V_{\varrho}(\varrho)}{\varrho}\right)\left|\nabla_{G}u\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}\left(\frac{u_{\varrho}}{f(\varrho)}\right)\right|^{2}f^{2}(\varrho)\>{\rm
d}x\>{\rm d}t,$ (1.5)

where the $\varrho$ in suffix means the derivative with respect to the radial
part $\varrho$ and with the assumption

$\displaystyle(Q-5)\frac{V(\varrho)}{\varrho^{2}}+3\frac{V_{\varrho}(\varrho)}{\varrho}-V_{\varrho\varrho}(\varrho)\geq
0.$ (1.6)

######  Remark 1.3.

The assumption (1.6) is not too restrictive. In fact in the seminal work of
[21, Theorem 3.1-3.3], in the Euclidean space similar assumption was
introduced involving both $V$ and $W,$ and therefore (1.6) is less restrictive
compared to [21]. Our assumption provides a wide class of Bessel pairs for
which (1.6) is satisfied. However, such an assumption which only depends on
$V$ was first established in [2, Theorem 3.2] on the Hyperbolic space. More
recently, there has been a new development in the Euclidean setting in [8]
where the authors refined the assumption of [21] and proved several
interesting consequences of their results.

######  Remark 1.4.

We note that in Theorem 1.3 when we move from radial to non-radial function,
we break equality and ultimately prove an inequality in (1.3). It is,
therefore, reasonable to inquire as to whether we might demonstrate equality
in (1.3) for a specific selection of $V$ and $W.$

As an immediate consequence, we will see the Hardy-Rellich and Rellich
inequalities with non-negative remainder terms. We obtain the following
corollaries.

######  Corollary 1.1.

Let $Q\geq 5$ and $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$.
Then there holds

  * (a)

$(\text{Hardy-Rellich inequality})$

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle\geq\frac{Q^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u_{\varrho}\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t.$

  * (b)

$(\text{Rellich inequality})$

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle\geq\frac{Q^{2}(Q-4)^{2}}{16}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{4}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\frac{Q^{2}}{4}\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u\varrho^{\frac{Q-4}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u_{\varrho}\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t.$

Let us rewrite the result for the radial functions. Instead of inequality, we
will have an identity.

######  Corollary 1.2.

Let $Q\geq 3$. Then for all radial function $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, there holds

  * (a)

$(\text{Hardy-Rellich identity})$

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\frac{Q^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u_{\varrho}\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t.$

  * (b)

$(\text{Rellich identity})$

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\frac{Q^{2}(Q-4)^{2}}{16}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{4}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\frac{Q^{2}}{4}\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u\varrho^{\frac{Q-4}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u_{\varrho}\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t.$

The above results extensively make use of the notion of spherical harmonics
developed for $\mathcal{L}_{G}$ by Garafalo and Shen in [19] and the explicit
form of $\varrho$. In [33] spherical harmonics were developed for
$\mathcal{L}_{G}^{\alpha},\alpha,m\in\mathbb{N}$. Our methods work for the
case of any $\alpha\in\mathbb{N}$. Although the analysis performed in the
present article could be extended to the general case for
$\alpha\in\mathbb{N},m>1$, for the sake of clarity, we state our results for
the $\alpha=1=m$ case.

Given Remark 1.4, we shall define spherical vector fields corresponding to the
Grushin vector fields.

######  Definition 1.1.

Let for $j=1,2,\ldots,n$ we define

$L_{j}=\frac{\partial}{\partial x_{j}}-\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\quad\mbox{and}\quad
L_{n+1}=|x|\frac{\partial}{\partial t}-|x|\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial\varrho}.$

Recall that $\varrho$ is the homogeneous norm corresponding to the Grushin
operator $\mathcal{L}_{G}$. Using the expression of $\varrho$ we can compute
that $\frac{\partial\varrho}{\partial x_{j}}=\frac{x_{j}|x|^{2}}{\varrho^{3}}$
for $j=1,2,\ldots,n$ and $\frac{\partial\varrho}{\partial
t}=\frac{2t}{\varrho^{3}}$. In the Euclidean setting analogous vector fields
were introduced by [3, 4], also (see [12]). In Lemma 5.1 below, we shall
provide all necessary, (although simple) identities involving the vector
fields $L_{j}.$ In particular, one can immediately see the following identity:

$\mathcal{L}_{G}=\mathcal{L}_{\varrho,G}+\sum_{j=1}^{n+1}L_{j}^{2},$

where $\mathcal{L}_{\varrho,G}$ is the radial component of $\mathcal{L}_{G}$
and is defined in Section 2. The above identity leads us to the following
theorem which compares the Grushin operator with its radial and spherical
components.

######  Theorem 1.4.

Let $Q\geq 4$. For all $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, we have

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{\varrho,G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t+\int_{\mathbb{R}^{n+1}}\frac{|\sum_{j=1}^{n+1}L_{j}^{2}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\frac{Q(Q-4)}{2}\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\frac{|L_{j}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm
d}t+2\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\frac{\partial}{\partial\varrho}\left(L_{j}u\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t.$

######  Remark 1.5.

It is noteworthy that classical Hardy-Rellich and Rellich identities for
functions (not necessarily radial) are provided by Theorem 1.4. We may get the
required identities with the exact remainder term and sharp constant by using
Theorem 1.2 on $\mathcal{L}_{\varrho,G}$. One might compare the aforementioned
theorem with [36, Theorem 1.2].

######  Remark 1.6.

Later in Theorem 5.1, we compute the deficit, namely the $L^{2}$-norm of the
Laplacian and radial components of the Laplacian for the Grushin operator with
spherical harmonics. In fact, in Theorem 1.4 this deficit is computed in terms
of the $L^{2}$\- norm of the spherical derivatives and with exact remainder
terms. Later in theorem 5.1, we shall compute this deficit in terms of
spherical harmonics and using subsequent lemmas we shall quantitatively
connect spherical derivatives with spherical harmonics. We postpone this
discussion for a while and we refer to Section 5 for detailed statements and
proofs.

We end this section with the outline of the article below.

The organisation of the article is as follows:

  * Section 1:

Introduction section contains a brief background on the Hardy, Hardy-Rellich
and Rellich inequality for the Grushin operator, the main results of this
article, and the approach we follow in the article.

  * Section 2:

Contains the polar coordinates, spherical harmonics and spectral properties of
the Grushin operator which were developed by Garofalo-Shen [19]. Moreover, the
necessary preliminaries of the Bessel pair are also discussed.

  * Section 3:

Contains the proof of several Hardy identities via Bessel pairs and some
straightforward implications of the Hardy identities.

  * Section 4:

This section is devoted to the proof of the abstract Hardy-Rellich identity.
The proof is divided into several steps and as a consequence, Hardy-Rellich
and Rellich inequalities with exact remainder terms are deduced.

  * Section 5:

This section begins with the definition of spherical vector fields and
factorizes the Grushin operator into the radial and spherical parts through a
simple identity. Then the Rellich-type identity has been derived by isolating
the radial contribution and the spherical contribution of the $L^{2}$ norm of
the Grushin operator $\mathcal{L}_{G}$ (see Theorem 1.4). Furthermore, the
spherical derivatives have been computed via spherical harmonics.

  * Section 6:

This section deals with the symmetrization principle on the Grushin space. The
main theme here is to prove analogous Pólya-Szegö inequality for the second-
order operator. The Hardy-Rellich-type identity is crucially used to prove
Theorem 6.3.

  * Section 7:

This section deals with applications of our theorems. The second-order
uncertainty principle (USP), in particular, Heisenberg and Hydrogen USP has
been derived. Further employing certain scaling arguments second-order
Caffarelli-Kohn-Nirenberg inequalities with sharp constants have been derived
from USP.

##  2\. Preliminaries and functional analysis set-up

This section is devoted to the introduction of the Grushin space, in
particular, the setting up of the polar coordinates, spherical harmonics and
spectral properties. In the end, we shall also define the Bessel pair and the
corresponding ordinary differential equations.

Polar coordinate. First, let us introduce the polar coordinate structure of
this space. We can write
$(x,t)=(\varrho,\phi,\theta_{1},\theta_{2},\cdots,\theta_{n-1})$ as below

$\displaystyle
x_{1}=\varrho\,\sin^{1/2}\phi\,\sin\theta_{1}\cdots\sin\theta_{n-1},$
$\displaystyle
x_{2}=\varrho\,\sin^{1/2}\phi\,\sin\theta_{1}\cdots\cos\theta_{n-1},$
$\displaystyle\cdots$ $\displaystyle
x_{n}=\varrho\,\sin^{1/2}\phi\,\cos\theta_{1},$ $\displaystyle
t=\frac{\varrho^{2}}{2}\cos\phi,$

where $0<\phi<\pi$, $0<\theta_{i}<\pi$ for $i=1,2,\cdots,n-2$ and
$0<\theta_{n-1}<2\pi$. Now denoting $r=|x|$, we have

$\displaystyle
r=|x|=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}=\varrho\sin^{1/2}\phi\implies\sin\phi=\frac{|x|^{2}}{\varrho^{2}}=\psi.$

Now this will give

$\displaystyle|\nabla_{G}\varrho(x,t)|^{2}=|\nabla_{x}\varrho|^{2}+|x|^{2}(\partial_{t}\varrho)^{2}=\frac{|x|^{2}}{\varrho^{2}(x,t)}=\psi(x,t)=\sin\phi=\frac{r^{2}}{\varrho^{2}},$

and

$\displaystyle\mathcal{L}_{G}\varrho(x,t)=(Q-1)\frac{|x|^{2}}{\varrho^{3}(x,t)}=(Q-1)\frac{\psi(x,t)}{\varrho(x,t)}.$

Using polar coordinates, we obtain

$\displaystyle\>{\rm d}x=r^{n-1}{\rm d}r\,{\rm d}w,\text{ and }\>{\rm
d}x\>{\rm d}t=\frac{1}{2}\varrho^{n+1}(\sin\phi)^{\frac{n-2}{2}}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w,$

where $\>{\rm d}w$ is the spherical measure on the standard round sphere
$\mathbb{S}^{n-1}$ in the Euclidean space $\mathbb{R}^{n}$. Now consider the
$\varrho$-sphere on the Grushin space as follows

$\displaystyle\Omega=\biggl{\\{}(x,t)\in\mathbb{R}^{n+1}:\varrho(x,t)=1\biggr{\\}}.$

Moreover, using the standard argument there holds the following polar
integration. Let $u$ be some integrable function, and then we have

$\displaystyle\int_{\mathbb{R}^{n+1}}u(x,t)\>{\rm d}x\>{\rm d}t$
$\displaystyle=\int_{\Omega}\int_{0}^{\infty}u(\varrho,\phi,w)\frac{1}{2}\varrho^{n+1}(\sin\phi)^{\frac{n-2}{2}}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=\int_{\Omega}\int_{0}^{\infty}u(\varrho,\sigma)\frac{\varrho^{n+1}}{2\psi}\>{\rm
d}\varrho\>{\rm d}\Omega,$

where

${\rm d}\Omega=(\sin\phi)^{\frac{n}{2}}\>{\rm d}\phi\>{\rm d}w,\text{ and
}\sigma=(\phi,w),\text{ with }w\in\mathbb{S}^{n-1}.$

Operators and its radial part. Now we will write all the operators in terms of
polar coordinate structure. The Grushin gradient in polar coordinates may be
defined as follows

$\displaystyle\nabla_{G}=\psi^{1/2}\bigg{(}\partial_{\varrho},\frac{2}{\varrho}\partial_{\phi},\frac{1}{\varrho\sin\phi}\nabla_{w}\bigg{)},$

where $(x,t)=(\varrho,\phi,w)$, $w\in\mathbb{S}^{n-1}$, $\nabla_{w}$ is the
gradient on $\mathbb{S}^{n-1}$. Also, it can be written as follows

$\displaystyle\nabla_{G}=\psi^{1/2}\bigg{(}\partial_{\varrho},\frac{2}{\varrho}\nabla_{\sigma}\bigg{)},$

where $\nabla_{\sigma}$ is the gradient operator on the Grushin
$\varrho$-sphere. We refer to [13, Page. 11] for more details. In polar
coordinates, the Grushin operator takes the form

$\displaystyle\mathcal{L}_{G}=\psi\biggl{\\{}\frac{\partial^{2}}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial}{\partial\varrho}+\frac{4}{\varrho^{2}}\mathcal{L}_{\sigma}\biggr{\\}},$
(2.1)

where $\sigma=(\phi,w)$ and $\mathcal{L}_{\sigma}$ is the Laplacian on Grushin
$\varrho$-sphere and $\sigma=(\phi,w).$

To this end, we shall define the radial contribution of the operators. The
radial Grushin gradient and the radial Grushin operator are defined as

$\displaystyle\nabla_{\varrho,G}=\psi^{1/2}\big{(}\partial_{\varrho},0,0\big{)},\quad\text{
and
}\quad\mathcal{L}_{\varrho,G}=\psi\biggl{\\{}\frac{\partial^{2}}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial}{\partial\varrho}\biggr{\\}}.$

Spherical harmonics. Note that, for $k=0,1,\cdots,$ we form the function
$u_{k}(\varrho,\sigma):=\varrho^{k}g(\sigma)$. Then, $u_{k}(\varrho,\sigma)$
is a solution of $\mathcal{L}_{G}=0$ if and only if

$\displaystyle\mathcal{L}_{\sigma}g=-\frac{k(n+k)}{4}g.$ (2.2)

Before going further let us mention some key observations of spherical
harmonics in Grushin space. Let us recall some known facts :

######  Lemma 2.1.

[19, Lemma 2.6] Let $k$ be a nonnegative integer and $l\equiv k(\mbox{mod}\
2),$ with $0\leq l\leq k.$ Suppose that $Y_{l}$ is a spherical harmonic of
degree $l.$ Then

$g(\phi,w):=g(\sigma)=\sin^{l/2}(\phi)C_{\frac{k-l}{2}}^{\frac{l}{2}+\frac{n}{4}}(\cos\phi)Y_{l}(w)$

satisfies (2.2), where $C_{\frac{k-l}{2}}^{\frac{l}{2}+\frac{n}{4}}$ is a
Gegenbauer polynomial.

