Local and global results for wave maps I

We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in dimensions,. We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov space . Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in then the solutions stay bounded in the same space. In part II of this work we shall prove that the same result holds in dimensions n = 2,3.