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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 | import OAI.MeasureTheory.Falconer.Measures.WeightedIntegralSquare
namespace OAI
namespace PlanarFalconer
open MeasureTheory Filter
theorem integrable_coeff_norm_power {α : Type*} [MeasurableSpace α] (μ : Measure α)
(c F : α → ℂ) (hic : Integrable c μ) (hF : Measurable F)
{C : ℝ} (_ : 0 ≤ C) (hb : ∀ x, ‖F x‖ ≤ C) (n : ℕ) :
Integrable (fun x => ‖c x‖ * ‖F x‖^n) μ := by
apply (hic.norm.mul_const (C^n)).mono'
(hic.aestronglyMeasurable.norm.mul (hF.aestronglyMeasurable.norm.pow n))
filter_upwards with x
change ‖‖c x‖ * ‖F x‖^n‖ ≤ ‖c x‖ * C^n
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg (norm_nonneg _) _))]
exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hb x) n) (norm_nonneg _)
theorem continuous_coefficient_shell_square (μ : Measure ℝ) [IsFiniteMeasure μ]
(c : Plane → ℂ) (hc : Measurable c) (hic : Integrable c)
(F : Plane × ℝ → ℂ) (hF : Measurable F)
{C B : ℝ} (hC : 0 ≤ C) (hb : ∀ p, ‖F p‖ ≤ C)
(hB : ∀ ξ, (∫ r, ‖F (ξ,r)‖^2 ∂μ) ≤ B) :
(∫ r, ‖∫ ξ, c ξ * F (ξ,r)‖^2 ∂μ) ≤ (∫ ξ, ‖c ξ‖)^2 * B := by
let W := ∫ ξ, ‖c ξ‖
have hW : 0 ≤ W := integral_nonneg (fun _ => norm_nonneg _)
have hiJoint : Integrable (fun p : Plane × ℝ => ‖c p.1‖ * ‖F p‖^2)
(volume.prod μ) := by
apply ((hic.norm.comp_fst μ).mul_const (C^2)).mono'
((hic.aestronglyMeasurable.norm.comp_fst).mul (hF.aestronglyMeasurable.norm.pow 2))
filter_upwards with p
change ‖‖c p.1‖ * ‖F p‖^2‖ ≤ ‖c p.1‖ * C^2
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (norm_nonneg _) (sq_nonneg _))]
exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hb p) 2) (norm_nonneg _)
have hPoint (r : ℝ) : ‖∫ ξ, c ξ * F (ξ,r)‖^2 ≤
W * (∫ ξ, ‖c ξ‖ * ‖F (ξ,r)‖^2) := by
have hmF : Measurable (fun ξ => F (ξ,r)) := hF.comp (measurable_id.prodMk measurable_const)
have hi₁ := integrable_coeff_norm_power volume c (fun ξ => F (ξ,r)) hic hmF hC (fun ξ => hb (ξ,r)) 1
have hi₂ := integrable_coeff_norm_power volume c (fun ξ => F (ξ,r)) hic hmF hC (fun ξ => hb (ξ,r)) 2
exact norm_integral_coeff_square volume c (fun ξ => F (ξ,r)) hic
(by simpa only [pow_one] using hi₁) hi₂
have hm : Measurable (fun r => ∫ ξ, c ξ * F (ξ,r)) :=
((hc.comp measurable_fst).mul hF).stronglyMeasurable.integral_prod_left'.measurable
have hiBound : Integrable (fun r => W * (∫ ξ, ‖c ξ‖ * ‖F (ξ,r)‖^2)) μ :=
hiJoint.integral_prod_right.const_mul W
have hiMix : Integrable (fun r => ‖∫ ξ, c ξ * F (ξ,r)‖^2) μ :=
hiBound.mono' (hm.norm.pow_const 2).aestronglyMeasurable (by
filter_upwards with r
change ‖‖∫ ξ, c ξ * F (ξ,r)‖^2‖ ≤ _
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (‖∫ ξ, c ξ * F (ξ,r)‖))]
exact hPoint r)
calc
_ ≤ ∫ r, W * (∫ ξ, ‖c ξ‖ * ‖F (ξ,r)‖^2) ∂μ :=
integral_mono hiMix hiBound hPoint
_ = W * (∫ ξ, ‖c ξ‖ * (∫ r, ‖F (ξ,r)‖^2 ∂μ)) := by
rw [integral_const_mul, ← integral_integral_swap
(f := fun ξ r => ‖c ξ‖ * ‖F (ξ,r)‖^2) hiJoint]
simp_rw [integral_const_mul]
_ ≤ W * (∫ ξ, ‖c ξ‖ * B) := by
apply mul_le_mul_of_nonneg_left _ hW
apply integral_mono
· simpa only [integral_const_mul] using hiJoint.integral_prod_left
· exact hic.norm.mul_const B
· intro ξ
exact mul_le_mul_of_nonneg_left (hB ξ) (norm_nonneg _)
_ = _ := by rw [integral_mul_const]; dsimp [W]; ring
end PlanarFalconer
end OAI
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