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We prove that the Cauchy problem for the Dirac-Klein-Gordon system ofequations in 1D is globally well-posed in a range of Sobolev spaces of negativeindex for the Dirac spinor and positive index for the scalar field. The mainingredient in the proof is the theory of almost conservation law and I-methodintroduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof alsorelies on the null structure in the system, and bilinear spacetime estimates ofKlainerman-Machedon type.

Global Well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index