File size: 3,967 Bytes
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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProductCommutatorLinear
namespace OAI
/-! # Uniform continuity in the product commutator coefficient -/
open scoped ENNReal
namespace DefocusingNLS
theorem expandingOrderedFourierEnergy_norm_le (a L : ℝ) (N : ℕ) (hL : 1 ≤ L)
(j : Fin N → Fin 12) (f : FourierL2) :
‖expandingOrderedFourierEnergy a L N hL j f‖ ≤ ‖f‖ := by
apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
intro n
rw [expandingOrderedFourierEnergy_apply, norm_mul]
exact mul_le_of_le_one_left (norm_nonneg _) (expandingOrderedMultiplier_norm_le a L N hL j n)
theorem expandingProductCommutator_norm_le (a L : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ)) (hL : 1 ≤ L)
(j : Fin N → Fin 12) (q f : FourierL2) :
‖expandingProductCommutator a L N ha ha1 hN hL j q f‖ ≤
((2 * Real.sqrt (4 ^ (N : ℝ)) + 1) * expandingEmbeddingBound a N) * ‖q‖ * ‖f‖ := by
have hc := fourierConvolution_norm_le (expandingFourierCoefficient a N L q)
(summable_norm_expandingFourierCoefficient a N L ha ha1 hN hL q)
(expandingOrderedFourierEnergy a L N hL j f)
have hc' : ‖fourierConvolution (expandingFourierCoefficient a N L q)
(expandingOrderedFourierEnergy a L N hL j f)‖ ≤
(expandingEmbeddingBound a N * ‖q‖) * ‖f‖ :=
hc.trans (mul_le_mul (tsum_norm_expandingFourierCoefficient_le a N L ha ha1 hN hL q)
(expandingOrderedFourierEnergy_norm_le a L N hL j f) (norm_nonneg _)
(by unfold expandingEmbeddingBound; positivity))
have hp := (expandingOrderedFourierEnergy_norm_le a L N hL j
(expandingProduct a N L ha ha1 hN hL q f)).trans
(expandingProduct_norm_le a N L ha ha1 hN hL q f)
exact (norm_sub_le _ _).trans ((add_le_add hp hc').trans_eq (by ring))
theorem fourierConvolution_sub_left (v w : frequencyLattice → ℂ)
(hv : Summable (fun n => ‖v n‖)) (hw : Summable (fun n => ‖w n‖)) (f : FourierL2) :
fourierConvolution (v - w) f = fourierConvolution v f - fourierConvolution w f := by
unfold fourierConvolution
simp only [Pi.sub_apply, sub_smul]
exact (summable_fourierConvolution_terms v hv f).tsum_sub
(summable_fourierConvolution_terms w hw f)
theorem expandingProductCommutator_sub_left (a L : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ)) (hL : 1 ≤ L)
(j : Fin N → Fin 12) (q r f : FourierL2) :
expandingProductCommutator a L N ha ha1 hN hL j (q - r) f =
expandingProductCommutator a L N ha ha1 hN hL j q f -
expandingProductCommutator a L N ha ha1 hN hL j r f := by
have hp : expandingProduct a N L ha ha1 hN hL (q - r) f =
expandingProduct a N L ha ha1 hN hL q f - expandingProduct a N L ha ha1 hN hL r f := by
exact congrArg (fun B : FourierL2 →L[ℂ] FourierL2 => B f)
((expandingBilinearProduct a N L ha ha1 hN hL).map_sub q r)
have hv : expandingFourierCoefficient a N L (q - r) =
expandingFourierCoefficient a N L q - expandingFourierCoefficient a N L r := by
funext n
simp only [expandingFourierCoefficient, lp.coeFn_sub, Pi.sub_apply, mul_sub]
unfold expandingProductCommutator
rw [hp, map_sub, hv, fourierConvolution_sub_left _ _
(summable_norm_expandingFourierCoefficient a N L ha ha1 hN hL q)
(summable_norm_expandingFourierCoefficient a N L ha ha1 hN hL r)]
abel
theorem expandingProductCommutator_coefficient_difference (a L : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < (N : ℝ)) (hL : 1 ≤ L)
(j : Fin N → Fin 12) (q r f : FourierL2) :
‖expandingProductCommutator a L N ha ha1 hN hL j q f -
expandingProductCommutator a L N ha ha1 hN hL j r f‖ ≤
((2 * Real.sqrt (4 ^ (N : ℝ)) + 1) * expandingEmbeddingBound a N) * ‖q - r‖ * ‖f‖ := by
rw [← expandingProductCommutator_sub_left]
exact expandingProductCommutator_norm_le a L N ha ha1 hN hL j (q - r) f
end DefocusingNLS
end OAI
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