| import OAI.Geometry.Kahler.BaseDensityUpdate |
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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 density_central_envelope {W : Base → ℝ} (hW : Differentiable ℝ W) |
| (hpos : ∀ ξ : Sphere, 0 < W ξ) {H : ℝ} (hH : 0 ≤ H) (hd : HorizontalBound W H) |
| {k : ℕ} {R : ℝ} (hk : 0 < k) (hR : 0 < R) (hkR : 2*R ≤ k) |
| (p ξ : Sphere) (hξ : ξ ∈ peakPatch k R p) : |
| Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ) ≤ |
| Real.exp (H*(2*Real.sqrt (2*R)/Real.sqrt (k:ℝ))+2*R/k)*W ξ := by |
| have hh : densityHeight k p ξ < R := ((peakPatch_iff_height hk R p ξ).mp hξ).2 |
| have hk0 : (0:ℝ) < k := by exact_mod_cast hk |
| have hw := (central_comparison hW hpos hH hd hk hR hkR p ξ hξ).1 |
| calc |
| _ ≤ Real.exp (2*R/k)*(Real.exp (H*(2*Real.sqrt (2*R)/Real.sqrt (k:ℝ)))*W ξ) := by |
| apply mul_le_mul _ hw (le_of_lt (hpos ⟨centralPoint p ξ,by simpa only [Metric.mem_sphere,dist_zero_right] using centralPoint_norm p (norm_pos_iff.mp (peakPatch_norm_pos hξ))⟩)) (Real.exp_pos _).le |
| exact Real.exp_le_exp.mpr (by gcongr) |
| _ = _ := by rw [← mul_assoc,← Real.exp_add]; congr 2; ring |
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| def centralSphere (p ξ : Sphere) (hξ : bracket (ξ:Base) (p:Base) ≠ 0) : Sphere := |
| ⟨centralPoint p ξ,by simpa only [Metric.mem_sphere,dist_zero_right] using centralPoint_norm p hξ⟩ |
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| lemma densityCorrection_patch_value {k : ℕ} (hk : 0 < k) |
| (f b : ℝ → ℝ) (W : Base → ℝ) {R F B r H : ℝ} |
| (hF : 0 ≤ F) (hB : 0 ≤ B) (hr : 0 ≤ r) |
| (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 ξ : Sphere) (hξ : ξ ∈ peakPatch k R p) |
| (henv : Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ) ≤ H*W ξ) : |
| |densityCorrection k R f b W p ξ| ≤ H*(F+B*r)*W ξ := by |
| have hc : bracket (ξ:Base) (p:Base) ≠ 0 := norm_pos_iff.mp (peakPatch_norm_pos hξ) |
| change Real.exp (-R/k)<‖bracket (ξ:Base) (p:Base)‖ at hξ |
| rw [densityCorrection,ite_eq_left hξ] |
| have hh := densityCorrectionRaw_value_bound hk f b W p ξ hc hF hB hr |
| (hpos (centralSphere p ξ hc)).le (hT (centralSphere p ξ hc)) (hf _ (patchHeight_nonneg k p ξ)) (hb _ (patchHeight_nonneg k p ξ)) |
| exact hh.trans ((mul_le_mul_of_nonneg_right henv (by positivity)).trans_eq (by ring)) |
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| lemma densityCorrection_patch_direction {k : ℕ} (hk : 0 < k) |
| {f b : ℝ → ℝ} {W : Base → ℝ} (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) (hW : ContDiff ℝ ∞ W) |
| {R F B F₁ B₁ H D : ℝ} (hF : 0 ≤ F) (hB : 0 ≤ B) (hF₁ : 0 ≤ F₁) (hB₁ : 0 ≤ B₁) (hD : 0 ≤ D) |
| (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 ξ : Sphere) (hξ : ξ ∈ peakPatch k R p) (v : Base) |
| (hdir : ‖bracket v (p:Base)/bracket (ξ:Base) (p:Base)‖ ≤ D/k) |
| (henv : Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ) ≤ H*W ξ) : |
| |fderiv ℝ (densityCorrection k R f b W p) ξ v| ≤ D*H*(4*(F+B)+F₁+B₁)*W ξ := by |
| have hc : bracket (ξ:Base) (p:Base) ≠ 0 := norm_pos_iff.mp (peakPatch_norm_pos hξ) |
| rw [densityCorrection_fderiv_raw f b W p ξ hξ] |
| have hh := densityCorrectionRaw_direction_bound hk hD hF hB hF₁ hB₁ hf hb hW p ξ v hc hdir |
| (hpos (centralSphere p ξ hc)).le (hT (centralSphere p ξ hc)) (hTT (centralSphere p ξ hc)) (hfv _ (patchHeight_nonneg k p ξ)) (hbv _ (patchHeight_nonneg k p ξ)) (hfd _ (patchHeight_nonneg k p ξ)) (hbd _ (patchHeight_nonneg k p ξ)) |
| calc |
| _ ≤ D*(Real.exp (2*densityHeight k p ξ/k)*W (centralPoint p ξ))*(4*(F+B)+F₁+B₁) := by simpa only [mul_assoc] using hh |
| _ ≤ D*(H*W ξ)*(4*(F+B)+F₁+B₁) := by gcongr |
| _ = _ := by ring |
|
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| lemma fderiv_density_update {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) (P : Finset Sphere) (ξ v : Base) : |
| fderiv ℝ (fun z => W z+∑ p ∈ P, densityCorrection k R f b W p z) ξ v = |
| fderiv ℝ W ξ v+∑ p ∈ P, fderiv ℝ (densityCorrection k R f b W p) ξ v := by |
| have hw := hW.differentiable (by simp) ξ |
| have hp : ∀ p ∈ P, DifferentiableAt ℝ (densityCorrection k R f b W p) ξ := |
| fun p hp => (densityCorrection_smooth hk hER hf hb hW htail p).differentiable (by simp) ξ |
| rw [fderiv_fun_add hw (DifferentiableAt.fun_sum hp),fderiv_fun_sum hp] |
| simp only [add_apply,sum_apply] |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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