| import OAI.Analysis.AncientFlat.Evolution.RowIntegral |
| import OAI.Analysis.AncientFlat.Evolution.GammaBetaRow |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped SchwartzMap Topology ZeroAtInfty BoundedContinuousFunction ContDiff |
| namespace ParametrixVolterra |
| section IntegrableRowDataGamma |
| open SourceFrozenRows StrongWeightedVolterra |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| local instance : ContinuousENorm (V →ᵇ (V [×4]→L[ℝ] ℂ)) := |
| @SeminormedAddGroup.toContinuousENorm (V →ᵇ (V [×4]→L[ℝ] ℂ)) inferInstance |
| 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)) |
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| variable (γ : ℝ) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (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‖^γ) |
| include hγ hγ1 hL hH hLip hhold |
|
|
| lemma integrable_rowData_gamma (t : Time T) : Integrable (rowData K hα A j t) := by |
| obtain ⟨b,hbi,hb⟩ := exists_rowData_bound_gamma K hα A j γ hγ hγ1 L H hL hH hLip hhold t.val t.property.1 |
| apply hbi.mono' (f := rowData K hα A j t) |
| · exact ((continuous_rowData K hα A j).comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact Eventually.of_forall (hb t le_rfl) |
|
|
| lemma continuous_rowIntegralBCF_gamma : Continuous (rowIntegralBCF K hα A j) := by |
| apply continuous_iff_continuousAt.mpr |
| intro t |
| obtain ⟨b,hbi,hb⟩ := exists_rowData_bound_gamma K hα A j γ hγ hγ1 L H hL hH hLip hhold (t.val/2) (half_pos t.property.1) |
| apply continuous_subtype_val.continuousAt.smul |
| apply continuousAt_of_dominated |
| · exact Eventually.of_forall (fun r => ((continuous_rowData K hα A j).comp |
| (continuous_const.prodMk continuous_id)).aestronglyMeasurable) |
| · filter_upwards [continuous_subtype_val.continuousAt |
| (Ioi_mem_nhds (by linarith [t.property.1] : t.val/2<t.val))] with r hr |
| exact Eventually.of_forall (hb r hr.le) |
| · exact hbi |
| · exact Eventually.of_forall (fun u => ((continuous_rowData K hα A j).comp |
| (continuous_id.prodMk continuous_const)).continuousAt) |
|
|
| lemma rowIntegralBCF_apply_gamma (hT : 0 ≤ T) (t : Time T) (x : V) : |
| rowIntegralBCF K hα A j t x = SourceFrozenRows.integralRow K (extendedA A hT) |
| (fun s => (unweighted j s).toBCF) t.val 4 x := by |
| let ev := BoundedContinuousFunction.evalCLM ℝ x (β := V [×4]→L[ℝ] ℂ) |
| have hi := integrable_rowData_gamma K hα A j γ hγ hγ1 L H hL hH hLip hhold t |
| change t.val • ev (∫ u : UnitTime, rowData K hα A j t u) = _ |
| have hev := ev.integral_comp_comm (𝕜 := ℝ) (Fₗ := V [×4]→L[ℝ] ℂ) (φ := rowData K hα A j t) hi |
| rw [← hev] |
| let F : ℝ → (V [×4]→L[ℝ] ℂ) := fun s => |
| row K (extendedA A hT s) ((t.val-s)^(1/4 : ℝ)) (unweighted j s).toBCF 4 x |
| change t.val • (∫ u : UnitTime, rowData K hα A j t u x) = ∫ s in Ioo 0 t.val, F s |
| have hc := intervalIntegral.smul_integral_comp_mul_left (a := 0) (b := 1) F t.val |
| simp only [mul_zero,mul_one] at hc |
