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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Geometry.RCD.Coordinates.CoordinateHeatInput
import OAI.Geometry.RCD.Coordinates.LongBoundedSource
import OAI.Geometry.RCD.Harmonic.AveragedLpPairing
import OAI.Geometry.RCD.Heat.DyadicTimeIntegral
namespace OAI
/-! # Integrating the bounded curvature and endpoint source estimates -/
open Set Metric MeasureTheory
open scoped BigOperators ENNReal
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X]
(μ : Measure X) [μ.Regular] [IsLocallyFiniteMeasure μ] (hQ : QuadraticCheegerEnergy μ)
{ker : ℝ → X → X → ℝ} (hk : IsContinuousHeatKernel μ hQ ker)
include hk in
/-- The complete short-time bounded-source contribution has the square
of the averaging radius. -/
theorem exists_short_bounded_source_integral_bound (hg : HasCoordinateHeatBounds μ ker)
(n : ℕ) {A V W : ℝ} (hA : 0 ≤ A) (hV : 0 < V) (hW : 0 ≤ W) :
∃ C : ℝ, 0 < C ∧ ∀ (R : ℝ), 0 < R → R ≤ 1 → ∀ (i : ℕ) (x q : X),
μ (ball q (R / 2 ^ i)) ≠ ∞ →
V * (R / 2 ^ i) ^ n ≤ μ.real (ball q (R / 2 ^ i)) →
dist x q ≤ A * (R / 2 ^ i) →
(∀ j ≤ i, μ.real (ball x (2 ^ j * (R / 2 ^ i))) ≤ W * (2 ^ j * (R / 2 ^ i)) ^ n) →
(∀ t, 0 < t → t ≤ R ^ 2 → ∀ z ∈ ball x R,
V * (Real.sqrt t) ^ n ≤ μ.real (ball z (Real.sqrt t))) →
∀ (f : Lp ℝ 2 μ) (B : ℝ), 0 ≤ B → (∀ᵐ z ∂μ, |f z| ≤ B) →
(∀ᵐ z ∂μ, z ∉ ball x R → f z = 0) →
|∫ t in (0 : ℝ)..(R / 2 ^ i) ^ 2,
∫ z, kernelAverage μ ker t (ball q (R / 2 ^ i)) z * f z ∂μ| ≤
C * B * (R / 2 ^ i) ^ 2 := by
obtain ⟨C, hC, hb⟩ := exists_short_bounded_source_bound μ hQ hk hg n hA hV hW
refine ⟨C, hC, ?_⟩
intro R hR hR1 i x q hfinite hvol hcenter hupper hsource f B hB hfb hfzero
have hs : 0 < R / (2 : ℝ) ^ i := div_pos hR (by positivity)
have hsR : (R / (2 : ℝ) ^ i) ^ 2 ≤ R ^ 2 := by
apply pow_le_pow_left₀ hs.le
exact div_le_self hR.le (one_le_pow₀ (by norm_num))
apply short_time_integral_norm_le hs (by positivity)
intro t ht htime
have he := hb R hR hR1 i x q hfinite hvol hcenter hupper t ht htime
(hsource t ht (htime.trans hsR)) f B hB hfb hfzero
nlinarith only [mul_le_mul_of_nonneg_left he (sq_nonneg (R / (2 : ℝ) ^ i))]
include hk in
/-- All long-time bounded-source blocks sum to r times the working
radius. This is the gain needed for the curvature contribution. -/
theorem exists_long_bounded_source_integral_bound (hg : HasCoordinateHeatBounds μ ker)
(hdiff : HasGaussianHeatDifferenceBounds μ ker)
(n : ℕ) {A V W : ℝ} (hA : 0 ≤ A) (hV : 0 < V) (hW : 0 ≤ W) :
∃ C : ℝ, 0 < C ∧ ∀ (R : ℝ), 0 < R → R ≤ 1 → ∀ (i : ℕ) (x q : X),
μ (ball q (R / 2 ^ i)) ≠ ∞ → 0 < μ.real (ball q (R / 2 ^ i)) →
dist x q ≤ A * (R / 2 ^ i) →
(∀ j, μ.real (ball x (R / 2 ^ j)) ≤ W * (R / 2 ^ j) ^ n) →
