Estimates of the distribution of the maximum of a random field

Let $ \xi(t) $ be a random field with values in $ \mathbb R^1$, defined for $ t \in T$, $T$ an arbitrary set. In this paper two-sided exponential estimates are derived for probabilities $ P(T,u) = \mathbb P\{\sup_{t \in T} \xi(t) > u \} $:
$$ C_1 g_2(u) \l \log P(T,u) + g_1(u) \l C_2 g_2(u), $$
where $ g_1(u) $ is a convex function, $u \to \infty \Rightarrow \lim g_1'(u) = \infty$, $\lim [g_2(u)/g_1(u)] = 0$, $C_k$ are positive numbers independent of $u$.