Upper functions for Lp-norms of Gaussian random fields

In this paper, we are interested in finding upper functions for a collection of random variables {parallel to xi(-)(h)parallel to(p), (h) over right arrow is an element of H), 1 <= p < infinity. Here xi(<(h)over right arrow>) (x), x is an element of (-b, b)(d), d >= 1 is a kernel -type Gaussian random field and parallel to.parallel to(p) stands for L-p-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 {Psi(epsilon) ((h) over right arrow), (h) over right arrow is an element of H} such that E{sup((h) over right arrow is an element of H)[parallel to xi((h) over right arrow)parallel to(p) - Psi(epsilon) ((h) over right arrow)](+)}(q) <= epsilon(q), q >= 1where epsilon > 0 is prescribed level.