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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.MeasureTheory.Falconer.Shells.HighShellMainEnergy
import OAI.MeasureTheory.Falconer.Shells.SmoothShellDomination
import OAI.MeasureTheory.Falconer.Recursion.ParentHFBridge
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ENNReal
namespace PositiveDistanceData
noncomputable def inversePhasedHF (d : PositiveDistanceData) (v : ℝ) (Ψ : ℝ → ℝ)
(k : PairMaskIndex → ℕ) (ξ : Plane) : ℝ≥0∞ :=
sourceParentHF d.a v Ψ (d.Q.descendant d.X) (d.Q.descendant d.Y)
d.ρ d.ρ d.levelMass d.levelMass
(spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol k 0))
(spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol k 1))
theorem inversePhasedHF_finite_eq (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (j : ℕ) {b : ℂ × (PairMaskIndex → ℕ)}
(hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList))
(v : ℝ) (Ψ : ℝ → ℝ) {W : ℝ} (hW : 0 ≤ W)
(hΨW : ∀ x, Ψ x ≤ W) (hs : ∀ x, 4 < x → Ψ x = 0) (ξ : Plane) :
d.inversePhasedHF v Ψ b.2 ξ ≠ ⊤ ∧
(d.inversePhasedHF v Ψ b.2 ξ).toReal =
sourceRadialSquareEnergy d.a v Ψ (d.inversePhasedForm b.2 ξ) := by
have hC : 0 ≤ symbolDerivativeConstant (2*j)^(2*j) :=
pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _
apply sourceParentHF_finite_eq _ _ _ _ _ d.ρ d.ρ
(show 0 < d.levelMass by unfold levelMass; positivity)
(show 0 < d.levelMass by unfold levelMass; positivity) _ _
(spatialPhasedSymbol_measurable _ _ _ (d.inverseEndpointSymbol_measurable b.2 0))
(spatialPhasedSymbol_measurable _ _ _ (d.inverseEndpointSymbol_measurable b.2 1)) hC hC hW
_ _ hΨW hs
· intro x dir
rw [spatialPhasedSymbol_norm]
exact d.inverseEndpointSymbol_norm hQ j hb 0 x dir
· intro x dir
rw [spatialPhasedSymbol_norm]
exact d.inverseEndpointSymbol_norm hQ j hb 1 x dir
theorem normalizedHighShellF_parent_energy (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (hsep : d.StronglySeparated) (hsig : 0 ≤ d.sig)
(hq : 0 < d.q) (hqs : d.sig*d.N ≤ (d.q:ℝ)) (hw : d.a+100*d.q < d.t)
{v : ℝ} (hv : (d.t:ℝ)-d.sig*d.N ≤ v) (n : ℕ)
(Ψ : ℝ → ℝ) (hΨ : Measurable Ψ) (hpos : ∀ x, 0 ≤ Ψ x)
(hs : ∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4)
{W : ℝ} (hW : 0 ≤ W) (hΨW : ∀ x, Ψ x ≤ W)
(hone : ∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) {H : ℝ} (hH0 : 0 ≤ H)
(hH : ∀ j ∈ Finset.range (n+1),
∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ,
(d.inversePhasedHF v Ψ b.2 ξ).toReal ≤ H) :
(2:ℝ)^(-(d.a:ℝ)) *
(∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖d.normalizedHighShellF n v r‖^2) ≤
2*(∑ j ∈ Finset.range (n+1), spatialLeibnizCost 96 2004 (by norm_num) (by norm_num) j
(fun k => d.mass k-(k:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t)^2*H := by
have hsupport : ∀ x, 4 < x → Ψ x = 0 := by
intro x hx
by_contra hn
exact (not_lt_of_ge hx.le) (hs x hn).2
have hE := d.normalizedHighShellF_shell hQ hsep hsig hq hqs hw hv n
(B := ((2:ℝ)^v/(2:ℝ)^(2*(v-d.a)))*H) (by positivity) (by
intro j hj b hb ξ
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)
apply dyadic_shell_le_radial_energy d.a v Ψ hΨ hpos hs hW hΨW hone _ hm
(d.inversePhasedBound_nonneg j) (fun r => hd.2 (ξ,r))
rw [← (d.inversePhasedHF_finite_eq hQ j hb v Ψ hW hΨW hsupport ξ).2]
exact hH j hj b hb ξ)
apply (mul_le_mul_of_nonneg_left hE (show 0 ≤ (2:ℝ)^(-(d.a:ℝ)) by positivity)).trans_eq
let S := ∑ j ∈ Finset.range (n+1), spatialLeibnizCost 96 2004 (by norm_num) (by norm_num) j
(fun k => d.mass k-(k:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t
change (2:ℝ)^(-(d.a:ℝ))*(2*(2:ℝ)^(v-d.a)*S^2*
(((2:ℝ)^v/(2:ℝ)^(2*(v-d.a)))*H)) = 2*S^2*H
calc
_ = 2*S^2*H*((2:ℝ)^(-(d.a:ℝ))*(2:ℝ)^v*(2:ℝ)^(v-d.a)/(2:ℝ)^(2*(v-d.a))) := by ring
_ = _ := by rw [high_shell_energy_prefactor,div_self (by positivity),mul_one]
end PositiveDistanceData
end PlanarFalconer
end OAI
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