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 $\cS'_{\partial_-}(\R^n)$ which we have introduced in \cite{PullJMP}. Moreover, we show that every element of $\cS'_{\partial_-}(\R^n)$ appears as the ``tame'' restriction of a solution of the (homogeneous) Klein-Gordon equation.