$. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the Lie algebras of $G$ and $K $. Let also $\mathfrak{p}$ be the subspace of $\mathfrak{g}$ which is orthogonal to $\mathfrak{k}$ with respect to the Killing form. The Killing form induces a $K$-invariant scalar product on $\mathfrak{p}$ and hence a $G$-invariant metric on $X$. Denote by $\Delta $ the Laplace-Beltrami operator on $X$, by $d(.,.)$ the Riemannian distance and by $dx$ the associated Riemannian measure on $X$. Denote by $\left\vert B\left( x,r\right)
\right\vert $ the volume of the ball $B\left( x,r\right) $, $x\in X$, $r>0$, and recall that there is a $c>0$, such that $\left\vert B\left( x,r\right)
\right\vert \leq cr^{n}$ for all $r\leq 1$.

Fix $\mathfrak{a} $ a maximal abelian subspace of $\mathfrak{p}$ and denote by $\mathfrak{a}^{\ast }$ the real dual of $\mathfrak{a}$. If $\operatorname{dim}%
\mathfrak{a}=l$, we say that $X$ has rank $l$. We also say that $\alpha \in
\mathfrak{a}^{\ast }$ is a root vector, if the space $$\mathfrak{g}^{\alpha }=\left\{ X\in \mathfrak{g}:[H,X]=\alpha (H)X,\text{
for all }H\in \mathfrak{a}\right\} \neq \left\{ 0\right\} .$$

Let $A$ be the analytic subgroup of $G$ with Lie algebra $\mathfrak{a}.$ Let $\mathfrak{a}_{+}\subset \mathfrak{a}$ be a positive Weyl chamber and let $%
\overline{\mathfrak{a}_{+}}$ be its closure. Set $A^{+}=\exp \mathfrak{a}%
_{+} $. Its 