ActualRadialHarmonic.lean`

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Harmonic correction of the actual distance radial derivatives

theorem OAI.RegularPointCoordinates.actual_distance_radial_approximation line 21

Source documentation

The full Poisson mass calculation and localized Hessian input now construct a harmonic approximation to each actual coordinate radial derivative, with a squared error linear in the same source mass.

theorem actual_distance_radial_approximation {K : ℝ} {n : ℕ}
    (hX : NoncollapsedRCD K n X) [IsFiniteMeasureOnCompacts (hausdorff n X)]
    (hn : 2 ≤ n) (hhessian : LocalPoissonHessianInput hX)
    {κ r0 B J C A epsilon Cχ : ℝ} (hκ : 0 ≤ κ) (hK : -κ ≤ K)
    (hB : 0 ≤ B) (hJ : 0 ≤ J) (hA : 0 ≤ A) (hepsilon : 0 ≤ epsilon) (hCχ : 0 ≤ Cχ)
    (p : X) (S : DyadicDistanceSources (hausdorff n X) hX.quadratic n κ p r0)
    (T : DyadicDistanceReplacements (hausdorff n X) hX.quadratic S B J C) (i : ℕ)
    (hs1 : distanceSourceRadius r0 i ≤ 1)
    (u : Fin n → SobolevSpace (hausdorff n X) hX.quadratic)
    (hu : ∀ j, IsSobolevHarmonicOn (hausdorff n X) hX.quadratic
      (ball p (distanceSourceRadius r0 i / 4)) (u j))
    (hgrad : ∀ j, ∀ᵐ x ∂(hausdorff n X).restrict (ball p (distanceSourceRadius r0 i / 4)),
…

Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.LocalPoissonHessianInput, OAI.RegularPointCoordinates.DyadicDistanceSources, OAI.RegularPointCoordinates.DyadicDistanceReplacements, OAI.RegularPointCoordinates.distanceSourceRadius, OAI.RegularPointCoordinates.SobolevSpace, OAI.RegularPointCoordinates.IsSobolevHarmonicOn