| import OAI.Geometry.Kahler.BaseDensityPatchEstimates |
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| namespace OAI |
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| 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 |
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| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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| lemma patch_unique {P : Finset Sphere} {k : ℕ} {R : ℝ} |
| (hd : (P:Set Sphere).PairwiseDisjoint (peakPatch k R)) (ξ : Sphere) |
| (p : Sphere) (hp : p ∈ P) (q : Sphere) (hq : q ∈ P) |
| (hξp : ξ ∈ peakPatch k R p) (hξq : ξ ∈ peakPatch k R q) : p=q := by |
| by_contra hpq |
| exact Set.disjoint_left.mp (hd hp hq hpq) hξp hξq |
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| lemma density_update_value {k : ℕ} (hk : 0 < k) |
| (f b : ℝ → ℝ) (W : Base → ℝ) {R F B r H c : ℝ} |
| (hF : 0 ≤ F) (hB : 0 ≤ B) (hr : 0 ≤ r) (hc : 0 < c) (hc1 : 2*c ≤ 1) |
| (hpos : ∀ ξ : Sphere, 0 < W ξ) |
| (hT : ∀ ξ : Sphere, |phaseDerivative W ξ| ≤ (k:ℝ)*r*W ξ) |
| (hf : ∀ y, 0 ≤ y → |f y| ≤ F) (hb : ∀ y, 0 ≤ y → |b y| ≤ B) |
| (P : Finset Sphere) (hd : (P:Set Sphere).PairwiseDisjoint (peakPatch k R)) |
| (henv : ∀ p ∈ P, ∀ ξ ∈ peakPatch k R p, |
| Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ) ≤ H*W ξ) |
| (hsmall : H*(F+B*r) ≤ 1-2*c) (ξ : Sphere) : |
| let V := W ξ+∑ p ∈ P, densityCorrection k R f b W p ξ |
| 0 < V ∧ c*W ξ ≤ V ∧ V ≤ 2*W ξ := by |
| have ha : |∑ p ∈ P, densityCorrection k R f b W p ξ| ≤ (1-2*c)*W ξ := by |
| apply disjoint_sum_abs_bound P (fun p => ξ ∈ peakPatch k R p) (patch_unique hd ξ) _ (by nlinarith [hpos ξ]) |
| · intro p hp hn |
| exact ite_eq_right hn |
| · intro p hp ht |
| exact (densityCorrection_patch_value hk f b W hF hB hr hpos hT hf hb p ξ ht (henv p hp ξ ht)).trans |
| (mul_le_mul_of_nonneg_right hsmall (hpos ξ).le) |
| obtain ⟨hlo,hhi⟩ := abs_le.mp ha |
| dsimp only |
| have hp := hpos ξ |
| have hcp : 0 < c*W ξ := mul_pos hc hp |
| constructor |
| · nlinarith |
| constructor <;> nlinarith |
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| lemma density_update_direction {k : ℕ} (hk : 0 < k) {R E : ℝ} (hER : E < R) |
| {f b : ℝ → ℝ} {W : Base → ℝ} (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) (hW : ContDiff ℝ ∞ W) |
| (htail : ∀ y, E ≤ y → f y=0 ∧ b y=0) |
| {F B F₁ B₁ H D : ℝ} (hF : 0 ≤ F) (hB : 0 ≤ B) (hF₁ : 0 ≤ F₁) (hB₁ : 0 ≤ B₁) (hD : 0 ≤ D) (hH : 0 ≤ H) |
| (hpos : ∀ ξ : Sphere, 0 < W ξ) |
| (hT : ∀ ξ : Sphere, |phaseDerivative W ξ| ≤ (k:ℝ)*W ξ) |
| (hTT : ∀ ξ : Sphere, |phaseDerivative (phaseDerivative W) ξ| ≤ (k:ℝ)^2*W ξ) |
| (hfv : ∀ y, 0 ≤ y → |f y| ≤ F) (hbv : ∀ y, 0 ≤ y → |b y| ≤ B) |
| (hfd : ∀ y, 0 ≤ y → |deriv f y| ≤ F₁) (hbd : ∀ y, 0 ≤ y → |deriv b y| ≤ B₁) |
| (P : Finset Sphere) (hd : (P:Set Sphere).PairwiseDisjoint (peakPatch k R)) |
| (ξ : Sphere) (v : Base) |
| (hdir : ∀ p ∈ P, ξ ∈ peakPatch k R p → ‖bracket v (p:Base)/bracket (ξ:Base) (p:Base)‖ ≤ D/k) |
| (henv : ∀ p ∈ P, ξ ∈ peakPatch k R p → |
| Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ) ≤ H*W ξ) : |
| |fderiv ℝ (fun z => W z+∑ p ∈ P, densityCorrection k R f b W p z) ξ v| ≤ |
| |fderiv ℝ W ξ v|+D*H*(4*(F+B)+F₁+B₁)*W ξ := by |
| rw [fderiv_density_update hk hER hf hb hW htail] |
| apply (abs_add_le _ _).trans |
| apply add_le_add le_rfl |
| have hw0 := (hpos ξ).le |
| apply disjoint_sum_abs_bound P (fun p => ξ ∈ peakPatch k R p) (patch_unique hd ξ) _ (by positivity) |
| · intro p hp hn |
| rw [densityCorrection_fderiv_zero hk hER htail W p ξ hn] |
| rfl |
| · intro p hp ht |
| exact densityCorrection_patch_direction hk hf hb hW hF hB hF₁ hB₁ hD hpos hT hTT hfv hbv hfd hbd p ξ ht v (hdir p hp ht) (henv p hp ht) |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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