| import OAI.Analysis.AncientFlat.Evolution.MeasurableTime |
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| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| namespace EarlyIntegral |
| variable {V E : Type} [NormedAddCommGroup V] [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [CompleteSpace E] |
| omit [CompleteSpace E] in |
| lemma holder_of_bounds (F : ℝ → ℝ → V → E) (a b T γ : ℝ) |
| (_ : 0<a) (hb : 0<b) (hγ : 0<γ) (hγ1 : γ<1) |
| (w : ℝ → ℝ) (hw : IntegrableOn w (Ioo 0 b)) (hw₀ : ∀ r∈Ioo 0 b, 0≤w r) |
| (B D E₀ : ℝ) (hB : 0≤B) (hD : 0≤D) (hE : 0≤E₀) |
| (hi : ∀ t∈Icc a T, ∀ x, IntegrableOn (fun r => F t r x) (Ioo 0 b)) |
| (hbound : ∀ t∈Icc a T, ∀ r∈Ioo 0 b, ∀ x, ‖F t r x‖≤B*w r) |
| (hspace : ∀ t∈Icc a T, ∀ r∈Ioo 0 b, ∀ x y, ‖F t r x-F t r y‖≤D*w r*‖x-y‖) |
| (htime : ∀ s∈Icc a T, ∀ t∈Icc a T, s≤t → t-s≤b → ∀ r∈Ioo 0 b, ∀ x, |
| ‖F t r x-F s r x‖≤E₀*w r*(t-s)) : |
| ∃ C : ℝ, 0≤C ∧ |
| (∀ t∈Icc a T, ∀ x, ‖∫ r in Ioo 0 b, F t r x‖≤C) ∧ |
| (∀ t∈Icc a T, ∀ s∈Icc a T, ∀ x y, |
| ‖(∫ r in Ioo 0 b, F t r x)-(∫ r in Ioo 0 b, F s r y)‖≤ |
| C*(‖x-y‖^γ+|t-s|^(γ/4))) := by |
| let W := ∫ r in Ioo 0 b,w r |
| have hW : 0≤W := integral_nonneg_of_ae (by |
| filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr |
| exact hw₀ r hr) |
| have hn (t) (ht : t∈Icc a T) (x : V) : ‖∫ r in Ioo 0 b,F t r x‖≤B*W := by |
| have he := norm_integral_le_of_norm_le (hw.const_mul B) (by |
| filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr |
| exact hbound t ht r hr x) |
| simpa only [integral_const_mul] using he |
| have hs (t) (ht : t∈Icc a T) (x y : V) : |
| ‖(∫ r in Ioo 0 b,F t r x)-(∫ r in Ioo 0 b,F t r y)‖≤D*W*‖x-y‖ := by |
| rw [←integral_sub (hi t ht x) (hi t ht y)] |
| have he := norm_integral_le_of_norm_le ((hw.const_mul D).mul_const ‖x-y‖) (by |
| filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr |
| exact hspace t ht r hr x y) |
| simpa only [integral_mul_const,integral_const_mul] using he |
| let E₁ := E₀*W+2*B*W/b |
| have hE₁ : 0≤E₁ := by dsimp [E₁]; positivity |
| have htord (s) (hs : s∈Icc a T) (t) (ht : t∈Icc a T) (hst : s≤t) (x : V) : |
| ‖(∫ r in Ioo 0 b,F t r x)-(∫ r in Ioo 0 b,F s r x)‖≤E₁*(t-s) := by |
| by_cases he : t-s≤b |
| · have he' : ‖(∫ r in Ioo 0 b,F t r x)-(∫ r in Ioo 0 b,F s r x)‖≤E₀*W*(t-s) := by |
| rw [←integral_sub (hi t ht x) (hi s hs x)] |
| have hm := norm_integral_le_of_norm_le ((hw.const_mul E₀).mul_const (t-s)) (by |
| filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr |
