Endpoint Strichartz Estimates for the Schrödinger Equation

Question

The non-endpoint Strichartz estimates for the (linear) Schrödinger equation: $$ \|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)} $$ $$ 2 \leq q,r \leq \infty,\;\frac{2}{q}+\frac{d}{r} = \frac{d}{2},\; (q,r,d) \neq (2,\infty,2),\; q\neq 2 $$ are easily obtained using (mainly) the Hardy-Littlewood-Sobolev inequality, the endpoint case $q = 2$ is however much harder (see Keel-Tao for example.)

Playing around with the Fourier transform one sees that estimates for the restriction operator sometimes give estimates similar to Strichartz's. For example, the Tomas-Stein restriction theorem for the paraboloid gives: $$ \|e^{i t \Delta/2} u_0\|_{L^{2(d+2)/d}_t L^{2(d+2)/d}_x} \lesssim \|u_0\|_{L^2_x}, $$ which, interpolating with the easy bound $$ \|e^{i t \Delta/2} u_0\|_{L^{\infty}_t L^{2}_x} \lesssim \|u_0\|_{L^2_x}, $$ gives precisely Strichartz's inequality but restricted to the range $$ 2 \leq r \leq 2\frac{d+2}{d} \leq q \leq \infty. $$

As far as I know, the Tomas-Stein theorem (for the whole paraboloid) gives the restriction estimate $R_S^*(q'\to p')$ for $q' = \bigl(\frac{dp'}{d+2}\bigr)'$ (this $q$ is different from the one above), so I'm guessing that this cannot be strengthened (?).

So my question is: what's the intuition of what goes wrong when trying to prove Strichartz's estimates all the way down to the endpoints using only Fourier restriction theory?

Answer 1

From my less-than-expert (where's Terry when you need him?) point of view, a possible reason seems to be the following (I wouldn't call it something going wrong or even a difficulty):

The statement of restriction estimates only give you estimates where the left hand side is an isotropic Lebesgue space, in the sense that you get an estimate $L^q_tL^r_x$ with $q = r$. This naturally excludes the end-point, which requires $r > q$.

Why is this? The reason is that the restriction theorems only care about the local geometry of the hypersurface, and not its global geometry. (For example, the versions given in Stein's Harmonic Analysis requires either the hypersurface to have non-vanishing Gaussian curvature for a weaker version, or that the hypersurface to be finite type for a slightly stronger version. Both of these conditions are assumptions on the geometry of the hypersurface locally as a graph over a tangent plane.) Now, on each local piece, you do have something more similar to the classical dispersive estimates with $r > q$, which is derived using the method of oscillatory integrals (see, for example, Chapter IX of Stein's book; the dispersive estimate (15) [which has, morally speaking $q = r = \infty$ but with a weight "in $t$", so actually implies something with $q < \infty$] is used to prove Theorem 1, which is then used to derive the restriction theorem). But once you try to piece together the various "local" estimates to get an estimate on the whole function, you have no guarantee of what the "normal direction" is over the entire surface. (The normal direction, in the case of the application to PDEs, is the direction of the Fourier conjugate of the "time" variable.) So in the context of the restriction theorem, it is most natural to write the theorem using the $q = r$ version, since in the more general context of restriction theorems, there is no guarantee that you would have a globally preferred direction $t$.

(Note that Keel-Tao's contribution is not in picking out that time direction: that Strichartz estimates can be obtained from interpolation of a dispersive inequality and energy conservation is well known, and quite a bit of the non-endpoint cases are already available as intermediate consequences of the proof of restriction theorems. The main contribution is a refined interpolation method to pick out the end-point exponents.)