Schrodinger estimates

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms $L_{t}^{q}L_{x}^{r}$ or $L_{x}^{r}L_{t}^{q}$, or in $X^{s,b}$ spaces, defined by

$||u||_{X^{s,b}}=||u||_{s,b}:=||\langle \xi \rangle ^{s}\langle \tau -|\xi |^{2}\rangle ^{b}{\hat {u}}||_{L_{\tau ,\xi }^{2}}.$

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities. See also [Bo1993b], [KlMa1993].