Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Why the product boundary is unchanged

Yang–Feldman characterization and quantum order › Why the product boundary is unchanged · lines 1108–1142

The inherited X-B-ST nonextension statement remains in force XBST (:nonextension). Its auxiliary kernel

Δ_{F} = Δ_{+} - i E_{A}, P_{m, x} Δ_{F} = - i δ_{diag},

contains the full nonzero diagonal conormal in its wavefront set. Thus at every diagonal point it contains both

(x, x; ξ, - ξ) and (x, x; - ξ, ξ), ξ \neq = 0.

The ordinary wavefront product criterion therefore does not define $Δ_{F} \cdot Δ_{F}$ there Hormander (Chapter 8).

Equations (eq:second-map) and (eq:iteration) never perform that operation. The second Green propagation in (eq:iteration) acts on the smooth compact test $a_{c}$ . Sequential action on smooth tests is not an extension of an identified-endpoint square of singular kernels.

No $T_{2} (V, V)$ , $T_{3}$ , or general nonlinear-insertion $R_{2}$ is supplied. Failure of the sufficient product criterion is not a universal nonexistence theorem for renormalized extensions. Such extensions and their normalizations are additional analytical tasks BF,DFreference.

Displayed formulas in this passage

unlabeled · line 1113
$Δ_{F} = Δ_{+} - i E_{A}, P_{m, x} Δ_{F} = - i δ_{diag},$
unlabeled · line 1120
$(x, x; ξ, - ξ) and (x, x; - ξ, ξ), ξ \neq = 0.$