Section, remainder, and transverse estimates

If a continuous section $s$ with $\mathsf{Rec}s=\mathrm{id}$ has been constructed,

A descending map kills this remainder. An algebraic selection of representatives is not a continuity, uniformity, or intrinsic-symbol theorem.

For a restricted locally integrable transverse function satisfying

on compact tangential sets, choose integer $p\ge 0$ with $p>a-c-1$. The Taylor subtraction

gives the extension

The remainder is $O(\text{abs}{q}^{p+1})$, so radial integrability is $r^{p+c-a}$, with exponent greater than $-1$. This proves extension for that printed class. It does not prove a full characteristic extension from frequency homogeneity alone.

For general distributional transverse extensions, scaling and wavefront hypotheses must be checked against an applicable theorem Dang,BrunettiFredenhagen. One cannot cite such a theorem without establishing its domain, scaling, conormal-landing, and support conditions.