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7f37411 | 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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphSourceFixedGeometry
import OAI.MeasureTheory.FalconerCompletion.Graphs.SpherePlaneWaveSub
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
namespace CurrentFalconer
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
theorem graph_source_four_spatial_decay
{sig c₀ Ccap G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀)
(hCcap : 0 ≤ Ccap) (hG : 0 ≤ G) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (a p v L : ℝ), p-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p →
let R := (2:ℝ)^N
let δ := min c₀ (((2:ℝ)^(p-a))⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-p)*R^(30*sig)
let A := 1+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹
∀ (μ₁ μ₂ μ₃ μ₄ : Measure V),
∀ [IsFiniteMeasure μ₁] [IsFiniteMeasure μ₂] [IsFiniteMeasure μ₃] [IsFiniteMeasure μ₄],
∀ (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ : V) (K J₁ J₂ : ℝ)
(θ₁ θ₂ : V → ℂ) (b₁ b₂ b₃ b₄ : V → V → ℂ)
(P₁ P₂ B₁ B₂ B₃ B₄ : ℝ),
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 →
(∫ u : ℝ, ‖Ψ u‖) ≤ K → (∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) →
‖w₁‖=1 → ‖w₂‖=1 → 0 ≤ K → 0 ≤ J₁ → 0 ≤ J₂ →
Measurable θ₁ → Measurable θ₂ →
Measurable (Function.uncurry b₁) → Measurable (Function.uncurry b₂) →
Measurable (Function.uncurry b₃) → Measurable (Function.uncurry b₄) →
0 ≤ P₁ → 0 ≤ P₂ → 0 ≤ B₁ → 0 ≤ B₂ → 0 ≤ B₃ → 0 ≤ B₄ →
(∀ w, ‖w‖=1 → ‖θ₁ w‖ ≤ P₁) → (∀ w, ‖w‖=1 → ‖θ₂ w‖ ≤ P₂) →
(∀ x w, ‖w‖=1 → ‖b₁ x w‖ ≤ B₁) → (∀ x w, ‖w‖=1 → ‖b₂ x w‖ ≤ B₂) →
(∀ x w, ‖w‖=1 → ‖b₃ x w‖ ≤ B₃) → (∀ x w, ‖w‖=1 → ‖b₄ x w‖ ≤ B₄) →
(∀ᵐ z ∂(μ₁.prod μ₂).prod (μ₃.prod μ₄),
‖z.1.1-z.1.2‖ ≤ G*(2:ℝ)^(-a) ∧ ‖z.2.1-z.2.2‖ ≤ G*(2:ℝ)^(-a) ∧
‖(z.1.1-z.1.2)-d₁‖ ≤ G*(2:ℝ)^(-p) ∧ ‖(z.2.1-z.2.2)-d₂‖ ≤ G*(2:ℝ)^(-p)) →
¬(‖d₁-inner ℝ d₁ w₁ • w₁‖ ≤ τ ∧
‖d₂-inner ℝ d₂ w₂ • w₂‖ ≤ τ ∧ |inner ℝ w₁ d₁+inner ℝ w₂ d₂| ≤ τ) →
(∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ₁ b₁ b₂ z)) →
(∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ₂ b₃ b₄ z)) →
(∀ z j, j ≤ m → ∀ w, ‖w‖=1 →
‖iteratedFDeriv ℝ j (graphPairAmplitude θ₁ b₁ b₂ z) w‖ ≤ J₁*A^j) →
(∀ z j, j ≤ m → ∀ w, ‖w‖=1 →
‖iteratedFDeriv ℝ j (graphPairAmplitude θ₂ b₃ b₄ z) w‖ ≤ J₂*A^j) →
(∀ w, ‖w‖=1 → θ₁ w ≠ 0 → ‖w-w₁‖ ≤ Ccap*δ) →
(∀ w, ‖w‖=1 → θ₂ w ≠ 0 → ‖w-w₂‖ ≤ Ccap*δ) →
‖∫ u, Ψ u*graphPairMoment μ₁ μ₂ b₁ b₂ θ₁ ((2:ℝ)^v*u)*
graphPairMoment μ₃ μ₄ b₃ b₄ θ₂ ((2:ℝ)^v*u)‖ ≤
C*K*J₁*J₂*R^(-T)*((μ₁.prod μ₂).prod (μ₃.prod μ₄)).real univ := by
obtain ⟨m,C,hC,N₀,hN₀⟩ := graph_source_fixed_nonedge_decay V hsig hc₀ hCcap hG T
refine ⟨m,C,hC,N₀,?_⟩
intro N hN a p v L hh hL hq R δ τ A μ₁ μ₂ μ₃ μ₄ _ _ _ _
Ψ w₁ w₂ d₁ d₂ K J₁ J₂ θ₁ θ₂ b₁ b₂ b₃ b₄ P₁ P₂ B₁ B₂ B₃ B₄
hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hJ₁ hJ₂ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄
hP₁ hP₂ hB₁ hB₂ hB₃ hB₄ hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄ hgeo hne
ha₁ ha₂ hd₁ hd₂ hs₁ hs₂
have hiΨ : Integrable Ψ := hΨ.continuous.integrable_of_hasCompactSupport
(isCompact_Icc.of_isClosed_subset (isClosed_tsupport Ψ) hsΨ)
rw [graph_four_spatial_factorization μ₁ μ₂ μ₃ μ₄ hΨ.continuous.measurable hiΨ ((2:ℝ)^v) θ₁ θ₂ b₁ b₂ b₃ b₄ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄ hP₁ hP₂ hB₁ hB₂ hB₃ hB₄ hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄]
apply norm_integral_le_of_norm_le_const
filter_upwards [hgeo] with z hz
apply hN₀ N hN a p v L hh hL hq Ψ w₁ w₂ d₁ d₂
(z.1.1-z.1.2) (z.2.1-z.2.2) K J₁ J₂
(graphPairAmplitude θ₁ b₁ b₂ z.1) (graphPairAmplitude θ₂ b₃ b₄ z.2)
hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hJ₁ hJ₂ hz.1 hz.2.1 hz.2.2.1 hz.2.2.2 hne
(ha₁ z.1) (ha₂ z.2) (hd₁ z.1) (hd₂ z.2)
· intro w hw hne
apply hs₁ w hw
intro hz
apply hne
simp [graphPairAmplitude,hz]
· intro w hw hne
apply hs₂ w hw
intro hz
apply hne
simp [graphPairAmplitude,hz]
end CurrentFalconer
end
end
end OAI
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