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8e4d533 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 | import Mathlib
import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear
import OAI.Geometry.CAT0Fillings.Gradient.Multiplier
import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
{hT : IsMetricCurrent T} (q : ChartGeometry hT)
lemma sobolev_tendsto {P : ℕ → q.Sobolev} {Q : q.Sobolev}
(hv : Tendsto (fun j => q.inclusion (P j)) atTop (𝓝 (q.inclusion Q)))
(hg : Tendsto (fun j => q.closedGradient (P j)) atTop (𝓝 (q.closedGradient Q))) :
Tendsto P atTop (𝓝 Q) := by
apply tendsto_subtype_rng.mpr
exact ((WithLp.prodContinuousLinearEquiv 2 ℝ
(Lp ℝ 2 (MassMeasure.currentMassMeasure hT)) (Lp (Euc (k+1)) 2 q.atlasMeasure)).symm.continuous.tendsto
(q.inclusion Q,q.closedGradient Q)).comp (hv.prodMk_nhds hg)
lemma closed_chain_in_closed {F F' : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
(hdF : ∀ t, HasDerivAt F (F' t) t) (hcF : Continuous F')
{C : ℝ} (hC : 0 ≤ C) (hbF : ∀ t, |F' t| ≤ C) (P : q.Sobolev)
{S : Set q.Sobolev} (hS : IsClosed S)
(hmem : ∀ (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), q.lipSobolev (hF.comp hu) ∈ S) :
∃ Q : q.Sobolev, Q ∈ S ∧
(q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
(fun x => F ((q.inclusion P) x)) ∧
(q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
(fun w => F' ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
obtain ⟨u,K,hu,hl⟩ := q.exists_lipschitz_approximants P
have hv := (q.valueProjection.continuous.tendsto (P : q.GraphAmbient)).comp hl
have hg := (q.gradientProjection.continuous.tendsto (P : q.GraphAmbient)).comp hl
change Tendsto (fun j => value (hu j)) atTop (𝓝 (q.inclusion P)) at hv
change Tendsto (fun j => q.gradient (hu j)) atTop (𝓝 (q.closedGradient P)) at hg
obtain ⟨s,hs,hae⟩ := (tendstoInMeasure_of_tendsto_Lp hv).exists_seq_tendsto_ae
have hnames : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, ∀ j,
(value (hT := hT) (hu (s j))) x = u (s j) x := by
apply ae_all_iff.mpr
intro j
exact ((Foundations.boundedLip_of_lipschitz (hu (s j))).memLp 2).coeFn_toLp
have hae' : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT,
Tendsto (fun j => u (s j) x) atTop (𝓝 ((q.inclusion P) x)) := by
filter_upwards [hae,hnames] with x hx he
simpa only [he] using hx
let b (j : ℕ) (w : ℕ × Euc (k+1)) := F' (u (s j) (q.atlasParam w))
let c (w : ℕ × Euc (k+1)) := F' ((q.inclusion P) (q.atlasParam w))
have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure :=
(hcF.measurable.comp ((hu (s j)).continuous.measurable.comp q.measurable_atlasParam)).aestronglyMeasurable
have hc : AEStronglyMeasurable c q.atlasMeasure :=
hcF.comp_aestronglyMeasurable ((Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving)
have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C :=
Eventually.of_forall fun point => hbF (u (s j) (q.atlasParam point))
have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C :=
Eventually.of_forall fun point => hbF ((q.inclusion P) (q.atlasParam point))
have hbc : ∀ᵐ w ∂q.atlasMeasure, Tendsto (fun j => b j w) atTop (𝓝 (c w)) := by
have ha : ∀ᵐ w ∂q.atlasMeasure,
Tendsto (fun j => u (s j) (q.atlasParam w)) atTop (𝓝 ((q.inclusion P) (q.atlasParam w))) := by
apply ae_of_ae_map (μ := q.atlasMeasure) (p := fun x =>
Tendsto (fun j => u (s j) x) atTop (𝓝 ((q.inclusion P) x))) q.measurable_atlasParam.aemeasurable
rwa [q.map_atlasMeasure]
filter_upwards [ha] with w hw
exact (hcF.tendsto _).comp hw
have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC hbc (hg.comp hs.tendsto_atTop)
have he (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.gradient (hu (s j))) =
q.gradient (hF.comp (hu (s j))) := by
apply Lp.ext
exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (q.gradient_chain_ae (hu (s j)) hF hdF).symm
simp only [Function.comp_apply,he] at hgl
have hvl := (q.continuous_composeValue hF).tendsto (q.inclusion P) |>.comp (hv.comp hs.tendsto_atTop)
simp only [Function.comp_def,q.composeValue_value] at hvl
