59 problems

Planchon's space-time conjecture. One has

Consider the three-dimensional wave equation associated with the inverse-square potential parameter $a$, with $-\frac{1}{4}<a<0$, and also the critical case $a=-\frac{1}{4}$. Let t…

Let $X_{s,b}$ denote the Bourgain space associated with the KdV dispersion relation, consisting of distributions whose space-time Fourier transform is weighted by…

Euclidean Strichartz conjecture. There holds

Euclidean Strichartz conjecture. The estimate

Let $S(t)$ denote the linear Schrödinger evolution on $[?]$ and let $[?]$ be frequency-localized to $[-N,N]$. For the test function … the associated Strichartz constant is conjectu…

Wang's conjecture. The range of $\beta$ in Proposition can be extended to

Endpoint logarithmic Strichartz conjecture. For $\lambda\gg1$,

Let $d$ and $j_0$ be as in the periodic hyperbolic Schrödinger setting, and let $\delta(j_0)$ denote the parameter in the Strichartz estimate. At the endpoint … the estimate may co…

Let $d\geq 2$, let $j_0\in\{0,1,\ldots,d\}$, and let $\delta(j_0)$ be the parameter appearing in the hyperbolic Strichartz estimate. For frequency scale $N\in 2^{\mathbb Z}$, the e…

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 $n=1$. Let $A$ denote the endpoint in the $(1/p,1/q)$-region for the frequency-global estimate, and let $\alpha$ be the summability exponent for the orthonormal system. Endpoin…

For the irrotational compressible Euler equation with a vacuum boundary, let $c_s$ denote the speed of sound and d_{Gamma_t the distance to the boundary of a particle. In the slo…

Let $r_0\geq 1$ and $r_1\geq 0$ be integers, with $r=r_0+r_1$, and let ${\bf b}=(b_1,\ldots,b_r)\in\mathbb{R}^r$. For $N>1$, define … Write ${\xi}_1=(n_{r_0+1},\ldots,n_r)$ and tak…

Periodic fifth-order Airy Strichartz conjecture. The estimate holds for $a=0$.

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…

Let $N$ be a dyadic frequency scale, let $P_N$ denote the corresponding frequency projection, and let $f\in L^2(\mathbb{T})$. Logarithmic short-time Strichartz conjecture. The esti…

Optimal-loss conjecture. The $L^{12}$ Strichartz estimate should hold with only $\varepsilon$ losses; consequently, the optimal value in the associated regularity-loss parameter is…

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…

Spherical exponential maximizer conjecture. For all $f$,