  In this paper we show the global existence for critical dissipative
quasi-geostrophic equations if $\|\widehat{\theta^0}\|_{L^1}$ is small enough;
among others we prove the analyticity of such a solution. If in addition the
initial condition verifies $|D|^{-\delta}\theta^0\in L^1(\mathbb R^2)$ with
$0<\delta<1$, then the solution remains regular and
$\lim_{t\rightarrow\infty}t^\delta\|\widehat{\theta}(t)\|_{L^1}=0$. Fourier
analysis and standard techniques are used.
