| import OAI.Geometry.Kahler.BaseProfiles |
|
|
| 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 |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| namespace PinchedHartogs.BaseConstruction |
|
|
| def phaseDerivative (W : Base → ℝ) (z : Base) : ℝ := fderiv ℝ W z (Complex.I • z) |
|
|
| def centralPoint (p : Sphere) (z : Base) : Base := |
| (bracket z (p:Base) / (‖bracket z (p:Base)‖:ℂ)) • (p:Base) |
|
|
| def densityHeight (k : ℕ) (p : Sphere) (z : Base) : ℝ := |
| -(k:ℝ)*Real.log ‖bracket z (p:Base)‖ |
|
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| def densityCorrectionRaw (k : ℕ) (f b : ℝ → ℝ) (W : Base → ℝ) (p : Sphere) (z : Base) : ℝ := |
| let c := bracket z (p:Base) |
| let phase := c/(‖c‖:ℂ) |
| let y := densityHeight k p z |
| (‖c‖^2)⁻¹ * (f y * W (centralPoint p z) * (phase^k).re - |
| b y / k * phaseDerivative W (centralPoint p z) * (phase^k).im) |
|
|
| def densityCorrection (k : ℕ) (R : ℝ) (f b : ℝ → ℝ) (W : Base → ℝ) (p : Sphere) (z : Base) : ℝ := |
| if Real.exp (-R/k) < ‖bracket z (p:Base)‖ then densityCorrectionRaw k f b W p z else 0 |
|
|
| def density (Q : ℕ) (R : ℝ) (f b : ℝ → ℝ) (P : ℕ → Finset Sphere) : ℕ → Base → ℝ |
| | 0 => fun _ => 1 |
| | j+1 => fun z => density Q R f b P j z + |
| ∑ p ∈ P (Q^(j+1)), densityCorrection (Q^(j+1)) R f b (density Q R f b P j) p z |
|
|
| lemma phaseDerivative_smooth {W : Base → ℝ} (hW : ContDiff ℝ ∞ W) : |
| ContDiff ℝ ∞ (phaseDerivative W) := by |
| exact (hW.fderiv_right (by simp)).clm_apply ((Complex.I • (1 : Base →L[ℂ] Base)).restrictScalars ℝ).contDiff |
|
|
| lemma centralPoint_smoothAt (p : Sphere) {z : Base} (hz : bracket z (p:Base) ≠ 0) : |
| ContDiffAt ℝ ∞ (centralPoint p) z := by |
| have hc : ContDiff ℝ ∞ (fun z : Base => bracket z (p:Base)) := (bracket_analytic (p:Base)).contDiff.restrict_scalars ℝ |
| have hn : ContDiffAt ℝ ∞ (fun z : Base => ‖bracket z (p:Base)‖) z := hc.contDiffAt.norm ℂ hz |
| simpa only [centralPoint,div_eq_mul_inv,Function.comp_def,Complex.ofRealCLM_apply] using! |
| ((hc.contDiffAt.mul ((Complex.ofRealCLM.contDiff.comp_contDiffAt z hn).inv |
| (by simpa only [Function.comp_def,Complex.ofRealCLM_apply,ne_eq,Complex.ofReal_eq_zero] using norm_ne_zero_iff.mpr hz))).smul (contDiffAt_const (c := (p:Base)))) |
|
|
| lemma densityHeight_smoothAt (k : ℕ) (p : Sphere) {z : Base} (hz : bracket z (p:Base) ≠ 0) : |
| ContDiffAt ℝ ∞ (densityHeight k p) z := by |
| have hc : ContDiff ℝ ∞ (fun z : Base => bracket z (p:Base)) := (bracket_analytic (p:Base)).contDiff.restrict_scalars ℝ |
| exact contDiffAt_const.mul ((hc.contDiffAt.norm ℂ hz).log (norm_ne_zero_iff.mpr hz)) |
|
|
| lemma densityCorrectionRaw_smoothAt (k : ℕ) {f b : ℝ → ℝ} {W : Base → ℝ} |
| (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) (hW : ContDiff ℝ ∞ W) |
| (p : Sphere) {z : Base} (hz : bracket z (p:Base) ≠ 0) : |
| ContDiffAt ℝ ∞ (densityCorrectionRaw k f b W p) z := by |
| have hc : ContDiff ℝ ∞ (fun z : Base => bracket z (p:Base)) := (bracket_analytic (p:Base)).contDiff.restrict_scalars ℝ |
| have hn := hc.contDiffAt.norm ℂ hz |
| have hn0 := norm_ne_zero_iff.mpr hz |
| have hphase : ContDiffAt ℝ ∞ (fun z : Base => bracket z (p:Base)/(‖bracket z (p:Base)‖:ℂ)) z := by |
