# Functional inequalities on symmetric spaces of noncompact type and
applications

Aidyn Kassymov  Aidyn Kassymov: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium and Institute of Mathematics
and Mathematical Modeling Almaty, Kazakhstan and Al-Farabi Kazakh National
University Almaty, Kazakhstan E-mail address<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>,  Vishvesh Kumar  Vishvesh Kumar: Department of
Mathematics: Analysis, Logic and Discrete Mathematics Ghent University,
Belgium E-mail address<EMAIL_ADDRESS>and<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael Ruzhansky: Department of Mathematics:
Analysis, Logic and Discrete Mathematics Ghent University, Ghent, Belgium and
School of Mathematical SciencesQueen Mary University of London, United Kingdom
E-mail address<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to begin a systematic study of functional
inequalities on symmetric spaces of noncompact type of higher rank. Our first
main goal of this study is to establish the Stein-Weiss inequality, also known
as a weighted Hardy-Littlewood-Sobolev inequality, for the Riesz potential on
symmetric spaces of noncompact type. This is achieved by performing delicate
estimates of ground spherical function with the use of polyhedral distance on
symmetric spaces and by combining the integral Hardy inequality developed by
Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on
symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence
of the Stein-Weiss inequality, we deduce Hardy-Sobolev, Hardy-Littlewood-
Sobolev, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on
symmetric spaces of noncompact type. The second main purpose of this paper is
to show the applications of aforementioned inequalities for studying nonlinear
PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg
inequality can be used to establish small data global existence results for
the semilinear wave equations with damping and mass terms for the Laplace-
Beltrami operator on symmetric spaces.

###### Key words and phrases:

Symmetric spaces of noncompact type, Hardy-Littlewood-Sobolev inequality,
Stein-Weiss inequality, Riesz potential, Laplace-Beltrami operator, Sobolev
inequality, Hardy inequality, Gagliardo-Nirenberg inequality, Caffarelli-Kohn-
Nirenberg inequality, nonlinear wave equations, global wellposedness.

###### 2010 Mathematics Subject Classification:

26D10, 45J05.

AK, VK and MR are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis
and Partial Differential Equations, the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). VK and MR are
supported by FWO Senior Research Grant G011522N. MR is also supported by EPSRC
grant EP/R003025/2. The authors are grateful to Jean-Philippe Anker for
inspiring this work by his minicourse on symmetric spaces at Ghent University,
and by further stimulating discussions. The authors would also like to thank
Hong-Wei Zhang for several fruitful discussions.

###### Contents

  1. 1 Introduction
  2. 2 Riemannian symmetric spaces and Helgason-Fourier transform
    1. 2.1 Riemannian symmetric spaces
    2. 2.2 Helgason-Fourier transform on Riemannian symmetric spaces
    3. 2.3 Integral Hardy inequalities on metric measure spaces
  3. 3 Stein-Weiss and Hardy-Littlewood-Sobolev inequalities on symmetric spaces of noncompact type
  4. 4 Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on symmetric spaces
  5. 5 Nonlinear wave equation with a damped term associated with the Laplace-Beltrami operator on symmetric spaces

##  1\. Introduction

The study of functional inequalities and weighted functional inequalities
plays a significant role in the investigation of problems in differential
geometry, harmonic analysis, partial differential equations and several other
areas of mathematics. In particular, these inequalities have been used
extensively to study existence results related to several important non-linear
equations. In this paper, we establish several important functional
inequalities including Stein-Weiss inequality, Hardy-Sobolev, Gagliardo-
Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on the higher rank
Riemannian symmetric spaces of noncompact type and present some applications
to study global existence of wave equations with dumping and mass term
associated to the Laplace-Beltrami operator on symmetric spaces. Riemannian
symmetric spaces are a very important class of non-positively curved
Riemannian manifolds. Recently, many researchers contributed to the
development of certain important functional inequalities on non-positively
curved manifolds using different suitable methods from the Fourier analysis
and geometric analysis. But most of them were only confined to rank one
symmetric space, e.g., real or complex hyperbolic spaces and their different
models and, in this case, one uses the Helgason-Fourier analysis. One
fundamental functional inequality in the Euclidean harmonic analysis is the
classical Stein-Weiss inequality established by Stein and Weiss [63]. It
states that:

######  Theorem 1.1.

Let $0<\lambda<N$, $1<p<\infty$, $\alpha<\frac{N(p-1)}{p}$,
$\beta<\frac{N}{q}$, $\alpha+\beta\geq 0$ and
$\frac{\lambda-\alpha-\beta}{N}=\frac{1}{p}-\frac{1}{q}$. For $1<p\leq
q<\infty$, we have

(1.1) $\||x|^{-\beta}I_{\lambda}u\|_{L^{q}(\mathbb{R}^{N})}\leq
C\||x|^{\alpha}u\|_{L^{p}(\mathbb{R}^{N})},$

where $C$ is a positive constant independent of $u$. Here the Riesz potential
$I_{\lambda}$ on $\mathbb{R}^{N}$ is defined as

(1.2)
$I_{\lambda}u(x)=\int_{\mathbb{R}^{N}}\frac{u(y)}{|x-y|^{N-\lambda}}dy,\,\,\,\,0<\lambda<N.$

The unweighted version of inequality (1.1) was proved by Sobolev [61] by
extending a multi-dimensional version of the Hardy-Littlewood inequality [36].
For this reason the unweighted case of (1.1) is called the Hardy-Littlewood-
Sobolev inequality. In other words, the Stein-Weiss inequality is a radially
weighted version of the Hardy-Littlewood-Sobolev inequality.

In the Euclidean space setting, many researchers have studied generalisations
of the Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities. There are
several works devoted to the analysis of the best constants and extremisers of
the aforementioned inequalities. The literature of this area is so vast that
it is practically impossible to fully review it, but we cite [22, 26, 25, 46,
49, 53, 1, 10, 11, 12] here just to mention a very few of them. In the
noncommuative setting (e.g. Heisenberg groups, homogeneous groups, general Lie
groups), the Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities are
also well-developed, see [27, 28, 34, 42, 56, 57, 58, 21, 16, 13, 14]. We
refer to [35] for a version of the Hardy-Littlewood-Sobolev inequality on
compact Riemannian manifolds.

One of the main objectives of this paper is to prove the Stein-Weiss
inequality on symmetric spaces of noncompact type. The strategy of this paper
follows that of [56], where the case of graded groups was considered, however
the argument on symmetric spaces is more geometrically involved. Now, we state
this result as follows.

######  Theorem 1.2.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$. Let
$0<\sigma<n$, $1<p<\infty$, $\alpha<\frac{n}{p^{\prime}}$,
$\beta<\frac{n}{q},$ $\alpha+\beta\geq 0$ and
$\frac{\sigma-\alpha-\beta}{n}=\frac{1}{p}-\frac{1}{q}$. Then, for $1<p\leq
q<\infty$ and for sufficiently large $\xi>0,$ there exists a positive constant
$C$ independent by $u$, such that

(1.3)
$\left(\int_{X}\left|\int_{X}G_{\xi,\sigma}(y^{-1}x)u(y)dy\right|^{q}\,|x|^{-\beta
q}\,dx\right)^{\frac{1}{q}}\leq C\||x|^{\alpha}u\|_{L^{p}(X)}.$

Here $G_{\xi,\sigma}$ denotes the Bessel-Green-Riesz Kernel on the Riemannian
symmetric space $X$ of any rank (see Section 2 for the definition), and $|x|$
denotes the distance between $0:=eK\in X$ and the point $x\in X$.

As in the Euclidean setting, the Stein-Weiss inequality (1.3) implies the
Hardy-Littlewood-Sobolev inequality on Riemannian symmetric spaces of
noncompact type by choosing $\alpha=0$ and $\beta=0.$ It states that:

######  Theorem 1.3.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$.
Suppose that $0<\sigma<n$, $1<p<\infty$ and
$\frac{\sigma}{n}=\frac{1}{p}-\frac{1}{q}$. Then for $1<p<q<\infty$ and $u\in
L^{p}(X)$, we have

(1.4)
$\left(\int_{X}\left|\int_{X}G_{\xi,\sigma}(y^{-1}x)u(y)dy\right|^{q}dx\right)^{\frac{1}{q}}\leq
C\left(\int_{X}|u(x)|^{p}dx\right)^{\frac{1}{p}},$

where $C$ is a positive constant and independent of $u$.

We remark here that the Hardy-Littlewood-Sobolev inequality on noncompact
symmetric spaces of noncompact type was also obtained by Anker [3] (see also
[64, Section 4]) by different methods.

The Hardy inequality is one of the well-known inequalities of G. H. Hardy
which basically says that:

(1.5)
$\left\|\frac{f}{|x|}\right\|_{L^{p}(\mathbb{R}^{N})}\leq\frac{p}{N-p}\|\nabla
f\|_{L^{p}(\mathbb{R}^{N})},\,\,\,\,1<p<N,$

where $f\in C^{\infty}_{0}(\mathbb{R}^{N})$ and $\nabla$ is the Euclidean
gradient. It is known that the constant $\frac{p}{N-p}$ is sharp.

There is a vast literature available on the Hardy inequalities on Euclidean
spaces, Lie groups and on manifolds. We refer to [18, 1, 39, 43, 58, 45, 44,
45, 70, 19, 55] and references therein for recent developments. Recently, in
[15], the authors proved a fractional analogue of the Hardy inequality on the
symmetric spaces of noncompact type using the fractional Poisson kernel on
symmetric spaces with the help of the extension problem for the fractional
Laplace-Beltrami operator on symmetric spaces [9].

One of our main aims is to show an analogue of the classical multi-dimensional
Hardy inequality on symmetric spaces of noncompact type. As a consequence, we
obtain the uncertainly principle in our setting. In fact, we first establish
the following inequality, an analogue of the Hardy-Sobolev inequality on
symmetric spaces of noncompact type.

######  Theorem 1.4.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$.
Suppose that $0<\sigma<n$, $1<p\leq q<\infty$ and $0\leq\beta<\frac{n}{q}$
such that $\frac{\sigma-\beta}{n}=\frac{1}{p}-\frac{1}{q}$. Then for $u\in
H^{\sigma,p}(X)$, we have

(1.6) $\left\|\frac{u}{|x|^{\beta}}\right\|_{L^{q}(X)}\leq
C\|u\|_{H^{\sigma,p}(X)},$

where $C$ is a positive constant independent of $u$.

Here $H^{\sigma,p}(X)$ denotes the Sobolev space on the symmetric space $X$
defined as in (2.7) in the next section.

If we take $q=p$ and $0<\sigma<\frac{n}{p}$ in Theorem 1.4, then inequality
(1.6) gives the the following Hardy inequality on symmetric spaces

(1.7) $\left\|\frac{u}{|x|^{\sigma}}\right\|_{L^{p}(X)}\leq
C\|u\|_{H^{\sigma,p}(X)}.$

The uncertainly principle on symmetric spaces can be derived from (1.7).
Indeed, we will show that

(1.8) $\left\|u\right\|_{L^{2}(X)}\leq
C\|u\|_{H^{\sigma,p}(X)}\||x|^{\sigma}u\|_{L^{\frac{p}{p-1}}(X)}.$

The classical Sobolev inequality (or a continuous Sobolev embedding) is one of
prominent functional inequalities widely used to study partial differential
equations. Let $\Omega\subset\mathbb{R}^{N}$ be a measurable set and let
$1<p<N$. Then the (classical) Sobolev inequality is formulated as

(1.9) $\|u\|_{L^{p^{*}}(\Omega)}\leq C\|\nabla u\|_{L^{p}(\Omega)},\,\,\,u\in
C^{\infty}_{0}(\Omega),$

where $C=C(N,p)>0$ is a positive constant, $p^{*}=\frac{Np}{N-p}$ and $\nabla$
is a standard gradient on $\mathbb{R}^{N}$. The best constant of this
inequality was obtained by Talenti in [67] and Aubin in [8].

E. Gagliardo [29] and L. Nirenberg [51] independently, obtained an
(interpolation) inequality, widely known as the Gagliardo-Nirenberg
inequality, which says that

(1.10) $\|u\|^{p}_{L^{p*}(\mathbb{R}^{N})}\leq C\|\nabla
u\|^{N(p^{*}-2)/2}_{L^{2}(\mathbb{R}^{N})}\|u\|^{(2p^{*}-N(p^{*}-2))/2}_{L^{2}(\mathbb{R}^{N})},\,\,u\in
H^{1}(\mathbb{R}^{N}),$

where

$\begin{cases}2\leq p^{*}\leq\infty\,\,\text{for}\,\,N=2,\\\ 2\leq
p^{*}\leq\frac{2N}{N-2}\,\,\text{for}\,\,N>2.\end{cases}$

Sobolev and Gagliardo-Nirenberg inequalities have several applications in PDEs
and variational principles. In this paper, we establish the Sobolev inequality
and the Gagliardo-Nirenberg inequality on symmetric spaces of noncompact type
which we will state as follows.

######  Theorem 1.5.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$.
Suppose $0<\sigma<n$, $\tau>0$, $p>1$, $\sigma p<n$, $\nu\geq 1$, $a\in(0,1]$
and

(1.11)
$\frac{1}{\tau}=a\left(\frac{1}{p}-\frac{\sigma}{n}\right)+\frac{(1-a)}{\nu}.$

Then there exists positive constant $C$ such that

(1.12) $\|u\|_{L^{\tau}(X)}\leq
C\|u\|^{a}_{H^{\sigma,p}}\|u\|^{1-a}_{L^{\nu}(X)}.$

By choosing $a=1$ in the above theorem we obtain the Sobolev inequality on
symmetric spaces of noncompact type. Indeed, for $0<\sigma<n$ and $1<p\leq
q<\infty$ such that $\frac{\sigma}{n}=\frac{1}{p}-\frac{1}{q}$ we obtain

(1.13) $\left\|u\right\|_{L^{q}(X)}\leq C\|u\|_{H^{\sigma,p}(X)}.$

We note that the Sobolev inequality on symmetric spaces and on the general Lie
groups was previously obtained by many authors; see [69, 64, 16, 57].

We will show applications of the Gagliardo-Nirenberg inequality to prove the
small data global existence of the solution for the semilinear PDEs on
symmetric spaces of noncompact type. For this, the following particular
version of the Gagliardo-Nirenberg inequality will be useful.

