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

Proposition: A compact balanced kernel with smooth tested kernels

G2: the chosen locus and the relation audit › Smooth residual testing can hide the relevant wavefront · lines 1117–1158 · proof included or linked

XIV-G@1#prop:testing, XIV-G@1#eq:testingexample

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There exists a compact distribution $T (a, b, z)$ on $A \times B \times Z$ such that

WF (T) \subset Z_{3} = {(a, ξ; b, η; z, ζ) : ξ + η + ζ = 0} ∖ 0,

and $T$ has nonzero wavefront directions on $a = z$ , while

L_{G} (a, b) = ⟨ T (a, b, z), G (z) ⟩

is smooth for every $G \in C_{c}^{\infty} (Z)$ .

Proof. Choose smooth compact functions $χ_{A}, χ_{B}, χ_{Z}$ , nonzero near the retained $a, b, z$ coordinates, with $χ_{A} χ_{Z}$ not identically zero. Set

T (a, b, z) = χ_{A} (a) χ_{B} (b) χ_{Z} (z) δ^{(4)} (a - z) .

Its support is compact. The conormal directions of the attachment diagonal are

(ξ, 0, - ξ), ξ \neq = 0.

They have zero covector sum. Wherever the smooth weight is nonzero, the diagonal delta has these nonzero wavefront directions; multiplication by that nonvanishing weight does not remove them.

Pairing the residual variable gives

L_{G} (a, b) = χ_{A} (a) χ_{B} (b) χ_{Z} (a) G (a),

which is smooth for every smooth compact test $G$ .

Displayed formulas in this passage

XIV-G@1#eq:testingexample · line 1137
$T (a, b, z) = χ_{A} (a) χ_{B} (b) χ_{Z} (z) δ^{(4)} (a - z) .$
unlabeled · line 1121
$WF (T) \subset Z_{3} = {(a, ξ; b, η; z, ζ) : ξ + η + ζ = 0} ∖ 0,$
unlabeled · line 1127
$L_{G} (a, b) = ⟨ T (a, b, z), G (z) ⟩$
unlabeled · line 1144
$(ξ, 0, - ξ), ξ \neq = 0.$
unlabeled · line 1153
$L_{G} (a, b) = χ_{A} (a) χ_{B} (b) χ_{Z} (a) G (a),$

Proposition: A compact balanced kernel with smooth tested kernels

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