Upper functions for $\mathbb{L}_{p}$-norms of Gaussian random fields

International audienceIn this paper we are interested in finding upper functions for a collection of random variables { ξ ⃗ h p , ⃗ h ∈ H } , 1 ≤ p < ∞. Here ξ ⃗ h (x), x ∈ (−b, b) d , d ≥ 1 is a kernel-type gaussian random field and ∥ · ∥p stands for Lp-norm on (−b, b) d. The set H consists of d-variate vector-functions defined on (−b, b) d and taking values in some countable net in R d +. We seek a non-random family { Ψε (⃗ h) , ⃗ h ∈ H } such that E { sup ⃗ h∈H [ ξ ⃗ h p − Ψε (⃗ h)] + } q ≤ ε q , q ≥ 1, where ε > 0 is prescribed level