The Hörmander condition for the Newtonian second-derivative kernel #
The Calderón--Zygmund treatment of the second Newtonian derivative
∂ᵢ∂ⱼN on ℝ³ (see paper/ckn.tex, §1, where the ambient space is ℝ³
with the Euclidean norm and the ball volume is (4/3)πr³; the classical
route chosen for the L^{3/2} bound on the pressure is recorded in the design
notes, docs/DESIGN_NOTES.md) needs the classical Hörmander kernel
condition: a kernel K whose gradient decays like |x|⁻⁴ satisfies
∫_{|x| > 2|y|} |K(x - y) - K(x)| dx ≤ C
with a constant C independent of y (Stein, Singular Integrals and
Differentiability Properties of Functions, Ch. II, §2). The main result
of this file, hormander_integral_bound, is the explicit form of that
bound: if ‖∇K‖ ≤ C₂ |x|⁻⁴ off the origin then the integral above is at
most 64 π C₂.
The proof has two elementary halves. The mean value inequality on the
segment [x - y, x] gives the pointwise bound
|K (x - y) - K x| ≤ 16 C₂ |y| |x|⁻⁴, because every point of the segment
stays at distance at least |x| / 2 from the origin; then the integral is
estimated on the geometric shells of ratio 4/3 around the origin, using
the exact annulus volumes.
Two modelling remarks. Vec3 carries the supremum norm by default, while
the decay hypothesis hgrad is stated with the operator norm on Vec3 and
the Euclidean radius vec3EuclideanNorm. No factor √3 appears: the
operator norm is only ever applied to ‖x - (x - y)‖, and since the
supremum norm is dominated by the Euclidean norm (space_norm_le_euclideanNorm)
the displacement contributes exactly vec3EuclideanNorm y. All balls and
shells are Euclidean (vec3Ball), and the volume normalization
volume_vec3Ball_eq is the one of the paper.
Hörmander condition. If K is differentiable off the origin with
|∇K| ≤ C₂ |x|⁻⁴, then for every y ≠ 0 the integral of |K (x - y) - K x|
over the region |x| > 2 |y| is at most 64 π C₂, uniformly in y.