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Assuming initial data have small weighted $H^4\times H^3$ norm, we proveglobal existence of solutions to the Cauchy problem for systems of quasi-linearwave equations in three space dimensions satisfying the null condition ofKlainerman. Compared with the work of Christodoulou, our result assumessmallness of data with respect to $H^4\times H^3$ norm having a lower weight.Our proof uses the space-time $L^2$ estimate due to Alinhac for some specialderivatives of solutions to variable-coefficient wave equations. It also usesthe conformal energy estimate for inhomogeneous wave equation $\Box u=F$. A newobservation made in this paper is that, in comparison with the proofs ofKlainerman and H\"ormander, we can limit the number of occurrences of thegenerators of hyperbolic rotations or dilations in the course of a prioriestimates of solutions. This limitation allows us to obtain global solutionsfor radially symmetric data, when a certain norm with considerably low weightis small enough.

Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight