Rules for generalized coefficients

For distributions, the following are sufficient operation rules.

1. Multiplication. The product $uv$ is defined by the usual wavefront criterion if no $(x,\xi )\in \mathrm{WF}(u)$ and $(x,-\xi )\in \mathrm{WF}(v)$ occur simultaneously.

2. Fixed pullback. For a smooth map $j$, $j^{\ast }u$ is defined if

   $$\mathrm{WF}(u)\cap N_j=\varnothing ,N_j=\{(j(x),\eta ):(\mathrm{d}{j}_x{)}^{\mathrm{T}}\eta =0\}.$$

   A fixed diffeomorphism has $N_j$ empty away from the zero section.

3. Kernel composition. Products of kernels on the intermediate space must satisfy D1, and the subsequent pushforward must have proper support or an independently proved convergent noncompact integration.

4. Newton inversion. Local elliptic distributional inversion does not by itself impose the global decaying boundary condition. We use Lemma lem:Newton, or a separately stated weighted domain, for that condition.

5. Extension. Failure of D1 or D2 does not prove that every extension is impossible. A punctured distribution with finite transverse scaling degree can instead be extended, subject to its normalization and symmetry conditions.

These are the microlocal operation principles used in the analytic frameworks cited above BF,HWlocal; their hypotheses are checked for the particular kernels below.