openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingCommutatorCoefficient.lean
| 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 | |