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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import Mathlib
import OAI.Geometry.CAT0Fillings.Gradient.Chart
import OAI.Geometry.CAT0Fillings.Gradient.HilbertAtlas
import OAI.Geometry.CAT0Fillings.Tangent.Coordinates
namespace OAI
section
open Set Filter MeasureTheory Matrix
open scoped Topology ENNReal NNReal
namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n}
{hT : IsMetricCurrent T} (q : ChartGeometry hT)
lemma gradientFunction_chain {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u)
{F F' : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
(hdF : ∀ t, HasDerivAt F (F' t) t) :
q.gradientFunction (F ∘ u) =ᵐ[q.atlasMeasure]
(fun w => F' (u (q.atlasParam w)) • q.gradientFunction u w) := by
apply Measure.ae_sum_iff.mpr
intro i
apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr
apply (withDensity_absolutelyContinuous _ _).ae_le
let C := q.chart i
obtain ⟨L,U,hL,hU⟩ := C.bilipschitz
obtain ⟨f,hf,hef⟩ := (C.scalar_lipschitzOn hL hu).extend_real
have hecomp : EqOn (C.scalar (F ∘ u)) (F ∘ f) C.domain := by
intro z hz
simp only [IntegerChart.scalar,dite_eq_left hz,Function.comp_apply]
rw [←hef hz]
simp only [IntegerChart.scalar,dite_eq_left hz]
filter_upwards [ae_fderivWithin_eq_fderiv_extension volume C.borel hf hef,
ae_fderivWithin_eq_fderiv_extension volume C.borel (hF.comp hf) hecomp,
hf.ae_differentiableAt.filter_mono ae_restrict_le,ae_restrict_mem C.borel] with z hu hcomp hdz hz
change z ∈ C.domain at hz
change q.normalizedCovector i (F ∘ u) z =
F' (u (C.paramExtended z)) • q.normalizedCovector i u z
have hname : f z = u (C.paramExtended z) := by
rw [←hef hz]
simp only [IntegerChart.scalar,IntegerChart.paramExtended,dite_eq_left hz]
have hd := (hdF (f z)).comp_hasFDerivAt z hdz.hasFDerivAt
dsimp only [normalizedCovector,covector]
rw [hcomp,hd.fderiv,hu,hname]
have he : differentialRow (F' (u (C.paramExtended z)) • fderiv ℝ f z) =
F' (u (C.paramExtended z)) • differentialRow (fderiv ℝ f z) := rfl
rw [he,Matrix.mulVec_smul,WithLp.toLp_smul]
lemma gradient_chain_ae {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u)
{F F' : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
(hdF : ∀ t, HasDerivAt F (F' t) t) :
q.gradient (hF.comp hu) =ᵐ[q.atlasMeasure]
(fun w => F' (u (q.atlasParam w)) • q.gradient hu w) := by
filter_upwards [(q.memLp_gradientFunction (hF.comp hu)).coeFn_toLp,
(q.memLp_gradientFunction hu).coeFn_toLp,q.gradientFunction_chain hu hF hdF] with w h1 h2 h3
exact h1.trans (h3.trans (congrArg (F' (u (q.atlasParam w)) • ·) h2.symm))
end CAT0Fillings.ChartGeometry
end
section
open Set Filter MeasureTheory
open scoped Topology ENNReal
namespace CAT0Fillings.ClosedCalculus
variable {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] [InnerProductSpace ℝ E]
lemma l2_norm_sq (g : Lp E 2 μ) : ‖g‖^2 = ∫ x, ‖g x‖^2 ∂μ := by
rw [←real_inner_self_eq_norm_sq,L2.inner_def]
simp only [real_inner_self_eq_norm_sq]
lemma l2_tendsto_zero_of_bound (f : ℕ → Lp E 2 μ) (g : α → ℝ)
(hg : Integrable (fun x => (g x)^2) μ)
(hb : ∀ j, ∀ᵐ x ∂μ, ‖f j x‖ ≤ g x)
(hl : ∀ᵐ x ∂μ, Tendsto (fun j => f j x) atTop (𝓝 0)) :
Tendsto f atTop (𝓝 0) := by
have hh : Tendsto (fun j => ∫ x, ‖f j x‖^2 ∂μ) atTop (𝓝 (∫ _ : α, (0:ℝ) ∂μ)) := by
apply tendsto_integral_of_dominated_convergence (fun x => g x^2)
(fun j => (Lp.aestronglyMeasurable (f j)).norm.pow 2) hg
· intro j
filter_upwards [hb j] with x hx
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
exact (sq_le_sq₀ (norm_nonneg _) ((norm_nonneg _).trans hx)).mpr hx
