  We prove new bilinear estimates for the X^{s, b}_\pm(R^2) spaces which are
optimal up to endpoints. These estimates are often used in the theory of
nonlinear Dirac equations on R^{1+1}. The proof of the bilinear estimates
follows from a dyadic decomposition in the spirit of Tao [21] and D'Ancona,
Foschi, and Selberg [11]. As an application, by using the I-method of
Colliander, Keel, Staffilani, Takaoka, and Tao, we extend the work of Tesfahun
[23] on global existence below the charge class for the Dirac-Klein- Gordon
equation on R^{1+1}.
