Observation coverage

The estimates are global in the Cartesian observation variables, so they cover the following regions.

1. Off $\mathrm{Rel}$, the residue vanishes on supported tests. Equations (eq:UV) and (eq:Ar) then give convergence without a logarithmic subtraction.

2. Near $\mathrm{Rel}\setminus {\Delta }_{13}$, the same estimates apply without using a lower bound on $|d|$.

3. Near $\mathrm{Rel}\cap {\Delta }_{13}$, the exact phase and density are (eq:intersectionphase) and (eq:density). The mass-phase bound (eq:dmass) is uniform through $d=0$.

4. The remaining ordinary diagonals and the region $\mathbf{x}=\mathbf{y}$, where the incidence chart is unavailable, are covered by the Cartesian Fourier argument. Moreover, $\mathrm{Rel}\cap {\Delta }_{12}=\mathrm{Rel}\cap {\Delta }_{23}=\varnothing$ in $X$, because the corresponding reflected spatial distances are $t+s>0$ and $t^{\prime }+s>0$.

These are assertions on joint tests supported in open observation neighborhoods, not tests confined to positive-codimension strata. A finite smooth partition on any compact gives the corresponding regional estimates with the same test order.