Bilinear Fourier restriction estimates related to the 2d wave equation

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.