| import OAI.Analysis.AncientFlat.Evolution.RowContinuity | |
| import OAI.Analysis.AncientFlat.Evolution.ClassicalSpatial | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped SchwartzMap Topology ZeroAtInfty BoundedContinuousFunction ContDiff | |
| namespace SourceFrozenRows | |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| variable {α β : ℝ} | |
| lemma exists_timeBCF_four_bound (K : SchwartzMap V ℂ) (hα : 0 < α) : | |
| ∃ C D : ℝ, 0 ≤ C ∧ 0 ≤ D ∧ | |
| ∀ (a : V →ᵇ Icc α β) (L : ℝ) (_hL : 0 ≤ L) | |
| (_hLip : ∀ x y, ‖(a x).val-(a y).val‖ ≤ L*‖x-y‖) | |
| (f : V →ᵇ ℂ) (H : ℝ) (_hH : 0 ≤ H) | |
| (_hhold : ∀ x y, ‖f x-f y‖ ≤ H*‖x-y‖^(1/4 : ℝ)) | |
| (q : ℝ) (hq : 0 < q), | |
| ‖timeBCF K hα a q hq f 4‖ ≤ H*C*q^(-15/16 : ℝ)+‖f‖*L*D*q^(-3/4 : ℝ) := by | |
| obtain ⟨C,D,hC,hD,hb⟩ := exists_row_holder_bound K 4 (by omega) α β hα (1/4) (by norm_num) (by norm_num) | |
| refine ⟨C,D,hC,hD,?_⟩ | |
| intro a L hL hLip f H hH hhold q hq | |
| apply (BoundedContinuousFunction.norm_le (by positivity)).mpr | |
| intro x | |
| rw [timeBCF_apply] | |
| have h := hb (fun y => (a y).val) (continuous_subtype_val.comp a.continuous) | |
| (fun y => (a y).property) L hL hLip f H hH hhold (q^(1/4 : ℝ)) (Real.rpow_pos_of_pos hq (1/4)) x | |
| apply h.trans_eq | |
| have h1 := radius_weight q hq 4 (1/4) | |
| have h2 := radius_weight q hq 4 1 | |
| norm_num only [Nat.cast_ofNat] at h1 h2 | |
| rw [Real.rpow_one] at h2 | |
| calc | |
| _ = H*C*((q^(1/4 : ℝ))⁻¹^4*(q^(1/4 : ℝ))^(1/4 : ℝ))+ | |
| ‖f‖*L*D*((q^(1/4 : ℝ))⁻¹^4*q^(1/4 : ℝ)) := by ring | |
| _ = _ := by rw [h1,h2]; norm_num | |
| end SourceFrozenRows | |
| namespace ParametrixVolterra | |
| open SourceFrozenRows StrongWeightedVolterra | |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| variable {T α β : ℝ} (K : SchwartzMap V ℂ) (hα : 0 < α) | |
| (A : C(Slab T,V →ᵇ Icc α β)) (j : Time T →ᵇ C₀(V,ℂ)) | |
| def rowData (t : Time T) (u : UnitTime) : V →ᵇ (V [×4]→L[ℝ] ℂ) := | |
| timeBCF K hα (A (now (lag t u))) (t.val*(1-u.val)) | |
| (mul_pos t.property.1 (sub_pos.mpr u.property.2)) | |
| (unweighted j (lag t u).val).toBCF 4 | |
| lemma continuous_rowData : Continuous (Function.uncurry (rowData K hα A j)) := by | |
| apply continuous_timeBCF K hα _ _ | |
| (fun p => mul_pos p.1.property.1 (sub_pos.mpr p.2.property.2)) | |
| · exact A.continuous.comp (continuous_now.comp continuous_lag) | |
| · exact (continuous_subtype_val.comp continuous_fst).mul | |
| (continuous_const.sub (continuous_subtype_val.comp continuous_snd)) | |
| · exact ZeroAtInftyContinuousMap.isometry_toBCF.continuous.comp | |
| ((continuous_unweighted_on_time j).comp continuous_lag) | |
| lemma integrable_beta_unit (σ p : ℝ) (hσ : σ < 1) (hp : p < 1) : | |
| Integrable (fun u : UnitTime => BetaWeights.density σ p 1 u.val) := by | |
| have hi := (intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num : (0:ℝ) ≤ 1)).mp | |
| (BetaWeights.intervalIntegrable σ p hσ hp 1 zero_lt_one) | |
| exact (integrableOn_iff_comap_subtypeVal measurableSet_Ioo).mp hi | |
