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 dam** and mass terms for the Laplace–Beltrami operator on symmetric spaces.
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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 in several other areas of mathematics (see [11, 33, 40]). In particular, these inequalities have been utilised extensively to study global wellposedness results related to several important nonlinear partial differential 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 dum** and mass terms associated to the Laplace–Beltrami operator on Riemannian symmetric spaces. Riemannian symmetric spaces represent a significant category of Riemannian manifolds that are non-positively curved and encompass hyperbolic spaces. An intriguing characteristic of Riemannian symmetric spaces is that each of them can be expressed as G/K for some noncompact, connected, semisimple Lie group G with a finite centre, and its maximal subgroup K (see [39]). This characteristic enables the utilisation of representation theory and consequently, Fourier analysis on semisimple Lie groups in the study of analysis of symmetric spaces [31].
As in the Euclidean space setting, establishing certain functional inequalities on Riemannian manifolds is interesting in itself and is tightly useful in the analysis of nonlinear partial differential equations (PDEs) [40]. The study of the best constants of functional inequalities on manifolds has led to the conclusion of many geometrical and topological properties of underlying manifolds. We refer to several interesting papers [7, 18, 24, 25, 44, 45, 47, 68] and references therein for more details. In this work, our focus is on establishing certain interesting functional inequalities on Riemannian symmetric spaces of noncompact type, which are useful for studying nonlinear PDEs on such spaces. Recently, many researchers have contributed to the development of certain important functional inequalities on non-positively curved manifolds using different suitable methods from Fourier analysis and geometric analysis. However, most of them were confined to rank one symmetric spaces, such as real or complex hyperbolic spaces and their different models (see [13] and references therein). In this case, one uses Helgason–Fourier analysis on hyperbolic spaces [31, 39].
One fundamental functional inequality in the Euclidean harmonic analysis is the classical Stein–Weiss inequality established by Stein and Weiss [64]. It states that:
Theorem 1.1
Let \(0<\lambda <N\), \(1<p<\infty \), \(\alpha <\frac{N(p-1)}{p}\), \(\beta <\frac{N}{q}\), \(\alpha +\beta \ge 0\) and \(\frac{\lambda -\alpha -\beta }{N}=\frac{1}{p}-\frac{1}{q}\). For \(1<p\le q<\infty \), we have
where C is a positive constant independent of u. Here, the Riesz potential \(I_\lambda \) on \(\mathbb {R}^N\) is defined as
The unweighted version of inequality (1.1) was proved by Sobolev [62] by extending a multi-dimensional version of the Hardy–Littlewood inequality [37]. 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 [10,11,12, 22, 24, 26, 27, 47, 50, 54] here just to mention a very few of them. In the noncommutative setting (e.g. Heisenberg groups, homogeneous groups, general Lie groups), the Hardy-Littlewood-Sobolev and the Stein–Weiss inequalities are also well-developed, see [13, 14, 16, 21, 28, 29, 35, 43, 57,58,59]. We refer to [36] 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 [57], 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 \(n\ge 3\). Let \(0<\sigma < n\), \(1<p<\infty \), \(\alpha <\frac{n}{p'}\), \(\beta <\frac{n}{q},\) \(\alpha +\beta \ge 0\) and \(\frac{\sigma -\alpha -\beta }{n}=\frac{1}{p}-\frac{1}{q}\). Then, for \(1<p\le q<\infty \) and for sufficiently large \(\xi >0,\) there exists a positive constant C independent of u, such that
Here, \(G_{\xi , \sigma }\) denotes the Bessel–Green–Riesz Kernel on the Riemannian symmetric space X of any rank (see Sect. 2 for the definition), and |x| denotes the distance between \(0:=eK \in X:=G/K\) and the point \(x:=gK \in X\). Here, G is a noncompact, connected, semisimple Lie group with a finite centre and K is a maximal compact subgroup of G. We also clarify here that the \(y^{-1}x\) is not defined on X but on the corresponding group G via the identification \(X:=G/K.\)
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 \(n\ge 3\). 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
where C is a positive constant independent of u.
We remark here that the Hardy–Littlewood–Sobolev inequality on noncompact symmetric spaces of noncompact type was also obtained by Anker [2] (see also [65, Section 4] and references therein) by different methods.
The Hardy inequality is one of the well-known inequalities of G. H. Hardy which basically says that
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, 19, 24, 40, 44,45,46, 46, 56, 59, 71] 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 \(n\ge 3\). Suppose that \(0<\sigma <n\), \(1<p\le q<\infty \) and \(0\le \beta < \frac{n}{q}\) such that \(\frac{\sigma -\beta }{n}=\frac{1}{p}-\frac{1}{q}\). Then, for \(u\in H^{\sigma , p}(X)\), we have
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 following Hardy inequality on symmetric spaces
The uncertainly principle on symmetric spaces can be derived from (1.7). Indeed, we will show that
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
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 [68] and Aubin in [7].
