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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.MeasureTheory.Falconer.Fourier.IntegratedSmoothing
import OAI.MeasureTheory.Falconer.Estimates.CriticalScale
import OAI.MeasureTheory.Falconer.Slicing.GaussianAmplification
namespace OAI
/-! # The critical half-dimensional estimate on one frequency shell -/
namespace Falconer
open Set Metric MeasureTheory
open scoped ENNReal
variable {d : ℕ}
noncomputable def halfDimensionalShellConstant (d : ℕ) : ℝ :=
((Real.pi/2)^((d:ℝ)/2))⁻¹ *
(Real.exp 1*8*(2:ℝ)^((d:ℝ)/2)*gaussianBallConstant (Nat.ceil ((d:ℝ)/2+1))) *
(16*Real.pi)^((d:ℝ)/2)*gaussianBallConstant (Nat.ceil ((d:ℝ)/2))
theorem halfDimensionalShellConstant_nonneg (d : ℕ) :
0 ≤ halfDimensionalShellConstant d := by
have h₁ := gaussianBallConstant_nonneg (Nat.ceil ((d:ℝ)/2+1))
have h₂ := gaussianBallConstant_nonneg (Nat.ceil ((d:ℝ)/2))
unfold halfDimensionalShellConstant
positivity
theorem damped_half_dimensional_shell (β : Measure (Space d)) [IsFiniteMeasure β]
(ν : Measure (Sphere d)) [IsFiniteMeasure ν]
{B R K L : ℝ} (hB : 1 ≤ B) (hBR : B ≤ R) (hK : 0 ≤ K) (hL : 1 ≤ L)
(hmass : ν univ ≤ 1)
(hβ : ∀ x : Space d, ∀ r : ℝ, 1/B ≤ r →
β (ball x r) ≤ ENNReal.ofReal (K*r^((d:ℝ)/2)))
(hν : ∀ x : Space d, ∀ r : ℝ, 1/R ≤ r → r ≤ 1 →
ν {w : Sphere d | dist (w:Space d) x < r} ≤ ENNReal.ofReal (L*r^((d:ℝ)/2))) :
(∫⁻ v, ENNReal.ofReal (‖PlanarFalconer.sourceFourier (gaussianDampedMeasure β) v‖^2)
∂radialFrequencyMeasure ν B) ≤
ENNReal.ofReal (halfDimensionalShellConstant d*β.real univ*K*L) := by
let s : ℝ := (d:ℝ)/2
let A : ℝ := 8*(2:ℝ)^s*L/B^s
let C : ℝ := ((Real.pi/2)^s)⁻¹
let D : ℝ := Real.exp 1*A*gaussianBallConstant (Nat.ceil (s+1))
have hB₀ : 0 < B := zero_lt_one.trans_le hB
have hs : 0 ≤ s := by dsimp [s]; positivity
have hA : 0 ≤ A := by dsimp [A]; positivity
have hD : 0 ≤ D := mul_nonneg (mul_nonneg (Real.exp_pos _).le hA) (gaussianBallConstant_nonneg _)
have hC : 0 ≤ C := by dsimp [C]; positivity
have hbase := gaussian_integrated_smoothing β (radialFrequencyMeasure ν B) hB hA
(Nat.le_ceil (s+1)) ((radial_frequency_support ν B).mono (fun _ h => h.2))
(radial_frequency_polynomial_balls ν hB hBR hL hs hmass hν)
have henergy := gaussian_shell_energy_lintegral β hB₀ hK (Nat.le_ceil s) hβ
calc
_ ≤ ENNReal.ofReal (C*D)*
∫⁻ u : Space d, ENNReal.ofReal (Real.exp (-‖u‖^2/(16*B^2)))*
ENNReal.ofReal (‖PlanarFalconer.sourceFourier β u‖^2) := hbase
