  We study Fermi liquid properties of a weakly interacting 2D gas of
single-component fermionic polar molecules with dipole moments $d$ oriented
perpendicularly to the plane of their translational motion. This geometry
allows the minimization of inelastic losses due to chemical reactions for
reactive molecules and, at the same time, provides a possibility of a clear
description of many-body (beyond mean field) effects. The long-range character
of the dipole-dipole repulsive interaction between the molecules, which scales
as $1/r^3$ at large distances $r$, makes the problem drastically different from
the well-known problem of the two-species Fermi gas with repulsive contact
interspecies interaction. We solve the low-energy scattering problem and
develop a many-body perturbation theory beyond the mean field. The theory
relies on the presence of a small parameter $k_Fr_*$, where $k_F$ is the Fermi
momentum, and $r_*=md^2/\hbar^2$ is the dipole-dipole length, with $m$ being
the molecule mass. We obtain thermodynamic quantities as a series of expansion
up to the second order in $k_Fr_*$ and argue that many-body corrections to the
ground-state energy can be identified in experiments with ultracold molecules,
like it has been recently done for ultracold fermionic atoms. Moreover, we show
that only many-body effects provide the existence of zero sound and calculate
the sound velocity.
