| import Mathlib |
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| namespace OAI |
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| /-! Compression. -/ |
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| noncomputable section |
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| open MeasureTheory |
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| open scoped InnerProductSpace |
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| namespace CrouzeixHilbert.Compression |
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| section Multiplication |
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| variable {α E : Type*} [MeasurableSpace α] {μ : Measure α} |
| [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
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| def fieldAction (L : α → E →L[ℂ] E) (hL : AEStronglyMeasurable L μ) |
| (C : ℝ) (hC : 0 ≤ C) (hLn : ∀ᵐ t ∂μ, ‖L t‖ ≤ C) : |
| Lp E 2 μ →L[ℂ] Lp E 2 μ := by |
| let mem : ∀ u : Lp E 2 μ, MemLp (fun t => L t (u t)) 2 μ := fun u => |
| ((Lp.memLp u).norm.const_mul C).mono |
| ((show Continuous (fun z : (E →L[ℂ] E) × E => z.1 z.2) by fun_prop).comp_aestronglyMeasurable (hL.prodMk (Lp.aestronglyMeasurable u))) |
| (hLn.mono fun t ht => by |
| simp only [Real.norm_eq_abs, abs_mul, abs_of_nonneg hC, abs_norm] |
| exact ((L t).le_opNorm _).trans (mul_le_mul_of_nonneg_right ht (norm_nonneg _))) |
| let M : Lp E 2 μ →ₗ[ℂ] Lp E 2 μ := { |
| toFun u := (mem u).toLp _ |
| map_add' u v := by |
| apply Lp.ext |
| filter_upwards [(mem (u + v)).coeFn_toLp, (mem u).coeFn_toLp, |
| (mem v).coeFn_toLp, Lp.coeFn_add u v, |
| Lp.coeFn_add ((mem u).toLp _) ((mem v).toLp _)] with t huv hu hv ha hb |
| simp only [huv, hb, hu, hv, ha, map_add, Pi.add_apply] |
| map_smul' c u := by |
| apply Lp.ext |
| filter_upwards [(mem (c • u)).coeFn_toLp, (mem u).coeFn_toLp, |
| Lp.coeFn_smul c u, Lp.coeFn_smul c ((mem u).toLp _)] with t hcu hu ha hb |
| simp only [hcu, hb, hu, ha, map_smul, Pi.smul_apply, RingHom.id_apply] } |
| apply M.mkContinuous C |
| intro u |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| filter_upwards [(mem u).coeFn_toLp, hLn] with t ht hn |
| change ‖((mem u).toLp _) t‖ ≤ C * ‖u t‖ |
| rw [ht] |
| exact ((L t).le_opNorm _).trans (mul_le_mul_of_nonneg_right hn (norm_nonneg _)) |
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| theorem fieldAction_apply_ae (L : α → E →L[ℂ] E) (hL : AEStronglyMeasurable L μ) |
| (C : ℝ) (hC : 0 ≤ C) (hLn : ∀ᵐ t ∂μ, ‖L t‖ ≤ C) (u : Lp E 2 μ) : |
| fieldAction L hL C hC hLn u =ᵐ[μ] fun t => L t (u t) := by |
| unfold fieldAction |
| dsimp only [LinearMap.mkContinuous_apply, LinearMap.coe_mk, AddHom.coe_mk] |
| exact MemLp.coeFn_toLp _ |
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| theorem norm_fieldAction_le (L : α → E →L[ℂ] E) (hL : AEStronglyMeasurable L μ) |
| (C : ℝ) (hC : 0 ≤ C) (hLn : ∀ᵐ t ∂μ, ‖L t‖ ≤ C) : |
| ‖fieldAction L hL C hC hLn‖ ≤ C := by |
| apply ContinuousLinearMap.opNorm_le_bound _ hC |
| intro u |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| filter_upwards [fieldAction_apply_ae L hL C hC hLn u, hLn] with t ht hn |
| rw [ht] |
| exact ((L t).le_opNorm _).trans (mul_le_mul_of_nonneg_right hn (norm_nonneg _)) |
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| variable [CompleteSpace E] |
