44 problems

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 $\mathcal S$…

Let $n\geq 3$, and let … with $x\in\mathbb{R}^{n-1}$, $t\in\mathbb{R}$ and $y\in\mathbb{R}^{n-1}$. For a smooth amplitude $a$ supported near the origin, define the associated Hörma…

Let $\phi(x,\xi)$ be a Hörmander-type phase function, and let $G(x,\xi)$ denote its associated Gauss map. Suppose that Bourgain's condition holds: … for some scalar function…

Hörmander's conjecture. If $\phi$ satisfies (H1) and (H2), then

Arnold's conjecture. The asymptotic behavior of this oscillatory integral is completely determined by the Newton polyhedron of $\phi$ in a so-called adapted coordinate system.

Guo–Wang–Zhang conjecture. If $\phi$ satisfies Bourgain's condition, then $\mathbf{H}_{p\to p}(\phi)$ holds for every

Sectional-curvature pinching conjecture. For curved spaces, one should assume that the sectional curvatures are pinched away from zero in order to obtain favorable bounds for Nikod…

Carleson–Sj"olin model-phase conjecture. The operator norm should satisfy

Folding-operator extension conjecture. This behavior should remain true for general oscillatory integral operators with folding canonical relations satisfying the elliptical curvat…

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

Consider a family of oscillatory integral operators associated with a phase function satisfying the natural non-degeneracy conditions appropriate to the variable-coefficient settin…

Let $n\geq 2$, let $p=2n/(n+1)$, and let the oscillatory sum and square function be the quantities denoted by and in the source. Endpoint oscillatory sum conjecture. One should be…

Let $n\geq 2$, and let the oscillatory expression and square function be the quantities denoted by and in the source. Let $p=2n/(n+1)$, and write $R$ for the scale parameter. Endpo…

Finite-type extension conjecture. The same bound holds for $r>3$. This would extend the stated sharp endpoint estimate beyond types one, two, and three; the source gives no resolut…

Let $T_\la$ be the oscillatory integral operator associated with a canonical relation whose left and right projections are denoted by $\pi_L$ and $\pi_R$, and let $S_{1_r,0}$ denot…

Let $M$ be a three-dimensional analytic manifold, let $R_{\varepsilon,N}f$ denote the operator defined in the source, and let $p$ be an exponent. Restriction conjecture. For every…

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 $\phi$ be a H"ormander phase function satisfying Bourgain's condition at every point. Let $p$ be the exponent in the restriction estimate … should hold for all … The conjecture…

Let $T_N$ be an oscillatory integral operator with a phase function satisfying Hörmander's non-degeneracy condition, acting on functions supported in $\mathbb{B}_1^{d-1}(0)$. Hörma…

Smoothed-projector expansion conjecture. Roughly, one should have

Branching-point contribution conjecture. The contribution should satisfy

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 $\zeta\in\mathcal{C}_c^{\infty}(\mathbb{R}^d)$,