Linear growth of the pressure potentials

Combining the far-field decay of Newtonian potentials of compactly supported data with the ball estimate for the radial weight ‖x‖ ^ (-3/2) gives, for each of the three potential shapes N * g, ∂ⱼN * g and ∂ᵢ∂ⱼN * g, exactly the pair of hypotheses consumed by the Liouville theorem CKN.Foundation.Heat.weaklyHarmonicOn_eq_zero_of_lpNorm_linear_growth: local L^{3/2} membership on every round ball about the origin, and a growth bound of the form C * (1 + ρ). This is the decay-at-infinity input to the uniqueness half of the Newtonian representation ext:newtonian of the paper.

The local L^{3/2} hypothesis near the origin is left to the consumer: for the zeroth-order potential it follows from local integrability of the kernel, and for the derivative potentials it is the Calderón–Zygmund bound.

theorem CKN.memLp_and_lpNorm_linear_growth_pressureNewtonianDerivativePotential {g : Foundation.Parabolic.Vec3 → ℝ} (hg : MeasureTheory.Integrable g MeasureTheory.volume) {x₀ : Foundation.Parabolic.Vec3} {R : ℝ} (hR : 0 < R) (hSupp : tsupport g ⊆ Metric.closedBall x₀ R) (i : Fin 3) (hmeas : MeasureTheory.AEStronglyMeasurable (pressureNewtonianDerivativePotential i g) MeasureTheory.volume) (hnear : MeasureTheory.MemLp (pressureNewtonianDerivativePotential i g) (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict (Metric.ball 0 (2 * (2 * R + 2 * ‖x₀‖))))) : (∀ (ρ : ℝ), 0 < ρ → MeasureTheory.MemLp (pressureNewtonianDerivativePotential i g) (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict (euclideanBall 0 ρ))) ∧ ∀ (ρ : ℝ), 0 < ρ → MeasureTheory.lpNorm (pressureNewtonianDerivativePotential i g) (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict (euclideanBall 0 ρ)) ≤ invNormGrowthConstant (pressureNewtonianDerivativePotential i g) (2 * ((4 * (4 * Real.pi)⁻¹ * ∫ (y : Foundation.Parabolic.Vec3), |g y|) / (2 * R))) (2 * R + 2 * ‖x₀‖) * (1 + ρ)

The first-derivative Newtonian potential of integrable data supported in the closed ball of radius R about x₀ satisfies the Liouville growth hypotheses.