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

Lemma: Product covariance estimate

A quantitative transfer estimate from advanced evolution › Conversion to the Unruh moment topology · lines 1296–1318 · proof included or linked

Em03-D@1#lem:productbound

For a smooth $u$ of finite $X$ norm,

∥ k (a u) ∥_{U} \leq L_{R} ∥ u ∥_{X} .

Proof. The one-dimensional Sobolev inequality gives, with a conservative constant,

t, x sup ∣ u (t, x) ∣, t, x sup ∣ \partial_{t} u (t, x) ∣ \leq 2∥ u ∥_{X} .

Therefore

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

Apply Proposition prop:compactcovariance.

Displayed formulas in this passage

unlabeled · line 1299
$∥ k (a u) ∥_{U} \leq L_{R} ∥ u ∥_{X} .$
unlabeled · line 1307
$t, x sup ∣ u (t, x) ∣, t, x sup ∣ \partial_{t} u (t, x) ∣ \leq 2∥ u ∥_{X} .$
unlabeled · line 1312
$∥ a u ∥_{1} \leq 2 V_{K} a_{0} ∥ u ∥_{X}, ∥ \partial_{t} (a u) ∥_{1} \leq 2 V_{K} (a_{t} + a_{0}) ∥ u ∥_{X} .$