Gagliardo [30] and Nirenberg [52] independently obtained an (interpolation) inequality, widely known as the Gagliardo–Nirenberg inequality, which says that
where
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 \(n\ge 3\). Suppose \(0<\sigma <n\), \(\tau >0\), \(p>1\), \(\sigma p<n\), \(\mu \ge 1\), \(a\in (0,1]\) and
Then, there exists a positive constant C such that
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 \(1<p\le q<\infty \) such that \(\frac{\sigma }{n}=\frac{1}{p}-\frac{1}{q}\), we obtain
We note that the Sobolev inequality on symmetric spaces and on the general Lie groups was previously obtained by many authors; see [16, 58, 65, 70].
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 \(n\ge 3\). Let \(\tau \in \left[ 2,\frac{2n}{n-2}\right] \) and \(a=\frac{n(\tau -2)}{\tau }.\) Then, we have
In one of their pioneering works, Caffarelli, Kohn and Nirenberg in [17] established the following inequality:
Theorem 1.7
Let \(N\ge 1\), and let \(l_1\), \(l_2\), \(l_3\), \(a, \, b, \, d,\, \delta \in \mathbb {R}\) be such that \(l_1, l_2 \ge 1\), \(l_3 > 0, \,\,0 \le \delta \le 1,\) and
Then,
if and only if
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 [45, 51] 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 \(n\ge 3\) 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 \(p\le \frac{a\tau q}{q-(1-a)\tau }\). Let b, c be real numbers such that \(0\le (c(1-a)-b)\le \frac{n(q-(1-a)\tau )}{q\tau }\) 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
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:
where \(b,m>0\) and \(f:\mathbb {R}\rightarrow \mathbb {R}\) satisfies the following conditions:
and
The existence and non-existence results for the semilinear wave equation (1.19) with or without dum** 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 [33, 41, 42, 63, 67]. 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 [61, 72, 73]. It is well-known that, in contrast with the Euclidean space, the Strauss conjecture type phenomena do 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 [5, 6, 48, 49]. Later, these results were extended to non-trap** asymptotically hyperbolic manifolds [61] and to the Riemannian manifolds with strictly negatively sectional curvature [60] using different method. Very recently, Anker and Zhang [1] investigated semilinear wave equations on general symmetric spaces of noncompact type and proved that a similar phenomenon holds for general symmetric spaces; see also [72, 73]. The Strichartz inequality played a very important role for investigating aforementioned results on the Riemannian symmetric spaces [1, 72, 73].
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 \(n\ge 3\) and let \(1\le p\le \frac{n}{n-2}\). 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 \(\Vert u_{0}\Vert _{H^{1,2}(X)}+\Vert u_{1}\Vert _{L^{2}(X)}<\varepsilon \). Then, there exists \(\varepsilon _{0}>0\) such that for all \(0<\varepsilon \le \varepsilon _{0}\) 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 the 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 Sobolev 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. In Sect. 4, we will derive several important functional (Hardy, Hardy–Sobolev, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg) inequalities on symmetric spaces of noncompact type. In Sect. 5, we present an application of the Gagliardo–Nirenberg inequality obtained in Sect. 4 to the global existence results of wave equations with dum** and mass terms associated with the 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 on noncompact type. The material presented in this section can be found in the excellent books and research papers [2, 3, 31, 38, 39, 65]. Finally, we discuss the weighted integral Hardy inequalities on metric measure spaces with general weights [56] 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 centre 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 algebra \(\mathfrak {g}\) 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 \(x_1=g_1K\) and \(x_2=g_2K\) 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
Then, the set of restricted root of \(\mathfrak {g}\) with respect to \(\mathfrak {a}\) is denoted by \(\Sigma \) and defined as
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 \(\alpha \ne 0\) 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 map \((k, a, n) \mapsto kan \) 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 \(M'\) be the normalizer of A in K. Then, M is the normal subgroup of \(M'\) and normalize N. The factor group \(W=M'/M\) 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
The Haar measure on G corresponding to the Iwasawa decomposition and the polar decomposition can be described as, for any \(f \in C_c(G),\)
and
respectively. Here, the density J(Y) for \(Y\in \overline{\mathfrak {a}^+}\) is given by
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 \(d'(xK, yK):=\langle \rho / |\rho |, (y^{-1}x)^{+} \rangle \) 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 [1] that the Riemannian distance d and the polyhedral distance \(d'\) 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
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
whenever the integral exists. At times, we also denote \(\mathcal {H}f\) by \(\widehat{f}.\) It is known that the map \(f \mapsto \mathcal {H}(f)\) extends to isometry of \(L^2(X)\) onto \(L^2(\mathfrak {a}_+ \times K, |c(\lambda )|^2 d\lambda dk),\) where \(c(\lambda )\) denotes Harish–Chandra’s c-function.
Let us introduce Harish–Chandra’s elementary spherical function in the following form:
The elementary spherical function \(\varphi _{0}\) satisfies the following global estimate [3, Proposition 2.2.1]:
Moreover, we have
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, [3]]).
Throughout this paper, the symbol \(A\asymp B\) means that \(\exists \,C_{1},C_{2}>0\) such that \(C_{1}A\le B\le C_{2}A\).
The Sobolev space on the symmetric space of noncompact type X for \(0<\sigma \in \mathbb {R}\) and \(1<p<\infty \) is defined as
endowed with the norm
Then, it follows from [65, Theorem 4.4] that
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 \({\mathbb {Y}}\) 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 \(J \in L^1_{loc}({\mathbb {Y}})\) such that for all \(f\in L^1({\mathbb {Y}})\) we have
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 metric measure spaces are Euclidean spaces, homogeneous Lie groups, and Riemannian symmetric spaces of noncompact type. We denote \(|x|_a:=d(x, a).\)
The class of such metric measure spaces was introduced in [56], where the following integral Hardy inequality was obtained.
Theorem 2.1
Let \(1<p\le q <\infty \) 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 \(v^{1-p'}\in L^1_{loc}({\mathbb {Y}})\). Denote
Then, the inequality
holds for all measurable functions \(f:\mathbb {Y}\rightarrow {{\mathbb {C}}}\) if and only if any of the following equivalent conditions holds:
-
(1)
\({\mathcal {D}}_{1}:=\sup \limits _{x\not =a} \bigg \{U^\frac{1}{q}(x) V^\frac{1}{p'}(x)\bigg \}<\infty .\)
-
(2)
\({\mathcal {D}}_{2}:=\sup \limits _{x\not =a} \bigg \{\int _{{\mathbb {Y}}\backslash {B(a,|x|_a )}}u(y)V^{q(\frac{1}{p'}-s)}(y)dy\bigg \}^\frac{1}{q}V^s(x)<\infty .\)
-
(3)
\({\mathcal {D}}_{3}:=\sup \limits _{x\not =a}\bigg \{\int _{B(a,|x|_a)}u(y)V^{q(\frac{1}{p'}+s)}(y)dy\bigg \}^{\frac{1}{q}}V^{-s}(x)<\infty \), provided that \(u,v^{1-p'}\in L^1(\mathbb {Y})\).
-
(4)
\({\mathcal {D}}_{4}:=\sup \limits _{x\not =a}\bigg \{\int _{B(a,\vert x \vert _a)}v^{1-p'}(y) U^{p'(\frac{1}{q}-s)}(y)dy\bigg \}^\frac{1}{p'}U^s(x)<\infty .\)
-
(5)
\({\mathcal {D}}_{5}:=\sup \limits _{x\not =a}\bigg \{\int _{{\mathbb {Y}}\backslash {B(a,\vert x \vert _a )}}v^{1-p'}(y)U^{p'(\frac{1}{q}+s)}(y)dy\bigg \}^\frac{1}{p'}U^{-s}(x)<\infty \), provided that \(u,v^{1-p'}\in L^1(\mathbb {Y})\).
Moreover, the constant C for which (2.11) holds and quantities \({\mathcal {D}}_{1}-{\mathcal {D}}_{5}\) are related by
and
Similarly, in [56], the authors obtained the adjoint integral Hardy inequality in the following form:
Theorem 2.2
Let \(1<p\le q <\infty \) 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 \(v^{1-p'}\in L^1_{loc}({\mathbb {X}})\). Denote
Then, the inequality
holds for all measurable functions \(f:\mathbb {Y}\rightarrow {{\mathbb {C}}}\) if and only if any of the following equivalent conditions holds:
-
(1)
\({\mathcal {D}}^{*}_{1}:=\sup \limits _{x\not =a} \bigg \{U^\frac{1}{q}(x) V^\frac{1}{p'}(x)\bigg \}<\infty .\)
-
(2)
\({\mathcal {D}}^{*}_{2}:=\sup \limits _{x\not =a} \bigg \{\int _{\mathbb {Y}\backslash {B(a,|x|_a )}}u(y)V^{q(\frac{1}{p'}-s)}(y)dy\bigg \}^\frac{1}{q}V^s(x)<\infty .\)
-
(3)
\({\mathcal {D}}^{*}_{3}:=\sup \limits _{x\not =a}\bigg \{\int _{\mathbb {Y}\setminus B(a,|x|_a)}u(y)V^{q(\frac{1}{p'}+s)}(y)dy\bigg \}^{\frac{1}{q}}V^{-s}(x)<\infty \), provided that \(u,v^{1-p'}\in L^1(\mathbb {Y})\).
-
(4)
\({\mathcal {D}}^{*}_{4}:=\sup \limits _{x\not =a}\bigg \{\int _{\mathbb {Y}\setminus B(a,\vert x \vert _a)}v^{1-p'}(y) U^{p'(\frac{1}{q}-s)}(y)dy\bigg \}^\frac{1}{p'}U^s(x)<\infty .\)
-
(5)
\({\mathcal {D}}^{*}_{5}:=\sup \limits _{x\not =a}\bigg \{\int _{{B(a,\vert x \vert _a )}}v^{1-p'}(y)U^{p'(\frac{1}{q}+s)}(y)dy\bigg \}^\frac{1}{p'}U^{-s}(x)<\infty \), provided that \(u,v^{1-p'}\in L^1(\mathbb {Y})\).
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. The class of general metric measure spaces having a polar decomposition was analysed in [8].
