| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.NormSqIntegral |
| import OAI.Analysis.HardSphere.PathBounds.ConfigurationTailLp |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HardSphere |
| open MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| lemma integral_configuration_cons {n : ℕ} {L : Length} {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] (F : Configuration (n+1) L → E) |
| (hF : Integrable F volume) : |
| ∫ X, F X = ∫ Y : Configuration n L, ∫ x : Torus L, F (Fin.cons x Y) := by |
| let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0 |
| have he := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm |
| have hi := (he.integrable_comp_emb e.symm.measurableEmbedding).mpr hF |
| rw [← he.integral_comp' F] |
| simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv, |
| Equiv.coe_fn_mk, Fin.insertNth_zero', Function.comp_def] at hi ⊢ |
| exact integral_prod_symm _ hi |
|
|
| lemma coe_weightedTail {n : ℕ} {L : Length} (h : Configuration (n+1) L → ℂ) |
| (hh : MemLp h ∞ volume) (u : ScalarL2 n L) : |
| L2Multiplier.clm h hh (configurationTailCLM n L u) =ᵐ[volume] fun X ↦ h X*u (Fin.tail X) := by |
| filter_upwards [coe_configurationTailCLM u, |
| L2Multiplier.coe_clm h hh (configurationTailCLM n L u)] with X h1 h2 |
| exact h2.trans (by rw [h1]) |
|
|
| lemma weightedTail_pairing {n : ℕ} {L : Length} |
| (f g : Configuration (n+1) L → ℂ) (hf : MemLp f ∞ volume) (hg : MemLp g ∞ volume) |
| (u v : ScalarL2 n L) : |
| inner ℂ (L2Multiplier.clm f hf (configurationTailCLM n L u)) |
| (L2Multiplier.clm g hg (configurationTailCLM n L v)) = |
| ∫ Y : Configuration n L, (conj (u Y)*v Y)* |
| ∫ x : Torus L, conj (f (Fin.cons x Y))*g (Fin.cons x Y) := by |
| have hfm : MemLp (fun X : Configuration (n+1) L ↦ f X*u (Fin.tail X)) 2 volume := |
| hf.mul (memLp_configurationTail (Lp.memLp u)) |
| have hgm : MemLp (fun X : Configuration (n+1) L ↦ g X*v (Fin.tail X)) 2 volume := |
| hg.mul (memLp_configurationTail (Lp.memLp v)) |
| have hi : Integrable (fun X : Configuration (n+1) L ↦ |
| conj (f X*u (Fin.tail X))*(g X*v (Fin.tail X))) volume := by |
| have h := (hfm.inner_const (𝕜 := ℂ) (1:ℂ)).integrable_mul hgm |
| exact h.congr (Eventually.of_forall fun X ↦ by |
| dsimp only [Pi.mul_apply] |
| rw [RCLike.inner_apply,one_mul]) |
| rw [L2.inner_def] |
| calc |
| _ = ∫ X : Configuration (n+1) L, conj (f X*u (Fin.tail X))*(g X*v (Fin.tail X)) := by |
| apply integral_congr_ae |
| filter_upwards [coe_weightedTail f hf u,coe_weightedTail g hg v] with X h1 h2 |
| simp only [h1,h2,RCLike.inner_apply'] |
| _ = ∫ Y : Configuration n L, ∫ x : Torus L, |
| conj (f (Fin.cons x Y)*u Y)*(g (Fin.cons x Y)*v Y) := by |
| simpa only [Fin.tail_cons] using integral_configuration_cons _ hi |
| _ = _ := by |
| apply integral_congr_ae |
| exact Eventually.of_forall fun Y ↦ by |
| dsimp only |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun x ↦ by simp only [map_mul]; ring |
| |
| lemma norm_weightedTail_sq {n : ℕ} {L : Length} |
| (f : Configuration (n+1) L → ℂ) (hf : MemLp f ∞ volume) (u : ScalarL2 n L) : |
| ‖L2Multiplier.clm f hf (configurationTailCLM n L u)‖^2 = |
| ∫ Y : Configuration n L, ‖u Y‖^2 * ∫ x : Torus L, ‖f (Fin.cons x Y)‖^2 := by |
| have hm : MemLp (fun X : Configuration (n+1) L ↦ f X*u (Fin.tail X)) 2 volume := |
| hf.mul (memLp_configurationTail (Lp.memLp u)) |
| rw [IndicatorStrong.norm_sq_integral] |
| calc |
| _ = ∫ X : Configuration (n+1) L, ‖f X*u (Fin.tail X)‖^2 := by |
| exact integral_congr_ae ((coe_weightedTail f hf u).mono fun _ h ↦ by dsimp only at h ⊢; rw [h]) |
| _ = ∫ Y : Configuration n L, ∫ x : Torus L, ‖f (Fin.cons x Y)*u Y‖^2 := by |
| simpa only [Fin.tail_cons] using integral_configuration_cons _ (hm.integrable_norm_pow (by decide : (2:ℕ)≠0)) |
| _ = _ := by |
| apply integral_congr_ae |
| exact Eventually.of_forall fun Y ↦ by |
| dsimp only |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun x ↦ by dsimp only; rw [norm_mul,mul_pow,mul_comm] |
| |
| lemma SmoothMultiplier.norm_insertionScalar_sq {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) |
| (hn : ∀ Y : Configuration n L, ∫ x : Torus L, ‖f.toFun (Fin.cons x Y)‖^2=1) |
| (u : ScalarL2 n L) : ‖f.insertionScalar u‖^2=‖u‖^2 := by |
| rw [SmoothMultiplier.insertionScalar,ContinuousLinearMap.comp_apply,SmoothMultiplier.onScalar, |
| norm_weightedTail_sq,IndicatorStrong.norm_sq_integral] |
| simp only [hn,mul_one] |
| |
| lemma SmoothMultiplier.norm_insertionScalar {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) |
| (hn : ∀ Y : Configuration n L, ∫ x : Torus L, ‖f.toFun (Fin.cons x Y)‖^2=1) |
| (u : ScalarL2 n L) : ‖f.insertionScalar u‖=‖u‖ := |
| (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp (f.norm_insertionScalar_sq hn u) |
| |
| lemma SmoothMultiplier.insertion_cross_zero {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) (p : Direction (n+1)) |
| (ho : ∀ Y : Configuration n L, ∫ x : Torus L, |
| conj (f.toFun (Fin.cons x Y))*coordDerivative (n+1) L f.toFun p (Fin.cons x Y)=0) |
| (u v : ScalarL2 n L) : |
| inner ℂ (f.insertionScalar u) |
| (L2Multiplier.clm _ (f.partial_memLp_top p) (configurationTailCLM n L v))=0 := by |
| rw [SmoothMultiplier.insertionScalar,ContinuousLinearMap.comp_apply,SmoothMultiplier.onScalar, |
| weightedTail_pairing] |
| simp only [ho,mul_zero,integral_zero] |
| |
| end HardSphere |
| |
| end |
| |
| end OAI |
| |
