| import OAI.Geometry.RCD.Heat.HeatKernelDifferenceInput | |
| namespace OAI | |
| /-! # The actual averaged long-time kernel difference -/ | |
| open Set Metric MeasureTheory Filter | |
| open scoped ENNReal | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] | |
| omit [MetricSpace X] in | |
| /-- Averaging preserves a uniform bound on the pointwise kernel difference. -/ | |
| theorem kernelAverage_difference_le (μ : Measure X) (ker : ℝ → X → X → ℝ) | |
| {s : Set X} (hfinite : μ s ≠ ∞) (hvol : 0 < μ.real s) | |
| (t : ℝ) (x z : X) (hf : IntegrableOn (fun y => ker t y z) s μ) | |
| {B : ℝ} (hbound : ∀ᵐ y ∂μ.restrict s, |ker t y z - ker t x z| ≤ B) : | |
| |kernelAverage μ ker t s z - ker t x z| ≤ B := by | |
| have hc : IntegrableOn (fun _ : X => ker t x z) s μ := integrableOn_const hfinite | |
| have hd := hf.sub hc | |
| have hμ : (μ.restrict s).real univ = μ.real s := by | |
| simp only [Measure.real, Measure.restrict_apply_univ] | |
| have he : kernelAverage μ ker t s z - ker t x z = | |
| (μ.real s)⁻¹ * ∫ y in s, (ker t y z - ker t x z) ∂μ := by | |
| rw [integral_sub hf hc, integral_const, smul_eq_mul, hμ] | |
| dsimp only [kernelAverage] | |
| field_simp [ne_of_gt hvol] | |
| rw [he, abs_mul, abs_of_nonneg (inv_nonneg.mpr hvol.le)] | |
| calc | |
| _ ≤ (μ.real s)⁻¹ * ∫ y in s, |ker t y z - ker t x z| ∂μ := | |
| mul_le_mul_of_nonneg_left (abs_integral_le_integral_abs) (inv_nonneg.mpr hvol.le) | |
| _ ≤ (μ.real s)⁻¹ * ∫ _y in s, B ∂μ := | |
| mul_le_mul_of_nonneg_left | |
| (integral_mono_ae hd.abs (integrableOn_const hfinite) hbound) (inv_nonneg.mpr hvol.le) | |
| _ = B := by rw [integral_const, smul_eq_mul, hμ]; field_simp | |
| variable [BorelSpace X] [ProperSpace X] | |
| (μ : Measure X) (hQ : QuadraticCheegerEnergy μ) [μ.Regular] | |
| {ker : ℝ → X → X → ℝ} (hk : IsContinuousHeatKernel μ hQ ker) | |
| include hk in | |
| /-- The published kernel gradient estimate yields the long-time averaged | |
| kernel bound with the exact displacement-to-heat-length factor. -/ | |
| theorem averaged_kernel_difference_bound | |
| {C D B : ℝ} (hC : 0 ≤ C) | |
| (hdifference : ∀ t : ℝ, 0 < t → t ≤ 1 → ∀ x y z : X, dist x y ≤ Real.sqrt t → | |
| |ker t x z - ker t y z| ≤ | |
| (C / (Real.sqrt t * μ.real (ball z (Real.sqrt t)))) * dist x y * | |
| Real.exp (-(dist x z) ^ 2 / (D * t) + B * t)) | |
| {t r A : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) (_hr : 0 < r) | |
| (x q z : X) (hfinite : μ (ball q r) ≠ ∞) (hvol : 0 < μ.real (ball q r)) | |
| (hcenter : dist x q ≤ A * r) (hscale : (A + 1) * r ≤ Real.sqrt t) : | |
| |kernelAverage μ ker t (ball q r) z - ker t x z| ≤ | |
| (C / (Real.sqrt t * μ.real (ball z (Real.sqrt t)))) * ((A + 1) * r) * | |
| Real.exp (-(dist x z) ^ 2 / (D * t) + B * t) := by | |
| have hi : IntegrableOn (fun y => ker t y z) (ball q r) μ := by | |
| apply (hk.integrable t ht z).integrableOn.congr | |
| exact Eventually.of_forall fun y => hk.symmetric t ht z y | |
| apply kernelAverage_difference_le μ ker hfinite hvol t x z hi | |
| filter_upwards [ae_restrict_mem measurableSet_ball] with y hy | |
| have hxy : dist x y ≤ (A + 1) * r := by | |
| have hyq : dist q y < r := by simpa only [mem_ball, dist_comm] using hy | |
| linarith [dist_triangle x q y] | |
| have hh := hdifference t ht ht1 x y z (hxy.trans hscale) | |
| rw [abs_sub_comm] at hh | |
| exact hh.trans (mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left hxy (div_nonneg hC (mul_nonneg (Real.sqrt_nonneg _) measureReal_nonneg))) | |
| (Real.exp_pos _).le) | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |