The derivatives of the heat kernel on symmetric spaces

TL;DR

This paper derives estimates for the derivatives of the heat kernel on symmetric spaces and uses these to analyze the boundedness of related operators on locally symmetric spaces.

Contribution

It provides new derivative estimates of the heat kernel on symmetric spaces and applies them to study operator boundedness in a broad class of locally symmetric spaces.

Findings

Established derivative estimates for the heat kernel on symmetric spaces

Proved $L^{p}$-boundedness of Littlewood-Paley-Stein operators

Analyzed the Laplacian of the heat operator on locally symmetric spaces

Abstract

We derive estimates of the derivatives of the heat kernel on noncompact symmetric spaces and on locally symmetric spaces. Applying these estimates we study the $L^{p}$-boundedness of Littlewood-Paley-Stein operators and the Laplacian of the heat operator on a wide class of locally symmetric spaces.

Equations

$$\left|\frac{{\partial }^i{h}_t}{\partial t^i}(\exp H)\right|\le c{t}^{-(n/2)-i}{e}^{-(1-ϵ)\left({\Vert \rho \Vert }^{2}t+⟨\rho ,H⟩+{\Vert H\Vert }^2/(4t)\right)},$$

$$\Vert \nabla h_t(\exp H)\Vert \le c{t}^{-(n+1)/2}{e}^{-(1-ϵ)\left({\Vert \rho \Vert }^{2}t+⟨\rho ,H⟩+\Vert H{\Vert }^2/(4t)\right)},$$

$$P_s(x,y)=\sum _{\gamma \in \Gamma }{e}^{-sd(x,\gamma y)},x,y\in X,$$

$$\delta (\Gamma )=\inf \{s\in (0,\infty ):P_s(x,y)<\infty \},$$

$${\rho }_m=\underset{H\in \overline{{\mathfrak{a}}_{+}},\Vert H\Vert =1}{\min }⟨\rho ,H⟩.$$

$$\begin{array}{rl}\left|\frac{{\partial }^i{h}_t^M}{\partial t^i}(\tilde{x},\tilde{y})\right|& \le \frac{c}{t^{n/2+i}}{e}^{-(1-ϵ)\left((1-{\alpha }_1){\Vert \rho \Vert }^{2}t+({\alpha }_2-\delta (\Gamma ))d_M(\tilde{x},\tilde{y})+(1-{\alpha }_3)\frac{d_M^2(\tilde{x},\tilde{y})}{4t}\right)}\\ & \times P_{ϵ+\delta (\Gamma )}(x,y),\end{array}$$

$$s(p)=2\min \left\{\frac{1}{p},\frac{1}{p^{\prime }}\right\}.$$

$$\Vert \ast |\kappa |{\Vert }_{L^p(M)\to L^p(M)}\le c\underset{G}{\int }|\kappa (g)|{\varphi }_{-i{\eta }_{\Gamma }}(g{)}^{s(p)}dg,$$

$$H_{\sigma }(f)(x)={\left({\int }_0^{\infty }{e}^{2\sigma t}\left(t^{2i-1}{\left|\frac{{\partial }^i}{\partial t^i}{H}_{t}f(x)\right|}^2+\Vert {\nabla }_x{H}_{t}f(x){\Vert }^2\right)\right)}^{1/2}.$$

$$\sigma <s(p)\left(\Vert \rho \Vert -\Vert {\eta }_{\Gamma }\Vert \right)\left(2\Vert \rho \Vert -s(p)\left(\Vert \rho \Vert -\Vert {\eta }_{\Gamma }\Vert \right)\right).$$

$$\Vert {\Delta }_M{e}^{-{\Delta }_{M}t}{\Vert }_{L^p(M)\to L^p(M)}\le \{\begin{array}{l}c{t}^{-1},\text{if}0\le t<1,\\ c{e}^{-ϵt},\text{if}t\ge 1,\end{array}$$

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

$$G=K\exp \overline{{\mathfrak{a}}^{+}}K.$$

$$\rho :=\frac{1}{2}\sum _{\alpha \in {\sum }^{+}}{m}_{\alpha }\alpha$$

$$d(x,y)=d(x_0,\exp H{x}_0)=\Vert H\Vert .$$

$${\int }_{G}f\left(g\right)dg=c{\int }_{K}d{k}_1{\int }_{{\mathfrak{a}}^{+}}\delta \left(H\right)dH{\int }_{K}f\left(k_1\left(\exp H\right)k_2\right)d{k}_2,$$

$$\delta \left(H\right)\le e^{2⟨\rho ,H⟩},H\in \overline{{\mathfrak{a}}^{+}}.$$

$${\int }_{G}f\left(g\right)dg=c{\int }_{\overline{{\mathfrak{a}}^{+}}}f\left(\exp H\right)\delta (H)dH.$$

$$0<{\varphi }_{-i\lambda }(\exp H)\le c(1+\Vert H\Vert {)}^a{e}^{\lambda (H)-\rho (H)},H\in \overline{{\mathfrak{a}}^{+}},\lambda \in \overline{{\mathfrak{a}}_{+}^{\ast }},$$

$$d_M(\tilde{x},\tilde{y})=\underset{\gamma \in \Gamma }{\min }d(x,\gamma y).$$

$$\begin{array}{rl}{h}_t(\exp H)& \asymp t^{-n/2}\left(\prod _{\alpha \in {\Sigma }_0^{+}}(1+⟨\alpha ,H⟩)(1+t+⟨\alpha ,H⟩{)}^{\frac{m_{\alpha }+m_{2\alpha }}{2}-1}\right)\\ & \times e^{-{\Vert \rho \Vert }^{2}t-⟨\rho ,H⟩-{\Vert H\Vert }^2/(4t)},\end{array}$$

