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In this paper, we prove that if the initial data $\theta_0$ and its Riesztransforms ($\mathcal{R}_1(\theta_0)$ and $\mathcal{R}_2(\theta_0)$) belong tothe space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2\alpha ,\infty}}$,where $\alpha \in ]1/2,1[$, then the 2D Quasi-Geostrophic equation withdissipation $\alpha$ has a unique global in time solution $\theta$. Moreover,we show that if in addition $\theta_0 \in X$ for some functional space $X$ suchas Lebesgue, Sobolev and Besov's spaces then the solution $\theta$ belongs tothe space $C([0,+\infty [,X).$

Global Existence of Solutions to the 2D subcritical dissipative Quasi-Geostrophic equation and persistency of the initial regularity