Kernel support and the location of the obstruction

For $x\in \{t>0\}$, let ${\Lambda }^{\mu ,\alpha \beta }(x,z)$ be the distribution kernel of $L$:

Equivalently, it is characterized by

The first geometric insertion has mixed kernel

so that

Its support satisfies

This is a compact subset of $M\times M$.

The boundary part is located at $z^0=0$, whereas $x^0>0$ on $\mathrm{supp}f$. It is therefore away from the ordinary diagonal $z=x$. Corollary cor:quadratic proves the failure of its required wavefront cone without needing to assert a complete wavefront equality for (eq:mixed-kernel).

For the contracted adjoint kernel

the transverse scaling degree of the delta term is one wherever its coefficient is nonzero. The step term has transverse scaling degree at most zero. These are finite-order distributions. Thus the obstruction is not excessive transverse scaling and is not the nonexistence of a distributional extension: it is incompatibility with the off-shell cone condition.