| 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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|
