6 problems

Mizohata–Takeuchi conjecture. If $\Sigma$ is convex, then for all $f\in L^2(\Sigma)$ and measurable weights $w\colon\mathbb{R}^n\to[0,\infty)$,

Mixed-norm x-ray transform conjecture. Together with the condition $r<\infty$, these necessary conditions are also sufficient for the estimate to hold. This conjecture asks for the…

Let $x'\in\mathbb R^{d-1}$, let $\alpha$ be a parameter, and let $\gamma(x',\alpha)$ define a line complex. Consider the X-ray transform … Assume that the support of $\chi$ is cont…

Sharp Sobolev regularity conjecture. The operator $\mathfrak R$ boundedly maps $L^p$ to $L_\alpha^p$ for $\alpha \le \alpha(p)$. The sharp $L^p$ regularity estimate is open in high…

Let $K \subset \mathbb R^n$ be a compact body with $C^{\infty}$ boundary $\partial K$. For $\xi \in \mathbb R^n$ with $|\xi|=1$ and $t \in \mathbb R$, let … be the sectional volume…

Let $d\geq 3$ and let $P:{\bf R}\to{\bf R}^{d-1}$ be a polynomial of degree $N$. Write … for affine arclength measure, and let $X^P$ denote the restricted X-ray transform associate…