Symmetry of the double Riesz pressure components #
The mixed second derivative realization identifies the two index orders.
theorem
CKN.Leray.rieszPressureOperator_ae_eq_index_swap
(r : ℝ)
(hr : 1 < r)
(i j : Fin 3)
(g : Foundation.Parabolic.Vec3 → ℝ)
(hg : MeasureTheory.MemLp g (ENNReal.ofReal r) MeasureTheory.volume)
(hg2 : MeasureTheory.MemLp g 2 MeasureTheory.volume)
:
(fun (x : Foundation.Parabolic.Vec3) =>
↑↑((rieszPressureOperator r hr i j) (MeasureTheory.MemLp.toLp g hg)) x) =ᵐ[MeasureTheory.volume]
fun (x : Foundation.Parabolic.Vec3) => ↑↑((rieszPressureOperator r hr j i) (MeasureTheory.MemLp.toLp g hg)) x
CKN's two all-exponent pressure components agree in their index order,
used by lem:riesz-duality.