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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 49 50 51 52 53 54 55 56 57 58 59 60 61 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingFilteredPhysicalEnergy
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactDerivativeLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactPhysicalLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusProductCommutator
namespace OAI
/-! # Compactness of the actual low-frequency product commutator -/
open Filter Topology
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem tendsto_expandingSample_low_commutator (a M R S : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ)) (hS : 0 < S)
(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 (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 → Fin 12) :
Tendsto (fun n => expandingProductCommutator a (L n) N ha ha1 hN (hL n) j
(schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K))
(expandingSmoothLow (L n) S hS (f n))) atTop (𝓝 0) := by
let q := fun n => schwartzTorusSample a N (L n) ha1 hN (hL n) (radianFourierKernel K)
let u := fun n => expandingSmoothLow (L n) S hS (f n)
have hfirst := tendsto_expandingSample_low_derivative a M R S N ha ha1 hN hS
L hL hLinf f hf hlocal K hK j
have hfirst' := (torusFourierIsometry.continuous.tendsto 0).comp hfirst
have hsecond := tendsto_expandingCompactPhysical_derivative a N M R S ha ha1 hN hS
N j L hL hLinf f hf hlocal K hK
rw [← tendsto_zero_iff_norm_tendsto_zero] at hsecond
have he (n : ℕ) : torusFourierIsometry
(expandingProductCommutator a (L n) N ha ha1 hN (hL n) j (q n) (u n)) =
torusFourierIsometry (expandingOrderedFourierEnergy a (L n) N (hL n) j
(expandingProduct a N (L n) ha ha1 hN (hL n) (q n) (u n))) -
torusPhysicalL2Value (L n) (expandingUnitTorusFunction a N (L n) (q n) *
expandingUnitTorusFunction a N (L n) (expandingSmoothDerivative (L n) S hS N j (f n))) := by
rw [torusProductCommutator_physical]
dsimp only [u]
rw [expandingOrderedPhysicalEnergy_smoothLow a (L n) S N ha ha1 hN (hL n) hS,
torusL2Product_physical]
rfl
have ht := hfirst'.sub hsecond
simp only [map_zero, sub_zero] at ht
rw [tendsto_zero_iff_norm_tendsto_zero]
change Tendsto (fun n => torusFourierIsometry
(expandingOrderedFourierEnergy a (L n) N (hL n) j
(expandingProduct a N (L n) ha ha1 hN (hL n) (q n) (u n))) -
torusPhysicalL2Value (L n) (expandingUnitTorusFunction a N (L n) (q n) *
expandingUnitTorusFunction a N (L n) (expandingSmoothDerivative (L n) S hS N j (f n))))
atTop (𝓝 0) at ht
simp_rw [← he] at ht
simpa only [LinearIsometryEquiv.norm_map, norm_zero] using ht.norm
end DefocusingNLS
end OAI
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