Manuscript IX-E-3@1 · IX-E-3.tex · section · 8c41417a

Sharp adjoint and support bounds

Copied inputs and conventions › Sharp adjoint and support bounds · lines 379–405

IX-E-3@1#eq:adjoint, IX-E-3@1#eq:Bsurface, IX-E-3@1#eq:upper

For $a$ supported in $M_{+}^{\circ}$ , the supplied VII-B identity is

L^{*} a = θ C^{*} b + δ (t) B (b ∣_{Σ}), b = E_{0}^{*} a,

where

B^{α β} (v) = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .

There is no $δ^{'} (t)$ term. The off-shell surface contribution is retained; an on-shell smoothing does not replace it VIIB.

For $F = supp f$ and $G = supp g$ , the copied bounds are

supp_{fun} B_{f} \subset J^{-} (F) \cap M_{+} \cup F, supp_{fun} W_{g} \subset G .

Displayed formulas in this passage

IX-E-3@1#eq:adjoint · line 382
$L^{*} a = θ C^{*} b + δ (t) B (b ∣_{Σ}), b = E_{0}^{*} a,$
IX-E-3@1#eq:Bsurface · line 389
$B^{α β} (v) = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .$
IX-E-3@1#eq:upper · line 399
$supp_{fun} B_{f} \subset J^{-} (F) \cap M_{+} \cup F, supp_{fun} W_{g} \subset G .$