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d5b562f | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactLowCommutator
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProductSmoothRemainder
import OAI.MathematicalPhysics.DefocusingNLS.Certificates.UniformCutoffLimit
namespace OAI
/-! # Strong compactness of sampled compact-coefficient commutators -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem tendsto_expandingSample_commutator (a M R : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < ((N + 1 : ℕ) : ℝ))
(L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop)
(f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M)
(hlocal : ∀ R ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ y : E, ‖y‖ ≤ R →
‖expandingTorusFunction a (N + 1 : ℕ) (L n) (f n)
(euclideanToTorus ((L n)⁻¹ • y))‖ < ε)
(K : 𝓢(E, ℂ)) (hK : ∀ y : E, R < ‖y‖ → K y = 0)
(j : Fin (N + 1) → Fin 12) :
Tendsto (fun n => expandingProductCommutator a (L n) (N + 1) ha ha1 hN (hL n) j
(schwartzTorusSample a (N + 1 : ℕ) (L n) ha1 hN (hL n) (radianFourierKernel K)) (f n))
atTop (𝓝 0) := by
obtain ⟨C, hC, hc⟩ := exists_expandingProductCommutator_smooth_remainder a N ha ha1 hN
(radianFourierKernel K)
have hM : 0 ≤ M := (norm_nonneg (f 0)).trans (hf 0)
let q := fun n => schwartzTorusSample a (N + 1 : ℕ) (L n) ha1 hN (hL n) (radianFourierKernel K)
apply tendsto_zero_of_uniform_cutoff
(fun n => expandingProductCommutator a (L n) (N + 1) ha ha1 hN (hL n) j (q n) (f n))
(fun S hS n => expandingProductCommutator a (L n) (N + 1) ha ha1 hN (hL n) j (q n)
(expandingSmoothLow (L n) S (by linarith) (f n))) (C * M) (mul_nonneg hC hM)
· intro S hS
exact tendsto_expandingSample_low_commutator a M R S (N + 1) ha ha1 hN (by linarith)
L hL hLinf f hf hlocal K hK j
· intro S hS n
calc
_ ≤ (C / S) * ‖f n‖ := hc (L n) S (hL n) hS (f n) j
_ ≤ (C / S) * M := mul_le_mul_of_nonneg_left (hf n) (div_nonneg hC (by linarith))
_ = (C * M) / S := by ring
end DefocusingNLS
end OAI
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