| import OAI.Geometry.RCD.Heat.KernelDensityTime | |
| import OAI.Geometry.RCD.Sobolev.CutoffErrorEstimates | |
| namespace OAI | |
| /-! # Genuine time integrability for compact L1 sources -/ | |
| open Set Metric MeasureTheory Filter | |
| open scoped ENNReal | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] | |
| (μ : Measure X) [μ.Regular] (hQ : QuadraticCheegerEnergy μ) | |
| {ker : ℝ → X → X → ℝ} (hk : IsContinuousHeatKernel μ hQ ker) | |
| include hk in | |
| /-- The spatial average makes any compact L1 source time-integrable at | |
| zero, using only its finite mass and the exact inverse-volume bound. -/ | |
| theorem average_density_intervalIntegrable_zero (A : Set X) | |
| (hA : MeasurableSet A) (hAt : μ A ≠ ∞) | |
| {g : X → ℝ} (hg : Integrable g μ) {K : Set X} (hK : IsCompact K) | |
| (hz : ∀ᵐ z ∂μ, z ∉ K → g z = 0) {T : ℝ} (hT : 0 ≤ T) : | |
| IntervalIntegrable (fun t => ∫ z, kernelAverage μ ker t A z * g z ∂μ) volume 0 T := by | |
| let f := fun t => ∫ z, kernelAverage μ ker t A z * g z ∂μ | |
| have hc : ContinuousOn f (Ioi 0) := average_density_continuousOn_time μ hQ hk A hA hAt hg hK hz | |
| let D := (μ.real A)⁻¹ * ∫ z, |g z| ∂μ | |
| have hb (t : ℝ) (ht : 0 < t) : |f t| ≤ D := by | |
| apply abs_integral_mul_le_of_bound_on μ hg (inv_nonneg.mpr (measureReal_nonneg)) | |
| (S := univ) (Eventually.of_forall fun _ hn => (hn (mem_univ _)).elim) | |
| intro z _ | |
| rw [abs_of_nonneg (kernelAverage_bounds μ hQ hk ht A z).1] | |
| exact (kernelAverage_bounds μ hQ hk ht A z).2 | |
| rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hT] | |
| apply IntegrableOn.of_bound measure_Ioc_lt_top | |
| ((hc.mono (fun _ ht => ht.1)).aestronglyMeasurable measurableSet_Ioc) D | |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht | |
| simpa only [Real.norm_eq_abs] using hb t ht.1 | |
| include hk in | |
| /-- The difference pairing is integrable on every compact positive-time interval. -/ | |
| theorem difference_density_intervalIntegrable (A : Set X) | |
| (hA : MeasurableSet A) (hbounded : Bornology.IsBounded A) (x : X) | |
| {g : X → ℝ} (hg : Integrable g μ) {K : Set X} (hK : IsCompact K) | |
| (hz : ∀ᵐ z ∂μ, z ∉ K → g z = 0) {s t : ℝ} (hs : 0 < s) (hst : s ≤ t) : | |
| IntervalIntegrable (fun τ => ∫ z, (kernelAverage μ ker τ A z - ker τ x z) * g z ∂μ) | |
| volume s t := | |
| ((difference_density_continuousOn_time μ hQ hk A hA hbounded x hg hK hz).mono | |
| (fun _ ht => hs.trans_le ht.1)).intervalIntegrable_of_Icc hst | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |