Strichartz estimate

Remarks

The proof combines two facts. The first is the dispersive estimate $\norm{e^{it\Delta} f}_{L^\infty} \le (4\pi\abs{t})^{-n/2} \norm{f}_{L^1}$, which follows from the explicit kernel of $e^{it\Delta}$; interpolation with the conservation of the $L^2$ norm gives the $L^{r'} \to L^r$ bounds. The second is the $TT^*$ argument, which reduces the estimate to a bound for a time convolution handled by the Hardy–Littlewood–Sobolev inequality when $q > 2$. The endpoint $q = 2$, $r = \frac{2n}{n - 2}$ for $n \ge 3$ was proved by Keel and Tao in 1998. The relation $\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$ is forced by scaling: if $f_\lambda(x) = f(\lambda x)$, then $e^{it\Delta} f_\lambda (x) = (e^{i\lambda^2 t\Delta} f)(\lambda x)$, and comparing powers of $\lambda$ on both sides gives the relation. Example: for $n = 1$ the pair $(q, r) = (4, \infty)$ is admissible, and for $n = 2$ the pair $(4, 4)$ is admissible, the case first proved by Strichartz.

Going deeper

For a student

Strichartz estimates are the basic tool for the local and global well-posedness of nonlinear dispersive equations, such as the nonlinear Schrödinger equation $i u_t + \Delta u = \pm\abs{u}^{p-1} u$. One writes the equation in Duhamel form and runs a fixed point argument in the spaces $L^q_t L^r_x$, using the inhomogeneous version of the estimate. Similar estimates hold for the wave and Klein–Gordon equations, with different admissibility conditions and a loss of derivatives. Abstract versions, due to Keel and Tao, need only an energy bound and a dispersive decay estimate for the evolution.