| import OAI.Geometry.RCD.Heat.KernelHeatL2Representation |
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| namespace OAI |
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| /-! # Approximating the bounded complement in the conservative heat kernel |
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| The complement of an L2 function need not be in L2 on an infinite-volume |
| space. Finite-ball truncations stay in L2 and their heat values converge |
| to the exact bounded complement, by mass conservation. |
| -/ |
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| open Set Filter MeasureTheory Metric |
| open scoped ENNReal NNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] (μ : Measure X) [μ.Regular] (Q : QuadraticCheegerEnergy μ) |
| {ker : ℝ → X → X → ℝ} (hk : IsContinuousHeatKernel μ Q ker) |
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| include hk in |
| theorem bounded_heat_complement_sequence (p : X) (f : Lp ℝ 2 μ) (A : ℝ≥0) |
| (hf0 : 0 ≤ f) (hfA : ∀ᵐ x ∂μ, |f x| ≤ A) : |
| ∃ g : ℕ → Lp ℝ 2 μ, |
| (∀ k, 0 ≤ g k) ∧ (∀ k, ∀ᵐ x ∂μ, |g k x| ≤ A) ∧ |
| ∀ (t : ℝ), 0 < t → ∀ x : X, |
| Tendsto (fun k => kernelHeat μ Q hk t (g k) x) atTop |
| (𝓝 ((A:ℝ)-kernelHeat μ Q hk t f x)) := by |
| let E (k : ℕ) := closedBall p ((k:ℝ)+1) |
| have hE (k : ℕ) : MeasurableSet (E k) := measurableSet_closedBall |
| let F (k : ℕ) := (E k).indicator (fun x => (A:ℝ)-f x) |
| have hm (k : ℕ) : MemLp (F k) 2 μ := by |
| have hconst : MemLp ((E k).indicator (fun _ : X => (A:ℝ))) 2 μ := |
| memLp_indicator_const 2 (hE k) _ (Or.inr (isCompact_closedBall p ((k:ℝ)+1)).measure_lt_top.ne) |
| have he : F k = (E k).indicator (fun _ : X => (A:ℝ))-(E k).indicator (fun x => f x) := by |
| ext x |
| by_cases hx : x ∈ E k <;> simp [F,indicator_of_mem,indicator_of_notMem,hx] |
| rw [he] |
| exact hconst.sub ((Lp.memLp f).indicator (hE k)) |
| let g (k : ℕ) := (hm k).toLp (F k) |
| have hg (k : ℕ) : g k =ᵐ[μ] F k := (hm k).coeFn_toLp |
| have hb (k : ℕ) : ∀ᵐ x ∂μ, 0 ≤ F k x ∧ F k x ≤ A := by |
| filter_upwards [(Lp.coeFn_nonneg _).mpr hf0,hfA] with x h0 hA |
| by_cases hx : x ∈ E k |
| · rw [show F k x=(A:ℝ)-f x from indicator_of_mem hx _] |
| exact ⟨sub_nonneg.mpr ((le_abs_self _).trans hA),sub_le_self _ h0⟩ |
| · simp only [F,indicator_of_notMem hx] |
| exact ⟨le_rfl,A.coe_nonneg⟩ |
| refine ⟨g,?_,?_,?_⟩ |
| · intro k |
| apply (Lp.coeFn_nonneg _).mp |
| filter_upwards [hg k,hb k] with x he hx |
| rw [he] |
| exact hx.1 |
| · intro k |
| filter_upwards [hg k,hb k] with x he hx |
| rw [he,abs_of_nonneg hx.1] |
| exact hx.2 |
| · intro t ht x |
| have hi : Integrable (fun y => ker t x y*(A:ℝ)) μ := (hk.integrable t ht x).mul_const A |
| have hmeas (k : ℕ) : AEStronglyMeasurable (fun y => ker t x y*F k y) μ := |
| (hk.continuous_right t ht x).aestronglyMeasurable.mul (hm k).aestronglyMeasurable |
| have hbound (k : ℕ) : ∀ᵐ y ∂μ, ‖ker t x y*F k y‖ ≤ ker t x y*(A:ℝ) := by |
| filter_upwards [hb k] with y hy |
| rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (hk.nonneg t ht x y) hy.1)] |
| exact mul_le_mul_of_nonneg_left hy.2 (hk.nonneg t ht x y) |
| have hconv : ∀ᵐ y ∂μ, Tendsto (fun k => ker t x y*F k y) atTop |
| (𝓝 (ker t x y*((A:ℝ)-f y))) := by |
| apply Eventually.of_forall |
| intro y |
| obtain ⟨N,hN⟩ := exists_nat_gt (dist y p) |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_ge_atTop N] with k hk |
| have hy : y ∈ E k := by |
| change dist y p ≤ (k:ℝ)+1 |
| have hn : (N:ℝ) ≤ k := by exact_mod_cast hk |
| linarith |
| rw [show F k y=(A:ℝ)-f y from indicator_of_mem hy _] |
| have hlim := tendsto_integral_of_dominated_convergence |
| (fun y => ker t x y*(A:ℝ)) hmeas hi hbound hconv |
| have hprod : Integrable (fun y => ker t x y*f y) μ := by |
| have hh := (hk.integrable t ht x).bdd_mul (Lp.aestronglyMeasurable f) |
| (by simpa only [Real.norm_eq_abs] using hfA) |
| simpa only [mul_comm] using hh |
| have he : (∫ y, ker t x y*((A:ℝ)-f y) ∂μ) = (A:ℝ)-kernelHeat μ Q hk t f x := by |
| simp_rw [mul_sub] |
| rw [integral_sub hi hprod,integral_mul_const,hk.mass_one t ht x,one_mul, |
| kernelHeat_eq_integral μ Q hk ht f x] |
| rw [he] at hlim |
| apply hlim.congr' |
| apply Eventually.of_forall |
| intro k |
| dsimp only |
| rw [kernelHeat_eq_integral μ Q hk ht] |
| exact integral_congr_ae ((hg k).mono fun y hy => congrArg (fun z : ℝ => ker t x y*z) hy.symm) |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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