######  Theorem 1.6.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$. Let
$\tau\in\left[2,\frac{2n}{n-2}\right]$ and $a=\frac{n(\tau-2)}{\tau}.$ Then we
have

(1.14) $\|u\|_{L^{\tau}(X)}\leq
C\|u\|^{a}_{H^{1,2}(X)}\|u\|^{1-a}_{L^{\tau}(X)}.$

In one of their pioneering works, L. Caffarelli, R. Kohn and L. Nirenberg in
[17] established the following inequality:

######  Theorem 1.7.

Let $N\geq 1$, and let $l_{1}$, $l_{2}$, $l_{3}$,
$a,\,b,\,d,\,\delta\in\mathbb{R}$ be such that $l_{1},l_{2}\geq 1$,
$l_{3}>0,\,\,0\leq\delta\leq 1,$ and

(1.15)
$\frac{1}{l_{1}}+\frac{a}{N},\,\,\,\frac{1}{l_{2}}+\frac{b}{N},\,\,\,\frac{1}{l_{3}}+\frac{\delta
d+(1-\delta)b}{N}>0.$

Then,

(1.16) $\||x|^{\delta d+(1-\delta)b}u\|_{L^{l_{3}}(\mathbb{R}^{N})}\leq
C\||x|^{a}\nabla
u\|^{\delta}_{L^{l_{1}}(\mathbb{R}^{N})}\||x|^{b}u\|^{1-\delta}_{L^{l_{2}}(\mathbb{R}^{N})},\,\,\,u\in
C^{\infty}_{c}(\mathbb{R}^{N}),$

if and only if

(1.17) $\frac{1}{l_{3}}+\frac{\delta
d+(1-\delta)b}{N}=\delta\left(\frac{1}{l_{1}}+\frac{a-1}{N}\right)+(1-\delta)\left(\frac{1}{l_{2}}+\frac{b}{N}\right),\\\
a-d\geq 0,\,\,\,\,\text{if}\,\,\,\delta>0,\\\ a-d\leq
1,\,\,\,\,\text{if}\,\,\,\delta>0\,\,\,\text{and}\,\,\,\frac{1}{l_{3}}+\frac{\delta
d+(1-\delta)b}{N}=\frac{1}{l_{1}}+\frac{a-1}{N},$

where $C$ is a positive constant independent of $u$.

It is worth noting that the Caffarelli-Kohn-Nirenberg inequality includes
aforementioned well-known inequalities, such as the Gagliardo–Nirenberg
inequality, Hardy–Sobolev inequality and Sobolev inequality. We refer to [50,
44] and references therein for the investigation regarding the effects of the
curvature of the Riemannian manifolds for the validity of the Hardy and
Caffarelli-Kohn-Nirenberg inequalities on these manifolds and their best
constants. In this paper, we obtain the following Caffarelli-Kohn-Nirenberg
inequality on symmetric spaces of noncompact type.

######  Theorem 1.8.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
$\sigma$ be such that $0<\sigma<n$. Suppose $p>1$, $0<q<\tau<\infty$ such that
$a\in\left(\frac{\tau-q}{\tau},1\right]$ and $p\leq\frac{a\tau
q}{q-(1-a)\tau}$. Let $b,c$ be real numbers such that
$0\leq(c(1-a)-b)\leq\frac{n(q-(1-a)\tau)}{q\tau}$ and
$\frac{\sigma-n}{n}-\frac{(c(1-a)-b)q}{an}+\frac{q-(1-a)\tau}{a\tau
q}-\frac{1}{p}+1=0$. Then there exists a positive constant and independent of
$u$ such that

(1.18) $\||x|^{b}u\|_{L^{\tau}(X)}\leq
C\|u\|^{a}_{H^{\sigma,p}(X)}\||x|^{c}u\|^{1-a}_{L^{q}(X)}.$

As an application of the established Gagliardo-Nirenberg inequality, we show
the small data global existence of the solution for the following nonlinear
Cauchy problem involving the shifted Laplace-Beltrami operator
$\Delta_{x}:=\Delta+|\rho|^{2}$ on symmetric space of noncompact type $X,$
where $\rho$ is the half sum of multiplicity of positive roots:

(1.19)
$\begin{split}\begin{cases}u_{tt}(x,t)-\Delta_{x}u(x,t)+bu_{t}(x,t)+mu(x,t)=f(u(x,t)),\,\,\,\,(x,t)\in
X\times(0,T),\,\,T>0,\\\ u(x,0)=u_{0}(x),\,\,\,\,x\in X,\\\
u_{t}(x,0)=u_{1}(x),\,\,\,\,x\in X,\end{cases}\end{split}$

where $b,m>0$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following
conditions:

(1.20) $f(0)=0$

and

(1.21) $|f(u)-f(v)|\leq C(|u|^{p-1}+|v|^{p-1})|u-v|.$

The existence and non-existence results for the semilinear wave equation
(1.19) with or without dumping and mass term have been studied by many
prominent researchers for Euclidean spaces and on certain Lie groups by
employing different methods. On the Euclidean space $\mathbb{R}^{N},$ the
small data global existence of semilinear wave equation associated with
Laplacian $\Delta_{{\mathbb{R}^{N}}}$ on $\mathbb{R}^{n}$ (with $b=0$ and
$m=0$) is closely related with the Strauss conjecture. For more details on the
Strauss conjecture on the Euclidean space, we cite [62, 66, 40, 41, 32]. On
the Riemannian manifolds of negative curvature, the small data global
existence for the semilinear wave equation (with $b=0$) (1.19) has been
investigated in details during the recent years [47, 48, 6, 7, 59, 60, 72, 71,
2]. It is well-known that, in contrast with the Euclidean space, the Strauss
conjecture type phenomena does not occur in the case of negatively curved
Riemannian manifolds. Particularly, this was observed in the setting of
hyperbolic spaces (also on Damek-Ricci spaces) in [47, 48, 6, 7]. Later, these
results were extended to non-trapping asymptotically hyperbolic manifolds [60]
and to the Riemannian manifolds with strictly negatively sectional curvature
[59] using different method. Very recently, Anker and Zhang [2] investigated
semilinear wave equations on general symmetric spaces of noncompact type and
proved that a similar phenomena holds for general symmetric spaces; see also
[72, 71]. The Strichartz inequality played a very important role for
investigating aforementioned results on the Riemannian symmetric spaces [71,
72, 2].

In this paper, we consider the semilinear dumped wave equation with a mass
term (1.19) on symmetric spaces of noncompact type and prove the following
result concerning the global existence and uniqueness of problem (1.19). We
use the Gagliardo-Nirenberg inequality (1.14) to establish the proof of the
next theorem instead of the Strichartz inequality on symmetric spaces.

######  Theorem 1.9.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and let
$1\leq p\leq\frac{n}{n-2}$. Suppose that $f$ satisfies the conditions
(1.20)-(1.21). Assume that $u_{0}\in H^{1,2}(X)$ and $u_{1}\in L^{2}(X)$ are
such that $\|u_{0}\|_{H^{1,2}(X)}+\|u_{1}\|_{L^{2}(X)}<\varepsilon$. Then
there exists $\varepsilon_{0}>0$ such that for all
$0<\varepsilon\leq\varepsilon_{0}$ the Cauchy problem (1.19) has a unique
global solution $u\in C(\mathbb{R}_{+},H^{1,2}(X))\cap
C^{1}(\mathbb{R}_{+},L^{2}(X))$.

We will organise this manuscript as follows: In the next section, we will
provide a brief overview of analysis on the Riemannian symmetric spaces and
the Helgason-Fourier transform. In addition to this, we will also recall some
useful tools such as the Sobelev spaces on symmetric spaces and the integral
Hardy inequalities on metric measure spaces. Section 3 will be mainly occupied
by the proof of the Stein-Weiss inequality on symmetric spaces of any rank. As
a consequence, we deduce the Hardy-Littlewood-Sobolev inequality on symmetric
spaces of noncompact type. In Section 4, we will derived several important
functional inequalities on symmetric spaces of noncompact type. To guide the
reader through different inequalities established in this section, let us
briefly summarise the obtained results.

  * •

Hardy inequality on symmetric spaces of noncompact type;

  * •

Hardy-Sobolev inequality on symmetric spaces of noncompact type;

  * •

Gagliardo-Nirenberg inequality on symmetric spaces of noncompact type;

  * •

Caffarelli-Kohn-Nirenberg inequality on symmetric spaces of noncompact type.

In Section 5, we present an application of the Gagliardo-Nirenberg inequality
obtained in Section 4 to the global existence results of wave equations with
dumping term associated to Laplace-Beltrami operator on symmetric spaces on
noncompact type.

##  2\. Riemannian symmetric spaces and Helgason-Fourier transform

In this section, we recall some basic definitions, notation and nomenclature
related with the higher rank Riemannian symmetric spaces of noncompact type.
We also present definitions and fundamental properties of the Helgason-Fourier
transform, Sobolev spaces and some kernel estimates on symmetric symmetric on
noncompact type. The material presented in this section can be found in the
excellent books and research papers [38, 37, 30, 3, 4, 64]. Finally, we
discuss the weighted integral Hardy inequalities on metric measure spaces with
general weights [55] which will be helpful to establish our results in the
subsequent sections.

###  2.1. Riemannian symmetric spaces

Let $G$ be a noncompact, connected, semisimple Lie group with finite center
and let $K$ be a maximal compact subgroup. The homogeneous space $X:=G/K$ is a
Riemannian symmetric space of noncompact type. Let us assume that $\theta$ is
a fixed Cartan involution on the Lie algbra $\mathfrak{g}$ of $G$ associated
with the Cartan decomposition $\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{p}$ at
the Lie algebra level, where $\mathfrak{t}$ and $\mathfrak{p}$ are $+1$ and
$-1$ eigenspaces of $\theta$ respectively. It is known that if $\mathfrak{B}$
is the Cartan killing form of $\mathfrak{g}$ then $\mathfrak{B}$ induces the
$G$-invariant metric $d$ on $X$ by identifying the tangent space at origin
$eK$ of $X$ with $\mathfrak{p}$ and by restricting $\mathfrak{B}$ to
$\mathfrak{p}.$ The distance between two points $x_{1}=g_{1}K$ and
$x_{2}=g_{2}K$ of $X$ will be denoted by $d(x_{1},x_{2}).$ We will also use
the notation $|x|$ to denote $d(0,x),$ the distance between $0=eK\in X$ and
the point $x\in X.$ Let $\mathfrak{a}$ be a maximal abelian subalgebra of
$\mathfrak{p}$ and let $\mathfrak{a}^{*}$ be its dual space. The dimension of
$\mathfrak{a}$ is called the rank of $X.$ We denote $\dim\mathfrak{a}=l.$ For
$\alpha\in\mathfrak{a}^{*},$ we define

$\mathfrak{g}_{\alpha}:=\\{Y\in\mathfrak{a}:[H,Y]=\alpha(H)Y,\,\,\text{for
all}\,\,H\in\mathfrak{a}\\}.$

Then the set of restricted root of $\mathfrak{g}$ with respect to
$\mathfrak{a}$ is denoted by $\Sigma$ and defined as

$\Sigma:=\\{\alpha\in\mathfrak{a}^{*}\backslash\\{0\\}:\mathfrak{g}_{\alpha}\neq
0\\}.$

We denote $m_{\alpha}=\dim(\mathfrak{g}_{\alpha})$ for
$\alpha\in\mathfrak{a}^{*}.$ Let us choose a connected component in
$\mathfrak{a}$ in a manner that $\alpha\neq 0$ for all $\alpha\in\Sigma.$
Denote by $\mathfrak{a}^{+}$ the connected component, called a positive Weyl
chamber. Now, with respect to $\mathfrak{a}^{+},$ we define positive roots and
positive indivisible roots by
$\Sigma^{+}=\\{\alpha\in\Sigma:\alpha>0\,\,\text{on}\,\,\mathfrak{a}^{+}\\}$
and $\Sigma^{+}_{0}=\\{\alpha\in\Sigma:\frac{\alpha}{2}\notin\Sigma^{+}\\}$
respectively. We set
$\mathfrak{n}=\oplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\alpha}.$ Then
$\mathfrak{n}$ is a nilpotent subalgebra of $\mathfrak{g}.$ We denote the half
sum of positive roots counted with multiplicities $m_{\alpha}$ by
$\rho:=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha.$ The dimension
and the pseudo-dimension of $X$ will be denoted by $n$ and $\nu$ respectively,
that is, $n=l+\sum_{\alpha\in\Sigma^{+}}m_{\alpha}$ and
$\nu=l+2|\Sigma^{+}_{0}|.$ The Iwasawa decomposition of $\mathfrak{g}$ is
given by $\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{a}\oplus\mathfrak{n}$ on
the Lie algebra level. On the Lie group level, if we write
$N=\exp\mathfrak{n}$ and $A=\exp\mathfrak{a},$ then we get the Iwasawa
decomposition of $G=KAN.$ This means every $g\in G$ can be uniquely written as
$g=k(g)\,\exp(H(g))\,n(g),$ where $k(g)\in K,H(g)\in\mathfrak{a}$ and $n(g)\in
N.$ The maps $(k,a,n)\mapsto kan$ is a global diffeomorphism of $K\times
A\times N$ onto $G.$

Let $\Delta$ be the Laplace-Beltrami operator on $X$ with respect to the
$G$-invariant Riemannian metric and $dx$ be the corresponding measure. It is
known that the $L^{2}$-spectrum of $\Delta$ is $(-\infty,-|\rho|^{2}).$ Let
$M$ be the centralizer of $A$ in $K$ and $M^{\prime}$ be the normalizer of $A$
in $K.$ Then $M$ is the normal subgroup of $M^{\prime}$ and normalize $N.$ The
factor group $W=M^{\prime}/M$ is a finite group of order $|W|$, called the
Weyl group of $X.$ The action of the Weyl group $W$ on $\mathfrak{a}$ is given
by an adjoint action. It acts as a group of orthogonal transformations
(preserving the Cartan-Killing form) on $\mathfrak{a}^{*}$ by
$(s\lambda)(H)=\lambda(s^{-1}\cdot H)$ for $H\in\mathfrak{a},$
$\lambda\in\mathfrak{a}^{*}$ and $s\in W,$ where $g.Y=\text{Ad}(g)(Y)$ for
$g\in G,\,Y\in\mathfrak{g}.$ We fix a normalized Haar measure on $dk$ on the
compact group $K$ and the Haar measure $dn$ on $N.$ We have the decompositions

$\displaystyle\begin{cases}\,G\,=\,N\,(\exp\mathfrak{a})\,K\quad&\textnormal{(Iwasawa)},\\\
\,G\,=\,K\,(\exp\overline{\mathfrak{a}^{+}})\,K\quad&\textnormal{(polar)}.\end{cases}$

The Haar measure on $G$ corresponding to the Iwasawa decomposition and the
polar decomposition can be described as, for any $f\in C_{c}(G),$

$\int_{G}f(g)\,dg=\int_{K}\int_{\mathfrak{a}}\int_{N}f(k(\exp
Y)n)\,e^{2\langle\rho,Y\rangle}dn\,dk\,dY,$

and

$\int_{G}f(g)\,dg=\int_{K}\int_{\overline{\mathfrak{a}^{+}}}\int_{K}f(k_{1}\exp{Y}k_{2})J(\exp
Y)dk_{1}\,dY\,dk_{2},$

respectively. Here the density $J(Y)$ for $Y\in\overline{\mathfrak{a}^{+}}$ is
given by

$\displaystyle\textstyle J(\exp
Y)\,=\,c\prod\limits_{\alpha\in\Sigma^{+}}\,(\sinh\langle{\alpha,H}\rangle)^{m_{\alpha}}\,\asymp\,\prod\limits_{\alpha\in\Sigma^{+}}\Big{\\{}\frac{\langle\alpha,H\rangle}{1+\langle\alpha,H\rangle}\Big{\\}}^{m_{\alpha}}\,e^{2\langle\rho,H\rangle},$

where $c$ is a normalizing constant. Using the polar (Cartan) decomposition we
can define another distance on $X$ called the polyhedral distance on $X$
defined as $d^{\prime}(xK,yK):=\langle\rho/|\rho|,(y^{-1}x)^{+}\rangle$ for
all $x,y\in G,$ where $(y^{-1}x)^{+}$ is the
$\overline{\mathfrak{a}^{+}}$-component of $y^{-1}x$ in the polar
decomposition. It was proved in [2] that the Riemannian distance $d$ and the
polyhedral distance $d^{\prime}$ are equivalent. Any function $f$ defined on
$X$ can be thought of as a function on $G$ which is right $G$-invariant under
the action of $K.$ Then it follows that we have a $G$-invariant measure $dx$
on $X$ such that

