  The 2D electrons trapped in vacuum near the atomically thin dielectric (ATD,
mono- or $N$-layer film of $h$-BN or transition metal dichalcogenide) are
considered. ATD is suspended above the back gate and forms the capacitor which
is controlled by the biased voltage determining 2D concentration, $n_{2D}$. It
is found that the leakage current through ATD is negligible and effect of the
polarizability of ATD is weak if $N\leq 5$. At temperatures $T=0.1\div$15 K and
$n_{2D}=5\times 10^8\div 10^{10}$ cm$^{-2}$, one deals with the Boltzmann
liquid of the macroscopic thickness $\sim$100 A. Due to bending of ATD the
quadratic dispersion law of the flexural vibrations is transformed into the
linear one at small wave vectors. The scattering processes of the electrons
caused by these phonons or the monolayer islands on ATD are examined and the
momentum and energy relaxation rates are analyzed based on the corresponding
balance equations. The momentum relaxation times varies over orders of
magnitude in the above region ($T$, $n_{2D}$) and $N$. The response may changed
from the polaron transport, for a perfect single-layer ATD at low $T$ and high
$n_{2D}$, to the high-mobility ($\geq 10^7$ cm$^2$/Vs) regime at high $T$ and
low $n_{2D}$. The quasi-elastic energy relaxation due to the phonon-induced
scattering is considered and the conditions for heating of electrons by a weak
in-plane electric field are found.
