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8e4d533 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 | 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
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