# Wave estimates

Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms $L^q_t L^r_x$, or in $X^{s,b}$ spaces, defined by $$\| u \|_{X^{s,b}} = \| \langle\xi\rangle^s \langle|\xi| - |\tau|\rangle^b \hat{u} ( \tau, \xi )\|_2 $$

Linear space-time estimates are known as Strichartz estimates. They are especially useful for the semilinear NLW without derivatives, and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The $X^{s,b}$ spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces first appear in one-dimension in RaRe1982 and in higher dimensions in Be1983 in the context of propagation of singularities; they were used implicitly for LWP in KlMa1993, while the Schrodinger and KdV analogues were developed in Bo1993, Bo1993b.