30 problems

Let $u$ solve the wave equation Cauchy problem … with $u(x,0)=f(x)$ and $\partial_tu(x,0)=0$ on $\mathbb{R}^d\times\mathbb{R}$. For $s_p=\frac{d-1}{2}-\frac{d-1}{p}$, Sogge's conje…

Let $0<\alpha\le 1$, and let $p,q$ satisfy … Suppose that $E\subset [1,2]$ has bounded $\alpha$-Assouad characteristic. Fractal local smoothing conjecture. The estimate … holds wit…

Compact-manifold local smoothing conjecture. For all $p\geq p_n^+$ and $\sigma<\frac1p$, one has

Fix $b1 eq 0$?

Let $d\geq2$. Let $T_a^{\phi_1,\phi_2}$ be a bilinear Fourier integral operator whose non-degenerate phase functions satisfy the cinematic curvature condition, and let…

Linear smoothing conjecture for FIOs. For any $p\geq\overline{p}_d$ and $\sigma<\frac{1}{p}$, one has

Wave local smoothing conjecture. The estimate holds for $r\ge 2d/(d-1)$. This is an epsilon-loss local smoothing prediction for the wave propagator; the supplied text gives no reso…

Local smoothing conjecture. The displayed inequality holds for all $\sigma < \sigma(p,n)$. This conjecture is solved affirmatively for $n=2$; for $n \geq 3$, it is known when…

Let $A_t$ denote the spherical averaging operator, let $\partial_t^\gamma$ be the fractional time derivative, and let $L^p_s(\mathbb{R}^d)$ denote the inhomogeneous Sobolev space.…

Let … where the symbol $a$ is supported in a compact set, the phase is homogeneous of degree one in $\xi$, and suppose that $\mathcal{F}$ satisfies the following conditions: (H1)…

Endpoint local smoothing conjecture. For every $p>\frac{2d}{d-1}$, the local smoothing estimate

Suppose that $\mu$ is an Ahlfors--David regular probability measure supported on $[1,2]$ and satisfying … for $r<\operatorname{diam}(\operatorname{supp}\mu)$. Rough local smoothing…

Let $E\subset [1,2]$ satisfy $\dim_M E=\dim_A E=\beta$. For each scale $j$, let $\mathcal{Z}_j(E)$ index a minimal collection of $\delta$-separated points $t_\nu$ whose intervals…

Let $d\geq 1$, let $2<p<\infty$, let $0<T<\infty$, and let $u$ solve the Hermite wave equation … with $u(0)=u_0$ and $\dot u(0)=0$. The local smoothing conjecture for the Hermite w…

The Klein–Gordon local smoothing conjecture. The frequency-localized estimate

Let $d\geq 1$, let $s_p=(d-1)\left|\frac{1}{2}-\frac{1}{p}\right|$, and let $v$ solve the Euclidean wave equation with initial data $v(0)$. The Euclidean wave local smoothing conje…

Local smoothing conjecture. For every $\varepsilon>0$, if $u_{0}\in W^{\sigma(p)+\varepsilon,p}(\mathbb{R}^{n})$ and $u_{1}\in W^{\sigma(p)-1+\varepsilon,p}(\mathbb{R}^{n})$, then

$L^p$-$L^q$ local smoothing conjecture. The inequality

Let $n\geq 3$, let $\alpha\in(0,1)\cup(1,\infty)$, and define the restriction-type operator … where $e(t):=e^{2\pi it}$, $\psi$ is compactly supported and smooth, and…

Local smoothing conjecture. Fix $\alpha\in(0,1)\cup(1,\infty)$. For $p>\frac{2n}{n-1}$, one has

FIO local smoothing conjecture. For every $p\ge\bar p_d$ and every $\sigma<\frac1p$, there is a constant $C_s$ such that

Compact-manifold local smoothing conjecture. For every $p\ge\bar p_{d,+}$ and every $\sigma<\frac1p$, one has

Suppose $M=\mathbb R^d$ has the flat metric, $u$ solves the Cauchy problem … and define $s_p=\frac{d-1}{2}-\frac{d-1}{p}$ and $p_d=\frac{2d}{d-1}$. Euclidean local smoothing conjec…

Let $u(x,t)=e^{it(-\Delta)^{\frac{\alpha}{2}}}f(x)$ solve the fractional Schrödinger equation with $\alpha>1$, where $f$ is a Schwartz function, and let $L^p_s(\mathbb R^n)$ denote…

Positive-curvature local smoothing conjecture. The optimal local smoothing property should hold for these operators whenever $p\geqslant p_{n,+}$. The conjecture proposes an improv…