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59d1c80 | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetNorm
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Analysis.Normed.Module.FiniteDimension
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology
universe u
namespace SmoothLocal.Analytic
open SmoothLocal.Geometry
variable {F : Type u} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
def compactParameterIntegral (H : Coord × ℝ → F) (p : Coord) : F :=
∫ s in Icc (0 : ℝ) 1, H (p,s)
def compactParameterPartial (H : Coord × ℝ → F) (q : Coord × ℝ) : Coord →L[ℝ] F :=
(fderiv ℝ H q).comp (ContinuousLinearMap.inl ℝ Coord ℝ)
omit [NormedSpace ℝ F] [CompleteSpace F] in
theorem parameter_slice_continuousOn
{H : Coord × ℝ → F} {U : Set Coord} {D : Set (Coord × ℝ)}
(hH : ContinuousOn H D) (hUD : U ×ˢ Icc (0 : ℝ) 1 ⊆ D)
{p : Coord} (hp : p ∈ U) : ContinuousOn (fun s => H (p,s)) (Icc (0 : ℝ) 1) :=
hH.comp (continuous_const.prodMk continuous_id).continuousOn (fun _ hs => hUD ⟨hp,hs⟩)
omit [NormedSpace ℝ F] [CompleteSpace F] in
theorem exists_local_parameter_norm_bound
{H : Coord × ℝ → F} {U : Set Coord} {D : Set (Coord × ℝ)}
(hH : ContinuousOn H D) (hU : IsOpen U)
(hUD : U ×ˢ Icc (0 : ℝ) 1 ⊆ D) {p : Coord} (hp : p ∈ U) :
∃ e B : ℝ, 0 < e ∧ 0 ≤ B ∧ Metric.ball p e ⊆ U ∧
∀ x ∈ Metric.ball p e, ∀ s ∈ Icc (0 : ℝ) 1, ‖H (x,s)‖ ≤ B := by
obtain ⟨e,he,hball⟩ := Metric.mem_nhds_iff.mp (hU.mem_nhds hp)
let K := Metric.closedBall p (e/2) ×ˢ Icc (0 : ℝ) 1
have hK : IsCompact K := (isCompact_closedBall p (e/2)).prod isCompact_Icc
have hKD : K ⊆ D := by
intro q hq
exact hUD ⟨hball (Metric.closedBall_subset_ball (half_lt_self he) hq.1),hq.2⟩
obtain ⟨B,hB⟩ := hK.exists_bound_of_continuousOn (hH.mono hKD)
refine ⟨e/2,max B 0,half_pos he,le_max_right _ _,?_,?_⟩
· intro x hx
exact hball ((Metric.ball_subset_ball (half_le_self he.le)) hx)
· intro x hx s hs
exact (hB (x,s) ⟨Metric.ball_subset_closedBall hx,hs⟩).trans (le_max_left _ _)
omit [CompleteSpace F] in
theorem compactParameterIntegral_continuousOn
{H : Coord × ℝ → F} {U : Set Coord} {D : Set (Coord × ℝ)}
(hH : ContinuousOn H D) (hD : IsOpen D) (hU : IsOpen U)
(hUD : U ×ˢ Icc (0 : ℝ) 1 ⊆ D) : ContinuousOn (compactParameterIntegral H) U := by
intro p hp
obtain ⟨e,B,he,hB,hball,hbound⟩ := exists_local_parameter_norm_bound hH hU hUD hp
have hmeas : ∀ᶠ x in 𝓝 p, AEStronglyMeasurable (fun s => H (x,s))
(volume.restrict (Icc (0 : ℝ) 1)) := by
filter_upwards [hU.mem_nhds hp] with x hx
exact ((parameter_slice_continuousOn hH hUD hx).integrableOn_compact isCompact_Icc).aestronglyMeasurable
have hdom : ∀ᶠ x in 𝓝 p, ∀ᵐ s ∂volume.restrict (Icc (0 : ℝ) 1), ‖H (x,s)‖ ≤ B := by
filter_upwards [Metric.ball_mem_nhds p he] with x hx
filter_upwards [ae_restrict_mem measurableSet_Icc] with s hs
exact hbound x hx s hs
have hint : Integrable (fun _ : ℝ => B) (volume.restrict (Icc (0 : ℝ) 1)) :=
integrableOn_const isCompact_Icc.measure_ne_top
have hpoint : ∀ᵐ s ∂volume.restrict (Icc (0 : ℝ) 1), ContinuousAt (fun x => H (x,s)) p := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with s hs
exact (hH.continuousAt (hD.mem_nhds (hUD ⟨hp,hs⟩))).comp
(continuous_id.prodMk continuous_const).continuousAt
exact (continuousAt_of_dominated hmeas hdom hint hpoint).continuousWithinAt
omit [CompleteSpace F] in
theorem compactParameterPartial_contDiffOn
{H : Coord × ℝ → F} {D : Set (Coord × ℝ)} (hD : IsOpen D) (n : ℕ)
(hH : ContDiffOn ℝ (n+1 : ℕ) H D) :
ContDiffOn ℝ n (compactParameterPartial H) D := by
have hd : ContDiffOn ℝ n (fderiv ℝ H) D := hH.fderiv_of_isOpen hD (by simp)
exact hd.clm_comp contDiffOn_const
