The norm of the Riemann‐Liouville operator on L p [0,1]: a probabilistic approach

We obtain explicit lower and upper bounds for the norm of the Riemann–Liouville operator V s on L p [0, 1] which are asymptotically sharp, thus completing previous results by Eveson. Similar statements are shown with respect to the norms || V s f || p , whenever f satisfies certain smoothness properties. It turns out that the correct rate of convergence of || V s f || p as s → ∞ depends both on the infimum of the support of f and on the degree of smoothness of f . We use a probabilistic approach which allows us to give unified proofs.