| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Tangent.Coordinates |
| import OAI.Geometry.CAT0Fillings.Tangent.Kernel |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open TangentDensity |
|
|
| 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 chart_kernel_blowup_le (i : ℕ) (U : X → ℝ) (hU : Measurable U) |
| (hU0 : ∀ x, 0 ≤ U x) (u : Euc n → ℝ) (hu : ∀ z, 0 ≤ u z) |
| (huU : ∀ z ∈ (q.chart i).domain, u z ≤ U ((q.chart i).paramExtended z)) |
| (z : (q.chart i).domain) {ε β κ : ℝ} (hε : 0 < ε) (hb : 0 ≤ β) : |
| let ρ := (q.chart i).domain.indicator (q.density i) |
| (∫⁻ h, ENNReal.ofReal ((q.chart i).domain.indicator |
| (fun x => (u x)^(2*(n:ℝ)*β)*ρ x / |
| (1+κ^2*(dist (chartExtension (q.chart i).param z x) ((q.chart i).param z)/ε)^2*(u x)^(2*β))^n) |
| ((z:Euc n)+ε • h))) ≤ |
| ENNReal.ofReal ((ε^n)⁻¹) * ∫⁻ x, ENNReal.ofReal |
| ((U x)^(2*(n:ℝ)*β)/(1+κ^2*(dist x ((q.chart i).param z)/ε)^2*(U x)^(2*β))^n) |
| ∂MassMeasure.currentMassMeasure hT := by |
| classical |
| let ρ := (q.chart i).domain.indicator (q.density i) |
| let C := q.chart i |
| let F : X → ℝ≥0∞ := fun x => ENNReal.ofReal |
| ((U x)^(2*(n:ℝ)*β)/(1+κ^2*(dist x (C.param z)/ε)^2*(U x)^(2*β))^n) |
| have hF : Measurable F := ((hU.pow_const _).div |
| ((measurable_const.add ((measurable_const.mul |
| (((measurable_id.dist measurable_const).div_const ε).pow_const 2)).mul |
| (hU.pow_const _))).pow_const n)).ennreal_ofReal |
| let g : Euc n → ℝ := fun x => (u x)^(2*(n:ℝ)*β)*ρ x / |
| (1+κ^2*(dist (chartExtension C.param z x) (C.param z)/ε)^2*(u x)^(2*β))^n |
| change (∫⁻ h, ENNReal.ofReal (C.domain.indicator g ((z:Euc n)+ε • h))) ≤ _ |
| rw [lintegral_affine volume (fun x => ENNReal.ofReal (C.domain.indicator g x)) (z:Euc n) hε] |
| simp only [finrank_euclideanSpace,Fintype.card_fin] |
| gcongr 1 |
| apply le_trans ?_ (q.lintegral_coordinate_le i F hF) |
| have hi : (fun x => ENNReal.ofReal (C.domain.indicator g x)) = |
| C.domain.indicator (fun x => ENNReal.ofReal (g x)) := by |
| ext x |
| by_cases hx : x ∈ C.domain <;> simp [hx] |
| change (∫⁻ x, ENNReal.ofReal (C.domain.indicator g x)) ≤ _ |
| rw [hi,lintegral_indicator C.borel] |
| apply lintegral_mono_ae |
| filter_upwards [ae_restrict_mem C.borel] with x hx |
| have he : chartExtension C.param z x = C.paramExtended x := by |
| simp only [chartExtension,IntegerChart.paramExtended,dite_eq_left hx] |
| have density_nonneg : 0 ≤ q.density i x := |
| mul_nonneg (abs_nonneg _) (Real.sqrt_nonneg _) |
| change ENNReal.ofReal (g x) ≤ ENNReal.ofReal (q.density i x)*F (C.paramExtended x) |
| dsimp only [F] |
| have kernel_nonneg : 0 ≤ |
| (U (C.paramExtended x))^(2*(n:ℝ)*β) / |
| (1+κ^2*(dist (C.paramExtended x) (C.param z)/ε)^2* |
| (U (C.paramExtended x))^(2*β))^n := by |
| apply div_nonneg (Real.rpow_nonneg (hU0 _) _) |
| exact pow_nonneg (add_nonneg zero_le_one |
| (mul_nonneg (mul_nonneg (sq_nonneg κ) (sq_nonneg _)) |
| (Real.rpow_nonneg (hU0 _) _))) _ |
| rw [←ENNReal.ofReal_mul' kernel_nonneg] |
| apply ENNReal.ofReal_le_ofReal |
| have hm := tangent_kernel_mono (hu x) (huU x hx) hb |
| (mul_nonneg (sq_nonneg κ) (sq_nonneg (dist (C.paramExtended x) (C.param z)/ε))) n |
| dsimp only [g] |
| rw [show ρ x = q.density i x from indicator_of_mem hx _,he] |
| convert mul_le_mul_of_nonneg_right hm density_nonneg using 1 <;> first | rfl | ring |
| end CAT0Fillings.ChartGeometry |
| end |
| |
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal |
| |
