| 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 |
| |
