  We study the sine-Gordon model in two dimensional space time in two different
domains. For beta > 8 pi and weak coupling, we introduce an ultraviolet cutoff
and study the infrared behavior. A renormalization group analysis shows that
the model is asymptotically free in the infrared. For beta < 8 pi and weak
coupling, we introduce an infrared cutoff and study the ultraviolet behavior. A
renormalization group analysis shows that the model is asymptotically free in
the ultraviolet.
