Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

Abstract

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x 0 + x n = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space \({{\mathcal S}'_{\partial_-}({\mathbb{R}}^n)}\) which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of \({{\mathcal S}'_{\partial_-}({\mathbb{R}}^n)}\) appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.