| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.FrameInverseApply |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff RealInnerProductSpace |
| namespace YauCounterexamples |
|
|
| def phaseImagVector (z : Fin 3 → ℂ) : PhaseSpace := WithLp.toLp 2 (fun i => (z i).im) |
|
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| def normalImagCovector (q : NormalWaveParameter) (z : Fin 3 → ℂ) : NormalWaveSpace →L[ℝ] ℝ := |
| Complex.imCLM.comp (normalPhaseCovector q z) |
|
|
| lemma normalImagCovector_apply (q : NormalWaveParameter) (z : Fin 3 → ℂ) (v : NormalWaveSpace) : |
| normalImagCovector q z v = inner ℝ (phaseImagVector z) (q.2.inverse v) := by |
| change (normalPhaseCovector q z v).im = _ |
| rw [normalPhaseCovector_apply] |
| simp only [Complex.im_sum,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im, |
| zero_mul,add_zero,EuclideanSpace.inner_eq_star_dotProduct,dotProduct, |
| phaseImagVector,Pi.star_apply,star_trivial] |
| rfl |
|
|
| lemma null_phase_imag_energy {z : Fin 3 → ℂ} (hz : ∑ i, z i*z i = -1) : |
| ‖phaseImagVector z‖^2 = 1+‖phaseRealVector z‖^2 := by |
| have h := congrArg Complex.re hz |
| simp only [Complex.re_sum,Complex.mul_re,Complex.neg_re,Complex.one_re, |
| Finset.sum_sub_distrib] at h |
| simp only [←pow_two] at h |
| rw [EuclideanSpace.real_norm_sq_eq,EuclideanSpace.real_norm_sq_eq] |
| change (∑ i, ((z i).im)^2) = 1+∑ i, ((z i).re)^2 |
| linarith |
|
|
| lemma normalImagCovector_comp_frame (g : SmoothMetric NormalWaveSpace NormalWaveSpace) |
| {K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K) |
| (z : Fin 3 → ℂ) : |
| (normalImagCovector q z).comp q.2 = InnerProductSpace.toDual ℝ PhaseSpace (phaseImagVector z) := by |
| obtain ⟨A,hA⟩ := metricFrameSet_equiv g hq |
| ext v |
| simp only [ContinuousLinearMap.comp_apply,normalImagCovector_apply,←hA, |
| ContinuousLinearMap.inverse_equiv,ContinuousLinearEquiv.coe_coe, |
| A.symm_apply_apply,InnerProductSpace.toDual_apply_apply] |
|
|
| lemma normalImagCovector_eq (q : NormalWaveParameter) (z : Fin 3 → ℂ) : |
| normalImagCovector q z = |
| (InnerProductSpace.toDual ℝ PhaseSpace (phaseImagVector z)).comp q.2.inverse := by |
| ext v |
| exact normalImagCovector_apply q z v |
|
|
| lemma euclidean_covector_squared_norm (β : PhaseSpace →L[ℝ] ℝ) : |
| ‖β‖^2 = ∑ i : Fin 3, (β (EuclideanSpace.basisFun (Fin 3) ℝ i))^2 := by |
| let v := (InnerProductSpace.toDual ℝ PhaseSpace).symm β |
| have hv : ‖v‖ = ‖β‖ := (InnerProductSpace.toDual ℝ PhaseSpace).symm.norm_map β |
| rw [←hv,EuclideanSpace.real_norm_sq_eq] |
| apply Finset.sum_congr rfl |
| intro i _ |
| have hi : v i = β (EuclideanSpace.basisFun (Fin 3) ℝ i) := by |
| rw [←EuclideanSpace.inner_basisFun_real] |
| exact InnerProductSpace.toDual_symm_apply |
| rw [hi] |
|
|
| theorem compact_normal_angular_energy (g : SmoothMetric NormalWaveSpace NormalWaveSpace) |
| {K : Set NormalWaveSpace} (hK : IsCompact K) : |
| ∃ c > 0, ∃ C > 0, ∀ q ∈ metricFrameSet g K, ∀ z : Fin 3 → ℂ, |
| (∑ i, z i*z i = -1) → |
| c*(1+‖phaseRealVector z‖^2) ≤ |
| ∑ i : Fin 3, (normalImagCovector q z (EuclideanSpace.basisFun (Fin 3) ℝ i))^2 ∧ |
| (∑ i : Fin 3, (normalImagCovector q z (EuclideanSpace.basisFun (Fin 3) ℝ i))^2) ≤ |
| C*(1+‖phaseRealVector z‖^2) := by |
| obtain ⟨R,hR,hb⟩ := metricFrameSet_bound g hK |
| obtain ⟨S,hS,hi⟩ := metricFrameSet_inverse_bound g hK |
| refine ⟨(R^2)⁻¹,inv_pos.mpr (sq_pos_of_pos hR),S^2,sq_pos_of_pos hS,?_⟩ |
| intro q hq z hz |
| have hlo : ‖phaseImagVector z‖ ≤ ‖normalImagCovector q z‖*R := by |
| have hh := ContinuousLinearMap.opNorm_comp_le (normalImagCovector q z) q.2 |
| rw [normalImagCovector_comp_frame g hq z,LinearIsometryEquiv.norm_map] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_left (hb q hq) (norm_nonneg _)) |
| have hhi : ‖normalImagCovector q z‖ ≤ S*‖phaseImagVector z‖ := by |
| rw [normalImagCovector_eq] |
| have hh := ContinuousLinearMap.opNorm_comp_le |
| (InnerProductSpace.toDual ℝ PhaseSpace (phaseImagVector z)) q.2.inverse |
| rw [LinearIsometryEquiv.norm_map] at hh |
| exact hh.trans (by nlinarith [hi q hq,norm_nonneg (phaseImagVector z)]) |
| rw [←euclidean_covector_squared_norm,←null_phase_imag_energy hz] |
| constructor |
| · have hs : ‖phaseImagVector z‖^2 ≤ ‖normalImagCovector q z‖^2*R^2 := by |
| nlinarith [sq_nonneg (‖normalImagCovector q z‖*R-‖phaseImagVector z‖), |
| norm_nonneg (phaseImagVector z),norm_nonneg (normalImagCovector q z)] |
| calc |
| _ ≤ (R^2)⁻¹*(‖normalImagCovector q z‖^2*R^2) := |
| mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr (sq_nonneg R)) |
| _ = _ := by field_simp |
| · nlinarith [sq_nonneg (S*‖phaseImagVector z‖-‖normalImagCovector q z‖), |
| norm_nonneg (phaseImagVector z),norm_nonneg (normalImagCovector q z)] |
| end YauCounterexamples |
| end |
|
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| end OAI |
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