| namespace CAT0Fillings.ChartGeometry |
| open TangentDensity |
| |
| 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 chart_kernel_aemeasurable (i : ℕ) (u : Euc n → ℝ) (hu : Measurable u) |
| (z : (q.chart i).domain) {ε : ℝ} (hε : 0 < ε) (p β κ : ℝ) : |
| let ρ := (q.chart i).domain.indicator (q.density i) |
| AEMeasurable (fun h : Euc n => ENNReal.ofReal ((q.chart i).domain.indicator |
| (fun x => u x^p*ρ x / |
| (1+κ^2*(dist (chartExtension (q.chart i).param z x) ((q.chart i).param z)/ε)^2*u x^(2*β))^n) |
| ((z:Euc n)+ε • h))) volume := by |
| classical |
| let C := q.chart i |
| let ρ := C.domain.indicator (q.density i) |
| let A : Euc n → Euc n := fun h => (z:Euc n)+ε • h |
| have hA : Measurable A := by |
| dsimp [A] |
| fun_prop |
| have hρ : Integrable ρ := IntegrableOn.integrable_indicator (q.density_integrable i) C.borel |
| have hρA := (integrable_affine volume ρ hρ (z:Euc n) hε.ne').aestronglyMeasurable.aemeasurable |
| have huA := hu.comp hA |
| have hdA : Measurable (fun h => dist (C.paramExtended (A h)) (C.param z)/ε) := |
| (((C.measurable_paramExtended.comp hA).dist measurable_const).div_const ε) |
| have hm : AEMeasurable (fun h => u (A h)^p*ρ (A h) / |
| (1+κ^2*(dist (C.paramExtended (A h)) (C.param z)/ε)^2*u (A h)^(2*β))^n) volume := |
| ((huA.pow_const p).aemeasurable.mul hρA).div |
| ((measurable_const.add ((measurable_const.mul (hdA.pow_const 2)).mul |
| (huA.pow_const (2*β)))).pow_const n).aemeasurable |
| have hi := (hm.indicator (C.borel.preimage hA)).ennreal_ofReal |
| convert hi using 1 |
| ext h |
| by_cases hh : A h ∈ C.domain |
| · dsimp [A,C] at hh |
| simp [A,C,ρ,chartExtension,IntegerChart.paramExtended,hh] |
| · dsimp [A,C] at hh |
| simp [A,C,ρ,hh] |
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology ENNReal |
|
|
| namespace CAT0Fillings.TangentDensity |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] {X : Type*} [MetricSpace X] {μ : Measure E} |
|
|
| lemma local_density_bound {s : Set E} {φ : s → X} {z : s} {d : Seminorm ℝ E} |
| (hd : MetricDifferentiation.HasCenteredMetricDifferentialWithin s φ d z) |
| {ε : ℕ → ℝ} (hε : ∀ j, 0 < ε j) (hε0 : Tendsto ε atTop (𝓝 0)) |
| (u ρ : E → ℝ) {κ β p : ℝ} (n : ℕ) (hu₀ : 0 < u z) |
| (hu : TendstoInMeasure μ (fun j h => u ((z:E)+ε j • h)) atTop (fun _ => u z)) |
| (hρ : TendstoInMeasure μ (fun j h => ρ ((z:E)+ε j • h)) atTop (fun _ => ρ z)) |
| (hχ : TendstoInMeasure μ (fun j h => s.indicator (fun _ => (1:ℝ)) ((z:E)+ε j • h)) |
| atTop (fun _ => 1)) |
| (hK : ∀ j, AEMeasurable (fun h => ENNReal.ofReal (s.indicator (fun x => u x^p*ρ x / |
| (1+κ^2*(dist (chartExtension φ z x) (φ z)/ε j)^2*u x^(2*β))^n) |
| ((z:E)+ε j • h))) μ) |
| {B : ℕ → ℝ≥0∞} {L : ℝ≥0∞} |
| (hB : ∀ j, (∫⁻ h, ENNReal.ofReal (s.indicator (fun x => u x^p*ρ x / |
| (1+κ^2*(dist (chartExtension φ z x) (φ z)/ε j)^2*u x^(2*β))^n) |
| ((z:E)+ε j • h)) ∂μ) ≤ B j) |
| (hL : Tendsto B atTop (𝓝 L)) : |
| (∫⁻ h, ENNReal.ofReal (u z^p*ρ z/(1+κ^2*(d h)^2*u z^(2*β))^n) ∂μ) ≤ L := by |
| obtain ⟨σ,hσ,ht⟩ := common_subsequence_three hu hρ hχ |
| apply lintegral_limit_le (fun j => hK (σ j)) (B := B ∘ σ) (L := L) ?_ |
| (fun j => hB (σ j)) (hL.comp hσ.tendsto_atTop) |
| filter_upwards [ht] with h hh |
| exact ENNReal.continuous_ofReal.continuousAt.tendsto.comp |
| (chart_kernel_tendsto hd (fun j => hε (σ j)) (hε0.comp hσ.tendsto_atTop) u ρ n hu₀ h |
| hh.1 hh.2.1 hh.2.2) |
| end CAT0Fillings.TangentDensity |
| end |
|
|
| end OAI |
|
|
