  We describe experiments with spin-polarized atomic hydrogen gas adsorbed on
liquid $^{4}$He surface. The surface gas density is increased locally by
thermal compression up to $5.5\times10^{12}$ cm$^{-2}$ at 110 mK. This
corresponds to the onset of quantum degeneracy with the thermal de-Broglie
wavelength being 1.5 times larger than the mean interatomic spacing. The atoms
were detected directly with a 129 GHz electron-spin resonance spectrometer
probing both the surface and the bulk gas. This, and the simultaneous
measurement of the recombination power, allowed us to make accurate studies of
the adsorption isotherm and the heat removal from the adsorbed hydrogen gas.
  From the data, we estimate the thermal contact between 2D hydrogen gas and
phonons of the helium film. We analyze the limitations of the thermal
compression method and the possibility to reach the superfluid transition in 2D
hydrogen gas.
