SeparatedSourceBounds.lean`

2 declarations, 0 contain sorry, 0 declared as axioms

Actual L1 source bounds away from the heat pole

theorem OAI.RegularPointCoordinates.exists_separated_short_source_bound line 24

Source documentation

Positive separation controls the whole short-time average of an actual integrable source. No L2 regularity of the source is required.

theorem exists_separated_short_source_bound (hg : HasCoordinateHeatBounds μ ker)
    (n : ℕ) {A V δ : ℝ} (hA : 0 ≤ A) (hV : 0 < V) (hδ : 0 < δ) :
    ∃ C : ℝ, 0 < C ∧ ∀ (r t : ℝ) (x q : X) (S : Set X) (g : X → ℝ),
      0 < r → r ≤ 1 → 0 < t → t ≤ r ^ 2 →
      μ (ball q r) ≠ ∞ → V * r ^ n ≤ μ.real (ball q r) → dist x q ≤ A * r →
      (∀ z ∈ S, δ ≤ dist x z) →
      (∀ z ∈ S, V * (Real.sqrt t) ^ n ≤ μ.real (ball z (Real.sqrt t))) →
      Integrable g μ → (∀ᵐ z ∂μ, z ∉ S → g z = 0) →
      |∫ z, kernelAverage μ ker t (ball q r) z * g z ∂μ| ≤ C * ∫ z, |g z| ∂μ := by
…

Mentions: OAI.RegularPointCoordinates.HasCoordinateHeatBounds, OAI.RegularPointCoordinates.kernelAverage

theorem OAI.RegularPointCoordinates.exists_separated_long_source_bound line 55

Source documentation

Above the averaging scale, separation leaves an extra factor of the averaging radius in the actual difference pairing.

theorem exists_separated_long_source_bound (hg : HasCoordinateHeatBounds μ ker)
    (hdiff : HasGaussianHeatDifferenceBounds μ ker)
    (n : ℕ) {A V δ : ℝ} (hA : 0 ≤ A) (hV : 0 < V) (hδ : 0 < δ) :
    ∃ C : ℝ, 0 < C ∧ ∀ (r t : ℝ) (x q : X) (S : Set X) (g : X → ℝ),
      0 < r → 0 < t → t ≤ 1 → r ^ 2 ≤ t →
      μ (ball q r) ≠ ∞ → 0 < μ.real (ball q r) → dist x q ≤ A * r →
      (∀ z ∈ S, δ ≤ dist x z) →
      (∀ z ∈ S, V * (Real.sqrt t) ^ n ≤ μ.real (ball z (Real.sqrt t))) →
      Integrable g μ → (∀ᵐ z ∂μ, z ∉ S → g z = 0) →
      |∫ z, (kernelAverage μ ker t (ball q r) z - ker t x z) * g z ∂μ| ≤
        C * r * ∫ z, |g z| ∂μ := by
…

Mentions: OAI.RegularPointCoordinates.HasCoordinateHeatBounds, OAI.RegularPointCoordinates.HasGaussianHeatDifferenceBounds, OAI.RegularPointCoordinates.kernelAverage