| import OAI.Geometry.RCD.Heat.GaussianIntegralEnvelope |
| import OAI.Geometry.RCD.Heat.KernelMeasureBounds |
|
|
| namespace OAI |
|
|
| /-! # Shell estimates for the actual averaged Bochner heat mass -/ |
|
|
| open Set Metric MeasureTheory |
| open scoped BigOperators 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) (hsχ : HasCompactSupport χ) |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| / |
| is preserved when the spatial kernel is averaged first. -/ |
| theorem cutoffAverageHeatMass_integral (M : PositiveBilinearMeasure X E) |
| {t : ℝ} (ht : 0 < t) (s : Set X) (hs : MeasurableSet s) |
| (hbounded : Bornology.IsBounded s) (v : E) : |
| ⟪v, cutoffAverageHeatMass μ hQ hk hχ hsχ M ht s hs hbounded v⟫_ℝ = |
| ∫ z, χ z * kernelAverage μ ker t s z ∂(M.measure v) := by |
| rw [cutoffAverageHeatMass, M.weightedOperator_energy] |
| rfl |
|
|
| / |
| pointwise source, on the original full Bochner measure. -/ |
| theorem scaled_cutoffAverageHeatMass_le_geometric (M : PositiveBilinearMeasure X E) |
| (m : ℕ) (p : X) {r t C c T : ℝ} (hr : 0 < r) (ht : 0 < t) |
| (s : Set X) (hs : MeasurableSet s) (hbounded : Bornology.IsBounded s) |
| (hC : 0 ≤ C) (hc : 0 < c) (hT : 0 ≤ T) (N : ℕ) |
| (hlocal : ∀ j ≤ N, ∀ z ∈ dyadicShell p r j, |
| χ z * kernelAverage μ ker t s z ≤ (C / r ^ (m + 2)) * gaussianShellFactor c j) |
| (htail : ∀ z ∉ ball p (2 ^ N * r), χ z * kernelAverage μ ker t s z ≤ T) (v : E) : |
| r ^ 2 * ⟪v, cutoffAverageHeatMass μ hQ hk hχ hsχ M ht s hs hbounded v⟫_ℝ ≤ |
| C * gaussianMassConvolutionConstant m c * |
| (∑ j ∈ Finset.range (N + 1), (3 / 4 : ℝ) ^ j * |
| (((2 ^ j * r) ^ 2 / (2 ^ j * r) ^ (m + 2)) * |
| (M.measure v).real (ball p (2 ^ j * r)))) + |
| r ^ 2 * T * (M.measure v).real univ := by |
| rw [cutoffAverageHeatMass_integral] |
| apply scaled_integral_le_geometric_mass (M.measure v) m p hr |
| ((hχ.mul (kernelAverage_continuous μ hQ hk ht hs hbounded)).integrable_of_hasCompactSupport |
| hsχ.mul_right) hC hc hT N hlocal htail |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
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|
