The local smoothing conjecture for fractional Schrödinger equations

Let $n\geq 3$, let $\alpha\in(0,1)\cup(1,\infty)$, and consider the fractional Schrödinger evolution

$$e^{it(-\Delta)^{\alpha/2}}g(x):=\int_{\mathbb{R}^{n-1}}\widehat{g}(\xi)e(x\cdot\xi+t|\xi|^\alpha)\,d\xi.$$

Here $e(t):=e^{2\pi it}$, $W^{\beta,p}$ denotes the standard Bessel potential space, and $\beta_c=\beta_c(n)$ is defined by

$$\frac{\beta_c}{\alpha}=(n-1)\left(\frac12-\frac1p\right)-\frac1p.$$

Local smoothing conjecture. Fix $\alpha\in(0,1)\cup(1,\infty)$. For $p>\frac{2n}{n-1}$, one has

$$\Big\|e^{it(-\Delta)^{\alpha/2}}g\Big\|_{L^p_{x,t}(\mathbb{R}^{n-1}\times[0,1])}\lesssim\|g\|_{W^{\beta,p}(\mathbb{R}^{n-1})}$$

for every $\beta>\beta_c$. This is the local smoothing estimate for the fractional Schrödinger equation; the paper studies improvements toward this range, while the conjectured estimate itself is not established here.