(2.1) $\int_{X}f(x)\,dx=\int_{K/M}\int_{\mathfrak{a}^{+}}f(k\exp Y)\,J(\exp
Y)\,dY\,dk_{M},$

where $dk_{M}$ is the $K$-invariant measure on $K/M.$

###  2.2. Helgason-Fourier transform on Riemannian symmetric spaces

Let $\mathfrak{a}_{\mathbb{C}}^{*}$ be the complexification of
$\mathfrak{a}^{*},$ that is, the set of the all complex-valued real linear
functionals on $\mathfrak{a}.$ The usual extension of the Killing form
$\mathfrak{B}$ on $\mathfrak{a}_{\mathbb{C}}^{*}$ by duality and conjugate
linearity is again denoted by $\mathfrak{B}.$ For a nice function $f$ the
Helgason Fourier transform of $\mathcal{H}f$ is a function on
$\mathfrak{a}_{\mathbb{C}}^{*}\times K/M$ defined by

(2.2) $(\mathcal{H}f)(\lambda,kM):=\int_{X}f(x)e^{\langle
i\lambda-\rho,H(g^{-1}k)\rangle}\,dx,\quad\lambda\in\mathfrak{a}_{\mathbb{C}}^{*},\,\,kM\in
K/M,$

whenever the integral exists. At times, we also denote $\mathcal{H}f$ by
$\widehat{f}.$

Let us introduce Harish-Chandra’s elementary spherical function in the
following form:

(2.3)
$\varphi_{\lambda}(g):=\int_{K}e^{-(i\lambda+\rho)}H(g^{-1}k)dk,\,\,\forall
k\in K,\,\,\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}.$

The elementary spherical function $\varphi_{0}$ satisfies the following global
estimate [4, Proposition 2.2.1]:

(2.4) $\varphi_{0}(\exp
H)\asymp\left(\prod_{\alpha\in\sum_{0}^{+}}(1+\langle\alpha,H\rangle)\right)e^{-\rho(H)},\,\,\,\forall
H\in\overline{\mathfrak{a}^{+}}.$

Moreover, we have

(2.5) $|\varphi_{\lambda}(g)|\leq\varphi_{0}(g)\leq
1,\,\,\,\,\forall\lambda\in\overline{\mathfrak{a}^{*}_{+}}.$

For $\xi\in[0,\infty),\sigma\in\mathbb{R}$, let $G_{\xi,\sigma}(x)$ be the
Schwartz kernel of the operator
$(-\Delta-|\rho|^{2}+\xi^{2})^{-\frac{\sigma}{2}}$ if it exists. We call
$G_{\xi,\sigma}(x)$ the Bessel-Green-Riesz kernel or simply the Riesz
potential. The Riesz potential satisfies the following estimate (see [Theorem
4.2.2, [4]]).

(2.6)
$\begin{split}G_{\xi,\sigma}(x)\asymp\begin{cases}\begin{rcases}|x|^{\sigma-n},\,\,\,&0<\sigma<n\\\
\log\left(\frac{1}{|x|}\right),\,\,\,&\sigma=n\\\
1,&\sigma>n\end{rcases},\,\,\,\,|x|<1,\\\
|x|^{\frac{\sigma-l-1}{2}-|\Sigma^{+}_{0}|}\varphi_{0}(x)e^{-\xi|x|},\,\,\xi>0,\,\,\sigma>0,\,\,|x|\geq
1.\,\,\,\end{cases}\end{split}$

Throughout this paper, the symbol $A\asymp B$ means that
$\exists\,C_{1},C_{2}>0$ such that $C_{1}A\leq B\leq C_{2}A$.

The Sobolev space on the symmetric space of noncompact type $X$ for
$0<\sigma\in\mathbb{R}$ and $1<p<\infty$ is defined as

(2.7) $H^{\sigma,p}(X):=\left\\{u:u\in
L^{p}(X),\,\,(-\Delta)^{\frac{\sigma}{2}}u\in L^{p}(X)\right\\},$

endowed with the norm

(2.8)
$\|u\|_{H^{\sigma,p}(X)}:=\|(-\Delta)^{\frac{\sigma}{2}}u\|_{L^{p}(X)}+\|u\|_{L^{p}(X)}.$

Then it follows from [64, Theorem 4.4] that

(2.9)
$\|u\|_{H^{\sigma,p}(X)}\approx\|(\xi^{2}-|\rho|^{2}-\Delta)^{\frac{\sigma}{2}}u\|_{L^{p}(X)},$

where $\xi$ is large enough.

Here after, whenever we deal with $G_{\xi,\sigma}$ we will always assume that
$\xi$ is large enough.

###  2.3. Integral Hardy inequalities on metric measure spaces

Let us consider metric measure spaces $\mathbb{X}$ with a Borel measure $dx$
allowing for the following polar decomposition at $a\in{\mathbb{Y}}$: we
assume that there is a locally integrable function $\lambda\in L^{1}_{loc}$
such that for all $f\in L^{1}(\mathbb{X})$ we have

(2.10)
$\int_{\mathbb{Y}}f(x)dx=\int_{0}^{\infty}\int_{\Sigma_{r}}f(r,\omega)\lambda(r,\omega)\,d\omega_{r}\,dr,$

for the set $\Sigma_{r}=\\{x\in\mathbb{Y}:d(x,a)=r\\}\subset\mathbb{Y}$ with a
measure on it denoted by $d\omega_{r}$, and $(r,\omega)\rightarrow a$ as
$r\rightarrow 0$. Examples of such measure measure spaces are Euclidean
spaces, homogeneous Lie groups and Riemannian symmetric spaces of noncompact
type. We can denote $|x|_{a}:=d(x,a).$

The class of such metric measure spaces was introduced in [55], where the
following integral Hardy inequality was obtained.

######  Theorem 2.1.

Let $1<p\leq q<\infty$ and let $s>0$. Let $\mathbb{Y}$ be a metric measure
space with a polar decomposition (2.10) at $a$. Let $u,v>0$ be measurable
functions positive a.e in $\mathbb{Y}$ such that $u\in
L^{1}(\mathbb{Y}\backslash\\{a\\})$ and $v^{1-p^{\prime}}\in
L^{1}_{loc}(\mathbb{Y})$. Denote

$\displaystyle
U(x):={\int_{\mathbb{Y}\backslash{B(a,|x|_{a})}}u(y)dy}\quad\text{and}\quad
V(x):=\int_{B(a,|x|_{a})}v^{1-p^{\prime}}(y)dy.$

Then the inequality

(2.11)
$\bigg{(}\int_{\mathbb{Y}}\bigg{(}\int_{B(a,|x|_{a})}|f(y)|dy\bigg{)}^{q}u(x)dx\bigg{)}^{\frac{1}{q}}\leq
C\bigg{\\{}\int_{\mathbb{Y}}{|f(x)|}^{p}v(x)dx\bigg{\\}}^{\frac{1}{p}}$

holds for all measurable functions $f:\mathbb{Y}\to{\mathbb{C}}$ if and only
if any of the following equivalent conditions holds:

  1. (1)

$\mathcal{D}_{1}:=\sup\limits_{x\not=a}\bigg{\\{}U^{\frac{1}{q}}(x)V^{\frac{1}{p^{\prime}}}(x)\bigg{\\}}<\infty.$

  1. (2)

$\mathcal{D}_{2}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{\mathbb{Y}\backslash{B(a,|x|_{a})}}u(y)V^{q(\frac{1}{p^{\prime}}-s)}(y)dy\bigg{\\}}^{\frac{1}{q}}V^{s}(x)<\infty.$

  2. (3)

$\mathcal{D}_{3}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{B(a,|x|_{a})}u(y)V^{q(\frac{1}{p^{\prime}}+s)}(y)dy\bigg{\\}}^{\frac{1}{q}}V^{-s}(x)<\infty$,
provided that $u,v^{1-p^{\prime}}\in L^{1}(\mathbb{Y})$.

  1. (4)

$\mathcal{D}_{4}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{B(a,|x|_{a})}v^{1-p^{\prime}}(y)U^{p^{\prime}(\frac{1}{q}-s)}(y)dy\bigg{\\}}^{\frac{1}{p^{\prime}}}U^{s}(x)<\infty.$

  2. (5)

$\mathcal{D}_{5}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{\mathbb{Y}\backslash{B(a,|x|_{a})}}v^{1-p^{\prime}}(y)U^{p^{\prime}(\frac{1}{q}+s)}(y)dy\bigg{\\}}^{\frac{1}{p^{\prime}}}U^{-s}(x)<\infty$,
provided that $u,v^{1-p^{\prime}}\in L^{1}(\mathbb{Y})$.

Moreover, the constant $C$ for which (2.11) holds and quantities
$\mathcal{D}_{1}-\mathcal{D}_{5}$ are related by

(2.12) $\mathcal{D}_{1}\leq
C\leq\mathcal{D}_{1}(p^{\prime})^{\frac{1}{p^{\prime}}}p^{\frac{1}{q}},$

and

$\mathcal{D}_{1}\leq\left(\max\left\\{1,{p^{\prime}}{s}\right\\}\right)^{\frac{1}{q}}\mathcal{D}_{2},\;\mathcal{D}_{2}\leq\left(\max\left\\{1,\frac{1}{p^{\prime}s}\right\\}\right)^{\frac{1}{q}}\mathcal{D}_{1},$
$\left(\frac{sp^{\prime}}{1+p^{\prime}s}\right)^{\frac{1}{q}}\mathcal{D}_{3}\leq\mathcal{D}_{1}\leq(1+sp^{\prime})^{\frac{1}{q}}\mathcal{D}_{3},$
$\mathcal{D}_{1}\leq(\max\left\\{1,qs\right\\})^{\frac{1}{p^{\prime}}}\mathcal{D}_{4},\;\mathcal{D}_{4}\leq\left(\max\left\\{1,\frac{1}{qs}\right\\}\right)^{\frac{1}{p^{\prime}}}\mathcal{D}_{1},$
$\left(\frac{sq}{1+qs}\right)^{\frac{1}{p^{\prime}}}\mathcal{D}_{5}\leq\mathcal{D}_{1}\leq(1+sq)^{\frac{1}{p^{\prime}}}\mathcal{D}_{5}.$

Similarly, in [55], the authors obtained the adjoint integral Hardy inequality
in the following form:

######  Theorem 2.2.

Let $1<p\leq q<\infty$ and let $s>0$. Let $\mathbb{Y}$ be a metric measure
space with a polar decomposition (2.10) at $a$. Let $u,v>0$ be measurable
functions positive a.e in $\mathbb{Y}$ such that $u\in
L^{1}(\mathbb{Y}\backslash\\{a\\})$ and $v^{1-p^{\prime}}\in
L^{1}_{loc}(\mathbb{X})$. Denote

$\displaystyle U(x):={\int_{{B(a,|x|_{a})}}u(y)dy}\quad\text{and}\quad
V(x):=\int_{\mathbb{Y}\backslash B(a,|x|_{a})}v^{1-p^{\prime}}(y)dy.$

Then the inequality

(2.13) $\bigg{(}\int_{\mathbb{Y}}\bigg{(}\int_{\mathbb{Y}\setminus
B(a,|x|_{a})}|f(y)|dy\bigg{)}^{q}u(x)dx\bigg{)}^{\frac{1}{q}}\leq
C\bigg{\\{}\int_{\mathbb{Y}}{|f(x)|}^{p}v(x)dx\bigg{\\}}^{\frac{1}{p}}$

holds for all measurable functions $f:\mathbb{Y}\to{\mathbb{C}}$ if and only
if any of the following equivalent conditions holds:

  1. (1)

$\mathcal{D}^{*}_{1}:=\sup\limits_{x\not=a}\bigg{\\{}U^{\frac{1}{q}}(x)V^{\frac{1}{p^{\prime}}}(x)\bigg{\\}}<\infty.$

  1. (2)

$\mathcal{D}^{*}_{2}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{\mathbb{Y}\backslash{B(a,|x|_{a})}}u(y)V^{q(\frac{1}{p^{\prime}}-s)}(y)dy\bigg{\\}}^{\frac{1}{q}}V^{s}(x)<\infty.$

  2. (3)

$\mathcal{D}^{*}_{3}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{\mathbb{Y}\setminus
B(a,|x|_{a})}u(y)V^{q(\frac{1}{p^{\prime}}+s)}(y)dy\bigg{\\}}^{\frac{1}{q}}V^{-s}(x)<\infty$,
provided that $u,v^{1-p^{\prime}}\in L^{1}(\mathbb{Y})$.

  1. (4)

$\mathcal{D}^{*}_{4}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{\mathbb{Y}\setminus
B(a,|x|_{a})}v^{1-p^{\prime}}(y)U^{p^{\prime}(\frac{1}{q}-s)}(y)dy\bigg{\\}}^{\frac{1}{p^{\prime}}}U^{s}(x)<\infty.$

  2. (5)

$\mathcal{D}^{*}_{5}:=\sup\limits_{x\not=a}\bigg{\\{}\int_{{B(a,|x|_{a})}}v^{1-p^{\prime}}(y)U^{p^{\prime}(\frac{1}{q}+s)}(y)dy\bigg{\\}}^{\frac{1}{p^{\prime}}}U^{-s}(x)<\infty$,
provided that $u,v^{1-p^{\prime}}\in L^{1}(\mathbb{Y})$.

######  Remark 2.3.

In this paper, for using integral Hardy inequalities, we use conditions
$\mathcal{D}_{1}$ and $\mathcal{D}^{*}_{1}$ in the previous theorems.

##  3\. Stein-Weiss and Hardy-Littlewood-Sobolev inequalities on symmetric
spaces of noncompact type

Let us show the Stein-Weiss inequality on symmetric space of noncompact type.
The proof is an adaption of the argument in [56] that was developed for the
graded Lie groups, however here, it depends on the induced geometry of the
space in a more substantial way.

######  Theorem 3.1.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
rank $l\geq 1$. Let $0<\sigma<n$, $1<p<\infty$, $\alpha<\frac{n}{p^{\prime}}$,
$\beta<\frac{n}{q},$ $\alpha+\beta\geq 0$ and
$\frac{\sigma-\alpha-\beta}{n}=\frac{1}{p}-\frac{1}{q}$. Then, for
sufficiently large $\xi>0$ and $1<p\leq q<\infty,$ we have

(3.1)
$\left(\int_{X}\left|\int_{X}G_{\xi,\sigma}(y^{-1}x)u(y)dy\right|^{q}\frac{dx}{|x|^{\beta
q}}\right)^{\frac{1}{q}}\leq C\||x|^{\alpha}u\|_{L^{p}(X)},$

where $C$ is a positive constant independent of $u$.

###### Proof.

Let us begin the proof with the quantity on the left hand side of (3.1) and
divide it into three part as follows:

(3.2)
$\begin{split}\int_{X}\left|\int_{X}G_{\xi,\sigma}(y^{-1}x)u(y)dy\right|^{q}|x|^{-\beta
q}dx\leq C\left(I^{q}_{1}+I^{q}_{2}+I^{q}_{3}\right),\end{split}$

where

$I^{q}_{1}=\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx,$ $I^{q}_{2}=\int_{X}\left(\int_{B\left(0,2|x|\right)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx,$ $I^{q}_{3}=\int_{X}\left(\int_{X\setminus
B\left(0,2|x|\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx.$

Now, we will estimate $I_{i}^{q},i=1,2,3.$ This will be completed in three
different steps.

Step 1. In this step we consider $I_{1}^{q}$. By using the triangle inequality
for the Riemannian distance with $2|y|\leq|x|$, we obtain

$|x|\leq|y^{-1}x|+|y|\leq|y^{-1}x|+\frac{|x|}{2},$

which implies that

$\frac{|x|}{2}\leq|y^{-1}x|.$

Let us now consider the two different cases.

Case (a). When $\frac{|x|}{2}\geq 1.$ Let us first find an estimate of
$G_{\xi,\sigma}(y^{-1}x)$ in terms of $|x|,$ using the fact
$\frac{|x|}{2}\leq|y^{-1}x|,$ in the range when $2|y|\leq|x|.$ Note that by
(2.6), we have

$G_{\xi,\sigma}(y^{-1}x)\asymp|y^{-1}x|^{\frac{\sigma-l-1}{2}-|\Sigma_{0}^{+}|}\varphi_{0}(y^{-1}x)e^{-\xi|y^{-1}x|}.$

Since the Riemannian distance and the polyhedral distance are equivalent and
$G_{\xi,\sigma}$ is $K$-biinvariant, we have

$G_{\xi,\sigma}(y^{-1}x):=G(\exp(y^{-1}x)^{+})\asymp|(y^{-1}x)^{+}|^{\frac{\sigma-l-1}{2}-|\Sigma_{0}^{+}|}\varphi_{0}(\exp(y^{-1}x)^{+})e^{-\xi|(y^{-1}x)^{+}|}.$

The estimate (2.4) of the ground spherical function
$\varphi_{0}(\exp(y^{-1}x)^{+})\lesssim|(y^{-1}x)^{+}|^{|\Sigma_{0}^{+}|}e^{-\langle\rho,(y^{-1}x)^{+}\rangle}$
yields,

$G_{\xi,\sigma}(y^{-1}x)\lesssim|(y^{-1}x)^{+}|^{\frac{\sigma-l-1}{2}-|\Sigma_{0}^{+}|}|(y^{-1}x)^{+}|^{|\Sigma_{0}^{+}|}e^{-\langle\rho,\,(y^{-1}x)^{+}\rangle}e^{-\xi|(y^{-1}x)^{+}|}.$

Now, by
$\langle\rho,x^{+}\rangle\leq\langle\rho,y^{+}\rangle+\langle\rho,(y^{-1}x)^{+}\rangle,$
$\frac{|x^{+}|}{2}\leq|(y^{-1}x)^{+}|\leq\frac{3|x^{+}|}{2}$ using
$|y^{+}|\leq\frac{|x^{+}|}{2},$ and Cauchy-Schwarz inequality, we get

$\displaystyle G_{\xi,\sigma}(y^{-1}x)\lesssim$
$\displaystyle|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{\langle\rho,y^{+}\rangle}e^{-\frac{\xi}{2}|x^{+}|}$
$\displaystyle\lesssim$
$\displaystyle|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{|\rho||y^{+}|}e^{-\frac{\xi}{2}|x^{+}|}\lesssim|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{|\rho|\frac{|x^{+}|}{2}}e^{-\frac{\xi}{2}|x^{+}|}.$

Therefore, we obtain

(3.3)
$\begin{split}I^{q}_{1}&=\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx\\\ &\leq
C\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left(|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{|\rho|\frac{|x^{+}|}{2}}e^{-\frac{\xi}{2}|x^{+}|}\right)^{q}|x|^{-\beta
q}dx.\end{split}$

Our aim is to show the following inequality,

(3.4)
$\displaystyle\left(\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left(|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{|\rho|\frac{|x^{+}|}{2}}e^{-\frac{\xi}{2}|x^{+}|}\right)^{q}|x|^{-\beta
q}dx\right)^{\frac{1}{q}}$ $\displaystyle\leq C\left(\int_{X}|y|^{\alpha
p}|u(y)|^{p}dy\right)^{\frac{1}{p}},$

and, for this purpose, we need to check the condition $\mathcal{D}_{1}$ in
Theorem 2.1 which turns out to be the following,

$\displaystyle\mathcal{D}_{1}$ $\displaystyle=\sup_{x\neq
0}\left(\int_{X\backslash
B\left(0,\frac{|x|}{2}\right)}|z^{+}|^{q\frac{\sigma-l-1}{2}}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dx\right)^{\frac{1}{q}}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}$ (3.5)
$\displaystyle=\sup_{R>0}\left(\int_{|z|\geq
R}|z^{+}|^{q\frac{\sigma-l-1}{2}}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dx\right)^{\frac{1}{q}}\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}},$

by setting $|x|/2=R\geq 1.$

By using the Cartan decomposition with $R\geq 1$ in this case we get

(3.6) $\begin{split}&\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\\\ &\leq C\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|\geq
R\\}}|H|^{q\frac{\sigma-l-1}{2}-\beta
q}\,e^{-q\langle\rho,\,H\rangle}e^{q|\rho|\frac{|H|}{2}}e^{-\frac{\xi}{2}|H|q}e^{2\langle\rho,H\rangle}dH\\\
&\leq Ce^{-\xi q\frac{R}{4}}\int_{\\{H\in\mathfrak{a}:|H|\geq
R\\}}|H|^{q\frac{\sigma-l-1}{2}-\beta
q}e^{-\left(\frac{\xi}{4}-\frac{|\rho|}{2}\right)|H|q}\,e^{-(q-2)\langle\rho,\,H\rangle}dH\\\
&\leq Ce^{-\xi q\frac{R}{4}},\end{split}$

when $q\geq 2$ and $\xi$ large enough (e.g. $\xi\geq 2|\rho|$). For $q<2,$
with same calculations as in (3.6) we get

(3.7) $\begin{split}&\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\\\ &\leq Ce^{-\xi q\frac{R}{4}}\int_{\\{H\in\mathfrak{a}:|H|\geq
R\\}}|H|^{q\frac{\sigma-l-1}{2}-\beta
q}e^{-\left(\frac{\xi}{4}-\frac{|\rho|}{2}\right)|H|q}\,e^{(2-q)\langle\rho,\,H\rangle}dH\\\
&\leq Ce^{-\xi q\frac{R}{4}}\int_{\\{H\in\mathfrak{a}:|H|\geq
R\\}}|H|^{q\frac{\sigma-l-1}{2}-\beta
q}e^{-\left(\frac{\xi}{4}-\frac{|\rho|}{2}\right)|H|q+(2-q)|\rho||H|}dH\\\
&\leq Ce^{-\xi q\frac{R}{4}}\int_{\\{H\in\mathfrak{a}:|H|\geq
R\\}}|H|^{q\frac{\sigma-l-1}{2}-\beta
q}e^{-\left(\frac{\xi}{4}+\frac{q}{2}|\rho|-2|\rho|\right)|H|}dH\leq Ce^{-\xi
q\frac{R}{4}},\end{split}$

whenever $\frac{\xi}{4}+\frac{q}{2}|\rho|-2|\rho|\geq 0,$ which holds as $\xi$
is large enough.

Let us consider the second integral,

(3.8) $\begin{split}\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dx&\stackrel{{\scriptstyle
p(1-p^{\prime})=-p^{\prime}}}{{=}}\int_{0<|z|<1}|z|^{-\alpha
p^{\prime}}dz+\int_{1\leq|z|\leq R}|z|^{-\alpha p^{\prime}}dz\\\ &\leq
C\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|<1\\}}|H|^{-\alpha
p^{\prime}}|H|^{n-l}dH+C\int_{\\{H\in\overline{\mathfrak{a}^{+}}:1\leq|H|\leq
R\\}}|H|^{-\alpha p^{\prime}}e^{2|\rho||H|}dH\\\ &\leq C\int_{0}^{1}r^{-\alpha
p^{\prime}}r^{n-l}r^{l-1}dr+Ce^{2|\rho|R}\int_{1}^{R}r^{-\alpha
p^{\prime}+l-1}dr\\\
&\stackrel{{\scriptstyle\alpha<\frac{n}{p^{\prime}}}}{{=}}C(1+e^{2|\rho|R}).\end{split}$

By combining the estimates (3.6), (3.7) and (3.8) with the fact that $\xi$ is
sufficiently large (e.g., $\xi\geq 8|\rho|$) and substituting back in (3), we
deduce that

$\displaystyle\mathcal{D}_{1}$ $\displaystyle=\sup_{R>0}\left(\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}$ $\displaystyle\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}$ $\displaystyle\leq
C\sup_{R>0}e^{-\xi\frac{R}{4}}(1+e^{2|\rho|R})^{\frac{1}{p^{\prime}}}\leq
C\sup_{R>0}e^{-\xi\frac{R}{4}}(1+e^{2|\rho|R})<+\infty,$

as $p^{\prime}>1$ and $\xi$ is sufficiently large.

Case (b). When $0<\frac{|x|}{2}<1.$ Again, we need to consider the integral

$I^{q}_{1}=\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx.$

Since $\frac{|x|}{2}\leq|y^{-1}x|,$ we can divide the study of this integral
into two parts: one when $|y^{-1}x|<1$ and the second when $|y^{-1}x|\geq 1.$
In fact, the case when $|y^{-1}x|\geq 1,$ is similar to the Case (a). So by
proceeding similar to case (a) with the obtained estimate of
$G_{\xi,\sigma}(y^{-1}x)$ when $|y^{-1}x|\geq 1,$ we have

(3.9)
$\begin{split}I^{q}_{1}&=\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx\\\ &\leq
C\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left(|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{|\rho|\frac{|x^{+}|}{2}}e^{-\frac{\xi}{2}|x^{+}|}\right)^{q}|x|^{-\beta
q}dx.\end{split}$

To establish the inequality

(3.10)
$\displaystyle\left(\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left(|x^{+}|^{\frac{\sigma-l-1}{2}}\,e^{-\langle\rho,\,x^{+}\rangle}e^{|\rho|\frac{|x^{+}|}{2}}e^{-\frac{\xi}{2}|x^{+}|}\right)^{q}|x|^{-\beta
q}dx\right)^{\frac{1}{q}}$ $\displaystyle\leq C\left(\int_{X}|y|^{\alpha
p}|u(y)|^{p}dy\right)^{\frac{1}{p}},$

we need to verify that the following holds,

(3.11) $\displaystyle\mathcal{D}_{1}=\sup_{R>0}\left(\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dx\right)^{\frac{1}{q}}\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}<\infty.$

Here $0<R=|x|/2<1.$

In order to check that $\mathcal{D}_{1}<\infty,$ using the Cartan
decomposition we get

(3.12) $\begin{split}&\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\\\
&=\int_{R\leq|z|<1}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\\\ &\quad\quad\quad\quad\quad+\int_{|z|\geq
1}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\\\ &\leq
C\Big{\\{}\int_{\\{H\in\overline{\mathfrak{a}^{+}}:R\leq|H|<1\\}}|H|^{\frac{\sigma-l-1}{2}q-\beta
q}\,e^{-q\langle\rho,\,H\rangle}e^{q|\rho|\frac{|H|}{2}}e^{-\frac{\xi}{2}|H|q}e^{2\langle\rho,H\rangle}dH\\\
&\quad\quad\quad+\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|\geq
1\\}}|H|^{\frac{\sigma-l-1}{2}q-\beta
q}\,e^{-q\langle\rho,\,H\rangle}e^{q|\rho|\frac{|H|}{2}}e^{-\frac{\xi}{2}|H|q}e^{2\langle\rho,H\rangle}dH\Big{\\}}=C\\{J_{1}+J_{2}\\}.\end{split}$

It is easy to see that $J_{1}<\infty$ is an integral of a continuous function
over a compact set. To see that $J_{2}<\infty$ one can argue verbatim as in
(3.6) and (3.7) by considering two case $q\geq 2$ and $q<2$ with $\xi$
sufficiently large (in this case, $\xi>2|\rho|$ will work). Therefore, we get

(3.13) $\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dz\leq C.$

Also, we compute the following integral for $0<R<1$ with
$\alpha<\frac{n}{p^{\prime}}$:

(3.14) $\begin{split}\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz&\stackrel{{\scriptstyle
p(1-p^{\prime})=-p^{\prime}}}{{\leq}}C\int_{0}^{R}r^{-\alpha
p^{\prime}+n-1}dr\\\ &=CR^{-\alpha p^{\prime}+n}\leq C<\infty.\end{split}$

Thus, from (3.13) and (3.14) we have

$\begin{split}\mathcal{D}_{1}=\sup_{R>0}\left(\int_{|z|\geq
R}|z^{+}|^{\frac{\sigma-l-1}{2}q}\,e^{-q\langle\rho,\,z^{+}\rangle}e^{q|\rho|\frac{|z^{+}|}{2}}e^{-\frac{\xi}{2}|z^{+}|q}|z|^{-\beta
q}dx\right)^{\frac{1}{q}}\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}\leq C<\infty.\end{split}$

Next, we consider the remaining case, when $|y^{-1}x|<1.$ In this case, using
(2.6) we have $G_{\xi,\sigma}(y^{-1}x)\asymp|y^{-1}x|^{\sigma-n}$ for
$0<\sigma<n.$ So, by using $\frac{|x|}{2}\leq|y^{-1}x|$ we get
$G_{\xi,\sigma}(y^{-1}x)\asymp|y^{-1}x|^{\sigma-n}\leq\frac{|x|^{\sigma-n}}{2}\asymp
G_{\xi,\sigma}(\frac{x}{2})$ and therefore,

(3.15)
$\begin{split}I^{q}_{1}=&\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx\\\ &\quad\quad\quad\quad\leq
C\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left(G_{\xi,\sigma}\left(\frac{x}{2}\right)\right)^{q}|x|^{-\beta
q}dx.\end{split}$

To show the inequality

$\displaystyle\left(\int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left(G_{\xi,\sigma}\left(\frac{x}{2}\right)\right)^{q}|x|^{-\beta
q}dx\right)^{\frac{1}{q}}\leq C\left(\int_{X}|y|^{\alpha
p}|u(y)|^{p}dy\right)^{\frac{1}{p}},$

it is enough to show that,

(3.16) $\displaystyle\mathcal{D}_{1}=\sup_{R>0}\left(\int_{|z|\geq
R}\left(G\left(\frac{z}{2}\right)\right)^{q}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}<\infty.$

