  Theory of superconductivity generated in one atomic layer thick two
dimensional electron gas by a single flat band of high energy longitudinal
optical phonons is considered. The polar dielectric $SrTiO_{3}$ (STO) exhibits
such an energetic phonon mode and the 2DEG is created both when one unit cell
$FeSe$ layer is grown on its $\left( 100\right) $ surface and on the interface
with another dielectric like $LaAlO_{3}$ (LAO). We obtain a quantitative
description of both systems solving the gap equation for $T_{c}$ without making
use of approximations like the Kirzhnits Ansatz for arbitrary chemical
potential $\mu $, electron-phonon coupling $\lambda $ and the phonon frequency
$\Omega $, and direct (RPA) electron-electron repulsion strength $\alpha $. The
high temperature superconductivity in 1UC$FeSe$/STO is possible due to a
combination of three factors: high LO phonon frequency, large electron-phonon
coupling $\lambda \sim 0.5$ and huge dielectric constant of the substrate
suppression the Coulomb repulsion. It is shown that very low density electron
gas in the interfaces is still capable of generating superconductivity of the
order of $0.1$ K in LAO/STO. Superconductivity persists even on the band edge
$\mu =0$.