_ ≤ ENNReal.ofReal (C*D)*ENNReal.ofReal ((16*Real.pi*B^2)^s*
(β.real univ*(K*(1/B)^s*gaussianBallConstant (Nat.ceil s)))) :=
mul_le_mul_right henergy _
_ = ENNReal.ofReal (C*D*((16*Real.pi*B^2)^s*
(β.real univ*(K*(1/B)^s*gaussianBallConstant (Nat.ceil s))))) :=
(ENNReal.ofReal_mul (mul_nonneg hC hD)).symm
_ = _ := by
congr 1
calc
C*D*((16*Real.pi*B^2)^s*(β.real univ*(K*(1/B)^s*gaussianBallConstant (Nat.ceil s)))) =
(C*(Real.exp 1*8*(2:ℝ)^s*gaussianBallConstant (Nat.ceil (s+1)))*
gaussianBallConstant (Nat.ceil s)*β.real univ*K*L)*
((16*Real.pi*B^2)^s*(1/B)^s/B^s) := by dsimp [D,A]; ring
_ = _ := by
rw [critical_gaussian_scale hB₀ (by positivity : (0:ℝ) ≤ 16*Real.pi)]
dsimp [halfDimensionalShellConstant,C,s]
ring
/-- One frequency shell for a bounded original probability, with no lower-scale ball bounds. -/
theorem half_dimensional_shell (μ : Measure (Space d)) [IsProbabilityMeasure μ]
(ν : Measure (Sphere d)) [IsFiniteMeasure ν]
{D B R K L : ℝ} (hD : 0 ≤ D) (hsupport : ∀ᵐ x ∂μ, ‖x‖ ≤ D)
(hB : 1 ≤ B) (hBR : B ≤ R) (hK : 0 ≤ K) (hL : 1 ≤ L) (hmass : ν univ ≤ 1)
(hμ : ∀ x : Space d, ∀ r : ℝ, 1/B ≤ r →
μ (ball x r) ≤ ENNReal.ofReal (K*r^((d:ℝ)/2)))
(hν : ∀ x : Space d, ∀ r : ℝ, 1/R ≤ r → r ≤ 1 →
ν {w : Sphere d | dist (w:Space d) x < r} ≤ ENNReal.ofReal (L*r^((d:ℝ)/2))) :
(∫⁻ v, ENNReal.ofReal (‖PlanarFalconer.sourceFourier μ v‖^2) ∂radialFrequencyMeasure ν B) ≤
ENNReal.ofReal (halfDimensionalShellConstant d*
(Real.exp ((Real.pi^2/2)*D^2))^2*K*L) := by
let E := Real.exp ((Real.pi^2/2)*D^2)
let := gaussianAmplifiedMeasure_finite μ hD hsupport
have hβ : (gaussianAmplifiedMeasure μ).real univ ≤ E := by
have hh := gaussianAmplifiedMeasure_le μ hD hsupport univ
simp only [Measure.smul_apply,smul_eq_mul,measure_univ,mul_one] at hh
have hh' := ENNReal.toReal_mono ENNReal.ofReal_ne_top hh
simpa only [measureReal_def,E,ENNReal.toReal_ofReal (Real.exp_pos _).le] using hh'
have h := damped_half_dimensional_shell (gaussianAmplifiedMeasure μ) ν hB hBR
(mul_nonneg (Real.exp_pos _).le hK) hL hmass
(gaussianAmplifiedMeasure_ball_bound μ hD hsupport hμ) hν
rw [gaussianDampedMeasure_amplified] at h
refine h.trans ?_
apply ENNReal.ofReal_le_ofReal
have hC := halfDimensionalShellConstant_nonneg d
calc
halfDimensionalShellConstant d*(gaussianAmplifiedMeasure μ).real univ*(E*K)*L ≤
halfDimensionalShellConstant d*E*(E*K)*L := by
gcongr
_ = _ := by dsimp [E]; ring
end Falconer
end OAI
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