Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Let T"d:L"2([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(logk)^d^-^1^/^2. 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@e^-^2|log@e|^2^d^-^1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.

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