Now we define,

$\mathcal{H}_{k}:=\,\mbox{Span}\left\\{\sin^{l/2}(\phi)C_{\frac{k-l}{2}}^{\frac{l}{2}+\frac{n}{4}}(\cos\phi)Y_{l,j}(w):\
j=1,2,\ldots,d_{l},\quad 0\leq l\leq k,\quad l\equiv k(\mbox{mod}\
2)\right\\},$

where $d_{l}=\frac{(n+2l-2)\Gamma(n+l-2)}{\Gamma(l+1)\Gamma(n-1)}.$

######  Lemma 2.2.

[19, Lemma 2.11] The following direct sum decomposition holds :

$\displaystyle L^{2}(\Omega,{\rm
d}\Omega)=\bigoplus_{k=0}^{\infty}\mathcal{H}_{k},$

where the spaces $\mathcal{H}_{k}$ are mutually orthogonal in
$L^{2}(\Omega,{\rm d}\Omega).$

Given the above lemmas, we have, $u(x,t)=u(\varrho,\sigma)\in
C_{c}^{\infty}(\mathbb{R}^{n+1})$, $\varrho\in({0},\infty)$ and
$\sigma\in\Omega$, thus we can write

$u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma)$

in $L^{2}(\mathbb{R}^{n+1})$, where $\\{\Phi_{k}\\}$ is an orthogonal system
of spherical harmonics in $L^{2}(\Omega)$ and

$d_{k}(\varrho)=\int_{\Omega}u(\varrho,\sigma)\Phi_{k}(\sigma)\ {\rm
d}\Omega\,.$

A spherical harmonic $\\{\Phi_{k}\\}$ of order $k$ satisfies

$-\mathcal{L}_{\sigma}\Phi_{k}=\lambda_{k}\Phi_{k},$

for all $k\in\mathbb{N}\cup\\{0\\}$, where $\lambda_{k}=\frac{k(k+n)}{4}$. In
this context, let us write the gradient and Grushin operators as follows :

$\displaystyle|\nabla_{G}u|^{2}=\psi\bigg{(}\left(\partial_{\varrho}u\right)^{2}+\frac{4}{\varrho^{2}}|\nabla_{\sigma}u|^{2}\bigg{)},$

and

$\displaystyle(\mathcal{L}_{G}u)^{2}$
$\displaystyle=\psi^{2}\left(\partial^{2}_{\varrho}u+\frac{(Q-1)}{\varrho}\partial_{\varrho}u\right)^{2}+\psi^{2}\frac{16}{\varrho^{4}}(\mathcal{L}_{\sigma}u)^{2}$
$\displaystyle+2\psi^{2}\left(\partial^{2}_{\varrho}u+\frac{(Q-1)}{\varrho}\partial_{\varrho}u\right)\frac{4}{\varrho^{2}}(\mathcal{L}_{\sigma}u)u.$

Along with this, the radial contribution of the operators will look as follows
:

$\displaystyle|\nabla_{\varrho,G}u|^{2}=\psi\left(\partial_{\varrho}u\right)^{2},$

and

$\displaystyle(\mathcal{L}_{\varrho,G}u)^{2}$
$\displaystyle=\psi^{2}\left(\partial^{2}_{\varrho}u+\frac{(Q-1)}{\varrho}\partial_{\varrho}u\right)^{2}\,.$

We can deduce the following integration by parts formula :

######  Lemma 2.3.

Let $\Phi_{k}\in\mathcal{H}_{k}$ and $\Phi_{j}\in\mathcal{H}_{j},$ then there
holds

$\int_{\Omega}\nabla_{\sigma}\Phi_{k}(\sigma)\nabla_{\sigma}\Phi_{j}(\sigma)\>{\rm
d}\Omega=-\int_{\Omega}\mathcal{L}_{\sigma}\Phi_{k}(\sigma)\Phi_{j}(\sigma)\>{\rm
d}\Omega,$

for all $k,\,j\in\mathbb{N}.$

###### Proof.

Let $f\in C^{\infty}_{c}(0,\infty)$ and for $k\in\mathbb{N}$, we define

$\displaystyle
u_{k}(x,t):=u_{k}(\varrho,\sigma)=f(\varrho)\Phi_{k}(\sigma)\quad\text{ for
all }(x,t)\in\mathbb{R}^{n+1}.$

Now, let us describe the explicitly known $\Phi_{k}(\sigma)$. Let $l\equiv
m(\text{mod}\,2)$. Then we know

$\displaystyle
g_{l,m}(\phi,\omega)=(\sin\phi)^{l/2}C^{\frac{l}{2}+\frac{n}{4}}_{\frac{m-l}{2}}(\cos\phi)Y_{l}(\omega),$

where $Y_{l}$ is an $l$ degree spherical harmonic on $\mathbb{S}^{n-1}$ w.r.t.
the standard Laplacian on $\mathbb{R}^{n}$. In the spherical polar coordinates
on the Grushin space, we have $|x|=\varrho(\sin\phi)^{1/2}$. Therefore, we can
write

$\displaystyle(\sin\phi)^{l/2}Y_{l}(\omega)=Y_{l}\left(\frac{x}{\varrho}\right).$

Note that when $\varrho$ lies in a compact subset of $(0,\infty)$, then

$\displaystyle(x,t)\longmapsto\varrho(x,t)^{\gamma},\quad\text{ for
}\gamma\in\mathbb{R},$

is a smooth function of $(x,t)$. Similarly,

$\displaystyle(x,t)\longmapsto
C^{\frac{l}{2}+\frac{n}{4}}_{\frac{m-l}{2}}\left(\frac{2t}{\varrho(x,t)^{2}}\right)\quad\text{
and }\quad(x,t)\longmapsto Y_{l}\left(\frac{x}{\varrho}\right)$

are smooth functions of $(x,t)$. As ${\Phi_{k}}^{\prime}s$ are of the form
$g_{l,m}$, for some $m\geq 0$ and $l\equiv m(\text{mod}\,2)$. Therefore, we
can apply integration by parts in $\mathbb{R}^{n+1}$ to $u_{k}$ and $u_{j}$.

Set

$\displaystyle
I=\int_{\mathbb{R}^{n+1}}(\nabla_{G}u_{k},\nabla_{G}u_{j})\>{\rm d}x\>{\rm
d}t.$

In polar coordinates, the above integral can be split into a sum of two terms
involving gradient w.r.t. radial and spherical coordinates as follows

$I=\int_{0}^{\infty}\int_{\Omega}f^{\prime}(\varrho)^{2}\Phi_{k}(\sigma)\Phi_{j}(\sigma)\varrho^{Q-1}\>{\rm
d}\Omega\>{\rm d}\varrho\\\
+\int_{0}^{\infty}\int_{\Omega}f(\varrho)^{2}4\varrho^{-2}\nabla_{\sigma}\Phi_{k}(\sigma)\nabla_{\sigma}\Phi_{j}(\sigma)\varrho^{Q-1}\>{\rm
d}\Omega\>{\rm d}\varrho.$

On integration by parts by treating $I$ as an integral on $\mathbb{R}^{n+1}$,
we get that

$\displaystyle I=-\int_{\mathbb{R}^{n+1}}\mathcal{L}_{G}(u_{k})u_{j}\>{\rm
d}x\>{\rm d}t.$

Again splitting the radial and angular part of $\mathcal{L}_{G}$, the above
expression can be written as

$I=-\int_{0}^{\infty}\int_{\Omega}\mathcal{L}_{\varrho,G}f(\varrho)f(\varrho)\Phi_{k}(\sigma)\Phi_{j}(\sigma)\varrho^{Q-1}\>{\rm
d}\Omega\>{\rm d}\varrho\\\
-\int_{0}^{\infty}\int_{\Omega}f(\varrho)^{2}4\varrho^{-2}\mathcal{L}_{\sigma}\Phi_{k}(\sigma)\Phi_{j}(\sigma)\varrho^{Q-1}\>{\rm
d}\Omega\>{\rm d}\varrho.$

On equating the above two expressions for all $f\in C^{\infty}_{c}(0,\infty)$
we get that

$\int_{\Omega}\nabla_{\sigma}\Phi_{k}(\sigma)\nabla_{\sigma}\Phi_{j}(\sigma)\>{\rm
d}\Omega=-\int_{\Omega}\mathcal{L}_{\sigma}\Phi_{k}(\sigma)\Phi_{j}(\sigma)\>{\rm
d}\Omega$

for all $k,j\in\mathbb{N}$. This completes the proof. ∎

Projection operators. For each $k\geq 0$, we define $\mathcal{H}_{k}$ as the
closed subspace spanned by spherical harmonics of order $k$ multiplied by
radial functions. Any function $u(x)\in C_{c}^{\infty}(\mathbb{R}^{n+1})$ can
be expressed as follows:

$u(x)=u(\varrho,\sigma)=\sum_{k=0}^{\infty}\mathcal{P}_{k}u,$

where
$\mathcal{P}_{k}u=d_{k}(\varrho)\Phi_{k}(\sigma)\in\mathcal{H}_{k}(\mathbb{R}^{n+1})$
and $\mathcal{P}_{k}$ is the projection operator on $\mathcal{H}_{k}$.

Subspace. We have already introduced the spherical harmonics decomposition for
a function $u\in C_{c}^{\infty}(\mathbb{R}^{n+1})$. We will introduce a
certain subspace of $C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$. First,
consider $u\in C_{c}^{\infty}(\mathbb{R}^{n+1})$ and we can write

$u(x)=u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma)=\sum_{k=0}^{\infty}\mathcal{P}_{k}(\varrho,\sigma)$

and now define for $j\in\mathbb{N}\cup\\{0\\}$ and
$\Omega\subset\mathbb{R}^{n+1}$,

$\displaystyle\mathcal{S}_{j}(\Omega)$ $\displaystyle=\biggl{\\{}u\in
C_{c}^{\infty}(\Omega):\mathcal{P}_{0}(\varrho,\sigma)=\mathcal{P}_{1}(\varrho,\sigma)=\cdots=\mathcal{P}_{j}(\varrho,\sigma)=0\biggr{\\}}$
$\displaystyle=\biggl{\\{}u\in
C_{c}^{\infty}(\Omega):u(x)=u(\varrho,\sigma)=\sum_{k=j+1}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma)\biggr{\\}}.$

For notational clarity, we want to mention that
$\mathcal{S}_{-1}(\Omega)=C_{c}^{\infty}(\Omega).$

######  Definition 2.1 (Bessel pair).

We say that a pair $(V,W)$ of $C^{1}$-functions is a Bessel pair on $(0,R)$
for some $0<R\leq\infty$ if the ordinary differential equation:

$(Vy^{\prime})^{\prime}+Wy=0$

admits a positive solution $f$ on the interval $(0,R)$.

######  Definition 2.2.

We say that a pair $(V,W)$ of $C^{1}$-functions is a $Q$-dimensional Bessel
pair on $(0,R)$ for some $0<R\leq\infty$ if the ordinary differential
equation:

$(\varrho^{Q-1}Vy^{\prime})^{\prime}+\varrho^{Q-1}Wy=0$

admits a positive solution $f$ on the interval $(0,R)$.

##  3\. Proof of Theorem 1.1 and corollaries

This section aims to establish Hardy-type identity via a Bessel pair, namely
Theorem 1.1. Moreover, we shall also establish a refined Hardy-type inequality
for functions in $\mathcal{S}_{j}$ and subsequently prove several improvements
of Hardy inequality, namely, the improvement of the optimality of the
classical Hardy constant and weighted version of the classical Hardy
inequality. We begin with the proof of Theorem 1.1.

Proof of Theorem 1.1: Let us begin with $u=fv$ and then from spherical
harmonic decomposition we can write

$\displaystyle
u(\varrho,\sigma)=\sum_{k=0}^{\infty}f(\varrho)d_{k}(\varrho)\Phi_{k}(\sigma).$

Next exploiting the polar coordinate structure, using orthogonal properties of
$\\{\Phi_{k}\\},$ and Lemma 2.3 we deduce

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm d}t$
$\displaystyle=\frac{1}{2}\sum_{k=0}^{\infty}\int_{\Omega}\int_{0}^{R}V\bigg{[}{(fd_{k})^{\prime}}^{2}\Phi_{k}^{2}+\frac{4(fd_{k})^{2}}{\varrho^{2}}|\nabla_{\sigma}\Phi_{k}|^{2}\bigg{]}\varrho^{n+1}\>{\rm
d}\varrho\>{\rm d}\Omega$
$\displaystyle=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}\bigg{[}Vf^{2}{d_{k}^{\prime}}^{2}\varrho^{n+1}+V{(d_{k}^{2})}^{\prime}ff^{\prime}\varrho^{n+1}+V{f^{\prime}}^{2}d_{k}^{2}\varrho^{n+1}+4\lambda_{k}Vf^{2}d_{k}^{2}\varrho^{n-1}\bigg{]}\>{\rm
d}\varrho.$

Now using integration by parts we deduce,

$\displaystyle\int_{0}^{R}V{(d_{k}^{2})}^{\prime}ff^{\prime}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle=-\int_{0}^{R}V^{\prime}d_{k}^{2}ff^{\prime}\varrho^{n+1}\>{\rm
d}\varrho-\int_{0}^{R}Vd_{k}^{2}{f^{\prime}}^{2}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle-\int_{0}^{R}Vd_{k}^{2}ff^{\prime\prime}\varrho^{n+1}\>{\rm
d}\varrho-(Q-1)\int_{0}^{R}Vd_{k}^{2}ff^{\prime}\varrho^{n}\>{\rm d}\varrho.$

Therefore, we have

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm d}t$
$\displaystyle=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}\bigg{[}Vf^{2}{d_{k}^{\prime}}^{2}\varrho^{n+1}-V^{\prime}d_{k}^{2}ff^{\prime}\varrho^{n+1}-Vd_{k}^{2}ff^{\prime\prime}\varrho^{n+1}$
$\displaystyle\quad\quad-(Q-1)Vd_{k}^{2}ff^{\prime}\varrho^{n}+4\lambda_{k}Vf^{2}d_{k}^{2}\varrho^{n-1}\bigg{]}\>{\rm
d}\varrho.$

Also, we have

$\displaystyle\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{G}\bigg{(}\frac{u}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}\bigg{[}Vf^{2}{d_{k}^{\prime}}^{2}\varrho^{n+1}+4\lambda_{k}Vf^{2}d_{k}^{2}\varrho^{n-1}\bigg{]}\>{\rm
d}\varrho.$

Finally, after substituting we obtain

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t-\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{G}\bigg{(}\frac{u}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}\bigg{[}-V^{\prime}f^{\prime}-Vf^{\prime\prime}-(Q-1)\frac{V}{\varrho}f^{\prime}\bigg{]}d_{k}^{2}f\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}Wd_{k}^{2}f^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle=\int_{B_{R}(o)}W(\varrho)u^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t.$

The second identity follows with similar steps and decomposition. ∎

The above theorem leads us to the following result.

