  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 and to belong to the union of balls in the isotropic/anisotropic
Nikolskii's spaces. We will show that the optimally adaptive estimators over
the collection of considered functional classes do no exist. Also, in the
framework of an abstract density model we present several generic lower bounds
related to the adaptive estimation of an arbitrary functional of a probability
density. These results having independent interest have no analogue in the
existing literature. In the companion paper Cleanthous et al (2024) we prove
that established lower bounds are tight and provide with explicit construction
of adaptive estimators of $\mathbb{L}_2$-norm of the density.