$$m=\sum _{a\in \Sigma {}_0{}^{+}}\left(\frac{m{}_{\alpha }+m{}_{2\alpha }}{2}-1\right)\text{ and }A=\sum _{a\in {\Sigma }_0^{+}}\frac{m_{\alpha }+m_{2\alpha }}{2}.$$

$$h_t(\exp H)\le c{t}^{-n/2}(1+t{)}^m(1+\Vert H\Vert {)}^A{e}^{-\left({\Vert \rho \Vert }^{2}t+⟨\rho ,H⟩+\Vert H\Vert {}^2/(4t)\right)},$$

$$h_t(x,x)\le \frac{1}{f(t)},\text{ for all }t>0\text{ and }x\in X,$$

$$\left|\frac{{\partial }^i{h}_t}{\partial t^i}\right|(x,y)\le \frac{1}{\sqrt{f(t)f_{2i}(t)}},\text{ for all }i\in \mathbb{N},t>0,x,y\in X,$$

$$f_0(t)=f(t)\text{ and }{f}_i(t)={\int }_0^t{f}_{i-1}(s)ds,i\ge 1.$$

$$h_t(\exp H)\le t^{-n/2}(1+t{)}^m.$$

$$f(t)=t^{n/2}(1+t{)}^{-m}.$$

$$f_i(t)\ge t^{(n/2)+i}(1+t{)}^{-m}.$$

$$\left|\frac{{\partial }^i{h}_t}{\partial t^i}(\exp H)\right|\le c{t}^{-(n/2)-i}(1+t{)}^m,\text{ for all }t>0,H\in \overline{{\mathfrak{a}}^{+}},$$

Full text

1. Introduction and statement of the results

Our main objective in this article is to prove estimates of the derivatives of the heat kernel on noncompact symmetric spaces. We then obtain a variety of applications.

More than the results themselves, it is the method in the proof of our main result that is nontrivial. More specifically, we are able to estimate time derivatives of the heat kernel by combining sharp heat kernel estimates with rough estimates of its time derivatives, and by improving step by step the resulting estimates using an iterative argument. The final estimates obtained this way are precise.

A symmetric space is a homogeneous space that can be described as a coset Riemannian manifold $X=G/K,$ where $G$ is a semisimple Lie group and $K$ is a maximal compact subgroup. From now on, $X$ will denote an $n$-dimensional symmetric space.

Let $\mathfrak{g}$ and $\mathfrak{k}$ be the Lie algebras of $G$ and $K$ respectively. Let also $\mathfrak{p}$ be the subspace of $\mathfrak{g}$ which is orthogonal to $\mathfrak{k}$ with respect to the Killing form. Let $\mathfrak{a}$ be an abelian maximal subspace of $\mathfrak{p}$, $\mathfrak{a}^{\ast}$ its dual and let $\Sigma\subset\mathfrak{a}^{\ast}$ be the root system of ($\mathfrak{g}$,$\mathfrak{a}$). Choose a set $\Sigma^{+}$ of positives roots. Let $\rho$ be the half-sum of positive roots counted with multiplicity. Let $\mathfrak{a}^{+}\subset\mathfrak{a}$ be the corresponding positive Weyl chamber and let $\overline{\mathfrak{a}_{+}}$ be its closure. We have the Cartan decomposition $G=K(\exp\overline{\mathfrak{a}_{+}})K$. If $x\in G$, then it is uniquely written as $x=k_{1}\left(\exp H\right)k_{2},$ with $k_{1},k_{2}\in K$ and $H\in\overline{\mathfrak{a}_{+}}$.

Let $\Delta$ be the Laplace-Beltrami operator on $X$. Then, the heat kernel $h_{t}$ of $X$ is the fundamental solution of the heat equation $\partial_{t}h_{t}=\Delta h_{t}$. Note that the heat kernel is a $K$-bi-invariant function, i.e., if $x=k_{1}\left(\exp H\right)k_{2}\in X$, then $h_{t}(x)=h_{t}(\exp H)$.

Our main result is the following theorem.

Theorem 1**.**

If $X$ is a symmetric space of noncompact type, then for all $\epsilon\in(0,1)$ and $i\in\mathbb{N}$ there is a constant $c>0$ such that

[TABLE]

$\text{ for all }t>0\text{ and }H\in\overline{\mathfrak{a}_{+}}.$**

There is a very rich and long literature concerning heat kernel estimates in various geometric contexts. See for example [2, 3, 5, 8], and the references therein. In particular, Davies and Mandouvalos in [5], obtained optimal estimates of the heat kernel in hyperbolic spaces and Anker and Ji in [2] and Anker and Ostellari in [3], obtained estimates of the heat kernel in the case of symmetric spaces. Estimates of the time derivatives of the heat kernel have been obtained in [12] for hyperbolic spaces, and in [4] and [6] on general geometric contexts.

Next, we shall find applications of Theorem 1. Firstly, we obtain gradient estimates of $h_{t}.$

Corollary 2**.**

If $X$ is a symmetric space of noncompact type, then for all $\epsilon\in(0,1)$ there is a constant $c>0,$ such that

[TABLE]

$\text{ for all }t>0\text{ and }H\in\overline{\mathfrak{a}_{+}}.$**

Let $\Gamma$ be a discrete torsion free subgroup of $G$. Then the locally symmetric space $M=\Gamma\backslash X$, equipped with the projection of the canonical Riemannian structure of $X$, becomes a Riemannian manifold. Denote by $\Delta_{M}$ the Laplacian on $M$ and by ${d_{M}}$ the Riemannian distance on $M$. We denote by $\pi:X\longrightarrow M$ the canonical projection and for $x\in X$ we set $\tilde{x}=\pi\left(x\right)$. Denote by

[TABLE]

and

[TABLE]

the Poincaré series and the critical exponent respectively.

