Occupation times and exact asymptotics of small deviations of Bessel processes for $L^p$-norms with $p>0$

We prove theorems on exact asymptotics of the distributions of integral functionals of the occupation time of Bessel processes. Using these results, we obtain exact asymptotics of small deviations for Bessel processes in the $L^p$-norm. We use Laplace's method for the occupation times of Markov processes with continuous time. Computations are carried out for $p=2$ and $p=1$. We also solve extremal problems for the action functional.