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

Product and distributional pairing

Standard microlocal verification interface › Product and distributional pairing · lines 613–641

XVI-A@1#eq:producttest, XVI-A@1#eq:productbound

For $v, w \in D^{'} (X)$ , define $- WF (w) = {(x, - ξ) : (x, ξ) \in WF (w)}$ . The standard product criterion is

WF (v) \cap (- WF (w)) = \emptyset .

It suffices for the product $v w$ . If

WF_{0} (v) := WF (v) \cup {(x, 0) : x \in supp v},

then

WF (v w) \subset {(x, ξ + η) : (x, ξ) \in WF_{0} (v), (x, η) \in WF_{0} (w), ξ + η \neq = 0} .

See BDHintro (Theorem 13). Pairing two distributions requires such a product criterion, or another explicitly justified construction, together with the appropriate support conditions.

Displayed formulas in this passage

XVI-A@1#eq:producttest · line 618
$WF (v) \cap (- WF (w)) = \emptyset .$
XVI-A@1#eq:productbound · line 629
$WF (v w) \subset {(x, ξ + η) : (x, ξ) \in WF_{0} (v), (x, η) \in WF_{0} (w), ξ + η \neq = 0} .$
unlabeled · line 624
$WF_{0} (v) := WF (v) \cup {(x, 0) : x \in supp v},$