Minicozzi–Sogge local smoothing conjecture for compact manifolds

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 2$, let $\bar{s}_p=n\lvert\frac12-\frac1p\rvert$, and let $L^p_s(M)$ denote the corresponding Sobolev space. Write $e^{it\sqrt{-\Delta_g}}$ for the half-wave propagator associated to the Laplace–Beltrami operator. Define

$$\bar{p}_{n,+}=\begin{cases}\frac{2(3n+1)}{3n-3},&n\text{ odd},\\[2pt]\frac{2(3n+2)}{3n-2},&n\text{ even}. \end{cases}$$

Local smoothing conjecture for compact manifolds. The inequality

$$\Big(\int_1^2\|e^{it\sqrt{-\Delta_g}}f\|^p_{L^p_{-\bar{s}_p+\sigma}(M)}\,dt\Big)^{1/p}\lesssim\|f\|_{L^p(M)}$$

holds for all $\sigma<1/p$ whenever $\bar{p}_{n,+}\leq p<\infty$. Minicozzi and Sogge constructed compact manifolds where the full $1/p$ gain fails below $\bar{p}_{n,+}$, so this conjecture formulates the expected sharp range for arbitrary compact manifolds; its resolution is not established by the supplied text.