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e4d9d2b | 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 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Geometry.RCD.Heat.CompactFactorTimeRatio
import OAI.Geometry.RCD.Coordinates.CompactFactorRegularity
import OAI.Geometry.RCD.Heat.HeatHamiltonPositive
namespace OAI
/-! # A normalized midpoint datum for each actual compact heat factor
The amplitude is the supremum of the earlier heat factor. Kernel
comparison controls this same supremum at every observation point.
-/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped NNReal ENNReal Topology BoundedContinuousFunction
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_midpoint :
∃ C D : ℝ, 0≤C ∧ 0<D ∧
∀ {S : Set X} [CompactSpace S] (ν : Measure S) [IsProbabilityMeasure ν]
(ρ : X → ℝ) (hρ : Measurable ρ) (hρ0 : ∀ x, 0≤ρ x) {B : ℝ≥0},
(∀ᵐ x ∂hausdorff n X, |ρ x|≤B) →
ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x)) →
∀ (F : C(S, ℝ)), (∀ a, 0<F a) → ∀ (t : ℝ) (ht : 0<t), t≤1/2 →
∃ (g : Lp ℝ 2 (hausdorff n X)) (A : ℝ≥0), 0<(A:ℝ) ∧ 0≤g ∧
(∀ᵐ x ∂hausdorff n X, |g x|≤A) ∧
(∀ (s : ℝ) (hs : 0<s) (x : X),
kernelHeat (hausdorff n X) hX.quadratic
(hX.canonicalHeatKernel_isContinuousHeatKernel hn p) s g x =
hX.compactHeatFactor hn p ν F (t+s) (add_pos ht hs) x) ∧
∀ (x : X) (R : ℝ), 0≤R → (∀ a : S, dist x a≤R) →
Real.log A-Real.log (hX.compactHeatFactor hn p ν F (2*t) (by positivity) x)≤
C+D*R^2/t := by
obtain ⟨Q,D,hQ,hD,hcomp⟩ := hX.compact_heat_factor_time_ratio hn p
refine ⟨|Real.log Q|,D,abs_nonneg _,hD,?_⟩
intro S _ ν _ ρ hρ hρ0 B hB hν F hF t ht ht1
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 hB hν F
obtain ⟨E,hE⟩ := hf
let H := hX.compactHeatFactor hn p ν F t ht
have hHb (x : X) : |H x|≤E := by
rw [show H x=kernelHeat μ q hk t f x from he t ht x]
exact (hX.kernelHeat_bounded_regularity hn p hk f E hE t ht).2 x
let V : X →ᵇ ℝ := BoundedContinuousFunction.ofNormedAddCommGroup H H.continuous E
(fun x => by simpa only [Real.norm_eq_abs] using hHb x)
let g := heatAt μ q t f
have hrep : H =ᵐ[μ] g :=
(show H =ᵐ[μ] kernelHeat μ q hk t f from .of_forall (he t ht)).trans
(kernelHeat_represents_L2 μ q hk ht f)
have hV0 : 0<‖V‖ := lt_of_lt_of_le (by
change 0<‖H p‖
rw [Real.norm_eq_abs,abs_of_pos (hX.compactHeatFactor_positive hn p ν F hF t ht p)]
exact hX.compactHeatFactor_positive hn p ν F hF t ht p) (V.norm_coe_le_norm p)
have hg0 : 0≤g := by
apply (Lp.coeFn_nonneg _).mp
filter_upwards [hrep] with x hx
rw [←hx]
exact (hX.compactHeatFactor_positive hn p ν F hF t ht x).le
have hgA : ∀ᵐ x ∂μ, |g x|≤‖V‖₊ := by
filter_upwards [hrep] with x hx
rw [←hx]
exact V.norm_coe_le_norm x
refine ⟨g,‖V‖₊,hV0,hg0,hgA,?_,?_⟩
· intro s hs x
rw [←kernelHeat_add μ q hk ht hs f x]
exact (he (t+s) (add_pos ht hs) x).symm
· intro x R hR hdist
have hv := hX.compactHeatFactor_positive hn p ν F hF (2*t) (by positivity) x
have hnorm : ‖V‖≤Q*Real.exp (D*R^2/t)*hX.compactHeatFactor hn p ν F (2*t) (by positivity) x := by
apply (BoundedContinuousFunction.norm_le (by positivity)).mpr
intro y
change ‖H y‖≤_
rw [Real.norm_eq_abs,abs_of_pos (hX.compactHeatFactor_positive hn p ν F hF t ht y)]
exact hcomp ν F (fun a => (hF a).le) t ht ht1 x R hR hdist y
have hl := Real.log_le_log hV0 hnorm
rw [Real.log_mul (mul_pos hQ (Real.exp_pos _)).ne' hv.ne',
Real.log_mul hQ.ne' (Real.exp_ne_zero _),Real.log_exp] at hl
change Real.log ‖V‖-Real.log _≤_
linarith only [hl,le_abs_self (Real.log Q)]
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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