######  Corollary 3.1.

Let $G$ be the Grushin space with dimension $Q$ with $Q\geq 3$ and $B_{R}(o)$
denotes the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. If
$(V,W)$ is a $Q$-dimensional Bessel pair on $(0,R)$, with $V\geq 0$, then for
$u\in C_{0}^{\infty}(B_{R}(o)\setminus\\{o\\})$ there holds

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t\geq\int_{B_{R}(o)}W(\varrho)u^{2}|\nabla_{G}\varrho|^{2}\>{\rm d}x\>{\rm
d}t,$

and

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{\varrho,G}u|^{2}\>{\rm
d}x\>{\rm d}t\geq\int_{B_{R}(o)}W(\varrho)u^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t.$

In the next theorem, we shall refine the above theorem for functions in
$\mathcal{S}_{j},$ a subspace of
$C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\}).$ We deduce that the optimal
Hardy constant depends on the parameter $j.$ Also, see [16] for related
results in the Riemannian symmetric manifolds, in particular, in the
hyperbolic space.

######  Theorem 3.1.

Let $G$ be the Grushin space with dimension $Q$ with $Q\geq 3$ and $B_{R}(o)$
denotes the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. If
$(V,W)$ is a $Q$-dimensional Bessel pair on $(0,R)$ with positive solution
$f$, then for $u\in\mathcal{S}_{j}(B_{R}(o)\setminus\\{o\\})$ there holds

$\displaystyle\int_{B_{R}(o)}V(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm d}t$
$\displaystyle\geq\int_{B_{R}(o)}W(\varrho)u^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+(j+1)(Q+j-1)\int_{B_{R}(o)}\frac{V(\varrho)}{\varrho^{2}}u^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{\varrho,G}\bigg{(}\frac{u}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm d}t.$

###### Proof.

The proof is exactly similar to the proof of Theorem 1.1. In that proof, the
term involving $\lambda_{k}$ was cancelling. But in this case, as the last
term is radial gradient, the term involving $\lambda_{k}$ will not be
cancelled out. Then using the estimate $4\lambda_{k}\geq(j+1)(Q+j-1)$ for all
$k\geq j+1$ on $\mathcal{S}_{j}(B_{R}(o)\setminus\\{o\\})$ the result follows.
∎

######  Remark 3.1.

Observe that $\big{(}1,\frac{(Q-2)^{2}}{4}\frac{1}{\varrho^{2}}\big{)}$ is a
$Q$-dimensional Bessel pair on $(0,\infty)$ with positive solution
$\varrho^{-\frac{Q-2}{2}}$. Then substituting this particular Bessel pair in
Theorem 3.1 we obtain the Hardy inequality with a constant larger than the
classical Hardy constant. This reads as follows : for
$u\in\mathcal{S}_{j}(B_{R}(o)\setminus\\{o\\})$ there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm d}t$
$\displaystyle\geq\bigg{[}\frac{(Q-2)^{2}}{4}+(j+1)(Q+j-1)\bigg{]}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}}{\varrho^{2}}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t.$

######  Corollary 3.2.

Let $G$ be the Grushin space with dimension $Q\geq 3$ and
$\alpha\in\mathbb{R}$. Then for all $u\in
C_{0}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$ there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{\alpha}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle-\frac{(Q-2-\alpha)^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{\alpha+2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\big{|}\nabla_{G}\big{(}u\varrho^{\frac{Q-2-\alpha}{2}}\big{)}\big{|}^{2}\>{\rm
d}x\>{\rm d}t,$

and

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{\varrho,G}u|^{2}}{\varrho^{\alpha}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle-\frac{(Q-2-\alpha)^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{\alpha+2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\big{|}\nabla_{\varrho,G}\big{(}u\varrho^{\frac{Q-2-\alpha}{2}}\big{)}\big{|}^{2}\>{\rm
d}x\>{\rm d}t.$

###### Proof.

We apply Theorem 1.1 to the $Q$-dimensional Bessel pair
$\left(\frac{1}{\varrho^{\alpha}},\frac{(Q-2-\alpha)^{2}}{4}\frac{1}{\varrho^{\alpha+2}}\right)$
on $(0,\infty)$ with $f(r)=\varrho^{\frac{2-Q+\alpha}{2}}$ to get the desired
results. ∎

######  Remark 3.2.

The above corollary can be compared with the identity similar to [35, Theorem
1] on the Euclidean space in terms of weighted version. Notice that we have
used the spherical harmonic technique, whereas in [35] a different approach
was used. Moreover, in the above identity by neglecting the non-negative
remainder term we obtain

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{\varrho,G}u|^{2}}{\varrho^{\alpha}}\>{\rm
d}x\>{\rm
d}t\geq\frac{(Q-2-\alpha)^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{\alpha+2}}\>{\rm
d}x\>{\rm d}t,$

holds for every $u\in C_{0}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$. Now
by noticing that $|\nabla_{G}u|^{2}\geq|\nabla_{\varrho,G}u|^{2}$, we deduce
that this weighted Hardy inequality immediately gives an improvement of the
original weighted Hardy inequality on Grushin space. Also, this implies that
the constant is the optimal one.

##  4\. Abstract Hardy-Rellich inequality: Proof of Theorem 1.2 and Theorem
1.3

Now we will prove the abstract Hardy-Rellich identity via the Bessel pair for
radial functions. We shall first prove our theorem for radial functions,
namely, Theorem 1.2. The proof is a straightforward application of integration
by parts, and expanding the square of the Grushin operator on the radial
function. We begin the proof of Theorem 1.2.

Proof of Theorem 1.2: Let us assume $\psi=\sin\phi=|\nabla_{G}\varrho|^{2}$.
Also observe that for the radial function $u$, we have
$|\nabla_{G}u|^{2}=\psi\big{(}\frac{\partial u}{\partial\varrho}\big{)}^{2}$.
Now using polar coordinate decomposition we have

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\frac{1}{2}\int_{\Omega}\int_{0}^{R}V(\varrho)\frac{\psi^{2}}{\psi}\bigg{[}\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}\bigg{]}^{2}\varrho^{n+1}(\sin\phi)^{\frac{n-2}{2}}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=\frac{1}{2}\int_{\Omega}\int_{0}^{R}V(\varrho)\bigg{[}\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}\bigg{]}^{2}\varrho^{n+1}(\sin\phi)^{\frac{n}{2}}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=\frac{1}{2}\int_{0}^{R}V(\varrho)\bigg{[}\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}\bigg{]}^{2}\varrho^{n+1}\>{\rm
d}\varrho\int_{\Omega}(\sin\phi)^{\frac{n}{2}}\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=\frac{1}{2}|\Omega|\int_{0}^{R}V(\varrho)\bigg{[}\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}\bigg{]}^{2}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle=\frac{1}{2}|\Omega|\int_{0}^{R}V(\varrho)\bigg{[}\left(\frac{\partial^{2}u}{\partial\varrho^{2}}\right)^{2}+\frac{(Q-1)^{2}}{\varrho^{2}}\left(\frac{\partial
u}{\partial\varrho}\right)^{2}+2\frac{(Q-1)}{\varrho}\frac{\partial^{2}u}{\partial\varrho^{2}}\frac{\partial
u}{\partial\varrho}\bigg{]}\varrho^{n+1}\>{\rm d}\varrho.$

We will deal with each integral separately. Let us define and apply Theorem
1.1, and we have

$\displaystyle I_{1}$
$\displaystyle:=\frac{1}{2}|\Omega|\int_{0}^{R}V(\varrho)\left(\frac{\partial^{2}u}{\partial\varrho^{2}}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle=\frac{1}{2}\int_{\Omega}\int_{0}^{R}V(\varrho)\left(\frac{\partial^{2}u}{\partial\varrho^{2}}\right)^{2}(\sin\phi)^{\frac{n}{2}}\varrho^{n+1}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=\int_{\Omega}\int_{0}^{R}V(\varrho)\psi\left(\frac{\partial^{2}u}{\partial\varrho^{2}}\right)^{2}\frac{1}{2}\varrho^{n+1}(\sin\phi)^{\frac{n-2}{2}}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}u_{\varrho}\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{B_{R}(o)}W(\varrho)u_{\varrho}^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm
d}t+\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{G}\bigg{(}\frac{u_{\varrho}}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm d}t$ $\displaystyle=\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm
d}t+\int_{B_{R}(o)}V(\varrho)\bigg{|}\nabla_{G}\bigg{(}\frac{u_{\varrho}}{f(\varrho)}\bigg{)}\bigg{|}^{2}f(\varrho)^{2}\>{\rm
d}x\>{\rm d}t.$

Next, consider

$\displaystyle I_{2}$
$\displaystyle:=\frac{1}{2}|\Omega|\int_{0}^{R}V(\varrho)\frac{(Q-1)^{2}}{\varrho^{2}}\left(\frac{\partial
u}{\partial\varrho}\right)^{2}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle=\frac{1}{2}\int_{\Omega}\int_{0}^{R}V(\varrho)\frac{(Q-1)^{2}}{\varrho^{2}}\left(\frac{\partial
u}{\partial\varrho}\right)^{2}(\sin\phi)^{\frac{n}{2}}\varrho^{n+1}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=(Q-1)^{2}\int_{\Omega}\int_{0}^{R}\frac{V(\varrho)}{\varrho^{2}}\psi\left(\frac{\partial
u}{\partial\varrho}\right)^{2}\frac{1}{2}\varrho^{n+1}(\sin\phi)^{\frac{n-2}{2}}\>{\rm
d}\varrho\>{\rm d}\phi\>{\rm d}w$
$\displaystyle=(Q-1)^{2}\int_{B_{R}(o)}\frac{V(\varrho)}{\varrho^{2}}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

Then, using by parts we have

$\displaystyle I_{3}$
$\displaystyle:=\frac{1}{2}|\Omega|\int_{0}^{R}2V(\varrho)\frac{(Q-1)}{\varrho}\frac{\partial^{2}u}{\partial\varrho^{2}}\frac{\partial
u}{\partial\varrho}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle=-\frac{1}{2}|\Omega|(Q-1)\int_{0}^{R}\frac{V_{\varrho}(\varrho)}{\varrho}\bigg{(}\frac{\partial
u}{\partial\varrho}\bigg{)}^{2}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle-\frac{1}{2}|\Omega|(Q-1)(Q-2)\int_{0}^{R}\frac{V(\varrho)}{\varrho^{2}}\bigg{(}\frac{\partial
u}{\partial\varrho}\bigg{)}^{2}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle=-(Q-1)\int_{B_{R}(o)}\frac{V_{\varrho}(\varrho)}{\varrho}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm
d}t-(Q-1)(Q-2)\int_{B_{R}(o)}\frac{V(\varrho)}{\varrho^{2}}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

Hence, we have

$\displaystyle
I_{2}+I_{3}=-(Q-1)\int_{B_{R}(o)}\frac{V_{\varrho}(\varrho)}{\varrho}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm
d}t+(Q-1)\int_{B_{R}(o)}\frac{V(\varrho)}{\varrho^{2}}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

Finally, summing all these we deduce the desired result. ∎

Next, we shall move on to prove the Theorem 1.3. The proof is quite delicate
and we shall make use of the Theorem 1.2 for the radial part of the function.
Indeed we exploit the spherical harmonics and the representations of functions
on each $\mathcal{H}_{k}$ to deduce the theorem. We shall begin the proof of
Theorem 1.3 below.

Proof of Theorem 1.3: Again assume $\psi=\sin\phi=|\nabla_{G}\varrho|^{2}$. By
spherical harmonics expansion, we can write
$u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma)$. Using
the orthogonality of $\\{\Phi_{k}\\}$ on $L^{2}(\Omega,\>{\rm d}\Omega)$ we
have

$\displaystyle\int_{\Omega}(\mathcal{L}_{G}u)^{2}\psi^{-2}\>{\rm d}\Omega$
$\displaystyle=\sum_{k=0}^{\infty}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}+16\sum_{k=0}^{\infty}\frac{d_{k}^{2}}{\varrho^{4}}\int_{\Omega}(\mathcal{L}_{\sigma}\Phi_{k})^{2}\>{\rm
d}\Omega$
$\displaystyle+8\sum_{k=0}^{\infty}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)\frac{d_{k}}{\varrho^{2}}\int_{\Omega}(\mathcal{L}_{\sigma}\Phi_{k})\Phi_{k}\>{\rm
d}\Omega.$

Also, recall that $\\{\Phi_{k}\\}$ is an orthonormal system of spherical
harmonics in $L^{2}(\Omega)$ and satisfies

$-\mathcal{L}_{\sigma}\Phi_{k}=\lambda_{k}\Phi_{k},$

for all $k\in\mathbb{N}\cup\\{0\\}$, where $\lambda_{k}=\frac{k(k+n)}{4}$. Now
we will divide the proof into a few steps.

