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We show a new Bernstein's inequality which generalizes the results ofCannone-Planchon, Danchin and Lemari\'{e}-Rieusset. As an application of thisinequality, we prove the global well-posedness of the 2D quasi-geostrophicequation with the critical and super-critical dissipation for the small initialdata in the critical Besov space, and local well-posedness for the largeinitial data.Comment: 18page

A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation