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namespace OAI
noncomputable section
open Set MeasureTheory Filter Function
open scoped Topology ENNReal
namespace Brennan
lemma schlicht_bracketSq_growth {f : ℂ → ℂ} (hf : Schlicht f) {z : ℂ} (hz : z ∈ disk) :
bracketSq (f z) ≤ 2*(1-‖z‖)^(-4:ℝ) := by
have hr : ‖z‖ < 1 := by simpa [disk] using hz
have hd : 0 < 1-‖z‖ := by linarith
have hg := schlicht_growth_upper hf hz
have hb : ‖f z‖ ≤ 1/(1-‖z‖)^2 := hg.trans (div_le_div_of_nonneg_right hr.le (sq_nonneg _))
have hs := pow_le_pow_left₀ (norm_nonneg _) hb 2
have hδ : (1-‖z‖)^4 ≤ 1 := pow_le_one₀ hd.le (by nlinarith [norm_nonneg z])
rw [bracketSq_eq,Real.rpow_neg hd.le]
norm_num only [Real.rpow_ofNat]
rw [inv_eq_one_div]
have hdiv : (1/(1-‖z‖)^2)^2 = 1/(1-‖z‖)^4 := by field_simp
rw [hdiv] at hs
have hh : 1 ≤ 1/(1-‖z‖)^4 := (le_div_iff₀ (pow_pos hd _)).mpr (by simpa using hδ)
linarith
lemma continuousOn_smoothPower_comp {f : ℂ → ℂ} (hf : Schlicht f) (a : ℝ) :
ContinuousOn (fun z => smoothPower a (f z)) disk :=
(smoothPower_contDiff a 0).continuous.comp_continuousOn hf.1.1.continuousOn
lemma schlicht_function_area {f : ℂ → ℂ} (hf : Schlicht f) {a : ℝ} (ha : a < 1/2) :
IntegrableOn (fun z => bracketSq (f z)^a) disk := by
obtain ⟨q,hq0,hq4⟩ := exists_between (show max (0:ℝ) (a-1/4) < 1/4 by
exact max_lt (by norm_num) (by linarith))
have hq : 0 < q := (le_max_left _ _).trans_lt hq0
have haq : a-q < 1/4 := by have hh := (le_max_right _ _).trans_lt hq0; linarith
obtain ⟨C,hC,hmean⟩ := schlicht_smooth_means hf hq hq4
have hc : ContinuousOn (fun z => bracketSq (f z)^a) disk := continuousOn_smoothPower_comp hf a
have hm (r : ℝ) (hr : 0 < r) (hr1 : r < 1) :
IntervalIntegrable (fun θ => bracketSq (f (circlePoint r θ))^a) volume (-Real.pi) Real.pi :=
(continuous_circle_smoothPower hf a r hr.le hr1).intervalIntegrable _ _
have hqn (r : ℝ) (hr : 0 < r) (hr1 : r < 1) :
IntervalIntegrable (fun θ => smoothPower q (f (circlePoint r θ))) volume (-Real.pi) Real.pi :=
(continuous_circle_smoothPower hf q r hr.le hr1).intervalIntegrable _ _
by_cases haq0 : a ≤ q
· apply integrableOn_disk_of_power_circle_bound hc (fun z _ => Real.rpow_nonneg (bracketSq_pos _).le _)
(C := C) (a := 0) hC.le (by norm_num)
intro r hr hr1
rw [circle_lintegral_eq (F := fun z => bracketSq (f z)^a)
(continuous_circle_smoothPower hf a r hr.le hr1)
(fun θ => Real.rpow_nonneg (bracketSq_pos _).le _),Real.rpow_zero,mul_one]
apply ENNReal.ofReal_le_ofReal
refine (intervalIntegral.integral_mono_on (a := -Real.pi) (b := Real.pi) (by linarith [Real.pi_pos]) (hm r hr hr1) (hqn r hr hr1) ?_).trans (hmean r hr hr1)
intro θ _
exact Real.rpow_le_rpow_of_exponent_le (by rw [bracketSq_eq]; nlinarith [sq_nonneg ‖f (circlePoint r θ)‖]) haq0
· have haqp : 0 < a-q := by linarith
apply integrableOn_disk_of_power_circle_bound hc (fun z _ => Real.rpow_nonneg (bracketSq_pos _).le _)
(C := (2:ℝ)^(a-q)*C) (a := -4*(a-q)) (by positivity) (by linarith)
intro r hr hr1
have hd : 0 < 1-r := by linarith
rw [circle_lintegral_eq (F := fun z => bracketSq (f z)^a)
(continuous_circle_smoothPower hf a r hr.le hr1)
(fun θ => Real.rpow_nonneg (bracketSq_pos _).le _)]
apply ENNReal.ofReal_le_ofReal
have hi : (∫ θ in -Real.pi..Real.pi, bracketSq (f (circlePoint r θ))^a) ≤
∫ θ in -Real.pi..Real.pi, ((2:ℝ)^(a-q)*(1-r)^(-4*(a-q)))*smoothPower q (f (circlePoint r θ)) := by
apply intervalIntegral.integral_mono_on (a := -Real.pi) (b := Real.pi) (by linarith [Real.pi_pos]) (hm r hr hr1) ((hqn r hr hr1).const_mul _)
intro θ _
have hz : circlePoint r θ ∈ disk := by simpa [disk,circlePoint_norm,abs_of_pos hr] using hr1
have hg := schlicht_bracketSq_growth hf hz
rw [circlePoint_norm,abs_of_pos hr] at hg
have hp := Real.rpow_le_rpow (bracketSq_pos _).le hg haqp.le
rw [Real.mul_rpow (by norm_num) (Real.rpow_nonneg hd.le _),← Real.rpow_mul hd.le] at hp
