Manuscript Em03-F@1 · Em03-F-v1.tex · section · 0d4d148b

Finite advanced estimates

A local scalar probe for the sine witness › Finite advanced estimates · lines 1724–1766

In reduced coordinates put

a_{p} = f (r) ρ = - Y_{0} b / r, A_{p} u = G_{A, D} (a_{p} u), v_{A} = G_{A, D} z_{red} .

On the coupling support, $- a_{p} v_{A} = b$ .

Use the time interval

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

and the norm

∥ 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}) .

Conservative explicit constants are

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

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

Let $X = 2 L + T$ , $V_{K} = 2 X Δ$ , and define

a_{0} = ∥ a_{p} ∥_{\infty}, a_{t} = ∥ \partial_{t} a_{p} ∥_{\infty},

L_{p} = \frac{2 V _{K} C _{X}}{2 π} A_{0} a_{0}^{2} + A_{1} (a_{t} + a_{0})^{2} .

Displayed formulas in this passage

unlabeled · line 1727
$a_{p} = f (r) ρ = - Y_{0} b / r, A_{p} u = G_{A, D} (a_{p} u), v_{A} = G_{A, D} z_{red} .$
unlabeled · line 1735
$I = [t_{m} - T - 1, t_{m} + T + 3], Δ = ∣ I ∣,$
unlabeled · line 1740
$∥ u ∥_{X}^{2} = t \in I sup j = 0 \sum^{2} (∥ \partial_{t} \partial_{x}^{j} u ∥_{2}^{2} + ∥ \partial_{x}^{j + 1} u ∥_{2}^{2} + ∥ \partial_{x}^{j} u ∥_{2}^{2}) .$
unlabeled · line 1747
$c_{2} = 64 (1 + j = 0 \sum^{2} ∥ \partial_{x}^{j} V ∥_{\infty}),$
unlabeled · line 1750
$M_{2} = 8 t \in I sup j = 0 \sum^{2} ∥ \partial_{x}^{j} a_{p} (t, \cdot) ∥_{\infty}, Λ = Δ e^{c_{2} Δ} M_{2} .$
unlabeled · line 1757
$a_{0} = ∥ a_{p} ∥_{\infty}, a_{t} = ∥ \partial_{t} a_{p} ∥_{\infty},$
unlabeled · line 1761
$L_{p} = \frac{2 V _{K} C _{X}}{2 π} A_{0} a_{0}^{2} + A_{1} (a_{t} + a_{0})^{2} .$