Bounds for the forcing of the short-time construction #
The forced heat response depends only on the almost-everywhere class of its
tensor and is additive at almost every point of the slab. The canonical
pressure is Lipschitz in every exponent r > 1 with the constant of the double
Riesz transform. These are the ingredients of the contraction estimate in
prop:pv-local-solution.
theorem
ESS.forcedHeat_congr_ae
{G G' : Fin 3 → Fin 3 → CKN.Foundation.Parabolic.ParabolicPoint → ℝ}
(h : ∀ (i j : Fin 3), G i j =ᵐ[MeasureTheory.volume] G' i j)
:
The forced heat response of a tensor depends only on its almost-everywhere class.
theorem
ESS.forcedHeat_sub_ae
{τ : ℝ}
(hτ : 0 < τ)
{G G' : Fin 3 → Fin 3 → CKN.Foundation.Parabolic.ParabolicPoint → ℝ}
(hG : ∀ (i j : Fin 3), MeasureTheory.MemLp (G i j) 2 MeasureTheory.volume)
(hG' : ∀ (i j : Fin 3), MeasureTheory.MemLp (G' i j) 2 MeasureTheory.volume)
(hsG : ∀ (i j : Fin 3) (z : CKN.Foundation.Parabolic.ParabolicPoint), G i j z ≠ 0 → 0 < z.2)
(hsG' : ∀ (i j : Fin 3) (z : CKN.Foundation.Parabolic.ParabolicPoint), G' i j z ≠ 0 → 0 < z.2)
:
∀ᵐ (z :
CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict (CKN.spaceTimeSet Set.univ (Set.Ioo 0 τ)), forcedHeat (G - G') z = forcedHeat G z - forcedHeat G' z
The forced heat response is additive at almost every point of the slab, for square-integrable tensors vanishing at nonpositive times.
theorem
ESS.rieszPressureSpaceTime_sub_eLpNorm_le
(r : ℝ)
(hr : 1 < r)
(F F' : Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ)
(hF : ∀ (i j : Fin 3), MeasureTheory.MemLp (F i j) (ENNReal.ofReal r) MeasureTheory.volume)
(hF' : ∀ (i j : Fin 3), MeasureTheory.MemLp (F' i j) (ENNReal.ofReal r) MeasureTheory.volume)
:
MeasureTheory.eLpNorm (CKN.Leray.rieszPressureSpaceTime r hr F hF - CKN.Leray.rieszPressureSpaceTime r hr F' hF')
(ENNReal.ofReal r) MeasureTheory.volume ≤ ENNReal.ofReal (CKN.Leray.rieszPressureOperatorBound r hr) * ∑ i : Fin 3, ∑ j : Fin 3, MeasureTheory.eLpNorm (F i j - F' i j) (ENNReal.ofReal r) MeasureTheory.volume
The canonical Riesz pressure is Lipschitz in space-time L^r.