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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 43 44 45 46 47 48 49 | import OAI.Geometry.RCD.Heat.KernelHeatInitialUniqueness
import OAI.Geometry.RCD.Metric.CompactFactorL2
namespace OAI
/-! # Identification of the initial datum of a compact heat factor -/
open Set Filter MeasureTheory TopologicalSpace
open scoped ENNReal NNReal
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_initial_value {S : Set X} [CompactSpace S]
(ν : Measure S) (ρ : X → ℝ) (hρ : Measurable ρ) {C : ℝ≥0}
(hC : ∀ᵐ x ∂hausdorff n X, |ρ x| ≤ C)
(F : C(S,ℝ)) (f : Lp ℝ 2 (hausdorff n X))
(hrep : ∀ t : ℝ, 0 < t → ∀ x,
kernelHeat (hausdorff n X) hX.quadratic
(hX.canonicalHeatKernel_isContinuousHeatKernel hn p) t f x =
∫ a, hX.canonicalHeatKernel hn p t x a*F a ∂ν)
(hν : ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x)))
(hρ0 : ∀ x, 0 ≤ ρ x) :
f =ᵐ[hausdorff n X] fun x => ρ x*compactFactorExtension F x := by
let m := hausdorff n X
let Q := hX.quadratic
let hk := hX.canonicalHeatKernel_isContinuousHeatKernel hn p
have hS : IsCompact S := isCompact_iff_compactSpace.mpr inferInstance
have hm := compact_weighted_density_memLp hS F m hρ hC
let g := hm.toLp _
have hg : g =ᵐ[m] fun x => ρ x*compactFactorExtension F x := hm.coeFn_toLp
have hfg : f = g := kernelHeat_initial_unique m Q hk f g (by
intro t ht x
rw [hrep t ht x,kernelHeat_eq_integral m Q hk ht g x]
rw [compact_weighted_density_integral hS.measurableSet m ν ρ hρ hρ0 hν F]
apply integral_congr_ae
filter_upwards [hg] with y hy
rw [hy])
exact hfg ▸ hg
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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