Set also

[TABLE]

Let $\delta(\Gamma)<\left\|\rho\right\|+\rho_{m}.$ Consider $\alpha_{2}\in(\delta(\Gamma),\left\|\rho\right\|+\rho_{m})$ and $\alpha_{1},\alpha_{3}\in[0,1]$ such that $\alpha_{1}\alpha_{3}\in\left[\left(\left(\alpha_{2}-\rho_{m}\right)/\left\|\rho\right\|\right)^{2},1\right]$.

Next, we obtain estimates of the heat kernel on a locally symmetric space.

Theorem 3**.**

Suppose that $M=\Gamma\backslash X,$ with $\delta(\Gamma)<\left\|\rho\right\|+\rho_{m}$. Then, for all $\epsilon\in(0,1)$ there is a constant $c>0,$ such that

[TABLE]

for all $t>0$ and $\tilde{x},\tilde{y}\in M$.

Observe that the above result extends the results of Weber in [13].

Let $\kappa$ be a $K$-bi-invariant function and let $\ast|\kappa|$ denote the convolution operator whose kernel is $|\kappa|$. Let also $p\in[1,\infty]$, denote by $p^{\prime}$ its conjugate and set

[TABLE]

Assume that the following version of the Kunze-Stein phenomenon holds,

[TABLE]

where $\phi_{\lambda}$ are the elementary spherical functions, $\eta_{\Gamma}$ is a vector of the euclidean sphere $S(0,(\left\|\rho\right\|^{2}-\lambda_{0})^{1/2})$ of $\mathfrak{a}^{\ast}$ and $\lambda_{0}$ is the bottom of the spectrum of the Laplacian $-\Delta_{M}$. For example, this is the case for $M=\Gamma\backslash G/K$, when (i) $\Gamma$ is a lattice, or (ii) $G$ possesses Kazhdan’s property (T) or (iii) $\Gamma\backslash G$ is non-amenable, see [11] for more details.

Denote by $H_{t}=e^{t\Delta_{M}}$ the heat semigroup on $M$. Fix $i\in\mathbb{N}$. Then, for all $\sigma\geq 0$ we consider as in [1] the Littlewood-Paley-Stein operator

[TABLE]

Next, we apply Corollary 2 and Theorem 3 in order to prove $L^{p}$-boundedness of various operators related to the heat semigroup $H_{t}=e^{t\Delta_{M}}$, on certain locally symmetric spaces $M$.

Theorem 4**.**

Suppose that $M$ satisfies (KS). Then, the operator ${H}_{\sigma}$ is bounded on $L^{p}(M),\;p\in(1,\infty),$ provided that

[TABLE]

The Littlewood-Paley-Stein operator was first introduced and studied by Lohoué [9], in the case of Riemannian manifolds of non-positive curvature. In a variety of geometric settings it has been proved that ${H}_{\sigma}$ is bounded on $L^{p},\;p\in(1,\infty),$ under some conditions on $\sigma$ (see for example [9]). In particular, in the case of a symmetric space $X$, Anker in [1] has shown that ${H}_{\sigma}$ is bounded in $L^{p}(X)$, provided $\sigma<4\left\|\rho\right\|^{2}/pp^{\prime}$. In the case of symmetric spaces, where $\eta_{\Gamma}=0,$ [11], we observe that the condition (4) on $\sigma$ becomes $\sigma<\left\|\rho\right\|^{2}s(p)(2-s(p))=4\left\|\rho\right\|^{2}/pp^{\prime},$ thus Theorem 4 extends the result of Anker in [1].

Finally, we obtain $L^{p}$-estimates for the operator $\Delta_{M}e^{t\Delta_{M}}$.

Theorem 5**.**

Suppose that $M$ satisfies (KS). Then, for all $p\in[1,\infty]$ there exists $\epsilon>0$ such that

[TABLE]

This operator has been studied for complete Riemannian manifolds with bounded geometry by Davies in [4]. In [4] the $L^{p}$-norm of the operator $\Delta_{M}e^{-t\Delta_{M}}$ is proved to be bounded by a constant for $t\geq 1.$ In our case, we prove that it decays exponentially as $t\rightarrow\infty,$ thus we extend the result of Davies.

Let us now outline the organization of the paper. In Section 2 we recall some basic definitions about symmetric spaces and the heat kernel. In addition, we recall some results providing estimates of the heat kernel and estimates of its derivatives. In Section 3 we prove Theorem 1. Next, in Section 4, we obtain a variety of applications. Firstly, as a direct application we prove gradient estimates of the heat kernel. Next, we prove estimates of the derivatives of the heat kernel for locally symmetric spaces. Finally, we study Littlewood-Paley-Stein operators and the Laplacian of the heat operator. We prove that they are bounded on $L^{p}(M),$ where $M$ is a locally symmetric space satisfying (KS).

Throughout this article the different constants will always be denoted by the same letter $c$. When their dependence or independence is significant, it will be clearly stated.

2. Preliminaries

2.1. Symmetric and locally symmetric spaces

We shall recall some basic facts on symmetric and locally symmetric spaces. For details, see [1, 11].

As it is already mentioned in the Introduction, a symmetric space $X$ is the Riemannian manifold $G/K$ where $G$ is a real semisimple Lie group, connected, noncompact, with finite center and $K$ is a maximal compact subgroup of $G$.

Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mathfrak{k}$ the Lie algebra of $K$ respectively. Denote by $\mathfrak{p}$ the orthogonal complement of $\mathfrak{k},$ then $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p},$ is the Cartan decomposition at the Lie algebra level. Let us choose a maximal abelian subspace $\mathfrak{a}$ of $\mathfrak{p}$. Let $\mathfrak{a}^{\ast}$ be the dual space of $\mathfrak{a}$. For any $\alpha\in\mathfrak{a}^{\ast}$, let $ad(X)(Y)=[X,Y]$ for all $X,Y\in\mathfrak{g}$ and set

[TABLE]

If $\mathfrak{g}_{\alpha}\neq\{0\},$ then $\alpha\in\mathfrak{a}^{\ast}\backslash\{0\}$ is called a root of the pair $(\mathfrak{g},\mathfrak{a})$ and $\mathfrak{g}_{\alpha}$ is called the root space. Denote by $\Sigma$ the set of all roots. If $\alpha$ is a root, then the only multiples of $\alpha$ that can also be roots are $\pm\frac{1}{2}\alpha,\ \pm 2\alpha$, and $-\alpha.$ A positive root $\alpha$ is called indivisible if $\frac{1}{2}\alpha$ is not a root. We call an $H\in\mathfrak{a}$ regular if $\alpha(H)\neq 0$ for all $\alpha\in\Sigma$. The set of regular elements of $\mathfrak{a}^{\ast}$ is the complement of a union of finitely many hyperplanes and the connected components are called Weyl chambers. Let us fix a Weyl chamber $\mathfrak{a}^{+}$. With respect to this Weyl chamber a root $\alpha$ is said to be positive if $\alpha(H)>0$ for all $H\in\mathfrak{a}^{+}$. We denote by $\Sigma^{+}$ the set of positive roots and by $\Sigma^{+}_{0}$ the set of indivisible positive roots. If $\overline{\mathfrak{a}^{+}}$ is the closure of $\mathfrak{a}^{+}$ then we denote by $\mathfrak{a}^{*}_{+}$ and $\overline{\mathfrak{a}^{*}_{+}}$ the cones corresponding to $\mathfrak{a}^{+}$ and $\overline{\mathfrak{a}^{+}}$ in $\mathfrak{a}^{\ast}$ (see [13] for more details).

We have the Cartan decomposition of $G$:

[TABLE]

Define $m_{\alpha}:=\dim\mathfrak{g}_{\alpha}$ to be the multiplicity of a root $\alpha$ and let

[TABLE]

be half the sum of the positive roots counted according to their multiplicity.

Let $x,y\in X$ and consider a base point $x_{0}\in X.$ Then, there are $g,h\in G$ such that $x=g(x_{0})$ and $y=h(x_{0}).$ Because of the Cartan decomposition, there are $k_{1},k_{2}\in K,$ $H\in\overline{{a}^{+}}$ such that $g^{-1}h=k_{1}\exp{H}k_{2}.$ Then, the distance $d(x,y)$ of $x,y\in X$ is given by

[TABLE]

Recall that by the Cartan decomposition, the Haar measure on $G$ is written as

[TABLE]

where, $dk$ is the normalised Haar measure of $K$ and the modular function $\delta\left(H\right)$ satisfies:

[TABLE]

From (8) it follows that if $f$ is $K$-bi-invariant, then

[TABLE]

Recall that there are positive constants $c$ and $\alpha$ such that

[TABLE]

see [11] for more details.

If $\Gamma\subset G$ is a discrete, torsion free subgroup of isometries of $X$, then the quotient space $M=\Gamma\backslash X$ equipped with the projection of the Riemannian metric of $X$ is a Riemannian manifold and is called locally symmetric space. If $\pi:X\rightarrow M$ is the canonical projection, we write $\tilde{x}=\pi(x)$. The distance $d_{M}$ on $M$ is given by

[TABLE]

2.2. The heat kernel on symmetric and locally symmetric spaces

Denote by $h_{t}$ the heat kernel on the symmetric space $X$. The heat kernel on symmetric spaces has been extensively studied [2, 3]. Sharp estimates of the heat kernel have been real hyperbolic space have been obtained in [5] while in [2] Anker and Ji and Anker and Ostellari in [3], generalized results of [5] to all symmetric spaces of noncompact type. They proved the following sharp estimate

[TABLE]

for all $H\in\overline{\mathfrak{a}^{+}}$ and all $t>0.$ Recall that we write $f\asymp h$ for functions $f$ and $h$ if there is a positive constant $c>0$ such that $\frac{1}{c}h\leq f\leq ch.$

Set

[TABLE]

Note that (12) implies the following estimate

[TABLE]

for all $H\in\overline{\mathfrak{a}^{+}}$ and all $t>0.$

In [5], Davies and Mandouvalos obtained heat kernel estimates on quotients of the hyperbolic spaces, and in [13] Weber generalized these results on locally symmetric spaces.

Estimates of the time derivatives of the heat kernel are obtained by Mandouvalos and Tselepidis in [12] for the case of real hyperbolic spaces. In [6], Grigory’an derived Gaussian upper bounds for all time derivatives of the heat kernel, under some assumptions on the on-diagonal upper bound for $h_{t}$ on an arbitrary complete non-compact Riemannian manifold $X$. More precisely, it is proved that if there exists an increasing continuous function $f(t)>0,$ $t>0,$ such that

[TABLE]

then,

[TABLE]

where the sequence of functions $f_{i}=f_{i}(t),$ is defined by

[TABLE]

We shall now apply the estimate (15) of Grigory’an in the case of symmetric spaces of noncompact type.

We note that according to (19),

[TABLE]

Let $\Sigma^{+}$ be the set of positive roots and $l$ be the rank of $X$. Then, it holds $n=l+\sum\limits_{\alpha\in\Sigma^{+}}m_{\alpha}$, [3]. It follows from (13) that $n/2>m$. Then,

[TABLE]

is an increasing function, thus we can invoke (15).

