We find exact upper and lower bounds for the distribution of the supremum of a homogeneous Gaussian random field with pyramidal covariance function. The upper bound comes from a reflection principle type argument. The lower bound is found by exploiting a relationship between this random field and a particular Banach space valued process in one-dimensional time.
"The Supremum of a Particular Gaussian Field." Ann. Probab. 12 (2) 436 - 444, May, 1984. https://doi.org/10.1214/aop/1176993299