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 [57] 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 \(n\ge 3\) and rank \(l\ge 1\). Let \(0<\sigma < n\), \(1<p<\infty \), \(\alpha <\frac{n}{p'}\), \(\beta <\frac{n}{q},\) \(\alpha +\beta \ge 0\) and \(\frac{\sigma -\alpha -\beta }{n}=\frac{1}{p}-\frac{1}{q}\). Then, for sufficiently large \(\xi >0\) and \(1<p\le q<\infty ,\) we have
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:
where
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 \(2|y|\le |x|\), we obtain
which implies that
Let us now consider the two different cases.
Case (a). When \(\frac{|x|}{2} \ge 1.\) Let us first find an estimate of \(G_{\xi , \sigma }(y^{-1}x)\) in terms of |x|, using the fact \(\frac{|x|}{2}\le |y^{-1}x|,\) in the range when \(2|y| \le |x|.\) Note that by (2.6), we have
Since the Riemannian distance and the polyhedral distance are equivalent and \(G_{\xi , \sigma }\) is K-biinvariant, we have
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
Now, by \( \langle \rho , x^+ \rangle \le \langle \rho , y^+ \rangle +\langle \rho , (y^{-1}x)^+ \rangle ,\) \( \frac{|x^+|}{2} \le |(y^{-1}x)^{+}| \le \frac{3|x^+|}{2}\) using \(|y^+| \le \frac{|x^+|}{2},\) and Cauchy-Schwarz inequality, we get
Therefore, we obtain
Our aim is to show the following inequality,
and, for this purpose, we need to check the condition \(\mathcal D_{1}\) in Theorem 2.1 which turns out to be the following,
by setting \(|x|/2=R \ge 1.\)
By using the Cartan decomposition with \(R \ge 1\) in this case, we get
when \(q\ge 2\) and \(\xi \) large enough (e.g. \(\xi \ge 2|\rho |\)). For \(q<2,\) with same calculations as in (3.6), we get
whenever \(\frac{\xi }{4}+\frac{q}{2}|\rho |-2|\rho | \ge 0,\) which holds as \(\xi \) is large enough.
Let us consider the second integral,
By combining the estimates (3.6), (3.7) and (3.8) with the fact that \(\xi \) is sufficiently large (e.g. \(\xi \ge 8|\rho |\)) and substituting back in (3.5), we deduce that
as \(p'>1\) and \(\xi \) is sufficiently large.
Case (b). When \(0<\frac{|x|}{2} < 1.\) Again, we need to consider the integral
Since \(\frac{|x|}{2} \le |y^{-1}x|,\) we can divide the study of this integral into two parts: one when \(|y^{-1}x| < 1\) and the second when \(|y^{-1}x| \ge 1.\) In fact, the case when \(|y^{-1}x| \ge 1,\) 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 \(|y^{-1}x| \ge 1,\) we have
To establish the inequality
we need to verify that the following holds,
Here, \(0<R=|x|/2 < 1.\)
In order to check that \(\mathcal {D}_1<\infty ,\) using the Cartan decomposition, we get
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 \(q\ge 2\) and \(q<2\) with \(\xi \) sufficiently large (in this case, \(\xi >2|\rho |\) will work). Therefore, we get
Also, we compute the following integral for \(0<R<1\) with \(\alpha <\frac{n}{p'}\):
Thus, from (3.13) and (3.14), we have
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 \(\frac{|x|}{2} \le |y^{-1}x|\), we get \(G_{\xi , \sigma }(y^{-1}x) \asymp |y^{-1}x|^{\sigma -n} \le \frac{|x|^{\sigma -n}}{2} \asymp G_{\xi , \sigma }(\frac{x}{2})\) and therefore,
To show the inequality
it is enough to show that,
Here, \(0<R=|x|/2 < 1.\) Then, we have
Also, we compute the following integral for \(0<R<1\) with \(\alpha <\frac{n}{p'}\):
Thus, from (3.17), (3.18) and \(\frac{\sigma -\alpha -\beta }{n}=\frac{1}{p}-\frac{1}{q}\), we have
Therefore, in this case, we get
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 \(|y|\ge 2|x|\), we get
therefore, we have \(|x| \le \frac{|y|}{2}\le |y^{-1}x|\). By arguing exactly in the same way as in Case (a) of Step 1, we estimate that
Thus, we obtain
If we show the following condition,
then by using the conjugate integral Hardy inequality (see Theorem 2.2), we get
Let us check the condition (3.21). Similarly to the previous step, we consider two cases \(0<R<1\) and \(R \ge 1\). Firstly, let us consider \(R\ge 1\) and by using the Cartan decomposition, we get, same as (3.6) and (3.7), that
for sufficiently large \(\xi \). Then, let us compute the first integral in (3.21), we get
By combining (3.24) and (3.23), we get
thanks to sufficiently large \(\xi \).