By an induction argument we get that

[TABLE]

Then, we get the following result.

Lemma 6**.**

Suppose that $X$ is a symmetric space of noncompact type. For all $i\in\mathbb{N}$ there is a constant $c>0$ such that

[TABLE]

where $m$ is defined by (13).

3. Estimates of the time derivatives of the heat kernel on symmetric spaces

Let $X$ be a noncompact symmetric space. Recall the heat kernel estimate

[TABLE]

for all $t>0$ and $H\in\overline{\mathfrak{a}^{+}}$, where $m$ and $A$ are defined in (13).

In this section we shall prove the main result, stated in Theorem 1. For the proof of (1) we need several lemmata. The following lemma is technical but important for the proof of Theorem 1. Roughly speaking, according to the following result, an estimate for a function and its second derivative implies an estimate for its first derivative.

Lemma 7**.**

Let

[TABLE]

and assume that for fixed $H\in\overline{\mathfrak{a}^{+}}$ the function $f_{H}:(0,+\infty)\rightarrow\mathbb{R}$, satisfies

[TABLE]

and

[TABLE]

Then, for all $\epsilon\in(0,1)$, there is a constant $c=c(\epsilon)>0$, such that for all $H\in\overline{\mathfrak{a}^{+}},$

[TABLE]

where $\lambda_{\epsilon}=\frac{1-\epsilon}{1+\epsilon}.$

Proof.

Firstly, for all $\delta>0$, the mean value theorem yields for some $\theta\in(t,t+\delta)$:

[TABLE]

Applying once again the mean value theorem, now on the function $\frac{df_{H}}{dt}$ on $[t,\theta]$ we have

[TABLE]

It follows that

[TABLE]

Note that $t\longrightarrow t^{-\alpha}(1+t)^{\beta}$ is a decreasing function of $t$, since $\alpha>\beta,$ therefore

[TABLE]

Note also that $\langle\rho,H\rangle\geq\rho_{m}\|H\|>0$, for $H\neq 0$. Thus (23) and the estimates (21) and (22) imply that

[TABLE]

Choose now

[TABLE]

Thus,

[TABLE]

From (20), it follows that $\delta\leq\epsilon t.$ Thus

[TABLE]

Consequently,

[TABLE]

and similarly

[TABLE]

Thus, from (24), (25) and (26) it follows that

[TABLE]

where $c(\epsilon)=c(1/\epsilon+\epsilon)$, with $c>0$ constant. ∎

Next, we use an inductive argument. More precisely, we apply Lemma 7 and we are improving step by step the resulting estimates by using an iterative argument.

Lemma 8**.**

Suppose that $X$ is a symmetric space of noncompact type. Let us fix $\epsilon\in(0,1)$ and set $\lambda_{\epsilon}=\frac{1-\epsilon}{1+\epsilon}$. Then, for all $i,\ell\in\mathbb{N},$ there are constants $c,\beta_{\ell}^{i}$ and $\gamma_{\ell}^{i}>0$ such that

[TABLE]

for all $t>0$ and $H\in\overline{\mathfrak{a}_{+}}$, where $m,A$ are defined in (13) and $c$ is a constant that depends on $\epsilon,i,\ell$. Furthermore, the sequences $\beta_{\ell}^{i},\gamma_{\ell}^{i}$ satisfy the iteration formulas

[TABLE]

the inequalities $\beta_{\ell}^{i-1}\geq\beta_{\ell}^{i+1}$, $\gamma_{\ell}^{i-1}\geq\gamma_{\ell}^{i+1}$ and the initial conditions

[TABLE]

Proof.

For every $\ell\in\mathbb{N}$, consider the following statement $L(\ell)$: for all $i\in\mathbb{N},$ $\frac{\partial^{i}h_{t}}{\partial t^{i}}(\exp H)$ satisfies the estimate (27) and the constants $\beta_{\ell}^{i},\gamma_{\ell}^{i}$ appearing in (27) satisfy the iteration formulas (28) with initial conditions (29), and $\beta_{\ell}^{i-1}\geq\beta_{\ell}^{i+1}$, $\gamma_{\ell}^{i-1}\geq\gamma_{\ell}^{i+1}$. We shall then prove by induction, that $L(\ell)$ holds for every $\ell\in\mathbb{N}$.

For $\ell=0$ we have to prove that for all $i\in\mathbb{N},$ $\frac{\partial^{i}h_{t}}{\partial t^{i}}(\exp H)$ satisfies the estimate (27) and that the constants $\beta_{0}^{i},\gamma_{0}^{i}$ satisfy $\beta_{0}^{i}=\gamma_{0}^{i}=0,\text{ for all }i\geq 1,\text{ and }\beta_{0}^{0}=\gamma_{0}^{0}=1.$ Also, we need to show that $\beta_{0}^{i-1}\geq\beta_{0}^{i+1}$, $\gamma_{0}^{i-1}\geq\gamma_{0}^{i+1}$.

Indeed, from Lemma 6 we get that for $i\geq 1$

[TABLE]

But, $1\leq(1+\left\|H\right\|)^{A}.$ Thus

[TABLE]

i.e. (27) holds true for all $i\geq 1$, with $\beta_{0}^{i}=\gamma_{0}^{i}=0.$ Furthermore, from the estimates of the heat kernel in (19), we obtain that

[TABLE]

Thus (27) holds true also for $i=0$ and $\beta_{0}^{0}=\gamma_{0}^{0}=1.$ Last, from Lemma 6, for $i\geq 1$, we have $\beta_{0}^{i-1}\geq 0=\beta_{0}^{i+1}$, $\gamma_{0}^{i-1}\geq 0=\gamma_{0}^{i+1}$. Therefore the statement $L(0)$ holds true.