Step 1. In this step we decompose the l.h.s. of (1.3) and then by polar
coordinate we have

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\bigg{[}\int_{0}^{R}V(\varrho)\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle+16\lambda_{k}^{2}\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\int_{0}^{R}V(\varrho)\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)d_{k}\varrho^{n-1}\>{\rm
d}\varrho\bigg{]}.$

On the other hand, for each radial function $d_{k}$, using Theorem 1.2, we
deduce

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}d_{k})^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\frac{1}{2}|\Omega|\int_{0}^{R}V(\varrho)\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho$ $\displaystyle=\int_{B_{R}(o)}W(\varrho)|\nabla_{G}d_{k}|^{2}\>{\rm
d}x\>{\rm
d}t+(Q-1)\int_{B_{R}(o)}\left(\frac{V(\varrho)}{\varrho^{2}}-\frac{V_{\varrho}(\varrho)}{\varrho}\right)\left|\nabla_{G}d_{k}\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}\left(\frac{d_{k}^{\prime}}{f(\varrho)}\right)\right|^{2}f^{2}(\varrho)\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\frac{1}{2}|\Omega|\bigg{[}\int_{0}^{R}W(\varrho){d_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho+(Q-1)\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle-(Q-1)\int_{0}^{R}V_{\varrho}(\varrho){d_{k}^{\prime}}^{2}\varrho^{n}\>{\rm
d}\varrho+\int_{0}^{R}V(\varrho)\left|\left(\frac{d_{k}^{\prime}}{f(\varrho)}\right)^{\prime}\right|^{2}f^{2}(\varrho)\varrho^{n+1}\>{\rm
d}\varrho\bigg{]}.$

Using this in the last identity and after simplifying we have

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\sum_{k=0}^{\infty}\bigg{[}\int_{0}^{R}W(\varrho){d_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle+(Q-1)\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho-(Q-1)\int_{0}^{R}V_{\varrho}(\varrho){d_{k}^{\prime}}^{2}\varrho^{n}\>{\rm
d}\varrho$
$\displaystyle+\int_{0}^{R}V(\varrho)\left|\left(\frac{d_{k}^{\prime}}{f(\varrho)}\right)^{\prime}\right|^{2}f^{2}(\varrho)\varrho^{n+1}\>{\rm
d}\varrho+16\lambda_{k}^{2}\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho$
$\displaystyle-8\lambda_{k}\int_{0}^{R}V(\varrho)d_{k}^{\prime\prime}d_{k}\varrho^{n-1}\>{\rm
d}\varrho-8\lambda_{k}(Q-1)\int_{0}^{R}V(\varrho)d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm
d}\varrho\bigg{]}.$

Step 2. In this step we compute r.h.s. of (1.3):

$\displaystyle\int_{B_{R}(o)}$ $\displaystyle
W(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t+\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}\left(\frac{u_{\varrho}}{f(\varrho)}\right)\right|^{2}f^{2}(\varrho)\>{\rm
d}x\>{\rm d}t$
$\displaystyle+(Q-1)\int_{B_{R}(o)}\left(\frac{V(\varrho)}{\varrho^{2}}-\frac{V_{\varrho}(\varrho)}{\varrho}\right)\left|\nabla_{G}u\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{[}\int_{0}^{R}W(\varrho){d_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho+4\lambda_{k}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle+\int_{0}^{R}V(\varrho)\left|\left(\frac{d_{k}^{\prime}}{f(\varrho)}\right)^{\prime}\right|^{2}f^{2}(\varrho)\varrho^{n+1}\>{\rm
d}\varrho+4\lambda_{k}\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle+(Q-1)\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+4\lambda_{k}(Q-1)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho$
$\displaystyle-(Q-1)\int_{0}^{R}V_{\varrho}(\varrho){d_{k}^{\prime}}^{2}\varrho^{n}\>{\rm
d}\varrho-4\lambda_{k}(Q-1)\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho\bigg{]}.$

Step 3. Subtracting the r.h.s. of the identities obtained in Step 1 and Step 2
we obtain the expression below that we denote by $\mathcal{R}$ the following
quantity:

$\displaystyle\mathcal{R}:=$
$\displaystyle\sum_{k=0}^{\infty}\bigg{[}4\lambda_{k}(4\lambda_{k}-Q+1)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\int_{0}^{R}V(\varrho)d_{k}^{\prime\prime}d_{k}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle-8\lambda_{k}(Q-1)\int_{0}^{R}V(\varrho)d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm
d}\varrho-4\lambda_{k}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle-4\lambda_{k}\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+4\lambda_{k}(Q-1)\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho\bigg{]}.$

In the steps below, we will show $\mathcal{R}$ is non-negative and this will
establish the result. To do this we need some identities.

Step 4. Set

$\displaystyle\mathcal{I}:=\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho,$
$\displaystyle\mathcal{I}_{1}:=\int_{0}^{R}V(\varrho)d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm
d}\varrho,$

and

$\displaystyle\mathcal{I}_{2}:=\int_{0}^{R}V(\varrho)d_{k}^{\prime\prime}d_{k}\varrho^{n-1}\>{\rm
d}\varrho.$

Now using the by-parts we have

$\displaystyle\mathcal{I}_{1}=-\frac{1}{2}\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho-\frac{(Q-4)}{2}\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho,$

and

$\displaystyle\mathcal{I}_{2}$
$\displaystyle=-\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{\prime}d_{k}\varrho^{n-1}\>{\rm
d}\varrho-\mathcal{I}-(Q-3)\mathcal{I}_{1}$
$\displaystyle=\frac{1}{2}\int_{0}^{R}V_{\varrho\varrho}(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho+\frac{(Q-3)}{2}\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho-\mathcal{I}-(Q-3)\mathcal{I}_{1}.$

Step 5. In this step we will deal with $\mathcal{I}$. Let
$b_{k}(\varrho):=\frac{d_{k}(\varrho)}{\varrho}$, by Leibniz rule, performing
the derivative w.r.t $\varrho$, we have $d_{k}^{\prime}=\varrho
b_{k}^{\prime}+b_{k}$. Using this and the by-parts formula, we obtain

$\displaystyle\mathcal{I}$
$\displaystyle=\int_{0}^{R}V(\varrho){d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle=\int_{0}^{R}V(\varrho){b_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho-\int_{0}^{R}V_{\varrho}(\varrho)b_{k}^{2}\varrho^{n}\>{\rm
d}\varrho-(Q-3)\int_{0}^{R}V(\varrho)b_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho.$
(4.1)

Applying Theorem 1.1, for the radial function $b_{k}$, we deduce

$\displaystyle\int_{0}^{R}V(\varrho){b_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho$ $\displaystyle=\int_{0}^{R}W(\varrho)b_{k}^{2}\varrho^{n+1}\>{\rm
d}\varrho+\int_{0}^{R}V(\varrho)f^{2}(\varrho)\left[\bigg{(}\frac{b_{k}}{f(\varrho)}\bigg{)}^{\prime}\right]^{2}\varrho^{n+1}\>{\rm
d}\varrho.$

Using this estimate into (4) and writing $b_{k}$ in terms of $d_{k}$, we have

$\displaystyle\mathcal{I}$
$\displaystyle=\int_{0}^{R}W(\varrho)b_{k}^{2}\varrho^{n+1}\>{\rm
d}\varrho+\int_{0}^{R}V(\varrho)f^{2}(\varrho)\left[\bigg{(}\frac{b_{k}}{f(\varrho)}\bigg{)}^{\prime}\right]^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle-\int_{0}^{R}V_{\varrho}(\varrho)b_{k}^{2}\varrho^{n}\>{\rm
d}\varrho-(Q-3)\int_{0}^{R}V(\varrho)b_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho$
$\displaystyle=\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho+\int_{0}^{R}V(\varrho)f^{2}(\varrho)\left[\bigg{(}\frac{d_{k}}{\varrho
f(\varrho)}\bigg{)}^{\prime}\right]^{2}\varrho^{n+1}\>{\rm d}\varrho$
$\displaystyle-\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho-(Q-3)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm d}\varrho.$

Step 6. Next, using all these integrals and rewriting $\mathcal{R}$ as
follows:

$\displaystyle\mathcal{R}=$
$\displaystyle\sum_{k=0}^{\infty}\bigg{[}4\lambda_{k}(4\lambda_{k}-Q+1)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\mathcal{I}_{2}-8\lambda_{k}(Q-1)\mathcal{I}_{1}$
$\displaystyle-4\lambda_{k}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho-4\lambda_{k}\mathcal{I}+4\lambda_{k}(Q-1)\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho\bigg{]}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{[}4\lambda_{k}(4\lambda_{k}-Q+1)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\biggl{\\{}\frac{1}{2}\int_{0}^{R}V_{\varrho\varrho}(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle+\frac{(Q-3)}{2}\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho-\mathcal{I}-(Q-3)\mathcal{I}_{1}\biggr{\\}}-8\lambda_{k}(Q-1)\mathcal{I}_{1}$
$\displaystyle-4\lambda_{k}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho-4\lambda_{k}\mathcal{I}+4\lambda_{k}(Q-1)\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho\bigg{]}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{[}4\lambda_{k}(4\lambda_{k}-Q+1)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-4\lambda_{k}\int_{0}^{R}V_{\varrho\varrho}(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle+8\lambda_{k}\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho+4\lambda_{k}\mathcal{I}-16\lambda_{k}\mathcal{I}_{1}-4\lambda_{k}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{[}4\lambda_{k}(4\lambda_{k}+Q-7)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-4\lambda_{k}\int_{0}^{R}V_{\varrho\varrho}(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle+16\lambda_{k}\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho-4\lambda_{k}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho+4\lambda_{k}\mathcal{I}\bigg{]}.$

Now we will substitute the value of $\mathcal{I}$, and we obtain after further
simplifying

$\displaystyle\mathcal{R}$
$\displaystyle=\sum_{k=0}^{\infty}\bigg{[}16\lambda_{k}(\lambda_{k}-1)\int_{0}^{R}V(\varrho)d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-4\lambda_{k}\int_{0}^{R}V_{\varrho\varrho}(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho$
$\displaystyle+12\lambda_{k}\int_{0}^{R}V_{\varrho}(\varrho)d_{k}^{2}\varrho^{n-2}\>{\rm
d}\varrho+4\lambda_{k}\int_{0}^{R}V(\varrho)f^{2}(\varrho)\left[\bigg{(}\frac{d_{k}}{\varrho
f(\varrho)}\bigg{)}^{\prime}\right]^{2}\varrho^{n+1}\>{\rm d}\varrho\bigg{]}$
$\displaystyle\geq\sum_{k=0}^{\infty}4\lambda_{k}\int_{0}^{R}\bigg{[}4(\lambda_{k}-1)\frac{V(\varrho)}{\varrho^{2}}-V_{\varrho\varrho}(\varrho)+3\frac{V_{\varrho}(\varrho)}{\varrho}\bigg{]}d_{k}^{2}\varrho^{n-1}\>{\rm
d}\varrho.$

In the above, we remove the non-negative term due to $V\geq 0$. Also, we can
notice that for $k\geq 1$ there holds

$\displaystyle 4(\lambda_{k}-1)=k(n+k)-4\geq(n+1)-4=(Q-5).$

With the given condition (1.6) and the above estimate we conclude the proof. ∎

Proof of Corollary 1.1 and Corollary 1.2: Applying
$\big{(}1,\frac{(Q-2)^{2}}{4}\frac{1}{\varrho^{2}}\big{)}$ on Theorem 1.3 as a
$Q$-dimensional Bessel pair on $(0,\infty)$ with positive solution
$\varrho^{-\frac{Q-2}{2}}$, part (a) follows. Now applying weighted Hardy
inequality from Corollary 3.2, with $\alpha=2$, we deduce

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle-\frac{(Q-4)^{2}}{4}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{4}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u\varrho^{\frac{Q-4}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t.$

Clubbing these with part (a) we obtain our required Rellich inequality in part
(b).

To prove the Corollary 1.2, we will follow the same approach as Corollary 1.1,
and utilize Theorem 1.2. This completes the proof. ∎

######  Remark 4.1.

Using the polar coordinate decomposition and by parts formula, we can have the
following identity: for the radial function $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, there holds

$\displaystyle\frac{Q^{2}}{4}\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}$
$\displaystyle\left|\nabla_{G}\left(u\varrho^{\frac{Q-4}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm
d}t+\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\nabla_{G}\left(u_{\varrho}\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\int_{\mathbb{R}^{n+1}}\bigg{|}\frac{\mathcal{L}_{G}u}{|\nabla_{G}\varrho|}+\frac{Q(Q-4)}{4}\frac{u|\nabla_{G}\varrho|}{\varrho^{2}}\bigg{|}^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\frac{Q(Q-4)}{2}\int_{\mathbb{R}^{n+1}}\bigg{|}\frac{|\nabla_{G}u|}{\varrho|\nabla_{G}\varrho|}+\frac{(Q-4)}{2}\frac{u}{\varrho^{2}}\bigg{|}^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t.$

We can compare the corresponding Rellich identity with the radial Grushin
operator inspired by [36, Theorem 1.1]. For the sake of brevity, we omit the
details.

##  5\. Rellich identity and spherical harmonics via spherical derivatives:
proof of Theorem 1.4

This section is devoted to studying spherical harmonics via spherical
derivatives. We shall explicitly compute the contribution of the radial
Laplacian and the spherical Laplacian of the Grushin operator. Let us recall
the spherical derivatives, for $j=1,\ldots,n$ we define
$L_{j}=\frac{\partial}{\partial x_{j}}-\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}$ and
$L_{n+1}=|x|\frac{\partial}{\partial t}-|x|\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial\varrho}.$ Using polar coordinates, it is easy to
check that

$\frac{\partial}{\partial\varrho}=\frac{1}{\varrho}\left(\sum_{j=1}^{n}x_{j}\frac{\partial}{\partial
x_{j}}+2t\frac{\partial}{\partial t}\right).$ (5.1)

Now, we will state some useful identities involving $L_{j}$’s in the following
lemma.

######  Lemma 5.1.

The following relations hold:

  * 1.

There holds

$\sum_{j=1}^{n}x_{j}|x|^{2}L_{j}+2t|x|L_{n+1}=0.$ (5.2)

  * 2.

For all $j=1,2,\ldots,n+1$, we have

$L_{j}\frac{\partial}{\partial\varrho}=\frac{\partial}{\partial\varrho}L_{j}+\frac{1}{\varrho}L_{j}.$
(5.3)

  * 3.

We have the following identities

$\mathcal{L}_{G}=\mathcal{L}_{\varrho,G}+\sum_{j=1}^{n+1}L_{j}^{2},\quad\text{
and
}\quad\sum_{j=1}^{n+1}L_{j}^{2}=\frac{4\psi}{\varrho^{2}}\mathcal{L}_{\sigma}.$
(5.4)

  * 4.

