Weak Solutions of the Massless Field Equations

We consider weak solutions of the zero-rest-mass (z.r.m.) equations described in Eastwood et al. (1981). The space of hyperfunctions, which contains the space of distributions, is defined and we consider hyper-function solutions of the equations on real Minkowski space M^I and its conformal compactification M. We define a hyperfunction z.r.m. field to be future or past analytic if it is the boundary value of a holomorphic z.r.m. field on the future or past tube of complex Minkowski space respectively; and we demonstrate that any field on M^I that is the sum of future and past analytic fields extends as a hyperfunction z.r.m. field to all of M. It is shown that any distribution solution on M^I splits as required and hence extends as a hyperfunction solution to M. Twistor methods are then used to show that the same applies in the more general case of hyperfunction solutions on M^I. This leads to an alternative proof of the main result of Wells (1981): a hyperfunction z.r.m. field on compactified real Minkowski space is a unique sum of future and past analytic solutions.