Lp Extension Pairing Potential #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
theorem
CKN.Foundation.Euclidean.pressureNewtonianPotential_memLp_six
{F : Parabolic.Vec3 → ℝ}
(hF : ContDiff ℝ (↑⊤) F)
(hFc : HasCompactSupport F)
:
The Newtonian potential of a smooth compactly supported datum on ℝ³ lies in
L⁶(ℝ³) (paper label: the endpoint L⁶ membership of the pressure potential used in the
elliptic L^p extension estimates). The potential is bounded on the closed ball of radius
2R containing the support and decays like C / ‖x‖ beyond it, so the single bound
|Φ x| ≤ C / (1 + ‖x‖) dominates it; since 6 > 3 the profile (1 + ‖x‖)⁻⁶ is integrable.