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| theorem adjoint_fieldAction (L M : α → E →L[ℂ] E) |
| (hL : AEStronglyMeasurable L μ) (hM : AEStronglyMeasurable M μ) |
| (C : ℝ) (hC : 0 ≤ C) |
| (hLn : ∀ᵐ t ∂μ, ‖L t‖ ≤ C) (hMn : ∀ᵐ t ∂μ, ‖M t‖ ≤ C) |
| (ha : ∀ᵐ t ∂μ, ContinuousLinearMap.adjoint (L t) = M t) : |
| ContinuousLinearMap.adjoint (fieldAction L hL C hC hLn) = |
| fieldAction M hM C hC hMn := by |
| symm |
| apply (ContinuousLinearMap.eq_adjoint_iff _ _).mpr |
| intro u v |
| simp only [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [fieldAction_apply_ae M hM C hC hMn u, |
| fieldAction_apply_ae L hL C hC hLn v, ha] with t hu hv ht |
| rw [hu, hv, ← ht, ContinuousLinearMap.adjoint_inner_left] |
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| end Multiplication |
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| section Isometry |
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| variable {α E : Type*} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] |
| [CompactSpace α] [SecondCountableTopology α] {μ : Measure α} [IsFiniteMeasure μ] |
| [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] |
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| def orbit (R : C(α, E →L[ℂ] E)) : E →ₗ[ℂ] C(α, E) where |
| toFun x := ⟨fun t => R t x, R.continuous.clm_apply continuous_const⟩ |
| map_add' x y := by ext t; exact (R t).map_add x y |
| map_smul' c x := by ext t; exact (R t).map_smul c x |
| |
| def lift (R : C(α, E →L[ℂ] E)) : E →ₗ[ℂ] Lp E 2 μ := |
| (ContinuousMap.toLp 2 μ ℂ).toLinearMap.comp (orbit R) |
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| omit [CompleteSpace E] in |
| theorem lift_apply_ae (R : C(α, E →L[ℂ] E)) (x : E) : |
| lift (μ := μ) R x =ᵐ[μ] fun t => R t x := |
| ContinuousMap.coeFn_toLp μ (orbit R x) |
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| theorem inner_lift (R : C(α, E →L[ℂ] E)) (x y : E) : |
| ⟪lift (μ := μ) R x, lift (μ := μ) R y⟫_ℂ = |
| ⟪x, (∫ t, star (R t) * R t ∂μ) y⟫_ℂ := by |
| have hi : Integrable (fun t => star (R t) * R t) μ := |
| (R.continuous.star.mul R.continuous).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| rw [L2.inner_def, ContinuousLinearMap.integral_apply hi, |
| ← integral_inner (𝕜 := ℂ) (f := fun t => (star (R t) * R t) y) |
| ((ContinuousLinearMap.apply ℂ E y).integrable_comp hi) x] |
| apply integral_congr_ae |
| filter_upwards [lift_apply_ae R x, lift_apply_ae R y] with t hx hy |
| rw [hx, hy] |
| exact (ContinuousLinearMap.adjoint_inner_right (R t) x (R t y)).symm |
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| def isometry (R : C(α, E →L[ℂ] E)) (hmass : ∫ t, star (R t) * R t ∂μ = 1) : |
| E →ₗᵢ[ℂ] Lp E 2 μ := |
| (lift R).isometryOfInner fun x y => by rw [inner_lift, hmass]; rfl |
| |
| @[simp] theorem isometry_apply (R : C(α, E →L[ℂ] E)) |
| (hmass : ∫ t, star (R t) * R t ∂μ = 1) (x : E) : |
| isometry R hmass x = lift R x := rfl |
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| theorem integral_eq_compression (R L : C(α, E →L[ℂ] E)) |
| (hmass : ∫ t, star (R t) * R t ∂μ = 1) (C : ℝ) (hC : 0 ≤ C) |
| (hLn : ∀ t, ‖L t‖ ≤ C) : |
| (∫ t, star (R t) * L t * R t ∂μ) = |
| (isometry R hmass).toContinuousLinearMap.adjoint.comp |
| ((fieldAction L L.continuous.aestronglyMeasurable C hC |
| (Filter.Eventually.of_forall hLn)).comp (isometry R hmass).toContinuousLinearMap) := by |
