12 problems

Modified Newton distance conjecture. If $S\in\mathfrak S^m\mathbb R^{n_X+n_Z}$, then

Let … be the graph of a smooth function $h$, and define the extension operator … Write $q'=\frac{q}{q-1}$. Stein's restriction conjecture. If the Gaussian curvature of $\mathcal S$…

Curved Kakeya conjecture. If $\phi$ satisfies Bourgain's condition, then every $\phi$-Kakeya set has Hausdorff dimension $n$.

Hörmander dichotomy conjecture. The operators $T_\phi^\lambda$ satisfy the operator boundedness estimate for $p>\frac{2n}{n-1}$ if and only if $\phi$ satisfies Bourgain's condition…

Carbery–Wright conjecture. There exists an absolute constant $C>0$ such that

Let $\mathcal T^\lambda$ be the operator defined earlier in the paper, let $B_\lambda$ be its spatial domain, and let $\{\Gamma_{\theta,v}^\lambda\}_{\theta,v}$ be the associated c…

Uniform two-thirds decay conjecture. For any $\zeta\in\mathcal{C}_c^{\infty}(\mathbb{R}^d)$,

Curvature-controlled Fourier decay conjecture. There exists an absolute constant $C_1>0$ such that

Let $n\geq 3$ and let $T^{\lambda}$ be a Hörmander-type operator satisfying the straight condition: for each $\xi$, the associated quantity $G(x,\xi)$ is invariant under changes in…

Real-line Green's function conjecture. For every $c>0$ and every $\alpha\in(2,4]$, $G_{\mathbb{R}}$ is not strictly positive on $\mathbb{R}$ and is not monotonically decreasing on…

Let $1\leq k\leq n$, and let $(T_1,\dots,T_k)$ be a $\nu$-transverse $k$-tuple of H"ormander-type operators of the same signature $\sigma$. Thus, if $G_j$ denotes the generalised G…

Let $1\leq k\leq n$, and let $\mathbf{T}^{\lambda}=(T_1^{\lambda},\dots,T_k^{\lambda})$ be a $\nu$-transverse $k$-tuple of H"ormander-type operators with positive-definite phase fu…