Quadratic data on the cutoff support

The shifted field and its specified weak derivatives have finite quadratic energy on the compact support of the short-time scalar cutoff.

theorem ESS.bu_short_support_memLp_data (scale R ε : ℝ) (hscale : 0 < scale) (hscale1 : scale ≤ 1) (hR : 0 < R) (hε : 0 < ε) (w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (Dw : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3) (D2w : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3) (Dtw : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (hcont : ContinuousOn w ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1)) (hweak : CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) w Dw D2w Dtw) (hL2 : ∀ S ⊆ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1), Bornology.IsBounded S → ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤) : have κ := buShortFullCutoff scale R hR ε; have K := buCutSupportSet κ; MeasureTheory.MemLp (buAffineField (-scale ^ 2 / 2) scale w) 2 (MeasureTheory.volume.restrict K) ∧ MeasureTheory.MemLp (buAffineDw (-scale ^ 2 / 2) scale Dw) 2 (MeasureTheory.volume.restrict K) ∧ MeasureTheory.MemLp (buAffineD2w (-scale ^ 2 / 2) scale D2w) 2 (MeasureTheory.volume.restrict K) ∧ MeasureTheory.MemLp (buAffineDtw (-scale ^ 2 / 2) scale Dtw) 2 (MeasureTheory.volume.restrict K)

The rescaled field and its three derivative fields belong to L² on the compact support of the smooth short-time cutoff.