| 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 |
|
|
