  The present article is devoted to the 3D dissipative quasi-geostrophic system
(QG). This system can be obtained as limit model of the Primitive Equations in
the asymptotics of strong rotation and stratification, and involves a
non-radial, non-local, homogeneous pseudo-differential operator of order 2
denoted by $\Gamma$ (and whose semigroup kernel reaches negative values). After
a refined study of the non-local part of $\Gamma$, we prove apriori estimates
(in the general L^p setting) for the 3D QG-model. The main difficulty of this
article is to study the commutator of $\Gamma$ with a Lagrangian change of
variable. An important application of these a priori estimates, providing bound
from below to the lifespan of the solutions of the Primitive Equations for
ill-prepared blowing-up initial data, can be found in a companion paper.
