對於1+3維的DKG系統，我們能夠利用對偶性的論點，來說明Klein-Gordon和Dirac的部分都擁有null的形式。

In this paper, we study the null structure of the coupled Dirac-Klein-Gordon system of equations in 1+3 dimentions.

We can use duality argument to find that the null bilinear forms occur not only in the Klein-Gordon part of the system , but also in the Dirac part.

[B] J. Bourgain, Fourier transform restriction phenomena for certain lattice
subsets and applications to nonlinear evolution equations I, II. Geom.
Funct. Anal. 3, 107-156 and 209-262(1993)
[Bo] N. Bournaveas, Low regularity solutions of the Dirac-Klein-Gordon equa
tions in two space dimensions. Comm. Partial Differential Equations 26,
1345-1366 (2001)
[Ch] J. M. Chadam, Global solutions of the Cauchy problem for the (classical)
coupled Maxwell-Dirac equations in one space dimension. J. Funct. Anal.
13, 173-184 (1973)
[F1] Y. F. Fang, On the Dirac-Klein-Gordon equations in one space dimen
sion. Differential Integral Equations 17, 1321-1346 (2004)
[F2] Y. F. Fang, M. Grillakis, On the Dirac-Klein-Gordon equations in three
space dimensions. Comm. Partial Differential Equations 30, 783-812
(2005)
[Fo] D. Foschi, S. Klainerman, Bilinear space-time estimates for homogeneous
wave equations. Ann. Sci. Ecole Norm. Sup. (4) 23, 211-274 (2000)
[K1] S. Klainerman, M. Machedon, Smoothing estimates for null forms and
applications. Duke Math. J. 81, 99-133 (1995)
[K2] S. Klainerman, S. Selberg, Bilinear estimates and applications to nonlin
ear wave equations. Commun. Contemp. Math. 4, 223-295 (2002)
[K3] S. Klainerman, M. Machedon, Space-time estimates for null forms and
the local existence theorem. Comm. Pure Appl. Math. 46, 1221-1268
(1993)
[Ke] C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the KdV equation
in Sobolev spaces of negative indices. Duke Math. J. 71, 1-21 (1993)
[S1] S. Selberg, Multilinear spacetime estimates and applications to local ex
istence theory for nonlinear wave equations. Ph.D. thesis, Princeton Univ.
(1999)
[S2] S. Selberg, On an estimate for the wave equation and applications to non
linear problems. Differential Integral Equations 2, 213-236 (2002)
[St] R. Strichartz, Restriction of Fourier transforms to quadratic surfaces and
decay of solutions of wave equations. Duke Math. J. 44, 705-714 (1977)