| exact htime s hs t ht hst he r hr x) |
| simpa only [integral_mul_const,integral_const_mul] using hm |
| apply he'.trans |
| apply mul_le_mul_of_nonneg_right _ (sub_nonneg.mpr hst) |
| dsimp [E₁]; exact le_add_of_nonneg_right (by positivity) |
| · have hb' : b≤t-s := (lt_of_not_ge he).le |
| calc |
| _ ≤ ‖∫ r in Ioo 0 b,F t r x‖+‖∫ r in Ioo 0 b,F s r x‖ := norm_sub_le .. |
| _ ≤ B*W+B*W := add_le_add (hn t ht x) (hn s hs x) |
| _ = (2*B*W/b)*b := by field_simp; ring |
| _ ≤ (2*B*W/b)*(t-s) := mul_le_mul_of_nonneg_left hb' (by positivity) |
| _ ≤ E₁*(t-s) := mul_le_mul_of_nonneg_right (by dsimp [E₁]; exact le_add_of_nonneg_left (by positivity)) (sub_nonneg.mpr hst) |
| have htime' (s) (hs : s∈Icc a T) (t) (ht : t∈Icc a T) (x : V) : |
| ‖(∫ r in Ioo 0 b,F t r x)-(∫ r in Ioo 0 b,F s r x)‖≤E₁*|t-s| := by |
| rcases le_total s t with hst|hts |
| · rw [abs_of_nonneg (sub_nonneg.mpr hst)]; exact htord s hs t ht hst x |
| · rw [norm_sub_rev,abs_of_nonpos (sub_nonpos.mpr hts),neg_sub]; exact htord t ht s hs hts x |
| let Cs := (2*B*W)^(1-γ)*(D*W)^γ |
| let Ct := (2*B*W)^(1-γ/4)*E₁^(γ/4) |
| have hCs : 0≤Cs := by dsimp [Cs]; positivity |
| have hCt : 0≤Ct := by dsimp [Ct]; positivity |
| refine ⟨B*W+Cs+Ct,by positivity,fun t ht x => (hn t ht x).trans (by linarith),?_⟩ |
| intro t ht s hs' x y |
| have hsp : ‖(∫ r in Ioo 0 b,F t r x)-(∫ r in Ioo 0 b,F t r y)‖≤Cs*‖x-y‖^γ := by |
| apply HolderInterpolation.scalar (norm_nonneg _) (by positivity) (by positivity) (norm_nonneg _) hγ hγ1 |
| · exact (norm_sub_le ..).trans ((add_le_add (hn t ht x) (hn t ht y)).trans_eq (by ring)) |
| · exact hs t ht x y |
| have hti : ‖(∫ r in Ioo 0 b,F t r y)-(∫ r in Ioo 0 b,F s r y)‖≤Ct*|t-s|^(γ/4) := by |
| apply HolderInterpolation.scalar (norm_nonneg _) (by positivity) hE₁ (abs_nonneg _) (by positivity) (by linarith) |
| · exact (norm_sub_le ..).trans ((add_le_add (hn t ht y) (hn s hs' y)).trans_eq (by ring)) |
| · exact htime' s hs' t ht y |
| calc |
| _ ≤ ‖(∫ r in Ioo 0 b,F t r x)-(∫ r in Ioo 0 b,F t r y)‖+ |
| ‖(∫ r in Ioo 0 b,F t r y)-(∫ r in Ioo 0 b,F s r y)‖ := norm_sub_le_norm_sub_add_norm_sub .. |
| _ ≤ Cs*‖x-y‖^γ+Ct*|t-s|^(γ/4) := add_le_add hsp hti |
| _ ≤ (B*W+Cs+Ct)*‖x-y‖^γ+(B*W+Cs+Ct)*|t-s|^(γ/4) := by gcongr <;> nlinarith |
| _ = _ := by ring |
| end EarlyIntegral |
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| end |
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| end OAI |
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