Sobolev promotion through the Laplacian #
The second-order Bessel identity promotes a field by two Sobolev orders when both the field and its distributional Laplacian have the lower order.
theorem
CKN.Leray.regularised_memSobolev_add_two_of_laplacian
(s : ℝ)
(D : TemperedDistribution Foundation.Parabolic.L2Vec3 ComplexVec3)
(hD : TemperedDistribution.MemSobolev s 2 D)
(hΔ : TemperedDistribution.MemSobolev s 2 (Laplacian.laplacian D))
:
TemperedDistribution.MemSobolev (s + 2) 2 D
A field and its distributional Laplacian in Hˢ give the field in Hˢ⁺².