Manuscript X-A-conv@1 · X-A-conv.tex · lemma · c76fffa3

Lemma: Pure-gauge response of the regulated operator

Regulator preflight and an exact positive defect · lines 1035–1070 · proof included or linked

X-A-conv@1#lem:preflight, X-A-conv@1#eq:preflight

For every $ξ \in g_{U}$ ,

L_{g, ε} (K ξ) := E_{0} C T_{g, ε} (K ξ) = S_{ε} ξ .

The right-hand side has zero Cauchy data near $Σ$ .

Proof. Since $g = 1$ on a neighborhood of $supp ξ$ ,

g K ξ = K ξ .

Constant-coefficient differentiation commutes with convolution, so

T_{g, ε} (K ξ) = K (S_{ε} ξ), C K (S_{ε} ξ) = □_{η} (S_{ε} ξ) .

The identity $C K = □_{η}$ follows directly from (eq:C): the derivative of the divergence of $ξ$ cancels the derivative of half the trace of $K ξ$ .

The support of $S_{ε} ξ$ lies in the Euclidean $ε$ -neighborhood of $supp ξ$ , which is separated from $Σ$ . Lemma lem:Cauchy therefore gives

E_{0} □_{η} (S_{ε} ξ) = S_{ε} ξ .

Displayed formulas in this passage

X-A-conv@1#eq:preflight · line 1038
$L_{g, ε} (K ξ) := E_{0} C T_{g, ε} (K ξ) = S_{ε} ξ .$
unlabeled · line 1051
$g K ξ = K ξ .$
unlabeled · line 1055
$T_{g, ε} (K ξ) = K (S_{ε} ξ), C K (S_{ε} ξ) = □_{η} (S_{ε} ξ) .$
unlabeled · line 1067
$E_{0} □_{η} (S_{ε} ξ) = S_{ε} ξ .$