| import OAI.Analysis.DirectCrouzeix.HarmonicReflection | |
| namespace OAI | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| noncomputable section | |
| open MeasureTheory Set Filter Metric | |
| open scoped Topology Interval ENNReal NNReal ComplexConjugate | |
| noncomputable section | |
| open Filter Metric Set | |
| open scoped Topology ComplexConjugate | |
| namespace DirectCrouzeix | |
| namespace Conformal | |
| open Function Complex | |
| open scoped Pointwise | |
| open InnerProductSpace Real | |
| def logModulusUpper (f : ℂ → ℂ) (z : ℂ) : ℝ := | |
| if 0 < z.im then Real.log ‖f z‖ else 0 | |
| theorem logModulusUpper_real {f : ℂ → ℂ} {z : ℂ} (hz : z.im = 0) : | |
| logModulusUpper f z = 0 := by simp [logModulusUpper, hz] | |
| theorem logModulusUpper_continuous {R : ℝ} {f : ℂ → ℂ} | |
| (hc : ContinuousOn f (closedBall 0 R ∩ {z | 0 < z.im})) | |
| (hn : ∀ z ∈ closedBall 0 R ∩ {z | 0 < z.im}, f z ≠ 0) | |
| (hl : ∀ z ∈ closedBall 0 R, z.im = 0 → | |
| Tendsto (fun w => ‖f w‖) (𝓝[{w : ℂ | 0 < w.im}] z) (𝓝 1)) : | |
| ContinuousOn (logModulusUpper f) (upperClosedBall R) := by | |
| unfold logModulusUpper | |
| apply ContinuousOn.if' | |
| · intro z hz | |
| have hf := Complex.continuous_im.frontier_preimage_subset (Ioi (0 : ℝ)) hz.2 | |
| have hz0 : z.im = 0 := by simpa only [frontier_Ioi, mem_preimage, mem_singleton_iff] using hf | |
| simp only [hz0, lt_self_iff_false, ↓reduceIte] | |
| have ht : Tendsto (fun w => Real.log ‖f w‖) (𝓝[{w : ℂ | 0 < w.im}] z) (𝓝 0) := by | |
| simpa only [Function.comp_def, Real.log_one] using (Real.continuousAt_log one_ne_zero).tendsto.comp (hl z hz.1.1 hz0) | |
| exact ht.mono_left (nhdsWithin_mono _ inter_subset_right) | |
| · intro z hz | |
| have hf := Complex.continuous_im.frontier_preimage_subset (Ioi (0 : ℝ)) hz.2 | |
| have hz0 : z.im = 0 := by simpa only [frontier_Ioi, mem_preimage, mem_singleton_iff] using hf | |
| simp only [hz0, lt_self_iff_false, ↓reduceIte] | |
| exact tendsto_const_nhds | |
| · apply (hc.norm.log (fun z hz => norm_ne_zero_iff.mpr (hn z hz))).mono | |
| intro z hz | |
| exact ⟨hz.1.1, hz.2⟩ | |
| · exact continuousOn_const | |
| theorem upperBall_convex (R : ℝ) : Convex ℝ (upperBall R) := | |
| (convex_ball (0 : ℂ) R).inter (convex_Ioi (0 : ℝ) |>.linear_preimage Complex.imCLM.toLinearMap) | |
| theorem upperBall_half_radius {R : ℝ} (hR : 0 < R) : | |
| ((R / 2 : ℝ) : ℂ) * Complex.I ∈ upperBall R := by | |
| constructor | |
| · simp only [mem_ball_zero_iff, norm_mul, Complex.norm_I, mul_one, | |
| Complex.norm_real, Real.norm_eq_abs] | |
| rw [abs_of_pos (half_pos hR)] | |
| linarith | |
| · simpa using half_pos hR | |
| theorem circleAverage_pos_upper {R : ℝ} (hR : 0 ≤ R) {u : ℂ → ℝ} | |
| (hu : ContinuousOn u (sphere 0 R)) | |
| (hn : ∀ z ∈ sphere 0 R, 0 ≤ u z) (hp : 0 < u ((R : ℂ) * Complex.I)) : | |
| 0 < Real.circleAverage u 0 R := by | |
| have hall (θ : ℝ) : circleMap 0 R θ ∈ sphere 0 R := by | |
| simpa only [abs_of_nonneg hR] using circleMap_mem_sphere' (0 : ℂ) R θ | |
| have hc : Continuous (fun θ => u (circleMap 0 R θ)) := | |
| hu.comp_continuous (continuous_circleMap 0 R) hall | |