Let us consider the case \(0<R<1\). Similarly to (3.24), for \(0<R<1\), we get
Next we will estimate the second integral of (3.21) for \(0<R<1\). Thus, we have, same as (3.13), that
Finally, by combining (3.25) and (3.26), we get
Therefore, (3.20) and (3.22) imply that
Step 3. Let us now focus on the remaining case of \(I_2.\) We need to show that
We rewrite \(I_{2}\) in the following form:
Since \(|x|^{\beta q}\) is non-decreasing with respect to |x| near the origin, there exists \(k_0 \in \mathbb {Z}\) with \(k_0 \le -3\) 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:
where
and
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,
since \(r(\sigma -n)+(n-1)>0\) by \(\frac{\sigma -n-\alpha -\beta }{n}-\frac{1}{p}+\frac{1}{q}+1=0\) with \(\alpha +\beta \ge 0\) 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 \(1 \le r \le \infty \) and by setting \(\tilde{u}(y)=|y|^{\alpha }u(y)\), we establish
Note that from the conditions \(2^{k} \le |x| \le 2^{k+1}\) and \(|x| \le 2|y| \le 4|x|\), we deduce that \(2^{k-1} \le |y| \le 2^{k+3}.\) Therefore, from (3.34), we further have
Finally, we estimate \(I_{2,1}^q.\) By combining the triangle inequality for the Riemannian distance with \(|x|\le 2|y|\le 4|x|,\) one can deduce that \(|y^{-1}x|\le |y|+|x|\le 3|x|.\) Since \(|x|\le 2|y|\le 4|x|\) and \(2^{k} \le |x| \le 2^{k+1}\) with \(k \le k_0\), it yields that
Thus, by using (3.36) with Young’s inequality for \(1+\frac{1}{q}= \frac{1}{r}+\frac{1}{p}\), we establish
Computing \(\Vert G_{\xi , \sigma }\chi _{\{0<|\cdot |\le 3\cdot 2^{k+1}\}}\Vert _{L^{r}(X)}^q\) with (3.36), we obtain
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
Putting (3.38) and (3.39) in (3.37) and recalling that \(k_0\le -3\), we establish
Then, by putting together (3.35) and (3.40), we have
completing the proof of (3.28). \(\square \)
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 \(n\ge 3\) and rank \(l\ge 1.\) Suppose that \(0<\sigma <n\), \(1<p\le q<\infty \) and \(0\le \beta < \frac{n}{q}\) are such that \(\frac{\sigma -\beta }{n}=\frac{1}{p}-\frac{1}{q}\). Then, for all \(u\in H^{\sigma , p}(X)\), we have
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,
Corollary 4.3
(Uncertainty principle) By Theorem 4.2, we have the uncertainly principle on symmetric spaces of noncompact type, that is,
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 \(1<p\le q<\infty \) such that \(\frac{\sigma }{n}=\frac{1}{p}-\frac{1}{q}\) we obtain
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
Then, by using this with Theorem 3.1, we obtain
completing the proof. \(\square \)
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 \(n\ge 3\) and rank \(l\ge 1\). Suppose \(0<\sigma <n\), \(\tau >0\), \(p>1\), \(\sigma p<n\), \(\mu \ge 1\), \(a\in (0,1]\) and
Then, there exists a positive constant C such that we have
where C is independent of u.
Proof
By using Hölder’s inequality with \(1=\frac{a\tau }{q}+\frac{(1-a)\tau }{\mu }\) where \(\frac{1}{q}=\frac{1}{p}-\frac{\sigma }{n}\) and Sobolev inequality, we have
completing the proof. \(\square \)
In the next, we give a particular case of the Gagliardo–Nirenberg inequality, for that, we need the case \(\mu =p=2\) 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 \(n\ge 3\) and rank \(l\ge 1\). Let \(\tau \in \left[ 2,\frac{2n}{n-2}\right] \) and \(a=\frac{n(\tau -2)}{\tau },\) then we have
Proof
By taking \(\sigma =1\), \(p=2\) and \(\nu =2\) in Theorem 4.5, we get (4.10). \(\square \)
Then, let us show the Caffarelli–Kohn–Nirenberg inequality in the following theorem:
Theorem 4.7
Let X be a symmetric space of noncompact type of dimension \(n\ge 3\) and rank \(l\ge 1\) such that \(0<\sigma <n\). Suppose \(p>1\), \(0<q<\tau <\infty \) such that \(a\in \left( \frac{\tau -q}{\tau },1\right] \) and \(p\le \frac{a\tau q}{q-(1-a)\tau }\). Let b, c be real numbers such that \(0\le (c(1-a)-b)\le \frac{n(q-(1-a)\tau )}{q\tau }\) 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
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
completing the proof. \(\square \)
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 [55] and [69], 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 [55]. 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.
where \(b,m>0\) and \(f:\mathbb {R}\rightarrow \mathbb {R}\) satisfies the following conditions:
and
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
and
Proof
By applying the Helgason–Fourier transform in (5.1) with respect to the variable x and using the fact that \(\widehat{(\Delta _x u)}=-|\lambda |^2 \widehat{u}\), we obtain
Therefore, the solution of (5.6) is given by
where \(\gamma =\sqrt{m+|\lambda |^{2}}\).
