  We prove several dispersive estimates for the linear part of the Full
Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some
shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of
these estimates combines the stationary phase method with sharp asymptotics on
asymmetric Bessel functions, which may be of independent interest. As a
consequence, we prove that the initial value problem associated to the Full
Dispersion Kadomtsev-Petviashvili is locally well-posed in $H^s(\mathbb R^2)$,
for $s>\frac74$, in the capillary-gravity setting.
