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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 | import OAI.Geometry.RCD.Metric.CompactFactorL2
namespace OAI
/-! # Time derivatives of the actual compact endpoint heat factors -/
open Set Filter MeasureTheory TopologicalSpace
open scoped ENNReal NNReal Topology
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X)
(hn : 2 ≤ n) (p : X) [SeparableSpace X]
[IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular]
theorem compact_heat_factor_derivative {S : Set X} [CompactSpace S]
(ν : Measure S) [IsProbabilityMeasure ν] (ρ : X → ℝ)
(hρ : Measurable ρ) (hρ0 : ∀ x, 0 ≤ ρ x) {C : ℝ≥0}
(hC : ∀ᵐ x ∂hausdorff n X, |ρ x| ≤ C)
(hν : ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x)))
(F : C(S, ℝ)) :
∃ f : Lp ℝ 2 (hausdorff n X), HasBoundedAmplitude f ∧
(∀ (t : ℝ) (ht : 0 < t) (x : X), hX.compactHeatFactor hn p ν F t ht x =
kernelHeat (hausdorff n X) hX.quadratic (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) t f x) ∧
∀ (t : ℝ), 0 < t → ∀ x : X,
HasDerivAt (fun s => ∫ a, hX.canonicalHeatKernel hn p s x a*F a ∂ν)
(kernelHeatGeneratorSource (hausdorff n X) hX.quadratic
(hX.canonicalHeatKernel_isContinuousHeatKernel hn p) t f x) t := by
obtain ⟨f,hf,he⟩ := hX.compact_heat_factor_L2 hn p ν ρ hρ hρ0 hC hν F
refine ⟨f,hf,he,?_⟩
intro t ht x
apply (kernelHeat_generator_hasDerivAt (hausdorff n X) hX.quadratic
(hX.canonicalHeatKernel_isContinuousHeatKernel hn p) f ht x).congr_of_eventuallyEq
filter_upwards [Ioi_mem_nhds ht] with s hs
exact he s hs x
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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