Products of space-time Sobolev families #
A field in L²(I; H^m(ℝ³)) whose H^m slices are uniformly bounded in time
multiplies L²(I; H^m(ℝ³)) into itself, with the space-time Leibniz family as
the family of weak derivatives of the product. On almost every time slice the
whole-space product estimate applies to the two slice families; the slice
estimates integrate in time by Tonelli, and the slice Leibniz rules integrate
to the space-time weak derivative identities by Fubini. This is the product
estimate for the nonlinear term in prop:lps-smoothing.
The reduction is stated once for any slice product estimate from H^{m₁}
times H^{m₂} to H^{m₂} with m₂ ≤ m₁.
On a time slice, the space-time Leibniz family is the spatial Leibniz family of the slices of the two factors.
Lifting a whole-space product estimate from slices to space time
(prop:lps-smoothing): if the product of an H^{m₁} family and an H^{m₂}
family is an H^{m₂} family with the Leibniz derivatives and
‖fg‖²_{H^{m₂}} ≤ C ‖f‖²_{H^{m₁}} ‖g‖²_{H^{m₂}}, then for f in
L²(I; H^{m₁}) with H^{m₁} slices bounded by K and g in
L²(I; H^{m₂}), the product is in L²(I; H^{m₂}) with the space-time Leibniz
family and ‖fg‖²_{L²H^{m₂}} ≤ C K ‖g‖²_{L²H^{m₂}}.
The space-time algebra estimate (prop:lps-smoothing): for m ≥ 2, a
field f in L²(I; H^m(ℝ³)) whose H^m slices have squared norm at most K
multiplies g ∈ L²(I; H^m(ℝ³)) into L²(I; H^m(ℝ³)); the product has the
space-time Leibniz family and ‖fg‖²_{L²H^m} ≤ C K ‖g‖²_{L²H^m}.