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fac6777 | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusLinearStep
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffCoordinateInverse
namespace OAI
/-! # Continuous normalized torus data from kernel decay and coordinate transfer -/
open scoped SchwartzMap NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
theorem continuous_torusLinearSteps_of_transfer {F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
(a k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(χ : 𝓢(E, ℂ)) (ρ : ℝ) (hρ : 0 < ρ)
(hχ : ∀ y : E, ‖y‖ ≤ ρ → χ y = 1)
(π : HomogeneousY a k →L[ℝ] F)
(A : ℝ≥0 → Radius → FourierL2 →L[ℝ] FourierL2)
(hkernel : ∀ β : ℝ, 0 < β → ∃ T₀ : ℝ, 0 ≤ T₀ ∧
∀ T : ℝ≥0, T₀ ≤ T → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 →
∀ f : FourierL2, ‖f‖ ≤ 1 →
π (homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ f) = 0 → ‖A T L f‖ < β)
(hcoord : ∀ T, HasContractingTorusCoordinateTransfer a k ha ha1 hk χ π T (A T))
(hA : ∀ T, ∃ C : ℝ, 0 ≤ C ∧ ∀ L, ‖A T L‖ ≤ C) :
∃ T₀ : ℝ, 0 ≤ T₀ ∧ ∀ T : ℝ≥0, T₀ ≤ T →
HasContinuousTorusLinearStep (F := π.range) T (A T) := by
have hsurj : Function.Surjective π.rangeRestrict := by
rintro ⟨v, u, hu⟩
exact ⟨u, Subtype.ext hu⟩
obtain ⟨C, hC, Lf, ζ, hζc, hζ⟩ := exists_continuous_boundedTorusFrame
a k ha ha1 hk χ ρ hρ hχ π.rangeRestrict hsurj
obtain ⟨T₀, hT₀, hT⟩ := hkernel ((1 / 8) / (C * C)) (by positivity)
refine ⟨T₀, hT₀, ?_⟩
intro T hTT
obtain ⟨Lk, hLk⟩ := hT T hTT
obtain ⟨D, R, hDR, hR, hdefect⟩ := hcoord T
obtain ⟨CA, hCA, hAb⟩ := hA T
obtain ⟨CJ, hCJ, hJ⟩ := exists_homogeneousLocalization_bound a k ha ha1 hk χ
let Cπ := ‖π.rangeRestrict‖ * CJ
have hCπ : 0 ≤ Cπ := mul_nonneg (norm_nonneg π.rangeRestrict) hCJ
apply continuous_torusLinearStep_of_frames T (A T) ζ
(expandingCoordinates a k ha ha1 hk χ π.rangeRestrict) D R hDR hR
(max Lf Lk) C Cπ CA hC.le hCπ hCA
· intro L hL
obtain ⟨he, hz, hp⟩ := hζ L ((le_max_left _ _).trans hL)
exact ⟨fun v => congrArg (fun B : π.range →L[ℝ] π.range => B v) he, hz,
expandingCoordinates_norm_le a k CJ ha ha1 hk hCJ χ π.rangeRestrict hJ L, hp⟩
· intro L hL
have hLf : Lf ≤ L.1 := (le_max_left _ _).trans hL
have hLR : Lf ≤ (torusEndpointRadius T L).1 := hLf.trans (torusEndpointRadius_ge T L)
have hb := stableProjectedBlock_of_kernel_bound (ζ L) (ζ (torusEndpointRadius T L))
(expandingCoordinates a k ha ha1 hk χ π.rangeRestrict L)
(expandingCoordinates a k ha ha1 hk χ π.rangeRestrict (torusEndpointRadius T L))
(A T L) C ((1 / 8) / (C * C)) hC.le (by positivity)
(fun v => congrArg (fun B : π.range →L[ℝ] π.range => B v) (hζ L hLf).1)
(hζ L hLf).2.2 (hζ _ hLR).2.2
have hunit : ∀ f : FourierL2, ‖f‖ ≤ 1 →
expandingCoordinates a k ha ha1 hk χ π.rangeRestrict L f = 0 →
‖A T L f‖ ≤ (1 / 8) / (C * C) := by
intro f hf hz
have he : π (homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ f) = 0 :=
congrArg (fun v : π.range => (v : F)) hz
exact (hLk L ((le_max_right _ _).trans hL) f hf he).le
apply (hb hunit).trans_eq
field_simp
· exact hAb
· exact hdefect
· apply hζc.mono
intro L hL
exact (le_max_left Lf Lk).trans hL
· exact continuous_expandingCoordinates_apply a k ha ha1 hk χ π.rangeRestrict
end DefocusingNLS
end OAI
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