Let $f,\,g\in C_{c}^{2}(\mathbb{R}^{n+1})$. Then the following by-parts
formula holds: when $j=1,\ldots,n$, we have

$\int_{\mathbb{R}^{n+1}}g(L_{j}f)\>{\rm d}x\>{\rm
d}t=-\int_{\mathbb{R}^{n+1}}(L_{j}g){f}\>{\rm d}x\>{\rm
d}t+(Q-1)\int_{\mathbb{R}^{n+1}}g\frac{x_{j}|x|^{2}}{\varrho^{4}}{f}\>{\rm
d}x\>{\rm d}t,$

and when $j=n+1$, we get

$\int_{\mathbb{R}^{n+1}}g(L_{n+1}f)\>{\rm d}x\>{\rm
d}t=-\int_{\mathbb{R}^{n+1}}(L_{n+1}g){f}\>{\rm d}x\>{\rm
d}t+(Q-1)\int_{\mathbb{R}^{n+1}}g\frac{2t|x|}{\varrho^{4}}{f}\>{\rm d}x\>{\rm
d}t.$

  * 5.

Assume $f\in C^{2}(\mathbb{R}^{n+1})$. Then for all $j$ and
$\alpha\in\mathbb{R}$, there holds

$L_{j}(\varrho^{\alpha}f)=\varrho^{\alpha}(L_{j}f).$

###### Proof.

Let us begin the proof of part (1). Using the definition of $L_{j}$’s and
(5.1), we have

$\displaystyle\sum_{j=1}^{n}x_{j}|x|^{2}L_{j}+2t|x|L_{n+1}$
$\displaystyle=\sum_{j=1}^{n}x_{j}|x|^{2}\frac{\partial}{\partial
x_{j}}-\sum_{j=1}^{n}x_{j}|x|^{2}\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}+2t|x|^{2}\frac{\partial}{\partial
t}-2t|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial\varrho}$
$\displaystyle=\varrho|x|^{2}\frac{\partial}{\partial\varrho}-|x|^{2}\left(\sum_{j=1}^{n}\frac{x_{j}^{2}|x|^{2}}{\varrho^{3}}+\frac{4t^{2}}{\varrho^{3}}\right)\frac{\partial}{\partial\varrho}=0.$

Now we will prove part (2). Using (5.1) and the fact that
$\frac{\partial\varrho}{\partial x_{j}};j=1,2,\ldots,n$, is independent of
$\varrho$, we can notice that

$\displaystyle
L_{j}\frac{\partial}{\partial\varrho}-\frac{\partial}{\partial\varrho}L_{j}$
$\displaystyle=\left(\frac{\partial}{\partial
x_{j}}\right)\left(\sum_{k=1}^{n}\frac{x_{k}}{\varrho}\frac{\partial}{\partial
x_{k}}+\frac{2t}{\varrho}\frac{\partial}{\partial
t}\right)-\left(\sum_{k=1}^{n}\frac{x_{k}}{\varrho}\frac{\partial}{\partial
x_{k}}+\frac{2t}{\varrho}\frac{\partial}{\partial
t}\right)\left(\frac{\partial}{\partial x_{j}}\right)$
$\displaystyle=\sum_{k=1}^{n}\frac{x_{k}}{\varrho}\frac{\partial^{2}}{\partial
x_{j}\partial x_{k}}+\frac{2t}{\varrho}\frac{\partial}{\partial
x_{j}}\frac{\partial}{\partial
t}-\sum_{k=1}^{n}\frac{x_{k}}{\varrho}\frac{\partial^{2}}{\partial
x_{k}\partial x_{j}}-\frac{2t}{\varrho}\frac{\partial}{\partial
t}\frac{\partial}{\partial x_{j}}$
$\displaystyle+\frac{1}{\varrho}\frac{\partial}{\partial
x_{j}}-\sum_{k=1}^{n}\frac{x_{k}x_{j}|x|^{2}}{\varrho^{5}}\frac{\partial}{\partial
x_{k}}-\frac{2tx_{j}|x|^{2}}{\varrho^{5}}\frac{\partial}{\partial t}$
$\displaystyle=\frac{1}{\varrho}\bigg{[}\frac{\partial}{\partial
x_{j}}-\frac{\partial\varrho}{\partial
x_{j}}\bigg{(}\sum_{k=1}^{n}\frac{x_{k}}{\varrho}\frac{\partial}{\partial
x_{k}}+\frac{2t}{\varrho}\frac{\partial}{\partial t}\bigg{)}\bigg{]}.$

Now, using (5.1), and writing back the definition of $L_{j}$, we obtain the
result. The argument for $j=n+1$ is left to the reader.

Now we will prove part (3). Let us calculate $L_{j}^{2}$ first. For
$j=1,2,\ldots,n$, we have

$\displaystyle L_{j}^{2}=\frac{\partial^{2}}{\partial
x_{j}^{2}}-\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial
x_{j}}-\frac{\partial}{\partial x_{j}}\left(\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\right)+\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\left(\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\right).$

Using the fact that $\frac{\partial\varrho}{\partial x_{j}};j=1,2,\ldots,n$,
and $|x|\frac{\partial\varrho}{\partial t}$ is independent of $\varrho$ and
further simplifying the above expression of $L_{j}^{2}$ we get

$\displaystyle L_{j}^{2}=\frac{\partial^{2}}{\partial
x_{j}^{2}}+\left(\frac{\partial\varrho}{\partial
x_{j}}\right)^{2}\frac{\partial^{2}}{\partial\varrho^{2}}-\frac{|x|^{2}}{\varrho^{3}}-\frac{2x_{j}^{2}}{\varrho^{3}}+\frac{3x_{j}^{2}|x|^{4}}{\varrho^{7}}\frac{\partial}{\partial\varrho}-\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial
x_{j}}\frac{\partial}{\partial\varrho}-\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial x_{j}}.$

Similarly, we can compute that

$\displaystyle L_{n+1}^{2}=|x|^{2}\frac{\partial^{2}}{\partial
t^{2}}+|x|^{2}\left(\frac{\partial\varrho}{\partial
t}\right)^{2}\frac{\partial^{2}}{\partial\varrho^{2}}-\frac{2|x^{2}|}{\varrho^{3}}\frac{\partial}{\partial\varrho}+\frac{12|x|^{2}t}{\varrho^{7}}\frac{\partial}{\partial\varrho}$
$\displaystyle-|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial
t}\frac{\partial}{\partial\varrho}-|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial
t}-\frac{|x|^{2}}{\varrho}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial t}.$

Then, taking the summation of the above identities from $j=1,2,\ldots,n+1$, we
get

$\sum_{j=1}^{n+1}L_{j}^{2}=\mathcal{L}_{G}+|\nabla_{G}\varrho|^{2}\frac{\partial^{2}}{\partial\varrho^{2}}-\frac{n+1}{\varrho}|\nabla_{G}\varrho|^{2}\frac{\partial}{\partial\varrho}-D,$
(5.5)

where

$\displaystyle D=\sum_{j=1}^{n}\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial
x_{j}}\frac{\partial}{\partial\varrho}+|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial
t}\frac{\partial}{\partial\varrho}+\sum_{j=1}^{n}\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial
x_{j}}+|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial
t}+\frac{|x|^{2}}{\varrho}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial t}.$

Using the identity (5.1), and the fact that $\frac{x_{j}}{\varrho}$, and
$\frac{|x|^{2}}{\varrho^{2}}$ are $\varrho$ independent, we obtain

$\displaystyle\sum_{j=1}^{n}\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial
x_{j}}\frac{\partial}{\partial\varrho}+|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial
t}\frac{\partial}{\partial\varrho}=|\nabla_{G}\varrho|^{2}\frac{\partial^{2}}{\partial\varrho^{2}},$

and

$\displaystyle\sum_{j=1}^{n}\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial
x_{j}}=\frac{|x|^{2}}{\varrho^{2}}\frac{\partial}{\partial\varrho}\left(\frac{\partial}{\partial\varrho}-\frac{2t}{\varrho}\frac{\partial}{\partial
t}\right)=|\nabla_{G}\varrho|^{2}\frac{\partial^{2}}{\partial\varrho^{2}}-\frac{\partial}{\partial\varrho}\left(|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial t}\right).$

Again noticing that $|x|\frac{\partial\varrho}{\partial t}$ is independent of
$\varrho$, and using (5.1) to compute $\frac{\partial}{\partial\varrho}(|x|)$,
we obtain

$\displaystyle\frac{\partial}{\partial\varrho}\left(|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial
t}\right)=\frac{|x|^{2}}{\varrho}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial t}+|x|^{2}\frac{\partial\varrho}{\partial
t}\frac{\partial}{\partial\varrho}\frac{\partial}{\partial t}.$

Combining all these in the expression of $D$, it simplifies to
$D=2|\nabla_{G}\varrho|^{2}\frac{\partial^{2}}{\partial\varrho^{2}}$. Putting
this expression of $D$ in (5.5) we get that

$\mathcal{L}_{G}=\mathcal{L}_{\varrho,G}+\sum_{j=1}^{n+1}L_{j}^{2}.$

Using the expression of the Grushin operator in (2.1), we can obtain the
second identity of (5.4) and this completes the proof of part (3).

Now we will prove the part (4) of the lemma. This follows from the definition
of $L_{j}$’s and polar coordinate decomposition formula. Exactly similar
arguments hold for $j=n+1$, and for simplicity, we are just writing the case
when $j=1,2,\ldots,n$. Exploiting integration by parts, first w.r.t. $x_{j}$,
and then w.r.t. $\varrho$, and using the fact that
$\frac{\partial\varrho}{\partial x_{j}}$ is independent of $\varrho$, we have

$\displaystyle\int_{\mathbb{R}^{n+1}}g(L_{j}f)\>{\rm d}x\>{\rm d}t$
$\displaystyle=-\int_{\mathbb{R}^{n+1}}\frac{\partial g}{\partial
x_{j}}\,f\>{\rm d}x\>{\rm
d}t-\int_{\Omega}\int_{0}^{\infty}g\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial f}{\partial\varrho}\frac{\varrho^{Q-1}}{2\psi}\>{\rm
d}\varrho\>{\rm d}\Omega$
$\displaystyle=-\int_{\mathbb{R}^{n+1}}(L_{j}g){f}\>{\rm d}x\>{\rm
d}t+(Q-1)\int_{\mathbb{R}^{n+1}}\frac{g}{\varrho}\frac{\partial\varrho}{\partial
x_{j}}f\>{\rm d}x\>{\rm d}t.$

In this part, we will prove part (5). Notice that for any $j=1,2,\ldots,n$, we
have

$\displaystyle
L_{j}(\varrho^{\alpha}f)=\frac{\partial\varrho^{\alpha}}{\partial
x_{j}}f+\varrho^{\alpha}\frac{\partial f}{\partial
x_{j}}-\frac{\partial\varrho}{\partial
x_{j}}\frac{\partial\varrho^{\alpha}}{\partial\varrho}f-\frac{\partial\varrho}{\partial
x_{j}}\varrho^{\alpha}\frac{\partial
f}{\partial\varrho}=\varrho^{\alpha}(L_{j}f).$

A similar argument holds for $j=n+1$. This finishes the proof of the lemma. ∎

Now we begin the proof of Theorem 1.4.

Proof of Theorem 1.4: Using (5.4) we can write

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{\varrho,G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t+\int_{\mathbb{R}^{n+1}}\frac{|\sum_{j=1}^{n+1}L_{j}^{2}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+2~{}\left(\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}|\nabla_{G}\varrho|^{-2}\mathcal{L}_{\varrho,G}u{L_{j}^{2}u}\>{\rm
d}x\>{\rm d}t\right).$

We know that

$|\nabla_{G}\varrho|^{-2}\mathcal{L}_{\varrho,G}u=\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}.$

Define $L_{j}u=f_{j}$ and we write
$\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}=g$, then using integration by parts gives

$\int_{\mathbb{R}^{n+1}}g{L_{j}f_{j}}\>{\rm d}x\>{\rm
d}t=-\int_{\mathbb{R}^{n+1}}L_{j}g{f_{j}}\>{\rm d}x\>{\rm
d}t+(Q-1)\int_{\mathbb{R}^{n+1}}g\frac{x_{j}|x|^{2}}{\varrho^{4}}{f_{j}}\>{\rm
d}x\>{\rm d}t$

for $j=1,2,\ldots,n$, and for $j=n+1$, we get

$\int_{\mathbb{R}^{n+1}}g{L_{n+1}f_{n+1}}\>{\rm d}x\>{\rm
d}t=-\int_{\mathbb{R}^{n+1}}L_{n+1}g{f_{n+1}}\>{\rm d}x\>{\rm
d}t+(Q-1)\int_{\mathbb{R}^{n+1}}g\frac{2t|x|}{\varrho^{4}}{f_{n+1}}\>{\rm
d}x\>{\rm d}t.$

Now summing over $j$ and using (5.2) gives

$\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}|\nabla_{G}\varrho|^{-2}\mathcal{L}_{\varrho,G}u{L_{j}u}\>{\rm
d}x\>{\rm d}t=-\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}L_{j}g{f_{j}}\>{\rm
d}x\>{\rm d}t.$ (5.6)

By the definition of $g$ above it can be rewritten as
$g=\varrho^{-(Q-1)}\frac{\partial}{\partial\varrho}\left(\varrho^{Q-1}\frac{\partial
u}{\partial\varrho}\right)$. Using (5.3) and part (5) of Lemma 5.1, we can
write

$L_{j}g=\frac{\partial^{2}f_{j}}{\partial\varrho^{2}}+\frac{Q+1}{\varrho}\frac{\partial
f_{j}}{\partial\varrho}+\frac{Q-1}{\varrho^{2}}f_{j}.$

For a fixed $j$ we have

$\displaystyle\int_{\mathbb{R}^{n+1}}L_{j}g{f_{j}}\>{\rm d}x\>{\rm
d}t=\int_{\mathbb{R}^{n+1}}\frac{\partial^{2}f_{j}}{\partial\varrho^{2}}f_{j}+\frac{Q+1}{\varrho}\frac{\partial
f_{j}}{\partial\varrho}f_{j}+\frac{Q-1}{\varrho^{2}}f_{j}^{2}\>{\rm d}x\>{\rm
d}t.$

Using integration by parts gives

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{\partial^{2}f_{j}}{\partial\varrho^{2}}f_{j}+\frac{Q+1}{\varrho}\frac{\partial
f_{j}}{\partial\varrho}f_{j}\>{\rm d}x\>{\rm
d}t=-\int_{\mathbb{R}^{n+1}}\left(\frac{\partial
f_{j}}{\partial\varrho}\right)^{2}\>{\rm d}x\>{\rm d}t$
$\displaystyle-\frac{(Q+1)(Q-2)}{2}\int_{\mathbb{R}^{n+1}}\frac{f_{j}^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm
d}t+\frac{(Q-1)(Q-2)}{2}\int_{\mathbb{R}^{n+1}}\frac{f_{j}^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm d}t.$

Summing over $j$ gives

$\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}L_{j}g{f_{j}}\>{\rm d}x\>{\rm
d}t=-\sum_{j=1}^{n+1}\left(\int_{\mathbb{R}^{n+1}}\left|\frac{\partial
f_{j}}{\partial\varrho}\right|^{2}\>{\rm d}x\>{\rm
d}t-\int_{\mathbb{R}^{n+1}}\frac{|f_{j}|^{2}}{\varrho^{2}}\>{\rm d}x\>{\rm
d}t\right).$

Therefore, (5.6) reduces to

$\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}|\nabla_{G}\varrho|^{-2}\mathcal{L}_{\varrho,G}u{L_{j}u}\>{\rm
d}x\>{\rm
d}t=\sum_{j=1}^{n+1}\left(\int_{\mathbb{R}^{n+1}}\left|\frac{\partial
f_{j}}{\partial\varrho}\right|^{2}\>{\rm d}x\>{\rm
d}t-\int_{\mathbb{R}^{n+1}}\frac{|f_{j}|^{2}}{\varrho^{2}}\>{\rm d}x\>{\rm
d}t\right).$

Using the identity for $\nabla_{\varrho,G}$ when $\alpha=0$, in Corollary 3.2
for each $L_{j}u|\nabla_{G}\varrho|^{-1}$, we get the desired equality. ∎

Now we want to measure the following deficit

$\displaystyle\bigg{|}\bigg{|}\frac{\mathcal{L}_{G}u}{\psi^{1/2}}\bigg{|}\bigg{|}_{L^{2}(\mathbb{R}^{n+1})}-\quad\bigg{|}\bigg{|}\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}}\bigg{|}\bigg{|}_{L^{2}(\mathbb{R}^{n+1})},$

where $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$. Here we will
calculate the exact identity in terms of spherical harmonics. This can be
compared with the second part of the result [4, Theorem 1]. Our result can be
stated as follows.

######  Theorem 5.1.

Let $Q\geq 3$ and $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$.
Then there holds

$\displaystyle\bigg{|}\bigg{|}\frac{\mathcal{L}_{G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}=\quad\bigg{|}\bigg{|}\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}+16\sum_{k=0}^{\infty}\lambda_{k}^{2}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle+8\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u_{\varrho})\psi^{1/2}}{\varrho}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}+8(Q-4)\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

In the above $\varrho$ in suffix means the derivative with respect to the
radial part $\varrho$.

###### Proof.

Consider the decomposition
$u(\varrho,\sigma)=\sum_{k=0}^{\infty}\mathcal{P}_{k}u$ where
$\mathcal{P}_{k}u=d_{k}(\varrho)\Phi_{k}(\sigma)$. Then let us write the
Grushin operator as follows

$\displaystyle\mathcal{L}_{G}u$
$\displaystyle=\psi\biggl{\\{}\frac{\partial^{2}u}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
u}{\partial\varrho}+\frac{4}{\varrho^{2}}\mathcal{L}_{\sigma}u\biggr{\\}}$
$\displaystyle=\mathcal{L}_{\varrho,G}u+\frac{4\psi}{\varrho^{2}}\mathcal{L}_{\sigma}\bigg{(}\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma)\bigg{)}$
$\displaystyle=\mathcal{L}_{\varrho,G}u-\frac{4\psi}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma).$

If we now expand it then we deduce using orthogonality properties that

$\displaystyle\bigg{|}\bigg{|}\frac{\mathcal{L}_{G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle=\bigg{|}\bigg{|}\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}+\bigg{|}\bigg{|}\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle-2\left\langle\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}},\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\right\rangle_{L^{2}(\mathbb{R}^{n+1})}.$

Now we will compute each term separately. Let us begin with

$\displaystyle\bigg{|}\bigg{|}\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle=\int_{\Omega}\int_{0}^{\infty}\bigg{(}\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\bigg{)}\bigg{(}\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\bigg{)}\frac{\varrho^{n+1}}{2\psi}\>{\rm
d}\varrho\>{\rm d}\Omega$
$\displaystyle=\sum_{k=0}^{\infty}8\lambda_{k}^{2}\int_{0}^{\infty}\varrho^{n-3}d_{k}^{2}\>{\rm
d}\varrho,$

and

$\displaystyle\left\langle\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}},\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\right\rangle_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle=\int_{\Omega}\int_{0}^{\infty}\bigg{(}\psi^{1/2}\sum_{k=0}^{\infty}\biggl{\\{}\frac{\partial^{2}d_{k}}{\partial\varrho^{2}}+\frac{Q-1}{\varrho}\frac{\partial
d_{k}}{\partial\varrho}\biggr{\\}}\Phi_{k}\bigg{)}\bigg{(}\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}\Phi_{k}\bigg{)}\frac{\varrho^{n+1}}{2\psi}\>{\rm
d}\varrho\>{\rm d}\Omega$
$\displaystyle=\sum_{k=0}^{\infty}2\int_{0}^{\infty}\varrho^{n-1}\lambda_{k}\big{[}d_{k}^{\prime\prime}+\frac{Q-1}{\varrho}d_{k}^{\prime}\big{]}d_{k}\>{\rm
d}\varrho$
$\displaystyle=2\sum_{k=0}^{\infty}\lambda_{k}\int_{0}^{\infty}\varrho^{n-1}d_{k}^{\prime\prime}d_{k}\>{\rm
d}\varrho+2(Q-1)\sum_{k=0}^{\infty}\lambda_{k}\int_{0}^{\infty}\varrho^{n-2}d_{k}^{\prime}d_{k}\>{\rm
d}\varrho.$

Notice that,

$\displaystyle\int_{0}^{\infty}\varrho^{n-2}d_{k}^{\prime}d_{k}\>{\rm
d}\varrho=-\frac{(Q-4)}{2}\int_{0}^{\infty}\varrho^{n-3}d_{k}^{2}\>{\rm
d}\varrho,$

and

$\displaystyle\int_{0}^{\infty}\varrho^{n-1}d_{k}^{\prime\prime}d_{k}\>{\rm
d}\varrho$
$\displaystyle=-\int_{0}^{\infty}\varrho^{n-1}(d_{k}^{\prime})^{2}\>{\rm
d}\varrho-(Q-3)\int_{0}^{\infty}\varrho^{n-2}d_{k}^{\prime}d_{k}\>{\rm
d}\varrho$
$\displaystyle=-\int_{0}^{\infty}\varrho^{n-1}(d_{k}^{\prime})^{2}\>{\rm
d}\varrho+\frac{(Q-3)(Q-4)}{2}\int_{0}^{\infty}\varrho^{n-3}d_{k}^{2}\>{\rm
d}\varrho.$

Combining all these we deduce

$\displaystyle\left\langle\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}},\frac{4\psi^{1/2}}{\varrho^{2}}\sum_{k=0}^{\infty}\lambda_{k}d_{k}(\varrho)\Phi_{k}(\sigma)\right\rangle_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle=-2\sum_{k=0}^{\infty}\lambda_{k}\int_{0}^{\infty}\varrho^{n-1}(d_{k}^{\prime})^{2}\>{\rm
d}\varrho-2(Q-4)\sum_{k=0}^{\infty}\lambda_{k}\int_{0}^{\infty}\varrho^{n-3}d_{k}^{2}\>{\rm
d}\varrho.$

Hence we derive the identity

$\displaystyle\bigg{|}\bigg{|}\frac{\mathcal{L}_{G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle-\quad\bigg{|}\bigg{|}\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle=\sum_{k=0}^{\infty}8\lambda_{k}^{2}\int_{0}^{\infty}\varrho^{n-3}d_{k}^{2}\>{\rm
d}\varrho+4\sum_{k=0}^{\infty}\lambda_{k}\int_{0}^{\infty}\varrho^{n-1}(d_{k}^{\prime})^{2}\>{\rm
d}\varrho$
$\displaystyle\qquad+4(Q-4)\sum_{k=0}^{\infty}\lambda_{k}\int_{0}^{\infty}\varrho^{n-3}d_{k}^{2}\>{\rm
d}\varrho$
$\displaystyle=16\sum_{k=0}^{\infty}\lambda_{k}^{2}\bigg{|}\bigg{|}\frac{d_{k}\Phi_{k}\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}+8\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{d_{k}^{\prime}\Phi_{k}\psi^{1/2}}{\varrho}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}$
$\displaystyle\qquad+8(Q-4)\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{d_{k}\Phi_{k}\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

Now writing back for the function $u,$ we obtain the identity. ∎

######  Remark 5.1.

Let $Q\geq 3$ and $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$.
Then writing in terms of the given function and using the estimate
$\lambda_{k}\geq(Q-1)/4$, there holds

$\bigg{|}\bigg{|}\frac{\mathcal{L}_{G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}\geq\quad\bigg{|}\bigg{|}\frac{\mathcal{L}_{\varrho,G}u}{\psi^{1/2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}+2(Q-1)\bigg{|}\bigg{|}\frac{u_{\varrho}\psi^{1/2}}{\varrho}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}\\\
+3(Q-3)(Q-1)\bigg{|}\bigg{|}\frac{u\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

######  Remark 5.2.

After neglecting the non-negative remainder terms in the identity of Theorem
5.1, we have the following comparison result. Let $Q\geq 4$ and $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{\varrho,G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t.$ (5.7)

######  Remark 5.3.

The above comparison (5.7), and neglecting non-negative terms in the identity
of the part (b) of Corollary 1.2, in terms of the radial operator, we can
retrieve our known Rellich inequality. For $Q\geq 5$ and $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, there holds

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{\varrho,G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\frac{Q^{2}(Q-4)^{2}}{16}\int_{\mathbb{R}^{n+1}}\frac{|u|^{2}|\nabla_{G}\varrho|^{2}}{\varrho^{4}}\>{\rm
d}x\>{\rm d}t.$

This gives the improvement of (1.4) and also settles the optimality issue of
the Rellich constant.

It is worth mentioning that Theorem 1.4 and Theorem 5.1 are equivalent in some
sense. Here we show a connection between the spherical derivatives and
projection operators mentioned earlier. The lemma reads as follows :

######  Lemma 5.2.

Let $Q\geq 3$ and $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$.
Then we have

  * (a)

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\sum_{j=1}^{n+1}L_{j}^{2}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=16\sum_{k=0}^{\infty}\lambda_{k}^{2}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})};$

  * (b)

$\displaystyle\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\frac{|L_{j}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm
d}t=4\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

###### Proof.

Let us first write the proof of part (a). Begin with the spherical harmonic
decomposition of $u$ and we can write

$u(x)=u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma)=\sum_{k=0}^{\infty}\mathcal{P}_{k}u,\quad\text{
and }-\mathcal{L}_{\sigma}\Phi_{k}(\sigma)=\lambda_{k}\Phi_{k}(\sigma).$

Now, using (5.4), we have

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\sum_{j=1}^{n+1}L_{j}^{2}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=16\int_{\mathbb{R}^{n+1}}\frac{\psi}{\varrho^{4}}|\mathcal{L}_{\sigma}u|^{2}\>{\rm
d}x\>{\rm
d}t=16\sum_{k=0}^{\infty}\lambda_{k}^{2}\int_{\mathbb{R}^{n+1}}\frac{\psi}{\varrho^{4}}(\mathcal{P}_{k}u)^{2}\>{\rm
d}x\>{\rm d}t.$

Using this we deduce part (a). For part (b), we use by parts formula mentioned
in part (4), and then part (5) of Lemma 5.1. Finally we use (5.2), and (5.4),
and obtain

$\displaystyle\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\frac{|L_{j}u|^{2}}{\varrho^{2}}\>{\rm
d}x\>{\rm
d}t=-\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\varrho^{-2}u(L_{j}^{2}u)\>{\rm
d}x\>{\rm
d}t=\sum_{k=0}^{\infty}4\lambda_{k}\int_{\mathbb{R}^{n+1}}\frac{\psi(\mathcal{P}_{k}u)^{2}}{\varrho^{4}}\>{\rm
d}x\>{\rm d}t.$

∎

######  Lemma 5.3.

Let $Q\geq 3$ and $u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$.
Then we have

$\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\frac{\partial}{\partial\varrho}\left(L_{j}u\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t\\\
=4\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u_{\varrho})\psi^{1/2}}{\varrho}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}-(Q-4)^{2}\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

###### Proof.

By using part (5) of Lemma 5.1 and repeatedly exploiting integration by parts,
we have

$\displaystyle\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\varrho^{2-Q}\left|\frac{\partial}{\partial\varrho}\left(L_{j}u\varrho^{\frac{Q-2}{2}}\right)\right|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}|(L_{j}u)_{\varrho}|^{2}\>{\rm
d}x\>{\rm
d}t+\frac{(Q-2)^{2}}{4}\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\varrho^{-2}(L_{j}u)^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+(Q-2)\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}\varrho^{-1}(L_{j}u)_{\varrho}(L_{j}u)\>{\rm
d}x\>{\rm d}t$
$\displaystyle=\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}|(L_{j}u)_{\varrho}|^{2}\>{\rm
d}x\>{\rm
d}t-(Q-2)^{2}\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

Now using (5.3), we have
$L_{j}(u_{\varrho})=(L_{j}u)_{\varrho}+\varrho^{-1}L_{j}u$, and using
integration by parts we obtain

$\sum_{j=1}^{n+1}\int_{\mathbb{R}^{n+1}}|(L_{j}u)_{\varrho}|^{2}\>{\rm
d}x\>{\rm d}t\\\
=4\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u_{\varrho})\psi^{1/2}}{\varrho}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}+4(Q-3)\sum_{k=0}^{\infty}\lambda_{k}\bigg{|}\bigg{|}\frac{(\mathcal{P}_{k}u)\psi^{1/2}}{\varrho^{2}}\bigg{|}\bigg{|}^{2}_{L^{2}(\mathbb{R}^{n+1})}.$

In the above, we use Lemma 5.2 twice. Now, substituting this in the above and
combining two terms we deduce the required identity. ∎

######  Remark 5.4.

By utilizing Lemma 5.2 and Lemma 5.3, we can compare all terms in Theorem 1.4
and Theorem 5.1. After combining the terms, we find that the Rellich identity,
expressed in terms of spherical derivatives and projection operators, is the
same.

