| import Mathlib |
| import OAI.Geometry.Cannon.ScalarLimits.Dirac |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set Metric Topology |
| open Set Topology unitInterval |
| open Set Filter Topology |
| open scoped BigOperators |
| open Set unitInterval |
| open Set MeasureTheory |
| open scoped NNReal ENNReal |
|
|
| namespace CannonAtomic |
|
|
| open Set MeasureTheory |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| theorem record_annular_bound {X ι : Type*} [MeasurableSpace X] [Fintype ι] |
| (p : ι → X) (w : ι → ℝ) (M M' E D L : ℝ) (hM : 0 < M) |
| (hE : 0 ≤ E) (_hL : 0 < L) (hrecord : M' ≤ M/L) |
| (S : Set X) (hS : MeasurableSet S) |
| (hbound : D^2 ≤ E*M'*(∑ i ∈ Finset.univ.filter (fun i => p i ∈ S), (w i)^2)) : |
| D^2 ≤ (E/L)*(atomic p w M hM.le S : ℝ) := by |
| classical |
| rw [atomic_apply p w M hM.le S hS] |
| have hs : 0 ≤ ∑ i ∈ Finset.univ.filter (fun i => p i ∈ S), (w i)^2 := |
| Finset.sum_nonneg (fun i _ => sq_nonneg _) |
| calc |
| D^2 ≤ E*M'*(∑ i ∈ Finset.univ.filter (fun i => p i ∈ S), (w i)^2) := hbound |
| _ ≤ E*(M/L)*(∑ i ∈ Finset.univ.filter (fun i => p i ∈ S), (w i)^2) := |
| mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hrecord hE) hs |
| _ = (E/L)*(M*(∑ i ∈ Finset.univ.filter (fun i => p i ∈ S), (w i)^2)) := by ring |
|
|
| end CannonAtomic |
|
|
| open Set Filter Metric Topology |
|
|
| namespace CannonAnnularCompactness |
|
|
| variable {X : Type*} [PseudoMetricSpace X] |
|
|
| def closedAnnulus (x : X) (R : ℝ) : Set X := |
| {y | R/4 ≤ dist y x ∧ dist y x ≤ 4*R} |
|
|
| lemma annuli_disjoint (x : X) (R S : ℝ) (h : 16*S < R) : |
| Disjoint (closedAnnulus x R) (closedAnnulus x S) := by |
| apply Set.disjoint_left.mpr |
| intro y hy hz |
| dsimp [closedAnnulus] at hy hz |
| linarith [hy.1, hz.2] |
|
|
| theorem disjoint_atomic_energy {ι κ : Type*} [Fintype ι] [Fintype κ] |
| (A : κ → Finset ι) (w : ι → ℝ) (hA : Pairwise (fun i j => Disjoint (A i) (A j))) : |
| (∑ k, ∑ i ∈ A k, (w i)^2) ≤ ∑ i, (w i)^2 := by |
| classical |
| rw [← Finset.sum_biUnion (fun _ _ _ _ hne => hA hne)] |
| apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) |
| exact fun i _ _ => sq_nonneg _ |
|
|
| theorem many_annuli_bound {ι κ : Type*} [Fintype ι] [Fintype κ] |
| (A : κ → Finset ι) (w : ι → ℝ) (hA : Pairwise (fun i j => Disjoint (A i) (A j))) |
| (D C E : ℝ) (hC : 0 ≤ C) (hmass : (∑ i, (w i)^2) ≤ E) |
| (hlocal : ∀ k, D^2 ≤ C * (∑ i ∈ A k, (w i)^2)) : |
| (Fintype.card κ : ℝ)*D^2 ≤ C*E := by |
| calc |
| _ = ∑ k : κ, D^2 := by simp |