calc
_ = bracketSq (f (circlePoint r θ))^(a-q)*bracketSq (f (circlePoint r θ))^q := by
rw [← Real.rpow_add (bracketSq_pos _),sub_add_cancel]
_ ≤ _ := mul_le_mul_of_nonneg_right hp (Real.rpow_nonneg (bracketSq_pos _).le _)
rw [intervalIntegral.integral_const_mul] at hi
calc
_ ≤ ((2:ℝ)^(a-q)*(1-r)^(-4*(a-q)))*(∫ θ in -Real.pi..Real.pi, smoothPower q (f (circlePoint r θ))) := hi
_ ≤ ((2:ℝ)^(a-q)*(1-r)^(-4*(a-q)))*C := mul_le_mul_of_nonneg_left (hmean r hr hr1) (by positivity)
_ = _ := by ring
end Brennan
end
noncomputable section
open Set MeasureTheory Filter Function
open scoped Topology ENNReal
namespace Brennan
lemma bracketSq_negative_integrable {η : ℝ} (hη : 0 < η) :
Integrable (fun z : ℂ => bracketSq z^(-1-η)) := by
have h := integrable_rpow_neg_one_add_norm_sq (E := ℂ) (μ := volume) (r := 2+2*η)
(by norm_num [Complex.finrank_real_complex]; linarith)
convert! h using 1
ext z
rw [bracketSq_eq]
congr 1
ring
lemma schlicht_weighted_derivative_integrable {f : ℂ → ℂ} (hf : Schlicht f) {η : ℝ} (hη : 0 < η) :
IntegrableOn (fun z => ‖deriv f z‖^2*bracketSq (f z)^(-1-η)) disk := by
have hc : ContinuousOn (fun z => ‖deriv f z‖^2*bracketSq (f z)^(-1-η)) disk :=
((continuousOn_deriv_univalent hf.1).norm.pow 2).mul (continuousOn_smoothPower_comp hf _)
have hn : ∀ z : ℂ, 0 ≤ ‖deriv f z‖^2*bracketSq (f z)^(-1-η) :=
fun z => mul_nonneg (sq_nonneg _) (Real.rpow_nonneg (bracketSq_pos _).le _)
refine ⟨hc.aestronglyMeasurable Metric.isOpen_ball.measurableSet,?_⟩
apply (hasFiniteIntegral_iff_ofReal (Eventually.of_forall hn)).mpr
have hh := conformal_lintegral_le Metric.isOpen_ball.measurableSet
(fun z hz => hf.1.1.differentiableAt (Metric.isOpen_ball.mem_nhds hz)) hf.1.2
(fun w => ENNReal.ofReal (bracketSq w^(-1-η)))
simp_rw [← ENNReal.ofReal_mul (sq_nonneg _)] at hh
refine lt_of_le_of_lt hh ?_
exact (hasFiniteIntegral_iff_ofReal (Eventually.of_forall
(fun z : ℂ => Real.rpow_nonneg (bracketSq_pos z).le (-1-η)))).mp
(bracketSq_negative_integrable hη).hasFiniteIntegral
lemma schlicht_positive_area {f : ℂ → ℂ} (hf : Schlicht f) {t : ℝ}
(ht : 0 < t) (ht2 : t < 2/3) : IntegrableOn (fun z => ‖deriv f z‖^t) disk := by
let q := t/2
let η := (1-3*q)/(4*q)
let a := (1+η)*q/(1-q)
have hq : 0 < q := by dsimp [q]; linarith
have hq3 : q < 1/3 := by dsimp [q]; linarith
have hq1 : q < 1 := by linarith
have hη : 0 < η := div_pos (by linarith) (by positivity)
have ha : a < 1/2 := by
dsimp [a]
apply (div_lt_iff₀ (by linarith : 0 < 1-q)).mpr
have he : (1+η)*q = (1+q)/4 := by dsimp [η]; field_simp; ring
rw [he]
linarith
let A : ℂ → ℝ := fun z => ‖deriv f z‖^2*bracketSq (f z)^(-1-η)
let B : ℂ → ℝ := fun z => bracketSq (f z)^a
apply integrableOn_geometric_product Metric.isOpen_ball.measurableSet
(A := A) (B := B) (schlicht_weighted_derivative_integrable hf hη)
(schlicht_function_area hf ha)
((continuousOn_deriv_norm_rpow hf.1 t).aestronglyMeasurable Metric.isOpen_ball.measurableSet)
hq.le hq1.le
· intro z _hz
exact mul_nonneg (sq_nonneg _) (Real.rpow_nonneg (bracketSq_pos _).le _)
· intro z _hz
exact Real.rpow_nonneg (bracketSq_pos _).le _
· intro z _hz
dsimp [A,B]
rw [Real.mul_rpow (sq_nonneg _) (Real.rpow_nonneg (bracketSq_pos _).le _),
← Real.rpow_natCast ‖deriv f z‖ 2,← Real.rpow_mul (norm_nonneg _),
← Real.rpow_mul (bracketSq_pos _).le,← Real.rpow_mul (bracketSq_pos _).le]
have he : (-1-η)*q+a*(1-q) = 0 := by
dsimp [a]
rw [div_mul_cancel₀ _ (by linarith : 1-q ≠ 0)]
ring
have htq : (2:ℝ)*q = t := by dsimp [q]; ring
norm_num only [Nat.cast_ofNat] at *
rw [htq]
calc
_ = ‖deriv f z‖^t*1 := by ring
_ = ‖deriv f z‖^t*(bracketSq (f z)^((-1-η)*q)*bracketSq (f z)^(a*(1-q))) := by
rw [← Real.rpow_add (bracketSq_pos _),he,Real.rpow_zero]
_ = _ := by ring
end Brennan
end
end OAI
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