The singular integration by parts for the Newtonian kernel #
For a smooth compactly supported F : Vec3 → ℝ, moving one spatial derivative across the
Newtonian kernel singularity,
∫ N(x - y) ∂ₗF(y) dy = ∫ ∂ₗN(x - y) F(y) dy,
and the resulting representation of the gradient of a smooth compactly supported v by the
Newtonian potential of its Laplacian gradient,
∂ₗv(x) = -∫ ∂ₗN(x - y) Δv(y) dy.
Both kernels here are absolutely, not merely improperly, integrable in three dimensions, so no
principal value or excision argument is needed. The route regularizes the kernel by
N_ε(z) = (4π)⁻¹ (|z|² + ε²)^{-1/2}, which is smooth on all of Vec3, applies the ordinary
(non-singular) integration by parts identity to N_ε, and passes to the limit ε → 0 along the
sequence ε_n = 1/(n+1) by dominated convergence, using that |N_ε| ≤ N and |∂ₗN_ε| ≤ |∂ₗN|
pointwise away from the singularity. It is the elliptic analogue of the heat-kernel integration
by parts in the
interior estimates of lem:aniso:annulus.