| 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] |
|
|
| / |
| 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 |
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|
| end |
|
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| end OAI |
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|