Here $0<R=|x|/2<1.$ Then we have

(3.17) $\begin{split}\int_{|z|\geq
R}\left(G_{\xi,\sigma}\left(\frac{z}{2}\right)\right)^{q}|z|^{-\beta
q}dz&=\int_{R\leq|z|\leq
1}\left(G_{\xi,\sigma}\left(\frac{z}{2}\right)\right)^{q}|x|^{-\beta
q}dz+\int_{|z|\geq
1}\left(G_{\xi,\sigma}\left(\frac{z}{2}\right)\right)^{q}|z|^{-\beta q}dz\\\
&\asymp\int_{R\leq|z|\leq 1}|z|^{q(\sigma-n)}|z|^{-\beta q}dz+\int_{|z|\geq
1}\left(|z|^{\frac{\sigma-l-1}{2}-|\Sigma^{+}_{0}|}\varphi_{0}(z)e^{-\xi|z|}\right)^{q}|z|^{-\beta
q}dz\\\ &\leq C\int_{\\{H\in\overline{\mathfrak{a}^{+}}:R\leq|H|\leq
1\\}}|H|^{(\sigma-n)q-\beta q}|H|^{n-l}dH\\\
&+C\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|\geq
1\\}}|H|^{\frac{\sigma-l-1}{2}q-|\Sigma^{+}|q-\beta q}e^{-\xi
q\frac{|H|}{2}}e^{2|\rho||H|}dH\\\
&\stackrel{{\scriptstyle\eqref{estlar1}}}{{\leq}}C\int_{2R}^{2}r^{(\sigma-n)q-\beta
q+n-1}dr+Ce^{-\xi\frac{R}{2}}\\\ &\leq C(R^{(\sigma-n)q-\beta
q+n}+e^{-\xi\frac{R}{2}}).\end{split}$

Also, we compute the following integral for $0<R<1$ with
$\alpha<\frac{n}{p^{\prime}}$:

(3.18) $\begin{split}\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz&\stackrel{{\scriptstyle\eqref{estlar2}}}{{\leq}}C\int_{0}^{2R}r^{-\alpha
p^{\prime}+n-1}dr\\\ &=CR^{-\alpha p^{\prime}+n}.\end{split}$

Thus, from (3.17), (3.18) and
$\frac{\sigma-\alpha-\beta}{n}=\frac{1}{p}-\frac{1}{q}$, we have

$\begin{split}\left(\int_{|z|\geq
R}G_{\xi,\sigma}\left(\frac{z}{2}\right)^{q}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}&\leq
C\left(R^{\frac{(\sigma-n)q-\beta q+n}{q}+\frac{{-\alpha
p^{\prime}+n}}{p^{\prime}}}+R^{\frac{{-\alpha
p^{\prime}+n}}{p^{\prime}}}e^{-\xi\frac{R}{2}}\right)\\\
&=C\left(1+R^{\frac{{-\alpha
p^{\prime}+n}}{p^{\prime}}}e^{-\xi\frac{R}{2}}\right).\end{split}$

Therefore, in this case we get

$\begin{split}\mathcal{D}_{1}=\sup_{R>0}\left(\int_{|z|\geq
R}G_{\xi,\sigma}\left(\frac{z}{2}\right)^{q}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}\left(\int_{|z|\leq R}|z|^{\alpha
p(1-p^{\prime})}dz\right)^{\frac{1}{p^{\prime}}}&\leq
C\sup_{R>0}\left(1+R^{\frac{{-\alpha
p^{\prime}+n}}{p^{\prime}}}e^{-\xi\frac{R}{2}}\right)\\\ &<\infty.\end{split}$

The application of Theorem 2.1 completes the proof of case (b).

Step 2. In this step, we consider $I_{3}$. By using the triangle inequality
for the Riemannian distance with $|y|\geq 2|x|$, we get

(3.19) $|y|\leq|y^{-1}x|+|x|\leq|y^{-1}x|+\frac{|y|}{2},$

therefore, we have $|x|\leq\frac{|y|}{2}\leq|y^{-1}x|$. By arguing exactly in
the same way as in Case (a) of Step 1 we estimate that

$G_{\xi,\sigma}(y^{-1}x)\lesssim|y|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,y\rangle}e^{|\rho|\frac{|y|}{2}}e^{-\frac{\xi}{2}|y|}.$

Thus, we obtain

(3.20) $\begin{split}I_{3}^{q}&=\int_{X}\left(\int_{X\setminus
B\left(0,2|x|\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx\\\ &\leq\int_{X}\left(\int_{X\setminus
B\left(0,2|x|\right)}\left(|y|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,y\rangle}e^{|\rho|\frac{|y|}{2}}e^{-\frac{\xi}{2}|y|}\right)u(y)dy\right)^{q}|x|^{-\beta
q}dx.\end{split}$

If we show the following condition,

$\displaystyle\mathcal{D}^{*}_{1}$ $\displaystyle=\sup_{x\neq
0}\left(\int_{B(0,2|x|)}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}\left(\int_{X\backslash
B(0,2|x|)}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p^{\prime}}|z|^{-\alpha
p^{\prime}}dz\right)^{\frac{1}{p^{\prime}}}$ (3.21)
$\displaystyle=\sup_{R>0}\left(\int_{|z|\leq R}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}\left(\int_{|z|\geq
R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p^{\prime}}|z|^{-\alpha
p^{\prime}}dz\right)^{\frac{1}{p^{\prime}}},$

then by using the conjugate integral Hardy inequality (see Theorem 2.2), we
get

(3.22) $\begin{split}\int_{X}\left(\int_{X\setminus
B\left(0,2|x|\right)}\left(|y|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,y\rangle}e^{|\rho|\frac{|y|}{2}}e^{-\frac{\xi}{2}|y|}\right)u(y)dy\right)^{q}|x|^{-\beta
q}dx\leq C\left(\int_{X}|x|^{\alpha
p}|u(x)|^{p}dx\right)^{\frac{q}{p}}.\end{split}$

Let us check the condition (3). Similarly to the previous step, we consider
two cases $0<R<1$ and $R\geq 1$. Firstly, let us consider $R\geq 1$ and by
using the Cartan decomposition, we get, same as (3.6) and (3.7), that

(3.23) $\begin{split}\int_{|z|\geq
R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p^{\prime}}|z|^{-\alpha
p^{\prime}}dz\leq Ce^{-\xi p^{\prime}\frac{R}{2}},\end{split}$

for sufficiently large $\xi$. Then let us compute the first integral in (3),
we get

(3.24) $\begin{split}\int_{|z|\leq R}|z|^{-\beta q}dz&=\int_{|z|\leq
1}|z|^{-\beta q}dz+\int_{1<|z|\leq R}|z|^{-\beta q}dz\\\ &\leq
C\int_{0}^{1}r^{-\beta q+n-1}dr+C\max\\{1,R^{-\beta q}\\}\int_{1<|x|\leq
R}dx\\\ &\stackrel{{\scriptstyle\beta<\frac{n}{q}}}{{\leq}}C+CR^{-\beta
q+n}e^{2|\rho|R}\\\ &\leq C(1+R^{-\beta q+n}e^{2|\rho|R}).\end{split}$

By combining (3.24) and (3.23), we get

$\displaystyle\mathcal{D}_{1}^{*}=\sup_{R>0}\left(\int_{|z|\leq R}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}$ $\displaystyle\left(\int_{|z|\geq
R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p^{\prime}}|z|^{-\alpha
p^{\prime}}dz\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq\sup_{R>0}Ce^{-\xi\frac{R}{2}}(1+R^{\frac{-\beta
q+n}{q}}e^{\frac{2|\rho|R}{q}})<\infty,$

thanks to sufficiently large $\xi$.

Let us consider the case $0<R<1$. Similarly to (3.24), for $0<R<1$ we get

(3.25) $\begin{split}\int_{|z|\leq R}|z|^{-\beta q}dz&\leq
C\int_{\\{H\in\overline{\mathbb{a}^{+}}:|H|\leq R\\}}|H|^{-\beta
q}|H|^{n-l}dH\\\ &\leq C\int_{0}^{R}r^{-\beta q}r^{n-1}dr\\\
&\stackrel{{\scriptstyle\beta<\frac{n}{q}}}{{=}}CR^{-\beta q+n}\leq
C.\end{split}$

Next we will estimate the second integral of (3) for $0<R<1$. Thus we have,
same as (3.13), that

(3.26) $\begin{split}\int_{|z|\geq
R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p^{\prime}}|z|^{-\alpha
p^{\prime}}dz\leq C<\infty.\end{split}$

Finally, by combining (3.25) and (3.26) we get

$\mathcal{D}_{1}^{*}=\sup_{R>0}\left(\int_{|z|\leq R}|z|^{-\beta
q}dz\right)^{\frac{1}{q}}\left(\int_{|z|\geq
R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle\rho,z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p^{\prime}}|z|^{-\alpha
p^{\prime}}dz\right)^{\frac{1}{p^{\prime}}}\leq C<\infty.$

Therefore, (3.20) and (3.22) implies that

(3.27) $I^{q}_{3}\leq C\left(\int_{X}|x|^{\alpha
p}|u(x)|^{p}dx\right)^{\frac{q}{p}}.$

Step 3. Let us now focus on the remaining case of $I_{2}.$ We need to show
that

(3.28) $I_{2}=\left(\int_{X}\left(\int_{B\left(0,2|x|\right)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta
q}dx\right)^{\frac{1}{q}}\leq C\||x|^{\alpha}u\|_{L^{p}(X)}.$

We rewrite $I_{2}$ in the following form:

(3.29)
$\begin{split}I_{2}^{q}=\sum_{k=-\infty}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}}.\end{split}$

Since $|x|^{\beta q}$ is non-decreasing with respect to $|x|$ near the origin,
there exists $k_{0}\in\mathbb{Z}$ with $k_{0}\leq-3$ such that
$x\mapsto|x|^{\beta q}$ is non-decreasing for all $x$ satisfying
$0<|x|<2^{k_{0}+1}.$ Thus, it makes sense to decompose $I_{2}^{q}$ into two
parts as follows:

(3.30) $I_{2}^{q}=I^{q}_{2,1}+I^{q}_{2,2},$

where

(3.31) $I^{q}_{2,1}=\sum_{k=-\infty}^{k_{0}}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}},$

and

(3.32) $I^{q}_{2,2}=\sum_{k=k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}}.$

First, we show that $G_{\xi,\sigma}\in L^{r}(X)$ for $r\in[1,\infty]$ such
that $1-\frac{1}{r}=\frac{1}{p}-\frac{1}{q},$ which will play a significant
role in our proof. Indeed,

(3.33) $\begin{split}&\int_{X}|G_{\xi,\sigma}(x)|^{r}dx\\\ &=\int_{|x|\leq
1}|G_{\xi,\sigma}(x)|^{r}dx+\int_{|x|>1}|G_{\xi,\sigma}(x)|^{r}dx\\\
&=\int_{|x|\leq
1}|x|^{r(\sigma-n)}dx+\int_{|x|>1}|x|^{\frac{r(\sigma-l-1)}{2}-r|\Sigma^{+}_{0}|}\varphi_{0}(x)^{r}e^{-\xi|x|r}dx\\\
&=\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|\leq
1\\}}|H|^{r(\sigma-n)}H^{n-l}dH+\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|\geq
1\\}}|H|^{\frac{r(\sigma-l-1)}{2}}e^{-r\langle\rho,H\rangle}e^{-\xi|H|r}e^{\langle\rho,H\rangle}dH\\\
&\lesssim\int_{0}^{1}s^{r(\sigma-n)+(n-1)}ds+\int_{\\{H\in\overline{\mathfrak{a}^{+}}:|H|\geq
1\\}}|H|^{\frac{r(\sigma-l-1)}{2}}e^{-(r-1)\langle\rho,H\rangle}e^{-\xi|H|r}dH<\infty,\end{split}$

since $r(\sigma-n)+(n-1)>0$ by
$\frac{\sigma-n-\alpha-\beta}{n}-\frac{1}{p}+\frac{1}{q}+1=0$ with
$\alpha+\beta\geq 0$ and $r\in[1,\infty]$.

Let us first estimate $I_{2,2}^{q}$. By using estimate (3.33), Young’s
inequality for $1+\frac{1}{q}=\frac{1}{r}+\frac{1}{p}$ with $1\leq
r\leq\infty$ and by setting $\tilde{u}(y)=|y|^{\alpha}u(y)$, we establish

(3.34)
$\begin{split}I_{2,2}^{q}&=\sum_{k=-k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}}\\\ &=\sum_{k=-k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|x|^{-\alpha}|x|^{\alpha}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}}\\\ &\leq C\sum_{k=-k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|y|^{\alpha}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\alpha
q+\beta q}}\\\ &\stackrel{{\scriptstyle\alpha+\beta\geq
0}}{{\leq}}C\sum_{k=-k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)|y|^{\alpha}u(y)|dy\right)^{q}dx\\\
&=C\sum_{k=-k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)\tilde{u}(y)|dy\right)^{q}dx.\end{split}$

Note that from the conditions $2^{k}\leq|x|\leq 2^{k+1}$ and $|x|\leq 2|y|\leq
4|x|$ we deduce that $2^{k-1}\leq|y|\leq 2^{k+3}.$ Therefore, from (3.34) we
further have

(3.35) $\begin{split}I_{2,2}^{q}&\leq
C\sum_{k=-k_{0}+1}^{+\infty}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{2^{k-1}\leq|y|\leq
2^{k+3}}|G_{\xi,\sigma}(y^{-1}x)\tilde{u}(y)|dy\right)^{q}dx\\\ &\leq
C\sum_{k=-k_{0}+1}^{+\infty}\int_{X}\left(\int_{2^{k-1}\leq|y|\leq
2^{k+3}}|G_{\xi,\sigma}(y^{-1}x)|\,|\tilde{u}(y)|dy\right)^{q}dx\\\
&=C\sum_{k=-k_{0}+1}^{+\infty}\||G_{\xi,\sigma}|*(|\tilde{u}||\chi_{\\{2^{k-1}\leq|\cdot|\leq
2^{k+3}\\}})\|^{q}_{L^{q}(X)}\\\ &\leq
C\sum_{k=-k_{0}+1}^{+\infty}\|G_{\xi,\sigma}\|^{q}_{L^{r}(X)}\|\tilde{u}\chi_{\\{2^{k-1}\leq|\cdot|\leq
2^{k+3}\\}}\|^{q}_{L^{p}(X)}\\\ &\leq
C\|G_{\xi,\sigma}\|^{q}_{L^{r}(X)}\sum_{k\in\mathbb{Z}}\|\tilde{u}\chi_{\\{2^{k-1}\leq|\cdot|\leq
2^{k+3}\\}}\|^{q}_{L^{p}(X)}\\\
&\stackrel{{\scriptstyle\eqref{esti1}}}{{\leq}}C\||x|^{\alpha}u\|_{L^{p}(X)}^{q}.\end{split}$

Finally, we estimate $I_{2,1}^{q}.$ By combining the triangle inequality for
the Riemannian distance with $|x|\leq 2|y|\leq 4|x|,$ one can deduce that
$|y^{-1}x|\leq|y|+|x|\leq 3|x|.$ Since $|x|\leq 2|y|\leq 4|x|$ and
$2^{k}\leq|x|\leq 2^{k+1}$ with $k\leq k_{0}$, it yields that

(3.36) $|y^{-1}x|\leq 3|x|\leq 3\cdot 2^{k+1}\leq 3\cdot
2^{k_{0}+1}<1,\,\,\text{for}\,k_{0}\in[-\infty,-3].$

Thus, by using (3.36) with Young’s inequality for
$1+\frac{1}{q}=\frac{1}{r}+\frac{1}{p}$, we establish

