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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Geometry.Kahler.BaseCircleLog
namespace OAI
open Complex
open scoped ContDiff Matrix Matrix.Norms.Elementwise
open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder
open scoped ContDiff ComplexOrder
open scoped ContDiff ENNReal
open Set Filter Topology
open scoped ContDiff ENNReal Pointwise
open scoped ContDiff
open Set Filter Topology MeasureTheory
noncomputable section
open Set Filter Topology MeasureTheory
namespace PinchedHartogs.BaseConstruction
def logFactor (q ε : ℝ) : ℝ :=
(1+q^2+ε^2+Real.sqrt ((1+q^2+ε^2)^2-4*q^2))/2
lemma logFactor_discriminant (q ε : ℝ) :
(1-q^2)^2 ≤ (1+q^2+ε^2)^2-4*q^2 := by
nlinarith [sq_nonneg ε,mul_nonneg (sq_nonneg ε) (sq_nonneg q),sq_nonneg (ε^2)]
lemma logFactor_one_le (q ε : ℝ) : 1 ≤ logFactor q ε := by
have hh := Real.sqrt_le_sqrt (logFactor_discriminant q ε)
rw [Real.sqrt_sq_eq_abs] at hh
have hl := le_abs_self (1-q^2)
unfold logFactor
nlinarith [sq_nonneg ε]
lemma logFactor_pos (q ε : ℝ) : 0 < logFactor q ε := lt_of_lt_of_le zero_lt_one (logFactor_one_le q ε)
lemma logFactor_gt {q ε : ℝ} (hε : ε ≠ 0) : q < logFactor q ε := by
have he := sq_pos_of_ne_zero hε
have hs := Real.sqrt_nonneg ((1+q^2+ε^2)^2-4*q^2)
unfold logFactor
nlinarith [sq_nonneg (q-1)]
lemma logFactor_equation (q ε : ℝ) :
logFactor q ε+q^2/logFactor q ε=1+q^2+ε^2 := by
have hd : 0 ≤ (1+q^2+ε^2)^2-4*q^2 := (sq_nonneg _).trans (logFactor_discriminant q ε)
have hs := Real.sq_sqrt hd
have hl := (logFactor_pos q ε).ne'
field_simp
unfold logFactor
nlinarith
lemma logFactor_norm_identity (q ε : ℝ) (z : Circle) :
‖1-(q:ℂ)*(z:ℂ)‖^2+ε^2 = logFactor q ε*‖1-(q/logFactor q ε:ℝ)*(z:ℂ)‖^2 := by
have hz : (z:ℂ).re^2+(z:ℂ).im^2=1 := by
have hh := Complex.sq_norm (z:ℂ)
rw [z.norm_coe] at hh
simp only [Complex.normSq_apply] at hh
nlinarith
have he := logFactor_equation q ε
have hl := (logFactor_pos q ε).ne'
simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.one_re,Complex.one_im,
Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,zero_sub,add_zero]
field_simp at he ⊢
nlinarith [congrArg (fun t => q^2*t) hz,congrArg (fun t => logFactor q ε*q^2*t) hz]
lemma logFactor_zero (q : ℝ) : logFactor q 0 = max 1 (q^2) := by
have he : (1+q^2+(0:ℝ)^2)^2-4*q^2=(1-q^2)^2 := by ring
unfold logFactor
rw [he,Real.sqrt_sq_eq_abs]
by_cases hq : q^2 ≤ 1
· rw [abs_of_nonneg (by linarith),max_eq_left hq]; ring
· rw [abs_of_nonpos (by linarith),max_eq_right (by linarith)]; ring
lemma logFactor_continuous : Continuous (fun p : ℝ × ℝ => logFactor p.1 p.2) := by
unfold logFactor
fun_prop
lemma circle_log_continuous {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) :
Continuous (fun z : Circle => Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)) := by
apply Continuous.log
· fun_prop
· intro z
apply pow_ne_zero
apply norm_ne_zero_iff.mpr
intro hz
have hh := congrArg norm (sub_eq_zero.mp hz)
rw [norm_one,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hr,z.norm_coe,mul_one] at hh
linarith
lemma regularized_circle_moments {q ε : ℝ} (hq : 0 ≤ q) (hε : ε ≠ 0) (f : ℝ) :
(∫ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ))*(1+f*(z:ℂ).re) ∂circleMeasure) =
Real.log (logFactor q ε)-Real.log (1+ε^2)-f*q/logFactor q ε := by
let r := q/logFactor q ε
have hr : 0 ≤ r := div_nonneg hq (logFactor_pos q ε).le
have hr1 : r < 1 := (div_lt_one (logFactor_pos q ε)).mpr (logFactor_gt hε)
have hmom := circle_log_moments hr hr1
have hi := compact_continuous_integrable (μ := circleMeasure) (circle_log_continuous hr hr1)
have hic : Integrable (fun z : Circle => Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)*(z:ℂ).re) circleMeasure := by
apply compact_continuous_integrable
exact (circle_log_continuous hr hr1).mul (by fun_prop)
have he : ∀ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ)) =
Real.log (logFactor q ε)-Real.log (1+ε^2)+Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2) := by
intro z
rw [regularizedLog_eq 1 ε _ hε]
simp only [Complex.ofReal_one,one_mul]
rw [logFactor_norm_identity,Real.log_mul (logFactor_pos q ε).ne' (by
have hh := logFactor_norm_identity q ε z
have hp : 0 < ‖1-(q:ℂ)*(z:ℂ)‖^2+ε^2 := by positivity
exact ne_of_gt (pos_of_mul_pos_right (hh ▸ hp) (logFactor_pos q ε).le))]
dsimp [r]
ring
let A := Real.log (logFactor q ε)-Real.log (1+ε^2)
let g : Circle → ℝ := fun z => Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)
have hz : (∫ z : Circle, (z:ℂ).re ∂circleMeasure)=0 := by
simpa [phaseChar] using circle_real_char_mean (by norm_num : (1:ℤ) ≠ 0)
have he' : ∀ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ))*(1+f*(z:ℂ).re) =
A+(A*f)*(z:ℂ).re+g z+f*(g z*(z:ℂ).re) := by
intro z
rw [he]
dsimp [A,g]
ring
simp_rw [he']
have hiz : Integrable (fun z : Circle => (z:ℂ).re) circleMeasure :=
compact_continuous_integrable (by fun_prop)
erw [integral_add (((integrable_const A).add (hiz.const_mul _)).add hi) (hic.const_mul _),
integral_add ((integrable_const A).add (hiz.const_mul _)) hi,
integral_add (integrable_const A) (hiz.const_mul _),integral_const_mul,integral_const_mul]
change (∫ z : Circle, A ∂circleMeasure)+(A*f)*_+_+f*_=_
rw [hmom.1,hmom.2,hz]
simp [A,r]
ring
end PinchedHartogs.BaseConstruction
end
end OAI
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