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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzNonlinearPeriodization
import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzDyadicCutoff
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.UnitTorusPhysical
namespace OAI
/-! # A scale- and derivative-order independent bound for a sampled cutoff -/
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "T" => UnitAddTorus (Fin 12)
theorem norm_tsum_le_of_disjoint {ι : Type*} (u : ι → ℂ) (M : ℝ) (hM : 0 ≤ M)
(hu : ∀ i, ‖u i‖ ≤ M) (hd : ∀ i j, i ≠ j → u i = 0 ∨ u j = 0) :
‖∑' i, u i‖ ≤ M := by
classical
by_cases h : ∃ i, u i ≠ 0
· obtain ⟨i, hi⟩ := h
have hz (j : ι) (hji : j ≠ i) : u j = 0 := (hd j i hji).resolve_right hi
rw [tsum_eq_single i hz]
exact hu i
· have hz : ∀ i, u i = 0 := by simpa only [not_exists, not_not] using h
simpa only [hz, tsum_zero, norm_zero] using hM
theorem sampledCutoff_pointwise_bound (a k L : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (M : ℝ) (hM : 0 ≤ M)
(hQB : ∀ y : E, ‖Q y‖ ≤ M) (x : T) :
‖expandingUnitTorusFunction a k L
(schwartzTorusSample a k L ha1 hk hL (radianFourierKernel
(cutoffProfileSchwartz L (by linarith) χ hχ Q hQ))) x‖ ≤
SchwartzMap.seminorm ℝ 0 0 χ * M := by
let ψ := cutoffProfileSchwartz L (by linarith : 0 < L) χ hχ Q hQ
have hψ (y : E) (hy : 2 * L < ‖y‖) : ψ y = 0 := by
rw [cutoffProfileSchwartz_apply, hχzero, zero_mul]
rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr (by linarith))]
apply (le_inv_mul_iff₀ (by linarith : 0 < L)).mpr
linarith
have hb (y : E) : ‖ψ y‖ ≤ SchwartzMap.seminorm ℝ 0 0 χ * M := by
rw [cutoffProfileSchwartz_apply, norm_mul]
exact mul_le_mul (SchwartzMap.norm_le_seminorm ℝ χ _) (hQB y) (norm_nonneg _) (by positivity)
obtain ⟨y, rfl⟩ := unitTorusProjection_isOpenQuotientMap.surjective x
rw [expandingUnitTorusFunction_projection a k L ha ha1 hk hL]
have hy : (2 * Real.pi) • y = L⁻¹ • ((2 * Real.pi * L) • y) := by
rw [smul_smul]
congr 1
field_simp
rw [hy, schwartzTorusSample_physical a k L ha ha1 hk hL]
exact norm_tsum_le_of_disjoint _ _ (by positivity) (fun n => hb _)
(cutoff_periodization_disjoint L (by linarith) ψ hψ _)
end DefocusingNLS
end OAI
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