Abstract
We study the regularity of the solutions of the surface quasi-geostrophic equation with subcritical exponent 1/2 < alpha <= 1. We prove that if the initial data is small enough in the critical space H2-2 alpha(R-2), then the regularity of the solution is of exponential growth type with respect to time and its H2-2 alpha(R-2) norm decays exponentially fast. It becomes then infinitely differentiable with respect to time and has value in all homogeneous Sobolev spaces H-s(R-2) for s >= 2 - 2 alpha. Moreover, we give some general properties of the global solutions.