| import OAI.Analysis.IntegralMeans.PositiveArea |
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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_area {f : ℂ → ℂ} (hf : Schlicht f) {t : ℝ} |
| (ht : -2 < t) (ht2 : t < 2/3) : IntegrableOn (fun z => ‖deriv f z‖^t) disk := by |
| rcases lt_trichotomy t 0 with h | h | h |
| · exact schlicht_negative_area hf ht h |
| · subst t |
| simp only [Real.rpow_zero] |
| exact integrableOn_const (by |
| apply ne_of_lt |
| exact Metric.isBounded_ball.measure_lt_top) |
| · exact schlicht_positive_area hf h ht2 |
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| def normalizeDisk (f : ℂ → ℂ) (z : ℂ) : ℂ := (f z-f 0)/deriv f 0 |
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| lemma normalizeDisk_hasDeriv {f : ℂ → ℂ} (hf : UnivalentOn f disk) {z : ℂ} (hz : z ∈ disk) : |
| HasDerivAt (normalizeDisk f) (deriv f z/deriv f 0) z := |
| ((hf.1.differentiableAt (Metric.isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const (f 0)).div_const _ |
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| lemma normalizeDisk_schlicht {f : ℂ → ℂ} (hf : UnivalentOn f disk) : Schlicht (normalizeDisk f) := by |
| have h0 : (0:ℂ) ∈ disk := by simp [disk] |
| have hn := univalent_deriv_ne_zero Metric.isOpen_ball hf h0 |
| refine ⟨⟨?_,?_⟩,?_,?_⟩ |
| · intro z hz |
| exact (normalizeDisk_hasDeriv hf hz).differentiableAt.differentiableWithinAt |
| · intro z hz w hw he |
| have hh : f z-f 0 = f w-f 0 := (div_left_inj' hn).mp he |
| exact hf.2 hz hw (sub_left_inj.mp hh) |
| · simp [normalizeDisk] |
| · rw [(normalizeDisk_hasDeriv hf h0).deriv,div_self hn] |
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| lemma univalent_disk_area {f : ℂ → ℂ} (hf : UnivalentOn f disk) {t : ℝ} |
| (ht : -2 < t) (ht2 : t < 2/3) : IntegrableOn (fun z => ‖deriv f z‖^t) disk := by |
| have h0 : (0:ℂ) ∈ disk := by simp [disk] |
| have hn := univalent_deriv_ne_zero Metric.isOpen_ball hf h0 |
| have hg := schlicht_area (normalizeDisk_schlicht hf) ht ht2 |
| apply IntegrableOn.congr_fun (hg.const_mul (‖deriv f 0‖^t)) _ Metric.isOpen_ball.measurableSet |
| intro z hz |
| have he : deriv f z = deriv f 0 * deriv (normalizeDisk f) z := by |
| rw [(normalizeDisk_hasDeriv hf hz).deriv] |
| field_simp |
| dsimp only |
| rw [he,norm_mul,Real.mul_rpow (norm_nonneg _) (norm_nonneg _)] |
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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 univalent_inverse_hasDerivAt {U : Set ℂ} (hU : IsOpen U) {φ : ℂ → ℂ} |
| (hφ : UnivalentOn φ U) {z : ℂ} (hz : z ∈ U) : |
| HasDerivAt (Function.invFunOn φ U) (deriv φ z)⁻¹ (φ z) := by |
| have ha := hφ.1.analyticOnNhd hU z hz |
| apply (ha.hasStrictDerivAt.to_local_left_inverse (univalent_deriv_ne_zero hU hφ hz) ?_).hasDerivAt |
| filter_upwards [hU.mem_nhds hz] with w hw |
| exact hφ.2.leftInvOn_invFunOn hw |
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| lemma bijOn_inverse_univalent {U V : Set ℂ} (hU : IsOpen U) {φ : ℂ → ℂ} |
| (hφ : DifferentiableOn ℂ φ U) (hb : BijOn φ U V) : |
| UnivalentOn (Function.invFunOn φ U) V := by |
| constructor |
| · intro w hw |
| obtain ⟨z,hz,rfl⟩ := hb.surjOn hw |
| exact (univalent_inverse_hasDerivAt hU ⟨hφ,hb.injOn⟩ hz).differentiableAt.differentiableWithinAt |
| · exact (Function.invFunOn_injOn_image φ U).mono hb.surjOn |
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| lemma bijOn_inverse_image {U V : Set ℂ} {φ : ℂ → ℂ} (hb : BijOn φ U V) : |
| Function.invFunOn φ U '' V = U := by |
| rw [← hb.image_eq] |
