File size: 4,057 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 75 76 77 78 79 80 81 | import OAI.MeasureTheory.Falconer.Shells.HighShellOrderForm
import OAI.MeasureTheory.Falconer.Shells.HighShellRadialBounds
import OAI.MeasureTheory.Falconer.Estimates.FiniteSquareIntegrable
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory Filter
namespace PositiveDistanceData
theorem normalizedHighShellF_shell (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (hs : 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) {B : ℝ} (hB0 : 0 ≤ B)
(hB : ∀ j ∈ Finset.range (n+1),
∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ,
(∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖d.inversePhasedForm b.2 ξ r‖^2) ≤ B) :
(∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖d.normalizedHighShellF n v r‖^2) ≤
2*(2:ℝ)^(v-d.a) *
(∑ 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*B := by
let μ := volume.restrict (Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)))
let u := fun j (r : ℝ) => ((((r/(2:ℝ)^v)⁻¹)^j:ℝ):ℂ)
let F := fun j r => u j r*d.inverseOrderForm j v r
let w := fun j => 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
have hu (j : ℕ) : Measurable (u j) := by dsimp [u]; fun_prop
have huw (j : ℕ) : ∀ᵐ r ∂μ, ‖u j r‖ ≤ (2:ℝ)^j := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr
exact high_shell_radial_inverse hr.1 j
have hwn (j : ℕ) : 0 ≤ w j := by
apply mul_nonneg (spatialFourierCost_nonneg ..)
apply mul_nonneg _ (by positivity)
apply List.sum_nonneg
intro x hx
obtain ⟨b,hb,rfl⟩ := List.mem_map.mp hx
positivity
have hm (j : ℕ) : Measurable (F j) := (hu j).mul (d.inverseOrderForm_data hQ j v).1
have hi (j : ℕ) : Integrable (fun r => ‖F j r‖^2) μ := by
obtain ⟨hm,C,hC,hCb⟩ := d.inverseOrderForm_data hQ j v
exact (shell_multiplier_square μ (u j) _ (hu j) hm
(integrable_norm_square_bounded μ _ hm hC (Eventually.of_forall hCb))
(by positivity) (huw j)).1
have hE (j : ℕ) (hj : j ∈ Finset.range (n+1)) :
(∫ r, ‖F j r‖^2 ∂μ) ≤ (w j)^2*B := by
obtain ⟨hm,C,hC,hCb⟩ := d.inverseOrderForm_data hQ j v
have he := (shell_multiplier_square μ (u j) _ (hu j) hm
(integrable_norm_square_bounded μ _ hm hC (Eventually.of_forall hCb))
(by positivity) (huw j)).2
apply he.trans
apply (mul_le_mul_of_nonneg_left
(d.inverseOrderForm_shell hQ hsig hq hqs hw hv j μ hB0 (hB j hj))
(sq_nonneg ((2:ℝ)^j))).trans_eq
dsimp [w,spatialLeibnizCost,inverseDescriptorList]
ring
let G := fun r => ∑ j ∈ Finset.range (n+1), F j r
have hGm : Measurable G := Finset.measurable_sum _ (fun j _ => hm j)
have hGi : Integrable (fun r => ‖G r‖^2) μ :=
finite_sum_square_integrable _ μ F (fun j _ => hm j) (fun j _ => hi j)
have hGE := finite_weighted_shell_square (Finset.range (n+1)) μ F w
(fun j _ => hwn j) (fun j _ => hi j) (fun j _ => (hm j).aestronglyMeasurable) hE
let p := fun r : ℝ => (((r*(2:ℝ)^(-(d.a:ℝ)))^(1/2:ℝ):ℝ):ℂ)
have hpm : Measurable p := by dsimp [p]; fun_prop
have hp : ∀ᵐ r ∂μ, ‖p r‖ ≤ Real.sqrt (2*(2:ℝ)^(v-d.a)) := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with r hr
exact high_shell_radial_sqrt hr
have hE' := (shell_multiplier_square μ p G hpm hGm hGi (Real.sqrt_nonneg _) hp).2
rw [Real.sq_sqrt (by positivity)] at hE'
have he (r : ℝ) : d.normalizedHighShellF n v r = p r*G r :=
d.normalizedHighShellF_finite_separation hQ hs n v r
simp_rw [he]
exact hE'.trans (by
simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hGE
(show 0 ≤ 2*(2:ℝ)^(v-d.a) by positivity))
end PositiveDistanceData
end PlanarFalconer
end OAI
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