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

Lemma: Advanced and covariance bounds

A local scalar probe for the sine witness › Finite advanced estimates · lines 1767–1803 · proof included or linked

Em03-F@1#lem:advanced

The preceding constants satisfy

∥ A_{p} u ∥_{X} \leq Λ∥ u ∥_{X}, ∥ k (a_{p} u) ∥_{U} \leq L_{p} ∥ u ∥_{X} .

Proof. Differentiate the spatial-derivative wave energies through order two. Integration by parts cancels the principal terms; the potential commutators involve only $V, V^{'}, V^{''}$ . Cauchy–Schwarz gives the backward-time inequality

\frac{d}{d t} E_{2} \leq c_{2} E_{2} + ∥ J ∥_{H^{2}} .

The product rule gives $∥ a_{p} u ∥_{H^{2}} \leq M_{2} E_{2} (u)$ . Backward Gronwall inequality with zero final data proves the advanced bound.

The one-dimensional Sobolev inequality bounds both $sup ∣ u ∣$ and $sup ∣ \partial_{t} u ∣$ by $2∥ u ∥_{X}$ . Hence

∥ a_{p} u ∥_{1} \leq 2 V_{K} a_{0} ∥ u ∥_{X},

∥ \partial_{t} (a_{p} u) ∥_{1} \leq 2 V_{K} (a_{t} + a_{0}) ∥ u ∥_{X} .

Lemma lem:compact proves the covariance bound.

Displayed formulas in this passage

unlabeled · line 1770
$∥ A_{p} u ∥_{X} \leq Λ∥ u ∥_{X}, ∥ k (a_{p} u) ∥_{U} \leq L_{p} ∥ u ∥_{X} .$
unlabeled · line 1783
$\frac{d}{d t} E_{2} \leq c_{2} E_{2} + ∥ J ∥_{H^{2}} .$
unlabeled · line 1795
$∥ a_{p} u ∥_{1} \leq 2 V_{K} a_{0} ∥ u ∥_{X},$
unlabeled · line 1798
$∥ \partial_{t} (a_{p} u) ∥_{1} \leq 2 V_{K} (a_{t} + a_{0}) ∥ u ∥_{X} .$

Lemma: Advanced and covariance bounds

Displayed formulas in this passage

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