Let us assume now that $L(\ell-1)$ holds true. Thus, for all $i\in\mathbb{N}$, there are constants $c,\;\beta_{\ell-1}^{i},\;\gamma_{\ell-1}^{i}>0$ such that $\frac{\partial^{i}h_{t}}{\partial t^{i}}(\exp H)$ satisfies the estimate (27). In addition, $\beta_{\ell-1}^{i-1}\geq\beta_{\ell-1}^{i+1}$, $\gamma_{\ell-1}^{i-1}\geq\gamma_{\ell-1}^{i+1}.$

We shall prove that $L(\ell)$ holds true. Indeed, from the estimates of the heat kernel in (19), we have that

[TABLE]

Thus (27) holds true for $i=0$ with $\beta_{\ell}^{0}=\gamma_{\ell}^{0}=1.$

For $i\geq 1$, consider the function

[TABLE]

From the validity of $L\left(\ell-1\right)$, we get that for $i-1$ and $i+1$ we have that

[TABLE]

with $\alpha=(n/2)+i-1,$ $\beta=m,$ $\gamma=A,$ $D=B=\beta_{\ell-1}^{i-1},$ $C=\gamma_{\ell-1}^{i-1}$ and $D_{\ast}=B_{\ast}=\beta_{\ell-1}^{i+1},$ $C_{\ast}=\gamma_{\ell-1}^{i+1}$. Note that $B\geq B_{\ast}$ and $C\geq C_{\ast}$, from $L(\ell-1)$.

Thus, by Lemma 7, applied for the function $f_{H}(t)$, it follows that

[TABLE]

for all $i\geq 1,$ where $\beta_{\ell}^{i}$ and $\gamma_{\ell}^{i}$ satisfy (28). Finally, from $L(\ell-1)$ and (28) it is straightforward that $\beta_{\ell}^{i-1}\geq\beta_{\ell}^{i+1}$ and $\gamma_{\ell}^{i-1}\geq\gamma_{\ell}^{i+1}$ . Thus, the statement $L(\ell)$ is valid and the proof of the lemma is complete. ∎

Remark 1**.**

The constant $c=c(i,\ell,\epsilon)$ in relation (27) of Lemma 8 depends on $i,\ell$ and $\epsilon$ and it increases to infinity (when either $i\rightarrow\infty$ or $\ell\rightarrow\infty$ or $\epsilon\rightarrow 0$), but we only need the fact that it is finite for fixed $i,\ell,\epsilon$.

Finally, we shall show that the estimates obtained are precise, by proving that if $\ell\rightarrow\infty$ and $\epsilon\rightarrow 0$ then the exponents $\gamma_{\ell}^{i}$ and $\beta_{\ell}^{i}$ converge to 1. More precisely, we shall prove the following result.

Lemma 9**.**

For any $i\in\mathbb{N},$

[TABLE]

Proof.

We shall deal only with $\gamma_{\ell}^{i}$. The proof that $\lim_{\ell\rightarrow\infty}\beta_{\ell}^{i}=1$ is similar, and we shall omit it.

Claim 1. For every $\ell\in\mathbb{N}$ consider the following statement $L(\ell)$: for all $i\in\mathbb{N},$

[TABLE]

We shall prove by induction that $L(\ell)$ is valid for all $\ell\in\mathbb{N}$.

For $\ell=0$ we have to prove that for all $i\in\mathbb{N},$ we have that $\gamma_{0}^{i}\leq 1.$ Indeed, this is a consequence of the initial conditions $\gamma_{0}^{i}=0$ and $\gamma_{0}^{0}=1.$ Thus $L(0)$ holds true.

Let us assume now that $L(\ell-1)$ holds true. Thus, for all $i\in\mathbb{N},$ we have that $\gamma_{\ell-1}^{i}\leq 1.$

We shall prove that $L(\ell)$ holds true. Recall that by the induction assumption, for all $i\in\mathbb{N}$, for $i-1$ and $i+1$ we have that $\gamma_{\ell-1}^{i-1}\leq 1$ and $\gamma_{\ell-1}^{i+1}\leq 1$. Thus, from (28) it follows that

[TABLE]

thus the statement $L(\ell)$ is valid and this completes the proof of Claim 1.

Claim 2. For every $\ell\in\mathbb{N}$ consider the following statement $L(\ell)$: for all $i\in\mathbb{N},$

[TABLE]

We shall prove that $L(\ell)$ is valid for all $\ell\in\mathbb{N}$. We proceed once again by induction in $\ell\in\mathbb{N}.$

For $\ell=0$ we have to prove that for all $i\in\mathbb{N},$ $\gamma_{0}^{i}=0\leq\gamma_{1}^{i}.$ Indeed, from (29) it follows that $\gamma_{0}^{i}=0\leq\gamma_{1}^{i},$ for all $i>0$ and $\gamma_{0}^{0}=1=\gamma_{1}^{0}.$ Therefore the statement $L(0)$ holds true.

Let us assume now that $L(\ell-1)$ holds true, i.e. that for all $i\in\mathbb{N},$ $\gamma_{\ell-1}^{i}\leq\gamma_{\ell}^{i}.$

We shall prove that $L(\ell)$ holds true, i.e. that for all $i\in\mathbb{N},$ $\gamma_{\ell}^{i}\leq\gamma_{\ell+1}^{i}.$ Recall that by (28) we have that

[TABLE]

Then by the induction assumption for $i-1$ and $i+1$ we have that $\gamma_{\ell-1}^{i-1}\leq\gamma_{\ell}^{i-1}$ and $\gamma_{\ell-1}^{i+1}\leq\gamma_{\ell}^{i+1}.$ Hence, from (34) we get that

[TABLE]

Thus the statement $L(\ell)$ is valid and the proof of Claim 2 is complete.

