OAI/Geometry/RCD/Sobolev/PoissonGradientConstruction.lean

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The Poisson gradient input from the retained RCD calculus

  1. theorem OAI.RegularPointCoordinates.NoncollapsedRCD.poissonGradientInput line 19

    Source documentation

    Local truncation, Bochner interpolation and a countable interior cover give the exact half-ball estimate at every permitted radius.

    theorem NoncollapsedRCD.poissonGradientInput {K : ℝ} {n : ℕ}
    (hX : NoncollapsedRCD K n X) [(hausdorff n X).Regular]
    {ker : ℝ → X → X → ℝ}
    (hk : IsContinuousHeatKernel (hausdorff n X) hX.quadratic ker)
    (hg : HasCoordinateHeatBounds (hausdorff n X) ker)
    (hess : LocalPoissonHessianInput hX) : LocalPoissonGradientInput hX := by
…

    Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.IsContinuousHeatKernel, OAI.RegularPointCoordinates.HasCoordinateHeatBounds, OAI.RegularPointCoordinates.LocalPoissonHessianInput, OAI.RegularPointCoordinates.LocalPoissonGradientInput