| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Radial.ChartMass |
| import OAI.Geometry.CAT0Fillings.Radial.IntegralDerivative |
| import OAI.Geometry.CAT0Fillings.Radial.MeasurableBound |
|
|
| namespace OAI |
|
|
| section |
| 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 CAT0Fillings.IntegerChart |
| open MeasureTheory Set Filter Metric Matrix |
| open scoped Topology NNReal Matrix.Norms.Elementwise |
|
|
| variable {X : Type*} [MetricSpace X] [CompactSpace X] {k : ℕ} (C : IntegerChart X k) |
| local instance : MeasurableSpace (Matrix (Fin k) (Fin k) ℝ) := _root_.borel _ |
| local instance : BorelSpace (Matrix (Fin k) (Fin k) ℝ) := ⟨rfl⟩ |
| theorem radial_normalized_integral_hasDerivWithinAt_zero |
| (p : Euc k → Seminorm ℝ (Euc k)) (hpm : ∀ v, Measurable (fun z => p z v)) |
| (hp : ∀ᵐ z ∂volume.restrict C.domain, |
| (∀ hz : z ∈ C.domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| C.domain C.param (p z) ⟨z,hz⟩) ∧ |
| (∀ u v, p z (u+v)^2+p z (u-v)^2 = 2*p z u^2+2*p z v^2) ∧ |
| (∀ v, p z v = 0 ↔ v = 0)) |
| (o : X) {g : X → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) |
| (μ : Measure (Euc k)) [IsFiniteMeasure μ] (hμ : μ ≪ volume.restrict C.domain) : |
| HasDerivWithinAt |
| (fun t => ∫ z, C.radialJacobian |
| (fun z => polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) |
| o g t z / Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det ∂μ) |
| (∫ z, -((k:ℝ)*C.scalar g z+C.scalar (dist o) z * |
| (differentialRow (fderivWithin ℝ (C.scalar (dist o)) C.domain z) ⬝ᵥ |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis)⁻¹.mulVec |
| (differentialRow (fderivWithin ℝ (C.scalar g) C.domain z)))) ∂μ) |
| (Ici 0) 0 := by |
| obtain ⟨M,hM⟩ := isCompact_univ.exists_bound_of_continuousOn hg.continuous.continuousOn |
| let B : ℝ := max 1 (max (K:ℝ) (max M (Metric.diam (univ : Set X)))) |
| have hB : 1 ≤ B := le_max_left .. |
| have hB0 : 0 ≤ B := le_trans (by norm_num) hB |
| have hK : (K:ℝ) ≤ B := (le_max_left ..).trans (le_max_right ..) |
| have hgb : ∀ x, |g x| ≤ B := by |
| intro x |
| have hm : |g x| ≤ M := by simpa only [Real.norm_eq_abs] using hM x (mem_univ x) |
| exact hm.trans ((le_max_left ..).trans ((le_max_right ..).trans (le_max_right ..))) |
| have hrb : ∀ x, dist o x ≤ B := by |
| intro x |
| exact (Metric.dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ o) (mem_univ x)).trans |
| ((le_max_right ..).trans ((le_max_right ..).trans (le_max_right ..))) |
| obtain ⟨L,U,hL,_⟩ := C.bilipschitz |
| obtain ⟨R,hR,heR⟩ := (C.scalar_lipschitzOn hL (LipschitzWith.dist_right o)).extend_real |
| obtain ⟨G,hG,heG⟩ := (C.scalar_lipschitzOn hL hg).extend_real |
| let P (z : Euc k) := polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis |
| let q : Euc k → MetricRadialParameters (Fin k) := fun z => |
| (P z,((C.scalar g z,C.scalar (dist o) z), |
| (differentialRow (fderiv ℝ R z),differentialRow (fderiv ℝ G z)))) |
| have hPm : Measurable P := by |
| have hh : Measurable (fun z i j => P z i j) := |
| measurable_pi_iff.mpr fun i => measurable_pi_iff.mpr fun j => |
| measurable_polarizationMatrix p (EuclideanSpace.basisFun (Fin k) ℝ).toBasis hpm i j |
| have heq : _root_.borel (Fin k → Fin k → ℝ) = MeasurableSpace.pi := |
| BorelSpace.measurable_eq.symm |
| change @Measurable (Euc k) (Fin k → Fin k → ℝ) _ (_root_.borel _) P |
| rw [heq] |
| exact hh |
| have hRd : Measurable (fun z => differentialRow (fderiv ℝ R z)) := by |
| apply measurable_pi_iff.mpr |
| intro i |
| simp only [differentialRow] |
| fun_prop |
| have hGd : Measurable (fun z => differentialRow (fderiv ℝ G z)) := by |
| apply measurable_pi_iff.mpr |
| intro i |
| simp only [differentialRow] |
| fun_prop |
| have hq : Measurable q := hPm.prodMk |
| ((C.measurable_scalar hg).prodMk (C.measurable_scalar (LipschitzWith.dist_right o)) |>.prodMk |
| (hRd.prodMk hGd)) |
| have hRae := ae_fderivWithin_eq_fderiv_extension volume C.borel hR heR |
| have hGae := ae_fderivWithin_eq_fderiv_extension volume C.borel hG heG |
| have hdata : ∀ᵐ z ∂volume.restrict C.domain, |
| |(q z).2.1.1| ≤ B ∧ |(q z).2.1.2| ≤ B ∧ |
| (∀ v : Fin k → ℝ, ((q z).2.2.1 ⬝ᵥ v)^2 ≤ B^2*(v ⬝ᵥ (q z).1.mulVec v)) ∧ |
| (∀ v : Fin k → ℝ, ((q z).2.2.2 ⬝ᵥ v)^2 ≤ B^2*(v ⬝ᵥ (q z).1.mulVec v)) := by |
| filter_upwards [hp,ae_restrict_mem C.borel,hRae,hGae, |
| C.ae_scalar_row_quadratic_bound p hp (LipschitzWith.dist_right o), |
| C.ae_scalar_row_quadratic_bound p hp hg] with z hpz hzs hRz hGz hRb hGb |
| have hP : (P z).PosDef := polarizationMatrix_posDef _ _ hpz.2.2 hpz.2.1 |
| have hPr (v : Fin k → ℝ) : 0 ≤ v ⬝ᵥ (P z).mulVec v := hP.posSemidef.dotProduct_mulVec_nonneg v |
| refine ⟨?_,?_,?_,?_⟩ |
| · simpa only [q,C.scalar_eq hzs] using hgb (C.param ⟨z,hzs⟩) |
| · simpa only [q,C.scalar_eq hzs,abs_of_nonneg dist_nonneg] using hrb (C.param ⟨z,hzs⟩) |
| · intro v |
| change (differentialRow (fderiv ℝ R z) ⬝ᵥ v)^2 ≤ B^2*(v ⬝ᵥ (P z).mulVec v) |
| rw [hRz] at hRb |
| have hh := hRb v |
| simp only [NNReal.coe_one,one_pow,one_mul] at hh |
| exact hh.trans (le_mul_of_one_le_left (hPr v) (by nlinarith)) |
| · intro v |
| change (differentialRow (fderiv ℝ G z) ⬝ᵥ v)^2 ≤ B^2*(v ⬝ᵥ (P z).mulVec v) |
| rw [hGz] at hGb |
| exact (hGb v).trans (mul_le_mul_of_nonneg_right |
| ((sq_le_sq₀ K.coe_nonneg hB0).2 hK) (hPr v)) |
| have hpos : ∀ᵐ z ∂μ, (q z).1.PosDef := by |
| filter_upwards [hμ.ae_le hp] with z hz |
| exact polarizationMatrix_posDef _ _ hz.2.2 hz.2.1 |
| have hh := integrated_intrinsicRadialJacobian_hasDerivWithinAt_zero μ q hq hB0 |
| (hμ.ae_le hdata) hpos |
| convert hh using 1 |
| · funext t |
| apply integral_congr_ae |
| filter_upwards [hμ.ae_le hRae,hμ.ae_le hGae] with z hRz hGz |
| simp only [radialJacobian,q,P,hRz,hGz] |
| · apply integral_congr_ae |
| filter_upwards [hμ.ae_le hRae,hμ.ae_le hGae] with z hRz hGz |
