| 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 |
|
|
