| import OAI.Geometry.Kahler.BaseDensityScales |
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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 scoped ContDiff |
| open Set Filter Topology |
| noncomputable section |
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| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
|
|
| structure DensityStage (W : ℕ → Base → ℝ) (Q : ℕ) (c K : ℝ) (j : ℕ) : Prop where |
| positive : ∀ ξ : Sphere, 0 < W j ξ |
| horizontal : HorizontalBound (W j) (K*Real.sqrt ((Q:ℝ)^j)) |
| phase : ∀ ξ : Sphere, |phaseDerivative (W j) ξ| ≤ K*(Q:ℝ)^j*W j ξ |
| step : ∀ i, j=i+1 → ∀ ξ : Sphere, c*W i ξ ≤ W j ξ ∧ W j ξ ≤ 2*W i ξ |
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|
| lemma density_induction {Q N : ℕ} (hQ : 2 ≤ Q) {R E F B F₁ B₁ D c K : ℝ} |
| (hR : 0 < R) (hER : E < R) (hQR : 2*R ≤ Q) |
| (hF : 0 ≤ F) (hB : 0 ≤ B) (hF₁ : 0 ≤ F₁) (hB₁ : 0 ≤ B₁) |
| (hD1 : 1 ≤ D) (hDR : 2*Real.sqrt (2*R) ≤ D) |
| (hc : 0 < c) (hc1 : 2*c ≤ 1) (hK : 0 ≤ K) (hN : 2/c ≤ (2:ℝ)^N) |
| {f b : ℝ → ℝ} (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) |
| (htail : ∀ y, E ≤ y → f y=0 ∧ b y=0) |
| (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 : ℕ → Finset Sphere) |
| (hdis : ∀ j, (P (Q^(j+1)):Set Sphere).PairwiseDisjoint (peakPatch (Q^(j+1)) R)) |
| (hH : Real.exp (K*D/Real.sqrt (Q:ℝ)+2*R/Q) ≤ 2) |
| (hsmall : Real.exp (K*D/Real.sqrt (Q:ℝ)+2*R/Q)*(F+B*(K/Q)) ≤ 1-2*c) |
| (hphase : K/Q ≤ 1) |
| (hsecond : (2*(Real.exp K*(2/c))*4^N*Real.exp 2)/(Q:ℝ)^2 ≤ 1) |
| (hderH : 2*D*(4*(F+B)+F₁+B₁)+K/Real.sqrt (Q:ℝ) ≤ K*c) |
| (hderT : 2*D*(4*(F+B)+F₁+B₁)+K/Q ≤ K*c) : |
| ∀ j, DensityStage (density Q R f b P) Q c K j := by |
| have hQ0 : 0 < Q := by omega |
| have hq : (0:ℝ) < Q := by exact_mod_cast hQ0 |
| have hD : 0 ≤ D := by linarith |
| have hC : 0 ≤ 4*(F+B)+F₁+B₁ := by positivity |
| let W := density Q R f b P |
| have hsm : ∀ j, ContDiff ℝ ∞ (W j) := density_smooth hQ0 hER hf hb htail P |
| have hfder : ∀ ξ : Base, fderiv ℝ (fun _ : Base => (1:ℝ)) ξ=0 := fun ξ => (hasFDerivAt_const (1:ℝ) ξ).fderiv |
| intro j |
| induction j using Nat.strong_induction_on with |
| | h j ih => |
| cases j with |
| | zero => |
| constructor |
| · intro ξ; norm_num [W,density] |
| · intro ξ v hv |
| simp only [density,hfder,zero_apply,abs_zero,pow_zero,Real.sqrt_one,mul_one] |
| positivity |
| · intro ξ |
| simp only [density,phaseDerivative,hfder,zero_apply,abs_zero,pow_zero,mul_one] |
| exact hK |
| · intro i hi; omega |
| | succ j => |
| have old := ih j (Nat.lt_succ_self j) |
| have hk : 0 < Q^(j+1) := pow_pos hQ0 _ |
| have hk0 : (0:ℝ) < (Q:ℝ)^(j+1) := pow_pos hq _ |
| have hsk : 0 < Real.sqrt ((Q:ℝ)^(j+1)) := Real.sqrt_pos.mpr hk0 |
| have hsQ : 0 < Real.sqrt (Q:ℝ) := Real.sqrt_pos.mpr hq |
| have hkR : 2*R ≤ (Q^(j+1):ℕ) := by |
| exact_mod_cast hQR.trans (density_degree_ge (by omega : 1 ≤ Q) j) |
| have henv : ∀ p ∈ P (Q^(j+1)), ∀ ξ ∈ peakPatch (Q^(j+1)) R p, |
| Real.exp (2*densityHeight (Q^(j+1)) p ξ/(Q^(j+1):ℕ))*W j (centralPoint p ξ) ≤ |
| Real.exp (K*D/Real.sqrt (Q:ℝ)+2*R/Q)*W j ξ := by |
| intro p hp ξ ht |
| have hh := density_central_envelope ((hsm j).differentiable (by simp)) old.positive |
| (by positivity : 0 ≤ K*Real.sqrt ((Q:ℝ)^j)) old.horizontal hk hR hkR p ξ ht |
| apply hh.trans |
| apply mul_le_mul_of_nonneg_right _ (old.positive ξ).le |
| apply Real.exp_le_exp.mpr |
| simpa only [Nat.cast_pow] using density_envelope_bound (by omega : 1 ≤ Q) hK hR.le hDR j |
| have henv2 : ∀ p ∈ P (Q^(j+1)), ∀ ξ ∈ peakPatch (Q^(j+1)) R p, |
| Real.exp (2*densityHeight (Q^(j+1)) p ξ/(Q^(j+1):ℕ))*W j (centralPoint p ξ) ≤ 2*W j ξ := by |
| intro p hp ξ ht |
| exact (henv p hp ξ ht).trans (mul_le_mul_of_nonneg_right hH (old.positive ξ).le) |
| have hTscaled : ∀ ξ : Sphere, |phaseDerivative (W j) ξ| ≤ (Q^(j+1):ℕ)*(K/Q)*W j ξ := by |
| intro ξ |
| simpa only [Nat.cast_pow,density_phase_scale hQ0 K j] using old.phase ξ |
| have hT : ∀ ξ : Sphere, |phaseDerivative (W j) ξ| ≤ (Q^(j+1):ℕ)*W j ξ := by |
| intro ξ |
| apply (hTscaled ξ).trans |
| simp only [Nat.cast_pow] |
| exact mul_le_mul_of_nonneg_right ((mul_le_mul_of_nonneg_left hphase hk0.le).trans_eq (mul_one _)) (old.positive ξ).le |
| have hTT : ∀ ξ : Sphere, |phaseDerivative (phaseDerivative (W j)) ξ| ≤ ((Q^(j+1):ℕ):ℝ)^2*W j ξ := by |
| intro ξ |
| have hh := phase_second_from_steps hQ hc (by linarith) hK hN W rfl |
| (fun i hi => hsm i) (fun i hi => (ih i (by omega)).positive) |
| (fun i hi => (ih i (by omega)).phase) |
| (fun i hi => (ih (i+1) (by omega)).step i rfl) |
| (density_bandwidth hQ0 hER hf hb htail P j) ξ |
| have he := density_second_scale hQ0 (2*(Real.exp K*(2/c))*4^N*Real.exp 2) j |
| rw [← he] at hh |
| apply hh.trans |
| simp only [Nat.cast_pow] |
| have hp := (old.positive ξ).le |
| exact mul_le_mul_of_nonneg_right ((mul_le_mul_of_nonneg_left hsecond (sq_nonneg _)).trans_eq (mul_one _)) hp |
| have hval : ∀ ξ : Sphere, 0 < W (j+1) ξ ∧ c*W j ξ ≤ W (j+1) ξ ∧ W (j+1) ξ ≤ 2*W j ξ := by |
| intro ξ |
| exact density_update_value hk f b (W j) hF hB (by positivity : 0 ≤ K/Q) hc hc1 old.positive hTscaled hfv hbv |
| (P (Q^(j+1))) (hdis j) henv hsmall ξ |
| constructor |
| · intro ξ; exact (hval ξ).1 |
| · intro ξ v hv |
| have hdir : ∀ p ∈ P (Q^(j+1)), ξ ∈ peakPatch (Q^(j+1)) R p → |
| ‖bracket v (p:Base)/bracket (ξ:Base) (p:Base)‖ ≤ (D*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖)/(Q^(j+1):ℕ) := by |
