BoundedSourceHeatBounds.lean`

1 declarations, 0 contain sorry, 0 declared as axioms

Averaged heat estimates for bounded compact source densities

theorem OAI.RegularPointCoordinates.exists_short_bounded_source_bound line 23

Source documentation

The short-time average of a bounded source has a scale-independent bound. All volume assumptions are confined to the working ball.

theorem exists_short_bounded_source_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 : ℝ), 0 < t → t ≤ (R / 2 ^ i) ^ 2 →
      (∀ z ∈ ball x R, V * (Real.sqrt t) ^ n ≤ μ.real (ball z (Real.sqrt t))) →
      ∀ (g : X → ℝ) (B : ℝ), 0 ≤ B → (∀ᵐ z ∂μ, |g z| ≤ B) →
      (∀ᵐ z ∂μ, z ∉ ball x R → g z = 0) →
      |∫ z, kernelAverage μ ker t (ball q (R / 2 ^ i)) z * g z ∂μ| ≤ C * B := by
…

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