| _ ≤ ∑ k : κ, C * (∑ i ∈ A k, (w i)^2) := |
| Finset.sum_le_sum (fun k _ => hlocal k) |
| _ = C * (∑ k : κ, ∑ i ∈ A k, (w i)^2) := (Finset.mul_sum ..).symm |
| _ ≤ C * (∑ i, (w i)^2) := |
| mul_le_mul_of_nonneg_left (disjoint_atomic_energy A w hA) hC |
| _ ≤ C*E := mul_le_mul_of_nonneg_left hmass hC |
|
|
| theorem equicontinuousAt_of_tail_quadratic (f : ℕ → X → ℝ) (x : X) (E : ℝ) |
| (hE : 0 ≤ E) (hf : ∀ n, ContinuousAt (f n) x) |
| (hb : ∀ m : ℕ, ∃ δ > 0, ∃ N : ℕ, |
| ∀ n, N ≤ n → ∀ y, dist y x < δ → |
| (m : ℝ)*(f n x-f n y)^2 ≤ E) : |
| EquicontinuousAt f x := by |
| rw [Metric.equicontinuousAt_iff_right] |
| intro ε hε |
| obtain ⟨m, hm⟩ := exists_nat_gt (E/ε^2) |
| have hmpos : (0:ℝ) < m := (div_nonneg hE (sq_nonneg ε)).trans_lt hm |
| have hsmall : E < (m:ℝ)*ε^2 := (div_lt_iff₀ (sq_pos_of_pos hε)).mp hm |
| obtain ⟨δ, hδ, N, hN⟩ := hb m |
| have heq : EquicontinuousAt (fun n : Fin N => f n) x := |
| equicontinuousAt_finite.mpr (fun n => hf n) |
| have hfin := Metric.equicontinuousAt_iff_right.mp heq ε hε |
| filter_upwards [hfin, Metric.ball_mem_nhds x hδ] with y hy hyr n |
| by_cases hn : n < N |
| · exact hy ⟨n, hn⟩ |
| · have hbound := hN n (le_of_not_gt hn) y hyr |
| rw [Real.dist_eq] |
| by_contra hh |
| have hge : ε ≤ |f n x-f n y| := le_of_not_gt hh |
| have hsq := pow_le_pow_left₀ hε.le hge 2 |
| rw [sq_abs] at hsq |
| have := mul_le_mul_of_nonneg_left hsq hmpos.le |
| linarith |
|
|
| end CannonAnnularCompactness |
|
|
| namespace CannonAnnularCompactness |
|
|
| open MeasureTheory |
| open scoped NNReal ENNReal |
|
|
| lemma finite_measure_disjoint_sum {X ι : Type*} [MeasurableSpace X] [Fintype ι] |
| (μ : FiniteMeasure X) (A : ι → Set X) (hmeas : ∀ i, MeasurableSet (A i)) |
| (hdis : Pairwise (fun i j => Disjoint (A i) (A j))) : |
| (∑ i, (μ (A i) : ℝ)) ≤ (μ.mass : ℝ) := by |
| classical |
| have h := sum_measure_le_measure_univ (μ := (μ : Measure X)) |
| (s := Finset.univ) (t := A) (fun i _ => (hmeas i).nullMeasurableSet) |
| (fun i _ j _ hij => (hdis hij).aedisjoint) |
| simp only [← FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure] at h |
| have h' : (∑ i, μ (A i) : ℝ≥0) ≤ μ univ := by exact_mod_cast h |
| exact_mod_cast h' |
|
|
| theorem equicontinuousAt_of_disjoint_measures {X Y : Type*} |
| [PseudoMetricSpace X] [MeasurableSpace Y] |
| (f : ℕ → X → ℝ) (x : X) (μ : ℕ → FiniteMeasure Y) (C E : ℝ) |
| (hC : 0 ≤ C) (hE : 0 ≤ E) (hmass : ∀ n, ((μ n).mass : ℝ) ≤ E) |
| (hf : ∀ n, ContinuousAt (f n) x) |
| (hb : ∀ m : ℕ, ∃ A : Fin (m+1) → Set Y, |
