| 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 |
|
|