(3.37)
$\begin{split}I^{q}_{2,1}&=\sum_{k=-\infty}^{k_{0}}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}}\\\ &=\sum_{k=-\infty}^{k_{0}}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|y|^{-\alpha}|y|^{\alpha}|G_{\xi,\sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta
q}}dx\\\ &\leq
C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{B(0,2|x|)\setminus
B\left(0,\frac{|x|}{2}\right)}|G_{\xi,\sigma}(y^{-1}x)\tilde{u}(y)|dy\right)^{q}dx\\\
&\leq C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\int_{2^{k}\leq|x|\leq
2^{k+1}}\left(\int_{2^{k-1}\leq 2|y|\leq
2^{k+2}}|G_{\xi,\sigma}(y^{-1}x)|\,|\tilde{u}(y)|dy\right)^{q}dx\\\
&\stackrel{{\scriptstyle\eqref{ocendisk3}}}{{\leq}}C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\int_{X}\left(\int_{X}[|G_{\xi,\sigma}(y^{-1}x)|\chi_{\\{0<|y^{-1}x|\leq
3\cdot 2^{k+1}\\}}][|\tilde{u}(y)|\chi_{\\{2^{k-1}\leq 2|y|\leq
2^{k+2}\\}}]dy\right)^{q}dx\\\
&=C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\int_{X}|(|G_{\xi,\sigma}|\chi_{\\{0<|\cdot|\leq
3\cdot 2^{k+1}\\}}*|\tilde{u}|\chi_{\\{2^{k-1}\leq 2|\cdot|\leq
2^{k+2}\\}})(x)|^{q}dx\\\
&=C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\|(|G_{\xi,\sigma}|\chi_{\\{0<|\cdot|\leq
3\cdot 2^{k+1}\\}}*|\tilde{u}|\chi_{\\{2^{k-1}\leq 2|\cdot|\leq
2^{k+2}\\}})\|^{q}_{L^{q}(X)}\\\ &\leq
C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\|G_{\xi,\sigma}\chi_{\\{0<|\cdot|\leq
3\cdot 2^{k+1}\\}}\|^{q}_{L^{r}(X)}\|\tilde{u}\chi_{\\{2^{k-1}\leq
2|\cdot|\leq 2^{k+2}\\}}\|^{q}_{L^{p}(X)}.\end{split}$

Computing $\|G_{\xi,\sigma}\chi_{\\{0<|\cdot|\leq 3\cdot
2^{k+1}\\}}\|_{L^{r}(X)}^{q}$ with (3.36), we obtain

(3.38) $\begin{split}\|G_{\xi,\sigma}\chi_{\\{0<|\cdot|\leq 3\cdot
2^{k+1}\\}}\|_{L^{r}(X)}^{q}&=\left(\int_{\\{x\in X:0<|x|\leq 3\cdot
2^{k+1}\\}}|G_{\xi,\sigma}(x)|^{r}dx\right)^{\frac{q}{r}}\\\
&\stackrel{{\scriptstyle
3.2^{k+1}<1}}{{=}}\left(\int_{\\{H\in\overline{\mathfrak{a}^{+}}:0<|H|\leq
3\cdot 2^{k+1}\\}}|H|^{r(\sigma-n)}H^{n-l}dH\right)^{\frac{q}{r}}\\\ &\leq
C\left(\int_{0}^{3\cdot 2^{k+1}}s^{r(\sigma-n)+n-1}ds\right)^{\frac{q}{r}}\\\
&\stackrel{{\scriptstyle
r(\sigma-n)+n>0}}{{\leq}}C2^{\frac{(r(\sigma-n)+n)kq}{r}}.\end{split}$

From assumptions $\frac{\sigma-n-\alpha-\beta}{n}-\frac{1}{p}+\frac{1}{q}+1=0$
and $1-\frac{1}{r}=\frac{1}{p}-\frac{1}{q}$, we have

(3.39) $(-\alpha-\beta)+(\sigma-n)+\frac{n}{r}=0.$

Putting (3.38) and (3.39) in (3.37) and recalling that $k_{0}\leq-3$ we
establish

(3.40) $\begin{split}I^{q}_{2,1}&\leq
C\sum_{k=-\infty}^{k_{0}}2^{(-\alpha-\beta)qk}\|G_{\xi,\sigma}\chi_{\\{0<|\cdot|\leq
3\cdot 2^{k+1}\\}}\|^{q}_{L^{r}(X)}\|\tilde{u}\chi_{\\{2^{k-1}\leq
2|\cdot|\leq 2^{k+2}\\}}\|^{q}_{L^{p}(X)}\\\
&\stackrel{{\scriptstyle\eqref{ocenGI2}}}{{\leq}}C\sum_{k=-\infty}^{k_{0}}2^{\left((-\alpha-\beta)+(\sigma-n)+\frac{n}{r}\right)qk}\|\tilde{u}\chi_{\\{2^{k-1}\leq
2|\cdot|\leq 2^{k+2}\\}}\|^{q}_{L^{p}(X)}\\\
&\stackrel{{\scriptstyle\eqref{sigmaalphabeta}}}{{\leq}}C\sum_{k=-\infty}^{k_{0}}\|\tilde{u}\chi_{\\{2^{k-1}\leq
2|\cdot|\leq 2^{k+2}\\}}\|^{q}_{L^{p}(X)}\\\ &\leq
C\sum_{k=-\infty}^{+\infty}\|\tilde{u}\chi_{\\{2^{k-1}\leq 2|\cdot|\leq
2^{k+2}\\}}\|^{q}_{L^{p}(X)}\\\ &=C\|\tilde{u}\|^{q}_{L^{p}(X)}.\end{split}$

Then by putting together (3.35) and (3.40), we have

(3.41) $I_{2}^{q}\leq I^{q}_{2,1}+I^{q}_{2,2}\leq
C\|\tilde{u}\|^{q}_{L^{p}(X)}=C\||x|^{\alpha}u\|_{L^{p}(X)},$

completing the proof of (3.28). ∎

We get the following Hardy-Littlewood-Sobolev inequality on symmetric space of
noncompact type as a consequence of Theorem 3.1 by taking $\alpha=0$ and
$\beta=0.$

######  Theorem 3.2.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
rank $l\geq 1$. Suppose that $0<\sigma<n$, $1<p<\infty$ and
$\frac{\sigma}{n}=\frac{1}{p}-\frac{1}{q}$. Then, for sufficiently large
$\xi>0$ and $1<p\leq q<\infty,$ we have

(3.42)
$\left(\int_{X}\left|\int_{X}G_{\xi,\sigma}(y^{-1}x)u(y)dy\right|^{q}dx\right)^{\frac{1}{q}}\leq
C\left(\int_{X}|u(x)|^{p}dx\right)^{\frac{1}{p}},$

where $C$ is a positive constant independent of $u$.

##  4\. Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg and Caffarelli-
Kohn-Nirenberg inequalities on symmetric spaces

In this section, we show the Hardy, Sobolev, Hardy-Sobolev, Gagliardo-
Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on symmetric spaces.
First, we show the Hardy-Sobolev inequality on symmetric spaces.

######  Theorem 4.1 (Hardy-Sobolev inequality).

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
rank $l\geq 1.$ Suppose that $0<\sigma<n$, $1<p\leq q<\infty$ and
$0\leq\beta<\frac{n}{q}$ are such that
$\frac{\sigma-\beta}{n}=\frac{1}{p}-\frac{1}{q}$. Then for all $u\in
H^{\sigma,p}(X)$, we have

(4.1) $\left\|\frac{u}{|x|^{\beta}}\right\|_{L^{q}(X)}\leq
C\|u\|_{H^{\sigma,p}(X)},$

where $C$ is a positive constant independent of $u$.

######  Corollary 4.2 (Hardy inequality).

For $q=p$ and $0<\sigma<\frac{n}{p}$ in Theorem 4.1, the inequality (4.1)
gives the Hardy inequality on symmetric space, that is,

(4.2) $\left\|\frac{u}{|x|^{\sigma}}\right\|_{L^{p}(X)}\leq
C\|u\|_{H^{\sigma,p}(X)}.$

######  Corollary 4.3 (Uncertainty principle).

By Theorem 4.2, we have the uncertainly principle on symmetric spaces of
noncompact type, that is,

(4.3) $\left\|u\right\|_{L^{2}(X)}\leq
C\|u\|_{H^{\sigma,p}(X)}\||x|^{\sigma}u\|_{L^{\frac{p}{p-1}}(X)}.$

######  Corollary 4.4 (Sobolev inequality).

By taking $\beta=0$ in Theorem 4.1, we obtain the Sobolev inequality on
symmetric space of noncompact type, that is, for $0<\sigma<n$ and $1<p\leq
q<\infty$ such that $\frac{\sigma}{n}=\frac{1}{p}-\frac{1}{q}$ we obtain

(4.4) $\left\|u\right\|_{L^{q}(X)}\leq C\|u\|_{H^{\sigma,p}(X)}.$

###### Proof of Theorem 4.1.

By (2.6) and (2.9), we have that, for $u\in H^{\sigma,p}(X)$ there is $f\in
L^{p}(X)$ such that

(4.5)
$\begin{split}u=(-\Delta-|\rho|^{2}+\xi^{2})^{-\frac{\sigma}{2}}f=G_{\xi,\sigma}*f.\end{split}$

Then by using this with Theorem 3.1, we obtain

(4.6)
$\begin{split}\left\|\frac{u}{|x|^{\beta}}\right\|^{q}_{L^{q}(X)}&=\int_{X}\left|(-\Delta-|\rho|^{2}+\xi^{2})^{-\frac{\sigma}{2}}f(x)\right|^{q}\frac{dx}{|x|^{\beta
q}}\\\ &=\int_{X}|(G_{\xi,\sigma}*f)(x)|^{q}\frac{dx}{|x|^{\beta q}}\\\ &\leq
C\|f\|^{q}_{L^{p}(X)}=C\|(-\Delta-|\rho|^{2}+\xi^{2})^{\frac{\sigma}{2}}u\|^{q}_{L^{p}(X)}\stackrel{{\scriptstyle\eqref{equiso}}}{{\leq}}C\|u\|^{q}_{H^{\sigma,p}(X)},\end{split}$

completing the proof. ∎

Let us now present the Gagliardo-Nirenberg inequality on symmetric space of
noncompact type.

######  Theorem 4.5 (Gagliardo-Nirenberg inequality).

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
rank $l\geq 1$. Suppose $0<\sigma<n$, $\tau>0$, $p>1$, $\sigma p<n$, $\nu\geq
1$, $a\in(0,1]$ and

(4.7)
$\frac{1}{\tau}=a\left(\frac{1}{p}-\frac{\sigma}{n}\right)+\frac{(1-a)}{\nu}.$

Then there exists positive constant $C$ such that we have

(4.8) $\|u\|_{L^{\tau}(X)}\leq
C\|u\|^{a}_{H^{\sigma,p}}\|u\|^{1-a}_{L^{\nu}(X)},$

where $C$ is independent of $n.$

###### Proof.

By using Hölder’s inequality with $1=\frac{a\tau}{q}+\frac{(1-a)\tau}{\nu}$
where $\frac{1}{q}=\frac{1}{p}-\frac{\sigma}{n}$ and Sobolev inequality, we
have

(4.9)
$\begin{split}\|u\|^{\tau}_{L^{\tau}(X)}&=C\int_{X}|u(x)|^{a\tau}|u(x)|^{(1-a)\tau}dx\\\
&\leq\left(\int_{X}|u(x)|^{q}dx\right)^{\frac{a\tau}{q}}\left(\int_{X}|u(x)|^{\nu}dx\right)^{\frac{(1-a)\tau}{\nu}}\\\
&\stackrel{{\scriptstyle\eqref{Sobolevin}}}{{\leq}}C\|u\|^{a\tau}_{H^{\sigma,p}(X)}\|u\|^{(1-a)\tau}_{L^{\nu}(X)},\end{split}$

completing the proof. ∎

In the next, we give particular case of the Gagliardo-Nirenberg inequality,
for that we need to the case $\nu=p=2$ and $\sigma=1$, which serves as a
useful tool in our proof of the global existence of the wave equation in the
next section.

######  Corollary 4.6.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
rank $l\geq 1$. Let $\tau\in\left[2,\frac{2n}{n-2}\right]$ and
$a=\frac{n(\tau-2)}{\tau},$ then we have

(4.10) $\|u\|_{L^{\tau}(X)}\leq
C\|u\|^{a}_{H^{1,2}(X)}\|u\|^{1-a}_{L^{\tau}(X)}.$

###### Proof.

By taking $\sigma=1$, $p=2$ and $\nu=2$ in Theorem 4.5, we get (4.10). ∎

######  Remark 4.7.

In Theorem 4.5, by taking $a=1$, we get Sobolev inequality (4.4).

Then let us show the Caffarelli-Kohn-Nirenberg inequality in the following
theorem:

######  Theorem 4.8.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$ and
rank $l\geq 1$ such that $0<\sigma<n$. Suppose $p>1$, $0<q<\tau<\infty$ such
that $a\in\left(\frac{\tau-q}{\tau},1\right]$ and $p\leq\frac{a\tau
q}{q-(1-a)\tau}$. Let $b,c$ be real numbers such that
$0\leq(c(1-a)-b)\leq\frac{n(q-(1-a)\tau)}{q\tau}$ and
$\frac{\sigma-n}{n}-\frac{(c(1-a)-b)q}{an}+\frac{q-(1-a)\tau}{a\tau
q}-\frac{1}{p}+1=0$. Then there exists a positive constant independent of $u$
such that

(4.11) $\||x|^{b}u\|_{L^{\tau}(X)}\leq
C\|u\|^{a}_{H^{\sigma,p}(X)}\||x|^{c}u\|^{1-a}_{L^{q}(X)}.$

###### Proof.

By Hölder’s inequality with $\frac{q-(1-a)\tau}{q}+\frac{(1-a)\tau}{q}=1$ and
Hardy-Sobolev inequality (4.1) with assumptions of this theorem, we obtain

(4.12)
$\begin{split}\||x|^{b}u\|^{\tau}_{L^{\tau}(X)}&=\int_{X}|x|^{b\tau}|u(x)|^{\tau}dx\\\
&=\int_{X}\frac{|u(x)|^{a\tau}}{|x|^{(c(1-a)-b)\tau}}\frac{|u(x)|^{(1-a)\tau}}{|x|^{-c(1-a)\tau}}dx\\\
&\leq\left(\int_{X}\frac{|u(x)|^{a\tau\frac{q}{q-(1-a)\tau}}}{|x|^{(c(1-a)-b)\tau\frac{q}{q-(1-a)\tau}}}dx\right)^{\frac{q-(1-a)\tau}{q}}\left(\int_{X}\frac{|u(x)|^{q}}{|x|^{-cq}}dx\right)^{\frac{(1-a)\tau}{q}}\\\
&=\left(\int_{X}\frac{|u(x)|^{\frac{a\tau
q}{q-(1-a)\tau}}}{|x|^{\frac{c(1-a)-b}{a}\frac{a\tau
q}{q-(1-a)\tau}}}dx\right)^{a\tau\frac{q-(1-a)\tau}{a\tau
q}}\left(\int_{X}\frac{|u(x)|^{q}}{|x|^{-cq}}dx\right)^{\frac{(1-a)\tau}{q}}\\\
&\stackrel{{\scriptstyle\eqref{HardySobolev}}}{{\leq}}C\|u\|_{H^{\sigma,p}(X)}^{a\tau}\||x|^{c}u\|^{(1-a)\tau},\end{split}$

completing the proof. ∎

##  5\. Nonlinear wave equation with a damped term associated with the
Laplace-Beltrami operator on symmetric spaces

In this section, we show small data global existence for the semilinear wave
equation with damped term involving the Laplace-Beltrami operator on symmetric
spaces of noncompact type. In [54] and [68], similar questions have been
treated for Rockland operators on graded Lie groups and for Dunkl Laplacian on
Euclidean spaces. The strategy here follows that of [54]. The main aim of this
section is to obtain the global existence result for the Cauchy problem
involving the shifted Laplace-Beltrami operator
$\Delta_{x}=\Delta+|\rho|^{2}.$ We consider the shifted Laplace-Beltrami
operator instead of usual Laplace-Beltrami operator just to make computations
clear and simple, otherwise the result still holds true for the usual Laplace-
Beltrami operator.