| lemma exists_rowData_bound (L H : ℝ) (hL : 0 ≤ L) (hH : 0 ≤ H) | |
| (hLip : ∀ s : Slab T, ∀ x y, ‖(A s x).val-(A s y).val‖ ≤ L*‖x-y‖) | |
| (hhold : ∀ s : Time T, ∀ x y, ‖unweighted j s.val x-unweighted j s.val y‖ ≤ | |
| H*s.val^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ)) | |
| (a : ℝ) (ha : 0 < a) : | |
| ∃ b : UnitTime → ℝ, Integrable b ∧ ∀ t : Time T, a ≤ t.val → | |
| ∀ u, ‖rowData K hα A j t u‖ ≤ b u := by | |
| obtain ⟨C,D,hC,hD,hb⟩ := exists_timeBCF_four_bound (β := β) K hα | |
| let b : UnitTime → ℝ := fun u => | |
| H*C*a^(-7/8 : ℝ)*a^(-15/16 : ℝ)*BetaWeights.density (7/8) (15/16) 1 u.val+ | |
| ‖j‖*L*D*a^(-3/4 : ℝ)*a^(-3/4 : ℝ)*BetaWeights.density (3/4) (3/4) 1 u.val | |
| refine ⟨b,?_,?_⟩ | |
| · exact ((integrable_beta_unit (7/8) (15/16) (by norm_num) (by norm_num)).const_mul _).add | |
| ((integrable_beta_unit (3/4) (3/4) (by norm_num) (by norm_num)).const_mul _) | |
| · intro t ht u | |
| have hs := (lag t u).property.1 | |
| have htpos := t.property.1 | |
| have hup := u.property.1 | |
| have hum := sub_pos.mpr u.property.2 | |
| have hq : 0 < t.val*(1-u.val) := mul_pos t.property.1 (sub_pos.mpr u.property.2) | |
| have hn := norm_unweighted_le j (lag t u).val hs | |
| rw [← ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] at hn | |
| have hrow := hb (A (now (lag t u))) L hL (hLip _) (unweighted j (lag t u).val).toBCF | |
| (H*(lag t u).val^(-7/8 : ℝ)) (by positivity) (hhold (lag t u)) _ hq | |
| have hnorm := mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hn hL) hD) | |
| (Real.rpow_nonneg hq.le (-3/4)) | |
| refine hrow.trans ((add_le_add le_rfl hnorm).trans ?_) | |
| dsimp only [lag, b, BetaWeights.density] | |
| rw [Real.mul_rpow t.property.1.le u.property.1.le, | |
| Real.mul_rpow t.property.1.le (sub_pos.mpr u.property.2).le, | |
| Real.mul_rpow t.property.1.le u.property.1.le, | |
| Real.mul_rpow t.property.1.le (sub_pos.mpr u.property.2).le] | |
| have ht1 := Real.rpow_le_rpow_of_nonpos ha ht (by norm_num : (-7/8 : ℝ) ≤ 0) | |
| have ht2 := Real.rpow_le_rpow_of_nonpos ha ht (by norm_num : (-15/16 : ℝ) ≤ 0) | |
| have ht3 := Real.rpow_le_rpow_of_nonpos ha ht (by norm_num : (-3/4 : ℝ) ≤ 0) | |
| norm_num only [neg_div] | |
| calc | |
| _ = H*C*(t.val^(-7/8 : ℝ)*t.val^(-15/16 : ℝ))* | |
| (u.val^(-7/8 : ℝ)*(1-u.val)^(-15/16 : ℝ))+ | |
| ‖j‖*L*D*(t.val^(-3/4 : ℝ)*t.val^(-3/4 : ℝ))* | |
| (u.val^(-3/4 : ℝ)*(1-u.val)^(-3/4 : ℝ)) := by ring_nf | |
| _ ≤ H*C*(a^(-7/8 : ℝ)*a^(-15/16 : ℝ))* | |
| (u.val^(-7/8 : ℝ)*(1-u.val)^(-15/16 : ℝ))+ | |
| ‖j‖*L*D*(a^(-3/4 : ℝ)*a^(-3/4 : ℝ))* | |
| (u.val^(-3/4 : ℝ)*(1-u.val)^(-3/4 : ℝ)) := by | |
| apply add_le_add | |
| · exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left | |
| (mul_le_mul ht1 ht2 (by positivity) (by positivity)) (by positivity)) (by positivity) | |
| · exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left | |
| (mul_le_mul ht3 ht3 (by positivity) (by positivity)) (by positivity)) (by positivity) | |
| _ = _ := by ring_nf | |
| end ParametrixVolterra | |
| end | |
| end OAI | |