| import OAI.MeasureTheory.Falconer.Shells.HighShellMainEnergy |
| import OAI.MeasureTheory.Falconer.Shells.SmoothShellDomination |
| import OAI.MeasureTheory.Falconer.Recursion.ParentHFBridge |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ENNReal |
| namespace PositiveDistanceData |
|
|
| noncomputable def inversePhasedHF (d : PositiveDistanceData) (v : ℝ) (Ψ : ℝ → ℝ) |
| (k : PairMaskIndex → ℕ) (ξ : Plane) : ℝ≥0∞ := |
| sourceParentHF d.a v Ψ (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 k 0)) |
| (spatialPhasedSymbol ((2:ℝ)^(-(d.a:ℝ))) ξ (d.inverseEndpointSymbol k 1)) |
|
|
| theorem inversePhasedHF_finite_eq (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] |
| (hQ : 0 < d.Q.side) (j : ℕ) {b : ℂ × (PairMaskIndex → ℕ)} |
| (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)) |
| (v : ℝ) (Ψ : ℝ → ℝ) {W : ℝ} (hW : 0 ≤ W) |
| (hΨW : ∀ x, Ψ x ≤ W) (hs : ∀ x, 4 < x → Ψ x = 0) (ξ : Plane) : |
| d.inversePhasedHF v Ψ b.2 ξ ≠ ⊤ ∧ |
| (d.inversePhasedHF v Ψ b.2 ξ).toReal = |
| sourceRadialSquareEnergy d.a v Ψ (d.inversePhasedForm b.2 ξ) := by |
| have hC : 0 ≤ symbolDerivativeConstant (2*j)^(2*j) := |
| pow_nonneg (zero_le_one.trans (symbolDerivativeConstant_one_le _)) _ |
| apply sourceParentHF_finite_eq _ _ _ _ _ d.ρ d.ρ |
| (show 0 < d.levelMass by unfold levelMass |
| (show 0 < d.levelMass by unfold levelMass |
| (spatialPhasedSymbol_measurable _ _ _ (d.inverseEndpointSymbol_measurable b.2 0)) |
| (spatialPhasedSymbol_measurable _ _ _ (d.inverseEndpointSymbol_measurable b.2 1)) hC hC hW |
| _ _ hΨW hs |
| · intro x dir |
| rw [spatialPhasedSymbol_norm] |
| exact d.inverseEndpointSymbol_norm hQ j hb 0 x dir |
| · intro x dir |
| rw [spatialPhasedSymbol_norm] |
| exact d.inverseEndpointSymbol_norm hQ j hb 1 x dir |
| |
| theorem normalizedHighShellF_parent_energy (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] |
| (hQ : 0 < d.Q.side) (hsep : 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) |
| (Ψ : ℝ → ℝ) (hΨ : Measurable Ψ) (hpos : ∀ x, 0 ≤ Ψ x) |
| (hs : ∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) |
| {W : ℝ} (hW : 0 ≤ W) (hΨW : ∀ x, Ψ x ≤ W) |
| (hone : ∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) {H : ℝ} (hH0 : 0 ≤ H) |
| (hH : ∀ j ∈ Finset.range (n+1), |
| ∀ b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList), ∀ ξ, |
| (d.inversePhasedHF v Ψ b.2 ξ).toReal ≤ H) : |
| (2:ℝ)^(-(d.a:ℝ)) * |
| (∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖d.normalizedHighShellF n v r‖^2) ≤ |
| 2*(∑ 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*H := by |
| have hsupport : ∀ x, 4 < x → Ψ x = 0 := by |
| intro x hx |
| by_contra hn |
| exact (not_lt_of_ge hx.le) (hs x hn).2 |
| have hE := d.normalizedHighShellF_shell hQ hsep hsig hq hqs hw hv n |
| (B := ((2:ℝ)^v/(2:ℝ)^(2*(v-d.a)))*H) (by positivity) (by |
| intro j hj b hb ξ |
| have hd := d.inversePhasedForm_data hQ j hb |
| have hm : Measurable (d.inversePhasedForm b.2 ξ) := |
| hd.1.comp (f := fun r : ℝ => (ξ,r)) (measurable_const.prodMk measurable_id) |
| apply dyadic_shell_le_radial_energy d.a v Ψ hΨ hpos hs hW hΨW hone _ hm |
| (d.inversePhasedBound_nonneg j) (fun r => hd.2 (ξ,r)) |
| rw [← (d.inversePhasedHF_finite_eq hQ j hb v Ψ hW hΨW hsupport ξ).2] |
| exact hH j hj b hb ξ) |
| apply (mul_le_mul_of_nonneg_left hE (show 0 ≤ (2:ℝ)^(-(d.a:ℝ)) by positivity)).trans_eq |
| let S := ∑ 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 |
| change (2:ℝ)^(-(d.a:ℝ))*(2*(2:ℝ)^(v-d.a)*S^2* |
| (((2:ℝ)^v/(2:ℝ)^(2*(v-d.a)))*H)) = 2*S^2*H |
| calc |
| _ = 2*S^2*H*((2:ℝ)^(-(d.a:ℝ))*(2:ℝ)^v*(2:ℝ)^(v-d.a)/(2:ℝ)^(2*(v-d.a))) := by ring |
| _ = _ := by rw [high_shell_energy_prefactor,div_self (by positivity),mul_one] |
| |
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
