  We investigate the effects of gas-disk gravity on the planetesimal dynamics
in inclined binary systems, where the circumprimary disk plane is tilted by a
significant angle ($i_B$) with respect to the binary disk plane. Our focus is
on the Lidov-Kozai mechanism and the evolution of planetesimal eccentricity and
inclination. Using both analytical and numerical methods, we find that, on one
hand, the disk gravity generally narrows down the Kozai-on region, i.e., the
Lidov-Kozai effect can be suppressed in certain parts of (or even the whole of)
the disk, depending on various parameters. In the Kozai-off region,
planetesimals would move on orbits close to the mid-plane of gas-disk, with the
relative angle ($i^{'}$) following a small amplitude periodical oscillation. On
the other hand, when we include the effects of disk gravity, we find that the
Lidov-Kozai effect can operate even at arbitrarily low inclinations ($i_B$),
although lower $i_B$ leads to a smaller Kozai-on region. Furthermore, in the
Kozai-on region, most planetesimals' eccentricities can be excited to extremely
high values ($\sim 1$), and such extreme high eccentricities usually accompany
orbital flipping, i.e., planetesimal orbit flips back and forth between
anterograde and retrograde. Once a planetesimal reaches very high orbital
eccentricity, gas drag damping will shrink the planetesimal orbit, forming a
"hot planetesimal" on a near circular orbit very close to the primary star.
Such a mechanism, if replacing the planetesimals and gas drag damping with
Jupiters and tidal damping respectively, may lead to frequent production of
hot-Jupiters.
