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cead56f | 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 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Geometry.RCD.Coordinates.CoordinateHeatInput
import OAI.Geometry.RCD.Heat.KernelGaussianDifference
import OAI.Geometry.RCD.Heat.HeatBallPowerVolume
import OAI.Geometry.RCD.Heat.GaussianAnnularGradient
namespace OAI
/-! # Uniform heat-kernel differences across a separated annulus -/
open Set Metric MeasureTheory
open scoped NNReal ENNReal
noncomputable section
namespace RegularPointCoordinates
private theorem inverse_volume_coefficient {n : ℕ} {s R V W C H : ℝ}
(hs : 0 < s) (hR : 0 < R) (hV : 0 < V) (hW : 0 < W) (hC : 0 ≤ C) (hH : 0 < H)
(hv : (s/(H*R))^n*V ≤ W) :
C/(s*W) ≤ (C*H^n/V)*((R/s)^n/s) := by
let D := H^n*(R/s)^n
have hD : 0 < D := by dsimp [D]; positivity
have he : D*(s/(H*R))^n = 1 := by
dsimp [D]
rw [← mul_pow,← mul_pow]
have hx : H*(R/s)*(s/(H*R)) = 1 := by field_simp
rw [hx,one_pow]
have hvol : V ≤ D*W := by
calc
V = D*((s/(H*R))^n*V) := by rw [← mul_assoc,he,one_mul]
_ ≤ D*W := mul_le_mul_of_nonneg_left hv hD.le
have hi : 1/W ≤ D/V := (div_le_div_iff₀ hW hV).mpr (by simpa only [one_mul] using hvol)
calc
C/(s*W) = (C/s)*(1/W) := by ring
_ ≤ (C/s)*(D/V) := mul_le_mul_of_nonneg_left hi (div_nonneg hC hs.le)
_ = _ := by dsimp [D]; ring
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X]
/-- Positive separation from a source annulus absorbs all small-time losses
in the Gaussian difference estimate. The constant precedes both the center
and the working radius. -/
theorem NoncollapsedRCD.annular_kernel_difference {K : ℝ} {n : ℕ}
(hX : NoncollapsedRCD K n X) [(hausdorff n X).Regular] (hn : 2 ≤ n)
{ker : ℝ → X → X → ℝ}
(hk : IsContinuousHeatKernel (hausdorff n X) hX.quadratic ker)
(hg : HasCoordinateHeatBounds (hausdorff n X) ker) :
∃ J : ℝ, 0 < J ∧ ∀ (p : X) (R t : ℝ), 0 < R → -1 ≤ K*R^2 →
0 < t → t ≤ 1 → t ≤ R^2 → ∀ x y z : X,
dist x p ≤ R/2 → R ≤ dist z p → dist z p ≤ 2*R → dist x y ≤ Real.sqrt t →
|ker t x z-ker t y z| ≤
J/(R*(hausdorff n X).real (ball p R))*dist x y := by
obtain ⟨C₀,B₀,hC₀,hB₀,hupper⟩ := hg.upper_five (hausdorff n X)
obtain ⟨C,D,B,hC,hD,hB,hdiff⟩ :=
hX.gaussianHeatDifferenceBounds_of_upper hn hk hC₀ hB₀ hupper
obtain ⟨Q,hQ,hbound⟩ := exists_annular_gaussian_gradient_bound n hD
let H := 4*Real.exp 8
let J := C*H^n*Q*Real.exp B
have hH : 0 < H := by dsimp [H]; positivity
have hQ0 : 0 < Q := zero_lt_one.trans_le hQ
refine ⟨J,by dsimp [J]; positivity,?_⟩
intro p R t hR hcurv ht ht1 htR x y z hxp hzp hzp2 hxy
let μ := hausdorff n X
let V := μ.real (ball p R)
let W := μ.real (ball z (Real.sqrt t))
have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht
have hV : 0 < V := ENNReal.toReal_pos (hX.fullSupport p R hR).ne'
(hX.finiteOnBounded _ isBounded_ball).ne
have hW : 0 < W := ENNReal.toReal_pos (hX.fullSupport z _ hs).ne'
(hX.finiteOnBounded _ isBounded_ball).ne
have hv : (Real.sqrt t/(H*R))^n*V ≤ W :=
hX.heat_ball_volume_lower hn p hR hcurv ht htR z hzp2
have hc := inverse_volume_coefficient hs hR hV hW hC.le hH hv
have hsep : R/2 ≤ dist x z := by
have hh := dist_triangle z x p
rw [dist_comm z x] at hh
linarith
have he : Real.exp (-dist x z^2/(D*t)+B*t) ≤
Real.exp (-(R/2)^2/(D*t))*Real.exp B := by
rw [← Real.exp_add]
apply Real.exp_le_exp.mpr
have hsq : (R/2)^2 ≤ dist x z^2 := pow_le_pow_left₀ (by positivity) hsep 2
have hh := div_le_div_of_nonneg_right hsq (by positivity : 0 ≤ D*t)
have hBt : B*t ≤ B := mul_le_of_le_one_right hB ht1
simp only [neg_div]
linarith
have hgauss := hbound R t hR ht htR
have hraw := hdiff t ht ht1 x y z hxy
calc
_ ≤ (C/(Real.sqrt t*W))*Real.exp (-dist x z^2/(D*t)+B*t)*dist x y := hraw.trans_eq (by dsimp [W,μ]; ring)
_ ≤ ((C*H^n/V)*((R/Real.sqrt t)^n/Real.sqrt t))*
(Real.exp (-(R/2)^2/(D*t))*Real.exp B)*dist x y := by
gcongr
_ = (C*H^n/V)*(((R/Real.sqrt t)^n/Real.sqrt t)*
Real.exp (-(R/2)^2/(D*t)))*Real.exp B*dist x y := by ring
_ ≤ (C*H^n/V)*(Q/R)*Real.exp B*dist x y := by gcongr
_ = _ := by dsimp [J]; ring
end RegularPointCoordinates
end
end OAI
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