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This commit is contained in:
Stefan Kebekus
parent
ad298459ee
commit
ae3e64c83b
@ -11,28 +11,30 @@ import Nevanlinna.meromorphicOn_divisor
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open Real
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theorem jensen_case_R_eq_one'
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theorem jensen
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{R : ℝ}
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(hR : 0 < R)
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(f : ℂ → ℂ)
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(h₁f : StronglyMeromorphicOn f (Metric.closedBall 0 1))
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(h₁f : StronglyMeromorphicOn f (Metric.closedBall 0 R))
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(h₂f : f 0 ≠ 0) :
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log ‖f 0‖ = -∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log (‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
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log ‖f 0‖ = -∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log (R * ‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
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have h₁U : IsConnected (Metric.closedBall (0 : ℂ) 1) := by
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have h₁U : IsConnected (Metric.closedBall (0 : ℂ) R) := by
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constructor
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· apply Metric.nonempty_closedBall.mpr (by simp)
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· exact (convex_closedBall (0 : ℂ) 1).isPreconnected
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· apply Metric.nonempty_closedBall.mpr
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exact le_of_lt hR
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· exact (convex_closedBall (0 : ℂ) R).isPreconnected
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
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isCompact_closedBall 0 1
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) R) :=
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isCompact_closedBall 0 R
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have h'₂f : ∃ u : (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
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use ⟨0, Metric.mem_closedBall_self (by simp)⟩
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have h'₂f : ∃ u : (Metric.closedBall (0 : ℂ) R), f u ≠ 0 := by
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use ⟨0, Metric.mem_closedBall_self (le_of_lt hR)⟩
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have h₃f : Set.Finite (Function.support h₁f.meromorphicOn.divisor) := by
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exact Divisor.finiteSupport h₂U (StronglyMeromorphicOn.meromorphicOn h₁f).divisor
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have h₄f: Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (‖s‖⁻¹)) ⊆ h₃f.toFinset := by
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have h₄f: Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (R * ‖s‖⁻¹)) ⊆ h₃f.toFinset := by
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intro x
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contrapose
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simp
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@ -63,7 +65,7 @@ theorem jensen_case_R_eq_one'
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rw [hs]
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simp
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have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
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have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) R, f z ≠ 0 → log ‖f z‖ = G z := by
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intro z h₁z h₂z
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rw [h₄F]
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@ -104,40 +106,41 @@ theorem jensen_case_R_eq_one'
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simpa
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have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
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have int_logAbs_f_eq_int_G : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 R x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 R x) := by
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rw [intervalIntegral.integral_congr_ae]
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rw [MeasureTheory.ae_iff]
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apply Set.Countable.measure_zero
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simp
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have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
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⊆ (circleMap 0 1)⁻¹' (h₃f.toFinset) := by
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have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 R a)‖ = G (circleMap 0 R a)}
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⊆ (circleMap 0 R)⁻¹' (h₃f.toFinset) := by
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intro a ha
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simp at ha
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simp
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by_contra C
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have t₀ : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
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circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
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have t₁ : f (circleMap 0 1 a) ≠ 0 := by
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let A := h₁f (circleMap 0 1 a) t₀
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have t₀ : (circleMap 0 R a) ∈ Metric.closedBall 0 R := by
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apply circleMap_mem_closedBall
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exact le_of_lt hR
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have t₁ : f (circleMap 0 R a) ≠ 0 := by
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let A := h₁f (circleMap 0 R a) t₀
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rw [← A.order_eq_zero_iff]
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unfold MeromorphicOn.divisor at C
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simp [t₀] at C
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rcases C with C₁|C₂
