| import OAI.MeasureTheory.Falconer.Measures.WeightedIntegralSquare |
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| namespace OAI |
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| namespace PlanarFalconer |
| open MeasureTheory Filter |
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| 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 _) |
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| 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 |
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| end PlanarFalconer |
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| end OAI |
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