Compact-core duality for the space-time pressure operators #
The spatial CKN L² pairing yields duality first for compact space-time data.
theorem
CKN.Leray.rieszPressureComponentCompactClass_pairing
(r : ℝ)
(hr : 1 < r)
(q : ℝ)
(hq : 1 < q)
(hHolder : r.HolderConjugate q)
(i j : Fin 3)
{F G : Foundation.Parabolic.Vec3 × ℝ → ℝ}
(hF : Continuous F)
(hFc : HasCompactSupport F)
(hG : Continuous G)
(hGc : HasCompactSupport G)
:
∫ (z : Foundation.Parabolic.Vec3 × ℝ), ↑↑(rieszPressureComponentCompactClass r hr i j hF hFc) z * G z = ∫ (z : Foundation.Parabolic.Vec3 × ℝ), F z * ↑↑(rieszPressureComponentCompactClass q hq j i hG hGc) z
The joint compact representative obeys the adjoint identity, used by
lem:riesz-duality.