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· assumption
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· let B := h₁f.meromorphicOn.order_ne_top' h₁U
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let C := fun q ↦ B q ⟨(circleMap 0 1 a), t₀⟩
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let C := fun q ↦ B q ⟨(circleMap 0 R a), t₀⟩
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rw [C₂] at C
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have : ∃ u : (Metric.closedBall (0 : ℂ) 1), (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
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use ⟨(0 : ℂ), (by simp)⟩
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let H := h₁f 0 (by simp)
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have : ∃ u : (Metric.closedBall (0 : ℂ) R), (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
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use ⟨(0 : ℂ), (by simp; exact le_of_lt hR)⟩
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let H := h₁f 0 (by simp; exact le_of_lt hR)
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let K := H.order_eq_zero_iff.2 h₂f
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rw [K]
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simp
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let D := C this
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tauto
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exact ha.2 (decompose_f (circleMap 0 1 a) t₀ t₁)
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exact ha.2 (decompose_f (circleMap 0 R a) t₀ t₁)
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apply Set.Countable.mono t₀
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apply Set.Countable.preimage_circleMap
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@ -147,14 +150,14 @@ theorem jensen_case_R_eq_one'
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simp
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have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
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= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
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+ ∑ᶠ x, (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
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have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 R x)
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= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 R x))))
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+ ∑ᶠ x, (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 R x_1 - ↑x)) := by
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dsimp [G]
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rw [intervalIntegral.integral_add]
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congr
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have t₀ {x : ℝ} : Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (Complex.abs (circleMap 0 1 x - s))) ⊆ h₃f.toFinset := by
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have t₀ {x : ℝ} : Function.support (fun s ↦ (h₁f.meromorphicOn.divisor s) * log (Complex.abs (circleMap 0 R x - s))) ⊆ h₃f.toFinset := by
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intro s hs
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simp at hs
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simp [hs.1]
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@ -164,8 +167,8 @@ theorem jensen_case_R_eq_one'
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intro x
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rw [finsum_eq_sum_of_support_subset _ t₀]
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rw [intervalIntegral.integral_finset_sum]
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let G' := fun x ↦ ((h₁f.meromorphicOn.divisor x) : ℂ) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - x))
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have t₁ : (Function.support fun x ↦ (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - x))) ⊆ h₃f.toFinset := by
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let G' := fun x ↦ ((h₁f.meromorphicOn.divisor x) : ℂ) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 R x_1 - x))
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have t₁ : (Function.support fun x ↦ (h₁f.meromorphicOn.divisor x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 R x_1 - x))) ⊆ h₃f.toFinset := by
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simp
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intro s
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contrapose!
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@ -182,14 +185,15 @@ theorem jensen_case_R_eq_one'
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intro i _
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apply IntervalIntegrable.const_mul
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--simp at this
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by_cases h₂i : ‖i‖ = 1
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by_cases h₂i : ‖i‖ = R
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-- case pos
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exact int'₂ h₂i
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sorry
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--exact int'₂ h₂i
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-- case neg
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
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have : (fun x => log (Complex.abs (circleMap 0 R x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 R x - ↑i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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@ -201,9 +205,10 @@ theorem jensen_case_R_eq_one'
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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rw [abs_circleMap_zero R x]
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simp
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tauto
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linarith
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--
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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@ -211,13 +216,18 @@ theorem jensen_case_R_eq_one'
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
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have : (fun x => log (Complex.abs (F (circleMap 0 R x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 R x) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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exact h₃F ⟨(circleMap 0 1 x), (by simp)⟩
