  We study bilinear $L^2$ Fourier restriction estimates which are related to
the 2d wave equation in the sense that we restrict to subsets of thickened null
cones. In an earlier paper we studied the corresponding 3d problem, obtaining
several refinements of the Klainerman-Machedon type estimates. The latter are
bilinear generalizations of the $L^4$ estimate of Strichartz for the 3d wave
equation. In 2d there is no $L^4$ estimate for solutions of the wave equation,
but as we show here, one can nevertheless obtain $L^2$ bilinear estimates for
thickened null cones, which can be viewed as analogues of the 3d
Klainerman-Machedon type estimates. We then prove a number of refinements of
these estimates, analogous to those we obtained earlier in 3d. The main
application we have in mind is the Maxwell-Dirac system.
