Blowup Slice Uniform

Spatial slice bounds and rescaling identities in the endpoint compactness argument.

theorem ESS.blowup_component_slice_two_essSup_le_of_three (v : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (J : Set ℝ) (c : CKN.Foundation.Parabolic.Vec3) (R : ℝ) (i : Fin 3) (M : ENNReal) (hslice : ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict J, MeasureTheory.AEStronglyMeasurable (fun (x : CKN.Foundation.Parabolic.Vec3) => v (x, t)) MeasureTheory.volume ∧ MeasureTheory.eLpNorm (fun (x : CKN.Foundation.Parabolic.Vec3) => CKN.Foundation.Parabolic.vec3EuclideanNorm (v (x, t))) 3 MeasureTheory.volume ≤ M) : essSup (fun (t : ℝ) => MeasureTheory.eLpNorm (fun (x : CKN.Foundation.Parabolic.Vec3) => v (x, t) i) 2 (MeasureTheory.volume.restrict (CKN.Foundation.Parabolic.vec3Ball c R))) (MeasureTheory.volume.restrict J) ≤ M * MeasureTheory.volume (CKN.Foundation.Parabolic.vec3Ball c R) ^ (1 / 6)

A uniform whole-space L³ velocity slice bound controls the essential supremum of every component's local L² slice norm.