| import Mathlib |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian |
| namespace FlatSimilarity |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
|
|
| def SymbolBounds (c : ℝ) (Af Bf : V → ℝ) : Prop := |
| 0<c ∧ (∃ κ : ℝ, 0<κ ∧ ∀ x, κ≤Af x) ∧ |
| ContDiff ℝ ∞ Af ∧ ContDiff ℝ ∞ Bf ∧ |
| ∀ n : ℕ, ∃ C : ℝ, 0≤C ∧ ∀ x, |
| ‖iteratedFDeriv ℝ n (fun y => Af y-c^2) x‖+‖iteratedFDeriv ℝ n Bf x‖≤ |
| C*(1+‖x‖)^(-(4:ℝ)-n) |
|
|
| def MovesClassically (Af Bf : V → ℝ) (d : C₀(V,ℂ)) (u : ℝ → C₀(V,ℂ)) : Prop := |
| u 0=d ∧ ContinuousOn u (Icc (0:ℝ) 1) ∧ |
| (∀ t ∈ Ioo (0:ℝ) 1, ContDiff ℝ 4 (fun x => u t x)) ∧ |
| (∀ t ∈ Ioo (0:ℝ) 1, ∀ n : ℕ, n≤4 → ∃ M : ℝ, ∀ x, |
| ‖iteratedFDeriv ℝ n (fun y => u t y) x‖≤M) ∧ |
| (∀ t ∈ Ioo (0:ℝ) 1, ∃ p : V →ᵇ ℂ, |
| HasDerivAt (fun r => (u r).toBCF) p t ∧ ∀ x, |
| p x=-((Real.exp (-t)/16*Af (Real.exp (t/4) • x) : ℝ) : ℂ)* |
| Δ (Δ (fun y => u t y)) x-(1/2+(Bf (Real.exp (t/4) • x) : ℂ))*u t x) |
| end FlatSimilarity |
|
|
| end |
|
|
| end OAI |
|
|
