Weighted compact cutoff data

For each fixed Carleman parameter the cutoff energy and weak heat energy have finite weighted integrals on the cutoff support.

theorem ESS.bu_short_weighted_data_integrable (scale R ε 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 κ := buShortFullCutoff scale R hR ε; have v := buAffineField (-scale ^ 2 / 2) scale w; have Dv := buAffineDw (-scale ^ 2 / 2) scale Dw; have D2v := buAffineD2w (-scale ^ 2 / 2) scale D2w; have Dtv := buAffineDtw (-scale ^ 2 / 2) scale Dtw; have W := buCutField κ v; have DW := buCutDw κ v Dv; have D2W := buCutD2 κ v Dv D2v; have DtW := buCutDt κ v Dtv; MeasureTheory.IntegrableOn (fun (z : CKN.Foundation.Parabolic.ParabolicPoint) => buShortCarlemanWeight a z * (CKN.Foundation.Parabolic.vec3EuclideanNorm (W z) ^ 2 + CKN.spatialGradientSq W DW z)) K MeasureTheory.volume ∧ MeasureTheory.IntegrableOn (fun (z : CKN.Foundation.Parabolic.ParabolicPoint) => buShortCarlemanWeight a z * CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2W DtW z) ^ 2) K MeasureTheory.volume

The weighted quadratic field and weak heat energies are integrable on the short-time cutoff support.