Two global results for the initial-value problem for the Schrödinger map equation, , on , , , , are proved. In order to formulate these results we have to introduce some notation. As usual, denotes the Sobolev space on . For , , . The space is the intersection of all spaces and for , is the homogeneous Sobolev norm.
The first theorem proved in the paper states that if and , then there exist and a constant such that for any with there exists a unique solution of the initial-value problem, and for any and , .
The second theorem provides uniform bounds (on ) for and asserts that the operator admits continuous extensions for any and some , where .
The first theorem was proved in dimensions in [I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Adv. Math. 215, No. 1, 263–291 (2007; Zbl 1152.35049)]. In order to overcome the difficulties which appear, especially in the case , the authors of the paper under review use a sum of Galilei transforms of the lateral spaces and the caloric gauge.