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.-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(-Cε-2|log ε|2d-1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.
|Number of pages||6|
|Journal||Comptes Rendus de l'Academie des Sciences - Series I: Mathematics|
|Publication status||Published - 1 Jan 1998|
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