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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 80 81 82 83 84 85 | import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ContinuousSampledFrame
import OAI.MathematicalPhysics.DefocusingNLS.Certificates.CorrectedTorusFrame
namespace OAI
/-! # Continuous, uniformly bounded corrected torus frames -/
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
theorem nearIdentity_ringInverse (A : F →L[ℝ] F)
(hA : ‖A - ContinuousLinearMap.id ℝ F‖ < 1 / 2) :
IsUnit A ∧ A.comp (Ring.inverse A) = ContinuousLinearMap.id ℝ F ∧
(Ring.inverse A).comp A = ContinuousLinearMap.id ℝ F ∧ ‖Ring.inverse A‖ ≤ 2 := by
let : CompleteSpace F := FiniteDimensional.complete ℝ F
obtain ⟨D, hDA, hAD, hD⟩ := exists_nearIdentity_inverse A hA
let U : (F →L[ℝ] F)ˣ := ⟨A, D, hAD, hDA⟩
have hi : Ring.inverse A = D := Ring.inverse_unit U
exact ⟨⟨U, rfl⟩, hi ▸ hAD, hi ▸ hDA, hi ▸ hD⟩
theorem exists_continuous_boundedTorusFrame
(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) (hπ : Function.Surjective π) :
∃ C : ℝ, 0 < C ∧ ∃ L₀ : ℝ, ∃ ζ : Radius → F →L[ℝ] FourierL2,
ContinuousOn ζ {L : Radius | L₀ ≤ L.1} ∧
∀ L : Radius, L₀ ≤ L.1 →
(expandingCoordinates a k ha ha1 hk χ π L).comp (ζ L) =
ContinuousLinearMap.id ℝ F ∧
‖ζ L‖ ≤ C ∧
‖stableFrameProjection (ζ L) (expandingCoordinates a k ha ha1 hk χ π L)‖ ≤ C := by
classical
let : CompleteSpace F := FiniteDimensional.complete ℝ F
obtain ⟨B, hB⟩ := exists_homogeneousSchwartz_frame a k ha ha1 hk π hπ
obtain ⟨CB, hCB, hbound⟩ := sampledSchwartzFrame_uniform_bound a k ha1 hk B
obtain ⟨L₀, hL₀⟩ := sampledSchwartzFrame_coordinate_approximation a k ha ha1 hk
χ ρ hρ hχ π B hB (1 / 2) (by norm_num)
let A := fun L : Radius => (expandingCoordinates a k ha ha1 hk χ π L).comp
(sampledSchwartzFrame a k ha1 hk B L)
have hA : Continuous A := continuous_sampledSchwartzFrame_coordinates a k ha ha1 hk χ π B
have hAi (L : Radius) (hL : L₀ ≤ L.1) := nearIdentity_ringInverse (A L) (hL₀ L hL)
let ζ := fun L : Radius => (sampledSchwartzFrame a k ha1 hk B L).comp (Ring.inverse (A L))
have hζc : ContinuousOn ζ {L : Radius | L₀ ≤ L.1} := by
intro L hL
obtain ⟨U, hU⟩ := (hAi L hL).1
have hi : ContinuousAt (fun R => Ring.inverse (A R)) L := by
exact (hU ▸ NormedRing.inverse_continuousAt U).comp hA.continuousAt
exact (((continuous_sampledSchwartzFrame a k ha1 hk B).continuousAt).clm_comp hi).continuousWithinAt
have hζb (L : Radius) (hL : L₀ ≤ L.1) : ‖ζ L‖ ≤ CB * 2 :=
(ContinuousLinearMap.opNorm_comp_le _ _).trans
(mul_le_mul (hbound L) (hAi L hL).2.2.2 (norm_nonneg _) hCB)
obtain ⟨CJ, hCJ, hJ⟩ := exists_homogeneousLocalization_bound a k ha ha1 hk χ
let CP := ‖π‖ * CJ
have hCP : 0 ≤ CP := mul_nonneg (norm_nonneg _) hCJ
let C := 1 + CB * 2 + CB * 2 * CP
refine ⟨C, by dsimp [C]; positivity, L₀, ζ, hζc, ?_⟩
intro L hL
refine ⟨?_, (hζb L hL).trans (by
have hp : 0 ≤ CB * 2 * CP := by positivity
dsimp [C]
linarith), ?_⟩
· dsimp only [ζ]
rw [← ContinuousLinearMap.comp_assoc]
exact (hAi L hL).2.1
· have hp : ‖expandingCoordinates a k ha ha1 hk χ π L‖ ≤ CP :=
expandingCoordinates_norm_le a k CJ ha ha1 hk hCJ χ π hJ L
calc
_ ≤ ‖ContinuousLinearMap.id ℝ FourierL2‖ +
‖(ζ L).comp (expandingCoordinates a k ha ha1 hk χ π L)‖ := norm_sub_le _ _
_ ≤ 1 + CB * 2 * CP := add_le_add ContinuousLinearMap.norm_id_le
((ContinuousLinearMap.opNorm_comp_le _ _).trans
(mul_le_mul (hζb L hL) hp (norm_nonneg _) (by positivity)))
_ ≤ C := by dsimp [C]; linarith
end DefocusingNLS
end OAI
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