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 Bo1993b, although the analogous spaces for the wave equation appeared earlier RaRe1982, Be1983 in the context of propagation of singularities. See also Bo1993, KlMa1993.