The double sum method for Gaussian fields with a parameter set in $l^p$

In the paper it is presented a method to compute asymptotics of the probability $\mathsf P\Bigl\{\,\sup\limits_{t\in T}X(t)>u\Bigr\}$, where $X(t)$ is a Gaussian random field with a compact parameter set in the space $l^p$, $1<p\leq2$. On the basis of obtained result it is found the exact asymptotics of tail distribution for the supremum of $l^q$-norm of $l^q$-valued Ornstein–Uhlenbeck process when $q>2$.