| intro p hp ht |
| simpa only [Nat.cast_pow] using horizontal_direction_ratio hk hR hkR hDR p ξ ht v hv |
| have hh := density_update_direction hk hER hf hb (hsm j) htail hF hB hF₁ hB₁ |
| (by positivity : 0 ≤ D*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖) (by norm_num : (0:ℝ) ≤ 2) |
| old.positive hT hTT hfv hbv hfd hbd (P (Q^(j+1))) (hdis j) ξ v hdir (fun p hp ht => henv2 p hp ξ ht) |
| have hx := old.horizontal ξ v hv |
| have he : K*Real.sqrt ((Q:ℝ)^j)*‖v‖*W j ξ+ |
| (D*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖)*2*(4*(F+B)+F₁+B₁)*W j ξ = |
| (2*D*(4*(F+B)+F₁+B₁)+K/Real.sqrt (Q:ℝ))*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖*W j ξ := by |
| rw [density_sqrt_scale hQ0 j] |
| field_simp |
| ring |
| calc |
| _ ≤ |fderiv ℝ (W j) ξ v|+(D*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖)*2*(4*(F+B)+F₁+B₁)*W j ξ := hh |
| _ ≤ K*Real.sqrt ((Q:ℝ)^j)*‖v‖*W j ξ+(D*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖)*2*(4*(F+B)+F₁+B₁)*W j ξ := add_le_add hx le_rfl |
| _ = _ := he |
| _ ≤ (K*c)*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖*W j ξ := by gcongr; exact (old.positive ξ).le |
| _ = K*Real.sqrt ((Q:ℝ)^(j+1))*‖v‖*(c*W j ξ) := by ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hval ξ).2.1 (by positivity) |
| · intro ξ |
| have hdir : ∀ p ∈ P (Q^(j+1)), ξ ∈ peakPatch (Q^(j+1)) R p → |
| ‖bracket (Complex.I • (ξ:Base)) (p:Base)/bracket (ξ:Base) (p:Base)‖ ≤ (D*(Q:ℝ)^(j+1))/(Q^(j+1):ℕ) := by |
| intro p hp ht |
| rw [phase_direction_ratio p ξ (norm_pos_iff.mp (peakPatch_norm_pos ht))] |
| simp only [Nat.cast_pow] |
| rw [mul_div_cancel_right₀ D hk0.ne'] |
| exact hD1 |
| have hh := density_update_direction hk hER hf hb (hsm j) htail hF hB hF₁ hB₁ |
| (by positivity : 0 ≤ D*(Q:ℝ)^(j+1)) (by norm_num : (0:ℝ) ≤ 2) |
| old.positive hT hTT hfv hbv hfd hbd (P (Q^(j+1))) (hdis j) ξ (Complex.I • (ξ:Base)) hdir (fun p hp ht => henv2 p hp ξ ht) |
| have hx := old.phase ξ |
| have he : K*(Q:ℝ)^j*W j ξ+(D*(Q:ℝ)^(j+1))*2*(4*(F+B)+F₁+B₁)*W j ξ = |
| (2*D*(4*(F+B)+F₁+B₁)+K/Q)*(Q:ℝ)^(j+1)*W j ξ := by |
| rw [pow_succ] |
| field_simp |
| ring |
| calc |
| _ ≤ |phaseDerivative (W j) ξ|+(D*(Q:ℝ)^(j+1))*2*(4*(F+B)+F₁+B₁)*W j ξ := hh |
| _ ≤ K*(Q:ℝ)^j*W j ξ+(D*(Q:ℝ)^(j+1))*2*(4*(F+B)+F₁+B₁)*W j ξ := add_le_add hx le_rfl |
| _ = _ := he |
| _ ≤ (K*c)*(Q:ℝ)^(j+1)*W j ξ := by gcongr; exact (old.positive ξ).le |
| _ = K*(Q:ℝ)^(j+1)*(c*W j ξ) := by ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hval ξ).2.1 (by positivity) |
| · intro i hi ξ |
| have he : i=j := by omega |
| subst i |
| exact (hval ξ).2 |
|
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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