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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 54 55 56 57 58 59 60 61 62 | import OAI.Geometry.RCD.Metric.CompactFactorL2
namespace OAI
/-! # Nonnegative initial data for positive compact heat factors -/
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_nonnegative_L2 {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)) (A : ℝ≥0), 0≤f ∧
(∀ᵐ x ∂hausdorff n X, |f x|≤A) ∧
∀ (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 m := hausdorff n X
have hS : IsCompact S := isCompact_iff_compactSpace.mpr inferInstance
let f₀ := fun x => ρ x*compactFactorExtension F x
have hm : MemLp f₀ 2 m := compact_weighted_density_memLp hS F m hρ hC
let f := hm.toLp f₀
have hf : f =ᵐ[m] f₀ := hm.coeFn_toLp
have hf0 : 0≤f := by
apply (Lp.coeFn_nonneg _).mp
filter_upwards [hf] with x hx
rw [hx]
apply mul_nonneg (hρ0 x)
by_cases hs : x∈S
· exact (compactFactorExtension_coe F ⟨x,hs⟩) ▸ hF ⟨x,hs⟩
· rw [compactFactorExtension_off F hs]
obtain ⟨A,hA⟩ := compact_weighted_density_bounded hS F m hC
refine ⟨f,A,hf0,?_,?_⟩
· filter_upwards [hf,hA] with x hx hax
exact hx ▸ hax
· intro t ht x
change (∫ a, hX.canonicalHeatKernel hn p t x a*F a ∂ν) = _
calc
_ = ∫ y, hX.canonicalHeatKernel hn p t x y*f₀ y ∂m :=
compact_weighted_density_integral hS.isClosed.measurableSet m ν ρ hρ hρ0 hν F _
_ = ∫ y, hX.canonicalHeatKernel hn p t x y*f y ∂m := by
apply integral_congr_ae
filter_upwards [hf] with y hy
rw [hy]
_ = _ := (kernelHeat_eq_integral m hX.quadratic
(hX.canonicalHeatKernel_isContinuousHeatKernel hn p) ht f x).symm
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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