| import OAI.Analysis.IntegralMeans.SmoothMeans |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory Filter Function |
| open scoped Topology ENNReal |
| namespace Brennan |
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| 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 |
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| 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 |
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| 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 |
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| end Brennan |
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| end |
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| noncomputable section |
| open Set MeasureTheory Filter Function |
| open scoped Topology ENNReal |
| namespace Brennan |
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| 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 |
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| 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 |
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| 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 |
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| end Brennan |
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| end |
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| end OAI |
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