obtain ⟨Q,hQ,hG⟩ := q.closedGradient_closed (q.composeValue hF (q.inclusion P))
(ClosedCalculus.mulLp hc hcC (q.closedGradient P))
(fun j => q.lipSobolev (hF.comp (hu (s j)))) hvl hgl
refine ⟨Q,?_,?_,?_⟩
· apply hS.mem_of_tendsto (q.sobolev_tendsto (P := fun j => q.lipSobolev (hF.comp (hu (s j)))) (by simpa only [q.inclusion_lipSobolev,hQ] using hvl) (by simpa only [q.closedGradient_lipSobolev,hG] using hgl))
exact Eventually.of_forall fun j => hmem _ _ (hu (s j))
· rw [hQ]
exact q.composeValue_ae hF (q.inclusion P)
· rw [hG]
exact ClosedCalculus.mulLp_ae hc hcC _
end CAT0Fillings.ChartGeometry
end
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
{hT : IsMetricCurrent T} (q : ChartGeometry hT)
lemma closed_chain_uniform_in_closed {F D : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
(hD : Measurable D) {C : ℝ} (hC : 0 ≤ C) (hbD : ∀ t, |D t| ≤ C)
(Fj Dj : ℕ → ℝ → ℝ) (K : ℕ → ℝ≥0)
(hFj : ∀ j, LipschitzWith (K j) (Fj j))
(hdj : ∀ j t, HasDerivAt (Fj j) (Dj j t) t)
(hcj : ∀ j, Continuous (Dj j)) (hbj : ∀ j t, |Dj j t| ≤ C)
(hdlim : ∀ t, Tendsto (fun j => Dj j t) atTop (𝓝 (D t)))
(a : ℕ → ℝ) (ha : Tendsto a atTop (𝓝 0))
(he : ∀ j t, |Fj j t-F t| ≤ a j) (P : q.Sobolev)
{S : Set q.Sobolev} (hclosed : IsClosed S)
(hmem : ∀ j (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), q.lipSobolev ((hFj j).comp hu) ∈ S) :
∃ Q : q.Sobolev, Q ∈ S ∧
(q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
(fun x => F ((q.inclusion P) x)) ∧
(q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
(fun w => D ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
choose Q hQS hQ hG using (fun j => q.closed_chain_in_closed (hFj j) (hdj j) (hcj j) hC (hbj j) P hclosed (hmem j))
let b (j : ℕ) (w : ℕ × Euc (k+1)) := Dj j ((q.inclusion P) (q.atlasParam w))
let c (w : ℕ × Euc (k+1)) := D ((q.inclusion P) (q.atlasParam w))
have hpm := (Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving
have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := (hcj j).comp_aestronglyMeasurable hpm
have hc : AEStronglyMeasurable c q.atlasMeasure := (hD.comp_aemeasurable hpm.aemeasurable).aestronglyMeasurable
have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C := Eventually.of_forall fun _ => hbj j _
have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := Eventually.of_forall fun _ => hbD _
have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC
(Eventually.of_forall fun point => hdlim ((q.inclusion P) (q.atlasParam point)))
(tendsto_const_nhds (x := q.closedGradient P))
have heG (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.closedGradient P) = q.closedGradient (Q j) := by
apply Lp.ext
exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (hG j).symm
simp only [heG] at hgl
let V := q.composeValue hF (q.inclusion P)
have hv : Tendsto (fun j => q.inclusion (Q j)) atTop (𝓝 V) := by
apply tendsto_iff_norm_sub_tendsto_zero.mpr
apply squeeze_zero (fun _ => norm_nonneg _)
(g := fun j => a j*‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖)
· intro j
apply Lp.norm_le_mul_norm_of_ae_le_mul
filter_upwards [Lp.coeFn_sub (q.inclusion (Q j)) V,hQ j,q.composeValue_ae hF (q.inclusion P),
((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (1:ℝ))).memLp
(μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x hs hq hv h1
change (value (hT := hT) (LipschitzWith.const (1:ℝ)) : X → ℝ) x = 1 at h1
rw [hs,Pi.sub_apply,hq,hv,h1,Real.norm_eq_abs,norm_one,mul_one]
exact he j _
· simpa only [zero_mul] using ha.mul_const ‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖
obtain ⟨R,hR,hRG⟩ := q.closedGradient_closed V (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) Q hv hgl
refine ⟨R,?_,?_,?_⟩
· apply hclosed.mem_of_tendsto (q.sobolev_tendsto (P := Q) (by simpa only [hR] using hv) (by simpa only [hRG] using hgl))
exact Eventually.of_forall hQS
· rw [hR]
exact q.composeValue_ae hF (q.inclusion P)
· rw [hRG]
exact ClosedCalculus.mulLp_ae hc hcC _
end CAT0Fillings.ChartGeometry
end
end OAI
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