| import OAI.Analysis.IntegralMeans.DomainArea | |
| import OAI.Analysis.IntegralMeans.KoebeReduction | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped ENNReal | |
| namespace Brennan.Sharp | |
| lemma koebe_inverse_areaMoment (s : ℝ) : | |
| areaMoment koebeInverse koebeDomain s = areaMoment koebeMap disk (2 - s) := by | |
| have hdisk : MeasurableSet disk := Metric.isOpen_ball.measurableSet | |
| have hopen : IsOpen disk := Metric.isOpen_ball | |
| unfold areaMoment koebeDomain | |
| rw [conformal_lintegral_eq (F := koebeMap) (s := disk) hdisk | |
| (fun z hz => (koebeMap_hasDerivAt hz).differentiableAt) | |
| koebeMap_schlicht.1.2] | |
| apply setLIntegral_congr_fun hdisk | |
| intro z hz | |
| change ENNReal.ofReal (‖deriv koebeMap z‖ ^ (2 : ℕ)) * | |
| ENNReal.ofReal (‖deriv koebeInverse (koebeMap z)‖ ^ s) = | |
| ENNReal.ofReal (‖deriv koebeMap z‖ ^ (2 - s)) | |
| have hderiv : deriv koebeInverse (koebeMap z) = (deriv koebeMap z)⁻¹ := | |
| (univalent_inverse_hasDerivAt hopen koebeMap_schlicht.1 hz).deriv | |
| rw [hderiv, norm_inv, Real.inv_rpow (norm_nonneg _), | |
| ← ENNReal.ofReal_mul (sq_nonneg _)] | |
| congr 1 | |
| have hn : 0 < ‖deriv koebeMap z‖ := | |
| norm_pos_iff.mpr (univalent_deriv_ne_zero hopen koebeMap_schlicht.1 hz) | |
| rw [← Real.rpow_natCast ‖deriv koebeMap z‖ 2, | |
| ← Real.rpow_neg hn.le, ← Real.rpow_add hn] | |
| rfl | |
| lemma not_integrable_of_areaMoment_top {f : ℂ → ℂ} {U : Set ℂ} {t : ℝ} | |
| (h : areaMoment f U t = ⊤) : | |
| ¬ IntegrableOn (fun z => ‖deriv f z‖ ^ t) U volume := by | |
| intro hi | |
| have hf := (hasFiniteIntegral_iff_ofReal (Eventually.of_forall | |
| (fun z : ℂ => Real.rpow_nonneg (norm_nonneg (deriv f z)) t))).mp hi.hasFiniteIntegral | |
| change areaMoment f U t < ⊤ at hf | |
| rw [h] at hf | |
| exact lt_irrefl _ hf | |
| end Brennan.Sharp | |
| end | |
| end OAI | |