Both internal placements of K 5

to the predecessor's regular charts

The same internal module gives:

1. the ordinary noncharacteristic chart by replacing the full multiplier with its actual value $-1/p$ on an open set avoiding $p=0$;

2. the separated-cone chart by inverse Fourier reconstruction of the full pole expression, giving the exact retarded cone distribution;

3. the transition itself by the distributional expression (eq:pole), with the uniform bounds proved through the pole.

The copied regular conormal orders remain upper orders in their own charts. The distributional degree $2$ of the degree-four amplitude in (eq:A5) is not a replacement for either order.

In particular, there is no contradiction between a regular diagonal upper order $1/2$, a separated two-cone upper order one, and the internal distributional dilation estimate. They describe different symbol presentations and different normal variables.

joint internal wavefront bound

Let ${{\mathcal{C}}_{\mathrm{R}}}^0$ consist of:

together with

and the nonzero forward-cone directions

Here $r^{♭}=\eta (r,\cdot )$. This is the usual retarded-kernel wavefront upper bound, with zero covectors on its support adjoined.

In either adapted chart, define

The covectors are pulled back to $T^{\ast }W$ by the actual adapted coordinate maps.

The reconstructed kernels satisfy

The bounds retain the internal diagonal and the separated flow-out, as well as the directions where one tensor factor has zero covector.

They are uniform bounds on the untested kernel, not merely bounds after smooth residual testing.

Proof. At a nonzero point of the retarded cone, (eq:R) is delta-type on its smooth defining hypersurface, so its wavefront is conormal there. At the vertex the asserted upper bound includes every nonzero covector and is therefore valid.

The tensor-product wavefront bound, with zero covectors adjoined on supports, gives (eq:Gamma5) for the two independent factors in (eq:coefficientform). The coordinate pullbacks are diffeomorphisms. Differentiation and smooth multiplication do not enlarge these bounds. The finite sum gives the result Hormander (Section 8.2).

The $b$ covector in the outer cone component is not set to zero. The balanced-kernel testing example of XIV-G is not used to model these directions.