  We prove that the Cauchy problem for the Dirac-Klein-Gordon system of
equations in 1D is globally well-posed in a range of Sobolev spaces of negative
index for the Dirac spinor and positive index for the scalar field. The main
ingredient in the proof is the theory of almost conservation law and I-method
introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also
relies on the null structure in the system, and bilinear spacetime estimates of
Klainerman-Machedon type.
