| import OAI.MeasureTheory.Falconer.Shells.HighShellParentEnergy |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set MeasureTheory |
| namespace PositiveDistanceData |
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| noncomputable def inverseShellEnergies (d : PositiveDistanceData) (n : ℕ) |
| (v : ℝ) (Ψ : ℝ → ℝ) : Set ℝ := |
| insert 0 {h | ∃ j ∈ Finset.range (n+1), |
| ∃ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), |
| ∃ ξ, h = (d.inversePhasedHF v Ψ b.2 ξ).toReal} |
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| noncomputable def inverseShellSupremum (d : PositiveDistanceData) (n : ℕ) |
| (v : ℝ) (Ψ : ℝ → ℝ) : ℝ := sSup (d.inverseShellEnergies n v Ψ) |
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| theorem inverseShellEnergies_bddAbove (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] |
| (hQ : 0 < d.Q.side) (n : ℕ) (v : ℝ) (Ψ : ℝ → ℝ) |
| {W : ℝ} (hW : 0 ≤ W) (hΨW : ∀ x, Ψ x ≤ W) |
| (hs : ∀ x, 4 < x → Ψ x = 0) : BddAbove (d.inverseShellEnergies n v Ψ) := by |
| let μ := sourceAngularShellMeasure v Ψ |
| let : IsFiniteMeasure μ := sourceAngularShellMeasure_finite v Ψ hW hΨW hs |
| let E : ℝ := (2:ℝ)^(2*(v-d.a)) |
| let C := ∑ j ∈ Finset.range (n+1), (d.inversePhasedBound j)^2 |
| have hC : 0 ≤ C := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| refine ⟨E*μ.real univ*C,?_⟩ |
| intro x hx |
| rcases hx with rfl | ⟨j,hj,b,hb,ξ,rfl⟩ |
| · exact mul_nonneg (mul_nonneg (by dsimp [E]; positivity) measureReal_nonneg) hC |
| · rw [(d.inversePhasedHF_finite_eq hQ j hb v Ψ hW hΨW hs ξ).2] |
| have hd := d.inversePhasedForm_data hQ j hb |
| have hm : Measurable (d.inversePhasedForm b.2 ξ) := |
| hd.1.comp (f := fun r : ℝ => (ξ,r)) (measurable_const.prodMk measurable_id) |
| have hi := integrable_norm_square_bounded μ _ hm (d.inversePhasedBound_nonneg j) |
| (Filter.Eventually.of_forall (fun r => hd.2 (ξ,r))) |
| have hbnd := integral_mono hi (integrable_const ((d.inversePhasedBound j)^2)) |
| (fun r => pow_le_pow_left₀ (norm_nonneg _) (hd.2 (ξ,r)) 2) |
| rw [integral_const,smul_eq_mul] at hbnd |
| have hjC : (d.inversePhasedBound j)^2 ≤ C := |
| Finset.single_le_sum (fun i _ => sq_nonneg (d.inversePhasedBound i)) hj |
| change E*(∫ r, ‖d.inversePhasedForm b.2 ξ r‖^2 ∂μ) ≤ _ |
| calc |
| _ ≤ E*(μ.real univ*(d.inversePhasedBound j)^2) := |
| mul_le_mul_of_nonneg_left hbnd (by dsimp [E]; positivity) |
| _ ≤ E*(μ.real univ*C) := mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left hjC measureReal_nonneg) (by dsimp [E]; positivity) |
| _ = _ := by ring |
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| theorem inverseShellSupremum_majorizes (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] |
| (hQ : 0 < d.Q.side) (n : ℕ) (v : ℝ) (Ψ : ℝ → ℝ) |
| {W : ℝ} (hW : 0 ≤ W) (hΨW : ∀ x, Ψ x ≤ W) |
| (hs : ∀ x, 4 < x → Ψ x = 0) : |
| 0 ≤ d.inverseShellSupremum n v Ψ ∧ |
| ∀ j ∈ Finset.range (n+1), |
| ∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ, |
| (d.inversePhasedHF v Ψ b.2 ξ).toReal ≤ d.inverseShellSupremum n v Ψ := by |
| have hb := d.inverseShellEnergies_bddAbove hQ n v Ψ hW hΨW hs |
| refine ⟨le_csSup hb (Or.inl rfl),?_⟩ |
| intro j hj b hb' ξ |
| exact le_csSup hb (Or.inr ⟨j,hj,b,hb',ξ,rfl⟩) |
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| end PositiveDistanceData |
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| end PlanarFalconer |
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| end OAI |
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