Weighted cutoff error at fixed parameters

The scalar cutoff heat error has a finite weighted integral on the compact support of the cutoff for every fixed Carleman parameter.

theorem ESS.bu_short_weighted_error_integrable (scale R ε c₁ a : 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 K := buCutSupportSet (buShortFullCutoff scale R hR ε); have v := buAffineField (-scale ^ 2 / 2) scale w; have Dv := buAffineDw (-scale ^ 2 / 2) scale Dw; MeasureTheory.IntegrableOn (fun (z : CKN.Foundation.Parabolic.ParabolicPoint) => buShortCarlemanWeight a z * buShortCutoffHeatErrorSize scale R hR ε c₁ v Dv z ^ 2) K MeasureTheory.volume

The weighted squared heat error is integrable on the compact short-time cutoff support.