| have hip : 0 < ∫ θ in (0 : ℝ)..2 * Real.pi, u (circleMap 0 R θ) := by | |
| apply intervalIntegral.integral_pos Real.two_pi_pos hc.continuousOn | |
| · intro θ _ | |
| exact hn _ (hall θ) | |
| · refine ⟨Real.pi / 2, ⟨by positivity, by nlinarith [Real.pi_pos]⟩, ?_⟩ | |
| simpa only [circleMap_pi_div_two, zero_add] using hp | |
| exact mul_pos (inv_pos.mpr Real.two_pi_pos) hip | |
| theorem herglotz_deriv_im_pos {R : ℝ} (hR : 0 < R) {u : ℂ → ℝ} | |
| (hu : ContinuousOn u (sphere 0 R)) | |
| (hn : ∀ z ∈ sphere 0 R, u z * z.im ≤ 0) | |
| (hp : u ((R : ℂ) * Complex.I) < 0) : | |
| 0 < (deriv (herglotzExtension R u) 0).im := by | |
| let d := fun ζ : ℂ => (2 / ζ) * (u ζ : ℂ) | |
| have hzne : ∀ ζ ∈ sphere 0 R, ζ ≠ (0 : ℂ) := by | |
| intro ζ hζ hz | |
| subst ζ | |
| exact hR.ne' (by simpa only [mem_sphere, dist_self, eq_comm] using hζ) | |
| have hd : ContinuousOn d (sphere 0 R) := | |
| (continuousOn_const.div continuousOn_id hzne).mul (Complex.continuous_ofReal.comp_continuousOn hu) | |
| have hder := hasDerivAt_circleAverage_herglotzRieszKernel_smul | |
| ((Complex.continuous_ofReal.comp_continuousOn hu).circleIntegrable hR.le) | |
| (by simpa [abs_of_pos hR] using hR.ne : (0 : ℂ) ∉ sphere 0 |R|) | |
| change HasDerivAt (herglotzExtension R u) _ _ at hder | |
| have hval : deriv (herglotzExtension R u) 0 = Real.circleAverage d 0 R := by | |
| rw [hder.deriv] | |
| apply Real.circleAverage_congr_sphere | |
| intro ζ hζ | |
| have hz : ζ ≠ 0 := hzne ζ (by simpa only [abs_of_pos hR] using hζ) | |
| dsimp [d] | |
| simp only [sub_zero] | |
| congr 1 | |
| field_simp | |
| have he (ζ : ℂ) : (d ζ).im = -2 * (u ζ * ζ.im) / Complex.normSq ζ := by | |
| dsimp [d] | |
| simp [Complex.mul_im, Complex.div_im] | |
| ring | |
| have hnon : ∀ ζ ∈ sphere 0 R, 0 ≤ (d ζ).im := by | |
| intro ζ hζ | |
| rw [he] | |
| exact div_nonneg (mul_nonneg_of_nonpos_of_nonpos (by norm_num) (hn ζ hζ)) | |
| (Complex.normSq_nonneg ζ) | |
| have hpos : 0 < (d ((R : ℂ) * Complex.I)).im := by | |
| rw [he] | |
| have him : ((R : ℂ) * Complex.I).im = R := by simp | |
| rw [him] | |
| apply div_pos | |
| · exact mul_pos_of_neg_of_neg (by norm_num) (mul_neg_of_neg_of_pos hp hR) | |
| · rw [Complex.normSq_pos] | |
| exact mul_ne_zero (Complex.ofReal_ne_zero.mpr hR.ne') Complex.I_ne_zero | |
| rw [hval] | |
| change 0 < Complex.imCLM (Real.circleAverage d 0 R) | |
| rw [← Complex.imCLM.circleAverage_comp_comm (hd.circleIntegrable hR.le)] | |
| exact circleAverage_pos_upper hR.le (Complex.continuous_im.comp_continuousOn hd) hnon hpos | |
| theorem holomorphic_reflection_modulus {R : ℝ} (hR : 0 < R) {f : ℂ → ℂ} | |
| (hf : AnalyticOnNhd ℂ f (closedBall 0 R ∩ {z | 0 < z.im})) | |
| (hn : ∀ z ∈ closedBall 0 R ∩ {z | 0 < z.im}, f z ≠ 0) | |
| (hb : ∀ z ∈ closedBall 0 R ∩ {z | 0 < z.im}, ‖f z‖ < 1) | |
| (hl : ∀ z ∈ closedBall 0 R, z.im = 0 → | |
| Tendsto (fun w => ‖f w‖) (𝓝[{w : ℂ | 0 < w.im}] z) (𝓝 1)) : | |
| ∃ F : ℂ → ℂ, AnalyticOnNhd ℂ F (ball 0 R) ∧ EqOn F f (upperBall R) ∧ | |
| (∀ z ∈ ball 0 R, z.im = 0 → ‖F z‖ = 1) ∧ deriv F 0 ≠ 0 ∧ | |
| (∀ z ∈ ball 0 R, ‖F z‖ < 1 ↔ 0 < z.im) := by | |
| let u := logModulusUpper f | |
| let v := oddUpper u | |
