Airy estimates

Solutions to the Airy equation and its perturbations are either estimated in mixed space-time norms L^q_t L^r_x, L^r_x L^q_t, or in X^{s,b} spaces, defined by

$\|u\|_{X^{s,b}}=\|<\xi >^{s}<\tau -\xi ^{3}>^{b}{\hat {u}}(\tau ,\xi )\|_{L_{\tau ,\xi }^{2}}.$

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates, but these estimates only play a minor role in the theory. A more important category of linear estimates are the smoothing estimates and maximal function estimates. The X^{s,b} spaces are used primarily for bilinear Airy estimates, although more recently multilinear Airy estimates. These spaces and estimates first appear in the context of the Schrodinger equation in references.html#Bo1993b Bo1993b, although the analogous spaces for the wave equation appeared earlier references.html#RaRe1982 RaRe1982, references.html#Be1983 Be1983 in the context of propogation of singularities. See also references.html#Bo1993 Bo1993, references.html#KlMa1993 KlMa1993.