AffineConstantBound.lean`

1 declarations, 0 contain sorry, 0 declared as axioms

Recovering the constant operator of a centered affine residual

theorem OAI.RegularPointCoordinates.LocalHarmonicL2MeanValueInput.affine_constant_operator_bound line 17

Source documentation

The published mean-value estimate recovers the constant coefficient operator because the same harmonic coordinate map vanishes at the center.

theorem LocalHarmonicL2MeanValueInput.affine_constant_operator_bound
    {K : ℝ} {n : ℕ} {hX : NoncollapsedRCD K n X}
    (hinput : LocalHarmonicL2MeanValueInput hX) (hn : 2 ≤ n)
    {κ : ℝ} (hκ : 0 ≤ κ) :
    ∃ C : ℝ, 1 ≤ C ∧ ∀ (p : X) (R : ℝ), 0 < R → -κ ≤ K * R ^ 2 →
      ∀ (u : Fin n → SobolevSpace (hausdorff n X) hX.quadratic) (V : X → Euclidean n),
      (∀ j, IsSobolevHarmonicOn (hausdorff n X) hX.quadratic (ball p R) (u j)) →
      ContinuousOn V (ball p R) → V p = 0 →
      (∀ j, SobolevSpace.toLp (hausdorff n X) hX.quadratic (u j) =ᵐ[
        (hausdorff n X).restrict (ball p R)] fun x => V x j) →
      ∀ η : SobolevSpace (hausdorff n X) hX.quadratic,
      (SobolevSpace.toLp (hausdorff n X) hX.quadratic η =ᵐ[
…

Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.LocalHarmonicL2MeanValueInput, OAI.RegularPointCoordinates.SobolevSpace, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.Euclidean, OAI.RegularPointCoordinates.IsSobolevHarmonicOn, OAI.RegularPointCoordinates.SobolevSpace.toLp