Claim 3. For all $i\in\mathbb{N},$

[TABLE]

Note that by Claim 2, the sequence $\gamma_{\ell}^{i}$ is increasing in $\ell$ and by Claim 1, $\gamma_{\ell}^{i}$ is bounded above. Thus $\lim_{\ell\rightarrow\infty}\gamma_{\ell}^{i}$ exists and since $0\leq\gamma_{\ell}^{i}\leq 1,$ then

[TABLE]

Note that $\gamma_{\ell}^{0}=1$, for all $\ell\in\mathbb{N},$ thus $\gamma_{0}=1.$

Now, taking limits in the iteration formula (34) we obtain that

[TABLE]

thus

[TABLE]

This is a homogeneous linear recurrence relation with constant coefficients and the solutions of this equation are given by

[TABLE]

where $r_{1},r_{2}$ are the roots of the equation

[TABLE]

Thus, we conclude that

[TABLE]

for some $C_{1},C_{2}\in\mathbb{R}.$

Since $0\leq\gamma_{i}\leq 1$, we get $C_{2}=0,$ otherwise $\lim_{i\rightarrow\infty}\gamma_{i}=\infty$. Also, since $\gamma_{0}=1$, we get $C_{1}=1.$ Thus, from (36) for $C_{1}=1,$ $C_{2}=0,$ we get (35) and the proof is complete. ∎

End of the proof of Theorem 1: To complete the proof of Theorem 1, notice that $\lim_{\epsilon\rightarrow 0}\left(1-\sqrt{1-\lambda{}_{\epsilon}}\right)^{i}=1$. Thus, taking $\ell\in\mathbb{N}$ sufficiently large and $\epsilon$ sufficiently close to zero, one has $\gamma_{\ell}^{i}\geq 1-\epsilon$ and $\beta_{\ell}^{i}\geq 1-\epsilon.$ Thus, from (27) and (31) it follows that

[TABLE]

Taking now into account that if $\alpha,\beta>0$, then there exists a constant $c=c(\alpha,\beta)$ such that $x^{\alpha}\leq ce^{\beta x}$ for all $x>0,$ we conclude that for every $\epsilon>0,$ there exists a constant $c>0$ such that

[TABLE]

and the proof of Theorem 1 is complete.

Remark 2**.**

If $X$ is a Cartan-Hadamard manifold then, [7], the heat kernel $h_{t}$ of $X$ satisfies pointwise bounds of the type

[TABLE]

for some positive constants $c,A,B,C$ and $\alpha$. Proceeding as in Section 3, one can prove the following estimate: for all $\epsilon\in(0,1)$ and $i\in\mathbb{N},$ there is a constant $c>0$ such that

[TABLE]

4. Applications

4.1. Estimates of the gradient of the heat kernel on symmetric spaces

As a direct application of Theorem 1, we obtain the gradient estimates of $h_{t}$ given in Corollary 2.

Proof of Corollary 2..

Let us recall that if $X$ is a complete, non-compact, $n$-dimensional Riemannian manifold, with Ricci curvature bounded from below by $-R^{2}$, then by [8] for $\gamma>1$, we have that

[TABLE]

for all $t>0,\;x,y\in X.$

Using the heat kernel estimate (19) as well as Theorem 1, and inequality (37), the result follows. ∎

4.2. Estimates of the time derivatives of the heat kernel on locally symmetric spaces

In this section we obtain estimates of the heat kernel time derivatives in the case of a locally symmetric space $M=\Gamma\backslash X$, and prove Theorem 3. Our results extend the estimates of Weber (see [13]).

Recall that

[TABLE]

Suppose that $\delta(\Gamma)<\left\|\rho\right\|+\rho_{m}$.

The heat kernel $h_{t}^{M}(\tilde{x},\tilde{y})$ on $M$ is given by the formula

[TABLE]

[13].

Recall also that

[TABLE]

Consider $\alpha_{2}\in(\delta(\Gamma),\left\|\rho\right\|+\rho_{m})$ and $\alpha_{1},\alpha_{3}\in[0,1]$ such that

[TABLE]

We shall now prove Theorem 3.

Proof of Theorem 3..

According to Theorem 1,

[TABLE]

Thus, taking into account that $d(x,y)=\left\|H\right\|,$ as well as that $\langle\rho,H\rangle\geq\rho_{m}\|H\|$, we obtain the estimate

[TABLE]

Note that

[TABLE]

since $\alpha_{1}\alpha_{3}\in\left[\left(\frac{\alpha_{2}-\rho_{m}}{\left\|\rho\right\|}\right)^{2},1\right],$ with equality when $\alpha_{2}=\rho_{m}>\delta(\Gamma)$ and $\alpha_{1}=\alpha_{3}=0$.

Thus, from (39) and (40) it follows that

[TABLE]

A summation argument implies that for every $\epsilon\in(0,1)$ we have

[TABLE]

and the proof of Theorem 3 is complete. ∎

Remark 3**.**

Recall that the kernel $p_{t}^{M}$ of the Poisson semigroup $e^{-t\sqrt{-\Delta}}$ on a Riemannian manifold $M$ is given by the following subordination formula:

[TABLE]

[2, p.1075]**, where $h_{u}^{M}(\tilde{x},\tilde{y})$ is the heat kernel of $M$. Using the estimates of Theorem 3, we obtain the following pointwise estimates for the Poisson kernel

[TABLE]

for all $\tilde{x},\tilde{y}\in M,$ $t>0.$ Similarly, one can prove that the kernel

[TABLE]

satisfies

[TABLE]

for all $\tilde{x},\tilde{y}\in M$, if $d_{M}(\tilde{x},\tilde{y})>1$, and

[TABLE]

for all $\tilde{x},\tilde{y}\in M$, if $d_{M}(\tilde{x},\tilde{y})\leq 1.$

4.3. Functions of the Laplacian

In this section we apply the estimates of the derivatives of the heat kernel and we obtain the $L^{p}$-boundedness of some operators related to the heat semigroup.

