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We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in n + 1 dimensions, n ≥ 4. We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov spaceḂ 2,1 n/2 ×Ḃ 2,1 n/2−1 . Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in H s × H s−1 , s > n/2, then the solutions stay bounded in the same space. In part II of thisdoi:10.1080/03605309808821400 fatcat:4n3i737a55by7hstac32vdi7fm