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have : (circleMap 0 R x) ∈ (Metric.closedBall 0 R) := by
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simp
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rw [abs_le]
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simp [hR]
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exact le_of_lt hR
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exact h₃F ⟨(circleMap 0 R x), this⟩
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-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
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apply ContinuousAt.comp
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@ -225,7 +235,8 @@ theorem jensen_case_R_eq_one'
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apply ContinuousAt.comp
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apply DifferentiableAt.continuousAt (𝕜 := ℂ)
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apply AnalyticAt.differentiableAt
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exact h₂F (circleMap 0 1 x) (by simp)
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apply h₂F (circleMap 0 R x)
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simp; rw [abs_le]; simp [hR]; exact le_of_lt hR
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-- ContinuousAt (fun x => circleMap 0 1 x) x
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apply Continuous.continuousAt
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apply continuous_circleMap
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@ -245,15 +256,16 @@ theorem jensen_case_R_eq_one'
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apply IntervalIntegrable.const_mul
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--have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
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--simp at this
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by_cases h₂i : ‖i‖ = 1
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by_cases h₂i : ‖i‖ = R
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-- case pos
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exact int'₂ h₂i
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--exact int'₂ h₂i
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sorry
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-- case neg
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--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
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have : (fun x => log (Complex.abs (circleMap 0 R x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 R x - ↑i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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@ -265,33 +277,34 @@ theorem jensen_case_R_eq_one'
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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rw [abs_circleMap_zero R x]
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simp
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tauto
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linarith
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
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have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 R x)‖) = 2 * Real.pi * log ‖F 0‖ := by
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let logAbsF := fun w ↦ Real.log ‖F w‖
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have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
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have t₀ : ∀ z ∈ Metric.closedBall 0 R, HarmonicAt logAbsF z := by
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intro z hz
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apply logabs_of_holomorphicAt_is_harmonic
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exact AnalyticAt.holomorphicAt (h₂F z hz)
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exact h₃F ⟨z, hz⟩
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apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
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apply harmonic_meanValue₁ R hR t₀
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simp_rw [← Complex.norm_eq_abs] at decompose_int_G
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rw [t₁] at decompose_int_G
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have h₁G' : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 1 x_1 - s‖) ⊆ ↑h₃f.toFinset := by
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have h₁G' : (Function.support fun s => (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 R x_1 - s‖) ⊆ ↑h₃f.toFinset := by
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intro s hs
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simp at hs
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simp [hs.1]
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rw [finsum_eq_sum_of_support_subset _ h₁G'] at decompose_int_G
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have : ∑ s ∈ h₃f.toFinset, (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 1 x_1 - s‖ = 0 := by
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have : ∑ s ∈ h₃f.toFinset, (h₁f.meromorphicOn.divisor s) * ∫ (x_1 : ℝ) in (0)..(2 * π), log ‖circleMap 0 R x_1 - s‖ = 0 := by
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apply Finset.sum_eq_zero
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intro x hx
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rw [int₃ _]
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@ -359,170 +372,3 @@ theorem jensen_case_R_eq_one'
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rw [h₂f] at hx
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tauto
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assumption
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lemma const_mul_circleMap_zero'
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{R θ : ℝ} :
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circleMap 0 R θ = R * circleMap 0 1 θ := by
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rw [circleMap_zero, circleMap_zero]
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simp
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theorem jensen'
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{R : ℝ}
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(hR : 0 < R)
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(f : ℂ → ℂ)
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(h₁f : StronglyMeromorphicOn f (Metric.closedBall 0 R))