##  6\. Symmetrization principle on Grushin space

This section aims to discuss the symmetrization type principle in the Grushin
space. The rearrangement type arguments, particularly, Pólya-Szegö inequality
are not true when dealing with the higher-order operator, even for the
Euclidean Laplacian. Here we shall discuss the symmetrization principle in the
Grushin space. The results obtained here can be compared with [8, Theorem
1.11] in the Euclidean setting and we also refer to [20] for the related open
problems. The space $C_{c,rad}^{\infty}$ denotes the space of all radial
functions. Let us begin with the following first-order inequality.

######  Theorem 6.1.

Let $G$ be the Grushin space with dimension $Q\geq 3$ and $B_{R}(o)$ denotes
the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. Assume $W$ be
any function on $(0,R)$. Then we have

$\displaystyle\int_{B_{R}(o)}|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t\geq\int_{B_{R}(o)}W(\varrho)|u|^{2}|\nabla_{G}\varrho|^{2}\>{\rm d}x\>{\rm
d}t\quad\text{ for all }u\in
C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$

if and only if

$\displaystyle\int_{B_{R}(o)}|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t\geq\int_{B_{R}(o)}W(\varrho)|u|^{2}|\nabla_{G}\varrho|^{2}\>{\rm d}x\>{\rm
d}t\quad\text{ for all }u\in
C_{c,rad}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right).$

###### Proof.

The one-way implication is trivial. So it is enough to show the other way. Let
us begin with the spherical harmonic decomposition of $u$ and we can write

$\displaystyle
u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma).$

Next exploiting the polar coordinate structure we deduce

$\displaystyle\int_{B_{R}(o)}|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}\bigg{[}{d_{k}^{\prime}}^{2}\varrho^{n+1}+4\lambda_{k}d_{k}^{2}\varrho^{n-1}\bigg{]}\>{\rm
d}\varrho,$ (6.1)

and

$\displaystyle\int_{B_{R}(o)}W(\varrho)|u|^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n+1}\>{\rm
d}\varrho.$ (6.2)

Now, using the result for each radial function $d_{k}(\varrho)$ we have

$\displaystyle\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho\geq\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n+1}\>{\rm d}\varrho.$
(6.3)

After subtracting (6.2) from (6.1), and then using (6.3), it only remains to
check for each $k$, the non-negativity of the following term
$4\lambda_{k}\int_{0}^{R}d_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho$. This is
immediate as for each $k\geq 1$, we have $4\lambda_{k}\geq(Q-1)$. This
completes the proof. ∎

Now we are in a situation to prove the second-order version of the above
theorem. We obtain the following symmetrization principle for the Second-order
operator :

######  Theorem 6.2.

Let $G$ be the Grushin space with dimension $Q\geq 3$ and $B_{R}(o)$ denotes
the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. Assume $W$ be
any function on $(0,R)$. Then we have

$\displaystyle\int_{B_{R}(o)}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t\geq\int_{B_{R}(o)}W(\varrho)|u|^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t\quad\text{ for all }u\in
C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$

if and only if

$\displaystyle\int_{B_{R}(o)}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t\geq\int_{B_{R}(o)}W(\varrho)|u|^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm d}t\quad\text{ for all }u\in
C_{c,rad}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right).$

###### Proof.

The one-sided implication follows trivially. So it is enough to show the other
way. Let us begin with the spherical harmonic decomposition of $u$ and we can
write

$\displaystyle
u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma).$

By exploiting the polar coordinate structure, we deduce

$\displaystyle\int_{B_{R}(o)}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\bigg{[}\int_{0}^{R}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle+16\lambda_{k}^{2}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\int_{0}^{R}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)d_{k}\varrho^{n-1}\>{\rm
d}\varrho\bigg{]},$ (6.4)

and

$\displaystyle\int_{B_{R}(o)}W(\varrho)|u|^{2}|\nabla_{G}\varrho|^{2}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n+1}\>{\rm
d}\varrho.$ (6.5)

Then, using the result for each radial function $d_{k}(\varrho)$ we have

$\displaystyle\int_{0}^{R}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho\geq\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n+1}\>{\rm d}\varrho.$
(6.6)

After subtracting (6.5) from (6), and then using (6.6), it only remains to
check for each $k$, the non-negativity of the following term

$\displaystyle\mathcal{M}:=16\lambda_{k}^{2}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\int_{0}^{R}d_{k}^{\prime\prime}d_{k}\varrho^{n-1}\>{\rm
d}\varrho-8\lambda_{k}(Q-1)\int_{0}^{R}d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm
d}\varrho.$

Now using the by-parts we have

$\displaystyle\int_{0}^{R}d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm
d}\varrho=-\frac{(Q-4)}{2}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm d}\varrho,$
(6.7)

and

$\displaystyle\int_{0}^{R}d_{k}^{\prime\prime}d_{k}\varrho^{n-1}\>{\rm
d}\varrho$ $\displaystyle=-\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho-(Q-3)\int_{0}^{R}d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm d}\varrho$
$\displaystyle=-\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+\frac{(Q-3)(Q-4)}{2}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho.$ (6.8)

Using (6.7) and (6) in $\mathcal{M}$, we obtain

$\displaystyle\mathcal{M}$
$\displaystyle=8\lambda_{k}\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+8\lambda_{k}\big{(}2\lambda_{k}+Q-4\big{)}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho.$

This follows as for each $k\geq 1$, we have
$2\lambda_{k}+Q-4\geq\frac{3}{2}(Q-3)$. Finally, using $Q\geq 3$, we obtain
$\mathcal{M}$ is non-negative and the result follows. ∎

Our next theorem deals with the Hardy-Rellich type symmetrization principle.
It is worth noting that while proving the Hardy-Rellich we need to assume an
additional assumption on the function $W.$ The inequality reads as follows :

######  Theorem 6.3.

Let $G$ be the Grushin space with dimension $Q\geq 3$ and $B_{R}(o)$ denotes
the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. Assume $W(t)$ be
any function on $(0,R)$ satisfying

$\displaystyle 2\int_{0}^{R}{f^{\prime}}^{2}(t)t^{Q-3}\>{\rm d}t$
$\displaystyle+2\big{(}2\lambda_{1}+Q-4\big{)}\int_{0}^{R}f^{2}(t)t^{Q-5}\>{\rm
d}t$ $\displaystyle\geq\int_{0}^{R}W(t)f^{2}(t)t^{Q-3}\>{\rm d}t\quad\text{
for all }f\in C_{c}^{\infty}(0,R),$ (6.9)

where $\lambda_{1}=(Q-1)/4$. Then we have

$\displaystyle\int_{B_{R}(o)}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t\geq\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t\quad\text{ for all }u\in
C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$

if and only if

$\displaystyle\int_{B_{R}(o)}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t\geq\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t\quad\text{ for all }u\in
C_{c,rad}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right).$

###### Proof.

The one-sided implication follows trivially. So it is enough to show the other
way. Let us begin with the spherical harmonic decomposition of $u$ and we can
write

$\displaystyle
u(\varrho,\sigma)=\sum_{k=0}^{\infty}d_{k}(\varrho)\Phi_{k}(\sigma).$

Next exploiting the polar coordinate structure we deduce

$\displaystyle\int_{B_{R}(o)}\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\bigg{[}\int_{0}^{R}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho$
$\displaystyle+16\lambda_{k}^{2}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-8\lambda_{k}\int_{0}^{R}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)d_{k}\varrho^{n-1}\>{\rm
d}\varrho\bigg{]},$ (6.10)

and

$\displaystyle\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t=\frac{1}{2}\sum_{k=0}^{\infty}\int_{0}^{R}W(\varrho)\bigg{[}{d_{k}^{\prime}}^{2}\varrho^{n+1}+4\lambda_{k}d_{k}^{2}\varrho^{n-1}\bigg{]}\>{\rm
d}\varrho.$ (6.11)

Then, using the result for each radial function $d_{k}(\varrho)$ we have

$\displaystyle\int_{0}^{R}\left(d_{k}^{\prime\prime}+\frac{(Q-1)}{\varrho}d_{k}^{\prime}\right)^{2}\varrho^{n+1}\>{\rm
d}\varrho\geq\int_{0}^{R}W(\varrho){d_{k}^{\prime}}^{2}\varrho^{n+1}\>{\rm
d}\varrho.$ (6.12)

After subtracting (6.11) from (6), and then using (6.12), it is only remain to
check for each $k\geq 1$, the non-negativity of the following term

$\displaystyle\mathcal{B}_{k}$
$\displaystyle:=4\lambda_{k}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-2\int_{0}^{R}d_{k}^{\prime\prime}d_{k}\varrho^{n-1}\>{\rm d}\varrho$
$\displaystyle-2(Q-1)\int_{0}^{R}d_{k}^{\prime}d_{k}\varrho^{n-2}\>{\rm
d}\varrho-\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho.$

Using (6.7) and (6) in $\mathcal{B}_{k}$, we obtain

$\displaystyle\mathcal{B}_{k}=2\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+2\big{(}2\lambda_{k}+Q-4\big{)}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho$

We will be done if we prove $\mathcal{B}_{k}\geq 0$ for all $k\geq 2$, as the
$k=1$ case is already done if we chose radial function $d_{1}$ in (6.3). Now
define

$\displaystyle\mathcal{B}_{1,k}:=2\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+2\big{(}2\lambda_{1}+Q-4\big{)}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho$

for each $k\geq 2$. Then it is easy to observe that by choosing $d_{k}$ in
(6.3), we have $\mathcal{B}_{1,k}\geq 0$ for each $k\geq 2$. Now we notice

$\displaystyle\mathcal{B}_{k}-\mathcal{B}_{1,k}$
$\displaystyle=2\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho+2\big{(}2\lambda_{k}+Q-4\big{)}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho-\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho$
$\displaystyle-2\int_{0}^{R}{d_{k}^{\prime}}^{2}\varrho^{n-1}\>{\rm
d}\varrho-2\big{(}2\lambda_{1}+Q-4\big{)}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho+\int_{0}^{R}W(\varrho)d_{k}^{2}\varrho^{n-1}\>{\rm d}\varrho$
$\displaystyle=4\big{(}\lambda_{k}-\lambda_{1}\big{)}\int_{0}^{R}d_{k}^{2}\varrho^{n-3}\>{\rm
d}\varrho.$

Finally, observing for each $k\geq 2$, we have
$4(\lambda_{k}-\lambda_{1})\geq(Q^{2}-3Q+1)\geq 0$, when $Q\geq 3$, we say
each $\mathcal{B}_{k}$ is nonnegative. This completes the proof. ∎

######  Remark 6.1.

The rearrangement type arguments are valid in Hardy-Rellich type inequalities
for Grushin spaces with every homogeneous dimension $Q\geq 3$ under an
additional assumption (6.3). It is worth noting that such an assumption has
far-reaching consequences in the Euclidean setting, and is related to the
symmetry-breaking phenomenon depending on the dimension of the ambient space.
For further details, we refer to [8, Remark 1.2].

##  7\. Applications: Second-order uncertainty Principles and Caffarelli-Kohn-
Nirenberg inequalities

This final section is devoted to the application of the results we obtained in
the previous sections. Recently, there has been a growing interest in
obtaining sharp uncertainty principles and the existence of extremals. To
begin with, we shall observe some immediate facts of our main theorems. One of
the important aspects of Corollary 3.1 is that the Hardy inequality holds for
both radial and non-radial functions. However, we observed that in Theorem 1.3
we require an additional condition (1.6) to obtain Hardy-Relich type
inequality. In this part, we shall examine the outcomes when the dimensions of
the Grushin space and Bessel pair differ. The following result can be compared
with [8, Theorem 1.3, Theorem 1.4]. It is important to note that we will
obtain the identity without any sign assumption on $V,$ where $V$ is a radial
function, while for the non-radial setting, we need a non-negative sign
assumption on $V.$ The result is as follows :

######  Proposition 7.1.

Let $G$ be the Grushin space with dimension $Q\geq 3$ and $B_{R}(o)$ denotes
the $\varrho-$gauge ball with radius $R$ for $0<R\leq\infty$. Assume that
$\big{(}V(r),W(r)+Q\frac{V_{r}(r)}{r}\big{)}$ is a $(Q+2)$-dimensional Bessel
pair on $(0,R)$ with positive solution $f(r)$. Then for all radial function
$u\in C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$ there holds

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$ $\displaystyle=\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t$
$\displaystyle+\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}\left(\frac{u_{\varrho}}{\varrho
f(\varrho)}\right)\right|^{2}\varrho^{2}f^{2}(\varrho)\>{\rm d}x\>{\rm d}t.$

Moreover, if $V\geq 0$ and it satisfies (1.6), then for all $u\in
C_{c}^{\infty}\left(B_{R}(o)\backslash\\{o\\}\right)$ there holds

$\displaystyle\int_{B_{R}(o)}V(\varrho)\frac{(\mathcal{L}_{G}u)^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm d}t$
$\displaystyle\geq\int_{B_{R}(o)}W(\varrho)|\nabla_{G}u|^{2}\>{\rm d}x\>{\rm
d}t$
$\displaystyle+\int_{B_{R}(o)}V(\varrho)\left|\nabla_{G}\left(\frac{u_{\varrho}}{\varrho
f(\varrho)}\right)\right|^{2}\varrho^{2}f^{2}(\varrho)\>{\rm d}x\>{\rm d}t,$

where the $\varrho$ in suffix means the derivative with respect to the radial
part $\varrho$.

###### Proof.

Notice that if $\big{(}V(r),W(r)+Q\frac{V_{r}(r)}{r}\big{)}$ is a
$(Q+2)$-dimensional Bessel pair with positive solution $f(r)$, then
$\bigg{(}V(r),W(r)-(Q-1)\biggl{\\{}\frac{V(r)}{r^{2}}-\frac{V_{r}(r)}{r}\biggr{\\}}\bigg{)}$
is a $Q$-dimensional Bessel pair on $(0,R)$ with positive solution $rf(r)$.
Now applying Theorem 1.2 and Theorem 1.3, we deduce our results. ∎

As a consequence of this result, we obtain the following double-weighted
Hardy-Rellich inequality.

######  Corollary 7.1.

Let $G$ be the Grushin space with dimension $Q\geq 5$ and $B_{R}(o)$ denotes
the $\varrho-$gauge ball with radius $R$ for $R>0$. Then for all $u\in
C_{c}^{\infty}(B_{R}(o)\setminus\\{o\\})$, we have

$\displaystyle\int_{B_{R}(o)}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\frac{Q^{2}}{4}\int_{B_{R}(o)}\frac{|\nabla_{G}u|^{2}}{|\varrho|^{2}\big{(}1-\big{(}\frac{R}{\varrho}\big{)}^{-Q}\big{)}^{2}}\>{\rm
d}x\>{\rm d}t.$

###### Proof.

Applying
$\left(1,\frac{Q^{2}}{4}r^{-2}\left(1-\left(\frac{r}{R}\right)^{Q}\right)^{-2}\right)$
as $(Q+2)$ dimensional Bessel pair in Proposition 7.1 and then removing the
non-negative remainder term, we obtain the result. ∎

######  Remark 7.1.

Recently, the above inequality has been obtained in the Cartan-Hadamard
manifolds and nilpotent Lie groups in [39] and [49] respectively.

We can also deduce several Caffarelli-Kohn-Nirenberg-type inequalities.
Recently, the sharp $L^{2}$-Caffarelli-Kohn-Nirenberg inequalities for Grushin
vector fields were investigated in [13], and in [5] the sharp second-order
uncertainty principles were studied in the Euclidean space. Now we establish a
few second-order uncertainty principles (USP) and among them, Heisenberg and
Hydrogen USP are the two main results.

######  Corollary 7.2.

Let $Q\geq 5$. For all $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, we have

  * (a)

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\int_{\mathbb{R}^{n+1}}(Q+2-\varrho^{2})|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

  * (b)

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\int_{\mathbb{R}^{n+1}}\bigg{(}\frac{Q+1}{\varrho}-1\bigg{)}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

  * (c)

for $b<1:$

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\int_{\mathbb{R}^{n+1}}\bigg{(}\frac{Q+1-b}{\varrho^{b+1}}-\frac{1}{\varrho^{2b}}\bigg{)}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

  * (d)

for $b>1:$

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\geq\int_{\mathbb{R}^{n+1}}\bigg{(}\frac{Q+b-1}{\varrho^{b+1}}-\frac{1}{\varrho^{2b}}\bigg{)}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

###### Proof.

We apply the second part of Proposition 7.1 with appropriate
$(Q+2)$-dimensional Bessel pairs and after deducting the remainder terms we
obtain the results. Here are the respective Bessel pairs: $(1,Q+2-r^{2})$,
$\big{(}1,\frac{Q+1}{r}-1\big{)}$,
$\big{(}1,\frac{Q+1-b}{r^{b+1}}-\frac{1}{r^{2b}}\big{)}$ for $b<1$, and
$\big{(}\frac{Q+b-1}{r^{b+1}}-\frac{1}{r^{2b}}\big{)}$ for $b>1$ on
$(0,\infty)$. ∎

We can also deduce the following second-order Caffarelli-Kohn-Nirenberg (CKN)
inequalities on Grushin space from the above Corollary 7.2. Also, see [25, 47]
for certain types of CKN inequalities.

######  Corollary 7.3.

Let $Q\geq 5$. For all $u\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, we have

  * (a)

$(\text{Second order Heisenberg Uncertainty Principle})$

$\displaystyle\bigg{(}\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\mathbb{R}^{n+1}}\varrho^{2}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\geq\frac{(Q+2)}{2}\int_{\mathbb{R}^{n+1}}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm d}t.$

Moreover, the sharp constant is $\frac{(Q+2)}{2}$, and achieved by the
functions of the form $u(x,t)=\alpha e^{-\beta\varrho^{2}(x,t)}$,
$\alpha\in\mathbb{R}$, $\beta>0$.

  * (b)

$(\text{Second order Hydrogen Uncertainty Principle})$

$\displaystyle\bigg{(}\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\mathbb{R}^{n+1}}|\nabla_{G}u|^{2}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\geq\frac{(Q+1)}{2}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho}\>{\rm
d}x\>{\rm d}t.$

Moreover, the sharp constant is $\frac{(Q+1)}{2}$, and achieved by the
functions of the form
$u(x,t)=\alpha(1+\beta\varrho(x,t))e^{-\beta\varrho(x,t)}$,
$\alpha\in\mathbb{R}$, $\beta>0$.

  * (c)

for $b<1:$

$\displaystyle\bigg{(}\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{2b}}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\geq\frac{(Q+1-b)}{2}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{b+1}}\>{\rm
d}x\>{\rm d}t.$

Moreover, the sharp constant is $\frac{(Q+1-b)}{2}$, and achieved by the
functions of the form
$u(x,t)=\alpha\int_{\varrho(x,t)}^{\infty}s\exp\left(-\beta\frac{s^{1-b}}{1-b}\right){\rm
d}s$, $\alpha\in\mathbb{R}$, $\beta>0$.

  * (d)

for $b>1:$

$\displaystyle\bigg{(}\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{2b}}\>{\rm
d}x\>{\rm
d}t\bigg{)}^{\frac{1}{2}}\geq\frac{(Q+b-1)}{2}\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u|^{2}}{\varrho^{b+1}}\>{\rm
d}x\>{\rm d}t.$

Moreover, the sharp constant is $\frac{(Q+b-1)}{2}$, and achieved by the
functions of the form
$u(x,t)=\alpha\int_{\varrho(x,t)}^{\infty}s\exp\left(-\beta\frac{s^{b-1}}{b-1}\right){\rm
d}s$, $\alpha\in\mathbb{R}$, $\beta>0$.

###### Proof.

Using the standard scaling argument the proof follows. For clarity, we will
write detailed proof here. We will only write the proof of part (a), and
others will follow similarly. Let us write the dilation for positive real
number $\lambda$, on Grushin space by $\delta_{\lambda}(x,t):=(\lambda
x,\lambda^{2}t)$, where $(x,t)\in\mathbb{R}^{n+1}\setminus\\{o\\}$. Now for
$u\in C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, let us define the
scaling of the function by
$u_{\lambda}(x,t):=\lambda^{\frac{Q-2}{2}}u(\delta_{\lambda}(x,t))$, and it is
easy to notice that $u_{\lambda}\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$. Now, consider the change of
variable $\lambda x=z$ and $\lambda^{2}t=\mu$ and we are denoting the Grushin
gradient and Laplacian operator as $\nabla_{g}$ and $\mathcal{L}_{g}$ in terms
of $(z,\mu)$ variable. Then we have the following

$\displaystyle\>{\rm d}x\>{\rm d}t=\lambda^{-Q}{\rm d}z{\rm d}\mu,$
$\displaystyle\varrho^{2}(x,t)=\big{(}|x|^{4}+4t^{2}\big{)}^{\frac{1}{2}}=\lambda^{-2}\varrho^{2}(z,\mu),$

and

$\displaystyle|\nabla_{G}\varrho(x,t)|^{2}=\frac{|x|^{2}}{\varrho^{2}(x,t)}=\frac{\lambda^{-2}|z|^{2}}{\lambda^{-2}\varrho^{2}(z,\mu)}=|\nabla_{g}\varrho(z,\mu)|^{2}.$

Moreover, we observe that

$\displaystyle|\nabla_{G}u_{\lambda}(x,t)|^{2}=\lambda^{Q}|\nabla_{g}u(z,\mu)|^{2},$

and

$\displaystyle\mathcal{L}_{G}u_{\lambda}(x,t)=\lambda^{\frac{Q-2}{2}}\lambda^{2}\mathcal{L}_{g}u(z,\mu).$

Applying Corollary 7.2, part (a), for $u_{\lambda}\in
C_{c}^{\infty}(\mathbb{R}^{n+1}\setminus\\{o\\})$, with $\lambda>0$, we have

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u_{\lambda}|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t+\int_{\mathbb{R}^{n+1}}\varrho^{2}|\nabla_{G}u_{\lambda}|^{2}\>{\rm
d}x\>{\rm d}t\geq(Q+2)\int_{\mathbb{R}^{n+1}}|\nabla_{G}u_{\lambda}|^{2}\>{\rm
d}x\>{\rm d}t.$

Then writing in terms of $u$ and performing the above change of variable we
deduce

$\displaystyle\lambda^{2}\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{g}u|^{2}}{|\nabla_{g}\varrho|^{2}}\,{\rm
d}z{\rm
d}\mu+\lambda^{-2}\int_{\mathbb{R}^{n+1}}\varrho^{2}|\nabla_{g}u|^{2}\,{\rm
d}z{\rm d}\mu\geq(Q+2)\int_{\mathbb{R}^{n+1}}|\nabla_{g}u|^{2}\,{\rm d}z{\rm
d}\mu.$

Finally, choose

$\displaystyle\lambda=\left(\frac{\int_{\mathbb{R}^{n+1}}\varrho^{2}|\nabla_{g}u|^{2}\,{\rm
d}z{\rm
d}\mu}{\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{g}u|^{2}}{|\nabla_{g}\varrho|^{2}}\,{\rm
d}z{\rm d}\mu}\right)^{\frac{1}{4}},$

and we get the desired result.

Now let us discuss the optimality for the part (c). Consider the smooth
function

$\displaystyle
u_{\alpha,\beta}(x,t)=\alpha\int_{\varrho(x,t)}^{\infty}s\exp\left(-\beta\frac{s^{1-b}}{1-b}\right){\rm
d}s,$

for $\alpha\in\mathbb{R}$, $\beta>0$, and $1-b>0$. We can easily notice that

$\displaystyle\frac{\partial
u_{\alpha,\beta}}{\partial\varrho}=-\alpha\varrho\exp\left(-\beta\frac{\varrho^{1-b}}{1-b}\right),$

and

$\displaystyle\frac{\partial^{2}u_{\alpha,\beta}}{\partial\varrho^{2}}=-\alpha\exp\left(-\beta\frac{\varrho^{1-b}}{1-b}\right)+\alpha\beta\varrho^{1-b}\exp\left(-\beta\frac{\varrho^{1-b}}{1-b}\right).$

Now using polar coordinate decomposition, change of variable, and properties
of Gamma functions, we have

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u_{\alpha,\beta}|^{2}}{\varrho^{b+1}}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}|\Omega|\frac{\alpha^{2}}{2\beta}\left(\frac{1-b}{2\beta}\right)^{\frac{Q}{1-b}}\left(\frac{Q}{1-b}\right)\Gamma\left(\frac{Q}{1-b}\right),$
$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u_{\alpha,\beta}|^{2}}{\varrho^{2b}}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}|\Omega|\frac{\alpha^{2}}{2\beta}\left(\frac{1-b}{2\beta}\right)^{\frac{Q}{1-b}+1}\left(\frac{Q+1-b}{1-b}\right)\left(\frac{Q}{1-b}\right)\Gamma\left(\frac{Q}{1-b}\right),$

and

$\displaystyle\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u_{\alpha,\beta}|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t=\frac{1}{2}|\Omega|\frac{\alpha^{2}Q}{2\beta}\left(\frac{1-b}{2\beta}\right)^{\frac{Q}{1-b}-1}\left(\frac{Q+1-b}{4}\right)\Gamma\left(\frac{Q}{1-b}\right).$

Notice that as all the terms of the inequality are finite, and hence
inequality holds for such functions, which can be justified by the density
arguments of smooth compactly supported functions. Therefore, we have

$\displaystyle\frac{\left(\int_{\mathbb{R}^{n+1}}\frac{|\mathcal{L}_{G}u_{\alpha,\beta}|^{2}}{|\nabla_{G}\varrho|^{2}}\>{\rm
d}x\>{\rm
d}t\right)\left(\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u_{\alpha,\beta}|^{2}}{\varrho^{2b}}\>{\rm
d}x\>{\rm
d}t\right)}{\left(\int_{\mathbb{R}^{n+1}}\frac{|\nabla_{G}u_{\alpha,\beta}|^{2}}{\varrho^{b+1}}\>{\rm
d}x\>{\rm d}t\right)^{2}}=\left(\frac{Q+1-b}{2}\right)^{2}.$

The proof of the optimality of the constant in part (c) has been concluded.
The other cases can be proved similarly by considering corresponding
optimizing functions. It should be noted that for parts (a) and (b), the
optimizing function is a special case of part (c) with $b=-1$ and $b=0$
respectively. ∎

######  Remark 7.2.

The cases (a) and (b) above of the above theorem can be compared to [5,
Theorem 2.1, 2.4]. Moreover, the inequalities obtained in parts (c) and (d)
are related to a special case of [10, Theorem 1.3]. It’s worth mentioning that
by exploiting the first part of Proposition 7.1 and the corresponding Bessel
pair, we can obtain the results for radial operators as well. These second-
order $L^{2}$-Caffarelli-Kohn-Nirenberg inequalities are a completely new
addition to the literature for Grushin vector fields, and provide a natural
extension of the first-order case presented in [13]. Also, certain uncertainty
principle related to Rellich inequality was studied in [29].

## Acknowledgments

The research of D. Ganguly is partially supported by the SERB MATRICS
(MTR/2023/000331). The author P. Roychowdhury is supported by the National
Theoretical Science Research Center Operational Plan (Project number:
112L104040).

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