Extend Wp (p > 1) density estimation results beyond Besov classes

Establish minimax-optimal rates and corresponding procedures for estimating probability densities under the Wasserstein distance Wp with p > 1 for functional classes beyond Besov spaces, thereby extending the scope of existing results that currently apply only to Besov classes.

Background

The chapter shows that smoothness assumptions on densities can mitigate the curse of dimensionality and yield improved estimation rates in Wasserstein distance. For p > 1, known optimal rates and procedures rely on density classes with Besov regularity.

The authors explicitly point out that extending these results to other smoothness classes is unresolved and identify this as an open direction.