· filter_upwards [hl] with x hx
simpa only [Pi.pow_apply,norm_zero,zero_pow (by decide : 2 ≠ 0)] using hx.norm.pow 2
simp only [integral_zero,←l2_norm_sq] at hh
have hn := hh.sqrt
simp only [Real.sqrt_sq (norm_nonneg _),Real.sqrt_zero] at hn
exact tendsto_zero_iff_norm_tendsto_zero.mpr hn
lemma memLp_smul_bounded {b : α → ℝ} (hb : AEStronglyMeasurable b μ)
{C : ℝ} (hC : ∀ᵐ x ∂μ, |b x| ≤ C) (g : Lp E 2 μ) :
MemLp (fun x => b x • g x) 2 μ := by
apply (Lp.memLp g).of_le_mul (c := C) (hb.smul (Lp.aestronglyMeasurable g))
filter_upwards [hC] with x hx
change ‖b x • g x‖ ≤ C*‖g x‖
rw [norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
noncomputable def mulLp {b : α → ℝ} (hb : AEStronglyMeasurable b μ)
{C : ℝ} (hC : ∀ᵐ x ∂μ, |b x| ≤ C) (g : Lp E 2 μ) : Lp E 2 μ :=
(memLp_smul_bounded hb hC g).toLp _
lemma mulLp_ae {b : α → ℝ} (hb : AEStronglyMeasurable b μ)
{C : ℝ} (hC : ∀ᵐ x ∂μ, |b x| ≤ C) (g : Lp E 2 μ) :
mulLp hb hC g =ᵐ[μ] (fun x => b x • g x) := (memLp_smul_bounded hb hC g).coeFn_toLp
lemma mulLp_sub_bound {b : α → ℝ} (hb : AEStronglyMeasurable b μ)
{C : ℝ} (hC : ∀ᵐ x ∂μ, |b x| ≤ C) (g h : Lp E 2 μ) :
‖mulLp hb hC g-mulLp hb hC h‖ ≤ C*‖g-h‖ := by
apply Lp.norm_le_mul_norm_of_ae_le_mul
filter_upwards [Lp.coeFn_sub (mulLp hb hC g) (mulLp hb hC h),mulLp_ae hb hC g,
mulLp_ae hb hC h,Lp.coeFn_sub g h,hC] with x hs hg hh hd hCx
rw [hs,hd,Pi.sub_apply,hg,hh,←smul_sub,norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_right hCx (norm_nonneg _)
lemma mulLp_tendsto {b : ℕ → α → ℝ} {c : α → ℝ}
(hb : ∀ j, AEStronglyMeasurable (b j) μ) (hc : AEStronglyMeasurable c μ)
{C : ℝ} (hC : 0 ≤ C)
(hbC : ∀ j, ∀ᵐ x ∂μ, |b j x| ≤ C) (hcC : ∀ᵐ x ∂μ, |c x| ≤ C)
(hbc : ∀ᵐ x ∂μ, Tendsto (fun j => b j x) atTop (𝓝 (c x)))
{g : ℕ → Lp E 2 μ} {h : Lp E 2 μ} (hgh : Tendsto g atTop (𝓝 h)) :
Tendsto (fun j => mulLp (hb j) (hbC j) (g j)) atTop (𝓝 (mulLp hc hcC h)) := by
have hfixed : Tendsto (fun j => mulLp (hb j) (hbC j) h-mulLp hc hcC h) atTop (𝓝 0) := by
apply l2_tendsto_zero_of_bound _ (fun x => 2*C*‖h x‖)
· have hi := (Lp.memLp h).integrable_norm_pow (by norm_num : (2:ℕ) ≠ 0)
convert hi.const_mul ((2*C)^2) using 1
funext x
ring
· intro j
filter_upwards [Lp.coeFn_sub (mulLp (hb j) (hbC j) h) (mulLp hc hcC h),
mulLp_ae (hb j) (hbC j) h,mulLp_ae hc hcC h,hbC j,hcC] with x hs hb hc hbC hcC
rw [hs,Pi.sub_apply,hb,hc,←sub_smul,norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_right ((abs_sub (b j x) (c x)).trans (by linarith)) (norm_nonneg _)
· have hall : ∀ᵐ x ∂μ, ∀ j, (mulLp (hb j) (hbC j) h-mulLp hc hcC h) x =
(b j x-c x) • h x := by
apply ae_all_iff.mpr
intro j
filter_upwards [Lp.coeFn_sub (mulLp (hb j) (hbC j) h) (mulLp hc hcC h),
mulLp_ae (hb j) (hbC j) h,mulLp_ae hc hcC h] with x hs hb hc
rw [hs,Pi.sub_apply,hb,hc,sub_smul]
filter_upwards [hall,hbc] with x he hx
simp_rw [he]
simpa only [sub_self,zero_smul] using (hx.sub_const (c x)).smul_const (h x)
have hpert : Tendsto (fun j => mulLp (hb j) (hbC j) (g j)-mulLp (hb j) (hbC j) h)
atTop (𝓝 0) := by
apply tendsto_zero_iff_norm_tendsto_zero.mpr
apply squeeze_zero (fun _ => norm_nonneg _) (fun j => mulLp_sub_bound (hb j) (hbC j) (g j) h)
simpa only [Real.norm_of_nonneg hC,sub_self,norm_zero,mul_zero] using
((hgh.sub_const h).norm).const_mul ‖C‖
have hh := hpert.add hfixed
simp only [sub_add_sub_cancel,add_zero] at hh
have hf := hh.add_const (mulLp hc hcC h)
simpa only [sub_add_cancel,zero_add] using hf
end CAT0Fillings.ClosedCalculus
end
end OAI
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