File size: 12,722 Bytes
ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 | 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
|