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(h₂f : f 0 ≠ 0) :
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log ‖f 0‖ = -∑ᶠ s, (h₁f.meromorphicOn.divisor s) * log (R * ‖s‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 R x)‖ := by
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let ℓ : ℂ ≃L[ℂ] ℂ :=
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{
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toFun := fun x ↦ R * x
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map_add' := fun x y => DistribSMul.smul_add R x y
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map_smul' := fun m x => mul_smul_comm m (↑R) x
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invFun := fun x ↦ R⁻¹ * x
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left_inv := by
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intro x
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simp
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rw [← mul_assoc, mul_comm, inv_mul_cancel₀, mul_one]
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simp
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exact ne_of_gt hR
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right_inv := by
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intro x
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simp
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rw [← mul_assoc, mul_inv_cancel₀, one_mul]
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simp
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exact ne_of_gt hR
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continuous_toFun := continuous_const_smul R
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continuous_invFun := continuous_const_smul R⁻¹
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}
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let F := f ∘ ℓ
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have h₁F : StronglyMeromorphicOn F (Metric.closedBall 0 1) := by
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sorry
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/-
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apply AnalyticOnNhd.comp (t := Metric.closedBall 0 R)
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exact h₁f
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intro x _
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apply ℓ.toContinuousLinearMap.analyticAt x
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intro x hx
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have : ℓ x = R * x := by rfl
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rw [this]
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simp
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simp at hx
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rw [abs_of_pos hR]
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calc R * Complex.abs x
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_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
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_ = R := by simp
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-/
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have h₂F : F 0 ≠ 0 := by
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dsimp [F]
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have : ℓ 0 = R * 0 := by rfl
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rw [this]
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simpa
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let A := jensen_case_R_eq_one' F h₁F h₂F
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dsimp [F] at A
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have {x : ℂ} : ℓ x = R * x := by rfl
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repeat
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simp_rw [this] at A
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simp at A
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simp
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rw [A]
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simp_rw [← const_mul_circleMap_zero']
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simp
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let e : (Metric.closedBall (0 : ℂ) 1) → (Metric.closedBall (0 : ℂ) R) := by
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intro ⟨x, hx⟩
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have hy : R • x ∈ Metric.closedBall (0 : ℂ) R := by
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simp
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simp at hx
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||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R * Complex.abs x
|
||||
_ ≤ R * 1 := by exact (mul_le_mul_iff_of_pos_left hR).mpr hx
|
||||
_ = R := by simp
|
||||
exact ⟨R • x, hy⟩
|
||||
|
||||
let e' : (Metric.closedBall (0 : ℂ) R) → (Metric.closedBall (0 : ℂ) 1) := by
|
||||
intro ⟨x, hx⟩
|
||||
have hy : R⁻¹ • x ∈ Metric.closedBall (0 : ℂ) 1 := by
|
||||
simp
|
||||
simp at hx
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
norm_num
|
||||
calc R⁻¹ * Complex.abs x
|
||||
_ ≤ R⁻¹ * R := by
|
||||
apply mul_le_mul_of_nonneg_left hx
|
||||
apply inv_nonneg.mpr
|
||||
exact abs_eq_self.mp (id (Eq.symm this))
|
||||
_ = 1 := by
|
||||
apply inv_mul_cancel₀
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
exact ⟨R⁻¹ • x, hy⟩
|
||||
|
||||
apply finsum_eq_of_bijective e
|
||||
|
||||
|
||||
apply Function.bijective_iff_has_inverse.mpr
|
||||
use e'
|
||||
constructor
|
||||
· apply Function.leftInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [inv_mul_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
· apply Function.rightInverse_iff_comp.mpr
|
||||
funext x
|
||||
dsimp only [e, e', id_eq, eq_mp_eq_cast, Function.comp_apply]
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
rw [← smul_assoc, smul_eq_mul]
|
||||
rw [mul_inv_cancel₀ (Ne.symm (ne_of_lt hR))]
|
||||
rw [one_smul]
|
||||
|
||||
intro x
|
||||
simp
|
||||
by_cases hx : x = (0 : ℂ)
|
||||
rw [hx]
|
||||
simp
|
||||
|
||||
rw [log_mul, log_mul, log_inv, log_inv]
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
simp
|
||||
left
|
||||
congr 1
|
||||
|
||||
dsimp [AnalyticOnNhd.order]
|
||||
rw [← AnalyticAt.order_comp_CLE ℓ]
|
||||
|
||||
--
|
||||
simpa
|
||||
--
|
||||
have : R = |R| := by exact Eq.symm (abs_of_pos hR)
|
||||
rw [← this]
|
||||
apply inv_ne_zero
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
exact Ne.symm (ne_of_lt hR)
|
||||
--
|
||||
simp
|
||||
constructor
|
||||
· assumption
|
||||
· exact Ne.symm (ne_of_lt hR)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user