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59d1c80 | 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 | import Mathlib
import OAI.Geometry.IntegralFillings.Model
import OAI.Geometry.IntegralFillings.Charts.DensityPush
import OAI.Geometry.IntegralFillings.Geometry.Polarization
import OAI.Geometry.IntegralFillings.Geometry.MatrixRows
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open Filter Set
open scoped Topology NNReal
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
namespace SharpIntegralFillings
open Set MeasureTheory Filter
open scoped ENNReal NNReal Topology
lemma densityPush_mono_ae {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
{μ : Measure α} {φ : α → β} (hφ : Measurable φ) {ρ σ : α → ℝ}
(h : ρ ≤ᵐ[μ] σ) : densityPush μ φ ρ ≤ densityPush μ φ σ := by
rw [densityPush_eq_map μ hφ, densityPush_eq_map μ hφ]
apply Measure.map_mono _ hφ
exact withDensity_mono (h.mono fun _ hx => ENNReal.ofReal_le_ofReal hx)
lemma densityPush_const_mul {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
(μ : Measure α) (φ : α → β) (ρ : α → ℝ) (c : ℝ≥0) :
densityPush μ φ (fun z => (c : ℝ)*ρ z) = c • densityPush μ φ ρ := by
by_cases hc : c = 0
· subst c
simp [densityPush]
unfold densityPush
have hfun : (fun z => ENNReal.ofReal ((c : ℝ)*ρ z)) =
(c : ENNReal) • (fun z => ENNReal.ofReal (ρ z)) := by
ext z
simp only [Pi.smul_apply,smul_eq_mul,ENNReal.ofReal_mul c.coe_nonneg,ENNReal.ofReal_coe_nnreal]
rw [hfun,withDensity_smul' _ _ ENNReal.coe_ne_top,
aemeasurable_smul_measure_iff (ENNReal.coe_ne_zero.mpr hc)]
by_cases hφ : AEMeasurable φ (μ.withDensity fun z => ENNReal.ofReal (ρ z))
· simp only [ite_eq_left hφ]
rw [Measure.map_smul _ hφ]
rfl
· simp only [ite_eq_right hφ, smul_zero]
lemma densityPush_sum {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
{ι : Type*} (μ : ι → Measure α) {φ : α → β} (hφ : Measurable φ) (ρ : α → ℝ) :
densityPush (Measure.sum μ) φ ρ = Measure.sum (fun i => densityPush (μ i) φ ρ) := by
simp only [densityPush_eq_map _ hφ,withDensity_sum,Measure.map_sum hφ.aemeasurable]
lemma restrict_eq_sum_of_ae_partition {α : Type*} [MeasurableSpace α]
{μ : Measure α} {s : Set α} {t : ℕ → Set α}
(ht : ∀ i, MeasurableSet (t i)) (hts : ∀ i, t i ⊆ s)
(hdisj : Pairwise (fun i j => Disjoint (t i) (t j))) (hnull : μ (s \ ⋃ i, t i) = 0) :
μ.restrict s = Measure.sum (fun i => μ.restrict (t i)) := by
have hsub : (⋃ i, t i) ⊆ s := iUnion_subset hts
have heq : s =ᵐ[μ] ⋃ i, t i := ae_eq_set.mpr ⟨hnull,by simp only [sdiff_eq_empty.mpr hsub,measure_empty]⟩
rw [Measure.restrict_congr_set heq,Measure.restrict_iUnion hdisj ht]
lemma densityPush_le_of_partition {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
{μ : Measure α} {ν : Measure β} {φ : α → β} (hφ : Measurable φ) (ρ : α → ℝ)
{s : Set α} {t : ℕ → Set α} (ht : ∀ i, MeasurableSet (t i)) (hts : ∀ i, t i ⊆ s)
(hdisj : Pairwise (fun i j => Disjoint (t i) (t j))) (hnull : μ (s \ ⋃ i, t i) = 0)
(him : ∀ i, MeasurableSet (φ '' t i)) (hinj : Set.InjOn φ s)
(hbound : ∀ i, densityPush (μ.restrict (t i)) φ ρ ≤ ν.restrict (φ '' t i)) :
densityPush (μ.restrict s) φ ρ ≤ ν := by
rw [restrict_eq_sum_of_ae_partition ht hts hdisj hnull,densityPush_sum _ hφ]
have hdisj' : Pairwise (fun i j => Disjoint (φ '' t i) (φ '' t j)) := by
intro i j hij
apply Set.disjoint_left.mpr
rintro _ ⟨x,hxi,rfl⟩ ⟨y,hyj,hxy⟩
have hxy' : y = x := hinj (hts j hyj) (hts i hxi) hxy
subst y
exact Set.disjoint_left.mp (hdisj hij) hxi hyj
calc
_ ≤ Measure.sum (fun i => ν.restrict (φ '' t i)) := by
apply Measure.le_iff.mpr
intro u hu
simp only [Measure.sum_apply _ hu]
exact ENNReal.tsum_le_tsum (fun i => hbound i u)
_ = ν.restrict (⋃ i, φ '' t i) := (Measure.restrict_iUnion hdisj' him).symm
_ ≤ ν := Measure.restrict_le_self
end SharpIntegralFillings
namespace SharpIntegralFillings
open Matrix MeasureTheory Set Filter
open scoped MatrixOrder NNReal Topology
lemma sqrt_polarization_quadratic {k : ℕ} (p : Seminorm ℝ (Euc k))
(hp : ∀ v, p v = 0 ↔ v = 0)
(hpara : ∀ u v, p (u+v)^2+p (u-v)^2 = 2*p u^2+2*p v^2) (v : Fin k → ℝ) :
Real.sqrt (v ⬝ᵥ (polarizationMatrix p (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).mulVec v)
= p (WithLp.toLp 2 v) := by
have hrepr : (EuclideanSpace.basisFun (Fin k) ℝ).toBasis.repr (WithLp.toLp 2 v) = v := by
ext j
simp
have hquad := polarizationMatrix_quadratic (EuclideanSpace.basisFun (Fin k) ℝ).toBasis p hp hpara (WithLp.toLp 2 v)
rw [hrepr] at hquad
rw [hquad,Real.sqrt_sq (apply_nonneg _ _)]
lemma sqrt_det_polarization_le {k : ℕ} (p q : Seminorm ℝ (Euc k))
(hp : ∀ v, p v = 0 ↔ v = 0) (hq : ∀ v, q v = 0 ↔ v = 0)
(hpara : ∀ u v, p (u+v)^2+p (u-v)^2 = 2*p u^2+2*p v^2)
(hqpara : ∀ u v, q (u+v)^2+q (u-v)^2 = 2*q u^2+2*q v^2)
{L : ℝ} (hL : 0 ≤ L) (h : ∀ v, p v ≤ L*q v) :
Real.sqrt (polarizationMatrix p (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det ≤
L^k * Real.sqrt (polarizationMatrix q (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det := by
apply sqrt_det_le_mul_sqrt_det _ _
(polarizationMatrix_posDef _ p hp hpara).posSemidef
(polarizationMatrix_posDef _ q hq hqpara) hL
intro v
simpa only [sqrt_polarization_quadratic p hp hpara,
sqrt_polarization_quadratic q hq hqpara] using h (WithLp.toLp 2 v)
end SharpIntegralFillings
end
end OAI
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