  This is the second part of the research project initiated in Cleanthous et al
(2024). We deal with the problem of the adaptive estimation of the
$\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, from
independent observations. The unknown density is assumed to be uniformly
bounded by unknown constant and to belong to the union of balls in the
isotropic/anisotropic Nikolskii's spaces. In Cleanthous et al (2024) we have
proved that the optimally adaptive estimators do no exist in the considered
problem and provided with several lower bounds for the adaptive risk. In this
part we show that these bounds are tight and present the adaptive estimator
which is obtained by a data-driven selection from a family of kernel-based
estimators. The proposed estimation procedure as well as the computation of its
risk are heavily based on new concentration inequalities for decoupled
$U$-statistics of order two established in Section 4. It is also worth noting
that all our results are derived from the unique oracle inequality which may be
of independent interest.