| have hi : Integrable (fun t => star (R t) * L t * R t) μ := |
| ((R.continuous.star.mul L.continuous).mul R.continuous).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| ext y |
| apply ext_inner_left ℂ |
| intro x |
| rw [ContinuousLinearMap.integral_apply hi, |
| ← integral_inner (𝕜 := ℂ) (f := fun t => (star (R t) * L t * R t) y) |
| ((ContinuousLinearMap.apply ℂ E y).integrable_comp hi) x] |
| simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.adjoint_inner_right, |
| LinearIsometry.coe_toContinuousLinearMap, isometry_apply, L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [lift_apply_ae R x, |
| fieldAction_apply_ae L L.continuous.aestronglyMeasurable C hC |
| (Filter.Eventually.of_forall hLn) (lift R y), lift_apply_ae R y] with t hx hL hy |
| rw [hx, hL, hy] |
| exact ContinuousLinearMap.adjoint_inner_right (R t) x (L t (R t y)) |
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| theorem norm_integral_le (R L : C(α, E →L[ℂ] E)) |
| (hmass : ∫ t, star (R t) * R t ∂μ = 1) (C : ℝ) (hC : 0 ≤ C) |
| (hLn : ∀ t, ‖L t‖ ≤ C) : |
| ‖∫ t, star (R t) * L t * R t ∂μ‖ ≤ C := by |
| rw [integral_eq_compression R L hmass C hC hLn] |
| let V := (isometry R hmass).toContinuousLinearMap |
| have hn : ‖V‖ ≤ 1 := (isometry R hmass).norm_toContinuousLinearMap_le |
| calc |
| _ ≤ ‖V.adjoint‖ * (‖fieldAction L L.continuous.aestronglyMeasurable C hC |
| (Filter.Eventually.of_forall hLn)‖ * ‖V‖) := |
| (ContinuousLinearMap.opNorm_comp_le _ _).trans |
| (mul_le_mul_of_nonneg_left (ContinuousLinearMap.opNorm_comp_le _ _) (norm_nonneg _)) |
| _ ≤ 1 * (C * 1) := by |
| rw [ContinuousLinearMap.adjoint.norm_map] |
| exact mul_le_mul hn (mul_le_mul (norm_fieldAction_le _ _ _ _ _) hn |
| (norm_nonneg V) hC) (mul_nonneg (norm_nonneg _) (norm_nonneg V)) (by norm_num) |
| _ = C := by ring |
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| def densityRoot (Λ : C(α, E →L[ℂ] E)) (hΛ : ∀ t, 0 ≤ Λ t) : C(α, E →L[ℂ] E) := |
| ⟨fun t => CFC.sqrt (Λ t), |
| CFC.continuousOn_sqrt.comp_continuous Λ.continuous (fun t => hΛ t)⟩ |
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| omit [MeasurableSpace α] [BorelSpace α] [CompactSpace α] [SecondCountableTopology α] in |
| @[simp] theorem densityRoot_star_mul (Λ : C(α, E →L[ℂ] E)) (hΛ : ∀ t, 0 ≤ Λ t) |
| (t : α) : star (densityRoot Λ hΛ t) * densityRoot Λ hΛ t = Λ t := by |
| change star (CFC.sqrt (Λ t)) * CFC.sqrt (Λ t) = _ |
| rw [(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg _)).star_eq, CFC.sqrt_mul_sqrt_self _ (hΛ t)] |
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| def densityIsometry (Λ : C(α, E →L[ℂ] E)) (hΛ : ∀ t, 0 ≤ Λ t) |
| (hmass : ∫ t, Λ t ∂μ = 1) : E →ₗᵢ[ℂ] Lp E 2 μ := |
| isometry (densityRoot Λ hΛ) (by simpa only [densityRoot_star_mul] using hmass) |
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| theorem densityIsometry_apply_ae (Λ : C(α, E →L[ℂ] E)) (hΛ : ∀ t, 0 ≤ Λ t) |
| (hmass : ∫ t, Λ t ∂μ = 1) (x : E) : |
| densityIsometry Λ hΛ hmass x =ᵐ[μ] fun t => CFC.sqrt (Λ t) x := |
| lift_apply_ae (densityRoot Λ hΛ) x |
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| end Isometry |
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| end CrouzeixHilbert.Compression |
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| end |
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| end OAI |
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