File size: 3,564 Bytes
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 | 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
|