The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D
incompressible Euler equations. When dissipation is included in the model,
then solutions always exist if the dissipation's wave number dependence is
super-linear. Below this critical power, the dissipation appears to be ins
ufficient. For instance, it is not known if the critical dissipative QG equ
ation has global smooth solutions for arbitrary large initial data. In this
paper we prove existence and uniqueness of global classical solutions of t
he critical dissipative QG equation for initial data that have small L-infi
nity norm. The importance of an L-infinity smallness condition is due to th
e fact that L-infinity is a conserved norm for the non-dissipative QG equat
ion and is non-increasing on all solutions of the dissipative QG, irrespect
ive of size.