The first-order identification of the slice first pressure potential #
The local decomposition eq:pk writes the first pressure term as the double
Riesz transform p₁ = -R_iR_j(η U_{ij}), a second-order object. Display (3.5)
of the pressure-gradient section instead needs the first-order form. In the
Lean convention
pressureNewtonianDerivativePotential i g = −(∂ᵢN * g), the divergence-form
source is
Vᵢ = −∂ⱼ(η Uᵢⱼ) = +∂ⱼ(η uᵢ(uⱼ − ⟨uⱼ⟩)).
Both sides pair with Δψ against the same second-order source η U_{ij}: the
left by the distributional identity of the pressure decomposition, the right by
the adjoint of the first-order Newtonian derivative potential using the
displayed divergence-form source. Their difference is therefore weakly
harmonic on ℝ³, and the whole-space Liouville theorem with local L^{3/2}
linear growth identifies them almost everywhere.
The coordinate sum of first-order Newtonian derivative potentials of an integrable compactly supported source is integrable against the Laplacian of a test function.
The Laplacian pairing of the coordinate sum of first-order Newtonian derivative potentials is the divergence pairing of its source.
The first-order identification of p₁. Both p₁ and the coordinate sum
of the first-order Newtonian derivative potentials of V pair with Δψ
against the same second-order source G, and their difference has local
L^{3/2} linear growth; the whole-space Liouville theorem identifies them.
Display (3.5) needs the identification on almost every slice of a suitable
weak solution. The pressure side of the pairing is unconditional; the two
named inputs are the divergence-form characterization of the slice source V
and the local L^{3/2} linear growth of the residual.
The local growth of the first-order potential #
The whole-space Liouville step needs the residual p₁ - ∑ᵢ ∂ᵢN * Vᵢ to have
local L^{3/2} norms growing at most linearly in the radius. The pressure
side of that residual is the global L^{3/2} bound of p₁ supplied by the
Calderón–Zygmund selection; the potential side is the inverse-square far-field
decay of the first-order Newtonian derivative potential.
The first-derivative Newtonian potential of a compactly supported L^{6/5}
datum is L^{3/2} on every ball about the origin, with local norms growing at
most linearly in the radius.
The coordinate sum of first-order Newtonian derivative potentials of a
compactly supported L^{6/5} source has local L^{3/2} linear growth.
The identification residual has local L^{3/2} linear growth once the
first pressure potential is globally L^{3/2} and the source is a compactly
supported L^{6/5} field.
Display (3.5) needs the identification on almost every slice of a suitable
weak solution. The pressure side of the pairing and the growth of the residual
are unconditional given the Calderón–Zygmund slice bound; the single named
input is the divergence-form characterization of the slice source V.