The optimal exponential-sum conjecture for one-dimensional Strichartz estimates

Let $T\geq \lambda^{1/3}$, let $\psi(hD_t)v_h(t,x)$ denote the frequency-localized solution under consideration, and assume that sharp exponential-sum bounds hold for every $\epsilon>0$. The conjectured dispersive estimate is

$$\left|\psi(hD_t)v_h(t,x)\right|\lesssim \frac{1}{h}h^{1/3-\epsilon},\qquad \forall\epsilon>0.$$

Optimal-loss conjecture. Under these sharp exponential-sum bounds, the Strichartz estimates should hold with loss $1/6+\epsilon$ for every $\epsilon>0$ and arbitrary $L^2(\mathbb R_+)$ data. The same estimates are expected for solutions of the semiclassical Schrödinger equation in the Friedlander domain in dimension $d\geq 2$ and in a ball.

A loss of $1/6$ derivatives is known to be unavoidable from gallery-mode initial data, so the conjectured $1/6+\epsilon$ loss would be optimal up to an arbitrarily small error. The paper establishes improved estimates using Van der Corput bounds, but the sharp exponential-sum estimates required here remain unavailable.