A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation

We show a new Bernstein's inequality which generalizes the results of Cannone-Planchon, Danchin and Lemari\'{e}-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data.