File size: 3,225 Bytes
7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.MeasureTheory.Falconer.Shells.HighShellParentEnergy
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
namespace PositiveDistanceData
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}
noncomputable def inverseShellSupremum (d : PositiveDistanceData) (n : ℕ)
(v : ℝ) (Ψ : ℝ → ℝ) : ℝ := sSup (d.inverseShellEnergies n v Ψ)
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
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⟩)
end PositiveDistanceData
end PlanarFalconer
end OAI
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