Manuscript IV@1 · IV.tex · section · 351cb2d7

Removing the initial-time edge

Model 1: the fixed vacuum TT plane › Removing the initial-time edge · lines 1103–1137

IV@1#eq:edge

Choose $0 \leq ϑ \leq 1$ , smooth, with $ϑ (s) = 0$ for $s \leq 0$ and $1$ for $s \geq 1$ . Set

g_{a, f, ε} (t, x) = ϑ ((t - t_{0}) / ε) b_{a, f} (t, x), a = e, r,

and leave $g_{0, f}$ unchanged. For $0 < ε \leq 1$ , these are genuine compact smooth spacetime smearings. Put

B_{a, f} = \frac{1}{2 m} t_{0} \leq t \leq t_{0} + 1 sup ∥ b_{a, f} (t) ∥_{2} .

Equation (eq:ug) and Minkowski's integral inequality give

∥ u_{a, f, ε} - u_{a, f} ∥ \leq B_{a, f} ε .

Let $A^{(1)}_{f, ε} (λ)$ use these regulated vectors. All these operators satisfy the preceding proof with a common constant

C = 16 [1 + ∥ u_{0, f} ∥ + ∣ λ ∣ (∥ u_{e, f} ∥ + ∥ u_{r, f} ∥ + B_{e, f} + B_{r, f})]

in the number estimates. Their difference has the same estimates with

D_{ε} = 16∣ λ ∣ a = e, r \sum ∥ u_{a, f, ε} - u_{a, f} ∥ \leq 16∣ λ ∣ (B_{e, f} + B_{r, f}) ε .

Displayed formulas in this passage

IV@1#eq:edge · line 1118
$∥ u_{a, f, ε} - u_{a, f} ∥ \leq B_{a, f} ε .$
unlabeled · line 1107
$g_{a, f, ε} (t, x) = ϑ ((t - t_{0}) / ε) b_{a, f} (t, x), a = e, r,$
unlabeled · line 1113
$B_{a, f} = \frac{1}{2 m} t_{0} \leq t \leq t_{0} + 1 sup ∥ b_{a, f} (t) ∥_{2} .$
unlabeled · line 1125
$C = 16 [1 + ∥ u_{0, f} ∥ + ∣ λ ∣ (∥ u_{e, f} ∥ + ∥ u_{r, f} ∥ + B_{e, f} + B_{r, f})]$
unlabeled · line 1131
$D_{ε} = 16∣ λ ∣ a = e, r \sum ∥ u_{a, f, ε} - u_{a, f} ∥ \leq 16∣ λ ∣ (B_{e, f} + B_{r, f}) ε .$