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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCompactFilterLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingPhysicalFourier
namespace OAI
/-! # Compact physical mass of an unfiltered locally vanishing sequence -/
open Filter Topology MeasureTheory
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "T" => UnitAddTorus (Fin 12)
noncomputable local instance compactMassLimitMeasure : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance compactMassLimitProbability : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
theorem tendsto_expandingSample_product_mass (a k M R : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(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 k (L n) (f n) (euclideanToTorus ((L n)⁻¹ • y))‖ < ε)
(K : 𝓢(E, ℂ)) (hK : ∀ y : E, R < ‖y‖ → K y = 0) :
Tendsto (fun n => ‖expandingPhysicalMassVector a k (L n) ha ha1 hk (hL n)
(expandingProduct a k (L n) ha ha1 hk (hL n)
(schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)) (f n))‖)
atTop (𝓝 0) := by
have hb (n : ℕ) (y : E) :
‖expandingPhysicalContinuous a k (L n) ha ha1 hk (hL n) (f n) y‖ ≤
expandingEmbeddingBound a k * M :=
(expandingPhysicalContinuous_norm_le a k (L n) ha ha1 hk (hL n) (f n) y).trans
(mul_le_mul_of_nonneg_left (hf n) (by unfold expandingEmbeddingBound; positivity))
have ht := tendsto_weightedL2_of_locally_uniform_zero (expandingEmbeddingBound a k * M)
K (K.memLp 2) (fun n => expandingPhysicalContinuous a k (L n) ha ha1 hk (hL n) (f n)) hb (by simpa only [expandingPhysicalContinuous_apply] using hlocal)
have he : ∀ᶠ n in atTop, ‖expandingPhysicalMassVector a k (L n) ha ha1 hk (hL n)
(expandingProduct a k (L n) ha ha1 hk (hL n)
(schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel K)) (f n))‖ ^ 2 =
∫ y : E, ‖K y * expandingPhysicalContinuous a k (L n) ha ha1 hk (hL n) (f n) y‖ ^ 2 := by
filter_upwards [hLinf.eventually (eventually_ge_atTop R)] with n hn
rw [← LinearIsometryEquiv.norm_map torusFourierIsometry,
expandingPhysicalMassVector_isometry, expandingUnitTorusFunction_product,
torusPhysicalL2Value_norm_sq]
simp only [ContinuousMap.mul_apply]
apply expandingCompactWeight_integral a k (L n) R ha ha1 hk (hL n) _ K hK
calc
2 * R ≤ 2 * L n := mul_le_mul_of_nonneg_left hn (by norm_num)
_ < 2 * Real.pi * L n := mul_lt_mul_of_pos_right
(by linarith [Real.pi_gt_three] : (2 : ℝ) < 2 * Real.pi) (by linarith [hL n])
have hs := ht.congr' (he.mono fun _ hn => hn.symm)
simpa only [Real.sqrt_sq (norm_nonneg _), Real.sqrt_zero] using hs.sqrt
end DefocusingNLS
end OAI
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