AffineCoefficientEnergy.lean`

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Coefficient bounds from exact normalization and harmonic energy

theorem OAI.RegularPointCoordinates.LocalHarmonicL2GradientInput.affine_coefficient line 20

Source documentation

Exact mean normalization on the half ball and the harmonic gradient estimate give affine nondegeneracy with a uniform constant. The affine constant is allowed to be arbitrary.

theorem LocalHarmonicL2GradientInput.affine_coefficient
    {K : ℝ} {n : ℕ} {hX : NoncollapsedRCD K n X}
    (hinput : LocalHarmonicL2GradientInput 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)
        (N : Euclidean n ≃L[ℝ] Euclidean n),
      IsExactMeanNormalization (hausdorff n X) hX.quadratic
        (sobolevColumnMap (hausdorff n X) hX.quadratic u)
        (ball p (R / 2)) measurableSet_ball N →
      ∀ (a : Euclidean n) (w : SobolevSpace (hausdorff n X) hX.quadratic) (c : ℝ),
      IsSobolevHarmonicOn (hausdorff n X) hX.quadratic (ball p R) w →
…

Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.LocalHarmonicL2GradientInput, OAI.RegularPointCoordinates.SobolevSpace, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.Euclidean, OAI.RegularPointCoordinates.IsExactMeanNormalization, OAI.RegularPointCoordinates.sobolevColumnMap, OAI.RegularPointCoordinates.IsSobolevHarmonicOn