| let H := herglotzExtension R v | |
| have hu0 : ∀ z, z.im = 0 → u z = 0 := fun _ hz => logModulusUpper_real hz | |
| have huc := logModulusUpper_continuous hf.continuousOn hn hl | |
| have hvc : ContinuousOn v (sphere 0 R) := oddUpper_continuous_circle huc hu0 | |
| have hH : AnalyticOnNhd ℂ H (ball 0 R) := herglotz_analytic hR.le hvc | |
| have hu : HarmonicOnNhd u (upperBall R) := by | |
| intro z hz | |
| have hzK : z ∈ closedBall 0 R ∩ {z | 0 < z.im} := ⟨ball_subset_closedBall hz.1, hz.2⟩ | |
| have he : u =ᶠ[𝓝 z] (fun w => Real.log ‖f w‖) := by | |
| filter_upwards [(isOpen_lt continuous_const Complex.continuous_im).mem_nhds hz.2] with w hw | |
| simp only [u, logModulusUpper, hw, ↓reduceIte] | |
| rw [harmonicAt_congr_nhds he] | |
| exact (hf z hzK).harmonicAt_log_norm (hn z hzK) | |
| have hRe := harmonic_reflection hR hu huc hu0 | |
| let k := fun z => f z * Complex.exp (-H z) | |
| have hk : DifferentiableOn ℂ k (upperBall R) := by | |
| apply (hf.differentiableOn.mono (by intro z hz; exact ⟨ball_subset_closedBall hz.1, hz.2⟩)).mul | |
| exact (hH.differentiableOn.mono inter_subset_left).neg.cexp | |
| have hkn : ∀ z ∈ upperBall R, ‖k z‖ = 1 := by | |
| intro z hz | |
| have hRe' : (H z).re = Real.log ‖f z‖ := by | |
| have hh : (H z).re = u z := hRe hz | |
| simpa only [u, logModulusUpper, ite_eq_left (show 0 < z.im from hz.2)] using hh | |
| have hnz := hn z ⟨ball_subset_closedBall hz.1, hz.2⟩ | |
| dsimp [k] | |
| rw [norm_mul, Complex.norm_exp, Complex.neg_re, hRe', Real.exp_neg, | |
| Real.exp_log (norm_pos_iff.mpr hnz), mul_inv_cancel₀ (norm_ne_zero_iff.mpr hnz)] | |
| let c : ℂ := ((R / 2 : ℝ) : ℂ) * Complex.I | |
| have hc : c ∈ upperBall R := upperBall_half_radius hR | |
| have hconst := Complex.eqOn_of_isPreconnected_of_isMaxOn_norm | |
| (upperBall_convex R).isPreconnected (upperBall_open R) hk hc | |
| (show IsMaxOn (norm ∘ k) (upperBall R) c from | |
| fun z hz => by | |
| change ‖k z‖ ≤ ‖k c‖ | |
| rw [hkn z hz, hkn c hc]) | |
| refine ⟨fun z => k c * Complex.exp (H z), analyticOnNhd_const.mul hH.cexp, ?_, ?_, ?_, ?_⟩ | |
| · intro z hz | |
| have hh : k z = k c := hconst hz | |
| rw [← hh] | |
| simp [k, Complex.exp_neg, Complex.exp_ne_zero] | |
| · intro z hz hz0 | |
| have hHz : (H z).re = 0 := by | |
| rw [show H z = herglotzExtension R v z from rfl, herglotz_re hR.le hvc hz] | |
| exact poisson_real_zero (oddUpper_conj hu0) hz0 | |
| rw [norm_mul, hkn c hc, Complex.norm_exp, hHz, Real.exp_zero, mul_one] | |
| · have hvsign : ∀ z ∈ sphere 0 R, v z * z.im ≤ 0 := by | |
| intro z hz | |
| by_cases hi : 0 ≤ z.im | |
| · by_cases hi0 : z.im = 0 | |
| · simp [hi0] | |
| · have hip : 0 < z.im := lt_of_le_of_ne hi (Ne.symm hi0) | |
| have hzK : z ∈ closedBall 0 R ∩ {z | 0 < z.im} := ⟨sphere_subset_closedBall hz, hip⟩ | |
| have huNeg : u z < 0 := by | |
| simpa only [u, logModulusUpper, ite_eq_left hip] using | |
| Real.log_neg (norm_pos_iff.mpr (hn z hzK)) (hb z hzK) | |
| simpa only [v, oddUpper, ite_eq_left hi] using mul_nonpos_of_nonpos_of_nonneg huNeg.le hi | |
| · have hip : 0 < (conj z).im := by simpa only [Complex.conj_im] using neg_pos.mpr (lt_of_not_ge hi) | |
