Manuscript IX-B-ST@1 · IX-B-ST.tex · section · f6af1523

Observation derivatives and smooth spectators

Observation derivatives and smooth spectators · lines 970–991

Let

D = ∣ β ∣ \leq d \sum b_{β} \partial^{β}

be a fixed differential operator on $X$ , with smooth coefficients, and let $a \in C^{\infty} (X)$ . The ordered operation is

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

Its transpose, relative to the displayed coordinate density, is

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

There is no coefficient conjugation in this transpose.

Displayed formulas in this passage

unlabeled · line 974
$D = ∣ β ∣ \leq d \sum b_{β} \partial^{β}$
unlabeled · line 980
$O_{a, D} U = a D U .$
unlabeled · line 984
$D^{t} Φ = ∣ β ∣ \leq d \sum (- \partial)^{β} (b_{β} Φ), ⟨ a D U, Φ ⟩ = ⟨ U, D^{t} (a Φ) ⟩ .$