This is an interesting and significant paper on the characteristic initial value problem of the vacuum Einstein equations. As in a pioneering theorem by A. D. Rendall [Proc. R. Soc. Lond., Ser. A 427, No. 1872, 221–239 (1990; Zbl 0701.35149)], this paper works with a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere. Rendall set smooth characteristic initial data on the null cones and proved that the Einstein equations can be solved in a small neighborhood of the intersecting sphere, to the future of the null cones. This paper takes the analysis a step further. It shows local existence in a neighborhood of the two null cones, not just in a neighborhood of the sphere, as long as the constraint equations are initially satisfied on the null cones. Furthermore, the size of the neighborhood can be made to depend only on the size of the initial data. The proof is based on error estimates for the Einstein equations in double null foliation, the null structure being crucial for the analysis.