Morrey improvement from actual localized sources #
The potential comparison uses almost-everywhere finiteness. Measurability of the potential and finiteness of the explicit Adams constants are derived, rather than supplied as extra inputs.
theorem
CKN.Core.Endgame.bootstrap_morrey_of_sources
{v g : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
{h : Fin 3 → Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
{z₀ : Foundation.Parabolic.ParabolicPoint}
{R : ℝ}
(hR : 0 < R)
(hg : ∀ (i : Fin 3), AEMeasurable (fun (w : Foundation.Parabolic.ParabolicPoint) => g w i) MeasureTheory.volume)
(hh : ∀ (j i : Fin 3), AEMeasurable (fun (w : Foundation.Parabolic.ParabolicPoint) => h j w i) MeasureTheory.volume)
(hgN :
(Foundation.Parabolic.Morrey.morreyNorm (6 / 5) (25 / 11) fun (w : Foundation.Parabolic.ParabolicPoint) =>
Foundation.Parabolic.vec3EuclideanNorm (g w)) < ⊤)
(hhN :
∀ (j : Fin 3),
(Foundation.Parabolic.Morrey.morreyNorm 3 (25 / 6) fun (w : Foundation.Parabolic.ParabolicPoint) =>
Foundation.Parabolic.vec3EuclideanNorm (h j w)) < ⊤)
(hgsupp : ∀ w ∉ Foundation.Parabolic.parabolicCylinder z₀.1 z₀.2 R, g w = 0)
(hhsupp : ∀ (j : Fin 3), ∀ w ∉ Foundation.Parabolic.parabolicCylinder z₀.1 z₀.2 R, h j w = 0)
(hrep : v =ᵐ[MeasureTheory.volume] Step3.duhamelPotential g h)
:
(Foundation.Parabolic.Morrey.morreyNorm 3 25 fun (z : Foundation.Parabolic.ParabolicPoint) =>
Foundation.Parabolic.vec3EuclideanNorm (v z)) < ⊤
The first velocity-exponent improvement follows from the real localized source norms and representation, without all-point integrability assumptions.