The Hessian of the Newtonian kernel

For n ≠ 2, the second derivatives of the Newtonian kernel Gₙ = TauCeti.newtonianKernel n on ℝⁿ are, away from the pole,

D²Gₙ(x)(v, w) = ωₙ⁻¹ ‖x‖⁻ⁿ⁻² ⟪x, v⟫ ⟪x, w⟫ - (n ωₙ)⁻¹ ‖x‖⁻ⁿ ⟪v, w⟫,

where ωₙ is the volume of the unit ball. This file proves that formula and the two estimates that make K(x, y) = D²Gₙ(x - y) a standard Calderón–Zygmund kernel:

• the size estimate ‖D²Gₙ(x)‖ ≤ (n + 1) (n ωₙ)⁻¹ ‖x‖⁻ⁿ;
• the smoothness estimate ‖D²Gₙ(a) - D²Gₙ(b)‖ ≤ 2ⁿ⁺¹ (n + 5) ωₙ⁻¹ ‖a - b‖ ‖b‖⁻ⁿ⁻¹ whenever 2 ‖a - b‖ ≤ ‖b‖; with a = x - y and b = x - y' it reads ‖K(x, y) - K(x, y')‖ ≤ C ‖y - y'‖ ‖x - y'‖⁻ⁿ⁻¹ for 2 ‖y - y'‖ ≤ ‖x - y'‖.

For n ≥ 3 the kernel is nonzero and homogeneous of degree -n, so it is not locally integrable, and the operator f ↦ D²(Gₙ ⋆ f) is a singular integral. (In dimension one the formula vanishes identically, and in dimension two the totalized kernel is zero; see below.) The two estimates are the kernel hypotheses of the Calderón–Zygmund theory of such operators; together with the L² bound of the operator, that theory gives the Lᵖ bound ‖D²u‖_p ≤ C ‖Δu‖_p, 1 < p < ∞, behind the W^{2,p} estimates for the Laplacian.

In dimension two TauCeti.newtonianKernel 2 is the zero function (its normalizing constant is 0⁻¹ = 0), so the formula needs n ≠ 2; the two estimates are stated for every n, and hold trivially at n = 2. The logarithmic planar kernel TauCeti.planarNewtonianKernel is not covered here.

Both estimates are instances of the general bounds for the Hessian of x ↦ ‖x‖ ^ p, TauCeti.norm_iteratedFDeriv_two_norm_rpow_le and TauCeti.norm_iteratedFDeriv_two_norm_rpow_sub_le, at p = 2 - n.

Main declarations

• TauCeti.iteratedFDeriv_two_newtonianKernel_apply: the Hessian formula, and its evenness TauCeti.iteratedFDeriv_two_newtonianKernel_neg.
• TauCeti.norm_iteratedFDeriv_two_newtonianKernel_le: the size estimate.
• TauCeti.norm_iteratedFDeriv_two_newtonianKernel_sub_le: the smoothness estimate.

References

• D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Section 2.4 (the derivatives of the fundamental solution) and Section 9.4.
• E. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter II, §2 and §3.

theorem TauCeti.iteratedFDeriv_two_newtonianKernel_apply {n : ℕ} (hn : n ≠ 2) {x : EuclideanSpace ℝ (Fin n)} (hx : x ≠ 0) (v w : EuclideanSpace ℝ (Fin n)) : (iteratedFDeriv ℝ 2 (newtonianKernel n) x) ![v, w] = (MeasureTheory.volume.real (Metric.ball 0 1))⁻¹ * ‖x‖ ^ (-(↑n + 2)) * inner ℝ x v * inner ℝ x w - (↑n * MeasureTheory.volume.real (Metric.ball 0 1))⁻¹ * ‖x‖ ^ (-↑n) * inner ℝ v w

The Hessian of the Newtonian kernel away from its pole: D²Gₙ(x)(v, w) = ωₙ⁻¹ ‖x‖⁻ⁿ⁻² ⟪x, v⟫ ⟪x, w⟫ - (n ωₙ)⁻¹ ‖x‖⁻ⁿ ⟪v, w⟫.

@[simp]

theorem TauCeti.iteratedFDeriv_two_newtonianKernel_neg {n : ℕ} (x : EuclideanSpace ℝ (Fin n)) : iteratedFDeriv ℝ 2 (newtonianKernel n) (-x) = iteratedFDeriv ℝ 2 (newtonianKernel n) x

The Hessian of the Newtonian kernel is even: D²Gₙ(-x) = D²Gₙ(x).

theorem TauCeti.norm_iteratedFDeriv_two_newtonianKernel_le {n : ℕ} {x : EuclideanSpace ℝ (Fin n)} (hx : x ≠ 0) : ‖iteratedFDeriv ℝ 2 (newtonianKernel n) x‖ ≤ (↑n + 1) * (↑n * MeasureTheory.volume.real (Metric.ball 0 1))⁻¹ * ‖x‖ ^ (-↑n)

The size estimate for the Hessian of the Newtonian kernel: away from the pole, ‖D²Gₙ(x)‖ ≤ (n + 1) (n ωₙ)⁻¹ ‖x‖⁻ⁿ.

theorem TauCeti.norm_iteratedFDeriv_two_newtonianKernel_sub_le {n : ℕ} {a b : EuclideanSpace ℝ (Fin n)} (hab : 2 * ‖a - b‖ ≤ ‖b‖) : ‖iteratedFDeriv ℝ 2 (newtonianKernel n) a - iteratedFDeriv ℝ 2 (newtonianKernel n) b‖ ≤ 2 ^ (n + 1) * (↑n + 5) * (MeasureTheory.volume.real (Metric.ball 0 1))⁻¹ * ‖a - b‖ * ‖b‖ ^ (-(↑n + 1))

The smoothness estimate for the Hessian of the Newtonian kernel. If 2 ‖a - b‖ ≤ ‖b‖, then

‖D²Gₙ(a) - D²Gₙ(b)‖ ≤ 2ⁿ⁺¹ (n + 5) ωₙ⁻¹ ‖a - b‖ ‖b‖⁻ⁿ⁻¹.

With a = x - y and b = x - y', this is the smoothness estimate ‖K(x, y) - K(x, y')‖ ≤ C ‖y - y'‖ ‖x - y'‖⁻ⁿ⁻¹ for 2 ‖y - y'‖ ≤ ‖x - y'‖ of the Calderón–Zygmund kernel K(x, y) = D²Gₙ(x - y).