| exact hb.injOn.invFunOn_image (Subset.refl _) |
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| lemma conformal_lintegral_eq {F : ℂ → ℂ} {s : Set ℂ} (hs : MeasurableSet s) |
| (hF : ∀ z ∈ s, DifferentiableAt ℂ F z) (hi : InjOn F s) (b : ℂ → ℝ≥0∞) : |
| (∫⁻ w in F '' s, b w) = ∫⁻ z in s, ENNReal.ofReal (‖deriv F z‖^2)*b (F z) := by |
| rw [lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hs |
| (fun z hz => (hF z hz).hasFDerivAt.restrictScalars ℝ |>.hasFDerivWithinAt) hi b] |
| apply setLIntegral_congr_fun hs |
| intro z hz |
| have hd := (hF z hz).hasFDerivAt.restrictScalars ℝ |
| have hf : (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ F z)) = fderiv ℝ F z := hd.fderiv.symm |
| dsimp only |
| have hd' : (fderiv ℝ F z).det = ‖deriv F z‖^2 := det_fderiv_complex (hF z hz) |
| rw [hf,hd',abs_of_nonneg (sq_nonneg _)] |
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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 conformal_domain_area {W : Set ℂ} (hW : IsOpen W) {φ : ℂ → ℂ} |
| (hφ : DifferentiableOn ℂ φ W) (hb : BijOn φ W disk) {s : ℝ} |
| (hs : 4/3 < s) (hs4 : s < 4) : IntegrableOn (fun z => ‖deriv φ z‖^s) W := by |
| let f := Function.invFunOn φ W |
| have hf : UnivalentOn f disk := bijOn_inverse_univalent hW hφ hb |
| have himage : f '' disk = W := bijOn_inverse_image hb |
| have hfi : IntegrableOn (fun w => ‖deriv f w‖^(2-s)) disk := univalent_disk_area hf (by linarith) (by linarith) |
| have hderiv (w : ℂ) (hw : w ∈ disk) : deriv φ (f w) = (deriv f w)⁻¹ := by |
| have hz : f w ∈ W := hb.surjOn.mapsTo_invFunOn hw |
| have hv : φ (f w) = w := hb.surjOn.rightInvOn_invFunOn hw |
| have hd := (univalent_inverse_hasDerivAt hW ⟨hφ,hb.injOn⟩ hz).deriv |
| change deriv f (φ (f w)) = (deriv φ (f w))⁻¹ at hd |
| rw [hv] at hd |
| rw [hd,inv_inv] |
| have hc : ContinuousOn (fun z => ‖deriv φ z‖^s) W := by |
| apply (hφ.deriv hW).continuousOn.norm.rpow_const |
| intro z hz |
| exact Or.inl (norm_ne_zero_iff.mpr (univalent_deriv_ne_zero hW ⟨hφ,hb.injOn⟩ hz)) |
| refine ⟨hc.aestronglyMeasurable hW.measurableSet,?_⟩ |
| apply (hasFiniteIntegral_iff_ofReal (Eventually.of_forall (fun z : ℂ => Real.rpow_nonneg (norm_nonneg _) s))).mpr |
| have he := conformal_lintegral_eq Metric.isOpen_ball.measurableSet |
| (fun z hz => hf.1.differentiableAt (Metric.isOpen_ball.mem_nhds hz)) hf.2 |
| (fun z => ENNReal.ofReal (‖deriv φ z‖^s)) |
| change (∫⁻ z in f '' disk, ENNReal.ofReal (‖deriv φ z‖^s)) = |
| ∫⁻ z in disk, ENNReal.ofReal (‖deriv f z‖^2)*ENNReal.ofReal (‖deriv φ (f z)‖^s) at he |
| rw [himage] at he |
| rw [he] |
| have hp : (∫⁻ w in disk, ENNReal.ofReal (‖deriv f w‖^2)*ENNReal.ofReal (‖deriv φ (f w)‖^s)) = |
| ∫⁻ w in disk, ENNReal.ofReal (‖deriv f w‖^(2-s)) := by |
| apply setLIntegral_congr_fun Metric.isOpen_ball.measurableSet |
| intro w hw |
| dsimp only |
| rw [hderiv w hw,norm_inv,Real.inv_rpow (norm_nonneg _),← ENNReal.ofReal_mul (sq_nonneg _)] |
| congr 1 |
| have hn : 0 < ‖deriv f w‖ := norm_pos_iff.mpr (univalent_deriv_ne_zero Metric.isOpen_ball hf hw) |
| rw [← Real.rpow_natCast ‖deriv f w‖ 2,← Real.rpow_neg hn.le,← Real.rpow_add hn] |
| rfl |
| rw [hp] |
| exact (hasFiniteIntegral_iff_ofReal (Eventually.of_forall |
| (fun w : ℂ => Real.rpow_nonneg (norm_nonneg (deriv f w)) (2-s)))).mp hfi.hasFiniteIntegral |
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| end Brennan |
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| end |
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| end OAI |
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