We study the question of well-posedness of the Cauchy problem for Schrödinger maps from ℝ1 x ℝ2 to the sphere double-struck S2 or to ℍ2, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well-posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well-posedness of the Schrödinger map itself from it. In proving well-posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L2-functions. © 2002 Wiley Periodicals, Inc.
