Manuscript VII-H@1 · VII-H.tex · proposition · d20d2a9f

Proposition: Conversion consistency and derivative transfer

The VII-D conversion and the inherited analytic boundary › What can be transported from a numerical conversion · lines 1617–1646 · proof included or linked

VII-H@1#prop:conversion

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If $r \neq = 0$ , the exact relation (eq:VII-D-conversion) implies

c_{ab} + c_{b c} = c_{a c} .

If a fixed linear response map $P$ has been constructed, its conversion is $c_{ab} P (r)$ .

Let $T$ be a differential operator of order $d$ with fixed smooth coefficients on a compact test stage. An estimate

∣ ⟨ v_{R}, F ⟩ ∣ \leq C R^{- 1} p_{K, 64} (F)

implies

⟨ T^{T} v_{R}, F ⟩ \leq C^{'} R^{- 1} p_{K, 64 + d} (F) .

Proof. Subtract the three distributions to obtain $(c_{ab} + c_{b c} - c_{a c}) r = 0$ , and use $r \neq = 0$ . The response statement is Proposition prop:propagation. For the last estimate, pair $v_{R}$ with $T F$ and use the compact-stage bound $p_{K, 64} (T F) \leq C_{T} p_{K, 64 + d} (F)$ .

Displayed formulas in this passage

unlabeled · line 1621
$c_{ab} + c_{b c} = c_{a c} .$
unlabeled · line 1629
$∣ ⟨ v_{R}, F ⟩ ∣ \leq C R^{- 1} p_{K, 64} (F)$
unlabeled · line 1633
$⟨ T^{T} v_{R}, F ⟩ \leq C^{'} R^{- 1} p_{K, 64 + d} (F) .$

Proposition: Conversion consistency and derivative transfer

Displayed formulas in this passage

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