The first-order Newtonian adjoint and the smooth pairing

The singular identity ∫ ∂ᵢN(x - y) ∂ⱼψ(x) dx = -∫ N(x - y) ∂ᵢ∂ⱼψ(x) dx is obtained by heat subordination: at each positive time the Gaussian kernel is smooth, so the identity is the ordinary compactly supported integration by parts, and the time integral of the Gaussian family reproduces the Newtonian kernel and its first derivative. Fubini in the time variable is legitimate because the test function has compact support.

Combining this with the Fubini pairing for potentials and with two classical integrations by parts gives, for smooth compactly supported data G,

∫ Pᵢ(G) ∂ⱼψ = ∫ ∂ᵢ∂ⱼ(N * G) ψ,

which is the smooth case of the distributional adjointness used for the L^(6/5) Calderón--Zygmund endpoint.

theorem CKN.Foundation.Euclidean.newtonian_derivative_kernel_adjoint {ψ : Parabolic.Vec3 → ℝ} {y : Parabolic.Vec3} (hψ : ContDiff ℝ (↑⊤) ψ) (hψc : HasCompactSupport ψ) (i j : Fin 3) : ∫ (x : Parabolic.Vec3), spatialDeriv Heat.newtonianKernel i (x - y) * spatialDeriv ψ j x = -∫ (x : Parabolic.Vec3), Heat.newtonianKernel (x - y) * mixedSecond ψ i j x

The first-order Newtonian kernel adjoint. Heat subordination reduces the singular identity to the ordinary compactly supported integration-by-parts identity at each positive time.

theorem CKN.Foundation.Euclidean.pressure_newtonian_derivative_potential_smooth_pairing {i j : Fin 3} {G ψ : Parabolic.Vec3 → ℝ} (hG : ContDiff ℝ (↑⊤) G) (hGc : HasCompactSupport G) (hψ : ContDiff ℝ (↑⊤) ψ) (hψc : HasCompactSupport ψ) : ∫ (x : Parabolic.Vec3), pressureNewtonianDerivativePotential i G x * spatialDeriv ψ j x = ∫ (x : Parabolic.Vec3), mixedSecond (pressureNewtonianPotential G) i j x * ψ x

For smooth compactly supported data the first derivative potential is adjoint to the classical Hessian of the Newtonian potential.