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b37e936 | 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 99 100 101 | import OAI.Geometry.RCD.Heat.KernelHeatL2Representation
namespace OAI
/-! # Approximating the bounded complement in the conservative heat kernel
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.
-/
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)
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)
end RegularPointCoordinates
end
end OAI
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