20 problems

Optimal-loss conjecture. Under these sharp exponential-sum bounds, the Strichartz estimates should hold with loss $1/6+\epsilon$ for every $\epsilon>0$ and arbitrary…

The Strichartz conjecture on $\mathbb{R}\times\mathbb{S}^3$. One has

Let $N\geq 1$, let $\varphi$ be the time cutoff appearing in the estimate, and let $\phi\in L^2(\mathbb{R}\times\mathbb{T})$. For $\xi_0\in\mathbb{R}\times\mathbb{Z}$, write…

Let $n\geq2$, $q\geq \frac{2(n+1)}{n-1}$, and let $(f_j)_j$ be an orthonormal system in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^n)$. For a sequence of coefficients…

Let $n\geq2$, let $(1/p,1/q)$ lie on the line segment $(O,A]$, and set $s$ by … For an orthonormal family $\{f_j\}_{j\in J}\subset\dot{H}^{s}(\mathbb{R}^n)$ and coefficients…

Let $n\geq1$, and let $A=(1/p,1/q)=((n-1)/(n+1),n/(n+1))$. For an orthonormal family $\{f_j\}_{j\in J}\subset L^2(\mathbb{R}^n)$ and a sequence…

Let $\tilde{\Delta}:=\frac{1}{2\pi}(\partial_{11}+\alpha\partial_{22})$ on $\mathbb{T}^2$, where $\alpha<0$ is irrational, and let $e^{it\tilde{\Delta}}f$ denote the solution to…

Planchon's space-time conjecture. One has

Let $d \geq 2$, $q \in (2,\infty)$, and $r \in [2,\infty)$ satisfy … Let $\beta_{d/2}(q,r)$ denote the exponent appearing in the conjectured estimate, and let the estimate … holds…

The optimal Strichartz-range conjecture. The estimate should hold on any compact Lie group for every $p>2+\frac{4}{d}$. The paper provides evidence for this assertion for class fun…

Let $(\cdot,\cdot)$ be a positive-definite quadratic form of integral coefficients in $r$ variables, let $B$ be a bounded region in $\mathbb{R}^r$, and let $I$ be a bounded interva…

Let $M$ be a symmetric space of compact type of dimension $d$ and rank $r$, and let $M_0$ range over its irreducible components, with dimension $d_0$ and rank $r_0$. For…

Restricted-type endpoint conjecture. The estimate

Improved symmetric-space Strichartz conjecture. Under these assumptions, this estimate holds for all $p>2+\frac{4}{r}$. This strengthens the preceding conjectured range…

Let $G/K$ be a Riemannian globally symmetric space of compact type, equipped with the negative of the Cartan--Killing form as the canonical Riemannian metric $g$. Let $d$ and $r$ b…

Let $G$ be a compact Lie group equipped with a rational metric $g$. Let $d$ be the dimension of $G$ and $r$ the rank of $G$. Let $I\subset\mathbb{R}$ be a finite time interval. Com…

Gaussian extremizer conjecture. A function $f(\boldsymbol{x})$ maximizes the Strichartz estimate if and only if it has the form

Generic Strichartz conjecture. For arbitrarily small $\varepsilon>0$, one has

Let $P_N$ denote the frequency projection onto frequencies $\lesssim N$, and let $e^{it\Delta_{\mathbb{T}^2}}$ be the Schrödinger propagator on $\mathbb{T}^2$. For all sufficiently…

Frequency-localized dispersive estimate. If