(∀ t, 0 < t → t ≤ R ^ 2 → ∀ z ∈ ball x R,
V * (Real.sqrt t) ^ n ≤ μ.real (ball z (Real.sqrt t))) →
∀ (f : Lp ℝ 2 μ) (B : ℝ), 0 ≤ B → (∀ᵐ z ∂μ, |f z| ≤ B) →
(∀ᵐ z ∂μ, z ∉ ball x R → f z = 0) →
|∫ t in (R / 2 ^ i) ^ 2..R ^ 2,
∫ z, (kernelAverage μ ker t (ball q (R / 2 ^ i)) z - ker t x z) * f z ∂μ| ≤
C * B * (R / 2 ^ i) * R := by
obtain ⟨C, hC, hb⟩ := exists_long_bounded_source_bound μ hQ hk hg hdiff n hA hV hW
refine ⟨2 * C, by positivity, ?_⟩
intro R hR hR1 i x q hfinite hvol hcenter hupper hsource f B hB hfb hfzero
let F := fun t => ∫ z, (kernelAverage μ ker t (ball q (R / 2 ^ i)) z - ker t x z) * f z ∂μ
have hs : 0 < R / (2 : ℝ) ^ i := div_pos hR (by positivity)
have hsR : (R / (2 : ℝ) ^ i) ^ 2 ≤ R ^ 2 := by
apply pow_le_pow_left₀ hs.le
exact div_le_self hR.le (one_le_pow₀ (by norm_num))
have hi := kernelAverage_difference_lp_intervalIntegrable μ hQ hk
(ball q (R / (2 : ℝ) ^ i)) measurableSet_ball hfinite x f (sq_pos_of_pos hs) hsR
have hblock : ∀ j < i,
|∫ t in (R / (2 : ℝ) ^ j) ^ 2 / 4..(R / (2 : ℝ) ^ j) ^ 2, F t| ≤
C * B * (R / (2 : ℝ) ^ i) * (R / (2 : ℝ) ^ j) := by
intro j hji
have hj : 0 < R / (2 : ℝ) ^ j := div_pos hR (by positivity)
have hrj : R / (2 : ℝ) ^ i ≤ (R / (2 : ℝ) ^ j) / 2 := by
rw [div_div, ← pow_succ]
exact div_le_div_of_nonneg_left hR.le (by positivity) (pow_le_pow_right₀ (by norm_num) hji)
have hjR : (R / (2 : ℝ) ^ j) ^ 2 ≤ R ^ 2 := by
apply pow_le_pow_left₀ hj.le
exact div_le_self hR.le (one_le_pow₀ (by norm_num))
apply dyadic_time_integral_norm_le hj (by positivity)
intro t htime
have ht : 0 < t := (div_pos (sq_pos_of_pos hj) (by norm_num : (0 : ℝ) < 4)).trans_le htime.1
have hroot : (2 : ℝ) ^ j * (R / (2 : ℝ) ^ j) = R := by field_simp
have he := hb R hR hR1 j (R / (2 : ℝ) ^ i) x q hs hrj hfinite hvol hcenter
(fun k hk => by
have he := outward_radius_eq_ancestor hroot (Nat.sub_le j k)
simp only [Nat.sub_sub_self hk] at he
rw [he]
exact hupper (j - k))
t ht htime (hsource t ht (htime.2.trans hjR)) f B hB hfb hfzero
have hm := mul_le_mul_of_nonneg_left he (sq_nonneg (R / (2 : ℝ) ^ j))
apply hm.trans_eq
field_simp
have he := dyadic_time_integral_abs_le F hR i hi _ hblock
have hsum : (∑ j ∈ Finset.range i, C * B * (R / (2 : ℝ) ^ i) * (R / (2 : ℝ) ^ j)) =
C * B * (R / (2 : ℝ) ^ i) * R * ∑ j ∈ Finset.range i, (1 / 2 : ℝ) ^ j := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j _
rw [div_eq_mul_inv, one_div_pow]
ring
rw [hsum] at he
have hbound := mul_le_mul_of_nonneg_left (sum_geometric_two_le i)
(by positivity : 0 ≤ C * B * (R / (2 : ℝ) ^ i) * R)
nlinarith only [he, hbound]
end RegularPointCoordinates
end
end OAI
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