We set \(B:=b^2- 4 \gamma ^2.\) We will now deduce a pointwise estimate for each case. First, let us consider the case when \(B>0\). We know that
and for any \(c>0\), we have
so that
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
Therefore, by using these facts, we establish
For the case \(B=0\), that is, \(b=2\gamma ,\) by using previous computation in (5.11), we get
Finally, for case when \(B<0,\) that is, \(b<2\gamma ,\) we perform a similar computation as in the case \(b>2\gamma \) to obtain
To get the Sobolev norm estimates of solutions, we consider again case by case using the previously established pointwise estimates. We begin with the case when \(b^2<4m,\) that is, \(B<0\) for all \(\lambda \in \mathfrak {a}^*\). In this case, the function \(\frac{\xi ^2+|\lambda |^{2}}{4m-b^{2}+|\lambda |^{2}}\) is bounded for all \(\lambda \in \mathfrak {a}^*\) as \(b^2<4m.\) Using the Plancherel formula for the Helgason–Fourier transform, we have
The case \(4m \le b^2\) is a bit tricky due to the presence of singularity at \(|\lambda |= \frac{\sqrt{b^2-2m}}{2}.\) To tackle this situation, we divide the integral over the set \(D=\{\lambda \in \mathfrak {a}^*: |B|<1\}\) and \(D^C:=\mathfrak {a}^* \backslash D.\) In other words, we will calculate
On \(\mathfrak {a}^* \times K,\) we are always in the case when \(B<0\) or \(B>0.\) Therefore, one can deduce the same estimate (5.12). Indeed, by the similar approach, since the function \(\lambda \mapsto \frac{\xi ^2+|\lambda |^{2}}{|4\,m-b^{2}+|\lambda |^{2}|}\) is bounded on \(D^C,\) we obtain that
Next, for the integral over the set \(D \times K.\) We can see that the estimate (5.13) holds in general situation for some constants \(C, \delta >0\) by analysing it case by case. For the case, when \(B=0\) has been proved in (5.13). For the case \(B>0\) and \(B<0,\) we showed that the estimate (5.12) holds, which by adjusting the constant will imply (5.13) for these two cases as well. In conclusion, we have the following estimate
Now, on the set D, we have \(|B|<1,\) which gives that
which says that on the set \(D \times K,\) the quantity \(\xi ^2+|\lambda |^2\) is bounded above by a constant independent of t. Thus, we get the estimate
Now, using (5.3) and (5.2) in (5.16), we obtained the desired estimate (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. \(\square \)
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 \(n\ge 3\). Let \(1\le p\le \frac{n}{n-2}\). 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 \(\Vert u_{0}\Vert _{H^{1,2}(X)}+\Vert u_{1}\Vert _{L^{2}(X)}<\varepsilon \). Then, there exists \(\varepsilon _{0}>0\) such that \(0<\varepsilon \le \varepsilon _{0}\) 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 on Banach fixed point theorem. First, let us consider the following Banach space
where M will be defined later and the norm is defined by
where \(\delta >0\) as in Lemma 5.1. Let us define the map** \(L:Z\rightarrow Z\),
where \(\psi \) is the solution of the linear problem (5.1) and Kv is the solution of the following Cauchy problem:
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
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
By using definition of the norm on the space Z, for all \(u\in Z\), we have
then we get
Putting this estimate in (5.25), we get
for all \(t>0\) and \(u,v\in Z\).
Then, via the equivalence of the norms on \(H^{1,2}\)-spaces, we have
Let us now compute the first norm that appears in the right hand side of the last estimate. First, we note that
By combining the above equality with the Cauchy–Schwarz inequality, Lemma 5.1 and (5.28), we obtain
We know that Kf(u)(x, t) is a solution of (5.23), then \(K(f(u))(x,0)=0\). By using this fact, we have
and therefore, similarly to (5.30), with the use of Lemma 5.1 (with \(s=0\)) and (5.28), we obtain
Putting (5.30) and (5.32) in (5.29), we have
From the last fact, we establish
and by taking supremum, we get
By choosing \(M^{p-1}=\frac{c}{C_{1}}\), where \(c\in (0,1)\), we show (5.24), it means
After that, let us prove
By taking \(v=0\) and \(f(0)=0\) in (5.35), we get
By Lemma 5.1 and assumption of this theorem, we have
By combining these facts with choosing \(\varepsilon \le \frac{M(1-c)}{C_{2}}=\varepsilon _{0}\) where c is defined in (5.36), we get
completing the proof the L is a contraction on Z. Therefore, Banach fixed point theorem guarantee ensures 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 (5.28), (5.30) and (5.32), we obtain that
So, from the definition of Z and using the equivalence of the norms (2.9), we see that the map \(t \mapsto \Vert u(s, \cdot )\Vert _{H^{1,2}(X)}^{2(p-1)}+\Vert v(s, \cdot )\Vert _{H^{1,2}(X)}^2 \) is continuous, and therefore, it is bounded on the compact set \([0, t^*].\) So inequality (5.41) takes the form
and, so by Gronwall’s lemma, we obtain
