| import OAI.Analysis.IrrationalRotation.Model.WeightedShift |
| import Mathlib.MeasureTheory.Function.L2Space |
| import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal lp Topology |
|
|
| namespace Invariant233.WeightedShift |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} |
|
|
| theorem diagonal_value (w : Weight) (v : Hilbert) (i : ℤ) : |
| diagonal w v i = w i * v i := rfl |
|
|
| theorem diagonal_bound (w : Weight) (C : ℝ) (hw : ‖w‖ ≤ C) (v : Hilbert) : |
| ‖diagonal w v‖ ≤ C * ‖v‖ := by |
| apply (diagonal w).le_of_opNorm_le |
| exact (lp.norm_mapCLM_le 2 _ (norm_nonneg w) _).trans hw |
|
|
| theorem diagonal_aestronglyMeasurable (a : X → Weight) (C : ℝ) (_hC : 0 ≤ C) |
| (_ha : ∀ x, ‖a x‖ ≤ C) |
| (hm : ∀ i, AEStronglyMeasurable (fun x => a x i) μ) |
| {f : X → Hilbert} (hf : AEStronglyMeasurable f μ) : |
| AEStronglyMeasurable (fun x => diagonal (a x) (f x)) μ := by |
| classical |
| let : Nonempty (Finset ℤ) := ⟨∅⟩ |
| let F : Finset ℤ → X → Hilbert := |
| fun s => ∑ i ∈ s, fun x => lp.single 2 i (a x i * f x i) |
| apply aestronglyMeasurable_of_tendsto_ae (atTop : Filter (Finset ℤ)) |
| (f := F) |
| · intro s |
| apply Finset.aestronglyMeasurable_sum |
| intro i hi |
| exact (lp.singleContinuousLinearMap ℂ (fun _ : ℤ => ℂ) 2 i).continuous.comp_aestronglyMeasurable |
| ((hm i).mul ((lp.evalCLM ℂ (fun _ : ℤ => ℂ) 2 i).continuous.comp_aestronglyMeasurable hf)) |
| · apply Filter.Eventually.of_forall |
| intro x |
| simpa only [F, Finset.sum_apply, diagonal_value, HasSum, SummationFilter.unconditional] using |
| lp.hasSum_single (by norm_num) (diagonal (a x) (f x)) |
|
|
| def fieldDiagonalVec (a : X → Weight) (C : ℝ) (hC : 0 ≤ C) |
| (ha : ∀ x, ‖a x‖ ≤ C) |
| (hm : ∀ i, AEStronglyMeasurable (fun x => a x i) μ) |
| (f : Lp (Hilbert) 2 μ) : Lp (Hilbert) 2 μ := |
| ((Lp.memLp f).of_le_mul |
| (diagonal_aestronglyMeasurable a C hC ha hm (Lp.aestronglyMeasurable f)) |
| (Filter.Eventually.of_forall fun x => diagonal_bound (a x) C (ha x) (f x))).toLp _ |
|
|
| theorem fieldDiagonalVec_ae (a : X → Weight) (C : ℝ) (hC : 0 ≤ C) |
| (ha : ∀ x, ‖a x‖ ≤ C) |
| (hm : ∀ i, AEStronglyMeasurable (fun x => a x i) μ) |
| (f : Lp (Hilbert) 2 μ) : |
| fieldDiagonalVec a C hC ha hm f =ᵐ[μ] |
| fun x => diagonal (a x) (f x) := |
| MemLp.coeFn_toLp _ |
|
|
| def fieldDiagonalLinear (a : X → Weight) (C : ℝ) (hC : 0 ≤ C) |
| (ha : ∀ x, ‖a x‖ ≤ C) |
| (hm : ∀ i, AEStronglyMeasurable (fun x => a x i) μ) : |
| Lp (Hilbert) 2 μ →ₗ[ℂ] Lp (Hilbert) 2 μ where |
| toFun := fieldDiagonalVec a C hC ha hm |
| map_add' f g := by |
| apply Lp.ext |
| filter_upwards [fieldDiagonalVec_ae a C hC ha hm (f + g), |
| fieldDiagonalVec_ae a C hC ha hm f, fieldDiagonalVec_ae a C hC ha hm g, |
| Lp.coeFn_add f g, Lp.coeFn_add |
| (fieldDiagonalVec a C hC ha hm f) (fieldDiagonalVec a C hC ha hm g)] |
| with x hfg hf hg hsum hout |
| simp only [Pi.add_apply] at hsum hout |
| rw [hfg, hout, hf, hg, hsum, map_add] |
| map_smul' c f := by |
| change fieldDiagonalVec a C hC ha hm (c • f) = c • fieldDiagonalVec a C hC ha hm f |
| apply Lp.ext |
| filter_upwards [fieldDiagonalVec_ae a C hC ha hm (c • f), |
| fieldDiagonalVec_ae a C hC ha hm f, Lp.coeFn_smul c f, |
| Lp.coeFn_smul c (fieldDiagonalVec a C hC ha hm f)] with x hcf hf hsmul hout |
| simp only [Pi.smul_apply] at hsmul hout |
| rw [hcf, hout, hf, hsmul, map_smul] |
|
|
| def fieldDiagonal (a : X → Weight) (C : ℝ) (hC : 0 ≤ C) |
| (ha : ∀ x, ‖a x‖ ≤ C) |
| (hm : ∀ i, AEStronglyMeasurable (fun x => a x i) μ) : |
| Lp (Hilbert) 2 μ →L[ℂ] Lp (Hilbert) 2 μ := |
| (fieldDiagonalLinear a C hC ha hm).mkContinuous C (by |
| intro f |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| change ∀ᵐ x ∂μ, ‖fieldDiagonalVec a C hC ha hm f x‖ ≤ C * ‖f x‖ |
| filter_upwards [fieldDiagonalVec_ae a C hC ha hm f] with x hx |
| rw [hx] |
| exact diagonal_bound (a x) C (ha x) (f x)) |
|
|
| end Invariant233.WeightedShift |
|
|
| end |
|
|
| end OAI |
|
|