| simpa only [div_eq_mul_inv,Function.comp_def,Complex.ofRealCLM_apply] using! hc.contDiffAt.mul ((Complex.ofRealCLM.contDiff.comp_contDiffAt z hn).inv |
| (by exact_mod_cast hn0 : (‖bracket z (p:Base)‖:ℂ) ≠ 0)) |
| have hy := densityHeight_smoothAt k p hz |
| have hs := centralPoint_smoothAt p hz |
| have hreal := Complex.reCLM.contDiff.comp_contDiffAt z (hphase.pow k) |
| have him := Complex.imCLM.contDiff.comp_contDiffAt z (hphase.pow k) |
| exact ((hn.pow 2).inv (pow_ne_zero 2 hn0)).mul |
| ((((hf.contDiffAt.comp z hy).mul (hW.contDiffAt.comp z hs)).mul hreal).sub |
| ((((hb.contDiffAt.comp z hy).div_const (k:ℝ)).mul ((phaseDerivative_smooth hW).contDiffAt.comp z hs)).mul him)) |
|
|
| lemma densityCorrection_zero_low {k : ℕ} (hk : 0 < k) {R E : ℝ} {f b : ℝ → ℝ} |
| (htail : ∀ y, E ≤ y → f y=0 ∧ b y=0) (W : Base → ℝ) (p : Sphere) (z : Base) |
| (hz : ‖bracket z (p:Base)‖ ≤ Real.exp (-E/k)) : densityCorrection k R f b W p z=0 := by |
| unfold densityCorrection |
| split_ifs with hc |
| · have hn : 0 < ‖bracket z (p:Base)‖ := lt_trans (Real.exp_pos _) hc |
| have hh := Real.log_le_log hn hz |
| rw [Real.log_exp] at hh |
| have hk0 : (0:ℝ) < k := by exact_mod_cast hk |
| have hhy : E ≤ densityHeight k p z := by |
| have hm := (le_div_iff₀ hk0).mp hh |
| unfold densityHeight |
| nlinarith |
| obtain ⟨hf,hb⟩ := htail _ hhy |
| simp [densityCorrectionRaw,hf,hb] |
| · rfl |
|
|
| lemma densityCorrection_smooth {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 : Sphere) : |
| ContDiff ℝ ∞ (densityCorrection k R f b W p) := by |
| rw [contDiff_iff_contDiffAt] |
| intro z |
| have hc : Continuous (fun z : Base => ‖bracket z (p:Base)‖) := (bracket_analytic (p:Base)).continuous.norm |
| by_cases hz : Real.exp (-R/k) < ‖bracket z (p:Base)‖ |
| · have he : densityCorrection k R f b W p =ᶠ[𝓝 z] densityCorrectionRaw k f b W p := by |
| filter_upwards [(isOpen_lt continuous_const hc).mem_nhds hz] with v hv |
| exact ite_eq_left hv |
| exact (densityCorrectionRaw_smoothAt k hf hb hW p (norm_pos_iff.mp (lt_trans (Real.exp_pos _) hz))).congr_of_eventuallyEq he |
| · have hk0 : (0:ℝ) < k := by exact_mod_cast hk |
| have hcut : Real.exp (-R/k) < Real.exp (-E/k) := Real.exp_lt_exp.mpr ((div_lt_div_iff_of_pos_right hk0).mpr (neg_lt_neg hER)) |
| have hzE : ‖bracket z (p:Base)‖ < Real.exp (-E/k) := lt_of_le_of_lt (le_of_not_gt hz) hcut |
| have he : densityCorrection k R f b W p =ᶠ[𝓝 z] (fun _ => 0) := by |
| filter_upwards [(isOpen_lt hc continuous_const).mem_nhds hzE] with v hv |
| exact densityCorrection_zero_low hk htail W p v hv.le |
| exact contDiffAt_const.congr_of_eventuallyEq he |
|
|
| lemma density_smooth {Q : ℕ} (hQ : 0 < Q) {R E : ℝ} (hER : E < R) |
| {f b : ℝ → ℝ} (hf : ContDiff ℝ ∞ f) (hb : ContDiff ℝ ∞ b) |
| (htail : ∀ y, E ≤ y → f y=0 ∧ b y=0) (P : ℕ → Finset Sphere) (j : ℕ) : |
| ContDiff ℝ ∞ (density Q R f b P j) := by |
| induction j with |
| | zero => exact contDiff_const |
| | succ j ih => |
| exact ih.add (ContDiff.sum (fun p hp => densityCorrection_smooth (pow_pos hQ _) hER hf hb ih htail p)) |
|
|
| end PinchedHartogs.BaseConstruction |
|
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| end |
|
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| end OAI |
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