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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 50 51 52 53 | import OAI.Geometry.RCD.Metric.CompactFactorL2
namespace OAI
/-! # Nonnegative initial data for the positive compact heat factors
Positivity at every positive heat time passes to the original L² datum by
strong continuity at zero.
-/
open Set Filter MeasureTheory TopologicalSpace
open scoped NNReal ENNReal 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_nonnegative_data {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)) (hF : ∀ a, 0<F a) :
∃ f : Lp ℝ 2 (hausdorff n X), 0≤f ∧ 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 := by
let μ := hausdorff n X
let Q := hX.quadratic
let hk := hX.canonicalHeatKernel_isContinuousHeatKernel hn p
obtain ⟨f,hf,he⟩ := hX.compact_heat_factor_L2 hn p ν ρ hρ hρ0 hC hν F
have htpos (t : ℝ) (ht : 0<t) : 0≤heatAt μ Q t f := by
apply (Lp.coeFn_nonneg _).mp
filter_upwards [kernelHeat_represents_L2 μ Q hk ht f] with x hx
rw [←hx,←he t ht x]
exact (hX.compactHeatFactor_positive hn p ν F hF t ht x).le
let e : ℕ → ℝ := fun i => 1/(i+1:ℝ)
have he0 : Tendsto e atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
have ht : Tendsto (fun i => heatAt μ Q (e i) f) atTop (𝓝 f) := by
have hh := ((heatAt_continuous μ Q f).tendsto 0).comp he0
rw [heatAt_zero,one_apply_eq_self] at hh
exact hh
have hf0 : 0≤f := ge_of_tendsto ht (.of_forall fun i => htpos (e i) (by dsimp only [e]; positivity))
exact ⟨f,hf0,hf,he⟩
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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