16 problems

Decay-extension conjecture. The results of Wirth for very slowly oscillating dissipation terms, including decay estimates of the same structure as in that work, should extend to ge…

Slow-dispersion conjecture. For every $\varepsilon>0$, there exists a choice of $\nu(t)$ such that the $L^1\to L^\infty$ decay rate is no faster than $t^{-\varepsilon}$; more preci…

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…

In the water wave problem, dispersive estimates concern the decay or spacetime bounds satisfied by solutions, and paralinearization is a method of replacing nonlinear operators by…

Weighted Lorentz dispersive estimates conjecture. For every $\alpha\ne0$ and $y\in\mathbb{R}^3$, one has

Let $H_{\alpha,y}$ be the Schrödinger operator with a point interaction of strength $\alpha$ at $y\in\mathbb{R}^3$, let $P_{ac}$ denote the projection onto its absolutely continuou…

Global optimal decay conjecture. A global optimal estimate

Conjecture on optimal estimates. Optimal estimates could be achieved through an asymptotic expansion of first order, following the approach used for the Klein–Gordon equation with…

Kernel boundedness conjecture. The condition

Optimal dispersive estimate conjecture. In dimension $n=3$, the estimate

Let … where … and … with $x,y\in{\bf R}^n$, $k\ge1$, $1\le m<n$, and $0<h\ll1$. The oscillatory-integral conjecture. If $V$ satisfies condition (1.6), then … with a constant $C>0$…

The critical-regularity dispersive conjecture. The perturbed wave group $e^{it\sqrt{G}}$ should satisfy optimal dispersive estimates under these assumptions, possibly requiring onl…

Weighted-potential conjecture. The condition

Let $S(t)$ be the linear propagator, let $\mathscr{A}f(t)=\int_{\mathbb{R}}S(t-\tau)f(\tau)\,d\tau$ be the associated full-time integral operator, and let $\Box_k$ be the frequency…

Let $n=2$, let $p=4/n$, and let … also holds in the case $p=4/n$ when $n=2$. This is an endpoint extension of the preceding estimate; the source gives no resolution of this conject…

Let $n\geq 4$, let $G=-\Delta+V$ be the Schrödinger operator considered in the paper, and assume that zero is a regular point for $G$. Suppose that the potential $V$ satisfies cond…