Manuscript III@2 · III.tex · section · 82dad3e5

Five distinct maps

Sign, inverse, boundary, and kernel audit › Five distinct maps · lines 2940–2991

Map	Identity used	Domain or prescription
Newton $N$	$Δ N s = s, N Δ u = u$	Compact or Schwartz sources; decaying inverse branch.
Finite-radius $N_{R}$	$N_{R} Δ u = u + B_{R} [u]$	Ball integral; explicit boundary traces.
Fourier $N_{ρ}$	$N_{ρ} Δ = χ_{ρ} (D)$	Smooth infrared/ultraviolet multiplier; not an exact inverse.
Retarded $R_{□}$	$□_{η} R_{□} F = F, R_{□} □_{η} a = a$	Future support from a compact set.
Canonical $M^{- 1}$	Inverse of $diag (- 4Δ, Δ I)$	Leading phase-space gauge conditions; decaying descriptors.

The canonical inverse uses $N$ , but is a map between different phase-space quantities with a different normalization. None of the five maps is identified merely because their formulas contain an inverse Laplacian or wave operator.

The compensation signs are

w_{P}^{i} [K ξ] = - ξ^{i}, w_{P}^{0} [K ξ] = ξ_{0} = - ξ^{0},

whereas

w_{R} (K ξ) = R_{□} □_{η} ξ = ξ

on $G_{R}$ .

A homogeneous wave is zero in the past-trivial class but need not be zero in the compact-Cauchy-data class. This is why the retarded section proof does not transplant to C1.

Displayed formulas in this passage

unlabeled · line 2978
$w_{P}^{i} [K ξ] = - ξ^{i}, w_{P}^{0} [K ξ] = ξ_{0} = - ξ^{0},$
unlabeled · line 2983
$w_{R} (K ξ) = R_{□} □_{η} ξ = ξ$