| import OAI.Geometry.Kahler.BaseCentralSecond |
|
|
| 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 Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
|
|
| def phaseSphere (z : Circle) (ξ : Sphere) : Sphere := |
| ⟨(z:ℂ) • (ξ:Base),by simp only [Metric.mem_sphere,dist_zero_right,norm_smul,z.norm_coe,sphere_norm,one_mul]⟩ |
|
|
| lemma densityBandwidth_le_twice {Q : ℕ} (hQ : 2 ≤ Q) (j : ℕ) : |
| densityBandwidth Q j ≤ 2*Q^j := by |
| cases j with |
| | zero => simp [densityBandwidth] |
| | succ j => |
| rw [densityBandwidth_succ] |
| have hh := densityBandwidth_lt hQ j |
| omega |
|
|
| lemma phase_second_from_steps {Q n N : ℕ} (hQ : 2 ≤ Q) |
| {c K : ℝ} (hc : 0 < c) (hc1 : c ≤ 1) (hK : 0 ≤ K) (hN : 2/c ≤ (2:ℝ)^N) |
| (W : ℕ → Base → ℝ) (hzero : W 0=fun _ => 1) |
| (hsm : ∀ j ≤ n, ContDiff ℝ ∞ (W j)) |
| (hpos : ∀ j ≤ n, ∀ ξ : Sphere, 0 < W j ξ) |
| (hder : ∀ j ≤ n, ∀ ξ : Sphere, |phaseDerivative (W j) ξ| ≤ K*(Q:ℝ)^j*W j ξ) |
| (hstep : ∀ j, j < n → ∀ ξ : Sphere, c*W j ξ ≤ W (j+1) ξ ∧ W (j+1) ξ ≤ 2*W j ξ) |
| (hband : PhaseBandwidth (W n) (densityBandwidth Q n)) (ξ : Sphere) : |
| |phaseDerivative (phaseDerivative (W n)) ξ| ≤ |
| (2*(Real.exp K*(2/c))*4^N*Real.exp 2)*((Q:ℝ)^n)^2*W n ξ := by |
| let A := Real.exp K*(2/c) |
| have hA : 0 ≤ A := by dsimp [A]; positivity |
| have hQ0 : (0:ℝ) < Q := by exact_mod_cast (by omega : 0 < Q) |
| have hg : ∀ u : ℝ, W n ((Circle.exp u:ℂ) • (ξ:Base)) ≤ A*(1+(Q:ℝ)^n*|u|)^N*W n ξ := by |
| intro u |
| let V : ℕ → ℝ → ℝ := fun j θ => W j ((Circle.exp θ:ℂ) • (ξ:Base)) |
| have hv0 : V 0=fun _ => 1 := by funext θ; simp [V,hzero] |
| have hvd : ∀ j ≤ n, ∀ θ, HasDerivAt (V j) (phaseDerivative (W j) ((Circle.exp θ:ℂ) • (ξ:Base))) θ := |
| fun j hj θ => phaseDerivative_curve ((hsm j hj).differentiable (by simp)) ξ θ |
| have hvder : ∀ j ≤ n, ∀ θ, |deriv (V j) θ| ≤ K*(Q:ℝ)^j*V j θ := by |
| intro j hj θ |
| rw [(hvd j hj θ).deriv] |
| exact hder j hj (phaseSphere (Circle.exp θ) ξ) |
| have hh := orbit_growth_nat hQ hc hc1 hK hN V hv0 |
| (fun j hj θ => hpos j hj (phaseSphere (Circle.exp θ) ξ)) |
| (fun j hj θ => (hvd j hj θ).differentiableAt) hvder |
| (fun j hj θ => hstep j hj (phaseSphere (Circle.exp θ) ξ)) 0 u |
| simp only [V,zero_add,Circle.exp_zero,Circle.coe_one,one_smul] at hh |
| refine hh.trans ?_ |
| apply mul_le_mul_of_nonneg_right _ (hpos n le_rfl ξ).le |
| apply mul_le_mul_of_nonneg_left _ hA |
| apply pow_le_pow_left₀ (by positivity) |
| exact max_le (by linarith [mul_nonneg (pow_nonneg hQ0.le n) (abs_nonneg u)]) (by linarith : (Q:ℝ)^n*|u| ≤ 1+(Q:ℝ)^n*|u|) |
| exact phase_second_weighted (hsm n le_rfl) hband (pow_pos hQ0 n) hA |
| (by exact_mod_cast densityBandwidth_le_twice hQ n) N ξ |
| (fun z => hpos n le_rfl (phaseSphere z ξ)) hg |
|
|
| end PinchedHartogs.BaseConstruction |
|
|
| end |
|
|
| end OAI |
|
|
