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9eeb9c7 | 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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
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