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. \(\square \)
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References
Anker, J.-Ph., Zhang, H.-W.: Wave equation on general noncompact symmetric spaces (to appear in) Am. J. Math. (2022). ar**v:2010.08467
Anker, J.-Ph.: Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65(2), 257–297 (1992)
Anker, J.-Ph., Ji, L.: Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9, 1035–1091 (1999)
Anker, J.-Ph., Pierfelice, V., Vallarino, M.: The wave equation on hyperbolic spaces. J. Differ. Equ. 252(10), 5613–5661 (2012)
Anker, J.-Ph., Pierfelice, V.: Wave and Klein-Gordon equations on hyperbolic spaces. Anal. PDE 7(4), 953–995 (2014)
Anker, J.-Ph., Pierfelice, V., Vallarino, M.: The wave equation on Damek–Ricci spaces. Ann. Mat. Pura Appl. 4(194), 731–758 (2015)
Aubin, Th.: Problèmes isoperimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)
Avetisyan, Zh., Ruzhansky, M.: A note on the polar decomposition in metric spaces. J. Math. Sci. 280, 73–82 (2024)
Banica, V., González, M., Sáez, M.: Some constructions for the fractional Laplacian on noncompact manifolds. Rev. Mat. Iberoam. 31(2), 681–712 (2015)
Beckner, W.: Pitt’s inequality and the uncertainty principle. Proc. Am. Math. Soc. 123, 1897–1905 (1995)
Beckner, W.: Weighted inequalities and Stein–Weiss potentials. Forum Math. 20, 587–606 (2008)
Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)
Beckner, W.: On Lie groups and hyperbolic symmetry-from Kunze–Stein phenomena to Riesz potentials. Nonlinear Anal. 126, 394–414 (2015)
Beckner, W.: Symmetry in Fourier analysis: Heisenberg group to Stein–Weiss integrals. J. Geom. Anal. 31(7), 7008–7035 (2021)
Bhowmik, M., Pusti, S.: An extension problem and Hardy’s inequality for the fractional Laplace–Beltrami operator on Riemannian symmetric spaces of noncompact type. J. Funct. Anal. 282(9), 109413 (2022)
Bruno, T., Peloso, M.M., Tabacco, A., Vallarino, M.: Sobolev spaces on Lie groups: embedding theorems and algebra properties. J. Funct. Anal. 276(10), 3014–3050 (2019)
Caffarelli, L.A., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53(3), 259–275 (1984)
Carron, G.: Inégalitès de Hardy sur les variétès riemanniennes non-compactes. J. Math. Pures Appl. (9) 76(10), 883–891 (1997)
Ciatti, P., Cowling, M.G., Ricci, F.: Hardy and uncertainty inequalities on stratified Lie groups. Adv. Math. 277, 365–387 (2015)
Chen, W., Li, C.: The best constant in a weighted Hardy–Littlewood–Sobolev inequality. Proc. Am. Math. Soc. 136, 955–962 (2008)
Chen, L., Lu, G., Tao, C.: Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group. J. Funct. Anal. 277(4), 1112–1138 (2019)
Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974)
Clerc, J.L., Stein, E.M.: \(L^p\)-multipliers for noncompact symmetric spaces. Proc. Natl. Acad. Sci. USA 71(10), 3911–3912 (1974)
D’Ambrosio, L., Dipierro, S.: Hardy inequalities on Riemannian manifolds and applications. Ann. Inst. Henri Poincaré Anal. Non Lin éaire 31, 449–475 (2014)
do Carmo, M.P., **a, C.: Complete manifolds with non-negative Ricci curvature and the Caffarelli–Kohn–Nirenberg inequalities. Compos. Math. 140(3), 818–826 (2004)
Dou, J.: Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space. Commun. Contemp. Math. 18, 1550067 (2016)
Fefferman, C., Muckenhoupt, B.: Two nonequivalent conditions for weight functions. Proc. Am. Math. Soc. 45, 99–104 (1974)
Folland, G.B., Stein, E.M.: Estimates for the \(\overline{\partial _{b}}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)
Frank, R.L., Lieb, E.H.: Sharp constants in several inequalities on the Heisenberg group. Ann. Math. 176, 349–381 (2012)
Gagliardo, E.: Ulteriori proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 8, 24–51 (1959)
Gangolli, R., Varadarajan, V.: Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin (1988)
Garetto, C., Ruzhansky, M.: Wave equation for sums of squares on compact Lie groups. J. Differ. Equ. 258(12), 4324–4347 (2015)
Georgiev, V., Lindblad, H., Sogge, C.: Weighted Strichartz estimates and global existence for semilinear wave equations. Am. J. Math. 119, 1291–1319 (1997)
Georgiev, V., Palmieri, A.: Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. J. Differ. Equ. 269, 420–448 (2020)
Han, X., Lu, G., Zhu, J.: Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group. Nonlinear Anal. 75, 4296–4314 (2012)
Han, Y., Zhu, M.: Hardy–Littlewood–Sobolev inequalities on compact Riemannian manifolds and applications. J. Differ. Equ. 260(1), 1–25 (2016)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. I. Math. Z. 27, 565–606 (1928)
Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1978)
Helgason, S.: Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI (1994)
Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics (1999)
John, F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr. Math. 28, 235–268 (1979)