| simp only [q,P,hRz,hGz,Fintype.card_fin] |
| end CAT0Fillings.IntegerChart |
|
|
| namespace CAT0Fillings.IntegerChart |
| open MeasureTheory Set Filter Metric Matrix |
| open scoped Topology NNReal Matrix.Norms.Elementwise |
|
|
| variable {X : Type*} [MetricSpace X] [CompactSpace X] {k : ℕ} (C : IntegerChart X k) |
|
|
| theorem radial_weighted_integral_hasDerivWithinAt_zero |
| (p : Euc k → Seminorm ℝ (Euc k)) (hpm : ∀ v, Measurable (fun z => p z v)) |
| (hp : ∀ᵐ z ∂volume.restrict C.domain, |
| (∀ hz : z ∈ C.domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| C.domain C.param (p z) ⟨z,hz⟩) ∧ |
| (∀ u v, p z (u+v)^2+p z (u-v)^2 = 2*p z u^2+2*p z v^2) ∧ |
| (∀ v, p z v = 0 ↔ v = 0)) |
| (hρ : Integrable (fun z => |(C.multiplicity z : ℝ)| * Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det) |
| (volume.restrict C.domain)) |
| (o : X) {g : X → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) : |
| HasDerivWithinAt |
| (fun t => ∫ z, |(C.multiplicity z : ℝ)| * C.radialJacobian |
| (fun z => polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) |
| o g t z ∂volume.restrict C.domain) |
| (∫ z, (|(C.multiplicity z : ℝ)| * Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det) * |
| -((k:ℝ)*C.scalar g z+C.scalar (dist o) z * |
| (differentialRow (fderivWithin ℝ (C.scalar (dist o)) C.domain z) ⬝ᵥ |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis)⁻¹.mulVec |
| (differentialRow (fderivWithin ℝ (C.scalar g) C.domain z)))) |
| ∂volume.restrict C.domain) |
| (Ici 0) 0 := by |
| let ν := volume.restrict C.domain |
| let ρ : Euc k → ℝ := fun z => |(C.multiplicity z : ℝ)| * Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det |
| let μ := ν.withDensity (fun z => ENNReal.ofReal (ρ z)) |
| have hρ0 (z) : 0 ≤ ρ z := mul_nonneg (abs_nonneg _) (Real.sqrt_nonneg _) |
| let : IsFiniteMeasure μ := isFiniteMeasure_withDensity_ofReal hρ.hasFiniteIntegral |
| have hμ : μ ≪ ν := withDensity_absolutelyContinuous _ _ |
| have hh := C.radial_normalized_integral_hasDerivWithinAt_zero p hpm hp o hg μ hμ |
| have hchange (f : Euc k → ℝ) : (∫ z, f z ∂μ) = ∫ z, ρ z*f z ∂ν := by |
| rw [integral_withDensity_eq_integral_toReal_smul₀ |
| hρ.aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| change (∫ z, (ENNReal.ofReal (ρ z)).toReal • f z ∂ν) = _ |
| apply integral_congr_ae |
| exact Eventually.of_forall fun z => by |
| change (ENNReal.ofReal (ρ z)).toReal • f z = ρ z * f z |
| rw [ENNReal.toReal_ofReal (hρ0 z),smul_eq_mul] |
| convert hh using 1 |
| · funext t |
| rw [hchange] |
| apply integral_congr_ae |
| filter_upwards [hp] with z hz |
| have hs : Real.sqrt |
| (polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det ≠ 0 := |
| ne_of_gt (Real.sqrt_pos.2 (polarizationMatrix_posDef _ _ hz.2.2 hz.2.1).det_pos) |
| dsimp only [ρ] |
| field_simp |
| · exact (hchange _).symm |
| end CAT0Fillings.IntegerChart |
|
|
| end |
| end |
|
|
| end OAI |
|
|
