Kolmogorov numbers of Riemann–Liouville operators over small sets and applications to Gaussian processes

Abstract

We investigate compactness properties of the Riemann–Liouville operator R α of fractional integration when regarded as operator from L 2[0,1] into C( K), the space of continuous functions over a compact subset K in [0,1]. Of special interest are small sets K, i.e. those possessing Lebesgue measure zero (e.g. fractal sets). We prove upper estimates for the Kolmogorov numbers of R α against certain entropy numbers of K. Under some regularity assumption about the entropy of K these estimates turn out to be two-sided. By standard methods the results are also valid for the (dyadic) entropy numbers of R α . Finally, we apply these estimates for the investigation of the small ball behavior of certain Gaussian stochastic processes, as e.g. fractional Brownian motion or Riemann–Liouville processes, indexed by small (fractal) sets.