| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.DyadicPowerSq |
|
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter Complex TopologicalSpace |
| open scoped Topology InnerProductSpace ENNReal |
| open MeasureTheory Filter Complex |
| open scoped Topology BigOperators ComplexConjugate FourierTransform SchwartzMap ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff SchwartzMap FourierTransform ENNReal |
| open MeasureTheory Filter |
| open scoped ContDiff InnerProductSpace Topology |
| open MeasureTheory Filter |
| open scoped ENNReal |
| namespace CoulombLT |
| open CoulombAtom Metric |
| variable {ι : Type*} |
| |
| def infiniteDensity (f : ι → Lp ℂ 2 (volume : Measure Space)) (x : Space) : ℝ≥0∞ := |
| ∑' j, ENNReal.ofReal (‖f j x‖^2) |
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| def fourierMoment (f : Lp ℂ 2 (volume : Measure Space)) : ℝ≥0∞ := |
| ∫⁻ ξ, ENNReal.ofReal (‖ξ‖^2 * ‖(𝓕 f : Lp ℂ 2 volume) ξ‖^2) |
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|
| lemma finite_fourierMoment [Fintype ι] (f : ι → Lp ℂ 2 (volume : Measure Space)) : |
| (∫⁻ ξ, ENNReal.ofReal (‖ξ‖^2*familyDensity (fun j => 𝓕 (f j)) ξ)) = |
| ∑ j, fourierMoment (f j) := by |
| have he (ξ : Space) : |
| ENNReal.ofReal (‖ξ‖^2*familyDensity (fun j => 𝓕 (f j)) ξ) = |
| ∑ j, ENNReal.ofReal (‖ξ‖^2*‖(𝓕 (f j) : Lp ℂ 2 volume) ξ‖^2) := by |
| rw [familyDensity, Finset.mul_sum] |
| exact ENNReal.ofReal_sum_of_nonneg (fun j _ => mul_nonneg (sq_nonneg _) (sq_nonneg _)) |
| simp_rw [he] |
| apply lintegral_finsetSum' |
| intro j _ |
| exact ((continuous_norm.pow 2).aestronglyMeasurable.mul |
| ((Lp.memLp (𝓕 (f j))).aestronglyMeasurable.norm.pow 2)).aemeasurable.ennreal_ofReal |
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| lemma infiniteDensity_rpow (f : ι → Lp ℂ 2 (volume : Measure Space)) (x : Space) : |
| infiniteDensity f x ^ (5/3:ℝ) = |
| ⨆ s : Finset ι, ENNReal.ofReal ((∑ j ∈ s, ‖f j x‖^2)^(5/3:ℝ)) := by |
| rw [infiniteDensity, ENNReal.tsum_eq_iSup_sum] |
| change (ENNReal.orderIsoRpow (5/3:ℝ) (by norm_num)) _ = _ |
| rw [OrderIso.map_iSup] |
| congr 1 |
| funext s |
| change (∑ j ∈ s, ENNReal.ofReal (‖f j x‖^2)) ^ (5/3:ℝ) = _ |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun j _ => sq_nonneg _)] |
| exact ENNReal.ofReal_rpow_of_nonneg (Finset.sum_nonneg (fun j _ => sq_nonneg _)) (by norm_num) |
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|
| theorem countable_lieb_thirring_lintegral [Countable ι] |
| {f : ι → Lp ℂ 2 (volume : Measure Space)} |
| (hf : ∀ s : Finset ι, BesselFamily (fun j : s => f j)) : |
| (∫⁻ x, infiniteDensity f x ^ (5/3:ℝ)) ≤ |
| 16*(∑' j, ENNReal.ofReal (‖f j‖^2)) + |
| ENNReal.ofReal (1024/3:ℝ)*(∑' j, fourierMoment (f j)) := by |
| classical |
| have hm (s : Finset ι) : AEMeasurable |
| (fun x => ENNReal.ofReal ((∑ j ∈ s, ‖f j x‖^2)^(5/3:ℝ))) volume := by |
| have hi : Integrable (fun x => ∑ j ∈ s, ‖f j x‖^2) volume := |
| integrable_finsetSum s (fun j _ => |
| (memLp_two_iff_integrable_sq_norm (Lp.memLp (f j)).aestronglyMeasurable).mp (Lp.memLp (f j))) |
| have h := hi.aestronglyMeasurable |
| exact ((Real.continuous_rpow_const (by norm_num : (0:ℝ)≤5/3)).comp_aestronglyMeasurable h).aemeasurable.ennreal_ofReal |
| have hmono (s t : Finset ι) (h : s ⊆ t) (x : Space) : |
| ENNReal.ofReal ((∑ j ∈ s, ‖f j x‖^2)^(5/3:ℝ)) ≤ |
| ENNReal.ofReal ((∑ j ∈ t, ‖f j x‖^2)^(5/3:ℝ)) := by |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.rpow_le_rpow (Finset.sum_nonneg (fun _ _ => sq_nonneg _)) _ (by norm_num) |
| exact Finset.sum_le_sum_of_subset_of_nonneg h (fun _ _ _ => sq_nonneg _) |
| simp_rw [infiniteDensity_rpow] |
| rw [lintegral_iSup_directed hm (fun s t => |
| ⟨s ∪ t, hmono s (s ∪ t) Finset.subset_union_left, |
| hmono t (s ∪ t) Finset.subset_union_right⟩)] |
| apply iSup_le |
| intro s |
| have hb := finite_lieb_thirring_lintegral (hf s) |
| rw [finite_fourierMoment, ENNReal.ofReal_sum_of_nonneg (fun j _ => sq_nonneg _)] at hb |
| have he (x : Space) : familyDensity (fun j : s => f j) x = ∑ j ∈ s, ‖f j x‖^2 := by |
| exact Finset.sum_coe_sort s (fun j => ‖f j x‖^2) |
| simp_rw [he] at hb |
| refine hb.trans ?_ |
| apply add_le_add |
| · gcongr |
| rw [Finset.sum_coe_sort s (fun j => ENNReal.ofReal (‖f j‖^2))] |
| exact ENNReal.sum_le_tsum s |
| · gcongr |
| rw [Finset.sum_coe_sort s (fun j => fourierMoment (f j))] |
| exact ENNReal.sum_le_tsum s |
| end CoulombLT |
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| end |
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| end OAI |
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