| have hzK : conj z ∈ closedBall 0 R ∩ {z | 0 < z.im} := | |
| ⟨by simpa only [mem_closedBall, dist_zero_right, Complex.norm_conj] using sphere_subset_closedBall hz, hip⟩ | |
| have huNeg : u (conj z) < 0 := by | |
| simpa only [u, logModulusUpper, ite_eq_left hip] using | |
| Real.log_neg (norm_pos_iff.mpr (hn _ hzK)) (hb _ hzK) | |
| simpa only [v, oddUpper, ite_eq_right hi] using | |
| mul_nonpos_of_nonneg_of_nonpos (neg_nonneg.mpr huNeg.le) (le_of_not_ge hi) | |
| have hvpole : v ((R : ℂ) * Complex.I) < 0 := by | |
| have hip : 0 < ((R : ℂ) * Complex.I).im := by simpa using hR | |
| have hzK : (R : ℂ) * Complex.I ∈ closedBall 0 R ∩ {z | 0 < z.im} := | |
| ⟨by simp [mem_closedBall, abs_of_pos hR], hip⟩ | |
| simpa only [v, oddUpper, ite_eq_left hip.le, u, logModulusUpper, ite_eq_left hip] using | |
| Real.log_neg (norm_pos_iff.mpr (hn _ hzK)) (hb _ hzK) | |
| have hdpos : 0 < (deriv H 0).im := herglotz_deriv_im_pos hR hvc hvsign hvpole | |
| have hdne : deriv H 0 ≠ 0 := by intro he; simp [he] at hdpos | |
| have hkc : k c ≠ 0 := by intro he; simpa [he] using hkn c hc | |
| have hderF := ((hH 0 (mem_ball_self hR)).differentiableAt.hasDerivAt.cexp).const_mul (k c) | |
| rw [hderF.deriv] | |
| exact mul_ne_zero hkc (mul_ne_zero (Complex.exp_ne_zero _) hdne) | |
| · intro z hz | |
| change ‖k c * Complex.exp (H z)‖ < 1 ↔ 0 < z.im | |
| by_cases hi : 0 < z.im | |
| · have hh : k z = k c := hconst ⟨hz, hi⟩ | |
| have he : k c * Complex.exp (H z) = f z := by | |
| rw [← hh] | |
| simp [k, Complex.exp_neg, Complex.exp_ne_zero] | |
| rw [he] | |
| exact iff_of_true (hb z ⟨ball_subset_closedBall hz, hi⟩) hi | |
| · by_cases hi0 : z.im = 0 | |
| · have hHz : (H z).re = 0 := by | |
| rw [show H z = herglotzExtension R v z from rfl, herglotz_re hR.le hvc hz] | |
| exact poisson_real_zero (oddUpper_conj hu0) hi0 | |
| simp only [norm_mul, hkn c hc, Complex.norm_exp, hHz, Real.exp_zero, mul_one, | |
| lt_self_iff_false, hi0] | |
| · have hic : 0 < (conj z).im := by | |
| simpa only [Complex.conj_im] using | |
| (neg_pos.mpr (lt_of_le_of_ne (le_of_not_gt hi) hi0)) | |
| have hzcb : conj z ∈ ball 0 R := by simpa only [mem_ball_zero_iff, Complex.norm_conj] using hz | |
| have hzK : conj z ∈ closedBall 0 R ∩ {z | 0 < z.im} := ⟨ball_subset_closedBall hzcb, hic⟩ | |
| have hRc : (H (conj z)).re = Real.log ‖f (conj z)‖ := by | |
| have hh : (H (conj z)).re = u (conj z) := hRe ⟨hzcb, hic⟩ | |
| simpa only [u, logModulusUpper, ite_eq_left hic] using hh | |
| have ho : (H (conj z)).re = -(H z).re := by | |
| change (herglotzExtension R v (conj z)).re = -(herglotzExtension R v z).re | |
| rw [herglotz_re hR.le hvc hzcb, herglotz_re hR.le hvc hz] | |
| exact poisson_odd (oddUpper_conj hu0) z | |
| have hpos : 0 < (H z).re := by | |
| have hnlog := Real.log_neg (norm_pos_iff.mpr (hn _ hzK)) (hb _ hzK) | |
| linarith | |
| have hlarge : 1 < ‖k c * Complex.exp (H z)‖ := by | |
| rw [norm_mul, hkn c hc, one_mul, Complex.norm_exp] | |
| exact Real.one_lt_exp_iff.mpr hpos | |
| exact iff_of_false (not_lt_of_ge hlarge.le) hi | |
| end Conformal | |
| end DirectCrouzeix | |
| end | |
| end | |
| end | |
| end OAI | |