4.3.1. Proof of Theorem 4

We shall consider separately the small time operator

[TABLE]

and the large time operator

[TABLE]

As noted in [1, p.276], the whole problem comes from the component $H_{\sigma}^{\infty}.$

Note that if $f\in C_{0}^{\infty}(M)$, then $H_{t}f=H_{t}^{X}f$, where $H_{t}^{X}$ is the heat semigroup on $X$, see for example [11, Lemma 5].

Let

[TABLE]

Then, the component $H_{\sigma}^{\infty}$ can be handled by estimating

[TABLE]

and applying the Kunze-Stein phenomenon.

To be more precise, Theorem 1 and the gradient estimates of Corollary 2 imply the following upper bound.

Lemma 10**.**

For all $\epsilon\in(0,1)$, there exists $c>0$ such that

[TABLE]

for all $H\in\overline{\mathfrak{a}^{+}}.$

According to Kunze-Stein phenomenon (KS) for locally symmetric spaces, it holds

[TABLE]

Thus, to prove Theorem 4, it is enough to show that the integral in (45) converges.

Using the estimates of ${k}_{\sigma}^{\infty}$ obtained in Lemma 10 and (10), we get that

[TABLE]

The integral above converges provided that

[TABLE]

Choosing $\epsilon$ small enough, it follows from (47) that the integral in (46) converges when

[TABLE]

Thus, $H_{\sigma}^{\infty}$ is bounded on $L^{p}(M),$ $p\in(1,\infty),$ if (48) holds true.

Next, it is left to show that the component $H_{\sigma}^{0}$ is bounded on $L^{p}(M),$ $p\in(1,\infty).$ We split the operator $H_{\sigma}^{0}$ into two parts using a smooth cut-off function $\psi\in C_{c}^{\infty}(K\backslash G/K)$, with $\psi\equiv 1$ near the origin and $\psi\equiv 0$ in $B\left(0,2\right)^{c}$. Then, let ${H}_{\sigma}^{0,0}$ and ${H}_{\sigma}^{0,\infty}$ be the part of the operator associated to $\ast(\psi h_{t})$ and $\ast(1-\psi)h_{t}$, respectively.

We observe that the operator $H_{\sigma}^{0,\infty}$ can be handled like $H_{\sigma}^{\infty}$, and the term ${H}_{\sigma}^{0,0}$ can be handled as in the Euclidean case (see for example [1, p.278]). In particular, Anker [1, p. 278] proves that $H_{\sigma}^{0,0}$ is bounded on $L^{p}(X)$ for all $p\in(1,\infty)$, by controlling $H_{\sigma}^{0,0}$ by a convolution operator that fits in singular integral theory. The same arguments give the continuity of $H_{\sigma}^{0,0}$ on $L^{p}(G).$ Proceeding as in [10, Proposition 13], we obtain the $L^{p}$ boundedness of $H_{\sigma}^{0,0}$ on $M$.

Remark 4**.**

Consider the Poisson operator $P_{t}=e^{-t(-\Delta)^{1/2}},$ whose kernel is given by

[TABLE]

(see [1] for more details).

Define the corresponding Littlewood-Paley-Stein operators. Then, in a similar way, one can prove that these operators are bounded on $L^{p}(M),$ for $M$ as in Theorem 4, provided that

[TABLE]

If $\eta_{\Gamma}=0$ then the condition (49) on $\sigma$ becomes $\sigma<2\left\|\rho\right\|/\sqrt{pp^{\prime}},$ thus we recover the result of Anker in [1].

4.3.2. Proof of Theorem 5

In this section we prove Theorem 5, which gives estimates of the norm

[TABLE]

From Theorem 1, it follows that for sufficiently small $\epsilon>0$, there exists $c>0$ such that

[TABLE]

Thus, for $t\geq 1,$ proceeding as in the proof of Theorem 4, the estimate (50) and the Kunze-Stein phenomenon, imply that

[TABLE]

The integral above converges whenever

[TABLE]

which holds true for sufficiently small $\epsilon>0$, since $\left\|\eta_{\Gamma}\right\|<\left\|\rho\right\|.$

Furthermore, from (51) we get that

[TABLE]

Next, it is left to show that for $t<1$ the operator $S_{t}$ is bounded on $L^{p}(M),$ $p\in[1,\infty].$ We split the operator $S_{t}$ into two parts using a smooth cut-off function $\psi\in C_{c}^{\infty}(K\backslash G/K)$, with $\psi\equiv 1$ near the origin and $\psi\equiv 0$ in $B\left(0,2\right)^{c}$. Let ${S}_{t}^{0}=\ast(\psi\frac{\partial h_{t}}{\partial t})$ and ${S}_{t}^{\infty}=\ast(1-\psi)\frac{\partial h_{t}}{\partial t}.$

First, note that the operator $S_{t}^{\infty}$ can be handled like $S_{t}$ for $t>1$. Next, it can be shown that the kernel of ${S}_{t}^{0}$ is in $L^{1}(M)$, using a summation argument and working as in the Euclidean case. Indeed, we have

[TABLE]

Using Theorem 1 and the fact that $\psi$ is supported around the origin,

[TABLE]

since the last sum is finite.