| import Mathlib |
| import OAI.Analysis.CoulombIonization.RadialBounds.TranslatedShortRange |
| import OAI.Analysis.CoulombIonization.FormDomain.FormRawPotential |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped BigOperators ENNReal |
|
|
| namespace CoulombAtom |
|
|
| def rawOneParticleLaw {N : ℕ} (μ : Measure (Configuration N)) : Measure Space := |
| Measure.sum (fun i : Fin N => μ.map (fun x => x i)) |
|
|
| lemma rawOneParticleLaw_finite {N : ℕ} (μ : Measure (Configuration N)) |
| [IsFiniteMeasure μ] : IsFiniteMeasure (rawOneParticleLaw μ) := by |
| rw [rawOneParticleLaw,Measure.sum_fintype] |
| infer_instance |
|
|
| lemma rawOneParticleLaw_ac {N : ℕ} (ψ : FormVector N) : |
| rawOneParticleLaw (formRawLaw ψ) ≪ volume := by |
| apply Measure.absolutelyContinuous_sum_left |
| intro i |
| exact ((withDensity_absolutelyContinuous volume _).map (measurable_pi_apply i)).trans |
| (Measure.quasiMeasurePreserving_eval (fun _ : Fin N => (volume : Measure Space)) i).absolutelyContinuous |
|
|
| def ordinaryDensity {N : ℕ} (ψ : FormVector N) (z : Space) : ℝ := |
| ((rawOneParticleLaw (formRawLaw ψ)).rnDeriv volume z).toReal |
|
|
| lemma ordinaryDensity_nonneg {N : ℕ} (ψ : FormVector N) (z : Space) : |
| 0 ≤ ordinaryDensity ψ z := ENNReal.toReal_nonneg |
|
|
| lemma ordinaryDensity_measurable {N : ℕ} (ψ : FormVector N) : |
| Measurable (ordinaryDensity ψ) := |
| (Measure.measurable_rnDeriv _ _).ennreal_toReal |
|
|
| lemma ordinaryDensity_integrable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) : |
| Integrable (ordinaryDensity ψ) := by |
| let : IsFiniteMeasure (formRawLaw ψ) := formRawLaw_finite hψ |
| let : IsFiniteMeasure (rawOneParticleLaw (formRawLaw ψ)) := rawOneParticleLaw_finite _ |
| exact Measure.integrable_toReal_rnDeriv |
|
|
| lemma rawOneParticleLaw_pairing {N : ℕ} (μ : Measure (Configuration N)) |
| {f : Space → ℝ} (hf : Measurable f) (hi : ∀ i : Fin N, Integrable (fun x => f (x i)) μ) : |
| (∫ z, f z ∂rawOneParticleLaw μ) = ∫ x, ∑ i, f (x i) ∂μ := by |
| have him (i : Fin N) : Integrable f (μ.map (fun x => x i)) := |
| (integrable_map_measure hf.aestronglyMeasurable (measurable_pi_apply i).aemeasurable).mpr (hi i) |
| rw [rawOneParticleLaw,Measure.sum_fintype,integral_finsetSum_measure (fun i _ => him i), |
| integral_finsetSum _ (fun i _ => hi i)] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact integral_map (measurable_pi_apply i).aemeasurable hf.aestronglyMeasurable |
|
|
| lemma ordinaryDensity_pairing {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| {f : Space → ℝ} (hf : Measurable f) |
| (hi : ∀ i : Fin N, Integrable (fun x => f (x i)) (formRawLaw ψ)) : |
| (∫ z, ordinaryDensity ψ z*f z) = ∫ x, ∑ i, f (x i) ∂formRawLaw ψ := by |
| let : IsFiniteMeasure (formRawLaw ψ) := formRawLaw_finite hψ |
| let : IsFiniteMeasure (rawOneParticleLaw (formRawLaw ψ)) := rawOneParticleLaw_finite _ |
| change (∫ z, ((rawOneParticleLaw (formRawLaw ψ)).rnDeriv volume z).toReal*f z) = _ |
| rw [integral_toReal_rnDeriv_mul (rawOneParticleLaw_ac ψ)] |
| exact rawOneParticleLaw_pairing _ hf hi |
|
|
| lemma ordinaryDensity_measure {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) : |
| volume.withDensity (fun z => ENNReal.ofReal (ordinaryDensity ψ z)) = |
| rawOneParticleLaw (formRawLaw ψ) := by |
| let : IsFiniteMeasure (formRawLaw ψ) := formRawLaw_finite hψ |
| let : IsFiniteMeasure (rawOneParticleLaw (formRawLaw ψ)) := rawOneParticleLaw_finite _ |
| calc |
| _ = volume.withDensity ((rawOneParticleLaw (formRawLaw ψ)).rnDeriv volume) := by |
| apply withDensity_congr_ae |
| filter_upwards [Measure.rnDeriv_lt_top (rawOneParticleLaw (formRawLaw ψ)) volume] with z hz |
| exact ENNReal.ofReal_toReal hz.ne |
| _ = _ := Measure.withDensity_rnDeriv_eq _ _ (rawOneParticleLaw_ac ψ) |
|
|
| lemma ordinaryDensity_mass {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) : |
| (∫ z, ordinaryDensity ψ z) = (N:ℝ)*formMass ψ := by |
| let : IsFiniteMeasure (formRawLaw ψ) := formRawLaw_finite hψ |
| have he := ordinaryDensity_pairing hψ (measurable_const (a := (1:ℝ))) |
| (fun _ => integrable_const 1) |
| simp only [mul_one,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at he |
| rw [he,integral_const,smul_eq_mul,measureReal_def,formRawLaw_univ hψ, |
| ENNReal.toReal_ofReal (formMass_nonneg ψ)] |
| ring |
|
|
| lemma ordinaryDensity_raw_local_potential {N : ℕ} {ψ : FormVector N} |
| (hψ : SobolevVector ψ) (y : Space) (q : ℝ) : |
| (∫ x, rawLocalPotential y q x ∂formRawLaw ψ) = |
| ∫ z in ball y q, ordinaryDensity ψ z/‖z-y‖ := by |
| let f : Space → ℝ := fun z => if ‖z-y‖ < q then 1/‖z-y‖ else 0 |
| have hf : Measurable f := Measurable.ite |
| (measurableSet_lt ((continuous_id.sub continuous_const).norm.measurable) measurable_const) |
| (measurable_const.div ((continuous_id.sub continuous_const).norm.measurable)) measurable_const |
| have hn (z : Space) : 0 ≤ f z := by dsimp [f]; split_ifs <;> positivity |
| have hi (i : Fin N) : Integrable (fun x : Configuration N => f (x i)) (formRawLaw ψ) := by |
| apply (rawPotential_form_integrable hψ y).mono' |
| (hf.comp (measurable_pi_apply i)).aestronglyMeasurable |
| apply ae_of_all |
| intro x |
| change ‖f (x i)‖ ≤ rawPotential y x |
| rw [Real.norm_of_nonneg (hn _)] |
| have hle : 1/‖x i-y‖ ≤ rawPotential y x := by |
| unfold rawPotential |
| exact Finset.single_le_sum (f := fun j : Fin N => (1:ℝ)/‖x j-y‖) |
| (fun _ _ => by positivity) (Finset.mem_univ i) |
| dsimp [f] |
| split_ifs |
| · exact hle |
| · exact rawPotential_nonneg y x |
| have hp := ordinaryDensity_pairing hψ hf hi |
| have he : (fun z => ordinaryDensity ψ z*f z) = |
| (ball y q).indicator (fun z => ordinaryDensity ψ z/‖z-y‖) := by |
| funext z |
| by_cases hz : ‖z-y‖ < q |
| · have hm : z ∈ ball y q := by simpa only [mem_ball,dist_eq_norm] using hz |
| rw [Set.indicator_of_mem hm] |
| simp [f,hz,div_eq_mul_inv] |
| · have hm : z ∉ ball y q := by simpa only [mem_ball,dist_eq_norm] using hz |
| rw [Set.indicator_of_notMem hm] |
| simp [f,hz] |
| rw [he,integral_indicator measurableSet_ball] at hp |
| exact hp.symm |
| |
| theorem ordinaryDensity_raw_local_holder {N : ℕ} {ψ : FormVector N} |
| (hψ : SobolevVector ψ) (y : Space) {q : ℝ} (hq : 0 < q) |
| (hp : MemLp (ordinaryDensity ψ) (5/3) (volume.restrict (ball y q))) : |
| (∫ x, rawLocalPotential y q x ∂formRawLaw ψ) ≤ |
| (∫ z in ball y q, (ordinaryDensity ψ z)^(5/3:ℝ))^(3/5:ℝ)* |
| (8*Real.pi)^(2/5:ℝ)*q^(1/5:ℝ) := by |
| rw [ordinaryDensity_raw_local_potential hψ] |
| have hh := CoulombAnalysis.coulomb_short_range_holder_centered hq y hp |
| simpa only [Real.norm_of_nonneg (ordinaryDensity_nonneg ψ _),norm_sub_rev y] using hh |
| |
| end CoulombAtom |
| |
| end |
| |
| end OAI |
| |
