\label{rightinv}$$

The spherical Fourier transform
-------------------------------

Denote by $S(K\backslash G/K)$ the Schwartz space of $K$-bi-invariant functions on $G$. For $f\in S(K\backslash G/K),$ the spherical Fourier transform $\mathcal{H}$ is defined by $$\ \mathcal{H}f(\lambda )=\int_{G}f(x)\varphi _{\lambda }(x)\;dx,\quad
\lambda \in \mathfrak{a^{\ast }},$$ where $\varphi _{\lambda }$ is the elementary spherical function of index $%
\lambda$ on $G$. Note that from [@HEL] we have the following estimate $$\label{phi0}
\varphi_0(\exp H) \leq c(1+|H|)^de^{-\rho(H)},$$ where $d$ is the cardinality of the set of positive indivisible roots.

Let $S(\mathfrak{a^{\ast }})$ be the usual Schwartz space on $\mathfrak{\
a^{\ast}}$. Denote by $W$ the Weyl group associated to the root system of $(%
\mathfrak{g},\mathfrak{a})$ and denote by $S(\mathfrak{a^{\ast }})^{W}$ the subspace of $W$-invariant functions in $S(\mathfrak{a^{\ast }})$. Then, by a celebrated theorem of Harish-Chandra, $\mathcal{H}$ is an isomorphism between $S(K\backslash G/K)$ and $S(\mathfrak{a^{\ast }})^W$ and its inverse is given by $$\ (\mathcal{H}^{-1}f)(x)=c\int_{\mathfrak{a\ast }}f(\lambda )\varphi
_{-\lambda }(x)\frac{d\lambda }{|\mathbf{c}(\lambda )|^{2}},\quad x\in
G,\quad f\in S(\mathfrak{a^{\ast }})^{W},$$ where $\mathbf{c}(\lambda )$ is the Harish-Chandra function.

The heat kernel on $X$
----------------------

Set $$w_t(\lambda )=e^{-t(\Vert \lambda \Vert ^{2}+\Vert \rho \Vert
    ^{2})},\quad t>0,\;\lambda \in \mathfrak{a^{\ast }},$$ Then the heat kernel $p_{t}(x)$ of $X$ is given by $(\mathcal{H}^{-1}w_t)(x)$ [@ANJ].

The heat kernel $p_t$ on symmetric spaces has 