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

Lemma: Advanced energy estimate

A quantitative transfer estimate from advanced evolution › Reduced advanced operator and explicit constants · lines 1183–1217 · proof included or linked

Em03-D@1#lem:energy

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The operator $A$ satisfies

∥ A u ∥_{X} \leq Λ∥ u ∥_{X} .

Proof. For a free inhomogeneous solution with zero final data, differentiate the sum of energies in the definition of $X$ . Spatial integration by parts cancels the principal wave terms. The remaining potential and commutator terms contain only $V, V^{'}, V^{''}$ . Cauchy–Schwarz and the product rule give

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

with the reversed-time version for advanced evolution. The stated $c_{2}$ dominates all the finite product-rule coefficients.

The spatial product estimate

∥ a (t) u (t) ∥_{H^{2}} \leq M_{2} (a) E_{2} (u; t)

follows by expanding the derivatives through order two. Backward Gronwall inequality on $I$ gives

∥ G_{A, D} (a u) ∥_{X} \leq Δ e^{c_{2} Δ} M_{2} (a) ∥ u ∥_{X} .

Displayed formulas in this passage

unlabeled · line 1186
$∥ A u ∥_{X} \leq Λ∥ u ∥_{X} .$
unlabeled · line 1198
$\frac{d}{d t} E_{2} (t) \leq c_{2} E_{2} (t) + ∥ J (t) ∥_{H^{2}},$
unlabeled · line 1207
$∥ a (t) u (t) ∥_{H^{2}} \leq M_{2} (a) E_{2} (u; t)$
unlabeled · line 1213
$∥ G_{A, D} (a u) ∥_{X} \leq Δ e^{c_{2} Δ} M_{2} (a) ∥ u ∥_{X} .$

Lemma: Advanced energy estimate

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