Manuscript XVI-A@1 · XVI-A.tex · section · 8177be5d

Pullback

Standard microlocal verification interface › Pullback · lines 589–612

XVI-A@1#eq:normalset, XVI-A@1#eq:pullbacktest, XVI-A@1#eq:pullbackbound

For a smooth map $a : X \to Y$ , define

N_{a} = {(a (x), η) : η \neq = 0, (d a_{x})^{T} η = 0} .

For $v \in D^{'} (Y)$ , a standard sufficient condition for $a^{*} v$ is

WF (v) \cap N_{a} = \emptyset .

Under this condition,

WF (a^{*} v) \subset {(x, (d a_{x})^{T} η) : (a (x), η) \in WF (v)} .

See BDHoperations (Proposition 5.1).

Displayed formulas in this passage

XVI-A@1#eq:normalset · line 592
$N_{a} = {(a (x), η) : η \neq = 0, (d a_{x})^{T} η = 0} .$
XVI-A@1#eq:pullbacktest · line 599
$WF (v) \cap N_{a} = \emptyset .$
XVI-A@1#eq:pullbackbound · line 604
$WF (a^{*} v) \subset {(x, (d a_{x})^{T} η) : (a (x), η) \in WF (v)} .$