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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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
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