| import OAI.MeasureTheory.Falconer.Shells.HighShellFourierError |
| import OAI.MeasureTheory.Falconer.Geometry.SeparatedSpatialDistanceForm |
| import OAI.MeasureTheory.Falconer.Graphs.ParentRetainedEnergy |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory Filter |
| namespace PositiveDistanceData |
|
|
| noncomputable def inverseDistanceEndpointKernel (d : PositiveDistanceData) (j : ℕ) |
| (k : PairMaskIndex → ℕ) (r : ℝ) (z : Plane × Plane) : ℂ := |
| ((((dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)))⁻¹)^j:ℝ):ℂ) * |
| sourceDistanceKernel |
| (sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) |
| (d.inverseEndpointSymbol k 0) (d.inverseEndpointSymbol k 1) r z.1 z.2 |
|
|
| theorem inverseDistanceEndpointKernel_measurable (d : PositiveDistanceData) (j : ℕ) |
| (k : PairMaskIndex → ℕ) (r : ℝ) : Measurable (d.inverseDistanceEndpointKernel j k r) := by |
| unfold inverseDistanceEndpointKernel |
| apply Measurable.mul |
| · fun_prop |
| · exact (measurable_sourceDistanceKernel _ _ _ (measurable_sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) |
| (d.inverseEndpointSymbol_measurable k 0) (d.inverseEndpointSymbol_measurable k 1)).comp |
| (f := fun z : Plane × Plane => (r,z)) (measurable_const.prodMk measurable_id) |
|
|
| theorem inverseDistanceEndpointKernel_integrable (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] |
| (hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (j : ℕ) |
| {b : ℂ × (PairMaskIndex → ℕ)} (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)) |
| (r : ℝ) : Integrable (d.inverseDistanceEndpointKernel j b.2 r) d.cellPair := by |
| let C := symbolDerivativeConstant (2*j)^(2*j) |
| have hC : 0 ≤ C := pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _ |
| refine Integrable.of_bound (d.inverseDistanceEndpointKernel_measurable j b.2 r).aestronglyMeasurable |
| (1*C*C*sourceArcLength.real univ) ?_ |
| filter_upwards [d.ae_cellPair_mem] with z hz |
| have hd := (d.separated_distance_ratio hQ hs (z := z) hz.1 hz.2).1 |
| have hdist : 1 ≤ dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)) := by linarith |
| have hinv : (dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)))⁻¹ ≤ 1 := inv_le_one_of_one_le₀ hdist |
| have hn : ‖((((dist z.1 z.2/(2:ℝ)^(-(d.a:ℝ)))⁻¹)^j:ℝ):ℂ)‖ ≤ 1 := by |
| rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (by positivity)] |
| exact pow_le_one₀ (by positivity) hinv |
| have hk := sourceDistanceKernel_norm_le _ _ _ (by norm_num : (0:ℝ) ≤ 1) hC hC |
| (sourceConeCircle_norm_le (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) (d.inverseEndpointSymbol_norm hQ j hb 0) |
| (d.inverseEndpointSymbol_norm hQ j hb 1) r z.1 z.2 |
| change ‖_ * _‖ ≤ _ |
| rw [norm_mul] |
| exact (mul_le_mul_of_nonneg_right hn (norm_nonneg _)).trans (by simpa only [one_mul] using hk) |
|
|
| theorem inverseDistanceEndpointKernel_form (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] |
| (hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (j : ℕ) |
| {b : ℂ × (PairMaskIndex → ℕ)} (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)) |
| (r : ℝ) : |
| (d.levelMass^(-2:ℤ):ℝ) • (∫ z, d.inverseDistanceEndpointKernel j b.2 r z ∂d.cellPair) = |
| ∫ ξ, spatialFourierCoeff 96 2004 (by norm_num) (by norm_num) j ξ * |
| sourceParentDistanceF (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 b.2 0)) |
| (spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol b.2 1)) r := by |
| have hC : 0 ≤ symbolDerivativeConstant (2*j)^(2*j) := |
| pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _ |
| have he := separatedSpatialDistanceForm 96 2004 (by norm_num) (by norm_num) j |
| (s := (2:ℝ)^(-(d.a:ℝ))) (by positivity) d.ρ d.ρ |
| (d.Q.descendant d.X).carrier (d.Q.descendant d.Y).carrier |
| (d.Q.descendant d.X).measurable (d.Q.descendant d.Y).measurable |
| (fun x hx y hy => d.separated_distance_ratio hQ hs (z := (x,y)) hx hy) |
| d.levelMass d.levelMass _ _ _ (measurable_sourceConeCircle (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) |
| (d.inverseEndpointSymbol_measurable b.2 0) (d.inverseEndpointSymbol_measurable b.2 1) |
| (by norm_num : (0:ℝ) ≤ 1) hC hC (sourceConeCircle_norm_le (sourceParentAxis (d.Q.descendant d.X) (d.Q.descendant d.Y))) |
| (d.inverseEndpointSymbol_norm hQ j hb 0) (d.inverseEndpointSymbol_norm hQ j hb 1) r |
| simpa only [sourceParentDistanceF, inverseDistanceEndpointKernel, cellPair, |
| zpow_neg,zpow_ofNat,pow_two,inv_pow,Complex.ofReal_inv,Complex.ofReal_pow] using he |
|
|
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