omit [CompleteSpace F] in
theorem compactParameter_slice_hasFDerivAt
{H : Coord × ℝ → F} {D : Set (Coord × ℝ)} (hH : ContDiffOn ℝ 1 H D)
(hD : IsOpen D) {p : Coord} {s : ℝ} (hp : (p,s) ∈ D) :
HasFDerivAt (fun x => H (x,s)) (compactParameterPartial H (p,s)) p := by
have hd := ((hH.contDiffAt (hD.mem_nhds hp)).differentiableAt (by norm_num)).hasFDerivAt
have hi : HasFDerivAt (fun x : Coord => (x,s)) (ContinuousLinearMap.inl ℝ Coord ℝ) p :=
hasFDerivAt_prodMk_left p s
exact hd.comp p hi
omit [CompleteSpace F] in
theorem compactParameterIntegral_hasFDerivAt
{H : Coord × ℝ → F} {U : Set Coord} {D : Set (Coord × ℝ)}
(hH : ContDiffOn ℝ 1 H D) (hD : IsOpen D) (hU : IsOpen U)
(hUD : U ×ˢ Icc (0 : ℝ) 1 ⊆ D) {p : Coord} (hp : p ∈ U) :
HasFDerivAt (compactParameterIntegral H)
(compactParameterIntegral (compactParameterPartial H) p) p := by
have hpart : ContinuousOn (compactParameterPartial H) D :=
(compactParameterPartial_contDiffOn hD 0 hH).continuousOn
obtain ⟨e,B,he,hB,hball,hbound⟩ := exists_local_parameter_norm_bound hpart hU hUD hp
apply hasFDerivAt_integral_of_dominated_of_fderiv_le
(μ := volume.restrict (Icc (0 : ℝ) 1)) (bound := fun _ => B)
(s := Metric.ball p e) (F' := fun x s => compactParameterPartial H (x,s))
(Metric.ball_mem_nhds p he)
· filter_upwards [hU.mem_nhds hp] with x hx
exact ((parameter_slice_continuousOn hH.continuousOn hUD hx).integrableOn_compact isCompact_Icc).aestronglyMeasurable
· exact (parameter_slice_continuousOn hH.continuousOn hUD hp).integrableOn_compact isCompact_Icc
· exact ((parameter_slice_continuousOn hpart hUD hp).integrableOn_compact isCompact_Icc).aestronglyMeasurable
· filter_upwards [ae_restrict_mem measurableSet_Icc] with s hs
intro x hx
exact hbound x hx s hs
· exact integrableOn_const isCompact_Icc.measure_ne_top
· filter_upwards [ae_restrict_mem measurableSet_Icc] with s hs
intro x hx
exact compactParameter_slice_hasFDerivAt hH hD (hUD ⟨hball hx,hs⟩)
theorem compactParameterIntegral_contDiffOn_finite (n : ℕ) :
∀ {F : Type u} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
{H : Coord × ℝ → F} {U : Set Coord} {D : Set (Coord × ℝ)},
ContDiffOn ℝ n H D → IsOpen D → IsOpen U →
U ×ˢ Icc (0 : ℝ) 1 ⊆ D → ContDiffOn ℝ n (compactParameterIntegral H) U := by
induction n with
| zero =>
intro F _ _ _ H U D hH hD hU hUD
simpa only [Nat.cast_zero,contDiffOn_zero] using
compactParameterIntegral_continuousOn hH.continuousOn hD hU hUD
| succ n ih =>
intro F _ _ _ H U D hH hD hU hUD
have hH1 : ContDiffOn ℝ 1 H D := hH.of_le (by exact_mod_cast (show 1 ≤ n+1 by omega))
have hd (p : Coord) (hp : p ∈ U) := compactParameterIntegral_hasFDerivAt hH1 hD hU hUD hp
have hpart : ContDiffOn ℝ n (compactParameterPartial H) D :=
compactParameterPartial_contDiffOn hD n hH
have hmeanPart : ContDiffOn ℝ n (compactParameterIntegral (compactParameterPartial H)) U :=
ih hpart hD hU hUD
have hs : ContDiffOn ℝ ((n : ℕ∞ω)+1) (compactParameterIntegral H) U := by
rw [contDiffOn_succ_iff_fderiv_of_isOpen hU]
refine ⟨fun p hp => (hd p hp).differentiableAt.differentiableWithinAt,by simp,?_⟩
exact hmeanPart.congr (fun p hp => (hd p hp).fderiv)
simpa only [Nat.cast_add,Nat.cast_one] using hs
theorem compactParameterIntegral_contDiffOn
{H : Coord × ℝ → F} {U : Set Coord} {D : Set (Coord × ℝ)}
(hH : ContDiffOn ℝ ∞ H D) (hD : IsOpen D) (hU : IsOpen U)
(hUD : U ×ˢ Icc (0 : ℝ) 1 ⊆ D) :
ContDiffOn ℝ ∞ (compactParameterIntegral H) U :=
contDiffOn_infty.mpr (fun n => compactParameterIntegral_contDiffOn_finite n
(hH.of_le (WithTop.coe_le_coe.mpr le_top)) hD hU hUD)
end SmoothLocal.Analytic
end
end OAI
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