15 problems

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

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 $u$ 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…

Fractal-time local smoothing conjecture. For every $s>s_E(p,q)$, there exists a constant $C_{s,p,q}>0$ such that

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…

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

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…

Signature-refined local smoothing conjecture. There is $1/p-$ local smoothing for $\mathcal{F}$, meaning that the local smoothing estimate holds for every $\sigma<1/p$, for all…

Local smoothing conjecture for compact manifolds. The inequality