| rw [intervalIntegral.integral_of_le t.property.1.le,integral_Ioc_eq_integral_Ioo] at hc |
| rw [← hc,intervalIntegral.integral_of_le zero_le_one,integral_Ioc_eq_integral_Ioo, |
| ← integral_subtype measurableSet_Ioo] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards with u |
| rw [rowData,timeBCF_apply] |
| dsimp [F] |
| have he : extendedA A hT (t.val*u.val) = fun y => (A (now (lag t u)) y).val := by |
| funext y |
| unfold extendedA |
| rw [projIcc_of_mem hT (show t.val*u.val ∈ Icc 0 T from |
| ⟨(lag t u).property.1.le,(lag t u).property.2⟩)] |
| rfl |
| rw [he,show t.val-t.val*u.val = t.val*(1-u.val) by ring] |
| rfl |
| end IntegrableRowDataGamma |
|
|
| open SourceFrozenRows StrongWeightedVolterra BiharmonicKernel |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable {T α β : ℝ} (hα : 0 < α) (A : C(Slab T,V →ᵇ Icc α β)) |
|
|
| lemma iteratedFDeriv_duhamel_four_gamma (hT : 0 ≤ T) (γ : ℝ) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (L : ℝ) (hL : 0 ≤ L) |
| (hLip : ∀ s r : Slab T, ∀ x z : V, |
| |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) |
| (j : Time T →ᵇ C₀(V,ℂ)) (H : ℝ) (hH : 0 ≤ H) |
| (hj : ∀ s : Time T, ∀ x y, |
| ‖unweighted j s.val x-unweighted j s.val y‖ ≤ H*s.val^(-7/8 : ℝ)*‖x-y‖^γ) |
| (t : Time T) : iteratedFDeriv ℝ 4 (fun x => duhamel hα A j t.val x) = |
| rowIntegralBCF (heatKernel 1 zero_lt_one) hα A j t := by |
| have hf : StronglyMeasurable (fun s => (unweighted j s).toBCF) := |
| ZeroAtInftyContinuousMap.isometry_toBCF.continuous.comp_stronglyMeasurable |
| (stronglyMeasurable_unweighted j) |
| have hn : ∀ s ∈ Ioo 0 t.val, ‖(unweighted j s).toBCF‖ ≤ ‖j‖*s^(-(3/4 : ℝ)) := by |
| intro s hs |
| rw [ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] |
| simpa only [neg_div,mul_comm] using norm_unweighted_le j s hs.1 |
| have hh : ∀ s ∈ Ioo 0 t.val, ∀ x y : V, |
| ‖(unweighted j s).toBCF x-(unweighted j s).toBCF y‖ ≤ |
| (H*s^(-(7/8 : ℝ)))*‖x-y‖^γ := by |
| intro s hs x y |
| change ‖unweighted j s x-unweighted j s y‖ ≤ _ |
| simpa only [neg_div] using hj ⟨s,hs.1,hs.2.le.trans t.property.2⟩ x y |
| have hl : ∀ s : Slab T, ∀ x y, ‖(A s x).val-(A s y).val‖ ≤ L*‖x-y‖ := by |
| intro s x y |
| simpa only [Real.norm_eq_abs,sub_self,abs_zero,add_zero] using hLip s s x y |
| rw [funext (duhamel_eq_sourceDuhamel A hT hα j t), |
| WeightedSourceFrozenRows.iteratedFDeriv_duhamel (heatKernel 1 zero_lt_one) (extendedA A hT) |
| (continuous_extendedA A hT) α β hα (extendedA_range A hT) L hL (extendedA_lipschitz A hT L hLip) |
| (fun s => (unweighted j s).toBCF) hf (3/4) (7/8) ‖j‖ H γ t.val |
| (by norm_num) (by norm_num) (norm_nonneg j) hH t.property.1 hγ hγ1 hn hh 4 le_rfl] |
| exact (funext (rowIntegralBCF_apply_gamma (heatKernel 1 zero_lt_one) hα A j γ hγ hγ1 L H hL hH hl hj hT t)).symm |
| end ParametrixVolterra |
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| end |
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| end OAI |
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