Manuscript Em03-D@1 · Em03-D-v1.tex · section · 450d878a

Reduced advanced operator and explicit constants

A quantitative transfer estimate from advanced evolution › Reduced advanced operator and explicit constants · lines 1135–1182

The s-wave coupled operator is

(D ϵ a ϵ a D), a (t, x) = F (r (x)) ρ (t, r (x)) .

In particular, because $ρ = - F$ and $F = Y_{0} f / (r F (r))$ ,

a = - Y_{0} f / r, - a v = f on K .

Here the first $F$ in $ρ = - F$ is the bulk test, while $F (r) = 1 - r^{- 1}$ is the metric coefficient.

Work on the finite time interval

I = [t_{-}, t_{+}] = [- T - 1, T + 3], Δ = t_{+} - t_{-} .

For a reduced function $u$ , define

∥ u ∥_{X}^{2} = t \in I sup j = 0 \sum^{2} (∥ \partial_{t} \partial_{x}^{j} u ∥_{2}^{2} + ∥ \partial_{x}^{j + 1} u ∥_{2}^{2} + ∥ \partial_{x}^{j} u ∥_{2}^{2}) .

Let $G_{A, D}$ be the reduced free advanced Green operator, and define

A u = G_{A, D} (a u) .

Its source is compactly supported inside the coupling slab.

The following deliberately conservative constants suffice:

c_{2} = 64 (1 + j = 0 \sum^{2} ∥ \partial_{x}^{j} V ∥_{\infty}),

M_{2} (a) = 8 t \in I sup j = 0 \sum^{2} ∥ \partial_{x}^{j} a (t, \cdot) ∥_{\infty}, Λ = Δ e^{c_{2} Δ} M_{2} (a) .

Displayed formulas in this passage

unlabeled · line 1138
$(D ϵ a ϵ a D), a (t, x) = F (r (x)) ρ (t, r (x)) .$
unlabeled · line 1146
$a = - Y_{0} f / r, - a v = f on K .$
unlabeled · line 1154
$I = [t_{-}, t_{+}] = [- T - 1, T + 3], Δ = t_{+} - t_{-} .$
unlabeled · line 1158
$∥ u ∥_{X}^{2} = t \in I sup j = 0 \sum^{2} (∥ \partial_{t} \partial_{x}^{j} u ∥_{2}^{2} + ∥ \partial_{x}^{j + 1} u ∥_{2}^{2} + ∥ \partial_{x}^{j} u ∥_{2}^{2}) .$
unlabeled · line 1167
$A u = G_{A, D} (a u) .$
unlabeled · line 1173
$c_{2} = 64 (1 + j = 0 \sum^{2} ∥ \partial_{x}^{j} V ∥_{\infty}),$
unlabeled · line 1176
$M_{2} (a) = 8 t \in I sup j = 0 \sum^{2} ∥ \partial_{x}^{j} a (t, \cdot) ∥_{\infty}, Λ = Δ e^{c_{2} Δ} M_{2} (a) .$