Which classes of (scalar or systems of) linear second-order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that for every $g\in S$ whose support is disjoint from some past causal cone, there is a unique retarded solution $f\in S$ of $Lf=g$, i.e., such that the support of $g$ is disjoint from the past cone of $x$ for every $x$ such that the support of $g$ is disjoint from the past cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?