Manuscript XIV-G@1 · XIV-G.tex · section · 791c6644

Individual relations and order convention

A factorwise pullback audit within G2 › Individual relations and order convention · lines 1258–1305

XIV-G@1#eq:orderconvention

Define

S_{R D} = {q_{R} = 0, a = z} \subset W, S_{12} = {q_{1} = q_{2} = 0} \subset W .

Near the retained point,

codim S_{R D} = 5, codim S_{12} = 2.

The first assertion follows because $d q_{R}$ has no $z$ component and is nonzero. The second follows from independence of $d q_{1}, d q_{2}$ .

Use the conormal order convention in which a normal Fourier amplitude of order $d$ on a codimension- $k$ submanifold of an $n$ -dimensional space gives conormal order

μ = d - \frac{n - 2 k}{4} .

Principal symbols are taken in the usual conormal symbol quotient, with half-density and Maslov factors and the fixed numerical fiber factors. Local defining functions and the fixed coordinate density trivialize those factors for the calculations here Wunsch (Section 8.1).

The individual relations are:

Λ_{U} = N^{*} S_{R D} ∖ 0,

and the selected positive sector

Λ_{V}^{+} = {(w, θ_{1} d q_{1} + θ_{2} d q_{2}) : w \in S_{12}, θ_{1}, θ_{2} > 0} .

When a joint normal order for $V$ is used, it is understood microlocally in a sector where both positive frequencies are nonzero and comparable. No statement about the axes of that normal sector is needed.

These are relations of the two factors in (eq:UV), not the relation of (eq:target).

Displayed formulas in this passage

XIV-G@1#eq:orderconvention · line 1276
$μ = d - \frac{n - 2 k}{4} .$
unlabeled · line 1261
$S_{R D} = {q_{R} = 0, a = z} \subset W, S_{12} = {q_{1} = q_{2} = 0} \subset W .$
unlabeled · line 1266
$codim S_{R D} = 5, codim S_{12} = 2.$
unlabeled · line 1287
$Λ_{U} = N^{*} S_{R D} ∖ 0,$
unlabeled · line 1291
$Λ_{V}^{+} = {(w, θ_{1} d q_{1} + θ_{2} d q_{2}) : w \in S_{12}, θ_{1}, θ_{2} > 0} .$

Individual relations and order convention

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