| import OAI.Geometry.Kahler.BaseCompositionJets |
| |
| namespace OAI |
|
|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ComplexOrder |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
| |
| lemma regularizedLog_positive_jet_bound (a : ℝ) {ε R A B : ℝ} (he : ε ≠ 0) |
| (hR : 0 < R) (hA : 0 ≤ A) (hB : 0 ≤ B) {n : ℕ} (hn : 1 ≤ n) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (f : Base → ℂ) (c : Base) (t : ℝ), |
| AnalyticOnNhd ℂ f (Metric.ball c R) → ‖f c‖ ≤ A → 0 < t → t ≤ 1 → |
| (∀ x ∈ Metric.ball c R, ‖f x-f c‖ ≤ B*t) → |
| ‖iteratedFDeriv ℝ n (regularizedLog a ε ∘ f) c‖ ≤ C*t := by |
| let radius : NNReal := ⟨A, hA⟩ |
| obtain ⟨C,hC,hCb⟩ := smooth_jet_bound_compact (regularizedLog_contDiff a he) |
| (isCompact_closedBall (0:ℂ) (radius : ℝ)) n |
| let D : ℝ := 1+∑ i ∈ Finset.range (n+1), cauchyJetConstant i*B/(R/2)^i |
| have hterm (i : ℕ) : 0 ≤ cauchyJetConstant i*B/(R/2)^i := by |
| exact div_nonneg (mul_nonneg (cauchyJetConstant_pos i).le hB) (by positivity) |
| have hD : 1 ≤ D := by |
| have hh := Finset.sum_nonneg (s := Finset.range (n+1)) (fun i _ => hterm i) |
| dsimp [D] |
| have hDb (i : ℕ) (hi : i ≤ n) : cauchyJetConstant i*B/(R/2)^i ≤ D := by |
| have hh := Finset.single_le_sum (s := Finset.range (n+1)) (fun j _ => hterm j) |
| (show i ∈ Finset.range (n+1) by simp |
| dsimp [D] |
| refine ⟨n.factorial*C*D^n,by positivity,?_⟩ |
| intro f c t hf hfc ht ht1 hb |
| have hfn : ContDiffOn ℝ n f (Metric.ball c R) := (hf.contDiffOn Metric.isOpen_ball.uniqueDiffOn).restrict_scalars ℝ |
| have hh := norm_composition_jet_linear Metric.isOpen_ball (Metric.mem_ball_self hR) hn hfn |
| ((regularizedLog_contDiff a he).of_le (by exact_mod_cast (le_top : (n:ℕ∞) ≤ ⊤))) hD ht ht1 |
| (fun i hi => hCb i hi (f c) (by |
| rw [Metric.mem_closedBall, dist_zero_right] |
| exact hfc)) |
| (fun i hi hin => ?_) |
| · exact hh |
| rw [analytic_real_jet_norm (hf c (Metric.mem_ball_self hR)) i] |
| have hbound := analytic_positive_jet_bound hR (mul_nonneg hB ht.le) hf hb (by omega : 0 < i) |
| apply hbound.trans |
| calc cauchyJetConstant i*(B*t)/(R/2)^i = (cauchyJetConstant i*B/(R/2)^i)*t := by ring |
| _ ≤ D*t := mul_le_mul_of_nonneg_right (hDb i hin) ht.le |
|
|
| end PinchedHartogs.BaseConstruction |
| |
| end |
|
|
| end OAI |
|
|
