  Let $T_1^{(\mu)}$ be the first hitting time of the point 1 by the Bessel
process with index $\mu\in \R$ starting from $x>1$. Using an integral formula
for the density $q_x^{(\mu)}(t)$ of $T_1^{(\mu)}$, obtained in Byczkowski,
Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the
density of $T_1^{(\mu)}$ which exibit the dependence both on time and space
variables. Our result provides optimal estimates for the density of the hitting
time of the unit ball by the Brownian motion in $\mathbb{R}^n$, which improve
existing bounds. Another application is to provide sharp estimates for the
Poisson kernel for half-spaces for hyperbolic Brownian motion in real
hyperbolic spaces.
