Commutator estimate in terms of partial derivatives of solutions for the dissipative quasi-geostrophic equation

Through Littlewood-Paley decomposition argument, a commutator estimate in terms of partial derivatives of solutions for the critical and supercritical dissipative 2D Quasi-Geostrophic equation is established. As an application of this estimate, we obtain some new a priori estimates and prove the existence and uniqueness of solution for the small initial data in critical Besov spaces.