The local smoothing conjecture for the Hermite wave equation

Let $d\geq 1$, let $2<p<\infty$, let $0<T<\infty$, and let $u$ solve the Hermite wave equation

$$\partial_t^2u=\Delta u-|x|^2u,$$

with $u(0)=u_0$ and $\dot u(0)=0$. The local smoothing conjecture for the Hermite wave equation. The estimate

$$\|u\|_{L^p_{t,x}([-T,T]\times\mathbb{R}^d)}\lesssim_T\|u_0\|_{L^p_s(\mathbb{R}^d)}$$

holds provided that

$$s\geq\max\left(d\left|\frac{1}{2}-\frac{1}{p}\right|-\frac{1}{2},0\right).$$

This is the paper’s main conjectural local smoothing statement for Hermite wave solutions, with a derivative threshold suggested by the preceding linearized Klein–Gordon analysis; the source does not provide a resolution of the full claim.