| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Tangent.Blowup |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology |
|
|
| namespace CAT0Fillings.TangentDensity |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
|
|
| lemma ae_blowup_inMeasure {f : E → F} (hf : Integrable f μ) |
| {ε : ℕ → ℝ} (hε : ∀ j, 0 < ε j) (hε0 : Tendsto ε atTop (𝓝 0)) : |
| ∀ᵐ z ∂μ, ∀ R : ℝ, 0 < R → |
| TendstoInMeasure (μ.restrict (closedBall (0:E) R)) |
| (fun j h => f (z+ε j • h)) atTop (fun _ => f z) := by |
| filter_upwards [IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub μ hf.locallyIntegrable (1:ℝ)] with z hz |
| intro R hR |
| apply tendstoInMeasure_blowup μ hf hε hR |
| apply hz (fun _ : ℕ => z) (fun j => ε j*R) |
| · apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨?_,Eventually.of_forall (fun j => mul_pos (hε j) hR)⟩ |
| simpa using hε0.mul_const R |
| · filter_upwards [] with j |
| exact mem_closedBall_self (by simpa using (mul_pos (hε j) hR).le) |
|
|
| lemma ae_all_blowup_inMeasure {f : E → F} (hf : Integrable f μ) : |
| ∀ᵐ z ∂μ, ∀ (ε : ℕ → ℝ), (∀ j, 0 < ε j) → Tendsto ε atTop (𝓝 0) → |
| ∀ R : ℝ, 0 < R → TendstoInMeasure (μ.restrict (closedBall (0:E) R)) |
| (fun j h => f (z+ε j • h)) atTop (fun _ => f z) := by |
| filter_upwards [IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub μ hf.locallyIntegrable (1:ℝ)] with z hz |
| intro ε hε hε0 R hR |
| apply tendstoInMeasure_blowup μ hf hε hR |
| apply hz (fun _ : ℕ => z) (fun j => ε j*R) |
| · apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨?_,Eventually.of_forall (fun j => mul_pos (hε j) hR)⟩ |
| simpa using hε0.mul_const R |
| · filter_upwards [] with j |
| exact mem_closedBall_self (by simpa using (mul_pos (hε j) hR).le) |
|
|
| lemma common_subsequence_three {α A B C : Type*} [MeasurableSpace α] |
| [NormedAddCommGroup A] [NormedAddCommGroup B] [NormedAddCommGroup C] |
| {μ : Measure α} {f : ℕ → α → A} {g : ℕ → α → B} {h : ℕ → α → C} |
| {f₀ : α → A} {g₀ : α → B} {h₀ : α → C} |
| (hf : TendstoInMeasure μ f atTop f₀) (hg : TendstoInMeasure μ g atTop g₀) |
| (hh : TendstoInMeasure μ h atTop h₀) : |
| ∃ σ : ℕ → ℕ, StrictMono σ ∧ ∀ᵐ x ∂μ, |
| Tendsto (fun j => f (σ j) x) atTop (𝓝 (f₀ x)) ∧ |
| Tendsto (fun j => g (σ j) x) atTop (𝓝 (g₀ x)) ∧ |
| Tendsto (fun j => h (σ j) x) atTop (𝓝 (h₀ x)) := by |
| obtain ⟨σ,hσ,hf⟩ := hf.exists_seq_tendsto_ae |
| obtain ⟨τ,hτ,hg⟩ := (hg.comp hσ.tendsto_atTop).exists_seq_tendsto_ae |
| obtain ⟨υ,hυ,hh⟩ := (hh.comp (hσ.comp hτ).tendsto_atTop).exists_seq_tendsto_ae |
| refine ⟨σ ∘ τ ∘ υ,hσ.comp (hτ.comp hυ),?_⟩ |
| filter_upwards [hf,hg,hh] with x hf hg hh |
| exact ⟨hf.comp (hτ.comp hυ).tendsto_atTop,hg.comp hυ.tendsto_atTop,hh⟩ |
|
|
| lemma indicator_one_eventually {α : Type*} {s : Set α} [DecidablePred (· ∈ s)] |
| {q : ℕ → α} (h : Tendsto (fun j => s.indicator (fun _ => (1:ℝ)) (q j)) |
| atTop (𝓝 1)) : ∀ᶠ j in atTop, q j ∈ s := by |
| filter_upwards [(tendsto_order.1 h).1 (1/2) (by norm_num)] with j hj |
| by_contra hn |
| simp only [indicator_of_notMem hn] at hj |
| norm_num at hj |
| end CAT0Fillings.TangentDensity |
| end |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal |
|
|
| namespace CAT0Fillings.TangentDensity |
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} |
|
|
| lemma lintegral_limit_le {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} |
| (hf : ∀ j, AEMeasurable (f j) μ) |
| (hg : ∀ᵐ x ∂μ, Tendsto (fun j => f j x) atTop (𝓝 (g x))) |
| {B : ℕ → ℝ≥0∞} {L : ℝ≥0∞} (hB : ∀ j, ∫⁻ x, f j x ∂μ ≤ B j) |
| (hL : Tendsto B atTop (𝓝 L)) : ∫⁻ x, g x ∂μ ≤ L := by |
| calc |
| ∫⁻ x, g x ∂μ = ∫⁻ x, liminf (fun j => f j x) atTop ∂μ := |
