Upper functions for $L_p$-norm of gaussian random fields

In this paper we are interested in finding upper functions for a collection of random variables $\big\{\big\|ξ_{\vec{h}}\big\|_p, \vec{h}\in\mathrm{H}\big\}, 1\leq p<\infty$. Here $ξ_{\vec{h}}(x), x\in(-b,b)^d, d\geq 1$ is a kernel-type gaussian random field and $\|\cdot\|_p$ stands for $L_p$-norm on $(-b,b)^d$. The set $\mathrm{H}$ 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 $\left\{Ψ_α\big(\vec{h}\big),\;\;\vec{h}\in\mathrm{H}\right\}$ such that $E\big\{\sup_{\vec{h}\in\mathrm{H}}\big[\big\|ξ_{\vec{h}}\big\|_p-Ψ_α\big(\vec{h}\big)\big]_+\big\}^q\leq α^q,\; q\geq 1,$ where $α>0$ is prescribed level.

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