| import OAI.Geometry.RCD.Heat.HeatMassTimeContinuity |
|
|
| namespace OAI |
|
|
| /-! # Integrability of the averaged positive source at the initial time -/ |
|
|
| open Set MeasureTheory Filter |
| open scoped InnerProductSpace |
|
|
| noncomputable section |
| namespace RegularPointCoordinates |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
| (μ : Measure X) [μ.Regular] [IsLocallyFiniteMeasure μ] (hQ : QuadraticCheegerEnergy μ) |
| {ker : ℝ → X → X → ℝ} (hk : IsContinuousHeatKernel μ hQ ker) |
| {χ : X → ℝ} (hχ : Continuous χ) (hc : HasCompactSupport χ) |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| /-- Averaging makes the full finite positive source integrable down to |
| zero time. The bound uses unit heat mass and retains singular source mass. -/ |
| theorem cutoffAverageHeatMassAt_intervalIntegrable_zero |
| (M : PositiveBilinearMeasure X E) (hχbounds : ∀ z, 0 ≤ χ z ∧ χ z ≤ 1) |
| (A : Set X) (hA : MeasurableSet A) (hbounded : Bornology.IsBounded A) |
| (v : E) {T : ℝ} (hT : 0 ≤ T) : |
| IntervalIntegrable (fun t => ⟪v, cutoffAverageHeatMassAt μ hQ hk hχ hc M A hA hbounded t v⟫_ℝ) |
| volume 0 T := by |
| let F := fun t => ⟪v, cutoffAverageHeatMassAt μ hQ hk hχ hc M A hA hbounded t v⟫_ℝ |
| have hcont : ContinuousOn F (Ioc 0 T) := |
| (cutoffAverageHeatMassAt_continuousOn μ hQ hk hχ hc M A hA hbounded v).mono |
| (fun _ ht => ht.1) |
| have hm : AEStronglyMeasurable F (volume.restrict (Ioc 0 T)) := |
| hcont.aestronglyMeasurable measurableSet_Ioc |
| let B := (μ.real A)⁻¹ * (M.measure v).real univ |
| have hb : ∀ᵐ t ∂volume.restrict (Ioc 0 T), ‖F t‖ ≤ B := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| rw [Real.norm_eq_abs, abs_of_nonneg (cutoffAverageHeatMassAt_nonneg μ hQ hk hχ hc M |
| (fun z => (hχbounds z).1) A hA hbounded t v)] |
| rw [cutoffAverageHeatMassAt_of_pos μ hQ hk hχ hc M A hA hbounded ht.1] |
| rw [cutoffAverageHeatMass_integral] |
| have hi : Integrable (fun z => χ z * kernelAverage μ ker t A z) (M.measure v) := |
| (hχ.mul (kernelAverage_continuous μ hQ hk ht.1 hA hbounded)).integrable_of_hasCompactSupport |
| hc.mul_right |
| calc |
| _ ≤ ∫ _z, (μ.real A)⁻¹ ∂(M.measure v) := by |
| apply integral_mono hi (integrable_const _) |
| intro z |
| exact (mul_le_of_le_one_left (kernelAverage_bounds μ hQ hk ht.1 A z).1 |
| (hχbounds z).2).trans (kernelAverage_bounds μ hQ hk ht.1 A z).2 |
| _ = B := by simp only [integral_const, smul_eq_mul, B]; ring |
| rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hT] |
| exact IntegrableOn.of_bound measure_Ioc_lt_top hm B hb |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