(5.1)
$\begin{split}\begin{cases}u_{tt}(x,t)-\Delta_{x}u(x,t)+bu_{t}(x,t)+mu(x,t)=f(u)(x,t),\\\
u(x,0)=u_{0}(x),\,\,\,\,x\in X,\\\ u_{t}(x,0)=u_{1}(x),\,\,\,\,x\in
X,\end{cases}\end{split}$

where $b,m>0$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following
conditions:

(5.2) $f(0)=0,$

and

(5.3) $|f(u)-f(v)|\leq C(|u|^{p-1}+|v|^{p-1})|u-v|.$

To show the global existence of a solution of Cauchy problem (5.1) we need the
following lemma.

######  Lemma 5.1.

Assume that $s\in\mathbb{R}$. Let $u$ be a solution of (5.1) with $f=0$,
$u_{0}\in H^{s,2}$ and $u_{1}\in H^{s-1,2}$. Then there exists $\delta>0$ such
that

(5.4) $\|u\|^{2}_{H^{s,2}}\leq Ce^{-\delta
t}\left(\|u_{0}\|^{2}_{H^{s,2}}+\|u_{1}\|^{2}_{H^{s-1,2}}\right),$

and

(5.5) $\|u_{t}\|^{2}_{H^{s,2}}\leq Ce^{-\delta
t}\left(\|u_{0}\|^{2}_{H^{s+1,2}}+\|u_{1}\|^{2}_{H^{s,2}}\right).$

###### Proof.

By applying Helgason’s Fourier transform in (5.1) with respect to the variable
$x$, we obtain

(5.6)
$\begin{split}\begin{cases}\hat{u}_{tt}+|\lambda|^{2}\hat{u}+b\hat{u}_{t}+m\hat{u}=0,\\\
\hat{u}(0)=\hat{u}_{0},\\\ \hat{u}_{t}(0)=\hat{u}_{1}.\end{cases}\end{split}$

Therefore, the solution of (5.6) is given by

(5.7)
$\begin{split}\hat{u}=\begin{cases}e^{-\frac{b}{2}t}\left[\hat{u}_{0}\left(\cosh\frac{\sqrt{b^{2}-4\gamma}}{2}t+\frac{b}{\sqrt{b^{2}-4\gamma}}\sinh\frac{\sqrt{b^{2}-4\gamma}}{2}t\right)+\frac{2}{\sqrt{b^{2}-4\gamma}}\hat{u}_{1}\sinh\frac{\sqrt{b^{2}-4\gamma}}{2}t\right],\,\,b>2\gamma,\\\
\
e^{-\frac{b}{2}t}\left[\hat{u}_{0}\left(1+\frac{b}{2}t\right)+t\hat{u}_{1}\right],\,\,b=2\gamma,\\\
e^{-\frac{b}{2}t}\left[\hat{u}_{0}\left(\cos\frac{\sqrt{4\gamma-b^{2}}}{2}t+\frac{b}{\sqrt{4\gamma-b^{2}}}\sin\frac{\sqrt{4\gamma-b^{2}}}{2}t\right)+\frac{2}{\sqrt{4\gamma-b^{2}}}\hat{u}_{1}\sin\frac{\sqrt{4\gamma-b^{2}}}{2}t\right],\,\,b<2\gamma,\end{cases}\end{split}$

where $\gamma=\sqrt{m+|\lambda|^{2}}$.

First, let us consider the case when $b>2\gamma$. We know that

(5.8) $\frac{1-e^{-t}}{t}\leq 1,\,\,\,\,t\geq 0,$

and for any $c>0$, we have

(5.9) $bct\leq e^{bct},\,\,\,\,t\geq 0,$

so that

(5.10) $(bct)e^{-\frac{bt}{2}+\frac{\sqrt{b^{2}-4\gamma}}{2}}\leq
e^{-\frac{b(1-c)t}{2}+\frac{\sqrt{b^{2}-4\gamma}}{2}t}\leq
e^{-\frac{b(1-c)t}{2}+\frac{\sqrt{b^{2}-4m}}{2}t},$

thanks to $b>2\gamma=2\sqrt{m+|\lambda|^{2}}>2\sqrt{m}.$ Thus, if we choose
$0<c<1-\frac{\sqrt{b^{2}-4m}}{b}$, we get

(5.11) $(bct)e^{-\frac{bt}{2}+\frac{\sqrt{b^{2}-4\gamma}}{2}}\leq
e^{-\frac{b(1-c)t}{2}+\frac{\sqrt{b^{2}-4m}}{2}t}\leq e^{-\delta t}.$

Therefore, by using these facts we establish

(5.12)
$\begin{split}|\hat{u}|&=\big{|}e^{-\frac{b}{2}t}\big{[}\hat{u}_{0}\left(\cosh\frac{\sqrt{b^{2}-4\gamma}}{2}t+\frac{b}{\sqrt{b^{2}-4\gamma}}\sinh\frac{\sqrt{b^{2}-4\gamma}}{2}t\right)\\\
&+\frac{2}{\sqrt{b^{2}-4\gamma}}\hat{u}_{1}\sinh\frac{\sqrt{b^{2}-4\gamma}}{2}t\big{]}\big{|}\\\
&\leq
Ce^{-\frac{b}{2}t-\frac{\sqrt{b^{2}-4\gamma}}{2}t}\left(|\hat{u}_{0}|\left(1+\frac{bt}{2}\right)+\frac{1}{\sqrt{b^{2}-4\gamma}}|\hat{u}_{1}|\right)\\\
&\leq Ce^{-\delta
t}\left(|\hat{u}_{0}|+\frac{1}{\sqrt{b^{2}-4\gamma}}|\hat{u}_{1}|\right).\end{split}$

The function $\frac{\xi^{2}+|\lambda|^{2}}{b^{2}-4(m+|\lambda|^{2})}$ is
bounded for all $\lambda\in\mathfrak{a}^{*}$, using the Plancherel formula for
the Helgason-Fourier transform, we have

(5.13)
$\begin{split}\|u\|^{2}_{H^{s,2}}&=\frac{1}{|W|}\int_{\mathbb{a}^{*}\times
K}(\xi^{2}+|\lambda|^{2})^{s}|\hat{u}|^{2}\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &\leq Ce^{-2\delta t}\int_{\mathbb{a}^{*}\times
K}(\xi^{2}+|\lambda|^{2})^{s}\left(|\hat{u}_{0}|^{2}+\frac{1}{b^{2}-4\gamma}|\hat{u}_{1}|^{2}\right)\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &=Ce^{-2\delta t}\int_{\mathbb{a}^{*}\times
K}\left((\xi^{2}+|\lambda|^{2})^{s}|\hat{u}_{0}|^{2}+\frac{(\xi^{2}+|\lambda|^{2})^{s}}{b^{2}-4(m+|\lambda|^{2})}|\hat{u}_{1}|^{2}\right)\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &\leq Ce^{-2\delta t}\int_{\mathbb{a}^{*}\times
K}\left((\xi^{2}+|\lambda|^{2})^{s}|\hat{u}_{0}|^{2}+(\xi^{2}+|\lambda|^{2})^{s-1}|\hat{u}_{1}|^{2}\right)\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &=Ce^{-2\delta t}\|u_{0}\|^{2}_{H^{s,2}}+Ce^{-2\delta
t}\|u_{1}\|^{2}_{H^{s-1,2}}.\end{split}$

Let us consider the case $b=2\gamma.$ By using previous computation in (5.11),
we get

(5.14)
$\begin{split}|\hat{u}|&=\left|e^{-\frac{b}{2}t}\left[\hat{u}_{0}\left(1+\frac{b}{2}t\right)+t\hat{u}_{1}\right]\right|\leq
Ce^{-\delta t}\left(|\hat{u}_{0}|+|\hat{u}_{1}|\right).\end{split}$

From $b=2\gamma$, we have $|\lambda|^{2}=\frac{b^{2}}{4}-m>0,$ and by using
the last calculation, we have

(5.15)
$\begin{split}\|u\|^{2}_{H^{s,2}}&=\frac{1}{|W|}\int_{\mathbb{a}^{*}\times
K}(\xi^{2}+|\lambda|^{2})^{s}|\hat{u}|^{2}\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &\leq Ce^{-2\delta t}\int_{\mathbb{a}^{*}\times
K}(\xi^{2}+|\lambda|^{2})^{s}\left(|\hat{u}_{0}|+|\hat{u}_{1}|\right)\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &=Ce^{-2\delta t}\int_{\mathbb{a}^{*}\times
K}\big{(}(\xi^{2}+|\lambda|^{2})^{s}|\hat{u}_{0}|\\\
&+(\xi^{2}+|\lambda|^{2})^{s-1}(\xi+|\lambda|^{2})|\hat{u}_{1}|\big{)}\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &=Ce^{-2\delta t}\int_{\mathbb{a}^{*}\times
K}\big{(}(\xi^{2}+|\lambda|^{2})^{s}|\hat{u}_{0}|\\\
&+(\xi+|\lambda|^{2})^{s-1}\left(\xi^{2}+\frac{b^{2}}{4}-m\right)|\hat{u}_{1}|\big{)}\frac{d\lambda
dk}{|c(\lambda)|^{2}}\\\ &\leq Ce^{-2\delta
t}\|u_{0}\|^{2}_{H^{s,2}}+Ce^{-2\delta
t}\|u_{1}\|^{2}_{H^{s-1,2}}.\end{split}$

Finally, we consider the case when $b<2\gamma.$ By using a similar computation
as in the case $b>2\gamma$, we get

(5.16)
$\begin{split}|\hat{u}|&=\big{|}e^{-\frac{b}{2}t}\big{[}\hat{u}_{0}\left(\cos\frac{\sqrt{4\gamma-b^{2}}}{2}t+\frac{b}{\sqrt{4\gamma-b^{2}}}\sin\frac{\sqrt{4\gamma-b^{2}}}{2}t\right)\\\
&+\frac{2}{\sqrt{4\gamma-b^{2}}}\hat{u}_{1}\sin\frac{\sqrt{4\gamma-b^{2}}}{2}t\big{]}\big{|}\\\
&\leq Ce^{-\delta
t}\left(|\hat{u}_{0}|+\frac{1}{\sqrt{4\gamma-b^{2}}}|\hat{u}_{1}|\right).\end{split}$

Similarly to the case $b>2\gamma$, we get

(5.17) $\|u\|^{2}_{H^{s,2}}\leq Ce^{-2\delta
t}\|u_{0}\|^{2}_{H^{s,2}}+Ce^{-2\delta t}\|u_{1}\|^{2}_{H^{s-1,2}},$

completing the proof of (5.4). To obtain (5.5), we need take the derivative
with respect to the variable $t$ of (5.7) and repeat similar computations and
estimates as above. Hence, the proof is completed. ∎

Here we show the global existence and uniqueness result for the solution of
problem (5.1).

######  Theorem 5.2.

Let $X$ be a symmetric space of noncompact type of dimension $n\geq 3$. Let
$1\leq p\leq\frac{n}{n-2}$. Suppose that $f$ satisfies the conditions
(5.2)-(5.3). Assume that $u_{0}\in H^{1,2}(X)$ and $u_{1}\in L^{2}(X)$ are
such that $\|u_{0}\|_{H^{1,2}(X)}+\|u_{1}\|_{L^{2}(X)}<\varepsilon$. Then
there exists $\varepsilon_{0}>0$ such that $0<\varepsilon\leq\varepsilon_{0}$
the Cauchy problem (5.1) has a unique global solution $u\in
C(\mathbb{R}_{+},H^{1,2}(X))\cap C^{1}(\mathbb{R}_{+},L^{2}(X))$.

###### Proof.

The proof of this theorem is based Banach fixed point theorem. First, let us
consider the following Banach space

(5.18) $Z:=\\{u:u\in C(\mathbb{R}_{+},H^{1,2}(X))\cap
C^{1}(\mathbb{R}_{+},L^{2}(X)),\|u\|_{Z}\leq M\\},$

where $M$ will be defined later and the norm is defined by

(5.19) $\|u\|_{Z}:=\sup_{t>0}(1+t)^{-\frac{1}{2}}e^{\delta
t}\left(\|\Delta^{\frac{1}{2}}u(\cdot,t)\|_{L^{2}(X)}+\|u(\cdot,t)\|_{L^{2}(X)}+\|u_{t}(\cdot,t)\|_{L^{2}(X)}\right),$

where $\delta>0$ as in Lemma 5.1. Let us define the mapping $L:Z\rightarrow
Z$,

(5.20) $(Lu)(x,t)=\psi(x,t)+\int_{0}^{t}[K(f(u))](x,t-s)ds,$

where $\psi$ is the solution of the linear problem $\eqref{problem1}$ and $Kv$
is the solution of the following Cauchy problem:

(5.21)
$\begin{split}\begin{cases}u_{tt}(x,t)-\Delta_{x}u(x,t)+bu_{t}(x,t)+mu(x,t)=0,\\\
u(x,0)=0,\,\,\,\,x\in X,\\\ u_{t}(x,0)=v(x),\,\,\,\,x\in
X.\end{cases}\end{split}$

For simplicity, let us denote $J(f(u))(x,t):=\int_{0}^{t}[Kf(u)](x,t-s)ds.$
Firstly, let us show that the

(5.22) $\|Lu-Lv\|_{Z}\leq c\|u-v\|_{Z},\,\,\,\text{for\,\,\,all}\,\,\,u,v\in
Z,$

where $c\in(0,1)$.

For all $t>0$, $u,v\in Z$ and by using (5.2), Hölder’s inequality with
$\frac{2(p-1)}{2p}+\frac{2}{2p}=1$, Gagliardo-Nirenberg’s inequality (4.10),
Young’s inequality and equivalence of the norms, we compute

(5.23)
$\begin{split}&\|f(u)(\cdot,t)-f(v)(\cdot,t)\|^{2}_{L^{2}(X)}\leq\int_{X}|f(u)(x,t)-f(v)(x,t)|^{2}dx\\\
&\stackrel{{\scriptstyle\eqref{f1}}}{{\leq}}C\int_{X}\left[|u(x,s)|^{2(p-1)}+|v(x,s)|^{2(p-1)}\right]|u(x,s)-v(x,s)|dx\\\
&\stackrel{{\scriptstyle
HI}}{{\leq}}C\left[\|u(\cdot,t)\|^{2(p-1)}_{L^{2p}(X)}+\|v(\cdot,t)\|^{2(p-1)}_{L^{2p}(X)}\right]\|u(\cdot,t)-v(\cdot,t)\|^{2}_{L^{2p}(X)}\\\
&\stackrel{{\scriptstyle\eqref{GNPDE}}}{{\leq}}C\left[\|u\|^{2(p-1)a}_{H^{1,2}(X)}\|u\|_{L^{2}(X)}^{2(p-1)(1-a)}+\|v\|^{2(p-1)a}_{H^{1,2}(X)}\|v\|_{L^{2}(X)}^{2(p-1)(1-a)}\right]\|u-v\|^{2a}_{H^{1,2}(X)}\|u-v\|_{L^{2}(X)}^{2(1-a)}\\\
&\stackrel{{\scriptstyle
YI}}{{\leq}}C\left[\|u\|^{2(p-1)}_{H^{1,2}(X)}+\|u\|_{L^{2}(X)}^{2(p-1)}+\|v\|^{2(p-1)}_{H^{1,2}(X)}+\|v\|_{L^{2}(X)}^{2(p-1)}\right]\left[\|u-v\|^{2}_{H^{1,2}(X)}+\|u-v\|_{L^{2}(X)}^{2}\right]\\\
&\leq
C\left[\|\Delta^{\frac{1}{2}}u\|^{2(p-1)}_{L^{2}(X)}+\|u\|_{L^{2}(X)}^{2(p-1)}+\|\Delta^{\frac{1}{2}}v\|^{2(p-1)}_{L^{2}(X)}+\|v\|_{L^{2}(X)}^{2(p-1)}\right]\\\
&\times\left[\|\Delta^{\frac{1}{2}}(u-v)\|^{2}_{L^{2}(X)}+\|u-v\|_{L^{2}(X)}^{2}\right]\\\
&\leq
C\left[\left(\|\Delta^{\frac{1}{2}}u\|_{L^{2}(X)}+\|u\|_{L^{2}(X)}\right)^{2(p-1)}+\left(\|\Delta^{\frac{1}{2}}v\|_{L^{2}(X)}+\|v\|_{L^{2}(X)}\right)^{2(p-1)}\right]\\\
&\times\left(\|\Delta^{\frac{1}{2}}(u-v)\|_{L^{2}(X)}+\|u-v\|_{L^{2}(X)}\right)^{2}.\end{split}$

By using definition of the norm on the space $Z$, for all $u\in Z$ we have

(5.24) $(1+t)^{-\frac{1}{2}}e^{\delta
t}(\|\Delta^{\frac{1}{2}}u\|_{L^{2}(X)}+\|u\|_{L^{2}(X)})\leq\|u\|_{Z}\leq M,$

then we get

(5.25)
$\|\Delta^{\frac{1}{2}}u\|_{L^{2}(X)}+\|u\|_{L^{2}(X)}\leq(1+t)^{\frac{1}{2}}e^{-\delta
t}M.$

Putting this estimate in (5.23), we get

(5.26) $\begin{split}\|f(u)(\cdot,t)-f(v)(\cdot,t)\|^{2}_{L^{2}(X)}&\leq
C\left[\left(\|\Delta^{\frac{1}{2}}u\|_{L^{2}(X)}+\|u\|_{L^{2}(X)}\right)^{2(p-1)}+\left(\|\Delta^{\frac{1}{2}}v\|_{L^{2}(X)}+\|v\|_{L^{2}(X)}\right)^{2(p-1)}\right]\\\
&\times\left(\|\Delta^{\frac{1}{2}}(u-v)\|_{L^{2}(X)}+\|u-v\|_{L^{2}(X)}\right)^{2}\\\
&\leq
C(1+t)^{p-1}e^{-2\delta(p-1)t}M^{2p-2}\left(\|\Delta^{\frac{1}{2}}(u-v)\|_{L^{2}(X)}+\|u-v\|_{L^{2}(X)}\right)^{2}\\\
&\leq C(1+t)^{p}e^{-2\delta pt}M^{2p-2}\|u-v\|^{2}_{Z}.\end{split}$

for all $t>0$ and $u,v\in Z$.

Then, via the equivalence of the norms on $H^{1,2}$-spaces, we have

(5.27) $\begin{split}&\|Lu-Lv\|_{L^{2}(X)}+\|\Delta^{\frac{1}{2}}(Lu-
Lv)\|_{L^{2}(X)}+\left\|\frac{\partial}{\partial t}(Lu-
Lv)\right\|_{L^{2}(X)}\\\
&=\|J(f(u))-J(f(v))\|_{L^{2}(X)}+\|\Delta^{\frac{1}{2}}(J(f(u))-J(f(v)))\|_{L^{2}(X)}\\\
&\quad\quad\quad+\left\|\frac{\partial}{\partial
t}\left[J(f(u))-J(f(v))\right]\right\|_{L^{2}(X)}\\\ &\leq
C\|J(f(u))-J(f(v))\|_{H^{1,2}(X)}+\left\|\frac{\partial}{\partial
t}\left[J(f(u))-J(f(v))\right]\right\|_{L^{2}(X)}.\end{split}$

Let us now compute the first norm that appears in the right hand side of the
last estimate. First, we note that

$Kf(u)(x,t)-Kf(v)(x,t)=K(f(u)-f(v))(x,t).$

By combining the above equality with the Cauchy-Schwarz inequality, Lemma 5.1
and (5.26), we obtain

(5.28)
$\begin{split}\|J(f(u))-J(f(v))\|^{2}_{H^{1,2}(X)}&=\int_{X}\left|\int_{0}^{t}(\xi^{2}-\Delta_{x})^{\frac{1}{2}}[K(f(u))-K(f(v))]ds\right|^{2}dx\\\
&\stackrel{{\scriptstyle
C-S\,I}}{{\leq}}Ct\int_{0}^{t}\int_{X}\left|(\xi^{2}-\Delta_{x})^{\frac{1}{2}}[K(f(u))-K(f(v))]\right|^{2}dxds\\\
&=Ct\int_{0}^{t}\int_{X}\left|(\xi^{2}-\Delta_{x})^{\frac{1}{2}}K[f(u)-f(v)]\right|^{2}dxds\\\
&\stackrel{{\scriptstyle\eqref{ocensol}}}{{\leq}}Cte^{-2\delta
t}\int_{0}^{t}e^{2\delta s}\|f(u)-f(v)\|^{2}_{L^{2}(X)}ds\\\
&\stackrel{{\scriptstyle\eqref{focen2}}}{{\leq}}CM^{2p-2}te^{-2\delta
t}\int_{0}^{t}(1+s)^{p}e^{2\delta s}e^{-2\delta p}\|u-v\|^{2}_{Z}ds\\\
&=CM^{2p-2}te^{-2\delta
t}\|u-v\|^{2}_{Z}\int_{0}^{t}(1+s)^{p}e^{-2\delta(p-1)s}ds\\\ &\leq
CM^{2p-2}te^{-2\delta t}\|u-v\|^{2}_{Z}.\end{split}$

We know that $Kf(u)(x,t)$ is a solution of (5.21), then $K(f(u))(x,0)=0$. By
using this fact, we have

(5.29) $\frac{\partial}{\partial t}J(f(u))(x,t-s)=\frac{\partial}{\partial
t}\int_{0}^{t}[K(f)](x,t-s)ds=\int_{0}^{t}\frac{\partial}{\partial
t}[K(f)](x,t-s)ds,$

and therefore, similarly to (5.28), with the use of Lemma 5.1 (with $s=0$) and
(5.26), we obtain

(5.30) $\begin{split}\left\|\frac{\partial}{\partial
t}(J(f(u))-J(f(v)))\right\|^{2}_{L^{2}(X)}&=\int_{X}\left|\int_{0}^{t}\frac{\partial}{\partial
t}[K(f(u))-K(f(v))]ds\right|^{2}dx\\\
&\leq\int_{X}t\int_{0}^{t}\left|\frac{\partial}{\partial
t}[K(f(u))-K(f(v))]\right|^{2}dsdx\\\
&=Ct\int_{0}^{t}\int_{X}\left|\frac{\partial}{\partial
t}[K(f(u))-K(f(v))]\right|^{2}dxds\\\
&=Ct\int_{0}^{t}\int_{X}\left|\frac{\partial}{\partial
t}K[f(u)-f(v)]\right|^{2}dxds\\\
&\stackrel{{\scriptstyle\eqref{ocensolt}}}{{\leq}}Cte^{-2\delta
t}\int_{0}^{t}e^{2\delta s}\|f(u)-f(v)\|^{2}_{L^{2}(X)}ds\\\
&\stackrel{{\scriptstyle\eqref{focen2}}}{{\leq}}CM^{2p-2}te^{-2\delta
t}\int_{0}^{t}(1+s)^{p}e^{2\delta s}e^{-2\delta p}\|u-v\|^{2}_{Z}ds\\\
&=CM^{2p-2}e^{-2\delta
t}t\|u-v\|^{2}_{Z}\int_{0}^{t}(1+s)^{p}e^{-2\delta(p-1)s}ds\\\ &\leq
CM^{2p-2}te^{-2\delta t}\|u-v\|^{2}_{Z}.\end{split}$

Putting (5.28) and (5.30) in (5.27), we have

(5.31) $\begin{split}&\|Lu-Lv\|_{L^{2}(X)}+\|\Delta^{\frac{1}{2}}(Lu-
Lv)\|_{L^{2}(X)}+\left\|\frac{\partial}{\partial t}(Lu-
Lv)\right\|_{L^{2}(X)}\\\ &\leq
C\|J(f(u))-J(f(v))\|_{H^{1,2}(X)}+\left\|\frac{\partial}{\partial
t}\left[J(f(u))-J(f(v))\right]\right\|_{L^{2}(X)}\\\
&\stackrel{{\scriptstyle\eqref{ocen55},\eqref{ocen555}}}{{\leq}}CM^{p-1}t^{\frac{1}{2}}e^{-\delta
t}\|u-v\|_{Z}.\end{split}$

From the last fact we establish,

(5.32) $\begin{split}&(1+t)^{-\frac{1}{2}}e^{\delta t}\left[\|Lu-
Lv\|_{L^{2}(X)}+\|\Delta^{\frac{1}{2}}(Lu-
Lv)\|_{L^{2}(X)}+\left\|\frac{\partial}{\partial t}(Lu-
Lv)\right\|_{L^{2}(X)}\right]\\\ &\leq C(1+t)^{-\frac{1}{2}}e^{\delta
t}M^{p-1}t^{\frac{1}{2}}e^{-\delta t}\|u-v\|_{Z}\\\
&=C\left(\frac{t}{1+t}\right)^{\frac{1}{2}}M^{p-1}\|u-v\|_{Z}\\\ &\leq
CM^{p-1}\|u-v\|_{Z}\\\ &=C_{1}M^{p-1}\|u-v\|_{Z},\end{split}$

and by taking supremum, we get

(5.33) $\|Lu-Lv\|_{Z}\leq C_{1}M^{p-1}\|u-v\|_{Z}.$

By choosing $M^{p-1}=\frac{c}{C_{1}}$, where $c\in(0,1)$, we show (5.22), it
means

(5.34) $\|Lu-Lv\|_{Z}\leq c\|u-v\|_{Z}.$

After that, let us prove

(5.35) $\|Lu\|_{Z}\leq M,\,\,\,\,\,\forall u\in Z.$

By taking $v=0$ and $f(0)=0$ in (5.33), we get

(5.36) $\|Lu-L0\|_{Z}=\|Lu-\psi\|_{Z}\leq C_{1}M^{p-1}\|u\|_{Z}\leq
C_{1}M^{p}.$

By Lemma 5.1 and assumption of this theorem , we have

(5.37) $\|\psi\|_{Z}\leq
C_{2}(\|u_{0}\|_{H^{1,2}(X)}+\|u_{1}\|_{L^{2}(X)})\leq C_{2}\varepsilon.$

By combining these facts with choosing
$\varepsilon\leq\frac{M(1-c)}{C_{2}}=\varepsilon_{0}$ where $c$ is defined in
(5.34), we get

(5.38)
$\begin{split}\|Lu\|_{Z}=\|Lu-\psi+\psi\|_{Z}&\leq\|Lu-\psi\|_{Z}+\|\psi\|_{Z}\\\
&\leq C_{1}M^{p}+C_{2}\varepsilon\\\ &\leq cM+C_{2}\varepsilon_{0}\\\
&=cM+C_{2}\frac{M(1-c)}{C_{2}}\\\ &=M,\end{split}$

completing the proof the $L$ is a contraction on $Z.$ Therefore, Banach fixed
point theorem guarantee ensure the existence of a global solution.

The uniqueness of the solution can be easily deduced using the above
estimates. Indeed, suppose that there are two solutions $u,v\in Z$ of the
Cauchy problem (5.1). Set $\eta=u-v$ and fix $t^{*}>0.$ Since $u$ and $v$ are
fixed point of $L$, using the previous computations (5.26), (5.28) and (5.30)
we obtain that

$\displaystyle\|\eta(t,\cdot)\|_{H^{1,2}(X)}^{2}+\|\partial_{t}\eta(t,\cdot)\|_{L^{2}(X)}^{2}$
$\displaystyle=\big{\|}J(f(u))(t,\cdot)-J(f(v))(t,\cdot)\big{\|}_{H^{1,2}(X)}^{2}$
$\displaystyle\quad+\big{\|}\partial_{t}(J(f(u))-J(f(v)))(t,\cdot)\big{\|}_{L^{2}(X)}^{2}$
$\displaystyle\leq Cte^{-2\delta t}\int_{0}^{t}e^{2\delta
s}\|f(u(s,\cdot))-f(v(s,\cdot))\|_{2}^{2}ds$ (5.39) $\displaystyle\leq
Cte^{-2\delta
t}\int_{0}^{t}\left(\|u(s,\cdot)\|_{H^{1,2}(X)}^{2(p-1)}+\|v(s,\cdot)\|_{H^{1,2}(X)}^{2}\right)\|\eta(s,\cdot)\|_{H^{1,2}(X)}^{2}ds.$

So, from the definition of $Z$ and using the equivalence of the norms (2.9) we
see that the map
$t\mapsto\|u(s,\cdot)\|_{H^{1,2}(X)}^{2(p-1)}+\|v(s,\cdot)\|_{H^{1,2}(X)}^{2}$
is continuous and therefore, it is bounded on the compact set $[0,t^{*}].$ So
inequality (5) takes the form

$\|\eta(t,\cdot)\|_{H^{1,2}(X)}^{2}+\|\partial_{t}\eta(t,\cdot)\|_{L^{2}(X)}^{2}\leq
Cte^{-2\delta
t}\int_{0}^{t}\|\eta(s,\cdot)\|_{H^{1,2}(X)}^{2}+\|\partial_{t}\eta(t,\cdot)\|_{L^{2}(X)}^{2}ds,$

and, so by Gronwall’s lemma, we obtain

$\|\eta(t,\cdot)\|_{H^{1,2}(X)}^{2}+\|\partial_{t}\eta(t,\cdot)\|_{L^{2}(X)}^{2}=0,$

for all $0<t<t^{*},$ which eventually shows that $\eta=0$ in $[0,t^{*}]\times
X.$ Finally, since $t^{*}$ is arbitrary it implies that $\eta\equiv 0$ and
hence $u=v.$ This completes the proof of the theorem. ∎

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