If the number of dimensions $n$ is odd, the flat-space wave operator $-\partial_t^2 + \nabla^2$ admits the fundamental solution $(-\sigma)^{1-n/2} \Theta (-\sigma)$, where $\sigma = - t^2 + |x|^2$. This has support inside the characteristic cones $t = \pm |x|$, but not outside of them. My question is: Does there exists another fundamental solution which instead vanishes inside the cones, but not outside of them? My first guess was $\sigma^{1-n/2} \Theta (\sigma)$, but that seems to be a homogeneous solution.
This does work in even dimensions (in nontrivial cases where the operator is modified so that it no longer satisfies Huygens' principle). What I have in mind are wave operators $g^{ab} \nabla_a \nabla_b$ on curved spacetimes. For sufficiently short distances in 4D, there are typically fundamental solutions like \begin{equation} G = U \delta(\sigma) + V \Theta(-\sigma), \end{equation} where $U$, $V$ are well-behaved, $\delta = \Theta'$ is the usual Dirac distribution, and $\Theta$ the step distribution. Then $V$ is a homogeneous solution and $G-V = U \delta(\sigma) - V \Theta(\sigma)$ is a fundamental solution with the desired support. A similar trick doesn't work in odd dimensions, of course.