File size: 2,994 Bytes
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 | import OAI.MeasureTheory.Falconer.Shells.ContinuousCoefficientShell
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
namespace OAI
namespace PlanarFalconer
open MeasureTheory Filter
theorem finite_weighted_shell_square {ι : Type*} (s : Finset ι)
(μ : Measure ℝ) (F : ι → ℝ → ℂ) (w : ι → ℝ) {B : ℝ}
(hw : ∀ i ∈ s, 0 ≤ w i)
(hi : ∀ i ∈ s, Integrable (fun r => ‖F i r‖^2) μ)
(hm : ∀ i ∈ s, AEStronglyMeasurable (F i) μ)
(hB : ∀ i ∈ s, (∫ r, ‖F i r‖^2 ∂μ) ≤ (w i)^2*B) :
(∫ r, ‖∑ i ∈ s, F i r‖^2 ∂μ) ≤ (∑ i ∈ s, w i)^2*B := by
classical
have hz (i : ι) (his : i ∈ s) (hwi : w i = 0) :
∀ᵐ r ∂μ, F i r = 0 := by
have he : (∫ r, ‖F i r‖^2 ∂μ) = 0 :=
le_antisymm (by simpa [hwi] using hB i his)
(integral_nonneg (fun _ => sq_nonneg _))
have hh := (integral_eq_zero_iff_of_nonneg (fun r => sq_nonneg (‖F i r‖)) (hi i his)).mp he
filter_upwards [hh] with r hr
simpa using hr
have hae : ∀ᵐ r ∂μ, ∀ i ∈ s, w i = 0 → F i r = 0 := by
exact (eventually_all_finset s).mpr (fun i his =>
(ae_all_iff).mpr (fun hwi => hz i his hwi))
let G : ℝ → ℝ := fun r => ∑ i ∈ s, ‖F i r‖^2 / w i
let W : ℝ := ∑ i ∈ s, w i
have hW : 0 ≤ W := Finset.sum_nonneg hw
have hgi : Integrable G μ := integrable_finsetSum s
(fun i his => (hi i his).div_const (w i))
have hpoint : ∀ᵐ r ∂μ, ‖∑ i ∈ s, F i r‖^2 ≤ W*G r := by
filter_upwards [hae] with r hr
apply (pow_le_pow_left₀ (norm_nonneg _) (norm_sum_le s _) 2).trans
apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul s hw
(fun i his => div_nonneg (sq_nonneg _) (hw i his))
intro i his
by_cases hwi : w i = 0
· simp [hwi,hr i his hwi]
· rw [mul_div_cancel₀ _ hwi]
have hsi : Integrable (fun r => ‖∑ i ∈ s, F i r‖^2) μ := by
have hsm : AEStronglyMeasurable (fun r => ∑ i ∈ s, F i r) μ := by
convert Finset.aestronglyMeasurable_sum s hm using 1
ext r
simp
apply (hgi.const_mul W).mono' (hsm.norm.pow 2)
filter_upwards [hpoint] with r hr
change ‖‖∑ i ∈ s, F i r‖^2‖ ≤ _
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
exact hr
have hBi (i : ι) (his : i ∈ s) :
(∫ r, ‖F i r‖^2 ∂μ) / w i ≤ w i*B := by
by_cases hwi : w i = 0
· simp [hwi]
· apply (div_le_iff₀ (lt_of_le_of_ne (hw i his) (Ne.symm hwi))).mpr
nlinarith [hB i his]
calc
_ ≤ ∫ r, W*G r ∂μ := integral_mono_ae hsi (hgi.const_mul W) hpoint
_ = W * ∑ i ∈ s, (∫ r, ‖F i r‖^2 ∂μ) / w i := by
rw [integral_const_mul,integral_finsetSum s (fun i his => (hi i his).div_const (w i))]
simp only [integral_div]
_ ≤ W * ∑ i ∈ s, w i*B :=
mul_le_mul_of_nonneg_left (Finset.sum_le_sum hBi) hW
_ = _ := by rw [← Finset.sum_mul]; dsimp [W]; ring
end PlanarFalconer
end OAI
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