Journal of Approximation Theory 101, 6377 (1999) Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Let Td: L2([0, 1]d) C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k&1(log k)d&12. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(&C=&2 |log =|2d&1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms. 1999 Academic Press 1