| 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 |
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| end OAI |
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