| lintegral_congr_ae (hg.mono fun x hx => hx.liminf_eq.symm) |
| _ ≤ liminf (fun j => ∫⁻ x, f j x ∂μ) atTop := lintegral_liminf_le' hf |
| _ ≤ liminf B atTop := liminf_le_liminf (Eventually.of_forall hB) |
| _ = L := hL.liminf_eq |
|
|
| lemma integral_limit_le {f : ℕ → α → ℝ} {g : α → ℝ} |
| (hf : ∀ j, Integrable (f j) μ) (hg : Integrable g μ) |
| (hf0 : ∀ j, ∀ᵐ x ∂μ, 0 ≤ f j x) (hg0 : ∀ᵐ x ∂μ, 0 ≤ g x) |
| (hfg : ∀ᵐ x ∂μ, Tendsto (fun j => f j x) atTop (𝓝 (g x))) |
| {B : ℕ → ℝ} {L : ℝ} (hB : ∀ j, ∫ x, f j x ∂μ ≤ B j) |
| (hL : Tendsto B atTop (𝓝 L)) : (∫ x, g x ∂μ) ≤ L := by |
| have hL0 : 0 ≤ L := ge_of_tendsto hL (Eventually.of_forall fun j => |
| (integral_nonneg_of_ae (hf0 j)).trans (hB j)) |
| have hh := lintegral_limit_le (f := fun j x => ENNReal.ofReal (f j x)) |
| (fun j => ENNReal.continuous_ofReal.measurable.comp_aemeasurable (hf j).aestronglyMeasurable.aemeasurable) |
| (hfg.mono fun x hx => ENNReal.continuous_ofReal.continuousAt.tendsto.comp hx) |
| (B := fun j => ENNReal.ofReal (B j)) (L := ENNReal.ofReal L) (fun j => ?_) |
| (ENNReal.continuous_ofReal.continuousAt.tendsto.comp hL) |
| · rw [←ofReal_integral_eq_lintegral_ofReal hg (hg0)] at hh |
| exact (ENNReal.ofReal_le_ofReal_iff hL0).mp hh |
| · rw [←ofReal_integral_eq_lintegral_ofReal (hf j) (hf0 j)] |
| exact ENNReal.ofReal_le_ofReal (hB j) |
| end CAT0Fillings.TangentDensity |
| end |
|
|
| section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology |
|
|
| namespace CAT0Fillings.TangentDensity |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {X : Type*} [MetricSpace X] |
|
|
| lemma kernel_tendsto {u ρ R : ℕ → ℝ} {u₀ ρ₀ R₀ κ β p : ℝ} (n : ℕ) |
| (hu₀ : 0 < u₀) (hu : Tendsto u atTop (𝓝 u₀)) |
| (hρ : Tendsto ρ atTop (𝓝 ρ₀)) (hR : Tendsto R atTop (𝓝 R₀)) : |
| Tendsto (fun j => u j^p*ρ j/(1+κ^2*R j^2*u j^(2*β))^n) atTop |
| (𝓝 (u₀^p*ρ₀/(1+κ^2*R₀^2*u₀^(2*β))^n)) := by |
| exact ((hu.rpow_const (.inl hu₀.ne')).mul hρ).div |
| ((tendsto_const_nhds.add ((tendsto_const_nhds.mul (hR.pow 2)).mul |
| (hu.rpow_const (.inl hu₀.ne')))).pow n) |
| (pow_ne_zero n (ne_of_gt (by positivity))) |
|
|
| noncomputable def chartExtension {s : Set E} (φ : s → X) (z : s) (x : E) : X := by |
| classical |
| exact if hx : x ∈ s then φ ⟨x,hx⟩ else φ z |
|
|
| lemma chartExtension_rescaled_tendsto {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)) (h : E) |
| (hs : ∀ᶠ j in atTop, (z:E)+ε j • h ∈ s) : |
| Tendsto (fun j => dist (chartExtension φ z ((z:E)+ε j • h)) (φ z)/ε j) |
| atTop (𝓝 (d h)) := by |
| classical |
| let q : ℕ → s := fun j => if hx : (z:E)+ε j • h ∈ s then ⟨(z:E)+ε j • h,hx⟩ else z |
| have hq : ∀ᶠ j in atTop, (q j:E) = (z:E)+ε j • h := by |
| filter_upwards [hs] with j hj |
| simp [q,hj] |
| have ht := MetricDifferentiation.rescaled_distance_tendsto hd hε hε0 h q hq |
| apply ht.congr' |
| filter_upwards [hs] with j hj |
| simp [q,chartExtension,hj] |
| |
| lemma chart_kernel_tendsto {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) (h : E) |
| (hu : Tendsto (fun j => u ((z:E)+ε j • h)) atTop (𝓝 (u z))) |
| (hρ : Tendsto (fun j => ρ ((z:E)+ε j • h)) atTop (𝓝 (ρ z))) |
| (hχ : Tendsto (fun j => s.indicator (fun _ => (1:ℝ)) ((z:E)+ε j • h)) |
| atTop (𝓝 1)) : |
| Tendsto (fun j => 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)) atTop |
| (𝓝 (u z^p*ρ z/(1+κ^2*(d h)^2*u z^(2*β))^n)) := by |
| classical |
| have hs := indicator_one_eventually hχ |
| have ht := kernel_tendsto (κ := κ) (β := β) (p := p) n hu₀ hu hρ |
| (chartExtension_rescaled_tendsto hd hε hε0 h hs) |
| apply ht.congr' |
| filter_upwards [hs] with j hj |
| rw [indicator_of_mem hj] |
| end CAT0Fillings.TangentDensity |
| end |
|
|
| end OAI |
|
|
