HeatMeanComparison.lean`

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Relative heat/mean comparison for the normalized RCD harmonic column

All errors come from the actual finite-scale splitting, the raw gradient bound and the cutoff kernel. The missing heat mass and scalar regularizer contribute explicitly to the estimate.

theorem OAI.RegularPointCoordinates.finite_scale_heat_mean_comparison line 23

Source documentation

The finite normalization data give relative heat Gram control in the same exact frame, with a small prefix and a vanishing Gaussian tail.

theorem finite_scale_heat_mean_comparison {K : ℝ} (m : ℕ) (hX : NoncollapsedRCD K (m + 1) X)
    [(hausdorff (m + 1) X).Regular]
    {a : ℝ} (ha : 0 ≤ a)
    (hBG : BallVolumeComparison (hausdorff (m + 1) X) (comparisonModelVolume m a))
    (u : Fin (m + 1) → SobolevSpace (hausdorff (m + 1) X) hX.quadratic)
    (p : X) {r ε : ℝ} (hr : 0 < r) (hε : 0 ≤ ε)
    (hsmall : ((m + 1 : ℕ) : ℝ) ^ 2 * (normalizationVolumeRatio m * ε) ≤ 1 / 2)
    (A : Euclidean (m + 1) ≃L[ℝ] Euclidean (m + 1))
    (hA : IsExactMeanNormalization (hausdorff (m + 1) X) hX.quadratic
      (sobolevColumnMap (hausdorff (m + 1) X) hX.quadratic u) (ball p r) measurableSet_ball A)
    (k : ℕ) (hscale : ∀ j ≤ k, a * (3 * (2 ^ j * r)) ≤ 1)
    (hnormal : ∀ j ≤ k, ∃ (S : Euclidean (m + 1) ≃L[ℝ] Euclidean (m + 1))
…

Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.BallVolumeComparison, OAI.RegularPointCoordinates.comparisonModelVolume, OAI.RegularPointCoordinates.SobolevSpace, OAI.RegularPointCoordinates.normalizationVolumeRatio, OAI.RegularPointCoordinates.Euclidean, OAI.RegularPointCoordinates.IsExactMeanNormalization, OAI.RegularPointCoordinates.sobolevColumnMap