Kato, T.: Blow-up of solutions of some nonlinear hyperbolic equations. Commun. Pure Appl. Math. 33, 501–505 (1980)
Kassymov, A., Ruzhansky, M., Suragan, D.: Hardy–Littlewood–Sobolev and Stein–Weiss inequalities on homogeneous Lie groups. Integr. Transform. Spec. Funct. 30(8), 643–655 (2019)
Kombe, I., Ozaydin, M.: Improved Hardy and Rellich inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 361, 6191–6203 (2009)
Kristaly, A.: Sharp uncertainty principles on Riemannian manifolds: the influence of curvature. J. Math. Pures Appl. 119, 326–346 (2018)
Kombe, I., Ozaydin, M.: Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 365, 5035–5050 (2013)
Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)
Metcalfe, J., Taylor, M.: Nonlinear waves on 3D hyperbolic space. Trans. Am. Math. Soc. 363, 3489–3529 (2011)
Metcalfe, J., Taylor, M.: Dispersive wave estimates on 3D hyperbolic space. Proc. Am. Math. Soc. 140, 3861–3866 (2012)
Muckenhoupt, B., Wheeden, R.L.: Weighted norm inequality for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)
Nguyen, V.H.: Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature. Proc. R. Soc. Edinb. Sect. A 152(1), 102–127 (2022)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 3(13), 115–162 (1959)
Palmieri, A.: On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups. J. Differ. Equ. 281, 85–104 (2021)
Perez, C.: Two weighted norm inequalities for Riesz potentials and uniform \(L^{p}\)-weighted Sobolev inequalities. Indiana Univ. Math. J. 39, 31–44 (1990)
Ruzhansky, M., Tokmagambetov, N.: Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups. J. Differ. Equ. 265(10), 5212–5236 (2018)
Ruzhansky, M., Verma, D.: Hardy inequalities on metric measure spaces. Proc. R. Soc. A. 475(2223), 20180310 (2019)
Ruzhansky, M., Yessirkegenov, N.: Hypoelliptic functional inequalities, to appear in Math. Z (2024). ar**v:1805.01064v1
Ruzhansky, M., Yessirkegenov, N.: Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg inequalities on general Lie groups, to appear in J. Geom. Anal. (2024). ar**v:1810.08845
Ruzhansky, M., Suragan, D.: Hardy inequalities on homogeneous groups. 100 years of Hardy inequalities. Progress in Mathematics, vol. 327. Birkhäuser/Springer, Cham, xvi+571 (2019)
Sire, Y., Sogge, C.D., Wang, C.: The Strauss conjecture on negatively curved backgrounds. Discret. Contin. Dyn. Syst. 39, 7081–7099 (2019)
Sire, Y., Sogge, C.D., Wang, C., Zhang, J.: Strichartz estimates and Strauss conjecture on non-trap** asymptotically hyperbolic manifolds. Trans. Am. Math. Soc. 373(11), 7639–7668 (2020)
Sobolev, S.L.: On a theorem of functional analysis, Mat. Sb. (N.S.), 4:471–479, 1938, English transl. in Amer. Math. Soc. Transl. Ser. 2, 34, 39–68 (1963)
Strauss, W.A.: Nonlinear scattering theory at low energy. J. Funct. Anal. 41, 110–133 (1981)
Stein, E.M., Weiss, G.: Fractional integrals on \(n\)-dimensional Euclidean space. J. Math. Mech. 7(4), 503–514 (1958)
Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)
Strömberg, J. O.: Weak type \(L^1\) estimates for maximal functions on noncompact symmetric spaces. Ann. Math. 115-126 (1981)
Tataru, D.: Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Trans. Am. Math. Soc. 353, 795–807 (2001)
Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110(1), 353–372 (1976)
Velicu, A., Yessirkegenov, N.: Rellich, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities for Dunkl operators and applications. Isr. J. Math. (2021). https://doi.org/10.1007/s11856-021-2261-7
Varopoulos, NTh.: Sobolev inequalities on Lie groups and symmetric spaces. J. Funct. Anal. 86(1), 19–40 (1989)
Yafaev, D.: Sharp constants in the Hardy–Rellich inequalities. J. Funct. Anal. 168, 121–144 (1999)
Zhang, H.-W.: Wave and Klein-Gordon equations on certain locally symmetric spaces. J. Geom. Anal. 30(4), 4386–4406 (2020)
Zhang, H.-W.: Wave equation on certain noncompact symmetric spaces. Pure Appl. Anal. 3, 363–386 (2021)
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The authors wish to thank the anonymous referee for his/her helpful comments and suggestions that helped to improve the quality of the paper. 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. AK, VK and MR are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). AK is supported by Science Committee of Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. BR20281002). VK and MR are supported by FWO Senior Research Grant G011522N. MR is also supported by EPSRC Grant EP/V005529/1.
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Kassymov, A., Kumar, V. & Ruzhansky, M. Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications. J Geom Anal 34, 208 (2024). https://doi.org/10.1007/s12220-024-01644-3
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DOI: https://doi.org/10.1007/s12220-024-01644-3
Keywords
- 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