Asymptotics of large deviations of Gaussian processes of Wiener type for $L^p$-functionals, $p>0$, and the hypergeometric function

A general result is obtained on exact asymptotics of the probabilities
$$\mathsf P\biggl\{\int_0^1|\xi(t)|^p\,dt>u^p\biggr\}$$
as $u\to\infty$ and $p>0$ for Gaussian processes $\xi(t)$.
The general theorem is applied for the calculation of these asymptotics in the cases of the following processes: the Wiener process $w(t)$, the Brownian bridge, and the stationary Gaussian process $\eta(t):=w(t+1)-w(t)$, $t\in\mathbb R^1$.
The Laplace method in Banach spaces is used. The calculations of the constants reduce to solving an extremum problem for the action functional and studying the spectrum of a differential operator of the second order of Sturm–Liouville type.