File size: 3,484 Bytes
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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Geometry.RCD.Metric.CompactFactorTimeTail
import OAI.Geometry.RCD.Heat.GaussianValueIntegrable
import OAI.Geometry.RCD.Metric.CompactFactorNonnegativeL2
import OAI.Geometry.RCD.Coordinates.KernelHeatBoundedRegularity
namespace OAI
/-! # A common integrable bound for compact heat factors on a time window -/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped NNReal ENNReal
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_factor_integrable_envelope {S : Set X} [CompactSpace S]
(ν : Measure S) [IsProbabilityMeasure ν] (ρ : X → ℝ)
(hm : Measurable ρ) (hρ : ∀ x, 0 ≤ ρ x) {B : ℝ≥0}
(hB : ∀ᵐ x ∂hausdorff n X, |ρ x| ≤ B)
(hν : ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x)))
(F : C(S,ℝ)) (hF : ∀ y, 0 < F y) (z : X) {R a b : ℝ}
(hR : 1 ≤ R) (hS : ∀ y : S, dist z y ≤ R)
(ha : 0 < a) (hab : a ≤ b) (hb : b ≤ 1) :
∃ H : X → ℝ, Integrable H (hausdorff n X) ∧ (∀ x, 0 ≤ H x) ∧
∀ (t : ℝ) (hat : a ≤ t) (_htb : t ≤ b) (x : X),
|hX.compactHeatFactor hn p ν F t (ha.trans_le hat) x| ≤ H x := by
let m := hausdorff n X
let Q := hX.quadratic
let hk := hX.canonicalHeatKernel_isContinuousHeatKernel hn p
obtain ⟨f,A,_hf0,hf,hval⟩ := hX.compact_heat_factor_nonnegative_L2 hn p ν ρ hm hρ hB hν F
(fun y => (hF y).le)
obtain ⟨C,hC,htail⟩ := hX.compact_factor_time_tail hn p ν F (fun y => (hF y).le)
z hR hS ha hab hb
obtain ⟨V,D,hV,hD,hvol⟩ := hX.exponential_volume K n hn z
have hb0 : 0 < b := ha.trans_le hab
let g := fun x : X => Real.exp (-(dist x z)^2/(20*b))
have hg : Integrable g m := by
have he : g = fun x => Real.exp (-(1/(20*b))*(dist z x)^2) := by
funext x
dsimp only [g]
rw [dist_comm x z]
congr 1
ring
rw [he]
simpa only [dist_comm] using
(gaussian_integrable m z (c := 1/(20*b)) hV hD (by positivity) hvol)
let H := fun x => (closedBall z (2*R)).indicator (fun _ => (A:ℝ)) x+C*g x
have hAi : IntegrableOn (fun _ : X => (A:ℝ)) (closedBall z (2*R)) m :=
continuous_const.continuousOn.integrableOn_compact (isCompact_closedBall z (2*R))
have hHi : Integrable H m :=
((integrable_indicator_iff measurableSet_closedBall).mpr hAi).add (hg.const_mul C)
have hH0 (x : X) : 0 ≤ H x := by
apply add_nonneg
· exact indicator_nonneg (fun _ _ => A.coe_nonneg) x
· exact mul_nonneg hC (Real.exp_pos _).le
refine ⟨H,hHi,hH0,?_⟩
intro t hat htb x
have ht := ha.trans_le hat
by_cases hx : x ∈ closedBall z (2*R)
· have hbound := (hX.kernelHeat_bounded_regularity hn p hk f A hf t ht).2 x
rw [← hval t ht x] at hbound
dsimp only [H]
rw [indicator_of_mem hx]
exact hbound.trans (le_add_of_nonneg_right (mul_nonneg hC (Real.exp_pos _).le))
· have hfar : 2*R ≤ dist x z := le_of_lt (lt_of_not_ge hx)
have hbound := htail t hat htb x hfar
rw [← abs_of_pos (hX.compactHeatFactor_positive hn p ν F hF t ht x)] at hbound
dsimp only [H,g]
rw [indicator_of_notMem hx,zero_add]
exact hbound
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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