Manuscript XX-D@1 · XX-D.tex · proposition · c83a526d

Proposition: Transverse Taylor extension

Explicit one-contraction normalization › An alternative extension of the punctured numerical kernel · lines 1247–1297 · proof included or linked

XX-D@1#prop:Taylor, XX-D@1#eq:contacts

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Suppose the punctured coefficient $β$ has the stated Hadamard bound with smooth dependence on $(t, x)$ . It has a distributional extension across $r = 0$ . In the class with transverse scaling degree at most six, the extension freedom is

∣ α ∣ \leq 3 \sum c_{α} (t, x) \partial_{r}^{α} δ (r) .

The coefficients are not claimed to form a finite-dimensional space without further restrictions.

Proof. Choose $θ \in C_{c}^{\infty} (R^{3})$ , equal to one near zero. For a compact test $η$ , put

T_{3} η = η - θ (r) ∣ α ∣ \leq 3 \sum \frac{r ^{α}}{α !} \partial_{r}^{α} η (t, x, 0) .

The remainder is $O (∣ r ∣^{4})$ . Thus

\int β T_{3} η d t d^{3} x d^{3} r

converges absolutely near zero: the radial majorant is a constant times $(1 + ∣ lo g r ∣)^{N} d r$ . It defines a continuous extension after specifying values on the subtracted jets.

The finite scaling-degree extension principle of BF (Theorems 5.2–5.3 and Section 6) gives the scaling-degree bound. A difference of such extensions is supported on $r = 0$ . Its allowed normal derivative order is at most $6 - 3 = 3$ , giving (eq:contacts).

Displayed formulas in this passage

XX-D@1#eq:contacts · line 1256
$∣ α ∣ \leq 3 \sum c_{α} (t, x) \partial_{r}^{α} δ (r) .$
unlabeled · line 1270
$T_{3} η = η - θ (r) ∣ α ∣ \leq 3 \sum \frac{r ^{α}}{α !} \partial_{r}^{α} η (t, x, 0) .$
unlabeled · line 1280
$\int β T_{3} η d t d^{3} x d^{3} r$

Proposition: Transverse Taylor extension

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