Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Observation derivatives and smooth spectators

Observation derivatives and smooth spectators · lines 824–840

Let $D = \sum_{∣ α ∣ \leq d} b_{α} \partial^{α}$ be a fixed differential operator on $X$ with smooth coefficients, and let $a \in C^{\infty} (X)$ . Its distributional action is defined by the formal transpose relative to the coordinate density:

⟨ D U, Φ ⟩ = ⟨ U, D^{t} Φ ⟩, D^{t} Φ = ∣ α ∣ \leq d \sum (- \partial)^{α} (b_{α} Φ) .

No complex conjugation occurs in this transpose. The ordered operation is

O_{a, D} U = a D U .

Displayed formulas in this passage

unlabeled · line 831
$⟨ D U, Φ ⟩ = ⟨ U, D^{t} Φ ⟩, D^{t} Φ = ∣ α ∣ \leq d \sum (- \partial)^{α} (b_{α} Φ) .$
unlabeled · line 837
$O_{a, D} U = a D U .$

Nearby in this section

theorem: Theorem: Derivative–spectator finite part
corollary: Corollary: Descendant conversion and scale dependence