2 problems

Local smoothing conjecture. The displayed inequality holds for all $\sigma < \sigma(p,n)$. This conjecture is solved affirmatively for $n=2$; for $n \geq 3$, it is known when…

Maximal-operator conjecture. The operator $M_\gamma$ maps $L^p(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedly if and only if $p>n$. This conjecture concerns the $L^p$ theory of geom…