Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Output geometry and contact ambiguity

Output geometry and contact ambiguity · lines 1773–1806

Em01-B@3#eq:pushforward

Let

π (a, b, z, y) = (z, y) .

For a compactly supported existing distribution,

WF (π_{*} t) \subset {(z, y; p, q) : \exists a, b, (a, b, z, y; 0, 0, p, q) \in WF (t)} .

This operation does not define a missing joint product.

Set

Z_{W} = {(w; ξ_{a}, ξ_{b}, ξ_{z}, ξ_{y}) : ξ_{a} + ξ_{b} + ξ_{z} + ξ_{y} = 0} ∖ 0,

Z_{2} = {(z, p; y, q) : p + q = 0} ∖ 0.

If $WF (t) \subset Z_{W}$ , then $WF (π_{*} t) \subset Z_{2}$ : the eliminated covectors in (eq:pushforward) are zero. Smooth weights and differentiation preserve this intermediate inclusion.

Write $π_{#} Γ$ for the corresponding cotangent projection relation, not for distributional pushforward.

Displayed formulas in this passage

Em01-B@3#eq:pushforward · line 1781
$WF (π_{*} t) \subset {(z, y; p, q) : \exists a, b, (a, b, z, y; 0, 0, p, q) \in WF (t)} .$
unlabeled · line 1777
$π (a, b, z, y) = (z, y) .$
unlabeled · line 1791
$Z_{W} = {(w; ξ_{a}, ξ_{b}, ξ_{z}, ξ_{y}) : ξ_{a} + ξ_{b} + ξ_{z} + ξ_{y} = 0} ∖ 0,$
unlabeled · line 1796
$Z_{2} = {(z, p; y, q) : p + q = 0} ∖ 0.$

Output geometry and contact ambiguity

Displayed formulas in this passage

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