The Newtonian Hessian has critical radial growth away from the origin

Proved

HunterPDE.Newtonian.norm_secondPartial_le

by Mazecto · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

newtonian-potentialpdesingular-integrals

Let $n\ge2$ and let $\Gamma$ be the Newtonian fundamental solution with the sign convention $-\Delta\Gamma=\delta$ . For any coordinate indices $i,j$ , there is a finite nonnegative constant $C$ such that

|\partial_{ij}\Gamma(z)|\le C|z|^{-n}\qquad\text{for every }z\ne0.

This is the punctured Hessian growth bound used in the cancellation integrability step of Hunter's Newtonian second-derivative representation.

Preamble

import Theorems.Thm_HunterPDE_Newtonian_fundamentalSolution_partial
import Mathlib.Analysis.SpecialFunctions.Pow.Real
open HunterPDE.Newtonian

Formal statement

theorem HunterPDE.Newtonian.norm_secondPartial_le (n : ℕ) (hn : 2 ≤ n) (i j : Fin n) :
    ∃ C : ℝ, 0 ≤ C ∧ ∀ z : EuclideanSpace ℝ (Fin n), z ≠ 0 →
      ‖secondPartial (fundamentalSolution n) i j z‖ ≤ C * ‖z‖ ^ (-(n : ℝ)) := by sorry

Source

Hunter, Notes on Partial Differential Equations, revised 6/18/2014, https://www.math.ucdavis.edu/~hunter/pdes/ch2.pdf, printed p. 33 Eq. (2.14) and p. 38 Theorem 2.26 Eq. (2.27); radial growth estimate obtained by differentiating the displayed Newtonian gradient.

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