| import OAI.Geometry.RCD.Coordinates.CoordinateHeatInput |
| import OAI.Geometry.RCD.Metric.ShortAverageMass |
| import OAI.Geometry.RCD.Heat.HeatBlockIntegrals |
|
|
| namespace OAI |
|
|
| /-! # Actual short-time source integrals -/ |
|
|
| open Set Metric MeasureTheory |
| open scoped InnerProductSpace ENNReal BoundedContinuousFunction |
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| 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) (hcompact : HasCompactSupport χ) |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| / |
| def cutoffAverageHeatMassAt (M : PositiveBilinearMeasure X E) |
| (s : Set X) (hs : MeasurableSet s) (hbounded : Bornology.IsBounded s) (t : ℝ) : E →L[ℝ] E := |
| if ht : 0 < t then cutoffAverageHeatMass μ hQ hk hχ hcompact M ht s hs hbounded else 0 |
|
|
| theorem cutoffAverageHeatMassAt_of_pos (M : PositiveBilinearMeasure X E) |
| (s : Set X) (hs : MeasurableSet s) (hbounded : Bornology.IsBounded s) |
| {t : ℝ} (ht : 0 < t) : |
| cutoffAverageHeatMassAt μ hQ hk hχ hcompact M s hs hbounded t = |
| cutoffAverageHeatMass μ hQ hk hχ hcompact M ht s hs hbounded := dite_eq_left ht |
|
|
| theorem cutoffAverageHeatMassAt_nonneg (M : PositiveBilinearMeasure X E) |
| (hχpos : ∀ z, 0 ≤ χ z) (s : Set X) (hs : MeasurableSet s) |
| (hbounded : Bornology.IsBounded s) (t : ℝ) (v : E) : |
| 0 ≤ ⟪v, cutoffAverageHeatMassAt μ hQ hk hχ hcompact M s hs hbounded t v⟫_ℝ := by |
| by_cases ht : 0 < t |
| · rw [cutoffAverageHeatMassAt_of_pos μ hQ hk hχ hcompact M s hs hbounded ht, |
| cutoffAverageHeatMass_integral] |
| exact integral_nonneg fun z => mul_nonneg (hχpos z) (kernelAverage_bounds μ hQ hk ht s z).1 |
| · simp only [cutoffAverageHeatMassAt, dite_eq_right ht, zero_apply, inner_zero_right, le_refl] |
|
|
| / |
| Bochner envelope, with the same uniform constant as the pointwise estimate. -/ |
| theorem exists_short_average_mass_integral_bound (hg : HasCoordinateHeatBounds μ ker) |
| (m : ℕ) {A V : ℝ} (hA : 0 ≤ A) (hV : 0 < V) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (M : PositiveBilinearMeasure X (Euclidean (m + 2))) |
| (R : ℝ) (_hR : 0 < R), R ≤ 1 → ∀ (x : X) (χ : X →ᵇ ℝ), |
| ∀ hc : HasCompactSupport (χ : X → ℝ), (∀ z, 0 ≤ χ z ∧ χ z ≤ 1) → |
| (∀ z ∉ ball x R, χ z = 0) → ∀ (i : ℕ) (q : X), |
| μ (ball q (R / 2 ^ i)) ≠ ∞ → |
| V * (R / 2 ^ i) ^ (m + 2) ≤ μ.real (ball q (R / 2 ^ i)) → |
| dist x q ≤ A * (R / 2 ^ i) → |
| (∀ t, 0 < t → t ≤ (R / 2 ^ i) ^ 2 → |
| ∀ z, χ z ≠ 0 → V * (Real.sqrt t) ^ (m + 2) ≤ μ.real (ball z (Real.sqrt t))) → |
| ∀ v : Euclidean (m + 2), |
| |∫ t in (0 : ℝ)..(R / 2 ^ i) ^ 2, |
| ⟪v, cutoffAverageHeatMassAt μ hQ hk χ.continuous hc M |
| (ball q (R / 2 ^ i)) measurableSet_ball isBounded_ball t v⟫_ℝ| ≤ |
| C * ⟪v, dyadicBochnerEnvelope M (m + 2) x R i v⟫_ℝ := by |
| obtain ⟨C, hC, hb⟩ := exists_short_average_mass_bound μ hQ hk hg m hA hV |
| refine ⟨C, hC, ?_⟩ |
| intro M R hR hR1 x χ hc hχ hzero i q hfinite hvol hcenter hsource v |
| have hs : 0 < R / (2 : ℝ) ^ i := div_pos hR (by positivity) |
| apply short_time_integral_norm_le hs |
| (mul_nonneg hC.le ((dyadicBochnerEnvelope_isPositive M (m + 2) x hR.le i).inner_nonneg_right v)) |
| intro t ht hti |
| rw [abs_of_nonneg (cutoffAverageHeatMassAt_nonneg μ hQ hk χ.continuous hc M |
| (fun z => (hχ z).1) _ measurableSet_ball isBounded_ball t v), |
| cutoffAverageHeatMassAt_of_pos μ hQ hk χ.continuous hc M _ measurableSet_ball isBounded_ball ht] |
| exact hb M R hR hR1 x χ hc hχ hzero i q hfinite hvol hcenter t ht hti (hsource t ht hti) v |
|
|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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|
