25 problems

Let a Toeplitz operator on the Bargmann space have a symbol that is an exponential of a complex quadratic form, and let its Weyl symbol be the corresponding Weyl symbol. Berger–Cob…

Folding-operator extension conjecture. This behavior should remain true for general oscillatory integral operators with folding canonical relations satisfying the elliptical curvat…

Finite-type extension conjecture. The same bound holds for $r>3$. This would extend the stated sharp endpoint estimate beyond types one, two, and three; the source gives no resolut…

Let $x'\in\mathbb R^{d-1}$, let $\alpha$ be a parameter, and let $\gamma(x',\alpha)$ define a line complex. Consider the X-ray transform … Assume that the support of $\chi$ is cont…

Let $T_\la$ be the oscillatory integral operator associated with a canonical relation whose left and right projections are denoted by $\pi_L$ and $\pi_R$, and let $S_{1_r,0}$ denot…

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

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)…

Let $FIO(S,M^{\infty,q}_{1\otimes v_s})$ denote the class of continuous linear operators whose Wigner kernels have the localization and modulation-space structure specified above,…

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

Boutet de Monvel's conjecture. There is a maximal $\epsilon_0>0$ such that, for every $\epsilon\in(0,\epsilon_0)$, the extended flows define a real-analytic diffeomorphism

Let $\mathcal{R}$ be an integral operator associated with a canonical relation whose left and right projections are denoted by $\pi_L$ and $\pi_R$. Suppose the only singularities o…

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

Finite-type extension conjecture. The conclusion of Theorem ftthm should remain true for every $p>4$, and the assumptions on $\pi_R$ should even be dispensable altogether. The curr…

Consider the operator $R_\beta$ and its adjoint $R_\beta^*$ in the model discussed in the source. For $\beta\ne0$, the right projection $\pi_R$ has only fold singularities, and The…

Let ${\mathcal M}\subset\Omega_L\times\Omega_R$ be a four-dimensional manifold whose projections are submersions, and let $({\mathcal N}^*{\mathcal M})'$ be the corresponding prime…

Higher-order Compton scattering smoothing conjecture. For each scattering order $i$, $g_i$ is approximated by a Fourier integral operator $\mathcal{L}_i(\cdot,n_e)$ of order

Higher-dimensional logarithmic-phase FUP conjecture. For each $\nu>0$ there exists $\beta=\beta(\nu)>0$ such that

Let $\mathscr F$ be a Fourier integral operator, write $z=(x,t)$, and define $\mathscr F_t f(x)=\mathscr Ff(x,t)$. For a compact interval $I$, with $Z=X\times I$ and compact $X$ an…

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…

Let $n\geqslant 2$ be even, and let the Fourier integral operators under consideration satisfy the cinematic curvature conditions. Write $p$ for the Lebesgue exponent in the local…

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…

Let $\mathcal{F}$ be a Fourier integral operator on $\mathbb{R}^n\times[1,2]$ with symbol of order $\mu$, satisfying the conditions H1) and H2), and let…

Modulation-space boundedness conjecture. The operator $T_\sigma$ is bounded from $\mathcal{M}^{r_1,r_2}$ to $\mathcal{M}^{r_2,r_1}$. This conjecture proposes an analogue of the cor…

Local $L_1$ wave propagator conjecture. If $\alpha>1/2$, then there exists a constant $C_{\alpha,c}$ such that, for all $\epsilon\leq1/4$ and sufficiently small $t$,