Minimax estimation of the integral of a power of a density

We construct an estimator of ∫fp(z)dz, based on a random sample of size nn from a density ff on the unit cube in RdRd. This estimator achieves the minimax rate for ff known to belong to a multiple of the unit ball in a Hölder space of order αα, where α≤d/4α≤d/4. We are mostly interested in the case that the power pp is larger than 2 and/or the dimension dd is large.