| (∀ i, MeasurableSet (A i)) ∧ |
| Pairwise (fun i j => Disjoint (A i) (A j)) ∧ |
| ∀ i, ∃ δ > 0, ∃ N : ℕ, ∀ n, N ≤ n → ∀ y, dist y x < δ → |
| (f n x - f n y)^2 ≤ C*(μ n (A i) : ℝ)) : |
| EquicontinuousAt f x := by |
| classical |
| apply equicontinuousAt_of_tail_quadratic f x (C*E) (mul_nonneg hC hE) hf |
| intro m |
| obtain ⟨A,hA,hdis,hb⟩ := hb m |
| choose δ hδ N hN using hb |
| let d := Finset.univ.inf' Finset.univ_nonempty δ |
| have hd : 0 < d := (Finset.lt_inf'_iff _).mpr (fun i _ => hδ i) |
| let n₀ := Finset.univ.sup N |
| refine ⟨d,hd,n₀,?_⟩ |
| intro n hn y hy |
| have hlocal (i : Fin (m+1)) : (f n x-f n y)^2 ≤ C*(μ n (A i) : ℝ) := by |
| apply hN i n ((Finset.le_sup (Finset.mem_univ i)).trans hn) y |
| exact hy.trans_le (Finset.inf'_le _ (Finset.mem_univ i)) |
| have he : ((m+1 : ℕ) : ℝ)*(f n x-f n y)^2 ≤ C*E := by |
| calc |
| _ = ∑ i : Fin (m+1), (f n x-f n y)^2 := by simp |
| _ ≤ ∑ i : Fin (m+1), C*(μ n (A i) : ℝ) := Finset.sum_le_sum (fun i _ => hlocal i) |
| _ = C*(∑ i : Fin (m+1), (μ n (A i) : ℝ)) := (Finset.mul_sum ..).symm |
| _ ≤ C*((μ n).mass : ℝ) := mul_le_mul_of_nonneg_left |
| (finite_measure_disjoint_sum (μ n) A hA hdis) hC |
| _ ≤ C*E := mul_le_mul_of_nonneg_left (hmass n) hC |
| exact (mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.le_succ m) |
| (sq_nonneg (f n x-f n y))).trans he |
|
|
| theorem equicontinuousAt_of_annular_tails {X : Type*} [MetricSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
| (f : ℕ → X → ℝ) (x : X) (μ : ℕ → FiniteMeasure X) (C E : ℝ) |
| (hC : 0 ≤ C) (hE : 0 ≤ E) (hmass : ∀ n, ((μ n).mass : ℝ) ≤ E) |
| (hf : ∀ n, ContinuousAt (f n) x) |
| (R : ℕ → ℝ) (hR : ∀ i j, i < j → 16*R j < R i) |
| (hb : ∀ i, ∃ δ > 0, ∃ N : ℕ, ∀ n, N ≤ n → ∀ y, dist y x < δ → |
| (f n x - f n y)^2 ≤ C*(μ n (closedAnnulus x (R i)) : ℝ)) : |
| EquicontinuousAt f x := by |
| apply equicontinuousAt_of_disjoint_measures f x μ C E hC hE hmass hf |
| intro m |
| refine ⟨fun i => closedAnnulus x (R i), ?_, ?_, fun i => hb i⟩ |
| · intro i |
| exact (isClosed_le continuous_const (continuous_id.dist continuous_const)).measurableSet.inter |
| (isClosed_le (continuous_id.dist continuous_const) continuous_const).measurableSet |
| · intro i j hij |
| rcases lt_or_gt_of_ne hij with h | h |
| · exact annuli_disjoint x (R i) (R j) (hR i j h) |
| · exact (annuli_disjoint x (R j) (R i) (hR j i h)).symm |
|
|
| end CannonAnnularCompactness |
|
|
| open Set Filter Topology |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
