| import OAI.Geometry.Kahler.BaseDensityTheorem |
|
|
| namespace OAI |
|
|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| namespace PinchedHartogs.BaseConstruction |
|
|
| lemma circle_pow_surjective {k : ℕ} (hk : 0 < k) : Function.Surjective (fun z : Circle => z^k) := by |
| intro z |
| obtain ⟨x,rfl⟩ := Circle.exp_surjective z |
| refine ⟨Circle.exp (x/k),?_⟩ |
| dsimp only |
| rw [← Circle.exp_nsmul,nsmul_eq_mul] |
| congr 1 |
| have hk0 : (k:ℝ) ≠ 0 := by exact_mod_cast hk.ne' |
| field_simp |
|
|
| lemma circle_pow_measurePreserving {k : ℕ} (hk : 0 < k) : |
| MeasurePreserving (fun z : Circle => z^k) circleMeasure circleMeasure := |
| (powMonoidHom (α := Circle) k).measurePreserving (continuous_pow k) (circle_pow_surjective hk) rfl |
|
|
| lemma circle_pow_integral {k : ℕ} (hk : 0 < k) {F : Circle → ℝ} (hF : Continuous F) : |
| (∫ z : Circle, F (z^k) ∂circleMeasure)=∫ z : Circle, F z ∂circleMeasure := by |
| have hp := circle_pow_measurePreserving hk |
| calc |
| _ = ∫ z : Circle, F z ∂Measure.map (fun z : Circle => z^k) circleMeasure := |
| (integral_map hp.measurable.aemeasurable hF.aestronglyMeasurable).symm |
| _ = _ := by rw [hp.map_eq] |
|
|
| lemma regularizedLog_rescale (a ε : ℝ) (v : ℂ) : regularizedLog a ε v=regularizedLog 1 ε ((a:ℂ)*v) := by |
| simp only [regularizedLog,Complex.ofReal_one,one_mul] |
|
|
| lemma circle_regularized_mean {q ε : ℝ} (hq : 0 ≤ q) (hε : ε ≠ 0) : |
| (∫ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ)) ∂circleMeasure)= |
| Real.log (logFactor q ε)-Real.log (1+ε^2) := by |
| simpa only [zero_mul,add_zero,mul_one,zero_div,sub_zero] using regularized_circle_moments hq hε 0 |
|
|
| lemma circle_regularized_power_moments {q ε : ℝ} (hq : 0 ≤ q) (hε : ε ≠ 0) |
| {k : ℕ} (hk : 0 < k) (f : ℝ) : |
| (∫ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ)^k)*(1+f*((z:ℂ)^k).re) ∂circleMeasure)= |
| Real.log (logFactor q ε)-Real.log (1+ε^2)-f*q/logFactor q ε := by |
| have hc : Continuous (fun z : Circle => regularizedLog 1 ε ((q:ℂ)*(z:ℂ))*(1+f*(z:ℂ).re)) := |
| ((regularizedLog_contDiff 1 hε).continuous.comp (by fun_prop)).mul (by fun_prop) |
| have he := circle_pow_integral hk hc |
| simp only [Circle.coe_pow] at he |
| rw [he,regularized_circle_moments hq hε] |
|
|
| lemma circle_regularized_complex_mean {ε : ℝ} (hε : ε ≠ 0) (v : ℂ) {k : ℕ} (hk : 0 < k) : |
| (∫ z : Circle, regularizedLog 1 ε ((z:ℂ)^k*v) ∂circleMeasure)= |
| Real.log (logFactor ‖v‖ ε)-Real.log (1+ε^2) := by |
| by_cases hv : v=0 |
| · subst v |
| simp only [mul_zero,regularizedLog_zero,integral_zero,norm_zero] |
| have hh : logFactor 0 ε=1+ε^2 := by simpa using logFactor_equation 0 ε |
| rw [hh,sub_self] |
| have hn := norm_pos_iff.mpr hv |
| let u : Circle := ⟨v/(‖v‖:ℂ),by |
| apply mem_sphere_zero_iff_norm.mpr |
| rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hn,div_self hn.ne']⟩ |
| have hu : (‖v‖:ℂ)*(u:ℂ)=v := by |
| dsimp [u] |
| have hnc : (‖v‖:ℂ) ≠ 0 := by exact_mod_cast hn.ne' |
| field_simp [hnc] |
| have hc : Continuous (fun z : Circle => regularizedLog 1 ε ((z:ℂ)*v)) := |
| (regularizedLog_contDiff 1 hε).continuous.comp (by fun_prop) |
| have he := circle_pow_integral hk hc |
| simp only [Circle.coe_pow] at he |
| rw [he] |
| have hm := integral_mul_right_eq_self (μ := circleMeasure) |
| (fun z : Circle => regularizedLog 1 ε ((‖v‖:ℂ)*(z:ℂ))) u |
| have heq : ∀ z : Circle, (‖v‖:ℂ)*((z*u:Circle):ℂ)=(z:ℂ)*v := by |
| intro z |
| rw [Circle.coe_mul,mul_left_comm,hu] |
| simp_rw [heq] at hm |
| rw [hm,circle_regularized_mean (norm_nonneg _) hε] |
|
|
| lemma logFactor_small_mean_bound {q ε : ℝ} (hq : q^2 ≤ 1/4) : |
| Real.log (logFactor q ε)-Real.log (1+ε^2) ≤ 2*ε^2*q^2 := by |
| let L := logFactor q ε |
| have hL : 1 ≤ L := logFactor_one_le q ε |
| have hp : 0 < L := logFactor_pos q ε |
| have hA : 0 < 1+ε^2 := by positivity |
| have he := logFactor_equation q ε |
| change L+q^2/L=1+q^2+ε^2 at he |
| have hle : q^2/L ≤ q^2 := div_le_self (sq_nonneg _) hL |
| have hAL : 1+ε^2 ≤ L := by linarith |
| have hpoly : (L-(1+ε^2))*(L-q^2)=q^2*ε^2 := by |
| field_simp at he |
| nlinarith |
| have hdiff : L-(1+ε^2) ≤ 2*ε^2*q^2 := by |
| nlinarith [mul_nonneg (sub_nonneg.mpr hAL) (show 0 ≤ L-q^2-1/2 by linarith)] |
| have hlog := Real.log_le_sub_one_of_pos (div_pos hp hA) |
| rw [Real.log_div hp.ne' hA.ne'] at hlog |
| apply hlog.trans |
| have hm : L-(1+ε^2) ≤ (2*ε^2*q^2)*(1+ε^2) := |
| hdiff.trans (le_mul_of_one_le_right (by positivity) (by nlinarith [sq_nonneg ε])) |
| rw [div_sub_one hA.ne'] |
| exact (div_le_iff₀ hA).mpr hm |
|
|
| end PinchedHartogs.BaseConstruction |
|
|
| end |
|
|
| end OAI |
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