File size: 3,564 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 69 70 71 72 73 74 | import OAI.MeasureTheory.Falconer.Shells.HighShellMixture
import OAI.MeasureTheory.Falconer.Estimates.ListBoundedFunctions
import OAI.MeasureTheory.Falconer.Estimates.SpatialLeibnizSubpower
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory Filter
namespace PositiveDistanceData
noncomputable def inverseOrderForm (d : PositiveDistanceData) (j : ℕ) (v r : ℝ) : ℂ :=
((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b =>
d.absorbedInverseCoefficient j v b*d.inverseSpatialMixture j b.2 r)).sum
theorem inverseOrderForm_data (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (j : ℕ) (v : ℝ) :
Measurable (d.inverseOrderForm j v) ∧
∃ C : ℝ, 0 ≤ C ∧ ∀ r, ‖d.inverseOrderForm j v r‖ ≤ C := by
apply list_sum_measurable_bounded
intro b hb
have hd := d.inverseSpatialMixture_data hQ j hb (Measure.dirac (0:ℝ))
refine ⟨hd.1.const_mul _,
‖d.absorbedInverseCoefficient j v b‖*
(spatialFourierCost 96 2004 (by norm_num) (by norm_num) j*d.inversePhasedBound j),?_,?_⟩
· exact mul_nonneg (norm_nonneg _) (mul_nonneg (spatialFourierCost_nonneg ..) (d.inversePhasedBound_nonneg j))
· intro r
rw [norm_mul]
exact mul_le_mul_of_nonneg_left (hd.2.1 r) (norm_nonneg _)
theorem inverseOrderForm_shell (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (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) (j : ℕ)
(μ : Measure ℝ) [IsFiniteMeasure μ] {B : ℝ} (_hB0 : 0 ≤ B)
(hB : ∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ,
(∫ r, ‖d.inversePhasedForm b.2 ξ r‖^2 ∂μ) ≤ B) :
(∫ r, ‖d.inverseOrderForm j v r‖^2 ∂μ) ≤
(spatialFourierCost 96 2004 (by norm_num) (by norm_num) j *
((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b => ‖b.1‖)).sum)^2*B := by
let C := spatialFourierCost 96 2004 (by norm_num) (by norm_num) j
have hC : 0 ≤ C := spatialFourierCost_nonneg ..
let L := inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)
let F := fun b : ℂ × (PairMaskIndex → ℕ) =>
fun r => d.absorbedInverseCoefficient j v b*d.inverseSpatialMixture j b.2 r
have ht (b : ℂ × (PairMaskIndex → ℕ)) (hb : b ∈ L) :
Measurable (F b) ∧ Integrable (fun r => ‖F b r‖^2) μ ∧
(∫ r, ‖F b r‖^2 ∂μ) ≤ (‖b.1‖*C)^2*B := by
have hd := d.inverseSpatialMixture_data hQ j hb μ
have hcoeff := d.absorbedInverseCoefficient_norm hsig hq hqs hw hv j hb
have hm := shell_multiplier_square μ (fun _ => d.absorbedInverseCoefficient j v b)
(d.inverseSpatialMixture j b.2) measurable_const hd.1 hd.2.2 (norm_nonneg b.1)
(Eventually.of_forall (fun _ => hcoeff))
refine ⟨hd.1.const_mul _,hm.1,?_⟩
apply hm.2.trans
apply (mul_le_mul_of_nonneg_left (d.inverseSpatialMixture_shell hQ j hb μ (hB b hb))
(sq_nonneg ‖b.1‖)).trans_eq
dsimp [C]
ring
have hh := list_weighted_shell_square L μ F (fun b => ‖b.1‖*C)
(fun b _ => mul_nonneg (norm_nonneg _) hC)
(fun b hb => (ht b hb).2.1) (fun b hb => (ht b hb).1.aestronglyMeasurable)
(fun b hb => (ht b hb).2.2)
have he : (L.map (fun b => ‖b.1‖*C)).sum = C*(L.map (fun b => ‖b.1‖)).sum := by
rw [List.sum_map_mul_right]
ring
rw [he] at hh
exact hh
end PositiveDistanceData
end PlanarFalconer
end OAI
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