L²_{t,x} transfer for the cutoff1 source contract #
From a slicewise bound ‖F(t)‖²_{L²} ≤ c² ‖∇T(t)‖²_{L²} (t ∈ (0,1)) to the left side of
Cutoff1SourceContract, in terms of the space-time gradient norm of T.
theorem
AVenhance.Infra.Section5.Contracts.sb_l2_timeL2_le
{F T : ℝ → Homogenization.Vec 2 → ℝ}
(hT : ContDiffOn ℝ (↑⊤) (fun (p : ℝ × Homogenization.Vec 2) => T p.1 p.2) (Set.Ici 0 ×ˢ Set.univ))
{c : ℝ}
(hc : 0 ≤ c)
(h : ∀ t ∈ Set.Ioo 0 1, l2NormSq (F t) ≤ c ^ 2 * gradNormSq fun (x : Homogenization.Vec 2) => spaceGrad (T t) x)
: